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Seismic energy propagation in highly scattering environments and constraints on lunar interior structure… Blanchette-Guertin, Jean-François 2014

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Seismic Energy Propagation in HighlyScattering Environments and Constraints onLunar Interior Structure from the ScatteredSignals of the Apollo Passive SeismicExperimentbyJean-Franc¸ois Blanchette-GuertinB.Sc., The University of British Columbia, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Geophysics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2014c Jean-Franc¸ois Blanchette-Guertin 2014AbstractMeteoroid impacts over hundreds of millions to billions of years can produce ahighly fractured and heterogeneous megaregolith layer on planetary bodies such asthe Moon that lack effective surface recycling mechanisms. The energy from seis-mic events occurring on these bodies undergoes scattering in the fractured layer(s)and this process generates extensive coda wave trains that follow major seismicwave arrivals. These long coda trains can obscure the secondary crustal, mantle orcore phases that are often crucial in assessing the interior structure of these plan-etary bodies when using more traditional seismological analyses. However, thedecay properties of these codas are affected by the interior velocity, intrinsic atten-uation and scattering structure of the planet or moon. As such, these decay proper-ties can contain valuable information regarding these aspects of interior structure.This thesis provides the first systematic analysis of scattering in the Apollo PassiveSeismic Experiment dataset, demonstrating that scattering in the Moon occurs overa wide range of frequencies, and dominantly in the near-surface megaregolith thatcomprises many more small scale heterogeneities than large ones. I also presenta new numerical modeling technique (referred to as PHONON1D) that modelsseismic energy propagation and integrates high levels of scattering. Using thismethod, I investigate the effects of various velocity, scattering and intrinsic atten-uation structures on the scattered coda. Results show that the main controls onthe coda generation and decay times are the seismic velocity profile, attenuationlevels, and the number density of scatterers. Thus these properties can be assessedby comparing predicted synthetic seismic coda with those observed in the ApolloPassive Seismic Experiment data. Finally, I use the PHONON1D method to showthat locations within young and large impact basins, away from the edges, havethe potential to minimize the scattering observed in the recorded seismic signals.These locations would be ideal for the emplacement of future seismic surveys onthe lunar surface.iiPrefaceThis thesis comprises four complementary studies that have been prepared to beindividually published in peer-reviewed scientific journals. Consequently, somebackground information is repeated in the introductory sections of each chapter.A version of Chapter 21 is published in Journal of Geophysical Research –Planets [Blanchette-Guertin et al., 2012]. The co-authors are Jean-Franc¸ois Blanchette-Guertin, Catherine L. Johnson, and Jesse F. Lawrence. I was the lead investigatorfor this study, responsible for all data selection and analysis, as well as for the ma-jority of the manuscript composition. Both Catherine Johnson and Jesse Lawrenceprovided extensive feedback for the data analysis method and interpretations, aswell as invaluable editorial comments for all drafts prior to publication.A version of Chapter 3 is in revision for publication in a major geophysicalresearch journal. The co-authors are Jean-Franc¸ois Blanchette-Guertin (first au-thor), Catherine L. Johnson, and Jesse F. Lawrence. The modified phonon methodused in this study is a refinement and extension of a similar method presented inShearer and Earle [2004]. Jesse Lawrence provided me with the basic FORTRANcode, which I have extensively improved and modified to implement isotropicscattering, user-defined scatterer scale-lengths distribution, as well as frequency-dependent and phase-dependent intrinsic attenuation. I am also responsible for thecode benchmarking, for the model selections, for the data analysis, as well as forthe majority of the manuscript composition. Jesse Lawrence provided support forthe code modifications as well as feedback and comments during the data analysisand writing stages. Catherine Johnson provided continuous support for the codemodifications and data analysis as well as editorial comments on all drafts beforesubmission. All computations of the synthetic seismograms were performed on the1Reprinted from Journal of Geophysical Research – Planets, c2012, with permission from JohnWiley and Sons.iiiPrefaceCEES HPC facility at Stanford University, for which access was provided by JesseLawrence. All post-processing and analysis of the synthetics was done in-house atUBC.A version of Chapter 4 will be submitted to a major geophysical research jour-nal. The co-authors are Jean-Franc¸ois Blanchette-Guertin (first author), CatherineL. Johnson, and Jesse F. Lawrence. I selected the models and ran the analyses,and I am responsible for most of the manuscript. Catherine Johnson and JesseLawrence provided insightful comments and feedback during the data analysis andinterpretation stages, as well as editorial comments for the manuscript.A version of Chapter 5 is ready for submission to a major geophysical re-search journal. The co-authors are Jean-Franc¸ois Blanchette-Guertin (first author),Catherine L. Johnson, and Jesse F. Lawrence. I am responsible for the code devel-opment, the analyses and for writing most of the manuscript. Catherine Johnsonand Jesse Lawrence provided feedback and editorial comments for all drafts priorto submission.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scientific Context and Background . . . . . . . . . . . . . . . . 61.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Investigation of Scattering in Lunar Seismic Coda . . . . . . . . . . 162.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 Data Processing Steps . . . . . . . . . . . . . . . . . . . 212.3.2 Analytical Solutions for the Coda Decay . . . . . . . . . 242.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Decay Parameters Measured on the Long-Period Channels(LP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29vTable of Contents2.4.2 Decay Parameters Measured on the Short-Period Channel(SPZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.3 Dependence of Decay Parameters on ↵ . . . . . . . . . . 372.5 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.5.1 Shallow Events (SMQ, NI and AI) . . . . . . . . . . . . 402.5.2 Deep Moonquakes . . . . . . . . . . . . . . . . . . . . . 462.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Modeling Seismic Energy Propagation in Highly ScatteringEnvironments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.1 Model Geometry . . . . . . . . . . . . . . . . . . . . . . 553.2.2 Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . 573.2.3 Seismic Source . . . . . . . . . . . . . . . . . . . . . . . 593.2.4 Scattering Layers . . . . . . . . . . . . . . . . . . . . . 603.2.5 Regular Velocity Interfaces . . . . . . . . . . . . . . . . 623.2.6 Surface Recorders . . . . . . . . . . . . . . . . . . . . . 643.2.7 Intrinsic Attenuation . . . . . . . . . . . . . . . . . . . . 663.3 Modeling Method Benchmarking . . . . . . . . . . . . . . . . . 663.4 Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . 673.4.1 Processing of Synthetics . . . . . . . . . . . . . . . . . . 713.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 733.5.1 Two Distinct Decay Regimes . . . . . . . . . . . . . . . 763.5.2 Effects of Interior Structure on Coda Decay Times . . . . 823.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904 Constraints on Lunar Interior Structure from the Scattered Signals ofthe Apollo Passive Seismic Experiment . . . . . . . . . . . . . . . . 934.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 Apollo Passive Seismic Experiment Data . . . . . . . . . . . . . 954.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.3.1 Modeling Ambient Noise and APSE Instrument Effects . 101viTable of Contents4.3.2 Analysis of Synthetic Seismograms . . . . . . . . . . . . 1044.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.4.1 Results fromModeling of Seismic Noise and APSE-InstrumentEffects . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.4.2 Modeling Results . . . . . . . . . . . . . . . . . . . . . 1074.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125 Effects of Lateral Variations in Megaregolith Thickness on RecordedSeismic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . 1206 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.2 The Science Questions Answered . . . . . . . . . . . . . . . . . 1266.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130AppendixA Coefficients for Solid-Solid, Free Surface and Solid-Liquid Interfaces 140A.1 P-SV at Free Surface . . . . . . . . . . . . . . . . . . . . . . . . 140A.2 Solid-Solid Interfaces . . . . . . . . . . . . . . . . . . . . . . . 142A.3 Solid-Liquid Interfaces . . . . . . . . . . . . . . . . . . . . . . . 143viiList of Tables2.1 Occurrence time and location of events used in Chapter 2’s study . 192.2 Frequency bands investigated in Chapter 2’s analysis . . . . . . . 242.3 Decay least-squares fits results . . . . . . . . . . . . . . . . . . . 303.1 Models investigated in Chapter 3’s study . . . . . . . . . . . . . . 704.1 Decay least-squares fits results from APSE data . . . . . . . . . . 964.2 Models investigated in Chapter 4’s study . . . . . . . . . . . . . . 984.3 Median epicentral distances and signal-to-noise ratios measured onthe APSE signals . . . . . . . . . . . . . . . . . . . . . . . . . . 102viiiList of Figures1.1 Lunar internal structure . . . . . . . . . . . . . . . . . . . . . . . 21.2 Lateral variations in crustal thickness . . . . . . . . . . . . . . . . 31.3 Lunar vs. terrestrial seismic signals . . . . . . . . . . . . . . . . . 41.4 Map showing the locations of the four APSE receivers . . . . . . 71.5 Examples of lunar seismic signals . . . . . . . . . . . . . . . . . 81.6 Schematic diagram of the APSE unit . . . . . . . . . . . . . . . . 91.7 Schematic cross-section of the megaregolith . . . . . . . . . . . . 112.1 Map showing the locations of the APSE stations and the epicentersof seismic events used in this work . . . . . . . . . . . . . . . . . 182.2 Instrument response and frequency response curves of bandpassfilters used in the analysis . . . . . . . . . . . . . . . . . . . . . . 222.3 APSE Data processing steps . . . . . . . . . . . . . . . . . . . . 262.4 Natural impact and shallow moonquake S-rise times for the longperiod bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Measured ⌧d for all types of events on the LP channels . . . . . . 312.6 Measured Qc for all types of events on the LP channels . . . . . . 322.7 Measured ⌧d for all types of events on the SPZ channel . . . . . . 352.8 Measured Qc for all types of events on the SPZ channel . . . . . . 362.9 Comparison between terrestrial and lunar Q1c values . . . . . . . 382.10 Schematic representation of seismic energy propagation showingthe effect of epicentral distance on ⌧d . . . . . . . . . . . . . . . . 412.11 Schematic representation of seismic energy propagation in the scat-tering layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.12 ⌧d and Qc values for DMQ events in the A001 source region . . . 47ixList of Figures3.1 Model geometry 1-D . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Model geometry 2-D . . . . . . . . . . . . . . . . . . . . . . . . 583.3 Source functions used in the PHONON1D modeling . . . . . . . 603.4 Scatterer length-scale probability distribution . . . . . . . . . . . 613.5 Scattered ray path example . . . . . . . . . . . . . . . . . . . . . 633.6 Time correction applied to phonons that do not hit the surface di-rectly beneath the receivers . . . . . . . . . . . . . . . . . . . . . 653.7 Model benchmarking . . . . . . . . . . . . . . . . . . . . . . . . 683.8 1-D velocity and density profiles used in this study . . . . . . . . 723.9 Qi(f) styles used in the modeling work . . . . . . . . . . . . . . 733.10 Examples of synthetic traces for all models presented in this study 753.11 Decay times as a function of epicentral distances for selected models 773.12 Regime 1 vs. Regime 2 . . . . . . . . . . . . . . . . . . . . . . . 793.13 Average ⌧d, maximum S-coda amplitudes andQc values for Regimes1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.14 Average ⌧d values for each models and frequency bands, for sur-face impacts and deep events . . . . . . . . . . . . . . . . . . . . 833.15 Effect of Qi(f) on ⌧d and Qc(f) . . . . . . . . . . . . . . . . . . 884.1 Qi(z) styles used in this modeling work . . . . . . . . . . . . . . 994.2 Qi(f) styles used in this modeling work . . . . . . . . . . . . . . 1004.3 APSE instrument effect implementation steps . . . . . . . . . . . 1034.4 Examples of summary ⌧d measurements . . . . . . . . . . . . . . 1054.5 Comparison between noise-free and APSE-corrected synthetic sig-nals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.6 Median ⌧d values on the LP bands . . . . . . . . . . . . . . . . . 1084.7 Intercepts and slopes on the SP bands . . . . . . . . . . . . . . . 1105.1 Synthetic seismograms: Impact versus deep event in small basins . 1185.2 Synthetic seismograms: Thick megaregolith and large basins . . . 119A.1 Notations used in the transmission and reflection coefficient equa-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141xAcknowledgementsAs it turns out, completing a PhD thesis is quite the endeavour. And as it also turnsout, it would not have been as fruitful, or even sometimes as enjoyable, without thesupport (technical, professional, moral or otherwise) of many, many people. Thelist below is far from exhaustive. So to those of you I do not mention, but withwhom I have crossed paths during these past few years of graduate school, pleaseknow that you also bask in the warm glow of my sincere gratitude.So, in no specific order, and first in French:Un merci tout absolument spe´cial a` ma famille: Marie`ve, Ame´lie, Alexandra,Danie`le, Michel, Miche`le, Micheline, Gaston et Ellen (ainsi que les beaux-fre`reset les neveux!). Ce qui ne devait eˆtre que quatre ans a` Vancouver est subtilementdevenu un se´jour de pre`s de dix ans. Votre support continuel et sans e´quivoquependant les bons et moins bons moments de ma carrie`re universitaire m’ont permisde naviguer dans les eaux parfois tumultueuses de ces dernie`res anne´es. Je vousaime.Philippe, Gabrie`le et Laurence. Tout d’abord, vous eˆtes beau, belles et ma-gnifiques! Votre amitie´ m’a aide´ a` conserver les pieds fortement ancre´s et la teˆtebien haute, et ce, meˆme a` plus de 5000 km de distance. Merci. Je vous aime, aussi.To my current and past Vancouver family: Ariel, Carolina, Gina, Shelley, Jenn,Marianne, Holly, Ayesha, Lush, Jon, Hannah, Geoff, Nicola, A.J., Farley, and ofcourse Lilu, to only name a few. Thank you a thousand times. You have madethese past few years a bundle of unforgettable memories. Love, always.To the UBC crew: Jenn Fohring, Marianne Haseloff, Reka Winslow, KathiUnglert, Shelley Oliver, Ayesha Ahmed, Kirsten Halverson, Lucy Porritt, LeanneSmar, Shawn Hood, Claire Poulton, Vicky Liu, Andreas Ritzer, Hideharu Uno,Surdas Mohit, among many others. Thank you for sharing this experience andfor making it that much better through your support, friendship, or otherwise. AxiAcknowledgementsspecial thank you to Reka for being my partner in academic crime for a few years.I admire your strong analytical mind, professional work ethics, and unfalteringperseverance. I am looking forward to the day our paths will cross again and whenwe will both work on the same fun and fantastic planetary exploration mission.To the members of my committee: Michael Bostock, Jesse Lawrence and MarkJellinek. Michael, I really appreciate you humouring my surprise office calls andall your help with my often unclear and poorly constructed questions. Your helpand experience proved invaluable at many steps during this project. Jesse, first ofall, many thanks to you and your most fabulous family for hosting me on my manyvisits to Stanford. Your down-to-earth and genial approach to life made these tripsof course useful, but mostly fun. Also, thank you for ALL your support, comments,advice, and for the few motivational (for me!) discussions we had throughout thisPhD. I am grateful for the circumstances that made us work together on this project.Mark, I cannot thank you enough for your help, analytical acumen, and knowledgeof what seems to be all things. Our discussions forced me to think about new andbetter things, and for that I am grateful. You bring a breath of fresh air to science! Iremember a time before I even started this thesis when you told me in a very seriousvoice that everyone going through the PhD process leaves it a different person. Ihad doubts then, I believe you now.And of course, my thesis advisor, Catherine Johnson.Catherine, you have been witness to quite the wide range of professional andpersonal struggles, during which you have always made your full support clear. Iwill always be extremely grateful for that support, for your academic mentorship,for your ever apparent faith in my abilities, for you unfailing professionalism, and,most of all, for your ferocious and highly contagious passion for all things science.Passion without which I would most assuredly be on a drastically different careerpath today. (Absolutely) No regrets! I could expand on this for a long while, butyou know me well enough by now to know what I am trying to convey with thissimple thank you. Catherine, Merci.Finally, this work was generously funded by the Natural Sciences and Engi-neering Research Council of Canada, a University of British Columbia PhD Four-Year Fellowship, a Lorntzsen Scholarship, the department of Earth, Ocean andAtmospheric Sciences and by the Faculty of Science.xiiPour Danie`le....xiiiChapter 1Introduction1.1 MotivationThe Moon is the only Solar System object, other than the Earth (and the Sun, ifone considers helioseismology), for which we have seismic data. The four seis-mometers installed at the start of the Apollo Passive Seismic Experiment (APSE,1969-1977) returned over seven years of continuous seismic recordings. More than40 years of analyses of the APSE data have yielded important discoveries regardingthe inner structure and properties of the Moon. These include a better understand-ing of interior seismic velocities to depths of ⇠1000 km (e.g., Nakamura [1983];Lognonne´ et al. [2003]; Gagnepain-Beyneix et al. [2006]; Garcia et al. [2011]),as well as constraints (albeit limited ones) on the depths of the major composi-tional layers forming the Moon’s interior (e.g., Vinnik et al. [2001]; Weber et al.[2011]). A series of orbital missions and earth-based radar studies have also pro-vided us with a more extensive knowledge of surface topography (e.g., Smith et al.[2010]), lateral variations in relief at the crust-mantle interface (e.g., Wieczoreket al. [2013]), crustal composition (e.g., Jolliff et al. [2000]; Klima et al. [2011]),and other geophysical properties such as crustal magnetic anomalies (e.g., Hoodet al. [2001]). The resulting understanding of the first order interior structure ofthe Moon is shown in Figures 1.1 and 1.2. The presence of a small metallic lu-nar core is supported by gravity, seismic and magnetic studies. Volumetricallythe Moon is dominated by the silicate mantle and crustal rocks (Figure 1.1). Thecurrent best-estimate for the globally-averaged crustal thickness is about 40 km[Khan and Mosegaard, 2002; Lognonne´ et al., 2003; Gagnepain-Beyneix et al.,2006; Wieczorek et al., 2013], however crustal thickness varies geographically asseen in Figure 1.2.Four different types of seismic events were identified in the APSE data: arti-11.1. MotivationFigure 1.1: Cross-section of the lunar interior showing the main compositional lay-ering and the approximate depths of tectonic sources. Schematic is modified fromWieczorek [2009], with permission from the Mineralogical Society of America.ficial impacts, natural impacts, deep moonquakes and shallow moonquakes. Thesignals are atypical of the majority of events recorded daily on Earth, suggest-ing inherently different properties of the lunar and terrestrial interiors. Some ofthe characteristics of the lunar seismograms are: long durations (⇠60 minutes ormore), low amplitudes, lack of identifiable secondary wave arrivals, gradual buildup of energy from the onset of the events, and a slow amplitude decay from themaximum amplitude to the end of the event (when the signal amplitude reachesthe background noise level, see Figure 1.3). These aspects of the lunar seismicrecordings, combined with the low number of stations and the APSE network poorgeographical coverage (Figure 1.4), mean that some well-established seismic dataanalysis techniques used routinely in terrestrial studies cannot be easily applied tothe lunar data set. For example, the locations of moonquakes typically have anuncertainty of >100 km (e.g., Nakamura et al. [1979]; Lognonne´ et al. [2003]),which is an order of magnitude or more larger than typical uncertainties of terres-trial teleseismic events.A long-standing proposed explanation for the unusual characteristics of the21.1.MotivationFigure 1.2: Lateral variation in lunar crustal thickness derived from the Gravity Recovery and Interior Laboratory (GRAIL)and Lunar Reconnaissance Orbiter (LRO) data. Figure is reprinted from Wieczorek et al. [2013], with permission fromScience.31.1. MotivationFigure 1.3: Lunar moonquake signal (A) and typical terrestrial signal (B), showingthe main wave arrivals in red. Note the different scales for the time axes. The shal-low moonquake signal shows the very long duration, emergent body wave arrivals,and very long decaying coda train that are typical of lunar seismic signals.41.1. MotivationAPSE data is that the seismic energy propagates in a highly scattering layer charac-terized by a very low intrinsic attenuation [Nakamura, 1977; Latham et al., 1970a].This scattering layer is thought to comprise the very fine-grained surface regolithlayer and the underlying highly fractured layer known as the megaregolith [Ho¨rz,1991]. Because the Moon has no surface recycling mechanism (e.g., plate tecton-ics), these layers are global and may extend to depths of several tens of km. Thesmall- to large-scale heterogeneities in these layers cause coherent wave fronts torandomly spread in a process known as seismic scattering. The concentration andsize distribution of the scatterers will control the intensity of the scattering and thefrequencies that are affected. The lunar seismic data exhibit intense scattering overthe whole range of recorded frequencies (approximately 0.25 to 10 Hz).Although scattering has been inferred to be the cause of the long lunar seismiccoda, a systematic study of these signals was never undertaken, and so a potentiallyimportant resource for understanding the structure of the near-surface megaregolithlayer, as well as the interior structure of the Moon, has hitherto never been exploredfully. The main goal of my research has been to investigate the scattering and atten-uation structure of the Moon and to build understanding of the processes governingthe transmission of seismic energy in highly scattering environments through mod-eling. The general questions I aim to address with this research work are:Q1: What constraints on the interior structure of the Moon or of other highlyscattering bodies can we infer from analyzing the scattered coda of recordedseismic signals?Q2: What are the effects of seismic velocity, intrinsic attenuation and scatteringstructures on the scattered coda of seismic signals?Q3: Are there locations on the lunar surface that could minimize the intensityof scattering in recorded signals, and maximize science returns of futureseismic surveys?To answer these questions, I have first analyzed the scattering properties of theApollo Passive Seismic Experiment data to gain insight into the first-order scatter-ing and attenuation structures of the Moon. However, the interpretations from thisstudy were limited by the lack of understanding of how seismic energy propagates51.2. Scientific Context and Backgroundin highly scattering environments. To address this, a major part of my researchwork has been to develop a new synthetic seismogram modeling capability thatcan incorporate the types of scattering expected on planets and moons with highlyfractured megaregolith layers. Using this method, I have investigated the effects ofseveral scattering and intrinsic attenuation structures on the coda of scattered sig-nals. I have also used this new method, together with the data constraints providedby the initial study, to identify a suite of lunar interior models that reproduce thescattered characteristics observed in the APSE dataset.As well as shedding light on the scattering structure of the Moon, this researchis important for understanding the limitations of the APSE data, as well as the limitsand potential of any future seismic survey on highly scattering bodies. In Chapter 5we examine whether future seismometer deployments in large lunar basins wouldrecord less scattering than the APSE seismic stations. The research presented inthis thesis thus not only gives new insight into the mechanisms controlling the prop-agation of seismic energy in the Moon, but it also allows us to make predictionsregarding the characteristics of the seismic signals that could one day be recordedon other Solar System objects (e.g., asteroids and Mars).1.2 Scientific Context and BackgroundThe Apollo Passive Seismic Experiment consisted of a network of four seismome-ter packages forming a nearly triangular array with Apollo stations 12 and 14 atone corner, and stations 15 and 16 each about 1100 km away (Figure 1.4). Eachseismometer package comprised a three component long-period seismometer (twohorizontal channels, LPX and LPY, and one vertical channel, LPZ) and a verticalshort-period instrument (SPZ). The LP components had an instrument responseranging from 0.004 to ⇠2 Hz and could be operated in one of two modes: flatmode, more sensitive to lower frequency signals (natural frequency of 0.07 Hz),and peak mode, with a sharp peak in ground motion sensitivity at 0.45 Hz, but witha reduced sensitivity to low-frequency signals (< 0.45 Hz). The SP instrumentresponse ranged from 0.05 to 20 Hz, peaking at approximately 8 Hz. The APSEpackage weighed approximately 11.5 kg, only about a third of the typical terres-trial single-axis seismometer at the time, and was wrapped in insulating aluminized61.2.ScientificContextandBackground1214151617ï8 ï6 ï4 ï2 0 2 4 6 8 10Elevation (km)LEGENDApollo station Artificial impactsLunar moduleSaturn IV Booster Shallow moonquakeNatural impactDeep moonquakesource regionAPSE receiverNear side outlineFigure 1.4: Map showing the locations of the four APSE receivers (Apollo stations 12, 14, 15 and 16), and the epicentersof the seismic events used in the study presented in Chapter 2. Most located events in the APSE dataset occurred on thelunar near side (outlined by the red dashed line). The location of Apollo station 17, where an active seismic experimentstudied near-surface seismic velocity structure, is also shown. Station 17, however, was not part of the APSE network.The background is a shaded relief map of the surface topography, acquired by the LOLA instrument aboard the LunarReconnaissance Orbiter. This is a Hammer projection centered on longitude 0 showing the entire lunar surface.71.2. Scientific Context and BackgroundFigure 1.5: Examples of good quality signals: deep moonquake (DMQ), shallowmoonquake (SMQ), natural impact (NI), and artificial impact (AI). P-wave arrivalsare fairly easily identified visually. S-wave arrivals are not as clear and are partiallyobscured by the P-wave coda. Signals of this quality make up only 3% of therecorded events. Most signals look like the signal in the bottom plot, which is aDMQ from the same source region as the top DMQ example (A001).81.2. Scientific Context and BackgroundFigure 1.6: Schematic diagram of the APSE unit, showing the location of theLP (top) and SP (bottom) instruments, as well as the insulating aluminized My-lar wrapping. Image reproduced from Latham et al. [1970a] with permission fromNASA.Mylar, which reduced temperature variations on the surface of the instruments asshown in Figure 1.6 [Latham et al., 1970a]. The recorded seismic data was teleme-tered to Earth, where occasional transmission errors resulted in various gaps and/orlarge amplitude spikes in in the data record [Nakamura et al., 1980]. Noise in thedata also originated from several other distinct sources (e.g. passage through theday/night terminator, temperature variations on the lunar surface, the lunar seismichum due constant regolith gardening, etc. [Dorman et al., 1978; Lognonne´ et al.,2009; Sens-Scho¨nfelder and Larose, 2010]). Some of these noise sources resultedin high amplitude spikes in the data that could be removed or reduced throughvarious filtering techniques (e.g., Bulow et al. [2005]).Four types of events were identified in the APSE data. Deep moonquakes(DMQ) have depths ranging from 700 km to 1100 km, and occur repeatedly invery localized source