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Simultaneous-source seismic data acquisition and processing with compressive sensing Wason, Haneet 2017

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Simultaneous-source seismic data acquisition and processing withcompressive sensingbyHaneet WasonB.Sc., University of Calgary, 2010a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Geophysics)The University of British Columbia(Vancouver)August 2017c© Haneet Wason, 2017AbstractThe work in this thesis adapts ideas from the field of compressive sensing (CS) that lead to newinsights into acquiring and processing seismic data, where we can fundamentally rethink how we de-sign seismic acquisition surveys and process acquired data to minimize acquisition- and processing-related costs. Current efforts towards dense source/receiver sampling and full azimuthal coverageto produce high-resolution images of the subsurface have led to the deployment of multiple sourcesacross survey areas. A step ahead from multisource acquisition is simultaneous-source acquisition,where multiple sources fire shots at near-simultaneous/random times resulting in overlapping shotrecords, in comparison to no overlaps during conventional sequential-source acquisition. Adoptionof simultaneous-source techniques has helped to improve survey efficiency and data density. Theengine that drives simultaneous-source technology is simultaneous-source separation — a methodol-ogy that aims to recover conventional sequential-source data from simultaneous-source data. Thisis essential because many seismic processing techniques rely on dense and periodic (or regular)source/receiver sampling. We address the challenge of source separation through a combinationof tailored simultaneous-source acquisition design and sparsity-promoting recovery via convex op-timization using `1 objectives. We use CS metrics to investigate the relationship between marinesimultaneous-source acquisition design and data reconstruction fidelity, and consequently assert theimportance of randomness in the acquisition system in combination with an appropriate choice fora sparsifying transform (i.e., curvelet transform) in the reconstruction algorithm. We also addressthe challenge of minimizing the cost of expensive, dense, periodically-sampled and replicated time-lapse surveying and data processing by adapting ideas from distributed compressive sensing. Weshow that compressive randomized time-lapse surveys need not be replicated to attain acceptablelevels of data repeatability, as long as we know the shot positions (post acquisition) to a sufficientdegree of accuracy. We conclude by comparing sparsity-promoting and rank-minimization recoverytechniques for marine simultaneous-source separation, and demonstrate that recoveries are compa-rable; however, the latter approach readily scales to large-scale seismic data and is computationallyfaster.iiLay SummaryAdapting ideas from the field of compressive sensing, we design economic compressive (or subsam-pled) randomized simultaneous-source acquisitions and develop processing techniques to addressthe challenge of source separation. We recover dense periodic (or regular) conventional sequential-source data from subsampled randomized simultaneous-source data via structure promotion, i.e.,curvelet-domain sparsity or low rank. Adapting ideas from distributed compressive sensing, weshow that compressive randomized time-lapse surveys need not be replicated to attain acceptablelevels of data repeatability, and since irregular spatial sampling is inevitable in the real world, itwould be better to focus on knowing what the shot positions were (post acquisition) to a suffi-cient degree of accuracy, than aiming to replicate them. In a nutshell, compressive randomizedsimultaneous-source acquisitions and subsequent processing techniques provide flexibility in acqui-sition geometries for better survey-area coverage, and speedup acquisition — effectively minimizingacquisition- and processing-related costs.iiiPrefaceThe work presented in this thesis has been carried out at the Department of Earth, Ocean andAtmospheric Sciences at The University of British Columbia, Vancouver, Canada, under the super-vision of Professor Felix J. Herrmann as part of the Seismic Laboratory for Imaging and Modelling(SLIM).A version of Chapter 2 has been published in a geophysical magazine (Herrmann, F. J., H.Wason, and T. T. Y. Lin, 2011, Compressive sensing in seismic exploration: an outlook on a newparadigm: CSEG Recorder, 36. Part 1 [April Edition], 19–33. Part 2 [June Edition], 34–39). Felixwrote the theoretical sections and parts of the experimental section. I conducted experiments forland and marine simultaneous-source academic case study and wrote the corresponding section.Tim was involved in the setup of these experiments.A version of Chapter 3 has been published in a geophysical journal (Mansour, H., H. Wason, T.T. Y. Lin, and F. J. Herrmann, 2012, Randomized marine acquisition with compressive samplingmatrices: Geophysical Prospecting, 60, 648–662). The manuscript was written jointly. Hassan ledthe theoretical sections and corresponding experiments, and I led the seismic experimental sections.Tim and Felix were involved in editing the manuscript.A version of Chapter 4 has been published as an expanded abstract (Wason, H., and F. J. Her-rmann, 2013, Time-jittered ocean bottom seismic acquisition: SEG Technical Program ExpandedAbstracts, 32, 1–6). I was the lead investigator and manuscript composer.A version of Chapter 5 has been published in a geophysical journal (Oghenekohwo, F., H. Wason,E. Esser, and F. J. Herrmann, 2017, Low-cost time-lapse seismic with distributed compressivesensing-Part 1: exploiting common information among the vintages, Geophysics, 82, P1–P13). Themanuscript was written jointly. I conducted experiments for the synthetic seismic case study andwrote the corresponding section. F. Oghenekohwo provided the relevant literature of the subject,conducted the stylized experiments and wrote the corresponding section. F. Herrmann wrote mostof the theoretical sections with significant inputs from F. Oghenekohwo and I. E. Esser was involvedin the algorithm evaluation. Given the equivalent contributions made by F. Oghenekohwo and I, wemutually agreed to include this chapter verbatim in both our dissertations. This chapter appearsas Chapter 2 in F. Oghenekohwo’s dissertation. F. Oghenekohwo has granted permission for thischapter to also appear in my dissertation.A version of Chapter 6 has been published in a geophysical journal (Wason, H., F. Oghenekohwo,and F. J. Herrmann, 2017, Low-cost time-lapse seismic with distributed compressive sensing-Part2: impact on repeatability, 82, P15–P30). I was the lead investigator and manuscript composer.F. Oghenekohwo simulated the monitor model for the SEAM Phase 1 model. F. Herrmann wasinvolved in editing the manuscript. Since this chapter is a necessary extension of Chapter 5, thischapter also appears verbatim as Chapter 3 in F. Oghenekohwo’s dissertation. I have grantedpermission for this chapter to also appear in F. Oghenekohwo’s dissertation.ivA version of Chapter 7 has been published in a geophysical journal (Kumar, R., H. Wason, andF. J. Herrmann, 2015, Source separation for simultaneous towed-streamer marine acquisition — Acompressed sensing approach, Geophysics, 80, WD73–WD88). I encountered the source-separationproblem for towed-streamer acquisitions during an internship. Subsequently, the problem wasinvestigated at SLIM. The manuscript was written jointly with equal contribution from Rajiv andI. I led the theoretical section on sparsity-promoting technique and conducted the correspondingexperiments. Rajiv led the theoretical section on rank-minimization technique and conducted thecorresponding experiments. Felix was involved in editing the manuscript. Given the equivalentcontributions made by R. Kumar and I, we mutually agreed to include this chapter verbatim inboth our dissertations. This chapter appears as Chapter 4 in R. Kumar’s dissertation. R. Kumarhas granted permission for this chapter to also appear in my dissertation.MATLAB and its parallel computing toolbox has been used to prepare all the examples inthis thesis. The SPG`1 solver is a public toolbox provided by Ewout van den Berg and MichaelP. Friedlander. The SPG-LR solver is coded by Aleksandr Y. Aravkin. The Curvelet toolbox isprovided by Emmanuel J. Cande`s, Laurent Demanet, David L. Donoho and Lexing Ying.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Seismic exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Static vs. dynamic geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Simultaneous-source acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Static vs. dynamic simultaneous-source acquisition . . . . . . . . . . . . . . . 71.3 Time-lapse seismic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Compressive sensing in seismic exploration: an outlook on a new paradigm . 162.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Inspiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Nyquist sampling and the curse of dimensionality . . . . . . . . . . . . . . . . 172.2.2 Dimensionality reduction by compressive sensing . . . . . . . . . . . . . . . . 172.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.4 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Basics of compressive sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 Recovery by sparsity-promoting inversion . . . . . . . . . . . . . . . . . . . . 212.3.2 Recovery conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21vi2.4 Compressive-sensing design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 Seismic wavefield representations . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.2 Subsampling of shots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 An academic case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Randomized marine acquisition with compressive sampling matrices . . . . . . 433.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Compressed sensing overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4.1 The sparse recovery problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4.2 Recovery conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Compressed sensing and randomized marine acquisition . . . . . . . . . . . . . . . . 483.5.1 Randomized marine acquisition as a CS problem . . . . . . . . . . . . . . . . 483.5.2 Designing the randomized operator . . . . . . . . . . . . . . . . . . . . . . . . 493.5.3 Assessment of the sampling operators . . . . . . . . . . . . . . . . . . . . . . 513.5.4 Economic considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.6 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.6.1 Simultaneous-source acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . 543.6.2 Random time-dithering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.6.3 Periodic time-dithering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 Simultaneous-source time-jittered marine acquisition . . . . . . . . . . . . . . . 674.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3 Compressive sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4 Time-jittered marine acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 Low-cost time-lapse seismic with distributed Compressive Sensing — exploit-ing common information amongst the vintages . . . . . . . . . . . . . . . . . . . 815.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3.1 Synopsis of compressive sensing . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3.2 Independent recovery strategy (IRS) . . . . . . . . . . . . . . . . . . . . . . . 845.3.3 Shared information amongst the vintages . . . . . . . . . . . . . . . . . . . . 845.3.4 Joint recovery method (JRM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.4 Stylized experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4.1 Experiment 1—independent versus joint recovery . . . . . . . . . . . . . . . . 875.4.2 Experiment 2—impact of degree of survey replicability . . . . . . . . . . . . . 885.5 Experimental setup—on-the-grid time-lapse randomized subsampling . . . . . . . . . 90vii5.6 Synthetic seismic case study—time-lapse marine acquisition via time-jittered sources 915.6.1 Time-jittered marine acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 915.6.2 Acquisition geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.6.3 Experiments and observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.6.4 Repeatability measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 Low-cost time-lapse seismic with distributed Compressive Sensing — impacton repeatability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2.1 Motivation: on-the-grid vs. off-the-grid data recovery . . . . . . . . . . . . . . 1086.2.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.3 Time-jittered marine acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.3.1 Acquisition geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3.2 From discrete to continuous spatial subsampling . . . . . . . . . . . . . . . . 1186.3.3 Nonequispaced fast discrete curvelet transform (NFDCT) . . . . . . . . . . . 1186.4 Time-lapse acquisition via jittered sources . . . . . . . . . . . . . . . . . . . . . . . . 1196.4.1 Joint recovery method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.5 Economic performance indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.6 Synthetic seismic case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.6.1 BG COMPASS model—simple geology, complex time-lapse difference . . . . 1216.6.2 SEAM Phase 1 model—complex geology, complex time-lapse difference . . . 1246.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337 Source separation for simultaneous towed-streamer marine acquisition — acompressed sensing approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.2.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.3.1 Rank-revealing “transform domain” . . . . . . . . . . . . . . . . . . . . . . . 1387.3.2 Hierarchical Semi-Separable matrix representation (HSS) . . . . . . . . . . . 1427.3.3 Large-scale seismic data: SPG-LR framework . . . . . . . . . . . . . . . . . . 1437.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.4.1 Comparison with NMO-based median filtering . . . . . . . . . . . . . . . . . 1507.4.2 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1527.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154viii8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1628.1 Compressive sensing in seismic exploration . . . . . . . . . . . . . . . . . . . . . . . . 1628.2 Compressive simultaneous-source marine acquisition . . . . . . . . . . . . . . . . . . 1628.3 Compressive simultaneous-source time-lapse marine acquisition . . . . . . . . . . . . 1638.4 Compressive simultaneous-source towed-streamer acquisition . . . . . . . . . . . . . 1648.5 Follow-up work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1658.6 Current limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1658.7 Future research directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167ixList of TablesTable 3.1 Summary of recovery results (S/N in dB) based on the 3D curvelet and the 3DFourier transforms for the three sampling schemes. . . . . . . . . . . . . . . . . . 55Table 5.1 Summary of recoveries in terms of S/N (in dB) for the stacked sections. . . . . . . 97Table 6.1 Summary of recoveries in terms of S/N (dB) for data recovered via JRM for asubsampling factor η = 2. The S/Ns show little variability in the time-lapsedifference recovery for different overlaps between the surveys offering a possibilityto relax insistence on replicability of time-lapse surveys. This is supported bythe improved recovery of the vintages as the overlap decreases. Note that thedeviations are average deviations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Table 6.2 Summary of recoveries in terms of S/N (dB) for data recovered via JRM for asubsampling factor η = 4. The S/Ns show little variability in the time-lapsedifference recovery for different overlaps between the surveys offering a possibilityto relax insistence on replicability of time-lapse surveys. This is supported bythe improved recovery of the vintages as the overlap decreases. Note that thedeviations are average deviations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Table 7.1 Comparison of computational time (in hours), memory usage (in GB) and averageS/N (in dB) using sparsity-promoting and rank-minimization based techniques forthe Marmousi model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Table 7.2 Comparison of computational time (in hours), memory usage (in GB) and averageS/N (in dB) using sparsity-promoting and rank-minimization based techniques forthe Gulf of Suez dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Table 7.3 Comparison of computational time (in hours), memory usage (in GB) and averageS/N (in dB) using sparsity-promoting and rank-minimization based techniques forthe BP model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151xList of FiguresFigure 1.1 Schematic of land seismic survey. Image courtesy ION (www.iongeo.com). . . . . 2Figure 1.2 Schematic of different marine seismic surveys. “1” illustrates a towed-streamergeometry, “2” an ocean-bottom geometry, “3” a buried seafloor array, and “4” aVSP (vertical seismic profile) geometry, where the receivers are positioned in awell. [Source: Caldwell and Walker] . . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.3 Schematic of ocean-bottom node survey. Remotely operated vehicles (ROVs) areused to deploy and recover sensor nodes. [Source: Caldwell and Walker] . . . . . 3Figure 1.4 Summary of the majority of different types of marine seismic surveys. The letter“D” represents dimension and the letter “C” represents component (Z, X, Y ).[Source: Caldwell and Walker] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 1.5 Illustration of difference between 2D and 3D survey geometry for same surveyarea. The dashed lines suggest subsurface structure contour lines. [Source: Cald-well and Walker] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 1.6 Four most common seismic trace display formats. [Source: SEG Wiki] . . . . . . 5Figure 1.7 Schematic of land simultaneous-source acquisition. (a) and (b) Individual shotsacquired in a conventional survey. (c) Simultaneous shot acquired in a simultaneous-source survey. Images courtesy ION (www.iongeo.com). . . . . . . . . . . . . . . 8Figure 1.8 Shot-time randomness (or variability) for static and dynamic marine simultaneous-source acquisitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 1.9 Schematic of dynamic over/under marine simultaneous-source acquisition. Si-multaneous data acquired in the field is separated in to individual source com-ponents using source-separation techniques. . . . . . . . . . . . . . . . . . . . . . 10Figure 1.10 Schematic of dynamic marine simultaneous-long acquisition. Simultaneous dataacquired in the field is separated in to individual source components using source-separation techniques. Note that the streamer length is 6 km and the secondsource vessel is deployed one spread-length (6 km) ahead of the main seismicvessel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 1.11 Schematic of static marine simultaneous-source acquisition. This also illustratesthe design of our proposed time-jittered marine acquisition. Source separationaims to recover densely sampled interference-free data by unravelling overlappingshot records and interpolation to a fine source grid (Chapters 4–6). . . . . . . . . 12xiFigure 2.1 Different (sub)sampling schemes and their imprint in the Fourier domain for asignal that is the superposition of three cosine functions. Signal (a) regularlysampled above Nyquist rate, (c) randomly three-fold undersampled according toa discrete uniform distribution, and (e) regularly three-fold undersampled. Therespective amplitude spectra are plotted in (b), (d) and (f). Unlike aliases, thesubsampling artifacts due to random subsampling can easily be removed usinga standard denoising technique, e.g., nonlinear thresholding (dashed line), effec-tively recovering the original signal (adapted from (Hennenfent and Herrmann,2008)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 2.2 Averaged recovery error percentages for a k-sparse Fourier vector reconstructedfrom n time samples taken (a) regularly and (b) uniformly-randomly. In eachplot, the curves from top to bottom correspond to a subsampling factor rangingfrom two to six (adapted from Hennenfent and Herrmann (2008)). . . . . . . . . 34Figure 2.3 Curvelets and seismic data. (a) 2D curvelets in the time-space domain and thefrequency-wavenumber domain. (b) Curvelets approximate curved singularities,i.e., wavefronts, with very few significant curvelet coefficients. . . . . . . . . . . . 35Figure 2.4 Real common-receiver gather from Gulf of Suez data set. . . . . . . . . . . . . . 36Figure 2.5 Signal-to-noise ratios (S/Ns) for the nonlinear approximation errors of the common-receiver gather plotted in Figure 2.4. The S/Ns are plotted as a function of thesparsity ratio ρ ∈ (0, 0.02]. The plots include curves for the errors obtainedfrom the analysis and one-norm minimized synthesis coefficients. Notice the sig-nificant improvement in S/Ns for the synthesis coefficients obtained by solvingEquation 2.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 2.6 Recovery from a compressively-sampled common-receiver gather with 50% (δ =0.5) of the sources missing. (a) Left: Receiver gather with sequential shotsselected uniformly at random. (a) Right: The same but for random simultaneousshots. (b) Left: Recovery from incomplete data in (a) left-hand-side plot. (b)Right: The same but now for the data in (a) right-hand-side plot. (c) Left:Difference plot between the data in Figure 2.4 and the recovery in (b) left-hand-side plot. (c) Right: The same but now for recovery from simultaneous datain (a) right-hand-side plot. Notice the remarkable improvement in the recoveryfrom simultaneous data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 2.7 S/Ns (cf. Equation 2.12) for nonlinear sparsity-promoting recovery from com-pressively sampled data with 20% − 80% of the sources missing (δ ∈ [0.2, 0.8]).The results summarize 25 experiments for 25 different values of δ ∈ [0.2, 0.8].The plots include estimates for the standard deviations. From these results, itis clear that simultaneous acquisition (results in the left column) is more con-ducive to sparsity-promoting recovery. Curvelet-based recovery seems to workbest, especially towards high percentages of data missing. . . . . . . . . . . . . . 38Figure 2.8 Oversampling ratio δ/ρ as a function of the sampling ratio δ (cf. Equation 2.13)for sequential- and simultaneous-source experiments. As expected, the overheadis smallest for simultaneous acquisition and curvelet-based recovery. . . . . . . . 39xiiFigure 2.9 Different acquisition scenarios. (a) Left: Impulsive sources for conventional se-quential acquisition, yielding 128 shot records for 128 receivers and 512 timesamples. (a) Right: Corresponding fully sampled sequential data. (b) Left:Simultaneous sources for ‘Land’ acquisition with 64 simultaneous-source exper-iments. Notice that all shots fire simultaneously in this case. (b) Right: Cor-responding compressively sampled land data. (c) Left: Simultaneous sourcesfor ‘Marine’ acquisition with 128 sources firing at random times and locationsduring a continuous total ’survey’ time of T = 262s. (c) Right: Corresponding‘Marine’ data plotted as a conventional seismic line. . . . . . . . . . . . . . . . . 40Figure 2.10 Sparsity-promoting recovery with δ = 0.5 with the 2D curvelet transforms. (a)2D curvelet-based recovery from ‘Land’ data (10.3 dB). (b) The correspondingdifference plot. (c) 2D curvelet-based recovery from ‘Marine’ data (7.2 dB). (d)Corresponding difference plot. Notice the improvement in recovery from ‘Land’data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 2.11 Sparsity-promoting recovery with δ = 0.5 with the 3D curvelet transforms. (a)3D curvelet-based recovery from ‘Land’ data (11.6 dB). (b) The correspondingdifference plot. (c) 3D curvelet-based recovery from ‘Marine’ data (11.1 dB). (d)Corresponding difference plot. Notice the improvement in recovery compared to2D curvelet based recovery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.1 Examples of (a) random dithering in source location and trigger times, (b) se-quential locations and random time-triggers, and (c) periodic source firing triggers. 50Figure 3.2 Example of (a) “ideal” simultaneous-source operator defined by a Bernouilli ma-trix, (b) operator that corresponds to the more realizable Marine acquisition bythe random time-dithering, and (c) sampling operator with periodic time-dithering. 56Figure 3.3 Gram matrices of example random time-dithering and constant time-ditheringoperators, top row, with Ns = 10 and Nt = 40 coupled with a curvelet trans-form. The resulting mutual coherence is 0.695 for random time-dithering com-pared with 0.835 for periodic time-dithering. The center plots show column thecenter column of the Gram matrices. The bottom row shows column 252 (onethird) of the Gram matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 3.4 Gram matrices of example random time-dithering and constant time-ditheringoperators, top row, with Ns = 10 and Nt = 40 coupled with a Fourier transform.The resulting mutual coherence is 0.738 for random time-dithering comparedwith 0.768 for periodic time-dithering. The center plots show column the centercolumn of the Gram matrices. The bottom row shows column 133 (one third) ofthe Gram matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 3.5 Comparison between the histograms of δˆΛ from 1000 realizations of AΛ, therandom time-dithering sampling matrices A = RMSH restricted to a set Λ ofsize k, the size support of the largest transform coefficients of a real (Gulf ofSuez) seismic image. The transform S is (a) the curvelet transform and (b) thenonlocalized 2D Fourier transform. The histograms show that randomized time-shifting coupled with the curvelet transform has better behaved RIP constant(δˆΛ = max{1− σmin, σmax − 1} < 1) and therefore promotes better recovery. . . . 59Figure 3.6 A common-shot gather from Gulf of Suez data set. . . . . . . . . . . . . . . . . . 60xiiiFigure 3.7 (a) Simultaneous-source marine data (γ = 0.5) shown as a section between 45to 50 seconds of the ”supershot”. (b) Recovery from simultaneous ‘marine’ data(S/N = 10.5 dB). (c) The corresponding residual plot. . . . . . . . . . . . . . . . 61Figure 3.8 (a) Random time-dithered “marine” data (γ = 0.5) shown as a section between 45and 50 seconds of the “supershot”. (b) Sparse recovery with curvelet transformand S/N = 8.06dB. (c) The corresponding residual plot. . . . . . . . . . . . . . . 62Figure 3.9 (a) Sparse recovery with 3D Fourier transform from the same data shown inFigure 3.8(a), S/N = 6.83dB. (b) The corresponding residual plot. . . . . . . . . 63Figure 3.10 (a) Data recovered by applying adjoint of the sampling operator RM and 2Dmedian filtering, from the same data shown in Figure 3.8(a), with S/N = 3.92dB.(b) The corresponding residual plot. . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 3.11 (a) Periodic time-dithered “marine” data (γ = 0.5) shown as a section between 45and 50 seconds of the “supershot”. (b) Sparse recovery with curvelet transformand S/N = 4.80dB. (c) The corresponding residual plot. . . . . . . . . . . . . . . 65Figure 3.12 (a) Data recovered by applying adjoint of the sampling operator RM and 2Dmedian filtering, from the same data shown in Figure 3.11(a), with S/N = 1.26dB.(b) The corresponding residual plot. . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 4.1 Schematic comparison between different subsampling schemes. η is the subsam-pling factor. The vertical dashed lines define the regularly subsampled spatialgrid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 4.2 Acquisition geometry. (a,c) Conventional marine acquisition with one sourcevessel and two air-gun arrays for a spatial sampling of 12.5 m and 6.25 m, re-spectively. (b,d) The corresponding time-jittered marine acquisition with η = 2and η = 4, respectively. Note the acquisition speedup during jittered acquisition,where the recording time is reduced to one-half and one-fourth the recording timeof the conventional acquisition, respectively. . . . . . . . . . . . . . . . . . . . . . 74Figure 4.3 Histogram of δˆΛ from 100 realizations of AΛ, restricted to a set Λ of size k, thesize support of the largest curvelet-domain coefficients of a real (Gulf of Suez)seismic image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 4.4 Conventional data for a seismic line from the Gulf of Suez. (a) Common-receivergather spatially sampled at 12.5 m. (b) Common-shot gather spatially sampledat 12.5 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 4.5 Simultaneous data for conventional data spatially sampled at (a) 12.5 m and (b)6.25 m. Note that only 100.0 s of the full simultaneous data volume is shown. . . 76Figure 4.6 Interferences (or source crosstalk) in a (a) common-receiver gather and (b) common-shot gather for data spatially sampled at 12.5 m; and in a (c) common-receivergather and (d) common-shot gather for data spatially sampled at 6.25 m. Sincethe subsampling factor η = 2 and η = 4 for a spatial sampling of 12.5 m and 6.25m, respectively, (a) and (c) also have missing traces. The simultaneous data areseparated and interpolated to their respective fine sampling grids. . . . . . . . . 76Figure 4.7 Recovered data for a subsampling factor η = 2. (a,e) Common-receiver gathersrecovered with 2D FDCT and 3D FDCT, respectively. (b,f) The correspondingdifference from conventional data. (c,g) Common-shot gathers recovered with2D FDCT and 3D FDCT, respectively. (d,h) The corresponding difference fromconventional data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77xivFigure 4.8 Zoom sections of recovered data for a subsampling factor η = 2. Note thatthe color axis has been clipped at one-tenth the color axis of Figure 4.7. (a,e)Common-receiver gathers recovered with 2D FDCT and 3D FDCT, respectively.(b,f) The corresponding difference from conventional data. (c,g) Common-shotgathers recovered with 2D FDCT and 3D FDCT, respectively. (d,h) The corre-sponding difference from conventional data. . . . . . . . . . . . . . . . . . . . . . 78Figure 4.9 Recovered data for a subsampling factor η = 4. (a,e) Common-receiver gathersrecovered with 2D FDCT and 3D FDCT, respectively. (b,f) The correspondingdifference from conventional data. (c,g) Common-shot gathers recovered with2D FDCT and 3D FDCT, respectively. (d,h) The corresponding difference fromconventional data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 4.10 Zoom sections of recovered data for a subsampling factor η = 4. Note thatthe color axis has been clipped at one-tenth the color axis of Figure 4.9. (a,e)Common-receiver gathers recovered with 2D FDCT and 3D FDCT, respectively.(b,f) The corresponding difference from conventional data. (c,g) Common-shotgathers recovered with 2D FDCT and 3D FDCT, respectively. (d,h) The corre-sponding difference from conventional data. . . . . . . . . . . . . . . . . . . . . . 80Figure 5.1 Left: Decay of curvelet coefficients of time-lapse data and difference. Right:Scatter plot of curvelet coefficients of the baseline and monitor data indicatingthat they share significant information. . . . . . . . . . . . . . . . . . . . . . . . 85Figure 5.2 From top to bottom: z0, z1, z2,x1,x2,x1 − x2. We are particularly interested inrecovering estimates for x1,x2 and x1 − x2 from y1 and y2. . . . . . . . . . . . . 87Figure 5.3 Recovery of (a) the jointly sparse signals x1 and x2, (b) x1−x2; with and withoutrepetition of the measurement matrices, using the independent recovery strategyversus the joint recovery method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 5.4 Recovery as a function of overlap between measurement matrices. Probability ofrecovering (a) x1 and x2, (b) x1 − x2, with joint recovery method. . . . . . . . . 89Figure 5.5 Schematic comparison between different random realizations of a subsampledgrid. The subsampling factor is 3. As illustrated, random samples are takenexactly on the grid. Moreover, the samples are exactly replicated wheneverthere is an overlap between the time-lapse surveys. . . . . . . . . . . . . . . . . . 90Figure 5.6 Reservoir zoom of the synthetic time-lapse velocity models showing the change invelocity as a result of fluid substitution. (a) Baseline model, (b) monitor model,(c) difference between (a) and (b). . . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 5.7 A synthetic receiver gather from the conventional (a) baseline survey, (b) monitorsurvey. (c) The corresponding 4D signal. (d) Color scale of the vintages. (e)Color scale of the 4D signal. Note that (e) is one-tenth the scale of (d). Thesecolor scales apply to all the corresponding figures for the vintages and the 4Dsignal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Figure 5.8 Acquisition geometry: (a) conventional marine acquisition with one source ves-sel and two airgun arrays; time-jittered marine acquisition (with η = 2) for (b)baseline, and (c) monitor. Note the acquisition speedup during jittered acquisi-tion, where the recording time is reduced to one-half the recording time of theconventional acquisition. (b) and (c) are plotted on the same scale as (a) in orderto make the jittered locations easily visible. . . . . . . . . . . . . . . . . . . . . . 94xvFigure 5.9 Simultaneous data for the (a) baseline and (b) monitor surveys (only 50.0 s ofthe full data is shown). Interferences (or source crosstalk) in a common-receivergather for the (c) baseline and (d) monitor surveys, respectively. Since the sub-sampling factor η = 2, (c) and (d) also have missing traces. The simultaneousdata is separated and interpolated to a sampling grid of 12.5 m. . . . . . . . . . . 95Figure 5.10 Receiver gathers (from monitor survey) recovered via IRS from time-jittered ma-rine acquisition with (a) 100%, (b) 50%, and (c) 25% overlap in the measurementmatrices (A1 and A2). (d), (e), and (f) Corresponding difference plots from theoriginal receiver gather (5.7(b)). . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Figure 5.11 Receiver gathers (from monitor survey) recovered via JRM from time-jitteredmarine acquisition with (a) 100%, (b) 50%, and (c) 25% overlap in the measure-ment matrices (A1 and A2). (d), (e), and (f) Corresponding difference plotsfrom the original receiver gather (5.7(b)). . . . . . . . . . . . . . . . . . . . . . . 99Figure 5.12 Recovered 4D signal for the (a) 100%, (b) 50%, and (c) 25% overlap. Top row:IRS, bottom row: JRM. Note that the color axis is one-tenth the scale of thecolor axis for the vintages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 5.13 Stacked sections. (a) baseline; (b) true 4D signal; reconstructed 4D signals viaIRS for 100% (c), 50%(e), and 25% (g) overlap; the reconstructed 4D signals viaJRM for 100%(d), 50%(f), and 25% (h) overlap. Notice the improvements forJRM where we see much less deterioration as the overlap between the surveysdecreases. Note that the color axis for the time-lapse difference stacks is one-tenth the scale of the color axis for the baseline stack. . . . . . . . . . . . . . . . 101Figure 5.14 Normalized root-mean-squares NMRS for each recovered trace of the stackedsection for (a) 50% and the (b) 25% overlap. Vintages obtained with the jointrecovery method are far superior to results obtained with the independent recov-ery strategy and the “unprocessed” stacked data. The latter are unsuitable fortime lapse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 6.1 Schematic of conventional acquisition and simultaneous, compressed (or time-jittered) acquisition. If the source sampling grid for conventional acquisitionis 25.0 m (or 50.0 m for flip-flop acquisition), then the time-jittered acquisitionjitters (or perturbs) shot positions on a finer grid, which is 1/4 th of the con-ventional flip-flop sampling grid, for a single air-gun array. Following the samestrategy, adding another air-gun array makes the acquisition simultaneous, andhence results in a compressed data volume with overlapping, irregular shots andmissing traces. The sparsity-promoting inversion then aims to recover denselysampled data by separating the overlapping shots, regularizing irregular tracesand interpolating missing traces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Figure 6.2 Synthetic receiver gathers from a conventional (a) baseline survey, (b) monitorsurvey. (c) Corresponding time-lapse difference. . . . . . . . . . . . . . . . . . . . 110Figure 6.3 Data recovery via the joint recovery method and binning. (a), (b) Binned vin-tages and (c) corresponding time-lapse difference. (d), (e), (f) Correspondingdifference plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111xviFigure 6.4 Data recovery via the joint recovery method and regularization. (a), (b) Vin-tages and (c) time-lapse difference recovered via sparsity promotion includingregularization of irregular traces. (d), (e), (f) Corresponding difference plots. Asillustrated, regularization is imperative for high-quality data recovery. . . . . . . 112Figure 6.5 Marine acquisition with one source vessel and two air-gun arrays. (a) Conven-tional flip-flop acquisition. Time-jittered acquisition with a subsampling factorη = 2 for the (b) baseline and (c) monitor. Note the acquisition speedup duringjittered acquisition, where the recording time is reduced to one-half the record-ing time of the conventional acquisition. (d) Zoomed sections of (a), (b) and (c),respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure 6.6 Simultaneous data for the (a) baseline and (b) monitor surveys. Only 40.0 s ofthe full data is shown. Time-jittered acquisition generates a simultaneous datavolume with overlapping shots and missing shots. . . . . . . . . . . . . . . . . . . 116Figure 6.7 Interferences (or source crosstalk) in a common-receiver gather for the (a) base-line and (b) monitor surveys, respectively. Receiver gathers are obtained viaMHy for the time-lapse surveys. For a subsampling factor η, (a) and (b) haveNsη irregular traces. (c), (d) Common-receiver gathers for the baseline and mon-itor surveys, respectively, after applying the adjoint of a 1D NFFT operator to(a) and (b). (e) Corresponding time-lapse difference. As illustrated, the recov-ery problem needs to be considered as a (sparse) structure-promoting inversionproblem, wherein the simultaneous data volume is separated, regularized andinterpolated to a finer sampling grid rendering interference-free data. . . . . . . . 117Figure 6.8 Subset of the BG COMPASS model. (a) Baseline model; (b) monitor model; (c)difference between (a) and (b) showing the gas cloud. . . . . . . . . . . . . . . . 122Figure 6.9 JRM recovered monitor receiver gathers from the BG COMPASS model for asubsampling factor η = 2. Recovered monitor data and residual with (a,b)100% overlap in the measurement matrices (A1 and A2); (c,d) 100% overlapand average shot-position deviation of 1 m; (e,f) 100% overlap and average shot-position deviation of 2 m; (g,h) 100% overlap and average shot-position deviationof 3 m; (i,j) < 15% overlap, respectively. . . . . . . . . . . . . . . . . . . . . . . . 125Figure 6.10 JRM recovered time-lapse difference receiver gathers from the BG COMPASSmodel for a subsampling factor η = 2. Recovered time-lapse difference andresidual with (a,b) 100% overlap in the measurement matrices (A1 and A2);(c,d) 100% overlap and average shot-position deviation of 1 m; (e,f) 100% overlapand average shot-position deviation of 2 m; (g,h) 100% overlap and average shot-position deviation of 3 m; (i,j) < 15% overlap, respectively. . . . . . . . . . . . . 126Figure 6.11 JRM recovered monitor receiver gathers from the BG COMPASS model for asubsampling factor η = 4. Recovered monitor data and residual with (a,b)100% overlap in the measurement matrices (A1 and A2); (c,d) 100% overlapand average shot-position deviation of 1 m; (e,f) 100% overlap and average shot-position deviation of 2 m; (g,h) 100% overlap and average shot-position deviationof 3 m; (i,j) < 5% overlap, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 127xviiFigure 6.12 JRM recovered time-lapse difference receiver gathers from the BG COMPASSmodel for a subsampling factor η = 4. Recovered time-lapse difference andresidual with (a,b) 100% overlap in the measurement matrices (A1 and A2);(c,d) 100% overlap and average shot-position deviation of 1 m; (e,f) 100% overlapand average shot-position deviation of 2 m; (g,h) 100% overlap and average shot-position deviation of 3 m; (i,j) < 5% overlap, respectively. . . . . . . . . . . . . . 128Figure 6.13 Subset of the SEAM model. (a) Baseline model; (b) monitor model; (c) differencebetween (a) and (b) showing the time-lapse difference. . . . . . . . . . . . . . . . 129Figure 6.14 Synthetic receiver gathers from the conventional SEAM (a) baseline survey, (b)monitor survey. (c) Corresponding time-lapse difference. The amplitude of thetime-lapse difference is one-tenth the amplitude of the baseline and monitor data. 130Figure 6.15 JRM recovered monitor and time-lapse difference receiver gathers from the SEAMmodel for a subsampling factor η = 2. Recovered monitor data and residual with(a,b) 100% overlap in the measurement matrices (A1 and A2); (c,d) < 15% over-lap, respectively. Recovered time-lapse difference and residual with (e,f) 100%overlap in the measurement matrices; (g,h) < 15% overlap, respectively. Notethat the amplitude of the time-lapse difference is one-tenth the amplitude of themonitor data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Figure 7.1 Monochromatic frequency slice at 5 Hz in the source-receiver (s-r) and midpoint-offset (m-h) domain for blended data (a,c) with periodic firing times and (b,d)with uniformly random firing times for both sources. . . . . . . . . . . . . . . . . 140Figure 7.2 Decay of singular values for a frequency slice at (a) 5 Hz and (b) 40 Hz of blendeddata. Source-receiver domain: blue—periodic, red—random delays. Midpoint-offset domain: green—periodic, cyan—random delays. Corresponding decay ofthe normalized curvelet coefficients for a frequency slice at (c) 5 Hz and (d) 40Hz of blended data, in the source-channel domain. . . . . . . . . . . . . . . . . . 141Figure 7.3 Monochromatic frequency slice at 40 Hz in the s-r and m-h domain for blendeddata (a,c) with periodic firing times and (b,d) with uniformly random firing timesfor both sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Figure 7.4 HSS partitioning of a high-frequency slice at 40 Hz in the s-r domain: (a) first-level, (b) second-level, for randomized blended acquisition. . . . . . . . . . . . . . 143Figure 7.5 (a,b,c) First-level sub-block matrices (from Figure 7.4(a)). . . . . . . . . . . . . . 144Figure 7.6 Decay of singular values of the HSS sub-blocks in s-r domain: red—Figure 7.5(a),black—Figure 7.5(b), blue—Figure 7.5(c). . . . . . . . . . . . . . . . . . . . . . . 144Figure 7.7 Original shot gather of (a) source 1, (b) source 2, and (c) the correspondingblended shot gather for simultaneous over/under acquisition simulated on theMarmousi model. (d, e) Corresponding common-channel gathers for each sourceand (f) the blended common-channel gather. . . . . . . . . . . . . . . . . . . . . 147Figure 7.8 Original shot gather of (a) source 1, (b) source 2, and (c) the correspondingblended shot gather for simultaneous over/under acquisition from the Gulf ofSuez dataset. (d,e) Corresponding common-channel gathers for each source and(f) the blended common-channel gather. . . . . . . . . . . . . . . . . . . . . . . . 148xviiiFigure 7.9 Original shot gather of (a) source 1, (b) source 2, and (c) the correspondingblended shot gather for simultaneous long offset acquisition simulated on the BPsalt model. (d, e) Corresponding common-channel gathers for each source and(f) the blended common-channel gather. . . . . . . . . . . . . . . . . . . . . . . . 149Figure 7.10 Separated shot gathers and difference plots (from the Marmousi model) of source1 and source 2: (a,c) source separation using HSS based rank-minimization and(b,d) the corresponding difference plots; (e,g) source separation using curvelet-based sparsity-promotion and (f,h) the corresponding difference plots. . . . . . . 152Figure 7.11 Separated common-channel gathers and difference plots (from the Marmousimodel) of source 1 and source 2: (a,c) source separation using HSS based rank-minimization and (b,d) the corresponding difference plots; (e,g) source separationusing curvelet-based sparsity-promotion and (f,h) the corresponding differenceplots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Figure 7.12 Separated shot gathers and difference plots (from the Gulf of Suez dataset) ofsource 1 and source 2: (a,c) source separation using HSS based rank-minimizationand (b,d) the corresponding difference plots; (e,g) source separation using curvelet-based sparsity-promotion and (f,h) the corresponding difference plots. . . . . . . 154Figure 7.13 Separated common-channel gathers and difference plots (from the Gulf of Suezdataset) of source 1 and source 2: (a,c) source separated using HSS based rank-minimization and (b,d) the corresponding difference plots; (e,g) source separationusing curvelet-based sparsity-promotion and (f,h) the corresponding differenceplots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Figure 7.14 Separated shot gathers and difference plots (from the BP salt model) of source1 and source 2: (a,c) source separation using HSS based rank-minimization and(b,d) the corresponding difference plots; (e,g) source separation using curvelet-based sparsity-promotion and (f,h) the corresponding difference plots. . . . . . . 156Figure 7.15 Separated common-channel gathers and difference plots (from the BP salt model)of source 1 and source 2: (a,c) source separation using HSS based rank-minimizationand (b,d) the corresponding difference plots; (e,g) source separation using curvelet-based sparsity-promotion and (f,h) the corresponding difference plots. . . . . . . 157Figure 7.16 Signal-to-noise ratio (dB) over the frequency spectrum for the separated datafrom the Marmousi model. Red, blue curves—source separation without HSS;cyan, black curves—source separation using second-level HSS partitioning. Solidlines—separated source 1, + marker—separated source 2. . . . . . . . . . . . . . 158Figure 7.17 Blended common-midpoint gathers of (a) source 1 and (e) source 2 for the Mar-mousi model. Source separation using (b,f) NMO-based median filtering, (c,g)rank-minimization and (d,h) sparsity-promotion. . . . . . . . . . . . . . . . . . . 159Figure 7.18 Blended common-midpoint gathers of (a) source 1, (e) source 2 for the Gulf ofSuez dataset. Source separation using (b,f) NMO-based median filtering, (c,g)rank-minimization and (d,h) sparsity-promotion. . . . . . . . . . . . . . . . . . . 160Figure 7.19 Blended common-midpoint gathers of (a) source 1, (e) source 2 for the BP saltmodel. Source separation using (b,f) NMO-based median filtering, (c,g) rank-minimization and (d,h) sparsity-promotion. . . . . . . . . . . . . . . . . . . . . . 