in co h eren t n o ise su p p ressio n a n d d eco n v o ... · !c v ish al k u m ar 2009. abs...

Incoherent noise suppression and

deconvolution using curvelet-domain

sparsityby

Vishal Kumar

B.Sc., Indian Institute of Technology, Kharagpur, 2006

A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

Master of Science

in

The Faculty of Graduate Studies

(Geophysics)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

June, 2009

c! Vishal Kumar 2009

Abstract

Curvelets are a recently introduced transform domain that belongs toa family of multiscale and also multidirectional data expansions. As such,curvelets can be applied to resolution of the issues of complicated seismicwavefronts. We make use of this multiscale, multidirectional and hencesparsifying ability of the curvelet transform to suppress incoherent noisefrom crustal data where the signal-to-noise ratio is low and to develop animproved deconvolution procedure. Incoherent noise present in seismic re-flection data corrupts the quality of the signal and can often lead to misin-terpretation. The curvelet domain lends itself particularly well for denoisingbecause coherent seismic energy maps to a relatively small number of sig-nificant curvelet coe!cients while incoherent energy is spread more or lessevenly amongst all curvelet coe!cients. Following standard processing ofcrustal reflection data, we apply our curvelet denoising algorithm to deepreflection data. In terms of enhancing the coherent energy and removingincoherent noise, curvelets perform better than the F-X prediction method.We also use the curvelet transform to exploit the continuity along reflectorsfor cases in which the assumption of spiky reflectivity may not hold. We showthat such type of seismic reflectivity is sparse in the curvelet-domain. Thiscurvelet-domain compression of reflectivity opens new perspectives towardssolving classical problems in seismic processing, including the deconvolu-tion problem. We present a formulation that seeks curvelet-domain spar-sity for non-spiky reflectivity. Comparing the results with those obtainedfrom sparse spike deconvolution, curvelets perform better than the latter byrecovering the frequency components, which get degraded by convolutionoperator and noise.

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Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Theme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Theoretical background . . . . . . . . . . . . . . . . . . . . 2

1.4.1 Sparsity-promoting inversion background . . . . . . . 21.4.2 Sparsifying transforms . . . . . . . . . . . . . . . . . 41.4.3 Sparse representation of seismic data . . . . . . . . . 4

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Incoherent noise suppression with curvelet-domain sparsity-promotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Elementwise hard and soft thresholding . . . . . . . . 132.2.2 One-norm . . . . . . . . . . . . . . . . . . . . . . . . 14

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Table of Contents

2.3 Parameter selection for synthetic data . . . . . . . . . . . . . 152.4 Application to synthetic data . . . . . . . . . . . . . . . . . . 162.5 Parameter selection for real data . . . . . . . . . . . . . . . . 212.6 Application to deep reflection data . . . . . . . . . . . . . . . 212.7 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 23

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Deconvolution with curvelet-domain sparsity . . . . . . . . 303.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 313.3 Parameter selection . . . . . . . . . . . . . . . . . . . . . . . 323.4 Application and results . . . . . . . . . . . . . . . . . . . . . 32

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1 Curvelet-domain sparsity . . . . . . . . . . . . . . . . . . . . 404.2 Application to incoherent noise suppression . . . . . . . . . . 404.3 Application to deconvolution . . . . . . . . . . . . . . . . . . 404.4 Open and future research . . . . . . . . . . . . . . . . . . . . 41

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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List of Tables

2.1 The cooling method with iterative soft thresholding to solveEq. 3.5. L is the number of inner iterations, K is the num-ber of outer iterations, k is the counter for outer iterations,!1 · · ·!K are the values of ! in decreasing order, other sym-bols are as explained in the text. . . . . . . . . . . . . . . . . 16

2.2 SNR comparison for pre-stack denoising . . . . . . . . . . . . 21

3.1 Pseudo code to solve the optimization problem shown in Eq. 4.4.K is maximum number of iterations, k is the counter for it-erations, other quantities are as stated in the text. . . . . . . 32

3.2 SNR comparison for deconvolution methods . . . . . . . . . . 34

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List of Figures

1.1 (a) A few curvelets in t-x, (b) Same curvelets in F-K domain.The colored arrows show the positions of some curvelets intime-space and their representation in the frequency-wavenumberdomain. Each curvelet function is spatially localized becauseits amplitude rapidly decays to zero outside a certain region.The curvelets are localized in a wedge in the F-K domain.The orientation of a curvelet in the physical domain is per-pendicular to its wedge in the F-K domain. (c) Curvelet (top)and F-K (bottom) decomposition of two dipping events (con-flicting dips). Note the localization of curvelet functions andglobal nature of F-K functions used to decompose the image.(Adapted from Neelamani et al. (2008)) . . . . . . . . . . . . 6

1.2 Discrete curvelet partitioning of the 2-D Fourier plane intosecond dyadic coronae and sub-partitioning of the coronaeinto angular wedges. The tiling corresponds to 5 scales, 8angles at the 2nd coarsest level and curvelets at the finest(fifth) scale. The angles double every other scale (Adaptedfrom Hennenfent and Herrmann (2006)). . . . . . . . . . . . . 7

1.3 Curvelet decomposition of a shot gather at di"erent frequency-band (scale) and angles (dip). Four scales are used for the de-composition. The centre (coarsest scale) shows the DC andlow frequency components of the shot gather. The 2nd coars-est scale has 16 angles. The number of angles is doubled to32 at the third scale and stays the same (32) for the fourthscale. Note the portions of shot gather captured at variousangles and scales. . . . . . . . . . . . . . . . . . . . . . . . . . 8

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List of Figures

1.4 Decay of coe!cients of synthetic models mentioned in thetext. The models are taken to various transform domains andthe coe!cients are sorted in their descending order. (a) Theshot gather, (b) Poststack model, (c) Decay rates of trans-form coe!cients for shot gather, (d) Decay rates of trans-form coe!cients for poststack image. The decay rate of co-e!cients is proportional to the sparsity of the model in thetransform domain. The curvelet coe!cients (solid pink) de-cay faster than wavelet coe!cients (dashed red) and Fouriercoe!cients (dashed blue). Hence curvelets are the most ap-propriate sparsifying transform for these two models. Sincecurvelet transform is redundant in nature; i.e., it producesmore coe!cients than the data size, the x-axis is chosen aspercentages of coe!cients rather than absolute number of co-e!cients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Partial reconstruction of the two models (Fig. 2.4(a) and Fig.2.4(b)) in di"erent transform domains. The model is takento the transform domain and reconstructed from 1 % of theamplitude-largest coe!cients. (a) Fourier reconstruction ofshot gather, (b) Fourier reconstruction of poststack image,(c) Wavelet reconstruction of shot gather, (c) Wavelet recon-struction of poststack image, (d) Curvelet reconstruction ofshot gather, (e) Curvelet reconstruction of poststack image.As can be observed, the curvelet reconstruction is the mostaccurate approximation in terms of resemblance of the recon-structed images with the true images. . . . . . . . . . . . . . 10

2.1 (a) A seismic signal; dashed lines show the threshold level(!), (b) Signal after hard thresholding (!=2), (c) Signal aftersoft thresholding (!=2). Both hard and soft thresholdingsets entries less than the threshold to zero. In addition, softthresholding shrinks all remaining entries by the threshold.This moves the signal toward zero. . . . . . . . . . . . . . . . 14

2.2 (a) Comparison of signal-to-noise ratio (SNR) vs. thresholdvalue for white noise for hard thresholding and soft thresh-olding of curvelet coe!cients, (b) Comparison of signal-to-noise ratio (SNR) vs. threshold value for colored noise forhard thresholding and soft thresholding of curvelet coe!-cients. The arrows point to the peak SNR values and thecorresponding threshold value (!) is chosen. . . . . . . . . . . 17

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List of Figures

2.3 (a) Noise-free data (model), (b) Model with white noise, (c)Model with colored noise. . . . . . . . . . . . . . . . . . . . . 18

