Analysis of Geophysical Potential Fields : A Digital Signal Processing Approach

by Naidu, Prabhakar S.; Mathew, M. P.

• ISBN13: 9780444828019
• ISBN10: 044482801X
• Format: Hardcover
• Copyright: 1998-06-01
• Publisher: Elsevier Science Serials

Summary

When some useful information is hidden behind a mass of unwanted information we often resort to information processing used in its broad sense or specifically to signal processing when the useful information is a waveform. In geophysical surveys, in particular in aeromagnetic and gravity surveys, from the measured field it is often difficult to say much about any one specific target unless it is close to the surface and well isolated from the rest. The digital signal processing approach would enable us to bring out the underlying model of the source, that is, the geological structure. Some of the tools of dsp such as digital filtering, spectrum estimation, inversion, etc., have found extensive applications in aeromagnetic and gravity map analysis. There are other emerging applications of dsp in the area of inverse filtering, three dimensional visualization, etc.The purpose of this book is to bring numerous tools of dsp to the geophysical community, in particular, to those who are entering the geophysical profession. Also the practicing geophysicists, involved in the aeromagnetic and gravity data analysis, using the commercially available software packages, will find this book useful in answering their questions on "why and how?". It is hoped that such a background would enable the practising geophysicists to appreciate the prospects and limitations of the dsp in extracting useful information from the potential field maps. The topics covered are: potential field signals and models, digital filtering in two dimensions, spectrum estimation and application, parameter estimation with error bounds.

