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This paper develops a novel approach to the problem of source currentidentification for the diffusion equation in connection with geophysicalself-potential measurements. The problem is split into two subproblems: (a) thescalar source identification, and (b) solution of the divergence equation. Forsubproblem (a), we design an algorithm for reconstructing the scalar sourcefunction, which does not require solving the Fredholm integral equation of thefirst kind. Instead, the problem is reformulated as a linear operator equation,which is solved by a projection method. The dimension of the subspace, in whichthe source function is sought, is independent of the dimension of the forwardproblem, leading to reduction of the size of the inverse problem. Numericalexperiments with exact and noisy data are presented. For subproblem (b), thedivergence equation was posed as a minimization problem, which, by means ofLagrangian formalism, was reduced to a system of partial differential equationof Stokes type with a unique solution. To demonstrate how this framework can beused in a practical application, we implemented the algorithm in thetwo-dimensional physical space, using a finite-different discretization onstaggered grids.

Source identification in the self-potential method and its connection to Stokes type systems