2.4 Data corrupted with white noise: (a) One-norm approach,(b) Noise removed by one-norm approach, (c) Hard thresholdapproach, (d) Noise removed by hard threshold approach, (e)Soft threshold approach, (f) Noise removed by soft thresholdapproach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Data corrupted with colored noise: (a) One-norm approach,(b) Noise removed by one-norm approach, (c) Hard thresholdapproach, (d) Noise removed by hard threshold approach, (e)Soft threshold approach, (f) Noise removed by soft thresholdapproach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Tectonic map of the acquisition area and location of SNOR-CLE near vertical incidence (NVI) reflection Line 1. Sedi-ments of the Phanerozoic Western Canada Sedimentary Basinoverlie the Precambrian domains west of the long dashed line.Short dashed black lines show political boundaries. CS -Coronation Supergroup, SD - Sleepy Dragon, GSLsz - GreatSlave Lake shear zone, AB - Alberta, BC - British Columbia,NWT - Northwest Territories, YK - Yukon. Inset (top-left)shows location of map within Canada. Inset (green box)shows the area for the data used. . . . . . . . . . . . . . . . 22

2.7 Shot gather from Line 1 segment highlighted in Fig. 3.6. (a)Noisy data without AGC, (b) Noisy data with AGC, (c) Re-sult of applying curvelet denoising (inversion approach) onthe shot gather with AGC, (d) Noise removed by curvelet de-noising. (e) The flowchart with important processing steps forpre-stack denoising. The curvelet denoising is applied beforethe bandpass filtering. . . . . . . . . . . . . . . . . . . . . . . 25

2.8 (a) The original stacked seismic section contaminated withrandom noise. Arrows indicate some of the features discussedin the text. (b) The flowchart with important processing stepsfor poststack denoising. The curvelet denoising is appliedas an alternative to F-X prediction. (c) Result of applyingcurvelet denoising (inversion approach) on the same section.(d) Noise removed by curvelet denoising, (e) Result of apply-ing F-X prediction on the same section, (f) Noise removed byF-X prediction. . . . . . . . . . . . . . . . . . . . . . . . . . 26

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List of Figures

2.9 (a) Stacked section after pre-stack curvelet denoising, (b)Stacked section after conventional processing (without F-Xprediction), (c) Stacked section after post-stack curvelet de-noising. The poststack curvelet denoising is observed to bebetter than the other two. . . . . . . . . . . . . . . . . . . . . 27

3.1 (a) True reflectivity model, (b) Noisy data obtained by con-volving the reflectivity model with Ricker wavelet and addi-tion of noise ("=50), (c) Estimated reflectivity with curveletdeconvolution, (d) Estimated reflectivity with sparse spikedeconvolution. The box shows the region of interest that isenlarged in Fig. 4.2. . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 (a) Zoom-in plot of true reflectivity model, (b) Zoom-in plotof noisy data, (c) Zoom-in plot of estimated reflectivity withcurvelet deconvolution, (d) Zoom-in plot of estimated reflec-tivity with sparse spike deconvolution. Note the resemblanceof curvelet-estimated reflectivity with the true reflectivity. . . 36

3.3 (a) True reflectivity model, (b) Noisy data obtained by con-volving the reflectivity model with Ricker wavelet and addi-tion of noise ("=.01), (c) Estimated reflectivity with curveletdeconvolution, (d) Estimated reflectivity with sparse spike de-convolution. The box shows the region of interest (reservoirarea) for which enlarged versions of the sections are displayedin Fig. 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 (a) Zoom-in plot of true reflectivity model, (b) Zoom-in plotof noisy data, (c) Zoom-in plot of estimated reflectivity withcurvelet deconvolution, (d) Zoom-in plot of estimated reflec-tivity with sparse spike deconvolution. Note the resemblanceof curvelet-estimated reflectivity with the true reflectivity. . . 38

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Preface

The results in this thesis was prepared with Madagascar; a reproducibleresearch software package available at rsf.sf.net and Claritas; a standardpackage for seismic processing. The reproducibility allows direct access tothe experiments performed in this work allowing transparency and dissem-ination of knowledge. The programs required to reproduce the reportedresults are Madagascar programs written in C/C++, MatlabR!, or Python.

The results in this thesis are entirely reproducible with the exception ofreal data examples in Chapter 2.

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Acknowledgments

I would first like to thank my supervisors Felix Herrmann and RonClowes, for their support since my first days at UBC. Without their vi-sion and courage none of this (and much, much more) would have beenpossible.

I would also like to thank the SLIM team (both past and present) for theirencouragement, commitment and spirit. It has been an honor to be part ofthis dynamic group and to witness the amazing process of development thatit has undergone.

I would like to thank Jounada Oueity for his help and support with ourcombined research.

I am very fortunate to have Michael Bostock as part of my supervisorycommittee. The knowledge, insight and inspiration that he has providedthroughout my time at UBC have been invaluable.

I would like to thank Geoscience BC, Society of Exploration Geophysi-cists and Canadian Society of Exploration Geophysicists for providing an-nual scholarship.

Without the support and love of my parents and the rest of my fam-ily, I certainly would never have accomplished any of this. They have myunending thanks and love.

Finally, I would like to thank my wife Khushboo Kumari for her supportand understanding that enabled me to produce results at my work.

All the work for this thesis was financially supported by the NSERC Dis-covery Grants 22R81254 (FH) and 22R87707(RC) and CRD Grant DNOISE334810-05 (FH). It was carried out as part of the project with support,secured through ITF, from the following organizations: BG Group, BP,Chevron, ExxonMobil and Shell.

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To my family.

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Statement of Co-Authorship

A version of Chapter 2 will be submitted for publication. I preparedthe manuscript with inputs from Jounada Oueity, Ron Clowes and FelixHerrmann.

A version of Chapter 3 is a published extended abstract for which the ref-erence is: V. Kumar and F. J. Herrmann, 2008, Deconvolution with curvelet-domain sparsity, In Expanded Abstracts, Society of Exploration Geophysi-cists, Tulsa. The manuscript was jointly written with Felix Herrmann. Heled the more theoretical sections, and I led the more applied ones.

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Chapter 1

Introduction

Processing of seismic reflection data faces the challenge of noise contam-ination (Olhovich, 1964) and loss of frequency components (Yilmaz, 2001).The noise contamination and missing frequency in seismic reflection datacorrupt the quality of the signal and can often lead to misinterpretation.

1.1 Theme

The main theme of this thesis is a transform-based approach to theseismic denoising and deconvolution problems. Two key features motivateour approach, namely:

• High dimensionality: Seismic data are typically five dimensional:time, two spatial coordinates for the sources, and two spatial coordi-nates for the receivers (for a 3D survey).

• Strong geometrical structure: Seismic data are a spatial-temporalsampling of the reflected wavefield that correspond to di"erent inter-actions of the incident wavefield with inhomogeneities in the Earth’ssubsurface. The wavefields have wavefront-like structure with spatialcontinuity that needs to be exploited.

To make the most of these features, our approach uses the curvelet trans-form (Candes and Donoho, 2000), which is data-independent, multiscale,and multidirectional. The basic elements of this transform, the curvelets,are localized in the frequency domain and are of rapid decay in space. Spa-tial curvelets look like localized plane waves. Because of these properties,curvelets are very e!cient at representing curve-like events (e.g., wavefronts,seismic images). The transform concentrates the energy signal amongst afew significant coe!cients. In other words, only a few curvelets are neededto represent the complexity of real seismic data. We use this sparsity tosolve the problem of noise suppression (also referred to as denoising) andfrequency enhancement (also referred to as deconvolution).

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Chapter 1. Introduction

1.2 Objectives

The objectives of this thesis are twofold:

• To utilize the curvelet domain as a sparse representation of seismicdata/models that exploits high dimensionality and spatial continuity.

• To formulate practical sparsity-promoting algorithms, which addressseismic denoising and deconvolution problems, using inversion theory.

1.3 Outline

In Chapter 1, we give a theoretical background regarding the sparsity-promoting inversion approach, various sparsifying transforms and curvelets.

In Chapter 2, we present an overview of the curvelet-transform-baseddenoising algorithms and their application to synthetic data. Three curvelet-based methods are tested and the most appropriate (with greater signal-to-noise ratio) is applied to the real deep crustal data. For the real data, resultsof our curvelet-based algorithm are compared with standard F-X prediction(also known as F-X deconvolution) results. The parameters are carefullyselected such that we don’t harm the coherent energy of the signal.