regions. DMQ occurrence times exhibit periodicities presentin the spectrum of the Earth-Moon tides [Lammlein et al., 1974]. Several thousandDMQs have been identified in the APSE data. Shallow moonquakes (SMQ) arethought to occur at depths of less than 150 km. Only 28 events were identified, but91.2. Scientific Context and Backgroundthe quality and clarity of the P- and S-wave arrivals are much higher than for DMQevents. It is still unknown whether SMQs occur in the crust or upper mantle [Naka-mura et al., 1974; Lognonne´ et al., 2003]. The focal mechanisms of both DMQ andSMQ events have not been identified because of the small number of seismic sta-tions and the poor quality of the P- and S- arrivals. Thus, unlike on Earth the typeof faulting giving rise to lunar seismicity is unknown. Furthermore, uncertainty inthe hypocenter locations can be large (⇠30-1000 km). Natural impacts (NI) werealso recorded in the APSE data. Seismic waves generated when a meteoroid hitsthe lunar surface are extremely useful for inferring the seismic velocity profile ofthe lunar crust and upper mantle because the source depth is known, reducing theuncertainty in their locations. Artificial impacts (AI) that occurred close to thestations when the Lunar Modules or the Saturn IV Boosters were dropped on thelunar surface are also useful because of their known times of impact, locations, anddepths.Good quality signals of the different types of seismic events are presented inFigure 1.5 (top 4 signals). Recordings of this quality make up only 3% of allrecorded APSE events. Most signals are of the quality shown on the bottom plotof Figure 1.5. The events’ long duration (⇠60 min) and the incoherent wave trainsare the subjects of study of this thesis, and results from the low intrinsic attenuationand high scattering in the lunar interior.The most probable origin of scattering in the lunar coda is the megaregolithlayer and the underlying fractured crust. The lunar megaregolith is defined as thehighly fragmented layer, composed of the ejecta of the large basin-forming impactevents, that lies on top of the fractured but in-situ crust (see Figure 1.7) [Hartmann,1974]. Several estimates of the megaregolith thickness have been published, basedeither on ejecta excavation and deposition models [McGetchin et al., 1973; Pike,1974; Housen et al., 1983; Petro and Pieters, 2004, 2008], or on radar and opti-cal data [Shkuratov and Bondarenko, 2001; Thompson et al., 2009]. However, therange of published estimates is large (from a few hundreds of meters up to⇠10 kmthick). Information stored in the scattered coda of lunar seismic events can poten-tially constrain parameters such as the thickness of the scattering layer beneath thestations (see Dainty et al. [1974]) and the mean free path between scatterers (widthof ejecta blocks).101.2. Scientific Context and BackgroundFigure 1.7: Schematic cross-section illustrating the make-up of the lunar megare-golith. The megaregolith layer goes from the upper finer grained regolith to thetop of the fractured, in situ crust. The estimated thickness of the entire structure,including the fractured crust, is in the order of 10’s of kilometers. Seismic veloci-ties are from Tokso¨z et al. [1973]. Schematic is modified from Ho¨rz [1991], withpermission from Cambridge University Press.111.2. Scientific Context and BackgroundThe literature on wave propagation in heterogeneous media on Earth is exten-sive. Energy from local earthquakes (for which the distance between source andreceiver is ⇠100 km or less), propagating through a laterally heterogenous litho-sphere, often results in seismic coda with qualitatively similar characteristics tothose observed in the APSE data (e.g., Aki [1969]; Aki and Chouet [1975]; Marg-erin et al. [1999]). Several theoretical approaches to explain those characteristicshave been developed and applied successfully to terrestrial seismograms. Theseall resort to a statistical treatment of scattering, which allows the dominant charac-teristics of the coda to be modeled using a small number of parameters. Differentmodels can be used depending on the effective scattering strength of the medium.Single-scatterer to multiple-scatterer models are applicable in the case of weak tomoderate scattering [Aki, 1969; Aki and Chouet, 1975]. These assume interactionsof the seismic waves with a given number of point-like scatterers. Energy diffu-sion and radiative transfer models are used for more intensively scattering media[Latham et al., 1970b; Aki and Chouet, 1975; Margerin et al., 1998]. Scattering onthe Moon is strong and so the latter models are expected to be more applicable tothe APSE data [Latham et al., 1970b; Dainty et al., 1974].Importantly, no scattering theory currently exists that is directly applicable tothe lunar or other non-terrestrial contexts. The approaches described above typ-ically model modest scattering of energy from local events, with the structuralmodels comprising a half space (e.g. Aki and Chouet [1975]) or a constant veloc-ity plane layer over a half space (e.g. Margerin et al. [1998, 1999]), or low-leveldeep mantle scattering (e.g. Earle and Shearer [2001]). The Moon is a small rockybody with low intrinsic attenuation and strong scattering. While scattering canoccur throughout the lunar interior, the major contribution likely arises from themegaregolith layer which is global in extent, though spatially variable in thicknessand structure. All events in the APSE data set exhibit a long coda, indicating thatscattering affects all types of events, from any depth and any epicentral distance.Thus seismic energy propagation in the Moon, or in any other bodies with highscattering levels (e.g. potentially asteroids, Mars), is inherently different from onEarth and further modeling and analytical work is warranted.121.3. Thesis Structure1.3 Thesis StructureThis thesis comprises four complementary studies that each helps build a better un-derstanding of the causes and effects of highly scattering environments on seismicenergy propagation. The studies from Chapters 2 and 4 directly address the lunarcontext, whereas the studies in Chapters 3 and 5 can be applied more generally toplanetary bodies with high scattering levels.In Chapter 2, I have made the first systematic analysis of scattering in theApollo lunar seismic data, characterized through the rise and decay parametersof the seismic coda. This was done by measuring the coda wave characteristic de-cay times (⌧d) and decay factor (Qc) for all the types of seismic events, in severalfrequency bands, at all four Apollo stations. ⌧d and Qc are described in detail inChapter 2 and are both measures of the integrated effects of the scattering encoun-tered by the seismic energy as it travels from the source to the receiver. Largevalues of ⌧d (slow decay times) indicates a high level of scattering. This study isthe first research work that uses the Apollo seismic dataset to build understandingof the effect of a highly scattering medium on the propagation of seismic energyon a global planetary scale. As part of this study, I have identified much greaterscattering for shallow (⇠0-200 km depth) versus deep (> ⇠700 km) sources andfor frequencies greater than 2 Hz. This confirms that the megaregolith is playinga major role in the scattering of seismic energy. The results also indicate that theeffect of scattering would be less at frequencies lower than those observed by theAPSE experiment (mostly sensitive to the 0.5-10 Hz range), which will be helpfulin the future design of lunar (or other planetary bodies) seismic surveys. Finally,the results and interpretations provide constraints and testable hypotheses for lunarwaveform models that include the effect of intense scattering and a near-surfacelow velocity layer, and this motivates the two subsequent studies of my thesis.In Chapter 3, I model the ground deformation caused by seismic energy travel-ling in highly scattering medium using a “phonon method”. This method tracks alarge number of seismic wavelets (the phonons) from a seismic source (quake, orsurface impact) and record the ground deformation each time they reach the sur-face. This method is favoured over other synthetic seismic modeling methods usingthe wave equation (e.g. spectral finite element, frequency-wavenumber) because it131.3. Thesis Structurecan reach much larger frequencies and signal durations in very highly heteroge-neous medium with much less computational power. My updated method providesthe first numerical modeling of 3-D scattering (with 1-D background velocity mod-els), with user-defined power-law distributions of scatterer length-scales, frequencydependent intrinsic attenuation, as well as global background scattering. As partof this study, I modeled the ground deformation for simple subsurface models andvaried each model parameter independently in order to asses their individual effecton the seismic coda. The main results show that the magnitude of the decay timesis most affected by the background velocity model in particular the presence ofshallow low velocity layers, the event source depth, and the intrinsic attenuationlevel. The decay times are also controlled to a lesser extent by the size-frequencydistribution of scatterers, the thickness of the scattering layer and the impedancecontrast at the scatterers.In Chapter 4, I establish some constraints on the scattering and attenuationstructures of the Moon using the APSE-derived decay parameters from the studyin Chapter 2, along with synthetic modeling. By investigating the decay charac-teristics of synthetic signals generated in varying scattering environments, we canidentify a suite of lunar interior models that produce rise and decay characteris-tics similar to those observed in the Apollo Passive Seismic Experiment data. Thisstudy is the first to constrain lunar interior scattering properties using my new com-bined approach of APSE data analysis and numerical modeling. I demonstrate thatin order to get lunar-like signals, we not only need high scattering levels and lowintrinsic attenuation (as suggested in the literature), but also seismic velocity struc-tures that tend to trap the seismic energy near the surface and in the scattering layer(e.g. very low velocity regolith, crust, sharp velocity gradient with depth). I alsouse this approach to independently constrain the intrinsic attenuation factor valuesnear the Apollo stations (Qi ⇠ 4000 6000).In Chapter 5, I determine locations on the lunar surface that minimize the effectof scattering on recorded seismic signals, and that have the potential to maximizescience returns of future seismic surveys. I used the phonon method detailed inChapter 3 to model the ground deformation in environments with laterally varyingmegaregolith thicknesses. I show that ideal locations for seismic receivers wouldbe in large impact structures (assumed to have a thinner megaregolith than the141.3. Thesis Structurehighlands), away from the basins’ edges in order to avoid noise from the surround-ing thicker megaregolith. Locations above thin crust and thin regolith are alsopreferred in order to minimize the focusing of seismic energy in the near-surfacemegaregolith.15Chapter 2Investigation of Scattering inLunar Seismic Coda2.1 IntroductionData collected during the Apollo Passive Seismic Experiment (APSE, 1969–1977)have been pivotal in understanding lunar interior structure. However, with a fewexceptions [Nakamura et al., 1975; Horvath et al., 1980; Vinnik et al., 2001; Weberet al., 2011; Garcia et al., 2011], analyses of seismic phases other than initial P-and S-wave arrivals has proved challenging due to the following characteristics oflunar seismograms: long durations, low amplitudes, lack of identifiable secondarywave arrivals, gradual build up of energy from the onset of the events, and a slowamplitude decay from the maximum amplitude to the time when the signal ampli-tude reaches the background noise level at the end of the event. The long coda(typically⇠60 minutes or more) result from transmission of seismic energy withina poorly attenuating, and highly scattering medium [Latham et al., 1970a; Naka-mura, 1977]. These coda represent a largely untapped source of information aboutthe lunar interior and near-surface layers.On the Moon, past studies have looked at the frequency content of the coda toinvestigate geological structure surrounding the receivers. Nakamura et al. [1975]used the peak frequency of the ratio of the horizontal and vertical componentsto estimate thicknesses of 3.7–4.4 m for the fine-grained regolith layer beneathApollo stations 11, 12 and 15. The study also identified a layer beneath the regolithwhose P-wave velocities of 250–400 m/s could correspond to those of welded tuffdeposited by major impact or extensively fractured rocks. Horvath et al. [1980]used a spectral approach to calculate a seismic velocity profile for the near-surface162.1. Introductionlayers, to a depth of ⇠200 m. The decay of the coda amplitude with time can alsobe used to investigate the properties and structure of the scattering layer in whichthe coda is generated. The parameter typically used to characterize the coda decayis the quality factor, Qc, where the inverse, Q1c , represents the fraction of energydissipated per cycle at a given frequency, f . Latham et al. [1970b] used diffusiontheory (applicable in cases of intense wave scattering) to model the energy envelopefunction of a Lunar Module impact at Apollo station 12. The study found a codaQc of 3600, up to an order of magnitude larger than that typical of terrestrial crustalmaterials (100–1000, e.g., Jin and Aki [1988]). However, there exists no systematicstudy of scattering for all event types at all four APSE stations, which is the mainmotivation behind this study.As explained in Chapter 1, scattering of seismic energy on the Moon prob-ably occurs in the near-surface megaregolith layer. Understanding the scatteringproperties of the megaregolith is important because it will provide constraints onthe shallow seismic structure of the Moon, as well as contribute to understand-ing megaregolith formation and evolution on other terrestrial bodies. In addition,a quantitative study of the effects of scattering in current lunar seismograms isneeded to be able to assess whether secondary seismic phases (from velocity con-trasts in the lunar interior) can be identified in the Apollo or future data sets.I conduct a systematic survey of the coda characteristics of impacts and moon-quakes recorded at all four APSE stations. I briefly describe the APSE dataset(Section 2.2), data processing steps and criteria used to identify higher quality seis-mograms (Section 2.3). These steps result in a total of 369 seismograms from 72events at 55 distinct locations usable for this study (Figure 2.1 and Table 2.1). Forthese seismograms, the coda are characterized via their rise times (time from initialP- or S-wave arrival to maximum P- or S-wave amplitude), their maximum codaamplitudes, and their characteristic decay times (⌧d), where ⌧d is the time takenfor the coda amplitude to drop by a factor of e. I investigate any dependence ofthese parameters on source type, on source depth, and on source-receiver epicentraldistance. Those dependences are evaluated in seven different frequency bands.Analysis of the results (sections 2.4 and 2.5) focuses on two specific ques-tions: (1) What is the importance of global versus local (near the seismic stations)structure on the overall characteristics of the seismic signals? (2) What is the con-172.1.Introduction−8 −6 −4 −2 0 2 4 6 8 10Elevation (km)Near side Far side LEGENDArtificial impactsLunar moduleSaturn IV BoosterShallow moonquakeNatural impactDeep moonquakesource regionAPSE receiver15161412Figure 2.1: Map showing the locations of the four APSE receivers (Apollo stations 12, 14, 15 and 16), and the epicenters ofthe seismic events used here. Most located events in the APSE dataset occurred on the lunar near side. The background is ashaded relief map of the surface topography, acquired by the LRO-LOLA instrument.182.1. Introductiontribution of intrinsic attenuation (Qi) and scattering attenuation (Qs) to the signals’coda decay? The results clearly demonstrate the need for future theoretical workand synthetic seismogram modeling.Table 2.1: Occurrence time and location of events used in thisstudy. LM and SIVB refer to Lunar Module and Saturn IVBooster, respectively. Source for event locations: AI from Naka-mura [1983] and Lognonne´ et al. [2003], NI from Lognonne´ et al.[2003], SMQ from Nakamura et al. [1979], and DMQ from Naka-mura [2005]. Source for event depths: (a) From Nakamura [2005],and (b) from Gagnepain-Beyneix et al. [2006].Type Year Day Hr:Min Lat. Long. Deptha Depthb(N) (E) (km) (km)Artificial impactsLM12 1969 324 22:17 -3.9 -21.2 0 0LM14 1971 038 00:47 -3.4 -19.7 0 0LM15 1971 215 03:07 26.4 0.3 0 0LM17 1972 350 06:52 20.0 30.5 0 0SIVB13 1970 105 01:09 -2.8 -27.9 0 0SIVB15 1971 210 20:40 -1.5 -11.8 0 0SIVN16 1972 110 21:03 1.3 -23.8 0 0SIVB17 1972 345 20:32 -4.2 -12.3 0 0Natural impacts1972 004 06:35 74.1 2.6 0 01972 134 08:46 1.5 -17.1 0 01972 199 21:56 32.8 137.6 0 01972 213 18:08 24 10.1 0 01973 269 20:48 28.7 41.1 0 01973 358 10:05 -24.8 -25.1 0 01974 109 18:34 7.4 -33.6 0 01974 198 12:05 20.3 6.5 0 01974 325 13:16 -7.3 19.9 0 01974 349 09:08 1.6 -8.2 0 01975 064 21:52 -52.4 4.2 0 01975 102 18:15 2 43.2 0 01975 124 10:05 -36.4 -121.3 0 01976 013 07:14 -39.4 62.8 0 0Continued on next page192.1. IntroductionTable 2.1 – continued from previous pageType Year Day Hr:Min Lat. Long. Deptha Depthb(N) (E) (km) (km)1976 149 06:03 -16.8 -10.0 0 01977 179 22:25 -13.5 -75.3 0 0Shallow moonquakes1971 107 07:04 48 35 - -1971 140 17:29 42 -24 - -1972 002 22:32 54 101 - -1972 341 23:10 51 45 - 101973 072 08:01 84 -134 - -1973 171 20:25 1 -71 - -1973 274 04:00 -37 -29 - -1974 086 09:11 -48 -106 - -1974 192 00:52 21 88 - -1975 003 01:47 29 -98 - 01975 012 03:17 75 40 - 01975 013 00:28 -2 -51 - -1975 044 22:05 -19 -26 - 211975 127 06:40 -49 -45 - -1975 147 23:32 3 -58 - -1975 314 07:56 -8 64 - -1976 004 11:20 50 30 - 1251976 012 08:22 38 44 - -1976 066 10:16 50 -20 - 1851976 068 14:44 -19 -12 - -1976 137 12:36 77 -10 - -Deep moonquakesA001 - - - 15.7 -36.6 867 917A006 - - - 43.5 55.5 844 860A007 - - - 25 53.2 893 900A015 - - - 0.7 -3.9 747 -A020 - - - 23.7 -31.4 945 1055A022 - - - 21.6 43.6 788 -A026 - - - 14.3 5.2 1122 -A040 - - - -1.3 -10.3 867 886A042 - - - 22.2 -50.7 907 1004A209 - - - -26.5 -35.1 - -202.2. Data2.2 DataFour main types of seismic events were recorded at four Apollo stations: artificialimpacts (AI, including the Lunar Module impacts and the Saturn-IV Booster im-pacts), natural impacts (NI), deep moonquakes (DMQ) and shallow moonquakes(SMQ). The data were recorded on three orthogonal long-period channels (LP)with an instrument frequency response of 0.004 to 2 Hz and on one vertical short-period channel (SP) with a frequency response of 0.05 to 20 Hz. The long-periodinstruments could be operated in one of two modes: flat mode, more sensitive tolower frequency signals (natural frequency of 0.07 Hz), and peak mode, with asharp peak in ground motion sensitivity at 0.45 Hz, but with a reduced sensitivityto low-frequency signals (< 0.45 Hz, see Figure 2.2). The different channels areidentified here as LPX, LPY and LPZ for the two horizontal and the vertical long-period channels and as SPZ for the vertical short-period channel. The LPZ compo-nent at station 14 worked properly only for a short time and only 3 recordings fromthis channel are used in this study. Also, the SPZ component at station 12 mal-functioned early on and did not record any events [Latham et al., 1970b]. The datawere obtained from the IRIS Data Management Center (www.iris.edu/dms/dmc/).2.3 Methodology2.3.1 Data Processing StepsThe data were initially bandpass filtered and despiked using a robust median de-spiking algorithm [Bulow et al., 2005]. The long-period data were filtered from0.25 to 3.31 Hz (the Nyquist frequency) and the short-period data were filteredfrom 2 to 10 Hz. A 2 minute running median filter was used to despike the databy cropping amplitudes larger than 5 times the median. These steps removed longterm fluctuations, as well as most of the thermal and noise spikes.All DMQ, SMQ, NI and AI events in the lunar seismic catalogue [Nakamura,1992] were visually inspected and a signal was selected if it met the followingcriteria: (i) a clear decaying coda is present, (ii) at least one of the P- or S-wavearrival times is identifiable, (iii) the ratio of the maximum amplitude of the S-212.3. MethodologyScaled Amplitude00.20.40.60.810 5 10 15 20 25SP Instrument and Bandpass Filter ResponsesFrequency (Hz)IRFiltersS1 S2 S3 S40 0.5 1 1.5 2 2.5 300.20.40.60.81LP Instrument and Bandpass Filter ResponsesFrequency (Hz)Peak Mode IRFlat Mode IRFiltersL1 L3L2Figure 2.2: Instrument response curves (IR, red) and frequency response curvesof bandpass filters used in the analysis (blue). The left plot shows the long-period(LP) instrument responses, both in flat and peak mode. The right plot shows theshort-period (SP) instrument response curve.coda to the root-mean-square (RMS) amplitude of the noise is larger than 3. Thenoise was measured in a 2 minute window preferably before the P-wave arrival,or once the coda amplitude has decayed to the background level in cases wherethe data before the signal was not available or not optimal (e.g. if the data hadlarge amplitude spikes). A total of 369 signals from 72 events were selected foranalysis. As some of the events are DMQ that occurred at the same source region,the number of distinct locations is 55 (Table 2.1 and Figure 2.1). Arrival timesfor DMQ were identified through cross-correlation of the individual events withtheir respective source region stack. The stacks were generated using the methoddescribed in Bulow et al. [2005]. SMQ, NI and AI arrival times were identifiedvisually.Each signal was corrected for the instrument response using the well estab-lished Seismic Analysis Code, applying prewhitening to flatten the spectral re-sponse [Goldstein et al., 2003; Goldstein and Snoke, 2005]. This allows us to com-pare analyses from signals recorded on long-period channels, whether they wererecorded while the instrument was in peak or in flat mode. Events recorded in peak222.3. Methodologymode account for 71% of the total events recorded on the long-period channelsused here.Signals were then bandpass filtered to allow the measurement of the coda char-acteristic decay time and the coda rise time in specific frequency bands. The bandsfor the LP signals were set to be 0.5 Hz wide and centered on the 0.5, 1.0 and1.5 Hz frequencies. The bands for the SPZ signals were set to a width of 2 Hz andcentered on the 3, 5, 7 and 9 Hz frequencies. Similar studies using earthquake datawill typically use octave-wide bandpass filters (centered at 1.5, 3, 6, 12 Hz, andso on, e.g. Tsujiura [1978]). However, due the limited bandwidth of the Apolloinstruments, the use of filters of constant width is necessary in order to allow in-vestigation of the dependence of decay and rise times on frequency. The differentbands used in this analysis are discussed in the text as L1, L2 and L3 for the long-period bands, and S1, S2, S3 and S4 for the short-period bands (Table 2.2 andFigure 2.2). Figure 2.2 shows that the SP instrument response extends past 10 Hz,however the signal-to-noise ratio of the seismic signal above 10 Hz is too low forthe decay to be measured accurately.Signal envelopes were generated using:E(t) = pD(t)2 + H(t)2 (2.1)where D(t) is the original data, H(t) is the Hilbert transform and E(t) is theresulting envelope. Envelopes of LPX and LPY channels were averaged using:EH(t) = pEX(t)2 + EY (t)2 (2.2)whereEH(t) is the resulting envelope of the averaged horizontal long-period chan-nel, andEX(t) andEY (t) are the envelopes of the LPX and LPY components. Thisstep increases the signal-to-noise ratio around the S-wave arrival. The averagingassumes horizontal isotropy, a step that is justified by the similar (but noisier) re-sults obtained individually for LPX and LPY. This step and the previous bandpassfiltering step generated a set of 842 analyzable seismic traces. The envelope wassmoothed using a 5 minute (LP) or 35 second (SPZ) running window (⇠2000 sam-pling intervals), keeping the 75th percentile. The window length and percentile232.3. MethodologyTable 2.2: The different frequency bands investigated in this analysis. fc is thecentral frequency of each band.In text Range (Hz) fc (Hz)Long-Period L1 0.25–0.75 0.5L2 0.75–1.25 1.0L3 1.25–1.75 1.5Short-period S1 2–4 3.0S2 4–6 5.0S3 6–8 7.0S4 8–10 9.0were empirically determined to retain the characteristic signal amplitude, while re-ducing the contribution from remaining noise spikes. Different window lengths (2to 7 minutes for LP, 15 seconds to 1 minute for SPZ) and percentile values (themedian and 99th percentile) were tested and yielded comparable results.2.3.2 Analytical Solutions for the Coda DecayAnalytical solutions for coda decay are typically of the formE(t) = 1t↵ et/⌧d (2.3)where ⌧d is the characteristic decay time. The value of ↵ can range from 0.75 to 2and depends on the model geometry (which affects the geometrical spreading) andon the scattering environment (see review in Shearer [2007]). The model that mostclosely resembles the lunar scattering environment is that of Margerin et al. [1998],in which a scattering layer overlies a more transparent half-space. However, thelunar and terrestrial contexts are inherently different in the following ways:1. Studies of terrestrial coda waves mostly use local crustal events (see exam-ples in Yoshimoto and Jin [2008]). In the formulation of Margerin et al.242.3. Methodology[1998], the point-like source resides in the scattering layer, close to the re-ceiver. On the Moon, these assumptions are only valid for a few impactevents local to the seismic stations. For most impacts and some or all shallowmoonquakes, the source resides on or in the near-surface scattering layer, butbecause the events are teleseismic much of the energy travels in the mantlebefore entering, or re-entering, the scattering layer en route to the receiver.This effectively results in an infinite number of sources along the base of thescattering layer, instead of a single source.2. In Margerin et al. [1998], the scattering layer lies above a weakly or non-scattering half-space. Scattered energy leaks into the half-space, and doesnot return to the scattering layer. TheMoon is a small spherical body with in-creasing velocity with depth, and low intrinsic attenuation. As such, seismicenergy that propagates through the mantle re-enters a near-surface highlyscattering layer elsewhere no more than 7 minutes later (the approximatetime it takes for a downgoing P-wave to travel through the lunar interior andreach the surface). Seismic energy can go through this process several timesdue to the low lunar intrinsic attenuation. The net result is that once the bodywaves have expanded through the majority of the scattering layer’s volume(about 12-14 minutes after the quake source time, or the time it takes foran S-wave to travel the lunar interior), the many multiply-scattered waveswithin a sphere reduce wave expansion within the scattering layer (↵ ! 0).Lunar coda last for 30 to over 60 minutes after the S- wave arrival, suggestingthat geometrical spreading, which is captured by the ↵ parameter in equation2.3, should have a reduced effect on the coda decay.Thus currently there is no published theoretical solution for the coda decay thatis directly applicable to the lunar environment, but the above discussion suggeststhat solutions with ↵ ! 