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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2022
A practical and insightful discussion of time-frequency analysis methods and technologies Time-frequency analysis of seismic signals aims to reveal the local properties of nonstationary signals. The local properties, such as time-period, frequency, and spectral content, vary with time, and the time of a seismic signal is a proxy of geologic depth. Therefore, the time-frequency spectrum is composed of the frequency spectra that are generated by using the classic Fourier transform at different time positions. Different time-frequency analysis methods are distinguished in the construction of the local kernel prior to using the Fourier transform. Based on the difference in constructing the Fourier transform kernel, this book categorises time-frequency analysis methods into two groups: Gabor transform-type methods and energy density distribution methods. This book systematically presents time-frequency analysis methods, including technologies which have not been previously discussed in print or in which the author has been instrumental in developing. In the presentation of each method, the fundamental theory and mathematical concepts are summarised, with an emphasis on the engineering aspects. This book also provides a practical guide to geophysicists who attempt to generate geophysically meaningful time-frequency spectra, who attempt to process seismic data with time-dependent operations for the fidelity of nonstationary signals, and who attempt to exploit the time-frequency space seismic attributes for quantitative characterisation of hydrocarbon reservoirs.
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Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
1 Nonstationary Signals and Spectral Properties
1.1 Stationary Signals
1.2 Nonstationary Signals
1.3 The Fourier Transform and the Average Properties
1.4 The Analytic Signal and the Instantaneous Properties
1.5 Computation of the Instantaneous Frequency
1.6 Two Groups of Time‐Frequency Analysis Methods
2 The Gabor Transform
2.1 Short‐time Fourier Transform
2.2 The Gabor Transform
2.3 The Cosine Function Windows
2.4 Spectral Leakage of Window Functions
2.5 The Gabor Limit of Time‐Frequency Resolution
2.6 Implementation of the Gabor Transform
2.7 The Inverse Gabor Transform
2.8 Application in Inverse
Filtering
Notes
3 The Continuous Wavelet Transform
3.1 Basics of the Continuous Wavelet Transform
3.2 The Complex Morlet Wavelet
3.3 The Complex Morse Wavelet
3.4 The Generalised Seismic Wavelet
3.5 The Pseudo‐frequency Representation
3.6 The Inverse Wavelet Transform
3.7 Implementation of the Continuous Wavelet Transform
3.8 Hydrocarbon Reservoir Characterisation
4 The
Transform
4.1 Basics of the
Transform
4.2 The Generalised
Transform
4.3 The Fractional Fourier Transform
4.4 The Fractional
Transform
4.5 Implementation of the
Transforms
4.6 The Inverse
Transforms
4.7 Application to Clastic and Carbonate Reservoirs
5 The
Transform
5.1 Basics of the
Transform
5.2 The Generalised
Transform
5.3 Implementation of Nonstationary Convolution
5.4 The Inverse
Transform
5.5 Application to Detecting Hydrocarbon Reservoirs
5.6 Application to Detecting Karst Voids
6 The Wigner‐Ville Distribution
6.1 Basics of the Wigner‐Ville Distribution (WVD)
6.2 Defining the WVD with an Analytic Signal
6.3 Properties of the WVD
6.4 The Smoothed WVD
6.5 The Generalised Class of Time‐Frequency Representations
6.6 The Ambiguity Function and the Generalised WVD
6.7 Implementation of the Standard and Smoothed WVDs
6.8 Implementation of the Ambiguity Function and the Generalised WVD
7 Matching Pursuit
7.1 Basics of Matching Pursuit
7.2 Three‐stage Matching Pursuit
7.3 Matching Pursuit with the Morlet Wavelet
7.4 The Sigma Filter
7.5 Multichannel Matching Pursuit
7.6 Structure‐adaptive Matching Pursuit
7.7 Three Applications
8 Local Power Spectra with Multiple Windows
8.1 Multiple Orthogonal Windows
8.2 Multiple Windows Defined by the Prolate Spheroidal Wavefunctions
8.3 Multiple Windows Constructed by Solving a Discretised Eigenvalue Problem
8.4 Multiple Windows Constructed by Gaussian Functions
8.5 The Gabor Transform with Multiple Windows
8.6 The WVD with Multiple Windows
8.7 Prospective of Time‐Frequency Analysis without Windowing
Appendices
A The Gaussian Integrals, the Gamma Function, and the Gaussian Error Functions
B Fourier Transforms of the Tapered Boxcar Window, the Truncated Gaussian Window, and the Weighted Cosine Window
C The Generalised Seismic Wavelet in the Time Domain
D Implementation of the Fractional Fourier Transform
E Marginal Properties and the Analytic Signal in the WVD Definition
F The Prolate Spheroidal Wavefunctions, the Associated and the Ordinary Legendre Polynomials
References
Author Index
Subject Index
End User License Agreement
List of Illustrations
Chapter 1
Figure 1.1 The principle of 'superposition'. Each wavelet is scaled by the r...