Chapter 3 deals with the application of the curvelet-based inversion ap-proach to the deconvolution problem and its application. The two-dimensionalstructure of reflectivity is exploited by curvelets. Results are compared withstandard trace-by-trace sparse spike deconvolution method.

Chapter 4 has conclusions and highlights the scope for future work.

1.4 Theoretical background

The fundamental concept behind the methods discussed in this thesis isbased on sparsity of seismic signal in a transform domain that facilitates theuse of sparsity-promoting methods. For this reason we need to understandthe underlying theory behind sparsity-promoting inversion.

1.4.1 Sparsity-promoting inversion background

From the perspective of inversion, observed data can be expressed as:

y = Km + n, (1.1)

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where y is the known data, K is the linearized modeling operator, m isthe unknown model and n is additive zero-centered Gaussian noise. Thenoise can be white (full-band) or colored (band-limited). Given K and y,our objective is to estimate m. In a transform domain, the model can beapproximated by the superposition of basic elements of the transform (basisor frames1) with appropriate weighting and thus the model can be writtenas:

m =!

i!I!i#i, (1.2)

where !i are the weighting of each basic elements #i. The transform coe!-cients !i are inner products of the basic elements and the model at a specificlocation and orientation. In equation form, this can be represented by:

m =!

i!I!m, #i"#i. (1.3)

In matrix-vector notation:

m = S†Sm = S†x, (1.4)

where S is the transform, S† is the inverse transform and x is the vector oftransform coe!cients. The columns of S† are the basic elements (discrete)and !i are the elements of x. For this work, we will seek a transformdomain that provides sparse representation of the model; i.e., the elementsof x should have few significant values (or fast decay). Assuming, we have asparsifying transform for the model; we can use transform domain sparsityas a priori information to formulate our inversion problem:

x = arg minx

||x||1 s.t. ||y #KS†x||2 $ $, (1.5)

where x represents the estimate of the transform coe!cient vector x, S† isthe inverse transform operator and % is proportional to the noise level. Bysolving Eq. 1.5, we find the sparsest set of transform coe!cients that explainthe data within the noise level. The final estimated model is given by m =S†x. For this work, we are interested in estimating two-dimensional mod-els and for this reason we need to understand the various two-dimensionalsparsifying transforms.

1As opposed to orthonormal basis, redundant frame expansions decompose a lengthK signal into a frame expansion with M > K elements. Consequently, the compositionmatrix (S†) is rectangular with the number of columns exceeding the number of rows.

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1.4.2 Sparsifying transforms

The Fourier Transform is by far the most important used in seismology.Fourier’s theory states that a given signal can be synthesized as a summationof sinusoidal waves of various amplitudes, frequencies and phases. The lim-itation of Fourier transform is that it needs a large number of coe!cients toaccount for any jump in the signal and hence not sparse for any signal thatis discontinuous. The solution to handling discontinuities (jumps) comes inthe form of wavelets that are local in nature and hence can approximate ajump with few wavelet coe!cients. Although, the wavelet transform is local(multiscale) in nature, it is poorly directional in higher dimensions. This isbecause the basis functions of 2D/3D wavelet are isotropic in nature (hasno sense of direction). Since seismic wavefields contain wavefronts movingin all directions, wavelets require a larger number of elements to representthe seismic data.

More recently, curvelets have been introduced to resolve the issues ofdirectionality of complicated wavefronts as they belong to a family of multi-scale, and also multidirectional data expansions. Curvelets obey a parabolicscaling principle with length2 % width, with the frequency of the oscillationsacross the curvelet increasing with increasing scales. Fig. 1.1 shows a fewcurvelets in the physical and Fourier domains. The comparison of curveletand F-K decomposition of two dipping events is also shown in Fig 1.1.The curvelets di"er from F-K in the sense that the curvelets are able todecompose conflicting dips into localized functions of di"erent frequency-wavenumber bands, while F-K functions are global in nature. The increas-ingly anisotropic nature of curvelets can be attributed to the tiling of thefrequency plane into a collection of angular wedges positioned at di"erentscales with the number of angular wedges doubling at every second scaleas seen in Fig. 1.2. These wedges in the frequency plane are strictly local-ized while the curvelets in the physical domain have a rapid spatial decay.Fig. 1.3 shows the decomposition of a shot gather at di"erent scales andangles in the curvelet domain (also referred to as a mosaic plot).

1.4.3 Sparse representation of seismic data

A transform’s ability to e!ciently represent a seismic signal can be judgedby examining the decay curve of magnitude-sorted coe!cients. In otherwords, the sparsifying transform is able to concentrate most of the signalinformation within a few large coe!cients. A decay curve with a rapid decayto near zero will represent a more parsimonious representation of the seismic

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image as the transform concentrates most of the signal information withinfew large coe!cients. Fig. 1.4 shows the decay of transform coe!cients fora synthetic shot gather and a synthetic poststack model. The shot gather(Fig. 1.4(a)) is taken from the work of Herrmann et al. (2007). The poststackmodel (Fig. 1.4(b)) is a subset of the Marmousi model (Versteeg, 2008),which is a function of depth and o"set. For the purpose of our experiments,we assume that the depth axis is the time axis with sampling rate of 4 ms.As we see from Fig. 1.4, the curvelet coe!cients have the fastest decay ofmagnitudes and hence the two models are relatively sparse in the curveletdomain.

Another way to measure sparsity is by partially reconstructing a sig-nal with few significant transform coe!cients. The signal is taken into thetransform domain, the transform coe!cients are sorted in descending orderof their amplitudes and then 1% of the largest amplitude sorted coe!cientsare kept and the remainder are discarded (put to zero). The transformcoe!cients are then inverse transformed to the physical domain and the re-constructed image is observed. The transform that closely approximates thesignal in the process of reconstruction is considered to be best-suited spar-sifying transform. Fig. 1.5 shows the partial reconstruction of two modelsin Fourier, wavelet and curvelet domain. As we observe, the curvelet re-construction shows a better approximation of the initial test models. Thus,both the test models are sparse in curvelet domain and provides motivationfor the development of e"ective sparsity-promoting algorithms in subsequentchapters of this thesis.

Considering the above-mentioned examples, we choose curvelets as oursparsifying transform. For this work, we choose the numerically tight FDCTvia wrapping as our curvelet transform (Candes et al., 2005). For this trans-form, the pseudo-inverse (denoted by the symbol †) equals the adjoint andwe have m = C†x = CHx implying CHC = I . Hence our inversion problem(Eq. 1.5) simplifies to:

P! :

"x = arg minx ||x||1 s.t. ||y #KCHx||2 $ $,

m = CH x.(1.6)

For the sake of simple notation, we will use A = KCH in the subsequentchapters of this thesis. Thus for a given modeling operator K, we can for-mulate our problem in the preceding manner for promoting curvelet-domainsparsity of models. The selection of parameters will be discussed in detailin subsequent chapters of this thesis.

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(a) (b)

(c)

Figure 1.1: (a) A few curvelets in t-x, (b) Same curvelets in F-K domain.The colored arrows show the positions of some curvelets in time-space andtheir representation in the frequency-wavenumber domain. Each curveletfunction is spatially localized because its amplitude rapidly decays to zerooutside a certain region. The curvelets are localized in a wedge in the F-Kdomain. The orientation of a curvelet in the physical domain is perpendic-ular to its wedge in the F-K domain. (c) Curvelet (top) and F-K (bottom)decomposition of two dipping events (conflicting dips). Note the localizationof curvelet functions and global nature of F-K functions used to decomposethe image. (Adapted from Neelamani et al. (2008))

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k1

k2angular

wedge2j

2j/2

Figure 1.2: Discrete curvelet partitioning of the 2-D Fourier plane into sec-ond dyadic coronae and sub-partitioning of the coronae into angular wedges.The tiling corresponds to 5 scales, 8 angles at the 2nd coarsest level andcurvelets at the finest (fifth) scale. The angles double every other scale(Adapted from Hennenfent and Herrmann (2006)).

