# Analysis of Geophysical Potential Fields

**A Digital Signal Processing Approach**

**By**

- P.S. Naidu, Dept of ECE, Indian Institute of Science, Bangalore 560012, India
- M.P. Mathew, 2 Church Street, Geological Survey of India, Bangalore 560001, India

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.

### Book information

- Published: June 1998
- Imprint: ELSEVIER
- ISBN: 978-0-444-82801-9

### Table of Contents

Preface.**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. References.

**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. 2D source models. Line source. Cylinder with polygonal cross-section. Dyke. Fault. Singularities. 3D source models. Potential field in frequency domain. Variable density/magnetization model. Uniform vertical prism. 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. References.

**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). 2D discrete Fourier transform. Properties of dft coeffficients. Statistical properties of dft coefficients. Estimation 2D spectrum. Bias and variance. Estimation of coherence. Spectral windows. Depth estimation from radial spectrum. Single layer model. Fractal models of susceptibility variations. Many layers. Depth variation of susceptibility/density. Interface model. 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. References.

**Chapter 4. Digital Filtering of Maps I**. 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. Correlation filtering. Wiener filters. Basic 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. References.

**Chapter 5. 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. References.

**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. Vertical prism. Damped sinusoids CR bounds. Least squares inverse (non-linear). Gauss-Newton method. Levenberg-Marquardt modification. References. Subject Index.