0 might be expected. In the analyses I set ↵ equal tozero in equation 2.3. In Section 2.4.3 I discuss further the implications of using adifferent choice of ↵.An exponential decay curve of the form et/⌧d was fit using least-squares tothe decay of the smoothed envelope function and ⌧d was retrieved. The beginningof the fit was identified visually, as the exponential decay does not always start at252.3. MethodologyFigure 2.3: Example of the different methodology steps for a SMQ. A) Originaldata, on the LPX channel. B) The 3 bandpass filtered envelope functions for theaveraged LPX and LPY channels (black), the smoothed envelopes (red) and theresulting fit (blue) for the highest frequency band. The characteristic decay times⌧d (labeled on figure) and S-rise times (gray band) are also shown. Note that thesevary with frequency.262.4. Resultsthe maximum coda amplitude. The end of the fit was determined numerically anddefined as the time at which the envelope amplitude is twice the RMS backgroundnoise. Only those fits for which ⌧d is less than the length of the fit were keptfor the final analysis, resulting in 641 decay times. Figure 2.3 summarizes themethodology steps discussed above. The envelope functions can also be fit in logspace to reduce the relative weight of the larger amplitudes at the beginning of thecoda for which the assumption ↵ = 0 may not be valid. The resulting decay times,not shown, did not differ significantly from the results presented here.The measured ⌧d were then converted to the frequency specific decay factor Qcusing: Qc = 2⇡fc⌧d (2.4)where fc is the central frequency of the band of interest and Qc is a dimensionlessparameter called the coda wave quality factor [Aki and Chouet, 1975]. Its inverse,Q1c , represents the fraction of energy dissipated after one period of oscillation.Results for both ⌧d andQc are presented in the following section as both parameterscan provide insight into the attenuation and scattering properties of the Moon. Forexample, a decrease in ⌧d with frequency for a series of given frequency bands,with similar Qc values for all bands, might indicate a constant intrinsic attenuation(Qi) over the frequency range. In this case, high frequencies lose as much energyper cycle than low frequency (constant Qc), but the coda amplitude will decayfaster (⌧d) because high frequencies go through more cycles per unit time.2.4 ResultsMaximum amplitude and rise time results show no clear dependence on epicentraldistance, , or on the source depth. The measured NI rise times in the LP bandsare mostly greater than the SMQ rise times, and they decrease with increasing fre-quency. The SMQ rise times are constant among all 3 frequency bands (Figure 2.4).Rise time values are lower by⇠15% if I halve the length of the smoothing window,but the relationships among frequency, NI and SMQ values remain the same. Theseresults support the observations reported in Latham et al. [1971]. ⌧d and Qc do not272.4. Results0 200 400 600 800210123  0 200 400 600 800210123450 200 400 600 8002101234CountRise time (s)0.75 - 1.25 HzMean Median stdSMQ:NI: 346198 197 44347 1211.25 - 1.75 HzMean Median stdSMQ:NI: 208187 189 30204 770.25 - 0.75 HzMean Median stdSMQ:NI: 589254 234 88590 150SMQNIL1L2L3Figure 2.4: NI (red) and SMQ (blue) S-rise times for the L1, L2 and L3 bands.NI rise times are larger than SMQ rise times, and also exhibit a dependence onfrequency. SMQ rise times are frequency independent. The mean, median andstandard deviation (std) in seconds are given. Only the S-wave rise times thatcould be clearly measured are shown.282.4. Resultsexhibit a dependence on the depth of individual events, but are distinct for eventsoccurring at the surface (impacts) and in the lunar interior (moonquakes). I referto this below as a dependence on event type, and present the ⌧d and Qc results foreach event type as a function of  and frequency (Table 2.3 and Figures 2.5–2.8).The results from the vertical and horizontal LP channels are generally very similarand so no distinction is made between the two channels in Figures 2.5–2.8.The median ⌧d andQc values for all events with greater than 20 are given inTable 2.3. This distance was empirically determined to avoid local effects, and isbased on the AI and NI results, which show either a change in ⌧d and Qc values for between 0 and 20 (AI), or large differences among stations at short distances(NI). For greater than 20, no clear dependence of ⌧d or Qc on was identifiedon the LP components for any event type (Figures 2.5 and 2.6). Table 2.3 liststhe median values for each individual frequency band, as well as for all the bandscombined, along with the median absolute deviation (MAD). A dependence of ⌧dand Qc on  is seen in the short-period NI and SMQ data. The results of robustleast-square fits to individual frequency bands are shown for ⌧d or Qc vs.  forthe NI and SMQ (Figures 2.7 and 2.8), and Table 2.3 gives the slope and intercept(⌧d and Qc value at 0) of each straight line fit. All robust least-square fits weredone using the bisquare weighting function. Other weighting functions (e.g. huber,andrews) yield similar results.For comparison, I show the measured lunar Q1c values and the correspondingQ1c values reported for the terrestrial lithosphere (Figure 2.9). Lower attenuationand/or higher scattering in the Moon are responsible for the much lower lunar Q1cvalues (higher Qc) compared with those obtained for Earth.Below I discuss the ⌧d and Qc results measured on the long-period and short-period channels in more detail. I focus on any dependence on distance, frequency,and any differences among stations.2.4.1 Decay Parameters Measured on the Long-Period Channels(LP)⌧d values decrease with increasing frequency for all event types on the LP chan-nels, consistent with faster decay of high frequency signals compared with low292.4. ResultsTable 2.3: Results from the long- and short-period bands. Long-period: Median⌧d and Qc values and median absolute deviation (MAD) for individual frequencyband (L1, L2, and L3) and for all the bands combined. The median values arecalculated from all events with epicentral distances () larger than 20, to avoidlocal effects. Short-period results: Interpolated ⌧d and Qc values at  = 0 (⌧d(0)or Qc(0)) and the slope of the best-fit straight line. N indicates the number of datapoints used to calculate the median values or the best-fit lines.Long-Period Short-PeriodMedian ⌧d Median Qc ⌧d(0) Qc(0)Band N (MAD) (MAD) Band N (d⌧d/d) (dQc/d)Artificial impactsL1 3 2732 (239) 8582 (750) - -L2 7 1625 (258) 10212 (1618) - -L3 2 932 (126) 8788 (1192) - -All LP 12 1652 (530) 8982 (1512) - -Natural impactsL1 47 2330 (327) 7322 (1030) S1 16 380 (1.4) 7164 (26)L2 62 1362 (131) 8556 (824) S2 11 234 (1.4) 7362 (44)L3 46 865 (199) 8438 (1876) - -All LP 155 1405 (492) 8364 (1302) - -Shallow moonquakesL1 30 2011 (433) 6316 (1360) S1 33 298 (0.8) 5616 (14)L2 32 1114 (80) 6998 (500) S2 35 193 (1.2) 6052 (36)L3 28 722 (55) 6804 (518) S3 33 145 (1.8) 6366 (78)All LP 90 1114 (504) 6866 (824) S4 34 144 (2.0) 8172 (114)Deep moonquakesL1 14 1373 (434) 4312 (1362) - -L2 15 978 (143) 6202 (898) - -L3 5 732 (122) 6902 (1146) - -All LP 34 1062 (365) 5882 (1296) - -302.4. Results0 50 100 150 0 5010 100 1500 50 100 150Epicentral Distance (°)0 50 100 1500100050015002000250030003500010005001500200025003000350001000500150020002500300035000100050015002000250030003500Epicentral Distance (°)Artificial Impacts Natural ImpactsDeep Moonquakes Shallow Moonquakes(s)(s)Station 12 Station 14 Station 15 Station 160.25 - 0.75 Hz (L1) 0.75 - 1.25 Hz (L2) 1.25 - 1.75 Hz (L3)Figure 2.5: Measured ⌧d for all types of events on the LP channels. The coloursrepresent the different stations and the symbols refer to each frequency band. Nodistinction is made between the vertical and horizontal channels, as the measured⌧d are similar for a given event. Measured ⌧d from a single DMQ source regionhave been averaged, and the errorbars correspond to one median absolute deviation.312.4. Results0 50 100 150Qc0 5010 100 15020004000600080001000012000140000 50 100 150Epicentral Distance (°)Qc0 50 100 150200040006000800010000120001400020004000600080001000012000140002000400060008000100001200014000Epicentral Distance (°)Station 12 Station 14 Station 15 Station 160.25 - 0.75 Hz (L1) 0.75 - 1.25 Hz (L2) 1.25 - 1.75 Hz (L3)Artificial Impacts Natural ImpactsDeep Moonquakes Shallow MoonquakesFigure 2.6: Measured Qc for all types of events on the LP channels. The coloursand symbols are the same as in Figure 2.5.322.4. Resultsfrequency signals. No dependence of ⌧d or Qc on , for  greater than 20, isobserved. In general, Qc values for greater than 20 are similar for all frequencybands for individual event types, i.e., the median values for each band are indistin-guishable from one another within the reported median absolute deviations (MAD)in Table 2.3. Examination of the median Qc for all LP bands combined shows thatthe Qc values for impact events are larger than the Qc values for tectonic events(8982 ± 1512 for AI and 8364 ± 1302 for NI, versus 6866 ± 824 for SMQ and5882 ± 1296 for DMQ).At epicentral distances less than about 20, both ⌧d and Qc for AI may in-crease with ; however this is difficult to verify, given the variability in ⌧d and Qcmeasurements, possible differences among the results for individual stations (Fig-ure 2.6, AI), and the short overall distance range sampled by the AI events. Thereare insufficient NI data at epicentral distances less than 20 to investigate any de-pendence of ⌧d and Qc on; however some NI results at short epicentral distancesreveal clear differences between decay parameters measured at individual stations.For the two impact events closer than 10, Qc values at station 15 are about twicethe Qc values measured at station 16 (Figure 2.6, NI). In addition, Qc values forthe NI events show greater variability (Qc = ⇠4000–12000) at short distances thanat large distances (Qc = ⇠7000–10000 for > 90); this is particularly evident inthe results for station 14.⌧d and Qc values on the L1 band for all event types, at all , show differencesamong stations. This is most easily seen in the Qc results (Figure 2.6). Qc valuesmeasured at stations 14 and 16 for moonquakes are typically smaller than the valuesfrom stations 12 and 15 at similar epicentral distances. This is also true of thenatural impacts at epicentral distances less than about 90 (corresponding to theepicentral distance range for most moonquake observations). Similarly, for the AI,Qc on the L1 band appears to be lower at station 14 compared with station 12,although in contrast to the results for moonquakes and NI, the values for station15 are similar to those at station 14. However the range of epicentral distancesspanned by the AI data is more limited than for the other event types.332.4. Results2.4.2 Decay Parameters Measured on the Short-Period Channel(SPZ)The bandwidth covered by the SPZ instrument is wider than that covered by theLP instrument, aiding investigation of any dependence of ⌧d and Qc on frequency.However, the frequency content of events other than SMQs is richer in lower fre-quencies (< 3 Hz). This results in relatively fewer high quality AI, NI and DMQevents on the SPZ component. This is especially true for DMQs, for which thetypically lower magnitudes and overall lower frequency content are responsible forthe low number of events recorded on the SPZ channels. The SMQ events, on theother hand, have a frequency content that extends into the 8–10 Hz band resultingin Qc estimates for all four bands that cover a wide range of (⇠20–120).The low number of AI and DMQ events does not allow the identification of adependence of the decay parameters on  or frequency. However, a dependenceof ⌧d and Qc on distance and frequency is seen in the NI and SMQ data.⌧d shows a correlation with  on the 2–4 Hz and 4–6 Hz frequency bands forthe NI events (correlation coefficients R = 0.61 and R = 0.78 respectively),and for all frequency bands for the SMQ (R = 0.45 for 2–4 Hz, 0.65 for 4–6 Hz,0.75 for 6–8 Hz and 0.75 for 8–10 Hz). ⌧d values for NI on the S1 band are twicethose on the S2 band, but the dependence on distance is the same within error(Table 2.3). ⌧d values for SMQ on band S1 are larger than those on all other bands,and the values on the other bands (S2, S3 and S4) are similar to each other. Thedependence on distance increases for increasing frequencies (seen as increasingslopes in Table 2.3).The ⌧d data result in Qc values that show a clear dependence on epicentraldistance and frequency (Figure 2.8 and Table 2.3). The slope of the straight-linefits to Qc versus  increases from bands S1 to S2 for the NI, and from S1 to S4for the SMQ. NI show similar Qc values for S1 and S2 at very short distances(⇡ 7200). The Qc values for SMQ at short epicentral distances in bands S1 andS2 are also similar to each other, but lower than those for NI (⇡ 5800), and theyincrease for bands S3 and S4.Finally, there is a marked difference in the SMQ ⌧d and Qc values betweenstation 14 and stations 15 and 16, on the 8–10 Hz band. ⌧d values are larger by342.4. Results0 50 100 150 0 50 100 1500 50 100 150Epicentral Distance (°)0 50 100 150Artificial Impacts Natural ImpactsDeep Moonquakes Shallow MoonquakesEpicentral Distance (°)03002001004005006007000300200100400500600700030020010040050060070003002001004005006007006 - 8 Hz8 - 10 Hz2 - 4 Hz2 - 4 Hz4 - 6 Hz4 - 6 Hz(s)(s)Station 15 Station 166 - 8 Hz (S3) 8 - 10 Hz (S4)Station 12 Station 142 - 4 Hz (S1) 4 - 6 Hz (S2)Figure 2.7: Measured ⌧d for all types of events on the SPZ channel. The coloursrepresent the different stations and the symbols refer to each frequency band. Noevents were recorded at station 12 due to an early malfunction of the instrument.Robust least-squares fit straight lines showing dependence of ⌧d on epicentral dis-tance are shown for the two NI lower frequency bands and for all SMQ bands.352.4. Results0 50 100 150Qc0 50 100 1500 50 100 150Epicentral Distance (°)Qc0 50 100 150Artificial Impacts Natural ImpactsDeep Moonquakes Shallow MoonquakesEpicentral Distance (°)4000800012000160002000024000400080001200016000200002400040008000120001600020000240004000800012000160002000024000Station 15 Station 166 - 8 Hz (S3) 8 - 10 Hz (S4)Station 12 Station 142 - 4 Hz (S1) 4 - 6 Hz (S2)6 - 8 Hz8 - 10 Hz2 - 4 Hz2 - 4 Hz4 - 6 Hz4 - 6 HzFigure 2.8: Measured Qc for all types of events on the SPZ channel. The coloursand symbols are the same as in Figure 2.7. Robust least-squares fit straight linesare also shown for NI and SMQ.362.5. Implications⇠100 and Qc values are larger by ⇠4000 at stations 15 and 16 compared to station14. The dependence of ⌧d and Qc on  for each station is similar.2.4.3 Dependence of Decay Parameters on ↵I investigated whether the observed coda decay can be better or equally well fit bythe more general form of equation 2.3 with ↵ 6= 0. I used ↵ = 1 and ↵ = 2, and foreach value I refit the coda decay for all events. I compared the RMS misfits of thefits with ↵ = 1 or ↵ = 2 with those for ↵ = 0. Results show that fits with ↵ = 0are better, for all event types. I found that using ↵ > 0 generally resulted in valuesof ⌧d that are two to three times larger than their value for ↵ = 0. However forsome seismograms, using ↵ > 0 led to unrealistically large ⌧d and Qc values.Importantly, I found that the SPZ dependence of Qc on epicentral distance andfrequency was robust with respect to the choice of ↵. Given these results, and thelimitations of the assumptions underlying existing theoretical work for the lunarapplication, I retain the simplest possible function form for the lunar decay thatrelies on only one parameter (⌧d).2.5 ImplicationsThe physical significance of ⌧d andQc is still not fully understood, especially in thelunar context. Theoretical work suggests that for single scatterer models (i.e., weakscattering), Qc is an averaged S-wave attenuation parameter, which includes bothintrinsic (Qi) and scattering (Qs) attenuations [Sato and Fehler, 1998]. In the caseof multiple scatterer models (stronger scattering), numerical and laboratory exper-iments, as well as theoretical studies indicate that Qc corresponds to the S-waveQi (see review in Yoshimoto and Jin [2008]). This assumes that Qc is measuredat a lapse time larger than the mean free time (defined as twice the time it takesfor an S-wave to go from the source to the receiver). However, the dependenceof Qc on epicentral distance, observed in the short-period NI and SMQ data, isdifficult to reconcile with simple geographical and/or depth variations in Qi. Thissuggests a contribution of Qs to the observed Qc, at least for frequencies larger372.5. ImplicationsFrequency (Hz)10−4100 101 10210−310−2QC−1DMQReported terrestrial valuesLunar valuesSMQAINIFigure 2.9: Comparison between terrestrial and lunar Q1c values. Lunar valuesare much lower than terrestrial values (i.e., larger Qc). Lunar mean Q1c values donot exhibit a strong dependence on frequency in the 0.5–4 Hz range, in contrastto the terrestrial data. The blue (SMQ) and yellow (NI) shaded regions show thevariations inQ1c due to its dependence on epicentral distance at frequencies largerthan 2 Hz. Q1c was plotted (and notQc) in order to better compare with previouslypublished terrestrial data. Figure modified from Yoshimoto and Jin [2008], withpermission from Elsevier.382.5. Implicationsthan 2 Hz. The dependence of Qc on epicentral distance could then reflect the factthat the seismic energy will encounter more scatterers along longer path lengths,resulting in increased ⌧d and Qc values. Thus, a major question is what are therelative contributions of Qi or Qs to the signal recorded in the seismic coda?The lunar interior is different from the Earth in at least two respects that areimportant to understanding the observations. First, the relative paucity of volatilesin the lunar interior leads to lower intrinsic attenuation (higher Qi) than on Earth.Studies of the decay of the maximum signal amplitude with distance yield lunarupper mantle S-wave Qi values ranging from 4000 at frequencies of 3 Hz to 8000at 7 Hz [Nakamura et al., 1976; Nakamura and Koyama, 1982], compared withterrestrial values of ⇠25–500 (e.g., Mitchell [1995]). Second, unlike on Earth,the Moon has a highly fractured megaregolith layer that is global in extent butwith regional variations in thickness and structure (e.g., McGetchin et al. [1973];Thompson et al. [2009]). Thus, scattering can be expected to occur globally inthe near-surface megaregolith layer, with regional or local variations in intensity.Scattering in such a global layer is supported by the observations that seismogramsfor all types of lunar events, from all depths (up to ⇠1100 km) and all epicentraldistances (up to 150), exhibit strong scattering. Scattering of seismic energy in thelunar mantle may also be occurring, as on Earth (e.g., Earle and Shearer [2001]),but its seismic signature is likely obscured by the relatively intense scattering inthe megaregolith.Below, I present a scenario that can explain the observations of ⌧d and Qc fordifferent event types, frequencies, epicentral distances and stations. Based on theabove discussion, I assume that scattering occurs in the megaregolith layer. Twoend-member processes can explain increases in ⌧d and Qc values: the seismic en-ergy encounters (i) more scatterers along its path (higher effective Qs), through ei-ther a higher scatterer number-density and/or a longer path, effectively both spread-ing and redistributing the energy, causing it reach the station over a longer periodof time (increasing ⌧d and Qc); and (ii) a lower intrinsic attenuation (higher Qi)in the scattering layer, increasing the amplitude of the scattered wave arrivals, andeffectively increasing ⌧d and Qc. Accordingly, lower ⌧d and Qc values indicateless scattering (lower effective Qs) and/or higher intrinsic attenuation (lower Qi). Inote that for direct waves, scattering will defocus the energy, resulting in decreased392.5. Implicationsamplitudes. Thus high scattering might be expected to imply low Qs. However,in the lunar context, scattering continues well after the direct wave arrivals, redis-tributing the seismic energy, resulting in longer ⌧d. Thus high scattering implieshigh Qs. This effect can be expected to be even stronger if the scattering layer isalso a low velocity layer, as is quite likely for the megaregolith.I discuss shallow moonquakes and impacts (Section 2.5.1) separately fromdeep moonquakes (Section 2.5.2), because the distribution of energy entering thescattering layer is likely to be quite different in these two cases, and has implica-tions for the observed ⌧d and Qc.2.5.1 Shallow Events (SMQ, NI and AI)Teleseismic Shallow Events ( > 20)I propose that the seismic energy forming the coda of teleseismic shallow eventsspends a large proportion of its total travel time in the scattering layer. As such,⌧d and Qc values are controlled by the scattering layer Qi, by the scattering effi-ciency of the layer (related toQs), and by the distance travelled, or equivalently thetime spent, in the layer. In addition, the low seismic velocities at shallow depths[Garcia et al., 2011; Kovach et al., 1971] will contribute to trapping energy in anear-surface layer. While the depth extent and exact seismic velocity structure ofthe low velocity near-surface region is unknown, it is reasonable to suppose that itencompasses some or all of the scattering (megaregolith) layer.For teleseismic events, the most direct path in the scattering layer has a distanceproportional to . Figure 2.10 shows that increasing  will increase the timeintervals between the arrival of the most direct waves (making up the coda fromthe first arrival up to its maximum), the partly scattered waves (which travel partof the way in the mantle and enter the scattering layer at some point between thesource and the receiver), and the fully scattered waves (which enter the scatteringlayer at, or near, the source). Assuming similar relative amplitudes between thesearriving waves for all , ⌧d and Qc will increase for increasing.Importantly, the energy entering the scattering layer along the source-receiverpath does not have a constant amplitude at all points. Seismic waves entering thelayer near the source will have a larger amplitude than waves entering the scatter-402.5.ImplicationsFigure 2.10: Schematic representation of seismic energy propagation showing the effect of epicentral distance on ⌧d. At shortepicentral distances (A), the time intervals between the arrival of the most direct waves (in red, forming the coda up to itsmaximum amplitude), the partially scattered waves which travel part of the way in the mantle and enter the scattering layerat some point between the source and the receiver (in grey) and the fully scattered waves which enter the scattering layerat, or near, the source (in blue) are small. As the epicentral distance increases (B), these time intervals increase, resulting inlarger ⌧d, for similar relative amplitudes between the waves.412.5. Implicationsing layer after travelling some distance in the mantle (Figure 2.11B). Geometricalspreading results in amplitude differences of an order of magnitude or more forthe waves entering the scattering layer at different points between the source andreceiver. Furthermore, the mantle S-wave Qi has been estimated to be <1500 atdepths greater than 300 km at frequencies of 3 Hz [Nakamura et al., 1976]. Thusenergy traveling in the mantle and entering the scattering layer close to the receiverwill have been attenuated relative to the energy that enters the scattering layer nearthe source. Energy entering the scattering layer close to the source will spend moretime in the layer and will be the most scattered, increasing ⌧d and Qc. In this sce-nario, we can expect ⌧d and Qc at a given frequency to increase with increasing.The results show an increase in ⌧d and Qc with increasing  for signals in theS1–S4 frequency bands. Because the increase is seen for all 3 stations for whichwe have observations, it is difficult to explain in terms of geographical variationsin Qi, and so I examine the scattering hypothesis above further. Assuming S-wavevelocities, (Vs), in the scattering layer range from⇠300 to 3,000 m/s [Garcia et al.,2011], the relevant wavelengths of scatterers are given by  = Vs/f , where fis the frequency. For the S4 band, this yields scatterer scale lengths of about 30- 400m, for the S1 band scale lengths are about 75m to over a km. For the LPbands (for which we observe no increase in ⌧d and Qc with ), relevant scattererscale lengths are on the order of 200m up to several km. Laboratory impact ejectaexperiments (e.g., Ryan and Melosh [1998]), suggest that the size-frequency dis-tribution of ejecta blocks in the megaregolith will follow a power-law distribution,with relatively more small scale blocks than large ones. This suggests that highfrequency seismic energy will encounter more scatterers compared with low fre-quency energy, predicting an increase in ⌧d and Qc with frequency at a given ,as seen in the lunar data. Furthermore, I propose that the lack of dependence of ⌧dand Qc on in the LP results (frequencies < 2 Hz) is because at these frequenciesscatterer scale lengths are on the order of the scattering layer thickness. If the largescatterers are not distributed uniformly laterally and vertically, or are not presentin sufficient number, we might expect no clear dependence of ⌧d and Qc on .At the LP frequencies, scattering is still occurring (i.e., the coda waves are stillpresent), but the measured Qc reflects a relatively stronger contribution from Qi422.5. Implicationsthan at higher frequencies.The source depth, above or beneath the scattering and low-velocity layer(s),will also affect the intensity and duration of the recorded scattering. Impact eventsoccur directly above the scattering layer and the seismic energy is scattered at leasttwice along the way (going through the layer once, travelling in the mantle, andgoing through the layer again to reach the station, see NI plots on Figure 2.11A). Alarge part of that energy may also get trapped in the scattering layer, in particularif this layer is also a low-velocity region, and will travel within the layer all theway to the station. I propose that ⌧d and Qc values for impact events are largerthan for moonquakes because the seismic energy spends more time overall in thescattering layer. This could also explain the longer rise times for NI compared withSMQ (Figure 2.4). The larger decay values for NI could also be explained by thefact that impact seismic source functions last longer due to the time it takes forthe ejecta blanket to fall back down on the lunar surface. Longer source functionswould result in larger ⌧d and Qc. Median ⌧d and Qc for shallow moonquakesare also slightly larger than for deep moonquakes. SMQ events occur closer tothe scattering layer then DMQ. As such, more SMQ energy will be trapped in thescattering layer on its way to the receiver, increasing the overall amount of scatteredenergy, and increasing ⌧d.The difference in ⌧d and Qc magnitudes between station 14 and stations 15and 16 on the 8–10 Hz band points to differences in the local structure and/or rockcomposition surrounding the receivers at those wavelengths. A lower Qc (station14) points to either a weaker scattering, or a higher intrinsic attenuation (lowerQi). Station 14 sits on top of a thick layer of mare basalt, whereas station 15 sitsat the edge of a large basin and station 16 overlies crustal ferroanorthosite. Thisdifference in composition and structure could lead to different scattering properties.As such, the scattering characteristics of a signal are not only controlled by theglobal scattering layer, but also by the local structure near the receivers.Local Shallow Events ( < 20)In contrast to teleseismic shallow events, local shallow events defined here as SMQ,NI and AI events having an epicentral distance of 20 or less, have ⌧d andQc values432.5. ImplicationsFigure 2.11: See caption on following page.442.5. ImplicationsFigure 2.11: (A) Schematic illustration of the scattering layer. Each line representsa scatterer, and its thickness indicates its effective length scale. A larger scatterer(thicker line) is seen by long-period waves, whereas small-scale scatterers (thinlines) affect higher frequency waves. (B) The signal decay measured at the re-ceiver is dominated by the energy entering the scattering layer with the largest rel-ative amplitude. For NI events (left column), this is the energy trapped in the layerdirectly after impact. For SMQ occurring below the scattering layer, this is theenergy that enters the scattering layer near the source (top right). The relative am-plitude of seismic waves entering the base of the scattering layer is shown in blue.For impacts and SMQ, increasing the source-receiver distance () increases thenumber of scatterers encountered and for a power-law size distribution of scatter-ers [Ryan and Melosh, 1998], the effect will be greater at higher frequency (shownfor NI in left column). Thus, ⌧d and Qc increase with increasing  and frequency(gray shading). (C) For DMQ, the energy entering the base of the scattering layer isapproximately uniformly distributed over a wide area. Because DMQ magnitudesare small, only energy entering the scattering layer within some region of influencearound the receiver is still detectable after scattering. ⌧d and Qc values measuredat all  are similar, and represent the scattering properties of regions surroundingthe stations.that may be controlled dominantly by the local scattering structure surrounding thereceivers. An example of the possible effects of local structure is the differencebetween the ⌧d and Qc values of the two NI events with epicentral distances lessthan 10. The two events were recorded at station 15 (Qc = 7882 on L1) andstation 16 (Qc ⇠ 4200 on L1) and their locations are shown as crossed yellowcircles on the location map (Figure 2.1). Station 15 is slightly to the west of theboundary between Mare Imbrium and Mare Serenitatis, and the local impact eventoccurred to the south of this boundary. Deep lateral structure related to the marginsof these two large basins may have trapped the seismic energy arriving from thesouth, increasing the decay time of the signal (and increasing ⌧d and Qc). Station16 is on the highlands to the south-west of Mare Tranquilitatis, and the impactevent hit the surface to the north-east of the receiver. There are no identifiablestructures in the vicinity of station 16 that could trap seismic energy as for station15, potentially explaining the lower ⌧d and Qc values for this particular event.452.5. Implications2.5.2 Deep MoonquakesDMQs have depths of 700–1100 km, and energy propagates from the source re-gions through the lunar mantle and crust, to the scattering layer. The amplitude ofseismic waves reaching a given point at the base of the scattering layer dependson the combined effects of geometrical spreading, intrinsic attenuation and seismicvelocity structure of the mantle and crust. This energy is then scattered, and soseismic waves that enter the base of the scattering layer need to have a sufficientlylarge amplitude to still be detectable by an instrument on the surface. Path lengthstraveled by energy between a DMQ source region and the base of scattering layerwill vary by less than a factor of about 3 for the epicentral distance range overwhich we have observations of DMQ ( < 80). Consequently, in the