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Figure 1.3: Curvelet decomposition of a shot gather at di"erent frequency-band (scale) and angles (dip). Four scales are used for the decomposition.The centre (coarsest scale) shows the DC and low frequency componentsof the shot gather. The 2nd coarsest scale has 16 angles. The number ofangles is doubled to 32 at the third scale and stays the same (32) for thefourth scale. Note the portions of shot gather captured at various anglesand scales.

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(a) (b)

(c) (d)

Figure 1.4: Decay of coe!cients of synthetic models mentioned in the text.The models are taken to various transform domains and the coe!cientsare sorted in their descending order. (a) The shot gather, (b) Poststackmodel, (c) Decay rates of transform coe!cients for shot gather, (d) Decayrates of transform coe!cients for poststack image. The decay rate of coe!-cients is proportional to the sparsity of the model in the transform domain.The curvelet coe!cients (solid pink) decay faster than wavelet coe!cients(dashed red) and Fourier coe!cients (dashed blue). Hence curvelets are themost appropriate sparsifying transform for these two models. Since curvelettransform is redundant in nature; i.e., it produces more coe!cients thanthe data size, the x-axis is chosen as percentages of coe!cients rather thanabsolute number of coe!cients.

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(a) (b)

(c) (d)

(e) (f)

Figure 1.5: Partial reconstruction of the two models (Fig. 2.4(a) and Fig.2.4(b)) in di"erent transform domains. The model is taken to the transformdomain and reconstructed from 1 % of the amplitude-largest coe!cients. (a)Fourier reconstruction of shot gather, (b) Fourier reconstruction of poststackimage, (c) Wavelet reconstruction of shot gather, (c) Wavelet reconstructionof poststack image, (d) Curvelet reconstruction of shot gather, (e) Curveletreconstruction of poststack image. As can be observed, the curvelet recon-struction is the most accurate approximation in terms of resemblance of thereconstructed images with the true images. 10

Bibliography

Candes, E. J., L. Demanet, D. L. Donoho, and L. Ying, 2005, Fast discretecurvelet transforms: SIAM Multiscale Model. Simul., 5, 861–899.

Candes, E. J. and D. L. Donoho, 2000, Curvelets – a surprisingly e"ectivenonadaptive representation for objects with edges. Curves and Surfaces.Vanderbilt University Press.

Hennenfent, G. and F. J. Herrmann, 2006, Seismic denoising with nonuni-formly sampled curvelets: Computing in Science and Engineering, 8, 16–25.

Herrmann, F. J., U. Boeniger, and D. J. Verschuur, 2007, Nonlinearprimary-multiple separation with directional curvelet frames: Geophysi-cal Journal International, 170, 781–799.

Neelamani, R., A. I. Baumstein, D. G. Gillard, M. T. Hadidi, and W. L.Soroka, 2008, Coherent and random noise attenuation using the curvelettransform: The Leading Edge, 27, 240–248.

Olhovich, V. A., 1964, The causes of noise in seismic reflection and refrac-tion work: Geophysics, 29, 1015–1030.

Versteeg, R., 2008, The marmousi experience: Velocity model determina-tion on a synthetic complex data set: The Leading Edge, 13, 927936.

Yilmaz, O., 2001, Seismic data analysis. Society of Exploration Geophysi-cists.

11

Chapter 2

Incoherent noise suppressionwith curvelet-domainsparsity-promotion

2.1 Background

In seismic data, recorded wavefronts (i.e., reflections) arise from the interac-tion of the incident wavefield with inhomogeneities in the Earth’s subsurface.The wavefronts can become contaminated with various types of noise duringacquisition or due to processing problems (Olhovich, 1964). Thus, separa-tion of signal and noise is an important issue in seismic data processing,particularly in crustal data where the signal-to-noise ratio is low. By noise,we refer to the incoherent (random) noise that is present in the data.

Because noise attenuation is a key step in seismic processing, manymethods have been developed to suppress such incoherent noise (Donoho,1995; Zhang and Ulrych, 2003; Jones and Levy, 1987; Ulrych et al., 1999).Predictive deconvolution is one of the most common techniques in seismicprospecting to attenuate noise. It is also known as F-X deconvolution orF-X prediction (Abma and Claerbout, 1995). The prediction filter works ina windowed sense, predicting one dip at a time, and tends to attenuate allother secondary dips in that window. Thus F-X prediction methods are notgood for curved events and for events with conflicting dips (e.g. caustics,pinch-outs).

More recently, curvelets have been introduced to resolve the issues ofdirectionality of complicated wavefronts as they belong to a family of multi-scale, and also multidirectional data expansions (Candes et al., 2005). Eachcurvelet is associated with a position, frequency and an angle. In seismicdata, the underlying geological signal di"ers from most noises in terms of at

1A version of this chapter will be submitted for publication. V. Kumar, J. Oueity, F. J.Herrmann and R. M. Clowes (2009) Curvelet denoising: application to crustal reflectiondata.

12

Chapter 2. Incoherent noise suppression with curvelet-domain sparsity-promotion

least one of the following features: angle, frequency, or location. For exam-ple, a linear dipping event will map to the coe!cients of specific angles inthe curvelet domain. Consequently, the signal maps to some specific largecoe!cients whereas noise maps everywhere as small coe!cients after curve-let transformation. This makes the curvelet transform an appropriate choicefor detecting wave fronts and suppressing noise as noise and signal naturallyseparate in the curvelet domain (Neelamani et al., 2008; Hennenfent andHerrmannn, 2006).

Our objective is to demonstrate the e"ectiveness of the curvelet trans-form in removing incoherent noise without harming the signal. We comparethree curvelet-based approaches to suppress random noise with applicationto synthetic data and deep reflection data. The real data were recorded alongLITHOPROBE’s SNorCLE Line 1 in the Northwest Territories, Canada(Cook et al., 1999).

2.2 Problem formulation

The forward problem of seismic denoising can be written as:

y = m + n, (2.1)

where y is the known noisy data, m is the unknown noise free data (themodel) and n is the zero-centered Gaussian noise. The noise can be white(full band) or colored (band-limited) depending on the previous processingapplied on the data. Our objective is to recover m. The curvelet-basedmethods for noise suppression are discussed next.

2.2.1 Elementwise hard and soft thresholding

The most common method for noise suppression in a transform domain ishard and soft thresholding of transform coe!cients (Donoho, 1995). Bothof them are element-wise operation. The estimated model m after hardthresholding is defined as:

H" :

#$%

$&

m = CHH"(Cy), whereH"(xi) = xi, if xi & !

H"(xi) = 0, otherwise(2.2)

The symbol H" represents the hard thresholding function, C is the forwardcurvelet transform operator and CH is the inverse curvelet transform oper-ator. The symbol H denotes conjugate transpose that is equivalent to the

13


inverse for this choice of curvelet transform. The estimated model m aftersoft thresholding is defined as:

S" :

"m = CHS"(Cy), whereS"(xi) = sgn(xi) · max(0, |xi|# |!|),

(2.3)

where S"(x) is the soft thresholding operator and ! is the threshold value.Fig.2.1 shows the application of hard and soft thresholding applied on asignal with ! = 2.

(a) (b)

(c)

Figure 2.1: (a) A seismic signal; dashed lines show the threshold level (!),(b) Signal after hard thresholding (!=2), (c) Signal after soft thresholding(!=2). Both hard and soft thresholding sets entries less than the thresholdto zero. In addition, soft thresholding shrinks all remaining entries by thethreshold. This moves the signal toward zero.

14


2.2.2 One-norm

The denoising problem with curvelet-domain sparsity can be cast into thefollowing constrained optimization problem (Elad et al., 2005):

P! :

"x = arg minx ||x||1 s.t. ||y #Ax||2 $ $,

m = CH x,(2.4)

where x is the curvelet transform vector, x represents the estimated curvelettransform coe!cient vector, m is the estimated model, % is proportional tothe noise level. A = CH

r is the curvelet synthesis operator with a restrictionthat allows only those coe!cients to enter the solution that survive thebest hard threshold. The reason for using the restricted curvelet transformis that we already know from the best hard thresholding (explained later)about the positions of those coe!cients that contain no useful signal and wewould not like those coe!cients to enter our solution while solving Eq. 2.4.Honoring this restriction, we find the sparsest set of curvelet coe!cients(by minimizing the one-norm) that explains the data within the noise level.The constrained optimization problem (Eq. 2.4) is solved by a series of thefollowing unconstrained optimization problems (Herrmann and Hennenfent,2008; Elad et al., 2005; Figueiredo and Nowak, 2003):

x = arg minx

12||y #Ax||22 + !||x||1, (2.5)

where ! is a regularization parameter that determines the trade-o" betweendata consistency and the sparsity in the curvelet domain. Eq. 2.5 can besolved by iterative soft thresholding (Daubechies et al., 2005). The solutionis found by updating x with x ' S"(x+AH

'y#Ax

(), where S" is the soft

thresholding operator defined in Eq. 2.3. We solve a series of such problems(Eq. 2.5) starting with high ! and decreasing the value of ! until ||y #Ax||2 $ %, which corresponds to the solution of our optimization problem.The algorithm is summarized in Table 2.1.

