Modern Singular Spectral-Based Denoising and Filtering Techniques for 2D and 3D Reflection Seismic Data 3030193039, 9783030193034

This book discusses the latest advances in singular spectrum-based algorithms for seismic data processing, providing an

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Table of contents :
Foreword
Preface
Acknowledgments
Contents
Chapter 1: Introduction to Denoising and Data Gap Filling of Seismic Reflection Data
1.1 Introduction
1.2 General Classification of Noise in Seismic Data
1.2.1 Random Noise
1.2.2 Coherent Noise
1.3 Noise Suppression Methods Used in the Seismic Data Processing
1.4 Data Gap Filling
1.5 Singular Spectrum Analysis
1.6 SSA Methods for Seismic Data
1.7 Skeleton of the Book
Chapter 2: Time and Frequency Domain Eigen Image and Cadzow Noise Filtering of 2D Seismic Data
2.1 Introduction
2.2 Time Domain Eigen Image Processing
2.2.1 Example 1
2.2.2 Example 2
2.3 Frequency Domain Eigen Image Processing
2.3.1 Example 3
2.4 Time and Frequency Domain Cadzow Filters
2.4.1 Pseudo Code of Time Domain Cadzow Filter
2.4.2 Pseudo Code of Frequency Domain Cadzow Filter
2.4.3 Example 4
2.5 Conclusion
Chapter 3: Singular Spectrum Analysis-Based Time Domain Frequency Filtering
3.1 Introduction
3.2 Methodology
3.3 Data Analysis
3.3.1 Example of Testing on Synthetic Data
3.3.2 Application to Reflection Field Data
3.4 Grouping from Weighted Eigen Spectrogram (WES)
3.5 Conclusion
Chapter 4: Frequency and Time Domain SSA for 2D Seismic Data Denoising
4.1 Introduction
4.2 Methodological Description
4.2.1 FXSSA/Fxy Eigen Image Pseudo Code
4.2.2 TXSSA Pseudo Code
4.3 Example 1: F-xy Eigen Image Noise Suppression (Trickett 2003)
4.4 Example 2: Comparison of FXSSA Denoising with f-x Deconvolution (After Sacchi 2009)
4.5 Example 3: FXSSA Denoising of Synthetic Data in Comparison with TXSSA Method
4.6 Conclusion
Chapter 5: Filtering 2D Seismic Data Using the Time Slice Singular Spectral Analysis
5.1 Introduction
5.2 Example of Crustal Stratification
5.3 Time Slice Singular Spectrum Analysis (TSSSA) Methodology
5.4 Selection of Window Length and Triplet Group
5.5 Application to Synthetic Data
5.6 Application of TSSSA and FXSSA on Pre and Post Stack Seismic Field Data
5.7 Application of the Method on Seismic Field Data from Singareni Coal Field, Telangana, India
5.8 Conclusion
Chapter 6: Robust and Fast Algorithms for Singular Spectral Analysis of Seismic Data
6.1 Introduction
6.2 Optimized SSA Method
6.2.1 Methodology
6.2.2 Coloured Noise Suppression Using Optimized SSA
6.3 Factorized Hankel SVD
6.3.1 Methodology
6.3.2 Testing on Synthetic Data
6.3.3 Low Frequency Preservation in Factorized Hankel Method
6.3.4 Computational Efficiency
6.3.5 Application of the Method to Post Stack Seismic Data
6.4 Randomized SVD (R-SVD)
6.4.1 Methodology/Algorithm
6.4.2 Application of R-SVD to Seismic Data
6.5 Windowed SSA
6.5.1 Methodology
6.5.2 Application to a Seismic Reflection Trace
6.6 Conclusion
Chapter 7: Denoising the 3D Seismic Data Using Multichannel Singular Spectrum Analysis
7.1 Introduction
7.2 Methodology
7.3 Synthetic Examples
7.3.1 Multichannel Time Slice SSA
7.3.2 Frequency Domain MSSA
7.4 Application to Field Data
7.5 Conclusion
Chapter 8: Seismic Data Gap Filling Using the Singular Spectrum Based Analysis
8.1 Introduction
8.2 Methodology
8.2.1 Pseudo Code
8.3 Examples
8.4 Frequency Domain MSSA Based 3D-Data Gap Filling
8.5 Time Domain MSSA Based Iterative Data Gap Filling
8.6 Conclusions
Chapter 9: Singular Spectrum vs. Wavelet Based Denoising Schemes in Generalized Inversion Based Seismic Wavelet Estimation
9.1 Introduction
9.2 Generalized Inversion Based Wavelet Estimation
9.3 Analysis and Results
9.4 Conclusion
Chapter 10: Singular Spectrum-Based Filtering to Enhance the Resolution of Seismic Attributes
10.1 Introduction
10.2 TSSSA-Based Filtering to Improve Post Stack 2D Attributes
10.2.1 Example 1: 2D Post Stack Attributes from TSSSA Filtered Data
10.3 MSSA-Based Pre-filtering of Horizon Time Structures and Amplitudes to Enhance the Resolution
10.3.1 Example 2: Synthetic Modeling
10.3.2 Example 3: Enhancing the Resolution of Curvature Attributes from Utsira Top (UT) Horizon
10.3.3 Example 4: MSSA-Based Pre-filtering of Horizon Amplitudes from 3D Volumes to Interpret the Physical Changes in Horizon
10.4 Conclusion
Chapter 11: Singular Spectrum Analysis with MATLAB®
11.1 Introduction
11.2 MATLAB® Coding: Description and Application
11.2.1 Signal Decomposition
11.2.2 Signal Reconstruction
11.2.3 Eigen Modes and their Separation
11.3 MATLAB® Function for Singular Spectrum Analysis of 1D Data Series
11.4 Application of SSA to High Resolution Seismic Trace Data
11.5 Summary
Appendix: Eigen Decomposition—Singular Value Decomposition
Introduction
Methodology
Examples of Eigen Decomposition
Singular Value Decomposition
Computational Steps Involved in SVD of Non-square Matrix A
References
Index
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R.K. Tiwari · R. Rekapalli

Modern Singular Spectral-Based Denoising and Filtering Techniques for 2D and 3D Reflection Seismic Data

Modern Singular Spectral-Based Denoising and Filtering Techniques for 2D and 3D Reflection Seismic Data

R. K. Tiwari • R. Rekapalli

Modern Singular Spectral-Based Denoising and Filtering Techniques for 2D and 3D Reflection Seismic Data

R. K. Tiwari CSIR-NGRI Hyderabad, India

R. Rekapalli CSIR-NGRI Hyderabad, India

ISBN 978-3-030-19303-4    ISBN 978-3-030-19304-1 (eBook) https://doi.org/10.1007/978-3-030-19304-1 Jointly published with Capital Publishing Company, New Delhi, India. The print edition is not for sale in SAARC countries. Customers from SAARC countries – Afghanistan, Bangladesh, Bhutan, India, the Maldives, Nepal, Pakistan and Sri Lanka – please order the book from: Capital Publishing Company, 7/28, Mahaveer Street, Ansari Road, Daryaganj, New Delhi 110 002, India. © Capital Publishing Company 2020 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

Recent advent of modern computer technologies and acquisition of huge data has made the data analyses and processing unavoidable for robust interpretations, not only in the earth sciences but also for various other branches of science. This inevitability requires knowledgeable, skilled, and proficient researchers who will be able to modernize, develop, and make anticipated advancement and progress in the underlying field to render precise interpretation of the geophysical data. This book demonstrates and discusses the singular spectrum-based algorithms for data-­adaptive processing of seismic signals in time and frequency domains. Among different aspects of signal processing, the book mainly focuses on removal of undesired signals/noises which are not useful for physical interpretations of seismic data, time domain frequency filtering, data gap filling, and diffraction wave field separation. Drs. Tiwari and Rajesh provide comprehensive details of the up-to-date and advanced methodologies and have demonstrated the complete mathematical background, testing of the algorithms on synthetic data, as well as real-time example on 1D, 2D, and 3D field data with physical interpretation aided by regional geological as well as borehole data. In view of recent advances and necessity of computer coding, the authors have provided a chapter with simple MATLAB coding for singular spectrum analysis with simple examples. They have also incorporated useful materials on SVD computations with examples as appendix. This volume would serve as a liberal resource for understanding and practicing singular spectrum-based seismic data processing. Attempt has been made to comprehend the developments and application of singular spectrum-based methods in seismic data processing, and systematically up-to-date information have been put forth.

