Integral Transforms in Geophysics

For functions of a complex variable there is a known Cauchy integral theorem which is formulated as follows. Let L be a smooth closed curve (contour). Hereinafter the term smooth curve (contour) covers a simple (i.e., having no points of self-intersection) closed or open line with a slip tangent and with no cusps. If the function f(ζ) is analytical in the domain D bounded by the contour L and continuous on the contour itself (i.e., continuous in the closed domain D̄ = D∪L), we deal with the following Cauchy formula: where CD̄ is the complement of the closed domain D̄ with respect to the total complex plane. (From now on it is assumed that the contour is traversed in the positive sense for which the inner domain is always on the left.)

This chapter is devoted to two-dimensional (plane) potential fields. The main geopotential fields dealt with in geophysics are the gravitational and constant magnetic fields. The physical nature of these fields is outlined comprehensively in many treatises on general and applied geophysics. That is why, within the scope of this book, we will dwell only on the problems of mathematical theory of analysis, integral transforms, and interpretation of geopotential fields. Moreover, we shall confine ourselves just to the issues resolved most efficiently in terms of the Cauchy-type integral. How wide the range of these issues is will be shown below.

This chapter is concerned with application of the body of the Cauchy-type integral to the solution of one of the oldest problems encountered in geopotential field theory, namely separation of the field into external and internal parts. This problem was first tackled as early as the last century in Gauss’ classical works on geomagnetism. Gauss was interested in the location of sources of the terrestrial magnetic field, whether they are found inside or outside the Earth. Having employed his own concept of spherical harmonic analysis, the great German mathematician demonstrated that the external part of the magnetic potential is negligible, compared to the internal part. The contributions of the magnetic field made by external and internal sources were analyzed once again toward the end of the nineteenth century by the German physicist and mathematician Schmiedt (Chapman and Bartels 1940). Having refined appropriately the technique of spherical harmonic analysis and employing more precise field data on the magnetic field, Schmiedt succeeded in evaluating the contribution of external sources into the terrestrial magnetic field, which proved to be negligible too.

Now let us take up one of the most important concepts of the theory of a plane potential field, namely the concept of analytical continuation. As we have implied, in a two-dimensional case a plane field is described unequivocally by complex intensity. Obviously, the problem of analytical continuation of a plane field reduces to the well-known problem of analytical continuation of complex functions. A detailed treatment of this problem is offered in many treatises and monographs on the theory of functions of a complex variable. Those wishing to delve more deeply into this particular topic will no doubt start with the above literature. In this book, however, we will confine ourselves to the basic theorems and principles underlying the theory of analytical continuation, particularly to those needed for geophysical applications.

The previous section was concerned with the main concepts of analytical theory of two-dimensional geopotential fields based on the body of the Cauchy-type integral. But two-dimensional (plane) fields can serve just as an approximate model of actual geophysical fields dependent on three space coordinates. In view of this, an extremely important problem of theoretical geophysics is how to extend the results of a two-dimensional theory to a three-dimensional case. This section aims precisely at solving this problem.

The first part of this book has revealed the efficiency of theoretical analysis of plane gravimetric and magnetometric problems when the gravitational and the magnetic fields are represented in terms of the Cauchy-type integrals. In this connection, it seems tempting to derive similar expressions for three-dimensional geopotential fields. Solution to this problem, as well as some important applications of Cauchy integral analogs to the geopotential field theory, are taken up in the present chapter.

In Chap. 4 we recalled the fundamentals underlying the theory of analytical continuation of functions of a complex variable. It turns out that a greater part of the concepts of this theory also apply to three-dimensional Laplace vector fields described by real vector functions of a real variable. The basis for this relationship is provided by the analytical nature of Laplace vector fields.

As indicated earlier, the body of three-dimensional Cauchy integral analogs developed in the previous part permits the extension of many ideas and methods established in the theory of functions of a complex variable to three-dimensional situations important in the study of geopotential fields. It seems tempting to apply these methods to other geophysical fields, particularly to the electromagnetic field. A natural extension of the concept of the Cauchy-type integral to the electromagnetic field theory involves, as will be shown later, the Stratton-Chu type integrals introduced on the basis of the known integral formulas of Stratton-Chu (Stratton 1941).

Among the most important types of geophysical fields are elastic wave fields or simply wave fields. These fields are studied in seismology and in seismic prospecting and they yield extremely significant information about the deep structure of the Earth which could not be gained using other geophysical methods. In view of this, the problem of extending the body of Cauchy-type integrals to wave fields is of critical value in theoretical geophysics. In this chapter we will derive the classical Kirchhoff integral and a generalized Kirchhoff integral formula for the elastic displacement vector. The Kirchhoff-type integrals introduced on the basis of the above integral formulas are a natural extension of the Cauchy-type integrals to wave fields. They can be used to develop the theory of seismic field migration which presently underlies dynamic seismology.

It was already mentioned in Chap. 10 that one of the most effective methods used to interpret seismic data is that of seismic migration or so-called seismoholography (Hemon 1971; Timoshin 1972, 1978; Petrashen and Nakhamkin 1973; Vasiliev 1975; Clearbout 1976, 1985; Berkhout 1980, 1984). Behind these methods is the idea of time reversal of an elastic oscillation field and continuation of the reversed wave field toward sources. For instance, in diffraction transformation the Fresnel-Huygens principle for a time-reversed field is employed (Timoshin 1978). In migration by the Clearbout method, a boundary-value problem for wave equation is numerically solved. The physical prerequisite for developing all these methods is the fact that the distribution of transformed amplitudes of elastic waves over a medium yields, just as in the case of optical holography, an image of the studied medium. However, as a rule, it is not the true field but only a certain mapping of it that is reconstructed in a medium. At the same time, in many seismic prospecting and seismology problems it is of interest to analyze the real distribution of the elastic wave field in the interior of the Earth. This analysis facilitates, on the one hand, a more complete understanding of the space-time structure of the wave field, and, as will be shown later, it is highly instrumental in the solution of inverse problems of dynamic seismics, on the other hand.