absenceof large lateral variations in seismic velocity and intrinsic attenuation, the ampli-tude of the wave entering the scattering layer will vary by less than an order ofmagnitude for points along the base of the scattering layer (Figure 2.11C). DMQmagnitudes are small [Latham et al., 1973], and so the amplitude of the wave thatreaches any point along the base of the layer is also small. Hence energy enteringthe scattering layer further from the stations may decay to imperceptible levels be-fore reaching the receivers. The measured ⌧d and Qc are then representative of theproperties of the scattering layer in a region surrounding each station (the regionof influence). The extent of this region, and hence the measured decay parameters,should be insensitive to the source-receiver distance (Figure 2.11C), but might beexpected to increase for larger magnitude DMQs.No dependence of the decay parameters on  is observed in the data (Fig-ures 2.5 and 2.6), which is consistent with the above scenario. Source region A001exhibits the largest number of DMQ in the lunar event catalogue [Latham et al.,1971; Bulow et al., 2005], and contributes the greatest number of DMQ events tothe analyses here, so I use it to examine any dependence of the decay parameterson amplitude at a given station. DMQs from source region A001 do not show anydependence of ⌧d andQc on amplitude (Figure 2.12), but this may simply be due tothe low amplitudes, the small amplitude range for most events, and the fact that therecorded amplitudes also include local station effects. While the actual extent ofthe region of influence is unknown, ⌧d and Qc for band L1 for source region A001462.5. Implications400800600100012001400160018002000(s)Qc1000150030002500200035004000450050000 0.2 0.4 0.6 0.8 1Normalized Amplitude0 0.2 0.4 0.6 0.8 1Normalized AmplitudeStation 12 Station 14 Station 15 Station 160.25 - 0.75 Hz 2 - 4 Hz0.75 - 1.25 Hz 4 - 6 Hz1.25 - 1.75 HzFigure 2.12: ⌧d and Qc values for DMQ events in the A001 source region (samefor a given station).472.6. Conclusionsare larger at station 12 ( = 18.2) than at station 14 ( = 22.3), and station 16( = 51.2). This suggests localized differences in the scattering characteristicsaround station 12, compared with those around station 14, and might offer an upperbound on the size of the region of influence for DMQ type events of similar magni-tudes. This limit has to be smaller than the epicentral distance between stations 12and 14 ( = 180 km). I note that the observation of larger ⌧d and Qc for band L1at station 12 compared with stations 14 and 16 is supported by my global data setof NI, DMQ and SMQ as summarized in Section 2.4.1, suggestive of lateral vari-ations in scattering properties at scale lengths corresponding to the L1 frequencyband.Scattering of DMQ energy near the source region may also occur. If scatteringtakes place in the deep Moon, seismic waves entering the near-surface scatteringlayer will already comprise a long coda (with its own ⌧d). This initial ⌧d would beaffected by the strong scattering in the megaregolith, and would be increased evenfurther. However, a higher intrinsic attenuation in DMQ source regions (lowerQi, which is plausible due to the temperature increase with depth) would act tomitigate this effect. It is not possible to confirm or deny source-side scattering forDMQ with this data set: one test of source-side scattering might be differences inQc for different DMQ source regions, and I do not observe such differences. It islikely that the DMQ coda are dominated by scattering and intrinsic attenuation inthe megaregolith.2.6 ConclusionsThe long decay times of lunar seismograms, recorded for all types of lunar events,at all stations, indicate that scattering of seismic energy occurs in a global layer.This layer is likely the near-surface megaregolith, which is global in extent, withrecognized regional variations in thickness and structure. Both intrinsic attenuationand the scattering properties of the megaregolith contribute to coda decay times. Inaddition, low seismic velocities in (and possibly extending below) the megaregolithwill focus seismic energy into this scattering layer. While it is not possible touniquely identify the contributions of intrinsic attenuation and scattering, variationsof coda ⌧d andQc with event type, distance and frequency allow us to identify when482.6. Conclusionsthe relative contribution of scattering might outweigh that of intrinsic attenuation.The dependence of ⌧d andQc on epicentral distance at frequencies larger than 2 Hzsuggests that Qs dominates the coda decay at high frequencies and large distances.Differences in decay times between shallow events (SMQ and NI) and DMQsindicate that the attenuation and scattering properties of the scattering layer areexpressed differently in the seismograms of shallow and deep events. Shallowevents show an increase of ⌧d and Qc with increasing source-receiver distancesfor frequencies larger than 2 Hz. I propose that this is because energy travelinglonger distances in the scattering layer encounters more scatterers. This inferenceis further supported by the observation that ⌧d and Qc also increase with increasingfrequency, consistent with the presence of relatively more small-scale versus large-scale scatterers (Figure 2.11B). In contrast, the small magnitude and large depths ofDMQs may mean that the seismic energy recorded at the receiver from those deepevents samples the scattering layer in a distinct region (region of influence) aroundthe receiver (Figure 2.11C). This hypothesis is supported by the observation that⌧d and Qc for DMQs are independent of the source-receiver distance, and less thanthose of shallow events. Differences in ⌧d and Qc for DMQ recorded at stations 12and 14 suggest that the region of influence for those stations is less than 180 km(the epicentral distance between the two stations).Local structure can also be important, in particular for signals from local events( < 20). We see increased scattering in the coda of a NI event recorded close tostation 15, compared with one that occurred at a similar distance to station 16, per-haps reflecting the trapping of seismic energy by large scale impact basin-relatedstructures near station 15. Lower Qc values in the L1 band for stations 14 and 16,compared to stations 12 and 15, recorded for all event types (Figures 2.6 and 2.12),reflect either relative differences in intrinsic attenuation (at L1 frequencies) or inscattering (at scale lengths of several hundred meters to a few km) between thosetwo groups of stations. Stations 12 and 14 are both located on the southern basalticplains of Oceanus Procellarum and so the differences in L1 decay times for thosetwo stations point to local differences in structure near the receivers. Stations 14sits on a crater ejecta blanket (itself overlying the older basaltic layers) and asso-ciated low-velocities, and fractured and welded structure might be responsible forthese differences in attenuation and scattering properties in the L1 frequency band.492.6. ConclusionsIn contrast, at higher frequencies (S4), Qc values are lower at station 14 than atstations 15 and 16, which once again points to differences in the intrinsic atten-uations near the receivers or to relatively less scatterers corresponding to the S4frequencies at station 14 than at stations 15 and 16.The results demonstrate that the scattering layer comprises scatterers with arange of length scales, and they indicate that scattering will also be important at fre-quencies higher than those observed by the APSE experiment. In contrast, record-ings of lower frequency signals might yield data that contain minimal scatteringand would greatly facilitate the exploration of the Moon’s interior. The resultsand interpretations provide constraints and testable hypotheses for future wave-form modeling that includes the effects of very low intrinsic attenuation, intensescattering and near-surface low velocity layer(s) in a small rocky body. Such stud-ies can investigate suites of models that predict seismograms with coda propertiesthat match those reported here from the APSE data set. These in turn can constrainglobally averaged and possible regional variations in properties of the scatteringlayer such as intrinsic attenuation, velocity structure, scatterer size distribution andlayer thickness.50Chapter 3Modeling Seismic EnergyPropagation in Highly ScatteringEnvironments3.1 IntroductionThe study of seismic energy propagation in highly scattering bodies is a barelytapped field of planetary seismology research. Scattering of seismic energy oc-curs when coherent seismic wavefronts are randomly dispersed by a large num-ber of small-scale heterogeneities. In environments with high levels of scattering,this process can obscure the arrivals of waves other than the P- and S-waves andthus limit the use of traditional analysis techniques that make use of body wavearrival times. The best, and only known non-terrestrial, example of this processis observed in the lunar Apollo Passive Seismic Experiment (APSE, 1969-1977)data. The recorded seismic waveforms are characterized by very long durations,by emergent onsets and by the presence of very slowly decaying coda waves. Thesesignals result from the interaction of seismic energy with the various velocity, atten-uation and scattering structures within the Moon and as such have the potential toreveal information regarding the lunar interior that is not accessible through moreestablished seismological analyses. I showed in Chapter 2 that the manner in whichscattered seismic signals decay can be used to assess first-order properties of thelunar interior structure. My analyses of the APSE data indicated decay propertiesthat are consistent with a shallow scattering layer comprising frequency-dependentscatterer number-densities and scale-lengths. However, these interpretations wererestricted in part by the limitations of the APSE instruments (10-bit digitization513.1. Introductionand narrow bandwidth), but mostly by the lack of understanding of the processesgoverning seismic energy propagation in highly scattering media. Thus, in order tobetter appraise the interior structure of theMoon and other highly scattering bodies,we need to first understand the effects of varying velocity, attenuation and scatter-ing structures on seismic signal decay characteristics. In this paper, I address thefollowing general questions: What impact does the seismic wave velocity profilehave on the signals’ coda (e.g. presence of a crust and of a very low velocity layer,analogous to the surface regolith)? Can the decay properties of the seismic signalsbe used to assess the various scale-lengths involved in the scattering process (e.g.,scattering layer thickness, scale-lengths of scatterers)? Do deep structures like asmall core have an impact on the coda wave trains? What happens to the seismiccoda when scattering is not restricted to a near-surface highly scattering layer?I present results from modeling of seismic coda in a highly scattering me-dia, using a modified version of the Monte Carlo simulation method presented inShearer and Earle [2004], in which a large number of seismic phonons are trackedas they travel through a planetary interior. The phonon method was first used tomodel the envelopes of seismic waves of local earthquakes with epicentral dis-tances of 30–50 km [Gusev and Abubakirov, 1987]. These quakes exhibit scat-tering characteristics that are much weaker than, but qualitatively similar to thoseobserved in lunar seismic events. Adapted phonon methods were also used to in-vestigate depth dependent attenuation of coda waves (e.g. Hoshiba [1994]), tostudy the effect of increasing velocities with depth on the coda shape (e.g. Yoshi-moto [2000]), to study the partitioning of wave energy in multiple scattering events(e.g. Margerin et al. [2000]), as well as to model the global short period terrestrialwavefield (i.e. deep Earth scattering, see Shearer and Earle [2004]). However,this method has not been used to simulate and investigate the effects of highlyscattering environments such as the Moon on seismic signals, which is the mainmotivation behind this study. Our chosen modeling method effectively addressesthe problem outlined in Nakamura [1977] regarding the synthesizing of seismicsignals in highly scattering environments by providing a method to model both thediffusivity of seismic energy (high scattering levels) and simple body wave energytransmission.I have improved upon the Shearer and Earle [2004] method in two ways that523.1. Introductionare important for highly scattering environments: (i) 2.5-D ray tracing is usedfor scattered phonons, rather than pure two-dimensional (2-D) ray-tracing, withprobabilistic three-dimensional (3-D) heterogeneities. This is important for propercharacterization of the scattering. (ii) Predetermined velocity and density hetero-geneities can be incorporated directly in the model, in contrast to only stochasticperturbations. This can better characterize site effects due to scattering near thereceiver when waveforms from all events undergo similar scattering from the samenear-receiver structure.An important aspect of this method that differs from previous modeling work(e.g. Sato and Fehler [1998]; Przybilla et al. [2009]) is that scatterers are modeledas interfaces, and not as points or small volumes. The orientation of each scatterer,as well as the impedance contrast at the interface, determines by how much thephonons are deviated from their original path (i.e., by how much they are scat-tered). This method allows us to model the effect of a wide range of impedancecontrasts. Other established techniques for modeling scattering (e.g. using theBorn approximation) require that the velocity contrasts at the scatterers relative tothe background surrounding velocity are small [Sato and Fehler, 1998]. Scatteringof seismic energy on the Moon mostly takes place in the near-surface scatteringlayer, known as the megaregolith. This layer is the product of billions of years ofmeteorite impacts on the lunar surface [Ho¨rz, 1991]. In that layer, scattering occurspredominantly either at (i) the welded contacts between adjacent ejecta blocks, (ii)the interfaces between blocks of different lithologies (e.g. upper crust vs. uppermantle, major faults bounding large craters and impact basins), (iii) when seismicenergy encounters high porosity where interfaces between blocks are not welded.High porosity in the lunar uppermost crust is supported by GRAIL data and therecent work of Wieczorek et al. [2013]. In this context, the impedance contrastscaused by porosity or by adjacent varying lithologies are not always weak.Our phononmethod, dubbed PHONON1D because of its use of one-dimensional(1-D) velocity profiles, is described in detail in Section 3.2. I have benchmarkedthe code against the TTBox [Knapmeyer, 2004] and the CRFL [Fuchs, 1968; Fuchsand Mu¨ller, 1971] packages for travel times, as well as against the CRFL packagefor amplitudes for synthetics computed in non-scattering environments (see Sec-tion 3.3).533.2. MethodologyThe goal of this work is not to produce synthetics that match the scatteredproperties of the APSE data. Our purpose is to build an understanding of the gen-eral consequences of interior structures of highly scattering bodies, like the Moon,on surface seismograms. Accordingly, I used the PHONON1D method to gener-ate synthetic seismograms from shallow and deep sources in 16 different interiormodels with varying velocity, attenuation and scattering properties. The resultinghighly scattered waveforms can be characterized by their decay times (⌧d) and thefrequency-dependent equivalent decay factor (Qc). I describe in Section 3.4 theprocessing steps I followed to analyze the decay properties of the synthetic signals.I then present and compare examples of synthetic traces and decay properties forall the different models, and I follow with discussions on the effect of each modelparameters (Section 3.5).3.2 MethodologyIn the PHONON1D method, background P- and S-wave velocities (vp and vs),density (⇢), scattering probabilities (psc) and intrinsic attenuations (Qi) are pre-scribed from 1-D (i.e., depth-dependent) models. The phonons travel in a 2-D disk(i.e., a sagittal plane), and at any given time a phonon’s position is specified by itsdepth (z) and epicentral distance (). In order to capture the 3-D nature of scatter-ing, randomly oriented scatterers can scatter phonons out of the plane of the 2-Ddisk by altering their azimuthal angle (). In this case, only the projection of thephonons’ trajectories onto the 2-D disk are recorded. This method assumes lon-gitudinal symmetry of the background models and the statistics of the scattering,but allows us to model longitudinally scattered energy while only needing to trackphonons on a 2-D disk.543.2. Methodology3.2.1 Model GeometryThe phonons travel in a disk of radius RM , where RM is the radius of the plan-etary body under study. I use lunar-like dimensions (radius RM = 1737 km) inthis work, mostly to save on computational requirements (the larger the body, thegreater the number of phonons that have to be released), but also because the lunarseismic dataset is to date the only one showing evidence of such high scatteringintensity. Each phonon is released from a source located at  = 0, at a specifiedsource depth zs. At each time step, the phonon’s depth (z), epicentral distance (),and out-of-the-disk azimuth () are recorded.  is initially set to 0 and changesonly through the phonon’s interaction with a scatterer. Such interactions can addan out-of-the-plane (longitudinal) component to the phonon’s trajectory. In thatcase, for each time step, I project the phonon’s path increment back onto the trans-mission plane such that only the colatitudinal (d) and radial (dz) components ofdisplacement are recorded. For example, a phonon with an instantaneous azimuthof 90 (or 270) has no colatitudinal displacement (i.e. it remains at a constant over that time step, and until it is scattered again). This approach is justified bythe assumption of longitudinal symmetry. In this context, any phonon that leavesthe transmission plane ( 6= 0) is assumed to be replaced by a phonon enteringthe plane from the opposite direction. Thus, any phonon with a non-zero azimuthcan be considered equivalently as scattered energy that enters or exits the plane.This aspect of the model is required to capture the 3-D nature of scattering. Non-scattered phonons have a constant  = 0 and they propagate purely in the trans-mission disk (2-D propagation).The model geometry is illustrated in Figures 3.1and 3.2.In a sphere, out-of-plane energy will sample velocities at a different radius thanenergy projected back into the plane of the 2-D disk. This is important if the energytravels a large distance out of the plane, and/or if the curvature of the spherical shellat that radius/depth is large, i.e., the difference between the projected in-plane andthe out-of-plane radii and hence the difference in velocities increases with depth.However, one can minimize this effect by only using small time steps (short dis-tances) before re-evaluating the phonon’s velocity. In this case, the phonon willtravel a maximum distance of 10 km out of the plane (and typically only tens to553.2. Methodology1-D Models}0 102 10-5 14 6 820040020040060080010001200140016005000100000 102 10-5 14 6 8 5000 10000Depth (km)Top 40 km[km/s] [g/cm3] [-]Low background scatteringScattering layerModified VPREMOONFigure 3.1: See caption on following page.563.2. MethodologyFigure 3.1: Example of a 1-D layered model: P- and S-wave velocities (vp, vs),density (⇢), intrinsic attenuation for 1 Hz waves (Qi(1 Hz)), and the scatteringprobability (psc). The modified VPREMOON model shown here combines thecrustal and mantle velocities from Garcia et al. [2011] with the core velocities fromWeber et al. [2011]. This example model has a 10 km thick scattering layer (psc =1 near surface) and low overall background scattering (psc = 105 elsewhere).The top plot shows an expanded view of the top 40 km.hundreds of meters given the distribution of distances between scatterers) before Ireassess its radial position and the corresponding velocity. This results in a max-imum error in radius of 10 km, if the phonon travels directly perpendicular to thetransmission disk ( = 90), and very close to the core. This error decreases dra-matically at shallower depths, and is negligible at depths throughout the lunar crustand mantle. In this work, most of the scattering occurs within the megaregolithlayer, and for models that include background scattering beneath the main scatter-ing layer, the background scattering is limited to depths shallower than 1000 km.3.2.2 Velocity ProfilesThe velocity profiles are specified as individual layers of up to 10 km thickness,separated by interfaces at which the layer properties (wave velocities, intrinsic at-tenuations, density, scattering probability) are set. The properties of each layers aredefined at the overlying interface. Layers of 0 km thickness define sharp bound-aries (e.g., crust-mantle and core-mantle boundaries).I applied the exact Earth flattening transformation to each model interface[Bhattacharya, 1996, 2005]: r/RM = ezf/RM (3.1)vp = (r/RM )v?p, vs = (r/RM )v?s (3.2)⇢ = (RM/r)p2⇢? (3.3)where r is the distance from the center of the planetary body to the interface, RMis the radius of the planetary body, zf is the transformed depth coordinate, v?p andv?s are the original P- and S-wave velocities, vp and vs are the transformed P- and573.2. Methodology2-D Phonon Transmission DiskNSourcelocation1737.1 kmFigure 3.2: Phonons travel in a 2-D disk of radius RM (RM = 1737.1 km).The phonons’ depth (z) and epicentral distance () are tracked at each time step.Phonons that reach the surface are recorded by receivers (blue triangles) spacedat regular intervals along the surface away from the source (located at  = 0).Phonons are allowed to travel to distances greater than 180. They are howeverreflected back into the 0-180 half-disk if they reach the surface.583.2. MethodologyS-wave velocities, ⇢? and ⇢ are the original and transformed densities, and p is anarbitrary constant that varies with wave type [Bhattacharya, 1972, 1996]. In thiswork, p is set to 2, as in Bhattacharya [2005].The flattening transformation results in a singularity at the core (r = 0 km).To avoid numerical error during the calculation of the phonon path, I assume thatany phonon that travels to within 0.1 km of the center of the planetary body travelsthrough the center point, and I adjust the travel time (dt = 0.2/vcore, where vcoreis the seismic wave speed in the core in km/s) and epicentral distance (d = ⇡R)accordingly.3.2.3 Seismic SourceI use a modified simple analytical two-sided pulse [Dahlen, 2005] as a source func-tion for quakes (source depth zs > 0 km), defined as:m(t) = 4⇡2P20 (t/3 P0/2)e2⇡2(t/3P01/2)2 , (3.4)where m is the source function, t is time, P0 = 4t and t is the signal sam-pling interval. The equation was modified from the original in order to introducemore low frequency energy. The focal mechanisms of non-terrestrial quakes arestill poorly understood, and so I use a circular radiation pattern in this modelingwork. However, the radiation pattern could be adjusted to fit any particular fo-cal source. For a circular radiation pattern, the release angle is a uniform randomnumber between ⇡/2 and ⇡/2, where 0 is horizontal, ⇡/2 is upward, and ⇡/2is downward. The phonon’s polarity at the source is randomly determined basedon a energy partioning ratio of 1:10:10, for P-, SV- and SH-waves respectively[Boatwright and Fletcher, 1984]. In the case of impact events (zs = 0 km), thesource function is a delta function with release angles randomly selected between0 and ⇡/2 (downward direction only). Only P-wave energy is released for suchevents, i.e., the energy partitioning ratio is set to 1:0:0. The initial ray parameter(p) is set at this stage, based on the release angle, polarization and correspondingvelocity at the source depth. If the scattering probability (psc) is set to 0, then premains unchanged as the phonon travels throughout the Moon. p will change ifthe phonon is scattered by a non-horizontal scatterer. Sources used for the synthet-593.2. Methodology0 0.80.60.40.200.51-100.5-0.510 5 10 15 20−34−33−32−31−30−29−28Power (dB)−120−100−80−60−40−20Power (dB)Amplitude (DU)Amplitude (DU)Power SpectrumSource FunctionFrequency (Hz)Time (s)Surface impactsQuakesFigure 3.3: Source functions, in digital units (DU), used in the modeling for bothimpact events (top) and quakes (bottom). The right plots are the correspondingpower spectra. A sampling frequency of 40 Hz was used throughout.ics shown in the modeling section (Section 3.4) are presented in Figure 3.3, alongwith the corresponding power spectra. I used a sampling frequency of 40 Hz (i.e.t =0.025 s) for all synthetic signals presented in this article.3.2.4 Scattering LayersIf a phonon reaches a layer in which the scattering probability is greater than 0(psc > 0), it will travel within that layer, from scatterer to scatterer, until it reachesthe next velocity interface or until it is sent back to the initial interface if it is back-scattered. The distance between scatterers (sc) can be set to a constant value (e.g., Iuse 10 km for global background scattering), or can be stochastically selected froma power-law probability distribution (nsc ) where small inter-scatterer distanceshave a higher probability of occurrence than large distances (as in Figure 3.4). Apower-law distribution for megaregolith blocks is motivated by the heterogeneitylength-scales generated by impactors of different sizes (e.g. O’Keefe and Ahrens603.2. Methodology0 2 4 6 8 1000.020.040.060.080. 1ProbabilityScatterer Length-Scale Probability Distribution  [km]Figure 3.4: Scatterer length-scale probability distribution. The distance a phononwill travel between two consecutive scatterers is randomly picked from the powerlaw distribution plotted above, such that a phonon will travel a greater number ofshort distances (dsc(min)) than large distances (dsc(max)).[1987]). In this work n is always equal to 0.5. Further investigations could studythe effect of various n values, or different length-scale probability distributions(e.g. uniform distribution, only one sc values, etc.)Once a phonon reaches a scatterer, it is scattered, or not, based on the scat-tering probability. Let rx be a random number between 0 and 1 taken from anuniform distribution, where different subscripts, x, indicate different random num-bers as described below. If r1 > psc, the phonon is not scattered; if r1 < psc,scattering occurs. In that case, a random scatterer orientation and velocity/densityperturbations are generated and the phonon will be reflected or transmitted with aprobability based on the reflection/transmission coefficients as for conventional raytheory at a solid-solid interface (Figure 3.5B, and section 3.2.5). The orientationof the scatterer (dip between 0 and ⇡/2, where 0 is locally horizontal and ⇡/2 isvertical, and strike between 0 and 2⇡), as well as the magnitude of the perturba-tions (±r2vp/s and ±r2⇢, where v and ⇢ are the maximum change in velocityand density at the scattering interface, see Figure 3.5B) will define the orientation613.2. Methodologyof the scattered phonon via its incident and azimuth angles. In my models theorientations of the scatterers are selected such that the vectors normal to the scat-tering interfaces are uniformly distributed (strike is defined as r32⇡, and the dip assin1 r4). Also, the maximum v and ⇢ perturbation levels are kept equal and isreferred to as v,⇢. Once the phonon has interacted with the scatterer, it travels anew random distance sc to the next scatterer. If sc is larger than the distance be-tween the phonon and the next velocity interface, the phonon travels directly to thevelocity interface. Because of the 3-D geometry of the scatterers, and dependingon the phonon’s incident and azimuthal angles, scattering can convert the phonon’spolarity to P-, SV- or SH- energy (e.g., Figure 3.5B). The transmission and reflec-tion coefficients are calculated with the same equations that govern the behaviourof phonons at regular interfaces (next section).In the modeling work presented in Section 3.4, the scattering probability in thescattering layer (pSL) is always set to 1, such that phonons will always interactwith the scatterers. In this case, the intensity and overall direction of scatteringare controlled by the impedance contrast at the scatterer (i.e. the differences invelocity and density between the scatterers and the background values, (v,⇢). Asthe average impedance contrast approaches 0 (r2v,⇢ ! 0), most phonons willbe scattered forward (transmission is favored over reflection) with only a smalldeviation from their original path, regardless of the orientation of the scatterers.Scattering outside of the near-surface scattering layer can be set by setting theglobal scattering probability (pG) to a non-zero value. Global scattering is limitedto depths shallower than 1000 km in the models presented in section 3.4.3.2.5 Regular Velocity InterfacesWhen a phonon reaches an internal boundary, the likelihood that it reflects or trans-mits in a particular polarization is proportional to the square of the amplitude re-flection or transmission coefficient normalized by the sum of the squares of allcoefficients. A random number is generated to determine the state of the scatteredphase. The phonon will then travel to the next interface if the layer’s scatteringprobability is 0 (psc = 0), or to the next scatterer, if psc > 0.The coefficients are calculated based on the incoming phonon’s incident angle623.2.MethodologySurface receiverRay pathPotential Transmissions/ReflectionsScattering layerVelocity/density anomaliesRegular interfaceScattererDistance between scatterers ( )LEGEND1 km (not to scale)01011.927.9Depth (km)UPPER CRUSTLOWER CRUSTMANTLESURFACE HITS ARE RECORDEDSURFACE HITS ARE NOT RECORDEDA) C)SCATTERING LAYER3D ScatteringB)NExample of scattering stepsi) A phonon, with , moves toward a 3-D scatterer:ii) The phonon reflects off the scatterer.  