Director Virendra M. Tiwari CSIR-National Geophysical Research Institute Hyderabad, India v

Preface

Signal analyses and data processing have become unavoidable not only for geophysics but also for various other branches of science. This inevitability essentially requires knowledgeable, skilled, and proficient researchers who will be able to modernize, develop, and make anticipated advancement and progress in the underlying field to render precise interpretation of the geophysical data. The formulation of singular value decomposition (SVD) by Lanczos in 1961 led to the design and development of several SVD-based algorithms to decompose the data using data adaptive eigenvectors. The analysis of singular spectrum is a robust and promising method being used in the geosciences for trend and principal component analysis, filtering, data gap filling, etc. for over the past 50 years. These methods use the virtue of data adaptive-basis function for decomposing signals into linearly independent components of physical significance. The advances in these methods attracted the attention of several branches of geosciences (e.g., astronomical data gap filling; principal component and trend analysis of climate data; 2D, 3D, and 5D seismic data filtering; seismological data processing; gravity data processing; etc.) to improve the accuracy of interpretation. In the beginning of singular spectrum-based seismic data processing, SVD was applied onto 2D seismic data matrix in time domain to suppress noise and multiples. The SVD decomposes the data matrix into eigen-weighted sum of orthogonal matrices of rank one, which are also known as eigenimages of data matrix. T-x SSA frequency filtering was developed for noise suppression, data gap filling, wave field separation, frequency filtering, etc. Employing fixed basis functions in conventional filtering of spatiotemporal nonlinear and non-stationary seismic data creates the artifacts. Therefore, singular spectrum-based data-adaptive decomposition using the eigenvectors of data as basic functions gained importance as an alternative to avoid artifacts. In view of this, several fast computational algorithms are also developed for the easy adaption of singular spectrum methods into the seismic data processing. In this book, attempt has been made to comprehend the up-to-date developments and application of singular spectrum-based methods for seismic data processing. Beginning with the introduction to noises in seismic data, conventional and unconventional methods of denoising/filtering, and state-of-the-art information in vii

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Preface

the subject area, we have attempted to discuss the role of data-adaptive noise suppression in the reflection seismic methods in the first chapter. The time and frequency domain eigenimage and Cadzow filtering techniques are provided in the second chapter with complete description of algorithm and appropriate examples of filtering/denoising. Chapter 3 of this book deals with SSA-based time domain frequency filtering technique along with the eigen-weighted spectrum-based triplet grouping procedure as an alternative user-friendly method for filtering the non-­ stationary data to surmount the problems associated with filtering-related artifacts. The novelty of the method is discussed using synthetic data and then provided an example of its application to real data in comparison with conventional FFT-based filtering. The application of SSA on 2D seismic data in time and frequency domain is provided in the fourth chapter with illustrative examples demonstrating the basic difference between time and frequency domain applications in comparison with f-x deconvolution filter. Spatial/time slice SSA (SSSA/TSSSA) algorithm for noise suppression is discussed in Chap. 5 with testing on the synthetic data with different kinds of noises compared with f-x SSA and f-x deconvolution methods and an example case study from Durgapur, West Bengal, India. The accuracy of the denoised output is discussed, integrated with regional geology and litho-log data. In another case study, the method was applied to pre- and post-stack field data from Singareni coalfield, Telangana, India, to discuss the colored noise suppression and scattered primary amplitude recovery for accurate identification of intrinsic fault structures associated with coalfield. Chapter 6 provides comprehensive details about the advanced methodologies: (i) optimized SSA for colored noise suppression for seismic reflection data, (ii) randomized SVD, (iii) factorized Hankel technique, and (iv) windowed SSA for fast and robust application of singular value decomposition over the existing conventional methods on synthetic and real seismic data sets. The seventh chapter is destined to present the methodologies of time and frequency domain applications of multichannel SSA for simultaneous seismic data denoising and gap filling. The methods are tested on 3D seismic synthetic data and applied to seismic post-stack data from Sleipner CO2 storage site. The SSA-based 1D, 2D, and 3D data gap filling strategies are discussed in the next chapter. The detailed description of iterative frequency and time domain MSSA data gap filling strategies are discussed with appropriate examples. The influence of noise and comparative effectiveness of TSSSA over the multivariate wavelet method in deterministic wavelet estimation via generalized inversion is tested on synthetic data in Chap. 9. The singular spectrum-­based noise suppression was tested to enhance the accuracy of horizon and 2D seismic data for geologically consistent geophysical interpretation in the tenth chapter. The basic definition and classification of attributes as well as the impact of noise-/processing-related artifacts on derived attributes are discussed in this chapter. Then, we compile appropriate synthetic and real data examples to discuss the role and necessity for singular spectrum-based pre-filtering in attribute studies. Finally, a MATLAB-based singular spectrum analysis tutorial is presented in the 11th chapter for beginner to start coding the singular spectrum-based algorithms for better and artifact-free processing of nonlinear and non-stationary

Preface

ix

g­ eophysical data sets. The mathematical description, basic definition of eigenvectors, eigenvalues, and their physical significance, along with simple examples of SVD are provided as appendix for the use of beginners to understand the concept of SVD. Hence, this book as a whole serves as an ample volume of the modern development and application of singular spectrum-based methods for the improvement of 1D, 2D, and 3D seismic data quality, thereby accuracy of interpretations. This book may also serve as a liberal resource for understanding and practicing singular spectrum-­based seismic data processing algorithms of wide spectrum. Hyderabad, India 

R. K. Tiwari R. Rekapalli

Acknowledgments

The book is based on singular spectrum-based analysis, a robust method for analyzing the time series data to separate linearly independent processes for physical interpretation. Over the past decade, there is a lot of advancement in its applications to various branches of sciences. Among these, singular spectrum analysis of seismic data laid the roadway for rank reduction-based denoising, data gap filling, filtering, and wave field separation. We wrote this book to provide an integrated material for understanding the time and frequency domain singular spectrum-based methods that have been developed and used for seismic data processing. We have tried to provide an unbiased coverage of the subject developments by the global research community. We would like to mention that in order to present unbiased integrated picture and self-sufficiency of the book, repetition of some material is inevitable. There are some illustrations and mathematical descriptions taken from the work of various researchers which we have sincerely acknowledged. We, particularly, acknowledge the Society of Exploration Geophysics, Springer, Elsevier, Indian Geophysical Union, and Society of Petroleum Geophysicists, India, for using their figures modified after their published materials. We also acknowledge Dr. C. Lanczos, Dr. T.J. Ulrych, and Dr. S.R. Trickett for their pioneering contributions in the field of singular spectrum-based seismic data processing algorithms from which we have immensely benefited. We also acknowledge Dr. S.R. Trickett, Prof. Mauricio Sacchi, and Dr. Vicente Oropeza for further advancements in frequency domain singular spectrum-based methods for 2D and 3D seismic data. We thank Prof. Mrinal K. Sen, former Director, CSIR-NGRI, Hyderabad, India, for his support and encouragement. We thank Dr. V.M.  Tiwari, Director, CSIR-NGRI, Hyderabad, India, for his moral and scientific support and encouragement to complete this book. Our special thanks to all the members of Engineering Geophysics group of CSIR-NGRI, Hyderabad, India, for data support. We are also indebted to all those who generously supported us in this endeavor. Dr. R.K.  Tiwari acknowledges the Department of Atomic Energy for Raja Ramanna Fellowship. He is also grateful to his wife, Rukmini, and daughters, Rashmi and Supriya, for their enduring support during the preparation of this book. xi

