Resistivity imaging with controlled-source electromagnetic data: depth and data weighting

We discuss some computational aspects of resistivity imaging by inversion of offshore controlled-source electromagnetic data. We adopt the classic approach to imaging by formulating it as an inverse problem. A weighted least-squares functional measures the misfit between synthetic and observed data. Its minimization by a quasi-Newton algorithm requires the gradient of the functional with respect to the model parameters. We compute the gradient with the adjoint-state technique. Preconditioners can improve the convergence of the inversion. Diagonal preconditioners based on a Born approximation are commonly used. In the context of CSEM inversion, the Born approximation is not really accurate. This limits the possibility of estimating a correct approximation of the Hessian in a smooth medium or, in fact, in any reference background that does not roughly account for the resistors. We hence rely on the limited memory BFGS approximation of the inverse of the Hessian and we improve the inversion convergence with the help of heuristic data and depth weighting. Based on a numerical example, we show that a simple exponential depth weighting combined with offset or frequency data weighting significantly improves the convergence rate of deep-water controlled-source electromagnetic data inversion.