Figure 1.2 A stationary convolution process, in which a vector of seismic tr...
Figure 1.3 The nonstationary convolution process. The wavelet matrix
is fo...
Figure 1.4 The Fourier transform is the cross‐correlation between the seismi...
Figure 1.5 The concept of the instantaneous frequency. (a) A synthetic trace...
Figure 1.6 (a) A two‐dimensional seismic profile. (b) The instantaneous freq...
Chapter 2
Figure 2.1 Two boxcar window functions and their corresponding frequency spe...
Figure 2.2 Two tapered boxcar window functions and their corresponding spect...
Figure 2.3 The Gaussian function (with
), and the corresponding spectrum. ...
Figure 2.4 The Hanning window function (with
) and the corresponding frequ...
Figure 2.5 The Hamming window function (with
) and the corresponding frequ...
Figure 2.6 The Blackman window function (with
) and the corresponding freq...
Figure 2.7 A Blackman‐Harris window function with four terms (with
) and t...
Figure 2.8 The Blackman‐Harris window functions, with
and
, and the corre...
Figure 2.9 The amplitude spectra (in
) of four window functions: (a) the b...
Figure 2.10 The amplitude spectra (in
) of four cosine‐function windows: (...
Figure 2.11 A field seismic trace (a) and the corresponding time‐frequency s...
Figure 2.12 The field seismic trace (a) and the corresponding time‐frequency...
Figure 2.13 The field seismic trace (a) and the corresponding time‐frequency...
Figure 2.14 The field seismic trace (a) and the corresponding time‐frequency...
Figure 2.15 The synthesis scaling factor for the inverse Gabor transform. Ea...
Figure 2.16 The inverse
filtering based on the Gabor transform. (a) A synt...
Figure 2.17 The inverse
filtering based on the Gabor transform. (a) Synthe...
Figure 2.18 The effectiveness of inverse
filtering. (a) An example of a fi...
Chapter 3
Figure 3.1 The Gabor wavelets and the dilated wavelets. (a) The Gabor wavele...
Figure 3.2 The Morlet wavelet, defined with the central frequency
and the ...
Figure 3.3 The standard zeroth‐order Morse wavelet, defined with the paramet...
Figure 3.4 The modified zeroth‐order Morse wavelet, defined with parameters
Figure 3.5 The frequency spectra of the modified Morse wavelets, defined wit...
Figure 3.6 The modified Morse wavelets (real part) of the first (a), second ...
Figure 3.7 The generalised seismic wavelets, defined with the reference freq...
Figure 3.8 (a) The generalised seismic wavelets defined with fractional orde...
Figure 3.9 An example of the generalised seismic wavelet, defined with fract...
Figure 3.10 (a) The example field seismic trace. (b) The time‐frequency spec...
Figure 3.11 The depth structure (a) and an iso‐frequency spectral slice (b) ...
Figure 3.12 The sand channels (ellipses) within the target layer. (a) An exa...
Figure 3.13 The estimated thickness of the sand bodies distributed over the ...
Chapter 4
Figure 4.1 The time‐frequency spectrum produced by the standard
transform,...
Figure 4.2 The time‐frequency spectrum produced by the generalised
transfo...
Figure 4.3 The fractional Fourier transform
is a rotation from the time ax...
Figure 4.4 The fractional Fourier transforms. (a) The amplitude spectra prod...
Figure 4.5 The time‐frequency spectrum produced by the fractional
transfor...
Figure 4.6 The inverse fractional
transform. (a) The original seismic trac...
Figure 4.7 A seismic profile and the iso‐frequency profiles. Gas reservoirs ...
Figure 4.8 Detection of potential reservoirs within the carbonate formation ...
Chapter 5
Figure 5.1 Comparison between the
transform and the
transform. (a) A syn...
Figure 5.2 Time‐frequency spectra of a field seismic trace. (a) The field se...
Figure 5.3 The Gaussian spread parameter
versus the trend factor
. For
,...
Figure 5.4 The generalised window function
versus the trend factor
. (a) ...