becomes greater than :iii) The phonon travels a distance  to the next scatterer (red), but only the projected distance on the transmission plane is recorded (black). The total incremental time is recorded: CROSS-SECTIONPLAN VIEWAny phonon that  leaves the transmission plane  is assumed to be replaced by a similar phonon entering the plane from the opposite direction (longitudinal symmetry).Figure 3.5: A) Example of a scattered ray path in a 10 km thick surface scattering layer.The red line shows the path ofthe phonon as it enters and leaves the scattering layers. The white dots represent the 3-D scatterers. Inset shows angle which is non-zero when the phonon has an out-of-plane component. B) Randomly oriented 3-D scatterers, representedhere as interfaces with different impedance values from the background values. The probability that the incident ray willbe reflected or transmitted, as P, SV or SH energy, is proportional to the square of the reflection/transmission coefficient.Due to the 3-D nature of the scatterer, the phonon can acquire an out-of-the-plane component ( 6= 0, not shown here). C)Example scattering steps showing a change in the azimuthal angle ().633.2. Methodologyand polarity. Coefficients for solid-solid interfaces are based on Aki and Richards[2002], whereas those for the free-surface reflections and solid-liquid interfacescome from Ben-Menahem and Singh [2000]. The explicit form for the coefficientsis given in Appendix A. Benchmarking of wave amplitude with the CRFL package(Section 3.3) showed that results are better when energy is conserved rather thanamplitude at the interfaces. Hence, I use the square of the amplitude coefficients,normalized by the sum of the squares.All coefficients acquire an imaginary component if any of the transmission orreflection angles are supercritical (v(p,s)p > 1, where p is the ray parameter). Inthis case, I set the coefficient(s) with the supercritical angle to 0 (based on Ben-Menahem and Singh [2000]) and use the moduli of the other complex coefficients.3.2.6 Surface RecordersSeismometers are distributed at specific intervals along the surface and record theradial, transverse, and vertical ground displacements each time a phonon comeswithin a prescribed distance, set to 1 km in this work, from a receiver. I correctfor time if the phonon does not hit the surface directly beneath the receiver (Fig-ure 3.6). The applied correction, t, is:t = Lv = x sin ✓v = px, (3.5)where L is the additional distance the phonon would have travelled if it had hitthe surface directly beneath the receiver, v is the wave speed, x is the distancebetween the receiver and the arrival position of the phonon on the surface, ✓ is theray incident angle, and p is the ray parameter. t is negative when the phononhits the surface before the receiver (Figure 3.6A) and positive when it hits thesurface past the receiver (Figure 3.6B). The resulting t is also used to calculatethe correction to the phonon’s attenuation.643.2. MethodologyRay pathEquivalentray path1 km1 kmSurfaceWave frontAEquivalentray pathWave frontRay path1 km1 kmBFigure 3.6: All phonons that pass within 1 km horizontally (A) or vertically (B) of areceiver before hitting the surface are recorded, whether they hit the surface within1 km of the receiver (A), or not (B). In all cases, a time correction (t) is applied.I assume that the phonon’s path (red arrow) is perpendicular to the correspondingplanar wavefront. This phonon travels a distance that is different by L km fromthe distance travelled by a similar phonon (gray arrow) hitting the surface directlyunderneath the receiver at a velocity of v km/s. The resulting time difference (t)is equal to the horizontal slowness (p) multiplied by the distance difference at thesurface (x) and must be subtracted or added from the arrival time of the phononthat reaches the surface away from the receiver.653.3. Modeling Method Benchmarking3.2.7 Intrinsic AttenuationI calculate the attenuation of the phonon’s amplitude as follows:A(!) = A0e!t⇤ , (3.6)t⇤ = X tQi(z) (3.7)where A is the recorded amplitude at time t, A0 is the phonon’s initial amplitude(typically 1 digital unit (DU)), ! is the angular frequency, and Qi(z) is the depth-dependent intrinsic attenuation factor. I track each phonon’s t?, which is the sumof all time increments travelled divided by the attenuation factor (Qi) at the depthat which a given travel time increment was spent. For each time step, I recordthe incremental time spent in a layer with a given Qi, and t⇤ is the sum of theseincremental attenuations.3.3 Modeling Method BenchmarkingAs there are no standard numerical benchmarks against which to compare my scat-tering phonon code, I benchmarked the travel times and amplitude of syntheticsignals generated in non-scattering environments. The assumption is that synthet-ics generated in highly scattering media will have coda with realistic rise and de-cay characteristics if the amplitudes and times for non-scattered signals have beenshown to be appropriate.I benchmarked the PHONON1D code against the TTBox package [Knapmeyer,2004] for travel times, and against the reflectivity CRFL code [Fuchs, 1968; Fuchsand Mu¨ller, 1971] for both travel times and wave amplitudes. The PHONON1Dsynthetics match the TTBox and CRFL travel times, for both terrestrial (Figure 3.7A)and lunar (Figure 3.7B) models. Wave amplitudes (Figure 3.7B and 3.7C) com-pare well for direct waves, but small discrepancies between synthetics from thePHONON1D and CRFL codes are seen at larger times, after the seismic energyhas undergone multiple reflections and refractions. I briefly explain below whythese discrepancies arise and why the benchmarking results indicate that this codeis adequate to be used for its intended purpose of computing synthetics in highly663.4. Modeling Approachscattering media.The reflectivity method allows a quick estimation of the reflected and trans-mitted wave field as a function of frequency, ray parameters and distances [Fuchs,1968]. The CRFL code was constructed using wave theory in layered media, whichis intrinsically different from the PHONON1D approach, itself based on general-ized ray theory. For example, CRFL computes all wave phase shifts at interfaces,whereas PHONON1D only tracks the 90 shifts occurring at caustic points (e.g., Ido not track diffuse waves). I chose to not implement extensive phase shift trackingin order to keep the computation costs to a minimum. Nonetheless, I have optedto benchmark my code with CRFL because it allows us to compare synthetics athigher frequencies than other codes easily allow (e.g. SPECFEM, MINEOS). It isthus reasonable to expect an imperfect match between synthetics from both codes,especially for more complicated velocity profiles (e.g. terrestrial profiles vs. thesesimple lunar profiles). The criteria I used to decide if benchmarking was satisfac-tory were: (i) good visual fit of travel times, and (ii) good visual fit of primary waveamplitudes. All benchmarking runs met these two criteria.The PHONON1D code was not built to study seismic waves in a non-scatteringenvironment. CRFL, or other packages, are better suited to do so. The code’s mainpurpose is to study the propagation of seismic energy in highly scattering media atlarger frequencies (up to ⇠ 10 Hz in this work). In such an environment, most ofthe non-primary wave types (PP, SS, SP, PcS, and so on) will be very low amplitudeand are hidden by the P- and S-wave scattered coda, as observed in the APSE data.3.4 Modeling ApproachUsing the PHONON1D code, I computed synthetic signals in 16 distinct highlyscattering planetary interiors. I present this modeling work and the results inthis section. I focused my investigations on 7 different model parameters: spe-cific 1-D velocity and density profiles, the thickness of the near-surface scatteringlayer (TSL), the maximum velocity and density perturbation levels at each scat-terer (v,⇢), the reference intrinsic attenuation (Qi(1 Hz)), the intrinsic attenuationfrequency dependence (Qi(f)), the probability of low-level global scattering (pG),as well as the minimum and maximum scatterer length-scales (sc). Most of the673.4. Modeling ApproachFigure 3.7: See caption on following page.683.4. Modeling ApproachFigure 3.7: A) PHONON1D synthetics (blue) vs. TTBox travel times (black), fora PREM model, with a 100 km depth point-source. The logarithmic of the waveamplitudes is plotted in order to show weaker wave arrivals at later times. I onlyshow travel times that TTBOX can compute (i.e. not all multiple reflection phasesat later times are identified). B) Comparisons between PHONON1D traces (blue)and CRFL traces (red), showing the transverse component (no P-wave energy) inthe 0.10-0.15 Hz frequency band. This is for a simple two-layered Moon, with a700 km depth point-source. Black lines are TTBOX arrival times. C) Comparisonsbetween PHONON1D traces (blue) and CRFL traces (red), for a PREM model,with a 100 km depth source, showing the vertical component in the 0.10-0.15 Hzfrequency band. The PREM model has many interfaces and the discrepancies be-tween the two codes get larger at larger times (e.g. 160). Note for that both B andC, CRFL plots on top of the phonon code traces, so the absence of blue means agood fit.velocity profiles used are very simple, the majority having constant P- and S-wavevelocities throughout the body. Some of these velocity profiles are somewhat un-realistic, but were chosen for their simplicity and to allow us to easily investigatethe individual effects of each model parameter on the seismic coda.The parameters for all models are outlined in Table 3.1. Figures 3.1 and 3.8show the five 1-D velocity and density profiles used. Model A is defined as thebaseline model, and all other models implement simple variations from it. Othermodels have been grouped using sequential model names (e.g. C1, C2 and C3)in order to indicate which models are used to demonstrate the effect of a specificvelocity, attenuation or scattering structure on the coda. Note that the backgroundintrinsic attenuation factor, Qi(f), is high when compared to terrestrial values (i.e.much lower attenuation levels), but this is necessary to obtain synthetics with longscattered coda. Such high Qi values can be expected in very dry planetary environ-ments (e.g. Latham et al. [1970a], Nakamura et al. [1976]). Qi(f) is kept constantat 2000, 4500 or 6500 in most models, but increase with increasing frequencies inmodels C3 and F (Figure 3.9).I computed synthetics for events at three distinct depths: surface events (zs = 0 km),shallow events (zs = 30 km), and deep events (zs = 1000 km). These are analo-gous to surface impacts, lunar shallow quakes and lunar deep quakes, respectively.The resulting decay properties of the shallow events are very similar to these of the693.4. Modeling ApproachTable 3.1: Models investigated in this study. TSL is the scattering layer thickness,v,⇢ is the maximum velocity and density perturbation level at each scatterer, andQi(1 Hz) is the reference intrinsic attenuation. The different Qi(f) styles are plot-ted on Figure 3.9. pG is the low-level global scattering probability, and sc sets theminimum and maximum scatterer length-scales, as defined in Figure 3.4. ModelA is the baseline model, with constant P- and S-wave velocities, a scattering layerthickness of 30 km, maximum velocity and density perturbations at the scatterersof ±0.75 %. For model C3, Qi = Qi(f, z) and Qi(1 Hz) is depth-dependent andis plotted on Figure 3.1.1-D Velocity TSL v,⇢ Qi(1 Hz) Qi(f) pG sc [km]Model Model [km] [%] [-] style [%] min. max.A Constant Velocity 30 0.75 4500 1 0 0.05 10B Constant Velocity 15 0.75 4500 1 0 0.025 5C1 Crust 30 0.75 4500 1 0 0.05 10C2 VLVL 30 0.75 4500 1 0 0.05 10C3 VPREMOON 30 0.75 Qi(z) 2 0 0.05 10D1 Constant Velocity 5 0.25 4500 1 0 0.05 10D2 Constant Velocity 5 0.75 4500 1 0 0.05 10D3 Constant Velocity 5 0.95 4500 1 0 0.05 10D4 Constant Velocity 60 0.25 4500 1 0 0.05 10D5 Constant Velocity 60 0.75 4500 1 0 0.05 10D6 Constant Velocity 60 0.95 4500 1 0 0.05 10E Solid Core 30 0.75 4500 1 0 0.05 10F Constant Velocity 30 0.75 4500 2 0 0.05 10G1 Constant Velocity 30 0.75 2000 1 0 0.05 10G2 Constant Velocity 30 0.75 6500 1 0 0.05 10H Constant Velocity 30 0.75 4500 1 0.01 0.05 10703.4. Modeling Approachsurface impacts. This is a result of the sources being located in, or very near thesurface scattering layer in both cases. Thus, I only present the synthetic traces anddecay analysis results from the surface impacts and deep events. As mentionedearlier, I used lunar-like dimensions for this particular work (planetary radius of1737 km). However, this method could be applied to much larger (i.e. Mars) orsmaller (e.g. asteroids) bodies.3.4.1 Processing of SyntheticsI used a method similar to the one described in Chapter 2 to measure the character-istic decay time (⌧d) and decay factor (Qi) of the synthetic signals. All syntheticdata were bandpass filtered to allow the measurement of the decay properties inspecific frequency bands. For this study, I investigated decay properties in five2-Hz wide bands centered on 1, 3, 5, 7 and 9 Hz. I then computed the signals’envelope functions and smoothed them in order to reduce the contribution of noisespikes in the decay fits. Smoothing was done by using a 35 second running win-dow, keeping the 75th percentile. The decaying part of the smoothed curves werefitted in a least-squares sense with an exponential decay curve of the form et/⌧d ,and ⌧d was retrieved (See Figure 2.3). The coda wave decay factor Qc was thencalculated using: Qc = 2⇡fc⌧d (3.8)where fc is the central frequency of the band of interest. Qc is useful to comparedecay properties in different frequency bands. Its inverse, Q1c , represents thefraction of energy dissipated after one period of oscillation. In a closed systemwith uniformly distributed energy, the signal amplitude decay factor Qc should becontrolled by, and equal to, the intrinsic attenuation factor Qi. Qc values that areless or more than the local Qi suggest leaking or focusing of seismic energy awayfrom or toward the receiver.The beginning of the fits were determined automatically as the time at whichthe energy starts to decay exponentially after the theoretical S-wave arrival time. Insome models with lower levels of scattering not all the energy from the impulsivewave arrivals is converted into the scattered coda. Fitting only the exponentially713.4. Modeling Approach02004006008001000120014001600Depth (km)Depth (km)010203040500 5 10/0 5 10/0 5 10/0 5 10km/s or g/cm3 km/s or g/cm3 km/s or g/cm3 km/s or g/cm30 5 10/0 5 10/0 5 10/0 5 10km/s or g/cm3 km/s or g/cm3 km/s or g/cm3 km/s or g/cm3Constant Velocity Crust VLVL Solid CoreFigure 3.8: Four of the five 1-D velocity and density profiles used in this study,showing the S-wave velocity (red), the P-wave velocity (blue) and the density(black). The modified VPREMOON profile used in C3 is shown in Figure 3.1.723.5. Results and Discussion0 2 4 6 8 100 2 4 6 8 10010002000300040005000600070000.511.52x 1040Frequency (Hz) Frequency (Hz)Style 1 Style 2Model G2Model G1Models C3 & FAll other modelsFigure 3.9: The two different Qi(f) styles used in the modeling work. Style 1 hasa constant Qi over the entire frequency range. Most models use Qi(f) = 4500,except for models G1 (Qi(f) = 2000) and G2 (Qi(f) = 6500). Style 2 hasincreasing Qi for increasing frequency: Qi(1 Hz) = 4500, up to Qi(9 Hz) =15300. Style 2 is used in models C3 and F.decaying part of the coda ensures that I do not fit the larger amplitude impulsivearrivals and retrieve artificially low decay times. It follows that the decay times anddecay factors discussed in the following sections are the S-wave coda decay timesand factors. All S-wave travel times were computed using the TTBOX package[Knapmeyer, 2004]. The ⌧d values for individual radial, vertical, and transversecomponents were similar in all cases and only the average ⌧d values from all threechannels are shown here. Fits were of an arbitrarily determined length of 2500s, and were only retained if ⌧d was less than 2500 s. Most fits fell within thatconstraint and only a small number of fits for models with high scattering levelswere dropped.3.5 Results and DiscussionFigure 3.10 presents examples of synthetic traces from all models, for surfaceimpacts and deep events, recorded at an epicentral distance of 50. All traceshave been normalized by their maximum amplitude for better visualization and arealigned on the theoretical P-wave arrival times. The thin red lines indicate the time733.5. Results and Discussionof the first ±107 DU hit, i.e., the first observable P-wave arrival. 107 DU is theminimum amplitude output by the code. Thus, the red line does not indicate thetheoretical P-wave arrival, but the time at which enough energy hits the receiverto become observable. In a non-scattering environment, this would coincide withthe theoretical P-wave arrival time (Figure 3.7A). Increased levels of scattering candistribute wave energy over a long time, generating very emergent wave arrivals.This can impede precise wave arrival picks as seen in the traces for models C2,C3, D5 and D6, because the initial direct P-wave energy can be too weak to be ob-served. The uncertainty in picks would be even greater in the presence of ambientand instrument noise. The effect is greater for impact events than for deep sourcesbecause rays from impacts tend to hit the base of the scattering layer at a shallowerangle, and this energy thus travels a longer distance in the scattering layer beforereaching the receivers.The substantial visual differences among some of the traces are good initialindicators of the influence that the seismic velocities, scattering and attenuationenvironment each have on surface recordings. Traces from models with lowerscattering such as models D1, D2 and D3, overall have distinct P- and S-wavecodas. Traces from models with higher levels of scattering such as C3 and D6have traces that show a blended P- and S-wave coda. Another important differenceamong the traces is the apparent signal length. The total signal length of scatteredsignals is dependent on the initial energy released at the source and on how fastthis energy decays near the receivers. For all events I released the same numberof phonons (i.e., same total energy at the source), but some recordings, especiallythose from models with high scattering levels, show a much longer duration signal.I present an example of detailed results from the coda decay fits in Figure 3.11for models D4, E and F. I show ⌧d as a function of epicentral distance, for thefive investigated frequency bands, for both impacts (left) and deep events (right).All impact events exhibit a sharp rise in ⌧d values at short epicentral distances,tend to stabilize at larger distances, and may display another short rise at verylarge distances. Deep events show similar behavior but without the sharp rise ⌧dvalues at short epicentral distances. The effect of Qi(f) on the coda decays can beaddressed by observing ⌧d in different frequency bands. ⌧d clearly decreases withincreasing frequency for models with a constant Qi(f) (e.g., models D4 and E in743.5.ResultsandDiscussionFigure 3.10: See caption on following page.753.5. Results and DiscussionFigure 3.10: Examples of synthetic traces for all models presented in this work, forboth surface impacts and deep event, at  = 50. Note that amplitudes can varygreatly amongmodels, so the traces have been normalized such that their maximumamplitudes are equal. All traces are aligned on the theoretical P-wave arrival times.The top traces are from the baseline model (model A). For each other models (B toH), the the annotation on the right indicates the main difference(s) from the baselinemodel. A detailed description of all the models, including the symbol definitions,can be found in Table 3.1. The red lines indicate the time of the first 107 DU hitfor each traces, which is not the theoretical P-wave arrival time, but the first time atwhich enough phonons hit the receiver to trigger the minimum 107 DU amplitudesignal. The choice of label colors is arbitrary and designed only to accentuate thedifferent model groupings. Models with white labels are those that can be usedindividually to show the effect of a particular model parameter.Figure 3.11). On the other hand, models with an increasing Qi(f) (e.g., model F),shows a clustering of ⌧d values at high frequencies.Two decay regimes have been identified. Regime 1 (larger outlined circles inFigure 3.11) is characterized by overall shorter decay times, whereas Regime 2(smaller circles) has much longer decay times. These two regimes were identifiedin all models and so I first discuss them in more detail below. We shall see thatin practice Regime 1 is likely to be the regime that is relevant to seismogramsrecorded on a planetary surface, and so I then proceed to discuss the results forcoda decay times for Regime 1 from the suite of interior structure models studiedhere. I note that the level of scattering in model D1, with a 5 km scattering layer(TSL = 5 km) and lower maximum velocities and density perturbation levels, wastoo low to generate codas adequate for fitting and so no further results are shownfrom that model.3.5.1 Two Distinct Decay RegimesFigure 3.11 shows that the transition between Regimes 1 (shorter decay times) and2 (longer decay times) is gradual and occurs over a fairly large range of epicentraldistances. This range varies depending on the internal structure model and on thefrequency band. High frequency bands transition to Regime 2 at smaller epicentraldistances than lower frequencies. In fact, in most of the models under investigation,763.5. Results and Discussion Epicentral distance (˚)  Epicentral distance (˚)Surface impacts Deep events0 20 40 60 80 100 120 140 160 1800 20 40 60 80 100 120 140 160 180 (s)101102103104101102103104Fit length(2500s)0 - 2 Hz 2 - 4 Hz 4 - 6 Hz 6 - 8 Hz 8 - 10 HzLEGEND6 - 8 Hz 8 - 10 HzRegime 20 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 (s)1011021031041011021031040 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 (s)101102103104101102103104Effect ofsolid coreEffect ofsolid coreModel D4SOLID 300km COREModel EIncreasingModel FFigure 3.11: See caption on following page.773.5. Results and DiscussionFigure 3.11: A) PHONON1D synthetics (blue) vs. TTBox travel times (black), fora PREM model, with a 100 km depth point-source. The logarithmic of the waveamplitudes is plotted in order to show weaker wave arrivals at later times. I onlyshow travel times that TTBOX can compute (i.e. not all multiple reflection phasesat later times are identified). B) Comparisons between PHONON1D traces (blue)and CRFL traces (red), showing the transverse component (no P-wave energy) inthe 0.10-0.15 Hz frequency band. This is for a simple two-layered Moon, with a700 km depth point-source. Black lines are TTBOX arrival times. C) Comparisonsbetween PHONON1D traces (blue) and CRFL traces (red), for a PREM model,with a 100 km depth source, showing the vertical component in the 0.10-0.15 Hzfrequency band. The PREM model has many interfaces and the discrepancies be-tween the two codes get larger at larger times (e.g. 160). Note for that both B andC, CRFL plots on top of the phonon code traces, so the absence of blue means agood fit.scattered codas in the 8-10 Hz band are in Regime 2, at least at larger epicentraldistances. On the other hand, scattered codas in the 0-2 Hz band always remainin Regime 1. This frequency-dependent transition is demonstrated in the top rightplot of Figure 3.11 (model D4, deep events). The transition occurs first in the 8-10Hz band, starting at around 40 and ending around 80, and is followed by the 6-8Hz band with a transition between 90 and 140.Figure 3.12 presents synthetic traces and decay fits for traces in Regime 1, inthe transition zone, and in Regime 2. I also show the change in ⌧d and maximum S-coda amplitude values as a function of epicentral distance. The synthetic traces, ⌧dand amplitude values were measured in the 6-8 Hz band of a model D6 deep event.It is evident that Regime 1 is characterized by much larger amplitudes than Regime2. The transition between the two regimes occur when the change in maximum S-coda amplitude as a function of epicentral becomes very small, i.e. the maximumS-coda amplitude reaches a near-constant low value for the remaining epicentraldistances.The differences between decays in Regime 1 and 2 are clearly expressed in Fig-ure 3.13 which shows histograms of the averaged ⌧d, maximum S-coda amplitudesand Qc values for impacts and deep events. The average ⌧d values are calculatedover the distance range in which the ⌧d values are approximately constant for ei-ther Regime 1 or Regime 2. Results from all frequency bands are included, from783.5. Results and DiscussionFigure 3.12: Traces and decay times for a deep source event in model D6, showingthe transition between regimes 1 and 2. Only traces and results form the 6-8 Hzfrequency band are shown. Top 3 panels show the envelope functions (black), thesmoothed envelope functions (red) and the least-squares fits (blue) for traces atepicentral distances  = 12, 60 and 148. The lower panel shows the change in⌧d (blue) and maximum S-coda amplitude (red) values as a function of epicentraldistance.793.5. Results and Discussionall models with a constant Qi(f) = 4500. There is a distinct difference in themean decay time and mean S-coda amplitude values between the two regimes.Models with lower intrinsic attenuation levels will have less energy transitioningto Regime 2. For example codas from model F, which has an increasing Qi (lowerattenuation) for increasing frequency, have no decay in Regime 2 (Figure 3.11).The maximum amplitude coda in Regime 2 is close to the average signal’s codaamplitude, and typically there are only two characteristic decay times in the 2500 slong fits, as compared with ten or more in Regime 1. As such, the decay timesfor Regime 2 decays are less well estimated and in fact, Regime 2 may not bebest-estimated by an exponential decay. Further modeling work with longer traceswould be necessary to investigate this, which is beyond the scope of the workpresented here. However, it is still valid to state that at some distance exponen-tially decaying scattered signals (Regime 1) become so scattered that the recordedground deformations are similar to very slowly decaying and very low amplitudeseismic noise (Regime 2). This occurs at high frequencies first, because of thelarger number of small-scale scatterers than large-scale ones, and because high fre-quencies go through more cycles over same time period than low frequencies andare thus more attenuated. Regime 2 decays thus occur when most of the P- andS-wave impulsive signals have been reduced to very low amplitudes and very longduration source functions through intrinsic attenuation and scattering.In Figure 3.13 I also compare the measured Qc values with Qi (= 4500 for themodels shown). Qc values for Regime 1 decays are on average slightly less thanQi. I propose that in high scattering environments, energy reaching the receiver ispartially trapped locally. This is mainly due to the scattering layer reflecting theenergy back towards the receiver, effectively mimicking a nearly closed system.In this case, intrinsic attenuation is the main process by which energy levels arereduced. The fact that individual Qc values are slightly less or more than Qi im-plies that some leakage or focusing of energy occurs, due to the particular interiorstructure of each model. Still, I conclude that Qc values of Regime 1 decays aregood first-order indicators of intrinsic attenuation levels (i.e. good first order ap-proximation of the local Qi). Qc values from Regime 2, on the other hand, are alllarger than the models Qi(f), suggesting high frequency energy is focused nearthe surface. However, the low amplitudes and less well-constrained least-squares803.5. Results and Discussion048121620048121620 (s)Counts100 101 102 103 104 105 100 101 102 103 104 105Surface impacts Deep eventsRegime 1 Regime 2 (s)1314 40Mean std (%)234 69 202 731072 36Mean std (%)030402010030402010Counts106101 102 103 104 105 106101 102 103 104 1054500 45005.60e4 43Mean std (%)6.56e4 44Mean std (%)3788 37 3165 24030252015105030252015105Amplitude (DU)Counts10010-10 10-8 10-6 10-4 10-2Amplitude (DU)3.26e-5 162Mean std (%)5.05e-5 161Mean std (%)1.89e-3 227 1.95e-3 14610010-10 10-8 10-6 10-4 10-2Figure 3.13: Histograms showing the average ⌧d (top row), maximum S-coda am-plitudes (middle row) andQc values (bottom row) for Regimes 1 (blue) and 2 (red),and for both impact events (left) and deep events (right). The mean values for eachregime are indicated, along with the standard deviation given as a percentage of themean. Only models for which Qi(f) = 4500 are compiled here. Results from allfrequency bands are included. The dashed lines in the Qc plots show the Qi valueused. Qc