Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Methods for the treatment of acoustic and absorptive/dispersive wave field measurements Innanen, Kristopher Albert Holm 2003

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2003-859347.pdf [ 8.93MB ]
Metadata
JSON: 831-1.0052683.json
JSON-LD: 831-1.0052683-ld.json
RDF/XML (Pretty): 831-1.0052683-rdf.xml
RDF/JSON: 831-1.0052683-rdf.json
Turtle: 831-1.0052683-turtle.txt
N-Triples: 831-1.0052683-rdf-ntriples.txt
Original Record: 831-1.0052683-source.json
Full Text
831-1.0052683-fulltext.txt
Citation
831-1.0052683.ris

Full Text

METHODS FOR THE TREATMENT OF ACOUSTIC AND ABSORPTIVE/DISPERSIVE WAVE FIELD MEASUREMENTS by Kristoplior Albert Holm lunation B.Se., York University, 1996 M . S c , York University, 1998 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S FOR T H E D E G R E E O F D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Earth and Ocean Sciences) Wo accept, this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A .July 11, 2003 © Kristoplior Albert Holm Innanon, 2003 In presenting this thesis i n partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the L i b r a r y shall make it freely available for reference and study. 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without, my written permission. Department of E a r t h and Ocean Sciences The University Of B r i t i s h Columbia Vancouver, Canada D a t e 2 " c * * ' * Abstract ii Abstract Many recent methods of seismic wave field processing and inversion concern themselves with the fine detail of the amplitude and phase characteristics of measured events. Processes of absorption and dispersion have a strong impact on both; the impact is particularly deleterious to the effective resolution of images created from the data. There is a need to understand the dissipation of seismic wave energy as it affects such methods. I identify: algorithms based on the inverse scattering series, algorithms based on multiresolution analysis, and algorithms based on the estimation of the order of the singularities of seismic data, as requiring this kind of study. As it turns out, these approaches may be cast such that they deal directly with issues of attenuation, to the point where they can be seen as tools for viscoacoustic forward modelling, Q estimation, viscoacoustic inversion, and/or Q compensation. In this thesis I demonstrate these ideas in turn. The forward scattering series is formulated such that a viscoacoustic wave field is represented as an expansion about an acoustic reference; analysis of the con-vergence properties and scattering diagrams are carried out, and it is shown that (i) the attenuated wave field may be generated by the nonlinear interplay of acoustic reference fields, and (ii) the cumulative effect of certain scattering types is responsible for macroscopic wave field properties; also, the basic form of the absorptive/dispersive inversion problem is predicted. Following this, the impact of Q on measurements of the local regularity of a seismic trace, via Lipschitz exponents, is discussed, with the aim of using these exponents as a means to estimate local Q values. The problem of inverse scattering based imaging and inversion is treated next: I present a simple, computable form for the simultaneous imaging and wavespeed inversion of ID acoustic wave field data. This method is applied to ID, normal incidence synthetic data; its sensitivity with respect to contrast, complexity, noise and bandlimited data are concurrently surveyed. I next develop and test a Born inversion for simultaneous contrasts in wavespeed and Q, distinguishing between the results of a pure Born inversion and a further, bootstrap, approach that improves the quality of the linear results. The nonlinear inversion subseries of the inverse scattering series is then cast for simplified viscoacoustic media, to understand the behaviour and implied capabilities of the series/subseries to handle Q. The "communication between events" of the inversion subseries is developed in theory and with numeric examples; it is shown that terms which contain cumulative information from all portions of the data dominate over local terms in determining correct, local, model amplitudes. Finally, I consider the use of a wavelet-based regularization of the operator for viscoacoustic downward continuation. The inclusion of absorption and dispersion in the theory that underlies many seismic methods leads to processing and inversion methods that estimate attenuation parameters and compensate for unwanted effects. These methods are sensitive to amplitude and phase information (by design) and so require low noise, often broadband data; however the methods have responded very favourably to synthetic data tests, and tend to be forgiving to bandlimited data with small amounts of error. Contents i i i Contents Abstract i i Contents i i i List of Tables v i List of Figures v i i Acknowledgements ix 1 Introduction 1 1.1 Thesis Philosophy and Scope 1 1.2 Thesis Overview 2 1.3 A Selective Review of the Theory of Wave Attenuation 3 1.3.1 Waves in an Absorptive Medium 4 1.3.2 Approach I: Imposition of Causality 4 1.3.3 Approach II: Causal Relaxation/Creep Functions 5 1.3.4 Q Estimation and Compensation 6 1.4 General Issues of Terminology 7 2 Viscoacoustic Born Series 9 2.1 Introduction 9 2.2 Viscoacoustic Scattering Potentials 10 2.3 The Terms and Diagrams of the Born Series 11 2.4 Convergence of the Viscoacoustic Born Series 17 2.5 Toward Scattering-based Q Compensation/Estimation 20 2.6 Remarks 23 3 Lipschitz Exponents and Q 24 3.1 Introduction 24 3.2 Lipschitz Exponents and Their Estimation 25 3.3 A Smoothed-Singularity Signal Mode l 26 3.4 Non-linear Inversion for Lipschitz Regularity 28 3.5 The Regularity of the Constant-CJ Impulse Response 32 3.6 Q Estimation from Lipschitz Exponents 37 3.7 The Effect of Bandlimitation 38 3.8 Remarks 42 4 Simultaneous Imaging and Inversion 44 4.1 Introduction and a Useful Notation 44 4.1.1 The Inverse Scattering Series 46 4.1.2 A Useful Notation , 48 4.1.3 Imaging and Inversion 50 4.2 Simultaneous Imaging and Inversion 52 4.2.1 A Quantity Related to Imaging and Inversion 52 Contents iv 4 .2 .2 Mapping Between KndnHn/dzn and an 52 4 .3 Inherent Simplicities and Approximations 54 4 .3 .1 Deriving KndnHn/dzn: Drop a Term, F ind a Pattern 55 4 .3 .2 Simplifying Groups of Terms 56 4 .4 Associated Subseries 58 4 .4 .1 Leading Order Imaging Subseries 58 4 . 4 . 2 Inversion Subseries 5 8 4.5 Computation: Models and Numerical Issues 59 4 .5 .1 Synthetic I D Models, Data and Born Approximations 59 4 . 5 . 2 Brute Implementation of KndnHn/dzn 60 4 .5 .3 Stabilizing the n'th Derivative 64 4 .5 .4 Numerical Examples of Simultaneous Imaging and Inversion 65 4.6 Numeric Effect of Noise and Bandlimitation 70 4 .6 .1 Robustness to Incoherent Noise 70 4 .6 .2 Spectral Extrapolation: Computing a± from Bandlimited Data 74 4 . 7 A Signal-Processing View of KndnHn/dzn 79 4 .8 Remarks 80 5 Linear and Nonlinear Viscoacoustic Inversion 82 5.1 Introduction • 82 5.2 A Viscoacoustic Scattering Potential 84 5.3 I D Normal-Incidence Inversion for Q/Wavespeed 85 5.3.1 A Complex, Frequency Dependent Reflection Coefficient 86 5.3 .2 Numeric Examples 88 5.4 Linearized Estimation of c , Q Profiles 94 5.4.1 Numeric Examples 9 7 5.4 .2 A Bootstrap Approach to Interval Q Estimation 103 5.5 Linear Viscoacoustic Inversion Using Offset I l l 5.5.1 General Linearized Inversion Using Offset I l l 5 .5 .2 Viscoacoustic Inversion Using Offset 113 5.6 Nonlinear Inversion and Interevent Communication 114 5.7 Towards Nonlinear Q Estimation 1 1 7 5.7.1 Another Complex Reflection Coefficient 118 5.7 .2 A Non-attenuating Viscoacoustic Data Set 118 5.7 .3 Implications for Q-Processing in the Imaging Subseries 126 5.8 Remarks 1 2 6 6 Multiresolution Wave Field Continuation 127 6.1 Introduction 127 6.2 Downward Continuation in Attenuating Media 129 6.3 Propagation Depth is a Scaling Parameter 130 6.3.1 Numerical Illustration 131 6.4 Multiresolution Wave Field Reconstruction • 132 6.4.1 Numerical Illustration 133 6.5 The Localization Properties of Inverse Filters 134 6.5.1 The Inverse Filtering Problem 1 3 4 6.5 .2 A Framework for Comparison of Regularization Schemes 136 6.5 .3 Localization Properties of Inverse Filters 1 3 7 6.5.4 Comparison of Regularized Inverse Kernels 138 6.5 .5 Comparison of Constructed Models 141 6.6 Remarks 149 6.6.1 Issues in Wavelet Regularization 149 Contents v 7 Conclusions 1 5 0 7 . 1 Summary • 1 5 0 7 . 2 Continuing Work 1 5 1 Bibliography 1 5 3 A Inverse Scattering Terms 1 5 7 List of Tables vi List of Tables 3.1 Comparison between a and a from the Full and Supp. IR cases 34 3.2 Estimated a vs. Q values: full and bandlimited cases 40 4.1 Earth models used in the following imaging/inversion examples 60 5.1 Test models used for the single interface c, CJ linear inversion 89 5.2 Test models used for the single layer c, Q linear inversion 97 List of Figures vii List of Figures 2.1 ID transmission model and framework for scattering diagrams 12 2.2 Scattering diagrams, for the ID transmission example 13 2.3 Born series terms with associated scattering diagrams 15 2.4 Scattering-types associated with specific aspects of the wave field 16 2.5 Viscoacoustic Born series convergence illustration 20 2.6 Schematic of two models for a ID reflected wave field 21 3.1 The extraction and tabulation of the W T M M 29 3.2 Illustrations of the objective function for a estimation 30 3.3 Illustration of accuracy degradation in a recovery for an impulse 31 3.4 Illustration of accuracy degradation in a recovery for a step function 32 3.5 Components of the time domain approximation for the constant-Q IR 33 3.6 Illustration of inputs for a-recovery investigation 35 3.7 Results of the Q impulse response component test 36 3.8 Logarithmic plot for empirical map Q(a) 37 3.9 Accuracy of a(Q) map at high and low Q 38 3.10 Wavelet transform scales admitted by a Ricker wavelet 40 3.11 The effect of bandlimitation on the distribution of signal energy over scale 41 3.12 Illustration of input used to generate Q(a) empirical map 42 3.13 Bandlimited vs. full bandwidth comparison 42 4.1 Perturbations (XINVI&PINV VS. RI 60 4.2 Synthetic data and its integrals for Model 1 61 4.3 The third-order correction (using Model 1) 62 4.4 The fourth-order correction (using Model 1) 63 4.5 Regularization windows for various parameter choices ' 64 4.6 Regularized derivative operators in the k domain 65 4.7 The cumulative sum of « 100 terms from equation (4.42), using Model 1 66 4.8 The cumulative sum of « 100 terms from equation (4.42), using Model 2 67 4.9 The cumulative sum of « 100 terms from equation (4.42), using Model 3 68 4.10 The cumulative sum of « 100 terms from equation (4.42), using Model 4 69 4.11 Imaging and inversion results with noise (Model 2), realization 1 71 4.12 Imaging and inversion results with noise (Model 2), realization 2 72 4.13 Imaging and inversion results with noise (Model 2), realization 3 73 4.14 Spectrally-extrapolated data • 76 4.15 Imaging and inversion results from extrapolated data (Model 2) 77 4.16 Imaging and inversion results from extrapolated data (Model 4) 78 4.17 The effect of the operator dn(-)n/dzn on the n'th power of the input 81 5.1 The real and imaginary parts of R(k) 87 5.2 Recovered parameters from the Born inversion (Model 1) 90 5.3 Recovered parameters from the Born inversion (Model 2) 91 5.4 Recovered parameters from the Born inversion (Model 3) 92 5.5 Recovered parameters from the Born inversion (Model 4) 93 5.6 Example data set used to demonstrate the c, Q profile inversion 95 List of Figures viii 5.7 Linear c, Q profile inversion (Model 1 ) 98 5.8 Linear c, Q profile inversion (Model 2 ) 99 5.9 Linear c, Q profile inversion (Model 3 ) 100 5.10 Linear c, CJ profile inversion (Model 4 ) 101 5.11 Linear c, Q profile inversion (Model 5 ) 102 5.12 Bootstrap c, Q profile inversion (Model 1) 105 5.13 Bootstrap c, Q profile inversion (Model 2) 106 5.14 Bootstrap c, Q profile inversion (Model 3) ' 107 5.15 Bootstrap c, Q profile inversion (Model 4) 108 5.16 Bootstrap c, Q profile inversion (Model 5) 109 5.17 Bootstrap c, Q profile inversion (Model 6) 110 5.18 Inversion subseries results (contrasts fa = 0.15, fa = 0.25) 120 5.19 Inversion subseries results (contrasts fa = 0.35, fa = 0.55) 121 5.20 Inversion subseries results (contrasts fa = 0.55, fa — 0.35) 122 5.21 Inversion subseries results in profile (contrasts fa = 0.15, fa = 0.25) 123 5.22 Inversion subseries results in profile (contrasts fa = 0.35, fa = 0.55) 124 5.23 Inversion subseries results in profile (contrasts fa = 0.55, fa = 0.35) 125 6.1 Amplitude spectrum \K(kx,ky)\ for an acoustic medium 130 6.2 Amplitude spectrum \K(kx, ky)\ for a viscoacoustic medium 130 6.3 1-D synthetic example of viscoacoustic propagation 132 6.4 1-D multiresolution reconstruction example 134 6.5 The localization of a function and its Fourier transform 138 6.6 Inverse kernel approximations in the kz domain 139 6.7 Inverse kernel approximations in the z domain 141 6.8 Four reconstructions from noisy data (LSQ) 142 6.9 Four reconstructions from noisy data (MRWR) 143 6.10 Four reconstructions from noisy data (TSVD) . . . 144 6.11 Four reconstructions from data with "remote noise" (LSQ) 146 6.12 Four reconstructions from data with "remote noise" (MRWR) 147 6.13 Four reconstructions from data with "remote noise" (TSVD) 148 Acknowledgements ix Acknowledgements This thesis is dedicated to my parents, Kim and Sandra Innanen. It is impossible to spend time around Tad Ulrych and Arthur Weglein and not have their enthusiasm, desire to understand, and devotion to science rub off. Never have any of the usual graduate student symptoms of worry, nerves, frustration, or lack of motivation lasted more than ten seconds into a discussion with either of them. They have enriched (and funded!) my life and work over these past five years; they presented me with a limitless well of ideas to learn about and work on, and then they congratulated me for my cleverness when I did! I first and foremost acknowledge and thank these gentlemen, and my lucky stars for having met them. I am grateful for the guidance, inspired instruction, and encouragement of my committee members Michael Bostock, Bruce Buffett, and Doug Oldenburg; also for those who have lent their expertise and time to teaching me and commenting on work I have done: Ken Matson, Shougen Song, Mauricio Sacchi, Felix Herrmann, Gus Correa, Don Russell, Matt Yedlin, Bill Nickerson, and Neil Hargreaves. I owe a tremendous amount to my M-OSRP friends, including and especially the incomparable Simon Shaw, and to my CDSST mates, Dan Trad, Sam Kaplan, and Rongfeng Zhang, for their input and advice and help on all levels. This department is and was full to overflowing with magnificent people and great friends: Gwenn Flowers, Len Pasion, Kevin Kingdon, Meg Sheffer, Sean Walker (and honourary dept. member Tanya Boughtflower), Kim Welford, Colin Farquharson, Camille L i , Chris Trotter, Parisa Jourabchi, Nigel Phillips, Phil Hammer, Nicolas L'Homme, Tim Creyts, and many others. These people make it easy to come to work early and leave late. The final thanks goes to my wife Kate, who has kept the ship afloat in all ways, with love, support, and the ability to comprehend concepts such as bill payment. She has had faith in this adventure, and is prepared to admit that "all that math" was worthwhile. Chapter 1. Introduction 1 Chapter 1 Introduction The amplitude and phase characteristics of echoes in a reflection seismogram contain a great deal of infor-mation regarding the medium through which they have propagated. The concurrent advancement of seismic data acquisition techniques and preprocessing technology permits more and more sophisticated analysis of these characteristics - and hence, a greater ability to determine the nature of the subsurface - sometimes with less and less reliance on a priori information. When we look closely at amplitude and phase, however, we take on the burden of understanding all of the important effects of the medium on these characteristics of the echo. Amongst these effects are phenomena of absorption and dispersion, which dampen and distort waves as they propagate through any real medium. This thesis concerns itself in large part with the incorporation of attenuation into processing and inversion methods. Almost five years ago as of this writing, Shougen Song and Tad Ulrych introduced me to the concept of attenuation and its effect on the resolution of seismic data, and we worked on a means to stabilize the re-institution of this resolution in downward continuation. Since then, I have been further tantalized by two approaches to seismic data analysis and processing/inversion: the formalism of the inverse scattering series, and the characterization of seismic events by singularity analysis. Both are very much in their infancy as means to treat primaries in reflection data; both still need to be understood in, amongst other things, how they interrogate data, and what they require of the data in terms of noise, bandwidth, aperture, and so on. In considering these approaches, being pioneered in our field by people such as Arthur Weglein and Felix Herrmann, it seemed natural (given my recent interests) to think of attenuation. These methods rely delicately upon the amplitude and/or phase of the events of the data: mustn't we then include the effects of dissipation - so important to the amplitude and phase of the wave field - within the framework of each? Or, employing a more positive spin, how might these frameworks be cast to estimate and compensate for absorption and dispersion effects in our data? Consider a plucked violin string: a note sounds for a few seconds and then dies away. Waves attenuate in time. If this were not so, we would all still be listening to the note. Consider a long string, sinusoidally forced at one end: near the other end the amplitude of the oscillation is greatly reduced. Waves attenuate in space. These physical situations are not difficult to visualize, or reproduce. Suppose we excite a propagating wave packet in the string. How does the natural attenuation of the medium imprint itself on the packet? Does the character of the packet tell us more about the medium because it bears this imprint? Or does it tell us less about the medium because the imprinting has decayed the amplitudes? Let us take the attitude that much information is there to be interpreted (and so we'll use it), and much of the "lost" information has merely been reduced (and so we'll try to partially restore it). With apologies to William Shakespeare, I adopt the mind-set that Nothing of him that doth fade But doth suffer a sea-change Into something rich and strange. 1.1 Thesis Philosophy and Scope The goal of this research is, first, to develop and understand the role of dissipation models within the frameworks of inverse scattering, and wavelet and singularity analysis. Secondly it is to develop and test basic algorithms for the processing and inversion of data. The approach I take is to adopt the simplest possible mathematical description of wave propagation, alter it to include absorption and dispersion, and proceed to investigate the effects. This usually involves ID, normal incidence physical configurations, the behaviour of Chapter 1. Introduction 2 which are assumed to be governed by the constant-density acoustic wave equation. Alternatively it involves assuming the convolutional model of seismic data (i.e., a reflectivity series convolved with a wavelet), and altering the impulse response to impose absorption and dispersion on the trace. In many cases this approach has suggested forms for algorithms, and formulations for linear and non-linear inverse methods. The practical-minded reader might well ask: I add dissipation to the acoustic wave equation, but ignore, amongst other things, multidimensional behaviour, elastic behaviour (e.g., conversions) and anisotropy. Do I really think I am that much closer to modelling the true Earth? The answer is no - not as such. But the point here is to gain insight into the basic way dissipation enters and affects these methods. Algorithms developed herein are meant to either be used when dissipation and/or acoustic behaviour dominate in the data, or, when necessary, be included as part of a more complicated description. I go so far as to generate ID normal-incidence numerical results of simultaneous imaging and inversion, using the inverse scattering series (see chapter 4 for development and review). This too requires a word of explanation, since there are other ways to invert ID normal incidence wave fields. For instance, a different formulation of the inverse scattering problem, requiring the Gel'fand Levitan as well as the Lippmann-Schwinger integral representation of the wave field (Ware and Aki, 1968; Weglein, 1985; Stolt and Jacobs, 1981), is possible in ID via a change of variables that alters the effective dependence of the scattering potential. The reason I consider ID normal incidence configurations is so that, again, the underlying mechanisms of the methods as they apply to acoustic/viscoacoustic wave fields are easier to understand and analyse. The behaviour of the inverse scattering series in ID is especially interesting, however, only because it generalizes to multiple dimensions. The Gel'fand Levitan method, as applied to acoustic/elastic equations, does not. The generalization of the results of chapters 4 and 5 to multiple dimensions will not be immediate or trivial, nor are they guaranteed to be as simple or work as well, but nevertheless, this research is done with that ultimate step in mind. 1.2 Thesis Overview The chapters in this thesis are organized more or less to reflect the transition from viscoacoustic (Q) forward modelling, to Q estimation and/or inversion, and finally to Q compensation. A brief overview of each chapter follows. Viscoacoustic Born Series The Born series, or forward scattering series, as applied to wave theory, is an expression for the wave field due to a given source and medium; it is concerned with forward modelling. The representation is based on perturbation theory, i.e. it involves explicit solutions of the wave equation only for simple reference media. The true wave field is an expansion in series about these reference solutions. Physically the series terms have interpretations as propagations from point to point in the reference medium, with true, non-reference behaviour occurring only as a consequence of the cumulative, nonlinear, interplay of these propagations. I show in this chapter, using ID examples, that an attenuated wave field may be synthesized using acoustic reference media. This is compelling in the sense that the correct phase and amplitude distortions are produced via a formalism in which no attenuation ever occurs. I proceed to address the questions: what are the convergence properties of such a series? Do the different interaction types implied by the terms in the series combine to alter macroscopic properties of the true wave field? What can the Born series tell us about the eventual inclusion of attenuation in the corresponding inverse problem? Lipschitz Exponents and Q I then turn from scattering theory, temporarily, to summarize a largely empirical investigation into the rela-tionship between absorption/dispersion and regularity estimates from seismic events. Local signal regularity, characterized via Lipschitz or Holder exponents, is a generalized measure of the local differentiability of a function. As such it has the ability to characterize edges (or discontinuities, or singularities) by their order. Intuitively the lengthening of the tail, and the changing of the shape, of a causally attenuated echo suggests that the regularity of these echoes should change with time. I use arguments based on existing Q models Chapter 1. Introduction 3 to develop a mapping between local Lipschitz exponents, computable from the wavelet transform of a trace, and local Q values. This may prove useful as a framework for Q estimation. Simultaneous Imaging and Inversion In this chapter I consider the inverse scattering series and its application to the imaging and inversion of the primaries of seismic reflection data. I do not yet deal with attenuation. Again using ID normal incidence solutions for the constant density acoustic wave equation, I consider the consequences to imaging/inversion resulting from certain approximations (i.e., the dropping of classes of series terms). I deduce a simple form for the n'th term in a series which corrects the amplitude and location of the Born inverse, and compute numeric examples for a range of test models that differ in contrast and complexity. I use this computable form, which simultaneously carries out tasks of imaging and inversion, as a platform upon which to test the sensitivity of the series to incoherent noise, and to test the use of spectral extension and inversion techniques to compensate for bandlimitation. I speculate, by considering the simultaneous imaging and inversion formula in its capacity as an operator which acts on the Born approximation, on what basic properties it has which might transfer, conceptually and practically, to a multidimensional form. Linear and Nonlinear Viscoacoustic Inversion I next consider application of inverse scattering to the inversion of a viscoacoustic wave field. The first part of the chapter involves casting a ID normal incidence problem such that a linearized (Born) inversion for simultaneous contrasts in Q and wavespeed may be carried out. This is first done for the simple case of a single reflection from a single interface; it is then extended so that it permits multiple interfaces (i.e. interval Q and wavespeed inversion) to be treated. In both cases overdetermined problems, involving multiple frequency components of the data, are solved; the framework is made tractable by assuming a spatial form for the contrasts dictated by the data. I then alter the interval wavespeed-Q inversion to produce a bootstrap inversion scheme; this makes the most of the linear inversion as a standalone algorithm for viscoacoustic inversion. The second part of the chapter involves going "beyond Born", and utilizing the higher order terms of the so-called inversion subseries of the inverse scattering series. Synthetic data sets based on contrasts in a friction-type Q model are used as input, and the results are compared with the predictions of the forward scattering series investigation (chapter 2). Multiresolution Wave Field Continuation Finally I consider the problem of Q compensation, the re-institution of the resolution that is naturally suppressed in viscoacoustic wave propagation. I make some broad comments on the resemblance of the upward continuation operator for viscoacoustic media to the scale function of multiresolution theory - both act to reduce the finer scales of a signal. I then review the idea of a multiresolution wave field reconstruction, that uses a wavelet basis to restore resolution in a manner that mirrors the scale-operator like loss (while naturally stabilizing the process). This work is due to myself and Shougen Song, and is based almost entirely on his ideas. I then take this framework for wave field reconstruction and compare it against two other well known ways of regularizing ill-conditioned deconvolution problems. I show that the wavelet method balances the localization properties of inverse filtering schemes in the space/wavenumber domains. As such it appears to be robust to a variety of forms of data inaccuracy to which other methods more easily fall prey. 1.3 A Selective Review of the Theory of Wave Attenuation I make regular use of two quite different absorption/dispersion laws in this thesis. I include here a brief overview of these models, which I refer to as the friction model and the constant Q model. For a more complete development of some of the ideas discussed here, the reader is referred to Einar Kjartansson's thesis (1979), Aki and Richards (2002), Cerveny (2001) (in the context of ray theory), and the classic paper of Futterman (1962). Chapter 1. Introduction 4 1.3.1 Waves in an Absorptive Medium Morse and Feshbach (1953) show that for a vibrating string, the addition of a single term to the wave equation, proportional to the displacement velocity, results in dampened solutions in space and time. Such a force has the physical interpretation of being due to viscous friction inherent to the medium in which the string vibrates. In other words, with the addition of one further parameter describing the medium (the constant of proportionality for this new term), a dissipative element is added to the problem of wave motion. The equation for displacement if) becomes, in free space: This kind of equation of motion is, conceptually, the basis for the description of the attenuative propagation of seismic wavelets. This wave equation has the frequency-domain solution = e - ^ V ^ * (1.2) where to is the angular frequency, CQ is the wavespeed, z is the propagation distance, and a is the attenuation parameter that arises from the friction-like term R. In most developments the form for the solution ip given by equation (1.2) is assumed (i.e. the relationship between a and R implied by these introductory comments is not pursued); the frequency dependence of a is then determined via physical arguments more apropos for seismic studies. The parameter Q, or the Quality factor is, like the attenuation parameter a, an inherent property of the medium, and is probably the most often used descriptor of attenuation. It is defined as where A is the maximum amplitude of a wave field for a particular frequency, and AA is the change in that amplitude per cycle. By this definition, Q and a are related by Knopoff (1964) notes that for most solids, within the measured frequency band Q ^ Q(u), or is at most weakly dependent on ui. From equation (1.4) this requires that a be linear in frequency. The model encapsulated by equations (1.2) and (1.4), which I refer to as the friction model in this thesis, is attractive in its simplicity, and in its applicability (it reproduces the empirical pulse broadening seen in seismic data, and is based on a constant Q). With this simplicity, however, comes inaccuracy; the friction model fails to stand up to more detailed scrutiny both physically and with respect to empirical studies of seismic data. White (1983) and Aki and Richards (2002) summarize the theoretical argument that a model like that of equation (1.2) is non-physical because it predicts noncausal impulses. That is to say, measurement of the waveform of equation (1.2) is theoretically possible before a sufficient amount of time has elapsed - in fact, for short propagation distances, this solution predicts a non-zero response prior to the activation of the source. The friction model also predicts a "rise time", a measure of the time from first onset of the response to its peak, that is exactly equal to the "decay time" of the pulse - i.e. it is symmetric. This disagrees with empirical results (Aki and Richards, 2002), which suggest that the rise time is much faster; responses tend to be long-tailed. Therefore in the context of seismic studies, much Q modelling work is involved with adapting the basic form of equation (1.2) to (i) produce a causal response, and (ii) keep the associated Q more or less constant within the frequency range of measurement. 1.3.2 Approach I: Imposition of Causality The "trouble" with the friction model arises from the chosen form for the propagation wavenumber which allows absorption through the parameter a: Chapter 1. Introduction 5 7 W / \ Ul k = -—h ia(ui) = — Co c 0 1 + 2 Q (1.5) The pulse associated with this propagation wavenumber is symmetric about the arrival time Z/CQ. One approach for correcting this model (while maintaining much of its basic character), is to appeal to the analogous problem in electromagnetics, in which Kramers (1927) and Kronig (1927) developed the theory of causal absorption and dispersion pairs. The idea is to allow Co = CQ(UJ), and demand that the real and imaginary components of a be such that the effective impulse response is zero before t = 0. A new wavenumber (Aki and Richards, 2002) is defined: K LO C0(LO) + ia(u>), which, by the wave theory analogue to the Kramers-Kronig relations, implies LO LO C0(LO) C 0 + H[a(to)}, (1.6) (1.7) or K co •H[a(co)} + ia(w), (1-where H[-] is the Hilbert transform. Any a(u) that honours this Hilbert transform pair will produce a causal waveform, but the interested researcher is still left with the problem of choosing an a(u>) that results in an almost constant Q. The requisite linear a(u>) does not lead to a convergent Hilbert transform (Aki and Richards, 2002), which is another way of saying that no linear attenuation parameter can result in a causal response. The final tack, as espoused in, for instance, Azimi et al. (1968) tends to be choosing a Hilbert transform pair that closely corresponds to a constant Q. It is possible to do so in such a way that Q is constant over a reasonable seismic frequency band (e.g. Futterman (1962)). Aki and Richards (2002) note that many approaches result in combined absorption and dispersion pairs that, in the context of Q, and on a reasonable seismic frequency band, amount to the replacement of the wavenumber k = LO/CQ by co 1 + 2 Q 1 ln (1.9) where cor is a reference frequency, at which the wave field propagates with the phase velocity CQ. 1.3.3 Approach II: Causal Relaxation/Creep Functions A related approach (Aki and Richards, 2002, e.g.) for constructing appropriate Q models for seismic studies gives rise to a particular model, that of Kjartansson (1979), which is attractively simple and reproduces many important aspects of observed attenuation phenomena. The approach begins by defining the stress-strain relations such that (i) the stress response to strain is causal (and vice versa), and (ii) the associated creep and relaxation functions have an assumed mathematical form that guarantees a constant Q. The constitutive relations that give rise to a ID elastic wave equation are a re-statement of Hooke's Law, namely that the time dependence of the stress a(t) of a medium is related to the time dependence of the strain e(t) by a(t) = Me(t), (1.10) where M is the modulus. Hooke's law states that the instantaneous stress is proportional to the instantaneous strain. A more general relationship would allow the current value of the stress to depend on the current and past values of the strain, and vice versa. Maintaining linearity, a relationship embodying these ideas is a(t) = m(t) * e(t), (1.11) Chapter 1. Introduction 6 where * denotes convolution. (Taking the Fourier transform of m(t) produces a modulus M(UJ) that, in contrast to Hooke's Law, is complex and frequency dependent.) This modulus is the time derivative of a quantity called the "relaxation function", whose convolutional inverse is the "creep function". The creep function is the strain that results from a unit step in stress, and the relaxation function is the stress that results from a unit step in strain. Kjartansson (1979) chooses a creep function that is (i) zero prior to the application of the unit step in stress, and (ii) based on a power law in time. The argument is that the latter implies a constant loss per cycle, which in turn implies a constant Q. The result is a form for the frequency dependence of the generalized modulus function: M(UJ) = M0 where MQ and wr are reference quantities, and ILO 2 7 (1.12) 1 _ i / 1 7 = - tan — TT \Q (1.13) Letting MQ imply a reference wavespeed CQ , this finally results in a constant Q transfer function B(oS) of B(OJ) = exp | — Kjartansson (1979) remarks that provided Co 1 - 7 u> 0JR tan (JY) + «sgn(w) 1 ( UJ -75 In - ) « 1 , TTQ \U!R (1.14) (1.15) neglecting terms quadratic in 1/Q allows the transfer function of equation (1.14) to be due to a wavenumber of the form K that is, in the bandlimited approximation, it is equivalent to many other models of attenuative propagation. The agreement of the propagation wavenumber of equation (1.16) with many models of attenuation and dispersion makes it a reasonable one to adopt in the estimation/inversion/compensation methods developed in this thesis. Even though in the form of equation (1.16) it more closely resembles the so-called "nearly constant-Q" models, hereafter I refer to propagation according to the wavenumber of equation (1.16) as the constant Q model. 1.3.4 Q Estimation and Compensation The treatment of problems of absorption and dispersion in seismic data processing is roughly divisible into three tasks: modelling, estimation, and compensation. The first task, modelling, is side-stepped in this thesis by adopting particular Q models: the constant-Q model, and the friction model, discussed above. It is worth mentioning, in addition to this approach of modelling of causal friction mechanisms, that compelling arguments have been made that apparent attenuation of seismic data can occur due to medium heterogeneity at scales finer than the wavelet length scale (O'Doherty and Anstey, 1971; Schoenberger and Levin, 1974; Spencer et al., 1982). The issue of efficient modelling of both dispersive and absorptive/dispersive seismic data is addressed by Varela et al. (1993). The practical problem of Q estimation varies strongly depending on the experimental milieu. There is a large body of literature regarding laboratory measurement (see, for example, the review of Johnston (1979)). For VSP data and reflection data, methods tend to be based on various incarnations of spectral ratio computations (Tonn, 1991; Dasgupta and Clark, 1998; Zhang and Ulrych, 2002). Tonn (1991) also discusses Q estimation via rise-time measurements. Chapter 1. Introduction 7 Q-compensation may be accomplished in the Fourier domain, e.g. Hargreaves and Calvert (1991), or may be the result of a stationary inverse filter applied to stretched data (Bickel, 1993); also the seismic deconvolution problem, e.g. (Robinson and Treitel, 1980), may be re-formulated to handle non-stationary wavelets (Clarke, 1968; Margrave, 1997). Q compensation is strongly dependent on Q estimation, as it is an inherently ill-conditioned problem that is sensitive to noise and input Q factors (Song et al., 1999; Song and Innanen, 2002). One might consider accurate Q-estimation an indispensable precursor to the compensation step. 1.4 General Issues of Terminology When one draws ideas and formalisms from different fields of study, often there is a clash of terms. This is certainly the case here. Rather than solve potential problems by engaging in wholesale re-definition of the terms found in the literature, not to mention using the entire Greek alphabet, I have chosen to permit terms to have different meanings from chapter to chapter. For instance, in the chapters based on scattering theory the term a denotes the perturbation of the medium away from a reference state; whereas, in the chapter on Lipschitz exponents for Q estimation a is the local regularity of the signal. This is done to adhere to the standard terminology associated with these fields of study, (perhaps) at the expense of a certain continuity. Within chapters the terminology is self-consistent, and at the beginning of each chapter a table of terms and their significance, mathematically and/or physically, is included. I have adopted the term viscoacoustic in many of the chapters to follow, because I apply the constant Q propagation law to wave theoretic methods that are based on the constant density acoustic wave equation. The variability in terminology spanning scattering theory, wave field continuation, and the attenuation parameter/Q terminology discussed here, as well as the vagaries of the mathematics of individual components of this thesis, makes it necessary to (often subtly) alter the way the two absorption/dispersion models are written down. I finish this section with a listing of any altered attenuation parameter terminology used in the chapters to follow. In most cases a parameter (3, closely related to the reciprocal of Q, is introduced. Viscoacoustic Born Series To investigate the convergence properties of the viscoacoustic series I make use of scattering potentials that are based on propagation wavenumbers in which (3 = J3R + i(3j is a complex quantity. With the choice /3R = 0 and (3i = 1/2Q, equation (1.17) corresponds to the friction model of propagation; with the choice /3R = — 1/TTQ ln(w/w r) and /3j = 1/2Q it corresponds to the constant Q model. Linear and Nonlinear Viscoacoustic Inversion To linearly invert for viscoacoustic parameters from seismic data I consider the propagation wavenumber in which F(LO) is a complex, frequency dependent quantity, and, this time, f3 is a real positive number. This terminology corresponds to the constant Q model when f3 = 1/Q and F(w) = i/2 — l / i ln (w/w r ) . Later in the chapter I revert to the simpler friction model as I investigate a nonlinear viscoacoustic inversion approach. This is a dispersion-free model, and amounts to adopting equation (1.18) and setting F(to) = i/2. Multiresolution Wave Field Continuation k = (1.17) A = - [ 1 + F H J 3 ] (1.18) The ill-conditioned nature of the problem of viscoacoustic downward continuation of wave fields is due to the absorptive part of attenuation - "dispersion compensation" per se is not an unstable problem. Therefore, to Chapter 1. Introduction 8 explore the ill-conditioned problem, it is essentially equivalent to consider the friction model and the constant Q model. Here the wavenumber k = —[l + i0], (1.19) co where (3 = 1/2Q, is adopted for the first part of the chapter, in which the correspondence between the prop-agation operator for viscoacoustic media is related to the scaling kernel in the multiresolution representation of signals. The numerical examples in the latter part of the chapter utilize the full constant Q model. Chapter 2. Viscoacoustic Born Series 9 Chapter 2 V i s c o a c o u s t i c B o r n S e r i e s Term Signifies 4>s(z\zs; k) Wave field at z due to source at zs GQ(Z\ZS; k) Green's function at z dure to source at zs c 0, c(z) Reference, non-reference wavespeed profiles k0, k(z) Reference, non-reference wavenumber profiles ki Non-reference wavenumber u>/c\ a(z) General perturbation acq{z) Perturbation due to coincident Q, c contrasts aq(z) Perturbation due to Q contrast only eti Perturbation amplitude, single interface model V(k0,z) General ID (visco) acoustic scattering potential for acoustic reference tpj(z\zs;k) Wave field component: j ' t h order in the model ipjk(z\zs ;k) Wave field component: fc'th scattering geometry, j ' t h order P(">,z) General attenuation parameter Q Quality factor 0Jr Reference frequency, intrinsic to constant Q model H(z-zx) "Right-opening" Heaviside function, step at z\ R Reflection coefficient, single interface model T Transmission coefficient, single interface model 7 Ratio of non-reference/reference wavenumbers 1ppr2(z\z s j k) 2nd primary in the two-interface model R2 Reflection coefficient, lower interface in the two-interface model TQI, TW Downward, upward transmission coefficients, two-interface model 2.1 Introduction This chapter involves refining our understanding of how the forward scattering series, or Born series, functions in media which attenuate the wave field. The first section develops some ideas of how various types of scattering diagram conspire to construct aspects of the viscoacoustic wave field - for instance, the absorptive propagation effects, and the negative of the direct wave. This portion in essence recapitulates the work of Matson (1996), showing it to hold for absorptive/dispersive media. The second section is more focused on predicting the nature of an inverse scattering-based scheme for Q compensation and estimation. In it I consider the role of the so-called "separated" and "self-interacting" scatter-type comparatively, contrasting a purely acoustic case with a purely absorptive one. The results suggest where in the inverse series one must look for tools to accomplish these tasks of estimation and compensation. To begin, I review the Born series wave field representation, and construct some appropriate scattering potentials. The development follows that of Weglein (1985). The Born series representation of a wave field arises from a perturbation of the coefficients of the wave equation around a reference value. For instance, in a ID constant density acoustic medium, the equation tp(z\zs\uj) = 5(z - z3), (2.1) ~ d2 | OJ dz 2 c2(z) Chapter 2, Viscoacoustic Born Series 10 which describes the behaviour of the wave field IP(Z\ZS;LO), measured at z and due to an impulsive source at za, in a medium characterized by the wavespeed profile c(z), is re-written dz2 + fcg(l-a(*)) ip(z\zs;k0) = 6(z - zs), (2.2) where k\= to1 jc\. Usually the reference model, here represented by the constant wavespeed Co, is assumed 2 to be known, so the perturbation ot(z) = 1 — is the de facto model. The scattering potential V(k0,z) is the difference between the "true" and reference wave operators: V(k0,z) - d2 LO2 1 " d2 LU2] dz2 ~ h c2(z)\ dz2 " = kQa(z). (2.3) The acoustic Born series is a representation of the solution of equation (2.1) in orders of V(ko,z). A straightforward derivation involves placing the term V(ko, z)ip(z\zs; ko), onto the right-hand side of equation (2.2), multiplying these "sources" by the Green's function Go{z\zs\ ko), which satisfies dz2 + k20 G0(z\zs;u>) = 5{z - zs), and integrating: /oo G0(z\z'; k0)V(k0, z')ip(z'\zs; k0)dz'. -OO (2-4) (2.5) Finally, equation (2.5), which is the ID version of the Lippmann-Schwinger equation, is expanded to produce the Born series: ip(z\zs; k0) = il>o(z\zs; k0) + ipi(z\zs; k0) + ip2(z\zs; k0) + i>3(z\z3; k0) + (2-6) where ipo(z\zs;k0) = G0(z\zs;k0), poo ipi(z\zs; k0) = / G0{z\z';ko)V(kQ,z')Go(z'\zs;k0)dz', — OO OO (2.7) i>2(z\zt / POO G0(z\z'; k0)V(k0, z') / G0(z'\z"; k0)V(k0, z")G0(z"\zs; k0)dz"dz\ - oo ./ — oo etc. Clearly, tj)\ is first order in V, whereas V>2 is second order in V, and so forth. Because the Green's function GQ(Z\Z'\ ko) describes propagation in the reference medium from point z' to point z, the term ip^ may be interpreted as a wave field which has propagated in the reference medium N + 1 times, and has ./V times interacted with the perturbation a(z) via the scattering potential. Since the reference medium is characterized by constant wavespeed CQ , the Green's function is, e.g. (DeSanto, 1993): G0(z\zs-k0) aika\z — zs\ 2ikn (2.8) Therefore, having defined V(ko,z) via some desired Earth model, and knowing Go, one may compute as many terms as desired in equation (2.6) to approximate the solution. 