Figure 5.5 The generalised window function
with different trend factors,
Figure 5.6 The generalised
transform with different
values. (a) The synt...
Figure 5.7 Comparison between the ordinary and the generalised
transforms....
Figure 5.8 The Rényi energy of the
transform spectra. The dashed curve and...
Figure 5.9 The time‐frequency spectrum produced by the
transform. (a) The ...
Figure 5.10 The time‐frequency spectrum produced by the
transform. (a) The...
Figure 5.11 An anomalous zone (marked by the ellipse) in a seismic profile a...
Figure 5.12 A selected seismic trace and the spectra. (a) The seismic trace ...
Figure 5.13 The amplitude spectra produced by the
transform. (a‐d) The iso...
Figure 5.14 Cores reveal near‐surface geological anomalies, that correspond ...
Chapter 6
Figure 6.1 The joint time‐frequency energy density
. The horizontal band is...
Figure 6.2 WVDs defined by the real and the analytic signals. (a) A linear c...
Figure 6.3 The cross‐term interference in the WVD. The location of the cross...
Figure 6.4 The smoothing effect in the WVD. (a) A synthetic seismic trace co...
Figure 6.5 Selecting the parameter pair
for the smoothing filter. The colo...
Figure 6.6 An example signal, consistin of two Gabor wavelets, and the ambig...
Figure 6.7 The generalised ambiguity function and the generalised WVD. The g...
Figure 6.8 An analytic trace
, formed by a synthetic trace (black curve) wi...
Figure 6.9 The analytic trace
, and the WVD implemented with the definition...
Figure 6.10 The WVD implemented using the WVD definition in the frequency do...
Figure 6.11 The smoothed WVD. The two‐dimensional smoothing process can effe...
Figure 6.12 The example of a field seismic trace, the standard WVD
, and th...
Figure 6.13 The ambiguity function
, which is complex. (a) The analytic sig...
Figure 6.14 The generalised WVD. (a) The analytic signal
. (b) The real par...
Figure 6.15 The analytic signal
, and the final generalised WVD, which is t...
Figure 6.16 The field seismic trace, the ambiguity function
, and the final...
Chapter 7
Figure 7.1 Time‐frequency spectra generated by different methods. (a) A fiel...
Figure 7.2 A synthetic seismic trace, the wavelets decomposed by matching pu...
Figure 7.3 The Gabor transform with different window sizes. The spread param...
Figure 7.4 An example of a field seismic trace, decomposed wavelets with dif...
Figure 7.5 Sigma
filtering. The time‐frequency spectrum, corresponding wav...
Figure 7.6 Completeness of multichannel matching pursuit. (a) A field seismi...
Figure 7.7 Structurally adaptive matching pursuit. (a) A group of synthetic ...
Figure 7.8 Comparison between the continuous wavelet transform and the multi...
Figure 7.9 Extracting strong coal‐seam reflections using multichannel matchi...
Figure 7.10 Detecting low‐frequency shadows of gas reservoirs by multichanne...
Chapter 8
Figure 8.1 Multiple windows defined by the prolate spheroidal wavefunctions....
Figure 8.2 Multiple windows constructed by solving a discretised eigenvalue ...
Figure 8.3 Comparison between the window functions defined in terms of the p...
Figure 8.4 The amplitude spectra of multiple windows constructed by the disc...
Figure 8.5 Multiple windows constructed by Gaussian functions, with the (hal...
Figure 8.6 Multiple windows constructed by Gaussian functions, with the (hal...
Figure 8.7 Spectral analysis with multiple windows, at a given time
. The o...
Figure 8.8 The Gabor transform using a single or multiple window function. (...
Figure 8.9 The WVD with multiple windows. Each of the multiple orthogonal wi...
Figure 8.10 Comparison between the smoothed WVD and the multi‐window WVD. (a...
Appendices
Figure A.1 The Gaussian error function,
. Three characteristic values of th...
Figure C.1 The generalised seismic wavelets, corresponding to the fractional...
Figure F.1 Solving the coefficients of the prolate spheroidal wavefunctions ...
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
Preface
Begin Reading
Appendices
References
Author Index
Subject Index
End User License Agreement
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Time–Frequency Analysis of Seismic Signals
Yanghua Wang
Imperial College London
This edition first published 2023
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Library of Congress Cataloging‐in‐Publication Data applied for
[HB ISBN: 9781119892342]
Cover Design: Wiley
Cover Image: Courtesy of Yanghua Wang
This book is dedicated to my wife, Guo‐ling, and my two children, Brian and Claire.
