Abstract
Besides field applications to geophysical prospecting and subsurface hydrology, electrical resistivity tomography can be applied to finite-scale blocks in the laboratory to characterize the resistivity structure of the blocks and to monitor internal physical and chemical processes. This requires a fast and accurate calculation of the sensitivity matrix to perform a successful resistivity inversion for such blocks. However, the complex geometric shape and boundary and the finite size of the block limit the application of field-suitable sensitivity calculation methods to these blocks. As blocks and finite columns are often used in the laboratory experiments, this paper develops practical analytic expressions, based on the method of image charges, for the sensitivity matrix for these two types of homogenous bodies. The corresponding formulae for the electric potential distribution and theelectrode array coefficient are also presented. As a result of the theory, the effects of placing limits on the sum index in the electric-potential calculation can be analyzed, and a comparison of the theoretical and the numerically simulated electric potential is shown. The results demonstrate the correctness of the theory and indicate that even the addition of only one set of mirror current sources greatly reduces the effects of the block boundary on the electric-potential calculation. Finally, several interesting sensitivity distributions for cross-surface arrays on blocks, and for circular and vertical arrays on columns, are given. Although the formulae developed here are only valid for homogeneous blocks and columns, and an element of relatively small volume is required to permit a good approximation to the sensitivity, the theory is useful in the verification of numerically simulated results, in sensitivity-analysis for optimum probing-scheme design, and in successful resistivity inversion calculation for finite bodies.
