Discontinuous parameter estimates with least squares estimators

We discuss weighted least squares estimates of ill-conditioned linear inverse problems where weights are chosen to be inverse error covariance matrices. Least squares estimators are the maximum likelihood estimate for normally distributed data and parameters, but here we do not assume particular probability distributions. Weights for the estimator are found by ensuring its minimum follows a χ2χ2 distribution. Previous work with this approach has shown that it is competitive with regularization methods such as the L-curve and Generalized Cross Validation (GCV) [20]. In this work we extend the method to find diagonal weighting matrices, rather than a scalar regularization parameter. Diagonal weighting matrices are advantageous because they give piecewise smooth least squares estimates and hence are a mechanism through which least squares can be used to estimate discontinuous parameters. This is explained by viewing least squares estimation as a constrained optimization problem. Results with diagonal weighting matrices are given for a benchmark discontinuous inverse problem from [13]. In addition, the method is used to estimate soil moisture from data collected in the Dry Creek Watershed near Boise, Idaho. Parameter estimates are found that combine two different types of measurements, and weighting matrices are found that incorporate uncertainty due to spatial variation so that the parameters can be used over larger scales than those that were measured.