REGULARIZED SOLUTIONS OF AN INTEGRAL EQUATION OF GRAVIMETRY

Abstract

We consider the problem of determining the relative density ρ of a body in the interior of the earth from surface gravity anomalies created by this body. Let ρ_1, be the mass density, and ρ_2 be the density of the surrounding medium, the relative density of the body is ρ=ρ_1-ρ_2. The earth is represented by a half-space (x,z), -∞ <z≤ H, H > 0. The body Ω is represented by Ω = {(x; y): 0<x<1 ; 0<z< σ(x)} where σ: [0, 1] → IR is piecewise C1 function such that với 0<x<1; α>0. In general, with , the problem of determining the relative density p of a body in the interior of the earth from surface gravity anomalies is no uniqueness. In the case of , the above problem admits at most one solution. Then ρ satisfies a nonlinear integral equation of the first kind. In the case of , the nonlinear integral equation is changed to the convolution equation. The solution of the convolution equation is regularized by the Tikhonov method.