Properties of the c-response function for conductivity distributions of class Formula+

Abstract

SUMMARY The properties of the c-response function c(ω) arising from a very extensive class of conductivity distributions (class s+) are studied. A generalized Helmholtz equation for the complex electric field, that involves a formal differential operator incorporating the slowness function as the Lebesgue-Stieltjes measure; is obtained and its equivalent formulation in terms of a Lebesgue-Stieltjes integral equation is developed. Two fundamental solutions w1(z, ω) and w2(z, ω) to the generalized Helmholtz equation are obtained and their relationship to the c-response function is derived. The analytic properties of c(ω) are deduced from those of w1(z, ω) and w2(z, ω). The relationship between cc(ω) and c(zm, ω) is investigated, where cc(ω) is the c-response function observed at the Earth's surface, for a conductivity profile σ∈s+ supported on the interval [O, zm] and terminated in the basement half-space at z = zm by a arbitrary conductivity distribution with c-response function given by c(zm, ω). The two response functions, namely cc(ω) and c(zm, ω), are shown to be related through a Mobius transformation. The analytic properties of c(ω) are utilized to develop two representations for the response function. First, an infinite product expansion based on the Weierstrass factor theorem is obtained and second, an integral representation based on Cauer's representation theorem is developed. This integral representation is the same as that obtained by Weidelt for conductivity distributions of class Co+. Furthermore, an integral representation of exponential form is derived for the apparent resistivity function. Finally, the properties of the Mobius transformation which relate cc(ω) to c(zm, ω) are exploited to provide a quantitative characterization of the critical depth below which the conductivity distribution may be arbitrarily specified without affecting the surface response function. This corresponds to the region of indeterminancy first studied by Parker. The result of this analysis is applied to the Larsen response data set.