2.3 Parameter selection for synthetic data

For simplicity of our experiments, the real-valued curvelet transform is used.The curvelet transform has 5 scales, 16 angles at the 2nd coarsest level andcurvelets at the finest (fifth) scale. To choose the best threshold level forhard and soft thresholding, we did an experiment to plot a graph of signal-to-noise ratio (SNR) vs. threshold for both approaches. The SNR is calculated

15


Choose: L, KInitialize: k = 1,(AHy(" > !1 > · · · > !K ,x = 0

while (y #Ax(2 > % and k $ K dofor l = 1 to L

x = S"k

'x + AH (y #Ax)

(

end fork = k + 1

end while)m = CHx

Table 2.1: The cooling method with iterative soft thresholding to solveEq. 3.5. L is the number of inner iterations, K is the number of outeriterations, k is the counter for outer iterations, !1 · · ·!K are the values of !in decreasing order, other symbols are as explained in the text.

by the following formula:

SNR = 20 ) log10||m||2

||m# )m||2, (2.6)

where m is the true model. As we see from Figs. 2.2, the SNR shows agradual increase and then starts to decrease. This is expected as the initialthreshold level discards the noisy component of the data resulting in anincrease of the SNR value. After a certain point (the arrows in Fig. 2.2)the threshold level starts harming the signal components and thus we seea decrease in SNR value. Based on this experiment, the threshold valueis chosen for the peak value of SNR as shown by the arrows in Fig. 2.2.Since we know the noise, exact value of noise level (% = ||n||2) is used forthe one-norm denoising. We used two outer iterations (K=2) and one inneriteration (L=1) for the algorithm (Table 2.1). We choose the value of Kand L after intensive experiments in order to get the best possible SNRvalue. We observe that increasing the number of iterations cause a decreasein SNR value because iterative soft thresholding started introducing bais inthe amplitudes of estimated curvelet coe!cients. This bias corresponds toharming the coherent components. Since our criterion is to avoid coherentdamage, we limit our number of iterations.

16


0 20 40 60 80 100 120 140 160 180 2002

4

6

8

10

12

14

16

threshold value

SN

R

Hard

Soft

(a)

0 20 40 60 80 100 120 140 160 180 2002

4

6

8

10

12

14

16

threshold value

SN

R

Hard

Soft

(b)

Figure 2.2: (a) Comparison of signal-to-noise ratio (SNR) vs. thresholdvalue for white noise for hard thresholding and soft thresholding of curveletcoe!cients, (b) Comparison of signal-to-noise ratio (SNR) vs. thresholdvalue for colored noise for hard thresholding and soft thresholding of curveletcoe!cients. The arrows point to the peak SNR values and the correspondingthreshold value (!) is chosen.

2.4 Application to synthetic data

A synthetic shot gather is used for our experiments (same shot gather asdescribed in Chapter 2). The model (Fig. 2.3(a)) lives in the frequency range5-60 Hz. Noisy data (Fig. 2.3(b)) are obtained by addition of zero-centredwhite Gaussian noise ("=50) to the model. We also created a colored noiserealization by doing a band-pass filter (5-60 Hz) on the same white noiserealization. Fig. 2.3(c) shows the noisy data for additive colored noise. Allthree curvelet denoising approaches are applied to the noisy data and thesignal-to-noise ratio (SNR) is calculated for the denoised data.

The estimated models and the di"erences are shown in Fig. 2.4 andFig. 2.5. For the one-norm denoising, the di"erence for white noise (Fig. 2.4(b))has almost no coherent energy while the di"erence for colored noise (Fig. 2.5(b))has some coherent energy. In terms of the di"erence, both one-norm (Fig. 2.4(b),2.5(b)) and hard thresholding (Fig. 2.4(d), 2.5(d)) show that they have neg-ligible impact on the signal energy. The di"erence image of soft thresholding(Fig. 2.4(f), 2.5(f)) shows a significant amount of coherent energy. Becausesoft thresholding causes biasness in the amplitudes of curvelet coe!cients,the noise removed (di"erence) contains a lot of coherent energy. This is awell-known e"ect and may not be important for image denoising. Howeverthis is important to us as we do not want to harm any coherent energy.

17


In terms of SNR (Table 2.2), one-norm denoising has the highest SNRcompared to hard and soft thresholding of curvelet coe!cients (for bothwhite and colored noise). As we are dealing with redundant transform, one-norm puts a constraint on the coe!cients and thus stabilizes the solution.Thus, one-norm gives the highest SNR.

Table 2.2: SNR comparison for pre-stack denoisingNoise Data One-norm Hard SoftWhite 3.44 14.69 14.44 12.77Color 7.67 15.44 15.20 14.01

(a) (b)

(c)

Figure 2.3: (a) Noise-free data (model), (b) Model with white noise, (c)Model with colored noise.

18


(a) (b)

(c) (d)

(e) (f)

Figure 2.4: Data corrupted with white noise: (a) One-norm approach, (b)Noise removed by one-norm approach, (c) Hard threshold approach, (d)Noise removed by hard threshold approach, (e) Soft threshold approach, (f)Noise removed by soft threshold approach.

19


(a) (b)

(c) (d)

(e) (f)

Figure 2.5: Data corrupted with colored noise: (a) One-norm approach,(b) Noise removed by one-norm approach, (c) Hard threshold approach, (d)Noise removed by hard threshold approach, (e) Soft threshold approach, (f)Noise removed by soft threshold approach.

20


2.5 Parameter selection for real data

For the case of real data, we kept the same number of scales and anglesfor curvelets as for synthetic data. Observing the success of one-norm overhard and soft thresholding, we apply one-norm to real data. Choosing thevalue of noise level (%) is always a challenge for real data. For this work,we looked at various threshold levels for hard threshold and observed thenoise removed for a series of threshold values. Out of these experiments wepicked the result that caused reasonable noise attenuation with di"erence asrandom noise. We call this best hard threshold result. Based on this besthard thresholding, we defined the restricted curvelet transform CH

r and %.In other words, we allow only those curvelet coe!cients that survive besthard thresholding to enter the solution. As we will see from the examples,choosing the parameters this way caused minimal harm to the signal.

2.6 Application to deep reflection data

Near-vertical incidence crustal seismic data were recorded along SNorCLELine 1 as part of LITHOPROBE multidisciplinary studies in the Paleoproterozoic-Archean domains of the Northwest Territories, Canada (Fig. 2.6). Line 1is 725 km long and follows the highway from east of Yellowknife to west ofFort Simpson. For the seismic data acquisition, four to five vibroseis truckswere used with a 90 m shot point spacing, 60 m receiver spacing and a totalof 404 channels. The sweep frequencies ranged between 10 and 80 Hz witha record length of 32 s (Cook et al., 1999).

Fig. 2.7(a) shows that raw shot gather is contaminated with ground rolland random noise, which mask many features, especially deeper reflectors.As observed, the shot gathers are characterized by a gradual decrease inreflectivity with time. The decrease is due to the weak reflections fromdeeper levels. Some weak reflections are observed at 11.0 s that can berelated to the crust-mantle transition. We first balanced the amplitudes ofthe shot gathers to deal with high amplitude groundroll and bad traces.The balancing is done with an automatic gain control (AGC with 500 mswindow, see Fig. 2.7(b)) and subsequently ran the data through the curveletdenoising (Fig. 2.7(e)). Fig. 2.7(c) and Fig. 2.7(d) show the result of applyingcurvelet denoising to the data of Fig. 2.7(b) and the noise that is removed,respectively.