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Acknowledgments

Dr. Rajesh Rekaplli acknowledges CSIR for Research Associate Fellowship. He is also grateful to his brother, Vikram; wife, Satyavani; and daughter, Rohitha, for their extended support to complete this book. Last but not least, he also owes his sincere gratitude to all his teachers and thanks to colleagues.

Contents

1 Introduction to Denoising and Data Gap Filling of Seismic Reflection Data ����������������������������������������������������������������������������������������    1 1.1 Introduction��������������������������������������������������������������������������������������    1 1.2 General Classification of Noise in Seismic Data������������������������������    1 1.2.1 Random Noise����������������������������������������������������������������������    2 1.2.2 Coherent Noise���������������������������������������������������������������������    3 1.3 Noise Suppression Methods Used in the Seismic Data Processing��������������������������������������������������������������������������������    4 1.4 Data Gap Filling��������������������������������������������������������������������������������    5 1.5 Singular Spectrum Analysis��������������������������������������������������������������    5 1.6 SSA Methods for Seismic Data��������������������������������������������������������    6 1.7 Skeleton of the Book������������������������������������������������������������������������    8 2 Time and Frequency Domain Eigen Image and Cadzow Noise Filtering of 2D Seismic Data ��������������������������������������������������������   11 2.1 Introduction��������������������������������������������������������������������������������������   11 2.2 Time Domain Eigen Image Processing��������������������������������������������   12 2.2.1 Example 1 ����������������������������������������������������������������������������   13 2.2.2 Example 2 ����������������������������������������������������������������������������   14 2.3 Frequency Domain Eigen Image Processing������������������������������������   15 2.3.1 Example 3 ����������������������������������������������������������������������������   17 2.4 Time and Frequency Domain Cadzow Filters����������������������������������   18 2.4.1 Pseudo Code of Time Domain Cadzow Filter����������������������   19 2.4.2 Pseudo Code of Frequency Domain Cadzow Filter��������������������������������������������������������������������������������������   19 2.4.3 Example 4 ����������������������������������������������������������������������������   20 2.5 Conclusion����������������������������������������������������������������������������������������   21 3 Singular Spectrum Analysis-Based Time Domain Frequency Filtering����������������������������������������������������������������������������������������������������   23 3.1 Introduction��������������������������������������������������������������������������������������   23 3.2 Methodology ������������������������������������������������������������������������������������   24 xiii

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3.3 Data Analysis������������������������������������������������������������������������������������   25 3.3.1 Example of Testing on Synthetic Data����������������������������������   25 3.3.2 Application to Reflection Field Data������������������������������������   26 3.4 Grouping from Weighted Eigen Spectrogram (WES)����������������������   28 3.5 Conclusion����������������������������������������������������������������������������������������   30 4 Frequency and Time Domain SSA for 2D Seismic Data Denoising ����������������������������������������������������������������������������������������   33 4.1 Introduction��������������������������������������������������������������������������������������   33 4.2 Methodological Description��������������������������������������������������������������   33 4.2.1 FXSSA/Fxy Eigen Image Pseudo Code ������������������������������   34 4.2.2 TXSSA Pseudo Code������������������������������������������������������������   35 4.3 Example 1: F-xy Eigen Image Noise Suppression (Trickett 2003)����������������������������������������������������������������������������������   35 4.4 Example 2: Comparison of FXSSA Denoising with f-x Deconvolution (After Sacchi 2009)������������������������������������   36 4.5 Example 3: FXSSA Denoising of Synthetic Data in Comparison with TXSSA Method������������������������������������������������   38 4.6 Conclusion����������������������������������������������������������������������������������������   41 5 Filtering 2D Seismic Data Using the Time Slice Singular Spectral Analysis��������������������������������������������������������������������������������������   43 5.1 Introduction��������������������������������������������������������������������������������������   43 5.2 Example of Crustal Stratification������������������������������������������������������   44 5.3 Time Slice Singular Spectrum Analysis (TSSSA) Methodology ������������������������������������������������������������������������������������   45 5.4 Selection of Window Length and Triplet Group������������������������������   48 5.5 Application to Synthetic Data����������������������������������������������������������   49 5.6 Application of TSSSA and FXSSA on Pre and Post Stack Seismic Field Data������������������������������������������������������������������   55 5.7 Application of the Method on Seismic Field Data from Singareni Coal Field, Telangana, India������������������������������������   58 5.8 Conclusion����������������������������������������������������������������������������������������   65 6 Robust and Fast Algorithms for Singular Spectral Analysis of Seismic Data��������������������������������������������������������������������������   67 6.1 Introduction��������������������������������������������������������������������������������������   67 6.2 Optimized SSA Method��������������������������������������������������������������������   68 6.2.1 Methodology ������������������������������������������������������������������������   69 6.2.2 Coloured Noise Suppression Using Optimized SSA ��������������������������������������������������������������������   70 6.3 Factorized Hankel SVD��������������������������������������������������������������������   71 6.3.1 Methodology ������������������������������������������������������������������������   72 6.3.2 Testing on Synthetic Data ����������������������������������������������������   73 6.3.3 Low Frequency Preservation in Factorized Hankel Method����������������������������������������������������������������������   75

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6.3.4 Computational Efficiency ����������������������������������������������������   75 6.3.5 Application of the Method to Post Stack Seismic Data ������������������������������������������������������������������������   77 6.4 Randomized SVD (R-SVD)��������������������������������������������������������������   77 6.4.1 Methodology/Algorithm ������������������������������������������������������   78 6.4.2 Application of R-SVD to Seismic Data��������������������������������   79 6.5 Windowed SSA��������������������������������������������������������������������������������   80 6.5.1 Methodology ������������������������������������������������������������������������   81 6.5.2 Application to a Seismic Reflection Trace����������������������������   82 6.6 Conclusion����������������������������������������������������������������������������������������   82 7 Denoising the 3D Seismic Data Using Multichannel Singular Spectrum Analysis��������������������������������������������������������������������   85 7.1 Introduction��������������������������������������������������������������������������������������   85 7.2 Methodology ������������������������������������������������������������������������������������   86 7.3 Synthetic Examples��������������������������������������������������������������������������   89 7.3.1 Multichannel Time Slice SSA����������������������������������������������   89 7.3.2 Frequency Domain MSSA����������������������������������������������������   90 7.4 Application to Field Data������������������������������������������������������������������   91 7.5 Conclusion����������������������������������������������������������������������������������������   93 8 Seismic Data Gap Filling Using the Singular Spectrum Based Analysis������������������������������������������������������������������������������������������   95 8.1 Introduction��������������������������������������������������������������������������������������   95 8.2 Methodology ������������������������������������������������������������������������������������   96 8.2.1 Pseudo Code�������������������������������������������������������������������������   96 8.3 Examples������������������������������������������������������������������������������������������   97 8.4 Frequency Domain MSSA Based 3D-Data Gap Filling ������������������   98 8.5 Time Domain MSSA Based Iterative Data Gap Filling��������������������   99 8.6 Conclusions��������������������������������������������������������������������������������������  101 9 Singular Spectrum vs. Wavelet Based Denoising Schemes in Generalized Inversion Based Seismic Wavelet Estimation��������������  103 9.1 Introduction��������������������������������������������������������������������������������������  103 9.2 Generalized Inversion Based Wavelet Estimation����������������������������  104 9.3 Analysis and Results ������������������������������������������������������������������������  105 9.4 Conclusion����������������������������������������������������������������������������������������  108 10 Singular Spectrum-Based Filtering to Enhance the Resolution of Seismic Attributes������������������������������������������������������  109 10.1 Introduction������������������������������������������������������������������������������������  109 10.2 TSSSA-Based Filtering to Improve Post Stack 2D Attributes����������������������������������������������������������������������������������������  110 10.2.1 Example 1: 2D Post Stack Attributes from TSSSA Filtered Data����������������������������������������������������������������������  111 10.3 MSSA-Based Pre-filtering of Horizon Time Structures and Amplitudes to Enhance the Resolution������������������������������������  115