Table of Contents

Preface	p. v
Geophysical Potential Fields	p. 1
Potential field surveys for mineral and hydrocarbon exploration	p. 1
Brief description of GandM surveys	p. 1
Information content in potential fields	p. 5
Role of digital signal processing (dsp)	p. 6
Digital filtering	p. 8
Parameter estimation	p. 9
Inverse filtering	p. 9
Spectrum analysis	p. 10
Image processing	p. 11
Some reservations	p. 11
A comparison with seismic signal processing	p. 12
Prologue	p. 13
Notation	p. 16
Conventions	p. 16
References	p. 17
Potential Field Signals and Models	p. 19
Potential field in source free space	p. 19
Fourier transform	p. 20
Potential field	p. 21
Poisson relation	p. 23
Hilbert transform	p. 24
Singularities of potential field	p. 26
Potential field in source filled space
Gravity potential	p. 27
Magnetic potential	p. 29
2D source models	p. 30
Line source	p. 30
Cylinder with polygonal cross-section	p. 31
Dyke	p. 33
Fault	p. 34
Singularities	p. 35
3D source models	p. 36
Potential field in frequency domain	p. 36
Variable density/magnetization model	p. 39
Uniform vertical prism	p. 42
Singularities	p. 44
Prism with polygonal cross-section	p. 45
Stochastic models I: random interface	p. 49
Stochastic field	p. 49
Random interface	p. 53
Magnetic field	p. 56
Prism model	p. 57
Layered strata	p. 58
Stochastic model II: random medium	p. 62
Thin layer	p. 62
Thick layer	p. 64
Half space	p. 66
Undulating layer with random density or magnetization	p. 67
Relation between gravity and magnetic fields	p. 70
References	p. 72
Power Spectrum and its Applications	p. 75
Introduction	p. 75
Spectrum of random fields	p. 75
Random functions (2D)	p. 76
Autocorrelation and cross-correlation	p. 76
Spectrum and cross-spectrum	p. 78
Radial and angular spectrum	p. 79
Coherence	p. 81
Transfer function	p. 82
Discrete potential fields	p. 83
Sampling theorem	p. 83
Folding of spectrum and aliasing error	p. 83
Generalized sampling	p. 86
Quantization errors	p. 91
Estimation of power spectrum	p. 92
Discrete Fourier transform (dft)	p. 92
Fast Fourier transform (FFT)	p. 97
2D discrete Fourier transform	p. 99
Properties of dft coefficients	p. 101
Statistical properties of dft coefficients	p. 105
Estimation 2D spectrum	p. 106
Bias and variance	p. 109
Estimation of coherence	p. 111
Spectral windows	p. 112
Depth estimation from radial spectrum	p. 114
Single layer model	p. 114
Fractal models of susceptibility variations	p. 116
Many layers	p. 120
Depth variation of susceptibility/density	p. 121
Interface model	p. 124
Physical significance of 'spectral' depths	p. 125
Estimation of radial spectrum	p. 126
Effect of quantization	p. 128
Angular spectrum	p. 128
Angular spectrum of uniformly magnetized layer	p. 129
Estimation of angular spectrum	p. 130
Orientation of a fault	p. 131
Application to real data	p. 133
Coherence analysis	p. 136
Stochastic model for the density and susceptibility	p. 137
Isostatic compensation	p. 142
References	p. 142
Digital Filtering of Maps I	p. 145
Two-dimensional digital filters	p. 145
Lowpass filters	p. 146
Polygonal support	p. 147
Gibb's oscillations	p. 149
Design of an finite 2D filter	p. 153
Polygonal filter	p. 156
Transformation of 1D filters	p. 158
Elliptical pass band	p. 162
Implementation of digital filters	p. 166
Spatial and frequency domain approaches	p. 166
Fast convolution	p. 166
Relative speed	p. 168
Additional refinements	p. 169
Filtering for signal enhancement	p. 169
Lowpass filtering for removal of regional fields	p. 170
Directional filtering	p. 172
Digital filters for analytical operations	p. 174
Analytic continuation	p. 175
Derivative maps	p. 176
Total field	p. 181
Continuation of field for enhancing deep seated anomalies	p. 181
Reduction to pole and equator	p. 183
Reduction to pole	p. 183
Low latitude effect	p. 185
Reduction to equator	p. 186
Pseudogravity	p. 188
Distortion analysis	p. 190
Reduction to a plane surface	p. 193
Least squares approach	p. 194
Iterative filtering	p. 195
Removal of the terrain effect	p. 200
Filters to remove terrain effect	p. 202
Correlation filtering	p. 203
Wiener filters	p. 210
Basic theory	p. 210
Extraction of potential field signal	p. 211
Signal distortion	p. 213
Wiener filter for reduction-to-pole	p. 215
Wiener filter for separation of fields from different levels	p. 216
Matched filter	p. 218
References	p. 219
Digital Filtering of Maps II	p. 223
Inverse filtering	p. 223
Irregular interface	p. 223
Density maps	p. 227
Susceptibility maps	p. 228
Undulating layer	p. 229
Least squares inversion (2D distribution)	p. 233
Discrete model	p. 233
Least squares solution	p. 234
Measurement error	p. 239
Backus-Gilbert inversion	p. 239
Resolution	p. 242
Least squares inversion (3D distribution)	p. 243
Discrete model (3D)	p. 243
Constraint least squares	p. 245
Linear programming	p. 246
Texture analysis	p. 248
Non-linear transformations	p. 248
Textural spectrum	p. 251
Textural features	p. 253
References	p. 254
Parameter Estimation	p. 257
Maximum likelihood (ML) estimation	p. 257
Basic detection theory	p. 258
Parameter estimation	p. 260
Cramer-Rao bound	p. 261
Properties of ML estimates	p. 262
ML estimation and Gaussian noise	p. 262
ML estimation source parameters	p. 264
Point mass	p. 264
Point mass - location parameters	p. 269
Point mass CR bounds	p. 273
Dipole	p. 275
Dipole CR bounds	p. 277
Vertical prism	p. 279
Damped sinusoids CR bounds	p. 285
Least squares inverse (non-linear)	p. 288
Gauss-Newton method	p. 289
Levenberg-Marquardt modification	p. 291
References	p. 294
Subject index	p. 295
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