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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Extra resources for Integral Transforms in Geophysics
_-_ ...... 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.