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10.3.1 Example 2: Synthetic Modeling����������������������������������������  117 10.3.2 Example 3: Enhancing the Resolution of Curvature Attributes from Utsira Top (UT) Horizon������������������������������������������������������������������������������  119 10.3.3 Example 4: MSSA-Based Pre-filtering of Horizon Amplitudes from 3D Volumes to Interpret the Physical Changes in Horizon��������������������������������������  121 10.4 Conclusion��������������������������������������������������������������������������������������  122 11 Singular Spectrum Analysis with MATLAB®��������������������������������������  125 11.1 Introduction������������������������������������������������������������������������������������  125 11.2 MATLAB® Coding: Description and Application ������������������������  126 11.2.1 Signal Decomposition ������������������������������������������������������  126 11.2.2 Signal Reconstruction ������������������������������������������������������  128 11.2.3 Eigen Modes and their Separation������������������������������������  130 11.3 MATLAB® Function for Singular Spectrum Analysis of 1D Data Series����������������������������������������������������������������������������  132 11.4 Application of SSA to High Resolution Seismic Trace Data ��������������������������������������������������������������������������������������  134 11.5 Summary ����������������������������������������������������������������������������������������  138 Appendix: Eigen Decomposition—Singular Value Decomposition ������������  139 References ��������������������������������������������������������������������������������������������������������  145 Index������������������������������������������������������������������������������������������������������������������  153

Chapter 1

Introduction to Denoising and Data Gap Filling of Seismic Reflection Data

1.1  Introduction Seismic data is a mixture of several wanted (reflections, refractions) and unwanted (ground roll, diffractions, airwave etc.) signals. Different wave fields recorded by the seismic receiver can be seen from Fig. 1.1. Separation of wanted signal from such unwanted noises is therefore fundamental in geophysical signal processing. Especially, it is very difficult for a visual inspection to completely distinguish primary reflections and noise in the raw data of active seismic experiment for the interpretation of subsurface layers and their discontinuities. The primary reflections in the raw seismic data are always hindered by the wave fields arising from several unwanted sources such as: diffractions, ground roll, airwave etc. and unknown random signals. Separation of signal from the noise is an important task in geophysical signal processing industry. The accuracy of seismic data interpretation mainly depends on the quality of the data i.e., Signal-to-Noise Ratio (S/N). Therefore, separation of unwanted signals from the seismic field data is almost essential and a challenging task in seismic industry for accurate and geologically consistent physical interpretation of primary reflections for understanding the study area. In the following section, we begin with the basic classification of different kinds of noises based on their statistical and spectral characteristics before proceeding to discuss about denoising techniques used in seismic industry.

1.2  General Classification of Noise in Seismic Data As discussed above, it is not possible to discern all the sources of noise in the seismic data to discuss their characteristics completely. ‘What is noise?’ is highly subjective (Scales and Snieder 1998). Simple visual inspection of the data cannot assure © Capital Publishing Company 2020 R. K. Tiwari, R. Rekapalli, Modern Singular Spectral-Based Denoising and Filtering Techniques for 2D and 3D Reflection Seismic Data, https://doi.org/10.1007/978-3-030-19304-1_1

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1  Introduction to Denoising and Data Gap Filling of Seismic Reflection Data

Fig. 1.1 (a) Schematic illustration of different types of waves in seismic reflection experiment. (b) Different kinds of seismic wave fields in the shot gather. (A Airwave, D Direct wave, R Reflected wave, R1 Refracted wave, S Surface wave, St Surface wave (traffic), Dt Dead channel)

the identification of noise and its nature. However, based on the information and literature available today, depending on the statistical characteristics, the noises that corrupt the seismic data are broadly categorized into two types, namely (i) random/incoherent and (ii) coherent noises (Telford and Sheriff 2010).

1.2.1  Random Noise The disturbances in the seismic signal amplitudes that lack the trace-to-trace spatial coherency are considered as the random noise. The random/white noise does not result from the sources that generate the seismic energy. Near-surface scattering,

1.2 General Classification of Noise in Seismic Data

3

wind and rain are some examples of possible sources of random noises in land seismic reflection data. The white noise occupies all the frequencies with the same spectral power results with constant shift in the spectral power at all frequencies. Therefore, constant scaling (i.e., dividing the spectral power of each frequency component in the data by constant number) will help reducing the random noise. In general, these random noise sources are non-stationary, i.e., they do not add to the original reflections in the same manner, if we acquire data at two different times (4D manner).

1.2.2  Coherent Noise The seismic source is the origin of some spatially distributed coherent noises. It is an undesirable additive energy to the primary reflections. Such energy shows consistent phase from trace to trace. The multiple reflections/multiples, surface waves like ground roll and airwaves, coherent scattered waves, dynamite ghosts etc., are the coherent noises that are commonly present with the seismic data (Fig. 1.1b). Improper removal of coherent noise affects nearly all the subsequent data processing and complicates the interpretation of geological structures. The land seismic reflection data mainly contains ground roll and it is obvious to know its characteristics for appropriate removal. The ground roll is the low velocity (  … σi … > σr. Andrews and Hunt (1977) designated the outer dot product of Ui and ViT as ith Eigen image of the data matrix. Because of the ortho-normality of Eigenvectors, the Eigen images form set of basis functions, which are used in the reconstruction of processed data matrix Dr. The decomposition of data matrix into Eigen images can be pictorially represented as shown in Fig. 2.1. The Eigen spectrum which depicts the cross plot between singular component numbers and respective singular value enables us to analyze the contribution of each Eigen image to the data. Figure 2.2 shows the Eigen spectrum.

Fig. 2.1  Eigen image decomposition of data matrix. (Modified after Ulrych et al. 1999)

2.2  Time Domain Eigen Image Processing

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Fig. 2.2 Eigen/Singular spectrum

Generally, the first few Eigen images contribute most of the signal variance. The random and/or incoherent signal energies generally contribute small variance to the signal and, therefore, they appear in the tail of the singular spectrum. Hence, the denoised data (Dr) can be reconstructed from p number of selected Eigen triplets with considerably significant Eigenvalue percentage using the following equation: r



Dr = ∑σi U i ViT i =1

(2.2)

The Eigen image method is successfully applied to the seismic data denoising (Ulrych et al. 1988, 1999).

