Inverse resistivity problems in computational geoscience

Abstract

We study coefficient inverse problems arising in modeling of resistivity prospecting problems. Numerical simulations are investigated in the cases of vertically and cylindrically layered medium. Conductivity coefficients are assumed to be sufficiently smooth 1D functions. The model leads to an inverse problem of identification of an unknown coefficient (conductivity) in an elliptic equation in R2 inside a slab or in a cylinder. The direct problem is formulated as a mixed BVP in R2. Measured data are assumed to be available on the upper boundary of the medium or along the axis of the well. A logarithmic transformation is applied to the unknown coefficient, and the inverse problem is studied as a minimization problem for the residual functional. A numerical method is discussed for interpreting the data of a resistivity prospecting in both considered models of layered medium. The method is implemented for realistic conductivity distributions, with both noise-free and noisy data.