Fig. 2.8(a) shows the stacked section after standard processing (Fig. 2.8(b))for a 60-km-long segment. The clarity of the recorded seismic data is highly

21


YK

BC

DeformedAncestral

NorthAmerica

SlaveWopmayCordillera

Seismic Profile

120

128

CS

1112 1109

1107

1104

1101

FortSimpson

1113

1100

Extent of Phanerozoic cover

Shot Points

Fig. 1. Oueity and Clowes

Paleoproterozoic slab subduction

39

Seismic Profile Extent of Phanerozoic cover

Figure 2.6: Tectonic map of the acquisition area and location of SNORCLEnear vertical incidence (NVI) reflection Line 1. Sediments of the Phanero-zoic Western Canada Sedimentary Basin overlie the Precambrian domainswest of the long dashed line. Short dashed black lines show political bound-aries. CS - Coronation Supergroup, SD - Sleepy Dragon, GSLsz - GreatSlave Lake shear zone, AB - Alberta, BC - British Columbia, NWT - North-west Territories, YK - Yukon. Inset (top-left) shows location of map withinCanada. Inset (green box) shows the area for the data used.

22


degraded by the low signal-to-noise ratio that precludes detailed interpreta-tion of reflectivity images. The stacked seismic profile shows a decrease inreflectivity between the crust and upper mantle; the crust-mantle transitionis identified at 11.0 s (*34 km). On the southwestern end of the segmentshown, the crust-mantle transition becomes more di"use due to high back-ground noise (arrow at 11 s). Some northeast-dipping events can also beidentified on the southwestern end at 4.0 and 6.5 s but can only be followedover very short distances (arrow at 7 s Fig. 2.8(a)).

We applied curvelet denoising to the pre-stack and the post-stack datato compare our results with the conventional processing. Processing to stackfollowed standard procedures used in Lithoprobe studies (Cook et al., 1999)with curvelet denoising before the bandpass filter for the pre-stack dataas shown in Fig. 2.7(e). Curvelet denoising is applied on post-stack dataas an alternative to F-X prediction as shown in Fig. 2.8(b). Fig. 2.8(c)shows the result from denoising with curvelet processing while Fig. 2.8(e)shows results from F-X prediction. We do quality control of our results byobserving the di"erences (noise removed) between the original data and thedenoised data for each of the denoising applications. The noise removed isshown in (Fig. 2.8(d) for curvelet denoising, Fig. 2.8(f) for F-X prediction)

2.7 Results and discussion

In the case of pre-stack data, the results demonstrate that curvelet denois-ing successfully suppressed both the groundroll energy and random noise,thereby enhancing deep reflections. The Moho is clearly visible in our resultsat around 11 s. Individual reflectors throughout the crust are enhanced andmore clearly visible (Fig. 2.7(c)). The di"erence plot between the originaldata and the curvelet result (Fig. 2.7(d)) shows that the noise removed bycurvelet processing contains groundroll, corrupted traces and random noise.

In the case of post-stack data, the improvement in image resolution isclear when comparing the two data sets before and after the curvelet de-noising (Fig. 2.8). The curvelet denoised seismic image (Fig. 2.8(c)) showsa highly reflective crust with significant dipping reflectivity, a relatively flatMoho with an o"set of about 1s at about CDP 1100 and a transparentupper mantle. The lower crust contains a series of discrete, east-dippingreflections within a zone that is 2.0 s to 3.0 s thick (about 6.5 km to 10km) and flattens into the Moho. Below about 9.5 s at the southwesternend of the section, the reflectivity becomes sub-horizontal to about the po-sition of Moho o"set. These events are not visible on the original stacked

23


image. The Moho is characterized by a sharp, narrow band of reflections(200 - 300 ms) that are piece-wise continuous, except for the o"set, overthe 60-km-long segment. The di"erence plot between the original data andthe curvelet result (Fig. 2.8(d)) shows that the noise removed by curve-let processing is random in nature. Thus, the technique achieves excellentnoise attenuation without removing any coherent events. The F-X predictedimage (Fig. 2.8(e)) shows most of the features observed on the curvelet de-noised image, but the northeast-dipping and sub-horizontal reflectivity isnot clearly expressed. The noise removed by F-X prediction (Fig. 2.8(f))also contains some coherent reflectivity.

Fig. 2.9(a) shows the stack produced by the denoised shot gather. Thestacked image did produce a better stack compared to the stack created byconventional processing (Fig. 2.9(b)) but the overall quality of the stack isnot as good as what we got after post stack curvelet denoising (Fig. 2.9(c)).The possible reason for this is that stacking is a powerful operation thatitself is used to attenuate noise and errors creep in during subsequent pro-cessing steps. Nevertheless, the pre-stack denoising produces clearer reflec-tion hyperbolae that are advantageous for pre-stack processing (e.g. velocityanalysis, deconvolution, Normal Moveout correction).

24


(a) (b)

(c) (d)

Crooked line geometry

Static correction

Curvelet Denoising

Bandpass filter

Deconvolution

NMO

Stack

(e)

Figure 2.7: Shot gather from Line 1 segment highlighted in Fig. 3.6. (a)Noisy data without AGC, (b) Noisy data with AGC, (c) Result of applyingcurvelet denoising (inversion approach) on the shot gather with AGC, (d)Noise removed by curvelet denoising. (e) The flowchart with importantprocessing steps for pre-stack denoising. The curvelet denoising is appliedbefore the bandpass filtering.

25


SW NE

(a)

Crooked line geometry

Static correction

Bandpass filter

Deconvolution

NMO

Stack

F-X Prediction Curvelet Denoising

(b)

SW NE

(c)

SW NE

(d)

SW NE

(e)

SW NE

(f)

Figure 2.8: (a) The original stacked seismic section contaminated with ran-dom noise. Arrows indicate some of the features discussed in the text. (b)The flowchart with important processing steps for poststack denoising. Thecurvelet denoising is applied as an alternative to F-X prediction. (c) Resultof applying curvelet denoising (inversion approach) on the same section. (d)Noise removed by curvelet denoising, (e) Result of applying F-X predictionon the same section, (f) Noise removed by F-X prediction.

26


(a)

SW NE

(b)

SW NE

(c)

Figure 2.9: (a) Stacked section after pre-stack curvelet denoising, (b)Stacked section after conventional processing (without F-X prediction), (c)Stacked section after post-stack curvelet denoising. The poststack curveletdenoising is observed to be better than the other two.

27

Bibliography

Abma, R. and J. Claerbout, 1995, Lateral prediction for noise attenuationby t-x and f-x techniques: Geophysics, 60, 1887–1896.

Candes, E. J., L. Demanet, D. L. Donoho, and L. Ying, 2005, Fast discretecurvelet transforms: SIAM Multiscale Model. Simul., 5, 861–899.

Cook, F., A. van der Velden, K. Hall, and B. Roberts, 1999, Frozen sub-duction in canada’s northwest territories: lithoprobe deep lithospheric re-flection profiling of the western canadian shield: Tectonics, 18, 124.

Daubechies, I., M. Defrise, and C. de Mol, 2005, An iterative thresholdingalgorithm for linear inverse problems with a sparsity constraint: Comm.Pure Appl. Math., 57, 1413–1457.

Donoho, D., 1995, De-noising by soft thresholding: IEEE Trans. Inform.Theory, 41, 613–627.

Elad, M., J. L. Starck, P. Querre, and D. L. Donoho, 2005, Simulataneouscartoon and texture image inpainting using morphological component anal-ysis (MCA): Appl. Comput. harmon. Anal., 19, 340–358.

Figueiredo, M. and R. Nowak, 2003, An EM algorithm for wavelet-basedimage restoration: IEEE Trans. Image Processing, 12, 906–916.

Hennenfent, G. and F. J. Herrmannn, 2006, Seismic denoising with nonuni-formly sampled curvelets: Computing in Science and Engineering, 8, 16–25.

Herrmann, F. J. and G. Hennenfent, 2008, Non-parametric seismic datarecovery with curvelet frames: Geophysical Journal International, 173,233–248.