2.2.1  Example 1 An example (modified after Ulrych et al. 1999) of Eigen image denoising is presented in Fig.  2.3. The synthetic data of fault model contaminated with random noise (Fig. 2.3a) is decomposed using SVD and then recovered the denoised data (Fig. 2.3d) from the first two Eigen images (Fig. 2.3b, c). It can be noted that the number of dips are equal to number of EI considered for reconstruction. There is a clear enhancement of SNR in Fig. 2.3d compared to Fig. 2.3a.

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Fig. 2.3  Eigen image processing for denoising seismic synthetic data: (a) Synthetic data with 20% random noise, (b) First Eigen image, (c) second Eigen image and (d) denoised data reconstructed from first and second Eigen images. (Modified after Ulrych et al. 1999)

2.2.2  Example 2 In the second example, the time domain Eigen image algorithm was applied on the synthetic data generated from a reverse fault model with 51 traces. Figures 2.4 and 2.5 respectively depict the noise-free synthetic data and synthetic data contaminated with 30% of noise. The Eigen spectrum computed by applying the Singular Value Decomposition on the noisy synthetic data (Fig. 2.4) is shown in Fig. 2.6. The Eigen spectra of first seven Eigen images are showing decreasing trend. After the seventh Eigen image the spectra is flat. Therefore, the denoised signal which is shown in Fig. 2.7 was reconstructed from the first seven Eigen images. The denoised data shows good match with original pure synthetic data. The residual of reconstruction is also computed by subtracting the denoised data from noisy synthetic data (Fig. 2.8).

2.3  Frequency Domain Eigen Image Processing Fig. 2.4  Pure synthetic data resembling the reverse fault

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2.3  Frequency Domain Eigen Image Processing In frequency domain Eigen image processing, the Fast Fourier Transform (FFT) of the data matrix D was decomposed using SVD. The pseudo code of the methodology can be written as follows.

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Fig. 2.6  Eigen spectrum of noisy synthetic data in time domain

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Step 1: Take the 2D FFT of data matrix D to form m × n complex valued matrix Df Step 2: Decompose Df into Eigen images (similar to 2.1) Step 3: Reconstruct the Df r from the selected Eigen images Step 4: Apply 2D Inverse FFT (IFFT) on Df r to obtain the denoised output in time domain

2.3  Frequency Domain Eigen Image Processing Fig. 2.8  Residual noisy data (Fig. 2.5)-Denoised data (Fig. 2.7) of time domain Eigen image processing

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2.3.1  Example 3 In the second example we have presented frequency domain Eigen image processing. F-xy Eigen image method applied to the noisy synthetic data is shown in Fig. 2.5. The Eigen spectrum of the F-xy Eigen image method is shown in Fig. 2.9. As one can see here, it is almost similar as of the Eigen spectrum of time domain Eigen image method, which was unnoticed earlier by several researchers. The

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Fig. 2.10  Filtered output of noisy data (Fig. 2.5) using frequency domain Eigen image processing

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denoising in the Eigen image processing (time and frequency) depends mainly on the Eigen spectrum. Physically, the similarity of Eigen spectra of time and frequency domain Eigen image techniques suggests that decomposition of the data is domain independent (i.e., number of Eigen images is same either in time domain or frequency domain). As can be seen, after first seven Eigen images there is gradual decrease in Eigenvalue percentage in the frequency domain case also (Fig. 2.10). Therefore, first seven Eigen images are considered to reconstruct the denoised data using the method discussed above. It is interesting to see that there is no change in the denoised output compared to the output of time domain Eigen image method (Fig. 2.7). We can also notice the same by comparing the reconstruction residuals from time (Fig. 2.8) and frequency (Fig. 2.11) domain Eigen image methods. The present comparison suggests the domain insensitivity of SVD for Eigen image analysis.

2.4  Time and Frequency Domain Cadzow Filters Cadzow filtering technique was first introduced by Cadzow (1988) for image filtering. The basic methodology involves the formulation of trajectory matrix from all channel data of a particular time (time slice). Then the trajectory matrix was decomposed using SVD to analyze the Eigenvalue percentages. Finally the denoised data are reconstructed from specific Eigen components. Simply, the trajectory matrix is subjected to time domain Eigen image processing followed by diagonal averaging. The process is repeated for all time slices to recover the data.

2.4  Time and Frequency Domain Cadzow Filters Fig. 2.11  Residual (noisy data (Fig. 2.5)-Denoised data (Fig. 2.10)) of frequency domain Eigen image processing

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2.4.1  Pseudo Code of Time Domain Cadzow Filter For each time slice 1. Form the n by n matrix A whose elements are the reflection amplitudes of each trace for this time. 2. Apply SVD on A to analyze its Eigen spectrum. 3. Reconstruct A from k significant Eigen images and apply diagonal averaging.

2.4.2  Pseudo Code of Frequency Domain Cadzow Filter Apply FFT on each trace of the 2D data. For each frequency 1. Form the n by n matrix A whose elements are the reflection amplitudes of each trace for this frequency. 2. Apply SVD on A to analyze its Eigen spectrum. 3. Reconstruct A from k significant Eigen images and apply diagonal averaging. Similar to the difference between time and frequency domain Eigen image methods, the individual frequency slice of 2D seismic data in the frequency domain (after applying the FFT on individual trace) are processed using the above pseudo code to get the denoised output Fx-Cadzow filter or Fx-Eigen image. If there are N sampling times, it is required to analyze N Eigen spectra ­independently to recover the denoised data. However, it is not possible to show all

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the Eigen spectra here. As suggested by Trickett (2002), the Fx-Eigen image filter/ frequency domain Cadzow filter is dip dependent. Accordingly, if there are five dipping events, the denoised data can be reconstructed from first five Eigen images. In the next example (Example 4), we discuss the comparison of time and frequency domain Cadzow filters by applying the methods to the noisy synthetic data as shown in Fig. 2.5.

2.4.3  Example 4 The outputs of the time and frequency domain Cadzow filters are shown respectively in Fig.  2.12a, b. In time domain Cadzow filtering, the denoised data was reconstructed by analyzing the Eigen spectrum. Number of Eigen images considered for signal reconstruction purely depends on first Eigen image contribution. Here, the Eigen image selection criterion is based on Eigenvalue percentage. If the Eigenvalues percentage of first Eigen image is greater than 18, then first six Eigen images are selected for reconstruction of the filtered data. First four Eigen images are used if the Eigenvalue percentage of first Eigen image is between 15 and 18. Otherwise first two Eigen images are used for signal recovery. These numbers are relative and vary from data to data. In frequency domain Cadzow, the data was initially padded with zeros to see that the number of data points is 2n for fulfilling the criteria for FFT.  Accordingly, 2n data points were prepared in each trace before applying FFT. Then the data was processed using the algorithm presented in pseudo code. The denoised signal was recovered from first three Eigen images as the output with less or more number of Eigen images show more deviation from pure synthetic a

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Fig. 2.12  Denoised output of (a) time domain Cadzow filter and (b) frequency domain Cadzow filter

2.5 Conclusion

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Fig. 2.13  Reconstruction residual of (a) time domain Cadzow and (b) frequency domain Cadzow filters

data. It can be seen from the figure that the reflections are more smoothened in the frequency domain Cadzow output. The reconstruction residuals are computed from both the data sets. The reconstruction residuals of time and frequency domain Cadzow filters are shown respectively in Fig. 2.13a, b. Comparatively, the frequency domain residual contains strong positive or negative residual indicating either boosting of signals or loss of signal (at green circles) in the reconstructed data. Although the denoised signal was reconstructed using more number of Eigen images in time domain, our code shows that the time domain method is faster than the frequency domain method. This time difference is mainly connected with the domain conversion procedure. The reconstruction time are computed on a laptop with Inter (R) Core (TM) 2Duo process (2.24G. Hz) on 64 bit windows operating system in sequential mode. The time and frequency domain Cadzow filtering discussed above are applied to 1D data (of a constant time or frequency) and are extended to more dimension by formulating the trajectory matrix from the trajectory matrices of 1D data as discussed by Trickett (2003, 2008). However, these methods are identical to the TXSSA and FXSSA discussed in the next chapter.

