Fourier Transform Methods in Geophysics

1. Fourier Transforms Fourier transform are use in many areas of geophysics such as image processing, time series analysis, and antenna design. Here we focus on the use of fourier transforms for solving linear partial differential equations (PDE). Some examples include: Poisson's equation for problems in gravity and magnetics; the biharmonic equation for problems in linear visco-elasticity; and the diffusion equation for problems in heat conduction. We do not treat the wave equation in this book because there are already many excellent books on seismology. For each of these problems we search for the Green's function that represents the response of the model to a point load. There are two approaches to solving this class of problem. In some cases one can derive a fully analytic solution, or Green's function, to the point-load problem. Then a more general model can be constructed by convolving the actual distribution of sources with the Green's function. A familiar example is the case of constructing a gravity anomaly model given a 3-D density anomaly structure. The second semi-analytic approach can be used to solve more complicated problems where the development of a fully analytic Green's function is impossible. This involves using the derivative property of the fourier transform to reduce the PDE and boundary conditions to algebraic equations that can be solved exactly in the transform domain. A more general model can be constructed by taking the fourier transform of the source, multiplying by the transform domain solution, and performing the inverse transform numerically. Indeed the only difference between the two methods is that in the first case the final model is generated by direct convolution while in the second case the convolution theorem is used for model generation. When dealing with spatially complex models the second approach can be sometimes orders of magnitude more computationally efficient because of the efficiency of the fast fourier transform algorithm. In this chapter we introduce the minimum amount of fourier analysis needed to understand the solutions to the PDE's provided in the following chapters. If the reader is not familiar with fourier transforms and analysis, they should first study any of the excellent books on the topic. We recommend the first 6 chapters of the book by Bracewell [1978] for a more complete discussion of the material presented in Chapter 1.