Geophysical Interpretation using Integral Equations

About this book

The development of numerical methods, together with the advent of fast digital computers, have facilitated the application of integral equations in geophysical modelling. This is also due to the successful derivation of integral equations that are applicable to the modelling of complex structures and efficient numerical algorithms for their solution. The purpose of this work is to give the principles by which boundary value problems describing geophysical models can be converted into integral equations. "Geophysical Interpretation and Integral Equations" introduces Fredholm integral equations that are well suited to the numerical solution of boundary value problems representing the electrical, magnetic, electromagnetic and seismic models of geophysics. These methods form a most efficient class of techniques for the numerical modelling of geophysical phenomena. The geophysical methods are briefly described, their mathematical expressions are given in the form of boundary value problems and, by applying the Green's functions, these boundary value problems are then converted into integral equations that can then be solved by standard numerical methematics. The end results are integral formulae and integral equations that form the theoretical framework for model calculations associated with practical geophysical interpretation. The approach is physical rather than mathematical, ie the physical phenomenon is represented by the integral formulae explained in detail, with the mathematical analysis confined to a minimum. Numerical algorithms for solving the integral equations are discussed in connection with some illustrative examples involving numerical modelling results. This work seeks to provide a reference source for all geophysicists and engineers concerned with geophysical phenomenon.

Contents

Part 1 General matters concerning integral equations: demonstration of an integral equation solution; classification of integral equations; numerical solution. Part 2 Elements of electrostatics and potential theory: differential representation of electrical potential; integral representation of electrical potential; primary current electrode; volume distribution of simple sources; surface distribution of simple sources; surface distribution of double sources. Part 3 Electrical methods: resistivity of rocks; resistivity method; magnetometric resistivity; mis-a-la-masse method; surface polarization; induced polarization; self-potential; electrical anisotropy. Part 4 Elements of magnetrostatics: integral representation of magnetic potential; volume distribution of simple poles; surface distribution of simple poles; volume distribution of dipoles. Part 5 Magnetic methods: magnetic properties of rocks; high-susceptibility models; demagnetization and low-susceptibility models; numerical applications; effect of remanence. Part 6 Electromagnetic methods: boundary value problems for electromagnetic fields; Green's dyadics for electromagnetic boundary value problems; volume integral equations for 3-dimensional electromagnetic field; volume integral equations for 2-dimensional electromagnetic fields; surface integral equations for electromagnetic fields; integral equation solution for electromagnetic fields in a thin conductor model. Part 7 Integral formulae for elastic wave fields in an anisotropic medium; integral formulae for elastic wave fields in an isotropic medium; separation of elastic wave fields into a compressional and a rotational mode; integral formulae for acoustic wave fields in the frequency domain; integral formulae for acoustic wave fields in the time domain; applications. Appendices: Green's function for scalar potential in a two-layer half-space; Green's function for scalar potential in a half-space with a vertical contact; Green's function for scalar potential in an anisotropic half-space; electric Green's dyadic for a half-space below the ground surface.

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