Time-frequency Analysis of Seismic Signals

Description

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. Cover Title Page Copyright Contents 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 Q Filtering 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 S Transform 4.1 Basics of the S Transform 4.2 The Generalised S Transform 4.3 The Fractional Fourier Transform 4.4 The Fractional S Transform 4.5 Implementation of the S Transforms 4.6 The Inverse S Transforms 4.7 Application to Clastic and Carbonate Reservoirs 5 The W Transform 5.1 Basics of the W Transform 5.2 The Generalised W Transform 5.3 Implementation of Nonstationary Convolution 5.4 The Inverse W 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 B.1 The Fourier Transform of the Tapered Boxcar Window B.2 The Fourier Transform of the Truncated Gaussian Window Function B.3 The Fourier Transform of the Weighted Cosine Window Function 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 E.1 Marginal Properties of the WVD Definition E.2 The WVD Definition Using an Analytic Signal or a Real Signal F The Prolate Spheroidal Wavefunctions, the Associated and the Ordinary Legendre Polynomials F.1 Prolate Spheroidal Wavefunctions F.2 The Associated Legendre Polynomials F.3 The Ordinary Legendre Polynomials References Author Index Subject Index EULA