By Michael S. Zhdanov, Tamara M. Pyankova
Integral Transforms of Geophysical Fields function one of many significant instruments for processing and reading geophysical info. during this booklet the authors current a unified therapy of this concept, starting from the concepts of the transfor- mation of 2-D and 3-D capability fields to the speculation of se- paration and migration of electromagnetic and seismic fields. Of curiosity essentially to scientists and post-gradu- ate scholars engaged in gravimetrics, but additionally worthwhile to geophysicists and researchers in mathematical physics.
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Concerning the ProductPublished via the yank Geophysical Union as a part of the Geophysical Monograph sequence. content material:
Additional info for Integral Transforms in Geophysics
30) dx . The Hilbert operator finds much favor in techniques used to separate geophysical fields (see Chap. 3). 1 Plane Potential Fields and Their Governing Equations 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.
The question arises whether there exists an analytical function in the domain D(CD) for which the specified function tp(O will be its limit value on the contour L. Generally, this Cauchy-Type Integral 18 question should be answered negatively. Indeed, on the basis of the values of the real part, Re qJ(O, we can construct a function U(x, z) harmonic in the domain D (or CD) whose limit values on the contour L will coincide with Re qJ(O (the known Dirichlet problem). Similarly, from the imaginary part 1m qJ(O we can construct another function V(x, z) harmonic in D (or CD).
Proof of this is provided in the book by N. I. Muskhelishvily (1962). In concluding the present subsection, we should like to call the reader's attention to the following circumstance. In view of the Sokhotsky-Plemelj formulas, the limit values of the Cauchy-type integral (the C+«(o) and C-«(o) functions) are continuous functions of (0 everywhere on a piecewise smooth contour L. /(O satisfies the Holder condition with an indicator A on L, the limit values of the Cauchy-type integral, C+«(o) and C-«(o) will also satisfy this condition indicator being the same for A < 1 or lower than A by an arbitrarily small value for A = 1.