161xixGlossaryBPDN - Basis pursuit denoiseCS - Compressive sensingDCS - Distributed compressive sensingFDCT - Fast discrete curvelet transformHSS - Hierarchical semi-separableIRS - Independent recovery strategyJRM - Joint recovery model (or method)LASSO - Least absolute shrinkage and selection operatorNFDCT - Nonequispaced fast discrete curvelet transformNRMS - Normalized root-mean-squareOBC - Ocean-bottom cableOBN - Ocean-bottom nodeRIP - Restricted isometry propertySLO - Simultaneous long offsetS/N - Signal-to-noise ratioSPG - Spectral projected gradientSVD - Singular value decompositionxxAcknowledgmentsI would first and foremost like to thank my advisor, Professor Dr. Felix J. Herrmann, for givingme a position in the SLIM group, where I have tackled some of the most interesting problems andlearnt a great deal from the diverse nature of these problems. I am grateful to him for his supportthroughout my time in graduate school. I am very fortunate to have been a part of the SLIMgroup. This experience has most definitely made me a better student.I would also like to thank Professor Dr. Eldad Haber and Professor Dr. Michael P. Friedlanderfor serving on my advisory committee and for generously offering their time, support and advice.My journey through graduate school would not have been memorable if it were not for thepeople I have worked with. I would like to thank each and every member of the SLIM group,students and postdoctorate fellows, for being a part of my journey that has been nothing short ofa roller-coaster ride. The exciting and stimulating interactions with my colleagues has increasedmy knowledge of scientific research tremendously. I wish them all the very best for their careers.The SLIM group would not be able to function smoothly without the support of Miranda Joyceand Henryk Modzelewski. A very special thanks to them for always being so concerned, generousand helpful. I truly value their friendship.I am very grateful to PGS (Weybridge, UK) for the hospitality during my internship. I wouldlike to thank Sverre Brandsberg-Dahl and Andreas Klaedtke for giving me this opportunity. Specialthanks to Andreas Klaedtke and Rolf Baardman for great technical discussions. This experienceimproved my understanding of my research topic and its relevance to the industry immensely.I would like to thank Dr. Eric Verschuur for providing the Gulf of Suez dataset, which I use inChapters 2, 3, 4 and 7, and the BG group for providing the synthetic 3D COMPASS velocity modelthat I use in Chapters 5 and 6. Many thanks to the authors of SPG`1, CurveLab, Madagascar,IWAVE, SPG-LR, the Marmousi velocity model and the BP salt model, which I use throughoutthis work.This work was financially supported in part by the Natural Sciences and Engineering ResearchCouncil of Canada Collaborative Research and Development Grant DNOISE II (CDRP J 375142-08). This research was carried out as part of the SINBAD II project with the support of the memberorganizations of the SINBAD Consortium. I also wish to acknowledge the SENAI CIMATECSupercomputing Center for Industrial Innovation, with support from BG Brasil and the BrazilianAuthority for Oil, Gas and Biofuels (ANP), for the provision and operation of computationalfacilities and the commitment to invest in Research and Development.xxiChapter 1Introduction1.1 Seismic explorationGeophysical surveys determine characteristics of the earth’s subsurface by measuring the physicaldifferences between rock types or physical discontinuities without seeing them directly by digging,drilling or tunnelling. Geophysical surveys are classified as seismic and non-seismic surveys thatinclude magnetic and electromagnetic surveys, gravitational surveys, seismic surveys, etc. Seismicis perhaps the most commonly used geophysical technique to locate potential oil and natural gasdeposits in the geologic structures within the earth. Seismic exploration techniques involve thecollection of massive data volumes, where regularly sampled wavefields exhibit up to a 5-dimensionalstructure (1D for the time dimension × 2D for the receiver positions × 2D for the source positions),and their exploitation during processing.1.1.1 Static vs. dynamic geometriesSeismic surveys can be conducted on onshore (land) and offshore (marine). Land and marinesurveys operate on the same basic principles but differ operationally. Both land and marine surveysinclude a source and a receiver but may differ in the geometry of the receiver system, the densityof measurements made over a given area, and the type of source and receiver (or sensor) used(Caldwell and Walker, 2011).LandLand seismic data acquisition uses primarily two types of seismic sources — vibroseis vehicles(vibrators mounted on trucks) or a low-impact explosive charge — that generate acoustic waves,which propagate deep into the earth. Each time an acoustic wave encounters a change in the rockformation, part of the wave is reflected back to the surface where an array of sensors records thereturning sound waves. The receivers are typically geophones, which are like small microphonespushed into the soil to measure the ground motion (Caldwell and Walker, 2011). Since the receiversare fixed on the earth’s surface, we refer to this acquisition geometry as the “static” geometry.Figure 1.1 illustrates a land seismic survey.1Figure 1.1: Schematic of land seismic survey. Image courtesy ION (www.iongeo.com).MarineIn water, the energy source is typically an array of air guns, i.e., guns with different sized air-chambers filled with compressed air. The source is towed behind a seismic survey vessel andreleases bursts of high pressure energy (acoustic pulses) into the water. The returning soundwaves are detected and recorded by sensors that are either hydrophones (measure wave pressure)spaced out along a series of cables, i.e., streamers towed behind a survey vessel, or arrays placedon the seafloor, i.e., ocean-bottom seismic sensors comprising of hydrophones and/or geophones.Figure 1.2 illustrates the different receiver geometries used in marine seismic surveying (Caldwelland Walker, 2011). Similar to land acquisition, we refer to the ocean-bottom surveys as “static”marine surveys, while the towed-streamer surveys are referred to as “dynamic” marine surveys.Ocean-bottom surveys can further be classified as ocean-bottom cable (OBC) surveys, whereindata are recorded using cables laid on the seabed (acquisition geometry 2 in Figure 1.2), or ocean-bottom node (OBN) surveys, wherein sensor nodes (which may or may not be connected by cables)are placed on the seafloor (Figure 1.3).Figure 1.4 provides a list of the different types of marine surveys (Caldwell and Walker, 2011).2D surveys comprise of a single source vessel towing a single streamer along a single line, called a sailline, over a survey area. In 3D surveying, groups of sail lines are acquired, i.e., 3D acquisition is theacquisition of many 2D lines closely spaced over the area. Surveys that are acquired repeatedly overthe same area, particularly on established (or producing) fields to monitor changes in the reservoirover time due to production, are known as 4D or time-lapse surveys. The duration between surveyscan be on the order of months or years. Figure 1.5 illustrates a 2D and 3D survey geometry.Seismic data volumes are a collection of seismic traces. A seismic trace represents the response ofthe elastic wavefield to velocity and density contrasts across interfaces of layers of rock or sedimentsas energy travels from a source through the subsurface to a receiver or receiver array [SchlumbergerOilfield Glossary]. The convention adopted by the Society of Exploration Geophysicists (SEG) fordisplay of (zero-phase) seismic data is as follows: If the signal arises from a reflection that indicates2Figure 1.2: Schematic of different marine seismic surveys. “1” illustrates a towed-streamergeometry, “2” an ocean-bottom geometry, “3” a buried seafloor array, and “4” a VSP(vertical seismic profile) geometry, where the receivers are positioned in a well. [Source:Caldwell and Walker]Figure 1.3: Schematic of ocean-bottom node survey. Remotely operated vehicles (ROVs) areused to deploy and recover sensor nodes. [Source: Caldwell and Walker]3Figure 1.4: Summary of the majority of different types of marine seismic surveys. The letter“D” represents dimension and the letter “C” represents component (Z, X, Y ). [Source:Caldwell and Walker]Figure 1.5: Illustration of difference between 2D and 3D survey geometry for same surveyarea. The dashed lines suggest subsurface structure contour lines. [Source: Caldwelland Walker]4Figure 1.6: Four most common seismic trace display formats. [Source: SEG Wiki]an increase in acoustic impedance (product of density and seismic velocity), the polarity is positiveand is displayed as a peak. If the signal arises from a reflection that indicates a decrease in acousticimpedance, the polarity is negative and is displayed as a trough [Schlumberger Oilfield Glossary].Figure 1.6 illustrates the four most common trace display formats [SEG wiki].1.2 Simultaneous-source acquisitionMost of the commonly used processing algorithms, e.g., amplitude-versus-offset (AVO) analysis,surface-related multiple elimination (SRME) (Verschuur et al., 1992), estimation of primaries bysparse inversion (EPSI) (van Groenestijn and Verschuur, 2009; Lin and Herrmann, 2013), wave-equation based inversion techniques such as reverse-time migration (RTM) and full-waveform in-version (FWI) need dense and periodic (or regular) coverage of the survey area to produce high-resolution images of the subsurface. The need for dense sampling and full azimuthal coveragehave led to the use of multiple source vessels, and simultaneous-source acquisition techniques. Inseismic-acquisition literature, the term “azimuth” is defined as the angle at the source locationbetween the sail line and the direction to a given receiver. The “simultaneous-source acquisition”methodology is also referred to as “blended acquisition” (Beasley et al., 1998; de Kok and Gillespie,2002; Beasley, 2008; Berkhout, 2008; Moldoveanu, 2010; Abma et al., 2013; Mosher et al., 2014).Long restricted to land acquisition, simultaneous-source methodology has now been provenin a marine environment (see references below). In principle, simultaneous-source methodologyinvolves firing multiple sources at near-simultaneous/random-dithered times, hence, resulting inoverlaps between shot records, as opposed to no overlaps during conventional (periodically-sampled)acquisition. In land seismic acquisition, simultaneous sources have revolutionized the way surveysare acquired, providing much greater sampling and consequently better imaging (Bagaini, 2010, and5the references therein; Krohn and Neelamani, 2008; Neelamani et al., 2008). Marine simultaneous-source acquisition was invented in the year 1998-99 by the researchers at WesternGeco, however,its implementation had to overcome some fairly significant hurdles. The invention came just beforethe big downturn in the seismic industry that lasted until 2004, resulting in little or no interestfor this new technology. Cost of additional vessels was another big hurdle to surmount. Adventof wide-azimuth (WAZ) surveys, fortunately, made multivessel operations more common, and asmore companies saw its benefits, the simultaneous source concept began to gain momentum (Duey,2012).Since 2010-11, marine simultaneous-source acquisition has been an emerging technology stim-ulating both geophysical research and commercial efforts. The benefits of this methodology aresubstantial since seismic data is acquired in an economic and environmentally more sustainableway, i.e., data is acquired in less time, as compared to conventional acquisition, by firing multiplesources at near simultaneous/random times, or more data is acquired within the same time or acombination of both. Seismic acquisition literature contains a whole slew of works that have ex-plored the concept of simultaneous- or blended-source activation (Beasley et al., 1998; de Kok andGillespie, 2002; Beasley, 2008; Berkhout, 2008; Hampson et al., 2008; Moldoveanu, 2010; Berkhout,2012; Abma et al., 2013; Mosher et al., 2014). However, there are challenges associated withsimultaneous-source acquisition. Since many subsurface attribute inversion schemes (e.g., AVOanalysis, SRME, EPSI, FWI, RTM, etc.) still rely on single-source prestack data volumes, oneof the main challenges of simultaneous-source acquisition is to recover conventional sequential (orperiodic) data from simultaneous data, i.e., estimate interference-free shot (and receiver) gathers,and particularly recover subtle late reflections of low amplitudes that can be overlaid by interfer-ing seismic responses from other shots. This is known as source separation, also referred to as“deblending”.Stefani et al. (2007), Moore et al. (2008) and Akerberg et al. (2008) have observed that aslong as the sources are fired at suitably randomly-dithered times, the resulting interferences (insimultaneous data) will appear noise-like in specific gather domains such as common-offset andcommon-receiver, turning the separation into a typical (random) noise removal procedure. Appli-cation to land acquisition is reported in Bagaini and Ji (2010). Subsequent source-separation (ordeblending) techniques, which aim to remove noise-like source crosstalk, vary from vector-medianfilters (Huo et al., 2009) to inversion-type algorithms (Moore, 2010; Abma et al., 2010; Mahdadet al., 2011; Doulgeris et al., 2012) to a combination of both (Baardman and van Borselen, 2013).The former are mostly “processing” techniques where the interfering energy (i.e., source crosstalk) isremoved and not mapped back to coherent energy, at least not in a single step alone, while the latterare designed to take advantage of sparse representations of coherent seismic signals, which is ad-vantageous because they exploit inherent structure in seismic data. Maraschini et al. (2012), Chengand Sacchi (2013) and Kumar et al. (2015b) use matrix rank-reduction scheme for source separa-tion. Recent success of simultaneous-source field trials, its implementation and source-separationtechniques, have increased the industry’s confidence in this technology (Beasley et al., 2012; Abmaet al., 2012, 2013; Mosher et al., 2014), where it has been observed that simultaneous-source surveyscan be acquired faster and at lower costs.Theoretical results from Compressive Sensing (CS, Donoho (2006); Cande`s et al. (2006c)) sug-gest that there is a direct relationship between the acquisition design and the expected fidelity of theachieved structure-promoting recovery, however, the aforementioned works did not investigate thislink between the specific properties of the acquisition system and the sparsity-based recovery, espe-6cially in marine acquisition. Compressive sensing is a novel nonlinear sampling paradigm effectivefor signals that have a sparse representation in some transform domain. It is a novel paradigm inthe sense that it provides theory for the link between the acquisition design, sparsity-promoting sig-nal recovery and signal reconstruction quality. For land-based acquisition, the CS-related schemes(suitable for forward-modelling in the computer) presented by Neelamani et al. (2008) and Neela-mani et al. (2010) suggest the use of noise-like signals as sweeps, Lin and Herrmann (2009a) userandomly phase-encoded vibroseis sweeps, and Herrmann et al. (2009) use impulsive sources thatrequire modulation of each source by a randomly determined scaling in amplitude.Analysis of pragmatic marine acquisition schemes in terms of CS arguments remains challengingbecause, while the success of CS hinges on an incoherent (or random) sampling technique, adaptingthis approach to real-life problems in exploration seismology is subject to physical constraints on theplacement, type, and number of (simultaneous) sources, and number of receivers. Therefore, one ofthe objectives of this thesis is to propose a pragmatic sampling technique for marine simultaneous-source acquisition that adapts ideas from CS and no longer relies on the Nyquist sampling criteria.This technique, termed time-jittered marine, aims to achieve increased source sampling density andshorter acquisition times, and thus mitigate acquisition-related costs. Pioneered by Hennenfent andHerrmann (2008) and Herrmann (2010), this thesis presents a pragmatic CS simultaneous-sourceacquisition technique, specifically for marine, and addresses the challenge of source separation bysparsity-promoting recovery via convex optimization using `1 objectives. Over the course of thisresearch, Mosher et al. (2014) reported successful field application of randomized CS surveys andalso showed the advantage of recovery via structure promotion in contrast to simply “processing”the acquired data. This encouraged further research on CS acquisition designs and processing, andtheir implication on time-lapse seismic.1.2.1 Static vs. dynamic simultaneous-source acquisitionBased on the acquisition geometry, i.e., static or dynamic, there are different ways of acquiringsimultaneous data.Land simultaneous-source acquisitionAn instance of land simultaneous-source acquisition is shown in Figure 1.7, wherein multiple vi-broseis vehicles fire shots simultaneously. Specifically, sequential impulsive sources are replaced byimpulsive simultaneous ‘phase-encoded’ sources. Chapter 2 presents this scenario in more details.After simultaneous data is acquired, the aim is then to recover individual sequential shot recordsas acquired during conventional acquisition.Marine simultaneous-source acquisitionAs mentioned above, and also explained in this thesis, to render possible artifacts induced bysimultaneous-source firing incoherent, the key for simultaneous-source acquisition is the inclusion ofrandomization in the acquisition design, e.g., randomizing shot-firing times, randomizing source/re-ceiver positions, randomizing distance between sail lines, etc. Randomization of shot-firing timesdepends on whether data is acquired with dynamic towed streamers or static receivers (OBCs orOBNs). Figure 1.8 illustrates the variability (or randomness) in shot-firing times for static and dy-namic marine acquisition geometries. For dynamic towed streamers, randomness in shot-firing time7(a)(b)(c)Figure 1.7: Schematic of land simultaneous-source acquisition. (a) and (b) Individual shotsacquired in a conventional survey. (c) Simultaneous shot acquired in a simultaneous-source survey. Images courtesy ION (www.iongeo.com).8Figure 1.8: Shot-time randomness (or variability) for static and dynamic marinesimultaneous-source acquisitions.is small, on the order of a few seconds, i.e., sources fire within 1 or 2 second(s) of each other. This isbecause receivers are in motion, and therefore the returning sounds waves from the subsurface needto be captured within the time-frame of one shot record, which is typically 10 seconds. Moreover,moving arrays can only be compensated for a couple of meters. In contrast, static geometries enjoylarge degrees of randomness, on the order of tens of seconds, in shot-firing times since the receiversare fixed.Two instances of dynamic simultaneous-source acquisitions are over/under (or multilevel) sourceacquisition (Hill et al., 2006; Moldoveanu et al., 2007; Lansley et al., 2007; Long, 2009; Hegna andParkes, 2012; Hoy et al., 2013) and Simultaneous-Long Offset (SLO) acquisition (Long et al., 2013,and the references therein). Over/under source acquisition extends the recorded bandwidth at thelow and high ends of the spectrum because the depths of the sources produce complementary ghostfunctions, avoiding deep notches in the spectrum, while SLO acquisition provides a longer coveragein offsets without the need to tow very long streamer cables that can be problematic to deal within the field (Chapter 7). Both acquisitions generate simultaneous data volumes that need to beseparated in to corresponding individual data volumes for further processing (Figures 1.9 and 1.10).The proposed CS simultaneous-source acquisition scheme, i.e., time-jittered marine acquisitionis an instance of static simultaneous-source acquisition, wherein a single (and/or multiple) sourcevessel(s) sail(s) across an ocean-bottom array firing air guns at randomly jittered-time instances,which translate to (sub-Nyquist or subsampled) jittered shot positions for a given (fixed) speed ofthe source vessel. The basic idea of jittered subsampling is to regularly decimate the interpolationgrid and subsequently perturb the coarse-grid sample points on the fine grid while controlling themaximum gap size between adjacent sample points (i.e., shot locations). Figure 1.11 illustrates thisacquisition scheme for a single source vessel and two air gun arrays with receivers (OBC) recordingcontinuously, resulting in a continuous subsampled simultaneous time-jittered data volume. Theproposed acquisition scheme leads to improved spatial sampling of recovered (or separated) data,and speedup in acquisition compared to conventional periodically-sampled acquisition (Chapters 4-6). The improvement in spatial sampling is a result of separation and interpolation of simultaneousdata, which is acquired on (relatively) coarsely-sampled spatial grids, to finely-sampled spatial9Figure 1.9: Schematic of dynamic over/under marine simultaneous-source acquisition. Si-multaneous data acquired in the field is separated in to individual source componentsusing source-separation techniques.grids. Since static geometries provide better control over receiver positioning compared to dynamictowed streamers, the latter is a relatively more challenging scenario for source separation (Chapters4-7). Additional challenges include processing of massive (especially 3D) simultaneous data volumesin computationally efficient ways, i.e., to reduce computational time at each step of the recoveryalgorithm, and efficient ways to store recovered data volumes in memory. Chapters 4 and 7 providedetails on how we address these challenges.1.3 Time-lapse seismicTime-lapse (or 4D) seismic techniques involve acquisition, processing and interpretation of multiple2D or 3D seismic surveys over a producing field with the aim of understanding the changes in thereservoir over time, particularly its behaviour during production (Lumley, 2001; Fanchi, 2001). Theneed for high degrees of data repeatability in time-lapse seismic has lead to the need to replicatedense surveys that are mostly OBC/OBN surveys, since these surveys provide better control overreceiver positioning compared to towed streamers. Densely sampled and replicated surveys areexpensive, and generate dense time-lapse data volumes whose processing is also computationallyexpensive. Therefore, the challenge is to minimize the cost of time-lapse surveying and data pro-cessing without impacting data repeatability. Owing to the positive impact of simultaneous-sourceacquisition on the industry, i.e., improved survey efficiency and data density, two key questionsarise: “What are the implications of randomization on the attainable repeatability of time-lapseseismic?”, and “Should randomized time-lapse surveys be replicated?” These questions are of greatimportance because the incorporation of simultaneous-source acquisition in time-lapse seismic can10Figure 1.10: Schematic of dynamic marine simultaneous-long acquisition. Simultaneous dataacquired in the field is separated in to individual source components using source-separation techniques. Note that the streamer length is 6 km and the second sourcevessel is deployed one spread-length (6 km) ahead of the main seismic vessel.significantly change the current paradigm of time-lapse seismic that relies on expensive dense pe-riodic sampling and replication of the baseline and monitor surveys (Lumley and Behrens, 1998).Therefore, another objective of this thesis is to analyze the effects of simultaneous-source (or ran-domized) surveys in time-lapse seismic by comparing repeatability of data recovered from fullyreplicated randomly-subsampled surveys and nonreplicated randomly-subsampled surveys. To thisend, we present a new approach that explicitly exploits common information shared by the differenttime-lapse vintages. Note that we refer to the baseline and monitor data as the time-lapse vintages.The presented joint-recovery method (JRM), which is derived from distributed compressive sensing(DCS, Baron et al. (2009)), inverts for the common component and innovations with respect tothis common component.1.4 ObjectivesThe main purpose of this thesis is to develop practical compressive randomized marine simultaneous-source acquisitions and source-separation techniques by adapting ideas from compressive sensing.The objectives can be summarized as follows:1. Adapt ideas from CS to design pragmatic marine simultaneous-source acquisition that ac-quires data economically with a reduced environmental imprint, whereby cost of surveysdepends on certain inherent structure in seismic data rather than on the Nyquist samplingcriteria. Investigate the relationship between the acquisition design and recovery quality.11(a)(b)Figure 1.11: Schematic of static marine simultaneous-source acquisition. This also illustratesthe design of our proposed time-jittered marine acquisition. Source separation aims torecover densely sampled interference-free data by unravelling overlapping shot recordsand interpolation to a fine source grid (Chapters 4–6).2. Address the challenge of source separation through a combination of tailored simultaneous-source acquisition design and sparsity-promoting recovery via convex optimization using `1objectives.3. Compare sparsity-promoting and rank-minimization recovery techniques for static and dy-namic marine simultaneous-source acquisitions. Develop a thorough understanding of theperformance and limitations of the two techniques.4. Adapt ideas from distributed compressive sensing to analyze the implications of simultaneous-source (or randomized) surveys on the attainable repeatability of time-lapse seismic.121.5 ContributionsTo the best of our knowledge, this work represents a first step towards a comprehensive studyof designing marine simultaneous-source acquisitions by adapting ideas from compressive sensing.We use CS metrics, such as mutual coherence and restricted isometry property, to investigate therelationship between the acquisition design and data reconstruction fidelity. Consequently, we areable to assert the importance of randomness in the acquisition system in combination with anappropriate choice for a sparsifying transform in the reconstruction algorithm. This leads to newinsights into acquiring and processing seismic data where we can fundamentally rethink on how wedesign acquisition surveys. Compressive (or subsampled) randomized (or jittered) simultaneous-source acquisitions and subsequent recovery techniques lead to improved wavefield reconstructionby increasing source-sampling density, and also speedup acquisition.This work also presents a first instance of adapting ideas from CS and DCS to assess the effects(or risks) of random or irregular spatial sampling (i.e., samples that do not lie on a regular orperiodic grid) in the field on time-lapse data, and demonstrate that high-quality data recoveriesare the norm and not the exception. The main finding that compressive randomized time-lapsesurveys need not be replicated to attain similar/acceptable levels of repeatability is significant sinceit can potentially change the current paradigm of time-lapse seismic that relies on expensive denseperiodic sampling and replication of time-lapse surveys. Using a joint-recovery model (JRM) toprocess compressive randomized time-lapse data, we observe that recovery of the vintages improveswhen the time-lapse surveys are not replicated, since independent surveys give additional structuralinformation. Moreover, since irregular spatial sampling is inevitable in the real world, it would bebetter to focus on knowing what the shot positions were (post acquisition) to a sufficient degree ofaccuracy, than aiming to replicate them. Recent successes of randomized surveys in the field (see,e.g., Mosher et al. (2014)) show that this can be achieved in practice.1.6 OutlineThe thesis comprises of eight chapters, including this introduction. We begin by presenting asynopsis of compressive sensing in Chapter 2, where we outline its three key principles: (i) Findrepresentations that reveal structure of data, e.g, sparse (few nonzero entries) or compressible(can be well-approximated by a sparse signal). Examples of such representations are sparsifyingtransform domains, e.g., Fourier, curvelets, etc. (ii) Design a randomized subsampling scheme,which turns subsampling related artifacts into incoherent noise that is not sparse or compressible.(iii) Recover artifact-free fully sampled data by promoting structure, i.e., sparse recovery via one-norm minimization. Motivated by the challenges of the Nyquist’s sampling criterion and “curse ofdimensionality” (exponential increase in volume when adding extra dimensions to data collection)in exploration seismology, we discuss how we adapt ideas from CS to land and (static) marinesimultaneous-source seismic acquisitions. We demonstrate the benefits of random subsamplingover periodic subsampling by conducting of 1D synthetic experiments using random realizations ofharmonic signals. We also empirically demonstrate that curvelets lead to compressible (real-worldsignals are not strictly sparse) representation of seismic data compared to wavelets, wave atoms,etc. Hence, we use curvelets for recovery via sparsity promotion.Since the primary objective of this thesis to develop practical compressive randomized marinesimultaneous-source acquisitions, the focus henceforth is on marine simultaneous-source acquisi-tions only. In Chapter 3, we identify (static) marine simultaneous-source acquisition as a linear13subsampling system, which we subsequently analyze using metrics from compressive sensing, suchas mutual coherence and restricted isometry property. Importance of sparsity-based recovery forsimultaneous-source acquisition has been noticed by many authors. However, few have thoroughlyinvestigated the underlying interaction between acquisition design and reconstruction fidelity, es-pecially in the marine setting. According to CS, a sparsifying transform that is incoherent withthe CS matrix can significantly impact the reconstruction quality. We demonstrate that the CSmatrix resulting from our proposed sampling scheme is incoherent with the curvelet transform.We quantitatively verify the importance of randomness in the acquisition system and more com-pressible transforms by comparing recoveries from synthetic seismic experiments (using data froma real seismic line) for three different simultaneous acquisition schemes, namely “ideal” simulta-neous acquisition, random time-dithering and periodic time-dithering acquisition, with differentsubsamplings for each.While the observations made from Chapters 2 and 3 on adapting CS ideas to simultaneous-source acquisitions are very encouraging, it is important to not forget that compressive marinesimultaneous-source acquisitions are beneficial to the seismic industry only if they are physicallyrealizable in the field. We learnt this lesson while presenting the work at a conference meeting,where the nonrealistic nature of our acquisition scheme was revealed to us by communications withindustry experts. This lead to a detailed investigation of the proposed compressive acquisitionscheme to render it practical, and the findings of which are reported in Chapter 4. We developa pragmatic compressive marine simultaneous-source acquisition scheme, termed time-jittered ma-rine, wherein a single (and/or multiple) source vessel(s) sails across an ocean-bottom array firingair guns at jittered-time instances, which translate to jittered shot positions for a given (fixed)speed of the source vessel. The simultaneous data are time compressed, and are therefore acquiredeconomically with a small environmental imprint. We demonstrate that all the observations madein Chapters 2 and 3 hold true for the pragmatic time-jittered marine acquisition. We conductsynthetic seismic experiments to recover interference-free densely sampled data from compressive(or subsampled) simultaneous data via sparsity promotion using 2D and 3D curvelet transforms(Cande`s et al., 2006a; Ying et al., 2005).Chapter 5 delves in to the implications of compressive randomized acquisitions on the attain-able repeatability of time-lapse seismic. Adapting ideas from CS and DCS, we aim to address thechallenges of the current time-lapse seismic paradigm, such as, reliance on expensive dense periodicand replicated time-lapse surveys. We demonstrate that under certain assumptions, high-qualityprestack data can be obtained from randomized subsampled measurements that are observed fromnonreplicated surveys. We present a joint-recovery method (JRM) that exploits common informa-tion among the vintages leading to significant improvements in recovery quality of the time-lapsevintages when the same on-the-grid shot locations are not revisited. We compare joint recoveryof time-lapse data with independent recovery, where each vintage is recovered independently. Theacquisition is low cost because the measurements are subsampled. We conduct numerous 1D syn-thetic stylized experiments to test the JRM. We also confirm that high degrees of repeatability areachievable from the proposed time-jittered marine acquisition scheme. We assume measurementsare taken on-the-grid (i.e., a discrete grid where the measurements lie “exactly” on the grid) andignore errors related to taking measurements off the grid, a more realistic scenario that is dealtwith in Chapter 6.Since irregular or off-the-grid spatial sampling of sources and receivers is inevitable in fieldseismic acquisitions, it is important to analyze the implications of randomization on time-lapse14seismic in this realistic setting. In Chapter 6, we extend our time-jittered marine acquisition totime-lapse surveys by designing acquisition on irregular spatial grids that render simultaneous,subsampled, and irregular measurements. We adapt the JRM to incorporate a regularizationoperator that maps traces from an irregular spatial grid to a regular periodic grid. The recoverymethod is therefore a combined operation of regularization, interpolation (estimating missing fine-grid traces from subsampled coarse-grid data), and source separation (unraveling overlapping shotrecords). We introduce the nonequispaced fast discrete curvelet transform (NFDCT, Hennenfentet al. (2010)) and its application to recover periodic densely sampled seismic lines from simultaneousand irregular measurements via sparsity-promoting inversion. We conduct a series of syntheticseismic experiments with different random realizations of the time-jittered marine acquisition withirregular sampling.Chapter 7 addresses the source-separation problem for the more challenging dynamic towed-streamer acquisitions. We formulate the problem as a CS problem, which we subsequently solveby promoting two types of structure in seismic data, i.e., sparse and low rank. We simulatetwo simultaneous towed-streamer acquisitions, namely over/under and SLO. For recovery via rankminimization, we adopt the hierarchical semiseparable (HSS) matrix representation method pro-posed by Chandrasekaran et al. (2006) to exploit low-rank structure at high frequencies. We alsocombine the singular-value-decomposition-free matrix factorization approach recently developed byLee et al. (2010) with the Pareto curve approach proposed by Berg and Friedlander (2008) thatrenders this framework suitable for large-scale seismic data because it avoids expensive singularvalue decompositions (SVDs), a necessary step in traditional rank-minimization-based methods.We compare recovery via sparsity promotion and rank minimization in terms of separation qual-ity, computational time, and memory usage. We also make comparisons with the NMO-basedmedian-filtering-type technique proposed by Chen et al. (2014).In Chapter 8, we summarize the work done in this thesis, discuss certain associated limitationsand propose ideas to address them by means of future research directions.15Chapter 2Compressive sensing in seismicexploration: an outlook on a newparadigm2.1 SummaryMany seismic exploration techniques rely on the collection of massive data volumes that are subse-quently mined for information during processing. While this approach has been extremely successfulin the past, current efforts toward higher resolution images in increasingly complicated regions ofthe Earth continue to reveal fundamental shortcomings in our workflows. Chiefly amongst these isthe so-called “curse of dimensionality” exemplified by Nyquist’s sampling criterion, which dispro-portionately strains current acquisition and processing systems as the size and desired resolution ofour survey areas continues to increase. We offer an alternative sampling method leveraging recentinsights from compressive sensing towards seismic acquisition and processing for data that, from atraditional point of view, are considered to be undersampled. The main outcome of this approachis a new technology where acquisition and processing related costs are decoupled the stringentNyquist sampling criterion.At the heart of our approach lies randomized incoherent sampling that breaks subsampling-related interferences by turning them into harmless noise, which we subsequently remove by pro-moting sparsity in a transform-domain. Acquisition schemes designed to fit into this regime nolonger grow significantly in cost with increasing resolution and dimensionality of the survey area,but instead its cost ideally only depends on transform-domain sparsity of the expected data. Ourcontribution is twofold. First, we demonstrate by means of carefully designed numerical exper-iments that ideas from compressive sensing can be adapted to seismic acquisition. Second, weleverage the property that seismic data volumes are well approximated by a small percentage ofcurvelet coefficients. Thus curvelet-domain sparsity allows us to recover conventionally-sampledseismic data volumes from compressively-sampled data volumes whose size exceeds this percentageby only a small factor. Because compressive sensing combines transformation and encoding by asingle linear encoding step, this technology is directly applicable to seismic acquisition and thereforeA version of this chapter has been published in CSEG Recorder, 2011, vol. 36, Part 1 [April Edition]: pp. 19–33,Part 2 [June Edition]: pp. 34–39.16constitutes a new paradigm where acquisitions costs scale with transform-domain sparsity insteadof with the gridsize. We illustrate this principle by showcasing recovery of a real seismic line fromsimulated compressively sampled acquisitions.2.2 Inspiration2.2.1 Nyquist sampling and the curse of dimensionalityThe livelihood of exploration seismology depends on our ability to collect, process, and imageextremely large seismic data volumes. The recent push towards full-waveform approaches onlyexacerbates this reliance, and we, much like researchers in many other fields in science and en-gineering, are constantly faced with the challenge to come up with new and innovative ways tomine this overwhelming barrage of data for information. This challenge is especially daunting inexploration seismology because our data volumes sample wavefields that exhibit structure in up tofive dimensions (two coordinates for the sources, two for the receivers, and one for time). When ac-quiring and processing this high-dimensional structure, we are not only confronted with Nyquist’ssampling criterion but we also face the so-called “curse of dimensionality”, which refers to theexponential increase in volume when adding extra dimensions to our data collection.These two challenges are amongst the largest impediments to progress in the application ofmore sophisticated seismic methods to oil and gas exploration. In this chapter, we introduce a newmethodology adapted from the field of “compressive sensing” or “compressive sampling” (CS inshort throughout the article, Cande`s et al., 2006c; Donoho, 2006; Mallat, 2009), which is aimed atremoving these impediments via dimensionality reduction techniques based on randomized subsam-pling. With this dimensionality reduction, we arrive at a sampling framework where the samplingrates are no longer scaling directly with the gridsize, but by transform-domain compression; morecompressible data requires less sampling.2.2.2 Dimensionality reduction by compressive sensingCurrent nonlinear data-compression techniques are based on high-resolution linear sampling (e.g.,sampling by a CCD chip in a digital camera) followed by a nonlinear encoding technique thatconsists of transforming the samples to some transformed domain, where the signal’s energy isencoded by a relatively small number of significant transform-domain coefficients (Mallat, 2009).Compression is accomplished by keeping only the largest transform-domain coefficients. Becausethis compression is lossy, there is an error after decompression. A compression ratio expresses thecompressed-signal size as a fraction of the size of the original signal. The better the transformcaptures the energy in the sampled data, the larger the attainable compression ratio for a fixedloss.Even though this technique underlies the digital revolution of many consumer devices, includingdigital cameras, music, movies, etc., it does not seem possible for exploration seismology to scalein a similar fashion because of two major hurdles. First, high-resolution data has to be collectedduring the linear sampling step, which is already prohibitively expensive for exploration seismology.Second, the encoding phase is nonlinear. This means that if we select a compression ratio thatis too high, the decompressed signal may have an unacceptable error, in the worst case making itnecessary to repeat collection of the high-resolution samples.17By replacing the combination of high-resolution sampling and nonlinear compression by a singlerandomized subsampling technique that combines sampling and encoding in one single linear step,CS addresses many of the above shortcomings. First of all, randomized subsampling has the distinctadvantage that the encoding is linear and does not require access to high-resolution data duringencoding. This opens possibilities to sample incrementally and to process data in the compresseddomain. Second, encoding through randomized sampling suppresses subsampling related artifacts.Coherent subsampling related artifacts—whether these are caused by periodic missing traces or bycross-talk between coherent simultaneous-sources—are turned into relatively harmless incoherentGaussian noise by randomized subsampling (see e.g. Herrmann and Hennenfent, 2008; Hennenfentand Herrmann, 2008; Herrmann et al., 2009, for seismic applications of this idea).By solving a sparsity-promoting problem (Cande`s et al., 2006c; Donoho, 2006; Herrmann et al.,2007; Mallat, 2009), we reconstruct high-resolution data volumes from the randomized samplesat the moderate cost of a minor oversampling factor compared to data volumes obtained afterconventional compression (see e.g. Donoho et al., 1999a, for wavelet-based compression). Withsufficient sampling, this nonlinear recovery outputs a set of largest transform-domain coefficientsthat produces a reconstruction with a recovery error comparable with the error incurred duringconventional compression. As in conventional compression this error is controllable, but in case ofCS this recovery error depends on the sampling ratio—i.e., the ratio between the number of samplestaken and the number of samples of the high-resolution data. Because compressively sampled datavolumes are much smaller than high-resolution data volumes, we reduce the dimensionality andhence the costs of acquisition, storage, and possibly of data-driven processing.We mainly consider recovery methods that derive from compressive sampling. Therefore ourmethod differs from interpolation methods based on pattern recognition (Spitz, 1999), plane-wavedestruction (Fomel et al., 2002) and data mapping (Bleistein et al., 2001), including parabolic,apex-shifted Radon and DMO-NMO/AMO (Trad, 2003; Trad et al., 2003; Harlan et al., 1984; Hale,1995; Canning and Gardner, 1996; Bleistein et al., 2001; Fomel, 2003; Malcolm et al., 2005). Tobenefit fully from this new sampling paradigm, we will translate and adapt its ideas to explorationseismology while evaluating their performance. Here lies our main contribution. Before we embarkon this mission we first share some basic insights from compressive sensing in the context of a well-known problem in geophysics: recovery of time-harmonic signals, which is relevant for missing-traceinterpolation.Compressive sensing is based on three key elements: randomized sampling, sparsifying trans-forms, and sparsity-promotion recovery by convex optimization. By themselves, these elementsare not new to geophysics. Spiky deconvolution and high-resolution transforms are all based onsparsity-promotion (Taylor et al., 1979; Oldenburg et al., 1981; Ulrych and Walker, 1982; Levy et al.,1988; Sacchi et al., 1994) and analyzed by mathematicians (Santosa and Symes, 1986; Donoho andLogan, 1992); wavelet transforms are used for seismic data compression (Donoho et al., 1999a);randomized samples have been shown to benefit Fourier-based recovery from missing traces (Tradet al., 2003; Xu et al., 2005; Abma and Kabir, 2006; Zwartjes and Sacchi, 2007). The novelty of CSlies in the combination of these concepts into a comprehensive theoretical framework that providesdesign principles and performance guarantees.182.2.3 ExamplesPeriodic versus uniformly-random subsamplingBecause Nyquist’s sampling criterion guarantees perfect reconstruction of arbitrary bandwidth-limited signals, it has been the leading design principle for seismic data acquisition and processing.This explains why acquisition crews go at length to place sources and receivers as finely and asregularly as possible. Although this approach spearheaded progress in our field, CS proves thatperiodic sampling at Nyquist rates may be far from optimal when the signal of interest exhibitssome sort of structure, such as when the signal permits a transform-domain representation with fewsignificant and many zero or insignificant coefficients. For this class of signals (which includes nearlyall real-world signals) it suffices to sample randomly with fewer samples than that determined byNyquist.Take any arbitrary time-harmonic signal. According to compressive sensing, we can guaranteeits recovery from a very small number of samples drawn at random times. In the seismic situation,this corresponds to using seismic arrays with fewer geophones selected uniformly-randomly from anunderlying regular sampling grid with spacings defined by Nyquist (meaning it does not violate theNyquist sampling theorem). By taking these samples randomly instead of periodically, the majorityof artifacts directly due to incomplete sampling will behave like Gaussian white noise (Hennenfentand Herrmann, 2008; Donoho et al., 2009) as illustrated in Figure 2.1. We observe that for thesame number of samples the subsampling artifacts can behave very differently.In the geophysical community, subsampling-related artifacts are commonly known as “spectralleakage” (Xu et al., 2005), where energy from each frequency is leaked to other frequencies. Under-standably, the amount of spectral leakage depends on the degree of subsampling: the higher thisdegree the more leakage. However, the characteristics of the artifacts themselves depend on theirregularity of the sampling. The more uniformly-random our sampling is, the more the leakagebehaves as zero-centered Gaussian noise spread over the entire frequency spectrum.Compressive sensing schemes aim to design acquisition that specifically create Gaussian-noiselike subsampling artifacts (Donoho et al., 2009). As opposed to coherent subsampling relatedartifacts (Figure 2.1(f)), these noise-like artifacts (Figure 2.1(d)) can subsequently be removed by asparse recovery procedure, during which the artifacts are separated from the signal and amplitudesare restored. Of course, the success of this method also hinges on the degree of subsampling, whichdetermines the noise level, and the sparsity level of the signal.By carrying out a random ensemble of experiments, where random realizations of harmonicsignals are recovered from randomized samplings with decreasing sampling ratios, we confirm thisbehavior empirically. Our findings are summarized in Figure 2.2. The estimated spectra are ob-tained by solving a sparsifying program with the Spectral Projected Gradient for `1 solver (SPGL1- Berg and Friedlander, 2008) for signals with k non-zero entries in the Fourier domain. We definethese spectra by randomly selecting k entries from vectors of length 600 and populating these withvalues drawn from a Gaussian distribution with unit standard deviation. As we will show below,the solution of each of these problems corresponds to the inversion of a matrix whose aspect ratio(the ratio of the number of columns over the number of rows) increases as the number of samplesdecreases.To get reasonable estimates, each experiment is repeated 100 times for the different subsamplingschemes and for varying sampling ratios ranging from 1/2 to 1/6. The reconstruction error is thenumber of vector entries at which the estimated spectrum and the true spectrum disagree by19more than 10−4. This error counts both false positives and false negatives. The averaged resultsfor the different experiments are summarized in Figures 2.2(a) and 2.2(b), which correspond toregular and random subsampling, respectively. The horizontal axes in these plots represent therelative underdeterminedness of the system, i.e., the ratio of the number k of nonzero entries inthe spectrum to the number of acquired data points n. The vertical axes denote the percentage oferroneous entries. The different curves represents the different subsampling factors. In each plot,the curves from top to bottom correspond to sampling ratios of 1/2 to 1/6.Figure 2.2(a) shows that, regardless of the subsampling factor, there is no range of relativeunderdeterminedness for which the spectrum, and hence the signal, can accurately be recoveredfrom regular subsamplings. Sparsity is not enough to discriminate the signal components from thespectral leakage. The situation is completely different in Figure 2.2(b) for the random sampling.In this case, one can observe that for a subsampling ratio of 1/2 exact recovery is possible for0 < k/n . 