Jones, I. and S. Levy, 1987, Signal-to-noise enhancement in multichannelseismic data via the karhunen-love transform: Geophysical Prospecting,35, 12–32.


28

Bibliography


Ulrych, T., M. Sacchi, and J. Graul, 1999, Signal and noise separation, artand science: Geophysics, 64, 1648– 1656.

Zhang, R. and T. J. Ulrych, 2003, Physical wavelet frame denoising: Geo-physics, 68, 225–231.

29

Chapter 3

Deconvolution withcurvelet-domain sparsity

3.1 Background

Seismic deconvolution is one of the most widely researched seismic signalprocessing tools. The primary goal of seismic deconvolution is to removethe characteristics of the source wavelet from the recorded seismic timeseries (data), so that one is ideally left with only the reflection coe!cients(reflectivity). The reflection coe!cients identify and quantify the impedancemismatches between di"erent geological layers that are of great interest tothe geophysicist. The forward problem of seismic deconvolution can bewritten as:

y = s & m + n, (3.1)

where y is the data, s is the source wavelet, m is the reflectivity modeland n is zero-centered white Gaussian noise. The symbol & denotes linearconvolution. The only quantity we know is the data and we have to estimatethe reflectivity and source signature (wavelet). In other words, we have oneequation and two unknowns. In practice, we make assumptions that helpus to estimate both the reflectivity and source wavelet from data (Jurkevicsand Wiggins, 1984). For this work, we assume that we have an estimateof source wavelet and our goal is to estimate the reflectivity given the dataand the source wavelet. The source wavelet is also assumed to be stationary,i.e., it does not change with time or o"set. In matrix-vector notation theforward problem (Eq. 3.1) can be written as:

y = Km + n, (3.2)

where K is the convolution operator (Toeplitz matrix). The columns of Kcontain the shifted versions of s. Given K and y, we need to find m.

1A version of this chapter has been published. V. Kumar and F. J. Herrmann (2008)Deconvolution with curvelet-domain sparsity, In Expanded Abstracts, Society of Explo-ration Geophysicists, Tulsa.

30

Chapter 3. Deconvolution with curvelet-domain sparsity

Since the early 1980’s, researchers have cast this problem as a '1-normminimization (Taylor et al., 1979; Oldenburg et al., 1981, 1988; Yarlagaddaet al., 1985), where the reflectivity is assumed to comprise spikes. Themethod is often referred to as sparse-spike deconvolution or spiky deconvo-lution. The optimization problem solved for spiky deconvolution is givenby:

m = arg minm

||m||1 s.t. ||y #Km||2 $ $, (3.3)

where m represents the estimate of model vector and % is proportional to thenoise level. In recent work, Herrmann (2005) showed that the assumptionof spiky reflectivity is too limited to describe seismic reflectivity.

Our objective is to use the curvelet transform in solving the classicaldeconvolution problem to invert for a non-spiky reflectivity. We show thatcurvelet sparsity can be used to estimate the non-spiky reflectivity and hencethe Earth model. We start with an introduction to our algorithm followedby application to synthetic data for a shot gather and a poststack model.

3.2 Problem formulation

The deconvolution problem with curvelet-domain sparsity can be cast intothe following constrained optimization problem:

x = arg minx

||x||1 s.t. ||y #Ax||2 $ $, (3.4)

where A = KCH , CH is the curvelet transform synthesis operator, x rep-resents the estimated transform coe!cient vector, and % is proportional tothe noise level. By solving Eq. 3.4, we find the sparsest set of transform co-e!cients that explains the data within the noise level (Herrmann and Hen-nenfent, 2008). The final estimated reflectivity is given by m = CH x. Theconstrained optimization problem (Eq. 3.4) is solved by a series of the follow-ing unconstrained optimization problem (Herrmann and Hennenfent, 2008;Hennenfent and Herrmannn, 2006; van den Berg and Friedlander, 2008):

x = arg minx

12||y #Ax||22 + !||x||1, (3.5)

where ! is a regularization parameter that determines the trade-o" betweendata consistency and the sparsity. To speed up the convergence, we solvethese problems (Eq. 3.5) starting with a high ! and decreasing its valueuntil ||y#Ax||2 $ %, which corresponds to the solution of our optimizationproblem. The lowering of ! is done in a controlled way so that we reach the

31


optimum ! rapidly using the spectral projected-gradient algorithm (SPG'1)(van den Berg and Friedlander, 2008; Hennenfent et al., 2008). Details aboutthe algorithm are beyond the scope of this thesis but can be found in van denBerg and Friedlander (2008). The algorithm to solve Eq. 3.4 is summarizedin Table 3.1. By putting A = K, Eq. 3.4 simplifies to the problem of spikydeconvolution (Eq. 3.3).

Choose: K, !1 > · · · > !K

Initialize: k = 1,while (y #Ax(2 > % and k $ K do

x = arg minx12 ||y #Ax||22 + !||x||1

k = k + 1end while

)m = CHx

Table 3.1: Pseudo code to solve the optimization problem shown in Eq. 4.4.K is maximum number of iterations, k is the counter for iterations, otherquantities are as stated in the text.

3.3 Parameter selection

The curvelet transform used for this work has 5 scales, 16 angles at the2nd coarsest level and curvelets at the finest (fifth) scale. In the case ofadditive white Gaussian noise with standard deviation ", the square normof error ||n||22 is a chi-square random variable with mean "2N and standarddeviation "2

+2N , where N is the total number of data points (Candes et al.,

2005). For this work, we assume that the probability of ||n||22 exceeding itsmean by two standard deviations is small. The maximum ||n||22 within twostandard deviations is given by "2(N +2

+2N). Thus, we solve Eq. 3.4 with

%2 = "2(N + 2+

2N).

3.4 Application and results

The reflectivity models used for this work are the same models as describedin Chapter 1. Noisy data are obtained by convolving a Ricker wavelet (cen-tral frequency=25 Hz) with the reflectivity models followed by the additionof zero-centred Gaussian noise. We apply our curvelet deconvolution to the

32


noisy data to estimate the reflectivity. For comparison, we also include re-sults obtained by sparse spike deconvolution on the same data. For bothapproaches, we keep the same % (noise-level estimate) to enable a fair com-parison. We also calculate signal-to-noise ratio (SNR) for the estimatedmodel given by the expression:

SNR = 20 ) log10||m||2

||m# )m||2, (3.6)

where m is the true reflectivity model.First, we start with the prestack reflectivity model as shown in Fig. 3.1(a).

Fig. 3.1(b) shows the data that are degraded by the convolution operatorand the noise. Fig. 3.1(c) and Fig. 3.1(d) show the estimated reflectivityobtained by both algorithms. Fig. 3.2 shows an enhancement for the regionwithin the box of Fig. 3.1. The reflectivity model (Fig. 3.2(a)) is not madeup of spikes (does not have zero-order discontinuity). Fig. 3.1(b) shows thereflectivity model after convolution and with added noise. We observe thatthe coherent reflectivity is degraded. The curvelet estimated reflectivity(Fig. 3.2(c)) is very similar to the true reflectivity (Fig. 3.2(a)). The spikyestimate (Fig. 3.2(d)) consists of sharp spikes that is inconsistent with thetrue reflectivity (Fig. 3.2(a)).

We also apply the algorithm for the poststack model as shown in Fig. 3.3(a).Usually the poststack model has sharp boundaries but to show that our al-gorithm is suited for fractional order singularities, we did a fractional (half)integration of the model in the frequency domain to obtain a non-spikymodel. The non-spiky reflectivity is still sparse in the curvelet domain be-cause of its smooth variation in the o"set direction. Fig. 3.3(b) shows thedata that were input to the two deconvolution procedures. Fig. 3.3(c) andFig. 3.3(d) shows the estimated reflectivity obtained by curvelet deconvolu-tion and spiky deconvolution applied to the noisy data shown in Fig. 3.3(b).Fig. 3.4 shows a zoom-in wiggle plot of the reservoir area of Fig. 3.3. Asobserved, the model (Fig. 3.4(a)) is non-spiky in nature; convolution andadditive noise degrades the coherent reflectivity as seen in Fig. 3.4(b). Thecurvelet-estimated reflectivity (Fig. 3.4(c)) has a close similarity to the truereflectivity (Fig. 3.4(a)). In particular, the curvelet result (Fig. 3.4(c)) im-proves the resolution of the pinch-out at 1.1 seconds, which was degraded bythe convolution operator and noise. The spiky estimate (Fig. 3.4(d)) consistsof sharp spikes that do not resemble the true reflectivity (Fig. 3.4(a)).