2.2 Viscoacoustic Scattering Potentials One may define a wide variety of scattering potentials, differing in what the "true" medium properties are with respect to the reference medium. Here I consider two variants on the acoustic case, each utilizing wavenumbers which permit attenuation to be modelled in addition to acoustic behaviour. This requires moving away from the acoustic k0 = LO/CQ, and adopting for the true medium: Chapter 2. Viscoacoustic Born Series 11 k(z) c(z) [l + /3{w,z)}, (2.9) where (3(UJ,Z), a complex number, is the spatial distribution of an attenuation parameter which instills absorption and dispersion character into the wave field. From equation (2.9), and guided by equation (2.3), two related scattering potentials are defined. The first corresponds to media in which both wavespeed contrasts and attenuation contrasts are permitted: acq(z) = 1 k 2(z) K, 0 c 2(z) [1 + 2(3(UJ,Z) + (3 2(UJ,Z)} , (2.10) The second corresponds to a medium in which the wavespeed is constant throughout, and contrasts are only permitted in /?. Since in such a case c(z) = CQ , from equation (2.10) the form is aq(z) = -2(3{oj,z)-l32{uj,z). (2.11) The remarkable simplicity of this perturbation and its association with the actual value of (3 arises partly because there is no attenuation in the reference medium, and also because the attenuating wavenumber of equation (2.9) already resembles a perturbation away from the acoustic case. In this chapter I consider only cases in which the reference medium is acoustic, and non-attenuative. If I choose the constant Q model, for instance, j3 becomes P(OJ,Z) In 2Q(z) irQ(z) V < 4 where ur is a chosen reference frequency. This arises due to the form of a ID constant Q wavenumber: (2.12) OJ k= — CO 1 + * ^ In ( W 2Q TTQ \ojr (2.13) 2.3 The Terms and Diagrams of the Born Series The Born series, equation (2.6), is a decomposition of the full wave field into components distinguished by the number of times they have interacted with the non-reference medium; each interaction is separated by a propagation in the reference medium. Scattering diagrams, which, as used here, are the wave-theoretic analogues of the Feynman diagrams of quantum field theory, e.g. (Weglein et al., 2002), arise from further decomposition of the terms in this series. In this section, the scattering diagrams associated with a simple ID transmission case are developed from the form of the Born series integrals, and linked to the explicitly computed terms in the series. The integrals which give rise to the terms tpn in equation (2.6) subdivide because of the geometric constraints on the propagation imposed by the Green's functions (equation 2.8). To see this, consider equations (2.7) with the condition that the source zs is less than all z for which the perturbation is non-zero. Then: p i f c 0 ( z - z „ ) ^(4? 8 ; fco) = — • (2-14) ZIKQ Also, 1 Cz 1 f°° i M*W, ko) = -±eifc°<*-*"> J_ a(z')dz' - - e - * > ( ^ ) ^  ei2fc„* a { z ! ) d z > = ^11 + ^12-The integrals are likewise broken up in the computation of 4>2-1p2{z\zs; ko) = V»21 + 1p22 + ^23 + i>24, (2-16) Chapter 2. Viscoacoustic Born Series 12 where 'tp2i(z\zs;k0) ip22{z\zs; k0) ip23(z\zs; k0) ip24(z\zs;k0) In the "new" series, 1p(z\za; k0) =tpO+ Ipll + tpl2 + tp21 + ^ 2 2 + ^ 2 3 + 1p24 + ••; (2.18) the terms have been divided into, first, the number of interactions, and second, the relative location of the interactions. For instance, the term tp23 represents the totality of second order interactions in which z' > z and z' > z"; whence arise the scattering diagrams, symbolizing this particular scattering geometry. Since the addition of a further order of interaction, e.g. going from second to third order, involves the inclusion of one further Green's function, which must be subdivided into two cases, in general the n'th order term produces 2™ sub-terms. In this way, the eight terms of tps, followed by the 16 terms of ip4, and so forth may be produced. Scattering diagrams may now be drawn, one for each term in equation (2.18). This is preceded by a description of the chosen ID model. reference (§> Zs (source) Z medium i • c(z) =Co Zi non-reference ® z (receiver) medium c(z),M m Zs (source) z 1 reference \ T medium \ Zi \ non-reference \ 0 z (receiver) medium \ y Z Figure 2.1: ID transmission model and framework for scattering diagrams: (a) a homogeneous acoustic wholespace is chosen as the reference medium, in which the source is located; a homogeneous (visco-)acoustic half-space is chosen as the non-reference medium, in which the receiver is located; the step-like interface is located at zi; (b) an example (^z) of the form and construction of the scattering diagrams is superimposed on the chosen transmission model; the labels included in this example are assumed but omitted in subsequent diagrams. The model is a ID homogeneous acoustic whole-space, characterized by constant density and the wavespeed = \ik0e iko(z — z<< \ikoeika{z~z-"] f a(z') j a(z")dz"dz\ J — oo J — oo /Z POO e~i2k"z'a(z') / ei2knZ"a(z")dz"dz', -oo Jz' 1 f°° I fz' - i f c 0 e - i f e o ( z + z ' ) / ei2k"z a(z') / a(z")dz"dz', 8 Jz J-oo -| POO POO ^ f c 0 e - i f c o ( z + z ' ) J a(z') J ei2knZ"a(z")dz"dz'. (2.17) Chapter 2. Viscoacoustic Born Series 13 Co- Overlaying this reference whole-space is the perturbation, a homogeneous half-space in which the medium parameters, i.e. c(z) and/or (3(z), are constant and may or may not differ from that of the reference medium. A source is located at zs = 0, in the reference medium, and a receiver is located at z, in the non-reference medium, thus mimicking a transmission experiment; the interface between the reference medium and the non-reference medium is at z\. The model is illustrated in Figure 2.1a. This configuration is geometri-cally identical to one used by Matson (1996), such that the mapping developed therein, from Born series to closed-form, may be utilized. Figure 2.1b illustrates, as an example, the scattering diagram associated with ^ 2 3 > within the context of the chosen model. In later diagrams the labels are omitted; nevertheless, all propagations go from zs to z. Further, it is worth mentioning that in this ID model the lateral separation of scattering points has no meaning other than as an aid to visualization. Figure 2.2 contains the scattering diagrams associated with ipo (one diagram), ipi (two diagrams), i/j2 (four diagrams), and -03 (eight diagrams), pictured without the ID model in the background. As in Figure 2.1b, the "top-left" endpoint is za, and the "bottom-right" is z. The numbering, for instance in the second order 21, 22, 23, and 24 is not significant in the sense of the meaning of the diagrams; it is merely an enumeration of distinct terms. Figure 2.2: Scattering diagrams, for the ID transmission example of Figure la, are illustrated up to third order in the perturbation a. (a) O'th order, ipoi (b) 1st order, ipn - i/>i2,' (c) 2nd order, tp2i - i/>24,' (d) the eight diagrams of the 3rd order. Those diagrams associated with terms that have been explicitly computed in this paper are labelled as in the calculations. The next step is to evaluate these integrals, given the chosen transmission model of Figure 2.1, and to use the Matson (1996) approach to produce the closed form expression for the transmitted wave field, all the while tracking the diagrams through the organization and collapsing of series. To remain general in the sense of acoustic/viscoacoustic perturbation, the perturbation a(z) is assigned the spatial dependence a{z) = a1H(z-z1), (2.19) such that the details of the contrast (i.e. acoustic or viscoacoustic) are hidden in the amplitude c*i, which could have a form like either of equations (2.10) or (2.11). The step-like behaviour is explicitly present in the Heaviside function H(z — z\). Substituting equation (2.19) and zs — 0 into the terms of equation (2.18) results in: Chapter 2. Viscoacoustic Born Series 14 2iko ' o.\ e ikoZ •0n(z|O; k0) = — - — — 2 i k o ( z - zi), 4 AlKo ipn(z\0;ko) ai e ikoZ T 2ik0 ' a? e lkaz tf>2i(*|0; k0) = -Yhko~[lko{z ~ Zl)]2' (2'20) a? e ik" z ^ 2 2 ( ^ | 0 ; fc 0 ) = - — — — 2ik0(z - z i ) , l b liko a ? e i f c , ) 2 fe(^|0; fc0) = - — — — 2iko(z - zx), l b Z l f e n ^24(^0 ; / c 0 ) 16 2ik o Figure 2.3 contains an organization of the terms comprising the Born series representation of the wave field ip(z\0; ko), including those of equations (2.20); above the low order terms, the associated scattering diagram is included. These terms are, together, ip(z\0;k0) = „ikaz 2iko — 2iko(z — Zi) -kl{z-zif eikaz f 1 ik0 \i ai + - a x 128 1 1 , 5 , 8 a i + l 6 a i + 128" 1 + -(2.21) 1 -A + 16 1 64 1 256 To "collapse" equation (2.21) to closed form, two definitions are made: and R = 2 { 1 ' ^ 2 - ^ - ^ 1 / 2 ) , 7 =(l-a 1 ) 1 / 2 = ^ 1 ko (2.22) (2.23) where ki is the wavenumber in the non-reference medium (which can describe acoustic or viscoacoustic propagation). Noting that 7 may be expanded in Taylor series: 7 = (1 - a i ) 1/2 l - \ a i 16 -a. 128 and keeping in mind the terms found in Figure 2.3, one may write 1 16 5 3 _ R 128 Q l + - - 2 ' 9 1 6 a i + 6 4 a i + 2 5 6 a I 7 ) R (2.24) (2.25) (2.26) and so on. The series in powers of ai, i.e. the rows of Figure 2.3, therefore collapse into single terms in R and 1 — 7. In fact: Chapter 2. Viscoacoustic Born Series 15 V ( z l O ; k „ ) = , ik„z 2ik„ iko - a , ^(z-z,) -kbl(z-zi): TV -^-(z-^)l 128 ' 1 __2 - a , + —or + — a . + 8 1 16 1 128 1 - a . + — a , +... 16 1 64 1 ... 1 } 32 1 J J Figure 2.3: T7ie Born series terms for the ID transmission case of equation (2.SI) are illustrated with their associated scattering diagrams. *<**> + - " - „ - M ( 1 . 7 ) » + . . . } . (, 2 7 ) These collapsed series in ot\ are therefore coefficients of a further series in orders of —iko(z — z\). Not ice that one m a y expand as a Taylor's series: „ - t f c o ( z - Z i ) ( l - 7 ) _ ( X + 7 ) 2 - ikQ(z - Zl)(l - 7 ) - f (z - * i ) 2 ( l - 7 ) 2 + - (2-28) = HZJ) _ ifco(z _ Zl)(i _ 7 ) - M(Z _ Zlf(l - 7)2 + ... T h i s is the form of the series in equation (2.27). Subst i tut ing equation (2.28) into equation (2.27) produces the closed-form expression: ( 1 + 7 ) 2 Chapter 2. Viscoacoustic Born Series 16 #z|0;fco) 2iko gifeoz 2iko k0 Oikoz R iko 1 fc0 e - i f c o ( z - Z i ) ( l - 7 ) ( 1 + 7 ) 7 fco + fci ik0 0ik\ (z — zt) -i/coz 2iko (2.29) oifeozj fc0 + fci ik0 i f e i ( z -z i ) using the definitions of equations (2.22) and (2.23). This expression for the transmitted wave field, generalized to accommodate viscoacoustic or acoustic propagation, propagates according to fci, which is the non-reference wavenumber. Observing the process of collapsing the series, from the form of Figure 2.3 to that of equation (2.29), the roles of the types of scattering interaction in the construction of the eventual wave field become clear. The wave field, propagating correctly in the non-reference medium via the term eik0ze- i f c » ( z - z i ) ( l - 7 ) e t f c o * e - ifco(z- •Zl) etfc()(fel/fe())(z- Z l ) 0ik\(z - Z i ) e - i / c ( ) z i (2.30) is generated in the Born series by a weighted series in orders of —iko(z — z\), i.e. each row of equation (2.21) and Figure 2.3. Notice from Figure 2.3 that the leading terms in each order n > 0 of [—iko(z — zi)]n, arise due to the same scattering interaction-type, that is, those with no direction-change from source to receiver (see Figure 2.4a). It is therefore justifiable to attribute much of the burden of alteration of propagation (wavespeed and/or attenuation) to this type of scattering interaction in the Born series. Of course, to correctly alter ko to fci requires 7 = fci/fco; which in turn requires terms at all orders of a.\ \ nevertheless, the leading terms are the most significant, especially for small a\. Figure 2.4: Certain scattering-types are seen to be associated with the generation of macroscopic properties of the transmitted wave field, (a) These interaction types are responsible for the leading order terms in the alteration of the prop-agation wavenumber, from ko (reference medium) to fci (true medium); (b) These interaction types contain all information necessary to properly alter the amplitude of the transmitted wave field; also, these interaction types are solely responsible for producing the negative of the direct wave. Next consider the amplitude of the transmitted wave field, fco/(fco + fci) = R/(l ~ 7)> which is correctly produced by the series of O'th order in —iko{z — z{), i.e. the first "row" of equation (2.21) and Figure 2.3. While it is true that every order of — iko(z — z{) has such a series embedded in it, so in truth the whole series produces the amplitude coefficient, the information required for its correct computation is laid down by this first row of equation (2.21). Inspection of Figure 2.3 reveals that this series is also characterized by scattering interactions of common type (see Figure 2.4b). This amplitude produces the expected transmission coefficient, as was noted in Matson (1996) for the acoustic case: Chapter 2. Viscoacoustic Born Series 17 Mz\0;ko)\ 2k0 |Go(z|0;feo)| fco + fci' { ' and so one is justified in looking to these types of interaction as being central to amplitude adjustment. In "mixed" terms, of order higher than n = 1 in [a\] n and n = 0 in [—iko(z — Zi)]n, more than one type of scattering diagram is associated with each term. In other words, certain scattering interaction-types are not distinct in the solution. Inspection of Figure 2.3 suggests that these indistinct components of the solution are related in that they share the same number of "up-" and "down-" scattering directions. For instance, the term — k^(z — Zi) 2(3/64)af has three contributing diagrams: from left to right, "down-down-down-up", "down-up-down-down", and "down-down-up-down". Since the source is fixed to be above all interactions, the first direction must be "down"; therefore, these three diagrams represent all permutations of "two downs + one up-" interaction type. Finally, consider the third key task of the Born series: the elimination of the direct wave i/'o = e tk" z/2iko-The series accomplishes this, in equation (2.28), concurrently to the wave field construction, by creating the negative of the direct wave such that they destructively interfere. This is also a conclusion of Matson (1996). This "task" is accomplished by the terms which are O'th order in —iko(z — z\)\ the unit first term is split into two parts, (1 +j)/2 and 1 — (1 + 7)/2, the former of which ultimately becomes the negative of the direct wave (equation 2.28). This direct wave eliminator, unlike the amplitude term, owes its existence solely to scattering interactions of the type seen in Figure 2.4b. Lastly, I might underscore a remarkable aspect of the propagations and interactions which constitute the terms in the Born series. The reference Green's function propagates from interaction point to interaction point in every term - no other type of propagation ever occurs in this formalism. The reference Green's function is, here, the solution to the acoustic wave equation, yet the final wave field is viscoacoustic: this means that an attenuated and dispersed wave field is being correctly generated by a sophisticated interplay of non-attenuating wave propagations. In this section the scattering potentials, generalized to permit viscoacoustic wave propagation, have been confirmed as producing the expected wave field expression for a simple ID transmission case. Concurrently, scattering diagrams, which are a byproduct of the subdivision of the Born series terms into computable units, are carried through the calculations. Thus, when the mapping of Matson (1996) is used to produce the closed form expression for the wave field, the "scattering interaction-types" that produced each term may be categorized as to their contribution to overall wave field properties, such as amplitude, phase (and attenuation), and the destruction of unwanted wave field components. 2.4 Convergence of the Viscoacoustic Born Series In order to describe the convergence properties of the solutions, recall that the closed-form solutions for the forward scattering series were found by using two series: and ( 1 - , ) . / > . 1 - ^ - ^ ( 2 . 3 2 ) i+* + |r + lr + ¥ + -' (2-33) for some x. The convergence of the scattering solutions depends completely on the convergence properties of these two series. Consider equation (2.33) first. Since it converges on -oo < x < oo, in the context of equation (2.28) no numerical values of fcrj, 2> zi> o r 7 c a n alter the convergence of the portion of the solution represented by this series. Therefore this series drops out of the present analysis. Not so the other series, equation (2.32), which converges for |x| < 1. In terms of a\, the convergence of this series, and therefore the entire solution, rests on the condition This may be equivalently expressed as (2.34) Chapter 2. Viscoacoustic Born Series 18 K| 2 < 1. (2.35) For an acoustic perturbation associated with a medium contrast in which the wavespeed jumps from CQ to Ci, this may be written v i - | ) ( i - ^ < i . P J » ) Expanding this expression results in c o / c o 2 < 0, (2.37) and since c 2 / c f > 0, this amounts to ,2 § - 2 < 0 , (2.38) and so the results of Matson (1996) are reproduced: co < \/2ci. (2.39) Equation (2.39) limits the contrast between the reference medium wavespeed Co and the non-reference wavespeed Ci . It is worth emphasizing that this condition arises due to the ratio test for convergence; there are others. Nita et al. (2003) explore the forward scattering series for, amongst other things, post-critical wave phenomena, and comment in particular on other tests for convergence. In the following, I will express the convergence properties of the Born series solution for viscoacoustic media, exemplified by two Q models. I isolate the impact of Q on convergency by considering single interface models involving contrasts in Q (or attenuation parameter) only, i.e. using ctq(z) from equation (2.11). Consider a ID normal incidence forward problem, in which the reference medium is acoustic, and the non-reference medium is a homogenous viscoacoustic half-space characterized by the attenuation parameter (3\. Calling aq the amplitude of the perturbation for the single interface model, the condition expressed by equation (2.35) becomes K l 2 < i , n n 2.40 |2 /3 i+/?i 2| 2<l. The attenuation parameter is complex, i.e., Pi=PiR + iPu, (2-41) so aq = - 2 ( /3 IR + ipu) - {P1R + if3u)2 = aqR + iaqi, where aqR = -2l31R-(ftR-P1I), (2.43) c v = -2pu - 2p1Rpu. Therefore the condition (2.40) is Chapter 2. Viscoacoustic Born Series 19 Kl 2 < i <*2qR + a2qI<l (2.44) B*R + 20lRp2j + pi, + A(3\R + 4/? l f i/? 2 7 + 4/3?* + 4(32u < 1. This condition for attenuative contrasts may now be explored for various Q models. I consider two. First I consider the friction model for attenuation, where P l R = °' (2.45) /3u = /3. y 1 This model is too simplistic to produce dispersion, and hence is non-causal; in this sense it is often seen as unedifying physically. However it captures the amplitude behaviour of attenuating media quite well, and corresponds to a Q that is independent of frequency. Its simplicity makes it a good candidate to begin this look at convergence. Substituting equation (2.45) into the convergence condition of equation (2.44) results in (32{(32+4) <l (2.46) as the condition of convergence for the friction-model attenuation parameter. Since (3 is usually considerably less than unity, I let f32 + 4 « 4, replacing equation (2.46) with 4/32 < 1, (2.47) Hence for the Born series expression for a ID, normal incidence, reflected or transmitted wave field to converge for a contrast in (3 from an acoustic reference requires that (3 be less than 1/2. Since this amounts to a loss per cycle of the order of unity, I conclude that the Born series is convergent for any Earth-like attenuation parameter. Complicating matters are the dispersion elements of more realistic models, which make the convergence condition frequency dependent. I choose a constant Q model such that PiR = — 7 : log (•j-*Q \fry ( 2 4 8 ) 0 1 1 = 2Q' where / is the frequency in Hz, and fr is the reference frequency. I will assume a reference frequency of 120 Hz for this convergence investigation. Again consider the resulting convergence condition. Equation (2.48) is contingent on the constant Q model: (£)«!• ( " » ) Equation (2.49) says that this propagation law breaks down at low Q and low frequency. Since this "break-down" produces rapid growth of the perturbation, divergence of the series may be expected to occur where equation (2.49) no longer holds. Using the definitions of equation (2.48) I plot the value of the quantity on the left-hand side of equation (2.44) for a range of Q and frequency values (see Figure 2.5a). It is clear that the convergence indeed comes into question in regions of low Q and low frequency. Figure 2.5b illustrates a mask, black meaning that the quantity is below unity, white meaning it is above; white regions correspond to combinations of Q and frequency for which the series is divergent. Again, this is seen for Q values which are quite low, on the order of unity, if the frequency is kept above approximately 5 Hz, so it is concluded that for most Earth-like contrasts and experimental bandwidths the series converges. Chapter 2. Viscoacoustic Born Series •20 Q Q Figure 2.5: Convergence illustration for ID normal incidence, acoustic reference, Q contrast configuration; (a) quantity on left hand side of equation (2.44) * s plotted for a range of Q and frequency values (Hz); (b) a mask is defined to be unity if (a) is greater than 1, nil if (a) is less than 1. Hence white components signify divergence of the series for those combinations of Q and f. 2.5 Toward Scattering-based Q Compensation/Estimation In this section I utilize a different categorization of scattering diagram, that of "separated" versus "self-interaction" (Weglein et al., 2002). The insight gained from such a categorization directly guides the search in the inverse series for task-oriented subseries. The promise of the inverse scattering series, convergence issues notwithstanding, is to compute the model a(z) via certain nonlinear operations carried out upon the data. If the measured wave field is distorted and smoothed because it propagated through an attenuating medium that has sharp transitions, then the reconstruction of these sharp transitions must include some process of Q-compensation. Furthermore, since Q can be cast as part of the perturbation, the computation of the model must include Q-estimation. To determine how and where such processes take place in the inversion, it is, as ever, useful to turn to the forward scattering series. It is particularly compelling to consider a ID case in which only Q contrasts, and no wavespeed contrasts, exist. These contrasts produce reflections, but the wave never travels at a speed different from that of the reference medium. So, in the inverse, no beyond-Born imaging per se will be required. All events will be correctly located by imaging with the reference wavespeed. The only processing step necessary will be to remove the smoothing and distortion due to Q. Weglein et al. (2002) have identified a separable subseries responsible for imaging in the presence of wavespeed contrasts - will this subseries shut down in a Q-only case? How then will Q compensation occur? Let us next address these questions by looking at the forward analogue of the imaging subseries. Consider the two models shown in Figures 2.6a and b, both of which represent ID media. Both are geometrically identical, but the first (2.5a) represents a purely acoustic variation: the wave field propagates at wavespeed c 0 for z < z\ and z > z2, but at C\ in the layer between ( 2 1 , 2 2 ) . Meanwhile, the second Chapter 2. Viscoacoustic Born Series 21 corresponds to a situation akin to that described above: the wavespeed never changes, c(z) = CQ everywhere. The absorption parameter Q, which I assume obeys the dispersion relation associated with the bandlimited constant Q model, undergoes a contrast in this case (Figure 5b), from co outside (zi,Z2) to Q\ within. a C = CO Q = oo C = CO Q = oo Figure 2.6: Schematic of two models for a ID reflected wave field, (a) The acoustic case: consider the "second primary", i.e. that which has reflected at the Z2 interface, which in this case corresponds to a wavespeed contrast, (b) The viscoacoustic case: an identical geometry to that of (a), here a contrast in Q only is considered. This latter wave field travels everywhere with the wavespeed co, and hence no alteration of the Born approximate arrival time is required; the idea is to ascertain what, if any, effect the "mover", or timing-related terms in the series have on the construction of this field. Matson (1996) finds an expression for the primary reflection associated with the lower interface (z^) in a model of this sort (whose ray-path is illustrated in Figures 2.6a and b). Previous work of this chapter has shown that the general form of Matson's expressions are the same for both the acoustic and viscoacoustic examples. After the amplitude series have been collapsed, this primary, ippr2, is 1ppr2(z < Z!\0;kO) e - i f c ( )Z e i2 fcoZ2 2ik0 R2T10T01 [1 + ik02(z2 - zi)(7 - 1) - 1 4 ( ^ - ^ ( 7 - I ) 2 - | f 8 ( z 2 * i ) 3 ( 7 - l ) 3 + which further collapses to i>pri{z < zi\0; k0) e-ik0z ei2k0z2 2iko ii^TioToie , i 2 f e ( ) ( z 2 - z i ) ( 7 - l ) (2.50) (2.51) recalling that 7 = fei/feo, i - e - the ratio of the reference and non-reference wavenumbers. The reflection and transmission coefficients (R2, and T\o and TQI, for the reflection, transmission from medium 1 to 0 and transmission from medium 0 to 1 respectively) have been produced similarly to the amplitude of the transmitted wave field in the previous section. The terms in the Born series which have conspired to produce the bulk of the e l 2 M z 2 - z i ) ( 7 - i ) component in equation (2.50) correspond to "separated" forms (Weglein et al., 2000), by virtue of the presence of powers of (z2 — z\); hence this component is due to the forward analogue of the imaging subseries. One expects, Chapter 2. Viscoacoustic Born Series 22 therefore, that in the acoustic case, this term will do much of the work required to take the incorrectly-timed arrival of the Born approximation, and alter it such that it arrives having travelled everywhere at the correct wavespeed. Let us first see that this is the case. In the acoustic case _ fci _ Cp_ fco c i ' so, expanding equation (2.51), (2.52) VV2(* < *i|0; fco) = ^r-R2TwT01 [e*2fc„,2e-*2fc„z2n ^ f c o z ^ M ^ - z O ^ ( 2 5 3 ) Consider first equation (2.51). The incorrect arrival time of the Born approximation appears in the uncor-rected leftmost factor -—\fko " 2 ; specifically in the part which produces a phase delay over the distance 2z2 with wavenumber fco: e l2k" Z2. Next consider equation (2.53). Notice that the —1 portion of ( 7 — 1) in the correction produces a term opposing this arrival (both are in square brackets [•]) when the expression is expanded. So in the acoustic case, the first task of the series "corrector" term, e l 2 f c ° ( 2 2 _ z , ^ 7 ~ 1 ) , is to delete the incorrectly timed event. Next, p^r2(z < Zl\0;ko) = e-^R2TwTmei2k^e i2[^^-^ = ^ — — R 2 T w T 0 1 ei 2 k n Z ' e i 2 k l i z 2 - z ^ . 2ifco Here, the second task of the series corrector is engaged: the remainder of ( 7 — 1), namely co/ci, multiplies the reference wavenumber fco in the square brackets, deleting the incorrect wavespeed and replacing it with the correct wavespeed and thus the correct wavenumber fci. In their totality, the "separated diagram" type terms therefore (i) delete the Born approximate arrival, and (ii) replace it with the correctly-timed true arrival. Thus arises the interpretation, in the inverse analogue, of these separated diagram terms as being "movers". One might indeed expect that, in the event of a true medium with no wavespeed variation, these terms would shut down. There are a number of problems with this expectation, however, not least of which is: they don't. I proceed by examining the purely-Q contrast case: fci 7 1 + rQi \VT J 2Qi n i 1 1 1 1 W r \ , , i \ . ( UJ fco % 2Qi TTQI \U>.. = l + dr-^rk - • (2-55) Once again expanding equation (2.51), but this time with the viscoacoustic 7, , / In 1 \ e - i k o Z e i 2 k o Z 2 i 2 f c o ( z 2 - 2 l ) [ l + * 4 - l n ( i ) - l l ippr2(z < zi 0;fco) = —. i? 2Ti 0Toie u l ^ ^ J = c- ik° Zf 2kaZ> R 2 T 1 0 T 0 1 e ^ - ^ - ^ -(»)]. 2ifcn (2.56) So the correcting term doesn't become unity, or by any means vanish, even though there are no timing changes to be made - the imaging subseries analogue stays alive. What happens instead is that the form of the viscoacoustic 7 extinguishes the —1 from ( 7 — 1 ) (see equation (2.56)). This was the mechanism in the acoustic case that deleted the incorrect arrival. The Born arrival time of this primary is kept! What is left in place of a "mover" term, that deletes and replaces wave field events, is an operator, e t 2 f c ( ) ' 2 2 2 : 1 ^[^T » Q i l n ( u . , . ) ] ; that distorts the amplitude and phase of the Born primary according to the Q model. One arrives at VV2(* < *i|0;fco) = ^ ^ f l 2 T i 0 r 0 i e i 2 f e H Z 2 [ 1 + ^ (2.57) Chapter 2. Viscoacoustic Born Series 23 Here, via the term e l 2 k o Z 2 [1+iQi » « I '"(«,.)] ^ ^ e w a v e field propagates the entire distance 2z2, attenuating with Q\. This is the correct arrival time, but the incorrect amount of attenuation - too much. The rightmost term, e t 2 f c ( , Z l [ 3 ^ T ^7 l n (",-)] ; corrects this by deconvolving the attenuation (not the timing) associated with the distance 2z\. The final result is that the second primary has experienced the expected amount of attenuation, that is, through the distance 2(z2 — z\). The important point here is that the "separated diagram"-type terms have played an enormously impor-tant role in the Q contrast only case, in spite of the fact that no "moving" was required. It seems clear that a redefinition of the forward analogue of the imaging subseries is in order. I surmise that these terms are responsible for generating propagation effects rather than simply timing changes - with the former reducing to the latter in the acoustic case. The "movers", i.e. the imaging subseries terms in the inverse case, must be generalized to "de-propagators". 2.6 Remarks The work described in this chapter reflects the course of the initial investigation into the use of scattering theory as a means to process data with non-negligible attenuation present. It contains the results of computing terms in the forward (or Born) series, and following the scattering diagram associated with each of these terms. The results are compelling in two respects. First, the attenuated and dispersed wave field is generated (for acoustic reference media) via nothing but the non-linear interplay of acoustic wave fields. Second, some macroscopic properties of the viscoacoustic wave field are related to scattering interactions (symbolized by the diagrams) of like kind. These include the negative of the direct wave, the creation of the wave field amplitude, and the leading order terms in the alteration of the propagation wavenumber term. This chapter also concerns itself with the impact on a reflected primary of terms that involve "separated" interactions. These are the forward series analogues of the imaging terms of the inverse scattering series. I reach the conclusion that the meaning of the "moving" terms, those whose diagrammatical representation in-volves separated scattering interactions, must be generalized to accommodate the inclusion of all propagation effects into the true wave field. The consequences with regards to how the inverse scattering series must treat attenuation are clear - one must look to the imaging subseries to remove the smoothing and distortion of Q. In an example created with no wavespeed changes, it may indeed be found that the imaging series does only this. Beyond ascertaining that this is the case, a clear future avenue of research is to cast the inverse scattering series problem with at least two parameters (perhaps c and Q), and investigate both the imaging (de-propagation) and inversion (c and Q estimation) subseries. In chapter 5 I present the latter. Chapter 3. Lipschitz Exponents and Q 24 Chapter 3 L i p s c h i t z E x p o n e n t s a n d Q Term Signifies Q Quality factor a Lipschitz/Holder exponent a Smoothness parameter A Amplitude parameter Ba{t) Gaussian of variance a 2 f(t) Arbitrary function, measured hit) Arbitrary singular function Mt) Wavelet at scale s Wsf(t) Continuous wavelet transform of / at scale s Wf(u,s) Wavelet transform of / at scale s, location u 4> Objective function aj j ' t h wavelet transform modulus maximum b(t) Constant Q impulse response (CQIR) B(OJ) Constant Q transfer function (Fourier transform of CQIR) 1,P Parameters internal to CQIR ba{t) Approximate CQIR LOR Reference frequency CO Wavespeed Q(a) Empirical mapping between a and Q r(t) Reflectivity x(t) Seismic trace xb(t) Bandlimited seismic trace wit) Ricker wavelet 3.1 Introduction Having introduced the application of the forward scattering series as a means to model the propagation of a wave field in viscoacoustic media, and prior to the application of inverse methods to such media, I here consider a purely signal-processing based means for the treatment of attenuated data. This chapter is concerned with the manner in which absorption and dispersion alter the local regularity of the events of a seismic trace, with an emphasis on utilizing this as a novel means to estimate Q. The theory and results included here are also found in (Innanen, 2002, 2003). At present the connection between singularity analysis, in this study, and the inverse scattering series is in its specific application to Q problems only. But as will become clear in chapter 4, the nature of the discontinuities - the singularities - of seismic data are of paramount importance to the series in how it constructs the desired Earth model; it is difficult not to wonder how these two might eventually relate. In signal (and image) processing, much of a signal's information is contained in regions of abrupt change. Such regions are often considered to be the discrete expression of singularities in an underlying continuous function. Explicit study of the characteristic singularities of a seismic trace is an as-yet little used means of seismic data analysis, with broad potential for the extraction of useful information (Herrmann, 1999). I begin with a review of the characterization of singularities by Lipschitz exponents, and the use of Fourier Chapter 3. Lipschitz Exponents and Q 25 and wavelet coefficients to estimate these exponents. In particular the localization properties of the wavelet method are emphasized as useful for singularity analysis of non-stationary data. Secondly I discuss a signal model of which Lipschitz exponents are one parameter, and a nonlinear regression by which these parameters may be estimated from the data set. I then apply this model and parameter inversion to a simple ID non-stationary trace, whose impulse response corresponds to a constant-Q model of pulse attenuation. The inversion results are explained via theoretical arguments regarding the local regularity of a pulse for the end member cases in which Q —> 0 and Q —> oo. A further linear regression is suggested as a means to empirically generate a map between regularity and Q. This suggests a possible means for Q estimation. Finally, I discuss the effect of bandlimitation on this process of Q-estimation. 3.2 Lipschitz Exponents and Their Estimation The Lipschitz exponent (also called the Holder exponent)1, is a generalized measure of the differentiability of a function. The number of times a function /(£) is continuously differentiable may be decided by monitoring the difference between f(t) and its Taylor's series approximation pt0,n(t) (of order n about to) as n grows. A classical result bounding the error et(un{t) of a Taylor's series approximation of /(t) is (Mallat, 1998) Cto.nW = /(*) - Pt0,n(t) < K\t - to\n, (3.1) where Ptn,n\z) — / J m! m=0 dt(r (3.2) and where K is a function of n and the n'th derivative of f(t). Equation (3.1) may be used to estimate the number of times f(t) is continuously differentiable: n is increased until this upper bound on the error is violated. One has located at that n the differentiability of /(t). The Lipschitz exponent is a generalization of n in equation (3.1) to the set of real numbers, indirectly thereby invoking the concepts of fractional differentiation and integration. In direct analogy to equation (3.1), the Lipschitz regularity is the largest a such that the Taylor's series approximation error eta,n(t) obeys et0,n(t) <K\t-t0\a, (3.3) for some K > 0. The definition of equation (3.3) can be cast in the frequency domain: a may be shown, e.g. (Daubechies, 1992), to be the largest value such that |/H| (1 + \u\a) dw <oo (3.4) holds. In the sense that this permits brute force estimation of a, this Fourier domain expression can be thought of as a more profitable casting of the definition of regularity. Note, however, that the localized reference point to, about which the Taylor's series expansion is carried out, is not present in equation (3.4). Equation (3.4) says that the regularity a is related to the decay of the Fourier coefficients of /(t) with increasing frequency to. This is intuitively reasonable, since in the Fourier domain the n'th order derivative operator is (—iuj)n. It is important to note that the generalization to a permits the regularity of a distribution to be determined also. That is, a may be less than unity, or indeed it may be negative, and therefore be descriptive of delta functions, Heaviside functions, and so on. It is particularly in this ability of a to characterize singularities that makes it of interest to signal processing, and ultimately, to this research. However, the coefficients f(oj) are, as ever, very poor at describing temporally local properties of f(t), since the kernels of the Fourier transform are of infinite extent in time. If one is dealing with a non-stationary signal a brute Fourier transform will produce coefficients which in some unhelpful way characterize the bulk regularity of the signal. Some kind of judiciously applied windowing, or short-time Fourier transform must be applied. 1In this chapter, I use all three terms Lipschitz exponent, Lipschitz regularity, and regularity. I do not intend any distinction between them. Chapter 3. Lipschitz Exponents and Q 26 Another possibility is to resort to an appropriate wavelet analysis, in which the basis functions as they stand interrogate the signal for its local characteristics. I proceed in this manner for the current research. Conceptually, the process is similar to that of the use of Fourier coefficients - the decay of wavelet coefficients (those associated with a particular event or interval of interest in the signal) with scale determines the exponent a at that event or on that interval.2 In practice, it comes about somewhat differently. If a wavelet has n vanishing moments, then the wavelet transform of a polynomial of degree n or lower is zero. This means that, with the correctly-chosen wavelet, the wavelet transform of a function /(£) is equal to the wavelet transform of the approximation error of the 71,'th order Taylor's series expansion of /(£). The wavelet transform is denoted W such that Wf(u, s) = 1 J°° f (r> {—^j dt (3.5) is the normalized wavelet transform of f(t). The parameters s and u are the scale and position of the wavelet respectively. Since Wpt„,n = 0 (for a wavelet with n vanishing moments), the polynomial pt0,n(t) may be added to f(t) without altering the result: Wf(u, s) = -aj" [f{t) - pUun{t)\ ^ ( i ^ J dt. (3.6) Since the integrand of the transform now contains the Taylor's series truncation error etn,n(t), from equations (3.1) and (3.3), the modulus \Wf(u, s)\ may be re-written as: \Wf(u,s)\ < - K\t-t0 S J-00 then, with some manipulation (Mallat, 1998), \Wf\ < Asa, (3.8) where A is not a function of s. Equation (3.8) says that the modulus of the wavelet coefficients of f(t) associated with the point to vary with the scale s according to the Lipschitz regularity a of f(t) at to. The modulus of Wf(t0,s) for a fixed t0 describes a path in scale space that is perpendicular to the u axis, but because the wavelets dilate at larger scales, the single point to at small scales influences a "cone", and therefore to has a footprint in scale space that grows with scale s. The Lipschitz regularity a is usually estimated by measuring the decay of the modulus maxima of the wavelet transform within this cone (Mallat, 1998). Taking logarithms of equation (3.8), and following the path of one of the maxima of the wavelet transform modulus, one has \og\W f (t0,s)\< log A+ a\ogs, (3-9) so finding the slope of the linear relationship in equation (3.9) produces an estimate of a. The modulus maxima perform three useful tasks in the context of signal processing. First, the existence of local maxima marks the existence of a singularity (or discontinuity, or edge) in the signal. In this sense the wavelet transform is similar to well-known edge-detectors in image processing. Second, these maxima, within the cone of influence of an edge, form paths u(s) which at fine scales locate the edge in the original function. Third, as discussed, the modulus of these maxima can characterize the edge via its regularity, i.e. estimate the order of singularity which has led to the detected signal edge. 3.3 A Smoothed-Singularity Signal Model The preceding review suggests that, given some discretized version of a signal with several localized dis-continuities, the discontinuities may be located and characterized (via a) using the maxima of the signal's 2 F o r a function or signal with non-isolated singularities, a pointwise Lipschitz regularity is well-defined. For a function with isolated singularities, which is assumed to be the case here, the Lipschitz regularity is also well-defined on a chosen interval (Daubechies, 1992). 