results to the left of that line indicate that the codas decay faster than theywould if intrinsic attenuation was the only process by which energy was reduced.813.5. Results and Discussionfits prohibits any further interpretations of Regime 2.In the following sections, all interpretation regarding the effects of the velocityprofile, and attenuation and scattering structure on the coda decays uses the Regime1 decays. This is the regime likely to be measured with surface instruments giventhe much larger amplitudes.3.5.2 Effects of Interior Structure on Coda Decay TimesGiven the large amount of data, I characterize the ⌧d as a function of epicentral dis-tance and frequency by an average Regime 1 ⌧d calculated over the distance rangein which the ⌧d values are approximately constant. I summarize these average ⌧dvalues for all models and frequency bands in Figure 3.14. Models A, C1, C2 andC3 show the effect of 1-D velocity profiles that increasingly tend to trap seismicenergy in lower velocity layers near the surface and in the scattering layer. ModelsA and D2 to D6 illustrate the changes in coda decay for increasing levels of scat-tering (implemented using increasingly thicker scattering layers and by varying themaximum velocity and density perturbation levels). Models A, G1 and G3 empha-size the effect of reducing or increasing the reference intrinsic attenuation factorQi(1 Hz). Model B halves the scattering layer thickness (TSL = 15 km) and theminimum and maximum scatterer length scales, such that the seismic energy inter-acts with the same amount of scatterers, but in a thinner scattering layer. Model Eshows the effects of having a small solid core on the averaged ⌧d. Model F illus-trates the effect of a constant versus an increasing Qi with frequency (i.e. lowerintrinsic attenuation at high frequency). Finally, model H is used to show the effectof low level global scattering on the coda.Sharp rise in impacts ⌧d times at short distancesFigure 3.11 shows an increase in ⌧d values for impact events between 0 and ap-proximately 10. This increase is seen in all models, including those not presentedin Figure 3.11. Furthermore, a similar trend is observed in the APSE seismic lunardata (see Figure 2.5). The rate of increase, as well as the exact distance at which ⌧dbecomes near constant appear to be model dependent, suggesting that these valuescould be used to infer first-order near-surface interior structure properties such as823.5. Results and Discussion102103 (s)102103 (s)Surface impactsDeep eventsVelocity Profile Scattering intensity Individual modelsD2 D3 D4 D5 D6 B E F HAG1 G2AC1 C2 C3A AAPSE range0 - 2 Hz 2 - 4 Hz 4 - 6 Hz 6 - 8 Hz 8 - 10 HzLEGENDFigure 3.14: See caption on following page.833.5. Results and DiscussionFigure 3.14: Plots showing the average ⌧d values for each models and each fre-quency bands, for surface impacts (top), and deep events (bottom). Each symbolstand for a different frequency band and the error bars represent one standard de-viation about the mean. Only the ⌧d from Regime 1 are shown. In some model,⌧d were in Regime 2 for the entire range of epicentral distances and are thus notincluded here. The pink bars show the range of ⌧d present in the Apollo PassiveSeismic Experiment (APSE) dataset as comparison (see Chapter 2). Models havebeen grouped based on which model parameters they help investigate, and eachare plotted against the baseline model A. Models C1, C2 and C3 help identify theeffect of the velocity profiles that increasingly trap seismic energy near the surface.The scattering intensity (combining both TSL and v,⇢) is investigated with modelsD2 to D6. The effect of a lower and larger Qi(1 Hz) is shown with models G1and G2. Model B shows the effect of halving the scattering layer thickness andthe scatterer length scales (i.e. as many interaction with scatterers as in model A).Model E demonstrates the lack of effect that a small solid core has on the average⌧d values. Model F has increasing Qi values for increasing frequencies (Qi style2 compared to style 1 in model A). Model H shows the effect of a low scatteringlevel in the interior.the thickness of the scattering layer and scattering intensity. However, the receiversampling distance used in this modeling (2) is too large to assess precisely differ-ences between models. Further modeling work could investigate this near-sourceeffect, its potential as a geophysical tool and any specific application to the Apolloseismic data.I propose that the distance at which the sharp rise in decay times ends relatesto the distance at which the seismic energy traveling between the source and thereceivers stops interacting only with the scattering layer and starts to have a sub-stantial fraction of the travel path in the underlying non-scattering (or less highlyscattering) lower crust or mantle. Our model impact source function is an impul-sive delta function (⌧d ! 0). This initial impulse spreads out as it travels in thescattering layer. Each adjacent receiver is hit by an increasingly diffused sourcefunction that travelled primarily within the scattering layer, with closer receiversseeing a more impulsive arrival (shorter ⌧d) than the more distant ones (longer ⌧d).Distant receivers (> 10) all see a similar source wave function, that has left thescattering layer near the source and travelled in the mantle to reenter the scatteringlayer at all point near the receivers.843.5. Results and DiscussionVelocity profileThe velocity profile appears to be one of the parameters that has the biggest impacton the characteristics of the scattered codas, especially for impact events. Mod-els A, C1, C2 and C3 have velocity profiles that increasingly tend to trap or focusseismic energy near the surface and in the scattering layer. This results in in-creasingly large decay times, for example from ⌧d(1 Hz) = 380 s in model A to⌧d(1 Hz) = 1170 s in Model C3 for impact events. Benchmark model A has onlymantle velocities and scattering in the scattering layer is the only process by whichthe energy remains near the surface. Model C1 introduces a crust/mantle bound-ary, which will reflect some downgoing rays up into the scattering layer. ModelC2 not only has a crust, but also has a near-surface, thin, very-low-velocity layeranalogous to a regolith layer. This layer is particularly effective in trapping energyfrom impact events near the surface and in the scattering layer. Model C3 has acrust, a thin low velocity layer, and a more realistic velocity gradient (velocities in-crease with depth) which causes the rays to turn at depth, sending them back nearthe surface sooner than in models without a gradient.This shows that velocity profiles that are conducive to focusing seismic en-ergy near the surface and / or in the scattering layer can drastically increase thecoda decay times. I found that for all other attenuation and scattering parametersremaining equal, the velocity profile has the biggest impact on the decay times.Scattering layer thickness and impedance contrastModels A and D2 to D6 show the effect of increasing the scattering layer thicknessand the impedance contrast on the coda decays. The impedance contrast is incor-porated in the model as the maximum variations from the background density andvelocities at each scatterers (v,⇢). A small v,⇢ means that most phonons will hitscatterers with low velocity and density contrast and that most will be transmittedwith only small variations to their original path and so weak forward scattering ispreferred. On the other hand, a large v,⇢ will cause more phonons to be widelyscattered off path resulting in more isotropic scattering. Low impedance contrasts(e.g. v,⇢ = 0.25 %) may be analogous to interaction with annealed fractures orinterfaces between two similar lithologies, whereas large impedance contrasts (e.g.853.5. Results and Discussionv,⇢ = 0.95 %) may be analogous to high porosity levels. Models D2 and D3 havea 5 km thick scattering layer, with v,⇢ values of 0.75 and 0.95 % respectively.Models D4, D5 and D6 all have a much thicker scattering layer (TSL = 60 km) andv,⇢ equal to 0.25, 0.75 and 0.95 %. These are compared to the baseline model Athat has TSL = 30 km, v,⇢ = 0.75 %. All models used the same scatterer length-scale distribution. None of these models had a velocity profile that was conduciveto trapping the energy near the surface and I infer that this is why changing thescattering layer structure has a bigger effect for impact events than for deep events.⌧d(f) values for deep events tend to be within one standard deviation of the meanvalues of all other models and do not seem much affected by variations in TSL andv,⇢. In fact, most of the deep event energy will hit the scattering layer at a fairlysteep angle and spend less time overall in the layer than energy from impact events.For impact events both the scattering layer thickness and the average impedancecontrast affect the level of scattering and the resulting decay times. Models D3, Aand D5 all have v,⇢ set at 0.75 %, but have successively increasing scattering layerthicknesses. It follows that ⌧d values from model D5 (TSL = 60 km) are largerthan those from model A (TSL = 30 km) and from model D3 (TSL = 5 km),with ⌧d of 541, 380, and 370 s in the 0-2 Hz band, respectively. The same is trueof increasing impedance values: ⌧d values increase from models D4 to D6 whichall have a similar scattering layer thickness (421 s, 541 s and 583 s in the 0-2 Hzband). Note that ⌧d values for the 4 to 10 Hz frequency bands are not shown formodel D6, as the high levels of scattering caused these bands to be in Regime 2 atall epicentral distances.Both increasing impedance contrasts at the scatterers and an increasing scat-tering layer thickness will result in longer coda decay times. However, a highimpedance contrast does not necessarily generate long scattered coda if the scatter-ing layer is insufficiently thick because there will not be enough interactions withscatterers. An example is model D3, which has TSL = 5 km with a maximumimpedance contrast of 95 %.863.5. Results and DiscussionIntrinsic attenuationModels G1, A, and G2 show the effect of various levels of intrinsic attenuationwithQi(1Hz) = 2500, 4500 and 6000, respectively. This corresponds to decreasinglevels of intrinsic attenuation. In all models, Qi(f) is constant, resulting in highfrequencies that attenuate faster than lower frequencies, because they go throughmore cycles for similar time periods. Results show that decay times increase forincreasing Qi(1Hz), at all frequency levels. Decay times in the 0-2 Hz band forimpact events increase from 154 s in model G1, to 380 s in model A, and up to535 s in model G2. In addition the higher attenuation level of model G1 causesmore frequency bands to switch to regime 2 scattering and only the 0-2 and 2-4 Hzbands have codas in the Regime 1. The effect is similar for deep events. Overall,results show that an increase in Qi by a factor of 2 or 3 can yield similar increasein ⌧d, depending on the frequency.The behaviour of ⌧d(f) and Qc(f) can be used to assess the properties ofQi(f), as can be seen with the results frommodels A, C3 and F. Model A has a con-stant Qi(f), whereas models C3 and F have an increasing Qi(f), with Qi = 4500in the 0-2 Hz band as in model A, 7200 in the 2-4 Hz band, 9900 in the 4-6 Hzband, 12600 Hz in the 6-8 Hz band, and 15300 in the 8-10 Hz band. The attenu-ation differences are reflected in the decay times and factors. The ⌧d in model Adecreases as a function of frequency (as seen on Figures 3.14 and 3.15), the corre-sponding Qc values, however, tend to cluster, at least for frequencies above 2 Hz.The opposite is true for models C3 and F, where the ⌧d values cluster in the 2-10 Hzband, and where Qc(f) increases with frequency. This observation is particularlyimportant as a similar behaviour is seen in the Apollo lunar seismic data (clusteringof Qc values at low frequencies and clustering of ⌧d values at high frequencies, asshown in Chapter 2. This suggests a near constant Qi(f) for lower lunar seismicfrequencies (0.5-1.5 Hz) and an increasing Qi with increasing frequency for thehigher frequency range (2-10 Hz).Scatterer density and scattering timeModel B is similar to model A, except that the scattering layer thickness andthe minimum and maximum scatterer length-scales are halved. This means that873.5. Results and DiscussionF C3A F C3A102103103104 (s)0 - 2 Hz 2 - 4 Hz 4 - 6 Hz6 - 8 Hz 8 - 10 HzFigure 3.15: ⌧d (left) and Qc (right) for impact events from models A, F and C3.Model A has a constant Qi(f) (style 1 of Figure 3.9). Models F and C3 have anincreasingQi(f) with increasing frequencies (style 2 of Figure 3.9). Changes in ⌧dand Qc as a function of frequency is dependent on Qi(f). For model A, ⌧d valuesdecrease as a function of frequency, whereas for models F and C3, ⌧d values tendto cluster around a constant value (except for the 0-2 Hz band). The opposite istrue for Qc(f). Model A values, except for the 0-2 hz band, cluster around 1800,whereas Qc values increase with increasing frequency for models F and C3. Assuch, comparing ⌧d(f) and Qc(f) can be useful to assess the properties of Qi(f).883.5. Results and Discussionphonons in both models interact with the same amount of scatterers. In otherwords, the scatterer density in model B is twice that of model A, but the scat-tering layer thickness in B is half that of model A. One would expect similar decaytimes for both models, given that they interact with as many scatterers. However,⌧d values from model B are overall less than ⌧d from model A. This may be duefact that although the seismic energy in model B interacts with as many scatterersas the energy in model A, the path length between those scatterers will be half aslong, such that scattered energy will be delayed less relatively to the geometricpath than in model A. This leads to overall shorter decay times. One might expecta more drastic drop in decay times in model B (close to half of that of model A),but because the energy in model B spends less time in the scattering layer, it willalso be less attenuated, and this effect acts to increase the decay times.Presence of a solid coreThe presence of a small solid core (model E) has little effect on the average decaytimes for either surface impact or deep events (as seen in Figure 3.14). We observea slight decrease in ⌧d for the deep events when compared to model A values, butall ⌧d remain within one standard deviation of each other. This lack of effect on theaverage ⌧d is mostly due to the fact that most of the direct P- and S-wave energyleaving the sources does not actually interact with the core. The effect of the coredoes show up however at larger distances, where no direct P- or S-wave energyhits the surface (the core shadow zone, see middle plots of Figure 3.11 showingincreased ⌧d in shadow zone). Decay times in this region are much larger. Ourresults also indicate that maximum S-coda amplitudes at stations within the shadowzone are smaller by about 2 orders of magnitude than coda amplitudes at stationsjust adjacent to the shadow zone. I propose that in high scattering environments,the energy that makes it to surface recorders in the core shadow zone has leakedfrom the surrounding scattering layers. This scattered leaked energy dominates thesignal, in contrast to the other stations where energy from the direct P- and S-wavearrivals dominate.893.6. ConclusionsLow level mantle scatteringModel H introduces low level scattering in the planetary body’s interior. The scat-terer length-scale outside of the scattering layer was set to 10 km, with a scatteringprobability of 0.01. This means that a phonon interacts with a scatterer approx-imately every 1000 km of travel distance in the interior. v,⇢ was set to 0.75 %,similar to that in the scattering layer. Low level mantle scattering slightly increasesthe average ⌧d for both impact and deep events, as observed by comparing modelsA and H values in Figure 3.14. This increase in average ⌧d is caused by a slowlyincreasing ⌧d with increasing  (not shown here). This in turn is due to the factthat travelling phonons hitting the base of the scattering layer at large distances willhave been scattered more in the interior than those hitting the scattering layer nearcloser receivers. Another effect of interior scattering can observed by comparingtraces from model A and H in Figure 3.10. For the impact events, low level man-tle scattering will effectively blend the P- and S-wave codas, effectively obscuringthe S-wave arrival (bottom left trace). Low level mantle scattering as implementedin this model appears to have a negligent effect on deep event traces, except forlengthening the decay time and slightly accentuating the P-wave coda.3.6 ConclusionsI used a phonon method to model the propagation of seismic energy in highly scat-tering environments. I have generated synthetic traces and measured their char-acteristic decay times and decay factors to assess the effects of various velocity,attenuation and scattering parameters on the seismic signals. The main observa-tions and conclusions of this study are:1. High levels of seismic energy scattering will generate ground deformationwith amplitudes that decay exponentially (Regime 1), up to a point at whichintrinsic attenuation and scattering transform the traveling source functionand the resulting ground deformation into very low amplitude, slowly de-caying noise-like signals (Regime 2). Our modeling work favored higherscattering levels and faster attenuation for high frequencies and as such, high903.6. Conclusionsfrequencies transition to Regime 2 at shorter epicentral distances than lowerfrequencies.2. I observed a sharp rise in characteristic decay times at short epicentral dis-tances (0 to⇠10) for impact events, after which ⌧d values tend to stabilize.The distance at which ⌧d() stabilizes and the rate of increase appear to bemodel-dependent, suggesting these values, together with further modeling,could be used to infer first-order near-surface interior structure propertiessuch as the thickness of the scattering layer and the scattering intensity.3. Velocity structures that tend to trap and focus seismic energy near the sur-face and in the scattering layer have considerable effect on the characteristicdecay times. A crust-mantle boundary, the presence of a thin very low ve-locity layer near the surface (e.g. the lunar regolith), or a sharp velocitygradient with depth increased the decay times by up to at least a factor of 4in the models investigated here. This effect is also larger for shallow events– impacts or quakes – than for deep events.4. Both the impedance contrast at the scattering interfaces and the scatteringlayer thickness have an effect on the resulting seismic codas. Larger impedancecontrasts and a thicker scattering layer will lead to longer decay times. How-ever, a thick scattering layer with lower levels of impedance can scatter en-ergy more than a thin scattering layer with high levels of impedance, at leastin the manner in which impedance was modeled in this study.5. Our modeling showed that higher attenuation levels (lower Qi) will lead toshorter decay times, and conversely that lower attenuation levels result inlonger decay times. An increase in Qi by a factor of 2 or 3 can yield similarincrease in ⌧d, depending on the frequency band. Qc values of Regime 1signals are good first-order indicators of the near-station intrinsic attenuationlevel, Qi. Similarly, the change of Qc as a function of frequency can be usedto assess Qi(f): Near constant Qc(f) suggests a constant Qi(f), whereasQc values that are increasing with frequency indicate an increasing Qi withfrequency.913.6. Conclusions6. Deep structures such as a small core have little impact on the decay times.Any effect seems to be spatially limited to the core’s direct P- and S-waveshadow zone. Energy is observed in the shadow zone even if no direct wavecan propagate into the region, as the energy leaks from the surrounding scat-tering layer. At these epicentral distances, the maximum S-coda amplitudesare much lower, and the decay times much longer.7. Increasing ⌧d as a function of epicentral distance suggests low levels of seis-mic scattering in the interior. Interior scattering also tends to blend the P-and S-wave codas, making it harder to identify with precision the location ofthe S-wave arrival.The velocity profile, intrinsic attenuation and scattering structures of highlyscattering planetary bodies have a considerable effect on the duration and formof the resulting ground deformations. Our ability to model and understand theseeffects will lead to better interpretations of current and future seismic datasets andwill provide us with a clearer understanding of the interiors of planetary objectsunder investigation.92Chapter 4Constraints on Lunar InteriorStructure from the ScatteredSignals of the Apollo PassiveSeismic Experiment4.1 IntroductionSeismic energy propagation in the Moon is dominated by two processes particularto the lunar environment: extreme levels of seismic scattering and very low intrin-sic attenuation [Latham et al., 1970a; Nakamura, 1977]. The resulting recordedseismic waveforms are characterized by very long durations, by emergent onsetsand by the presence of slowly decaying coda waves that mostly obscure the ar-rivals of waves other than the P- and S-waves. These long coda typically hin-der more traditional analysis techniques that make use of seismic wave arrivals. Ishowed in Chapter 2 that the decay properties of these coda can be used to assessfirst-order subsurface scattering and attenuation structures. The results indicatedhigher levels of scattering for shallow events (impacts and shallow moonquakes)and for increasing frequencies, supporting the idea that scattering mostly occurs inthe near-surface megaregolith layer made up of much more small-scale scatterersthan large-scale ones.Nonetheless, further interpretations of the APSE coda observations were hin-dered by limitations in modeling of seismic energy propagation in a highly scatter-ing, spherical body. I have addressed this issue by developing a method to modelseismic energy transmission in lunar-like environments (Chapter 3). My adapted934.1. Introductionversion of the phonon method described in Shearer and Earle [2004] tracks a largenumber of seismic wavelets (the phonons) as they travel in the planetary body.These phonons are redirected by randomly oriented scatterers in the megaregolithlayer, and ground deformation is recorded each time the phonon packets hit thesurface near a receiver. The method has been demonstrated to generate syntheticsignals that reproduce the general characteristics of the seismograms observed inthe APSE dataset. Modeling results from the initial study in Chapter 3 showthat several model parameters affect the decay properties of the scattered coda.These include the velocity profile with depth, the megaregolith layer thickness, theimpedance contrast at the scatterers, the scatterer length-scales, as well as the in-trinsic attenuation levels. Furthermore, I showed that there is a tradeoff betweenthe scattering intensity and the intrinsic attenuation level on the decay rate of thescattered seismic signals.The main motivation behind this work is to build on the more general results ofChapter 3 and use the APSE data to constrain the interior scattering structure andintrinsic attenuation levels in the Moon. Accordingly, I use the phonon methoddeveloped in Chapter 3, along with the coda decay constraints from Chapter 2 toattempt to identify suites of lunar interior models that produce signals with simi-lar decay characteristics to those observed in the APSE data. The APSE receiversrecorded with a narrower bandwidth (⇠0.5–20 Hz for the long-period and short-period instruments combined) and poorer digitization (10-bit) than typical modernreceivers. Thus as part of this study I investigate the limitations of the APSE in-strumentation and the resulting data by implementing the APSE instrumentationeffects on the synthetic signals.Early studies with the APSE data provided some constraints on shallow sub-surface structure and intrinsic attenuation. Active seismic experiments on the lunarsurface (e.g. Kovach et al. [1972]), as well as inversions of seismic signal traveltimes lead to a series of seismic velocity profiles constraining seismic velocitiesin the crust and mantle, down to depths of ⇠1100 km (depth of the deepest lo-cated events, see e.g., Tokso¨z et al. [1971]; Nakamura et al. [1975]; Nakamura[1983]; Khan et al. [2000]; Kuskov et al. [2002]; Lognonne´ et al. [2003]; Garciaet al. [2011]; Weber et al. [2011]). All of these models included a thin and shallowvery low-velocity layer (i.e. the lunar regolith) and a crust-mantle interface. These944.2. Apollo Passive Seismic Experiment Datastructures will tend to focus seismic energy into the megaregolith layer, increas-ing the overall scattering observed in the data. Other studies inferred limits on thelevels of intrinsic attenuation (as measured by the quality factor, Qi, where a lowQi corresponds to high attenuation and vice versa) in the crust and upper mantle.Reported Qi values vary from 3000 to 6750 in the upper crust, from 4000 to 9000in the upper mantle, and from ⇠1000 to 1500 in the upper mantle (see Nakamuraand Koyama [1982]; Garcia et al. [2011] and references therein). Nakamura andKoyama [1982] also reported an increase in the shear wave Qi from about 4000 at3 Hz, to at least 7000 near 8 Hz. In addition, modeling and radar studies of the lunarmegaregolith indicate a power-law distributions of block sizes for blocks in rangeof meters to about 1% of the largest basins diameters (or about 10 km, see Schultzand Mendell [1978]), and a scattering layer thickness of a few hundred meters toa few kilometers [McGetchin et al., 1973; Pike, 1974; Housen and Schmidt, 1991;Petro and Pieters, 2004, 2008; Shkuratov and Bondarenko, 2001; Thompson et al.,2009]. I use these results collectively to motivate the general classes of modelsconsidered here: models with near-surface low-velocity layers (regolith and crust),with power-law distributions of block sizes, and with intrinsic attenuation levelsbased on published results.I first briefly review the APSE dataset in section 4.2 and follow with the model-ing approach and data analysis steps in section 4.3. Modeling results are presentedin section 4.4.4.2 Apollo Passive Seismic Experiment DataThe Apollo Passive Seismic Experiment consisted of four seismic recorders, form-ing a triangular array with stations 12 and 14 at one apex, approximately 1100 kmfrom each of stations 15 and 16. Four main types of seismic events were recorded:artificial impacts (AI, including the LunarModule impacts and the Saturn-IV Boosterimpacts), natural impacts (NI), deep moonquakes (DMQ) and shallowmoonquakes(SMQ).The seismic recorders comprised three orthogonal long-period channels (LP)with an instrument frequency response of 0.004 to 2 Hz and one vertical short-period channel (SP) with a frequency response of 0.05 to 20 Hz. In Chapter 2, I954.2. Apollo Passive Seismic Experiment DataTable 4.1: Least-squares fit results from the long- and short-period bands, as mea-sured in Chapter 2.Long-period: Median ⌧d values (in seconds) and median ab-solute deviation (MAD) for individual frequency band (L1, L2, and L3) and forall the bands combined. Short-period results: Interpolated ⌧d values at  = 0(⌧d(0)) and the slope of the best-fit straight line (seconds per degree), with the cor-responding standard errors on the coefficient estimates (s.e.). Refer to Figure 2.2and Table 2.2 to see the frequency range covered by each band.Long-Period Short-PeriodMedian ⌧d ⌧d(0) d⌧d/dBand (MAD) Band (s.e.) (s.e.)Artificial impactsL1 2732 (239) - -L2 1625 (258) - -L3 932 (126) - -Natural impactsL1 2330 (327) S1 380 (23) 1.4 (0.5)L2 1362 (131) S2 234 (26) 1.4 (0.4)L3 865 (199) - -Shallow moonquakesL1 2011 (433) S1 298 (19) 0.8 (0.3)L2 1114 (80) S2 193 (16) 1.2 (0.2)L3 722 (55) S3 145 (19) 1.8 (0.3)Deep moonquakesL1 1373 (434) - -L2 978 (143) - -L3 732 (122) - -964.3. Methodologyselected 72 events, from 55 distinct source locations, and measured their character-istic decay times (⌧d) and decay factor (QC) in 7 frequency bands: 3 bands for thesignals recorded on the LP components, and 4 bands on the SP component. Thebands for the LP signals were set to be 0.5 Hz wide and were centered on frequen-cies of 0.5, 1.0 and 1.5 Hz (L1, L2, and L3 respectively). The bands for the SPZsignals were set to a width of 2 Hz and centered on frequencies of 3, 5, 7 and 9Hz (S1 to S4, respectively). Results from the LP bands showed no dependence of⌧d or Qc on epicentral distance, whereas higher frequency signals recorded on theSP component have a clear dependence on epicentral distance. I summarize the ⌧dresults in Table 4.1 for each type of seismic event. I quote results from Chapter 2for the median ⌧d values for the LP bands, and the intercepts (value of ⌧d at 0 epi-central distance) and slopes (d⌧d/d) of the linear fits with epicentral distance forthe SP bands. In this chapter, I investigate how well I can match these propertiesof the APSE data with synthetic seismograms generated using the method detailedin Chapter 3 and using interior structure models that have a range of scattering andintrinsic attenuation properties.4.3 MethodologyThe modified phonon method used here is described in detail in Chapter 3 and sum-marized in the introduction above. In this chapter I investigate 32 models for whichI varied seven different model parameters: the thickness of the near-surface scatter-ing layer (TSL), the probability of low-level global scattering (pG), the minimumand maximum scatterer length-scales (sc) and their power-law distribution (nsc ),the maximum velocity and density perturbation levels at each scatterer (v,⇢), thedepth-dependent 1-D reference intrinsic attenuation (Qi(1 Hz)), as well as the in-trinsic attenuation frequency dependence (Qi(f)). The choices of parameters foreach model are given in Table 4.2.The different Qi(z) and Qi(f) profiles are plotted in Figures 4.1 and 4.2, re-spectively. I investigated four distinct Qi(z) profiles: one given by Garcia et al.