The SNR values are summarized in Table. 3.2. In terms of SNR, weobserve that the curvelet-regularized deconvolution has high SNR comparedto sparse spike deconvolution for both models.

33


Table 3.2: SNR comparison for deconvolution methodsData Curvelet decon Spiky decon

Poststack -3.32 12.01 8.05Pre-stack -3.22 14.09 8.27

By exploiting the continuity along reflectors, our curvelet algorithm en-hances the degraded frequency components. Sparse spike deconvolution is atrace-by-trace operation, and doesn’t account for the two-dimensional struc-ture of the reflectivity model. On the other hand, curvelets, because of theirlocalization and similarity with curved events, exploit the structure in bothtime and spatial coordinate.

34


(a) (b)

(c) (d)

Figure 3.1: (a) True reflectivity model, (b) Noisy data obtained by convolv-ing the reflectivity model with Ricker wavelet and addition of noise ("=50),(c) Estimated reflectivity with curvelet deconvolution, (d) Estimated reflec-tivity with sparse spike deconvolution. The box shows the region of interestthat is enlarged in Fig. 4.2.

35


(a) (b)

(c) (d)

Figure 3.2: (a) Zoom-in plot of true reflectivity model, (b) Zoom-in plotof noisy data, (c) Zoom-in plot of estimated reflectivity with curvelet de-convolution, (d) Zoom-in plot of estimated reflectivity with sparse spikedeconvolution. Note the resemblance of curvelet-estimated reflectivity withthe true reflectivity.

36


(a) (b)

(c) (d)

Figure 3.3: (a) True reflectivity model, (b) Noisy data obtained by convolv-ing the reflectivity model with Ricker wavelet and addition of noise ("=.01),(c) Estimated reflectivity with curvelet deconvolution, (d) Estimated reflec-tivity with sparse spike deconvolution. The box shows the region of interest(reservoir area) for which enlarged versions of the sections are displayed inFig. 4.4.

37


(a) (b)

(c) (d)

Figure 3.4: (a) Zoom-in plot of true reflectivity model, (b) Zoom-in plotof noisy data, (c) Zoom-in plot of estimated reflectivity with curvelet de-convolution, (d) Zoom-in plot of estimated reflectivity with sparse spikedeconvolution. Note the resemblance of curvelet-estimated reflectivity withthe true reflectivity.

38

Bibliography

Candes, E., J. Romberg, and T. Tao, 2005, Stable signal recovery fromincomplete and inaccurate measurements: Comm. Pure Appl. Math., 59,1207–1223.Hennenfent, G. and F. J. Herrmannn, 2006, Seismic denoising with nonuni-formly sampled curvelets: Computing in Science and Engineering, 8, 16–25.Hennenfent, G., E. van den Berg, M. P. Friedlander, and F. J. Herrmann,2008, New insights into one-norm solvers from the pareto curve: Geo-physics, 73, A23–A26.Herrmann, F. J., 2005, Seismic deconvolution by atomic decomposition: aparametric approach with sparseness constraints: Integr. Computer-AidedEng., 12, 69–91.Herrmann, F. J. and G. Hennenfent, 2008, Non-parametric seismic datarecovery with curvelet frames: Geophysical Journal International, 173,233–248.Jurkevics, A. and R. Wiggins, 1984, A critique of seismic deconvolutionmethods: Geophysics, 49, 2109–2116.Oldenburg, D. W., S. Levy, and K. P. Whittall, 1981, Wavelet estimationand deconvolution: Geophysics, 46, 1528–1542.Oldenburg, D. W., T. Scheuer, and S. Levy, 1988, Recovery of the acousticimpedance from reflection seismograms: Inversion of geophysical data, 245–264.Taylor, H. L., S. C. Banks, and J. F. McCoy, 1979, Deconvolution withl1-norm: Geophysics, 44, 39–52.van den Berg, E. and M. P. Friedlander, 2008, Probing the pareto frontierfor basis pursuit solutions: SIAM Journal on Scientific Computing, 31,890–912.Yarlagadda, R., J. B. Bednar, and T. L. Watt, 1985, Fast algorithms for lpdeconvolution: IEEE Trans. on Acoustics, Speech, and Signal Processing,174–182.

39

Chapter 4

Conclusions

4.1 Curvelet-domain sparsity

The purpose of this thesis is to explore the ability of sparsity-promotingmethods as applied to noise suppression and deconvolution. As shown inexamples, curvelets provide a sparse representation of seismic data. Thissparsity in the curvelet domain provides a scope for new sparsity-promotingmethods that can be utilized and applied to seismic data, such as the twoalgorithms discussed in this thesis. As we see from the presented examples,curvelet-based-sparsity-promotion helps us to overcome the limitations oftraditional methods for denoising and deconvolution.

4.2 Application to incoherent noise suppression

We showed how the ability of curvelets to detect wavefronts can be usedto suppress incoherent noise. The inversion approach (one-norm denoising)with curvelet-domain sparsity not only recovers the features that are de-graded by noise, but also gives the highest SNR for both white and colorednoise without compromising the signal. We demonstrate that incoherentnoise in deep crustal reflection data can be suppressed using curvelet pro-cessing procedures and that the results are better than those from moreconventional methods, including F-X prediction. The algorithm is able toremove the incoherent noise and improve the resolution of the seismic image(especially near the Moho) both in pre-stack and post-stack data withoutcompromising the coherent reflectivity.

4.3 Application to deconvolution

We demonstrate that non-spiky reflectivity can be e"ectively recovered us-ing curvelet-regularized deconvolution and that the results show a betterfrequency enhancement than those from more conventional methods, in-cluding sparse spike deconvolution. The assumption of spiky reflectivity is

40

Chapter 4. Conclusions

too limited and may not be applicable in all cases; leading to deterioratedperformance of the traditional algorithms. Our algorithm is able to enhancethe frequency components that get degraded by noise and a convolution op-erator. Thus, sparsity of such type of reflectivity in the curvelet-domain is astrong characteristic that is exploited within our deconvolution algorithm.

4.4 Open and future research

The introduction of curvelet processing adds two new tools to the grow-ing family of curvelet-based approaches that includes primary-multiple sep-aration, groundroll removal, seismic data regularization, imaging and inver-sion. The use of the curvelet domain for seismic processing will most likelycontinue to expand as new techniques and methods are developed to takeadvantage of the sparsity. The curvelet domain also allows access to specificscales, angles and locations that have yet to be fully utilized.

Another avenue for future work is the influx of new transforms. Whilethe curvelet transform is the most parsimonious of the transforms discussed,many other similar systems are being developed. Shearlets (Labate et al.,2005) are a good example of this. New developments may also lead to atransform that is sparser for seismic data than the curvelet transform.

Careful depth and angular weighting within the curvelet domain mayfurther enhance the denoising procedure. The deconvolution algorithm canbe incorporated to do simultaneous migration and deconvolution. Both thealgorithms can be extended to three dimensions using 3D-curvelets so thatcontinuity of structure in three dimensions can be exploited.

One of the most challenging parts of the methods that were describedcenter on parameter selection (especially for the real data). There is cur-rently no fully automated way of selecting optimal parameters for the meth-ods described in this thesis. While fully automated parameter selection maybe a bit ambitious, schemes that automatically adjust the parameters forthe algorithms described may be possible. A good example of automati-cally setting the threshold level is the Generalized Cross-Validation (GCV)criterion (Jansen et al., 1997).

Another challenge posed by the curvelets is the curvelet vector lengththat is more than the number of data points. Curvelets are around eighttimes redundant for two-dimensional data and around twenty-four timesredundant in three-dimensions. While two-dimensional data generally aresmall enough that vector storage is not a problem, problems could be en-countered for large three-dimensional data cubes. Thus a parallel version of

41

Chapter 4. Conclusions

curvelet algorithm is favorable to handle three-dimensional and even four-dimensional (time-lapse) seismic data.

42

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