The successive linear estimator: A revisit

This paper examines the theoretical basis of the successive linear estimator (SLE) that has been developed for the inverse problem in subsurface hydrology. We show that the SLE algorithm is a non-linear iterative estimator to the inverse problem. The weights used in the SLE algorithm are calculated based on conditional covariances, and yield estimates that satisfy the minimum-mean-square-error criterion. Furthermore, the weights for well-posed or deterministic inverse problems are equivalent to the inverse of Jacobian matrices of the classical Newton-Raphson (NR) algorithm for the non-linear forward problem. For ill-posed or stochastic inverse problems, the weights are smooth interpreted quantities of the inverse of the Jacobian at data locations, based on the spatial covariance of parameters. For both deterministic and stochastic inverse problems, the SLE algorithm converges as in the NR scheme for the forward non-linear problem. The SLE approach is verified with a simple forward exponential model and compared to the exact lognormal conditional mean estimates. Results show that for the deterministic inverse problem, this approach can yield an exact solution, whereas the estimate of the SLE approach for the stochastic inverse problem is exact up to a known residual term related to the conditional estimation variance.