1/4. The main purpose of these plots is to qualitatively show the transition fromsuccessful to failed recovery. The quantitative interpretation for these diagrams of the transitionis less well understood but also observed in phase diagrams in the literature (Donoho and Tanner,2009; Donoho et al., 2009). A possible explanation for the observed behavior of the error lies inthe nonlinear behavior of the solvers and on an error not measured in the `2 sense.2.2.4 Main contributionsWe propose and analyze randomized sampling schemes, termed compressive seismic acquisition.Under specific conditions, these schemes create favourable recovery conditions for seismic wavefieldreconstructions that impose transform-domain sparsity in Fourier or Fourier-related domains (seee.g. Sacchi et al., 1998; Xu et al., 2005; Zwartjes and Sacchi, 2007; Herrmann et al., 2007; Hennen-fent and Herrmann, 2008; Tang et al., 2009). Our contribution is twofold. First, we demonstrateby means of carefully designed numerical experiments on synthetic and real data that compressivesensing can successfully be adapted to seismic acquisition, leading to a new generation of random-ized acquisition and processing methodologies where high-resolution wavefields can be sampled andreconstructed with a controllable error. We introduce a number of performance measures that allowus to compare wavefield recoveries based on different sampling schemes and sparsifying transforms.Second, we show that accurate recovery can be accomplished for compressively sampled data vol-umes sizes that exceed the size of conventional transform-domain compressed data volumes by asmall factor. Because compressive sensing combines transformation and encoding by a single linearencoding step, this technology is directly applicable to seismic acquisition and to dimensionalityreduction during processing. We verify this claim by a series of experiments on real data. We alsoshow that the linearity of CS allows us to extend this technology to seismic data processing. Ineither case, sampling, storage, and processing costs scale with transform-domain sparsity.2.2.5 OutlineFirst, we briefly present the key principles of CS, followed by a discussion on how to adapt theseprinciples to the seismic situation. For this purpose, we introduce measures that quantify recon-struction and recovery errors and expresses the overhead that CS imposes. We use these measuresto compare the performance of different transform domains and sampling strategies during re-construction. We then use this information to evaluate and apply this new sampling technologytowards acquisition and processing of a 2D seismic line.202.3 Basics of compressive sensingIn this section, we give a brief overview of CS and concise recovery criteria. CS relies on specificproperties of the compressive-sensing matrix and the sparsity of the to-be-recovered signal.2.3.1 Recovery by sparsity-promoting inversionConsider the following linear forward model for samplingb = Ax0, (2.1)where b ∈ Rn represents the compressively sampled data consisting of n measurements. Supposethat a high-resolution data f0 ∈ RN , with N the ambient dimension, has a sparse representationx0 ∈ RN in some known transform domain. For now, we assume that this representation is theidentity basis—i.e., f0 = x0. We will also assume that the data is noise free. According to thismodel, measurements are defined as inner products between rows of A and high-resolution data.The sparse recovery problem involves the reconstruction of the vector x0 ∈ RN given incompletemeasurements b ∈ Rn with n  N . This involves the inversion of an underdetermined system ofequations defined by the matrix A ∈ Rn×N , which represents the sampling operator that collectsthe acquired samples from the model, f0.The main contribution of CS is to come up with conditions on the compressive-sampling matrixA and the sparse representation x0 that guarantee recovery by solving a convex sparsity-promotingoptimization problem. This sparsity-promoting program leverages sparsity of x0 and hence over-comes the singular nature of A when estimating x0 from b. After sparsity-promoting inversion,the recovered representation for the signal is given byx˜ = arg minx||x||1 subject to b = Ax. (2.2)In this expression, the symbol ˜ represents estimated quantities and the `1 norm ‖x‖1 is definedas ‖x‖1 def=∑Ni=1 |x[i]|, where x[i] is the ith entry of the vector x.Minimizing the `1 norm in equation 2.2 promotes sparsity in x and the equality constraintensures that the solution honors the acquired data. Among all possible solutions of the (severely)underdetermined system of linear equations (n N) in equation 2.1, the optimization problem inequation 2.2 finds a sparse or, under certain conditions, the sparsest (i.e., smallest `0 norm (Donohoand Huo, 2001)) possible solution that exactly explains the data.2.3.2 Recovery conditionsThe basic idea behind CS (see e.g. Cande`s et al., 2006c; Mallat, 2009) is that recovery is possibleand stable as long as any subset S of k columns of the n×N matrix A—with k ≤ N the number ofnonzeros in x—behave approximately as an orthogonal basis. In that case, we can find a constantδˆk for which we can bound the energy of the signal from above and below —i.e.,(1− δˆk)‖xS‖2`2 ≤ ‖ASxS‖2`2 ≤ (1 + δˆk)‖xS‖2`2 , (2.3)where S runs over sets of all possible combinations of columns with the number of columns |S| ≤ k(with |S| the cardinality of S). The smaller δˆk, the more energy is captured and the more stable21the inversion of A becomes for signals x with maximally k nonzero entries.The key factor that bounds the restricted-isometry constants δˆk > 0 from above is the mutualcoherence amongst the columns of A—i.e.,δˆk ≤ (k − 1)µ (2.4)withµ = max1≤i 6=j≤N|aHi aj |, (2.5)where ai is the ith column of A and H denotes the Hermitian transpose.Matrices for which δˆk is small contain subsets of k columns that are incoherent. Randommatrices, with Gaussian i.i.d. entries with variance n−1 have this property, whereas deterministicconstructions almost always have structure.For these random Gaussian matrices (there are other possibilities such as Bernouilli or restrictedFourier matrices that accomplish approximately the same behavior, see e.g. Cande`s et al., 2006c;Mallat, 2009), the mutual coherence is small. For this type of CS matrices, it can be proven thatEquation 2.3 holds and Equation 2.2 recovers x0’s exactly with high probability as long as thisvector is maximally k sparse withk ≤ C · nlog2(N/n), (2.6)where C is a moderately sized constant. This result proves that for large N , recovery of k nonzerosonly requires an oversampling ratio of n/k ≈ C · log2N , as opposed to taking all N measurements.The above result is profound because it entails an oversampling with a factor C ·log2N comparedto the number of nonzeros k. Hence, the number of measurements that are required to recover thesenonzeros is much smaller than the ambient dimension (n N for large N) of high-resolution data.Similar results hold for compressible instead of strictly sparse signals while measurements can benoisy (Cande`s et al., 2006c; Mallat, 2009). In that case, the recovery error depends on the noise leveland on the transform-domain compression rate—i.e., the decay of the magnitude-sorted coefficients.In summary, according to CS (Cande`s et al., 2006b; Donoho, 2006), the solution x˜ of equation2.2 and x0 coincide when two conditions are met, namely 1) x0 is sufficiently sparse, i.e., x0 hasfew nonzero entries, and 2) the subsampling artifacts are incoherent, which is a direct consequenceof measurements with a matrix whose action mimics that of a Gaussian matrix.Unfortunately, most rigorous results from CS, except for work by Rauhut et al. (2008), arevalid for orthonormal measurement and sparsity bases only and the computation of the recoveryconditions for realistically sized seismic problems remains computational prohibitive. To overcomethese important shortcomings, we will in the next section introduce a number of practical andcomputable performance measures that allow us to design and compare different compressive-seismic acquisition strategies.2.4 Compressive-sensing designAs we have seen, the machinery that supports sparse recovery from incomplete data depends onspecific properties of the compressive-sensing matrix. It is important to note that CS is not meant tobe blindly applied to arbitrary linear inversion problems. To the contrary, the success of a samplingscheme operating in the CS framework hinges on the design of new acquisition strategies that areboth practically feasible and lead to favourable conditions for sparse recovery. Mathematically22speaking, the resulting CS sampling matrix needs to both be realizable and behave as a Gaussianmatrix. To this end, the following key components need to be in place:1. a sparsifying signal representation that exploits the signal’s structure by mapping theenergy into a small number of significant transform-domain coefficients. The smaller thenumber of significant coefficients, the better the recovery;2. sparse recovery by transform-domain one-norm minimization that is able to handlelarge system sizes. The fewer the number of matrix-vector evaluations, the faster and morepractically feasible the wavefield reconstruction;3. randomized seismic acquisition that breaks coherent interferences induced by determin-istic subsampling schemes. Randomization renders subsampling related artifacts—includingaliases and simultaneous source crosstalk—harmless by turning these artifacts into incoherentGaussian noise;Given the complexity of seismic data in high dimensions and field practicalities of seismic acquisi-tion, the mathematical formulation of CS outlined in the previous section does not readily applyto seismic exploration. Therefore, we will focus specifically on the design of source subsamplingschemes that favor recovery and on the selection of the appropriate sparsifying transform. Becausetheoretical results are mostly lacking, we will guide ourselves by numerical experiments that aredesigned to measure recovery performance.During seismic data acquisition, data volumes are collected that represent discretizations ofanalog finite-energy wavefields in two or more dimensions including time. We recover the discretizedwavefield f by inverting the compressive-sampling matrixA :=restriction︷︸︸︷R M︸︷︷︸measurementsynthesis︷︸︸︷SH (2.7)with the sparsity-promoting program:f˜ = SH x˜ with x˜ = arg minx‖x‖1 :=P−1∑p=0|x[i]| subject to Ax = b. (2.8)This formulation differs from standard compressive sensing because we allow for a wavefield repre-sentation that is redundant—i.e., S ∈ CP×N with P ≥ N . Aside from results reported by Rauhutet al. (2008), which show that recovery with redundant frames is determined by the RIP constantδˆ of the restricted sampling and sparsifying matrices that is least favorable, there is no practicalalgorithm to compute these constants. Therefore, our hope is that the above sparsity-promoting op-timization program, which finds amongst all possible transform-domain vectors the vector x˜ ∈ RPthat has the smallest `1-norm, recovers high-resolution data f˜ ∈ RN .2.4.1 Seismic wavefield representationsOne of the key ideas of CS is leveraging structure within signals to reduce sampling. Typically,structure translates into transform-domains that concentrate the signal’s energy in as few as possible23significant coefficients. The size of seismic data volumes, along with the complexity of its high-dimensional and highly directional wavefront-like features, makes it difficult to find a transformthat accomplishes this task.To meet this challenge, we only consider transforms that are fast (at the most N logN with Nthe number of samples), multiscale (splitting the Fourier spectrum into dyadic frequency bands),and multidirectional (splitting Fourier spectrum into second dyadic angular wedges). For reference,we also include separable 2D wavelets in our study. We define this wavelet transform as theKronecker product (denoted by the symbol ⊗) of two 1D wavelet transforms: W = W1⊗W1 withW1 the 1D wavelet-transform matrix.Separable versus non-separable transformsThere exists numerous signal representations that decompose a multi-dimensional signal with re-spect to directional and localized elements. For the appropriate representation of seismic wavefields,we limit our search to non-separable curvelets (Cande`s et al., 2006a) and wave atoms (Demanet andYing, 2007). The elements of these transforms behave approximately as high-frequency asymptoticeigenfunctions of wave equations (see e.g. Smith, 1998; Cande`s and Demanet, 2005; Cande`s et al.,2006a; Herrmann et al., 2008), which makes these two representations particularly well suited forour task of representing seismic data parsimoniously.Unlike wavelets, which compose curved wavefronts into a superposition of multiscale “fat dots”with limited directionality, curvelets and wave atoms compose wavefields as a superposition ofhighly anisotropic localized and multiscale waveforms, which obey a so-called parabolic scalingprinciple. For curvelets in the physical domain, this principle translates into a support with itslength proportional to the square of the width. At the fine scales, this scaling leads to curvelets thatbecome increasingly anisotropic, i.e., needle-like. Each dyadic frequency band is split into a numberof overlapping angular wedges that double in every other dyadic scale. This partitioning results inincreased directionality at the fine scales. This construction makes curvelets well adapted to datawith impulsive wavefront-like features. Figure 2.3(a) shows the multiscale and multidirectional 2Dcurvelets in the time-space domain and the frequency-wavenumber domain. Curvelets approximatecurved singularities, i.e., wavefronts, in a nonadaptive manner with very few significant curveletcoefficients (Figure 2.3(b)). Wave atoms, on the other hand, are anisotropic because it is theirwavelength, not the physical length of the individual wave atoms, that depends quadratically ontheir width. By construction, wave atoms are more appropriate for data with oscillatory patterns.Because seismic data sits somewhere between these two extremes, we include both transforms inour study.Approximation errorFor an appropriately chosen representation magnitude-sorted transform-domain coefficients oftendecay rapidly–i.e., the magnitude of the jth largest coefficient is O(j−s) with s ≥ 1/2. For orthonor-mal bases, this decay rate is directly linked to the decay of the nonlinear approximation error (seee.g. Mallat, 2009). This error is expressed byσ(k) = ‖f − fk‖ = O(k1/2−s), (2.9)with fk the reconstruction from the largest k - coefficients. Notice that this error does not accountfor discretization errors (cf. Equation 2.16), which we ignore.24Unfortunately, this relationship between the decay rates of the magnitude-sorted coefficientsand the decay rate of the nonlinear approximation error does not hold for redundant transforms.Also, there are many coefficient sequences that explain the data f making them less sparse–i.e.,expansions with respect to this type of signal representations are not unique. For instance, analysisby the curvelet transform of a single curvelet does not produce a single non-zero entry in thecurvelet coefficient vector.To address this issue, we use an alternative definition for the nonlinear approximation error,which is based on the solution of a sparsity-promoting program. With this definition, the k-termnonlinear-approximation error is computed by taking the k−largest coefficients from the vectorthat solvesminx‖x‖1 subject to SHx = f . (2.10)Because this vector is obtained by inverting the synthesis operator SH with a sparsity-promotingprogram, this vector is always sparser than the vector obtained by applying the analysis operatorS directly.To account for different redundancies in the transforms, we study signal-to-noise ratios (S/Ns)as a function of the sparsity ratio ρ = k/P (with P = N for orthonormal bases) defined asS/N(ρ) = −20 log ‖f − fρ‖‖f‖ . (2.11)The smaller this ratio, the more coefficients we ignore, the sparser the transform-coefficient vectorbecomes, which in turn leads to a smaller S/N. In our study, we include fρ that are derived fromeither the analysis coefficients or from the synthesis coefficients. The latter coefficients are solutionsof the above sparsity-promoting program (Equation 2.10).Empirical approximation errorsThe above definition gives us a metric to compare recovery S/Ns of seismic data for wavelets,curvelets, and wave atoms. We make this comparison on a common-receiver gather (Figure 2.4)extracted from a Gulf of Suez data set. Because the current implementations of wave atoms(Demanet and Ying, 2007) only support data that is square, we padded the 178 traces with zerosto 1024 traces. The temporal and spatial sampling interval of the high-resolution data are 0.004sand 25m, respectively. Because this zero-padding biases the ρ, we apply a correction.Our results are summarized in Figure 2.5 and they clearly show that curvelets lead to rapidimprovements in S/N as the sparsity ratio increases. This effect is most pronounced for synthesiscoefficients, benefiting remarkably from sparsity promotion. By comparison, wave atoms benefitnot as much, and wavelet even less. This behavior is consistent with the overcompleteness of thesetransforms, the curvelet transform matrix has the largest redundancy (a factor of about eight in2D) and is therefore the tallest. Wave atoms only have a redundancy of two and wavelets areorthogonal. Since our method is based on sparse recovery, this experiment suggests that sparserecovery from subsampling would potentially benefit most from curvelets. However, this is not theonly factor that determines the performance of our compressive-sampling scheme.252.4.2 Subsampling of shotsAside from obtaining good reconstructions from small compression ratios, breaking the periodicityof coherent sampling is paramount to the success of sparse recovery—whether this involves selectionof subsets of sources or the design of incoherent simultaneous-source experiments. To underline theimportance of maximizing incoherence in seismic acquisition, we conduct two experiments wherecommon-source gathers are recovered from subsets of sequential and simultaneous-source experi-ments. To make useful comparisons, we keep for each survey the number of source experiments,and hence the size of the collected data volumes, the same.Coherent versus incoherent samplingMathematically, sequential and simultaneous acquisition only differ in the definition of the measure-ment basis. For sequential-source acquisition, this sampling matrix is given by the Kronecker prod-uct of two identity bases—i.e., I := INs⊗INt , which is a N×N identity matrix with N = Nt×Ns,the product of the number of time samples Nt and the number of shots Ns. For simultaneousacquisition, where all sources fire simultaneously, this matrix is given by M := GNs ⊗ INt withGNs a Ns × Ns Gaussian matrix with i.i.d. entries. In both cases, we use a restriction operatorR := Rns ⊗ INt to model the collection of incomplete data by reducing the number of shots tons  Ns. This restriction acts on the source coordinate only.Roughly speaking, CS predicts superior recovery for compressive-sampling matrices with smallercoherence. According to Equation 2.5, this coherence depends on the interplay between the restric-tion, measurement, and synthesis matrices. To make a fair comparison, we keep the restrictionmatrix the same and study the effect of having measurement matrices that are either given by theidentity or by a random Gaussian matrix. Physically, the first CS experiment corresponds to sur-veys with sequential shots missing. The second CS experiment corresponds to simultaneous-sourceexperiments with simultaneous source experiments missing. Examples of both measurements forthe real common-receiver gather of Figure 2.4 are plotted in Figure 2.6. Both data sets have 50%of the original size. Remember that the horizontal axes in the simultaneous experiment no longerhas a physical meaning. Notice also that there is no observable coherent crosstalk amongst thesimultaneous sources.Multiplication of orthonormal sparsifying bases by random measurement matrices turns intorandom matrices with a small mutual coherence amongst the columns. This property also holds(but only approximately) for redundant signal representations with synthesis matrices that arewide and have columns that are linearly dependent. This suggests improved performance usingrandom incoherent measurement matrices. To verify this statement empirically, we compare sparserecoveries with Equation 2.8 from data plotted in Figure 2.6(a).Despite the fact that simultaneous acquisition with all sources firing simultaneously may not beeasily implementable in practice1, this approach has been applied successfully to reduce simulationand imaging costs (Herrmann et al., 2009; Herrmann, 2009; Lin and Herrmann, 2009a,b). In the“eyeball norm”, the recovery from the simultaneous data is as expected clearly superior (cf. Fig-ures 2.6(b)). The difference plots (cf. Figures 2.6(c)) confirm this observation and show very littlecoherent signal loss for the recovery from simultaneous data. This is consistent with CS, whichpredicts improved performance for sampling schemes that are more incoherent. Because this quali-1Although one can easily imagine a procedure in the field where a “supershot” is created by some stackingprocedure.26tative statement depends on the interplay between the sampling and the sparsifying transform, weconduct an extensive series of experiments to get a better idea on the performance of these twodifferent sampling schemes for different sparsifying transforms. We postpone our analysis of thequantitative behavior of the recovery S/Ns to after that discussion.Sparse recovery errorsThe examples of the previous section clearly illustrate that randomized sampling is important,and that randomized simultaneous acquisition leads to better recovery compared to randomizedsubsampling of sequential sources. To establish this observation more rigorously, we calculateestimates for the recovery error as a function of the sampling ratio δ = n/N by conducting a seriesof 25 controlled recovery experiments. For each δ ∈ [0.2, 0.8], we generate 25 realizations of therandomized compressive-sampling matrix. Applying these matrices to our common-receiver gather(Figure 2.4) produces 25 different data sets that are subsequently used as input to sparse recoverywith wavelets, curvelets, and wave atoms. For each realization, we calculate the S/N(δ) withS/N(δ) = −20 log ‖f − f˜δ‖‖f‖ , (2.12)wheref˜δ = SH x˜δ and x˜δ = arg minx‖x‖1 subject to Aδx = b.For each experiment, the recovery of f˜δ is calculated by solving this optimization problem for 25different realizations of Aδ with Aδ := RδMδSH , where Rδ := Rns ⊗ INt with δ = ns/Ns. Foreach simultaneous experiment, we also generate different realizations of the measurement matrixM := GNs ⊗ INt .From these randomly selected experiments, we calculate the average S/Ns for the recovery error,S/N(δ), including its standard deviation. By selecting δ evenly on the interval δ ∈ [0.2, 0.8], weobtain reasonable reliable estimates with error bars. Results of this exercise are summarized inFigure 2.7. From these plots it becomes immediately clear that simultaneous acquisition greatlyimproves recovery for all three transforms. Not only are the S/Ns better, but the spread in S/Nsamongst the different reconstructions is also much less, which is important for quality assurance.The plots validate CS, which predicts improved recovery for increased sampling ratios. Althoughsomewhat less pronounced as for the approximation S/Ns in Figure 2.5, our results again showsuperior performance for curvelets compared to wave atoms and wavelets. This observation isconsistent with our earlier empirical findings.Empirical oversampling ratiosThe key factor that establishes CS is the sparsity ratio ρ that is required to recover wavefields witherrors that do not exceed a predetermined nonlinear approximation error (cf. Equation 2.11). Thelatter sets the fraction of largest coefficients that needs to be recovered to meet a preset minimalS/N for reconstruction.Motivated by Mallat (2009), we introduce the oversampling ratio δ/ρ ≥ 1. For a given δ, weobtain a target S/N from S/N(δ). Then, we find the smallest ρ for which the nonlinear recoveryS/N is greater or equal to S/N(δ). Thus, the oversampling ratio δ/ρ ≥ 1 expresses the sampling27overhead required by compressive sensing. This measure helps us to determine the performanceof our CS scheme numerically. The smaller this ratio, the smaller the overhead and the moreeconomically favorable this technology becomes compared to conventional sampling schemes.We calculate for each δ ∈ [0.2, 0.8]δ/ρ with ρ = inf{ρ˜ : S/N(δ) ≤ S/N(ρ˜)}. (2.13)When the sampling ratio approaches one from below (δ → 1), the data becomes more and moresampled leading to smaller and smaller recovery errors. To match this decreasing error, the sparsityratio ρ→ 1 and consequently we can expect this oversampling ratio to go to one, δ/ρ→ 1.Remember that in the CS paradigm, acquisition costs grow with the permissible recovery S/Nthat determines the sparsity ratio. Conversely, the costs of conventional sampling grow with the sizeof the sampling grid irrespective of the transform-domain compressibility of the wavefield, whichin higher dimensions proves to be a major difficulty.The numerical results of our experiments are summarized in Figure 2.8. Our calculations useempirical S/Ns for both the approximation errors of the synthesis coefficients as a function of ρand the recovery errors as a function of δ. The estimated curves lead to the following observations.First, as the sampling ratio increases the oversampling ratio decreases, which can be understoodbecause the recovery becomes easier and more accurate. Second, recoveries from simultaneousdata have significantly less overhead and curvelets outperform wave atoms, which in turn performsignificantly better than wavelets. All curves converge to the lower limit (depicted by the dashedline) as δ → 1. Because of the large errorbars in the recovery S/Ns (cf. Figure 2.7), the results forthe recovery from missing sequential sources are less clear. However, general trends predicted byCS are also observable for this type of acquisition, but the performance is significantly worse thanfor recovery with simultaneous sources. Finally, the observed oversampling ratios are reasonablefor both curvelet and wave atoms.2.5 An academic case studyNow that we established that high S/N’s are achievable with modest oversampling ratios, we studythe performance of our recovery algorithm on a seismic line from the Gulf of Suez by comparingtwo simultaneous-source scenarios with coincident source-receiver positions:• ‘Land’ acquisition with random amplitude encoding: Here, sequential impulsive sourcesare replaced by impulsive simultaneous ‘phase-encoded’ sources. Mathematically, simultane-ous measurements are obtained by replacing the sampling matrix for the sources—normallygiven by identity matrix—by a measurement matrix obtained by phase encoding along thesource coordinate. Following Romberg (2008) and Herrmann et al. (2009), we define themeasurement matrix by the following Kronecker productM :=I ⊗Gaussian matrix︷ ︸︸ ︷diag (η)F∗s diag(eiˆθ)Fs⊗I . (2.14)In this expression, conventional sampling, which corresponds to the action of the identitymatrix I, is replaces by a ’random phase encoding’ consisting of applying a Fourier trans-28form along the source coordinate (Fs), followed by uniformly drawn random phase rotationsθ ∈ [0, pi], an inverse Fourier transform (F∗s ), and a multiplication by a a random sign vec-tor (i.e., multiplication by diag (η) with (η) ∈ N(0, 1)). As shown by Romberg (2008), thecombined action of these operations corresponds to the action of a Gaussian matrix at re-duced computational costs (see also Herrmann et al., 2009). Application of this matrix to aconventionally-sampled seismic line turns sequential impulsive source into a simultaneous ‘su-pershot’ where all sources fire simultaneously with weights drawn from a single Gaussian dis-tribution. As before, the restriction operator selects a subset of n′s ‘supershots’ generated bydifferent randomly-weighted simultaneous sources. After restriction along the source coordi-nate, the sampling matrix has an aspect(or undersampling) ratio of δ = n′s/ns. An example ofthis type of sampling, resulting in a seismic line consisting of n′s  ns supershots, is included inFigure 2.9. In this Figure, ns single impulsive-source experiments (2.9(a) left-hand-side plot)become n′s simultaneous-source experiments (juxtapose Figure 2.9(a) left-hand-side plot and2.9(b) left-hand-side plot). While this sort of sampling is perhaps physically unrealizable—i.e., we typically do not have large numbers of vibroseis trucks available—it gives us the mostfavorable recovery conditions from the compressive-sensing perspective. Therefore, our ‘Land’acquisition will serve as a benchmark with which we can compare alternative and physicallymore realistic acquisition scenarios.• ‘Marine’ acquisition with random-time dithering: Here, sequential acquisition with asingle air gun is replaced by continuous acquisition with multiple air guns that fire continu-ously at random times and at random locations. In this scenario, a seismic line is mappedinto a single long ‘supershot’. Mathematically, this type of acquisition is represented by thefollowing sampling operatorRM := [I ⊗T] . (2.15)In this expression, the linear operator T turns sequential recordings (Figure 2.9(a) right-hand-side plot) with synchronized impulsive shots (Figure 2.9(a) left-hand-side plot) into continuousrecordings with n∗s impulsive sources firing at random positions (Figure 2.9(c) left-hand-side plot), selected uniformly-random from [1 · · ·ns] discrete source indices and from discreterandom time indices, selected uniformly from (0 · · · (n∗s − 1) × nt)] time indices. Note thatT acts both on the shot and the time coordinate. The resulting data is one long supershot’that contains a superposition of n∗s impulsive shots. For plotting reasons, we reshaped inFigure 2.9(c) (right-hand-side plot) this long record into multiple shorter records. Notice thatthis type of ‘Marine’ acquisition is physically realizable as long as the number of simultaneoussources involved is limited.Aside from mathematical factors, such as the mutual coherence (cf. Equation 2.5) that deter-mines the recovery quality, there are also economical factors to consider. For this purpose, Berkhout(2008) proposed two performance indicators, which quantify the cost savings associated with simul-taneous and continuous acquisition. The first measure compares the number of sources involved inconventional and simultaneous acquisition and is expressed in terms of the source-density ratioSDR =number of sources in the simultaneous surveynumber of sources in the conventional survey. (2.16)For ‘Land data’ acquisition, this quantity equals SDRLand = (ns × n′s)/ns = n′s and for ‘Marinedata’ SDRMarine = n∗s/ns. Remember that the number of sources refers the number of sources firing29and not the number of source experiments. Clearly, ‘Land’ acquisition has a significant higher SDR.Aside from the number of sources, the cost of acquisition is also determined by survey-timeratioSTR =time of the conventional sequential surveytime of the continuous and simultaneous recording. (2.17)Ignoring overhead in sequential shooting, this quantity equals STRLand = ns/n′s in the first andSTRMarine = ns × T0/T with T0 the time of a single sequential experiment and T the total surveytime of the continuous recording. The overall economic performance is measured by the productof these two ratios. For ‘Land’ acquisition this product is proportional to ns and for ‘Marine’acquisition proportional to n∗s × T0/T .As we have seen from our discussion on compressive sensing, recovery depends on the mutualcoherence of the sampling matrix. So, the challenge really lies in the design of acquisition scenariosthat obtain the lowest mutual coherence while maximizing the above two economic performanceindicators. To get a better insight in how these factors determine the quality of recovered data,we conduct a series of experiments by simulating possible alternative acquisition strategies on aperviously traditionally recorded real seismic line.First, we simulate ‘Land’ data for δ = 0.5 (64 simultaneous source experiments with all sourcesfiring) and study the recovery based on 2D and 3D curvelets. The former is based on a 2D discretecurvelet transform along the source and receiver coordinates, and the discrete wavelet transformalong the remaining time coordinate:S := C2 ⊗W. (2.18)We conduct a similar experiment for the ‘Marine case’. In this case, we randomly select 128 shotsfrom the total survey time T = δ × (ns − 1)× T0, yielding the same aspect ratio for the samplingmatrix.Figures 2.10 and 2.11 summarize the results for ‘Land’ and ‘Marine’ acquisition using recoveriesbased on the 2D and 3D curvelet transform. The following observations can be made. First, it isclear that accurate recovery is possible by solving an `1 optimization problem using SPG`1 (Bergand Friedlander, 2008) while limiting the number of iterations for the 2D case to 500 and the 3Dcase to 200. Second, the recovery results for 3D recovery of ‘Land’ data show and improvement1.3 dB by exploiting 3D structure of the wavefronts. Similarly, we find an improvement of 3.9 dB forthe ‘Marine’ case. Both observations can be explained by the fact that the 3D curvelet transformsattains higher sparsity because it explores continuity of the wavefield along all three coordinateaxes. Second, ‘Land’ acquisition clearly favors recovery by curvelet-domain sparsity promotioncompared to ‘Marine’ acquisition. This is true despite the fact that the subsampling ratio, i.e.,the aspect ratio of the sampling matrices, are the same. Clearly this difference lies in the mutualcoherence of the sampling matrix. The columns of the sampling matrix for ‘Land’ acquisition aremore incoherent and hence more independent and this favors recovery. These observations areconfirmed by the S/Ns, which for ‘Land’ acquisition equal 10.3 dB and 11.6dB, for the 2D/3Drecovery, respectively, and 7.2 dB and 11.1 dB, for ‘Marine’ acquisition.Unfortunately, recovery quality is not the only consideration. The economics expressed by theSDR and STR also play a role. In the above setting, the ‘Land’ acquisition has a SDR = 64 andSTR = 2 while the ‘Marine’ acquisition has SDR = 1 and STR = 2. Clearly, the SDR for landacquisition may not be realistic.302.6 DiscussionThe presented results illustrate that we are at the cusp of exciting new developments where ac-quisition workflows are no longer impeded by subsampling related artifacts. Instead, we arrive atacquisition schemes that control these artifacts. We accomplish by applying the following designprinciples: (i) randomize—break coherent aliases by introducing randomness, e.g. by designingrandomly perturbed acquisition grids, or by designing randomized simultaneous sources; and (ii)sparsify—utilize sparsifying transforms in conjunction with sparsity-promoting programs that sep-arate signal and subsampling artifacts and that restore amplitudes. The implications of randomizedincoherent sampling go far beyond the examples presented here. For instance, our approach is ap-plicable to land acquisition for physically realizable sources (Krohn and Neelamani, 2008; Romberg,2008) and can be used to compute solutions to wavefield simulations (Herrmann et al., 2009) andto compute full waveform inversion (Herrmann et al., 2009) faster. Because randomized sampling islinear (Bobin et al., 2008), wavefield reconstructions and processing can be carried out incrementallyas more compressive data becomes available.Indeed, compressive sensing offers enticing perspectives towards the design of future Land andMarine acquisition systems. In order for this technology to become successful the following issuesneed to be addressed, namely the performance of recovery• from field data including all its idiosyncrasies. This will require an concerted effort from prac-titioners in the field and theoreticians. For Marine acquisition, recent work by Moldoveanu(2010) has shown early indications that randomized jittered sampling leads to improved imag-ing.• from discrete data with quantization errors. Addressing this issue calls for integration ofdigital-to-analog conversion into compressive and recent progress has been made in this area(see e.g. Gu¨ntu¨rk et al., 2010);• from Land data that has the imprint of statics. Addressing this issue will be essential becausesevere static effects may adversely affect transform-domain sparsity on which recovery fromcompressive-sampled data relies.2.7 ConclusionsFollowing ideas from compressive sensing, we made the case that seismic wavefields can be recon-structed with a controllable error from randomized subsamplings. By means of carefully designednumerical experiments on synthetic and real data, we established that compressive sensing can in-deed successfully be adapted to seismic data acquisition, leading to a new generation of randomizedacquisition and processing methodologies.With carefully designed experiments and the introduction of performance measures for nonlinearapproximation and recovery errors, we established that curvelets perform best in recovery, closelyfollowed by wave atoms, and with wavelets coming in as a distant third, which is consistent with thedirectional nature of seismic wavefronts. This finding is remarkable for the following reasons: (i) itunderlines the importance of sparsity promotion, which offsets the “costs” of redundancy and (ii)it shows that the relative sparsity ratio effectively determines the recovery performance rather thanthe absolute number of significant coefficients. Our observation of significantly improved recoveryfor simultaneous-source acquisition also confirms predictions of compressive sensing. Finally, our31analysis showed that accurate recoveries are possible from compressively sampled data volumesthat exceed the size of conventionally compressed data volumes by only a small factor.The fact that compressive sensing combines sampling and compression in a single linear encodingstep has profound implications for exploration seismology that include: a new randomized samplingparadigm, where the cost of acquisition are no longer dominated by resolution and size of theacquisition area, but by the desired reconstruction error and transform domain sparsity of thewavefield, and a new paradigm for randomized processing and inversion, where dimensionalityreductions will allow us to mine high-dimensional data volumes for information in ways, whichpreviously, would have been computationally infeasible.32(a) (b)(c) (d)(e) (f)Figure 2.1: Different (sub)sampling schemes and their imprint in the Fourier domain for asignal that is the superposition of three cosine functions. Signal (a) regularly sampledabove Nyquist rate, (c) randomly three-fold undersampled according to a discrete uni-form distribution, and (e) regularly three-fold undersampled. The respective amplitudespectra are plotted in (b), (d) and (f). Unlike aliases, the subsampling artifacts due torandom subsampling can easily be removed using a standard denoising technique, e.g.,nonlinear thresholding (dashed line), effectively recovering the original signal (adaptedfrom (Hennenfent and Herrmann, 2008)).33(a)(b)Figure 2.2: Averaged recovery error percentages for a k-sparse Fourier vector reconstructedfrom n time samples taken (a) regularly and (b) uniformly-randomly. In each plot,the curves from top to bottom correspond to a subsampling factor ranging from twoto six (adapted from Hennenfent and Herrmann (2008)).34(a)(b)Figure 2.3: Curvelets and seismic data. (a) 2D curvelets in the time-space domain and thefrequency-wavenumber domain. (b) Curvelets approximate curved singularities, i.e.,wavefronts, with very few significant curvelet coefficients.35Figure 2.4: Real common-receiver gather from Gulf of Suez data set.Figure 2.5: Signal-to-noise ratios (S/Ns) for the nonlinear approximation errors of thecommon-receiver gather plotted in Figure 2.4. The S/Ns are plotted as a functionof the sparsity ratio ρ ∈ (0, 0.02]. The plots include curves for the errors obtainedfrom the analysis and one-norm minimized synthesis coefficients. Notice the significantimprovement in S/Ns for the synthesis coefficients obtained by solving Equation 2.10.36(a)(b)(c)Figure 2.6: Recovery from a compressively-sampled common-receiver gather with 50% (δ =0.5) of the sources missing. (a) Left: Receiver gather with sequential shots selecteduniformly at random. (a) Right: The same but for random simultaneous shots. (b)Left: Recovery from incomplete data in (a) left-hand-side plot. (b) Right: The samebut now for the data in (a) right-hand-side plot. (c) Left: Difference plot between thedata in Figure 2.4 and the recovery in (b) left-hand-side plot. (c) Right: The samebut now for recovery from simultaneous data in (a) right-hand-side plot. Notice theremarkable improvement in the recovery from simultaneous data.37Figure 2.7: S/Ns (cf. Equation 2.12) for nonlinear sparsity-promoting recovery from compres-sively sampled data with 20%−80% of the sources missing (δ ∈ [0.2, 0.8]). The resultssummarize 25 experiments for 25 different values of δ ∈ [0.2, 0.8]. The plots includeestimates for the standard deviations. From these results, it is clear that simultaneousacquisition (results in the left column) is more conducive to sparsity-promoting recov-ery. Curvelet-based recovery seems to work best, especially towards high percentagesof data missing.38Figure 2.8: Oversampling ratio δ/ρ as a function of the sampling ratio δ (cf. Equation 2.13)for sequential- and simultaneous-source experiments. As expected, the overhead issmallest for simultaneous acquisition and curvelet-based recovery.39(a)(b)(c)Figure 2.9: Different acquisition scenarios. (a) Left: Impulsive sources for conventional se-quential acquisition, yielding 128 shot records for 128 receivers and 512 time sam-ples. (a) Right: Corresponding fully sampled sequential data. (b) Left: Simultaneoussources for ‘Land’ acquisition with 64 simultaneous-source experiments. Notice thatall shots fire simultaneously in this case. (b) Right: Corresponding compressivelysampled land data. (c) Left: Simultaneous sources for ‘Marine’ acquisition with 128sources firing at random times and locations during a continuous total ’survey’ time ofT = 262s. (c) Right: Corresponding ‘Marine’ data plotted as a conventional seismicline.40(a) (b)(c) (d)Figure 2.10: Sparsity-promoting recovery with δ = 0.5 with the 2D curvelet transforms.(a) 2D curvelet-based recovery from ‘Land’ data (10.3 dB). (b) The correspondingdifference plot. (c) 2D curvelet-based recovery from ‘Marine’ data (7.2 dB). (d)Corresponding difference plot. Notice the improvement in recovery from ‘Land’ data.41(a) (b)(c) (d)Figure 2.11: Sparsity-promoting recovery with δ = 0.5 with the 3D curvelet transforms.(a) 3D curvelet-based recovery from ‘Land’ data (11.6 dB). (b) The correspondingdifference plot. (c) 3D curvelet-based recovery from ‘Marine’ data (11.1 dB). (d)Corresponding difference plot. Notice the improvement in recovery compared to 2Dcurvelet based recovery.42Chapter 3Randomized marine acquisition withcompressive sampling matrices3.1 SummarySeismic data acquisition in marine environments is a costly process that calls for the adoption ofsimultaneous-source or randomized acquisition - an emerging technology that is stimulating bothgeophysical research and commercial efforts. Simultaneous marine acquisition calls for the devel-opment of a new set of design principles and post-processing tools. In this chapter, we discussthe properties of a specific class of randomized simultaneous acquisition matrices and demonstratethat sparsity-promoting recovery improves the quality of the reconstructed seismic data volumes.We propose a practical randomized marine acquisition scheme where the sequential sources fire airguns at only randomly time-dithered instances. We demonstrate that the recovery using sparseapproximation from random time-dithering with a single source approaches the recovery fromsimultaneous-source acquisition with multiple sources. Established findings from the field of com-pressive sensing indicate that the choice of the sparsifying transform that is incoherent with thecompressive sampling matrix can significantly impact the reconstruction quality. Leveraging thesefindings, we then demonstrate that the compressive sampling matrix resulting from our proposedsampling scheme is incoherent with the curvelet transform. The combined measurement matrixexhibits better isometry properties than other transform bases such as a non-localized multidimen-sional Fourier transform. We illustrate our results with simulations of “ideal” simultaneous-sourcemarine acquisition, which dithers both in time and space, compared with periodic and randomizedtime-dithering.3.2 IntroductionData acquisition in seismic exploration forms one of the bottlenecks in seismic imaging and in-version. It involves the collection and processing of massive data volumes, which can be up to5-dimensional in nature (2D for the source positions × 2D for the receiver positions × 1D for thetime dimension). Constrained by the Nyquist sampling rate, the increasing sizes of these data vol-umes pose a fundamental shortcoming in the traditional sampling paradigm as the size and desiredA version of this chapter has been published in Geophysical Prospecting, 2012, vol. 60, pp. 648–662.43resolution of the survey areas continue to grow.Conventional marine acquisition is carried out as single-source experiments of the subsurfaceresponse. This means that the size of the field recording is a product of the number of source loca-tions, the number of receiver locations active per source experiment, and the number of discretizedtime samples proportional to the length of the reflection series. To reduce the threat of aliasing,the source and receiver locations are preferably bound to a grid spacing of less than 50 meters,therefore the number of receivers and sources are in turn direct functions of the size of the surveyarea.Geological considerations, such as the presence of salt bodies in the Gulf of Mexico, create aneed for wide azimuth coverage (i.e., receivers at large offsets), which pushes survey areas to coverthousands of square kilometres, with transverse lengths of 50 kilometres or more. However, adheringto a conventional single-source recording scheme requires recording vessels to move at a pace nomore than 10 kilometres per hour to maintain the desired source spacing. This represents a directconflict to the productivity of surveys, and makes large area acquisition particularly expensive. Onthe other hand, it is entirely possible and even desirable in terms of streamer stability for surveyvessels to move faster (up to operational limits such as streamer drag), but doing so will not usuallyallow sufficient time for the single-source seismic responses to completely decay before the vesselshave reached the next source position.Several works in the seismic imaging literature have explored the concept of simultaneous orblended source activation to account for this situation Beasley et al. (1998); de Kok and Gillespie(2002); Beasley (2008); Berkhout (2008); Hampson et al. (2008). When sources are fired simultane-ously, the main issue is the resulting interference between the responses of the different sources thatmakes it difficult to estimate interference-free shot gathers. Therefore, the challenge is to recoversubtle late reflections that can be overlaid by interfering seismic response from other shots. Wewill show that this challenge can be effectively addressed through recovery by sparsity promotion.Recently, “compressed sensing” (Donoho, 2006; Cande`s and Tao, 2006) has emerged as analternate sampling paradigm in which randomized sub-Nyquist sampling is used to capture thestructure of the data with the assumption that it is sparse or compressible in some transformdomain. A signal is said to admit a sparse (or compressible) representation in a transform domainif only a small number k of the transform coefficients are nonzero (or if the signal can be wellapproximated by the k largest-in-magnitude transform coefficients). In seismic exploration, dataconsists of wavefronts that exhibit structure in multiple dimensions. With the appropriate datatransformation, we capture this structure by a small number of significant transform coefficientsresulting in a sparse representation of data.We rely on the compressed sensing literature to analyze a physically realizable simultaneous-source marine acquisition technology where acquisition related costs are no longer determined bythe Nyquist sampling criteria. We also propose a random time-dithered acquisition scheme whoseperformance using a single source approaches that of simultaneous-source acquisition. Under thisparadigm, data are no longer collected as separate shot records with single-source experiments.Instead, we continuously record over the whole acquisition process, collecting a single long “super-shot” record that is acquired over a time interval shorter than the cumulative time of conventionalmarine acquisition (excluding downtime and overhead such as vessel turning). We then recoverthe canonical sequential single-source shot record by solving a sparsity promoting problem. Thecontributions of this work can be summarized as follows:• We develop the relation between simultaneous-source sampling that is physically realizable44in a marine setting and curvelet-based sparse recovery.