2.5  Conclusion The methodologies of time and frequency domain Eigen image and Cadzow techniques were discussed in this chapter. Results of time and frequency domain Eigen image methods are compared on the synthetic data for denoising. The results s­ uggest

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that there is no change in the time and frequency domain Eigen image process. Staring from the Eigen Spectrum, the denoised data and reconstruction residuals obtained from time and frequency domain Eigen image methods agree well with each other. The denoised data of time and frequency domain Eigen image methods show 70% (mean correlation coefficient 0.69) match with the original synthetic data. In case of Cadzow filtering, there is, however, mismatch in the time and frequency domain denoised outputs. Although the output looks more clean in frequency domain, the residual analysis indicate the presence of artefacts (gain or loss in signal amplitudes). On an average the outputs from both the methods show nearly 65% correlations with the original data. Thus, it may be suggested that there is no added advantage in frequency domain in both Eigen image and Cadzow filtering methods.

Chapter 3

Singular Spectrum Analysis-Based Time Domain Frequency Filtering

3.1  Introduction The SVD-based time and frequency domain Eigen image and Cadzow filters were compared and discussed in the previous chapter for random noise attenuation based on coherency of signal. The time domain frequency filtering of seismic traces using Singular Spectrum Analysis (SSA) method, a data adaptive time domain filtering mechanism, is presented in this chapter. Frequency filtering methods are in use for filtering the band limited seismic reflection data using transforms like Fourier, Wavelet, and Curvlet (Canales 1984; Bracewell 1986; Yilmaz 1987; Foufoula and Kumar 1994; Abma and Claerbout 1995; Yilmaz 2001; Karsli et al. 2006; Hennenfent and Herrmann 2006; Rekapalli et al. 2014). But the decomposition of seismic signals containing abrupt changes/ sudden jumps in amplitude and phase using fixed (like sine and cosines) or semi fixed (mother wavelets) basis functions and then suppressing the coefficients corresponding to the frequencies to be filtered could lead to the artifact generation in the processed output (Claerbout 1976; Sheriff and Geldart 1983; Rekapalli et  al. 2014). For example, the Fourier transform-based analysis of sudden change like step function requires infinite number of harmonics, which limits the accurate reconstruction of such changes (Bath 1974; Dimri 2013; Bansal and Dimri 2001, 2005). However, such abrupt jumps and/or discontinuities in signal can be reconstructed precisely using a single pair of Eigen modes in SSA (Ghil and Taricco 1997; Rekapalli et al. 2014). As the shape and spectral content of different Eigen modes of seismic data differ significantly, usage of data adaptive basis function (i.e., Eigenvectors) in SSA technique helps for more precise signal reconstruction (Yiou et  al. 2000; Rekapalli et al. 2014). The SSA technique have been applied for filtering the geophysical data and astronomical images (Zotov 2012), frequency interpretation and filtering (Harris © Capital Publishing Company 2020 R. K. Tiwari, R. Rekapalli, Modern Singular Spectral-Based Denoising and Filtering Techniques for 2D and 3D Reflection Seismic Data, https://doi.org/10.1007/978-3-030-19304-1_3

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3  Singular Spectrum Analysis-Based Time Domain Frequency Filtering

and Yuan 2010; Rekapalli et al. 2014). According to synthetic modelling of Bozzo et al. (2010), the filtering output of non-noisy periodic data using SSA and Fourier-­ based methods are comparable. Their study indicates that the filtering of noisy signal using SSA is more accurate and appropriate than using Fourier transform-based methods. Signal decomposition using data-adaptive Eigen modes or basis functions of the trajectory matrix in SSA provides a robust way to artifact free filtering in time domain. A technique based on SVD has been used (Trickett 2003) to decompose the seismic images to suppress noise. In a recent study, Oropeza and Sacchi (2011) have applied multichannel SSA (MSSA) on frequency domain seismic data for the suppression of random noise. In both the studies, the methods (SVD or MSSA) are applied on the seismic data converted into the frequency domain using Fourier transform. As discussed before, the artefacts generated due to the domain conversion may be further enhanced in the process of singular spectral de-noising/filtering. However, domain conversion related artefacts could be successfully eliminated by applying the SSA on the time domain seismic data (Rekapalli et al. 2014). Here we have demonstrated the time domain SSA-based frequency-filtering technique to alleviate effect of artefacts and to facilitate better reconstruction of signal with abrupt changes in seismic reflection data. Some test examples of its application on the noisy synthetic seismic data as well as on real seismic reflection data from Singareni coalfields are illustrated to judge efficacy of the method. The new Weighted Eigen Spectrogram based triplet selection procedure is also discussed for user-friendly triplet grouping process in SSA-based frequency filtering for its wider applicability.

3.2  Methodology The SSA-based frequency-filtering starts with the formation of trajectory matrix using appropriate window length. Here, individual traces from the 2D data are used to form the trajectory matrix. The selection of optimal window length (L) is essential in SSA or SSA frequency filtering. Within the classical limit (i.e., 2  140 Hz. The Eigen triplets from 5 to 17 with frequency > 30 Hz and   20  Hz and   85 Hz as shown in Fig. 6.11. However, the matching of low frequency component of the WSSA filtered data and original data demonstrate the efficacy of WSSA method for preserving the low frequency content.

6.6  Conclusion In this chapter, robust algorithms are demonstrated to (i) remove the coloured noise to enhance the signal to noise ratio and (ii) reduce the computational cost in SSA based seismic data processing method. These algorithms are faster and robust in comparison to the existing conventional methods. The inclusion of optimization process considerably suppresses the erratic as well as coloured noises. The R-SVD method effectively reduces the computational cost and is useful in SSA and MSSA based seismic data processing. The factorized Hankel SVD and Windowed SSA

6.6 Conclusion

83

Fig. 6.10  Original along with the windows on which the WSSA methods were applied and WSSA reconstructed seismic traces. (Modified after Rajesh and Tiwari (2015))

Power (amp^2/Hz)

1010

108

106

104

102

PSD of WSSA reconstructed trace PSD of original trace

100

101 Frequency (Hz)

Frequnecy cutoff 102

Fig. 6.11  Power spectral density of reconstructed and original seismic trace. (Modified after Rajesh and Tiwari (2015))

84

6  Robust and Fast Algorithms for Singular Spectral Analysis of Seismic Data

methods reduce the computational cost without affecting the low frequency content in signal reconstruction. Comparative study performed to reduce the coloured noise from the post stack seismic data shows that the FHOTSSSA method is efficient among the all. The optimization process in the FHOSSSA algorithm serves to recover the true curvature of the reflectors (real feature) in the denoising. The usage of R-SVD in factorized Hankel SVD and Windowed SSA would be more prospective future approach to reduce the computational time.

