ANALYSIS OF GEOPHYSICAL POTENTIAL FIELDS A Digital Signal Processing Approach

Transcription

1 ADVANCES IN EXPLORATION GEOPHYSICS 5 ANALYSIS OF GEOPHYSICAL POTENTIAL FIELDS A Digital Signal Processing Approach PRABHAKAR S. NAIDU Indian Institute of Science, Bangalore , India AND M.P. MATHEW 2 Church Street, Geological Survey of India, Bangalore , India 1998 ELSEVIER Amsterdam - Lausanne - New York - Oxford - Shannon - Singapore - Tokvo TLNGEN

2 Preface v Chapter 1. Geophysical Potential Fields Potential field surveys for mineral and hydrocarbon exploration Brief description of G&M surveys Information content in potential fields : Role of digital signal processing (dsp) Digital filtering Parameter estimation Inverse filtering Spectrum analysis Image processing Some reservations A comparison with seismic signal processing Prologue Notation Conventions 16 References 17 Chapter 2. Potential Field Signals and Models Potential field in source free space Fourier transform : Potential field ' Poisson relation Hilbert transform Singularities of potential field Potential field in source filled space Gravity potential Magnetic potential D source models Line source Cylinder with polygonal cross-section Dyke Fault Singularities D source models Potential field in frequency domain Variable density/magnetization model Uniform vertical prism 42

3 viii Contents Singularities Prism with polygonal cross-section Stochastic models I: random interface Stochastic field Random interface Magnetic field Prism model Layered strata Stochastic model II: random medium Thin layer Thick layer Half space Undulating layer with random density or magnetization Relation between gravity and magnetic fields 70 References 72 Chapter 3. Power Spectrum and its Applications Introduction Spectrum of random fields Random functions (2D) Autocorrelation and cross-correlation Spectrum and cross-spectrum Radial and angular spectrum Coherence Transfer function Discrete potential fields Sampling theorem Folding of spectrum and aliasing error Generalized sampling Quantization errors Estimation of power spectrum Discrete Fourier transform (dft) Fast Fourier transform (FFT) D discrete Fourier transform Properties of dft coefficients Statistical properties of dft coefficients Estimation 2D spectrum Bias and variance Estimation of coherence Ill Spectral windows Depth estimation from radial spectrum Single layer model.: Fractal models of susceptibility variations Many layers Depth variation of susceptibility/density: Interface model 124

4 ix Physical significance of 'spectral' depths Estimation of radial spectrum Effect of quantization Angular spectrum Angular spectrum of uniformly magnetized layer Estimation of angular spectrum Orientation of a fault Application to real data Coherence analysis Stochastic model for the density and susceptibility Isostatic compensation 142 References 142 Chapter 4. Digital Filtering of Maps Two-dimensional digital filters Lowpass filters Polygonal support Gibb's oscillations Design of an finite 2D filter Polygonal filter Transformation of ID filters Elliptical pass band Implementation of digital filters Spatial and frequency domain approaches Fast convolution Relative speed Additional refinements Filtering for signal enhancement Lowpass filtering for removal of regional fields Directional filtering Digital filters for analytical operations Analytic continuation Derivative maps Total field Continuation of field for enhancing deep seated anomalies Reduction to pole and equator Reduction to pole Low latitude effect Reduction to equator Pseudogravity Distortion analysis Reduction to a plane surface Least squares approach Iterative filtering Removal of the terrain effect Filters to remove terrain effect 202

5 Correlation filtering Wiener filters Bas'ic theory Extraction of potential field signal Signal distortion, Wiener filter for reduction-to-pole Wiener filter for separation of fields from different levels Matched filter 218 References 219 Chapter S. Digital Filtering of Maps II Inverse filtering Irregular interface Density maps Susceptibility maps Undulating layer Least squares inversion (2D distribution) Discrete model Least squares solution Measurement error Backus-Gilbert inversion Resolution Least squares inversion (3D distribution) Discrete model (3D) Constraint least squares Linear programming Texture analysis Non-linear transformations Textural spectrum Textural features 253 References '. 254 Chapter 6. Parameter Estimation Maximum likelihood (ML) estimation Basic detection theory Parameter estimation Cramer-Rao bound Properties of ML estimates ML estimation and Gaussian noise ML estimation source parameters Point mass! Point mass - location parameters Point mass CR bounds Dipole Dipole CR bounds 277

6 xi Vertical prism Damped sinusoids CR bounds Least squares inverse (non-linear) Gauss-Newton method Levenberg-Marquardt modification 291 References 294 Subject index 295