• We propose a random time-dithering marine acquisition scheme which can deliver even in thecase of a single source a sparse recovery performance that approaches that of simultaneous-source acquisition with multiple sources with random time and space dithering.• Through Monte-Carlo simulations, we estimate quantitative measures introduced in Com-pressive Sensing that predict the performance of sparse recovery for a particular acquisitiondesign.• Through simulated experiments, we demonstrate the performance of the proposed samplingmatrices in recovering real prestack seismic data.3.3 Related workThe earliest works on simultaneous acquisition were formulated with land acquisition in mind, wherethe prevalent use of vibroseis sweeps have allowed the freedom of employing sophisticated codesin source signatures as a way to differentiate between the responses due to different simultaneoussources (Allen et al., 1998; Beasley et al., 1998; Bagaini, 2006; Lin and Herrmann, 2009a). Marineacquisition, on the other hand, rarely employs the marine vibrators that are the analog of thevibroseis due to poor signal-to-noise ratio and low-frequency content. Consequently, the marinecase of simultaneous acquisition was less well-explored, as the more commonly used impulsive air-gun sources are considerably more rigid when it comes to manipulating its signature. Literaturedirectly discussing the marine case topic did not appear until Beasley et al. (1998), where the source“encoding” are limited to dithered activation times and locations. Near-simultaneous marine caseswere also discussed in de Kok and Gillespie (2002) and Hampson et al. (2008). The main messagesin these works seem to be that interferences due to simultaneous firing are often ignorable afterstacking and simple filtering, and thus do not seriously impact the imaging step.However, many subsurface attribute inversion schemes still rely on single-source prestack data.Recovering these volumes from simultaneous marine recordings did not truly become feasible untilthe recognition that, as long as the shot timings are suitably randomly delayed, the resultinginterferences will appear noise-like in specific gather domains such as common-offset and common-receiver. This property differentiates these events from responses due to single-source experimentsthat remain coherent in these gather domains. This observation was reported in Stefani et al.(2007); Moore et al. (2008) and Akerberg et al. (2008) with application to land acquisition inBagaini and Ji (2010). Subsequent processing techniques, which aim to remove noise-like sourcecrosstalk, vary from simple filters (Huo et al., 2009) to more sophisticated inversion-type algorithms(Moore, 2010; Abma et al., 2010; Mahdad et al., 2011). The latter are designed to take advantageof sparse representations for coherent seismic signals.The aforementioned works did not investigate the link between sparsity-based recovery and thespecific properties of the acquisition system, but theoretical results from compressive sensing dosuggest a direct relationship between acquisition design and the expected fidelity of the achievablerecovery. An analysis for practical acquisition systems exist in terms of incoherency arguments(Blacquie`re et al., 2009, which interestingly also considers receiver-side blending), but analysis interms of compressive sensing arguments remain challenging, as most existing mathematical resultsin compressive sensing deal with rather abstract acquisition systems. Evidently, existing worksrelating to seismic acquisition and compressive sensing only seem to suggest schemes suitable for45forward-modelling in the computer. Neelamani et al. (2008, 2010) suggested an acquisition thatuses noise-like signals as sweeps for land-based acquisition. Herrmann et al. (2009) on the otherhand uses impulsive sources but requires modulation of each source by a randomly-determinedscaling in amplitude. These papers also suggested recovery using sparse inversion of the curveletrepresentation of the data. We will keep the sparse inversion technique but consider more practicalacquisition systems suitable for field marine acquisition.3.4 Compressed sensing overviewCompressive sensing (abbreviated as CS throughout the chapter) is a process of acquiring randomlinear measurements of a signal and then reconstructing it by utilizing the prior knowledge thatthe signal is sparse or compressible in some transform domain. One of the main advantages ofCS is that it combines sampling and compression in a single linear step, thus reducing the cost oftraditional Nyquist sampling followed by dimensionality reduction through data encoding. A directapplication which can benefit from this feature of CS is seismic acquisition where the acquisitioncosts are now quantified by the transform-domain sparsity of seismic data instead of by the gridsize.3.4.1 The sparse recovery problemSuppose that x0 is an P dimensional vector with at most k  P nonzero entries. The sparserecovery problem involves solving an underdetermined system of equationsb = Ax0, (3.1)where b ∈ Cn, n < P represents the compressively sampled data of n measurements, and A ∈ Cn×Prepresents the measurement matrix. When x0 is sparse—i.e., when there are only k < n nonzeroentries in x0— sparsity-promoting recovery can be achieved by solving the `0 minimization problemx˜ = arg minx∈CP‖x‖0 subject to b = Ax, (3.2)where x˜ represents the sparse approximation of x0, and the `0 norm ‖x‖0 is the number of non-zeroentries in x0. Note that if the `0 minimization problem were solvable in practice and every n × nsubmatrix of A is invertible, then x˜ = x0 when k < n/2 (Donoho and Elad, 2003).However, `0 minimization is a combinatorial problem and quickly becomes intractable as thedimensions increase. Instead, the basis pursuit (BP) convex optimization problem shown belowcan be used to recover an estimate x˜ at the cost of decreasing the level of recoverable sparsity k —e.g. k . n/ log(N/n) < n/2 when A is a Gaussian matrix with independent identically distributed(i.i.d.) entries (Cande`s et al., 2006c; Donoho, 2006). The BP problem is given byx˜ = arg minx∈CP‖x‖1 subject to b = Ax, (3.3)where x˜ represents the sparse (or compressible) approximation of x0, and the `1 norm ‖x‖1 is thesum of absolute values of the elements of a vector x. The BP problem typically finds a sparse or(under some conditions) the sparsest solution that explains the data exactly.Finally, we note that x0 can be the sparse expansion of a physical domain signal f0 ∈ CN in46some transform domain characterized by the operator S ∈ CP×N with P ≥ N . In this case, A canbe composed of the product of a sampling operator RM with the sparsifying operator S such thatA = RMSH , where H denotes the Hermitian transpose. Consequently, the acquired measurementsb are given byb = Ax0 = RMf0.We will elaborate more on this concept of sparse recovery in the randomized marine acquisitionsection.3.4.2 Recovery conditionsNext, we discuss some conditions that make unique recovery possible despite the fact that the linearsystem we are solving is underdetermined, meaning that we have fewer equations than unknowns.We present two sets of conditions, those that guarantee recovery of any arbitrary signal x0, andthose that are specialized for a particular class of signals x0.Suppose that the vector x0 is an arbitrary signal that can be well approximated by the vectorxk which contains only the largest k coefficients of x0, i.e., the largest k  P nonzero entries of xkcontain most of the energy of x0. Let a be a number larger than 1. As long as any subset Λ of akcolumns of the n × P matrix A are linearly independent and constitute a submatrix AΛ which isinvertible and has a small condition number (close to 1), there exists some algorithm that recoversx0 exactly.To quantify this property of A, Cande`s and Tao (2005) define the restricted isometry property(RIP) which states that there exists a constant 0 < δak < 1 for which(1− δak)‖u‖22 ≤ ‖AΛu‖22 ≤ (1 + δak)‖u‖22, (3.4)where Λ is any subset of {1 . . . P} of size |Λ| ≤ ak, AΛ is the submatrix of A whose columns areindexed by Λ, and u is an arbitrary k-dimensional vector. The definition above indicates that ifevery submatrix AΛ has an RIP constant that is close to zero, then its condition number approaches1. More precisely, let σmin and σmax be the smallest and largest singular values of AΛ, respectively,the RIP constantδak = supΛ∈{1,...P}max{1− σmin, σmax − 1}. (3.5)That is, the RIP constant is the smallest upper bound on the maximum of {1−σmin, σmax− 1} forall subsets Λ ∈ {1, . . . P} of size ak.The RIP constant is difficult to compute since it requires evaluating δak for every subset ofak columns of A and there are(Pak)of such subsets. However, it is possible to find theoreticalupper bounds on the RIP constant for matrices whose entries are drawn i.i.d from sub-Gaussiandistributions. Otherwise, Monte Carlo simulations are used to approximate the value of δak. Itwas shown in (Cande`s et al., 2006c) that if δ(a+1)k <a−1a+1 (e.g. δ3k <13), then the BP problem canrecover an approximation x˜ to x0 with an error bounded by‖x˜− x0‖2 ≤ C(δak)√k‖xk − x0‖1, (3.6)where C(δak) is a well-behaved constant. This error bound indicates that if the matrix A hasthe RIP for a specific sparsity level k, then the recovery error is bounded by the best k-term47approximation error of the signal. Finally, we note that the condition on the RIP constant waslater improved by Cande`s (2008) to δ2k <√2− 1.An easier to compute but less informative characterization of A is the mutual coherence µ(A).The mutual coherence, which measures correlations between the columns of A, provides an upperbound on δk < (k − 1)µ(A) and is given byµ(A) = max1≤i 6=j≤P|aHi aj |(‖ai‖2 · ‖aj‖2) , (3.7)where ai is the ith column of A. It is evident from equation (3.7) that the mutual coherence ismuch easier to compute than the RIP constant. If we normalize the columns of A and form theGram matrix G = AHA, the mutual coherence is then simply equal to the maximum off-diagonalelement of G. Therefore, the mutual coherence is the largest absolute normalized inner productbetween different columns of A (Bruckstein et al., 2009). Because near orthogonal matrices havesmall correlations amongst their columns, matrices with small mutual coherence favor recovery.3.5 Compressed sensing and randomized marine acquisitionOur focus in this section is on the design of source subsamplingschemes that favor recovery in com-bination with the selection of the appropriate sparsifying transform. To illustrate the importanceof transform-domain sparsity and mutual coherence, we include sparse recovery by the curvelettransform and the 3D Fourier transform in our simulations. Note that this Fourier transform isa simple 3D transform and should not be confused with windowed Fourier transforms that aretypically used in seismic data processing.3.5.1 Randomized marine acquisition as a CS problemConsider marine data organized in a seismic line with Ns sources, Nr receivers, and Nt time sam-ples. For simplicity, we assume that all sources see the same receivers, which makes our methodapplicable to marine acquisition with ocean-bottom cables. The seismic line can be reshaped intoan N dimensional vector f , where N = NsNrNt. It is well known that seismic data admit sparserepresentations by curvelets that capture “wavefront sets” efficiently (see e.g. Smith, 1998; Cande`sand Demanet, 2005; Cande`s et al., 2006a; Herrmann et al., 2008, and the references therein). There-fore, we wish to recover a sparse approximation f˜ of the discretized wavefield f from measurementsb = RMf .Let S be a sparsifying operator that characterizes a transform domain of f , such that S ∈ CP×Nwith P ≥ N . When S is an orthonormal basis, i.e. P = N and SSH = SHS = I where I is theidentity matrix, the signal f admits a unique transform domain representation x = Sf . On theother hand, if S is a tight frame with P ≥ N and SHS = I, as in the case of the redundantcurvelet transform (Cande`s et al., 2006a), then the expansion of f in S is not unique. The sparseapproximation f˜ is obtained by solving the inverse problemA := RMSH (3.8)with the basis pursuit sparsity-promoting programx˜ = arg minx‖x‖1 subject to Ax = b, (3.9)48yielding f˜ = SH x˜. To solve this one-norm optimization problem, we use the SPG`1 solver (Bergand Friedlander, 2008).By solving a sparsity-promoting problem (Cande`s and Tao, 2006; Donoho, 2006; Herrmann et al.,2007; Mallat, 2009), it is possible to reconstruct high-resolution data volumes from the randomizedsamples at moderate sampling overhead compared to data volumes obtained after conventionalcompression (see e.g., Donoho et al. (1999b) for wavelet-based compression, and Herrmann (2010)for empirical oversampling rates for seismic problems). As in conventional compression, the recoveryerror is controllable, but in the case of CS this recovery error depends on the sampling ratio γ = nN .This ratio is given by the number of compressive samples and the number of conventionally acquiredsamples. From a simultaneous-source marine seismic perspective, this is the ratio between the size ofthe continuous and simultaneous recordings and the size of the conventional sequential data. Fromthe perspective of the proposed single-source randomly time-dithered marine acquisition scheme,this is the ratio between the size of the randomly overlapping sequential source recordings and thesize of the conventional non-overlapping sequential data. Consequently, in both the simultaneous-source and the randomized acquisition scenarios, the survey time is reduced for a fixed numberof shots. Conversely, the number of shots recorded can also be increased given the same amountof recording time as a conventional survey, which is useful for projects afflicted with poor shotcoverage.3.5.2 Designing the randomized operatorThe design of the linear sampling operator RM is critical to the success of the recovery algorithm.RM may in some cases be separable and composed of an n×N restriction matrix R multiplyingan N×N mixing matrix M. This is not true in the case of simultaneous-source or the single sourcerandom time-dithered marine acquisition where, as we will illustrate, the sampling operator RMis nonseparable. In the simultaneous marine acquisition scenario, the classic sequential acquisitionwith a single air gun is replaced with continuous acquisition with multiple air guns firing at randomtimes and at random locations that span the entire survey area. This “ideal” simultaneous-sourcesampling scheme is illustrated in Figure 3.1(a) where the circles indicate the firing times andlocations of the multiple sources. Such simultaneous acquisition schemes require an air gun to belocated at each source location throughout the survey, which is infeasible. Alternatively, it maybe possible yet costly to send out several vessels with air guns that swarm over an ocean-bottomarray.We present a new alternative which requires a very small number of vessels (possibly one) thatmap the entire survey area while firing sequential shots at randomly time-dithered instances. In therandom time-dithered acquisition scheme a single air gun or multiple air guns are fired sequentiallywith random lag intervals between shots. This random time-dithered marine acquisition scheme isillustrated in Figure 3.1(b) where, similar to the simultaneous source scheme, the firing times arestill random but the source positions are sorted with respect to survey time. Remember that theordered acquisition is still random by virtue of the random time shifts as opposed to the case of aperiodic time-dithering scheme where we simply decrease the intershot time delays as depicted inFigure 3.1(c). In the remainder of this chapter, we use the CS criteria of the previous section toanalyze the efficacy of this random time-dithered scheme combined with the appropriate transformS and sparse approximation algorithm in recovering the discretized wavefield f from measurementsb = RMf .First we develop the structure of the sampling operator RM. Suppose as before we have Ns49500 1000 1500 2000 2500 300020406080100120Source position (m)Time (s)(a)500 1000 1500 2000 2500 300020406080100120Source position (m)Time (s)(b)500 1000 1500 2000 2500 300020406080100120Source position (m)Time (s)(c)Figure 3.1: Examples of (a) random dithering in source location and trigger times, (b) se-quential locations and random time-triggers, and (c) periodic source firing triggers.shots, Nr receivers, and every shot decays after Nt time samples. We first map the seismic line intoa series of sequential shots f of total length N = Ns × Nt × Nr and apply the sampling operatorRM to reduce f to a single long “supershot” of length n N that consists of a superposition of Nsimpulsive shots. Again, to make the analysis more tractable, we ignore varying detector coverageby assuming a fixed receiver spread.Since the subsampling/mixing is performed in the source-time domain, the resulting sampling50operator is defined as followsRM := [I⊗T], (3.10)where ⊗ is the Kronecker product, I is an Nr×Nr identity matrix, and T is a combined random shotselector and time shifting operator. This structure decouples the receiver axis from the source-timeaxis in the sampling operator allowing T to operate on the vectorized common receiver gathers.Taking the Kronecker product of T with I simply repeats the operation of T on every availablereceiver. The operator T turns the sequential-source recordings into continuous recordings with Nsimpulsive sources, and firing at time instances selected uniformly at random from {1, . . . (Ns−1)Nt}discrete times1. Consequently, the operator T subsamples the NsNt samples recorded at eachreceiver to m NsNt samples resulting in a total number n = mNr compressive samples2.In the marine case with air-gun sources, we can only work with binary (0, 1)-matrices becausewe have virtually no control over the source signature and energy output of air-gun arrays. In theconventional sequential acquisition scheme where no overlap exists between the source responses,the operator T would be a block diagonal matrix of Ns blocks, each block being an Nt×Nt identitymatrix. Each Nt×Nt identity matrix corresponds to the time taken for a source response to decay.This results in a large identity matrix of size NsNt × NsNt. The simultaneous-source acquisitionscheme destroys the block diagonal structure by placing the Nt ×Nt identity matrices at randompositions inside the matrix T. An example of the corresponding operator is shown in Figure 3.2(a).In the case of random time-dithering, the Nt ×Nt identity matrices are situated in an overlappedblock diagonal structure as illustrated in Figure 3.2(b). The effect of random time-dithering onthe structure of the operator T can be seen when we look at the operator in Figure 3.2(c) whichcorresponds to the periodic time-dithering scheme where we simply decrease the intershot timedelays.3.5.3 Assessment of the sampling operatorsIn this section, we limit the assessment to the 2D case due to the large dimensionality of thedata in the 3D case. Consequently, the sampling operator A that we analyze here is constructedas A = TSH , where S is a 2D curvelet or Fourier transform. The operator A is then the lowestdimensional nonseparable component of the 3D Kronecker structure. These results constitute worstcase bounds for the 3D case since the performance of the Kronecker structure is bounded by theworst case performance of its components (Duarte and Baraniuk, 2011).Mutual-coherence based assessmentThe randomized time-dithering operator results in a measurement matrix A that exhibits a smallermutual coherence compared to the periodic time-dithering sampling operator (cf. Figures 3.2(b)and Figure 3.2(c)). Notice that in both cases we fixed the number of source experiments and thenumber of collected samples. Aside from the randomization of the sampling operator, the choice of1It is possible to subsample the number of sources such that ns < Ns shots are selected uniformly at random from{1, . . . Ns} source locations. Such a configuration requires a modified operator T in which the number of Nt × Ntidentity submatrices is equal to ns.2It is also possible to subsample the receiver axis or equivalently to randomize the locations of the availablereceivers in order to produce a higher resolution receiver grid. This is achieved by replacing the Nr × Nr identitymatrix in RM by an nr ×Nr restriction matrix that selects the nr < Nr physical receiver coordinates from the highresolution grid. Consequently, the number of collected CS measurements would be n = mnr.51sparsifying transform also determines the mutual coherence. To study this combined effect, let usconsider the Gram matrix G = AHA of deterministic versus random sampling matrices A usingeither curvelets or Fourier as sparsifying transforms.Recall that the mutual coherence µ(A) is given by the largest off-diagonal element of G. AsFigures 3.3 and 3.4 indicate, there is big difference between the coherences for the deterministicversus the randomized acquisitions. In fact, the mutual coherence of the curvelet based operator inthis example is 0.695 for random time-dithering compared with 0.835 for periodic time-dithering.Similarly, the mutual coherence of the Fourier based operator is 0.738 for random time-ditheringcompared with 0.768 for periodic time-dithering. For the Fourier and curvelet-based CS-samplingthere is only a slight difference between the coherences despite significant differences in appearancesof the Gram matrices. Therefore, the mutual coherence is a crude measure, which is confirmed inour experimental section.RIP-based assessmentWhile the calculation of the mutual coherence is straight forward, it can only give us a pessimisticupper bound on the recoverable sparsity k of a signal (Bruckstein et al., 2009). A better bound(i.e. a guarantee for larger k) can be achieved by evaluating the restricted isometry property (RIP)of A. Bounding this RIP constant allows us to guarantee recovery of less sparse (larger k) signalsthan what is guaranteed by the mutual coherence. The RIP constant δk of A is evaluated for allsubmatrices of A of size n× k. Let Λ be a set of column indices of A of size k. For any matrix AΛthe following property holdsσ2min‖u‖22 ≤ ‖AΛu‖22 ≤ σ2max‖u‖22,where σmin and σmax are the smallest and largest singular values of the matrix AΛ, respectively.The RIP constant δk is the smallest upper bound on δˆΛ := max{1−σmin, σmax−1}, i.e. δk = maxΛδˆΛ.Consequently, if we can show that for all sets Λ ∈ {1, . . . P}, δˆΛ < 1 then there exists some solverwhich can recover any k-sparse signal. Moreover, if δˆΛ <√2 − 1 or δˆΛ < a−1a+1 for some integera > 1, then with high probability we can guarantee that the BP program (4.2) can recover anysparse signal with sparsity less than or equal to k/2 or k/a, respectively.Unfortunately, there are(Pk)combinations of the AΛ submatrices in A, which makes evaluatingδˆΛ computationally infeasible in realistic settings. To overcome this difficulty, and since the trans-form coefficients of seismic images are often not strictly sparse, we first identify the appropriate kas the smallest number of transform coefficients that capture say 90% of the signal energy. Thisallows us to bound the recovery error in terms of the best k-term approximation of the signal. Weare unaware of theoretical results that lead to a bound on δˆΛ for our particular choice of A. Toovercome this, we estimate this quantity by Monte-Carlo sampling over different realizations of thesampling matrix and different realizations of the support. In our simulations, we generate 1000realizations of the random time-dithering sampling matrix. For each of these matrices, we evaluateδˆΛ for 100 random realizations of the set Λ. The RIP constant δk is estimated as the maximum ofthe computed values of δˆΛ.We plot the results of these simulations in Figure 3.5, which shows the histograms of δˆΛ for thecurvelet and Fourier transforms. These simulations show that the δˆΛ for the curvelet transform areless than one, which means that this matrix has RIP, while the matrix based on Fourier may nothave RIP for certain realizations of Λ. As a consequence, we can expect higher fidelity for curvelet-52based recovery. Notice that in the simulations for the curvelet case, the estimate for δˆΛ ≈ 0.76is larger than the theoretical bound of√2 − 1 which guarantees stable recovery using BP withrespect to the best k/2-term approximation x. This is mainly due to the choice of k we used inour simulations. By choosing a smaller value for k, it would be possible to achieve the√2 − 1mark at the expense of increasing the best k-term approximation error. For example, the conditionδ9s < 7/9 guarantees stable recovery with respect to the best s-term approximation of the signal,where 8s = k and k = |Λ| is the same as in the simulations above.On the other hand, the RIP constant of the Fourier-based operator may exceed one. Therefore,it is not possible to find an s for which the RIP-based recovery conditions hold. We believe thatthis observation reflects the poorer recovery results of the Fourier-based operator compared to thecurvelet-based operator as will be shown in the experimental results section.3.5.4 Economic considerationsAside from mathematical factors, such as the mutual coherence that determines the recovery qual-ity, there are also economical factors to consider. For this purpose, Berkhout (2008) proposed twoperformance indicators, which quantify the cost savings associated with simultaneous and contin-uous acquisition. The first measure compares the number of sources involved in conventional andsimultaneous acquisition and is expressed in terms of the source-density ratioSDR =number of sources in the simultaneous surveynumber of sources in the conventional sequential survey. (3.11)In the marine data acquisition setting, the SDR = nsNs . Aside from the number of sources, the costof acquisition is also determined by survey-time ratioSTR =time of the conventional sequential surveytime of the continuous and simultaneous recording. (3.12)The survey time ratio is therefore given by STR = NsNtm which is equal to the aspect ratio of theoperator T. The overall economic performance is measured by the product of these two ratios.3.6 Experimental resultsWe illustrate the effectiveness of our simultaneous source acquisition approach by studying the per-formance of the three sampling schemes; simultaneous-source acquisition, random time-dithering,and periodic time-dithering, on a seismic line from the Gulf of Suez (Figure 3.6 shows a common-shot gather). The fully sampled sequential data is composed of Ns = 128 shots, Nr = 128 receiversand Nt = 512 time samples with 12.5 m source-receiver sampling interval. Prestack data fromsequential sources is recovered using `1 minimization with 3D curvelets as the sparsifying trans-form. For comparison, we perform sparse recovery with a 3D Fourier transform and with the morerudimentary median filtering, which can also be used to suppress the crosstalk. Note that thisFourier transform is a simple 3D transform and not a windowed Fourier transform that is typicallyused in seismic data processing. We also perform linear recovery using the adjoint of the samplingoperator followed by 2D median filtering in the offset domain for additional comparison.We evaluate the recovery performance in terms of the signal-to-noise ratio (S/N) which is53computed as follows for a signal x and its estimate x˜:S/N(x, x˜) = −20 log10‖x− x˜‖2‖x‖2 . (3.13)3.6.1 Simultaneous-source acquisitionWe simulate simultaneous-source marine acquisition by randomly selecting 128 shots from the totalsurvey time t = Ns × Nt with a subsampling ratio γ = mNs×Nt = 1STR = 0.5. The subsampling isperformed only in time so that the length of the “supershot” is half the length of the conventionalsurvey time of sequential-source data. Figure 3.7(a) represents the “supershot” plotted as conven-tional survey by applying the sampling operator RM to sequential-source data. Notice that thistype of “marine” acquisition is physically realizable only with a limited number of simultaneoussources, although truly random positioning of sources may still prove impractical depending on themanoeuvrability of source vessels.To exploit continuity of the wavefield along all three coordinate axes, we use the 3D curvelettransform Ying et al. (2005). Figures 3.7(b) and 3.7(c) show the recovery and residual results,respectively. The recovered data volume has an S/N of 10.5 dB and was obtained with 200 iterationsof solving the BP problem using SPG`1.3.6.2 Random time-ditheringTo overcome the limitation in the number of simultaneous sources required by the “ideal” simultaneous-source approach, we propose the random time-dithering scheme. Under this scheme, we allow all128 shots to be fired sequentially with adjacent shots firing before the previous shot fully decays.We impose a random overlap between the shot records created by a random time lag between thefiring of each shot. Therefore, the start time of each shot is chosen uniformly at random betweenthe starting time of the previous shot and the time by which the previous shot record decays.In our simulations, we apply a sampling operator with subsampling ratio γ = 0.5. A section ofthe “supershot” obtained by random time-dithering is shown in Figure 3.8(a). Using the 3D curvelettransform, a recovery of 8.06 dB is achieved (Figure 3.8(b)). Figure 3.8(c) shows the correspondingresidual plot. Figure 3.9 summarizes the results for recovery based on the 3D Fourier transform.The recovered data volume has an S/N = 6.83 dB, which agrees with our predictions for RIPconstants estimated in the previous section.We also demonstrate the effectiveness of sparse recovery compared with linear recovery usingthe adjoint of the sampling operator RM followed by 2D median filtering in the midpoint-offsetdomain. The recovery results are shown in Figure 3.10. The resulting S/N, 3.92 dB, is considerablylower than the S/Ns achieved by sparse recovery.3.6.3 Periodic time-ditheringThe importance of the “randomness” in time-dithering becomes evident when we compare therecovery of this sampling operator with that of a periodic time-dithering operator. Under thesame subsampling conditions and sparsifying transform, the periodic time dither operator can onlyachieve an S/N = 4.80 dB. Figure 3.11 shows the periodic time-dithered “supershot”, the recovereddata volume and the corresponding residual. This poor performance is consistent with predictions54of CS that require randomness in the design of sampling matrices. Furthermore, comparing sparserecovery with linear recovery of 1.26 dB (Figure 3.12), demonstrates the effectiveness of the former.Finally, we illustrate the recovery for five subsampling ratios (γ = 0.75, 0.50, 0.33, 0.25, 0.10)for each of the schemes described above. Table 3.1 summarizes the S/Ns for the three samplingschemes based on the 3D curvelet and 3D Fourier transforms.SubsampleratioSimultaneousacquisitionRandom time-ditheringPeriodic time-dithering1/STR Curvelet Fourier Curvelet Fourier Curvelet Fourier0.75 13.0 10.2 11.2 9.44 6.93 4.930.50 10.5 7.06 8.06 6.83 4.80 2.420.33 8.31 4.50 5.33 4.10 7.32 1.370.25 6.55 2.93 4.35 2.88 2.85 0.890.10 2.82 0.27 1.14 0.20 1.60 0.19Table 3.1: Summary of recovery results (S/N in dB) based on the 3D curvelet and the 3DFourier transforms for the three sampling schemes.3.7 ConclusionsRecovering single-source prestack data volumes from simultaneously acquired marine data essen-tially involves removing noise-like crosstalk from coherent seismic responses. Many authors havenoticed the important role of sparsity-based recovery for this problem, but few have thoroughlyinvestigated the underlying interaction between acquisition design and reconstruction fidelity, es-pecially in the marine setting. In contrast, we identify simultaneous marine acquisition as a linearsubsampling system, which we subsequently analyze by using metrics from Compressive Sensing.We also propose a randomized time-dithering scheme which can match with a single source theperformance of simultaneous-source acquisition. With the introduction of methods to calculatethe mutual coherence and restricted isometry constants we are able to assert the importance ofrandomness in the acquisition system in combination with the appropriate choice for the sparsify-ing transform in the reconstruction. By comparing reconstructions on a real seismic marine linewith different sparsifying transforms and sampled with different synthetic acquisitions, we quanti-tatively verified that more randomness in the acquisition system and more compressible transformsimprove the mutual coherence and restricted isometry constants, which predict a higher reconstruc-tion quality. As such this work represents a first step towards a comprehensive theory that predictsthe reconstruction quality as a function of the type of acquisition.55NsNtm50 100 150 200 250 300 350 400 450 50020406080100120140160(a)NsNtm50 100 150 200 250 300 350 400 450 50020406080100120140160(b)NsNtm50 100 150 200 250 300 350 400 450 50020406080100120140160(c)Figure 3.2: Example of (a) “ideal” simultaneous-source operator defined by a Bernouilli ma-trix, (b) operator that corresponds to the more realizable Marine acquisition by therandom time-dithering, and (c) sampling operator with periodic time-dithering.56Random timeïshiftGram matrix200 400 6002004006000 200 400 60000.20.40.60.81Center column0 200 400 60000.20.40.60.81252th columnConstant timeïshiftGram matrix200 400 6002004006000 200 400 60000.20.40.60.81Center column0 200 400 60000.20.40.60.81252th columnFigure 3.3: Gram matrices of example random time-dithering and constant time-ditheringoperators, top row, with Ns = 10 and Nt = 40 coupled with a curvelet transform. Theresulting mutual coherence is 0.695 for random time-dithering compared with 0.835 forperiodic time-dithering. The center plots show column the center column of the Grammatrices. The bottom row shows column 252 (one third) of the Gram matrices.57Random timeïshiftGram matrix100 200 300 4001002003004000 100 200 300 40000.20.40.60.81Center column0 100 200 300 40000.20.40.60.81133th columnConstant timeïshiftGram matrix100 200 300 4001002003004000 100 200 300 40000.20.40.60.81Center column0 100 200 300 40000.20.40.60.81133th columnFigure 3.4: Gram matrices of example random time-dithering and constant time-ditheringoperators, top row, with Ns = 10 and Nt = 40 coupled with a Fourier transform. Theresulting mutual coherence is 0.738 for random time-dithering compared with 0.768 forperiodic time-dithering. The center plots show column the center column of the Grammatrices. The bottom row shows column 133 (one third) of the Gram matrices.580.74 0.745 0.75 0.755 0.76 0.765 0.77 0.775 0.78020406080100120Number of occurrencesbR(a)0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50100200300400500600700800900Number of occurrencesbR(b)Figure 3.5: Comparison between the histograms of δˆΛ from 1000 realizations of AΛ, therandom time-dithering sampling matrices A = RMSH restricted to a set Λ of size k,the size support of the largest transform coefficients of a real (Gulf of Suez) seismicimage. The transform S is (a) the curvelet transform and (b) the nonlocalized 2DFourier transform. The histograms show that randomized time-shifting coupled withthe curvelet transform has better behaved RIP constant (δˆΛ = max{1 − σmin, σmax −1} < 1) and therefore promotes better recovery.59Figure 3.6: A common-shot gather from Gulf of Suez data set.60(a) (b)(c)Figure 3.7: (a) Simultaneous-source marine data (γ = 0.5) shown as a section between 45 to50 seconds of the ”supershot”. (b) Recovery from simultaneous ‘marine’ data (S/N =10.5 dB). (c) The corresponding residual plot.61(a) (b)(c)Figure 3.8: (a) Random time-dithered “marine” data (γ = 0.5) shown as a section between45 and 50 seconds of the “supershot”. (b) Sparse recovery with curvelet transform andS/N = 8.06dB. (c) The corresponding residual plot.62(a) (b)Figure 3.9: (a) Sparse recovery with 3D Fourier transform from the same data shown inFigure 3.8(a), S/N = 6.83dB. (b) The corresponding residual plot.63(a) (b)Figure 3.10: (a) Data recovered by applying adjoint of the sampling operator RM and 2Dmedian filtering, from the same data shown in Figure 3.8(a), with S/N = 3.92dB. (b)The corresponding residual plot.64(a) (b)(c)Figure 3.11: (a) Periodic time-dithered “marine” data (γ = 0.5) shown as a section between45 and 50 seconds of the “supershot”. (b) Sparse recovery with curvelet transformand S/N = 4.80dB. (c) The corresponding residual plot.65(a) (b)Figure 3.12: (a) Data recovered by applying adjoint of the sampling operator RM and 2Dmedian filtering, from the same data shown in Figure 3.11(a), with S/N = 1.26dB.(b) The corresponding residual plot.66Chapter 4Simultaneous-source time-jitteredmarine acquisition4.1 SummaryAdapting ideas from the field of compressive sensing, we show how simultaneous- or blended-source acquisition can be setup as a compressive sensing problem. This helps us to design a prag-matic simultaneous-source time-jittered marine acquisition scheme where multiple source vesselssail across an ocean-bottom array firing air guns at jittered source locations and instances in time,resulting in better spatial sampling, and speedup acquisition. Furthermore, we can significantlyimpact the reconstruction quality of conventional seismic data from simultaneous jittered data anddemonstrate successful recovery by sparsity promotion. In contrast to random (sub)sampling, ac-quisition via jittered (sub)sampling helps in controlling the maximum gap size, which is a practicalrequirement of wavefield reconstruction with localized sparsifying transforms. Results are illus-trated with simulations of time-jittered marine acquisition on a seismic line, which translates tojittered source locations for a given speed of the source vessel.4.2 IntroductionConstrained by the Nyquist sampling rate, the increasing sizes of the conventionally acquired marineseismic data volumes pose a fundamental shortcoming in the traditional sampling paradigm andmake large area acquisition particularly expensive. Physical constraints on the speed of a sourcevessel during acquisition, on the minimal time interval between adjacent shots (to avoid overlaps),and on the minimal spatial shot sampling further aggravate the acquisition related costs. Severalworks in the seismic acquisition literature have explored the concept of simultaneous- or blended-source activation to account for these situations (Beasley et al., 1998; de Kok and Gillespie, 2002;Beasley, 2008; Berkhout, 2008; Hampson et al., 2008; Moldoveanu and Fealy, 2010).For simultaneous-source acquisition, the challenge is to estimate interference-free shot gathers(source separation or deblending) and recover small subtle late reflections that can be overlaidby interfering seismic responses from other shots. Stefani et al. (2007), Moore et al. (2008) andA version of this chapter has been published in SEG Technical Program Expanded Abstracts, 2013, vol. 32, pp.1–6.67Akerberg et al. (2008) have observed that the interferences in simultaneous (or blended) data willappear noise-like in specific gather domains such as common offset and common receiver, turningthe separation into a typical (random) noise removal procedure. Application to land acquisitionis reported in Bagaini and Ji (2010). Subsequent processing techniques vary from vector-medianfilters (Huo et al., 2009) to inversion-type algorithms (Moore, 2010; Abma et al., 2010; Mahdadet al., 2011). The former are mostly ”processing” techniques where the interfering energy (i.e.,source crosstalk) is removed and not mapped back to coherent energy, at least not in a singlestep alone, while the latter (inversion-type algorithms) are designed to take advantage of sparserepresentations of coherent seismic signals, which is advantageous because they exploit inherentstructure in seismic data. In this chapter, we show that the challenge of source separation can beeffectively addressed through a combination of tailored simultaneous-source acquisition design andcurvelet-based sparsity-promoting recovery, where we map noise-like or incoherent source crosstalkto coherent seismic responses.Recently, compressive sensing (CS, Donoho, 2006; Cande`s and Tao, 2006) has emerged as analternate sampling paradigm in which randomized sub-Nyquist sampling is used to capture thestructure of the data with the assumption that it is sparse or compressible in some transform do-main. Seismic data consists of wavefronts that exhibit structure across different scales and amongstdifferent directions. With the appropriate data transformation, we can capture this structure by asmall number of significant transform coefficients resulting in a sparse representation of data. In ourwork, we rely on the CS literature to analyze a physically realizable time-jittered marine acquisi-tion scheme, and recover the canonical sequential single-source (interference-free/nonsimultaneous)data by solving a sparsity-promoting problem (Mansour et al., 2012a; Wason and Herrmann, 2012).Hence, we develop a relation between simultaneous-source acquisition design and (curvelet-based)sparse recovery, within the CS framework.4.3 Compressive sensingCompressive sensing is a signal processing technique that allows a signal to be sampled at sub-Nyquist rate and reconstructs it (from relatively few measurements) by utilizing the prior knowledgethat the signal is sparse or compressible in some transform domain, i.e., if only a small number kof the transform coefficients are nonzero or if the signal can be well approximated by the k largest-in-magnitude transform coefficients. For high resolution data represented by the N -dimensionalvector f0 ∈ RN , which admits a sparse representation x0 ∈ CP in some transform domain charac-terized by the operator S ∈ CP×N with P ≥ N , the sparse recovery problem involves solving anunderdetermined system of equationsb = Ax0, (4.1)where b ∈ Cn, n  N ≤ P , represents the compressively sampled data of n measurements, andA ∈ Cn×P represents the measurement matrix. We denote by x0 a sparse synthesis coefficientvector of f0. When x0 is strictly sparse (i.e., only k < n nonzero entries in x0), sparsity-promotingrecovery can be achieved by solving the `0 minimization problem, which is a combinatorial problemand quickly becomes intractable as the dimension increases. Instead, the basis pursuit (BP) convexoptimization problemx˜ = arg minx∈CP‖x‖1 subject to b = Ax, (4.2)68can be used to recover x˜, where x˜ represents the estimate of x0, and the `1 norm ‖x‖1 is thesum of absolute values of the elements of a vector x. The BP problem typically finds a sparse or(under some conditions) the sparsest solution that explains the measurements exactly. The matrixA can be composed of the product of a restriction operator (subsampling matrix) R ∈ Rn×N , anN ×N mixing matrix M, and the sparsifying operator S such that A := RMSH , here H denotesthe Hermitian transpose. Consequently, the measurements b are given by b = Ax0 = RMf0.A seismic line with Ns sources, Nr receivers, and Nt time samples can be reshaped into an Ndimensional vector f , where N = Ns ×Nr ×Nt. For simplicity, we assume that all sources see thesame receivers, which makes our method applicable to marine acquisition with ocean-bottom cablesor nodes (OBC or OBN). We wish to recover a sparse approximation f˜ of the discretized wavefieldf from measurements b = RMf (jittered data). This is done by solving the BP sparsity-promotingprogram (Equation 4.2), using the SPG`1 solver (Berg and Friedlander, 2008), yielding f˜ = SH x˜.4.4 Time-jittered marine acquisitionThe success of CS hinges on randomization of the acquisition, since random subsampling renderscoherent aliases (e.g., interferences due to overlapping shot records in simultaneous-source acquisi-tion) into harmless incoherent random noise, effectively turning the interpolation problem, whichis also a source separation problem in our case, into a simple denoising problem (Hennenfent andHerrmann, 2008). Given limited control over the source signature of the air guns and their rechargetime between shots (typically, a minimal time interval of 10.0 s is required), the only way to invokerandomness is to work with sources that fire at random times that map to random shot locationsfor a given speed of the source vessel. Unfortunately, random (sub)sampling does not provide con-trol on the maximum gap size between adjacent measurements (Figure 4.1), which is a practicalrequirement of wavefield reconstruction with localized sparsifying transforms such as curvelets. Jit-tered (sub)sampling, on the other hand, shares the benefits of random sampling and offers controlon the maximum gap size (Figure 4.1) (Hennenfent and Herrmann, 2008). Since we are still on thegrid, this is a case of discrete jittering. A jittering parameter, dictated by the type of acquisitionand parameters such as the minimum distance (and/or minimum recharge time for the air guns)required between adjacent shots, relates to how close and how far the jittered sampling point canbe from the regular coarse grid, effectively controlling the maximum acquisition gap.The design of the sampling operator M is critical to the success of the recovery algorithm. Notethat we overwrite our notation of the sampling operator from Chapter 3 and define the operatorM as the n×N acquisition operator. We present a pragmatic marine acquisition scheme whereina single (and/or multiple) source vessel(s) maps the survey area while firing shots at jittered timeinstances, which translate to jittered shot locations for a fixed (conventional) speed of the sourcevessel. Conventional acquisition with one source vessel and two air gun arrays where each air-gunarray fires at every alternate periodic location is called flip-flop acquisition. If we wish to acquire10.0 s-long shot records at every 12.5 m, the speed of the source vessel would have to be reducedto about 1.25 m/s (≈ 2.5 knots). The conventional speed of seismic source vessels is about 2.0–2.5m/s (≈ 4–5 knots). Figure 4.2(a) illustrates a conventional acquisition scheme with one sourcevessel travelling at about 1.25 m/s carrying two air-gun arrays, where each air-gun array fires every20.0 s (or 25.0 m) in a flip-flop manner, resulting in nonoverlapping shot records. In time-jitteredacquisition, the source vessel travels at a conventional speed of about 2.5 m/s with air-gun arraysfiring every 20.0 s (or 50.0 m) jittered time instances (or shot locations), i.e., the minimum