A computationally efficient discrete Backus-Gilbert footprint-matching algorithm

A computationally efficient discrete Backus-Gilbert (BG) method is derived that is appropriate for resolution-matching applications using oversampled data. The method builds upon existing BG methods and approximation techniques to create a modified set of BG coefficients. The method in its current form is restricted to a resolution-only minimization constraint, but in the future could be extended to use a simultaneous noise minimization constraint using a generalized singular value decomposition (GSVD) approach. A theoretical one-dimensional intercomparison is performed using a hypothetical sensor configuration. A comparison of the discrete BG method with a nondiscrete BG method shows that the new approach can be 250% more efficient while maintaining similar accuracies. In addition, an SVD approximation increases the computational efficiencies an additional 43%-106%, depending upon the scene. Several quadrature methods were also tested. The results suggest that accuracy improvements are possible using customized quadrature in regions containing known physical data discontinuities (such as along coastlines in microwave imagery data). The ability to recompute the modified BG coefficients dynamically at lower computational cost makes this work applicable toward applications in which noise may vary, or where data observations are not available consistently (e.g. in radio frequency interference contaminated environments).