Chapter 7

Denoising the 3D Seismic Data Using Multichannel Singular Spectrum Analysis

7.1  Introduction Interpreting the out of plane seismic reflections is more appropriate in 3D seismic studies, which confuses the interpretation of 2D data. As discussed for 2D data in previous chapters, the noises and data gaps also complicate the 3D seismic data processing and subsequently cause difficulties in the geological interpretation. Several methods have been reported in literature filtering the random noise from seismic data (Canales 1984; Abma and Claerbout 1995; Chakraborthy and Okaya 1995; Yilmaz 2001; Karsli et al. 2006; Herrmann et al. 2008; Sacchi 2009; Oropeza and Sacchi 2011; Tiwari et al. 2014; Rekapalli and Tiwari 2015b; Rekapalli et al. 2017) Majority of the noise filtering algorithms involve conversion of data into frequency domain using Fourier transform or Radon transform (Darche 1990; Sacchi et al. 1998; Duijndam et al. 1999; Trad et al. 2002; Liu and Sacchi 2004; Herrmann et al. 2008; Sacchi 2009; Oropeza and Sacchi 2011). In general, however, conversion of data into frequency domain is done assuming that the data are stationary in nature. But the seismic data exhibit spatio-temporal non-stationary and non-linear behaviour. As discussed in previous chapters, the Singular Spectrum Analysis (Broomhead and King 1986a, b; Vautard and Ghil 1989; Golyandina et al. 2001) has been found to be robust tool to examine such non-stationary and non-linear seismic amplitudes (Rekapalli and Tiwari 2015b; Rekapalli et al. 2015a; Rekapalli et al. 2016) in time domain. Researchers have successfully denoised 3D seismic data using frequency and time domain Multichannel SSA methods (Oropeza and Sacchi 2011; Rekapalli et al. 2017). However, similar to the problems seen in the frequency domain methods discussed in previous chapters, original seismic amplitudes are corrupted by the domain conversion related artefacts in the frequency domain MSSA. The time domain MSSA based Multichannel Time Slice SSA (MTSSSA) method (Rekapalli et al. 2015a, 2017), which filters the individual time slices of 3D volume, s­ urmounts © Capital Publishing Company 2020 R. K. Tiwari, R. Rekapalli, Modern Singular Spectral-Based Denoising and Filtering Techniques for 2D and 3D Reflection Seismic Data, https://doi.org/10.1007/978-3-030-19304-1_7

85

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7  Denoising the 3D Seismic Data Using Multichannel Singular Spectrum Analysis

such domain conversion related issues. In addition to noise filtering, Oropeza and Sacchi (2011) and Rekapalli et al. (2017) have demonstrated iterative algorithms for missing data reconstruction, which will be discussed in Chap. 8. The MTSSSA method relies on (i) spatial data series of constant time (loosely from single layer) holds high correlation, and (ii) noise can be identified in the tails of the Eigen/ Singular spectrum with insignificant Eigenvalues (Rekapalli et al. 2017). Whereas the frequency domain method involves the application of MSSA method on the constant frequency slices of 3D seismic data. The frequency domain MSSA and MTSSSA methods are simple extension of FXSSA and TXSSA using the advantage of Multichannel SSA (MSSA) (Ghil et al. 2002; Oropeza and Sacchi 2011; Rekapalli et al. 2017). In this chapter, the Multi-channel Singular Spectrum Analysis (MSSA) based time and frequency domain algorithms are discussed for 3D seismic data denoising in time and frequency domains. Similar to the 2D data processing using SVD and SSA based methods discussed in the previous chapters, noises from the individual time or frequency slices of 3D volumes are suppressed through the analysis of Eigen triplets of the trajectory matrix. Thus, we aim to demonstrate the frequency domain MSSA (Oropeza and Sacchi 2011) and MTSSSA (Rekapalli et al. 2017) algorithm for 3D seismic data denoising on synthetic data contaminated with mixed noise. Some relevant examples of application of algorithm to 3D field data from Sleipner offshore CO2 storage site are demonstrated.

7.2  Methodology Figure 7.1 depicts the analogy between 2D and 3D seismic data. The 2D data and 3D data volume (D) and Time Slice are shown in Fig. 7.1. As discussed above the Multichannel SSA is applied on individual time slices of the 3D data in the proposed Multichannel Time Slice SSA. The X, Y and Z directions in Fig. 7.1 respectively represent the inline, cross line and depth axis directions. The total length of the recording time is a synonym here for depth along z direction. The application of Multichannel SSA enables us to identify the coherency among the data from a fixed time slice or horizon to filter out noises as they tend to map onto the tail of the Eigen spectrum. Individual time slice of 3D seismic record given by S(x, y, t) is subjected to MSSA. Although the dimensionality is higher, the methodology comprises the same trajectory formulation, singular value decomposition, grouping and diagonal averaging steps as discussed in Chap. 4 in the case of FXSSA and TXSSA. The only difference is that here the method was applied to the time slices instead of depth slices. Accordingly, for 3D case, let us consider pth time slice as given by

7.2 Methodology

87

Fig. 7.1  Schematic depicting the analogy between 2D and 3D data along with 3D synthetic data. Physical representation of the time slice used in the present study (red colour rectangle)



 s (11  s (1,N y ,p )  , ,p ) s (1,2,p )      S ( x,y,p )( N × N ) =   x y    s ( N x ,1,p ) s ( N x ,2,p )  s ( N x ,N y ,p ) 

(7.1)

The elements s(i, j, p) represent the reflection signal amplitudes at time p (0 > L=6; >> N=length(x); >> K=N-L+1; >>for i=1:K T(:,i)=x(i:i+L-1,1); end

The above ‘for loop’ will create the trajectory matrix ‘T’ of size L × K. (ii) Singular Value Decomposition: The trajectory matrix is decomposed into Eigenvectors (Ui, Vi) and Eigenvalues using Singular Value λi

( )

Decomposition (SVD) operation. Follow the mathematical representation given below d



i.e.T = ∑ λ i U i Vi T i =1

(11.2)

Here the group (√λi, Ui, Vi) is called the ith Eigen triplet and λ1  >  λi  >  0 for 1 ≤ i ≤ d. Here ‘d’ represents the number of Eigen triplets with non-zero Eigenvalue. Now, we can apply SVD on trajectory matrix ‘T’ obtained in the workspace (from the previous MATLAB exercise) to compute Eigenvectors and Eigenvalues of T as follows. For more details, refer appendix for mathematical description on SVD with examples. >> [U S V] =svd(T);

After executing the above command, the Eigenvectors U and V of sizes 6 × 6 and 7 × 7 and the Eigenvalues matrix S of size 6 × 7 are computed and stored in the work space. Now we can plot the Eigenvalues percentages or Singular spectrum as follows.

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>> plot(1:l,100∗diag(S)/sum(diag(S)),’-rd’) >> title (‘Eigen Spectrum’, ‘FontName’, ‘Times New Roman’, ‘FontWeight’, ‘bold’, ‘FontSize’, 12); >>xlabel(‘Eigen Triplet Number’, ‘FontName’, ‘Times New Roman’, ‘FontWeight’, ‘bold’ ,‘FontSize’,12); >>ylabel(‘Eiegn Value Percentage’,‘FontName’,‘Times New Roman’, ‘FontWeight’, ‘bold’, ‘FontSize’,12); >>set(gca,‘FontName’, ‘TimeNewRoman’,‘FontSize’,12, ‘ FontWeight’, ‘bold’);

One can notice from the Eigen spectrum that the first two eigen modes are only with non-zero Eigenvalue percentage.