intervalbetween the jittered times (or shots) is maintained at 10.0 s (or 25.0 m, a practical requirement)69Figure 4.1: Schematic comparison between different subsampling schemes. η is the subsam-pling factor. The vertical dashed lines define the regularly subsampled spatial grid.and the maximum interval is 30.0 s (or 75.0 m). Figure 4.2(b) depicts this scenario resulting inoverlapping shot records. This corresponds to (η =) 2× subsampled jittered acquisition grid for aconventional acquisition with nonoverlapping shot records at every 12.5 m. η is the subsamplingfactor, which is calculated asη =1number of air-gun arrays× jittered spatial grid intervalconventional spatial grid interval=12× 50.0 m12.5 m= 2. (4.3)Note that the source vessel travels at a fixed conventional speed during the time-jittered acquisition,i.e., it does not accelerate or decelerate while firing at jittered instances in time (cf. Chapter 3),which would render this scenario impractical.With a fixed (conventional) speed of the source vessel, if conventional acquisition could becarried out at a shot interval of 6.25 m then (following Equation 4.3) acquisition on the 50.0 mjittered grid would be a result of a subsampling factor of 4 (Figures 4.2(c) and 4.2(d)). Hence,in order to recover data at finer source (and/or receiver) sampling intervals of 12.5 m, 6.25 m,etc., from simultaneous jittered data, the recovery problem becomes a joint source separation andinterpolation problem. Since subsampling is performed in the source-time domain, the acquisitionoperator is defined asM := [I⊗T], (4.4)where ⊗ is the Kronecker product, I is an Nr×Nr identity matrix, and T is a combined jittered-shotselector and time-shifting operator. Taking the Kronecker product of T with I simply repeats theoperation of T on every available receiver. Note that it is also possible to subsample the receiveraxis or equivalently randomize/jitter positions of the ocean-bottom transducers (as in the case ofOBN acquisition).Following the same methods of estimating the RIP (restricted isometry property) constant, δak,70as used in Chapter 3, we analyze the performance of the proposed physically-realizable (or realistic)time-jittered acquisition scheme by estimating δak. In summary, if σmin and σmax are the smallestand largest singular values of AΛ, respectively, then the RIP constantδak = supΛ∈{1,...P}max{1− σmin, σmax − 1}. (4.5)That is, the RIP constant is the smallest upper bound on the maximum of {1 − σmin, σmax − 1}for all subsets Λ ∈ {1, . . . P} of size ak. The RIP constant is difficult to compute since it requiresevaluating δak for every subset of ak columns of A and there are(Pak)of such subsets. Therefore,we use Monte Carlo simulations to approximate the value of δak. Figure 4.3 shows the histogram ofthe estimated RIP constant, δˆΛ from 100 realizations of AΛ, for the realistic time-jittering matrixA = MSH restricted to a set Λ of size k, the size support of the largest transform-domain (i.e.,curvelet-domain) coefficients of a seismic image. For each of these 100 realizations, we evaluate δˆΛfor 100 random realizations of the set Λ. For the proposed time-jittering matrix, the estimated RIPconstant, δˆΛ = max{1−σmin, σmax−1} < 1, illustrating that the design of this realistic time-jitteredmarine acquisition will favor recovery by curvelet-domain sparsity promotion.4.5 Experimental resultsWe illustrate the performance of our time-jittered marine acquisition scheme on simultaneous(time-jittered) data generated from a real seismic line from the Gulf of Suez. We use two setsof conventional data, one sampled at a source (and receiver) sampling of 12.5 m and the othersampled at a source (and receiver) sampling of 6.25 m, with Ns = 128 shots, Nr = 128 receiversand Nt = 1024 time samples each. Figures 4.4(a) and 4.4(b) show a common-receiver and acommon-shot gather, respectively, from conventional data sampled at 12.5 m. We recover (conven-tional) dense periodically-sampled seismic lines from simultaneous data via `1 minimization using2D curvelets (Fast discrete curvelet transform (FDCT), Cande`s et al. (2006a)) Kroneckered with1D wavelets as the sparsifying transform. We also compare recoveries with 3D curvelets (Yinget al., 2005). It is well known that seismic data admit sparse representations by curvelets thatcapture “wavefront sets” efficiently (Smith, 1998; Cande`s and Demanet, 2005; Cande`s et al., 2006a;Herrmann et al., 2008).Figures 4.5(a) and 4.5(b) display 100.0 s of the simultaneous jittered data volumes for the12.5 m and 6.25 m spatial sampling, respectively, where the periodic coarse 50.0 m grid is jitteredusing our jitter subsampling scheme (Figure 4.1) resulting in overlapping shot records. If wesimply apply the adjoint of the acquisition operator to the simultaneous data, i.e., MHy, theinterferences (or source crosstalk) due to overlaps in the shot records appear as random noise, i.e.,incoherent and nonsparse, in common-receiver gathers (Figures 4.6(a) and 4.6(c)) and coherentoverlaps in common-shot gathers (Figures 4.6(b) and 4.6(d)). Our aim is to recover conventional,nonoverlapping shot records from simultaneous data by working with the entire simultaneous datavolume, and not on a shot-by-shot basis. We compare recoveries by computing a signal-to-noiseratio:S/N(f , f˜) = −20 log10‖f − f˜‖2‖f‖2 , (4.6)Sparsity-promoting recovery results in a S/N of 11.5 dB for 2D curvelets and 12.4 dB for 3Dcurvelets, effectively separating the jittered data and interpolating it to a fine 12.5 m grid. The71slight improvement in S/N with 3D curvelets is attributed to exploiting sparse structure of seismicdata with curvelets, which represent curve-like singularities optimally (Cande`s and Demanet, 2005),in all three dimensions, and therefore lead to sparser representations. The improvement in recoveryis small potentially due to small size of the data volumes. Moreover, this improvement comesat the expense of increased computational costs because the 3D FDCT is about 24× redundant,in contrast to the 8× redundant 2D FDCT, rendering large-scale processing extremely memoryintensive, and hence impractical. Figures 4.7(a) – 4.7(d) show a recovered common-receiver andcommon-shot gather and the corresponding residual, respectively, for recovery with 2D FDCT.The corresponding recoveries for 3D FDCT are shown in Figures 4.7(e) – 4.7(h). Figures 4.8(a)– 4.8(h) show the corresponding zoom sections where most of the weak late-arriving events arewell recovered. Similarly, for data with the spatial sampling of 6.25 m, recovery with 2D FDCTresults in a S/N of 4.9 dB while the S/N with 3D FDCT is 5.7 dB. Figures 4.9(a) – 4.9(h) show thecorresponding recoveries with the zoom sections shown in Figures 4.10(a) – 4.10(h). The artifactsnear the bottom of the zoom sections in Figures 4.10(e) – 4.10(h) can possibly be reduced byrunning more iterations of the recovery algorithm. Note that the S/N’s decrease for increasedsubsampling factor η = 4 (for 6.25 m spatial sampling grid). This observation is in accordance withCS theory, where the recovery quality decreases for increased subsampling.To quantify the cost savings associated with simultaneous-source acquisition, we measure theperformance of the proposed acquisition design and subsequent recovery strategy in terms of animproved spatial-sampling ratio (ISSR), defined asISSR =number of shots recovered via sparsity-promoting inversionnumber of shots in simultaneous-source acquisition. (4.7)For time-jittered marine acquisition, a subsampling factor η = 2, 4, etc., implies a gain in thespatial sampling by a factor of 2, 4, etc. In practice, this corresponds to an improved efficiency ofthe acquisition by the same factor. Recently, Mosher et al. (2014) have shown that factors of twoor as high as ten in efficiency improvement are achievable in the field.One of the performance indicators proposed by Berkhout (2008) is the survey-time ratio (STR):STR =time of conventional acquisitiontime of simultaneous-source acquisition, (4.8)If we wish to acquire 10.0 s-long shot records at every 12.5 m, the speed of the source vessel wouldhave to be reduced to about 1.25 m (≈ 2.5 knots). As mentioned previously, in simultaneous-sourceacquisition, speed of the source vessel is approximately maintained at (the standard) 2.5 m/s (≈5.0 knots). Therefore, for a subsampling factor of η = 2, 4, etc., there is an implicit reduction inthe survey time by 1η .4.6 ConclusionsSimultaneous-source time-jittered marine acquisition is an instance of compressive sensing, whichshares the benefits of random sampling while offering control on the maximum acquisition gap size.The results vindicate the importance of randomness in the acquisition scheme, wherein the morerandomness we have in terms of the air-gun firing times/shot locations (as shown here) and/orreceiver locations, the more readily we can adapt ideas from CS to sample data economically (i.e.,acquire subsampled data) and recover dense periodically-sampled data via structure promotion.72Using 3D FDCT for sparsity-promoting recovery results in slightly improved recoveries comparedto 2D FDCT, since it exploits curvelet-domain sparsity in all three dimensions, but at the expenseof increased computational costs. The redundancy of 3D FDCT (about 24 ×) renders large-scaleprocessing extremely memory intensive, and hence impractical. The combination of randomizedsampling and sparsity-promoting recovery technique aids in improved source separation coupledwith interpolation to finer and finer sampling grids, mitigating acquisition-related costs in theincreasingly complicated regions of the earth to produce images of desired resolution. Future workincludes working with nonuniform sampling grids.73(a) (b)(c) (d)Figure 4.2: Acquisition geometry. (a,c) Conventional marine acquisition with one source ves-sel and two air-gun arrays for a spatial sampling of 12.5 m and 6.25 m, respectively.(b,d) The corresponding time-jittered marine acquisition with η = 2 and η = 4, respec-tively. Note the acquisition speedup during jittered acquisition, where the recordingtime is reduced to one-half and one-fourth the recording time of the conventional ac-quisition, respectively.74(a)Figure 4.3: Histogram of δˆΛ from 100 realizations of AΛ, restricted to a set Λ of size k, thesize support of the largest curvelet-domain coefficients of a real (Gulf of Suez) seismicimage.(a) (b)Figure 4.4: Conventional data for a seismic line from the Gulf of Suez. (a) Common-receivergather spatially sampled at 12.5 m. (b) Common-shot gather spatially sampled at 12.5m.75(a) (b)Figure 4.5: Simultaneous data for conventional data spatially sampled at (a) 12.5 m and (b)6.25 m. Note that only 100.0 s of the full simultaneous data volume is shown.(a) (b) (c) (d)Figure 4.6: Interferences (or source crosstalk) in a (a) common-receiver gather and (b)common-shot gather for data spatially sampled at 12.5 m; and in a (c) common-receivergather and (d) common-shot gather for data spatially sampled at 6.25 m. Since thesubsampling factor η = 2 and η = 4 for a spatial sampling of 12.5 m and 6.25 m,respectively, (a) and (c) also have missing traces. The simultaneous data are separatedand interpolated to their respective fine sampling grids.76(a) (b) (c) (d)(e) (f) (g) (h)Figure 4.7: Recovered data for a subsampling factor η = 2. (a,e) Common-receiver gathersrecovered with 2D FDCT and 3D FDCT, respectively. (b,f) The corresponding differ-ence from conventional data. (c,g) Common-shot gathers recovered with 2D FDCT and3D FDCT, respectively. (d,h) The corresponding difference from conventional data.77(a) (b) (c) (d)(e) (f) (g) (h)Figure 4.8: Zoom sections of recovered data for a subsampling factor η = 2. Note that thecolor axis has been clipped at one-tenth the color axis of Figure 4.7. (a,e) Common-receiver gathers recovered with 2D FDCT and 3D FDCT, respectively. (b,f) Thecorresponding difference from conventional data. (c,g) Common-shot gathers recoveredwith 2D FDCT and 3D FDCT, respectively. (d,h) The corresponding difference fromconventional data.78(a) (b) (c) (d)(e) (f) (g) (h)Figure 4.9: Recovered data for a subsampling factor η = 4. (a,e) Common-receiver gathersrecovered with 2D FDCT and 3D FDCT, respectively. (b,f) The corresponding differ-ence from conventional data. (c,g) Common-shot gathers recovered with 2D FDCT and3D FDCT, respectively. (d,h) The corresponding difference from conventional data.79(a) (b) (c) (d)(e) (f) (g) (h)Figure 4.10: Zoom sections of recovered data for a subsampling factor η = 4. Note that thecolor axis has been clipped at one-tenth the color axis of Figure 4.9. (a,e) Common-receiver gathers recovered with 2D FDCT and 3D FDCT, respectively. (b,f) Thecorresponding difference from conventional data. (c,g) Common-shot gathers recov-ered with 2D FDCT and 3D FDCT, respectively. (d,h) The corresponding differencefrom conventional data.80Chapter 5Low-cost time-lapse seismic withdistributed Compressive Sensing —exploiting common informationamongst the vintages5.1 SummaryTime-lapse seismic is a powerful technology for monitoring a variety of subsurface changes dueto reservoir fluid flow. However, the practice can be technically challenging when one seeks toacquire colocated time-lapse surveys with high degrees of replicability amongst the shot locations.We demonstrate that under “ideal” circumstances, where we ignore errors related to taking mea-surements off the grid, high-quality prestack data can be obtained from randomized subsampledmeasurements that are observed from surveys where we choose not to revisit the same randomlysubsampled on-the-grid shot locations. Our acquisition is low cost since our measurements are sub-sampled. We find that the recovered finely sampled prestack baseline and monitor data actuallyimprove significantly when the same on-the-grid shot locations are not revisited. We achieve thisresult by using the fact that different time-lapse data share information and that nonreplicated(on-the-grid) acquisitions can add information when prestack data are recovered jointly. Wheneverthe time-lapse data exhibit joint structure—i.e., are compressible in some transform domain andshare information—sparsity-promoting recovery of the “common component” and “innovations”,with respect to this common component, outperforms independent recovery of both the prestackbaseline and monitor data. The recovered time-lapse data are of high enough quality to serveas input to extract poststack attributes used to compute time-lapse differences. Without jointrecovery, artifacts—due to the randomized subsampling—lead to deterioration of the degree of re-peatability of the time-lapse data. We support our claims by carrying out experiments that collectreliable statistics from thousands of repeated experiments. We also confirm that high degrees of re-peatability are achievable for an ocean-bottom cable survey acquired with time-jittered continuousrecording.A version of this chapter has been published in Geophysics, 2017, vol. 82, pp. P1–P13.815.2 IntroductionTime-lapse (4D) seismic techniques involve the acquisition, processing and interpretation of multiple2D or 3D seismic surveys, over a particular time period of production (Lumley, 2001). While thistechnology has been applied successfully for reservoir monitoring (Koster et al., 2000; Fanchi, 2001)and CO2 sequestration (Lumley, 2010), it remains a challenging and expensive technology becauseit relies on finely sampled and replicated surveys each of which have their challenges (Lumley andBehrens, 1998). To improve repeatability of the combination of acquisition and processing, variousapproaches have been proposed varying from more repeatable survey geometries (Beasley et al.,1997; Porter-Hirsche and Hirsche, 1998; Eiken et al., 2003; Brown and Paulsen, 2011; Eggenbergeret al., 2014) to tailored processing techniques (Ross and Altan, 1997) such as cross equalization(Rickett and Lumley, 2001), curvelet-domain processing (Beyreuther et al., 2005) and matching(Tegtmeier-Last and Hennenfent, 2013).We present a new approach that addresses these acquisition- and processing-related issues byexplicitly exploiting common information shared by the different time-lapse vintages. To this end,we consider time-lapse acquisition as an inversion problem, which produces finely sampled colocateddata from randomly subsampled baseline and monitor measurements. The presented joint recoverymethod, which derives from distributed compressive sensing (DCS, Baron et al., 2009), inverts forthe “common component” and “innovations” with respect to this common component. As duringconventional compressive sensing (CS, Donoho, 2006; Cande`s and Tao, 2006), which has successfullybeen adapted and applied to various seismic settings (Hennenfent and Herrmann, 2008; Herrmann,2010; Mansour et al., 2012b; Wason and Herrmann, 2013b) including actual field surveys (seee.g., Mosher et al., 2014), the proposed method exploits transform-based (curvelet) sparsity incombination with the fact that randomized acquisitions break this structure and thereby createfavorable recovery conditions.While the potential advantages of randomized subsampling on individual surveys are relativelywell understood (see e.g., Wason and Herrmann, 2013b), the implications of these randomized sub-sampling schemes on time-lapse seismic have not yet been studied, particularly regarding achievablerepeatability of the prestack data after recovery and processing. Since the different surveys containthe common component and their respective innovations, the question is how the proposed jointrecovery model performs on the vintages and the time-lapse differences, and what is the impor-tance of replicating the surveys. Our analyses will be carried out assuming our observations lie ona discrete grid so that exact survey replicability is in principle achievable. In this situation, weignore any errors associated with taking measurements from an irregular grid. Our approach makesour time-lapse acquisition low-cost since our measurements are always subsampled and we do notnecessarily replicate the surveys. In the next chapter, we demonstrate how we deal with the effectsof non-replicability of the surveys, particularly when we take measurements from an irregular grid.Since the observations are subsampled and on the grid for this chapter (off the grid for the nextchapter), the aim is to recover vintages on a colocated fine grid.We also ignore complicating factors—such as tidal differences and seasonal changes in watertemperature—that may adversely affect repeatability of the time-lapse surveys. Since one of thegoals of 4D seismic data processing is to obtain excellent 3D seismic images for each data set(Lumley, 2001), and since time-lapse changes are mostly derived from poststack attributes (Landrø,2001; Spetzler and Kvam, 2006), we will be mainly concerned with the quality of the prestackvintages themselves rather than the prestack time-lapse differences.The chapter is organized as follows. First, we summarize the main findings of CS, its governing82equations, and its main premise that structured signals can be recovered from randomized mea-surements sampled at a rate below Nyquist. Next, we set up the CS framework for time-lapsesurveys, and we discuss an independent recovery strategy, where the baseline and monitor data arerecovered independently. We juxtapose this approach with our joint recovery method, which pro-duces accurate estimates for the common component—i.e., the component that is shared amongstall vintages—and innovations with respect to this common component. To study the performanceof these two recovery strategies, we conduct a series of stylized experiments for thousands of ran-dom realizations that capture the essential features of randomized seismic acquisition. From theseexperiments, we compute recovery probabilities as a function of the number of measurements andsurvey replicability, the two main factors that determine the cost of seismic acquisitions. Next, weconduct a series of synthetic experiments that involve time-lapse ocean-bottom surveys with time-jittered continuous recordings and overlapping shots as recently proposed by Wason and Herrmann(2013b). Aside from computing signal-to-noise ratios measured with respect to finely sampledtrue baseline, monitor, and time-lapse differences and their stacks, we also use Kragh and Christie(2002)’s root-mean-square (NRMS) metric to quantify the repeatability of the recovered data.5.3 Methodology5.3.1 Synopsis of compressive sensingCompressive sensing (CS) is a sampling paradigm that aims to reconstruct a signal x ∈ RN (Nis the fully sampled ambient dimension) that is sparse (only a few of the entries are non-zero) orcompressible (can be well approximated by a sparse signal) in some transform domain, from fewmeasurements y ∈ Rn, with n N . According to the theory of CS (Cande`s et al., 2006c; Donoho,2006), recovery of x is attained from n linear subsampled measurements given byy = Ax, (5.1)where A ∈ Rn×N is the sampling matrix.Finding a solution to the above underdetermined system of equations involves solving the fol-lowing sparsity-promoting convex optimization program :x˜ = arg minx‖x‖1 :=N∑i=1|xi| subject to y = Ax. (5.2)where x˜ is an approximation of x. In the noise-free case, this (`1-minimization ) problem findsamongst all possible vectors x, the vector that has the smallest `1-norm and that explains theobserved subsampled data. To arrive at this solution, we use the software package SPG`1 (Bergand Friedlander, 2008). The main contribution of CS is to design sampling matrices that guar-antee solutions to the recovery problem in Equation 6.1, by providing rigorous proofs in specificsettings. Furthermore, a key highlight in CS is that favorable conditions for recovery is attained viarandomized subsampling rather than periodic subsampling. This is because random subsamplingintroduces incoherent, and therefore non-sparse, subsampling related artifacts that are removedduring sparsity-promoting signal recovery. Basically, CS is an extension of the anti-leakage Fouriertransform (Xu et al., 2005; Schonewille et al., 2009), where random sampling in the physical domain83renders coherent aliases into incoherent noisy crosstalk (leakage) in the spatial Fourier domain. Inthis case, the signal is sparse in the Fourier basis.For details on precise recovery conditions in terms of the number of measurements n, allowablerecovery error, and construction of measurement/sampling matrices A, we refer to the literatureon compressive sensing (Donoho, 2006; Cande`s and Tao, 2006; Cande`s and Wakin, 2008). For ourapplication to time-lapse seismic, we follow adaptations of this theory by Herrmann et al. (2008)and Herrmann and Hennenfent (2008), and use curvelets as the sparsifying transform in the seismicexamples that involve randomized marine acquisition (Mansour et al., 2012b; Wason and Herrmann,2013b; Wason et al., 2015). The latter references involve marine acquisition with ocean-bottomnodes and time-jittered time-compressed firing times with single or multiple source vessels. Asshown by Wason and Herrmann (2013b), this type of randomized acquisition and processing leadsto better wavefield reconstructions than the processing of regularly subsampled data. Furthermore,because of the reduced acquisition time, it is more efficient economically (Mosher et al., 2014).5.3.2 Independent recovery strategy (IRS)To arrive at a compressive sensing formulation for time-lapse seismic, we describe noise-free time-lapse data acquired from the baseline (j = 1) and monitor (j = 2) surveys asyj = Ajxj for j = {1, 2}. (5.3)In this CS formulation, which can be extended to J > 2 surveys, the vectors y1 and y2 represent thecorresponding subsampled measurement vectors; A1 and A2 are the corresponding flat (n  N)measurement matrices, which are not necessarily equal. As before, finely sampled vintages can inprinciple be recovered under the right conditions by solving Equation 5.3 with a sparsity-promotingoptimization program (cf. Equation 6.1) for each vintage separately. We will refer to this approachas the independent recovery strategy (IRS). In this context, we compute the time-lapse signal bydirectly subtracting the recovered vintages.5.3.3 Shared information amongst the vintagesAside from invoking randomizations during subsampling, CS exploits structure residing within seis-mic data volumes during reconstruction—the better the compression the better the reconstructionbecomes for a given set of measurements. If we consider the surveys separately, curvelets are goodcandidates to reveal this structure because they concentrate the signal’s energy into few large-magnitude coefficients and many small coefficients (see left-hand side plot in Figure 5.1). Curveletshave this ability because they decompose seismic data into multiscale and multi-angular localizedwaveforms. As the cross plot in Figure 5.1 reveals (right-hand side plot), the curvelet transform’sability to compress seismic data and time-lapse difference (left-hand side plot Figure 5.1) is not theonly type of structure that we can exploit. The fact that most of the magnitudes of the curvelet co-efficients of two common-receiver gathers from a 2D OBS time-lapse survey (see Figure 5.8) nearlycoincide indicate that the data from the two vintages shares lots of information in the curvelet do-main. Therefore, we can further exploit this complementary structure during time-lapse recoveryfrom randomized subsampling in order to improve the repeatability.84Figure 5.1: Left: Decay of curvelet coefficients of time-lapse data and difference. Right:Scatter plot of curvelet coefficients of the baseline and monitor data indicating thatthey share significant information.5.3.4 Joint recovery method (JRM)Baron et al. (2009) introduced and analyzed mathematically a model for distributed CS wherejointly sparse signals are recovered jointly. Aside from permitting sparse representations individu-ally, jointly sparse signals share information. For instance, sensor arrays aimed at the same objecttend to share information (see Xiong et al. (2004) and the references therein) and time-lapse seismicsurveys are no exception.There are different ways to incorporate this shared information amongst the different vintages.We found that we get the best recovery result if we exploit the common component amongst thebaseline and monitor data explicitly. This means that for two-vintage surveys we end up with threeunknown vectors. One for the common component, denoted by z0, and two for the innovations zjfor j ∈ 1, 2 with respect to this common component that is shared by the vintages. In this model,the vectors for the vintages are given byxj = z0 + zj , j ∈ 1, 2. (5.4)As we can see, the vintages contain the common component z0 and the time-lapse difference iscontained within the difference between the innovations zj for j ∈ 1, 2. Because z0 is part of bothsurveys, the observed measurements are now given by[y1y2]=[A1 A1 0A2 0 A2]z0z1z2 , ory = Az.(5.5)In this expression, we overloaded the symbol A, which from now on refers to the matrix linkingthe observations of the time-lapse surveys to the common component and innovations pertainingto the different vintages. The above joint recovery model readily extends to J > 2 surveys, yielding85a J × (number of vintages + 1) system.Contrary to the IRS, which essentially corresponds to setting the common component to zeroso there is no communication between the different surveys, both vintages share the commoncomponent in Equation 5.5. As a result correlations amongst the vintages will be exploited if wesolve insteadz˜ = arg minz‖z‖1 subject to y = Az. (5.6)As a result, we seek solutions for the common component and innovations that have the smallest`1-norm such that the observations explain both the incomplete recordings for both vintages. Es-timates for the finely sampled vintages are readily obtained via Equation 6.4 with the recovered z˜while the time-lapse difference is computed via z˜1 − z˜2.Albeit recent progress has been made (Li, 2015), precise recovery conditions for JRM are notyet very well studied. Moreover, the JRM was also not designed to compute differences between theinnovations. To gain some insight on our formulation, we will first compare the performance of IRSand JRM in cases where the surveys are exactly replicated (A1 = A2), partially replicated (A1 andA2 share certain fractions of rows), or where A1 and A2 are statistically completely independent.To get reliable statistics on the recovery performance for the different recovery schemes, we repeat aseries of small stylized problems thousands of times. These small stylized examples serve as proxiesfor seismic acquisition problems that we will discuss later.5.4 Stylized experimentsTo collect statistics on the performance of the different recovery strategies, we repeat several seriesof small experiments many times. Each random time-lapse realization is represented by a vectorwith N = 50 elements that has k = 13 nonzero entries with Gaussian distributed weights that arelocated at random locations such that the number of nonzero entries in each innovation is two—i.e.,k1 = k2 = 2. This leaves 11 nonzeros for the common component. For each random experiment, n ={10, 11, · · · , 40} observations y1 and y2 are collected using Equation 5.3 for Gaussian matrices A1and A2 that are redrawn for each repeated experiment. These Gaussian matrices have independentidentically distributed Gaussian entries and serve as a proxy for randomized acquisitions in the field.An example of the time lapse vectors z0, z1, z2,x1,x2, and x1−x2 involved in these experiments isincluded in Figure 5.2. Our goal is to recover estimates for the vintages and time-lapse signals—i.e., we want to obtain the estimates x˜1 and x˜2, and their difference x˜1 − x˜2 from subsampledmeasurements y1 and y2. When using the joint recovery model, we compute estimates for thejointly sparse vectors via x˜1 = z˜0 + z˜1, and x˜2 = z˜0 + z˜2, where z˜ is found by solving Equation 6.5.To get reliable statistics on the probability of recovering the vectors representing the vintagesand the time-lapse differences, we choose to perform M = 2000 repeated time-lapse experimentsgenerating M different realizations for y1 and y2 from different realizations of x1 and x2. Next,we recover x˜1 and x˜2 from these measurements using the IRS or JRM. From these estimates, wecompute empirical probabilities of successful recovery viaP (x) =Number of times‖x− x˜‖2‖x‖2 < ρM. (5.7)We set the relative error threshold to ρ = 0.1. The vector x either represents the vintages or the86Figure 5.2: From top to bottom: z0, z1, z2,x1,x2,x1 − x2. We are particularly interested inrecovering estimates for x1,x2 and x1 − x2 from y1 and y2.difference. In case of the vintages, we multiply the probabilities.5.4.1 Experiment 1—independent versus joint recoveryTo reflect current practices in time-lapse acquisition—where people aim to replicate the surveys—we run the experiments by drawing the same random Gaussian matrices of size n × N for n ={10, 11, · · · , 40} and N = 50 for A1 and A2—i.e., A1 = A2. We conduct the same experimentswhere the surveys are not replicated by drawing statistically independent measurement matricesfor each repeated experiment, yielding A1 6= A2. For each series of experiments, we recoverestimates x˜1, x˜2, and x˜1−x˜2 from which we compute the corresponding recovery probabilities usingEquation 5.7. The results are plotted in Figure 5.3 for the recovery of the vintages (Figure 5.3(a))and time-lapse difference (Figure 5.3(b)).The results of these experiments indicate that regardless of the number of measurements, JRMleads to improved recovery compared to IRS because it exploits information shared by the twojointly sparse vectors representing the vintages. The recovery probabilities for JRM (solid linesin Figure 5.3) show an overall improvement for both the time-lapse vectors and the time-lapsedifference vector—all probability curves are to the left compared to those from IRS meaning thatrecovery is more likely for fewer measurements. For the time-lapse vectors, this improvementis much more pronounced for measurement matrices that are statistically independent—i.e., notreplicated (A1 6= A2). This observation is consistent with distributed compressive sensing, whichpredicts significant improvements when the time-lapse vectors share a significant common compo-nent. In that case, the shared component benefits most from being observed by both surveys (via87(a) (b)Figure 5.3: Recovery of (a) the jointly sparse signals x1 and x2, (b) x1−x2; with and withoutrepetition of the measurement matrices, using the independent recovery strategy versusthe joint recovery method.the first column of A, cf. Equation 5.5). The IRS results for the time-lapse vectors are much lessaffected whether the survey is replicated or not, which makes sense because the processing is donein parallel and independently. This suggests that for time-lapse seismic, independent surveys giveadditional information on the sparse structure of the vintages that is reflected in their improvedrecovery quality. Another likely interpretation is that time-lapse data obtained via JRM has betterrepeatability compared to data obtained via IRS.While independent surveys improve recovery with JRM, the recovery probability of the time-lapse difference vectors improves drastically when the experiments are replicated exactly. Thereason for this is that the JRM simplifies to the recovery of the time-lapse differences alone in caseswhere the time-lapse measurements are exactly replicated. Since these time-lapse differences aresparser than the vintage vectors themselves, the time-lapse difference vectors are well recoveredwhile the time-lapse vectors themselves are not. This result is not surprising since the error inreconstructing the vintages cancels out in the difference. This means that in CS, if one is interestedin the time-lapse difference, exact repetition of the survey is preferred. However, this approach doesnot provide any additional structural information in the vintages. We will revisit this observationin Experiment 2 to see how the recovery performs when we have varying degrees of repeatabilityin the measurements.5.4.2 Experiment 2—impact of degree of survey replicabilitySo far, we explored only two extremes, namely recovery of vintages with absolutely no replication(A1 6= A2 and statistically independent) or exact replication (A1 = A2). To get a better under-standing of how replication factors into the recovery, we repeat the experiments where we vary thedegree of dependence between the surveys by changing the number of rows the matrices A1 andA2 have in common. When all rows are in common, the survey is replicated and the percentage ofoverlap between the surveys is a measure for the degree of replicability of the surveys. Since JRMclearly outperformed IRS, we only consider recovery with JRM.As before we compute recovery probabilities from M = 2000 repeated time-lapse experiments88(a) (b)Figure 5.4: Recovery as a function of overlap between measurement matrices. Probability ofrecovering (a) x1 and x2, (b) x1 − x2, with joint recovery method.generating M different realizations for the observations. We summarize the recovery probabilitycurves for varying degrees of overlap in Figure 5.4. These curves confirm that the recovery of thetime-lapse vectors improves when the surveys are not replicated. As soon as the surveys are nolonger replicated, the recovery probabilities for the time-lapse vectors improve. These improvementsbecome less prominent when large percentages do not overlap and as expected reaches its maximumwhen the surveys become independent. Recovery of the time-lapse differences on the other handsuffers drastically when the surveys are no longer 100% replicated. When less then 80% of thesurveys are no longer replicated, the recovery probabilities no longer benefit from replicating thesurveys. Recovery of the time-lapse vectors, on the other hand, already improves significantly atthis point.While these experiments are perhaps too idealized and small to serve as a strict guidance onhow to design time-lapse surveys, they lead to the following observations. Firstly, the recoveryprobabilities improve when we exploit joint sparsity amongst the time-lapse vectors via JRM.Secondly, since the joint component is observed by all surveys recovery of the common componentand therefore vintages improves if the surveys are not replicated. Thirdly, the time-lapse differencesbenefit from high degrees of replication of the surveys. In that case, the JRM degenerates to recoveryof the time-lapse difference alone and as a consequence the time-lapse vectors are not well recovered.Even though the quality of the time-lapse difference is often considered as a good means ofquality control, we caution the reader to draw the conclusion that we should aim to replicate thesurveys. The reason for this is that time-lapse differences are generally computed from poststackattributes computed from finely sampled, and therefore recovered, prestack baseline and monitordata and not from prestack differences. Therefore, recovery of time-lapse difference alone may notbe sufficient to draw firm conclusions. Our observations were also based on very small idealizedexperiments that did not involve stacking and permit exact replication, which may not be realisticin practice.89Figure 5.5: Schematic comparison between different random realizations of a subsampledgrid. The subsampling factor is 3. As illustrated, random samples are taken exactlyon the grid. Moreover, the samples are exactly replicated whenever there is an overlapbetween the time-lapse surveys.5.5 Experimental setup—on-the-grid time-lapse randomizedsubsamplingOne of the main parts of the experimental setup for the synthetic seismic case study is how wedefine the underlying grid on which samples are taken. In context of this chapter, we assumethat the samples are taken on a discrete grid—i.e., samples lie “exactly” on the grid. It is alsoimportant to note that we randomly subsample the grid. As mentioned in the compressive sensingsection above, randomized subsampling introduces incoherent subsampling related artifacts thatare removed during sparsity-promoting signal recovery. Figure 5.5 shows a schematic comparisonbetween different random realizations of a subsampled grid. As illustrated in the schematic, randomsamples are taken exactly on the grid. We define the term “overlap” as the percentage of on-the-grid shot locations exactly replicated between two (or more) time-lapse surveys. For the syntheticseismic case study, whenever there is an overlap between the surveys (e.g., 50%, 33%, 25%, etc.) theon-the-grid shot locations are exactly replicated for the baseline and monitor surveys. Similarly, forthe stylized experiments, when two rows of the Gaussian matrices are the same it can be interpretedas if we hit the same shot location for both the baseline and monitor surveys. Therefore, weeither assume that the experimental circumstances are ideal or alternatively we can think of thisassumption as ignoring the effects of being off the grid. The next chapter analyses the effects ofthe more realistic off-the-grid sampling. In summary, we consider the case where measurementsare exactly replicated whenever we choose to visit the same shot location for the two surveys.However, because we are subsampled we need not choose to revisit all the shot locations of thebaseline survey.90(a) (b) (c)Figure 5.6: Reservoir zoom of the synthetic time-lapse velocity models showing the changein velocity as a result of fluid substitution. (a) Baseline model, (b) monitor model, (c)difference between (a) and (b).5.6 Synthetic seismic case study—time-lapse marine acquisitionvia time-jittered sourcesTo study a more realistic example, we carry out a number of experiments on 2D seismic linesgenerated from a synthetic velocity model—the BG COMPASS model (provided by BG Group).To illustrate the performance of randomized subsamplings—in particular the time-jittered marineacquisition—in time-lapse seismic, we use a subset of the BG COMPASS model (Figure 6.8(a)) forthe baseline. We define the monitor model (Figure 6.8(b)) from the baseline via a fluid substitutionresulting in a localized time-lapse difference at the reservoir level as shown in Figure 6.8(c).Using IWAVE (Symes, 2010) time-stepping acoustic simulation software, two acoustic datasetswith a conventional source (and receiver) sampling of 12.5 m are generated, one from the baselinemodel and the other from the monitor model. Each dataset has Nt = 512 time samples, Nr = 100receivers, and Ns = 100 sources. Subtracting the two datasets yields the time-lapse difference,whose amplitude is small in comparison to the two datasets (about one-tenth). Since no noiseis added to the data, the time-lapse difference is simply the time-lapse signal. A receiver gatherfrom the simulated baseline data, the monitor data and the corresponding time-lapse difference isshown in Figure 5.7. In order to make the time-lapse difference visible, the color axis for the figuresshowing the time-lapse difference is one-tenth the scale of the color axis for the figures showing thebaseline and the monitor data. This colormap applies for the remainder of the chapter. Also, thefirst source position in the receiver gathers—labeled as 0 m in the figures—corresponds to 725 m inthe synthetic velocity model.5.6.1 Time-jittered marine acquisitionWason and Herrmann (2013b) presented a pragmatic single vessel, albeit easily extendable to mul-tiple vessels, simultaneous marine acquisition scheme that leverages CS by invoking randomnessin the acquisition via random jittering of the source firing times. As a result, source interfer-ences become incoherent in common-receiver gathers creating a favorable condition for separatingthe simultaneous data into conventional nonsimultaneous data (also known as “deblending”) viacurvelet-domain sparsity promotion. Like missing-trace interpolation, the randomization via jitter-ing turns the recovery into a relatively simple “denoising” problem with control over the maximum91(a) (b) (c)(d) (e)Figure 5.7: A synthetic receiver gather from the conventional (a) baseline survey, (b) monitorsurvey. (c) The corresponding 4D signal. (d) Color scale of the vintages. (e) Colorscale of the 4D signal. Note that (e) is one-tenth the scale of (d). These color scalesapply to all the corresponding figures for the vintages and the 4D signal.gap size between adjacent shot locations (Hennenfent and Herrmann, 2008), which is a practi-cal requirement of wavefield reconstruction with localized sparsifying transforms such as curvelets(Hennenfent and Herrmann, 2008). The basic idea of jittered subsampling is to regularly decimatethe interpolation grid and subsequently perturb the coarse-grid sample points on the fine grid.A jittering parameter, dictated by the type of acquisition and parameters such as the minimumdistance (or minimum recharge time for the airguns) required between adjacent shots, relates tohow close and how far the jittered sampling point can be from the regular coarse grid, effectivelycontrolling the maximum acquisition gap. Since we are still on the grid, this is a case of discretejittering. In this chapter, we limit ourselves to the discrete case but this technique can relativelyeasily be taken off the grid as we discuss in the next chapter.A seismic line with Ns sources, Nr receivers, and Nt time samples can be reshaped into an Ndimensional vector f , where N = Ns × Nr × Nt. For simplicity, we assume that all sources seethe same receivers, which makes our method applicable to marine acquisition with ocean-bottomcables or nodes (OBC or OBN). As stated previously, seismic data volumes permit a compressiblerepresentation x in the curvelet domain denoted by S. Therefore, f = SHx, where H denotes theHermitian transpose (or adjoint), which equals the inverse curvelet transform. Since curvelets area redundant frame (an over-complete sparsifying dictionary), S ∈ CP×N with P > N , and x ∈ CP .With the inclusion of the sparsifying transform, the matrix A can be factored into the productof a n × N (with n  N) acquisition matrix M and the synthesis matrix SH . The design of theacquisition matrix M is critical to the success of the recovery algorithm. From a practical point of92view, it is important to note that matrix-vector products with these matrices are matrix free—i.e.,these matrices are operators that define the action of the matrix on a vector. Since the marineacquisition is performed in the source-time domain, the acquisition operator M is a combinedjittered-shot selector and time-shifting operator. Note that in this framework it is also possible torandomly subsample the receivers.Given a baseline data vector f1 and a monitor data vector f2, x1 and x2 are the correspondingsparse representations—i.e., f1 = SHx1, and f2 = SHx2. Given the measurements y1 = M1f1 andy2 = M2f2, and A1 = M1SH , A2 = M2SH , our aim is to recover sparse approximations f˜1 andf˜2 by solving sparse recovery problems for the scenarios (IRS and JRM) as described above fromwhich the time-lapse signal can be computed.5.6.2 Acquisition geometryIn time-jittered marine acquisition, source vessels map the survey area firing shots at jitteredtime-instances, which translate to jittered shot locations for a given speed of the source vessel.Conventional acquisition with one source vessel and two airgun arrays—where each airgun arrayfires at every alternate periodic location—is called flip-flop acquisition. If we wish to acquire10.0 s—long shot records at every 12.5 m, the speed of the source vessel would have to be about1.25 m/s (approximately 2.5 knots). Figure 5.8(a) illustrates one such conventional acquisitionscheme, where each airgun array fires every 20.0 s (or 25.0 m) in a flip-flop manner, traveling atabout 1.25 m/s, resulting in nonoverlapping shot records of 10.0 s every 12.5 m. In time-jitteredacquisition, Figure 5.8(b), each airgun array fires at every 20.0 s jittered time-instances, traveling atabout 2.5 m/s (approximately 5.0 knots), with the receivers (OBC) recording continuously, resultingin overlapping (or blended) shot records (Figure 5.9(a)). Since the acquisition design involvessubsampling, the acquired data volume has overlapping shot records and missing shots/traces.Consequently, the jittered flip-flop acquisition might not mimic the conventional flip-flop acquisitionwhere airgun array 1 and 2 fire one after the other—i.e., in Figures 5.8(b) and 5.8(c), a circle(denoting array 1) may be followed by another circle instead of a star (denoting array 2). Theminimum interval between the jittered times, however, is maintained at 10.0 s (typical intervalrequired for airgun recharge) and the maximum interval is 30.0 s. For the speed of 2.5 m/s, thistranslates to jittering a 50.0 m source grid with a minimum (and maximum) interval of 25.0 m (and75.0 m) between jittered shots. Both arrays fire at the 50.0 m jittered grid independent of eachother.Two realizations of the time-jittered marine acquisition are shown in Figures 5.8(b) and 5.8(c),one each for the baseline and the monitor survey. Acquisition on the 50.0 m jittered grid results inan subsampling factor,η =1number of airgun arrays× jittered spatial grid intervalconventional spatial grid interval=12× 50.0 m12.5 m= 2. (5.8)Figures 5.9(a) and 5.9(b) show the corresponding randomly subsampled and simultaneous measure-ments for the baseline and monitor surveys, respectively. Note that only 50.0 s of the continuouslyrecorded data is shown. If we simply apply the adjoint of the acquisition operator to the simulta-neous data—i.e., MHy, the interferences (or source crosstalk) due to overlaps in the shot recordsappear as random noise—i.e., incoherent and nonsparse, as illustrated in Figures 5.9(c) and 5.9(d).Our aim is to recover conventional, nonoverlapping shot records from simultaneous data by work-93(a) (b) (c)Figure 5.8: Acquisition geometry: (a) conventional marine acquisition with one source vesseland two airgun arrays; time-jittered marine acquisition (with η = 2) for (b) baseline,and (c) monitor. Note the acquisition speedup during jittered acquisition, where therecording time is reduced to one-half the recording time of the conventional acquisition.