[2011] (VPREMOON), one with a 50% increase in Qi(z) in the VPREMOONmodel (VPREMOON⇥1.5), and two with constant Qi(z). The two constant Qi(z)models were chosen to have values equal to 6000 and 9000 (Q6000 and Q9000974.3. MethodologyTable 4.2: Models investigated in this study. TSL is the scattering layer thickness,pG is the low-level global scattering probability (outside of the scattering layer), scsets the minimum and maximum scatterer length-scales and n defines the power-law probability distribution (n) as outlined in Chapter 3. v,⇢ is the maximumvelocity and density perturbation level at each scatterer. The different choices ofQi(z) and Qi(f) are plotted on Figures 4.1 and 4.2. In the table VP stands forVPREMOON. TSL pG sc [km] n v,⇢ Qi(z) Qi(f)Model [km] [%] min. max. [-] [%] model style1 5 0 0.05 10 -0.5 0.75 VP 32 30 0 0.05 10 -0.5 0.75 VP 33 30 0.01 0.05 10 -0.5 0.75 VP 34 15 0 0.05 10 -0.5 0.95 VP 45 15 0 0.05 10 -0.5 0.25 VP 46 15 0 0.05 10 -0.5 0.6 VP 47 30 0 0.05 10 -0.5 0.95 VP 48 30 0 0.05 10 -0.5 0.25 VP 49 30 0 0.05 10 -0.5 0.6 VP 410 30 0.05 0.05 10 -0.5 0.75 VP 411 40 0 0.05 10 -0.5 0.95 VP 412 40 0 0.05 10 -0.5 0.25 VP 413 20 0.0025 0.05 10 -0.5 0.6 Q6000 414 20 0.0025 0.05 10 -0.5 0.6 VP 515 20 0 0.05 10 -0.5 0.6 Q6000 516 20 0.0025 0.05 10 -0.5 0.6 Q6000 517 20 0.0025 0.025 5 -0.5 0.6 Q6000 518 20 0.0025 0.05 10 -0.5 0.9 Q6000 519 20 0.005 0.05 10 -0.5 0.6 Q6000 520 20 0.0025 0.05 10 -0.5 0.6 VP 621 30 0.0025 0.05 10 -0.5 0.6 VP 622 30 0.0025 0.05 10 -0.3 0.6 VP 623 30 0.0025 0.05 10 -0.6 0.6 VP 624 30 0.0025 0.05 2.5 -0.5 0.6 VP 625 40 0.0025 0.05 10 -0.5 0.6 VP 626 30 0.0025 0.05 10 -0.5 0.6 VP⇥1.5 627 30 0.005 0.05 10 -0.5 0.6 VP⇥1.5 628 20 0.0025 0.05 10 -0.5 0.6 Q6000 629 30 0.0025 0.05 10 -0.5 0.6 Q6000 630 30 0.0025 0.05 2.5 -0.5 0.6 Q6000 631 40 0.0025 0.05 10 -0.5 0.6 Q6000 632 30 0.0025 0.05 10 -0.5 0.6 Q9000 6984.3. Methodology02004006008001000120014001600Depth (km)0 7.5 15/0 7.5 15/0 7.5 15/0 7.5 15[-] [-] [-] [-]VPREMOON VPREMOON × 1.5 Q6000 Q9000( x1e4 )Figure 4.1: The four depth-dependent 1-D Qi(1 Hz) profiles used in this study.respectively), close to those for the crust and upper mantle in Garcia et al. [2011],and reasonable bounds on averages along the entire source–receiver path for deep,shallow and surface events. Each of the Qi(f) styles presented in Figure 4.2 werechosen based on results from Chapter 3 in order to generate synthetics with similar⌧d(f) and Qc(f) as the ones measured with the APSE data. The Qi(f) style 5 wascomputed so that Qi(f) equals the mean Qc(f) measured at the APSE stations.Qi(f) style 6 is similar to style 5, but with a larger increase in Qi at high frequen-cies. Lower attenuation at high frequencies is consistent with Qi values reportedin Nakamura and Koyama [1982]).I used a single velocity and density model in which I combined the crustal andmantle velocities from the VPREMOONmodel in Garcia et al. [2011] with the corevelocities from Weber et al. [2011] (see Figure 3.1A). The presence of a thin, low-velocity regolith, a crust, and a velocity gradient with depth is essential to obtainingdecay times in the range of these observed in the APSE data (see Chapter 3). I havenot tested the effect of a deeper low-velocity zone that is observed in some earliermantle velocity models (e.g. Nakamura [1983]), as such a layer is not required bythe travel time observations; however the effects of a mantle low-velocity zone can994.3. Methodology0 2 4 6 8 10012345Frequency (Hz) Frequency (Hz)0 2 4 6 8 100123450 2 4 6 8 10012345Frequency (Hz) Frequency (Hz)0 2 4 6 8 10012345Style 3 Style 4Style 5 Style 6Figure 4.2: The four different Qi(f) styles used in this analysis. The plots showthe ratios between Qi(f) and Qi(1 Hz), such that a ratio greater than 1 for a givenfrequency indicates a Qi greater than Qi(1 Hz) at that frequency (i.e. lower atten-uation). Styles 1 and 2 were used in the analysis presented in Chapter 3.1004.3. Methodologyeasily be tested in future modeling work. The source functions used here are thesame as the ones used in Chapter 3 (Figure 3.3). Future work could also investigatethe effect of different source functions on the signals, however this was outside thescope of this work.In section 4.3.1 I describe how I implemented the effects of ambient seismicnoise and of the APSE instruments in my modeling. The steps used to measuredcoda decay times are the same as in Chapter 2 and are reviewed in section 4.3.2.4.3.1 Modeling Ambient Noise and APSE Instrument EffectsNoise in the APSE dataset originated from several distinct sources. Among theseare increased noise levels recorded during passage through the terminator [Dormanet al., 1978] and also resulting from temperature variations on the lunar surface(e.g. [Sens-Scho¨nfelder and Larose, 2010]), the lunar seismic hum (meteoritic[Lognonne´ et al., 2009] or tectonic), and the long-range transmission of data backto Earth [Nakamura et al., 1980]. Some of these noise sources resulted in highamplitude spikes in the data that could be removed or reduced through variousfiltering techniques (e.g., Bulow et al. [2005]).Here I focused on the low amplitude noise that is present throughout most ofthe recordings. Due to the limited dynamic range of the instruments, this noise isexpressed as pervasive 1-bit fluctuations. To investigate the effect of this ambientnoise I added white gaussian noise to the synthetic signals, convolved the noisysignals with the APSE instrument responses, and 10-bit-digitized the resulting timeseries. I explain here how this was implemented.I measured the maximum signal-to-noise ratio (SNR) for all events analyzedin Chapter 2. I defined the SNR as the ratio between the root-mean-square (RMS)of a 10 s window centered on the time of the maximum S-coda amplitude, and theRMS of 10 s time window of noise measured before the P-wave arrival or afterthe signal amplitude has decreased to the background noise level. The SNR valuesrange from 1:1 to about 50:1. I then picked the median epicentral distance, and themedian SNR for each event type and sampling frequency. These median SNRs andepicentral distances for all types of events, on both the LP and SP components, arelisted in Table 4.3.1014.3. MethodologyTable 4.3: Median epicentral distances and signal-to-noise ratios measured on theAPSE signals Median EpicentralType Component Distance () Median SNRDMQ LP 34 4.98NI SP 42 8.01NI LP 42 10.46SMQ SP 68 15.08SMQ LP 68 14.28The magnitude ranges for individual impact or quake events are very poorlyconstrained. As a result, there is no reliable scaling relationship one can use forcharacterizing the appropriate quake mechanics to model nor the appropriate re-sulting measured amplitudes. In fact, such a relationship would depend on anaccurate attenuation relationship, which I attempt to calibrate here. For this rea-son, I scale the synthetic waveforms to the appropriate amplitudes relative to thebackground 1-bit noise. I make the assumption that the median SNR at the mediansource-receiver distance is a reasonable proxy for the appropriate SNR with whichto scale synthetics of each type.I defined the amplitude of one bit in the synthetics such that the synthetic sig-nals have a SNR similar to that of the data at an epicentral distance correspondingto the median epicentral distance identified for that type of event in the APSE data.I next added noise to all the synthetic recordings, where the noise was taken froma gaussian distribution with a mean of zero and a standard deviation equal to thatof the amplitude for 1-bit for each type of event. I then convolved each signal withthe LP or SP instrument responses, and bit-limited the resulting traces. For exam-ple, to add noise to a set of deep source synthetics, I measured the RMS of a 10 swindow centered on the maximum S-wave coda amplitude (RS) for the syntheticsignal recorded at 34 epicentral distance. RS/4.98 is the amplitude of 1-bit for allsignals related to this event. I then added noise with root-mean-square amplitudeof 1 bit to all signals originating this source. Examples of noise-free versus noisysignals, for an impact and a deep event are shown in Figure 4.3.1024.3.MethodologyFigure 4.3: Noise-free synthetic signals (left), and resulting signals after addition of 1-bit noise and 10-bit digitization (right).The top signal is from an impact event, recorded at 10 epicentral distance. The middle and bottom signals are both fromthe same deep event (source at 1000 km depth) and were recorded at epicentral distances of 34 and 110 respectively.1034.3. Methodology4.3.2 Analysis of Synthetic SeismogramsBefore measuring the coda decay time of each synthetic signal, I followed theanalysis steps used in Chapter 2 and for each synthetic seismogram I:1. Deconvolved the APSE instrument response,2. Band pass filtered the signal into three LP bands and four SP bands: 0.25-0.75 Hz, 0.75-1.25 Hz, 1.25- 1.75 Hz, 2-4 Hz, 4-6 Hz, 6-8 Hz and 8-10Hz,3. Generated the signal envelope,4. Smoothed the envelope function using a 5 minute (LP) or 35 second (SP)running window, keeping the 75th percentile,5. Determined when the amplitude started decaying exponentially (the begin-ning of the fit),6. Least-squares fit an exponential decay curve of the form et/⌧d to the decay-ing section of the smoothed envelope. ⌧d is the characteristic decay time. Allfits were of an arbitrary length of 2500 s for SP signals, and 4000 s for LPsignals.⌧d can be converted to the frequency-specific decay factor Qc = 2⇡fc⌧d, wherefc is the central frequency of the frequency band analyzed. Qc is useful to com-pare decay properties between different frequency bands. However, because Qcis directly proportional to ⌧d I use only the APSE ⌧d values as model constraints.Accordingly, only summary modeling ⌧d results are shown below.To compare the results for synthetic seismograms with the APSE decay valuesfrom Chapter 2, I plotted ⌧d estimated from the synthetics as a function of epicen-tral distance. I measured the median ⌧d (LP bands) or the slope and intercept of abest fit of ⌧d as a function of epicentral distance (SP bands), as shown in Figure 4.4.In all cases, I calculated the medians, or the slopes and intercepts using only databetween ⇠20  130 epicentral distance to avoid the near-source effect and theeffect of the core on the measured ⌧d (see Chapter 3).1044.3. Methodology0 20 40 60 80 100 120 140 160 180020040060080010001200 Epicentral distance (˚)Model 65 (s)Near-source effect Core effect1.25-1.75 Hz7-9 HzMeanSlope &InterceptFigure 4.4: Examples of the summary ⌧d values measured in the 1.25-1.75 Hz band(squares) and in the 7-9 Hz band (stars). The long period bands show no to onlya weak increase of ⌧d with epicentral distance and so only a median ⌧d value ismeasured. For short period bands, a least-squares robust fit of the data yields a⌧d intercept (⌧d(0)) and the slope (d⌧d/d). The measurements are made overepicentral distances of approximately 20–130 to avoid the effect of the core atlarge distances and of the sharply increasing ⌧d at short distances.1054.4. Results4.4 Results4.4.1 Results from Modeling of Seismic Noise and APSE-InstrumentEffectsI have measured the characteristic decay times from noise-free signals, as well asfrom the corresponding APSE-corrected signals. Figure 4.5 compares decay timesfrom a 50 km deep event for noise-free signals (Figure 4.5A), APSE-corrected sig-nals (Figure 4.5B), and shows the percentage difference in ⌧d between the two (Fig-ure 4.5C), for frequency bands corresponding to the APSE short-period compo-nent. Results indicate that the exponential decay times measured from the APSE-corrected synthetic signals match the decay times measured from the noise-freesynthetic signals, within⇠10%, as long as the SNR is approximately greater than 1.Below this threshold, the added noise and limited bit-digitization either totally ob-scures the original signal, or affects the signals’ decay such that they lose theirexponential behavior. In this case, the automatic fitting method fails and tend togenerate artificially large ⌧d values (for SNR < 1). The distance at which themeasured ⌧d on the APSE-corrected synthetic signals diverge from the noise-freesynthetic signals depends on the epicentral distance at which I implemented theaverage SNR (68 in Figure 4.5B). Higher frequencies decay times start divergingearlier due to their lower overall amplitudes (i.e., they reach SNR < 1 at shorterdistances).All signals used in the analysis presented in Chapter 2 were selected based onthe quality of the seismic trace, and all selected signals had a SNR > 1. Impor-tantly, the results presented here indicate the ⌧d and Qc values obtained in Chap-ter 2 were not affected by the APSE instruments bit-limitation or by the presence ofseismic noise, and that they are representative of the lunar interior structure. Basedon this conclusion, I now show only ⌧d values measured from noise-free syntheticsin order to examine any variations in ⌧d over the entire epicentral distance range(0-180).1064.4. Results0 30 60 90 120 150 180 0 30 60 90 120 150 18050010001500200025000 Epicentral distance (˚)  Epicentral distance (˚) (s)0 30 60 90 120 150 180Noise-free signalsA) APSE-corrected signalsB)2 - 4 HzPercentage difference in decay times (100×(B - A)/A)4 - 6 Hz 6 - 8 Hz 8 - 10 HzLEGENDDifference in decay times (%)C)SNR = 15.08Source depth = 50 km020406080100Figure 4.5: Characteristic decay times (⌧d) measured from noise-free signals (A),and from the same signals after being corrected for the 1-bit seismic noise andthe APSE-SPZ instrument effects (B). The percentage difference in decay times,(100⇥(B-A)/A)%, is shown in C. The orange line in B shows the distance at whichthe amplitude corresponding to 1 bit was measured ( = 68). The increase ⌧d atlarge distances in A is due to the small core included in the model.4.4.2 Modeling ResultsFigures 4.6 shows the median ⌧d measured in the long-period frequency bands(0.5-1.75 Hz) for both shallow and deep events, for all 32 investigated models. Asa comparison, the median ⌧d values from the APSE data are shown. Results showthat variations in the scattering structure (i.e. the scattering layer thickness, theimpedance contrast at the scatterers, the scatterer length-scales) have little effect onthe median ⌧d (e.g. models 1–20, except for model 10, have near constant median⌧d). Only model 10, which has a high global scattering probability (pg = 5%),shows larger median ⌧d values.The main control on the median ⌧d appears to be the intrinsic attenuation levelQi(f). Results indicate that only the models using somewhat lower attenuationlevels (i.e. higher Qi) than that proposed in the VPREMOON model produce me-dian ⌧d values that match the APSE data, for both the shallow and deep event cases.For example, models 26 and 27 with a Qi(z) profile like that of VPREMOON, but1.5⇥ larger in magnitude), or model 32 that has a constant Qi(z) of 9000 produce1074.4.Results05001000150020002500300005001000150020002500Mean (s)Mean (s)Models1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32V V V V V V V VV6 6 6 6 6 6 6 96 6 6VVVVVVVVVVModel APSE V61.25 -1.75 Hz0.75 -1.25 Hz0.25 - 0.75 Hz Style 3Style 4Style 5Style 6VPREMOONQ6000VPREMOON × 1.5Q90009LEGENDShallow eventsDeep eventsFigure 4.6: Median ⌧d values measured in the low frequency bands (0.5-1.5 Hz) for shallow events (top) and deep events(bottom) for the 32 models investigated in this analysis. The red bars indicate the median ⌧d measured with the APSE datain Chapter 2 within one median absolute deviation. Note that for overlapping APSE value ranges, the ranges are plotted ononly either half of the plot, as in the deep event case. However, these values apply to all models. Also, in some cases theuncertainty in ⌧d derived from the synthetics is less then the height of the symbol and is not shown.1084.5. Discussionsynthetic seismograms that yield median ⌧d that match the APSE data.In Figure 4.7 I present the measured slopes and intercepts for a selection ofmodels that best represent the effects of the scattering and intrinsic attenuationparameters on the data, as well as the models that best match the data. Resultsfor surface impacts and shallow events are presented, and measurements from theAPSE data are plotted for comparison (only the 2-4 and 4-6 Hz bands are availablefor the impact events for the APSE data). In the case of the ⌧d(0) intercepts(Figure 4.7A and B), the models with lower intrinsic attenuation values (models26, 27 and 32) perform best in matching the APSE data in the lower frequencybands (2-6 Hz). None of the models investigated properly reproduce the d⌧d/dslopes observed in the APSE data (Figure 4.7C and D). However, in contrast to themedian ⌧d values and the intercepts, the slopes for impact events are more stronglyaffected by the scattering structure (Figure 4.7C) and not only controlled by theintrinsic attenuation.4.5 DiscussionThe crust and thin surficial low-velocity layer (analogous to the lunar regolith) in-cluded in the velocity profile used in the analysis focus seismic energy near thesurface and into the scattering layer. As seen in Chapter 3, such velocity profilesgenerate coda that are much more scattered (i.e., longer ⌧d) than models without acrust or low-velocity regolith. In such very highly scattering environments, furthervariations in scattering properties do not appear to be the main control on the decaycharacteristics of the scattered codas. For example, both a thin and a thick scatter-ing layer will generate similar decay times (e.g. model 1 with TSL = 5 km andmodel 11 with TSL = 40 km). For models using the VPREMOON or the Q6000Qi(z) profiles, no combination of scattering property values produced decay timesthat match those observed in the APSE data, at least not in the lower frequencybands. The one exception is model 10 that has a high probability of global scat-tering (pg = 5%) and produced median decay times in the lower frequency bandsthat compare with the APSE values, but resulted in slopes d⌧d/d at higher fre-quencies that are too steep. I found that the best way to affect the decay times isto change the intrinsic attenuation levels in the models (Qi(f, z)). Models using1094.5. Discussion010020030040050001002003004005001 2 3 7 8 10 21 26 27 32 1 2 3 7 8 10 21 26 27 32012345678012345678V V 9VVVVV V V 9VVVVVModels ModelsAC DBModel APSE6 - 8 Hz8 - 10 Hz4 - 6 Hz2 - 4 Hz V6Style 3Style 4Style 5Style 6VPREMOONQ6000VPREMOON × 1.5Q90009LEGENDShallow eventsSurface impactsFigure 4.7: Intercepts (A and B) and slopes (C and D) of least-squares fits onthe short-period bands (2-10 Hz), for surface impacts and shallow events. Similarmeasurements from the APSE data are given (red), within one standard error. Sameformat as Figure 4.6.1104.5. DiscussionQi(z) values that are 50% larger than the ones in the VPREMOON model (mod-els 26 and 27), or models using a constant Qi(z) = 9000 (Q9000), produce decaytimes that compare well with the APSE data at low (median) and high (intercept)frequencies, at least up to 6 Hz. I propose that there are conditions under whichscattering is so extreme (e.g. when a crust and a regolith focus scattered energyback into the scattering layer) that variations in the scattering properties do notsignificantly affect the decay times of scattered coda. In such situations, anelasticprocesses exert the greatest control on the decay of seismic energy (intrinsic at-tenuation). As such, measuring median decay times and intercept values may notbe the best tool to assess the length scales of scattering of very highly scatteringbodies like the Moon, but they can give us some first order information regardingthe levels of intrinsic attenuation.Based on the modeling observations from Chapter 3, I found that a near con-stant Qi(f) at lower frequencies (0.5–1.5 Hz) and an increasing Qi(f) at higherfrequencies (2–10 Hz) is necessary to reproduce the corresponding APSE decaytimes dependence on frequency (⌧d(f)). Based on the models investigated, Qi(f)style 6 provides the best match when combined with the VPREMOON⇥1.5 or theQ9000 models. However, more investigations of the parameter space are needed totest which Qi(f) works with the depth-dependent Qi(z), either the styles alreadymodeled, or a different style for a better overall fit with the APSE data.One potential way to investigate scattering structures would be to make use ofthe slope of ⌧d as a function of epicentral distance in the high frequency bands.None of the models investigated in this study produced slopes that compare wellwith the ones from the APSE data (see Figures 4.7C and 4.7D). However, theslopes seem to be particularly affected by the model scattering parameters, espe-cially for surface impact events (Figures 4.7C). We see an increase in slope formodels with high maximum velocity and density perturbation level (e.g., model31), as well as for models with a higher probability of global scattering (models 10and 27). Slopes also seem to increase with increasing scattering layer thicknesses(e.g., model 1 with a 5 km thick scattering layer, versus model 2 with a 30 kmthick scattering layer). On the other hand, change in intrinsic attenuation levelshave little impact on the slopes, especially for shallow events (Figure 4.7D). Fu-ture work will need to further explore parameter space in order to find models with1114.6. Conclusionsslopes that better match the APSE data and achieve improved constraints on thescattering structure of the lunar interior.The main goal of this analysis was to investigate a suite of scattering structuresin order to constrain the length scales of scattering in the lunar interior (e.g., TSL,sc, v,⇢, pg). However, as discussed above, the median ⌧d values for models 1–25 (except model 10) show that changes in the scattering properties at the levelsimplemented in this work (e.g., changes in TSL, sc of a factor of a few) do not havea significant effect the coda. Further modeling (models 26–32) suggests that mostchanges in coda decay times are captured by the intrinsic attenuation (Qi(f, z)). Aseach model run takes approximately 3 weeks to complete on a multi-core system(computation of the synthetics and data analysis, including the smoothing of thesignals), I saved computational time for models 26 and above by not basing mychoice of model parameters on a systematic search of the parameter space, butrather attempting to explore changes to the models that experience suggested couldbe important. More modeling work is thus needed to put limits on the length scalesof scattering using Qi(f) styles 5 and 6, and the VPREMOON⇥1.5 and Q9000Qi(z) profiles.4.6 ConclusionsThe main objective of this analysis was to identify a suite of lunar interior modelsthat produce synthetic seismograms that share the decay properties of the scatteredseismic signals in the APSE dataset. I have first demonstrated that the ubiquitousseismic noise in the lunar data, along with the bit-limited nature of the APSE in-struments, did not significantly affect the decay times for lunar seismic signals witha SNR greater than 1. As such, I conclude that the decay measurements presentedin Chapter 2 are representative of the lunar interior structures and can be used asconstraints for the modeling work.The combination of a thin shallow regolith, a crust, and a scattering layer gen-erates a very highly scattering environment on the Moon. In such a context, thescattered coda decay times are mostly controlled by the intrinsic attenuation lev-els Qi(f, z), and not by the length scales of scattering. Consequently, the median⌧d values are not sufficient to identify the lunar scattering structure. This could1124.6. Conclusionsbe achieved by investigating the variations in the d⌧d/d slopes. However, themedian decay times can provide some constraints on intrinsic attenuation levels. Ihave observed in the modeling a need for higher Qi than proposed in Garcia et al.[2011], as well other published Qi values based on amplitude decay with traveltime and distance (e.g. Nakamura and Koyama [1982] and references therein).Models with a 50% increase in Qi values from the VPREMOON models, or with aconstant Qi(z) = 9000 at 1 Hz, combined with an increasing Qi(f) with increas-ing frequencies, produce scattered synthetics with similar decay times to thoseobserved in the APSE data.More modeling work is needed to systematically investigate the parameterspace of plausible intrinsic attenuation and scattering structures compatible withthe APSE coda decay times. I need to test the effect of varying the scattering prop-erties in higher Qi scenarios, and to test different velocity profiles to see if thedecay times then vary or if they remain within the range of values measured withthe APSE data. Further work will also investigate whether the slopes of the ⌧das a function of epicentral distance are a better tool to assess the length scales ofscattering.113Chapter 5Effects of Lateral Variations inMegaregolith Thickness onRecorded Seismic Signals5.1 IntroductionSeismic signals recorded during the Apollo Passive Seismic Experiment (APSE)were strongly affected by the scattering of seismic energy in the Moon’s interior,mainly within the near-surface megaregolith layer (e.g., Latham et al. [1970a];Blanchette-Guertin et al. [2012]). A consequence of the scattering is that sec-ondary seismic phases, which could be used to detect major compositional andphase boundaries within the Moon, are almost entirely obscured in the APSE dataset. In this work, I use a synthetic seismogram phonon method (Shearer and Earle[2004] and Chapter 3) to investigate the effects of laterally varying megaregoliththickness on the propagation of seismic energy, and on the resulting seismic signalsrecorded at various epicentral distances from the moonquake or impact source. Amajor objective of this study is to identify conditions under which seismogramsrecorded at the surface are less likely to be affected by high levels of scattering,and for which secondary arrivals containing important information about interiorstructure can be more readily identified. Such studies will help optimize the designof future non-terrestrial seismic surveys to maximize scientific return.Several past studies investigated the distribution of large basin ejecta on the lu-nar surface (e.g., Short and Forman [1970]; McGetchin et al. [1973]; Pike [1974];Petro and Pieters [2008]). Ejecta from major impact events is one of the majorcomponents of the megaregolith layer, along with the underlying fractured crust1145.1. Introduction[Ho¨rz, 1991]. Estimates of megaregolith thicknesses resulting from the differentejecta models presented in these studies vary from hundreds of meters to severalkilometers. However, all studies agree that the thickness of the megaregolith willvary geographically, from thicker near the edges of the basins, to thinner at loca-tions far from the impact structures. Other studies using Earth-based radar imageryshowed variations in megaregolith thickness between regions in the northern andsouthern hemisphere on the near side [Thompson et al., 2009], and a systematicdifference in the depth to the base of the megaregolith between the highlands andthe large mare basins [Thompson et al., 1979], the latter having substantially thin-ner megaregolith. It is assumed that the megaregolith will be thinner beneath thebasins themselves, due to ejection of material radially outward from the impactsite. In addition, the megaregolith beneath younger basins is expected to be thinnerbecause of less cumulative ejecta from subsequent large impacts in the surround-ing regions. The APSE stations were located on highlands near the edges of largebasins (station 16), or within old basin structures, very close to the basin edge (sta-tion 15), or near surface topography with high scattering potential (stations 12 and14). All seismic signals recorded at these locations were affected by similarly highlevels of scattering [Blanchette-Guertin et al., 2012].Using the phonon method developed in Chapter 3 I assess the effects of later-ally varying megaregolith thicknesses, occurring in association with impact basins,on the seismic signals recorded at various locations along the surface. I investigatehere two simple basin models: A) 4-diameter (⇠120 km) basins, centered at epi-central distances,, of 20 and 40; and B) a 40-diameter (⇠1200 km) basin cen-tered at equal to 60. I have tested several megaregolith thickness models, wherethe impact structures are surrounded by a 30 or 60 km thick megaregolith, and un-derlain by either no megaregolith or by a 5 km thick megaregolith. These scenar-ios are intended as proxies for younger impact basins with less subsequent impactresurfacing and megaregolith production than in the older surrounding terrain. Ihave computed synthetics for surface events (impacting on top of the megaregolith)as well as for 1000 km deep events (analogous to lunar deep quakes). In all cases, Iused a modified VPREMOON velocity model [Garcia et al., 2011] to which I haveadded the velocities for the lunar core from