Abstract
In the present paper the closed expressions of a class of non tabulated Fourier integrals are derived. These integrals are associated with a group of functions at space domain, which represent the electric potential of a distribution of elongated dipoles which are perpendicular to a flat surface. It is shown that the Fourier integrals are produced by the Fourier transform of the Green’s function of the potential of the dipole distribution, times a definite integral in which the distribution of the polarization is involved. Therefore the form of this distribution controls the expression of the Fourier integral. Introducing various dipole distributions, the respective Fourier integrals are derived. These integrals may be useful in the quantitative interpretation of electric potential anomalies produced by elongated dipole distributions, at spatial frequency domain.
Highlights
In earth sciences it is well known that the electric field which is produced by a fault with a geothermal activity may be simulated by a system of dipoles oriented perpendicularly to the surface of the fault (Corwin and Hoover 1979, Fitterman 1984)
A vertical fault with an infinite horizontal dimension is considered. h is the depth of the roof of the fault
T is its vertical dimension. t is the distance of a certain dipole from the roof of the fault and it takes values between 0 and T. x is the location of a point at ground surface
Summary
In earth sciences it is well known that the electric field which is produced by a fault with a geothermal activity may be simulated by a system of dipoles oriented perpendicularly to the surface of the fault (Corwin and Hoover 1979, Fitterman 1984). The subject of the present paper is the derivation of non tabulated expressions of the Fourier integral of V(x) of equation (2), for various forms of m(t). Combining equations (2) and (3) and changing the order of integration, U(u) takes the form: Z∞ UðuÞ 1⁄4 UðuÞ 1⁄4 −iπ expð−huÞ mðtÞ expð−utÞdt ð6Þ
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