(b) and (c) are plotted on the same scale as (a) in order to make the jittered locationseasily visible.ing with the entire (simultaneous) data volume, and not on a shot-by-shot basis. For the presentscenario, since η = 2, the recovery problem becomes a joint deblending and interpolation problem.In contrast to conventional acquisition at a source sampling grid of 12.5 m (Figure 5.8(a)), time-jittered acquisition takes half the acquisition time (Figures 5.8(b) and 5.8(c)), and the simultaneousdata is separated into its individual shot records along with interpolation to the 12.5 m samplinggrid. The recovery problem is solved by applying the independent recovery strategy and the jointrecovery method, as we will describe in the next section.5.6.3 Experiments and observationsTo analyze the implications of the time-jittered marine acquisition in time-lapse seismic, we followthe same sequence of experiments as conducted for the stylized examples—i.e., we compare theindependent (IRS) and joint recovery methods (JRM) for varying degrees of replicability in theacquisition. Given the 12.5 m spatial sampling of the simulated (conventional) time-lapse data,applying the time-jittered marine acquisition scheme results in a subsampling factor, η = 2 (Equa-tion 5.8). In practice, this corresponds to an improved efficiency of the acquisition with the samefactor. Recent work (Mosher et al., 2014) has shown that factors of two or as high as ten inefficiency improvement are achievable in the field. With this subsampling factor, the number ofmeasurements for each experiment is fixed—i.e., n = N/2, each for y1 and y2 albeit other scenariosare possible.We simulate different realizations of the time-jittered marine acquisition with 100%, 50%, and25% overlap between the baseline and monitor surveys. Because we are in a discrete setting,these overlaps translate one-to-one into percentages of replicated on-the-grid shot locations for thesurveys. Since η = 2, and by virtue of the design of the blended acquisition, it is not possible to94(a) (b)(c) (d)Figure 5.9: Simultaneous data for the (a) baseline and (b) monitor surveys (only 50.0 s ofthe full data is shown). Interferences (or source crosstalk) in a common-receiver gatherfor the (c) baseline and (d) monitor surveys, respectively. Since the subsampling factorη = 2, (c) and (d) also have missing traces. The simultaneous data is separated andinterpolated to a sampling grid of 12.5 m.95have two completely different (0% overlap) realizations of the time-jittered acquisition. In all cases,we recover the deblended and interpolated baseline and monitor data from the blended data y1 andy2, respectively, using the independent recovery strategy (by solving Equation 6.1) and the jointrecovery method (by solving Equation 6.5). As stated previously, the inherent time-lapse differenceis computed by subtracting the recovered baseline and monitor data.We perform 100 experiments for the baseline measurements, wherein each experiment has adifferent random realization of the measurement matrix A1. Then, for each experiment, we fix thebaseline measurement and subsequently work with different random realizations for the monitorsurvey, each corresponding to the 50% and the 25% overlap. The purpose of doing this is to examinethe impact of degree of replicability of acquisition in time-lapse seismic. Table 5.1 summarizes therecovery results for the stacked sections, in terms of the signal-to-noise ratio defined asS/N(f , f˜) = −20 log10‖f − f˜‖2‖f‖2 , (5.9)for different overlaps between the baseline and monitor surveys—i.e., measurement matrices A1and A2. Each S/N value is an average of 100 experiments including the standard deviation.Figure 5.10 shows the recovered receiver gathers and difference plots for the monitor survey(for the different overlaps) using the independent recovery strategy (IRS), and Figure 5.11 showsthe corresponding result using the joint recovery method (JRM). As illustrated in these figures,JRM leads to significantly improved recovery of the vintage compared to IRS because it exploitsthe shared information between the baseline and monitor data. Moreover, the recovery improveswith decrease in the overlap. The IRS and JRM recovered time-lapse differences for the differentoverlaps are shown in Figure 5.12, which shows that recovery via JRM is still significantly betterthan IRS, however, the recovery is slightly improved with increase in the overlap. The edge artifactsin Figures 5.10, 5.11 and 5.12 are related to missing traces near the edges that curvelets are unableto reconstruct.The S/Ns for the stacked sections indicate a similar trend in the observations as made from thestylized experiments—i.e., (i) JRM performs better than IRS because it exploits information sharedbetween the baseline and monitor data. Note that the S/N value, which is an average of the 100experiments, for recovery of the baseline dataset via IRS is repeated for all three cases of overlapbecause we work with the same 100 realizations of the jittered acquisition throughout. However,for each of the 100 experiments, different realizations are drawn for the monitor survey, whichexplains the variations in the S/Ns for the recovery via IRS. Similar fluctuations were observed byHerrmann (2010). (ii) Replication of surveys hardly affects recovery of the vintages via IRS (notesimilar S/Ns), since the processing is done in parallel and independently. (iii) Recovery of thebaseline and monitor data with JRM is better when there is a small degree of overlap between thetwo surveys, and it decreases with increasing degrees of overlap. As explained earlier, this behaviorcan be attributed to partial independence of the measurement matrices that contribute additionalinformation via the first column of A in Equation 6.5, i.e., for time-lapse seismic, independentsurveys give additional structural information leading to improved recovery quality of the vintages.(iv) The converse is true for recovery of the time-lapse difference, wherein it is better if the surveysare exactly replicated. Again, as stated previously, the reason for this is the increased sparsityof the time-lapse difference itself and apparent cancelations of recovery errors due to the exactlyreplicated geometry.In addition to the above observations, we find that for 100% overlap, good recovery of the96Overlap Baseline Monitor 4D signalIRS JRM IRS JRM IRS JRM100% 23.1 ± 1.2 24.8 ± 1.2 23.1 ± 1.3 24.8 ± 1.2 21.4 ± 1.8 23.4 ± 2.150% 23.1 ± 1.2 32.8 ± 1.6 23.4 ± 1.2 32.8 ± 1.6 9.1 ± 1.2 20.2 ± 1.325% 23.1 ± 1.2 35.3 ± 1.5 22.0 ± 1.1 35.0 ± 1.5 7.8 ± 1.3 18.0 ± 1.1Table 5.1: Summary of recoveries in terms of S/N (in dB) for the stacked sections.stacks for IRS and JRM is possible with S/Ns that are similar for the time-lapse difference and thevintages themselves. The standard deviations for the two recovery methods are also similar. Onecould construe that this is the ideal situation but unfortunately it is not easily attained in practice.As we move to more practical acquisition schemes where we decrease the overlap between thesurveys, we see a drastic jump downwards in the S/Ns for the time-lapse stack obtained with IRS.The results from JRM, on the other hand, decrease much more gradually with standard deviationsthat vary slightly from those for IRS, however, drops off with decrease in the overlap. In contrast,we actually see significant improvements for the S/Ns of the stacks of both the baseline and monitordata with slight variations in the standard deviations.Remember, that the number of measurements is the same for all experiments and the observeddifferences can be fully attributed to the performance of the recovery method in relation to theoverlap between the two surveys encoded in the measurement matrices. Also, the improvements inS/Ns of the vintages are significant as we lower the overlap, which goes at the expense of a relativelysmall loss in S/N for the time-lapse stack. However, given the context of randomized subsampling, itis important to recover the finely sampled vintages and then the time-lapse difference. In addition,time-lapse differences are often studied via differences in certain poststack attributes computed fromthe vintages, hence, reinforcing the importance of recovering prestack baseline and monitor data asopposed to recovering the time-lapse difference alone. While some degree of replication seeminglyimproves the prestack time-lapse difference, we feel that quality of the vintages themselves shouldprevail in the light of the above discussion. In addition, concentrating on the quality of the vintagesgives us the option to compute prestack time-lapse differences in alternative ways (Wang et al.,2008).All these observations are corroborated by the plots of the recovered (monitor) receiver gathersand their differences from the original (idealized) gather in Figures 5.10 and 5.11, and the recoveredtime-lapse differences in Figure 5.12. Stacked sections of the IRS and the JRM recovered time-lapsedifference are shown in Figure 5.13.5.6.4 Repeatability measureAside from measuring S/Ns, researchers have introduced repeatability measures expressing thesimilarity between prestack and poststack time-lapse datasets. One of the most commonly usedmetrics, which gives an intuitive understanding of the data repeatability, is the normalized root-mean-square (NRMS, Kragh and Christie, 2002):NRMS =2 RMS(f˜2 − f˜1)RMS(f˜1) + RMS(f˜2), (5.10)97(a) (b) (c)(d) (e) (f)Figure 5.10: Receiver gathers (from monitor survey) recovered via IRS from time-jitteredmarine acquisition with (a) 100%, (b) 50%, and (c) 25% overlap in the measurementmatrices (A1 and A2). (d), (e), and (f) Corresponding difference plots from theoriginal receiver gather (5.7(b)).98(a) (b) (c)(d) (e) (f)Figure 5.11: Receiver gathers (from monitor survey) recovered via JRM from time-jitteredmarine acquisition with (a) 100%, (b) 50%, and (c) 25% overlap in the measurementmatrices (A1 and A2). (d), (e), and (f) Corresponding difference plots from theoriginal receiver gather (5.7(b)).99(a) (b) (c)(d) (e) (f)Figure 5.12: Recovered 4D signal for the (a) 100%, (b) 50%, and (c) 25% overlap. Top row:IRS, bottom row: JRM. Note that the color axis is one-tenth the scale of the coloraxis for the vintages.100(a) (b)(c) (d)(e) (f)(g) (h)Figure 5.13: Stacked sections. (a) baseline; (b) true 4D signal; reconstructed 4D signals viaIRS for 100% (c), 50%(e), and 25% (g) overlap; the reconstructed 4D signals via JRMfor 100%(d), 50%(f), and 25% (h) overlap. Notice the improvements for JRM wherewe see much less deterioration as the overlap between the surveys decreases. Notethat the color axis for the time-lapse difference stacks is one-tenth the scale of thecolor axis for the baseline stack.101with RMS(f˜) being the root-mean-square of either vintage. This formula implies that the lowerthe NRMS, the higher the repeatability between the recovered datasets. Usually, lower levels ofNRMS are observed for stacked data compared to prestack data since stacking reduces uncorrelatedrandom noise. A NRMS ratio of 0 is achievable only in a perfectly repeatable world. In practice,NRMS ratios between 0.2 and 0.3 are considered as acceptable; ratios less than 0.2 are consideredexcellent. To further evaluate the results of our synthetic seismic experiment, we compute theNRMS ratios from stacked sections before and after recovery via IRS and JRM.To compute this quantity, we extract time windows from stacked sections around two-waytravel time between 0.5 s and 1.3 s, where we know there is no time-lapse signal present. We obtainthe stacked sections before and after processing by either applying the adjoint of the samplingmatrix (see discussion under Equation 5.8) to the observed data or by solving a sparsity-promotingprogram. The former serves as a proxy for acquisition scenarios where one relies on the fold tostack out acquisition related artifacts. Results of this exercise for 50% overlap and 25% overlapare included in Figures 5.14(a) and 5.14(b). These plots clearly show that (i) simply applying theadjoint, followed by stacking, leads to poor repeatability, and therefore is unsuitable for time-lapsepractices; (ii) sparse recovery improves the NRMS; (iii) exploiting shared information amongst thevintages leads to near optimal values for the NMRS despite the subsampling; and finally (iv) highdegrees of repeatability of recovered data are achievable from data collected with small overlaps inthe acquisition geometry.5.7 DiscussionObtaining useful time-lapse seismic is challenging for many reasons, including cost, the need tocalibrate the surveys, and the subsequent processing to extract reliable time-lapse information.Meeting these challenges in the field has resulted in acquisitions which aim to replicate the geometryof the previous survey(s) as precisely as possible. Unfortunately, this replication can be bothdifficult to achieve and expensive. Post acquisition, processing aims to improve the repeatabilityof the data such that certain (poststack) attributes can be derived reliably from the baseline andmonitor surveys. Within this context, our aim is to reduce the cost and improve the quality of theprestack baseline and monitor data without relying on expensive fine sampling and high degreesof replicability of the surveys. Our methodology involves a combination of economical randomizedsamplings and sparsity-promoting data recovery. The latter exploits (curvelet-domain) sparsity andcorrelations amongst different vintages. To the authors’ knowledge, this approach is among the firstto address time-lapse seismic problems in which the common component amongst vintages—andinnovations with respect to this shared component—is made explicit.The presented synthetic seismic case study, supported by the findings from the stylized examplesand theoretical results from the distributed compressive sensing literature (Baron et al., 2009),represents a proof of concept for how sharing information amongst the vintages can lead to high-fidelity vintages and 4D signals (with minor trade-offs) in a cost effective manner. This approachcreates new possibilities for meeting modern survey objectives, including cost and environmentalimpact considerations, and improvements in spatial sampling. In this chapter, even though ourmeasurements are taken on the grid, allowing us to ignore errors related to sampling off the grid,our proposed time-lapse acquisition is low-cost since we are always subsampled in the surveys.Our joint recovery model produces finely sampled data volumes from these subsampled, and notnecessarily replicable, randomized surveys. These data volumes exhibit better repeatability levels(in terms of NRMS ratios) compared to independent recovery, where correlations amongst the102(a)(b)Figure 5.14: Normalized root-mean-squares NMRS for each recovered trace of the stackedsection for (a) 50% and the (b) 25% overlap. Vintages obtained with the joint recoverymethod are far superior to results obtained with the independent recovery strategyand the “unprocessed” stacked data. The latter are unsuitable for time lapse.103vintages are not exploited.In the next chapter, we demonstrate how we deal with the effects of non-replicability of thesurveys when we take measurements from an irregular grid. We demonstrate that errors related tobeing off the grid cannot be ignored. The “bad news” is that replication is unattainable becausesmall inevitable deviations in the shot locations amongst the time-lapse surveys negate the benefitof replication for the time-lapse signal itself. However, the good news is that a slightly deviatedmeasurement already adds information that improves recovery of the vintages. This implies that anargument can be made to not replicate the surveys as long as we know sufficiently accurately wherewe fired in the field. Please remember that the claims of this chapter relate to the unnecessaryrequirement to visit the same randomly subsampled on-the-grid shot locations during the two, ormore, surveys.Furthermore, we did not consider surveys that have been acquired in situations where there aresignificant variations in the water column velocities amongst the different surveys. As long as thesephysical changes can be modelled, we do not foresee problems. As expected using standard CS, ourrecovery method should be stable with respect to noise (Cande`s et al., 2006c), but this needs to beinvestigated further. Moreover, recent successes in the application of compressive sensing to actualland and marine field data acquisition (see e.g. Mosher et al. (2014)) support the fact that thesetechnical challenges with noise and calibration can be overcome in practice. Our future researchwill also involve working with towed-streamer surveys where other challenges like the sparse andirregular crossline sampling will be investigated.In this study, we concentrated our efforts on producing high-quality baseline and monitor sur-veys from economic randomized acquisitions. There are areas of application for the joint recoverymodel that have not yet been explored in detail, such as imaging and full-waveform inversion prob-lems. Early results on these applications suggest that our joint recovery model extends to sparsity-promoting imaging (Tu et al., 2011; Herrmann and Li, 2012) including imaging with surface-relatedmultiples, and time-lapse full-waveform inversion (Oghenekohwo et al., 2015). In all applications,the use of shared information amongst vintages improves the inversion results even for acquisitionswith large gaps. Finally, none of the other recently proposed approaches in this research area—e.g., double differences (Yang et al., 2014) and total-variation norm minimization on time-lapseearth models (Maharramov and Biondi, 2014)—use the shared information amongst the vintagesexplicitly.5.8 ConclusionsWe considered the situation of recovering time-lapse data from on-the-grid but randomly subsam-pled surveys. In this idealized setting, where we ignore the effects of being off the grid, we foundthat it is better not to revisit the on-the-grid shot locations amongst the time-lapse surveys whenthe vintages themselves are of prime interest. This result is a direct consequence of introducinga common component, which contains information shared amongst the vintages, as part of ourproposed joint recovery method. Compared to independent recoveries of the vintages, we obtaintime-lapse data exhibiting a higher degree of repeatability in terms of normalized root-mean-squareratios. Under the above stated idealized setting and ignoring complicating factors such as tidal dif-ferences, our proposed method lowers the cost and environmental imprint of acquisition becausefewer shot locations are visited. It also allows us to extend the survey area or to increase the data’sresolution at the same costs as conventional surveys. Our improvements concern the vintages andnot the time-lapse difference itself, which would benefit if we choose to use the same shot locations104during the surveys. Because we are generally interested in “poststack” attributes derived from thevintages, their recovery took prevalence. So, we make the argument not to replicate—i.e., revisiton-the-grid shot locations during randomized surveys in cases where poststack time-lapse attributesare of interest only.105Chapter 6Low-cost time-lapse seismic withdistributed Compressive Sensing —impact on repeatability6.1 SummaryIrregular or off-the-grid spatial sampling of sources and receivers is inevitable in field seismic ac-quisitions. Consequently, time-lapse surveys become particularly expensive since current practicesaim to replicate densely sampled surveys for monitoring changes occurring in the reservoir due tohydrocarbon production. We demonstrate that under certain circumstances, high-quality prestackdata can be obtained from cheap randomized subsampled measurements that are observed fromnonreplicated surveys. We extend our time-jittered marine acquisition to time-lapse surveys bydesigning acquisition on irregular spatial grids that render simultaneous, subsampled and irregularmeasurements. Using the fact that different time-lapse data share information and that nonrepli-cated surveys add information when prestack data are recovered jointly, we recover periodic denselysampled and colocated prestack data by adapting the recovery method to incorporate a regular-ization operator that maps traces from an irregular spatial grid to a regular periodic grid. Therecovery method is, therefore, a combined operation of regularization, interpolation (estimatingmissing fine-grid traces from subsampled coarse-grid data), and source separation (unraveling over-lapping shot records). By relaxing the insistence on replicability between surveys, we find thatrecovery of the time-lapse difference shows little variability for realistic field scenarios of slightlynonreplicated surveys that suffer from unavoidable natural deviations in spatial sampling of shots(or receivers) and pragmatic compressed-sensing based nonreplicated surveys when compared to the“ideal” scenario of exact replicability between surveys. Moreover, the recovered densely sampledprestack baseline and monitor data improve significantly when the acquisitions are not replicated,and hence can serve as input to extract poststack attributes used to compute time-lapse differences.Our observations are based on experiments conducted for an ocean-bottom cable survey acquiredwith time-jittered continuous recording assuming source equalization (or same source signature)for the time-lapse surveys and no changes in wave heights, water column velocities or temperatureand salinity profiles, etc.A version of this chapter has been published in Geophysics, 2017, vol. 82, pp. P15–P30.1066.2 IntroductionSimultaneous marine acquisition is being recognized as an economic and environmentally moresustainable way to sample seismic data and speedup acquisition, wherein single or multiple sourcevessels fire shots at random, compressed times resulting in overlapping shot records (de Kok andGillespie, 2002; Beasley, 2008; Berkhout, 2008; Hampson et al., 2008; Moldoveanu and Quigley,2011; Abma et al., 2013), and hence generating compressed seismic data volumes. The aim thenis to separate the overlapping shot records into individual shot records, as acquired during conven-tional acquisition, but with denser source sampling while preserving amplitudes of the late, oftenweak, arrivals. This leads to recovering densely sampled data economically, which is essential forproducing high-resolution images of the subsurface.Mansour et al. (2012b), Wason and Herrmann (2013b) and Mosher et al. (2014) have showedthat compressed sensing (CS, Cande`s et al., 2006c; Donoho, 2006) is a viable technology to sam-ple seismic data economically with low environmental imprint—by reducing numbers of shots (orinjected energy in the subsurface) or compressing survey times. Mansour et al. (2012b) and Wa-son and Herrmann (2013b) proposed an alternate sampling strategy for simultaneous acquisition(“time-jittered” marine), addressing the separation problem through a combination of tailored (si-multaneous) acquisition design and sparsity-promoting recovery via convex optimization using `1objectives. This separation technique interpolates sub-Nyquist jittered shot positions to a fine reg-ular grid while unraveling the overlapping shots. The time-jittered marine acquisition is designedfor continuous recording, fixed-receiver (or “static”) geometries, which is different from the caseof towed-streamer (or “dynamic”) geometries, wherein multiple sources fire shots within a timeinterval of (0, 1) or (0, 2) s generating overlapping shot records that need to be separated into itsconstituent sources, i.e., a data volume for each individual source (Kumar et al., 2015b). Ourapproach for conventional data recovery from simultaneous data from static geometries can equallyapply to other settings including static land and other static marine geometries.The implications of randomization in time-lapse (or 4D) seismic, however, are less well-understoodsince the current paradigm relies on dense sampling and replicability amongst the baseline and mon-itor surveys (Lumley and Behrens, 1998). These requirements impose major challenges because theinsistence on dense sampling may be prohibitively expensive and variations in acquisition geome-tries (between the surveys) due to physical constraints do not allow for exact replication of thesurveys. In Chapter 5, we presented a new approach (the “joint recovery method”) that addressesthese acquisition- and processing-related issues by explicitly exploiting common information sharedby the different time-lapse vintages. Our analyses were carried out assuming that the observationslied on a discrete grid so that exact survey replicability is in principle achievable. We also as-sume sources to have the same source signature for the time-lapse surveys. While assuming sourceequalization in this chapter, we extend our work on simultaneous time-jittered marine acquisitionto time-lapse surveys for more realistic field acquisitions that lie on irregular spatial grids, wherethe notion of exact replicability of the surveys is inexistent. This is because the “real” world suf-fers from unavoidable deviations between pre- and post-acquisition shot (and receiver) positions,rendering regular, periodic spatial grids irregular, and hence exact replication of the surveys im-possible. As mentioned later in the chapter, accounting for the irregularity of seismic data is keyto recovering densely sampled data. Moreover, while we do not insist that we actually visit pre-designed (irregular) shot positions, but it is important to know these positions to sufficient accuracyafter acquisition for high-quality data recovery. Recent successes in the application of compressedsensing to land and marine field data acquisition (see e.g., Mosher et al., 2014) show that this can107be achieved in practice.Simultaneous time-jittered marine acquisition generates compressed and subsampled data vol-umes, therefore, extending this to time-lapse surveys generates compressed and subsampled base-line and monitor data. Consequently, we are interested in recovering densely sampled vintages andtime-lapse difference. Moreover, time-lapse differences are often studied via differences in certainpoststack attributes computed from the vintages (Landrø, 2001; Spetzler and Kvam, 2006), hence,we prioritize on recovering the prestack vintages. In this chapter, we push this technology to re-alistic settings of off-the-grid acquisitions and demonstrate that we actually gain if we relax theinsistence to replicate surveys since even the smallest known deviations from the grid can lead tosignificant improvements in the recovery of the vintages with minimal compromise with the recoveryof the time-lapse difference.6.2.1 Motivation: on-the-grid vs. off-the-grid data recoveryChapter 5 illustrated that the joint recovery method gives better recoveries of time-lapse data andtime-lapse difference than the independent recovery strategy, since the former approach exploitsthe common information shared by the vintages. It also showed that “exact” replication of thebaseline and monitor surveys lead to good recovery of the time-lapse difference but not of thevintages. These analyses, however, were carried out assuming that the observations lied on adiscrete grid so that exact survey replicability is achievable. Realistic field acquisitions, on thecontrary, lie off the grid—i.e., have irregular spatial sampling—where exact replicability of thesurveys is inexistent. Figure 6.1 shows a comparison between conventional periodic acquisitionwhich generates data with nonoverlapping shot records, and simultaneous time-jittered acquisitionwhich generates compressed recordings with overlapping shots. Note that the sampling grid forconventional acquisition “in the field” would be slightly irregular, however, this in contrast tothe jittered acquisition which by virtue of its design is aperiodic and lies on an irregular samplinggrid. Since the time-jittered acquisition scheme leverages compressed sensing—the success of whichhinges on randomized subsampling—additional and unavoidable deviations in the field add to therandomization of the designed irregular shot positions, and helps in sparsity-promoting inversionas long as we know the final shot positions to sufficient precision.Figures 6.2(a)-6.2(c) show receiver gathers from a conventional (synthetic) time-lapse dataset and the corresponding time-lapse difference. To recover periodic densely sampled data fromsimultaneous, compressed and irregular data, we could implicitly rely on binning, however, failureto account for irregularity of seismic traces can adversely affect the recovery as shown in Figure 6.3.This is because binning does not represent accurate positions of irregular traces. Note that thisexample corresponds to a time-jittered acquisition scheme for the baseline that is exactly replicatedfor the monitor. The results show that binning offsets all the gains of exact survey replication andalso of the joint recovery method. Figure 6.4 illustrates the importance of regularization of irregulartraces for high-quality data recovery. In this chapter, therefore, we extend our work on simultaneoustime-jittered acquisition to time-lapse surveys by acknowledging the irregularity of field seismic dataand incorporating sparsifying transforms that exploit this irregularity to recover periodic denselysampled time-lapse data.108Figure 6.1: Schematic of conventional acquisition and simultaneous, compressed (or time-jittered) acquisition. If the source sampling grid for conventional acquisition is 25.0 m(or 50.0 m for flip-flop acquisition), then the time-jittered acquisition jitters (or per-turbs) shot positions on a finer grid, which is 1/4 th of the conventional flip-flop sam-pling grid, for a single air-gun array. Following the same strategy, adding another air-gun array makes the acquisition simultaneous, and hence results in a compressed datavolume with overlapping, irregular shots and missing traces. The sparsity-promotinginversion then aims to recover densely sampled data by separating the overlappingshots, regularizing irregular traces and interpolating missing traces.6.2.2 ContributionsThe contributions of this work can be summarized as follows. First, we present an extension of oursimultaneous time-jittered marine acquisition for time-lapse surveys by working on more realisticfield acquisition scenarios by incorporating irregular spatial grids. Second, we leverage ideas fromcompressed sensing and distributed compressed sensing to develop an algorithm that separatessimultaneous data, regularizes irregular traces and interpolates missing traces—all at once. Third,through simulated experiments, we show that insistence on replicability of time-lapse surveys canbe relaxed since small known deviations in shot positions from a regular grid (or deviations in shotpositions of the monitor survey from those in the baseline survey) lead to significant improvementsin the recovery of the vintages, without drastic modifications in the recovery of the time-lapsedifference.6.2.3 OutlineThe chapter is organized as follows. We begin with the description of the simultaneous time-jittered marine acquisition design, where we explain how subsampled and irregular measurementsare generated. Next, we introduce the nonequispaced fast discrete curvelet transform (NFDCT) andits application to recover periodic densely sampled seismic lines from simultaneous and irregular109(a) (b) (c)Figure 6.2: Synthetic receiver gathers from a conventional (a) baseline survey, (b) monitorsurvey. (c) Corresponding time-lapse difference.measurements via sparsity-promoting inversion. We then extend this framework to time-lapsesurveys where we modify the measurement matrices in the joint recovery method to include theoff-the-grid information—i.e., the irregular shot positions and jittered times. Note that we do notdescribe the independent recovery strategy since it is clear in Chapter 5 that the joint recoverymethod outperforms the former approach. We conduct a series of synthetic seismic experimentswith different random realizations of the simultaneous time-jittered marine acquisition to assessthe effects (or risks) of irregular sampling in the field on time-lapse data and demonstrate thathigh-quality data recoveries are the norm and not the exception. We show this by generating 2Dseismic lines using two different velocity models—one with simple geology and complex time-lapsedifference (BG COMPASS model), and the other with complex geology and complex time-lapsedifference (SEAM Phase 1 model with simulated time-lapse difference). Aside from computingsignal-to-noise ratios measured with respect to densely sampled true baseline, monitor, and time-lapse differences, we also measure the economic and environmental performance of the proposedacquisition design and recovery strategy by computing the improvement in spatial sampling.6.3 Time-jittered marine acquisitionThe objective of CS is to recover densely sampled data from (randomly) subsampled data by exploit-ing sparse structure in the data during sparsity-promoting recovery. Mansour et al. (2012b), Wasonand Herrmann (2013b) presented a pragmatic simultaneous marine acquisition scheme, termed astime-jittered marine, that leverages ideas from compressed sensing by invoking randomness andsubsampling—i.e., sample randomly with fewer samples than required by Nyquist sampling criteriain the acquisition via random jittering of the source firing times. The success of CS hinges onrandomized subsampling since it renders subsampling related artifacts incoherent, and therefore110(a) (b) (c)(d) (e) (f)Figure 6.3: Data recovery via the joint recovery method and binning. (a), (b) Binned vin-tages and (c) corresponding time-lapse difference. (d), (e), (f) Corresponding differenceplots.111(a) (b) (c)(d) (e) (f)Figure 6.4: Data recovery via the joint recovery method and regularization. (a), (b) Vintagesand (c) time-lapse difference recovered via sparsity promotion including regularizationof irregular traces. (d), (e), (f) Corresponding difference plots. As illustrated, regular-ization is imperative for high-quality data recovery.112nonsparse, favouring data recovery via structure-promoting inversion. Consequently, source inter-ferences (in simultaneous acquisition) become incoherent in common-receiver gathers creating afavorable condition for separating simultaneous data into conventional nonsimultaneous data viacurvelet-domain sparsity promotion. The CS paradigm, however, assumes signals to be sampledon a periodic discrete grid—i.e., signals with sparse representation in finite discrete dictionaries.Data volumes collected during seismic acquisition represent discretization of analog finite-energywavefields in up to five dimensions including time—i.e., we acquire an analog spatiotemporal wave-field f¯(t, x) ∈ L2((0, T ]× [−X,X]4), two dimensions for receivers and two dimensions for sources,with time T in order of seconds and length X in order of kilometers. In an ideal world, signalswould perfectly lie on a periodic, regular grid. Hence, with a linear high-resolution analog-to-digital converter Φ¯s, the discrete signal is represented as f [q] = f¯ ? Φ¯s(q), for 0 ≤ q < N (Mal-lat, 2008), where the samples lie on a grid. Typically, these samples are organized into a vectorf = f [q]q=0,...,N−1 ∈ RN . Signals we encounter in the real world, however, are usually not uniformlyregular and do not lie on a regular grid. Therefore, it is imperative to define an irregular samplingadapted to the local signal regularity (Mallat, 2008). For irregular sampling, the discretized irreg-ular signal is represented as f [qn] = f¯ ? Φ¯s(qn), for n = 0, ...,M − 1 and M ≤ N , where qn areirregular points (or nonequispaced nodes) randomly chosen from the set {0, ..., N − 1}. Its vectorrepresentation is f = f [qn]n=0,...,M−1.For a signal f0 ∈ RN that admits a sparse representation x0 in some transform domain—i.e., f0is sparse with respect to a basis or redundant frame S ∈ CP×N , with P ≥ N , such that f0 = SHx0(x0 sparse), whereH denotes the Hermitian transpose—the goal in CS is to reconstruct the signalf0 from few random linear measurements, y = Af0, where A is an n×N measurement matrix withn  N . Utilizing prior knowledge that f0 is sparse with respect to a basis or redundant frame Sand assuming the signal to be sampled on a periodic discrete grid, CS aims to find an estimate x˜for the underdetermined system of linear equations: y = Af0. This is done by solving the basispursuit (BP, Cande`s et al., 2006c; Donoho, 2006) convex optimization problem:x˜ = arg minx‖x‖1 :=N∑i=1|xi| subject to y = Ax. (6.1)In the noise-free case, this problem finds amongst all possible vectors x, the vector that has thesmallest `1-norm and that explains the observed subsampled data.Mathematically, a seismic line with Ns sources, Nr receivers, and Nt time samples can bereshaped into an N dimensional vector f , where N = Ns×Nr×Nt. Since real-world signals are notexactly sparse but compressible—i.e., can be well approximated by a sparse signal—a compressiblerepresentation, x, of the seismic line in the curvelet domain, S, is represented as f = SHx. Sincecurvelets are a redundant frame (an over-complete sparsifying dictionary), S ∈ CP×N with P > N ,and x ∈ CP . With the inclusion of the sparsifying transform, the measurement matrix A can befactored into the product of a n×N (with n N) acquisition matrix M and the synthesis matrixSH—i.e., A = MSH . For the real-world irregular signals, however, we need to account for theacquired unstructured measurements for high-resolution data recovery. We do this by introducingan operator in the recovery algorithm (by modifying the measurement operator A—see details inthe next sections) that acknowledges the irregularity of seismic traces and uses this information torender regularized and interpolated data.1136.3.1 Acquisition geometryIn time-jittered marine acquisition, source vessels map the survey area firing shots at jittered timeinstances, which translate to jittered shot positions for a given (fixed) speed of the source vessel.The simultaneous data is time compressed, and therefore acquired economically with low environ-mental imprint. The recovered separated data is periodic and dense. For simplicity, we assumethat all shot positions see the same receivers, which makes our method applicable to marine ac-quisition with ocean bottom cables or nodes (OBC or OBN). The receivers record continuouslyresulting in simultaneous shot records. Randomization via jittered subsampling offers control overthe maximum gap size (on the acquisition grid), which is a practical requirement of wavefield re-construction with localized sparsifying transforms such as curvelets (Hennenfent and Herrmann,2008). For simultaneous time-jittered acquisition, parameters such as the minimum distance re-quired between adjacent shots and minimum recharge time for the air guns help in controlling themaximum acquisition gap while maintaining the minimum realistic acquisition gap.Conventional acquisition with one source vessel and two air-gun arrays where each air-gun ar-ray fires at every alternate periodic location is called flip-flop acquisition. If we wish to acquire10.0 s-long shot records at every 12.5 m, the speed of the source vessel would have to be about1.25 m/s (≈ 2.5 knots). Figure 6.5(a) illustrates one such conventional acquisition scheme, whereeach air-gun array fires every 20.0 s (or 25.0 m) in a flip-flop manner traveling at about 1.25 m/s,resulting in nonoverlapping shot records of 10.0 s every 12.5 m. In time-jittered acquisition, Fig-ures 6.5(b) and 6.5(c), each air-gun array fires on average at every 20.0 s jittered time-instancestraveling at about 2.5 m/s (≈ 5.0 knots) with the receivers (OBC/OBN) recording continuously,resulting in overlapping shot records (Figures 6.6(a) and 6.6(b)). Note that the acquisition designinvolves jittered subsampling—i.e., regular decimation of the (fine) interpolation grid and subse-quent perturbation of the coarse-grid points completely off the fine grid. The idealized discretejittered subsampling, by contrast, perturbs the coarse-grid points on the fine grid, as presented inChapter 5. The subsampling factor is represented by η. Therefore, the acquired data volume hasoverlapping shots and missing shots/traces (Figure 6.6(a) and 6.6(b)). For this reason, the jitteredflip-flop acquisition might not mimic the conventional flip-flop acquisition where air-gun array 1 and2 fire one after the other—i.e., in the center and right-hand plots of Figure 6.5(d) a circle (denotingarray 1) may be followed by another circle instead of a star (denoting array 2), and vice versa.However, the minimum interval between the jittered times is maintained at 10.0 s (typical intervalrequired for air-gun recharge), while the maximum interval is 30.0 s. For the speed of 2.5 m/s, thistranslates to jittering a 50.0 m source grid with a minimum (and maximum) interval of 25.0 m (and75.0 m) between jittered shots. Both arrays fire at the 50.0 m jittered grid independent of eachother.In time-jittered marine acquisition, the acquisition operator M is a combined jittered-shot se-lector and time-shifting operator. Since data is acquired on an irregular grid, it is imperative toincorporate operators in the design of the acquisition matrix M that account for and hence reg-ularize the irregularity in the data. This is critical to the success of the recovery algorithm. Theoff-the-grid acquisition design is different from that presented by Li et al. (2012), wherein an inter-polated restriction operator accounts for irregular points by incorporating Lagrange interpolationinto the restriction operator—i.e., the measurements are approximated using a kth-order Lagrangeinterpolation. In time-jittered acquisition, the jittered time instances are put on a time grid (definedby a time-sampling interval) where each jittered time instance is placed on the point closest to iton the regular time grid. The difference between the true jittered time and the regular grid point,114(a) (b) (c)(d)Figure 6.5: Marine acquisition with one source vessel and two air-gun arrays. (a) Conven-tional flip-flop acquisition. Time-jittered acquisition with a subsampling factor η = 2for the (b) baseline and (c) monitor. Note the acquisition speedup during jitteredacquisition, where the recording time is reduced to one-half the recording time of theconventional acquisition. (d) Zoomed sections of (a), (b) and (c), respectively.∆t, is corrected by shifting the traces by e−iω∆t, where ω is the angular frequency. The irregularityin the shot positions is corrected by including the nonequispaced fast Fourier transform, NFFT(Potts et al., 2001; Kunis, 2006), in the sparsifying operator S (Hennenfent and Herrmann, 2006;Hennenfent et al., 2010), as described in the next section. The NFFT evaluates a Fourier expansionat nonequispaced locations defined by the time-jittered acquisition. Note that in this framework itis also possible to randomly subsample the receivers.Randomly subsampled and simultaneous measurements for the baseline and monitor surveys areshown in Figures 6.6(a) and 6.6(b), respectively. Note that only 40.0 s of the continuously recordeddata is shown. If we simply apply the adjoint of the acquisition operator to the correspondingsimultaneous data—i.e., MHy—the interferences (or source crosstalk) due to overlapping shotsappear as incoherent and nonsparse in the receiver gathers (Figures 6.7(a) and 6.7(b)). Moreover,115(a) (b)Figure 6.6: Simultaneous data for the (a) baseline and (b) monitor surveys. Only 40.0 s ofthe full data is shown. Time-jittered acquisition generates a simultaneous data volumewith overlapping shots and missing shots.since regularization (and interpolation) is performed by the NFFT inside a nonequispaced curveletframework (see next section), Figures 6.7(a) and 6.7(b) have Nsη irregular traces, where η > 1 is thesubsampling factor. Since the baseline and monitor surveys have different irregular shot positions,the corresponding time-lapse difference cannot be computed unless both time-lapse data are re-aligned to a common spatal grid. For this purpose, if we apply the adjoint of a 1D NFFT operatorN—i.e., NHMHy—the time-lapse data are realigned to a common fine spatial grid (Figures 6.7(c)and 6.7(d)). The corresponding time-lapse difference is shown in Figure 6.7(e). As illustrated bythese figures, in order to eventually remove the interferences and interpolate missing traces it isimportant to consider the recovery problem as an inversion problem. Since the time-jittered acqui-sition generates simultaneous, irregular data with missing traces, the recovery problem becomes ajoint source separation, regularization and interpolation problem.116(a) (b)(c) (d) (e)Figure 6.7: Interferences (or source crosstalk) in a common-receiver gather for the (a) base-line and (b) monitor surveys, respectively. Receiver gathers are obtained via MHyfor the time-lapse surveys. For a subsampling factor η, (a) and (b) have Nsη irreg-ular traces. (c), (d) Common-receiver gathers for the baseline and monitor surveys,respectively, after applying the adjoint of a 1D NFFT operator to (a) and (b). (e)Corresponding time-lapse difference. As illustrated, the recovery problem needs to beconsidered as a (sparse) structure-promoting inversion problem, wherein the simulta-neous data volume is separated, regularized and interpolated to a finer sampling gridrendering interference-free data.1176.3.2 From discrete to continuous spatial subsamplingSubsampling schemes that are based on an underlying fine interpolation grid incorporate the dis-crete (spatial) subsampling schemes, since the subsampling is done on the grid. This situationtypically occurs when binning continuous randomly-sampled seismic data into small bins that de-fine the fine grid used for interpolation (Hennenfent and Herrmann, 2008). For such cases, thewrapping-based fast discrete curvelet transform, FDCT via wrapping (Cande`s et al., 2006a) canbe used to recover the fully sampled data since the inherent fast Fourier transform (FFT) as-sumes regular sampling along all coordinates. For the interested reader, the curvelet transformis a multiscale, multidirectional, and localized transform that corresponds to a specific tiling ofthe f-k domain into dyadic annuli based on concentric squares centered around the zero-frequencyzero-wavenumber point. Each annulus is subdivided into parabolic angular wedges—i.e., length ofwedge ∝ width2 of wedge. The architecture of the analysis operator (or forward operation) of theFDCT via wrapping is as follows: (1) apply the analysis 2D FFT; (2) form the angular wedges;(3) wrap each wedge around the origin; and (4) apply the synthesis 2D FFT to each wedge. Thesynthesis/adjoint operator—also the inverse owing to the tight-frame property—is computed byreversing these operations (Cande`s et al., 2006a).Seismic data, however, is usually acquired irregularly, typically nonuniformly sampled along thespatial coordinates. Simultaneous time-jittered marine acquisition, mentioned above, is an instanceof acquiring seismic data on irregular spatial grids. Hence, binning can lead to a poorly-jitteredsubsampling scheme, which will not favor wavefield reconstruction by sparsity-promoting inversion.Moreover, failure to account for the nonuniformly sampled data can adversely affect seismic pro-cessing, imaging, etc. Therefore, we should work with an extension to the curvelet transform forirregular grids (Hennenfent et al., 2010). Using this extension for the simultaneous time-jitteredmarine acquisition will produce colocated fine-scale time-lapse data. Continuous random samplingtypically leads to improved interpolation results because it does not create coherent subsamplingartifacts (Xu et al., 2005).6.3.3 Nonequispaced fast discrete curvelet transform (NFDCT)For irregularly acquired seismic data, the (FFT inside) FDCT (Cande`s et al., 2006a) assumes regularsampling along all (spatial) coordinates. Ignoring the nonuniformity of the spatial sampling nolonger helps in detecting the wavefronts because of a lack of continuity. Hennenfent and Herrmann(2006) addressed this issue by extending the FDCT to nonuniform (or irregular) grids via thenonequispaced fast Fourier transform, NFFT (Potts et al., 2001; Kunis, 2006). The outcome wasthe ‘first generation NFDCT’ (nonequispaced fast discrete curvelet transform), which relied onaccurate Fourier coefficients obtained by an `2-regularized inversion of the NFFT.The NFDCT handles irregular sampling, thus, exploring continuity along the wavefronts byviewing seismic data in a geometrically correct way—typically nonuniformly sampled along thespatial coordinates (source and/or receiver). In Hennenfent et al. (2010), the authors introduceda ‘second generation NFDCT’, which is based on a direct, `1-regularized inversion of the operatorthat links curvelet coefficients to irregular data. Unlike the first generation NFDCT, the secondgeneration NFDCT is lossless by construction—i.e., the curvelet coefficients explain the data atirregular locations exactly. This property is important for processing irregularly sampled seismicdata in the curvelet domain and bringing them back to their irregular recording locations with highfidelity. Note that the second generation NFDCT is lossless for regularization not interpolation.118The NFDCT framework as setup in Hennenfent et al. (2010) basically involves a Kronecker product(⊗) of a 1D FFT operator Ft, used along the temporal coordinate, and a 1D NFFT operator Nx,used along the spatial coordinate, followed by the application of the curvelet tilling operator Tthat maps curvelet coefficients to the Fourier domain—i.e., Bdef= T(Nx ⊗ Ft). Therefore, B isthe NFDCT operator that links the curvelet coefficients to nonequispaced traces. The 1D NFFToperator (Nx) replaces the 1D FFT operator (Fx) that acts along the spatial coordinate in FDCT.Note that the NFDCT operator described above is written differently than in Hennenfent et al.