Preface
The aim of this book, Time–Frequency Analysis of Seismic Signals, is to reveal the local properties of nonstationary seismic signals. Their local properties, such as time period, frequency, and spectral content, vary with time, and the time of seismic signals is a proxy for geologic depth.
Time–frequency analysis is a generalisation of classical Fourier spectral analysis for stationary signals, by extending it to spectral analysis of nonstationary signals. All methods of time–frequency analysis can be treated as a series of implementations of the classical Fourier transform. The kernel of each Fourier transform is formed from a segment of the original seismic signal. Therefore, the time–frequency spectrum is composed of the Fourier spectra generated at different time positions, where the time position corresponds to the centre of each segment.
Different methods of time–frequency analysis differ in the construction of the local kernel, before the Fourier transform is applied. Because of the differences in the construction of the Fourier transform kernel, the methods of time–frequency analysis in this book are divided into two groups.
The first group of methods are the Gabor transform‐type methods, including the Gabor transform itself (Chapter 2), the continuous wavelet transform (Chapter 3), the S transform (Chapter 4), and the W transform (Chapter 5). This group of methods focuses on the properties of the different forms of window functions. The window function, which acts as a convolution operator, is applied to the nonstationary signal in a sliding manner. The windowed signal segment is the local kernel for the Fourier transform.
The second group of methods are the energy density distribution methods, including the Wigner–Ville distribution (WVD) (Chapter 6), the matching pursuit‐based WVD (Chapter 7), and the local power spectrum with multiple orthogonal windows (Chapter 8). While the resulting time– frequency representation is the energy density distribution, this second group of methods focuses on manipulating and reformulating the original signal segment to form a time‐varying kernel for the Fourier transform.
An essential difference between these two groups of methods is that the Gabor transform‐type methods produce complex spectra in the time– frequency plane, while the energy density distribution methods produce real spectra that do not contain phase information. The Gabor transform‐ type methods have an inverse transform counterpart to reconstruct the original nonstationary signal, and therefore the time–frequency spectra produced by the Gabor transform‐type methods can be used in signal processing. However, the energy density distribution methods cannot recover the original signal because the phase information in the time– frequency plane is missing. Consequently, the time–frequency representation of the energy density distribution methods can only be used as a seismic attribute for geophysical characterisation.
With respect to the collective groupings, this book systematically presents various time–frequency analysis methods, including some techniques that have not yet been published or that the author has been instrumental in developing. In presenting each method, the basic theory and mathematical concepts are summarised, with emphasis on the technical aspects.
As such, this book is a practical guide for geophysicists seeking to produce geophysically meaningful time–frequency spectra, process seismic data with time‐dependent operations to faithfully represent nonstationary signals, and use the seismic attributes of time–frequency space for the quantitative characterisation of hydrocarbon reservoirs.
Yanghua Wang
25 July 2022
1Nonstationary Signals and Spectral Properties
Stationary signals are any idealised signals that have time‐independent properties, such as time period, frequency, and spectral content. Seismic signals, however, are nonstationary signals that violate the stationary rule described above. When seismic waves propagate through the anelastic media in the Earth's subsurface, seismic signals have variable properties that vary with the propagation path and travel time.
A stationary signal can be represented mathematically by a stack of sinusoids, using the Fourier transform. In contrast, a nonstationary signal cannot be properly represented by the Fourier transform because the amplitude and frequency of a sinusoidal representation can change dynamically as a function of travel time. To investigate the local properties of a nonstationary signal, an analytic signal‐based analysis method can be used instead.
1.1 Stationary Signals
For seismic signals, an idealised stationary model is a stationary convolution process. The physical meaning of the convolution process is 'superposition', which is a fundamental principle in the analysis of seismic signals.
Consider an earth model with a layered structure (Figure 1.1). Each layer has a different acoustic impedance, which is the product of velocity and density. The contrast of this physical property between two layers causes seismic reflections at the interface.
Figure 1.1 The principle of 'superposition'. Each wavelet is scaled by the reflectivity, which is the contrast in the acoustic impedance between two layers. A seismic trace is formed by summing all scaled wavelets, which are reflected from various interfaces.
The contrast of acoustic impedance is called reflectivity, or reflection coefficient. The seismic reflectivity includes primary and multiple , where is the travel time. The reflectivity series in time consists of both types, . Each reflectivity serves as a scaling factor to scale a particular wavelet . A recorded seismic trace is formed by summing all scaled wavelets reflected from different interfaces.
