Integral Transforms in Geophysics by Michael S. Zhdanov, Tamara M. Pyankova

By Michael S. Zhdanov, Tamara M. Pyankova

Integral Transforms of Geophysical Fields function one of many significant instruments for processing and reading geophysical facts. during this booklet the authors current a unified therapy of this concept, starting from the suggestions of the transfor- mation of 2-D and three-D strength 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 in addition valuable to geophysicists and researchers in mathematical physics.

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_-_ ...... 12)] . 12), one is readily convinced of the following. Let (0 be a corner of a piecewise smooth curve Land a be an angle made by two lines tangent to L at a point (0 and counted off on the left of the curve L (inside the domain D if L is a closed contour) (Fig. 5). 23) and . hm J-d(- = J-d( b- (0 . 24) a - (0 seem to hold good. -qJ«(o) . 25) for a = b. The Sokhotsky-Plemelj formulas for corners of the curve L are changed accordingly. /«(o) . 28) remain valid also for cusps, under the assumption that a = or a = 2 n, depending on whether the cusp faces the right or the left side (outward or inward) of the curve L.

5) It is stated in Sect. -'>oo and F(O-'>O when 1(1-'>00. 2 Complex Intensity Equations Now we will write equations fitting the complex intensity of a field. 8) Indeed, multiplying the first Eq. 1) by -1 and the second by -i, and summing them up we obtain x aFz) x - = ( -+1 . z) = -q(x, z) aF+ a . -a) (-Fx+iF - (- - I. 8) we write aF~O = _~ q(O . 10) We have arrived at a complex differential equation fitting the complex intensity of a plane field. Complex Intensity and Potential of a Plane Field 37 It is noteworthy that once If/(O is an arbitrary function analytical in the domain D, the Cauchy-Riemann relations imply that alf/(o/a~ = 0 .

24) (xo• zo) where the integral is taken around any path L connecting, within cf', a fixed point (xo, zo) with a variable point (x, z), and a is an arbitrary constant. 29 Logarithmic Potentials and the Cauchy-Type Integral Let us consider a line M of equal values of the function = C = const. 23) it coincides with the vector F: T = (Fx, F z) = F. In other words, the line M is a vector line of the field F or a current line. That is why the function V(x, z) is given the name of a flow function of a plane vector field F.

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