1> dt, (3.7) Chapter 3. Lipschitz Exponents and Q 27 wavelet coefficients: the so-called "wavelet transform modulus maxima" (WTMM). In particular, measuring the slope of the logarithm of the W T M M associated with a singularity produces an estimate of a. Mallat and Hwang (1992) note, however, that in practice modulus maxima are often produced in association with signal structures which are effectively non-singular because of gradual onset. They appear, in fact, to be smoothed. For instance, the wavelet transform of a Gaussian, a continuous and differentiable input, generates WTMMs. How one views such WTMMs depends on how one models the signal. Here I make use of a signal model which regards such edges to be important and gives them a framework (Mallat and Hwang, 1992; Mallat and Zhong, 1992). A signal is considered, in this model, to be a discretized version of a function /(£) which behaves, over intervals in t, like fit) = ea(t) * h(t), (3.io) where ea{t) = (3.11) is a Gaussian of variance cr2, * denotes convolution, and h{t) is an underlying function composed of un-smoothed singularities. The variance a2 need not be the same for all t; it is free to vary from discontinuity to discontinuity. It is perhaps best thought of as being piecewise constant on intervals, each of which contains one discontinuity. Equation (3.10) says that f(t) consists of some series of singularities which, individually, may be strongly smoothed, and have slow onset, or be completely unsmoothed. Locally this is controlled by the parameter a. This model fits snugly into the analysis of regularity, because wavelets with vanishing moments may be written using differential operators applied to Gaussians (Mallat and Zhong, 1992). For instance, a wavelet with a single vanishing moment may be written Mt) = sJtds(t), (3.12) and the continuous wavelet transform of f(t) using ij}s{t) is Wsf(t) = f(t)*^s(t). (3.13) Substituting the signal model of equation (3.10) into the wavelet transform of equation (3.13) produces wsf(t) = [ea(t) * h(t)\ * * h(t) (3-14) = -Ws„h(t), so where So = Vcr2 + s2. In other words, the wavelet transform of f(t) at scale s is equal to a scalar times the wavelet transform of h(t) at a different scale s0. Through the wavelet coefficients of the measured signal f(t) one has access to the wavelet coefficients of the underlying function h(t), even though h(t) itself is not measured. The variation of the W T M M of smoothed singularities, with the regularity a, may therefore also be readily written down using equation (3.8): \WSoh(t)\ = S-f\Wsf(t)\<As^ (3.15) or s so S0Jt9sn(t) \WJ(t)\ < sAs^1. (3.16) Chapter 3. Lipschitz Exponents and Q 28 Equation (3.16) relates the regularity of a signal discontinuity to the wavelet transform modulus, similarly to equation (3.8). However, in this smoothed singularity model the regularity pertains to a signal with discontinuities (h(t)) and not the possibly smoothed one that was measured (/(*)). To use this signal model is to insist that transient signal events are due to inherent singularities, in spite of some existing level of smoothness. To estimate a, one uses the maxima of the wavelet transform modulus associated with a sin-gularity of interest, as the scale-varying input to a regression based on equation (3.16). Of course, equation (3.16) is not as simple as equation (3.8). The parameter a is derived from the variation of log |W s /( t) | versus log so rather than the wavelet scale logs. But so is unknown - we know the wavelet transform, we just don't know at what scales we are looking. This scale ambiguity is intuitively reasonable when one considers incorporating smoothness into a multiscale singularity model: a Gaussian looks like a delta function if one backs away from it far enough. To continue, the three parameters a, a (from SQ), and A combine to vary the moduli |W s /( t) | in a complicated way that is not linearized by taking logarithms. In Mallat and Hwang (1992) a nonlinear regression is recommended, in which the residual || log |W s /(t) | — log(sAso _ 1 ) | | 2 is minimized to determine these parameters from the W T M M of f(t) on intervals of t of interest. A dyadic wavelet transform is carried out on the input signal, over the dyadic scales s = 2 J, j = 1,2, 3 , J , and with no downsampling in the shift (time) direction. That is, The W T M M a,j are located and tabulated at scales j = 1..J in the transform space (s,t), and the objective function is minimized to obtain estimates of A , o, and a. 3.4 Non-linear Inversion for Lipschitz Regularity In this section I demonstrate the use of the W T M M to invert for the Lipschitz exponent a using some simple inputs. This provides a framework for discussion regarding the accuracy of the inversion. In particular, I alter the steepest-descent approach suggested by Mallat and Hwang (1992) to a conjugate gradient approach to speed up convergence. Figure 3.1 provides details regarding the extraction of W T M M produced from a unit impulse input (Figure 3.1a). The wavelet transform is as recommended in Mallat and Hwang (1992), i.e. a dyadic wavelet transform performed over scales s = 2 J, j = (1, J). The transform is accomplished using an "a trous" algorithm, based on a spline wavelet which approximates the first derivative of a Gaussian. This wavelet has a single vanishing moment, and is therefore appropriate for characterizing a < 1. Figures 3.1b - 3.1e constitute the wavelet transform. Since this is effectively the impulse response of the wavelet transform, the wavelet itself is seen dilating as the scale increases downward. A set of modulus maxima are identified via arrows, which forms a path diverging from the location of the input unit impulse as scale increases. The set is tabulated as [ai, a 2 , as, a,4, ...]T; these values form the input for the minimization of the objective function of equation (3.18). The path one either side is suitable to characterize the singularity (which, since it approximates a delta function, has a = -1), and can be thought of as being the wavelet interrogating the singularity from either side. Wvf(t) = Mt)*f(t). (3.17) j=i (3-18) Chapter 3. Lipschitz Exponents and Q 29 CO c W Q . C CO CO o c/) E i o "w c co 1 _CD > co Ar t(s) Figure 3.1: TTie extraction and tabulation of the wavelet transform modulus maxima (WTMM) output from a unit impulse input, (a) The input unit impulse, (b) - (e) represents the wavelet transform across 4 scales without downsampling in time, and the tabulation [a\, a%, a3, 04, . . . ] T ofthe WTMM. In practice, the parameter A, which is related to the amplitude of the sharp variation (Mallat and Hwang, 1992), is problematic, and I treat it separately from the other parameters in the inversion. The inversion converges slowly in A, and appears to be sensitive to the initial value of A in the iterative minimization of <j>. To deal with this efficiently, A is held fixed, and a two-parameter inversion based on the gradients dab/da, dab j da is carried out. This is done several times for a range of A values that is empirically determined to be stable, and a low-order polynomial <j>(A) is minimized to obtain the final utilized value for A. This procedure is merely a careful approach to minimizing equation (3.18) that allows the parameter estimation to be guided and yet still efficient, even if some trial-and-error variation of A is necessary at the outset. Figures 3.2a-d illustrate the objective function over a range of a and a values, for fixed values of A and using the W T M M associated with a unit impulse input. Figure 3.2a is the objective function with incorrectly chosen A, and Figure 3.2b is the objective function with A found such that the objective function <f> is minimized. This illustrates the broad impact of changing A, which is to gradually shift the structure of the objective function in the a dimension, along with the minimum. The objective function changes slightly but maintains its overall character in the vicinity of the correct A. Both Figures 3.2a and 3.2b also illustrate the iterative inversion process, with paths from the initial parameter values to the minimum in a and a. In these cases the recommended steepest descent optimization has been used (Mallat and Hwang, 1992). Figures 3.2c and 3.2d illustrate precisely the same initial setup as in the previous two diagrams respectively, but with the optimization being accomplished using a conjugate gradient (CG) algorithm. The relative convergence rates seen here are typical - generally the number of required iterations are greatly reduced using the CG method. Empirically, CG convergence in a and a is found to occur in most cases when A is in the neighbourhood of the minimizing value. Large a may result in instability. Further, an accurate recovery of a usually occurs by the second CG step - beyond this, a changes very little as the minimum with respect to a is sought. Since a is the major target of this inversion, this means that an accurate inversion is possible with little computation. Mallat and Hwang (1992) predict Chapter 3. Lipschitz Exponents and Q 30 the accuracy of the Lipschitz exponent estimation to be within %10, with the main source of error stemming from the spline approximation in the wavelet transform. Initial a and a (with A near its true value) by and large do not alter the result, with one major exception. The objective function is symmetrical about a = 0, and any initial a < 0 will converge to a minimum corresponding to the negative of the "correct" a. For this reason, initial a is as a rule kept positive. Figure 3.2: Illustrations of the objective function for fixed A (i.e. for a range of a, a), (a): Incorrectly chosen A, and using input from a unit impulse input, (b): as (a), but with A correctly chosen, i.e. such that (f> is minimized, (a) and (b) are both minimized using the Mallat and Hwang (1992)-recommended steepest descent method, (c)-(d) are identical in every respect to (a)-(b) respectively, but are minimized using a conjugate gradient algorithm. Convergence is much faster. Good a accuracy is usually produced in 2 CG steps. For other WTMM inputs the objective function, which is symmetric about a = 0, can become double-lobed, with two minima mirroring each other about a = 0. For this reason, initial a is always chosen positive, so that a minimum corresponding to a > 0 is ensured. The recovered variance a2 is small for the unit impulse input (after a number of iterations that is large compared to that necessary to accurately recover a), but not exactly zero as one expects for a delta function. To understand the capability of the estimation method to accurately reproduce the absolute smoothing parameters, I use several Gaussian functions of increasing variance a2 as input to the W T M M scheme. Based on the smoothed-singularity model, in which smoothing and regularity are postulated to be independent signal characteristics, the inputs should be interpreted as delta-like singularities with increasing a. Of course, as the fine resolution is increasingly filtered out of the data, one expects the fine scale wavelet coefficients to fall below any reasonable noise level, eventually decreasing the accuracy of the estimation of the parameters A, a, and a. The Gaussian input experiment will serve to explore this question also, setting a limit on the smoothness a of the input, beyond which the signal is too coarse for fine-scale analysis. Figure 3.3 summarizes the results of this test. Using a time vector with time-step of At = 0.004 s, a = 0.015 (such that the input is f(t) = exp[— (t — to) 2/2a 2]) is found to be approximately the greatest level of smoothness from which an a with %10 accuracy may be extracted. The tabulated results in Figure 3.3 illustrate the limited capacity of the model to predict absolute values of smoothness a. Comparing ar and ai, the recovered and input variances respectively (Figures 3.3a - e), it is clear that ar is a poor estimate of cr,, the two often differing by several orders of magnitude. The explanation for this may be related to the "shallowness" of the objective Chapter 3. Lipschitz Exponents and Q 31 function in the o direction (see Figure 3.2) relative to the a-direction. On the other hand, or grows with CTJ approximately in proportion, so as a relative measure of smoothness the o estimate may have some value. The current interest in o is, in any case, less that an accurate absolute o is produced, but rather that the model correctly attributes specific signal behaviour (smoothness) to <r, considering the Gaussian to be, in essence, a modified <5-like singularity. The ability of this algorithm to correctly identify, via a, a Heaviside (step) function is also a necessary topic of study. To this end, an investigation into inverting for the local signal parameters from the W T M M of gradually smoothed discrete step functions is also carried out. The results, illustrated in Figure 3.4, are as follows. Given a Gaussian similar to those used in the previous experiment, convolving it with the step function, and normalizing the output amplitude to a unit maximum, results similar to the smoothed delta function were found. That is, o < 0.015, illustrated in Figure 3.4, produces a estimates which are within %10 accuracy. = 1.8 = -1.08 = 0.001 = 5.13 = -1.07 = 0.005 = 6.62 = -0.98 •  0.0075 = 7.87 = -0.93 = 0.01 = 10.7 = -0.89 = 0.015 t(S) Figure 3.3: Illustration of loss of accuracy in a, a-recovery with smoothness of input; (a)-(e) the recovered parameters or and ar, and the input standard deviation cr, are tabulated at the left. Chapter 3. Lipschitz Exponents and Q 32 = 3.13 = -0.04 = 0.001 o =4.73 r a = -0.09 a. = 0.005 = 5.64 = -0.09 = 0.0075 = 6.23 = 0.09 = 0.01 a = 6.4 a. = -0.29 a. = 0.0015 t(s) Figure 3.4: Illustration of loss of accuracy in a, o-recovery with smoothness of input; (a)-(e) the recovered parameters oy and Oir, and the input standard deviation Oi are tabulated at the left. 3.5 The Regularity of the Constant-Q Impulse Response The smoothed-singularity signal model views a signal event as being characterized by an amplitude A, and an order of singularity (with local Lipschitz regularity a) that has been smoothed by convolution with a Gaussian (of variance cr). In this section I use a time-domain approximation of the constant-Q impulse response to make some arguments regarding the limiting behaviour of a with respect to Q on these responses. These arguments, being based on an approximation only, are made broadly, and are meant as a theoretical prelude to the numerical tests to follow. The localization of the method of the Mallat and Hwang (1992) regularity estimator make it particularly well-suited for analysing a single trace whose non-stationary events follow the constant-Q attenuation law. I begin with a statement of the time-domain approximation of Strick (1970) in the terminology of Kjar-tansson (1979). The impulse response b(t) is b{t) = (iyPt7V+y)/2y \2^1{l-1)-^Y'/\-^-^)'^-U\ (3.19) where ts = t(z/cs)~13, (3 — 1/(1 - 7 ) , 7 — tan_1(l/<5), z is the propagation distance, and cs is the wavespeed scaled by a function of the reference frequency. Focusing on the time-dependence of the approximation in equation (3.19), it is clear that the impulse response has three essential components, an amplitude, a power-law time dependence, and an exponential time dependence: b(t) = A(z,ca,Q)tJ^ef'^\ (3.20) where A{z,c„Q) = ( 2 / c s ) - / 3 / y / 2 7 r 7 ( l - 7 ) -/i(Q) = (l + 7)/27. / 2 ( i s , Q ) = - 7 ( l - 7 ) ( 1 - 7 ) / 7 ^ 1 / 7 -1 /7 (3.21) Chapter 3. Lipschitz Exponents and Q 33 Also, time t in this framework is delayed such that t = 0 corresponds to the arrival time of the pulse. Figure 3.5 illustrates the division of equation (3.19) into these three components, and their multiplicative synthesis into the expected form for the full-bandwidth impulse response b(t). The power law term dictates much of the ultimate shape of the response (Figure 3.5c). The amplitude term obviously does not contribute (Figure 3.5a), and the exponential smooths and delays the peak of the response for small Q and/or large propagation distance z and/or slow wavespeed c. I postulate that these three components of the impulse response are, in their dominant effect on a signal, related one-to-one with the parameters of the smoothed singularity model of Mallat and Hwang (1992): the amplitude A(z, cs,Q) with the amplitude A of equation (3.17), the exponential time dependence e^2^"'^ ^ with the signal smoothing parameter cr, and the power-law time dependence ts with the regularity or Lipschitz exponent a. I then suggest that the most robustly-estimated of these parameters, the Lipschitz exponent a, be used to estimate the parameter Q, upon which it depends. I b r Multiply r d Figure 3.5: The three essential components of the time domain impulse response approximation for the constant-Q model. (a) the amplitude term, (b) the exponential term, (c). the power-law term, combining multiplicatively to produce the full approximation (d). I take it as self-evident that the amplitude of the impulse response A(z,cs,Q) does not change the Lipschitz exponent associated with it, simply because multiplying a function by a scalar does not alter its differentiability. What is needed is evidence that the exponential term e^2^"^ (Figure 3.5b), which qualitatively produces smoothness in the response and drives the phenomenon of rise-time, is explained, in the context of Mallat and Hwang (1992), fully by the smoothness parameter o. This is not guaranteed theoretically, since the component is not a Gaussian and is by no means convolved with the other two impulse response components. I explore this with a simple experiment. I create two data sets, each one a set of impulse responses spanning a range of Q values. One set suppresses the exponential component, i.e. uses as input signals of the form of Figure 3.5c (called Supp. IR), and the other leaves it untouched, i.e. uses as input signals of the form of Figure 3.5d, (Full IR). The signals are normalized to a peak amplitude of unity to de-emphasize the effect of A, and the parameters a, and a are inverted-for with each. The inputs to this test are illustrated in Figure 3.6. The results of the test are illustrated in Figure 3.7 and tabulated in Table 3.1. The results indicate that the algorithm is indeed "explaining" the exponential component by varying the recovered parameter a, i.e. the smoothness of the input. Notice that the recovered a over the Q range 500-70 shows very similar variation for both input data sets, indicating that the a recovery is largely insensitive to the addition of the exponential component (although, to be sure, the accuracy of the a estimate will be decreased at lower Q values). Substantiating this further, the recovered o varies between approximately 12-3 in the case of the full response (Figure 3.7c) but is effectively flat, varying between 2.4-3 Chapter 3. Lipschitz Exponents and Q 34 Q value a (Full IR) a (Full IR) a (Supp. IR) 0~ (Supp. IR) 500 -1.00 3.5 -1.00 2.4 300 -1.00 5.1 -0.99 2.8 200 -0.88 5.3 -0.90 2.9 100 -0.60 7.9 -0.70 3.0 90 -0.54 11.2 -0.62 2.7 70 -0.43 12.0 -0.49 2.3 Table 3.1: Parameter estimation results: comparison between regularity and smoothness from the full impulse response (Full IR) and the response with ostensible smoothness component suppressed (Supp. IR). in the case of the isolated power-law time dependence approximation (Figure 3.7c). In other words, the component tj^ 1^ in the impulse response of equation (3.20) dominates in dictating the local regularity of the constant-CJ impulse response. The absolute recovered a values are difficult to interpret, although they display the expected trend with decreasing Q. This is not crucial to the current research. What is important is that the degree of freedom o in the model is being exploited properly, and the smooth attenuated seismic response is yet interpreted as being based on an underlying singularity. Notice too that there appears to be a reasonably simple relationship between recovered a and Q. The limiting behaviour of a with respect to Q may be explained using the above empirical/numerical evidence that one component of b(t) determines a. Based on the results of Figure 3.7, consider an approximation to the b(t) of equation (3.19): in which all time-dependence apart from £_(1+T)/2~>' has been suppressed, and all amplitudes have been folded into the factor CQ. The approximation of equation (3.22) is a bad one in most respects. However, since the power-law component ts* 1^ dictates the regularity of the response, insofar as the algorithm of Mallat and Hwang (1992) is concerned, ba(t) and b(t) have the same Lipschitz exponent. Recalling that 7 = (l/7r) tan_1(l/<5), in the acoustic limit as Q —> 00 one has 7 —> 0, and hence the combined exponents in equation (3.22), —(1 +7)/27 —> —00. The approximation ba(t) therefore becomes delta-like at large Q and hence has a Lipschitz exponent of a = — 1, which is in agreement with the trend seen in Figures 3.7a and b. Consider next the viscous limit of small Q. If one momentarily disregards two moderating terms in ba(t), by setting the time dependence i - 1 / 2 = 1 and replacing the non-linearity 7 = tan _ 1 ( l /Q) with its large Q approximation 7 = 1/TTQ, equation (3.22) becomes It is clear that as Q becomes small there is a tendency for this function to become Heaviside-like or step-like, which suggests a Lipschitz exponent of a = 0. By re-introducing the non-linearity 7 = (I/71") t&n~ 1(l/Q), which limits the growth of 7 with decreasing Q, and the factor t - 1 / 2 , which dampens the step-like nature of equation (3.23), this Heaviside-like behaviour of the approximation is maintained but mitigated by decay with time. Hence one expects the impulse response to be characterized by a Lipschitz exponent a which approaches a — 0 from below as Q —* 0, without reaching it. This also agrees with the trend seen in Figure 3.7a and 3.7b. Chapter 3. Lipschitz Exponents and Q 35 Q = 500 Q = 300 Q = 100 Q = 90 Q = 70 Q = 50 Figure 3.6: Illustration of a selection of the inputs for a numerical test to determine the component of the constant-Q impulse response which dominates in determining the resulting signal's Lipschitz exponent a. (a)-(f) right column: the full impulse response for decreasing values of Q. (a)-(f) left column: the power-law component of the response (See Figure 3.5c) for the same set of Q values. Chapter 3. Lipschitz Exponents and Q 36 0 100 200 300 400 500 0 100 200 300 400 500 100 200 300 400 500 15 10 T 0 100 200 300 400 500 Q -> Figure 3.7: Results of the Q impulse response component test, (a) the recovered Lipschitz exponents a plotted vs. Q for the full impulse response, (b) the recovered a vs. Q for the power-law term only, (c) the recovered smoothness a vs. Q for the full impulse response, and (d) the recovered a vs. Q for the power-law term only. Chapter 3. Lipschitz Exponents and Q 37 3.6 Q Estimation from Lipschitz Exponents In addition to the end-member behaviour, Figure 3.7a suggests a simple relationship between a and Q, and therefore that by finding a simple map Q(a) it is possible to use recovered a from a trace to estimate Q. I compute empirically a map between a and Q by performing a linear regression on ln(a + 1) vs. Q, and, based on the limiting arguments for small Q, forcing the line to go through the origin. This is illustrated in Figure 3.8. Before discussing it further, issues of accuracy must first be revisited. 0.5 0 -0.5 -1.5 C "2-5 -3 -3.5 -4 -4.5 -5 0 50 100 150 200 250 300 Q-> Figure 3.8: Logarithmic plot for empirical map Q(a). There are two dominant sources of inaccuracy facing the use of this Q(a) map, (i) at small Q, inaccuracy in a-recovery due to increasing smoothness a, discussed previously, and (ii) at large Q the increasing insen-sitivity of a to changes in Q. The latter is a consequence of a asymptotically approaching — 1 as Q —> oo. This source of inaccuracy is easily visualized in Figure 3.7a: at Q w 150, small changes in a correspond to small changes in Q. At Q > 250, the same small changes in a correspond to much larger changes in Q. Hence a constant error Ac* produces a growing error in ln(a + 1) as a —> —1. Figures 3.9a and b repeat the plot of ln(a + 1) vs. Q, with error bars associated with the large-Q insensitivity, and a threshold at low Q w 50 beyond which the smoothness of the impulse response leads to stability problems in the a-estimation. The error bars are based on a constant error in a of A a « %5, which is between that predicted by Mallat and Hwang (1992) (%10) and that empirically found in this investigation (of the order of %1). Both these sources of error have regimes in which they seriously degrade the accuracy of the Q(a) map. Fortunately, between large and small Q there is a region where neither source of inaccuracy is grossly present. Chapter 3. Lipschitz Exponents and Q 38 0 50 100 150 200 250 300 0 50 100 150 200 250 300 0 50 100 150 200 25D 300 Q-* Q -Figure 3.9: Accuracy associated with (a) insensitivity of log(a + 1) to Q as a + 1 —> 0, (b) increased smoothing at small Q, which leads to accuracy worse than %10 for Q less than the dotted line, (c) Between the dotted lines is the proposed region within which a estimation maintains accuracy needed to characterize Q. Figure 3.9c illustrates this region, and a linear regression is applied to the a-estimates within the accept-able range. A high correlation coefficient R2 = 0.96 is obtained. The regression produces a Q(a) map of the form Q(a) = - ^ l n ( a + l) , (3.24) where, using a sampling interval of A i = 0.004s, \i = 10 is found. The regression would have to be reproduced prior to any usage with a different time-step. This range over which the Q(a) map maintains acceptable levels of accuracy is an amenable range for seismic studies (Q = 50-250). 3.7 The Effect of Bandlimitation The examples used previously were full-bandwidth, in the sense that all low- and high-frequency components necessary to capture the character of the impulse response were present. In field data the bandwidth is limited by source/receiver effects. Since broadband data is an achievable goal, but full-bandwidth data is not, a practical understanding of how the seismic wavelet alters previous development of Q{a) will be crucial to its practical application. In this section, bandlimitation and the practical application of a local-regularity based processing method is considered. In many practical instances, the complexity of the source wave field, directional and frequency-dependent, coupled with the receiver effects, are adequately modelled via the convolution of a reflectivity series with a stationary wavelet. For the ID reflectivityjQ impulse response model considered here, this approach is used. Thus far I have considered a seismic event to be locally generated by the impulse response b(t) of equation (3.19), a pulse having propagated a distance z at speed cn in a medium characterized by absorption Q. In the context of a reflectivity series model, locally a trace x(t), like those inputs of Figure 3.6a - e, can be written x(t)=b{t)*r{t), (3.25) where r(t) = R5(t — t0). Then this impulse would be seen to arise at time t0 with an amplitude dictated by the reflectivity R and the propagation law constituting b(t). At present I consider a single event rather than an r(t) made up of many lagged delta functions. This is again to avoid the issue of acquisition, and the concern over distinguishing between Q, c 0, and z. The bandlimited case is modelled by considering a wavelet w(t) with a spectrum which suppresses high and low frequencies in a manner consistent with the seismic experiment. Then the model of equation (3.25) would be replaced by xb(t) = w{t) * b(t) * r(t). (3.26) Chapter 3. Lipschitz Exponents and Q 39 Although w(t) is stationary, the apparent result of equation (3.26) is a reflectivity series convolved with a non-stationary wavelet w(t) * b(t). To proceed I consider a specific type of source wavelet w(t), and, to be sure, the results are strictly specific to this form of the source. Nevertheless, the conclusions regarding the basic impact of such an operator upon the data will (i) show that the presence of source/receiver effects are not fatal to computing Q(a), and (ii) produce a theoretical framework for compensating for w(t) in this regard. In all instances, the greater the prior knowledge of w(t), the better for the current Q-estimation scheme. The Ricker wavelet is a non-causal approximation to a source/receiver function which often provides a mathematically straightforward way to represent the bandlimited nature of seismic data. The Ricker wavelet is the second derivative of a Gaussian, the variance of which determines its frequency content. In the context of the smoothed-singularity model discussed throughout this chapter, it may be expressed as w(t) = Wr^9ar(t), (3.27) where the Gaussian 0ar(t) is characterized by variance a 2,, and where Wr is the amplitude. I continue by setting Wr = 1; there is no loss of generality, since, as discussed, a scalar amplitude does not change the regularity of an event. Equation (3.10) describes the smoothed-singularity signal model through which the Lipschitz exponent a etc. is estimated. The full-bandwidth trace x(t) of equation (3.25), in the framework of this model, is locally describable as x(t)=6a(t)*h(t), (3.28) where the smoothing parameter a and the regularity of an underlying, non-smooth, signal h(t) combine to determine the local behaviour of x(t). The results and Q(a) map of the previous section apply immediately to x(t). The associated bandlimited trace, by making use of equation (3.26) and (3.28), is xb(t) = w(t) * x(t) d2 dT2^ * [0a(t) * h(t)\ ( 3 2 9 ) d 2 dt 2' U(t)*^Ht), where C o = \f°~ 2 + c 2- The difference, between a signal bandlimited by a Ricker wavelet, and a full bandwidth signal, is the difference between £&(£) and x(t), or equations (3.29) and (3.28). In other words, two things have happened to the trace: (i) the smoothing a is increased to ao, and (ii) the underlying signal h(t) has been replaced with its second derivative. This will the estimated Lipschitz exponent a in a predictable way. First, it will be globally transposed to a — 2. Second, the accuracy of the numerical estimation of a will be compromised by the increased smoothness ao- Nevertheless, in principle the true Lipschitz exponent may be recovered atrUe = ctest +  awaveiet, where a e s t is the recovered exponent and awaVeiet is the order of the wavelet if it is considered to be a differential operator. In the case of the Ricker wavelet a w a v e [ e t = —2. There is an apparent conflict in the above statements. It has been pointed out (Herrmann, 1999) that in bandlimiting seismic data, say, via convolution with a Ricker wavelet, one is essentially taking the wavelet transform of the full-bandwidth signal at a fixed scale, and as such the data will be effectively monoscale. In the context of equation (3.29), the component #CT(i) suppresses high frequencies, and the operator d 2/dt 2 suppresses low frequencies, so the available information is limited to the single scale implied by the remaining passband. Chapter 3. Lipschitz Exponents and Q 40 .a •a b 1 \ / 1 / I 1 \ 1 \ 1 \ 1 \ J \ 1 V Figure 3.10: Illustration of wavelet transform scales admitted by a Richer wavelet; (a)-(e) illustrate the wavelet scales (solid lines) which contribute to a signal bandlimited by the Richer wavelet (dashed lines). While this is certainly true, there is a relativity to the term 'monoscale' amongst wavelet transforms. Numerical tests show that, in a Ricker wavelet, multiscale information is still discernable across several dyadic scales. This is illustrated in Figure 3.10. Within the passband of a Ricker wavelet, are non-negligible contributions to 5 scales for the same wavelet transform W2i used previously. In Figure 3.11b this can be seen to translate as W T M M which, when compared to the full-bandwidth case (Figure 3.11a) are reduced at the extreme scales, but still measureable. Q value a (Fullband) a (Bandlimited) 300 -1.00 -3.08 200 -0.88 -2.60 150 -0.79 -2.40 100 -0.60 -2.20 90 -0.54 -2.17 70 -0.43 -2.00 Table 3.2: Estimated a for a range of Q values: comparison of full bandwidth vs. bandlimited cases. If the signal is discernable across scales, and one may predict the cross-scale behaviour associated with this bandlimitation, then the a-estimation procedure is viable even in bandlimited cases. I repeat the experiment Chapter 3. Lipschitz Exponents and Q 41 t(s) t(s) Figure 3.11: The effect of bandlimitation on the distribution of signal energy across a selection of scales; (a) full bandwidth distribution of WTMM of unit impulse input; (b) bandlimited distribution of WTMM, Ricker input. of Figure 3.7 on the Q impulse response, now having been convolved with a Ricker wavelet, shown in Figure 3.13. The a vs. Q results are depicted in Figure 3.13, and tabulated in Table 3.2. As in the full-bandwidth examples, the W T M M are taken from the "early" side of the arrival (the left side of the illustration), as signal strength is consistently higher here. Because there are now more maxima paths (4 rather than 2), the larger of the two early maxima are utilized. In principle, any one of the paths is sufficient to characterize the local a at the event (Mallat, 1998). Within a Q range of approximately (70,200) a global transpose of the non-bandlimited results is noted, with an average value of —1.7. The reduction in the estimated a is in broad agreement with the theoretical remarks; the increase in error due to the higher level of smoothing from the bandlimiting wavelet is a new and important source of inaccuracy for the recovery of a e s t . Indeed, the gradual increase in recovered a with diminishing Q is not seen in this example, and large variation in the amplitude parameter has been required to "explain" the input. It is clear that for bandlimited data one is much closer to the boundary wherein (Mallat and Hwang, 1992) is reliable. Nevertheless the result is important in that it demonstrates that bandlimitation is not a fatal flaw for a method based on numerical estimation of Lipschitz exponents. In order to use a to estimate Q in such a bandlimited case, either (i) the stationary source wavelet must be known, and may then be deconvolved leaving only the effects of the constant-Q impulse response, or (ii) the regularity of the source wavelet (insofar as it approximates a linear differential operator) must be known or estimated, such that a transpose of the kind described here may be performed. Chapter 3. Lipschitz Exponents and Q 42 Q = 300 Q = 250 Q = 200 Q = 150 Q = 100 Q = 90 t(s) Figure 3.12: Illustration of the form of input used to generate Q{a) empirical map: (a) - (f) Ricker wavelet convolved with constant-Q impulse responses for range of Q values. Figure 3.13: Comparison between recovered a vs. Q for (circle) full-bandwidth case, and (filled circle) bandlimited case. 3.8 Remarks The goal of this chapter has been to understand the impact of absorption and dispersion on the characteristic singularities of a seismic trace; concurrently it has been to develop a tool for the extraction of local Q Chapter 3. Lipschitz Exponents and Q 43 values from a nonstationary seismic trace, by using measures of regularity derived through wavelet analysis. Numerical sensitivities, and the alterations to such extraction, necessary due to bandlimitation, have been investigated: results indicate that, especially in experiments with large bandwidth, and, it is assumed here, low noise, such analysis can be made practical. It is hoped that a practical implementation of this type of algorithm could become a useful member of the general Q-estimation toolbox. With respect to other Q estimation techniques, this regularity-based method resembles, in its output, those of the "rise-time" ilk, in that they are inherently local, i.e. pertaining to a single event. In this sense it also bears resemblance to methods based on the scaling laws developed by Kjartansson (1979) - it might be speculated that these two types of approach are more deeply linked, since both are derived at least indirectly through measures of scale. The caveat is that the multiscale nature of the Q estimation discussed in this paper is indirect, a wavelet-based means to measure local differentiability, which in turn is linked to Q. Nevertheless, investigation of such relationships might well constitute a fruitful line of inquiry. Chapter 4. Simultaneous Imaging and Inversion 44 Chapter 4 S i m u l t a n e o u s I m a g i n g a n d I n v e r s i o n Term Signifies L, L 0 Wave operators G , Go Green's operators i>* Scattered field V Perturbation operator Vj Perturbation component, j ' th order in the data G0(z\zs;k) ID Green's function, source at zs, receiver at z a(z) Perturbation for ID constant density acoustic case aj(z) a(z) component, j ' t h order in the data c 0 / c{z) Reference / non-reference wavespeed profile k / k{z) Reference / non-reference wavenumbers H(z - z') "Left-opening" Heaviside function n{-} "Left-opening" Heaviside convolution operator « " { • } "Right-opening" Heaviside convolution operator H Simplified notation for H { a i } In n'th term of simultaneous imaging and inversion (SII) formula n'th coefficient of SII formula Reflection coefficient of j ' t h layer in ID model OtINV Totality of inversion subseries terms CtpiNV Totality of partial (SII) inversion subseries terms 4.1 Introduction and a Useful Notation The inverse scattering series is the only known method for multidimensional direct inversion of the seismic wave field. That is why it is pursued. Its superficial promise of "black-box"-like transformation of the data into the model, however, has not been realized, numerically, because of divergence for all but the lowest-contrast examples (Carvalho, 1992). Whether this divergence is confined to numerical examples, or is due to more fundamental convergence issues, is not known. Experience has shown that to use the inverse scattering series as a means to process and invert seismic data requires a clear understanding of its inner workings, and an informed use of the data operations it espouses. This involves, but is not necessarily limited to: appropriate casting of the series, choice of reference medium, identification and separation of meaningful subseries, further choice of leading order terms from these subseries, and application of corrective strategies for numerical application, and application to field data. (Some of these steps are described in detail in this chapter.) In other words, intuition and empiricism are unavoidable components of the inverse scattering series' successful development and use. The idea of task separation has been the critical conceptual leap in the success of the inverse scattering series to date (Weglein et al., 1997, 2000, 2002). It hinges on the apparent willingness, or even predisposition, of the series to compartmentalize the entire inversion into portions that greatly resemble existing steps in seismic data processing. The removal of free-surface multiples is a prime example. Task separation is twice beneficial, because (i) separately accomplishing any one task disengages the user from the other, often far more complex (and possibly divergent) problems of inversion, and (ii) the output of this task may be a valuable product in its own right. Chapter 4. Simultaneous Imaging and Inversion 45 This chapter is the only one in the thesis in which the developed theory is not eventually applied to wave fields which have propagated in absorptive/dispersive media. The simple reason for this is that it represents an approach to non-linear inversion that has required a lot of development based on the simpler problem of ID acoustic wave propagation. Longer term plans include casting the viscoacoustic inversion problem along the lines of the acoustic case exercised here; as it stands, this chapter may be viewed as a project on acoustic imaging and wavespeed inversion, or as the groundwork to an eventual generalization to viscoacoustic media. The chapter is concerned with the two key tasks of inversion beyond those of free-surface and internal multiple removal and attenuation: imaging and target identification (or inversion proper). As amazing and counter-intuitive as the multiple examples are - i.e. that the multiples are eliminated or suppressed with no information regarding the subsurface or assumptions about its structure - internalizing the promise of these last two tasks requires an even greater leap. The possibility of correctly locating regions of sharp change in the Earth (reflectors) without knowing the wavespeed structure of the medium is not easily, or sensibly, envisioned from the standpoint of Green's theorem and wave field continuation. This is because the interrelationships amongst the events of the data which permit them to be correctly located are not predictable as a consequence of Green's theorem (which are cast only in terms of the true medium wavespeed). Yet it is possible: methods based on scattering theory are a casting of the relationship between Earth parameters and seismic data that naturally accommodates imperfect prior knowledge of the wavespeed structure of the medium. The true Earth model is mathematically related to an incorrect (reference) Earth model and the incorrect (reference) wave field. Candidate subseries of the inverse scattering series which are involved with, individually, only imaging and only inversion have been identified and are the subject of intensive research at present. Both methods are considered to act upon data which are made up of primaries only, in other words, one assumes that the multiples have been successfully removed. This is characteristic of the task separation approach as well: a processing step (e.g. internal multiple attenuation) is accomplished, and then the problem is recast as if multiples never existed. Amongst other advances, a leading-order imaging subseries has been identified (Shaw et al., 2003) in the ID normal incidence case for which (i) a term of arbitrary order may be immediately written down, and (ii) a closed-form expression exists. This subseries has been shown to locate reflectors with a high-degree of accuracy without concerning itself with the issue of perturbation amplitude (i.e. the ID version of the inversion task). Furthermore, the second term in the inversion subseries has been shown to improve the estimation of density and bulk modulus (and thereby P-wave velocity etc.) beyond the linearized form for a ID case with offset. This amounts to a method for nonlinear AVO (Zhang and Weglein, 2003). Hence, early evidence is strongly suggestive of the value of task separation. Nevertheless there is no absolute indication that the separation of the tasks of inversion and imaging is a requisite step. This is a strong theme of this chapter: I note that with certain approximations, combinations of terms can result in forms that are simply computable and readily stabilized numerically, and which amount to simultaneous imaging and inversion of the input data. It is likely that, ultimately, understanding which of combined imaging/inversion and separated imaging/inversion is the more appropriate will only result from treating and investigating both exhaustively. Indeed it is probably most prudent to assume that either approach could, situation to situation, be the most suitable. In this chapter, following a review of existing theory and method, I outline the formulation of the si-multaneous imaging and inversion approach. I initially use the formulation to reproduce terms from the ID normal incidence inverse series as a means to illustrate the extent of the approximations made. I then retrace my steps and detail how, making these approximations, the form of the terms was deduced, and, in particu-lar, how these approximations result in reducing the effect of entire classes of terms in the inverse series to simple alterations of the sign of the output. I then discuss the numerical implementation of this framework; specifically, the dampening of the derivative operators necessary for the computations to be stable. Results are shown for several ID normal incidence examples, including examples with greater and lesser structure, and greater and lesser contrast. I also illustrate the sensitivity of these examples to incoherent noise, and further apply a form of spectral extrapolation to deal with the bandlimitation. Finally, I discuss the implications of this work. The difference between ID examples, which are focused on here, and the multidimensional algorithms which are the ultimate aim of such research, is not trivial. One must be clear on the benefits gained from exploring ID aspects of this theory - and what insights into the Chapter 4. Simultaneous Imaging and Inversion 46 ID case can provide regarding an eventual multidimensional implementation. ID examples, in more well-developed methods of inverse scattering theory, are seen to speak to the multi-D case "mechanically". By mechanically, I mean that the way algorithms based on the inverse scattering series numerically interrogate the data for the required information tends to be the same in all dimensions. For instance, prediction of the timing of a free surface multiple from the timing of two primaries is independent of dimensionality: it is always done via the autoconvolution of the data. Levels of complexity change, to be sure, but the basic numerical mechanism is retained. I show how a simultaneous imaging/inversion operator, whose form is dictated by analysis of ID normal incidence acoustic models and the inverse scattering series, performs imaging via identification and correction of discontinuities in the measured data, and inversion via an alteration of the amplitudes that comes from the exponentiation of the integrals of the measured signal. 