This physical process is the 'superposition' and can be written as
where is the vector of the reflectivity series, is the vector of the seismic wavelet, and is the vector of the seismic trace. If the length of the reflectivity series is , and the length of the seismic wavelet is , the length of the resulting seismic trace is .
In the superposition process of Eq. (1.1), the wavelet is time shifted and scaled by the reflectivity. Combining all the time‐shifted wavelets to form a wavelet matrix, , the superposition process can be expressed in a matrix‐vector form, , and written explicitly as
in which the size of the wavelet matrix is .
In the matrix‐vector form of Eq. (1.2), the wavelet matrix is a Toeplitz matrix, because each diagonal of is a constant, . Then, if we look at the wavelet matrix row by row, we can see that each row of the matrix consists of the discretised wavelet samples in time‐reversed order:
Therefore, the matrix‐vector form of superposition, , is the discretised form of 'convolution'. The corresponding continuous form for the convolution process can be expressed as
where is the seismic wavelet, is the subsurface reflectivity series, is the convolution operator, and is the seismic trace. In this convolution process, the convolution kernel is the stationary wavelet , which has a constant time period, frequency, and spectral content, with respect to time variation.
A wavelet is a 'small wave', for which the time period is relatively short compared to the time duration of the reflectivity series and the resulting seismic trace. With the stationary wavelet , the stationary convolution process of Eq. (1.4) can be depicted as in Figure 1.2.
The stationary wavelet, that is used in Figure 1.2 for the demonstration, is the Ricker wavelet defined as (Ricker, 1953)
where is the dominant frequency (in ), and is the central position of the wavelet. The Ricker wavelet is symmetrical with respect to time and has a zero mean, . Therefore, it is often known as Mexican hat wavelet in the Americas for its sombrero shape.
Figure 1.2 A stationary convolution process, in which a vector of seismic trace is generated by the multiplication of a wavelet matrix and a vector of reflectivity series . Each column vector of the wavelet matrix is formed by a stationary seismic wavelet .
1.2 Nonstationary Signals
The stationary convolution process from the previous section is an idealised model for seismic signals. In reality, however, seismic signals are nonstationary due to the dissipation effect when seismic waves propagate through the subsurface anelastic media.
The nonstationary convolution process can be expressed as follows,
where is a nonstationary seismic wavelet, in place of the stationary wavelet . The nonstationary seismic wavelet evolves continually according to a dissipation model:
where is the dissipation coefficient, and acts on the idealised stationary wavelet .
In the convolution expression of Eq. (1.7) for the nonstationary wavelet , the dissipation coefficient , within which indicates the time dependency, is nonstationary. At any given time position , the dissipation coefficient can be defined by
where f is the frequency, is the frequency‐dependent attenuation coefficient, and is the associated dispersion, i.e. the phase delay of different frequency components (Futterman, 1962).
The attenuation coefficient of seismic waves propagating through the subsurface anelastic media is (Wang & Guo, 2004)
and the associated dispersion is
where is a reference frequency,
and is the quality factor of the subsurface anelastic media (Kolsky, 1956; Futterman, 1962; Wang, 2008). Because seismic signals have a relatively narrow frequency band, it is reasonable to assume to be frequency independent. But is time dependent, and the time here is a proxy for geologic depth.
For numerical calculation, the nonstationary convolution process of Eq. (1.6) may also be expressed in a matrix‐vector form as
where is the nonstationary wavelet matrix. Figure 1.3 depicts this matrix‐vector form of the nonstationary convolution process.
Figure 1.3 The nonstationary convolution process. The wavelet matrix is formed by column vectors, each of which is a nonstationary seismic wavelet , and thus the seismic trace is a nonstationary signal.
The essential distinction between stationary and nonstationary convolution models is the nonstationary wavelet matrix . While the example symmetrical wavelet in Figure 1.2 is a zero‐phase wavelet, the form of the wavelet in Figure 1.3 changes continuously. The amplitude decreases and the phase varies with the travel time. The form of the wavelet gradually changes from symmetrical to asymmetrical.
In general, the differences between stationary and nonstationary signals can be captured in three characteristics.
Time period: The time period for a stationary signal always remains constant, whereas the time period for a nonstationary signal is not constant and varies with time.
Frequency: The frequency of a stationary signal remains constant throughout the process, while the frequency of a nonstationary signal changes continuously during the process.
Spectral content: The spectral content of a stationary signal is constant, while the spectral content of a nonstationary signal varies continuously with respect to time.