(2010) because the latter defines the synthesis FFT operator as F, whereas F is the analysis FFToperator in this chapter. This also ensures consistency of notation and terminology with Chapter5.For the proposed simultaneous acquisition, the joint problem of source separation, regularizationand interpolation is addressed by using a sparsifying operator (S) that handles the multidimen-sionality of this problem. Therefore, Sdef= C⊗W, where C is a 2D NFDCT operator and W is a1D wavelet operator. The NFDCT operator is modified asCdef= T(Nxs ⊗ Fxr), (6.2)where the 1D NFFT operator Nxs acts along the jittered shot coordinate and the 1D FFT operatorFxr acts along the regular receiver coordinate. The 1D wavelet operator is applied on the timecoordinate. As mentioned previously, the measurement matrix A = MSH . From a practical pointof view, it is important to note that matrix-vector products with all the matrices are matrix free—i.e., these matrices are operators that define the action of the matrix on a vector, but are neverformed explicitly.In summary, recovery of nonoverlapping, periodic and densely sampled data from simultaneous,irregular and compressed data is achieved by incorporating an NFFT operator inside the curveletframework that acts along the irregular spatial coordinate(s) and applying time shifts to the traceswherever necessary. Note that the NFFT operator is incorporated in the 2D NFDCT operatorC, which is incorporated in the sparsifying operator S, and the time shift ∆t is incorporated inthe acquisition operator M. The NFFT computes (fine grid) 2D Fourier coefficients by mappingthe coarse nonuniform spatial grid to a fine uniform grid. The curvelet coefficients are computeddirectly from the 2D Fourier coefficients.6.4 Time-lapse acquisition via jittered sourcesIn Chapter 5, we extended the time-jittered marine acquisition to time-lapse surveys where theshot positions were jittered on a discrete periodic grid. In this chapter, we extend the framework tomore realistic field acquisition scenarios by incorporating irregular grids. Figure 6.5(a) illustrates aconventional marine acquisition scheme and two realizations of the off-the-grid time-jittered marineacquisition are shown in Figures 6.5(b) and 6.5(c), one each for the baseline and the monitor survey.Remember that these surveys generate simultaneous, irregular and subsampled measurements. Weassume no significant variations in the water column velocities, wave heights or temperature andsalinity profiles, etc., amongst the different surveys. The source signature is also assumed to be thesame.We describe noise-free time-lapse data acquired from a baseline and a monitor survey asyj = Ajxj for j = {1, 2}, where y1 and y2 represent the subsampled, simultaneous measure-119ments for the baseline and monitor surveys, respectively; A1 and A2 are the corresponding flat(n  N < P ) measurement matrices. Note that both the measurement matrices incorporate theNFDCT operator, as described above, to account and correct for the irregularity in the observedmeasurements of the baseline (y1) and monitor surveys (y2). Recovering densely sampled vintagesfor each vintage independently (via Equation 6.1) is referred to as the independent recovery strat-egy (IRS). Since in Chapter 5 we demonstrated that recovery via IRS is inferior to recovery via thejoint recovery method, we work only with the latter in this chapter.6.4.1 Joint recovery methodThe joint recovery method (JRM) performs a joint inversion by exploiting shared informationbetween the vintages. The joint recovery model (DCS, Baron et al., 2009) is formulated as[y1y2]=[A1 A1 0A2 0 A2]z0z1z2 , ory = Az.(6.3)In this model, the vectors y1 and y2 represent observed measurements from the baseline and monitorsurveys, respectively. The vectors for the vintages are given byxj = z0 + zj , j ∈ 1, 2, (6.4)where the common component is denoted by z0, and the innovations are denoted by zj for j ∈1, 2 with respect to this common component that is shared by the vintages. The symbol A isoverloaded to refer to the matrix linking the observations of the time-lapse surveys to the commoncomponent and innovations pertaining to the different vintages. The above joint recovery modelcan be extended to J > 2 surveys, yielding a J × (number of vintages + 1) system.Since the vintages share the common component in Equation 6.3, solvingz˜ = arg minz‖z‖1 subject to y = Az, (6.5)will exploit the correlations amongst the vintages. Equation 6.5 seeks solutions for the commoncomponent and innovations that have the smallest `1-norm such that the observations explainthe incomplete recordings for both vintages. The densely sampled vintages are estimated viaEquation 6.4 with the recovered z˜ and the time-lapse difference is computed via z˜1 − z˜2.Given a baseline data vector f1 and a monitor data vector f2, x1 and x2 are the correspondingsparse representations. Given the measurements y1 = M1f1 and y2 = M2f2, and A1 = M1SH1 ,A2 = M2SH2 , our aim is to recover the wavefields (or sparse approximations) f˜1 and f˜2 by solvingthe sparse recovery problem as described above from which the time-lapse signal can be computed.Note that Sdef= C⊗W, where C is the NFDCT operator (see Equation 6.2) and W is a 1D waveletoperator. The reconstructed wavefields f˜1 and f˜2 are obtained as: f˜1 = SH x˜1 and f˜2 = SH x˜2, wherex˜1 and x˜2 are the recovered sparse representations and the operator S is overwritten to representthe Kronecker product between the standard FDCT operator and the 1D wavelet operator. Thestandard FDCT operator is used because the recovered sparse representations x˜1 and x˜2 correspondto the coefficients of the regularized wavefields. Since we are always subsampled in both the120baseline and monitor surveys, have irregular traces and cannot exactly repeat, which is inherentof the acquisition design and due to natural environmental constraints, we would like to recoverthe periodic densely sampled prestack vintages and time-lapse difference. For the given recoveryproblem, the vintages and time-lapse difference are mapped to one colocated fine regular periodicgrid.6.5 Economic performance indicatorsTo quantify the cost savings associated with simultaneous acquisition, we measure the performanceof the proposed acquisition design and recovery scheme in terms of an improved spatial-samplingratio (ISSR), defined asISSR =number of shots recovered via sparsity-promoting inversionnumber of shots in simultaneous acquisition. (6.6)For time-jittered marine acquisition, a subsampling factor η = 2, 4, ..., etc., implies a gain in thespatial sampling by factor of 2, 4, ..., etc. In practice, this corresponds to an improved efficiency ofthe acquisition by the same factor. Recently, Mosher et al. (2014) have shown that factors of twoor as high as ten in efficiency improvement are achievable in the field.The survey-time ratio (STR)—a performance indicator proposed by Berkhout (2008)—comparesthe time taken for conventional and simultaneous acquisition:STR =time of conventional acquisitiontime of simultaneous acquisition. (6.7)As mentioned previously, if we wish to acquire 10.0 s-long shot records at every 12.5 m, the speedof the source vessel would have to be about 1.25 m/s (≈ 2.5 knots). In simultaneous acquisition,the speed of the source vessel is approximately maintained at (the standard) 2.5 m/s (≈ 5.0 knots).Therefore, for a subsampling factor of η = 2, 4, ..., etc., there is an implicit reduction in the surveytime by 1η .6.6 Synthetic seismic case studyTo illustrate the performance of our proposed joint recovery method for off-the-grid surveys, wecarry out a number of experiments on 2D seismic lines generated from two different velocitymodels—first, the BG COMPASS model (provided by BG Group) that has simple geology withcomplex time-lapse difference; and second, the SEAM Phase 1 model (provided by HESS) that hascomplex geology with complex time-lapse difference due to the complexity of the overburden. Notethat for the SEAM model, we generate the time-lapse difference via fluid substitution as shownbelow. Also, the geology of the BG COMPASS model is relatively simpler than the SEAM model,although it does have vertical and lateral complexity.6.6.1 BG COMPASS model—simple geology, complex time-lapse differenceThe synthetic BG COMPASS model has a (relatively) simple geology but a complex time-lapsedifference. Figures 6.8(a) and 6.8(b) display the baseline and monitor models. Note that thisis a subset of the BG COMPASS model, wherein the monitor model includes a gas cloud. The121(a) (b) (c)Figure 6.8: Subset of the BG COMPASS model. (a) Baseline model; (b) monitor model; (c)difference between (a) and (b) showing the gas cloud.time-lapse difference in Figure 6.8(c) shows the gas cloud.Using IWAVE (Symes, 2010), a time-stepping simulation software, two acoustic data sets witha conventional source (and receiver) sampling of 12.5 m are generated, one from the baseline modeland the other from the monitor model. Each data set has Nt = 512 time samples, Nr = 260 receiversand Ns = 260 sources. The time sampling interval is 0.004 s. Subtracting the two data sets yieldsthe time-lapse difference. Since no noise is added to the data, the time-lapse difference is simplythe time-lapse signal. A receiver gather from the simulated baseline data, the monitor data andthe corresponding time-lapse difference is shown in Figure 6.2(a), 6.2(b) and 6.2(c), respectively.The first shot position in the receiver gathers—labeled as 0 m in the figures—corresponds to 1.5 kmin the synthetic velocity model. Given the spatial sampling of 12.5 m, the subsampling factor ηfor the time-jittered acquisition is 2. Hence, the number of measurements for each experiment isfixed—i.e., n = N/η = N/2, each for y1 and y2. We also conduct experiments for η = 4.To reflect current practices in time-lapse acquisition—where people aim to replicate the surveys—we simulate 10 different realizations of the time-jittered marine acquisition with 100% overlapbetween the baseline and monitor surveys. The term “overlap” refers to the percentage of shotpositions from the baseline survey revisited (or replicated exactly) for the monitor survey, andtherefore rows in the measurement matrices A1 and A2 are exactly the same. Note that these shotpositions are irregular, and hence off the grid. However, since exact replication of the surveys in thefield is not possible, we conduct experiments to study the impact of deviations in the shot positionsthat would occur naturally in the field. We introduce small deviations of average ±(1, 2, 3) m in theshot positions of the baseline surveys to generate the shot positions for the monitor surveys. Forinstance, given a realization of the time-jittered baseline survey, deviating each shot position by≈ ±1 m generates shot positions for the corresponding monitor survey. Note that these deviationsare average deviations in the sense that for a given realization of the time-jittered baseline survey,the shot positions are deviated by random real numbers resulting in average deviations of ±1 m,±2 m or ±3 m. One of our aims is to analyze the effects of nonreplication of the time-lapse surveyson time-lapse data—i.e., when A1 6= A2. By virtue of the design of the simultaneous acquisitionand based upon the subsampling factor (η), it is not possible to have two completely different (0%overlap) realizations of the time-jittered acquisition. Therefore, we compare recoveries from theabove cases with the acquisition scenarios that have least possible (or unavoidable) overlap betweenthe time-lapse surveys. In all cases, we recover periodic densely sampled baseline and monitor datafrom the simultaneous data y1 and y2, respectively, using the joint recovery method (by solvingEquation 6.5). The inherent time-lapse difference is computed by subtracting the recovered baselineand monitor data.122We conduct 10 experiments for the baseline measurements, wherein each experiment has adifferent random realization of the measurement matrix A1. Then, for each experiment, we fixthe baseline measurement and subsequently work with different realizations of the monitor surveygenerated by introducing small deviations in the shot positions and jittered firing times from thebaseline survey, resulting in slightly different overlaps between the surveys. To get better insighton the effects of nonreplication of the time-lapse surveys, we also conduct experiments for the caseof least possible overlap between the surveys. Tables 6.1 and 6.2 summarize the recovery resultsfor the time-lapse data for η = 2 and 4, respectively, in terms of the signal-to-noise ratio definedasS/N(f , f˜) = −20 log10‖f − f˜‖2‖f‖2 . (6.8)Each table compares recoveries for different overlaps between the baseline and monitor surveys,with and without position deviations. Each S/N value is an average of 10 experiments includingthe standard deviation. Note that for time-jittered acquisition with η = 2, the least possible overlapbetween the surveys is observed to be greater than 0% and less than 15%. Hence, Table 6.1 showsthe S/Ns for the overlap of < 15%. Similarly, for time-jittered acquisition with η = 4, Table 6.2shows the S/Ns for the overlap of < 5%.We recover periodic densely sampled data from simultaneous, subsampled and irregular data bysolving Equation 6.5. The recovered time-lapse data is colocated, regularized and interpolated to afine uniform grid since both the measurement matrices A1 and A2 incorporate a 2D nonequispacedfast discrete curvelet transform that handles irregularity of traces by viewing the observed data ina geometrically correct way. The S/Ns of the recovered time-lapse data lead to some interestingobservations. First, there is little variability in the recovery of the time-lapse difference from (theideal) 100% overlap between the surveys to the more realistic scenarios of in-the-field acquisitionsthat have natural deviations or irregularities in the shot positions. Second, time-lapse differencerecovery from the least possible overlap (between the surveys) is similar to the recovery of 100%overlap with and without deviations. This is significant because it indicates a possibility to relaxthe insistence on replication of the time-lapse surveys, which makes this technology challenging andexpensive. The small standard deviations for each case suggest little variability in the recovery fordifferent random realizations. Moreover, the standard deviations are greater for cases other thanthe minimum overlap. The above observations hold for both subsampling factors, η = 2 and 4, asillustrated in Figures 6.10 and 6.12.Third, increasing deviations or irregularities in shot positions improve recovery of the vintages(Figures 6.9(c), 6.9(e), 6.9(g)), with the minimum overlap between surveys giving the best recov-ery (Figure 6.9(i)). This is due to the (partial) independence of the measurement matrices thatcontribute additional information via the first column of A in Equation 6.3 connecting the com-mon component to observations of both vintages—i.e., for time-lapse seismic, independent surveysgive additional structural information leading to improved recovery quality of the vintages. Theimprovement in the recoveries is better visible through the corresponding difference plots in Fig-ures 6.9(d), 6.9(f), 6.9(h), 6.9(j). This observation is important because, as mentioned previously,time-lapse differences are often studied via differences in certain poststack attributes computedfrom the (recovered) prestack vintages. Hence, as the quality of the recovered prestack vintagesimproves with decrease in the overlap, they serve as better input to extract the poststack attributes.Moreover, the small standard deviations for each overlap indicate little variability in the recoveryfrom one random realization to another. This is desirable since it offers a possibility to relax the123Overlap ± avg. deviation Baseline Monitor 4D signal100% 19.8 ± 1.0 19.7 ± 1.0 11.3 ± 2.2100% ± 1.0 m 19.7 ± 1.0 19.6 ± 1.0 10.3 ± 1.5100% ± 2.0 m 20.3 ± 1.1 20.2 ± 1.0 10.7 ± 1.1100% ± 3.0 m 20.8 ± 1.2 20.7 ± 1.1 11.0 ± 1.4< 15% 23.8 ± 1.4 23.6 ± 1.4 10.2 ± 1.2Table 6.1: Summary of recoveries in terms of S/N (dB) for data recovered via JRM for asubsampling factor η = 2. The S/Ns show little variability in the time-lapse differencerecovery for different overlaps between the surveys offering a possibility to relax insis-tence on replicability of time-lapse surveys. This is supported by the improved recoveryof the vintages as the overlap decreases. Note that the deviations are average deviations.insistence on replication of the time-lapse surveys along with embracing the naturally occurringrandom deviations in the field. The standard deviations for different overlaps also do not fluctu-ate as much as compared to those of the time-lapse difference. Recovery of the vintages and thecorresponding difference plots for a subsampling of η = 4 are shown in Figure 6.11.An increase in the subsampling factor leads to decrease in the S/Ns of the recovered time-lapsedata, however, the recoveries are reasonable as shown in Figures 6.11 and 6.12. This observation isin accordance with the CS theory wherein the recovery quality decreases for increased subsampling.Note that recovery of weak late-arriving events can be further improved by rerunning the recoveryalgorithm using the residual as input, using weighted one-norm minimization that exploits corre-lations between locations of significant transform-domain coefficients of different partitions—e.g.,shot records, common-offset gathers, or frequency slices—of the acquired data (Mansour et al.,2013), etc. This needs to be carefully investigated. Remember that for a given subsampling factorthe number of measurements is the same for all experiments and the observed differences can befully attributed to the performance of the joint recovery method in relation to the overlap betweenthe two surveys encoded in the measurement matrices. Also, given the context of randomizedsubsampling and irregularity of the observed data, it is important to recover the densely sampledvintages and then the time-lapse difference. Moreover, as mentioned previously, while we do notinsist that we actually visit predesigned irregular (or off-the-grid) shot positions for the time-lapsesurveys, however, it is important to know these positions to sufficient accuracy after acquisition forhigh-quality data recovery. This can be achieved in practice as shown by Mosher et al. (2014).6.6.2 SEAM Phase 1 model—complex geology, complex time-lapse differenceThe SEAM model is a 3D deepwater subsalt earth model that includes a complex salt intrusivein a folded Tertiary basin. We select a 2D slice from the 3D model to generate a seismic line.Figure 6.13(a) shows a subset of the 2D slice used as the baseline model. We define the monitormodel, Figure 6.13(b), from the baseline model via fluid substitution resulting in a time-lapsedifference under the overburden as shown in Figure 6.13(c).Using IWAVE (Symes, 2010), two acoustic data sets with a conventional source (and receiver)sampling of 12.5 m are generated, one from the baseline model and the other from the monitormodel. Each data set has Nt = 2048 time samples, Nr = 320 receivers and Ns = 320 sources. Thetime sampling interval is 0.004 s. Subtracting the two data sets yields the time-lapse difference.124(a) (b) (c) (d)(e) (f) (g) (h)(i) (j)Figure 6.9: JRM recovered monitor receiver gathers from the BG COMPASS model for asubsampling factor η = 2. Recovered monitor data and residual with (a,b) 100%overlap in the measurement matrices (A1 and A2); (c,d) 100% overlap and averageshot-position deviation of 1 m; (e,f) 100% overlap and average shot-position deviationof 2 m; (g,h) 100% overlap and average shot-position deviation of 3 m; (i,j) < 15%overlap, respectively.125(a) (b) (c) (d)(e) (f) (g) (h)(i) (j)Figure 6.10: JRM recovered time-lapse difference receiver gathers from the BG COMPASSmodel for a subsampling factor η = 2. Recovered time-lapse difference and residualwith (a,b) 100% overlap in the measurement matrices (A1 and A2); (c,d) 100%overlap and average shot-position deviation of 1 m; (e,f) 100% overlap and averageshot-position deviation of 2 m; (g,h) 100% overlap and average shot-position deviationof 3 m; (i,j) < 15% overlap, respectively.126(a) (b) (c) (d)(e) (f) (g) (h)(i) (j)Figure 6.11: JRM recovered monitor receiver gathers from the BG COMPASS model for asubsampling factor η = 4. Recovered monitor data and residual with (a,b) 100%overlap in the measurement matrices (A1 and A2); (c,d) 100% overlap and averageshot-position deviation of 1 m; (e,f) 100% overlap and average shot-position deviationof 2 m; (g,h) 100% overlap and average shot-position deviation of 3 m; (i,j) < 5%overlap, respectively.127(a) (b) (c) (d)(e) (f) (g) (h)(i) (j)Figure 6.12: JRM recovered time-lapse difference receiver gathers from the BG COMPASSmodel for a subsampling factor η = 4. Recovered time-lapse difference and residualwith (a,b) 100% overlap in the measurement matrices (A1 and A2); (c,d) 100%overlap and average shot-position deviation of 1 m; (e,f) 100% overlap and averageshot-position deviation of 2 m; (g,h) 100% overlap and average shot-position deviationof 3 m; (i,j) < 5% overlap, respectively.128Overlap ± avg. deviation Baseline Monitor 4D signal100% 14.3 ± 0.6 14.2 ± 0.6 6.4 ± 0.7100% ± 1.0 m 14.9 ± 0.8 14.8 ± 0.8 6.5 ± 1.0100% ± 2.0 m 15.6 ± 1.0 15.5 ± 1.0 6.4 ± 1.3100% ± 3.0 m 16.4 ± 0.9 16.3 ± 0.9 6.4 ± 0.7< 5% 18.4 ± 0.7 18.2 ± 0.7 5.8 ± 0.4Table 6.2: Summary of recoveries in terms of S/N (dB) for data recovered via JRM for asubsampling factor η = 4. The S/Ns show little variability in the time-lapse differencerecovery for different overlaps between the surveys offering a possibility to relax insis-tence on replicability of time-lapse surveys. This is supported by the improved recoveryof the vintages as the overlap decreases. Note that the deviations are average deviations.(a) (b) (c)Figure 6.13: Subset of the SEAM model. (a) Baseline model; (b) monitor model; (c) differ-ence between (a) and (b) showing the time-lapse difference.Since no noise is added to the data, the time-lapse difference is simply the time-lapse signal. Areceiver gather from the simulated baseline data, the monitor data and the corresponding time-lapsedifference is shown in Figures 6.14(a), 6.14(b) and 6.14(c), respectively. Note that the amplitude ofthe time-lapse difference is one-tenth the amplitude of the baseline and monitor data. Therefore,in order to make the time-lapse difference visible, the color axis for the figures showing the time-lapse difference is one-tenth the color axis for the figures showing the baseline and monitor data.This colormap applies for the remainder of the chapter. Given the spatial sampling of 12.5 m, thesubsampling factor η for the time-jittered acquisition is 2. The number of measurements for eachexperiment is fixed—i.e., n = N/η = N/2, each for y1 and y2.We simulate a realization of the time-jittered marine acquisition with 100% overlap betweenthe baseline and monitor surveys. Since our main aim is to analyze the effects of nonreplicationof the time-lapse surveys on time-lapse data—i.e., when A1 6= A2—we compare recovery from the129(a) (b) (c)Figure 6.14: Synthetic receiver gathers from the conventional SEAM (a) baseline survey,(b) monitor survey. (c) Corresponding time-lapse difference. The amplitude of thetime-lapse difference is one-tenth the amplitude of the baseline and monitor data.above case with the acquisition scenario that has least possible (or unavoidable) overlap betweenthe time-lapse surveys only. Given the bigger size of the data set and limited computationalresources, we restrict ourselves to one experiment for each case and a subsampling of η = 2.Periodic densely sampled baseline and monitor data is recovered from the simultaneous data y1and y2, respectively, by solving Equation 6.5. The inherent time-lapse difference is computed bysubtracting the recovered baseline and monitor data.The recovered time-lapse data is colocated, regularized and interpolated to a fine uniform grid.We note that all the observations made for the BG COMPASS model, which is a relatively sim-pler model, hold true for the more complex SEAM model. Minimum overlap (or nonreplication)between time-lapse surveys improves recovery of the vintages since independent surveys give ad-ditional structural information. Hence, they serve as better input to extract certain poststackattributes used to study time-lapse differences. Figures 6.15(a), 6.15(b), 6.15(c) and 6.15(d) showthe corresponding monitor data recovery. The S/N for the vintage recovery for minimum overlapbetween the surveys is 30.2 dB—a significant improvement from the 19.5 dB recovery for 100%overlap between the surveys. Moreover, as seen in Figures 6.15(e), 6.15(f), 6.15(g) and 6.15(h),there is little variability in the recovery of the time-lapse difference from (the ideal) 100% overlapbetween the surveys to the more realistic almost nonreplicated surveys. The corresponding S/Nsfor the recovered time-lapse difference are 9.6 dB for 100% overlap and 4.1 dB for minimum overlap130between the surveys. We note that the S/N for the minimum overlap between the surveys is biaseddue the presence of incoherent noise—between 3.5 s to 5.0 s—above the main time-lapse difference.If we compute the S/Ns for the lower-half of the data that contains the time-lapse difference—i.e.,after 4.5 s—the S/N for minimum overlap between the surveys increases to 6.8 dB. More impor-tantly, if we look at the plots themselves, we see that there is not much difference in the tworecoveries. We are able to recover the primary arrivals and some reverberations below. Recall thatthe amplitude of the time-lapse difference is one-tenth the amplitude of the vintages. It is quiteremarkable that we get good results given the complexity of the model and the low amplitude ofthe time-lapse difference. Recovery of the vintages and the time-lapse difference for a subsamplingof η = 4 follows the same trend as above.6.7 DiscussionRealistic field seismic acquisitions suffer, amongst other possibly detrimental external factors, fromirregular spatial sampling of sources and receivers. This poses technical challenges for the time-lapse seismic technology that currently aims to replicate densely sampled surveys for monitoringchanges due to production. The experiments and synthetic results shown in the previous sectionsdemonstrate favourable effects of irregular sampling and nonreplication of surveys on time-lapsedata—i.e., decrease in replicability of the surveys leads to improved recovery of the vintages withlittle variability in the recovery of the time-lapse difference itself—while unraveling overlappingshot records. Note that we do not insist on replicating the irregular spatial positions in the field,however, the above observations hold as long as we know the irregular sampling positions afteracquisition to a sufficient degree of accuracy, which is attainable in practice (see e.g., Mosheret al., 2014). Furthermore, we assume that there are no significant variations in the water columnvelocities, wave heights or temperature and salinity profiles amongst the different surveys while thesource signature is also assumed to be the same. As long as these physical changes can be modeled,we do not foresee major problems. For instance, we expect that our approach can relatively easilybe combined with source equalization (see e.g., Rickett and Lumley, 2001) and curvelet-domainmatched filtering techniques (Beyreuther et al., 2005; Tegtmeier-Last and Hennenfent, 2013).The proposed methodology involves a combination of economical randomized samplings with lowenvironmental imprint and sparsity-promoting data recovery that aims to reduce cost of surveys andimprove quality of the prestack time-lapse data without relying on expensive dense sampling andhigh degrees of replicability of the surveys. The combined operation of source separation, regular-ization and interpolation renders periodic densely sampled time-lapse data from time-compressed,and therefore economical, simultaneous, subsampled and irregular data. While the simultaneousdata are separated reasonably well, recovery of the weak late-arriving events can be further im-proved by rerunning the recovery algorithm using the residual as input, using weighted one-normminimization that exploits correlations between locations of significant transform-domain coeffi-cients of different partitions—e.g., shot records, common-offset gathers, or frequency slices—of theacquired data (Mansour et al., 2013), etc. This needs to be examined in detail. Effects of noise andother physical changes in the environment also need to be carefully investigated. Nevertheless, asexpected using standard CS, our recovery method should be stable with respect to noise (Cande`set al., 2006c). Moreover, recent successes in the application of compressed sensing to land and ma-rine field data acquisition (see e.g., Mosher et al., 2014) support the fact that technical challengeswith noise and calibration can be overcome in practice.131(a) (b) (c) (d)(e) (f) (g) (h)Figure 6.15: JRM recovered monitor and time-lapse difference receiver gathers from theSEAM model for a subsampling factor η = 2. Recovered monitor data and resid-ual with (a,b) 100% overlap in the measurement matrices (A1 and A2); (c,d) < 15%overlap, respectively. Recovered time-lapse difference and residual with (e,f) 100%overlap in the measurement matrices; (g,h) < 15% overlap, respectively. Note thatthe amplitude of the time-lapse difference is one-tenth the amplitude of the monitordata.1326.8 ConclusionsWe present an extension of our simultaneous time-jittered marine acquisition to time-lapse sur-veys for realistic, off-the-grid acquisitions where the sample points are known but do not coincidewith a regular periodic grid. We conduct a series of synthetic seismic experiments with differ-ent random realizations of the simultaneous time-jittered marine acquisition to assess the effectsof irregular sampling in the field on time-lapse data and demonstrate that dense, high-qualitydata recoveries are the norm and not the exception. We achieve this by adapting our proposedjoint recovery method—a new and economic approach to randomized simultaneous time-lapse dataacquisition that exploits transform-domain sparsity and shared information among different time-lapse recordings—to incorporate a regularization operator that maps traces from an irregular gridto a regular periodic grid. The recovery method is a combined operation of source separation,regularization and interpolation, wherein periodic densely sampled and colocated prestack data isrecovered from time-compressed, and therefore economical, simultaneous, subsampled and irregulardata.We observe that with decrease in replication between the surveys—i.e., shot points are not repli-cated amongst the vintages—recovery of time-lapse data improve significantly with little variabilityin recovery of the time-lapse difference itself. We make this observation assuming source equaliza-tion and no significant changes in wave heights, water column velocities or temperature and salinityprofiles, etc., amongst the different surveys. We also demonstrate the delicate reliance on exactreplicability (between surveys) by showing that known deviations as small as average ±(1, 2, 3) min shot positions of the monitor surveys from the baseline surveys vary recovery quality of the time-lapse difference—expressed as slight decrease or increase in the signal-to-noise ratios—and hencenegate the efforts to replicate. Therefore, it would be better to focus on knowing what the shot po-sitions were (post acquisition) than aiming to replicate. Moreover, since irregular spatial samplingis inevitable in the real world, the requirement for replicability in time-lapse surveys can perhapsbe relaxed by embracing or better purposefully randomizing the acquisitions to maximize collectionof information by effectively doubling the number of measurements for the common component,leading to surveys acquired at low cost and environmental imprint.133Chapter 7Source separation for simultaneoustowed-streamer marine acquisition —a compressed sensing approach7.1 SummarySimultaneous marine acquisition is an economic way to sample seismic data and speedup acquisition,wherein single and/or multiple source vessels fire sources at near-simultaneous or slightly randomtimes, resulting in overlapping shot records. The current paradigm for simultaneous towed-streamermarine acquisition incorporates “low-variability” in source firing times—i.e., 0 ≤ 1 or 2 seconds,since both the sources and receivers are moving. This results in low degree of randomness in simulta-neous data, which is challenging to separate (into its constituent sources) using compressed sensingbased separation techniques since randomization is the key to successful recovery via compressedsensing. We address the challenge of source separation for simultaneous towed-streamer acquisitionsvia two compressed sensing based approaches—i.e., sparsity-promotion and rank-minimization. Weillustrate the performance of both the sparsity-promotion and rank-minimization based techniquesby simulating two simultaneous towed-streamer acquisition scenarios—i.e., over/under and simul-taneous long offset. A field data example from the Gulf of Suez for the over/under acquisitionscenario is also included. We observe that the proposed approaches give good and comparablerecovery qualities of the separated sources, but the rank-minimization technique outperforms thesparsity-promoting technique in terms of the computational time and memory. We also comparethese two techniques with the NMO-based median filtering type approach.7.2 IntroductionThe benefits of simultaneous source marine acquisition are manifold—it allows the acquisition ofimproved-quality seismic data at standard (conventional) acquisition turnaround, or a reducedturnaround time while maintaining similar quality, or a combination of both advantages. In simul-taneous marine acquisition, a single or multiple source vessels fire sources at near-simultaneous orslightly random times resulting in overlapping shot records (de Kok and Gillespie, 2002; Beasley,A version of this chapter has been published in Geophysics, 2015, vol. 80, pp. WD73–WD88.1342008; Berkhout, 2008; Hampson et al., 2008; Moldoveanu and Quigley, 2011; Abma et al., 2013), asopposed to nonoverlapping shot records in conventional marine acquisition. A variety of simultane-ous source survey designs have been proposed for towed-streamer and ocean bottom acquisitions,where small-to-large random time delays between multiple sources have been used (Beasley, 2008;Moldoveanu and Fealy, 2010; Mansour et al., 2012b; Abma et al., 2013; Wason and Herrmann,2013b; Mosher et al., 2014).An instance of low-variability in source firing times—e.g., 0 ≤ 1 (or 2) second, is the over/under(or multi-level) source acquisition (Hill et al., 2006; Moldoveanu et al., 2007; Lansley et al., 2007;Long, 2009; Hegna and Parkes, 2012; Hoy et al., 2013). The benefits of acquiring and processingover/under data are clear, the recorded bandwidth is extended at both low and high ends of thespectrum since the depths of the sources produce complementary ghost functions, avoiding deepnotches in the spectrum. The over/under acquisition allows separation of the up- and down-goingwavefields at the source (or receiver) using a vertical pair of sources (or receivers) to determinewave direction. Simultaneous long offset acquisition (SLO) is another variation of simultaneoustowed-streamer acquisition, where an extra source vessel is deployed, sailing one spread-lengthahead of the main seismic vessel (Long et al., 2013). The SLO technique is better in comparisonto conventional acquisition since it provides longer coverage in offsets, less equipment downtime(doubling the vessel count inherently reduces the streamer length by half), easier maneuvering, andshorter line turns.Simultaneous acquisition (e.g., over/under and SLO) results in seismic interferences or sourcecrosstalk that degrades quality of the migrated images. Therefore, an effective (simultaneous)source separation technique is required, which aims to recover unblended interference-free data—asacquired during conventional acquisition—from simultaneous data. The challenge of source sepa-ration (or deblending) has been addressed by many researchers (Stefani et al., 2007; Moore et al.,2008; Akerberg et al., 2008; Huo et al., 2009), wherein the key observation has been that as longas the sources are fired at suitably randomly dithered times, the resulting interferences (or sourcecrosstalk) will appear noise-like in specific gather domains such as common-offset and common-receiver, turning the separation problem into a (random) noise removal procedure. Inversion-typealgorithms (Moore, 2010; Abma et al., 2010; Mahdad et al., 2011; Doulgeris et al., 2012; Baardmanand van Borselen, 2013) take advantage of sparse representations of coherent seismic signals. Wa-son and Herrmann (2013a); Wason and Herrmann (2013b) proposed an alternate sampling strategyfor simultaneous acquisition (time-jittered marine) that leverages ideas from compressed sensing(CS), addressing the source-separation problem through a combination of tailored (blended) acqui-sition design and sparsity-promoting recovery via convex optimization using one-norm constraints.This represents a scenario of high-variability in source firing times—e.g., > 1 second, resulting inirregular shot locations.One of the source separation techniques is the normal moveout based median filtering, where thekey idea is as follows: i) transform the blended data into the midpoint-offset domain, ii) performsemblance analysis on common-midpoint gathers to pick the normal moveout (NMO) velocitiesfollowed by NMO corrections, iii) perform median filtering along the offset directions and thenapply inverse NMO corrections. One of the major assumptions in the described workflow is thatthe seismic events become flat after NMO corrections, however, this can be challenging when thegeology is complex and/or with the presence of noise in the data. Therefore, the above processalong with the velocity analysis is repeated a couple of times to get a good velocity model toeventually separate simultaneous data.135Recently, rank-minimization based techniques have been used for source separation by Maras-chini et al. (2012) and Cheng and Sacchi (2013). The general idea is to exploit the low-rank structureof seismic data when it is organized in a matrix. Low-rank structure refers to the small numberof nonzero singular values, or quickly decaying singular values. Maraschini et al. (2012) followedthe rank-minimization based approach proposed by Oropeza and Sacchi (2011), who identified thatseismic temporal frequency slices organized into a block Hankel matrix, in ideal conditions, is amatrix of rank k, where k is the number of different plane waves in the window of analysis. Oropezaand Sacchi (2011) showed that additive random noise increase the rank of the block Hankel matrixand presented an iterative algorithm that resembles seismic data reconstruction with the methodof projection onto convex sets, where they use a low-rank approximation of the Hankel matrixvia the randomized singular value decomposition (Liberty et al., 2007; Halko et al., 2011) to in-terpolate seismic temporal frequency slices. While this technique may be effective the approachrequires embedding the data into an even larger space where each dimension of size n is mapped toa matrix of size n× n. Consequently, these approaches are applied on small data windows, whereone has to choose the size of these windows. Although mathematically desirable due to the seismicsignal being stationary in sufficiently small windows, Kumar et al. (2015a) showed that the actof windowing from a matrix-rank point of view degrades the quality of reconstruction in the caseof missing-trace interpolation. Choosing window sizes a priori is also a difficult task, as it is notaltogether obvious how to ensure that the resulting subvolume is approximately a plane wave.7.2.1 MotivationThe success of CS hinges on randomization of the acquisition, as presented in our previous workon simultaneous source acquisition (Mansour et al., 2012b; Wason and Herrmann, 2013b), whichrepresents a case of high-variability in source firing times—e.g., within a range of 1-20 seconds, re-sulting in overlapping shot records that lie on irregular spatial grids. Consequently, this made ourmethod applicable to marine acquisition with ocean bottom cables/nodes. Successful separation ofsimultaneous data by sparse inversion via one-norm minimization, in this high-variability scenario,motivated us to analyze the performance of our separation algorithm for the low-variability, simul-taneous towed-streamer acquisitions. We address the challenge of source separation for two typesof simultaneous towed-streamer marine acquisition—over/under and simultaneous long offset. Wealso compare the sparsity-promoting separation technique with separation via rank-minimizationbased technique, since the latter is relatively computationally faster and memory efficient, as shownby Kumar et al. (2015a) for missing-trace interpolation.7.2.2 ContributionsOur contributions in this work are the following: first, we propose a practical framework for sourceseparation based upon the compressed sensing (CS) theory, where we outline the necessary con-ditions for separating the simultaneous towed-streamer data using sparsity-promoting and rank-minimization techniques. Second, we show that source separation using the rank-minimizationbased framework includes a “transform domain” where we exploit the low-rank structure of seismicdata. We further establish that in simultaneous towed-streamer acquisition each monochromaticfrequency slice of the fully sampled blended data matrix with periodic firing times has low-rankstructure in the proposed transform domain. However, uniformly random firing-time delays increasethe rank of the resulting frequency slice in this transform domain, which is a necessary condition136for successful recovery via rank-minimization based techniques.Third, we show that seismic frequency slices in the proposed transform domain exhibit low-rankstructure at low frequencies, but not at high frequencies. Therefore, in order to exploit the low-rank structure at higher frequencies we adopt the Hierarchical Semi-Separable matrix representation(HSS) method proposed by Chandrasekaran et al. (2006) to represent frequency slices. Finally, wecombine the (singular-value-decomposition-free) matrix factorization approach recently developedby Lee et al. (2010) with the Pareto curve approach proposed by Berg and Friedlander (2008).This renders the framework suitable for large-scale seismic data since it avoids the computation ofthe singular value decomposition (SVD), a necessary step in traditional rank-minimization basedmethods, which is prohibitively expensive for large matrices.We simulate two simultaneous towed-streamer acquisitions—over/under and simultaneous longoffset, and also use a field data example for over/under acquisition. We compare the recovery interms of the separation quality, computational time and memory usage. In addition, we also makecomparisons with the NMO-based median filtering type technique proposed by Chen et al. (2014).7.3 TheoryCompressed sensing is a signal processing technique that allows a signal to be sampled at sub-Nyquist rate and offers three fundamental principles for successful reconstruction of the originalsignal from relatively few measurements. The first principle utilizes the prior knowledge that theunderlying signal of interest is sparse or compressible in some transform domain—i.e., if only asmall number k of the transform coefficients are nonzero or if the signal can be well approximatedby the k largest-in-magnitude transform coefficients. The second principle is based upon a samplingscheme that breaks the underlying structure—i.e., decreases the sparsity of the original signal in thetransform domain. Once the above two principles hold, a sparsity-promoting optimization problemcan be solved in order to recover the fully sampled signal. It is well known that seismic data admitsparse representations by curvelets that capture “wavefront sets” efficiently (see e.g., Smith (1998),Cande`s and Demanet (2005), Hennenfent and Herrmann (2006) and the references therein).For high resolution data represented by the N -dimensional vector f0 ∈ RN , which admits asparse representation x0 ∈ CP in some transform domain characterized by the operator S ∈ CP×Nwith P ≥ N , the sparse recovery problem involves solving an underdetermined system of equations:b = Ax0, (7.1)where b ∈ Cn, n  N ≤ P , represents the compressively sampled data of n measurements, andA ∈ Cn×P represents the measurement matrix. We denote by x0 a sparse synthesis coefficientvector of f0. When x0 is strictly sparse—i.e., only k < n nonzero entries in x0, sparsity-promotingrecovery can be achieved by solving the `0 minimization problem, which is a combinatorial problemand quickly becomes intractable as the dimension increases. Instead, the basis pursuit denoise(BPDN) convex optimization problem:minimizex∈CP‖x‖1 subject to ‖b−Ax‖2 ≤ , (BPDN)can be used to recover x˜, which is an estimate of x0. Here,  represents the error-bound inthe least-squares misfit and the `1 norm ‖x‖1 is the sum of absolute values of the elements of avector x. The matrix A can be composed of the product of an n × N sampling (or acquisition)137matrix M and the sparsifying operator S such that A := MSH , here H denotes the Hermitiantranspose. Consequently, the measurements are given by b = Ax0 = Mf0. A seismic line with Nssources, Nr receivers, and Nt time samples can be reshaped into an N dimensional vector f , whereN = Ns×Nr×Nt. For simultaneous towed-streamer acquisition, given two unblended data vectorsx1 and x2 and (blended) measurements b, we can redefine Equation 7.1 asA︷ ︸︸ ︷[MT1SH MT2SH] x︷ ︸︸ ︷[x1x2]= b,(7.2)where T1 and T2 are defined as the firing-time delay operators which apply uniformly randomtime delays to the first and second source, respectively. Note that accurate knowledge of the firingtimes is essential for successful recovery by the proposed source separation techniques. We wishto recover a sparse approximation f˜ of the discretized wavefield f (corresponding to each source)from the measurements b. This is done by solving the BPDN sparsity-promoting program, usingthe SPG`1 solver (see Berg and Friedlander, 2008; Hennenfent et al., 2008, for details), yieldingf˜ = SH x˜ for each source.Sparsity is not the only structure seismic data exhibits where three- or five-dimensional seismicdata is organized as a vector. High-dimensional seismic data volumes can also be representedas matrices or tensors, where the low-rank structure of seismic data can be exploited (Trickettand Burroughs, 2009; Oropeza and Sacchi, 2011; Kreimer and Sacchi, 2012; Silva and Herrmann,2013; Aravkin et al., 2014). This low-rank property of seismic data leads to the notion of matrixcompletion theory which offers a reconstruction strategy for an unknown matrix X from its knownsubsets of entries (Cande`s and Recht, 2009; Recht et al., 2010). The success of matrix completionframework hinges on the fact that regularly sampled target dataset should exhibit a low-rankstructure in the rank-revealing “transform domain” while subsampling should destroy the low-rankstructure of seismic data in the transform