An analytic solution to the Laplace equation for potential distribution in response to current flow in a heterogeneous, two-dimensional semi-infinite domain is studied. Circular heterogeneities of varying sizes and electrical conductivities are considered. We investigate the response of the stream function, the potential field, and, in particular, the potential at the top boundary relative to the background as a function of the size, location, and electrical conductivity of circular inclusions taken singly or multiply. The analytic solution sets the basis for the application of sensitivity analysis to the electrical resistance tomography (ERT) method, as an initial step toward improving the application of the method to tracking rapid hydrological processes.

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