4.1.1 The Inverse Scattering Series This section follows closely the derivation and discussion of Weglein et al. (2003). To briefly review the theory of inverse scattering, it is useful to temporarily resort to an operator notation, whereby, for instance, a "true" wave field satisfies the equation L G = -<5 ( r - r s ) , (4.1) and a "reference" wave field satisfies L 0 G 0 = -<5 ( r - r s ) , (4.2) where L and Lo are the true and reference wave operators, and G and Go are the true and reference Green's operators respectively. The operators are general in the sense of model, and fully 3D. Equations (4.1) and (4.2) are in the space/temporal frequency domain. Two important quantities are associated with the difference of these operators: V = L - L 0 , (4.3) known as the perturbation operator, scattering potential, or scattering operator, and fl = G - G 0 , (4.4) known as the scattered wave field. The Lippmann-Schwinger equation used in chapter 2 is an operator identity in this framework: * s - G - Go = G 0 V G , (4.5) and, as in that chapter, it begets the Born series through self-substitution: tfs = G 0 V G o + G o V G o V G o + G 0 V G 0 V G o V G o + ... = ( * » ) l + ( * s ) 2 + ( * f l ) 3 + -In other words, the scattered field is represented as a series in increasing order in the scattering potential. As discussed, this formalism constitutes a forward modelling of the wave field; it is a nonlinear mapping between the perturbation and the wave field, the latter being written in increasing orders of the former. Inversion, or the solution for V from measurements of the scattered field outside of V , has no closed form. The approach taken here is that of Jost and Kohn (1952) and Moses (1956); it was formulated for the inversion of wave velocity by Razavy (1975), and discussed in the framework of seismic data processing and inversion by Stolt and Jacobs (1981) and Weglein et al. (1981). It is to represent the solution (the perturbation operator) as an infinite series: V = V 1 + V 2 + V 3 + . . . ) (4.7) Chapter 4. Simultaneous Imaging and Inversion 47 where V j is "j'th order in the data". This form is substituted into the terms of the Born series, and terms of like order in \I/S are equated (each term is considered to have been evaluated on the measurement surface m). This is the form of the inverse scattering series: ( * s ) m = ( G o V x G o ) ™ , 0 = ( G 0 V 2 G 0 ) m + ( G 0 V i G 0 V i G o ) m ) 0 = ( G 0 V 3 G 0 ) m + ( G o V i G o V x G o V i G o ) ™ + ( G 0 V 1 G 0 V 2 G o ) r a + (G0V2G0V1G0) The idea is that V i , the component of V that is linear in the data, is solved for with the first equation. This result is substituted into the second equation, leaving V 2 as the only unknown, which may then also be solved for. This continues until a sufficient set of V j are known to accurately approximate the desired result V . The form of the operators L , G , L 0 , Go, and V obviously vary depending on the desired form for the wave propagation (i.e. acoustic constant density, elastic, viscoacoustic, etc.), and, in the case of V , are particularly dependent on how propagation in the reference medium differs from that of the true medium. As in the case of the forward scattering series investigated in this thesis, I focus on the simplest possible cases: here that of ID constant density acoustic media. Reference media are kept homogeneous, and the scattering potential is considered to be confined to a finite region on one side of the source and receiver locations. This choice amounts to defining + 1 dz2 \ CQ L 0 in which case where k = OJ/CQ and a(z) = 1 + CQ/C2(Z). This simple physical framework also permits the use of the Green's function pik\z-z„\ G o { z l Z s ; k ) = ^ i k ~ ( 4 ' n ) (which becomes G 0 when it is included as part of the kernel of the integrals of the series). In this framework the inverse scattering series terms of interest (equation (4.7)) can be reduced to a(z) = ai{z) + a2(z) + 0:3(2) + ... (4.12) Finally, following the conventional physical interpretation of the "rightmost" Green's operator in every term of equation (4.8) as being the incident wave field, these are replaced with incident plane-waves ip0(z\zs; k). The terms of equation (4.8) become i>s(z\z3;k)= I G0{z\z';k)k2ai{z')ip0{z'\zs;k)dz', J — OO /OO Ga(z\z'\ k)k2a2{z')ip0{z'\zs; k)dz' -00 (4.13) /CO pOO G0{z\z'; k)k2ai(z') / G0{z'\z"; k)k2ai{z")W\zs; k)dz"dz', -OO J —OO Chapter 4. Simultaneous Imaging and Inversion 48 Consider these equations individually. Solving the first for ct\(z) is the ID equivalent of Born inversion, and amounts, in essence, to trace integration. The resemblance of the equation to a Fourier transform results in (Weglein et al., 2003) « i ( z ) =4 / ips{z')dz', (4.14) Jo where the depth variable here is the so-called "pseudo-depth", determined by the natural time variable of the measurement of the wave field tps and the reference wavespeed profile. Weglein et al. (2002) approach the subsequent orders of a(z) by casting them all in terms of the Born approximation 0:1(2;); because of its (comparatively) simple linear relationship with the data, a± is often referred to as the "data" in discussion. Also, choices for breaking the integrals up are made based on the resulting form of the terms in orders of ct\\ these forms are by no means the only way to solve the integrals of equation (4.13); they are a reasoned choice, the basis for the separation of tasks into those of inversion and of imaging. The formalism described above is based on the assumption that one measures the "scattered field" ips = ip — ipo- It is further assumed (as mentioned above) that all the scatterers occur "beneath" the source and receiver. These assumptions and others lead to a set of requirements on the data that characterize approaches, for the treatment of primary reflections, based on the inverse scattering series: 1. The source waveform has been compensated for. 2. The direct wave has been removed. 3. The source and receiver "ghosts" have been removed. 4. The free-surface multiples have been removed. 5. The internal (interbed) multiples have been removed. 4.1.2 A Useful Notation In this chapter, I describe various manipulations of the mathematics of the (previously discussed) casting of the terms of the inverse scattering series. Here I present a notation for these integrals which speeds up some of the manipulations based on the chain rule and integration by parts. Subsequently, I use this notation to derive the 2nd order terms of the inverse series, and later, to derive and simplify 3rd and 4th order terms as well. The terms in the inverse series are often profitably cast as an increasingly complex set of operations on ax(z), the Born approximate solution for a(z), which is linear in the data. Of particular importance is the operation f ax(z')dz', (4.15) J—00 and "nested" versions of the same, for instance / Z a i ( z ' ) f al{z")dz"dz', (4.16) and f a i ( z ' ) ^ a i ( z " ) f al(z" ,)dz"'dz"dz', (4.17) J—00 J—00 J—oo etc. Notice that equation (4.15) is a linear operator applied to cti(z), namely the convolution of ot\{z) with a "left-opening" Heaviside function. Define this operator, i.e. convolution with a left-opening Heaviside, as Ti {•}, such that /oo rz H(z-z')ai(z')dz' = / ai(z')dz'. (4.18) -OO J — 00 Chapter 4. Simultaneous Imaging and Inversion 49 Although it will be used sparingly, I further define the convolution of a\(z) with the time reverse of these Heaviside functions, that is, the "right-opening" kind. Let: /oo poo H(z' - z)a1(z')dz' = I ai{z')dz'. (4.19) •oo J z The following relationship between H and H~~ proves useful: for any f(z) and g(z), f(z')H-{g(z')}dz'= / g(z')H{f(z')}dz'. (4.20) - O O •' — oo It can be derived by substituting the explicit form of 7i~ into the left-hand side of equation (4.20) and switching variables. An explicit form of this manipulation is done regularly in the derivation of the terms of the inverse series. I have developed this method/notation simply to speed up the derivations. Since in this chapter the operator TC{} is often (but not exclusively) applied to ai(z), for convenience call H = H{al{z)}. (4.21) Since the operator Ti is essentially the antiderivative operator, it follows that for any f(z) that is confined to a finite region, The nesting seen in equations (4.16) and (4.17) is incorporated into this operator framework straightforwardly. Define H2{a1(z)} = H{a1(z)H{a1(z)}} = f a^z') f a1(z")dz"dz', (4.23) J — oo J — OO and /Z pZ t*Z ai(z') / ai(z") / a1(z"')dz"'dz"dz', (4.24) -oo J — oo J - co and so on. In general, Un {oi(z)} = H {ai(z)W n_i {a^z)}} . (4.25) This nesting notation, i.e. Hn {a\(z)} will not be retained. Since ^ = « i ( * ) . ( 4 - 2 6 ) equation (4.23) can be written n2{a1(z)} = r i ! [ ^ H y (4.27) Also, since 1 dH 2 dH 2 dz dz equation (4.22) can be used to write H, (4.28) Chapter 4. Simultaneous Imaging and Inversion 50 Using this result in the expression for H3, with similar arguments, and continuing on to 7in, the general relationship Hn{ai(z)} = - Hn (4.30) n! may be derived. The simplification implicit in equation (4.30) is not trivial - computing the right-hand side with a given ai is much simpler than computing the left. This terminology is used in the next section, to assist in the derivation and exposition of the terms in the inverse scattering series, and again (and more extensively) in subsequent sections to develop and analyse simultaneous imaging and inversion. 4.1.3 Imaging and Inversion Summing a.\ + a2 + ... implies the provision of the model a in terms of the measured wave field and the reference media; hence it is inversion, and imaging, without knowledge of, or determination of, the true wavespeed structure of the medium. Understanding the mechanisms of a series with this promise is made possible by study of the simplest possible models. In this section, I reproduce the 2nd order terms in the inverse series as cast by Weglein et al. (2002), using the operator notation defined above. The simplification achievable by the notation is not noticeable until later sections; here the idea is to show where in the process of deriving terms it becomes applicable. The second order terms in the inverse series form the equation /oo G0(z\z'; k)k 2a2(z')Mz'\zs; k)dz' = ,00 ,00 (4-31) - / G0(z\z';k)k 2ai(z') / G0(z'\z";k)k 2a1(z")ip0(z"\z3;k)dz"dz'. J— CO J—CO Upon substitution of the forms of the Green's functions etc. into equation (4.31), many like terms cancel. Furthermore, the left hand side of the equation, like in the Born approximate case, is a Fourier transform of the perturbation component (a2 in this case). Hence equation (4.31) may be written •t rCO pOO a2(-2k) = -~{i2k) / e ikz'ai(z') e ik\*'- z"\a1(z")e il""dz"dz', (4.32) ^ J — OO J —OO and, treating the two cases demanded by the absolute value bars, one has -1 poo pz a2(-2k) = ~-{i2k) / e ikz'ai(z') e ik^ z'- z"^ ax{z")e ikz" dz" dz' ^ J —OO J — OO -1 poo poo -j(i2k) / e ikz'ai(z') / e ik(z"- z,)ai(z")e ikz"dz"dz' ^ J — 00 J z •j pOO poo = -j (i2k)e i2kz'a^z') H(z' - z")ai(z")eikz"'dz"dz' J —00 J — 00 I pOO poo — / (i2fe)ai(z') / H{z" - z')ai(z")e i2kz"dz"dz'. ^ J—00 J—00 The operators defined previously are applicable at this stage, i.e. with the appearance of the convolutions of various quantities with Heaviside functions. Equation (4.33) can be written (4.33) Chapter 4. Simultaneous Imaging and Inversion 51 Q 2(-2fc) = --A\ (i2k)emz (z')dz' - - r (t2fe) [aiW- {e i 2 / c 2 ' a i} ] ( z ' ) ^ ' 1 f 0 0 = - - [i2k)e i2kz [aiH] (z')dz' ^ J—oo - - r (i2fc) [e i 2 f c z ' t t 1H] (*')<**'. ^ J — OO (4 .34) Because the Ti, Ti" operators are convolution operators, the totality of their action in the integrands in equation (4 .34) is an expression with z' dependence only. For convenience I therefore use the variable z' freely inside these operators even though strictly speaking it doesn't belong1. For instance, H~ { e i 2 k z ' a i } = ei2kz"ai(z")dz". Both terms in equation (4 .34) are Fourier transforms of derivatives with respect to z, so (4 .35 ) dai H + a\ Finally, therefore, a2(z)=a£\z)+a¥>(z) (2), 1 2/ x 1 " 2 a i ( z ) - 2 dai dz [ ai(z')dz'. Jo Using similar manipulations, the third order terms may be derived as (Weglein et al., 2 0 0 2 ) : a3(z) = a£\z) + a{2\z) + af{z) + a£\z) + af(z), where (4 .36) (4 .37) (4 .38) a. (2), af(z d 2ai J\i{z')dz^j , :cn{z) dai dz If ai(z')dz', a i \ I o a l { z ' ) d z ' 1_ fz fz \dai(z')] \dai(z") L6 70 Jo ai(z" + z' - z)dz"dz'. (4 .39 ) This specific casting of the inverse series, as in equations (4 .37) and ( 4 . 3 9 ) , naturally separates the full inversion process into tasks (Weglein et al., 2 0 0 3 ) . For instance, at each order there is a term which is a weighted power of the Born approximation. Finding these in equations (4 .37) and (4 .39) and summing produces the subseries J I t is reminiscent of the convention of discussing convolutions as h(z') = f(z') * g(z'), i.e. with seemingly careless use of the output variable z' in both input functions. Chapter 4. Simultaneous Imaging and Inversion 52 aiNv(z) = ct\(z) (4.40) This subseries has been identified (Weglein et al., 2003; Zhang and Weglein, 2003) and developed as the sub-series that is concerned with target identification, or inversion proper. In the following chapter, I investigate it as a means to invert for viscoacoustic parameters. Similarly, in each order are found terms that involve derivatives of the Born approximation ct\(z), weighted by integrals of the same; summing these produces ctLOisi{z) = ai(z) ^ ] / > - > - ' O f - ' ^ (4.41) This subseries (hereafter referred to as LOisi) turns out (Shaw et al., 2003) to be the "leading order imaging subseries", that is, the leading order set of terms which are concerned with the correct location of reflectors in the subsurface. This identification continues; a ^ \ z ) in equation (4.39) is the leading order internal multiple eliminator. In the following sections, I pursue a grouping of the inverse scattering terms (cast as reviewed in this sec-tion), which captures many imaging and inversion terms simultaneously in a compact and simply-computable form. Finally I explore a signal processing-based view of how the computation "processes" data. 4.2 Simultaneous Imaging and Inversion I begin this section with the statement of a formula for simultaneous imaging and inversion for this ID normal incidence acoustic framework. In section 4.3 I back-track somewhat to give an idea of how the expression was derived. The motivation stems ultimately from the fact that in the terms of the inverse scattering series, formulated as they are above, certain combinations of operations on the Born approximation repeatedly arise. It appears that many of these terms might be produced by a core "generating expression", that for this reason would be equivalent to an engine for the imaging and inversion of the input. Subsequently this formula is explored regarding (i) its ability to reproduce terms in the inverse series, (ii), the simplifications inherent in its neglect of a class of series terms, and (iii) its numeric use for a variety of inputs, including high contrast and noisy examples. 4.2.1 A Quantity Related to Imaging and Inversion Consider the quantity In(z) = Kna-—, (4.42) dz n where ( " I f " 1 ( 1 2("-i) fcl(n-fc-l)! (4.43) Lfe=0 To compute this quantity, the Born approximation ct\(z) is integrated once to get H — H {a\(z)}. The n ' th power is taken, followed by the n 'th derivative; finally it is weighted by Kn. This quantity appears to be intimately connected with the terms of the inverse scattering series which relate to imaging and inversion. In and of itself, it is merely an expression that specifies a combination of derivative orders and (effective) numbers of nested integrals of the Born approximation. 4.2.2 Mapping Between KndnHn/dzn and an One can best investigate equation (4.42) by carrying out the n 'th derivative on the n ' th power of H, without specifying ct\(z), and seeing what happens. In this section equation (4.42) is expanded in this way for n = 1, n — 2, n = 3, n = 4 and n = 5. The results are compared with existing derivations of the an(z) to clarify Chapter 4. Simultaneous Imaging and Inversion 53 which aspects of the inverse problem are addressed by computing them. For convenience I sometimes suppress the z dependence of the Born approximation. At all times c*i implies a\(z). Expansion for the n = 1 Term: If we set n = 1 then K\ = 1, and by equation (4.42), i.e. the Born approximation. Expansion for the n = 2 Term: For the n — 2 term we have 1 1 2 + 2 Meanwhile, d2H2 d dz2 dz Recalling the definition of H in equation (4.21), d2H2 1 [2Ha{\ =2a2 + 2 da\ dz H. h(z) = K2- dz2 da\ dz ai(z')dz'. (4.44) (4.45) (4.46) (4.47) Comparison of equations (4.37) and (4.47) demonstrates that, up to second order, the expression in equation (4.42) reproduces all of the expected inverse scattering series terms: Uz) a2(z). (4.48) Expansion for the n = 3 Term: Proceeding as before, the third term is found by computing and then 1 16 1 1 1 6 + 3 + 6 1 24' 1.49) Al l together, d3H3 dz3 dz2 3 dz 3H2ai 2Ha\ + H2 dai dz 2a\ 6a? + 18 4a i dai dz dai dz aiH + 3 H + 2aiH *d2ai dai dz2 dz H2. + H2 d2ai ~dz^ h{z) = Ki d3H3 dz3 3 dai ajH-t 1 ' d2ai + 4 dz " 8 dz2 (4.50) (4.51) or, explicitly, Chapter 4. Simultaneous Imaging and Inversion 54 h(z) = \al + l dai dz Oil iL^"*)+\ W\ (/L-^ )'- (452) These terms no longer match up one-to-one with the full set of inverse scattering series terms; the difference is due to approximations implied by equation (4.42), which are investigated in the next section. What is missing with this suppression is not merely the leading order internal multiple eliminator, but two other terms as well, including one which alters the coefficient of the inversion (af) term. This can be established by comparing equations (4.52) and (4.38). On the other hand, equation (4.42) has correctly incorporated the other components, including those due to - G o V i G o V i G o V i ^ o , and - ( G 0 V i G o V 2 V ; o + G u V 2 G 0 V 1 ' i / ' o ) , in one fell swoop. Expansion for the n = 4 Term: For the n = 4 case, we have 64 24 + 8 + 8 and d4H4 dz4 3 ? [ « ^ ] 4 ^ dz2 ZH2a2 dz 6Ha{ H dai dz dai dz aiH2 d?ai = 24a\ +144 dai dz a\H + 36 dai dz dz2 l2 1 24 1 192' *d2ai H2 + A8 dz2 axH2 + A So in total, dai dz dai dz H - -d2 ai dz2 ^ - 4 8 d3ai Itz3 d3ai avoiding the replacement of H with its explicit form this time around. Expansion for n = 5 Term: The n = 5 expansion is H3 H3 (4.53) (4.54) (4.55) w x 1 s 5 dai a\H-\ 15 dai dz " 32 . dz d2ai dz2 .2 u2 a\H 1 + 24 d3 Oil dz3 aiH3 _5_ 48 dz2 dai dz H3 + 384 d4ai dz4 H4 again omitting the replacement of H with its explicit form for convenience. (4.56) 4.3 Inherent Simplicities and Approximations Comparing the expansion of equation (4.42) with the derived terms of the inverse series, it is clear that some of the terms are missing, and others have the wrong coefficients. It is important to be very clear regarding what has been "kept" of the full inverse series, and what has been "rejected", in utilizing equation (4.42). This section is concerned with developing both a clear sense of the approximations made in this simultaneous imaging and inversion formulation, and explicitly demonstrating the simplified role of certain classes of terms in the series for the 3rd and 4th order (a role that is assumed to continue at all orders). Chapter 4. Simultaneous Imaging and Inversion 55 4.3.1 Deriving KndnHn/dzn: Drop a Term, Find a Pattern Equation (4.42) was deduced by noticing patterns in some components of the terms involving V i only; for instance, by paying attention to the G n V i G o V i G n V i ? / > o term in the 3rd order equation, and ignoring terms like GoViGoV2'0o- In fact, it was designed initially to compute these terms only. I will demonstrate by considering 0:3(2). Set a 3(z) = IP(z) + Is(z), (4.57) where •1 poo pz pz Ip(-2k) = jk 2 / e i2kz'ai(z') / ai(z") / a1(z"')dz"'dz"dz' ^ J — co J — C O - / - c o 1 poo poo pz" + -k 2 / ai(z') / e m'"ai{z") / a1(z'")dz'"dz"dz' T: J — OO J Z' J— C O ^ h^>) -I pOO pz' pOO + -k 2 e i2kz'ai(z') e~i2kz"ax(z") \ ei2kz"'a1(z"')dz"'dz"dz' ^ J— C O J—co J z" -1 y O O pOO pOO + -k 2 / ai(z') / ai(z") / e i 2 f c z "'a 1 (2 /")6?2"'d2"d2', ^ J— oo Jz' J z" i.e. Ip is due to G o V i G o V x G n V i ^ o ; Is is due to G o V i G 0 V 2 V ; o and G o ^ ^ G o V i ^ n , and is considered later. These terms are broken up based on the geometry of the scattering interactions, akin to the forward case of chapter 2. The third term in equation (4.58) involves a "down-up-down" scattering event, or a change in the directions of propagation; such expressions differ from those involving only upward scattering events, and don't immediately simplify in the same way. My approach has been to set this term to zero. Implementing the operator notation, the expression 1 f°° Jp(-2fc) = -fe2 / ei2kz o i ( z ' ) W { a i H { a i } } d z ' ^ J — OO + jk 2 ai(z')W" {ei2kz'ot{H{ai} W (4.59) J—00 + -k 2 r ai(z')H- [axH- lei2kz'aM dz' J —CO . is produced. I proceed by making repeated use of the relationship in equation (4.20) in the definitions section, to produce from equation (4.59) 1 f°° Ip(-2k) = - — (i2k)2 J e i2kz ai{z')H {aiH {a,}} dz' 1 r 0 0 -f-(i2k) 2 e i2kz ai(z')H2{ai}dz' (4.60) J — C O - ^(i2k) 2 J e i2kz ai(z')n {aiH {ai}} dz', noting that — (1/I6)(i2fc)2 = (l/4)fe2. Identifying these terms as Fourier transforms of second derivatives, equation (4.60) may be replaced with Chapter 4. Simultaneous Imaging and Inversion 56 Ip{z) = ~h>j>?{ai{z)n {a{H {ai}}) 1 d 2 ~ mdz* (ftl(z)H2 { Q l » 1 d 2 (4.61) and finally recalling that 7i2{cfi} = (1/2)H 2, this becomes (4.62) The preceding gives one a sense of how equation (4.42) was developed - doing this for a number of orders, and watching how the coefficient was generated. Interestingly, equation (4.62) is the negative of the associated h{z): K d Z R " r ( ^ and hence, using equation (4.51), is da\ dz axH d 2ai lb 2 H (4.63) (4.64) The difference between 73 and the third order series terms of equation (4.38) indicates the effect of dropping the term from equation (4.58). 4.3.2 Simplifying Groups of Terms Any reader who has taken the trouble to write down the inverse scattering series terms (even in general operator form), beyond the third or fourth order, say, would be excused if they looked at equation (4.42) with mounting skepticism. Its implied simplicity seems to contradict the growing complexity of the interactions of the series with increasing order. In this part of the chapter, I attempt to reconcile this apparent contradiction by explicitly considering terms in third and fourth order. I use the results of this investigation to deduce a pattern of behaviour for the series, a pattern that I assume holds at all orders. Shaw et al. (2003) point out that the LOISI terms are contributed-to from both terms that are in V . i only (e.g. G o V i G o V i G o V i ^ o ) ) and terms that are in V i , V 2 , . . . etc. (e.g. G o V x G o V a ^ o ) - Here I call these "primary" terms and "secondary" terms respectively2. The fact that the LOJSI is reproduced in the expan-sion of equation (4.42) for the third order (this can be seen by comparing equations (4.41) and (4.52)) suggests that simultaneous imaging and inversion correctly incorporates both primary and secondary terms. But I have just shown that equation (4.42) is almost exactly the same as the terms due to G o V i G o V i G o V i V ' o only - primary terms. The only way all these statements make sense is if the secondary terms are bringing about simple, predictable, alterations to their associated primary terms. Secondary terms for 3rd Order Consider again the 3rd order term in which a3(z) = Ip(z)+Is(z), (4.65) for primary and secondary terms respectively (that is, Ip(z) is due to — G r j V i G o V i G o V i V ' o and Is(z) is due to — GoViGoV2i/'o — GoV2GoVi'i/;o)- In the third order, the difference between the primary term 2 I will use the terms primary and secondary a lot in this section. The former shouldn't be confused with primaries as in "the primaries and multiples of seismic data". In this chapter, I never use the word to mean anything but those specific portions of the inverse scattering series defined here. Chapter 4. Simultaneous Imaging and Inversion 57 due to G n V i G o V i G o V i ^ o and the full expression of equation (4.42) is a minus sign - see equation (4.63). However, since the secondary terms only have the power to change the primary term by adding something to it, the simple process of changing a sign can only be achieved by constructing twice the negative of h(z) - a fair amount of work. We can see this by computing Is{z) from equation (4.57). Writing down equation (4.47), generalized to accommodate ct\ and a2, and substituting the expression for a2 therein, one has h{z) 1 2 1 ~ ~2 = -K2 = -K2 = -K2 = -K2 1 , dot\ dz dot\ dz 1 2 H{a2} OL\OL2 - -da2 n y<2 da i dz d 2H 2 dz 2 dH 2^ dz K2 H{ai} 2 H 2 dz 2 d 3H 2 1 dz 3 W { « i } dai dz da i dz dai dz da\ dz dz a\H + 1a\ axH + 2a\ + a 4 ^ + 12dt^H a\ + da i dz da\ a ^ + 2a\ + dz d2a\ Hz2' H axH H 2 + d dz d2a\ lb? a\ + da i dz H 2 + H da\ dz H axH axH d 2 ai dz 2 H 2 (4.66) which is twice —Ip(z). Note that the constant K2 may be brought out of the H.{-} operator. This result supports the postulate that the secondary terms effect some simple alteration to the primary terms. It seems reasonable to look for this behaviour in the other, higher order, terms as well. Secondary terms for 4th Order I proceed by considering the secondary components of the fourth order term. Since V 4 is solved-for through the equation G 0 V 4 V 0 = - G o V x G o V i G o V i G o V ^ o - ( G o V 1 G 0 V 1 G o V 2 V ' o + G o V i G o V 2 G 0 V i V ' o + G 0 V 2 G o V i G o V i ^ 0 ) (4.67) - ( G o V x G o V 3 V o + G 0 V 3 G 0 V i ^ o + G 0 V 2 G 0 V 2 ? / ; o ) , there are now six secondary components that need attention. Computation of the secondary terms in this fourth order set is different from that of the third order, because in this case incomplete terms are substituted in, namely the approximation .Z3 « 03. Previously, i.e. in equation (4.66), only substitution for a2, which is fully expressed by equation (4.42), was needed. Not so this time. Calling the inversion terms counterpart to those of equation (4.67) respectively a4= PRIMARY + /112 + /121 + /211 (4.68) + A 3 + -^31 + ^22, I use the same approach as that taken in equations (4.58) - (4.61) to find expressions for the six secondary terms above. At the risk of repeating myself, terms like the third of four in equation (4.58) are neglected in equation (4.68); this means that not only are they not included in the primary terms, but also they are not ever substituted into the secondary terms where they'll be expected, such as in a GnVaGoViGoViV'o-type expression. In appendix A, I show that, after neglecting such terms, Chapter 4. Simultaneous Imaging and Inversion 58 hn + I121 + hu + I13 + hi + I22 = 0. (4.69) That is, through a careful balancing, the fourth order secondary terms, none of which are zero on their own, sum to nil. To summarize, in spite of considerable effort, these "secondary" components of the inverse series with one quarter of the scattering interactions neglected (the neglecting occurs in going from equation (4.58) to (4.59)) only either change the sign of their associated primary terms, as in the third order, or do nothing, as in the fourth. This is why equation (4.42), although designed to compute only primary terms, produces the output of both primary and secondary terms with the simple inclusion of a factor which alters the sign of the output. At present I assume that the patterns seen explicitly, thus far, hold for all the orders of this portion of the inverse series. The correct reproduction of the LOISI subseries found by Shaw et al. (2003), from equation (4.42), would be a good check of this assumption. 4.4 Associated Subseries In this section I look more carefully at some of the subseries which arise when equation (4.42) is expanded for n orders. Some of these subseries have the form of the pure imaging and pure inversion type tasks, which have been developed and discussed elsewhere. 4.4.1 Leading Order Imaging Subseries In section 4.2.2 a set of In(z)'s were explicitly computed and/or listed. Notice that if we collect and sum the last term in each: Oil — 1 dai d 2ai n 7 ^ - 1 d3ai 2 dz dz 2 48 dz 3 the LOJSI series of Shaw et al. (2003) is reproduced: Equation (4.42) is therefore computing terms at all orders which are combinations of primary and secondary components. This supports the idea that (i) equation (4.42) is capturing much of the behaviour of the series, and (ii) therefore the net effect of secondary terms on this portion of the series is the aforementioned change (or not change) of sign. 4.4.2 Inversion Subseries Next, notice that if the first terms in each of the expansions of section 4.2.2 are summed, one gets a series which, because of the form of its constituents, must be devoted to inversion tasks. This is partial inversion only, since as discussed, some terms were neglected in the derivation of equation (4.42). Nevertheless there is a pattern: (4.71) (4.72) fe=i (4.73) The question is, what does this series do? The full inversion subseries is ajNv = on (4.74) Chapter 4. Simultaneous Imaging and Inversion 59 such that, for ID reflection over a single interface with a reflection coefficient of Ri, the inverted perturbation amplitude becomes, in terms of this Ri, a I N V = 4Rr (1 - 2RX + SR 2 - 4R* + ...) = 4 ^ (4.75) (i + K\y Clearly equation (4.73) is not this same series, therefore it will not produce the correct results. To find out what results are produced, I set a similar problem up, i.e. a ID reflection experiment over a single interface. No imaging will be required, so omitting all z dependence, a Born approximate amplitude for the contrast of a i = 4Ri (4.76) is produced. Substituting this into the partial inversion subseries, for api^y in equation (4.73) produces a P I N V = 4RX (1 - 2i?i + 4R\ - 8R\ + 16 R$ - ...) , (4.77) so up to second order in Ri the partial inversion subseries is the same as the full; compare equation (4.75) with (4.77). Beyond that they begin to differ, more-so with higher order in R\. This begins to suggest that the partial inversion subseries (OLPINV) is a low R\ approximation. Of course, without infinite terms, so is the full! This means that at low R\ both the full and partial inversion series are equivalent, but at higher R\ they differ, with the full inversion series performing better than the partial. This is demonstrated in a series of plots of aj^v and api^v against R\, in Figure 4.1. The partial subseries seems to capture the true reflection coefficient very well up until approximately Ri = 0.4 over 8 terms. It seems that apiNV converges to something very close to a/jvv, but more slowly as Ri increases. The point is that the portion of the full inverse scattering series that come from the neglected terms appear to supply the inversion subseries with higher-order components. At low order, the partial inversion subseries within equation (4.42) and the full inversion subseries are almost equivalent. To use equation (4.42) is to invert only partially, but with the help of Figure 4.1, the extent to which this partial inversion is accurate can be understood. In cases where the reflectivity is < 0.4 this partial inversion will be indistinguishable from the full after 8 terms; to put things into perspective, in order to produce a reflection coefficient of 0.4 you would need a wavespeed contrast of about 2 km/s to about 5 km/s. 4.5 Computation: Models and Numerical Issues With some clarity in place now regarding exactly what the "partial" nature of equation (4.42) means, the stability and appropriateness of use of this formula must be addressed. The full inverse series has appeared to be numerically unstable in tests done by others (Carvalho, 1992). Equation (4.42) represents a large portion of the entire series, and may well include whatever terms or operations that destabilizes the full series. Taking the n'th derivative of the n'th power of a discontinuous signal produces very large numbers, and without some regularization the series of equation (4.42) diverges. However, with some fairly straightforward alterations to the derivative operator, the series is stabilized and one may analyse the result. 4.5.1 Synthet ic I D Mode ls , D a t a and B o r n Approx imat ions In this section I use some simple Earth models to exercise the numerical computation of equation (4.42). I begin by generating simple ID models, pathologically designed so that, at the last value of the discrete z vector, H{ai} \z,„,,..,.. ~ 0, which avoids having the end of the signal behave like a strong reflector. This is done with no loss of generality, since any data set can have such an addendum included beyond the deepest point of interest. The first such model has 4 wavespeeds, the reference CQ = 1500m/s, and c\ = 1600m/s, C2 = 1650m/s, C3 = 1467m/s. (This last wavespeed value ensures that the above constraint holds.) These latter 3 wavespeeds correspond to layers which begin at depths 300m, 500m, and 700m respectively. This model is used to generate full-bandwidth data (D(z) at pseudo-depth z), then the Born approxi-mation cti(z), and its integral 7i{ai}. These are plotted respectively in Figure 4.2. This same process is carried out on a suite of Earth models, each having been chosen for (i) simplicity, (ii) high-contrast, and/or (iii) somewhat complex structure. Table 4.1 details the models used. Chapter 4. Simultaneous Imaging and Inversion 60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.3 0.4 0.5 0.6 Reflection coefficient R„ Figure 4.1: Perturbations otit^v/aPiNV  vs- R-i- Comparison of full (dashed) and partial (solid) inversion subseries for single interface experiment using progressively more terms (dotted line is the true model). Both are low Ri approximations, but we know that in its totality the full inversion converges to the true model, (a) 1 term, (b) 2 terms, (c) S terms, (d) 8 terms. The partial inversion subseries follows the full subseries well for R\ values under 0.4 after 8 terms. Depth (m) Model 1 (m/s) Model 2 (m/s) Model I ? (m/s) Model 4 (m/s) 300-500 1600 1600 2000 2000 500-700 1650 1650 1700 2200 700-750 1467 1600 1422 1423 750-800 - 1570 - -800-870 - 1530 - -870-910 - 1500 - -910-co - 1454 - -Table 4.1: All Earth models used in the following imaging/inversion examples. All have reference media (z < 300m,) char-acterized by wavespeed co = 1500m/s. Model 2 has structure deeper than the others: the dash - signifies that the model remains constant at the last given wavespeed value. For instance, Model 1 is constant at 1467m/s below 700m. 4.5.2 B r u t e Implementat ion of J 2 n KndnHn/dzn I begin by "naive" application of the formula. The first term returns a\. The result of computing and adding-in the second and third terms is seen in Figure 4.3. The figure is organized as follows: the top panel Chapter 4. Simultaneous Imaging and Inversion 61 0.05 & 40 £ - 2 0 1000 1000 400 500 600 Pseudo-depth z (m) 1000 Figure 4.2: Synthetic data (a) corresponding to Model 1 in Table 4-1 o,nd its integrals; the Born approximation a\{z) is in (b), and its integral H {ct\} is in (c). The latter is the main ingredient in computing the combined partial imaging and inversion. All three plots are against pseudo-depth z (m). (a) is the synthetic data; below this (b) consists of two functions, the Born approximation a\(z) (dashed), and the true perturbation a(z) (dotted). The required tasks of the inverse series are clear: the inversion must correct the amplitudes, and the imaging must correct the locations. In other words, the inversion must make the dashed be the same as the dotted in the up-down direction of the plots, and the imaging must make the dashed be the same as the dotted in the left-right direction. The next panel (c) again illustrates 0:1(2) (dashed), and includes the cumulative result of the terms in equation (4.42) beyond the first (solid). The lower panel (d) superimposes the full inversion results, a\ + cumulative result (solid), against the true perturbation a(z). In all figures of this kind that follow, "added value" associated with higher order terms in the series is demonstrated by having the plots in (d) become close to one another. One can see the disturbances created by the series at the discontinuities - clearly more terms are needed to correct the location of the discontinuities. It is interesting to note that, by the third term, the amplitude correction (inversion) has come, visually, close to accomplishing its task; away from the discontinuities, and following the Born structure of the model, the layers have found their desired amplitude. Continuing with the naive application of equation (4.42), compute the fourth term and add. This is plotted in Figure 4.4. Clearly the discontinuities on the higher derivative operators quickly create large oscillations over the whole signal. Shortly hereafter the sum "blows up". Something more sophisticated is required. Chapter 4. Simultaneous Imaging and Inversion 62 0.1 ~ 0 3 </5 O rr c o 0) 0 > c -0.1 200 200 300 300 400 500 600 700 400 500 600 Pseudo-depth z (m) 700 800 800 Figure 4.3: The third-order correction from equation (4-4®), using Model 1 from Table 4-1- (a) Data input; (b) Born approx-imation (dashed) vs. true perturbation (dotted); (c) Born approximation (dashed) vs. second-order correction (solid); (d) sum of Born approximation and correction (solid) vs. true perturbation. The inversion task is close to being done. Chapter 4. Simultaneous Imaging and Inversion 63 0.1 £ 0 -0.1 (/) CD rr c g > 200 200 200 300 300 400 400 300 500 600 700 800 500 600 700 800 400 500 600 Pseudo-depth z (m) 700 800 Figure 4.4: The cumulative sum up to fourth order from equation (4-4%), using Model 1 from Table 4-1- (a) Data input; (b) Born approximation (dashed) vs. true perturbation (dotted); (c) Born approximation (dashed) vs. second-order correction (solid); (d) sum of Born approximation and correction (solid) vs. true perturbation. Divergence occurs shortly hereafter. Chapter 4. Simultaneous Imaging and Inversion 64 4.5.3 Stab i l iz ing the n ' th Der ivat ive The uncontrolled oscillation of the unstable cumulative sum suggests that it is the high-frequency portions of the derivative operators which cause the problem. As will be developed in chapter 6, a simplistic regu-larization, amounting to a smoothed cutting-off of the highest frequencies, allows resolution to be traded-off for stability. The regularization works as follows. Windows in the k domain are constructed by convolving a gate function with a Gaussian; such a window is therefore defined with two parameters, the variance of the Gaussian, and the width of the gate. The latter controls the frequency cut-off, and the former controls the smoothness of the "shoulders" of the cutoff. Figure 4.5 illustrates these windows. -0.1 o 0.1 Wavenumber (k) -0.2 -0.1 0 0.1 Wavenumber (k) Figure 4.5: Four plots of regularization windows in the k domain for various parameters. A low variance Gaussian convolved with narrow gate is illustrated (a) vs. wide gate (b) as is a high variance Gaussian convolved with narrow gate (c) vs. wide gate (d). Character of the ensuing regularization may be altered by changing these parameters to suit specific situations. Chapter 4. Simultaneous Imaging and Inversion 65 Figure 4.6: Regularized derivative operators in the k domain (solid) vs. un-regularized operators (dotted), (a) d/dz, (b) d2/dz2, (c) d3/dz3, and (d) d4/dz4. "Levels" of regularization are dictated by the window parameters (see Figure 4.5). In these numerical examples the derivatives are computed in the wavenumber domain. Figure 4.6 illus-trates the un-regularized (dotted) and regularized derivative operators (solid) for orders 1 through 4. 4.5.4 Numerical Examples of Simultaneous Imaging and Inversion Using a sharp truncation (i.e. low variance) and a window that cuts off the high-frequency portion of the derivative operators on either end of the spectrum, I proceed to compute 100 terms in the series of equation (4.42) for the same input. The results are in Figure 4.7. The regularization parameters must be chosen for each example; I did this by trial and error for these examples. Clearly much of the character of the true model is captured here - see the additional examples in Figures 4.8 - 4.10. The main deviation is in the large contrast examples, in which inaccuracy in the imaging (reflector location) appears. In these examples, as evidenced by the poorer resolution, the derivative operators must also be more aggressively regularized (truncated); however, the missing bandwidth does not explain the inaccuracy. It is reasonable to postulate that it is due to the missing higher order imaging terms in the approximation. Recall (section 4.4.2) there was earlier evidence that the partial nature of this scheme meant inaccuracy at higher reflection coefficients. Chapter 4. Simultaneous Imaging and Inversion 66 0.1 2 • -0.1 200 300 400 500 600 700 800 I I 1 1 i 1 1 _ . . j 1 1 1 1 1 _: . i i i 0.2 0 -0.2 200 0.2 0 -0.2 300 400 500 600 700 800 1 1 1 r 1 ' ""'""Tl '' w 1 1 1 -NM 1 1 t 3 (/) 0) DC c g > -E 200 200 300 400 500 600 Pseudo-depth z (m) 700 800 Figure 4.7: The cumulative sum of RS 100 terms from equation (4-42), using Model 1 from Table 4-1- (a) Data input; (b) Born approximation (dashed) vs. true perturbation (dotted); (c) Born approximation (dashed) vs. (solid); (d) sum of Born approximation and correction (solid) vs. true perturbation. Using a low-variance Gaussian and a gate that cuts the derivative operator off on each end of the spectrum, 100 terms from equation (4-4%) are computed and summed. Results capture closely the desired result. Chapter 4. Simultaneous Imaging and Inversion 67 0.05 r i—v r - 0 .05 1 200 300 400 500 600 700 800 900 1000 0.2 i i i i r i i \ i i i o i \J \J 0.2 i i i i i i i i 200 300 400 500 600 700 800 900 1000 3 W CD rr c g CD > C 200 300 400 500 600 700 Pseudo-depth z (m) 800 900 1000 Figure 4.8: The cumulative effects of « 100 terms /rom equation (4-4%): using Model 2 from Table 4-1- (a) Data input; (b) Bom approximation (dashed) vs. true perturbation (dotted); (c) Born approximation (dashed) vs. correction (solid); (d) sum of Born approximation and correction (solid) vs. true perturbation. Using a low-variance Gaussian and a gate that cuts the derivative operator off on each end of the spectrum, 100 terms from equation (4-4®) are computed and summed. Results capture closely the desired result. Chapter 4. Simultaneous Imaging and Inversion 68 Q CO CD DC c o CO CD > c 0.2 0 -0.2 0.5 0 200 300 400 500 600 Pseudo-depth z(m) 700 -1 r " ~* i i \ i \ 1 1 1 1 1 1 800 Figure 4.9: The cumulative effects of « 100 terms from equation (4-4%), using Model 3 from Table 4.1. (a) Data input; (b) Born approximation (dashed) vs. true perturbation (dotted); (c) Born approximation (dashed) vs. correction (solid); (d) sum of Born approximation and correction (solid) vs. true perturbation. Using a low-variance Gaussian and a gate that cuts the derivative operator off on each end of the spectrum, 100 terms from equation (4-42) o-re computed and summed. Results capture closely the desired result, but some inaccuracy in the high-contrast correction is noticed. Chapter 4. Simultaneous Imaging and Inversion 69 3 w a> DC c g w a) > 0.2 0 -0.2 h 200 200 300 300 400 500 600 700 800 400 500 600 Pseudo-depth z (m) 700 800 Figure 4.10: The cumulative effects o / « 100 terms from equation (4-4^), using Model 4 from Table 4.1. (a) Data input; (b) Born approximation (dashed) vs. true perturbation (dotted); (c) Born approximation (dashed) vs. correction (solid); (d) sum of Born approximation and correction (solid) vs. true perturbation. Using a low-variance Gaussian and a gate that cuts the derivative operator off on each end of the spectrum, 100 terms from equation (4-4^) are computed and summed. Results capture closely the desired result, but greater inaccuracy in the high-contrast correction is noticed. Chapter 4. Simultaneous Imaging and Inversion 70 4.6 Numeric Effect of Noise and Bandlimitation In this section I consider the two key departures of real-world problems of seismic processing from theory: noise and bandlimitation. 