11.2.2  Signal Reconstruction The signal reconstruction of SSA involves the following two steps. (i) Grouping and Reconstruction: The second important aspect in the SSA based signal processing is appropriate selection of Eigen triplet group for the reconstruction of denoised signal or principal components. The estimate of weighted correlation (Wc) among principal components is useful to verify the accuracy of grouping at a specific window length (Ghil and Taricco 1997). According to Trickett (2003), the Eigen triplets can be selected based on their variance. Hassani et al. (2012) have discussed the selection of Eigen triplets using a new concept called r-separability for better separation of noise from signal. The weighted correlation method is a simple and reliable method for selecting appropriate window length for good separation of linearly independent Eigen modes. Once the group ‘I’ of linearly independent Eigen triplets are selected for required signal reconstruction, the reconstructed trajectory matrix (Tr) can be obtained using the following equation



Tr = ∑ λ i U i Vi T i∈I

(11.3)

Here ‘I’ represents the group or groups of selected Eigen triplets.



 y(1,1)  y(1,K )       Tr =    y  ( L,1)  y( L,K ) 

(11.4)

11.2 MATLAB® Coding: Description and Application

129

Here, y(i, j) denote the elements of reconstructed trajectory matrix where 1 ≤ i ≤ L and 1 ≤ j ≤ K. Coming to our MATLAB exercise, only first two eigen modes are selected to recover the clean signal as they are only having non-zero Eigenvalues. But in real data case there may be several Eigen modes with non-zero Eigenvalues. We form a vector comprising the Eigen mode numbers, which are used for signal reconstruction as follows. >> G=[1,2];

Then we reconstruct the trajectory matrix using the following loop. >> for i=1: length(G) if i==1 T_R=U(:,G(1,i))∗V(:,G(1,i))’∗S(G(1,i),G(1,i)); else T_R=T_R+U(:,G(1,i))∗V(:,G(1,i))’∗S(G(1,i),G(1,i)); end end

(ii) Diagonal averaging: Finally, the elements of reconstructed series Yr = {g1, g2, … gk … … .. gN}are obtained from the diagonal averaging of reconstructed trajectory matrix using the following equations 1 k +1 ∑ym,k −m + 2 for1 ≤ k < L k + 1 m =1

(11.5a)

1 L ∑ym,k −m + 2 forL ≤ k < K L m =1

(11.5b)

1 N − K +1 ∑ ym,k −m + 2 for K ≤ k ≤ N N − k m = k −K  2

(11.5c)

gk =



= =



Now, let us move to MATLAB workspace and command line to write simple coding for the above diagonal averaging. The reconstructed trajectory matrix T_R is now diagonally averaged as follows:

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11  Singular Spectrum Analysis with MATLAB®

>> g=zeros(N,1); >>Lp=min(L,K); >>Kp=max(L,K); >>for s=0:Lp-2 for m=1:s+1 g(s+1)=X(s+1)+(1/(s+1))∗T_R(m,s-m+2); end end >>for s=Lp-1:Kp-1 for m=1:Lp g(s+1)=X(s+1)+(1/(Lp))∗T_R(m,s-m+2); end end >>for s=Kp:N for m=s-Kp+2:N-Kp+1 g(s+1)=X(s+1)+(1/(N-s))∗T_R(m,s-m+2); end end

The above diagonal averaging will reproduce the original time series g  =  {1, 2,3,4,5,6,7,8,9,10,11,12}.

11.2.3  Eigen Modes and their Separation In this section, the ‘Weighted correlation method’ was discussed for its robustness in appropriate signal decomposition and reconstruction at a selected window length. As discussed in the previous chapters, appropriate caution should be exercised while choosing the window length. Smaller window length fails to provide well resolved Eigen spectrum and decomposes the data into fewer components. Hence, at smaller window length the decomposition would suffer from poor Eigen mode separation. Among the Eigen modes, some of them are linearly dependent and are grouped to reconstruct a single principal component for the interpretation of underlying dynamics (Ghil et  al. 2002; Serita 2005; Tiwari and Rajesh 2014; Rajesh and Tiwari 2015; Rekapalli and Tiwari 2015a;

11.2 MATLAB® Coding: Description and Application

131

Tiwari et  al. 2014; Rekapalli and Tiwari 2018). There are two fundamental ways for identifying such paired or individual Eigen modes. The first method relies on the analysis of Eigenvalues of the Eigen modes using the Eigen spectrum. The significant/major principal components appear in the first few Eigen modes with large variance in the Eigen spectrum. The linearly dependent or paired Eigen modes will have nearly same Eigenvalue. Therefore, analysis of Eigen/Singular values or Eigen/Singular spectrum of the trajectory matrix helps to group the Eigen modes for the reconstruction of principal component or denoised signal. The other method is the analysis of weighted correlation among the Eigen modes. The separation between the principal components and accuracy of grouping at a specific window length can also be inferred from the weighted correlation (Wc) between pairs of Eigen components using the Eq. (11.6) (Ghil and Taricco 1997).

( Pc ,Pc ) i

Wc =

j

w

Pc i w ⋅ Pc i

(11.6) w



where

( Pc ,Pc )

Pc i w =

i

( Pc ,Pc ) = ∑w .Pc Pc , w i



N

i

j

w

k

k =1

i k

j k

k

w

= min {k,L,N − k}



Here L is window length used in the trajectory matrix formulation (i = 1, 2, … d and j = 1, 2, … d). These components are said to be well resolved if the Wc is nearer to zero. The individual Eigen components can be reconstructed from the corresponding Eigen triplet. The weighted correlation discussed above is a post decomposition procedure, which is useful for triplet selection in grouping. But the same can also be used for selecting an optimal window length through an iterative computation of Wc. This will help to achieve maximum correlation along the principal diagonal of Wc matrix by varying the window length within the classical limit. The following function computes and plots the weighted correlation from a matrix of L Eigen modes.

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11  Singular Spectrum Analysis with MATLAB®

function weightedcorr(PC) %PC----> L number of Eigen modes computed from data (size of N×L) N=size(PC,1); L=size(PC,2); K=N-L+1; fori=1:N ifiK+1 w(i,1)=N-i; else w(i,1)=L; end end cc=zeros(L,L); fori=1:L for j=1:L cc(i,j)=sum(w.∗PC(:,i).∗PC(:,j))/sqrt((sum(w.∗PC(:,i).∗PC(:,i))∗ sum(w.∗PC(:,j).∗PC(:,j)))); end end imagesc(cc); title (‘Weighted Correlation’, ‘FontName’, ‘Times New Roman’, ‘FontWeight’, ‘bold’, ‘FontSize’, 12); xlabel(‘Eigen Triplet’, ‘FontName’, ‘Times New Roman’, ‘FontWeight’, ‘bold’ ,‘FontSize’,12); ylabel(‘Eiegn Triplet ‘,‘FontName’,‘Times New Roman’, ‘FontWeight’, ‘bold’, ‘FontSize’,12); set(gca,‘FontName’, ‘TimeNewRoman’,‘FontSize’,12, ‘FontWeight’, ‘bold’); set(gca, ‘xdir’,‘normal’); set(gca, ‘ydir’,‘normal’); colormap(‘gray’); caxis([0 1]); colorbar; end

11.3  M  ATLAB® Function for Singular Spectrum Analysis of 1D Data Series The following MATLAB function helps to perform SSA on 1D time/spatial series data.

11.3 MATLAB® Function for Singular Spectrum Analysis of 1D Data Series

133

function [X_R,Ev] = SSA1D(X,L) %UNTITLED Summary of this function goes here % Inputs: % X----> input data vector % L---->windowlength % Outputs: % X_R ----> Reconstructed series % Ev ----> Eigenvalue percentages of Eigen modes % This function allow the user to compute the Principal compoents % de-noised signals etc., using Eigen spectrum and Weighted correlations if nargin