domain.7.3.1 Rank-revealing “transform domain”Following the same analogy of CS, the main challenge in applying matrix completion techniques tothe source separation problem is to find a “transform domain” wherein: i) fully sampled conven-tional (or unblended) seismic data have low-rank structure—i.e., quickly decaying singular values;ii) blended seismic data have high-rank structure—i.e., slowly decaying singular values. Whenthese properties hold, rank-minimization techniques (used in matrix completion) can be used torecover the source-separated signal. Kumar et al. (2013) showed that the frequency slices of un-blended seismic data do not exhibit low-rank structure in the source-receiver (s-r) domain sincestrong wavefronts extend diagonally across the s-r plane. However, transforming the data into themidpoint-offset (m-h) domain results in a vertical alignment of the wavefronts, thereby reducingthe rank of the frequency slice matrix. The midpoint-offset domain is a coordinate transformationdefined as:xmidpoint =12(xsource + xreceiver),xoffset =12(xsource − xreceiver).138These observations motivate us to exploit the low-rank structure of seismic data in the midpoint-offset domain for simultaneous towed-streamer acquisition. Although the given problem does notcontain any missing traces and is a source-separation problem alone, incorporating this scenario inthe current framework is straightforward (Kumar et al., 2017). Seismic data processing literaturecontains numerous works done on missing-trace interpolation in midpoint-offset coordinates (seeTrad, 2009; Kreimer, 2013, and references therein).Figures 7.1(a) and 7.1(c) show a monochromatic frequency slice (at 5 Hz) for simultaneousacquisition with periodic firing times in the source-receiver (s-r) and midpoint-offset (m-h) domains,while Figures 7.1(b) and 7.1(d) show the same for simultaneous acquisition with random firing-timedelays. Note that we use the source-receiver reciprocity to convert each monochromatic frequencyslice of the towed-streamer acquisition to split-spread type acquisition, which is required by ourcurrent implementation of rank-minimization based techniques for 2D seismic acquisition. For3D seismic data acquisition, where seismic data exhibit 5D structure, we can follow the strategyproposed by Kumar et al. (2015a), where a simple permutation of matricized seismic data is usedas a transformation domain to exploit the low-rank structure of seismic data. Here, matricizationrefers to a process that reshapes a tensor into a matrix along specific dimensions. As shown inKumar et al. (2015a), 3D seismic data exhibits low-rank structure in the noncanonical matrixrepresentation, where the source and receiver coordinates in the x− and y− direction are lumpedtogether as opposed to the canonical matrix representation, where both the source coordinates (inthe x− and y− direction) are lumped together, and similarly for the receiver coordinates. Therefore,in 3D seismic data acquisition we do not have to work in the midpoint-offset domain which removesthe requirement of source-receiver reciprocity.As illustrated in Figure 7.1, simultaneously acquired data with periodic firing times preservescontinuity of the waveforms in the s-r and m-h domains, which inherently do not change therank of blended data compared to unblended data. Introducing random time delays destroyscontinuity of the waveforms in the s-r and m-h domains, thus increasing the rank of the blendeddata matrix drastically, which is a necessary condition for rank-minimization based algorithmsto work effectively. To illustrate this behaviour, we plot the decay of the singular values of a 5Hz monochromatic frequency slice extracted from the periodically and randomized simultaneousacquisition in the s-r and m-h domains, respectively in Figure 7.2(a). Note that uniformly randomfiring-time delays do not noticeably change the decay of the singular values in the source-receiver(s-r) domain, as expected, but significantly slow down the decay rate in the m-h domain.Similar trends are observed for a monochromatic frequency slice at 40 Hz in Figure 7.2(b).Following the same analogy, Figures 7.2(c) and 7.2(d) show how randomization in acquisitiondestroys the sparse structure of seismic data in the source-channel (or source-offset) domain—i.e.,slow decay of the curvelet coefficients, hence, favouring recovery via sparsity-promotion in thisdomain. Similarly, for simultaneous long offset acquisition, we exploit the low-rank structure ofseismic data in the m-h domain, and the sparse structure in the source-channel domain.Seismic frequency slices exhibit low-rank structure in the m-h domain at low frequencies, butthe same is not true for data at high frequencies. This is because in the low-frequency slices, thevertical alignment of the wavefronts can be accurately approximated by a low-rank representation.On the other hand, high-frequency slices include a variety of wave oscillations that increase the rank,even though the energy remains focused around the diagonal (Kumar et al., 2013). To illustratethis phenomenon, we plot a monochromatic frequency slice at 40 Hz in the s-r domain and them-h domain for over/under acquisition in Figure 7.3. When analyzing the decay of the singular139(a) (b)(c) (d)Figure 7.1: Monochromatic frequency slice at 5 Hz in the source-receiver (s-r) and midpoint-offset (m-h) domain for blended data (a,c) with periodic firing times and (b,d) withuniformly random firing times for both sources.140(a) (b)(c) (d)Figure 7.2: Decay of singular values for a frequency slice at (a) 5 Hz and (b) 40 Hz ofblended data. Source-receiver domain: blue—periodic, red—random delays. Midpoint-offset domain: green—periodic, cyan—random delays. Corresponding decay of thenormalized curvelet coefficients for a frequency slice at (c) 5 Hz and (d) 40 Hz ofblended data, in the source-channel domain.values for high-frequency slices in the s-r domain and the m-h domain (Figure 7.2(b)), we observethat the singular value decay is slower for the high-frequency slice than for the low-frequencyslice. Therefore, rank-minimization in the high-frequency range requires extended formulationsthat incorporate the low-rank structure.To exploit the low-rank structure of high-frequency data, we rely on the Hierarchical Semi-Separable matrix representation (HSS) method proposed by Chandrasekaran et al. (2006) to rep-resent frequency slices. The key idea in the HSS representation is that certain full-rank matrices,e.g., matrices that are diagonally dominant with energy decaying along the off-diagonals, can berepresented by a collection of low-rank sub-matrices. Kumar et al. (2013) showed the possibilityof finding accurate low-rank approximations of sub-matrices of the high-frequency slices by par-titioning the data into the HSS structure for missing-trace interpolation. Jumah and Herrmann(2014) showed that HSS representations can be used to reduce the storage and computational cost141Figure 7.3: Monochromatic frequency slice at 40 Hz in the s-r and m-h domain for blendeddata (a,c) with periodic firing times and (b,d) with uniformly random firing times forboth sources.for the estimation of primaries by sparse inversions. They combined the HSS representation withthe randomized SVD proposed by Halko et al. (2011) to accelerate matrix-vector multiplicationsthat are required for sparse inversion.7.3.2 Hierarchical Semi-Separable matrix representation (HSS)The HSS structure first partitions a matrix into diagonal and off-diagonal sub-matrices. The samepartitioning structure is then applied recursively to the diagonal sub-matrices only. To illustrate theHSS partitioning, we consider a 2D monochromatic high-frequency data matrix at 40 Hz in the s-rdomain. We show the first-level of partitioning in Figure 7.4(a) and the second-level partitioning inFigure 7.4(b) in their corresponding source-receiver domains. Figures 7.5(a) and 7.5(b) display thefirst-level off-diagonal sub-blocks, Figure 7.5(c) is the diagonal sub-block, and the corresponding142(a) (b)Figure 7.4: HSS partitioning of a high-frequency slice at 40 Hz in the s-r domain: (a) first-level, (b) second-level, for randomized blended acquisition.decay of the singular values is displayed in Figure 7.6. We can clearly see that the off-diagonalsub-matrices have low-rank structure, while the diagonal sub-matrices have higher rank. Furtherpartitioning of the diagonal sub-blocks (Figure 7.4(b)) allows us to find better low-rank approx-imations. The same argument holds for the simultaneous long offset acquisition. Therefore, forlow-variability acquisition scenarios, each frequency slice is first partitioned using HSS and thenseparated in its respective m-h domain, as shown for missing-trace interpolation by Kumar et al.(2013).One of the limitations of matrix completion type approaches for large-scale seismic data isthe nuclear-norm projection, which inherently involves the computation of SVDs. Aravkin et al.(2014) showed that the computation of SVD is prohibitively expensive for large-scale data suchas seismic, therefore, we propose a matrix-factorization based approach to avoid the need forexpensive computation of SVDs (see Aravkin et al., 2014, for details). In the next section, weintroduce the matrix completion framework and explore its necessary extension to separate large-scale simultaneous seismic data.7.3.3 Large-scale seismic data: SPG-LR frameworkLet X0 be a low-rank matrix in Cn×m and A be a linear measurement operator that maps fromCn×m → Cp with p n×m. Under the assumption that the blending process increases the rankof the matrix X0, the source separation problem is to find the matrix of lowest possible rank thatagrees with the above observations. The rank-minimization problem involves solving the followingproblem for A, up to a given tolerance :minimizeXrank(X) subject to ‖A(X)− b‖2 ≤ ,143(a) (b)(c)Figure 7.5: (a,b,c) First-level sub-block matrices (from Figure 7.4(a)).Figure 7.6: Decay of singular values of the HSS sub-blocks in s-r domain: red—Figure 7.5(a),black—Figure 7.5(b), blue—Figure 7.5(c).144where rank is defined as the maximum number of linearly independent rows or column of a ma-trix, b is a set of blended measurements. For simultaneous towed-streamer acquisition, we followequation 7.2 and redefine our system of equations asA︷ ︸︸ ︷[MT1SH MT2SH] X︷ ︸︸ ︷[X1X2]= b,where S is the transformation operator from the s-r domain to the m-h domain. Recht et al.(2010) showed that under certain general conditions on the operator A, the solution to the rank-minimization problem can be found by solving the following nuclear-norm minimization problem:minimizeX‖X‖∗ subject to ‖A(X)− b‖2 ≤ , (BPDN)where ‖X‖∗ = ‖σ‖1, and σ is a vector of singular values. Unfortunately, for large-scale data, solvingthe BPDN problem is difficult since it requires repeated projections onto the set ‖X‖∗ ≤ τ , whichmeans repeated SVD or partial SVD computations. Therefore, we avoid computing SVDs of thematrices and use an extension of the SPG`1 solver (Berg and Friedlander, 2008) developed forthe BPDN problem in Aravkin et al. (2013). We refer to this extension as SPG-LR in the restof the chapter. The SPG-LR algorithm finds the solution to the BPDN problem by solving asequence of LASSO (least absolute shrinkage and selection operator) subproblems:minimizeX‖A(X)− b‖2 subject to ||X||∗ ≤ τ, (LASSOτ )where τ is updated by traversing the Pareto curve. The Pareto curve defines the optimal trade-offbetween the two-norm of the residual and the one-norm of the solution (Berg and Friedlander,2008). Solving each LASSO subproblem requires a projection onto the nuclear-norm ball ‖X‖∗ ≤ τin every iteration by performing a singular value decomposition and then thresholding the singularvalues. For large-scale seismic problems, it becomes prohibitively expensive to carry out such alarge number of SVDs. Instead, we adopt a recent factorization-based approach to nuclear-normminimization (Rennie and Srebro, 2005; Lee et al., 2010; Recht and Re, 2013). The factorizationapproach parametrizes the matrix (X1, X2) ∈ Cn×m as the product of two low-rank factors (L1,L2) ∈ Cn×k and (R1, R2) ∈ Cm×k such that,X =[L1RH1L2RH2]. (7.3)Here, k represents the rank of the L and R factors. The optimization scheme can then be carriedout using the factors (L1,L2) and (R1,R2) instead of (X1,X2), thereby significantly reducing thesize of the decision variable from 2nm to 2k(n + m) when k  m,n. Rennie and Srebro (2005)showed that the nuclear-norm obeys the relationship:‖X‖∗ ≤ 12∥∥∥∥[L1R1]∥∥∥∥2F+12∥∥∥∥[L2R2]∥∥∥∥2F=: Φ(L1,R1,L2,R2), (7.4)145where ‖ · ‖2F is the Frobenius norm of the matrix—i.e., sum of the squared entires. Consequently,the LASSO subproblem can be replaced byminimizeL1,R1,L2,R2‖A(X)− b‖2 subject to Φ(L1,R1,L2,R2) ≤ τ , (7.5)where the projection onto Φ(L1,R1,L2,R2) ≤ τ is easily achieved by multiplying each factor(L1,L2) and (R1,R2) by the scalar√2τ/Φ(L1,R1,L2,R2). Equation 7.4, for each HSS sub-matrixin the m-h domain, guarantees that ‖X‖∗ ≤ τ for any solution of 7.5. Once the optimization problemis solved, each sub-matrix in the m-h domain is transformed back into the s-r domain, where weconcatenate all the sub-matrices to get the separated monochromatic frequency data matrices. Oneof the advantages of the HSS representation is that it works with recursive partitioning of a matrixand sub-matrices can be solved in parallel, speeding up the optimization formulation.7.4 ExperimentsWe perform source separation for two simultaneous towed-streamer acquisition scenarios—over/underand simultaneous long offset, by generating synthetic datasets on complex geological models usingthe IWAVE (Symes et al., 2011) time-stepping acoustic simulation software, and also use a fielddataset from the Gulf of Suez. Source separation for over/under acquisition is tested on two differ-ent datasets. The first dataset is simulated on the Marmousi model (Bourgeois et al., 1991), whichrepresents a complex-layer model with steeply dipping reflectors that make the data challenging.With a source (and channel/receiver) sampling of 20.0 m, one dataset is generated with a source-depth of 8.0 m (Figures 7.7(a) and 7.7(d)), while the other dataset has the source at 12.0 m depth(Figures 7.7(b) and 7.7(e)), resulting in 231 sources and 231 channels. The temporal length of eachdataset is 4.0 s with a sampling interval of 0.004 s. The second dataset is a field data example fromthe Gulf of Suez. In this case, the first source is placed at 5.0 m depth (Figures 7.8(a) and 7.8(d))and the second source is placed at 10.0 m depth (Figures 7.8(b) and 7.8(e)). The source (and chan-nel) sampling is 12.5 m, resulting in 178 sources and 178 channels with a time sampling interval of0.004 s.The simultaneous long offset acquisition is simulated on the BP salt model (Billette andBrandsberg-Dahl, 2004), where the presence of salt-bodies make the data challenging. The twosource vessels are 6.0 km apart and the streamer length is 6.0 km. Both the datasets (for source1 and source 2) contain 361 sources and 361 channels with a spatial interval of 12.5 m, where thesource and streamer depth is 6.25 m. The temporal length of each dataset is 6.0 s with a samplinginterval of 0.006 s. A single shot gather from each dataset is shown in Figures 7.9(a) and 7.9(b)and the corresponding channel gathers are shown in Figures 7.9(d) and 7.9(e). The datasets foreach source in both the acquisition scenarios are (simply) summed for simultaneous acquisitionwith periodic firing times, while uniformly random time delays between 0-1 second are applied toeach source for the randomized simultaneous acquisition. Figures 7.7(c), 7.8(c) and 7.9(c) showthe randomized blended shot gathers for the Marmousi, the Gulf of Suez and the BP datasets,respectively. As illustrated in the figures, both the sources fire at random times (independent ofeach other) within the interval of 0-1 second, hence, the difference between the firing times of thesources is always less than 1 second. The corresponding randomized blended channel gathers areshown in Figures 7.7(f), 7.8(f) and 7.9(f). Note that the speed of the vessels in both the acquisitionscenarios is no different than the current practical speed of the vessels in the field.146Channel (km)0 2 4Time (s)00.511.522.533.5(a)Channel (km)0 2 4Time (s)00.511.522.533.5(b)Channel (km)0 2 4Time (s)00.511.522.533.5(c)Source (km)0 2 4Time (s)00.511.522.533.5(d)Source (km)0 2 4Time (s)00.511.522.533.5(e)Source (km)0 2 4Time (s)00.511.522.533.5(f)Figure 7.7: Original shot gather of (a) source 1, (b) source 2, and (c) the correspondingblended shot gather for simultaneous over/under acquisition simulated on the Mar-mousi model. (d, e) Corresponding common-channel gathers for each source and (f)the blended common-channel gather.147Channel (km)0 1 2Time (s)00.511.522.533.5(a)Channel (km)0 1 2Time (s)00.511.522.533.5(b)Channel (km)0 1 2Time (s)00.511.522.533.5(c)Source (km)0 1 2Time (s)00.511.522.533.5(d)Source (km)0 1 2Time (s)00.511.522.533.5(e)Source (km)0 1 2Time (s)00.511.522.533.5(f)Figure 7.8: Original shot gather of (a) source 1, (b) source 2, and (c) the correspondingblended shot gather for simultaneous over/under acquisition from the Gulf of Suezdataset. (d,e) Corresponding common-channel gathers for each source and (f) theblended common-channel gather.148Channel (km)0 2 4 6Time (s)012345(a)Channel (km)0 2 4 6Time (s)012345(b)Channel (km)0 2 4 6Time (s)012345(c)Source (km)0 2 4 6Time (s)012345(d)Source (km)0 2 4 6Time (s)012345(e)Source (km)0 2 4 6Time (s)012345(f)Figure 7.9: Original shot gather of (a) source 1, (b) source 2, and (c) the correspondingblended shot gather for simultaneous long offset acquisition simulated on the BP saltmodel. (d, e) Corresponding common-channel gathers for each source and (f) theblended common-channel gather.149For source separation via rank-minimization, second-level of HSS partitioning, on each frequencyslice in the s-r domain, was sufficient for successful recovery in both the acquisition scenarios. Aftertransforming each sub-block into the m-h domain, source separation is then performed by solvingthe nuclear-norm minimization formulation (BPDN) on each sub-block, using 350 iterations ofSPG-LR. In order to choose an appropriate rank value, we first perform source separation forfrequency slices at 0.2 Hz and 125 Hz. For the over/under acquisition simulated on the Marmousimodel, the best rank value is 30 and 80 for each frequency slice, respectively. The best rank valuesfor the Gulf of Suez dataset are 20 and 100, respectively. For simultaneous long offset acquisition,the best rank value is 10 and 90 for frequency slices at 0.15 Hz and 80 Hz, respectively. Hence, weadjust the rank linearly within these ranges when moving from low to high frequencies, for eachacquisition scenario. For source separation via sparsity-promotion, we use the BPDN formulationto minimize the `1 norm (instead of the nuclear-norm) where the transformation operator S is the2D curvelet operator. Here, we run 350 iterations of SPG`1.For the over/under acquisition scenario simulated on the Marmousi model, Figures 7.10(a)and 7.10(c) show the separated shot gathers via rank-minimization and Figures 7.10(e) and 7.10(g)show the separated shot gathers via sparsity-promotion, respectively. The separated common-channel gathers via rank-minimization and sparsity-promotion are shown in Figures 7.11(a), 7.11(c)and Figures 7.11(e), 7.11(g), respectively. For the Gulf of Suez field dataset, Figures 7.12 and 7.13show the separated gathers and difference plots in the common-shot and common-channel domain,respectively. The corresponding separated gathers and difference plots in the common-shot andcommon-channel domain for the simultaneous long offset acquisition scenario are shown in Fig-ures 7.14 and 7.15.As illustrated by the results and their corresponding difference plots, both the CS-based ap-proaches of rank-minimization and sparsity-promotion are able to separate the data for the low-variability acquisition scenarios fairly well. In all the three different datasets, the average S/Nsfor separation via sparsity-promotion is slightly better than rank-minimization, but the differenceplots show that the recovery via rank-minimization is equivalent to the sparsity-promoting basedrecovery where it is able to recover most of the coherent energy. Also, rank-minimization outper-forms the sparsity-promoting technique in terms of the computational time and memory usage asrepresented in Tables 7.1, 7.2 and 7.3. Both the CS-based recoveries are better for the simultane-ous long offset acquisition than the recoveries from the over/under acquisition scenario. A possibleexplanation for this improvement is the long offset distance that increases randomization in thesimultaneous acquisition, which is a more favourable scenario for recovery by CS-based approaches.Figure 7.16 demonstrates the advantage of the HSS partitioning, where the S/Ns of the separateddata are significantly improved.7.4.1 Comparison with NMO-based median filteringWe also compare the performance of our CS-based source-separation techniques with deblendingusing the NMO-based median filtering technique proposed by Chen et al. (2014), where we work on acommon-midpoint gather from each acquisition scenario. For the over/under acquisition simulatedon the Marmousi model, Figures 7.17(a) and 7.17(e) show the blended common-midpoint gathersand source separation using the median filtering technique is shown in Figures 7.17(b) and 7.17(f).The corresponding separated common-midpoint gathers from the two CS-based techniques areshown in Figures 7.17(c,d,g,h). Figure 7.18 shows the blended and separated common-midpointgathers for the field data from the Gulf of Suez. We observe that recoveries via the proposed CS-150Marmousi modelTime Memory S/NSparsity 167 7.0 16.7, 16.7Rank 12 2.8 15.0, 14.8Table 7.1: Comparison of computational time (in hours), memory usage (in GB) and averageS/N (in dB) using sparsity-promoting and rank-minimization based techniques for theMarmousi model.Gulf of SuezTime Memory S/NSparsity 118 6.6 14.6Rank 8 2.6 12.8Table 7.2: Comparison of computational time (in hours), memory usage (in GB) and averageS/N (in dB) using sparsity-promoting and rank-minimization based techniques for theGulf of Suez dataset.based approaches are comparable to the recovery from the median filtering technique. Similarly,Figure 7.19 shows the results for the simultaneous long offset acquisition simulated on the BP saltmodel. Here, the CS-based techniques result in slightly improved recoveries.7.4.2 RemarkIt is important to note here that we perform the CS-based source separation algorithms only once,however, we can always perform a few more runs of the algorithms where we can first subtract theseparated source 1 and source 2 from the acquired blended data and then rerun the algorithmsto separate the energy in the residual data. Hence, the recovery can be further improved untilnecessary. Since separation via rank-minimization is computationally faster than the sparsity basedtechnique, multiple passes through the data is a computationally viable option for the former source-separation technique.BP modeltime memory S/NSparsity 325 7.0 32.0, 29.4Rank 20 2.8 29.4, 29.0Table 7.3: Comparison of computational time (in hours), memory usage (in GB) and averageS/N (in dB) using sparsity-promoting and rank-minimization based techniques for theBP model.151Channel (km)0 2 4Time (s)00.511.522.533.5(a)Channel (km)0 2 4Time (s)00.511.522.533.5(b)Channel (km)0 2 4Time (s)00.511.522.533.5(c)Channel (km)0 2 4Time (s)00.511.522.533.5(d)Channel (km)0 2 4Time (s)00.511.522.533.5(e)Channel (km)0 2 4Time (s)00.511.522.533.5(f)Channel (km)0 2 4Time (s)00.511.522.533.5(g)Channel (km)0 2 4Time (s)00.511.522.533.5(h)Figure 7.10: Separated shot gathers and difference plots (from the Marmousi model) ofsource 1 and source 2: (a,c) source separation using HSS based rank-minimization and(b,d) the corresponding difference plots; (e,g) source separation using curvelet-basedsparsity-promotion and (f,h) the corresponding difference plots.7.5 DiscussionThe above experiments demonstrate the successful implementation of the proposed CS-based ap-proaches of rank-minimization and sparsity-promotion for source separation in the low-variability,simultaneous towed-streamer acquisitions. The recovery is comparable for both approaches, how-ever, separation via rank-minimization is significantly faster and memory efficient. This is furtherenhanced by incorporating the HSS partitioning since it allows the exploitation of the low-rankstructure in the high-frequency regime, and renders its extension to large-scale data feasible. Notethat in the current implementation, we work with each temporal frequency slice and perform thesource separation individually. The separation results can further be enhanced by incorporatingthe information from the previously recovered frequency slice to the next frequency slice, as shownby Mansour et al. (2013) for seismic data interpolation.152Source (km)0 2 4Time (s)00.511.522.533.5(a)Source (km)0 2 4Time (s)00.511.522.533.5(b)Source (km)0 2 4Time (s)00.511.522.533.5(c)Source (km)0 2 4Time (s)00.511.522.533.5(d)Source (km)0 2 4Time (s)00.511.522.533.5(e)Source (km)0 2 4Time (s)00.511.522.533.5(f)Source (km)0 2 4Time (s)00.511.522.533.5(g)Source (km)0 2 4Time (s)00.511.522.533.5(h)Figure 7.11: Separated common-channel gathers and difference plots (from the Marmousimodel) of source 1 and source 2: (a,c) source separation using HSS based rank-minimization and (b,d) the corresponding difference plots; (e,g) source separationusing curvelet-based sparsity-promotion and (f,h) the corresponding difference plots.The success of CS hinges on randomization of the acquisition. Although, the low degree ofrandomization (e.g., 0 ≤ 1 second) in simultaneous towed-streamer acquisitions seems favourablefor source separation via CS-based techniques, however, high-variability in the firing times enhancesthe recovery quality of separated seismic data volumes, as shown in Wason and Herrmann (2013a);Wason and Herrmann (2013b) for ocean-bottom cable/node acquisition with continuous recording.One of the advantages of the proposed CS-based techniques is that it does not require velocityestimation, which can be a challenge for data with complex geologies. However, the proposedtechniques require accurate knowledge of the random firing times.So far, we have not considered the case of missing traces (sources and/or receivers), however,incorporating this scenario in the current framework is straightforward. This makes the problem ajoint source separation and interpolation problem. We can also extend our methods to separate 3Dblended seismic data volumes as shown in Kumar et al. (2017), wherein we address the problem of153Channel (km)0 1 2Time (s)00.511.522.533.5(a)Channel (km)0 1 2Time (s)00.511.522.533.5(b)Channel (km)0 1 2Time (s)00.511.522.533.5(c)Channel (km)0 1 2Time (s)00.511.522.533.5(d)Channel (km)0 1 2Time (s)00.511.522.533.5(e)Channel (km)0 1 2Time (s)00.511.522.533.5(f)Channel (km)0 1 2Time (s)00.511.522.533.5(g)Channel (km)0 1 2Time (s)00.511.522.533.5(h)Figure 7.12: Separated shot gathers and difference plots (from the Gulf of Suez dataset) ofsource 1 and source 2: (a,c) source separation using HSS based rank-minimization and(b,d) the corresponding difference plots; (e,g) source separation using curvelet-basedsparsity-promotion and (f,h) the corresponding difference plots.joint source separation and interpolation for time-lapse seismic. In reality, seismic data are typicallyirregularly sampled along spatial axes, and therefore future work includes working with nonuniformsampling grids.7.6 ConclusionsWe have presented two compressed sensing based methods for source separation for simultaneoustowed-streamer type acquisitions, such as the over/under and the simultaneous long offset acquisi-tion. Both the compressed sensing based approaches of rank-minimization and sparsity-promotiongive comparable source-separation results, however, the former approach is readily scalable to large-scale blended seismic data volumes and is computationally faster. This can be further enhancedby incorporating the HSS structure with factorization-based rank-regularized optimization formu-154Source (km)0 1 2Time (s)00.511.522.533.5(a)Source (km)0 1 2Time (s)00.511.522.533.5(b)Source (km)0 1 2Time (s)00.511.522.533.5(c)Source (km)0 1 2Time (s)00.511.522.533.5(d)Source (km)0 1 2Time (s)00.511.522.533.5(e)Source (km)0 1 2Time (s)00.511.522.533.5(f)Source (km)0 1 2Time (s)00.511.522.533.5(g)Source (km)0 1 2Time (s)00.511.522.533.5(h)Figure 7.13: Separated common-channel gathers and difference plots (from the Gulf of Suezdataset) of source 1 and source 2: (a,c) source separated using HSS based rank-minimization and (b,d) the corresponding difference plots; (e,g) source separationusing curvelet-based sparsity-promotion and (f,h) the corresponding difference plots.lations, along with improved recovery quality of the separated seismic data. We have combinedthe Pareto curve approach for optimizing BPDN formulations with the SVD-free matrix factor-ization methods to solve the nuclear-norm optimization formulation, which avoids the expensivecomputation of SVDs, a necessary step in traditional rank-minimization based methods. We findthat our proposed techniques are comparable to the commonly used NMO-based median filteringtechniques.155Channel (km)0 2 4 6Time (s)012345(a)Channel (km)0 2 4 6Time (s)012345(b)Channel (km)0 2 4 6Time (s)012345(c)Channel (km)0 2 4 6Time (s)012345(d)Channel (km)0 2 4 6Time (s)012345(e)Channel (km)0 2 4 6Time (s)012345(f)Channel (km)0 2 4 6Time (s)012345(g)Channel (km)0 2 4 6Time (s)012345(h)Figure 7.14: Separated shot gathers and difference plots (from the BP salt model) of source1 and source 2: (a,c) source separation using HSS based rank-minimization and(b,d) the corresponding difference plots; (e,g) source separation using curvelet-basedsparsity-promotion and (f,h) the corresponding difference plots.156Source (km)0 2 4 6Time (s)012345(a)Source (km)0 2 4 6Time (s)012345(b)Source (km)0 2 4 6Time (s)012345(c)Source (km)0 2 4 6Time (s)012345(d)Source (km)0 2 4 6Time (s)012345(e)Source (km)0 2 4 6Time (s)012345(f)Source (km)0 2 4 6Time (s)012345(g)Source (km)0 2 4 6Time (s)012345(h)Figure 7.15: Separated common-channel gathers and difference plots (from the BP saltmodel) of source 1 and source 2: (a,c) source separation using HSS based rank-minimization and (b,d) the corresponding difference plots; (e,g) source separationusing curvelet-based sparsity-promotion and (f,h) the corresponding difference plots.157Figure 7.16: Signal-to-noise ratio (dB) over the frequency spectrum for the separated datafrom the Marmousi model. Red, blue curves—source separation without HSS; cyan,black curves—source separation using second-level HSS partitioning. Solid lines—separated source 1, + marker—separated source 2.158Offset (km)0 2 4Time (s)00.511.522.533.5(a)Offset (km)0 2 4Time (s)00.511.522.533.5(b)Offset (km)0 2 4Time (s)00.511.522.533.5(c)Offset (km)0 2 4Time (s)00.511.522.533.5(d)Offset (km)0 2 4Time (s)00.511.522.533.5(e)Offset (km)0 2 4Time (s)00.511.522.533.5(f)Offset (km)0 2 4Time (s)00.511.522.533.5(g)Offset (km)0 2 4Time (s)00.511.522.533.5(h)Figure 7.17: Blended common-midpoint gathers of (a) source 1 and (e) source 2 for theMarmousi model. Source separation using (b,f) NMO-based median filtering, (c,g)rank-minimization and (d,h) sparsity-promotion.159Offset (km)0 1 2Time (s)00.511.522.533.5(a)Offset (km)0 1 2Time (s)00.511.522.533.5(b)Offset (km)0 1 2Time (s)00.511.522.533.5(c)Offset (km)0 1 2Time (s)00.511.522.533.5(d)Offset (km)0 1 2Time (s)00.511.522.533.5(e)Offset (km)0 1 2Time (s)00.511.522.533.5(f)Offset (km)0 1 2Time (s)00.511.522.533.5(g)Offset (km)0 1 2Time (s)00.511.522.533.5(h)Figure 7.18: Blended common-midpoint gathers of (a) source 1, (e) source 2 for the Gulfof Suez dataset. Source separation using (b,f) NMO-based median filtering, (c,g)rank-minimization and (d,h) sparsity-promotion.160Offset (km)0 2 4 6Time (s)012345(a)Offset (km)0 2 4 6Time (s)012345(b)Offset (km)0 2 4 6Time (s)012345(c)Offset (km)0 2 4 6Time (s)012345(d)Offset (km)0 2 4 6Time (s)012345(e)Offset (km)0 2 4 6Time (s)012345(f)Offset (km)0 2 4 6Time (s)012345(g)Offset (km)0 2 4 6Time (s)012345(h)Figure 7.19: Blended common-midpoint gathers of (a) source 1, (e) source 2 for the BPsalt model. Source separation using (b,f) NMO-based median filtering, (c,g) rank-minimization and (d,h) sparsity-promotion.161Chapter 8ConclusionAdapting ideas from the field of compressive sensing leads to new insights into acquiring and pro-cessing seismic data where we can fundamentally rethink on how we design acquisition surveys.Compressive randomized simultaneous-source acquisitions provide flexibility in acquisition geome-tries for better area coverage (i.e., improves data density) in surveys and speedup acquisition. Themain contributions of this thesis are summarized below.8.1 Compressive sensing in seismic explorationWe propose an alternative sampling method adapting insights from CS towards seismic acquisi-tion and processing for data that are subsampled. The main outcome of this approach is a newtechnology where acquisition and processing related costs are decoupled from the stringent Nyquistsampling criterion. Instead, these costs scale with the desired reconstruction error and transform-domain sparsity of the data. By means of carefully designed numerical experiments on syntheticand real data, we establish that CS can indeed successfully be adapted to seismic data acquisi-tion, wherein seismic wavefields can be reconstructed with a controllable error from randomizedsubsamplings. Specifically, three key components need to be in place: (i) a sparsifying signal (i.e.,structure revealing) representation that exploits the signal’s structure by mapping the energy intoa small number of significant transform-domain coefficients; (ii) a randomized subsampling schemethat turns subsampling related artifacts into incoherent noise that is not sparse or compressible;and (iii) recovery of artifact-free fully sampled data by promoting structure, i.e., sparse recoveryvia one-norm minimization. We also introduce performance measures for nonlinear approximationand recovery errors and empirically demonstrate that curvelets lead to compressible (real-world sig-nals are not strictly sparse) representation of seismic data compared to wave atoms, wavelets, etc.Hence, we use curvelets for recovery of densely sampled conventional data via sparsity promotion.In a nutshell, compressive sensing offers new perspectives towards the design of land and marineacquisition schemes.8.2 Compressive simultaneous-source marine acquisitionWe identify simultaneous-source marine acquisition as a linear subsampling system and analyze itusing CS metrics such as mutual coherence and restricted isometry property. We quantitativelyinvestigate the underlying interaction between acquisition design and reconstruction fidelity, and162show that more randomness in the acquisition system and more compressible transforms improve themutual coherence and restricted isometry constants, which predict a higher reconstruction quality.This is also true for our proposed pragmatic compressive marine simultaneous-source acquisitionscheme, termed time-jittered marine, wherein a single (and/or multiple) source vessel(s) sails acrossan ocean-bottom array firing air guns at jittered-time instances, which translate to jittered shotpositions for a given (fixed) speed of the source vessel. The simultaneous data are time compressedwith overlapping shot records, and are therefore acquired economically with a small environmentalimprint. The proposed acquisition scheme shares the benefits of random sampling while offeringcontrol on the maximum acquisition gap size since randomization via jittering turns the recoveryinto a relatively simple “denoising” problem with control over the maximum gap size betweenadjacent shot locations (Hennenfent and Herrmann, 2008), which is a practical requirement ofwavefield reconstruction with localized sparsifying transforms such as curvelets.According to CS, a sparsifying transform that is incoherent with the CS matrix can significantlyimpact the reconstruction quality. In other words, the mutual coherence between the sparsifyingtransform and the sensing (or measurement) matrix should be small for good signal reconstruc-tion. We show that the CS matrix resulting from our proposed sampling scheme is incoherentwith the curvelet transform. Recovering conventional nonsimultaneous prestack data volumes fromsimultaneous marine data essentially involves mapping noise-like or incoherent source crosstalk tocoherent seismic responses. Randomized simultaneous-source acquisitions render source crosstalk(or interferences) incoherent in common-receiver gathers creating favorable conditions for recoveringconventional nonsimultaneous data via curvelet-domain sparsity promotion. We recover conven-tional interference-free data using 2D and 3D FDCT (Fast Discrete Curvelet Transform, (Cande`set al., 2006a; Ying et al., 2005)). We observe that the 3D FDCT leads to slightly improved re-coveries compared to the 2D FDCT but at the expense of increased computational costs. Thisis because the 3D FDCT is about 24× redundant, in contrast to the 8× redundant 2D FDCT,rendering large-scale processing extremely memory intensive, and hence impractical. The combina-tion of randomized subsampling and sparsity-promoting recovery technique results in high-qualitynonsimultaneous data volumes recovered on fine periodic sampling grids. The results vindicate theimportance of randomness in the acquisition scheme. Recovery from the more realistic irregular oroff-the-grid subsamplings are reported in Chapter 6.8.3 Compressive simultaneous-source time-lapse marineacquisitionWe present a first instance of adapting ideas from CS and DCS to assess the effects of randomor irregular subsampling (in the field) on time-lapse data, and demonstrate that high-quality datarecoveries are the norm and not the exception. The main finding is that compressive randomizedtime-lapse surveys need not be replicated to attain similar/acceptable levels of data repeatabilitycompared to data acquired from (expensive) dense, periodically sampled and replicated time-lapsesurveys. This observation holds true when the vintages themselves are of prime interest, and inan idealized setting where we assume subsampled time-lapse measurements are taken on a discreteregular periodic grid, i.e., samples lie “exactly” on the grid (Chapter 5). This leads to “exact” repli-cation of sampling points between time-lapse surveys when the sampling points overlap (for differ-ent percentages of overlap). We achieve this result by using a joint-recovery model (JRM), derivedfrom DCS, that exploits common information shared between time-lapse vintages, and additional163structural information provided by nonreplicated (or independent) surveys. Therefore, whenevertime-lapse data exhibit joint structure — i.e., they are compressible in some transform domain andshare information — sparsity-promoting recovery of the “common component” and “innovations,”with respect to this common component, outperforms independent recovery of colocated (prestack)baseline and monitor data. Moreover, jointly recovered colocated (prestack) vintages exhibit ahigher degree of repeatability in terms of NRMS ratios compared to independent recovery. Ourproposed method lowers acquisition cost and environmental imprint because we have subsampledmeasurements, i.e., fewer shot locations are visited. This offers a possibility to extend the surveyarea or to increase the data’s resolution at the same costs as conventional surveys.Our findings in Chapter 6 corroborate the observations made above when we consider a morerealistic (field) scenario of taking measurements off the grid, i.e., irregular samples that do not lieon a discrete regular periodic grid. Extending our simultaneous time-jittered marine acquisitionto time-lapse surveys for realistic off-the-grid acquisitions, wherein off-the-grid sample points areknown, generates simultaneous subsampled and irregular measurements. Consequently, we adaptour proposed JRM to incorporate a regularization operator that maps traces from an irregular gridto a regular periodic grid. The recovery method is a combined operation of source separation, reg-ularization, and interpolation, in which periodic densely sampled and colocated prestack data arerecovered from time-compressed, and therefore economical, simultaneous subsampled and irregulardata. Similar to the observations made in Chapter 5, joint recovery of the vintages improves signif-icantly when the time-lapse surveys are not replicated, since independent surveys give additionalstructural information. We also show that realistic and inevitable off-the-grid sampling leads tolittle variability in recovery of the time-lapse difference for decreasing overlap between the surveys,and hence negates the efforts to replicate.Using two different geological velocity models, SEAM Phase 1 model that has a relativelycomplex geology and complex time-lapse difference compared to the BG COMPASS model (simplegeology and complex time-lapse difference), we show that the observations are quite universal.These observations are significant because they can potentially change the current paradigm oftime-lapse seismic that relies on expensive dense periodic sampling and replication of time-lapsesurveys. Although these observations are made assuming source equalization and no significantchanges in wave heights, water column velocities or temperature, and salinity profiles among thedifferent surveys, recent successes of randomized surveys in the field (see, e.g., Mosher et al. (2014))build our confidence in the success of pragmatic (field) compressive randomized simultaneous time-lapse surveys. Moreover, since irregular spatial sampling is inevitable in the real world, it wouldbe better to focus on knowing what the shot positions were (post acquisition) to a sufficient degreeof accuracy, than aiming to replicate them. Embracing randomness in surveys, whether naturalrandomness such as (streamer) cable feathering or randomness by design, maximizes collection ofdifferent/independent information leading to surveys acquired at low cost and a small environmentalimprint.8.4 Compressive simultaneous-source towed-streamer acquisitionWe present two CS-based techniques — sparsity promotion and rank minimization — for source sep-aration for dynamic simultaneous towed-streamer acquisitions, such as over/under and simultaneous-long offset acquisition. Recoveries from both techniques are comparable; however, the latter ap-proach readily scales to large-scale seismic data volumes and is computationally faster. Moreover,the SVD-free matrix factorization method used to solve the nuclear-norm optimization formula-164tion avoids expensive SVDs, a necessary step in traditional rank-minimization-based methods. Weaddress the challenge of processing high-frequency monochromatic slices, which do not exhibit low-rank structure in the midpoint-offset domain due to increase in wave oscillations away from thediagonal compared to low-frequency slices, by incorporating the hierarchical semiseparable struc-ture in the factorization-based rank-minimization framework leading to improved recovery qualityof the separated data volumes. We also find that both these techniques are comparable with thecommonly used NMO-based median-filtering techniques.8.5 Follow-up workMotivated by the successful implementation of CS to static and dynamic marine simultaneous-source acquisitions, and DCS to (static) simultaneous-source time-lapse surveys, we extend ourwork to derive a viable low-cost and low-environmental impact multi-azimuth towed-streamer time-lapse acquisition scheme. Initial findings of a simulation-based feasibility study for 3D randomizedtowed-streamer time-lapse surveys in a realistic field-scale setting have been reported in Kumaret al. (2017). In this acquisition scheme, we acquire economic, randomly subsampled (about 70%)and simultaneous towed-streamer time-lapse data without the need of replicating the surveys. Werecover densely sampled full-azimuth time-lapse data on one and the same periodic grid by using thejoint-recovery model (Chapters 5 and 6) coupled with the computationally cheap and scalable rank-minimization technique (Chapter 7). Our findings are consistent with those reported in this thesisthat indicate that acquisition efficiency can be improved significantly by adapting the principlesof CS. Furthermore, this new paradigm can also provide an appropriate framework for low-costtime-lapse wide-azimuth acquisition with towed arrays and multiple source vessels.Initial findings on the possible impact of unknown calibration errors, such as unknown deviationsbetween actual and post-plot acquisition geometry, on time-lapse data repeatability have beenreported by Oghenekohwo and Herrmann (2017). In this contribution, the authors show that fordata acquired via compressive, irregular and nonreplicated surveys, attainable recovery quality andrepeatability of time-lapse vintages and difference deteriorates gracefully as a function of increasingcalibration errors.8.6 Current limitationsSome limitations of the work presented in this thesis are as follows:1. Current implementation/code of the curvelet transform — the FDCT based on the wrappingof specially selected Fourier samples (Cande`s et al., 2006a) — is around 8× redundant in2D and around 24× redundant in 3D (Ying et al., 2005). This precludes tractable higher-dimensional FDCTs. Moreover, it renders large-scale computations infeasible especially whendealing with massive 3D seismic data volumes.2. For recovery of periodic densely sampled time-lapse data (Chapters 5 and 6), we assume thatmagnitude of the common component and innovations in the joint-recovery model is more orless similar. However, this is not the case in reality, and therefore certain scaling measuresare required that will lead to improved (sparse) recoveries (see point 4 in the next section).3. One of the limitations of the proposed SVD-free matrix-factorization approach (Chapter 7)is to find the rank parameter k associated with each low-rank factor. The cross-validation165techniques used for the proposed formulation are rendered impractical when dealing withlarge-scale (subsampled) seismic data volumes since it involves finding an appropriate smallvolume of data to run the cross-validation techniques — a computationally expensive process.The implementation of the HSS representation for 3D seismic data volumes also remains tobe investigated.8.7 Future research directionsSome ideas for future work are as follows:1. Develop a computationally faster and memory efficient implementation of the 2D and (moreimportantly) 3D curvelet transforms, since the curvelet transform is designed to representcurve-like singularities optimally (Cande`s and Demanet, 2005). Curvelets have different fre-quency content and dips to match wavefronts locally, leading to a sparse — arguably thesparsest — representation of seismic data.2. Conduct investigations for recovery of weak late-arriving events with high degrees of accuracyduring source separation. One approach to improve recovery of weak late-arriving events isto use weighted one-norm minimization (Mansour et al., 2013) that exploits correlationsbetween locations of significant transform-domain coefficients of different partitions —– e.g.,shot records, common-offset gathers or frequency slices —– of the acquired data.3. Develop robust algorithms to use simultaneous-source data directly in imaging and inversionwithout the need for separating simultaneous data. Some researchers have obtained goodsynthetic results for direct imaging of simultaneous-source data (Berkhout et al., 2012; Choiand Alkhalifah, 2012; Guitton and Daz, 2012; Gan et al., 2016; Xue et al., 2016) but successfulfield applications remain challenging.4. Investigate the effects of imposing a weight γ on the common component and innovationsthat turns the joint recovery via `1-norm minimization formulation (Chapters 5 and 6) in toa γ-weighted `1-norm formulation (DCS, Baron et al. (2009)):z˜ = arg minzγ0||z0‖1 + γ1||z1‖1 + γ2||z2‖1 subject to y = Az. (8.1)So far we have assumed γ0 = γ1 = γ2 = 1. However, in reality, magnitude of the commoncomponent and innovations of seismic data vary, and thus call for some enhancements in thereconstruction scheme by means of introducing appropriate weights on each element of z. 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