4.6.1 Robustness to Incoherent Noise It has been mentioned that, in principle, equation (4.42) should be unstable, and, indeed, it has been found that the derivative operators require high-frequency truncation, more-so with greater model contrast. Once this has been accomplished the recovered models capture the sharpness of the contrasts admirably, through the computation of « 100 terms. These results apply for full bandwidth data with no noise. In the next section I will address the problem of bandlimitation. Here I consider the results in the presence of varying levels of additive incoherent (Gaussian) noise. Consider the example of Figure 4.8. Three realizations of Gaussian noise (with variance of approximately %1 of the data amplitudes) are added to the data, and the imaging/inversion terms are recomputed using the H that comes from integrating the noisy data. Figures 4.11 - 4.13 contain the results: The results are clearly deteriorated by the presence of noise - in fact, realization to realization the same level of noise can produce quite different results. Both the location and the amplitude of the correction are corrupted, but a qualitative conclusion is that the inversion, or amplitude results are the most sensitive. In all cases the imaging component, i.e. the movement of the reflectors, still marks a great improvement over the Born approximation. The quality of these Earth models, recovered in the presence of noise, is heartening in the sense that one might expect the smallest amount of noise to render computation of equation (4.42) completely unstable. Nevertheless, it is clear that a very high-fidelity estimate of a\ and thereby H will be of tremendous value in an algorithm based on the development in this chapter. That might include edge-preserving noise reduction strategies, as well as (presumably) low frequency/wavenumber filling strategies. Chapter 4. Simultaneous Imaging and Inversion 7 1 0.05 ~ 0 Q -0.05 200 0.2 0 -0.2 200 3 w 0 rr c g 0 > c 0.2 0 - y j | -300 400 500 600 700 800 900 300 400 500 600 700 800 900 500 600 700 Pseudo-depth z (m) 1000 -I I I I I I I \ . . . . I J I I I ^ - - A — -\ I I I I I 1000 -I I I I I 1 \ 1 1 1 \r -1 I I I 1 — -\ ~ 1 1 1 Figure 4 . 1 1 : The data used to generate Figure 4-8 is corrupted with %1 noise, and the imaging/inversion results are recom-puted. Organization of results is as in Figure 4-8. This is the first of three realizations. Chapter 4. Simultaneous Imaging and Inversion 72 0.05 Q -0.05 -| y y y y-200 300 400 500 600 700 800 900 3 W 0 DC c g 0) i_ 0 > c 200 300 400 500 600 700 Pseudo-depth z (m) 1000 Figure 4.12: The data used to generate Figure 4-8 is corrupted with %1 noise, and the imaging/inversion results are recom-puted. Organization oj results is as in Figure 4-8- This is the second of three realizations. Chapter 4. Simultaneous Imaging and Inversion 73 Figure 4.13: The data used to generate Figure 4-8 is corrupted with %1 noise, and the imaging/inversion results are recom-puted. Organization of results is as in Figure 4-8. This is the third of three realizations. Chapter 4. Simultaneous Imaging and Inversion 74 4.6.2 Spectral Extrapolation: Computing a i from Bandlimited Data There are some striking computational similarities between these inverse scattering subseries methods and other well-studied problems of inversion of seismic wave field measurements. In particular, at its core are integral operations on a quantity which ideally should be realized over a full band of frequencies, including a DC component. Since this quantity is based on wave field measurements which are unavoidably bandlimited, however, it too is bandlimited in practice. The consequences to the output of the subseries are non-negligible. As with seismic inverse methods such as impedance inversion, methods for extension of the spectrum to zero frequency do exist. In this section an investigation of the applicability of such methods to our various subseries, in their current ID incarnations, is presented. I show some specific results and their application to examples akin to those previously presented; but the idea here is to broadly propose the use of a framework, long in the literature (see below for references), for tackling the important problem of bandlimited inversion. A measured signal, wavelet deconvolved, may be assumed to be the bandlimited expression of some reasonably simple full-bandwidth signal type. An example might be a series of lagged and scaled delta functions, i.e. a reflectivity series. The spectrum of the measured signal - which is the full bandwidth spectrum multiplied by a gate function - can be extended, in principle uniquely, guided by this assumption. The calculation of oti is linear inversion, and with the right assumptions it is precisely equivalent to ID inversion for acoustic impedance. So the root of the low-frequency trouble is in essence the same for the inverse series as it is for impedance inversion; strategies for coping with this lack in impedance recovery may therefore by readily applicable in the inverse series. Basic Concepts of Bandlimited Impedance Inversion I begin by discussing spectral extrapolation proper, as part of a brief review of methods for bandlimited inversion. A delta function in the time domain has a complex sinusoid as a spectrum. The frequency of this sinusoid is uniquely determined by the lag of the delta function. Since a sinusoid with additive noise is adequately described with autoregressive (and correctly modelled with ARMA) models, a spectrum of this kind that has been bandlimited can be effectively predicted and therefore extended beyond the bandwidth. Larger order A R M A and A R models may, further, predict the summation of many sinusoids (of different frequencies), and therefore extend the spectra of bandlimited delta series with many varied lags. Consider the reflectivity series that is this sum of delta functions, following Walker and Ulrych (1983): r(t) = J2bko(t-kA), (4.78) k where bk are the reflectivity coefficients and A is the time interval of the experiment. The Fourier transform, or the spectrum of the reflectivity, R(f), is: oo -i2irft «(/) - j e -oo ^ M ( t - f c A ) dt (4.79) if A is set to unity. So the R(f) is this sum of weighted complex sinusoids. With a finite bandwidth, the data R(f) may be extended to the lower (and, to some extent, the higher) frequencies via a prediction scheme. Although the noisy sum of sinusoids is correctly modelled via an A R M A model, Walker and Ulrych (1983) recommend, rather, a truncated AR prediction approach as being more stable. An autoregressive process has a predictable part and an unpredictable, or innovational, part. That is, the j ' t h element of a series yj can be written as a linear combination of previous elements, plus an error (innovation) term: v = ^2akVj-k + ej, (4.80) k=l or Chapter 4. Simultaneous Imaging and Inversion 75 p Vj = - ^SkVj-k + e. (4.81) fc=i where gk = —ak and go = 1 are the coefficients of the prediction error filter. With knowledge of the variance of the error ej (cre), and having in some appropriate way computed the autocorrelation matrix of yj ( R y y ) , the prediction error filter g is found by solving the system where i = [1,0,0, ...,0] T and g and R y y etc. all have complex elements. Associating bandlimited R(f) with yj, the filter g may be applied to existing values of R(f) to produce an extended spectrum data point, one discrete frequency step closer to zero. This point is thereafter treated as an existing value of the reflectivity spectrum, and the process is repeated. Spectral extension, which is essentially extrapolation via a difference equation, has a growing error in practical application as it predicts further from the known data values. There is in fact an overall tendency of extended spectra to decay. This can be avoided by an efficient alternate form of prediction based on the gap-filling algorithms of Wiggins and Miller (1972), in which the existence of known data values on both sides of the unknown region is utilized to avoid such accumulating error. Since, by conjugate symmetry, the negative frequencies are also known, the missing lower frequencies may be thought of as such a gap, on either side of which are known data. An unknown point in the spectrum is considered to be the weighted sum of predictions from the "left" and from the "right" data sets (and in which the weights are determined by minimizing the overall prediction error). Walker and Ulrych (1983); Ulrych and Walker (1984) show that a scheme of this kind finds all the missing data points (of the discrete spectrum) at once. One may extend to high frequencies in the same way, although, since the high frequency gap is considerably larger than the low frequency one, the predictions will be more highly attenuated. The problem may, secondly, be addressed by utilizing the formalism of inverse theory (Oldenburg, 1984; Oldenburg et al., 1983); the approach adopted here is the work of Ulrych (1989), which uses an entropic norm to solve for the whole spectrum with the existing spectrum as the input. The bandlimited signal is viewed as being non-unique, in the sense that an infinite number of models (i.e. time-domain signals) may be Fourier transformed and bandlimited to produce the data (i.e. the bandlimited Fourier transform). The method incorporates prior information garnered from the data (usually a threshold) as a constraint, which directs the model "choice", from the set of allowable models, to be that which is sparse, or spike-like. In such models the low and high frequencies tend to be present. Either of the two approaches discussed above is well-suited to the task of estimating the expected form of the full-bandwidth Born approximation. In the examples to follow, the latter is used, simply because it tends to estimate the high frequencies of the spectrum with greater accuracy. Numerical Examples Using Bandlimited Data In Figure 4.14, detail of the filled data is given close to a single reflection. The top panel (a) is the full bandwidth data, the second panel (b) is the bandlimited equivalent (assuming A i = 0.004s, these examples correspond to a band of 10-50Hz), the third is the spectrally-extrapolated output, and the fourth is a detail (zoom) of the extrapolated (solid) pulse vs. the full bandwidth pulse (dashed). In Figure 4.15, the results of Figure 4.8 are re-computed using spectrally-extrapolated estimates of the data, with only bandlimited traces as input. Further panels are added to the top of this figure; first is the bandlimited data, below that is the filled data, and below that are the panels illustrating the inversion/imaging as previously used. In Figures 4.15 and 4.16, the relatively spatially-complex data example and the high-contrast data exam-ple, both bandlimited are shown to respond very well to the spectral-extrapolation pre-processing. Of course, these examples have not specifically pushed the limits of spectral extrapolation technology. The key is to, in some quantitative way, address the issue of bandlimitation in these non-linear inversion schemes. The value of these examples is in the demonstration of working methods for compensating, often via some reasonable prior knowledge, for imperfections of the seismic experiment. R y y g = o~lh (4.82) Chapter 4. Simultaneous Imaging and Inversion 76 270 280 290 300 310 320 330 270. „ - 4 280 290 300 310 320 330 d 270 280 290 300 310 320 330 Pseudo-depth z (m) Figure 4.14: Detail of the filled data is given close to a single reflection, (a) full bandwidth data, (b) the bandlimited equivalent, (c) spectrally-extrapolated output, and (d) detail of the extrapolated (dashed) pulse vs. the full bandwidth pulse (solid). Chapter 4. Simultaneous Imaging and Inversion 77 TJ CO T3 CD k_ ^ CD N > Q 8 CD rr 3 03 CD DC c g 'to 1— CD > 0.2 0.1 0 200 300 -I / / I I i i i i \ I M i i i i 400 500 600 700 Pseudo-depth z (m) 1000 d Figure 4.15: Simultaneous imaging and inversion as applied to Model 2 from Table J,.l is repeated on spectrally-extrapolated estimates of the data, using only bandlimited traces as input. Further panels are added to the top of this figure (a) the input bandlimited data, (b) the spectrally-extrapolated (recovered) full-bandwidth data, (c) Born approximation (dashed) vs. true perturbation (dotted); (d) Born approximation (dashed) vs. second-order correction (solid); (e) sum of Born approximation and correction (solid) vs. true perturbation. Chapter 4. Simultaneous Imaging and Inversion 78 100 400 500 600 Pseudo-depth z (m) Figure 4.16: Simultaneous imaging and inversion as applied to Model 4 (high contrast) from Table 4-1 is repeated on spectrally-extrapolated estimates of the data, using only bandlimited traces as input. Further panels are added to the top of this figure (a) the input bandlimited data, (b) the spectrally-extrapolated (recovered) full-bandwidth data, (c) Born approximation (dashed) vs. true perturbation (dotted); (d) Born approximation (dashed) vs. second-order correction (solid); (e) sum of Born approximation and correction (solid) vs. true perturbation. Chapter 4. Simultaneous Imaging and Inversion 79 4.7 A Signal-Processing View of Knd nH n/dz .n One of the keys to pursuing (eventual) practical implementation of algorithms based on inverse scattering is to have a clear understanding of the operations being visited upon the data as a result of the theory. Such a working knowledge advances our general understanding of how amplitude and timing information is extracted and synthesized in the theoretical milieu of the inverse scattering series. In this section I explore the numerical action of equation (4.42) on a general input, with the aim of gaining a clearer signal processing-based view of what simultaneous imaging and inversion does. The engine of simultaneous imaging and inversion is where the quantity in brackets (•) is a discontinuous input akin to the second integral of a seismic trace (H in this chapter). In words, the quantity (•) is taken to the n'th power, and the ?i'th derivative with respect to z is carried out upon the result. This is done for a range (w 100) of values of n, and the various outputs are weighted and summed. The two cascaded operations, (1) take the n'th power and (2) take the n'th derivative, have a vari-able impact upon the integral of the input Born approximation. This helps explain, in an intuitive, signal processing-based way, how the processes of imaging and inversion can proceed simultaneously through a simple computation. I consider synthetic data due to a ID normal incidence model (for instance, Model 1 in Table 4.1). In pseudo-depth the resulting trace is a series of discrete impulses (e.g. Figure 4.2a). This is integrated once to produce ct\(z), and again to produce H (e.g. Figures 4.2b and c respectively). In general for piecewise constant impedance Earth models, the form of H tends, therefore, to be a piecewise linear signal. The key is to follow what the cascaded operation d nH n/dz n does to such an input H. H (Figure 4.2c) has two distinct types of structure, each of which reacts very differently to this operator. First consider an element of H away from all discontinuities, e.g. in Figure 4.2c between Z\ « 300 and 22 ~ 500. In general elements of H, called, say, Hnn(z), on such an interval may be described by where a and b are constants determined by the data at and above z\. (Notice that if z\ is the location of the shallowest interface, with a reflection coefficient of Ri, then a = 4i?i and b = 0.) Computing the first four orders of the operator Knd nH^n/dz n gives Hun(z) = az + b, (4.84) dHnn(z) dz a (4.85) = 4iZi) K2 dz 2 Adz 2 ~a 2z 2 + 2azb + b 2' (4.86) K3 dz 3 J_d?_ 24 dz 3 [a 3z 3 + ...] (4.87) Chapter 4. Simultaneous Imaging and Inversion 80 K4 (4.88) ( 32RJ), where in brackets the particular value a = AR\ is used. It is clear that, at all orders, the operator d n(-) n/dz n does not take these linear features Hun and create an output with any complicated spatial structure. In fact, the n'th power and the n'th derivative counteract each other almost completely for an input with a linear dependence. Knd n (Hun) n/dz n is the transformation from a linear function to a constant function, in which the only important task of n is in determining the weight - or the eventual value - of the constant output. So equation (4.42) in fact operates quite gently on the second integral of the data away from its discon-tinuities. It acts to alter its amplitudes in a way that corresponds exactly to the partial inversion subseries discussed earlier in this chapter. To see this, compare the bracketed results of equations (4.85) - (4.88), for the single interface case, with the terms of this partial inversion subseries for the same case in equation But the genteel act of transforming a linear function into a constant function belies the volatility of the operator that performs it. This volatility is manifested when the operator encounters the second structural type found in J7-like inputs, at and near its discontinuities. d n(-) n/dz n actually behaves like an edge-detector - it tends to do little to portions of a function that are well approximated by low-order polynomials, while strongly reacting to portions that resemble high-order polynomials, and especially signal edges and discontinuities. In the case of H-\ike functions, these discontinuities are at points where piecewise linear functions conjoin. In Figure 4.17 a test discontinuity (a) of this kind is operated on by d n(-) n/dz n, again for n = 1 - 4. Each of Figures 4.17a - d has 2 panels; the top is the function of Figure 4.17a taken to the n'th power, and the bottom is the numerical n'th derivative thereof. Interestingly, what is returned is a signal with the characteristics of weighted derivatives of the original function. This is not necessarily an intuitive result, since by eye, the discontinuities associated with Figures 4.17a - d, top panels, appear to be of changing order. However, numerically, the output continues reflect a "triangle"-like discontinuity, i.e. of order 1. What changes is that the differences in the slope on either side of the discontinuity become much larger, and this produces the effective weights on the output derivatives. So, near the discontinuities that typify the integral of the Born approximation, d n(-) n/dz n is outputting the sum of weighted derivatives of a piecewise constant function, O'th (Figure 4.17a, bottom panel) through 3rd (Figure 4.17d, bottom panel) in these examples. This is in agreement with the mechanics of the leading order imaging subseries as investigated in Shaw et al. (2003), and seen in equation (4.71) in this chapter. In summary, the simultaneous imaging and inversion formula can be seen to act as a flexible operator that mimics both the inversion subseries and the leading order imaging subseries. One dominates over the other based on the proximity of the operator to discontinuities in the integral of the Born approximation. In the simple physical framework of a ID constant density acoustic medium, the inverse scattering series may be cast such that it involves repeated integration and differentiation of the Born approximation. The patterns that result are such that a simple, computable formula (based on the n'th derivative of the n'th power of the integral of the Born approximation) reproduces many of the terms identified with the imaging and inversion of primaries. The formula is readily shown to produce good quality results on ID synthetic data, if the derivative operators are truncated at high frequency. With this done correctly, the results are very encouraging in even high contrast environments in the presence of low noise; furthermore, existing technology for spectral extrapolation may be implemented to overcome the sensitivity of the approach to bandlimited data. The main goal for future research is to generalize these investigations to 2D and 3D. To at least concep-tually broach this issue, I have included an analysis of the behaviour of the formula in the signal processing (4.77). 4.8 Remarks Chapter 4. Simultaneous Imaging and Inversion 81 Figure 4.17: Illustration of the effect of the operator dn(-)n/dzn on the n'th power of the characteristic discontinuities of the input H (or second integral of the data). Each of the four examples (a)-(d) has two panels. Top panel: n 'th power of input H; bottom panel: n 'th numerical derivative of the top panel. This is done for four example orders: (a) 1, (b) 2, (c), 3, and (d) 4- In spite ofthe increasing curvature on either side of the discontinuity, numerically the results consistently resemble weighted n 'th derivatives of order 1 discontinuities. sense, i.e. a determination of what it does, as an operator, to the input data. The results suggest that the method distinguishes between what might be called quiescent regions of the data and active regions of the data. In other words, the inversion, or amplitude-altering, component of the operator dominates where there are no discontinuities in the integrals of the data; whereas the imaging, or reflector-shifting component dominates where the singular events of the seismic experiment occur - its discontinuities. From the historical transition of inverse scattering based multiple technology from ID to 3D, one might expect that a comparable algorithm in multiple dimensions would be of this form. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 82 Chapter 5 L i n e a r a n d N o n l i n e a r V i s c o a c o u s t i c I n v e r s i o n Term Signifies k(z) Medium wavenumber, depth dependent Q(z) Medium Q, depth dependent c0/c(z) Reference/non-reference wavespeed kf Reference wavenumber F(k) Function inherent to constant Q model a(z) Acoustic (wavespeed) perturbation f3(z) Q perturbation V(z,k) General scattering potential D(k)/Dn(k) Data/n'th data event ips(zm\zs;k) Scattered field, measured at zm due to source at zs ^o(zm\zs; k) Incident field G0{zm\zs\k) Acoustic Green's function l{z)/ln{z) Viscoacoustic perturbation/n'th order vise. pert. ai{ki,k2)/f31(ki,k2) Born approximations for input frequencies fci, k2 ain(ki,k2)/(3ln{k1,k2) Born approximations for n 'th data event k(n)/C(n) Medium wavenumber/wavespeed of n 'th layer R(k) Frequency dependent reflection coefficient K(k)) Effective Reflection coeff. of (n — l) 'th interface kx/kxg Lateral wavenumbers K/Qzg Vertical wavenumbers e/er Angle/reference angle, away from vertical Local amplitude of Born approximation OLINV Result of acoustic/viscoacoustic inversion subseries 5.1 Introduction In acoustic/elastic/anelastic (etc.) wave theory, the parameters describing a medium are related non-linearly to the measurements of the wave field. Many forms of direct wave field inversion, including those used in the first part of this chapter, involve a linearization of the problem, in other words a solution for those components of the model which are linear in the measured data. There are two reasons for solving for the linear portion of the model. First, if the reference Green's function is sufficiently close to the true medium, then the linear portion of the model may be, in and of itself, of value as a close approximation to the true Earth. Second, a particular casting of the inverse scattering series uses this linear portion of the scattering potential (or model) as input for the solution of higher order terms. In either case, the quality and fidelity of the linear inversion results are of great importance; what defines that quality and fidelity, however, may differ. For instance, consider the imaging subseries of the inverse scattering series (Weglein et al., 2002; Shaw et al., 2003). Its task is to take the linearized inversion, which consists of contrasts that (i) are incorrectly located, and (ii) are of the incorrect amplitude (both in Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 83 comparison to the true model), and correct the locations of the contrasts without altering the amplitudes. This task is accomplished, in ID normal incidence examples, by weighting the derivatives of the linearized inversion result by factors which rely on the incorrect (Born) amplitudes. In other words, the imaging subseries relies on the correct provision of the incorrect model values. It doesn't matter that the linear inverse is not close to the true inverse - the inverse scattering series expects this. What matters is that the incorrect linear results are of high fidelity, so that a sophisticated nonlinear inversion procedure may function properly to correct them. A large component of this chapter involves solving for the linearized portion of a viscoacoustic Earth (i.e. the Born inverse). At the outset I make no distinction between which of the above two aims are being addressed. Under which circumstances the Born inverse is an adequate and useful approximation of the real Earth is investigated here by means of a survey of numerical examples. But it is useful to bear in mind that the decay of the proximity of the Born inverse to the real Earth does not, in methods based on inverse scattering, signal the end of the utility of the output. Rather, it marks the start of the necessity for inclusion, if possible, of higher order terms - terms "beyond Born". The presence of AVO effects in a seismic experiment suggests that the offset dimension is far more valuable than merely as a source of data redundancy for incoherent noise suppression, and even than as a source of velocity information via moveout. In fact, this dimension often provides the information necessary to correctly pose multi-parameter inversion problems. Seismic events are often better modelled as having been generated by changes in multiple Earth parameters than in a single one; for instance, density and wavespeed in an impedance-type description, or density and bulk modulus in a continuum mechanics-type description. In either case, the idea is that a single parameter velocity inversion (after that of Cohen and Bleistein (1977)) encounters problems because the amplitude of events is not reasonably explicable with a single parameter. In Clayton and Stolt (1981) and Raz (1981), density/bulk modulus and density/wavespeed models re-spectively are used with a single-scatterer approximation to invert linearly for profiles of these parameters. In both cases it is the variability of the data in the offset dimension that provides the information necessary to separate the two parameters. The key (Clayton and Stolt, 1981; Weglein, 1985) is to arrive at a relationship between the data and the linear model components in which, for each instance of an experimental variable, an independent equation is produced. For instance, in an AVO type problem, an overdetermined system of linear equations is produced (one equation for each offset), which may be solved for multiple parameters. In a physical problem involving dispersion, waves travel at different speeds depending on the frequency, which means that, at regions of sharp change of the inherent viscoacoustic properties of the medium, frequency-dependent reflection coefficients are found. This suggests that one might look to the frequency con-tent of the data, rather than the offset variability, as a means to similarly separate viscoacoustic parameters (i.e. wavespeed and Q). I begin this chapter by choosing an appropriate scattering potential, or perturbation operator, for the solution of the linear viscoacoustic problem. I then show how, by assuming a spatial form for the model contrast for a single interface, Q and wavespeed parameters may be found by solving an overdetermined linear system of equations. This "overdetermined-ness" is rooted in the fact that two parameters are solved for using many equations; one for every frequency involved in the experiment. I pursue this numerically for a number of single interface models encompassing varying levels of high- and low-contrast, and show that the result is very stable over a range of input frequency pairs. In the next section I extend this procedure such that the multiple layer case is treated. It is well established that the Born approximation is increasingly inaccurate as the number of layers increases. Because of the attenuation inherent to a viscoacoustic medium, this tendency is aggravated here. I gear a numerical survey of this linear profile-estimation method to test its limits as a means to provide a valuable standalone processing/inversion tool. I then consider a bootstrap means to iteratively compensate the linear Born inversion procedure, such that much more accurate interval wavespeed and Q may be estimated. I conclude that the "raw" Born inverse, sans bootstrap, is primarily of use as an input for higher order inverse scattering methods. These very methods are addressed next: the basic issues surrounding a "beyond Born" viscoacoustic inversion are explored. The inversion subseries of the inverse scattering series is considered, with an aim to understanding the basic means by which the deeper (and "more incorrect") layers are altered, by themselves and in concert with information from shallower layers. An acoustic (i.e. non-viscous) case at normal inci-Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 84 dence is used as a tool to develop this understanding. General formulae for the n'th term of the inversion subseries for lower layers are written down, and I use these to make general comments on how shallower layer information is used to correct deep amplitudes. Finally these formulae are used to address the viscoacoustic problem using a data set based on an elementary friction model for attenuation and an Earth which contains only Q contrasts. 5.2 A Viscoacoustic Scattering Potential As in previous chapters, I proceed by adopting the constant Q model (see the review of chapter 1), such that the dispersion relation is assumed, over a reasonable seismic bandwidth, to be given by i + * * \ J K (5.1) c(z) L 2Q(z) irQ(z) \krJ\' where fcr = ojr/co is a reference wavenumber, and fe = IO/CQ. In keeping with previous terminology, this is re-writeable using an attenuation parameter (3(z) = 1/Q(z) multiplied by a function F(k), of known form: ( 5 ' 2 > which utilizes (3(z) to correctly instill both the attenuation (i/2) and dispersion (—^ ln (fe/fer)). Notice that F(k) is frequency-dependent because of the dispersion term. Then k(z) = ~[l+f3(z)F(k)}- (5-3) The linearized Born inversion is based on a choice for the form of the scattering potential V, which is given by V = L - L 0 ) (5.4) or the difference of the wave operators describing propagation in the reference medium (Lo) and the true medium (L). For a constant density medium with a homogeneous reference this amounts to LO2 V = V(x,z,k) =k2(x,z) - —, (5.5) for a medium which varies in two dimensions, or V(z,k) = k2(z)-l4, (5.6) for a ID profile. Using equation (5.3), I specify the wavespeed/Q scattering potential to be 2 2 V(x,z,k) = [l + (3(x,z)F(k)]2 - ^ , (5.7) c'(x, z) eg and include the standard perturbation on the wavespeed profile c(x, z) in terms of ot(x, z), producing 2 w 2 V(x, z, k) = — [1 - a(x, z)} [1 + P(x, z)F(k)}2 - — (5.8) ^-^j[a(x,z)-2P(x,z)F(k)}, dropping all terms quadratic and higher in the perturbations a and (3. The ID profile version of this scattering potential is then, straightforwardly V(z,k) «-^[a(z) - 2(3(z)F(k)]. (5.9) co The scattering potential in equation (5.9) will be used regularly in this chapter. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 85 5.3 ID Normal-Incidence Inversion for Q/Wavespeed In general it is not possible to invert for two parameters from a ID normal incidence seismic experiment. However, if one assumes the spatial form of the Earth model (or perturbation from reference model), then this problem becomes tractable for a viscoacoustic dispersive Earth. Consider an experiment with coincident source and receiver zs = zg = 0. The linear data equation, in which the data are assumed to be the scattered field i})s measured at this source/receiver point, is /oo G0(0|z';fc)/c27l(2/)V'o(^|0;/c)^', - O O (5.10) which, following the substitution of the acoustic ID homogeneous Green's function and plane wave expressions Go and %po becomes D(k) 1 ifc7i(-2fc), (5..H) where the integral is recognized as being a Fourier transform of the linear portion of the perturbation, called 7i(z). The form for the perturbation is given by the difference between the wave operators for the reference medium (Lo) and the non-reference medium (L), as discussed in chapter 4. In this case, let the full scattering potential be due to a perturbation 7: 7(2) V(z,k) k 2 ' where V(z, k) is given by equation (5.9). Writing the linear portion of the overall perturbation as ll{z) = 2(3l{z)F{k)-ocl(z), taking its Fourier transform, and inserting it into equation (5.11), the data equations D(k) = -iifc[2/?i(-2fc)F(fc) - ai(-2fc)], (5.12) (5.13) (5.14) ai(-2fc)-2/3i(-2fc)F(fc) =4 D(k) i2k (5.15) are produced. Equation (5.15) as it stands cannot be used to separate ot\ and /3\. This is because at every wavenumber (k) one has a single equation and two unknowns. However, much of the information garnered from the data, frequency by frequency, is concerned with determining the spatial distribution of these parameters. If a specific spatial dependence is imposed on a\ and j3i the situation is different. Consider a constant density acoustic reference medium (a ID homogeneous whole space) characterized by wavespeed cn; let it be perturbed by a homogeneous viscoacoustic half-space, characterized by the wavespeed C i , and now also by the CJ-factor Cji. The contrast occurs at z = z\ > 0. Physically, this configuration amounts to probing a step-like interface with a normal incidence wave field, in which the medium above the interface (i.e. the acoustic overburden) is known. Data from an experiment over such a configuration are measurements of a wave field that has a delay of 2Z\/CQ and that is weighted by a complex, frequency dependent reflection coefficient: D(k) = R(k)e ,i2kz\ (5.16) This may be equated to the right-hand side of equation (5.14), in which the perturbation parameters are given the spatial form of a Heaviside function with a step at z\. This is the pseudo-depth, or the depth associated with the reference wavespeed cn and the measured arrival time of the reflection. The data equations become R(k)e %1kz\ -ik 2 CVi -ai2kz\ i2k 0V2kz\ (5.17) Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 86 or cn - 2piF(k) = 4i?(fc), (5.18) in which a.\ and /?i are constants. So having assumed a spatial form for the perturbations, the data equations (5.18) are now overdetermined, with two unknowns and as many equations as there are frequencies in the experiment. Notice that it is the frequency dependence of F(k) that ensures these equations are independent - hence, it is the dispersive nature of the attenuative medium that permits the inversion to take place. It is also necessary to be able to estimate R(k) from the measured data. This will become a more challenging issue in the next section, but it is worth emphasizing now that the phase and amplitude components of the response, not including the arrival time of the reflection (i.e. R{k), not R(k)el2kzi) are required. This is the equivalent, in this simple example, of having access to the reflectivity of the medium, as produced by imaging using the best available acoustic reference wavespeed model. This overdetermined set of linear equations may be solved using a least squares approach in a practical instance. To explore the separation of the wavespeed and Q further, I consider analytic data, in which case only two frequency components of the reflected data (R(k) above) are required. Notice that provided the source/receiver effects are known and compensated for, bandlimitation has no effect on this inversion apart from reducing the number of available frequencies for the least squares solution. Let these two frequencies be fci and k2. Then there are two data equations Oil Oil 2piF(ki) 2/?1F(fc2) AR(ki) 4R(k2), (5.19) through which estimates of Pi{ki,k2) and ai(k\,k2) are produced: /3i(fci,fc 2) =2 ai(ki,k2) = 4 R{k2) - R(h) F(ki)-F(k2Y R(k2)F(ki) - R(ki)F(k2) (5.20) F(ki)-F(k2) These equations may be solved using two frequency components of the reflected data as input. 5.3.1 A Complex, Frequency Dependent Reflection Coefficient The reflection coefficient predicted from constant Q theory, for a wavespeed/Q contrast from an acoustic reference medium, is R(k) Cl 1 + 5*T 1 + t 1 + (5.21) This is a complex, frequency dependent quantity that will alter the amplitude and phase spectra of the measured wave field. To successfully use the theory of this section to invert for c and Q contrasts, R(k) apparently needs to be recoverable in its entirety from these measurements. How much of the amplitude and phase information of R(k) in equation (5.21) is required? The amplitude spectrum of R(k) dominates in what is measured in an experiment like that of equation (5.16). To understand why this is so, and how this impacts the recovery of R(k), consider the numeric Fourier transform pairs of Figure 5.1. The illustration consists of two columns of three diagrams each. The left column is: (a) the real component of R(k) in the frequency (fc) domain, (b) the imaginary component of R(k) in the frequency domain, and (c) the inverse Fourier transform of R(k). The right column is: (d) the real component of R(k) in the frequency (fc) domain, (e) the imaginary component of R(k) in the frequency domain set to zero, and (f) the resulting inverse Fourier transform of R(k). The reflection coefficient is computed from equation (5.21) using the values cn = 1500 m/s, Ci = 1600 m/s, and Qi = 50 with a reference frequency of 120 Hz. The time domain response occurs about zero lag; for the purposes of illustration it has Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 8 7 been shifted to the centre of the signal. The large change in the reflection coefficient over a bandwidth of, say, 10-50 Hz, is somewhat exaggerated, and is due to the unusually large Q contrast used in the illustration. The idea is to demonstrate that this Q model does predict a non-negligible frequency-dependence in the response. 0.005 DC -0.005 | 0.04 I 0.03 CD c 'ro E o "D CD E 0.02 0.01 0 -100 -50 0 50 100 f(Hz) -100 -50 0 50 100 f(Hz) 0.005 0.2 t(s) 0.8 Figure 5.1: Illustration of the real and imaginary components of the Fourier transform pairs produced by the viscoacoustic reflection coefficient R(k). (a) the real component of R(k) in the frequency (k) domain, (b) the imaginary component of R(k) in the frequency domain, (c) the inverse Fourier transform of R(k). The right column is: (d) the real component of R(k) in the frequency (k) domain, (e) the imaginary component of R(k) in the frequency domain set to zero, and (f) the resulting inverse Fourier transform of R(k). The Q value used is 50, and the reference frequency is 120 Hz. First note that the imaginary component of the coefficient, for Q = 50, is an order of magnitude smaller than the real component (compare Figures 5.1a and 5.1b). This means that the real component of equation (5.21) will dominate in the measured signal. The phase spectrum of the reflection coefficient is due to the imaginary component. As a test of sensitivity, I also consider the ability of this linearized inversion procedure to perform in the absence of the phase, i.e. using the amplitude spectrum only. This is done by setting the imaginary component of R(k) to zero (Figures 5.Id - f), which also adds what might be considered biased error to the amplitude spectrum. The consequences of measuring only the real component of R(k), where R(k) is written as R(k) = Rr(k)+iRi(k), (5.22) Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 88 for the computation of a\ and /3\ are as follows. iRr(k2) + iRi(k2) - Rrih) - iRi(h) F(h)-F(k2) , Rr{k2) - Rrjh) ' ^ ( f e ) - i l n ( f e r ) + i , Ri(k2)-Ri(h) (5.23) The real component of /3i(fci, ^2) is therefore made up of only the real components of the reflection coefficient, so the solution for the Q perturbation felt by measuring only the real component of the reflection coefficient is unchanged. Any residual imaginary component is due to the linearization of the viscoacoustic scattering potential in equation (5.8). Figure 5.1b suggests that this imaginary component, based on the difference Ri(k2) — Ri(ki), will be small indeed. Next consider the acoustic (wavespeed) perturbation: a i ( f c i , f c 2 ) = 4 [Rr(k2) + iRiik^Fih) - [Rr(h) + iiJ i(fe1)]F(fc2) (5.24) Similarly to (5\, the imaginary component of OJ I should be, by definition, very small, or zero, and our main interest is whether the real component is dictated mostly by the real component of the reflection coefficient (i.e. the data). Equation (5.24) has a real component equal to 5ftWfci,fc2)} ARr(h) In ( £ ) - ARr(k2) In ( + 2i?i(fci) - 2Ri(k2) (5.25) This expression is not independent of the imaginary components of the reflection coefficient, as was the expression for $t{Pi(h,k2)}. The computation of the real component of a i therefore involves the imaginary components of the reflection coefficients. However, this portion of the computation is based on the difference of Ri(k2) and Ri(h), each of which, again, is a small number. I make the following approximation: a1(fci,fe2) 4iZ r (fci) ln(^) -ARr(k2)\n^ l n f e ) - l n ( > ) (5.26) In summary, the real components of the data are used to estimate the linear components of the Q (/?) and wavespeed (a) perturbations, as Pi{h,k2) ai(h,k2) Rr(k2)-Rr(h) F(h)-F(k2) ' 4 i ? r ( f c 1 ) l n ( ^ ) - i ? r ( f c 2 ) l n ( ^ ) n F(h) - F(k2) (5.27) 5.3.2 Numeric Examples Equation (5.27) may be used along with a chosen Earth model to numerically test the efficacy of this inversion. Table 5.1 contains the details of four models: Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 89 Model Reference CQ (m/s) Non-reference c\ (m/s) Non-reference Q\ 1 1500 1800 100 2 1500 1800 10 3 1500 2500 100 4 1500 2500 10 Table 5.1: Test models used for the single interface c, Q linear inversion. Figures 5.2 - 5.5 show sets of recovered parameters using the respective models in Table 5.1. For the sake of illustration, frequency pairs fci = fc2, for which the inversion equations are singular, are smoothed using averages of adjacent (fci ^ fc2) results. The recovered Q values from the the measured viscoacoustic wave field are in error on the order of %1; this is true for all realistic contrasts in Q (i.e., up to Q = 10 as tested here). As the wavespeed contrast increases, the recovered Q is in greater error, but even in the large contrast cases of Models 3 and 4, the error is under %10. In all cases the error increases at low frequency; it is particularly acute when both fcx and fc2 —> 0. This is because the attenuative wavenumber k(z) in equation (5.1) contains within it the approximation ^ In ^ 1> which fails as fc —> 0. It is encouraging to see that elsewhere, i.e. at larger fci, fc2, the nominal acoustic (non-attenuating) Born approximation for the wavespeed is attained. Compare the results of Figure 5.2 (Model 1), for instance, with the ID acoustic Born approximation associated with a wavespeed contrast of 1500m/s to 1800m/s (in which Ri « 0.091): C l ~ ( l - Q ! ) ^ " 1 / 3 = ( i - 4 ^ ) 1 / 2 m / s ~ 1880.3m/s (5.28) The commentary on measuring the real component of the reflection coefficient is borne out in. these examples; compare the c, Q pairs in the columns of each figure. The recovered Q values are not sensitive to such a change in input. The wavespeed perturbation is slightly altered, on the order of %1 percent of the wavespeed amplitude. I conclude that this inversion may be carried out either in the presence or absence of the phase information of the reflection. This makes the required local analysis of the viscoacoustic seismic events simpler. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 90 Figure 5.2: Recovered parameters from the normal incidence linearized Born inversion for Q and wavespeed. Top left: Q recovery using two (complex) reflection coefficients over a range of frequencies, fci and k2 (Hz). Top right: Q recovery using real component of the reflection coefficients. Bottom left: wavespeed recovery utilizing same (complex) reflection coefficients. Bottom right: wavespeed recovery using the real component of the reflection coefficient. Model parameters correspond to Model 1 in Table 5.1. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 91 1900 v Figure 5.3: Recovered parameters from the normal incidence linearized Born inversion for Q and wavespeed. Top left: Q recovery using two (complex) reflection coefficients over a range of frequencies, fci and ki (Hz). Top right: Q recovery using real component of the reflection coefficients. Bottom left: wavespeed recovery utilizing same (complex) reflection coefficients. Bottom right: wavespeed recovery using the real component of the reflection coefficient. Model parameters correspond to Model 2 in Table 5.1. Chapter 5. Linear and Nonlinear Viscoacoustic, Inversion 92 0 0 1 Figure 5.4: Recovered parameters from the normal incidence linearized Born inversion for Q and wavespeed. Top left: Q recovery using two (complex) reflection coefficients over a range of frequencies, k\ and &2 (Hz). Top right: Q recovery using real component of the reflection coefficients. Bottom left: wavespeed recovery utilizing same (complex) reflection coefficients. Bottom right: wavespeed recovery using the real component of the reflection coefficient. Model parameters correspond to Model 3 in Table 5.1. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 93 Figure 5 . 5 : Recovered parameters from the normal incidence linearized Born inversion for Q and wavespeed. Top left: Q recovery using two (complex) reflection coefficients over a range of frequencies, k\ and k2 (Hz). Top right: Q recovery using real component of the reflection coefficients. Bottom left: wavespeed recovery utilizing same (complex) reflection coefficients. Bottom right: wavespeed recovery using the real component of the reflection coefficient. Model parameters correspond to Model 4 in Table 5.1. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 94 5.4 Linearized Estimation of c, Q Profiles In section 5.31 assume that the data, which has a known form, is due to a single interface whose spatial form is also known. The relationship between the "spatial" form of the data and the spatial form of the contrast allows the wavespeed and the Q associated with this contrast to be simultaneously inverted-for. This is, in essence, a utilization of prior information: the model is assumed to have a structure whose frequency content - shape, in other words - is determined, modulo a contrast, by the structure of the data. In a single parameter normal incidence problem, i.e. in which acoustic wavespeed contrasts are linearly inverted for from the data by trace integration, profiles may be generated, not just a single interface contrast. A similar procedure may be developed for the ID normal incidence two parameter problem (c(z) and Q(z)) if one has access to an estimate of the local frequency content of each reflected event. With this information available, the data structure is once again used as prior information, and (therefore) a linear estimate of c(z) and Q(z) profiles may be determined from a single trace. I use the following model to describe the ID normal incidence data associated with a model with N interfaces, at each of which the wavespeed and Q values are assumed to alter. The data are N D(fc) = X>n(fc), (5.29) n = l such that Dn(k) = R^(k)exp < ^2i2kij_1)(zj - Zj-i) [3=1 n— 1 R'n(k)=Rn(k)Y[[l-R 2(k)}, ( 5- 3°) LO K ( J ) = -h L 1 i 1 (k 2Qj TTQJ \k where, as ever, k is the acoustic reference wavenumber LO/CQ, and where Rn{k) is the real component of the reflection coefficient of the n'th interface. The exponential functions imply an arrival time and a "shape' for each event. To develop the utility of local frequency analysis, without actually doing it, consider a two interface example with a source and receiver at the aforementioned zs = zg = 0: D{k) = Rx(k)ei2kZi + R'2{k)ei2kz'e i2k^- Zi). (5.31) Figure 5.6 shows an example data set of the form of equation (5.31) in the conjugate (pseudo-depth) domain. "Having access to estimates of local frequency content of the events" means that from the data, for instance that of equation (5.31), R\(k) and R'2{k) are distinguishable within D{k). To accomplish this, local frequency analysis of some kind is required. In this development, I assume that a windowed (or short-time) Fourier transform is applicable and that appropriate windows are derivable from the nature of the data themselves. Any local basis function (e.g. wavelet) analysis that can be engineered to estimate local spectra is suitable. Figure 5.6 illustrates a separation of the events in question into two parts, such that we, as analysts of measured data, have access to the individual components of the trace alluded to in equation (5.31): D(k)=D1(k)+D2(k), (5.32) where £>i(fc) = J?i(fc)e i 2 f c 2 1 D2{k) = R'2(k)e i2kzx g i 2 f c ( 1 ) ( 2 2 - z i ) (5.33) Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 95 0.02 0.01 h -0.01 I I / V I I 200 400 600 800 1000 1200 1400 1600 0.02 0.01 h I I I I -0.01 200 400 600 800 1000 1200 1400 1600 0.02 0.01 -0.01 200 400 600 800 1000 Pseudo-depth z (m) 1200 1400 1600 Figure 5.6: Example data set ofthe type used to validate/demonstrate the linearized c, Q profile inversion, (a) Full synthetic trace, consisting of two events, D\ + D2, plotted in the conjugate (pseudo-depth) domain. The first event corre-sponds to the contrast from acoustic reference medium to a viscoacoustic layer, and the second corresponds to a deeper viscoacoustic contrast; (b) first event D\ (plotted in the conjugate domain); (c) second event Di-in other words Ri(k) and R2{k) are known. In simplistic experimental configurations like the one presented here, the distinction between Dn and Rn is based on the reference-medium delay, or (homogeneous) acoustic propagation effect. In more complex situations, the idea of removing the effects of propagation from the data to achieve an estimate of the reflectivity is not so simple. However, the issues behind it are well-documented in the framework of "migration-inversion" (Stolt and Weglein, 1985; Weglein and Stolt, 1999). I do not pursue them further here. Since the linear components of the perturbation are the desired result, the total perturbation can be derived from a sum of the perturbations associated with each of these two events; I treat the trace components of Figure 5.6, for the time being, as individual experiments. It follows that the total perturbations are ai(-2fc) = a 1 1(-2fc) + ai 2(-2fe) p1(-2k) = pu(-2k)+012(-2k), in which a l n , j3\n are the perturbations associated with the event Dn(k) in equation (5.29). Consider first Dx(k): Di(fc) = R1(k)e i2kZl. (5.35) Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 96 In the previous section I developed a form for the data equations associated with perturbations in c and Q. Assuming that a n ( — 2k) and /3n(—2k) have the spatial form of a Heaviside function, the perturbations associated with the first interface may be solved for: Pn(h,k2) = 2 F { k i ) _ F { k 2 ) , ^ (5-36) Oiu(k1,k2) 4iJ1(fc1)ln(j^) -Rx(k2)\n fe,. ^( fc i ) - F (A: 2 ) from the two (arbitrary) frequency components of Ri, k\ = ui/co and k2 = UJ2/CQ. The application of this procedure to the deeper events is less straightforward. Nevertheless it is still quite correct to equate the data equations (5.14) with the corresponding data components; consider D2(k): D2(k) = R'2{k)ei2kziei2k^{z*-Zl) = ~^ik[2fil2(-2k)F{k) - a12{-2k)}. (5.37) With equation (5.37) the difficulties of the c, Q inversion for deeper layers become visible; they reside in the propagation wavenumber kci) (see equations (5.30)). Previously, enormous simplification was achieved by making the reasonable assumption that the perturbations were Heaviside-like; this tended to cancel with the spatial form associated with the data, leaving only one last task for the frequency-dependence of the data equations to take care of: estimate the amplitude of the perturbations. To follow in those footprints for this second event would be to assume that again, the perturbations (this time at z2) are due to a step-like interface at a point dictated by the arrival time of the event. This seems perfectly reasonable, or at least as reasonable as it was for the first layer. It would result in the relationship: R2{k)ei2k^ei2kw(-Z2~z^ = -Uk[2p\2F(k) - al2f & 2""^^ v v ~ i Z J i2k (5.38) 4 i ? 2 ( f c ) e i 2 ^ [ * F ( f c ) 1 ( 2 2 - Z l ) = Q 1 2 - 2f312F(k). Comparing equations (5.18) and (5.38), notice that not all of the propagation operator ef'l associated with the data is cancelled out by dint of this assumption. This is due to the difference between fc^j and u>/c\, or the attenuation of the wave field as it propagates between contrasts. In other words, if this was simply the acoustic problem, k^ would be identical to u/ci, and the second interface problem would be just as separable as the first interface one was (not to mention trivial in this formalism). As it is, the function el2^'QiF^k^z'1~z^ is unknown, and makes the use of equation (5.38) impossible. This is in fact an intuitively unsurprising situation, since the propagation effects associated with Q are nonlinear if the reference medium is acoustic: no linearized inversion could possibly compensate for them. The linearized estimate is distorted by such propagation effects for the same reason a linearized estimate is compromised by the presence of multiples; they are nonlinear in the current framework. I instead make the following proposition: that the natural spatial form for the linearized estimate of the c(z) and Q(z) profiles be due to that of the data themselves. For the second interface event, I set e t 2 f c z i e t 2 f c ( i ) ( z 2 - z i ) a12(-2k) = cv1 2 — l 2 k , (5.39) e i 2 k z i e i 2 k m ( Z 2 - z i ) / M - 2 * 0 = / ? i 2 ^ , in other words I assume that the spatial form of the perturbation is the same as the integral of the data (distorted under the attenuative propagation operator). This will amount to some smoothed step function which is a Heaviside in the limit as Q i —> oo. With this assumption equations (5.37) become AR'2(k) = a12 - 2f312F(k), (5.40) Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 97 precisely as before. Equation (5.40) may be solved, using two frequency components of R'2(k), similarly to the case of the first interface. This may be carried out on all further events. It is important to emphasize that, in using the form of the data to estimate the form of the linear inversion, we lose the insensitivity (in the single interface case) to bandlimitation. I proceed assuming that either (i) some means for estimation of low frequencies is available, or (ii) "band-limited inversion" is a sufficient goal. Whereas the results of the single interface examples indicate that the particular choice of input frequencies is not critical, there is now solid theoretical impetus to direct the choice of input frequencies in the profile estimation case. This is because the well-known propagation effects associated with an absorptive/dispersive medium preferentially decay the signal at higher frequencies. Such decay is not accounted for in a linearized inverse with an acoustic reference medium, and hence will throw off the accuracy of the inversion for lower layers. It is therefore straightforward to recommend at the outset that the lowest possible frequencies be utilized for the lower layers, since at these frequencies the deleterious effects of attenuative propagation are minimal. To summarize, taking two (or more) frequency components from individual events in a ID normal in-cidence data set involving simultaneous c(z) and Q(z) contrasts, and using them to estimate ct\ and (3i, amounts to assuming a spatial form for the linear portion of the model perturbations that includes the distortion associated with viscoacoustic propagation. The perturbation amplitudes, associated with individ-ual trace events, may be used to weight the spatial form of their corresponding perturbations, and may be summed together to arrive at the final form for the linear perturbations a\(z) and 0i(z), as in equation (5.34). 5.4.1 Numer i c Examples I present a series of numerical implementations of the profile estimation method described previously. In addition to broadly illustrating the procedure, I focus on two issues: (i) the onset and growth of the inaccuracy of lower layer inversions with increase in the contrast of the first layer, and (ii) the relative importance of the propagation component of transmission effects on (i). I utilize a single layer ID model involving step contrasts of varying size. Table 5.2 details the parameters used for all examples in this section. Model # Layer 1 c (m/s) Layer 1 Q Layer 2 c (m/s) Layer 2 Q 1 1550 10b 1600 300 2 1550 104 1600 300 3 1550 103 1600 300 4 1550 700 1600 300 5 1550 500 1600 300 6 1550 50 1600 300 Table 5.2: Test models used for the single layer c, Q linear inversion. The reference medium, z < 500m, is acoustic and characterized by co = 1500ra/s. Figures 5.7 - 5.11 are the results of the linearized inverse approach on the full synthetic data. The reconstruction of the deeper layer includes a slight spatial decay. This is the result of assuming that the perturbation has the spatial form of the first integral of the data; since the lower event is smoothed by attenuation, its first integral has an associated decay. Examined in turn from Figure 5.7 through to Figure 5.11, they make clear the importance of propagation (attenuation) on the Q perturbation results for the lower layer. The greater the attenuation in the first layer, the greater the adverse change to the effective reflection coefficient of the lower interface. Hence, the greater the error. The method does not impose similar degradation on the recovered wavespeed contrasts for these models: they remain at a constant accuracy level. This is simply because the attenuation in these examples is actually very small; these examples are more of a showcase of the sensitivity of the P\ profile inversion than the robustness of the cci inversion. It is clear that even a small amount of attenuation alters the ability of this approach to accurately recover Q from deeper intervals. Consider for example Figure 5.9 (Model 4). A contrast at 800 m depth, from Q = 700 to Q = 300 masquerades as no contrast at all because of the attenuative decay of the effective reflection Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 98 coefficient. A general conclusion is that interval Q estimation of this kind does not produce immediately useful results. In the last part of this chapter I consider the basic mechanisms by which a linear inversion, although not close enough to the true Earth to be immediately useful, can be improved upon by being used as input to a higher order, non-linear scheme. To finish this section, I consider a different strategy, a "patching up" of the linearized approach. 1600 1600 600 800 1000 Pseudo-depth (m) 1600 Figure 5.7: Linear c, Q profile inversion for a single layer model (Model 1 in Table 5.2). (a) Data used in inversion plotted against pseudo-depth z = crjt/2. (b) Recovered wavespeed perturbation ct\(z) (solid) plotted against true perturbation (dotted), (c) Recovered Q perturbation (3\(z) (solid) against true perturbation (dotted). Negligible attenuation in layer leads to good accuracy in inversion for second contrast (at z = 800m). Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 99 1600 1600 600 800 1000 Pseudo-depth (m) 1600 Figure 5.8: Linear c, Q profile inversion for a single layer model (Model 2 in Table 5.2). (a) Data used in inversion plotted against pseudo-depth z — cot/2, (b) Recovered wavespeed perturbation 0:1(2) (solid) plotted against true perturbation (dotted), (c) RecoveredQ perturbation f)\(z) (solid) against true perturbation (dotted). Small amount of attenuation in layer leads to good, but reduced, accuracy in Q inversion for second contrast (at z = 800m). Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 100 1600 1600 600 800 1000 Pseudo-depth (m) 1600 Figure 5.9: Linear c, Q profile inversion for a single layer model (Model 3 in Table 5.2). (a) Data used in inversion plotted against pseudo-depth z = cnt/2. (b) Recovered wavespeed perturbation cti(z) (solid) plotted against true perturbation (dotted), (c) Recovered Q perturbation /3i(z) (solid) against true perturbation (dotted). Attenuation in layer leads to decaying accuracy in Q inversion for second contrast (at z = 800mj. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 101 600 800 1000 Pseudo-depth (m) Figure 5.10: Linear c, Q profile inversion for a single layer model (Model 4 in Table 5.2). (a) Data used in inversion plotted against pseudo-depth z = crjt/2. (b) Recovered wavespeed perturbation a\(z) (solid) plotted against true perturbation (dotted), (c) Recovered Q perturbation (3i(z) (solid) against true perturbation (dotted). Attenuation in layer leads to strong inaccuracy in Q inversion for second contrast (at z = 800mJ. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 102 200 400 600 800 1000 Pseudo-depth (m) 1200 1400 1600 1600 1600 Figure 5.11: Linear c, Q profile inversion for a single layer model (Model 5 in Table 5.2). (a) Data used in inversion plotted against pseudo-depth z = cot/2, (b) Recovered wavespeed perturbation a\(z) (solid) plotted against true perturbation (dotted), (c) Recovered Q perturbation 0i(z) (solid) against true perturbation (dotted). Attenuation in layer leads to strong inaccuracy in Q inversion for second contrast (at z = 800m). Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 103 5.4.2 A Boots t rap Approach to Interval Q Es t ima t ion In considering the second event of the two-event data set of Figure 5.6 as an independent experiment, data equations based on the assumption of a Heaviside-like lower interface were considered: 4 R ' 2 ( k ) ei 2 ^ F { k ) ] { Z 2 - Z l ) = a i 2 - 2/312F(k). (5.41) It was pointed out that the operator ei2^F(k)](Z2-Zl) ( 5 _ 4 2 ) is unknown, and that therefore solution of this set of equations for the layer 2 wavespeed and Q perturbations was not possible. (And as a result, an alternative approach using the data to dictate the spatial form of the linear component of the perturbation was adopted.) But of course the second event is not from an independent experiment. Using the first event, we have determined C\ and Q\\ therefore, the operator given in equation (5.42) is known, and can be compensated-for. This suggests an event-by-event cascade of c, Q estimation, followed by compensation of the next deeper event. The explicit procedure is as follows. 1. Given the data D(k) — Dx(k) + D'2(k) + D'z(k) + perform local Fourier analysis to acquire estimates of the spectrum associated with each event R'n(k). 2. Assume that the first interface is a step-like contrast in c and Q, and, using Ri(k), solve 0 f l i ( fc 2 )- f l i ( fc i) /3n(fc1,fc2) = 2 F { k i ) _ F { k 2 ) , 4 i ? 1 ( f c 1 ) l n ( M - i ? 1 ( f c 2 ) l n ( ^ ) ( 5 " 4 3 ) a u ( f c l ' f c 2 ) = * F(kl)~F(k2) ' using two (fei, k2) or more of the frequency components of i?i(fc). The estimates a n and Pu lead to estimates of c\ and Q i , the wavespeed and Q of the first layer (or the shallowest perturbed region) of the Earth. 3. Use the estimates of the layer c\ and Qi to form the operator 02(k) = exp[i2^[-^F(k)}(z2 — z%)]. Apply the inverse of this operator to the spectrum of the second event, R'2(k): *<*> - Si- »•"> 4. Assume that the second interface is a step-like contrast in c and Q also. R2(k), now compen-sated for attenuation, permits the solution of a tu u\ 0^2(fc 2)--R 2(fci) M ^ k 2 ) =  2 F(kl)-F(k2) ' 4 i ? 2 ( k i ) l n ( ^ ) - W 2 ) l n ( j ^ ) ( 5 ' 4 5 ) ai2(h,k2) = — — ±-^L, TT F(ki) - F(k2) again using two ( f c i , k2) or more of the frequency components of R2(k). The estimates a 1 2 and P12 again immediately give equivalent c2 and Q2 estimates. 5. Use the estimates of the layer c2 and Q2 to form the operator 0^(k) = 02(k)exp[i2^[-^F(k)](z3-z2)\. Apply the inverse of this operator to the spectrum of the third event, found from D'3(k): Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 104 R3(k) = R'3(k) (5.46) o3(ky 6. Continue over desired number of events. This procedure has the added benefit that it works for bandlimited data (with the source signature deconvolved) as well. Figures 5.12 - 5.15 demonstrate this procedure using exactly the same models as those of Figures 5.7 - 5.11 respectively; following this one extra case is included. The difference is immediately noticeable on the inverted profiles: the effective reflection coefficients, now corrected, do a far better and more robust job of estimating the profiles, including the extra case (Figure 5.16), which uses a layer Q of 50. By excluding the use of the data as a means to determine the shape of the contrasts, the step-like shape of the recovered models (an assumption of this method) have a blocky, "perfect" look to them in the sense of structure. As long as the data are actually due to step-like interfaces, or something like a step-like interface, there is nothing disingenuous about this appearance. It is worth reiterating that this is a "patching up" of the linearized inverse - a true bootstrap procedure - designed to make the most of the linearization of the problem. The much less pretty linearized outputs of Figures 5.7 - 5.11 in fact capture the truly linear aspects of the model. As mentioned, with an acoustic reference medium no one should expect the linear inverse to compensate for attenuation. While this purity of linearized inverse degrades the immediate utility of the result, this is the input that higher order terms of the inverse series will be expecting to see and correct. The type of recovered models generated in Figures 5.7 -5.11 will be the type of input to any non-linear viscoacoustic inversion approach. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 105 600 800 1000 Pseudo-depth (m) 1600 1600 1600 Figure 5.12: Linear c, Q profile inversion for Model 1 in Table 5.2 with attenuative propagation effects compensated for. (a) Data used in inversion plotted against pseudo-depth z = cat/2, (b) Recovered wavespeed perturbation a\{z) (solid) plotted against true perturbation (dotted), (c) Recovered Q perturbation /?i(z) (solid) against true per-turbation (dotted). Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 106 to 1° (0 " 0.02 -0.02 1600 600 800 1000 Pseudo-depth (m) 1600. 1600 Figure 5.13: Linear c, Q profile inversion for Model 2 in Table 5.2 with attenuative propagation effects compensated for. (a) Data used in inversion plotted against pseudo-depth z = cot/2, (b) Recovered wavespeed perturbation 0:1(2) (solid) plotted against true perturbation (dotted), (c) Recovered Q perturbation (3\{z) (solid) against true per-turbation (dotted). Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 107 600 800 1000 Pseudo-depth (m) 1600 1600 1600 Figure 5.14: Linear c, Q profile inversion for Model 3 in Table 5.2 with attenuative propagation effects compensated for. (a) Data used in inversion plotted against pseudo-depth z = cot/2, (b) Recovered wavespeed perturbation ai(z) (solid) plotted against true perturbation (dotted), (c) Recovered Q perturbation 0i(z) (solid) against true per-turbation (dotted). Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 108 co co Q- d) «•£ » - Co £ c o o O CL O O 0.02 -0.02 1600 0.2 c o •S 0.1 CO u 0 a. -0.11 x 10" 200 400 600 800 1000 1200 200 400 600 800 1000 Pseudo-depth (m) 1200 1400 1400 1600 1600 Figure 5.15: Linear c, Q profile inversion for Model 4 in Table 5.2 with attenuative propagation effects compensated for. (a) Data used in inversion plotted against pseudo-depth z = crjt/2. (b) Recovered wavespeed perturbation a>i(z) (solid) plotted against true perturbation (dotted), (c) Recovered Q perturbation 0i(z) (solid) against true per-turbation (dotted). Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 109 & to °J 0.02 -Q. 0 ro •£ i - to to » o -O 0 O CL 00 E o O -0.02 -600 800 1000 Pseudo-depth z (m) Figure 5.16: Linear c, Q profile inversion for Model 5 in Table 5.2 with attenuative propagation effects compensated for. (a) Data used in inversion plotted against pseudo-depth z — cot/2, (b) Recovered wavespeed perturbation et\(z) (solid) plotted against true perturbation (dotted), (c) Recovered Q perturbation (3\{z) (solid) against true per-turbation (dotted). Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 110 600 800 1000 Pseudo-depth z (m) 1600 1600 1600 Figure 5.17: Linear c, Q profile inversion for Model 6 in Table 5.2 with attenuative propagation effects compensated for. (a) Data used in inversion plotted against pseudo-depth z = cot/2, (b) Recovered wavespeed perturbation 0:1(2) (solid) plotted against true perturbation (dotted), (c) Recovered Q perturbation Pi{z) (solid) against true per-turbation (dotted). Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 111 5.5 Linear Viscoacoustic Inversion Using Offset In spite of the simplicity of the experimental framework discussed above, it is generally not possible to acquire normal incidence data; if a small offset configuration does not closely-enough approximate normal incidence, the problem can be posed in the offset domain instead. In this section I develop the theoretical framework for such an approach. 5.5.1 General Linearized Inversion Using Offset The estimation of the ID contrast (i.e. in depth) of multiple parameters from seismic reflection data is considered, similar to, for instance, Clayton and Stolt (1981). For the sake of exposition I demonstrate how the problem is given the simplicity of a normal-incidence experiment by considering the bilinear form of the Green's function. In ID, for instance, the Green's function, which has the nominal form pik\z,,-z'\ G0{zg\z'-w)= 2 . k , (5.47) also has the bilinear form i roo eiK(zg-z') GQ{zg\z';uj) = - J Jk'z k 2 _ k / 2 , (5.48) where k — UJ/CQ. I consider the reference medium to be acoustic with constant wavespeed CQ. The scat-tered wave field (measured at xg, zg for a source at xs, zs), tps(xg,zg\xs,zs;u>), is approximated by model components that are linear in the data; these are denoted V\{z,w). This approximation is /oo roo dx' j dz'G0(xg,zg\x',z';ui)Vi(z',w)G0(x',z'\xs,zs;u>) (5.49) -oo J — oo where S is the source waveform. The function Go describes propagation in the acoustic reference medium, and can be written as a 2D Green's function in bilinear form: j /.oo <-oo eik'r(x,,-x') eik'.(z,,-z') G 0 ( * 9 ^ | * ' , . » = W?j_Jk'*j_jK fc2_(^ + ^) ' ( 5 ' 5 0 ) where k = OJ/CQ. Measurements over a range of xg will permit a Fourier transform to the coordinate kxg in the scattered wave field. On the right hand side of equation (5.49) this amounts to taking the Fourier transform of the left Green's function Go(xg,zg\x',z';w): I roo roo roo e-ik.r.,lx,leik'r{xll-x')eik'.(z,l-z') GQ(kxg,zg\x',z';uj) = J dk'x J dk'z J dxg  fc2_^fc/2 + fc/2^ • ( 5 - 5 1 ) Taking advantage of the sifting property of the Fourier transform: ^ roo roo roo ^{k*a-k\n)xlt p-ik'.^x' pik'^z^-z') (kxg, z9\x', z'-w) = dk'x dk'z dx9 k 2 _ { K 2 + K2) -y roo roo e-ik'rx'eik's(z,,-z') ' roo ^ = Loo ^  J-oo ^ k* - [k>2 + *i2) [J-oo  d X * R l 3 ^ roo roo e-ik'rx' eik'z(za-z') = 2n /_„  d K L d K fc2 - (kL* + fc^) 5 i K ~ ^ 1 . r°° PiK(z»-z') = J_e-ikr.,,x / dki t 2^ J - o o 2 q2 + K2 ' Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 112 where q2zg = k2 — kxg2, a vertical wavenumber. Notice that the remaining integral is a ID Green's function in bilinear form, as in equation (5.48). So equation (5.52) takes on the remarkably simplified form: 0iqz<l\za-z\ Go(kxg,zg\x',z';Lo) = e  ik'« x' The righthand Green's function in equation (5.49) may likewise be written /o \ 9 I dkXs I dkz-( 2 7 r) poo pc GQ(X ,Z \xS,Zs\L0) = rr>7r^2 J dkxs J un,zs — — 2 , u 2\ k2 {kxs -\- kzs ) and therefore the scattered wave field becomes (5.53) (5.54) 1 /*cc poo 4>s(kxg,zg\xs,zs\uj) =S(LU)——/ dx' dz' (2?r)^ 7_do J_CO /O O PC dkxs I , eio^,\z„-z'\ dkz % CC oo 2iqzg pik:,:„ (x' — X . , ) g i f c j . . ( 2 ' — 2.,) k2 [kXs "I- kzs ) Vi(z',u;). (5.55) 0. The seismic experiment is conducted along a surface, which for convenience may be set at zs = z Further, since the subsurface being considered has variation in z only, all "shot-record" type experiments are identical, and only one need be considered. I let this one shot be at xs = 0. This produces the simplified expression $8 rOO POO pOO PC (fcxs,0|0,0;w) =S(to)j^ J dx'J dz'J dkxs J e»c/=9z 2iq, 2 9 0ikz.,z' k2 — (kxs 4- kzs ) which, similarly to equation (5.52), becomes (5.56) ips(k 1 POO POO PC 0|0,0;w) = S(co)— / dz' / dkX3 / ^ J— oo J— oo J —<. -l POO POO = S(LO) — / dz' / dkz ^ J —OO J-co dk2 g » C / z , , 2 0 i f c s „ z ' 2iqzg k2 - {kxs2 + kzs2) 6(kxs - kxg)Vi(z',oj) pikz„z' 2iqzg k2 - {kxg2 + k 1 r°° Piqz„z' poo = S(UJ)— dz'- VifV.w) / dk 2TT J _ O Q 2iqzg J _ 0 0 nikzH z Qzo + kz (5.57) where again the vertical wavenumber q\ = k 2 — kxg appears. The integral over dkzs has the form of a ID Green's function. The data equations (one for each frequency), with the choice zs = zg = xs = 0, are now 1>, J Z ' ~ 2 ^ W ^ /OO dz'e'2''»JVi(z',w) -oo ^Qzg J - o o • | ^ V i ( - 2 , , s > w ) > (5.58) recognizing that the last integral is a Fourier transform of the scattering potential V\. Thus one has an expression of the unknown perturbation V\ (i.e. the model) that is linear in the data. Multiple parameters within V\ may be solved if, frequency by frequency, the data equations (5.58) are independent. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 113 5.5.2 Viscoacoustic Inversion Using Offset Estimation of multiple parameters from data with offset (i.e. using AVO), as developed in the previous parts of this section, requires that equations (5.58) be independent offset to offset. In the case of viscoacoustic inversion, I show in this section that it is the dispersive nature of an attenuation model which produces the independence of the data equations with offset, allowing the procedure to go forward. Consider the term F(k): As mentioned, the frequency dependence of F arises from the rightmost component in equation (5.59), the dispersion component. In ID wave propagation, this amounts to the "rule" by which the speed of the wave field alters, frequency by frequency, with respect to the reference wavenumber kr = uv/co, usually chosen using the largest frequency of the seismic experiment. In 2D wave propagation, F, which changes the propagation wavenumber k{z) in equation (5.3), now alters the wave field along its direction of propagation in (x,z). Let 8 represent the angle away from the downward, positive, z axis. A vertical wavenumber qz is related to k by qz = k cos 0; if one replaces the reference wavenumber kr with a reference angle 0r and reference vertical wavenumber qzr, then F becomes ! ) T / n , MO) i 1 / qz cos ( In 2 TT V Izr cos 0 Furthermore, the reference kr is chosen to correspond to the largest measured frequency; in keeping with this choice, it is reasonable to choose the reference angle 6r = 0, and choose qzr as the maximum available in the experiment. Then: F ( d , q z ) =l - - - \ n ( ^ - ) , (5.61) 2 TT \qzr cos 6 J so what remains is a function which, for a given vertical wavenumber, predicts an angle dependent alteration to the wave propagation. As such the scattering potential may be written as a function of angle and vertical wavenumber also: V(x, z,9,qz) = ~[a(x, z) - 2/3(x, z)F(6,qz)]. (5.62) co The angle dependence of F produces independent sets of data equations, since it alters the coefficient of j3(x,z) for different angles while leaving a(x,z) untouched. Using equation (5.62), one may write the linear component of a depth-dependent only scattering potential as: ViO.0,9*) = - ^ M * ) - 2/3i(z)F(9,qz)]. (5.63) co Recall from earlier in this section that the requisite data equations are ^s(kxg;w) = -^T-V1(-2qxg,uJ); (5.64) using equation (5.63), and considering the surface expression of the wave field to be the data, deconvolved of S(LJ), this becomes Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 114 D(qz,6) = -J-^- M - 2 9 z ) - 2p1(-2qz)F(9,qz)} Q z (5.65) g1{-2qz)-2(3l{-2qz)F{9,qz) 4 cos2 9 These data equations may be used in conjunction with an assumption regarding the spatial form of the contrast to solve for Q.\ and /?i using a fixed qz and several (at least two) angles. In the remainder of this section I show how this relationship may be used to invert for the contrast in c and Q associated with a single interface, above which is the reference medium. This heuristically corresponds to a simplified case in which the overburden of a medium is known. Here I assume that the overburden is acoustic, and characterized by wavespeed Co, and below the interface the medium is viscoacoustic, and characterized by c\ and Q\. Following Clayton and Stolt (1981), the data associated with a reflection of this kind is D(qz,9) = R(qz,9)^^ (5.66) izqz Assuming a step-like form for the parameters a i and ai(-2qz) = a i Pi{-2qz)=fr i2qz e %2qz again permits the data equations to become an overdetermined set in terms of the frequency components of the data R: a i - 2p1F{6,qz) = 4cos2 0R(6, qz); (5.68) Choosing a fixed qz = qzm, the data equations become «i - WiFqzm{6) = 4 cos2 9Rqzm{9), (5-69) where Rqzm{9) = R(9,qzm) and Fqzm(9) — F(9,qzm). This time the analytic inversion requires two inputs chosen from the experimental range of 9. Letting these be 9\ and 92, two equations are produced from equation (5.69) and the parameter estimates are o ,a () \ o C 0 S 2 92Rqzm{.92) - COS2 dxRgzmjei) . . Pi{Vu02)=2 j^- -rr- , (5.70) rqzrn\Pl) ~ rqzm{02) and ia a \ A FQzm(9i) cos2 92Rqzm(92) - Fqzm{62) cos2 9iRqzm(9i) ai{91,92) = 4 — . (5.71) This formulation in essence uses angle in an offset experiment as a proxy for frequency in a normal incidence experiment, and may be useful when a normal incidence experiment cannot easily be approximated with one's data. 5.6 Nonlinear Inversion and Interevent Communication This section involves a consideration of the inversion subseries of the inverse scattering series, i.e. those terms which appear to devote themselves strictly to tasks related to correcting the amplitude of the Born approximation; I investigate the apparent communication between layers or events that is a part of the inversion subseries, in which lower layers utilize information from above as a means to correct their amplitudes. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 115 It is tempting at first to regard the inversion subseries (see, for instance, the work on the second order inversion of Zhang and Weglein (2003)) as local, such that individual points in the inverted-for model rely only on powers of that same, co-located, point in the data. In fact, the powers of a ID acoustic Born approximation (which is a series of scaled Heaviside functions) have local values that rely not only on the single, co-located, value of the trace, but values at all points above it. In other words, in the computation of the inversion subseries, there are cross terms from shallow events which assist in the correction of the amplitude of deep events. It turns out that the cross terms, which can be thought of as communication between events, are much more important, in the sense of contributing to the final amplitude, than the local value of the approximation itself. The relative importance of the contribution of any combination of shallow events can be exactly predicted, in the ID acoustic case, by noting that the coefficients of the cross terms are those of the binomial and multinomial formulae. And because of these formulae (which, for instance, may be used to predict the outcome of random experiments, like coin tossings), it becomes clear that the largest coefficients are given to cross terms which utilize information that is maximally spread out over the shallower events. For instance, if we have a Born approximation made up of Heaviside functions with coefficients A\, A2, and A3, then in correcting A3, the inversion subseries relies tremendously on terms like the third order A\A2A3, and much less on terms like the third order "self help" term Af.. For the n'th order inversion term, the relative importance turns out to be, respectively, the same as that associated with the probability of throwing N/2 heads and N/2 tails rather than throwing N tails. This is a huge difference in importance between contributions. As Tom Stoppard puts it: The equanimity of your average tosser of coins depends upon a law, or rather a tendency, or let us say a probability, or at any rate a mathematically calculable chance, which ensures that he will not upset himself by losing too much nor upset his opponent by winning too often. This made for a kind of harmony and a kind of confidence. To see this in action in the inverse series, one benefits from an appeal again to the simplest possible physical framework. Consider the Born approximation for a ID normal incidence acoustic medium. If the data were given by D(t) = Rx5{t - ti) + R'28{t - t2) + (5.72) where Rn or R'n is the strength of the reflection at time tn, then the (acoustic) Born approximation is ai(z) = AxR(z - 21) + A2H{z - z2) + (5.73) where zn is the pseudo-depth associated with the arrival time tn and the reference wavespeed Co, and the An are coefficients which rely on the local reflection coefficient Rn and those at depths above. In fact n - l An = ARn]l(l-R 2). (5.74) j=i The inversion subseries, by which the amplitude of the Born approximation a.\ is corrected to ct/jvy, is elsewhere (Weglein et al., 2003; Zhang and Weglein, 2003) found to be aINV{z) = a i - \a\ + ^a* - ... = £ lt^a{(z). (5.75) Clearly the main feature of this series is that it involves repeated exponentiation of the Born approximation given by equation (5.73). If I exponentiate equation (5.73) I am very quickly going to have to figure out what happens when a Heaviside function is squared, a procedure which is, strictly speaking, not permitted. Consider the Heaviside function as a means by which to define a subset of the real line, and consider the product of Heaviside functions to be a refinement of this means. That is, H(z - z\)H{z2 - z) is a gate starting at z\ and ending at z2. Using this sense of the Heaviside - as a definer of boundaries - it is reasonable to Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 116 decide that the product H 2(z — z\) = H(z — z\)H{z — z\) is merely a redundant definition of the region beyond z\, so that, if zn > z„ H 2(z - zn) = H{z - zn), ^ H(z - zn)H(z - zm) = H(z - zn). This definition is valid in the sense that it describes the behaviour of manipulations of the numeric approxi-mation of a Heaviside of the form (..., 0 , 0 , 0 , 1 , 1 , 1 , . . . ) . But at the outset, regardless of how the Heavisides are treated, the exponentiation of a Born approximation like that of equation (5.73) will mimic the exponentiation of a multinomial; a binomial if one considers a two-interface case like ai(z) = AiH(z - Zl) + A2H(z - z2), (5.77) a trinomial if one considers the three-interface problem a , ( z ) = AiH{z - zi) + A2H{z - z2) + A3H(z - z3), (5.78) etc. The multinomial formula therefore provides the form of one of the terms that appear in the n ' th power of cti(z) with p interfaces: where the set {nk}k=(i,P) satisfies n i + n 2 + ... + np = n. The rest of the terms of this power of the Born approximation are produced by collecting all such sets of {nk}k=(i,p) and summing. For instance, the third power of a three-interface Born approximation like that of equation (5.78) is made up of the coefficients A\, A\, Al, 3 A 2 A 2 , 3AlAlt 3 A ? A 3 , 3A 23AU 3A 22A3, ZA\A2, 6AXA2A3. (5.80) The Heavisides, by the rules of equation (5.76), then apportion these coefficients out so that they alter different layers in the model. Because the deepest Heaviside in a product "wins out", coefficients are placed as part of the layer associated with their highest indexed A n . That is, al(z) = [Al] H{z - zi) + [A 3 + 3 A 2 A 2 + 3 ^ ] H(z - z2) + [A 33 + 3AlA3 + 3A 23AX + 3AJA3 + 3 A | A 2 + 6 A i A 2 A 3 ] H(z - z3); if it has an A 2 in it but not an A 3 , it goes with H(z — z2). The inversion subseries will sum weighted versions of expressions like that of equation (5.81) to create the correction to the amplitudes of the layers. Notice that structurally nothing has changed between a\(z) of equation (5.78) and the a3(z) of equation (5.81); it is still a set of two layers defined by interfaces at z\, z2, and Z3 - the Heavisides have ensured that. The only thing that is happening is that the coefficient of, say, H(z — z3) is changing. The inversion subseries is not trying to do anything fancy with the locations of the Born depths zn; it clearly does not think that such alterations are its task. This is handy for the current investigation, because it means that one may identify [Al + 3 A 2 A 3 + ZA\AX + ZA\A3 + 3 A 2 A 2 + 6 A i A 2 A 3 ] (5.82) as part of the third term in a series to correct the amplitude of the layer beyond the third interface at z3. This is pursued in the next section, in which the inversion of attenuated reflection data for Q is investigated. At present, consider the relative contributions, to a layer of a™(z), of terms that are "self-helping", like A 3 , versus terms which make more (e.g. A i A 2 A 3 ) or less (e.g. A 1 A 2 ) use of all shallower layers in constructing the amplitude correction. Notice in equation (5.81) that the term which makes simultaneous use of all the layers, A i A 2 A 3 , is, by the multinomial coefnents, six times as "important" as the term which uses only the one, A 3 . This is Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 117 characteristic of terms of the inversion subseries which evenly distribute reliance on the shallower events of the Born approximation - they are weighted heavily in comparison to those terms which rely on single events. Because the multinomial formula models the relative probabilities of the outcomes of random experiments, such as the throwing of dice, or the tossing of coins, there is a direct analogue between the reliance of inversion subseries terms on evenly versus unevenly distributions of shallow events, and the likelihood of tossing, for instance, equal numbers of heads and tails versus tossing large numbers of either. Consider this coin toss experiment: suppose the coin is tossed n times. Then the quantity (t + h) n = Y] • „ ^ ...t n- jh s (5.83) expresses the relative probabilities of throwing certain combinations of heads and tails. For instance, if the coin is tossed three times, (t + h) 3 = t3 + 3t 2h + 3th 2 + h 3, (5.84) then it is three times as likely that one will toss two heads and a tail than three tails. In the inversion analogue, equation (5.81) corresponds to the third-order term for a two-interface problem - in computing the correction for the second layer, the term with contributions most evenly distributed, e.g. A\A2 is weighted three times more than the least evenly distributed A3,. (The Heaviside rule apportions all but the first part of the analogue in equation (5.81) to the second interface.) In a problem with more interfaces, the analogous random experiment has additional possible outcomes - for instance, a six interface problem is mathematically similar to a dice-throwing experiment. There are two main points here: first, there is a straightforward means for predicting the distributions of communicating events in any term in the ID inversion subseries for a model with an arbitrary number of layers. Second, and perhaps most interesting, the multinomial formula tells us that in the inversion subseries, the terms which evenly distribute the contribution of all layers will dwarf those terms which use sparsely distributed information. This behaviour may be speaking of some intuitively reasonable feature of the inversion subseries, that in inverting for the amplitude of a deep layer, the global effect of what overlies it is far more important than any one (or few) local effect(s). 5.7 Towards Nonlinear Q Estimation In this section some properties of the inversion subseries as it addresses the reconstruction of attenuation contrasts from ID seismic data are considered. A peculiar data set is created for this purpose, one that is designed not to reflect reality, but rather to examine certain aspects of the inverse scattering series and only those aspects. Also, it involves the choice of a simplified model of attenuation. I begin with a description of the physical framework of this analysis. The attenuation in the wave field is modelled, for reasons of simplicity (i.e. frequency-independent wavespeed), by the friction model for attenuation of pulses. Furthermore, the reference medium in the following ID problem is a homogeneous acoustic whole space characterized by wavespeed CQ. For these reasons there is particular need for the following two dispersion relations: k = - , (5.85) fc(z) = ^ [ l + »/?