Weber et al. [2011]. In this chapter, Iuse the term basin to refer to both the smaller and the larger structures. Typically,1155.2. Methodologybasin refers to impact structures on the Moon that are 300 km or larger in diameter[Wilhelms, 1987].5.2 MethodologyThe phonon method used here is detailed in Chapter 3. The approach tracks alarge number of seismic wavelets as they travel through a planetary interior from asource located at  = 0, and it records the associated ground deformation eachtime a phonon packet hits the surface near a given receiver. In the megaregolithlayer the phonons encounter randomly oriented scatterers every sc m, where scis randomly sampled from a power-law distribution that has many more small-scale scatterers than large ones. The phonons are stochastically scattered, or not,based on the scattering probability and on the velocity and density perturbationsassociated with the scatterer. These perturbations are picked randomly and were onaverage±35% of the background values for the models presented here. This favorsforward scattering of the phonons vs. backscattering. The energy partitioning ratioat the source is 1:10:10 (P:SV:SH) in the case of deep event, and 1:0:0 for thesurface impacts.5.3 ResultsFigures 5.1 and 5.2 present synthetic seismograms for 5 models comprising threedistinct megaregolith structures and two different basin diameters. The basin di-ameters and megaregolith structures are shown in the schematics at the top of eachset of seismograms. The thickness of the megaregolith surrounding the impactbasin (TSL) and the thickness of the megaregolith beneath the basin itself (TB)are indicated for each model. Figure 5.1A shows signals derived from a surfaceimpact event (source depth of 0 km). All other models shown (Figure 5.1B andFigure 5.2) are for signals from deep sources (source depth of 1000 km). For allmodels the epicentral distance between each receivers is 0.5, or about 15 km, onthe lunar surface. The seismic traces shown beneath the receivers correspond tothe ground deformations recorded on the vertical channel at the receiver locations.The amplitudes in each trace have been normalized such that the root mean square1165.3. Resultsamplitudes of all traces are equal. In Figure 1 I compare synthetic signals recordedfor an impact and a deep event across a 120 km diameter basin, with TSL = 30 kmand TSL = 0 km. In Figure 2 I show synthetic signals for a deep event and differentmegaregolith structures and basin diameters.All impact-derived signals shown in Figure 5.1A exhibit strongly scattered co-das, including those recorded at stations in the basin even with no megaregolith(TB = 0 km). This makes it difficult to identify wave arrivals, even for the initialP-wave which is very emergent. This is mainly because all the energy is first scat-tered near the source. Other results, not shown here, indicate that signals recordedin basins with a thin to no megaregolith are much cleaner for all types of events forwhich there was no source-side scattering. These could be events near the surfacebut beneath the megaregolith (perhaps analogous to lunar shallow quakes) or im-pacts into a region with a very thin megaregolith. Synthetics are thus much cleanerin the case of deep events (Figure 5.1B and Figure 5.2), where most of the energyis allowed to first travel with minimal scattering in the mantle before reaching thereceivers. As a consequence, I now focus the investigations on deep events.In Figure 5.1B, I show signals derived from a deep source for the same megare-golith structure as in Figure 5.1A. In this case, both the P and S-wave arrivals areidentifiable, although the S-wave is more impulsive for signals recorded in thebasin than for those recorded at stations in the surrounding region where TSL =30 km. Importantly, the ScS core reflected phase is visible in the basin signals.Other core phases, which would have lower amplitudes, are lost in the coda gener-ated by the near-surface very low-velocity layer and in the seismic noise caused bythe energy leaked from the surrounding megaregolith.The deep event signals shown in Figure 5.2A demonstrate the effect of addinga megaregolith layer in the impact basin that is thin (TB = 5 km) compared withbeneath the surrounding terrain (TSL = 30 km). In this case, all signals exhibitstrong scattering. It is still possible to pick the P- and S-wave arrivals, however allsecondary phases are obscured by the coda.In Figure 5.2B, I investigate the effect of a thicker surrounding megaregolith(TSL = 60 km) on the seismic signals recorded near the basin edges. In Figure5.1B seismograms recorded at stations in the basin, but only 15 km from the basinedge, show much less scattering than the signals recorded on top of the megare-1175.3. ResultsFigure 5.1: (A) Vertical component of synthetic seismograms for a surface impactevent, and (B) a 1000 km depth event, recorded across the simplified structure rep-resenting a 120 km diameter basin. Note that I refer to both the large and smallimpact structures as basins for brevity in this chapter – see main text. The bluetriangles in the top schematic represent the seismic receivers at which the syntheticsignals shown below were recorded. Shaded regions under the stations representthe megaregolith layer. For these two events, the megaregolith thickness surround-ing the basin is 30 km (TSL = 30 km), and the megaregolith thickness beneath thebasin is 0 km (TB = 0 km). The distance between each station is 0.5 (15 km).The time is relative to the source origin time. Theoretical P (blue), S (red), and ScS(green) arrivals are plotted on B. The lines are slightly offset for better viewing ofthe arrivals.1185.3.ResultsFigure 5.2: Synthetic seismograms recorded for a deep source and three different megaregolith structures: (A) 120 km basinwith TSL = 30 km and TB = 5 km, (B) 120 km basin with TSL = 60 km and TB = 0 km, and (C) 1200 km basin (onlyshowing the edge) with TSL = 30 km and TB = 0 km, where TSL is the thickness of the megaregolith layer surrounding theimpact structure, and TB the thickness of the megaregolith in the structure itself. Figure format as in Figure 5.1. TheoreticalP (blue), S (red), and ScS (green) arrivals are plotted on B.1195.4. Discussion and Conclusionsgolith. In contrast, the signals shown in Figure 5.2B indicate that to record codawith substantially reduced scattering we now have to move at least 30 km, or halfthe scattering layer thickness (TSL/2) away from the basin edge, toward the basincenter. At stations beyond that distance, the P- and S-wave arrivals, as well as theScS core phase, are identifiable.For larger basins such as that shown in Figure 5.2C the records of deep eventsat stations in the basin interior are cleaner than those for the same megaregolithstructure but smaller basin diameter (Figure 5.1B). Note that only one edge ofthe basin is shown in Figure 5.2C. In particular, the basin is now large enoughthat the effects of leaked energy seen near the basin edges, and resulting fromthe surrounding thick megaregolith diminish significantly toward the center of thebasin.5.4 Discussion and ConclusionsThe results indicate that the total scattering observed in the recorded signals isminimized for seismic sources located beneath the megaregolith. Seismic energyfrom such sources does not undergo the source-side scattering that affects impactevents and any lunar quakes occurring within the scattering layer. Deep moon-quakes are known to occur in the lower lunar mantle, and occur repeatedly, withtidal periodicities at distinct locations [Lammlein et al., 1974; Nakamura, 2005;Bulow et al., 2005, 2007]. They are thus ideal sources for better characterizationof the lunar interior in future lunar seismic experiments. In addition some of theshallow moonquakes observed in the APSE data may occur beneath the scatteringlayer, either in the lowermost crust or in the upper mantle [Nakamura et al., 1979].These events are rare, only 28 shallow quakes were recorded over the 7 year APSEduration, however a few were of large magnitude and thus it would be desirable torecord such types of events during any new lunar seismic survey.Basins with a much thinner megaregolith layer than the surrounding region(TB << TSL) can yield seismograms at stations located in the basin that have sig-nificantly less scattered energy than signals recorded at stations on the surroundingterrain. Furthermore, the results indicate that the effect of leaked energy from thesurrounding thick megaregolith is greatly reduced for receivers located at a distance1205.4. Discussion and Conclusionsaway from the basin edge (toward the basin center) of at least half the thickness ofthe surrounding megaregolith, TSL/2. As a result, even small basins could allowidentification of direct wave arrivals, and potentially of secondary wave arrivals aslong as the megaregolith layer beneath the basin itself is thin. I note that in mysimulations I did not change the size-frequency distribution of scatterers, or theiroverall number density for the case of a thin megaregolith layer beneath the basin(Figure 5.2A). In reality both of these properties will be different in this layer, es-pecially for young basins where we would expect a thin megaregolith with a lowerscatterer number density overall. This effect will further act to reduce the scatter-ing recorded at stations located in the basin. However, large basins will be bettersuited to detect primary and secondary wave arrivals as it is easier to locate stationsaway from the basin edge yielding seismograms less affected by leaked, scatteredenergy.In my models, the effects of the dominantly forward-scattering vs. back-scattering are seen for stations recorded around the edges of a smaller basin (assees in Figure 5.1B and Figure 5.2B). The receivers located at epicentral distancesof 38 and 42 both sit on the edge of the basin, however, the seismogram at 38 ismore affected by forward-scattered energy from the edge of the basin (from energytraveling away from the source located at 0). In contrast the seismogram at the re-ceiver at 42 is less scattered because there is less back-scattered energy from thatedge of the basin. As noted earlier, in my models the average impedance contrastof the scatterers of ±35% of the background values favors forward scattering andis a value likely representative of that for annealed fractures or boundaries betweendifferent lithologies. Impedance contrasts representative of voids (correspondingto the high porosity inferred from GRAIL data for at least the uppermost megare-golith [Wieczorek et al., 2013; Besserer et al., 2013]) are larger, so the scatteringis more isotropic and the seismogram recorded at 42 would show more scatteringthan that seen in Figures 5.1B and 5.2B.The following effects were neglected in my models to minimize computationalrequirements, but they could have important effects on the scattering characteristicsof the recorded signals: (1) Surface topography. This could intensify or decreasescattering, especially near basin edges or near the seismic receivers (e.g., Schmerret al. [2011]). (2) Lateral variations in seismic velocities and densities. Large1215.4. Discussion and Conclusionsimpacts can affect the velocity profile beneath the impact site (e.g., crustal thinning,mantle uplifting, petrological evolution of melt material [Melosh, 1996; Ivanovet al., 2010]). These lateral impedance contrasts with the region surrounding thestructure could focus or scatter seismic energy into or out of the basins. (3) Noise(instrumental or seismic). The presence of background seismic noise could hinderthe identification of the emergent body wave arrivals as is the case with the APSEdata (e.g., Lognonne´ et al. [2003]).In conclusion, younger impact events, if large enough, can redistribute ejectaand reduce the thickness of the megaregolith at the site of impact. Seismic sig-nals recorded at stations located in the basin from deep moonquake sources orfrom shallow moonquakes located beneath the megaregolith show substantiallyreduced scattering relative to signals recorded at stations outside the basin. Theseismograms show clearer P and S-wave arrivals and can allow identification ofsecondary phases. Consequently, these impact sites should be considered as poten-tial seismometer locations in future seismic surveys of the lunar interior. One suchideal location for future lunar seismic surveys would be the far-side Schro¨dingerbasin. This basin is one of the youngest basins, only older than Orientale basin[Wilhelms, 1987], suggesting a minimal accumulation of ejecta material from sub-sequent large impacts. It has also already been identified as a high priority targetfor addressing several other aspects of lunar science such as lower crust and uppermantle compositions, impact chronology, etc. (e.g. Kohout et al. [2009]; Flahautet al. [2012]).122Chapter 6ConclusionsThe Apollo Passive Seismic Experiment was the first seismic survey installed onthe surface of a planetary body other than the Earth. It recorded more than sevenyears of continuous data, from which over 12, 000 individual seismic events wereidentified. One of the prominent features shared by the seismic signals generatedby these events is the very long-duration scattered coda train that obscures mostwave arrivals except for some of the higher amplitude P- and S-waves. Althoughmore than 40 years of analysis of this dataset has provided us with fundamentalunderstanding of the constitution and structure of the lunar interior, no study hassystematically looked at the information stored in these scattered codas. The workpresented in this thesis aimed at addressing this gap in analysis of the APSE data,as well as developing a new understanding of seismic energy propagation in highlyscattering bodies like the Moon, and potentially other terrestrial planets, moonsand asteroids.The combined conclusions of the four studies comprising this thesis show thatthe coda of scattered seismic signals can yield important information regardingthe scattering and attenuation properties of highly scattering planetary bodies, andthat coda investigations should be included in future analyses of scattered seismicdata. The modeling method I have developed can be used to assess the effects ofinternal structures on the seismic coda and to model synthetics seismograms forvarious surface environments. In this chapter, I summarize my research work andthe resulting conclusions (section 6.1). I then briefly review and address the mainscience questions I aimed to answer with this work (section 6.2). I conclude with ashort discussion of potential future research avenues that could build on the workpresented in this thesis (section 6.3).1236.1. Summary6.1 SummaryChapter 2 presents the first comprehensive analysis of scattering in the lunar seis-mic data. Whereas previous studies analyzed the amplitude decay of single impactevents (e.g. Latham et al. [1970b]), my work systematically looked at the decayproperties of all seismic event types (impacts and moonquakes), in several fre-quency bands, to more fully characterize scattering on the Moon as a function ofsource depth, frequency and epicentral distance. The results support the idea thatscattering dominantly occurs in the near surface megaregolith layer. I also showedthat all frequencies sampled by the APSE instruments are affected by high levelsof scattering, but that the observed scattering intensity increases with increasingfrequencies. This suggests that the megaregolith is composed of scatterers over awide range of scale lengths, with many more small scale scatterers than large scaleones. The study also underlines the importance of local structure on the recordedsignals for events with short epicentral distances. Station 16, which sits on top ofa thick crust with an assumed thicker megaregolith recorded signals with longerdecay times (i.e. more scattered signals) than station 15 for example, which is lo-cated in a basin with a thinner crust and megaregolith. The results from this studywill be useful for the development of future seismic surveys, which should focusthe instrument sensitivity on lower frequencies that are potentially less affected byscattering. The coda decay times measured in this study are also important con-straints for studies attempting to model lunar interior structures, as was done in thestudy presented in Chapter 4.Chapter 3 details the numerical method I have developed to model the trans-mission of seismic energy in highly scattering environments (the PHONON1Dmethod). This method is based on the phonon method of Shearer and Earle [2004],which I have adapted to the lunar context and refined by implementing isotropicscattering, user-defined scatterer scale-length distributions, as well as frequency-dependent and phase-dependent intrinsic attenuation. This chapter also presentsthe first study investigating the effects of large scale velocity, scattering and atten-uation structures on the generation and decay properties of scattered seismic coda.Using the PHONON1D method, I have generated synthetic traces for a range ofplanetary interiors with varying seismic velocity, scattering and attenuation struc-1246.1. Summarytures. Results show that the main controls on the coda decay times are the seismicvelocity profiles (i.e. substantially increased decay times when a crust and a thinlow-velocity regolith are present), the event source depth (larger decay times forshallower events), as well as the intrinsic attenuation as a function of depth andfrequency.In Chapter 4 I used the coda decay constraints from Chapter 2, along with thePHONON1D method detailed in Chapter 3, to identify a suite of lunar interiormodels that generate synthetic seismic signals that share the decay properties ofthe APSE data. I first modeled the APSE instrument effects (narrow frequencybands and 10-bit digitization), and the 1-bit lunar seismic noise in the data. Bycomparing decay times measured with noise-free signals with those measured onthe APSE-corrected signals I demonstrated that APSE instrument effects did notsignificantly alter the decay times. Consequently, the decay times from Chapter 2are representative of the lunar interior structures and can be used as constraints formodeling work. No other work to date has investigated the effect of 10-bit digiti-zation and of the 1-bit noise on the lunar seismograms relative to the signals thatcould be recorded by modern broadband 32-bit instrumentation (to which our cleansynthetic signals are equivalent). I then generated synthetic signals for 32 distinctlunar interior models, all of which used a seismic velocity profile with a crust and athin regolith, that focuses the scattered energy back towards the surface and into thescattering layer. This creates a very highly scattering environment which generatesdecay times on the order of the ones measured in the APSE data. Results indicatethat in such a very highly scattering environment, first order changes in the scatter-ing length scales do not produce an effect on the median decay time values. Thus,median ⌧d values are not the best tool for identifying the lunar scattering structurelength scales, but this could be best achieved by investigating the variations in thed⌧d/d slopes. In such environments, the main control on decay times appears tobe the intrinsic attenuation levels. Results show that only intrinsic attenuation val-ues larger then the ones previously published could yield decay times that matchedthe APSE decay times. However, a more systematic exploration of the parameterspace is needed to constrain lunar attenuation and scattering structures compatiblewith the APSE data.In Chapter 5, I have tested the effects of a laterally varying megaregolith thick-1256.2. The Science Questions Answeredness on the recorded seismic signals. The aim of this study was to identify locationson the lunar surface that minimize the scattering observed in the data, and maxi-mize the science returns of future lunar seismic surveys. Based on the results, Ipropose that the ideal receiver locations are in younger impact events, that haveless accumulated ejecta, a thinner megaregolith and likely thinned crust (as a re-sult of the impact process). Receivers should be placed at a distance away fromthe basin edge that is at least half of the thickness of the megaregolith surroundingthe basin. Synthetic seismograms showed much cleaner signals for seismic sourcedepths located beneath the megaregolith layer (deep moonquakes, and potentiallysome shallow moonquakes). Such events do not experience the same intensivesource-side scattering that affects surface impacts. In these cases, the P- and S-wave arrivals are much more impulsive and easier to pick, and useful secondaryphases (e.g. core phases) have a higher chance of being identified.6.2 The Science Questions AnsweredQ1: What constraints on the interior structure of the Moon or of other highlyscattering bodies can we infer from analyzing the scattered coda of recordedseismic signals?From the analysis of the APSE scattered codas, we can confirm that thescattering of seismic energy dominantly occurs in a near-surface global layer(i.e. the megaregolith). Results suggest that this layer has a size-distributionof ejecta blocks that has many more small scale than large scale scatterers.Using the decay properties of the APSE codas as constraint for models ofseismic energy propagation in highly scattering media, we can also show thatintrinsic attenuation levels in shallow lunar layers are probably lower thanwhat has been suggested in past literature. As such, the analysis of scatteredcoda of recorded seismic signals provide valuable information pertaining toshallow scattering and intrinsic attenuation structures of highly scatteringbodies.Q2: What are the effects of seismic velocity, intrinsic attenuation and scatteringstructures on the scattered coda of seismic signals?1266.3. Future WorkVelocity structures that tend to focus seismic energy near the surface andinto the scattering layer, will dramatically increase the intensity of the ob-served scattering (i.e. increased coda decay times). Such structures can benear-surface low velocity layers (e.g. thin regolith and/or crust), as well aspositive gradients of seismic velocities with depth. When such structuresare present, we found that the main control on the coda decay properties isthe intrinsic attenuation levels at shallow depths. Lower intrinsic attenua-tion will lead to longer decay times, whereas higher attenuation will lead toshorter decay times. Where no such structures are found (e.g. large undiffer-entiated asteroid), the thickness of the scattering layer, as well as the numberdensity of scatterers will also affect the coda. A thick scattering layer and ahigh number density lead to longer decay times.Q3: Are there locations on the lunar surface that could minimize the intensityof scattering in recorded signals, and maximize science returns of futureseismic surveys?Yes. Locations away from the edges of large impact structures, with a thincrust and a thin megaregolith, are ideal for seismic receivers. Younger im-pact structures are better suited candidates because less ejecta material fromsubsequent impacts has accumulated, leading to a thinner megaregolith layerin the structure. Modeling work shows that moving away from the basinedges by a distance that is equivalent to half the thickness of the surroundingmegaregolith is enough to minimize the effects due to leaked scattered en-ergy from the surrounding megaregolith. An example of such a location onthe Moon would be inside the far-side Schro¨dinger basin.6.3 Future WorkStudy of seismic scattering is not new, and a wide range of past studies have devel-oped theoretical approaches and methods to better understand scattering of seismicenergy in a terrestrial context. However no such study has investigated seismicscattering on a global level in environments where scattering processes dominate,as on the Moon. As such, my work provides a stepping stone for future studies1276.3. Future Workof highly scattered seismic data, and for further investigations to better understandthe relationship between surface and internal structures and seismic coda.For example, large impacts on the lunar surface can result in important subsur-face structure modifications (e.g. crustal thinning, mantle uplift, lithological phasechanges). The development of numerical models integrating surface topographyand radial and lateral variations in seismic velocities (2-D velocity profiles) couldtest the effects of these lateral structures on seismic coda and assess whether thesestructures are reflected in the recorded signals.Furthermore, a fully 3-D numerical model could be used to investigate seismicenergy propagation in smaller, non-spherical bodies (e.g. asteroids). An under-standing of scattering on such bodies will allow us to use future seismic studiesto better image and understand their interior structures. Methods already existthat simulate seismic energy transmission in 3-D media. However, similarly tothe 1-D case, computation costs and time for methods calculating the solution tothe wave equation (e.g. SPECFEM3D) at the short length-scales (down to meterlength-scales) and frequencies (10s of Hz) required to appropriately model the lu-nar context are still too high today (at least on the kind of computing tools used forthis thesis work). However, some ray tracing methods in a 3-D media do exist (e.g.Gjøystdal et al. [2002]) that are mostly used nowadays to image seismic data in oiland mineral exploration studies on local and regional scales. These methods couldbe adapted to model seismic energy propagation in highly scattering environmentsusing a similar approach to the one presented here.Finally, questions still remain about the full relationship between intrinsic at-tenuation (Qi), the scattering processes (often expressed as Qs), the resulting waveamplitudes, and coda decay times (Qc), either in low or high scattering environ-ments. For example, studies that assessed the levels of intrinsic attenuation onthe Moon mostly looked at the changes in the signals’ amplitude (or power) asa function of distance and frequency (e.g. Nakamura et al. [1976]; Nakamuraand Koyama [1982]). The main assumption was that variations in amplitude arecaused by intrinsic attenuation processes and geometrical spreading. However, Iobserved in my modeling work that signal amplitude as a function of epicentraldistance could vary greatly for models with similar velocity and intrinsic attenu-ation profiles, but with different scattering properties. This strongly suggests that1286.3. Future Workto properly measure intrinsic attenuations levels with seismic data in a lunar-likeenvironment, one must understand how scattering also affects the signals’ ampli-tudes. 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Equations for coefficients at solid-solid interfaces are from Akiand Richards [2002].In all equations, the first letter (P´ or S´) indicates the incident ray. An acuteaccent on the second letter (P´ or S´) indicates a transmitted ray, whereas a graveaccent (P` or S`) indicates a reflected ray. The letter (P or S) represents the polari-ties of the incident, transmitted and reflected rays (Figure A.1). Note that we keepthe same formulation for upward- and downward-incident rays.A.1 P-SV at Free SurfaceFrom Ben-Menahem and Singh [2000]:D1 = ✓1↵1◆ sin 2i1 sin 2j1 + cos2 2j1140A.1. P-SV at Free SurfaceIncident rays Reflected raysTransmitted raysInterface (or surface)(upward or downward)Figure A.1: Notations used in the transmission and reflection coefficient equations.Upward- and downward-incident rays are indicated with a first P´ or S´, dependingon the initial polarization. The second letter indicates transmitted rays (P´ or S´) orreflected rays (P` or S`).141A.2. Solid-Solid InterfacesP´ P` = D11 "✓1↵1◆2 sin 2i1 sin 2j1  cos2 2j1# (A.1)P´ S` = D11 2✓1↵1◆ sin 2i1 cos 2j1 (A.2)S´P` = D11 ✓1↵1◆ sin 4j1 (A.3)S´S` = P´ P` = D11 "✓1↵1◆2 sin 2i1 sin 2j1  cos2 2j1# (A.4)A.2 Solid-Solid InterfacesFrom Aki and Richards [2002]:P-SV Wavesa = ⇢2(1 222p2) ⇢1(1 221p2), b = ⇢2(1 222p2) + 2⇢121p2,c = ⇢1(1 221p2) + 2⇢222p2, d = 2(⇢222  ⇢121),E = bcos i1↵1 + ccos i2↵2 ,F = bcos j11 + ccos j22 ,G = a dcos i1↵1 cos j22 ,H = a dcos i2↵2 cos j11 ,D = EF + GHp2142A.3. Solid-Liquid InterfacesP´ P` = ✓bcos i1↵1  ccos i2↵2 ◆F  ✓a + dcos i1↵1 cos i12 ◆Hp2D (A.5)P´ S` = 2cos i1↵1 ✓ab + cdcos i2↵2 cos j22 ◆ p↵1/(1D) (A.6)P´ P´ = 2⇢1 cos i1↵1 F↵1/(↵2D) (A.7)P´ S´ = 2⇢1 cos i1↵1 Hp↵1/(2D) (A.8)S´P` = 2cos j11 ✓ab + cdcos i2↵2 cos j22 ◆ p1/(↵1D) (A.9)S´S` = ✓bcos j11  ccos j22 ◆E  ✓a + dcos i2↵2 cos j11 ◆Gp2D (A.10)S´P´ = 2⇢1 cos j11 Gp1/(↵2D) (A.11)S´S´ = 2⇢1 cos j11 E1/(2D) (A.12)SH Waves  = ⇢11 cos j1 + ⇢22 cos j2S´S` = ⇢11 cos j1  ⇢22 cos j2 (A.13)S´S´ = 2⇢11 cos j1 (A.14)A.3 Solid-Liquid InterfacesFrom Ben-Menahem and Singh [2000]:D = ✓↵◆ sin 2is sin 2js cos if + ✏✓↵f↵s◆ cos is + cos2 2js cos if✏ = ⇢f⇢s143A.3. Solid-Liquid InterfacesSolid to Liquid InterfaceP´ P` = D1 "✓ ↵s◆2 sin 2is sin 2js cos if + ✏✓↵f↵s◆ cos is  cos2 2js cos if#(A.15)P´ S` = D1 2✓ ↵s◆ sin 2is cos 2js cos 1f(A.16)P´ P´ = D1 [2 cos is cos 2jf ](A.17)S´P` = D1 ✓ ↵s◆ sin 4js cos if(A.18)S´S` = D1 "✓ ↵s◆2 sin 2is sin 2js cos if  ✏✓↵f↵s◆ cos is  cos2 2js cos if#(A.19)S´P´ = D1 2✓ ↵s◆ cos is sin 2js cos if(A.20)Liquid to Solid InterfaceP´ P` = D1 "✓ ↵s◆2 sin 2is sin 2js cos if + ✏✓↵f↵s◆ cos is  cos2 2js cos if#(A.21)P´ P´ = D1 2✏✓↵f↵s◆ cos 2js cos if(A.22)P´ S´ = D1 4✏✓↵f↵s◆ cos is sin js cos if(A.23)144

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