(*)], the first being that of the reference medium, and the second that of a non reference medium in which the wavespeed may vary, as may the parameter /3, which, like the wavespeed is considered an inherent property of the medium in which the wave propagates. It is responsible for not only altered reflection and transmission properties of a wave, but also for the amplitude decay associated with intrinsic friction. One could argue pretty reasonably that f3 is related to the better known parameter Q by f3 = 1 /2Q because of their similar roles Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 118 in changing the amplitude spectrum of an impulse. However, of course most Q models are more sophisticated in that they utilize dispersion to ensure the causality of the medium response. I make use of the friction model, which captures much of the key behaviour of attenuating media, to take advantage of its simplicity -in particular its associated frequency-independent Q. For this investigation, I restrict attention to models which contain variations in fi only. Waves propagate with wavespeed CQ everywhere. 5.7.1 Another Complex Reflection Coefficient Contrasts in j3 only will produce reflections with strength kn-\ — kn H.fin-1 — fin) ,K of!\ kn-i + kn 2 + l{pn-i + pn) and, since we assume an acoustic reference medium (/?o = 0), Notice that, as in the dispersive models used in this chapter, the reflection coefficient is complex. Without dispersion, the complex reflection coefficient implies the combination of a frequency independent phase ro-tation and a real reflection coefficient: R\ = \R\\e lB. This is akin to the reflection coefficients discussed in Born and Wolf (1999). If the frequency domain data from a reflection at an interface with such a reflection coefficient is given by D(k) = Riexp(i2kz\), then the expression of such a reflection in the measurement domain is (Aki and Richards, 2002, e.g.): D(t) = \R\cos9S (t- — ) + \R\sm6 I ^ r | . (5.88) Since the point of this particular section on the inversion subseries is insight, and not practical algorithms, I will in the main allow the use of the true complex coefficient as input to the data equations. I assume that the phase and amplitude spectra associated with D(k) may be adequately estimated from the information in D(t), equation (5.88). 5.7.2 A Non-attenuating Viscoacoustic Data Set A further simplification has to do with the measured data, rather than the medium itself. As in the acoustic case, true viscoacoustic data will be a series of transient pulses that return, delayed by an appropriate traveltime and weighted by a combination of reflection and (shallower) transmission coefficients. But unlike an acoustic medium, the propagation between contrasts also determines the character of the events. Each pulse that has travelled through a medium with fi ^ 0 will have had its amplitude suppressed at a cycle-independent rate, and therefore be broadened. Although this is a fundamental aspect of viscoacoustic wave propagation (and indeed one of the more interesting aspects for data processing), for the purposes of this investigation I suppress it in creating the data. The results to follow will help explain why. The perturbation 7 is based on an attenuative propagation law, and is itself complex: 7 ( 2 ) = 1 - = -2ifi(z) + fi 2(z), (5.89) If the propagation effects are suppressed (or have been corrected for already), then the Born approximation of 7 ( 2 ) , namely 7 1 ( 2 ) , is, from equation (5.73), 710) = AiH{z - Zl) + A2H(z - z2) + (5.90) where the A n are given by equation (5.74), and the reflection coefficients are those of equations (5.86) and (5.87). Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 119 In the previous section it was noted that the contributing terms to any layer in the inversion subseries can be found by invoking the multinomial formula; and the contributing terms to a two-layer model are produced by the binomial formula (which has a much more simply computable form). I assume a two-layer model, in which the reference medium (c = cn, /? = 0) is perturbed below z\ such that c = c 0 and P = Pi, and again below Z2 such that c = Cn and j3 = p2. The depths Z\ and z2 are chosen to be 300 m and 600 m respectively. This results in 7 1 ( 2 ) -4i/?i 2 + t/?i H(z-Zl) + -4W ~ fo) 1 ft H(z-z2). (5.91) L2 + »(&+ p\) V (2 + i/?i)2, The n'th power of the Born approximation for either layer of interest is computable by appealing to the binomial formula. The result of the inversion subseries on this two interface model is therefore, making use of equation (5.75), 1INV(z) = A[NVH{z - Zl) + AJ2"vH(z - z2) (5.92) where AINV _ \ ^ J'( 1)J' 1 Aj = 1 4 J ~ J _ 1 (5.93) AINV _ V ^ - 1 ) J 1 I V j ! ,lJ- f c,lfc and where 4 J - i ^ f c ! ( j _ f c ) r 2 > 2 + i / 3 1 ' (5 94) -4»(ft - P2) ( _ PI \ ^ 2 2 + i(p1+p2)\ (2 + iprfJ are the amplitudes of the Born approximation in equation (5.90). Figures 5.18 - 5.20 demonstrate numerically the convergence of these expressions (the real part, for the sake of illustration) towards the true value. There is clear oscillation over the first three or so terms for the second layer, but the oscillations settle down to close to the correct value with reasonable speed. This suggests an important aspect of going beyond the Born approximation in Q estimation - the deeper layers, whose amplitudes are related not only to their own Q value but to those above it, require the invocation of several, but not too many, nonlinear orders to achieve their correct value. Finally, Figures 5.21 - 5.23 illustrate the same inversion results as those seen in Figures 5.18 - 5.20, but in the context of the spatial distribution of the reconstructed perturbation. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 120 0.04 r-0.03 -b ' 0 5 10 15 Iteration Number/Inversion Order Figure 5.18: Convergence of the viscoacoustic inversion subseries for a two-interface model, consisting of contrasts in (friction-model) attenuation parameter. The real component of the perturbation is plotted against inversion order or iteration number (connected dots) and with respect to the true perturbation value (solid), (a) layer 1 inversion, (b) layer 1 inversion. For model contrasts fi\ = 0.15, 02 = 0.25, ~yiNv(z) is close to 7(2) for both layers. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 121 b Iteration Number/Inversion Order Figure 5.19: Convergence of the viscoacoustic inversion subseries for a two-interface model, consisting of contrasts in (friction-model) attenuation parameter. The real component of the perturbation is plotted against inversion order or iteration number (connected dots) and with respect to the true perturbation value (solid), (a) layer 1 inversion, (b) layer 2 inversion. For model contrasts f}\ = 0.35, 02 = 0.55, ~fiNv(z) differs slightly from 7(2) in the lower layer. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 122 a b Iteration Number/Inversion Order Figure 5.20: Convergence of the viscoacoustic inversion subseries for a two-interface model, consisting of contrasts in (friction-model) attenuation parameter. The real component of the perturbation is plotted against inversion order or iteration number (connected dots) and with respect to the true perturbation value (solid), (a) layer 1 inversion, (b) layer 2 inversion. For model contrasts f3\ = 0.55, @2 = 0.35, 7/jvv(z) again differs slightly from 7(2) in the lower layer. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 123 o ?-cn c g V-1 CC X3 D t 0 CL 100 200 300 400 500 600 Pseudo-depth (m) Figure 5.21: Convergence of the viscoacoustic inversion subseries for a two- interface model, consisting of contrasts in (friction-model) attenuation parameter. Depth profile of the real component of the constructed perturbation (solid) is plotted against the true perturbation (dotted) for 5 iterations, (a) iteration 1 (Born approximation) -(e) iteration 5. Model inputs: j3\ = 0.15, /32 = 0.25. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 124 O w c o CO r 100 200 300 400 500 600 Pseudo-depth (m) 900 900 900 900 900 Figure 5.22: Convergence of the viscoacoustic inversion subseries for a two- interface model, consisting of contrasts in (friction-model) attenuation parameter. Depth profile of the real component of the constructed perturbation (solid) is plotted against the true perturbation (dotted) for 5 iterations, (a) iteration 1 (Born approximation) -(e) iteration 5. Model inputs: (3\ = 0.35, @2 = 0.55. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 125 o (/} c o CO n t t CD -0.5 100 200 300 400 500 Pseudo-depth (m) 600 700 800 900 Figure 5.23: Convergence of the viscoacoustic inversion subseries for a two- interface model, consisting of contrasts in (friction-model) attenuation parameter. Depth profile of the real component of the constructed perturbation (solid) is plotted against the true perturbation (dotted) for 5 iterations, (a) iteration 1 (Born approximation) -(e) iteration 5. Model inputs: Pi = 0.55, 02 = 0.35. Chapter 5. Linear and Nonlinear Viscoacoustic Inversion 126 5.7.3 Impl icat ions for Q -Process ing in the Imaging Subseries The success of the inversion subseries in reconstructing the amplitudes of the attenuation contrasts in the examples of the previous section is worth pursuing, since the success was achieved by using a strange input data set; one in which the propagation effects of the medium had been stripped off a priori. How should this success be interpreted? It follows that if the inversion subseries correctly uses the above data set to estimate Q, then the subseries must have been expecting to encounter something of that nature. As a corollary, the other, non inversion subseries terms should be looked-to to provide just such an attenuation-compensated Born approximation. This is in agreement with the conclusions reached at the end of the forward scattering investigation of chapter 2, in which it appeared that the imaging subseries, in a Q-contrast only type case, would be involved exclusively in eliminating the effects of propagation on the wave field. It is also in agreement with the bootstrap alteration of the linearized results in the first part of this chapter, in which the quality of the profile inversion was seen to improve drastically in the absence of the attenuative effects of propagation. 5.8 R e m a r k s Both the linear and nonlinear processing and inversion ideas fleshed out in this chapter rely, in their detail, on specific models of absorption and dispersion. For instance, much of the formalism in the Born inversion for c and Q in the single interface case arises because the frequency dependence of the dispersion law (F(k)) is separable, multiplying a frequency-independent Q. Since the detail of the inversion relies on the good fortune of having a viscoacoustic model that behaves so accommodatingly, it is natural to ask: if I have tied myself to one particular model of dispersion, do I not then rise and fall with the (case by case) utility and accuracy of this one model? The practical answer is yes, of course; the specific formulae and methods in this chapter require that this constant Q model explain the propagation and reflection/transmission in the medium. But one of the core ideas of this chapter has been to root out the basic reasons why such an approach might work. The dispersion-based frequency dependence of the reflection coefficient as a means to separate Q from c is not a model dependent idea, regardless of model-to-model differences in detail and difficulty. It is also clear that these inversion strategies rely on the ability to detect fine variations in the data. This is particularly noticeable in the data examples of, for example, Figures 5.11 - 5.16; visually the reflections do not change a great deal model to model. (An example of a dispersive reflection whose features are more plainly visible, because of larger contrast, is Figure 5.16.) It is certainly worth emphasizing that high quality estimates of the event spectra will be required as input to these methods. There have been two concurrent aims in this chapter: first, the development of tools for application in viscoacoustic inversion, that is methods for c and Q estimation; second, the exploration of the basic mechanisms of linear and non-linear viscoacoustic inversion based on scattering theory. The two are not mutually exclusive. For instance, the investigation into the accuracy and applicability of the linearized viscoacoustic profile estimation both acts as an indicator of the limits of practical (immediate) use of such an approach, and as an indicator of the nature of the inversion as input to a higher-order non-linear scheme. As mentioned in the introduction, and as seen both in the latter part of this chapter and the previous chapter, the "wrongness" of the linearized inverse (compared to the correct model) is not a problem in the context of the inverse series; what is important is that this linearized inverse is wrong in the way the higher-order formulations expect. (In other words, one must be sure to provide the inverse series with the right wrong answer.) The first and second parts of this chapter appear to confirm the predictions of the forward series analysis in chapter 2; i.e. that prior to the inversion (target identification) step, the subseries terms which are nominally concerned with imaging, must also remove the effects of attenuative propagation. The confirmation in this chapter is circumstantial but compelling: the ubiquitous increase in quality of the inverse results (both low order constant Q and high order friction-based attenuation) in the absence of propagation effects strongly suggest that depropagated (Q compensated) inputs are expected in these inversion procedures. A clear future direction for research is in generalizing the imaging subseries to perform Q compensation simultaneously with reflector location; or indeed, investigating the feasibility of configuring the simultaneous imaging and inversion procedure of chapter 4 to perform all these tasks in one fell swoop. Chapter 6. Multiresolution Wave Field Continuation 127 Chapter 6 M u l t i r e s o l u t i o n W a v e F i e l d C o n t i n u a t i o n Term Signifies Wf(a,b) Wavelet transform of function / at scale a, shift b Wavelet at scale a, shift b K{z), K{kz), K Propagation kernel in z, kz, and arbitrary domains Attenuation parameter P(z,w) Pressure field at depth z <Ps Scaling function at scale s f,d General expressions for the measured signal p, m General expressions for the model Tab, rX Coefficients of the multiresolution reconstruction The u'th weight due to regularization parameter 7 Objective function for regularization parameter 7 Inverse kernel function approximated using 7 6.1 Introduction In this chapter the ability of the wavelet transform, in addressing the problem of downward continuation of a viscoacoustic wave field, to simultaneously (i) interrogate data scale by scale, and (ii) stabilize (regularize) this ill-conditioned linear inverse problem, is studied. The wavelet-based approach is formulated as in Song and Innanen (2002), and evaluated numerically in contrast to other well-known approaches as discussed in Song et al. (2000). The regularization discussed in this chapter is with regard to an inverse filtering problem, posed such that stability may be traded for resolution in the face of imperfect data. It therefore impacts other portions of this thesis with varying degrees of directness. In particular, since it involves Q compensation with known Q, it applies directly to the bootstrap linear interval Q estimation discussed in chapter 5. Further, since the problem amounts to a judicious windowing, or cutoff, of the singular values (Fourier coefficients) of the inverse operator, it explores potential (and more sophisticated) means by which to regularize/stabilize the numeric derivative operators which were used in all examples of chapter 4. Kernel functions for "lossy" convolutional integral equations often exhibit a decay in the time or frequency domains; because of this, the convolution process tends to strip signals of high spatial frequency information. In the viscoacoustic case, signal resolution is removed in a way largely determined by the propagation depth. The kernel function and depth of propagation, respectively, are here identified with the scale function and scale parameter in multiresolution analysis (for good reviews of multiresolution analysis, some more mathematical and some less, see Chui (1992a,b); Mallat (1998); Daubechies (1992)). In so doing, the propagation problem is cast as a multiresolution process, and the adoption of a multiresolution reconstruction method is rationalized as being one which adheres closely to the model of propagation in laterally homogeneous media. It is interesting to note that some of the fundamental ideas of multiresolution theory (i.e. wavelets) were introduced as means to analyse seismic data (Morlet et al., 1982). The idea, here espoused, of utilizing a multiresolution/multiscale framework as a sophisticated way of characterizing, and, automatically interpret-ing the nature of, signals is not new; a swell of research on scale space representations in the field of pattern Chapter 6. Multiresolution Wave Field Continuation 128 recognition (Witkin, 1983) led to the development of the multiresolution representation (Mallat, 1989), which incorporated the mathematical formalism of wavelet theory. Many regularization strategies for model construction can be seen as the result of two steps in the approximation of an inverse kernel function, in which: (i) a basis is chosen over which to expand the inverse kernel function in series; (ii) weights are prescribed, for the terms in this series, that are chosen via properties of the data. This is certainly the case for truncated singular value decomposition (TSVD), damped least-squares (LSQ) and the wavelet approach; hence a framework based on steps (i) and (ii) is general enough that these three approaches may be compared therewithin. In the first section of this chapter, I present this framework as a general expression for a regularized inverse kernel function (or regularized inverse filter). I then demonstrate how the specification of bases and weights described above lead to forms of the function that are consistent with LSQ, TSVD, and the wavelet regularization. A quasi-general scheme for comparison of the three techniques is then presented ("quasi", since regular-ization is a data-dependent process and, so, varies from example to example). The results of the comparison hinge on the localization properties, in space and wavenumber domains, of the downward continuation op-erator approximated by each regularization strategy. In fact, the wavelet approach is shown to provide an inverse kernel approximation that balances the more extreme localization properties of those of LSQ and TSVD. This balanced localization is linked, through simple theoretical arguments and numerical examples, to a robustness, in model construction, to specific types of data inaccuracy. In chapter 3, the regularity of a non-stationary seismic trace was computed using a specific wavelet transform. Such a "continuous" wavelet transform, involving a wavelet with a specified number of vanishing moments, was used because it was precisely suitable for regularity estimation. In this chapter the problem is quite different: the inverse kernel function is to be approximated by expansion in series, with a wavelet as the basis function. The stability imparted to the regularized inversion arises because of the frequency-domain localization properties of this wavelet. Consider the wavelet transform 1 r°° * _ /, Wf(a,b) = - r f(zM )dz. (6.1) V a J-oo a One can choose the so-called mother wavelet -ip(z) such that the wavelet basis function -^4>(^r) (where a G 3f+ and b 6 3t are the dilation and translation parameters respectively) has localization properties in both the space and wavenumber domains. For large a, the basis function becomes a stretched version of the mother wavelet, a low-wavenumber function; likewise for a -> 0 + the wavelet contracts, forming a high-wavenumber function. In this chapter, the wavelet ip(z) is chosen following Daubechies (1992): f < \kz\ < f (6.2) otherwise where = £ 4(35 - 84£ + 70£2 - 20£ 3). (6.3) Equation (6.2) gives the form of the so-called mother wavelet in the kz domain. It may be dilated (via a) and translated (via b) thus: ^ab(z)=2-a^(2az-b), (6.4) such that tpab(z) is an orthogonal basis for L2(5R). Clearly it is of compact support, i.e. beyond a certain frequency range the function is identically zero. The difference between a wavelet of this kind and one that has, for instance, fast decay in the frequency domain is in its ability to truncate the higher frequency components of any inverse filter, regardless of its rate of growth with kz. 1>(kz) = I ,ikJ2 pikz/2 e-"2sin\lv(^\kz COS\T:V\ 1)] 1)] Chapter 6. Multiresolution Wave Field Continuation 129 6.2 Downward Continuation in Attenuating Media Downward continuation of a wave field in a (laterally, at least) homogeneous medium with known wavespeed, and, in this case, Q, is expressible as a series of spatial deconvolution problems. In the space-frequency domain, one such deconvolution is required for each temporal frequency used (Berkhout, 1980, e.g.). In the frequency-wavenumber domain the problem is a matter of applying an appropriate phase shift to the Fourier components of the measured wave field (Gazdag, 1978). In this domain the phase shift is accomplished by the application of a diagonalized version of the downward continuation operator, which is a function of (de-)propagation depth, wavespeed, and Q (as well as u and kz). The inversion problem may be designed for viscoacoustic media. See also the work of Mittet et al. (1995). The simplicity of the operator, designed for homogeneous viscoacoustic media, belies its behaviour in the presence of imperfect data. The problem is most readily understood in terms of the operator's singular values (or Fourier coefficients, since this is a deconvolution problem); here I compare those of the acoustic case to those of the viscoacoustic case. In either case the upward continuation operator (the operator associated with forward propagation) is of the form K{kx,ky)=eik* z, (6.5) where z is the propagation distance. Figure 6.1 illustrates the modulus of the acoustic upward continuation operator in two wavenumber dimensions (Figure 6.1a) and in one wavenumber dimension (Figure 6.1b). In this case kz in equation (6.5) becomes V c o There is a subset of kxt ky in which the singular values of the operator are unchanging; in this region the condition number of the continuation operator is unity, and it may be inverted (to generate the downward continuation operator) without creating an ill-conditioned problem. It occurs at yjk 2, + k2 < LV/CQ; beyond this kz becomes imaginary and the operator decays with the wavenumber variables. The wave field associated with wavenumbers in this region of the spectrum is evanescent, and in downward continuation this region is normally suppressed (Stolt and Benson, 1986), because, when inverting this operator to downward continue data, the decay becomes growth. The problem is then ill-conditioned and extremely sensitive to inaccurate data. In contrast with this is the viscoacoustic version of the continuation operator, in which kz = ^ [ i + ip(u)]2-kZ-k*, (6.7) as illustrated in Figure 6.2a and b. The presence of the complex perturbation on the wavespeed (characterized by (3(u>)), i.e. the presence of absorption, imposes a decaying component on all regions of the spectrum. This is visible in the illustration of the modulus of the operator in Figure 6.2. There is no portion of the viscoacoustic spectrum for which the downward continuation operator is well-conditioned: any attempt to apply the inverse of equation (6.5) in which equation (6.7) holds will be sensitive to inaccurate data. The challenge of regularizing this problem is to appropriately balance the restoration of data resolution (via the Q compensation implicit in the viscoacoustic downward continuation operator) with maintaining stability in the face of noisy data. Chapter 6. Multiresolution Wave Field Continuation 130 Figure 6.1: Illustration of the amplitude spectrum of the kernel function \K(kx,kv)\ for 2-D wave field propagation m an acoustic medium; (a) in full as a surface in (kx,kv), (b) in cross-section at ky = 0. The standard reconstruction approach involves inversion of K for \kx\,\kv\ < W where it undergoes no decay. Figure 6.2: Illustration of the amplitude spectrum of the kernel function \K{kx,kv)\ for 2-D wave field propagation m a viscoacoustic medium; (a) in full as a surface in (kx,ky), (b) in cross-section at kv = 0. Kernel function decay exists at all kx and kv; inversion for reconstruction is therefore unstable at all values of kx and ky. 6.3 Propagation Depth is a Scaling Parameter In the frequency domain, the propagated wave field P, in ID, at the surface (z = 0) is, as discussed in the previous section, related to the interior wave field (at z = Az) by: Chapter 6. Multiresolution Wave Field Continuation 131 P(0,w) = P(Az,uj)eik'Az, (6.8) where kz — ^[l+if3(uj)} for a medium with wavespeed CQ and attenuation parameter (3(LO). This attenuation parameter may be chosen to reflect any one of many possible models of absorption and/or dispersion. I choose a friction-based model of attenuation, i.e. one in which the attenuation parameter is associated with a simple constant Q (f3(u>) = (3). The reasoning behind this choice lies in the fact that the problem I consider involves the downward continuation of attenuated data in which the medium parameters, including (3 (or alternatively Q) are known. Since compensation for dispersion is well-conditioned, attenuated and dispersed data can be made to resemble merely attenuated data via a dispersion compensation. The key multiscale aspects of absorptive/dispersive propagation, and the key difficulties in depropagating data in absorptive/dispersive media, are the same as those of merely absorptive media. There is, as discussed, a convolution relationship between propagated and unpropagated wave fields; in ID, and using the friction-based model of attenuation (3(w) = (3, it is characterized by the kernel function K(Az,co) = e i ^ A z e - ^ 0 A z . (6.9) Note the following property of the kernel function in equation (6.9): K(AZ,LU) = e - l ^ A z e ~ ^ A z = e - ^ A * ) e - ^ A * > = K(l,coAz). (6.10) In other words, the operator for depth Az and frequency to is equivalent to the operator for unit depth at a scaled frequency Az ui. Compare this behaviour with the general properties of the scale function cp from the theory of multiresolution analysis (see for example (Mallat, 1998)), for scaling parameter s: ips(w) = cp(ws), (6.11) in which the scaled version of the base function </?(o;), <ps(w), which is used to create the coarse component within the multiresolution decomposition, is equal to the base function itself with scaled frequency su. Since the viscoacoustic kernel function smooths the data, and the scaling seen in equations (6.10) and (6.11) holds, there is a correspondence between this kernel function and a multiresolution scaling function, where in the physical case propagation depth Az is seen as a parameter dictating scale in the evolving wave field. 6.3.1 Numerical Illustration Because the viscoacoustic kernel function is a lowpass-like filter, modelling the propagation of a wave field in a viscoacoustic medium in steps (for instance some propagation distance Z in N steps of Az , where Z — NAz), produces a chain of intermediate signals which strongly mirror the multiresolution decomposition. A simple 1-D numerical example will serve to illustrate. In Figure 6.3a, a set of three closely-spaced Ricker wavelets make up a synthetic signal in the Earth's interior. Figures 6.3b - 6.3e correspond to the waveforms after propagation. In these numerical examples I use the constant (with respect to frequency) attenuation parameter (3. This choice may be considered to be due to (i) the adoption of the simple friction-based model for attenuation, or, equivalently, (ii) a consideration of the filtering of the amplitude spectrum only for a propagation law based on both attenuation and dispersion. In either case, it is the attenuation that is of interest. This remains the approach for the companion example to Figure 6.3, the upcoming reconstruction of Figure 6.4. In later examples, testing the sensitivity of the reconstruction to data error, an absorptive/dispersive forward model is chosen, but the dispersion (a well-conditioned problem) is corrected as a pre-processing step. The propagation is done with c 0 = 1000 m/s, and using depth and absorption parameters such that (3Az = 2, 4, 8, and 12 m hold respectively. Such values might correspond, for instance, to an experiment in which (3 = 0.01, and depths range from 200 to 1200 m (such a value for (3 produces amplitude changes in the kernel corresponding to a Q-factor of approximately 50, assuming (3 « 1/2Q). The smearing action of the kernel function strips detail as propagation distance increases. In other words, as it propagates, the viscoacoustic wave field inhabits spaces mathematically akin to the coarse spaces of the multiresolution representation. In this sense the propagation depth is the scale parameter. Chapter 6. Multiresolution Wave Field Continuation 132 Propagation distance = 1200m e Propagation distance = 800m d Propagation distance = 400m c Propagation distance = 200m b Propagation distance = 0m a t(s) -» Figure 6.3: 1-D synthetic example of viscoacoustic propagation. An interior model of three closely-spaced point sources (ban-dlimited) (a), is propagated using depth and absorption parameter values such that (b) 0Az = 2 m, (c) 4 m, (d), 8 m, and (e) 12 m holds; the phase shift has been suppressed for illustration purposes. (This might correspond to an experiment in which 8 = 0.01, and depths range from 200 m to 1200 m. Such a value of 8 produces amplitude variations, in the kernel, corresponding to a Q-factor of approximately 50, assuming (3 = 1/2Q.) Wavespeed is 1000 m/s. The suppression of high spatial frequency information with depth motivates the view of propagation in which depth is seen as a scaling parameter. 6.4 Multiresolution Wave Field Reconstruction To discuss the inverse, or reconstruction problem, consider f(z) to represent the known or measured function, which will generally be realized as a set of discrete values {/(n)} and be contaminated by additive noise. The unpropagated, or unknown signal is designated p(z), and the kernel function is, as before, K(z), such that holds. Reconstruction for the continuation of the seismic wave field is routinely accomplished in the Fourier domain. In this regard it strongly resembles problems of continuation of potential fields, and like these, difficulties arise in the downward continuation, or un-smoothing process, when the recorded data contain some error or noise. This is rooted in the instability associated with the decay of the kernel function, and can be further seen by considering the solution of equation (6.12) in the Fourier domain: (6.12) P(kz) = (6.13) K(kzy The decay of K(kz) with kz leads to strong amplification of any noise present in f(kz): is therefore highly unstable in the presence of noise. Some regularization procedure, i.e. wavenumbers, is necessary. equation (6.13) a cutoff of high Chapter 6. Multiresolution Wave Field Continuation 133 Song and Innanen (2002) postulated formula for "multiresolution wave field reconstruction", or MRWR, that builds the image p(z) from a linear combination of data components at all scales, i.e. wavelet transforms of f(z) (using the wavelet defined in equation (6.2)). That is, one might anticipate a reconstruction formula of the form: oo oo P&= E E rab[f(z)* a^b(z)}. (6.14) a,— — oo b=—oo The rab are furthermore expected to depend on the inverse kernel function K ^(z). Substituting equation (6.14) into the propagation relationship, namely f(z)=K(z)*p(z), (6.15) one has rab = ( a^b(z),K-1(z)), (6.16) where (•, •) denotes the inner product in L 2(R). Equation (6.16) provides the coefficients, rab, then, so that the reconstruction formula of equation (6.14) becomes oo oo E ( ^ K - ^ l f ^ a ^ z ) . (6.17) a= — oo b= — oo For a practical implementation equation (6.17) would be replaced with A B P&= E Y,^{kz),K- l{kz))[f{z)*^ab(z)}, (6.18) a=-A b=-B with the scale limits A , in particular, chosen based on the desired aspects of the constructed model, and the levels of data error. 6.4.1 Numerical Illustration The use of the multiresolution reconstruction may be illustrated with a simple 1-D example, applied to synthetic data for a variety of scale approximations A. Equation (6.18) assumes scales to be cut off beyond |A|. In point of fact, the scale limits may have any reasonable lower and upper values (An, A\). AQ is chosen to maintain stability, and A \ is chosen only large enough to capture the all essential low-frequency signal components. Figure 6.4 depicts four multiresolution reconstructions of the synthetic signal of Figure 6.3e, i.e. a propagated version of the model of Figure 6.3a. Figure 6.4a corresponds to A 0 = 5, 6.4b to A 0 = 4, 6.4c to A 0 = 3, and 6.4d to A 0 = 2, i.e using increasingly high-resolution components of the signal. Note that at An = 2 the signal contains all of scale components necessary to completely capture the shape of the original waveform. Indeed, the sequential removal of low scales in the reconstruction (seen by viewing Figure 6.4 stepwise from 6.4d to a), markedly resembles the action of propagation through various depths (seen by viewing Figure 6.3 stepwise from a to e). The scale-by-scale reconstruction of the wave field resembles the depth-by-depth propagation of the field because of the kernel function-scale function correspondence discussed above. The exact character of the reconstruction, at various scales, is of course dictated by the wavelet and the corresponding scale function. Differences between the kernel function and the scale function explain differences in the evolution of signals in Figures 6.3 and 6.4. As mentioned, it is through the choice of lower and upper limits (A0,Ai) for a practical application of the reconstruction (equation (6.18)) that noise is kept under control (Song et al., 1999): analysis of noise in a particular data set leads to an estimate of scale limits which optimize the tradeoff between stability and resolution. One wishes to reconstruct using as low a An as possible, but at the same time not pushing into unstable regions of the (large) wavenumber domain. By admitting scale-information up to some AQ only, the ill-conditioned problem is regularized. Mathematically, the wavelet basis confers, on the singular values of Chapter 6. Multiresolution Wave Field Continuation 134 the inverse operator, a set of weights which extract the desired image components. These weights parse the image information into spatial scales. Scale limit A = 5 Scale limit A = 4 Scale limit A = 3 Scale limit A = 2 t(s) Figure 6.4: 1-D reconstruction example. Reconstructions occur using the general form of MRWR for a set of scale limits A: (a) A = 5, (b) A = 4, (c) A = 3, and (d) A = 2. yis in Figure 6.3, phase shifts have been suppressed for illustration purposes. As lower scales are incorporated, the unpropagated signal is reconstructed more completely. Sequential addition of lower scales, as seen in (a) through (d), mimics the reversal of the physical process of propagation. 6.5 The Localization Properties of Inverse Filters In chapter 4, it was found that high-order numerical derivative operators in the frequency domain could be approximated in a manner that produced high-quality inversion results while stabilizing the numerics. In this section I frame MRWR in a similar manner, as a stabilized approximation to the true operator that involves a specific (wavelet-based) truncation of the singular values (i.e. frequency-components) of the downward continuation kernel. The original motivation for the adoption of a wavelet-based method, as discussed in the previous sections, was to solve ill-conditioned deconvolution problems through the explicit use of scale information in the data; as such it is a model construction scheme that evokes the multiscale nature of the propagation operator. It is also a method wherein the relationship between regularization parameter and model type is not known. In the balance of this chapter, I develop a means to understand the behaviour of MRWR in the presence of inaccurate data, using comparisons to damped least squares and truncated singular value decomposition approaches. The results permit informed decisions to be made regarding case-by-case applicability of this wavelet-based method. 6.5.1 The Inverse Filtering Problem In this section I describe the regularized inverse filtering process within a framework general enough to include the wavelet method, as well as a variety of other inverse filtering methods. Chapter 6. Multiresolution Wave Field Continuation 135 I begin by considering the 1-D form of the problem, in which data d(z) are generated through the convolution of a model m(z) with some kernel function K(z): /oo K(z - z')m(z')dz' (6.19) -oo or simply d{z) = K{z) * m{z) (6.20) This holds for any linear, space-invariant system. In this section the 1-D kernel function and its inverse are expressed in three ways: as a function of the spatial variable z, the conjugate wavenumber kz, or neither (i.e. K(z), K(kz), and K respectively). The third notation is used in cases where the property or aspect of the function in question is not restricted to either domain. Furthermore, I freely transform in and out of the Fourier domain in this paper. For convenience, the current domain is signified only by the explicit dependence of K on either z or kz. Equation (6.20) is solved via m(z) = K~ 1{z)*d{z) (6.21) where K~ 1(z) is the convolutional inverse of K(z). Of course, equation (6.20) is never solved in this form, since K~ l is highly sensitive to imperfect data. Some regularization is required. I will continue by writing down a general expression for the regularized approximation of K~ x by K~ l, where 7 is a general regularization parameter. Consider an expression for the approximated inverse kernel, in which it is expressed as the expansion in series about an orthogonal set of basis functions <pu{z), in which each term is given a weight Wu(-y): K ~ \ z ) = ^ ( A - 1 , ^ > W M ( 7 ) ^ ( z ) , (6.22) where 4>u(z) is chosen to be an appropriate basis of L2(5R) (u may be a multiplicity of indices u = [u\, U2, •••]), and Wu(j) is a weight on the /j'th term in the series that will depend on the general regularization parameter 7-I next demonstrate how the three regularized inverse filtering approaches of TSVD, LSQ and the wavelet method are produced by specification of <j)(z) and the weights W. Truncated Singular Value Decomposition If (p^z) are chosen to be the singular vectors of linear, space-invariant operators, then the summation of equation (6.22) is a discrete inverse Fourier transform of the inverse kernel. If the weights, Wkz(kr), are chosen to be unity for \kz\ < kr, and zero elsewhere (with the regularization parameter 7 chosen to be some cutoff wavenumber kr), then the approximation corresponds to a truncated singular value decomposition (TSVD) inversion scheme. Damped Least-Squares Using the same 4>u{z), but setting the weights (again in the frequency-domain) to be then equation (6.22) again corresponds to the inverse Fourier transform of a kernel approximation. The weights here correspond to a smoothed dampening of the singular values that corresponds to a "damped least-squares" (LSQ) model construction, with 7 the regularization parameter. Chapter 6. Multiresolution Wave Field Continuation 136 Wavelet Regularization Reconsidering the wavelet-based regularization procedure discussed in Song and Innanen (2002), the approx-imation of equation (6.22) takes the form, for ID: A „ rrhnr(z) = ^(K-\l>ab)[d*il>ab](z), (6.24) a,6 where mmr is the MRWR reconstructed model. (The summation notation has been shortened for convenience: the only summation limit that has been explicitly shown is AQ, the minimum admitted scale.) The form of equation (6.24) remains true, so to speak, to the original motivation of explicit use of data scales, in that the extraction of scale-information via the term [d * tp](z) dominates in the expression. An equivalent representation of equation (6.24) is quickly achievable by simply switching the order of d and ?/>: mmr(z) = ^2(K~l, -ipab)^jk(z) * d(z) (6.25) a,b that is, mmr(z) = Kml(z) * d(z) (6.26) where Kmr(z) = ^(K-^MMz) (6.27) a,b which is the expansion of the inverse kernel function in the wavelet series. The function of the regularization parameter AQ here is clearly to truncate this series beyond a minimum scale such that a desired data misfit is achieved. In this form the wavelet based regularization approach may be easily fit into the framework of equation (6.22). To do so, let 4>ii.{z) n o w D e the orthogonal wavelet basis, in which p = (a,b), the scale and translation parameters discussed in the previous section: Mz) = Mz) (6-28) where tp(z) is particularly chosen to be of compact support in the wavenumber domain. Concurrently, consider a simple weighting function not unlike that of the TSVD regularization, i.e. Wab(Ao) is unity for a > AQ and zero elsewhere. Then A „ Kml(z) = Y,{K-\^ab)^k(z) (6.29) a,b which is the wavelet regularization discussed as in equation (6.27). To summarize, viewing regularized deconvolution methods as approximations of the inverse kernel via weights on its singular values, is a sufficiently general framework that the wavelet method may be contrasted with at least two well-known approaches. 6.5.2 A Framework for Comparison of Regularization Schemes In this section I describe a procedure which uses the aforementioned framework to allow contrast and com-parison among methods, given a particular model, operator, and data error type. The procedure focuses on the localization properties of various approximations of the inverse kernel function K~l in both the space and wavenumber domains, forming the theoretical basis for association of the use of local information in a particular domain with robustness to a particular type of data inaccuracy. Two conditions must be in place to usefully compare two or more inverse filters. First, a criterion must be met such that they comparably regularize the inverse problem; this is not a trivial matter, since the Chapter 6. Multiresolution Wave Field Continuation 137 properties of kernels, differently regularized, may differ quite drastically. Second, i.e. once the inverse filters can be said to be on equal footing, a criterion for the comparison itself must be decided upon. Each inverse filter discussed in the previous section is a deviation from the true inverse filter, in which the deviation is mediated by a single regularization parameter. A comparison will require the definition of not only the deterministic, but also the stochastic nature of the measured data; it cannot be done completely in the abstract. Consider that the stochastic properties of the data are known or estimated. A standard procedure in linear inverse theory is to vary 7 in equation (6.22) such that the so-called predicted data (the data produced by convolution of the constructed model with the forward operator) differs from the true data by a statistically expected amount. Specifically, for the case of noise drawn from a stationary Gaussian distribution with variance cr2, the expected data misfit <3>7 is equal to the number of data. In other words the predicted data should differ from the true data by approximately the number of data (Parker, 1994; Oldenburg, 1984). The choice of 7 which produces this misfit can be said to optimally balance the degree to which the model honours imperfect data. Expected misfit governs choice of regularization parameter, and provides a means to fix the degrees of freedom in the comparison problem. However, its calculation requires not only the kernel but the data. This is not a methodological weakness; rather, it is an expression of a truism in inverse theory - inversion itself is a highly situation-dependent process. What is available, in the stead of truly general comparison, is a guideline for fixing any 7 given a specific data set. The first condition for comparison is therefore as follows: two inverse filtering schemes, differing by basis for expansion and weights imposed on expansion terms, are considered equivalent when the data misfits for a specific case are the same. Achieving this condition permits reasonable comparison to occur. For convenience, I use the symbol = to denote the equivalence of two filters i f " 1 and if",,1 regularized in different ways: K- 1 ± K- 1 (6.30) if and only if their corresponding data misfits are equal: $ 7 1 = $ 7 2 . (6.31) Differences in the form of the two inverse kernel approximations remaining after the condition given by equation (6.31) is met, may be properly interpreted as inherent differences in regularization method. 6.5.3 Loca l iza t ion Proper t ies of Inverse F i l te rs The unregularized filter K~ 1{z) is a distribution, infinitely compact in the space domain, and of infinite extent in the wavenumber domain. In this sense it is an end-member of a "continuum of localization", the other end of which is occupied by functions like sin(z), which is infinite in spatial extent and infinitely compact in the wavenumber domain. Functions in L2(5R) are, strictly, in this continuum but not on its endpoints. The Uncertainty Principle as applied to signal analysis constrains the simultaneous localization properties of a signal in space and wavenumber (see for instance (Cohen, 1995) for a more detailed discussion of the Uncertainty Principle so applied). Hence the act of regularization of K{z), i.e. the approximation if 7 (z) , which is a bandlimiting operation, extends the support of the inverse filter in the space domain. I choose the relative positions of i f 7 i and i f 7 2 , where i f 7 i = i f 7 2 , on this localization continuum as the primary criterion for comparison of regularization methods. The localization of an inverse filter is defined to be the effective length of the filter in z coupled with the effective spread of the filter in kz. Consider for example the appropriately normalized Fourier transform pair given by e " ^ 6 e — ( 6 . 3 2 ) (depicted in Figure 6.5) as a proxy for such a filter. The localization ("spread") in kz is approximately s = (s 2 — S ] ) ; the localization ("length") in z is approximately / = (fa — h)- These designations are empirical, and not based on any calculation, because the moments of the inverse kernel approximations to be compared are very different. In the comparison scheme, the localization is also not designated with absolutes as above; instead inverse kernels are referred to only as being more or less localized than one another in any domain. Chapter 6. Multiresolution Wave Field Continuation 138 T 1 1 1 1 1 1 1 r Figure 6.5: The "localization" of a function and its Fourier transform illustrated using a Gaussian in (a) the z domain, and (b) the kz domain. The space and wavenumber vectors are normalized such that they lie between (0,1) and (—1/2,1/2) respectively. Localization in z is essentially the distance I2 — h over z for which the filter is non-negligible compared to its maximum amplitude. Localization in kz is the spread S2 — si over kz of the filter for which the amplitude is again non-negligible. This definition is chosen because of the suitability of "length" and "spread" in comparison, and because these properties can be linked to robustness to various types of data flaw. This choice arises because it can be linked to robustness of model estimate to various common data flaws. The link may be explained conceptually based on a consideration of the "mechanics of convolution". First consider the space domain. For inverse filtering, a model point is constructed via a weighted sum of the data, where the weights are provided by the inverse kernel function which is centred on the datum coincident with the model point. Hence if an inverse kernel approximation K~1(z) is characterized by length £7, then Z7 is implicitly a measure of the reliance of a model point on distant data points. In the limit Z7 —> 0, i.e. the non-regularized and singular inverse filter, clearly the operator relies only on data information precisely at the model point. Therefore increased localization in the space domain results in decreased reliance on remote data. Second consider the situation in the wavenumber domain. The convolution theorem determines the simple relationship between model, data and inverse filter here: every model point in wavenumber space is constructed by the product of the data and the inverse filter. If the inverse kernel approximation K~l{kz) is characterized by spread s 7, then s 7 is an indicator of the size of the window of permitted singular values, or Fourier coefficients of the reconstruction. Therefore localization in the wavenumber domain is a direct measure of the resolution (via highest wavenumber) of the constructed model. In the following section the localization properties of the various inverse kernels, equally regularized as per the discussion of the previous section, are investigated in this light. 6.5.4 Compar ison of Regular ized Inverse Kerne ls In order to fit the problem of downward continuation of viscoacoustic wave fields into the framework of previous sections, I associate the general inverse kernel function of equation (6.22) with the inverse e~~%k*z Chapter 6. Multiresolution Wave Field Continuation 139 of equation (6.5). The bases for expansion and the weighting terms are varied, permitting both theoretical comparison vis d vis localization in wavenumber and space domains, and empirical comparison vis d vis model estimate from the synthetic data sequences. In each instance of comparison across methods, regularization parameters are chosen such that data misfits are equal. Figure 6.6 illustrates three examples of the inverse filter for downward continuation, each regularized in a different way, such that K: -1 7 K, -1 7 Isq K: (6.33) The filters are displayed here in the Fourier domain. The top and bottom rows of Figure 6.6 represent the LSQ- and TSVD-based regularization methods respectively. Because the Fourier basis has been used to expand the inverse kernel function in series for these cases, one gets an explicit look at how the weighting for each method differs. The simple, sharp truncation of the TSVD weights are very clear in the bottom row, while the graduating effect of the cutoff using the LSQ weights is exemplified in the top row. LSQ clearly has a larger non-zero spread of energy in the wavenumber domain: sisq > Stsvd-high regularization med. regularization x 10 to c o Q-e co CC -1 § • • X 7 2 * a CL < CO rr § .. o I Q . < CO a J > CO {2.i r-"" O a. low regularization 0 co (rad/s) 0.5 -0.5 0.5 -0.5 0 co (rad/s) 10 x 10 0.5 -0.5 0 co (rad/s) 0.5 Figure 6.6: Inverse kernel approximations in the kz domain: three regularization levels for each of three regularization meth-ods. Rows correspond to method: (a), (b), (c) are LSQ; (d), (e), (f) are MRWR; and (g), (h), (i) are TSVD. Columns correspond to regularization level: (a), (d), (g) are low; (b), (e), (h) are mid-range; and (c), (f), (i) are high. These designations are qualitative, but each column's three filters are equivalent: K^SQ — K-TSVD ~ ^MV The kz axis is normalized to lie between (—1/2,1/2). TSVD and the wavelet method maintain a higher degree of localization in kz than LSQ over all regularization levels; the difference is most striking at high regularization. Chapter 6. Multiresolution Wave Field Continuation 140 Columns in Figure 6.6 correspond to filters with equivalent regularization levels in the sense of equation (6.30). Regularization levels themselves are qualitatively classified as "low", "midrange", and "high". Here low regularization might be used on essentially noise-free data, and high regularization for lower signal to noise ratios. The contrast in inverse filters at equivalent (high) regularization levels, seen in Figures 6.6a, 6.6d, and 6.6g is notable. Markedly higher wavenumbers are permitted "through" the LSQ inverse filter as compared to both TSVD and the wavelet method. The wavelet-based inverse filter has localization properties more clearly aligned with those of TSVD, but differing in the use of a more smooth roll-off, compared to the sharp discontinuity of TSVD. In general, higher levels of regularization correspond to less spread in inverse filters in the kz domain: Shigh > Smid > siow. This is visible in Figure 6.6 viewed column by column. Within a regularization level, however (meaning within each column of Figure 6.6), further differences in localization are visible, some limiting spread even further for a certain regularization level. TSVD in particular, but also the wavelet method do this relative to LSQ for all three regularization levels: stSvd,smr < sisq. Because this is most notable in the high regularization case, one expects to note related differences in model estimate in instances of high noise. In the next section this aspect of the filters is associated with robustness in the presence of high noise levels. Consider the same regularized filters in the conjugate (z) domain. Following the form of Figure 6.6, the inverse filters K~ x(z) are displayed in Figure 6.7. Two factors govern length or localization in the filters in this domain. First, the filters have an increase in spread as regularization level goes from low to high, i.e. moving column by column from left to right. But more important in this research are the differences noted within equivalently regularized filters, within columns of Figure 6.7, where the equivalence of equation (6.30) holds. Here the similarities noted in the kz domain between regularization approaches switch. Especially in low regularizations, the localization properties of LSQ and the wavelet method are now much more alike, whereas the TSVD-regularized filter has non-zero energy extending the length of the equivalent data record: hsqi L < hsvd- This aspect of the TSVD filter is simply due to the discontinuous cutoff in the wavenumber domain: the space domain filter "rings" in z. The roll-off, both of LSQ and the wavelet method, explain the more highly-localized inverse kernel function approximations. These space-domain properties of the kernels are more marked at medium range and low regularization levels. One expects, therefore, to note differences in model estimates due to z-domain behaviour in low noise cases. Furthermore, the localization of LSQ and the wavelet method in the space domain (opposite the lack thereof in TSVD) is expected to increase model quality in terms of the filter's reliance on remote data values. This is shown to be the case in the next section. Chapter 6. Multiresolution Wave Field Continuation 141 high regularization med. regularization X 10 low regularization 1 f 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 x10 5 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 t(5)^ t(s)^> Figure 6.7: Inverse kernel approximations in the pseudo-depth z domain: three regularization levels for each of three regu-larization methods. Rows correspond to method: (a), (b), (c) are LSQ; (d), (e), (f) are MRWR; and (g), (h), (i) are TSVD. Columns correspond to regularization level: (a), (d), (g) are low; (b), (e), (h) are mid-range; and (c), (f), (i) are high. These designations are qualitative, but each column's three filters are equivalent: K~^}.Q = Ky^VD = K^R. Especially in instances of mid-range and low regularization, LSQ and the wavelet method produce inverse kernel approximations with a much higher degree of localization than those of the TSVD method. 6.5.5 Compar ison of Const ruc ted Mode ls In the following, the regularized inverse filters are used to construct model estimates from noisy data. The results are inspected for confirmation of the behaviour predicted via the localization properties of the inverse filters. Model estimates vary widely in quality, not merely noise-level to noise-level, but realization to realization within a noise-level. For this reason, the quality of "robustness" to noise is ascribed to the methods, meaning that a method produces not just good model estimates but consistently good estimates over a reasonable number of trial realizations. The results of this section are presented in groups of four trials, such that the behaviour across a representative set of noise realizations is visible. The models are Ricker wavelets, corresponding to unpropagated wave fields (at depth). These model wave fields are propagated according to equation (6.5), for ID and using a (3 that corresponds to a constant Q medium (i.e. involving absorption and dispersion). The synthetic experiment uses a time interval At = 0.003s, wavespeed CQ = 1500m/s, a propagation distance of z = 1500m, and a low Q = 15. Figures 6.8 - 6.10 illustrate the results of model estimates from the data contaminated with stationary Chapter 6. Multiresolution Wave Field Continuation 142 noise of high variance. In each figure, four sets of results are presented which differ only in the realization of the added noise. In each set, the top panel is the noisy attenuated data, and the bottom panel contains the reconstructed wave field (solid) against the synthetic model (dotted). Figures 6.8a - d are models recovered via the LSQ regularized filter; figures 6.9a - d are models recovered via MRWR, figures 6.10a - d are those recovered via TSVD. Figure 6.8: Data and model-estimates for four (a - d) realizations of Gaussian noise. These examples are regularized using LSQ weights. Chapter 6. Multiresolution Wave Field Continuation 143 Chapter 6. Multiresolution Wave Field Continuation 144 Figure 6.10: Data and model-estimates for four (a - d) TSVD weights. realizations of Gaussian noise. These examples are regularized using Chapter 6. Multiresolution Wave Field Continuation 145 Although the noise level is very high, both TSVD and the wavelet method produce model estimates which rise above the ambient noise consistently across realization; whereas LSQ produces results are less consistent from realization to realization. (It is worth noting that the regularization parameter (3 used in this case, like the other two, was varied from the value achieving $p « N to test for more consistent results. The results of Figure 6.8 were not improved in this way.) In observing the inverse kernel function K~ 1(kz), in the wavenumber domain, one is inspecting the singular values of the inverse filtering operator, which completely describe its behaviour. One is justified, therefore, in looking to them to explain the results. The key difference in K^sld(kz) vs. K^(kz), and Kmr(kz) vs. KTlq(kz) is in their localization properties, the spread of non-zero energy (in kz). LSQ retains markedly higher wavenumbers than either TSVD or the wavelet method; the latter two have greater localization: Stsvd,Smr < sisq (see Figure 6.5). The noise in these high wavenumbers is permitted to "pass" through the LSQ inverse filter, where it may be seen to worsen the model estimate depending on the particular noise realization. The second data sequence is designed to test the use of local information in the conjugate domain (time t is used in these figures - depth z = coi/2 would also suffice). The data, containing remote pulse of noise, is operated on by the approximated inverse kernels corresponding to "low" regularization levels (Figure 6.7, right column). Figures 6.11 - 6.13 are organized similarly to Figures 6.8 - 6.10. Estimates are generated from each of four data sequences, where again the only difference amongst the data sequences is the realization of the noise. Here robustness of estimate is visible in both LSQ and the wavelet method from realization to realization, whereas TSVD produces inconsistent results, which are clearly extremely sensitive to the remote noise. This is again explainable given the previous localization arguments: the inverse kernel function of the TSVD approach constructs all model points using contributions from all of the data points, regardless of location in the record. This includes the regions of remote noise in the data sequences. LSQ and the wavelet method, in producing compact inverse kernels in z for low regularization levels, generate model estimates at any point using only local data values. These synthetic examples, simple as they are, serve to illustrate the impact on model estimates of the localization properties of the inverse filters. They confirm that equivalent regularization levels produce very different models, but that relative differences in these models may be predicted, to some degree, with an appropriate comparison of the effective inverse filters used. The wavelet expansion of the inverse kernel function exhibits an attractive balance in the consequent localization of the filter in both z and kz. Further, this balance may be linked to a robustness to some practical instances of data inaccuracy. Chapter 6. Multiresolution Wave Field Continuation 146 Figure 6.11: Data and model-estimates for four realizations of the remote noise "pulse" using the LSQ-regularized inverse kernel. Chapter 6. Multiresolution Wave Field Continuation 147 c o o i CO c o o 0 rr rr rr Q CO c o u—• o i_ -t—1 03 C o o 0 rr rr Q CM C g o < en c o o CD rr rr co D c o o i_ •«—» tt> c o o CD rr rr rr •i—• ca Q 0.02 -0.02 0.02 -0.02 t(s) t(s) Figure 6.12: Data and model-estimates for four realizations of the remote noise "pulse" using the MRWR-regularized inverse kernel. Chapter 6. Multiresolution Wave Field Continuation 148 Figure 6.13: Data and model-estimates for four realizations of the remote noise "pulse" using the TSVD-regularized inverse kernel. Chapter 6. Multiresolution Wave Field Continuation 149 6.6 Remarks The inversion of convolution-like operators is often regularized by using an approximation of the true inverse kernel function, which in its pure form is highly ill-conditioned. Any regularization scheme that amounts to the application of some stationary regularized inverse filter can be described as a two-part process, in which (i) a basis is chosen over which to expand the inverse kernel function, and (ii) a weight is prescribed for each term in the resulting series. In light of this, it is straightforward to write down a general expression for this inverse filter, and specify later to a particular "style" of regularization. In this chapter, truncated singular value decomposition, damped least-squares, and a wavelet strategy are each cast in this way. This formulation allows a convenient form of comparison to proceed in a maximally general way. First, inverse kernels are put on a "level playing field" for a general noisy data set by choosing regularization parameters such that data misfits are equal. Second, the localization of these filters is examined in both z and kz domains - in other words, the inverse filter is essentially given a position on the localization "continuum" defined by the Uncertainty Principle. Depending on the particulars of a data set, a method with greater localization in one or other domain may be beneficial - indeed, perhaps a balanced localization is most desirable. A downward continuation of viscoacoustic wave fields example is produced to exemplify these predictions. It is shown that strong localization of the inverse kernels in one domain delivers benefits in some scenarios of data inaccuracy, while leaving the method prey to other scenarios. The wavelet method is shown to strike a balance in localization for the approximated inverse filter (a reasonable result given the localization properties ascribed to these bases), which confers a robustness on the method in the chosen cases. 6.6.1 Issues in Wavelet Regularization Data misfit is achieved in the wavelet method through choice of minimum admitted scale. This scale corre-sponds to the "most compressed" version of the mother wavelet. Flexibility exists however in the choice of wavelet used, provided it is of compact support in the wavenumber domain. The Meyer wavelet, outstanding in this regard, is seen to increase in support in the wavenumber domain with decreasing scale; this lessens the flexibility of the regularization method, especially at lower scales. The integral parameter AQ cannot produce any desired data misfit, but only those corresponding to the addition of wavenumber-bands of data information. This issue should be addressed through the use of a wavelet packet as means to regularize the inverse filter, since it has the ability to more flexibly parse the time-frequency plane. Chapter 7. Conclusions 150 Chapter 7 C o n c l u s i o n s Phenomena of absorption and dispersion alter the character of the events of seismic data. This alteration, in spite of well-established and sometimes frankly brilliant theory, and field/laboratory work, is at best little agreed upon and at worst little understood. However, when a model describing the character and propa-gation of seismic waves becomes, by necessity, more complicated, it also introduces at least the possibility of extraction of more information regarding the subsurface. I prefer to see attenuation as an opportunity for such extraction. In this thesis I have, therefore, assumed one of two possible models for seismic dissipa-tion to be valid, and proceeded to investigate the resulting effects and opportunities these models present, with regard to some new methods of wave field processing and inversion. In particular I focus on inverse scattering theory, multiresolution analysis, and the characterization of singularities via regularity estimation. The results suggest that not only can these methods handle absorptive/dispersive behaviour, but they can be cast such that they actively involve themselves in Q estimation, compensation, forward modelling, and viscoacoustic inversion. During the course of the investigation I also have had occasion to develop some ideas behind possible inverse scattering methods for purely acoustic situations. These include the strategy of simultaneous imaging and inversion, and the development of the so-called inversion subseries, from the standpoint of what I have loosely referred to as "interevent communication". In spite of the number of somewhat disparate approaches to algorithm development presented here, all share some common conclusions. To a one, they offer increased value of output, whether this be the estimation of a further parameter (Q), the correctly imaged and inverted acoustic primary, or a stably reconstructed wave field. In return, they all demand input data of high quality (i.e. low noise, broad-band); however, most are shown to be forgiving to data which are imperfect (in these same senses). Methods based on amplitudes and (local) phase characteristics of the wave field, in contrast to methods more primarily based on travel-times only, are ambitious in this regard. In chapter 4 I discuss a method for imaging and inversion that uses the Born inverse as input; it is clear that the quality of that input is a high priority. The strategy, when faced with noisy, aperture-limited, bandlimited data must be to (i) understand what the higher order terms of the series expect of the Born approximation, and (ii) use whatever means are at hand to provide this, given imperfect and incomplete experiments. I have demonstrated the use of spectral extrapolation as an example; strategies will have to be extended and altered for multidimensional, multi-parameter versions of this and other inverse scattering based methods. In chapter 5, to offer another example, I develop an inversion for Q contrasts based on reflection coefficients; this differs from most Q estimation methods, which base themselves on measuring spectral changes due to propagation effects rather than reflection/transmission effects (the Q-Lipschitz exponent map of chapter 3 is an example of a method based on propagation effects). Although I have shown that, with certain reasonable assumptions, this method is not sensitive to bandlimited data, the method is based on subtle changes of the reflection coefficient with frequency, and will certainly require very low-noise data. The point is to develop methods that can return better inversion/estimation results if given better data. 7.1 Summary The Born series, or forward scattering series, as applied to wave theory, is an expression for the wave field due to a given source and medium; it is concerned with forward modelling. It is based on perturbation theory, i.e. it involves only explicit solutions of the wave equation for much simpler, reference, media. The true wave field is an expansion in series about these reference solutions. Physically the series terms have interpretations as propagations from point to point in the reference medium, with true, non-reference behaviour occurring only as a consequence of the cumulative, nonlinear, interplay of these propagations. I show, using ID examples, Chapter 7. Conclusions 151 that an attenuated wave field may be synthesized using acoustic reference media. This is compelling in the sense that the correct phase and amplitude distortions are produced via a formalism in which no attenuating propagation ever occurs. Local signal regularity, characterized via Lipschitz or Holder exponents, is a generalized measure of the local differentiability of a function. As such it has the ability to characterize edges or discontinuities or singularities in a mathematically fundamental way, by their order. Intuitively the lengthening of the tail of a causally attenuated echo suggests that, in its capacity as a transient, edge-like entity, attenuation alters the regularity of measured seismic data. I use arguments based on existing Q models to develop a mapping between local Lipschitz exponents, computable from the wavelet transform of a trace, and local Q values. I consider the inverse scattering series and its application to the imaging and inversion of the primaries of seismic reflection data, in an acoustic (non-attenuating) framework. Again using ID normal incidence solutions for the constant density acoustic wave equation, I consider the consequences to the computation of imaging/inversion solutions resulting from certain approximations (i.e. dropping of terms). I deduce a simple form for the n'th term in a series which corrects the amplitude and location of the Born inverse, and compute numeric examples for a range of test models that differ in contrast and complexity. I use this computable form, which simultaneously carries out tasks of imaging and inversion, as a platform upon which to test the sensitivity of the series to incoherent noise, and to test the use of gap-filling techniques to compensate for bandlimitation. I speculate, by considering the simultaneous imaging and inversion formula in its capacity as an operator which acts on the Born approximation as input, on what basic properties it has which might transfer, conceptually and practically, to a multidimensional form. I next consider application of inverse scattering to the inversion of a viscoacoustic wave field. The first part of the chapter involves casting a ID normal incidence problem such that a linearized (Born) inversion for simultaneous contrasts in Q and wavespeed may be carried out. This is first done for the simple case of a single reflection from a single interface; it is then extended so that it permits multiple interfaces (i.e. interval Q and wavespeed inversion) to be treated. In both cases overdetermined problems involving multiple frequency components of the data are solved; the framework is made tractable by assuming a spatial form for the contrasts dictated by the data. I then alter the interval wavespeed-Q inversion to produce a bootstrap inversion scheme; this makes the most of the linear inversion as a standalone algorithm for viscoacoustic inversion. The second part of the chapter involves going "beyond Born", and utilizing the higher order terms of the so-called inversion subseries of the inverse scattering series. I investigate the use of inter-event information implied by the terms of the inversion subseries, and suggest that terms involving information from all events simultaneously dominate over those from local/individual events. I pursue this numerically for some simple viscoacoustic data sets. Finally I consider the problem of Q compensation, the re-institution of the resolution that is naturally suppressed in viscoacoustic wave propagation. I make some broad comments on the resemblance of the upward continuation operator for viscoacoustic media to the scale function of multiresolution theory - both act to reduce the finer scales of a signal. I then review the idea of a multiresolution wave field reconstruction, that uses a wavelet basis to restore resolution in a manner that mirrors the scale-operator like loss (while naturally stabilizing the process). I compare the approach against two other deconvolution regularizations. The wavelet method is seen to balance the localization properties of the other inverse filtering schemes in the space/wavenumber domains. It appears to be robust to a variety of forms of data inaccuracies to which other methods more easily fall prey. 7.2 Continuing Work The two main steps forward for the methods developed herein are (i) extension to multiple dimensions, and (ii) testing on field data. The Q estimation technique of chapter 3, based on Lipschitz exponents, should be tested on some high-quality VSP data, or in some other experimental milieu where validation of the results is reasonably straight-forward. Subsequent discussions with Felix Herrmann have suggested that different expressions (in the sense of bandlimited nearly-constant Q models, and time-domain approximations) may yield very different predic-tions about the Q-regularity dependence; as also may different choices for interpretation of small and large scale wavelet coefficients (i.e. different "signal models"). Further investigation of more general expressions Chapter 7. Conclusions 152 for constant Q dispersion models, and synthetic tests thereof will be important augmentation to these ideas. Similarly simple reflection data approximating zero-offset should be used to test the wavespeed/CJ inversion of chapter 5. The simultaneous imaging/inversion (chapter 4), and the high-order terms of the inversion subseries (chapter 5), are to be developed in higher dimensions; a pre-stack ID configuration is a natural starting point. The low order terms of the inversion subseries are being closely investigated for ID prestack multi-parameter cases at present (Zhang and Weglein, 2003). For bandlimited data, the generalization of the work of chapter 4 to higher dimensions will also require extension, or, if necessary, re-formulation of the principles of bandwidth extension. Extending the multi-parameter viscoacoustic inversion to higher terms is also a priority. At present it appears that the casting of the scattering potential in chapter 5 will require alteration for this to proceed. Finally, the most interesting prediction of the forward scattering series investigation (chapter 2) has yet to be realized. The imaging subseries, or its multi-parameter counterpart, must involve itself with Q compensation. Given the stabilization required for the acoustic imaging/inversion, it must be anticipated that Q compensation within the series will require, again, either pristine data, or a carefully estimated Born approximation for input. The true linear viscoacoustic inversion (i.e. non-bootstrap) results of chapter 5 are the starting point for this investigation. Bibliography 153 B i b l i o g r a p h y Aki, K. and P. G. Richards. Quantitative Seismology. Second edition. USA: University Science Books, 2002. Azimi, S. A., A. V. Kalanin, V. V. Kalanin, and B. L. Pivovarov. "Impulse and Transient Characteristics of Media with Linear and Quadratic Absorption Laws." Izvestiya, Physics of the Solid Earth (1968): 88-93. Berkhout, A. J. Seismic Migration: Imaging of Acoustic Energy by Wave Field Extrapolation. First edition. Amsterdam: Elsevier Scientific Publishing Co., 1980. Bickel, S. H. "Similarity and the inverse Q filter: The Pareto-Levy stretch." Geophysics 58 (1993): 1629. Born, M . and E. Wolf. Principles of Optics. Seventh edition. Cambridge: Cambridge University Press, 1999. Carvalho, P. M . "Free-surface Multiple Elimination Method Based on Nonlinear Inversion of Seismic Data." PhD Thesis, Universidade Federal da Bahia, Brazil (in Portuguese) (1992). Cerveny, V. Seismic Ray Theory. Second edition. Cambridge: Cambridge University Press, 2001. Chui, C. K. An Introduction to Wavelets. First edition. New York: Academic Press, 1992. Chui, C. K. Wavelets: A Tutorial in Theory and Application. First edition. New York: Academic Press, 1992. Clarke, G. K. C. "Time-Varying Deconvolution Filters." Geophyics 33 (1968): 936. Clayton, R. W. and R. H. Stolt. "A Born-WKBJ Inversion Method for Acoustic Reflection Data." Geophyics 46 (1981): 1559. Cohen, J. K. and N. Bleistein. "An Inverse Method for Determining Small Variations in Propagation Speed." SIAM Journal of Applied Math 32 (1977): 784-799. Cohen, L. Time-Frequency Analysis. First edition. New Jersey: Prentice Hall Signal Processing Series, 1995. Dasgupta, R. and R. A. Clark. "Estimation of Q From Surface Seismic Reflection Data." Geophysics 63 (1998): 2120-2128. Daubechies, I. Ten Lectures on Wavelets. First edition. Philadelphia: Society for Industrial and Applied Mathematics, 1992. DeSanto, J. A. Scalar Wave Theory. First edition. Berlin: Springer-Verlab, 1993. Futterman, W. I. "Dispersive Body Waves." Journal of Geophysical Research 67 (1962): 5279-5291. Gazdag, J. "Wave Equation Migration with the Phase Shift Method." Geophysics 43 (1978): 1342. Hargreaves, N . D. and A. J. Calvert. "Inverse Q-filtering By Fourier Transform." Geophysics 56 (1991): 519-527. Herrmann, F. J. "Monoscale Analysis of Edges/Reflectors Using Fractional Differentiations/Integrations." 69th Annual Meeting of the Society of Exploration Geophysicists, Expanded Abstracts (1999). Bibliography 154 Innanen, K. A. "Local Signal Regularity as a Framework for Q Estimation." 12nd Annual Meeting of the Society of Exploration Geophysicists, Expanded Abstracts (2002). Innanen, K. A. "Local Signal Regularity and Lipschitz Exponents as a Means to Estimate Q." Journal of Seismic Exploration 12 (2003): 53-74. Johnston, D. H. "Attenuation: a State of the Art Summary." In: Seismic Wave Attenuation, Toksdz and Johnston, Eds. (1979): 123-135. Jost, R. and W. Kohn. "Construction of a Potential from a Phase Shift." Physical Review 87 (1952): 977-992. Kjartansson, E. "Constant-Q Wave Propagation and Attenuation." Journal of Geophysical Research 84 (1979): 4737-4748. Knopoff, L. "Q." Reviews of Geophysics 2 (1964): 625-660. Kramers, H. A. "La Diffusion de la Lumieres par les Atomes." Atti Congr. Intern. Fisica 2 (1927): 545-557. Kronig, R. "On the Theory of Dispersion of X-rays." Journal of the Optical Society of America 12 (1927): 547-557. Mallat, S. G. "A Theory for Multiresolution Signal Decomposition: The Wavelet Representation." IEEE Transactions on Pattern Analysis and Machine Intelligence 11 (1989): 674-693. Mallat, S. G. A Wavelet Tour of Signal Processing. Second edition. San Diego: Academic Press, 1998. Mallat, S. G. and W. L. Hwang. "Singularity Detection and Processing With Wavelets." IEEE Transactions on Information Theory 38 (1992): 607-643. Mallat, S. G. and S. Zhong. "Characterization of Signals From Multiscale Edges." IEEE Transactions on Pattern Analysis and Machine Intelligence 14 (1992): 710-732. Margrave, G. F. "Theory of nonstationary linear filtering in the Fourier domain with application to time-variant filtering." Geophysics 63 (1997): 244. Matson, K. H. "The Relationship Between Scattering Theory and the Primaries and Multiples of Reflection Seismic Data." Journal of Seismic Exploration 5 (1996): 63-78. Mittet, R., R. Sollie, and K. Hostad. "Prestack Depth Migration with Compensation for Absorption and Dispersion." Geophysics 60 (1995): 1485-1494. Morlet, J., G. Arens, I. Fourgeau, and D. Giard. "Wave Propagation and Sampling Theory." Geophysics 47 (1982): 203-236. Morse, P. M . and H. Feshbach. Methods of Theoretical Physics. First edition. New York: McGraw Hill, 1953. Moses, H. E. "Calculation of Scattering Potential From Reflection Coefficients." Physical Review 102 (1956): 559-567. Nita, B., K. H. Matson, and A. B. Weglein. "Forward Scattering Series Seismic Events: High Frequency Approximations, Critical and Postcritical Reflections." In Preparation (2003). O'Doherty, R. F. and N . A. Anstey. "Reflections on Amplitudes." Geophysical Prospecting 19 (1971): 430-458. Oldenburg, D. W. "An Introduction to Linear Inverse Theory." IEEE Transactions on Geoscience and Remote Sensing GE-22 (1984): 665-674. Bibliography 155 Oldenburg, D. W., T. Scheuer, and S. Levy. "Recovery of the Acoustic Impedance from Reflection Seismo-grams." Geophysics 48 (1983): 1318-1337. Parker, R. L. Geophysical Inverse Theory. First edition. Princeton University Press, 1994. Raz, S. "Direct Reconstruction of Velocity and Density Profiles from Scattered Field Data." Geophysics 46 (1981): 832. Razavy, M . "Determination of the Wave Velocity in an Inhomogeneous Medium from Reflection Data." Journal of the Acoustic Society of America 58 (1975): 956-963. Robinson, E. A. and S. Treitel. Geophysical signal analysis. First edition. Prentice-Hall Inc., 1980. Schoenberger, M . and F. K. Levin. "Apparent Attenuation Due to Intrabed Multiples." Geophysics 39 (1974): 278. Shaw, S. A., A. B. Weglein, D. J. Foster, K. H. Matson, and R. G. Keys. "Isolation of a Leading Order Depth Imaging Series and Analysis of its Convergence Properties." In Preparation (2003). Song, S. and K. A. Innanen. "Multiresolution Modelling and Wavefield Reconstruction in Attenuating Media." Geophysics 67 (2002): 1192-1201. Song, S., K. A. Innanen, and T. J. Ulrych. "Multiresolution Model Construction: Using Local Information." 70th Annual Meeting of the Society of Exploration Geophysicists, Expanded Abstracts (2000). Song, S., R. Zhang, and T. J. Ulrych. "High Resolution Wavefield Reconstruction in Seismic Migration for Viscoelastic Media." Journal of Seismic Exploration (1999). Spencer, T. W., J. R. Sonnad, and T. M . Butler. "Seismic Q - Stratigraphy or Dissipation." Geophysics 47 (1982): 16-24. Stolt, R. H. and A. K. Benson. Migration: Theory and Practice. First edition. London: Geophysical Press, 1986. Stolt, R. H. and B. Jacobs. "An Approach to the Inverse Seismic Problem." Stanford Exploration Project Report 25 (1981): 121-132. Stolt, R. H. and A. B. Weglein. "Migration and Inversion of Seismic Data." Geophysics 50 (1985): 2458-2472. Strick, E. "A Predicted Pedestal Effect for Pulse Propagation in Constant-Q Solids." Geophysics 35 (1970): 387-403. Tonn, R. "The Determination of the Seismic Quality Factor Q from VSP Data: a Comparison of Different Computational Methods." Geophysical Prospecting 39 (1991): 1-27. Ulrych, T. J. "Minimum Relative Entropy and Inversion." In: Geophysical Inversion, SIAM, J. B. Bednar, L. R. Lines, R. H. Stolt, A. B. Weglein Eds. (1989): 158. Ulrych, T. J. and C. Walker. "On a Modified Algorithm for the Autoregressive Recovery of the Acoustic Impedance." Geophysics 48 (1984): 2190-2192. Varela, C. L., A. L. R. Rosa, and T. J. Ulrych. "Modeling of Attenuation and Dispersion." Geophysics 58 (1993): 1167-1173. Walker, C. and T. J. Ulrych. "Autoregressive Recovery of the Acoustic Impedance." Geophysics 48 (1983): 1338-1350. Ware, J. A. and K. Aki. "Continuous and Discrete Inverse Scattering Problems in a Stratified Elastic Medium I. Plane Waves at Normal Incidence." Journal of the Acoustic Society of America 45 (1968): 911-921. Bibliography 156 Weglein, A. B. "The Inverse Scattering Concept and its Seismic Application." In: Dev. in Geophys. Explor. Meth., Fitch, A. A. Ed. 6 (1985): 111-138. Weglein, A. B., F. V. Araujo, P. M . Carvalho, R. H. Stolt, K. H. Matson, R. Coates, D. J. Foster, S. A. Shaw, and H. Zhang. "Topical Review: Inverse-scattering Series and Seismic Exploration." Inverse Problems, to appear (2003). Weglein, A. B., W. E. Boyse, and J. E. Anderson. "Obtaining Three-dimensional Velocity Information Directly from Reflection Seismic Data: An Inverse Scattering Formalism." Geophysics 46 (1981): 1116-1120. Weglein, A. B., D. J. Foster, K. H. Matson, S. A. Shaw, P. M . Carvalho, and D. Corrigan. "Predicting the Correct Spatial Location of the Reflectors without Knowing or Determining the Precise Medium and Wave Velocity: Initial Concept, and Analytic and Numerical Example." Journal of Seismic Exploration 10 (2002): 367-382. Weglein, A. B., F. A. Gasparotto, P. M . Carvalho, and R. H. Stolt. "An Inverse Scattering Series Method for Attenuating Multiples in Seismic Reflection Data." Geophysics 62 (1997): 1975-1989. Weglein, A. B., K. H. Matson, D. J. Foster, P. M . Carvalho, D. Corrigan, and S. A. Shaw. "Imaging and Inversion at Depth Without a Velocity Model." 70th Annual Meeting of the Society of Exploration Geophysicists, Calgary, Alberta, Expanded Abstracts (2000). Weglein, A. B. and R. H. Stolt. "Migration-Inversion Revisited." The Leading Edge Aug (1999): 950. White, J. E. Underground Sound: Application of Seismic Waves. First edition. Amsterdam: Elsevier, 1983. Wiggins, R. A. and S. P. Miller. "New Noise Reduction Techniques Applied to Long Period Oscillations of the Alaskan Earthquake." BSSA 62 (1972): 417-479. Witkin, A. P. "Scale Space Filtering." Proc. Int. Joint Conf. Artificial Intell. (1983): 1019-1022. Zhang, C. and T. J. Ulrych. "Estimation of Quality Factors from CMP Records." Geophysics 67 (2002): 1542. Zhang, H. and A. B. Weglein. "Target Identification Using the Inverse Scattering Series; Inversion of Large-Contrast, Variable Velocity and Density Acoustic Media." In Preparation (2003). Appendix A. Inverse Scattering Terms 157 Appendix A I n v e r s e S c a t t e r i n g T e r m s I show that the chosen terms from the fourth order inversion subseries of chapter 4 (for ID constant density acoustic imaging and wavespeed inversion) sum to nil. In the terminology of that chapter, these components are hu + i i2i + hn + lis + hi + hi- (A.l) I again make liberal use of the relationships listed in the definitions section of chapter 4. As in the derivations of that chapter, the operators H and the derivatives, that appear here, arise after substitution of the form of the Green's functions into the terms of the inverse scattering series. They are due, respectively, to the alterations of the integration limits and the appearance of powers of the wavenumber. To start, 1 d 2 Iii2 = ~YQ^2 [«2H{aiH{ai}} + axH {ai} H {a2} + a{H {aiH {ai2}}}, /i2i = -JQ-^2 la^H ia2n + A*H M N M + a * n ia2H ' ( A ' 2 ) 7 2 1 1 = ~ i ^ a % [ a i H { a i n { a 2 } } + a i H { a i } n { a 2 } + a 2 U { a i H { a i } } ] ' so 1 d 2 hn + + hn =hv = ~^2 [2a*n ia^n M ) + 2a^H ia^H M (A 3) + 2axH {axH {a2}} + 2a{H {a2H {«i}} + a2H {a^H {c*i}]. Substituting appropriate versions of equation (4.42) into the above produces Appendix A. Inverse Scattering Terms 158 hv l_d?_ 16 dz2 j 2 TJ2 \ Apr ( Al fj2 1 K 2 ^ ~ ) n { a i U { a i } } + ~iz~u { a i } n r 2 dz 2 dz 2 {dz ( 16 dz 2 ndH„, +2—H dz ' 16 dz 2 dz \ dz 2 d>H 2H2 + 2dHdjPH + 2dH 2dJLH dz AH dz dz d_ dz dH dz 2 ^ H dz dH dH 2 dz dz 16 dz 2 „dH AH dz \2dz 2 16 dz 2 K2CP_ 4 dz 2 K2f_ A dz 3 K2 d4H 4 d (dH dz V dz dH dz H-(A.4) 16 dz 4 The terms I3\ + h3 are of the same form as the secondary terms of the third order case, i.e. equation (4.66): ^31 + ^13 1 d_ 2 dz 2 dz 2 3K3 d2 2 dz 2 3K3 d2 T'dz 2 3K3 d2 2 dz 2 ^ H { a 3 } + a 3 H dz II dz2 + H 3d2H dz 2 dH * dz I dz \ dz dz ) dz \ dz 3K3 d4H 4 + 3K3 d2 8 dz 4 2 dz 2 dz dz (A.5) Similarly compute I22: Appendix A. Inverse Scattering Terms 159 '22 1 d ~2dz ld_ ~2dz Kjd_ 2 dz K 2^-K2dz* [a2H {a2}} d 2H 2 dH 2 dz 2 dz d 2H 2 dz 2 (A.6) Summing equations (A.5) and (A.6), we have dH dz H hi + hz + I; However, _ 3JY3 d 4H 4 3K3 d 2 2 ~ 8 + 3 K 3 d 4H 4 8 dz 4 + 2 dz 2 3 dB_ dz -H ^ K s - K 2 ) ^ d 2 dz 2 dH dz dz 2 2 dH dz -H -H (A.7) :Ko - K i 1 16 1 6 = ° ' (A.6 / 3 1 + I: 13 '22 = — " 3 X 3 d 4H 4 8 dz 4 Equations (A.4) and (A.9) now contain the totality of the fourth order secondary terms; these are and the result is added to the primary term (due to G O V I G O V I G Q V I G O V I ^ O ) to produce summing these secondary terms produces hv + hi +1 13 '22 d 4H 4 ' dz 4 K2 3K3 \ 16 8 ) d 4H 4 ' dz 4 1 JL_ "64 + 64 0. (A.9) summed, ot\. But (A.10) 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0052683/manifest

Comment

Related Items