How to Determine Spatial Resolution for an Inverse Problem

The information of solution's spatial resolution is important for model appraisal in an inversion, however, the computation to determine a spatial resolution is nontrivial and often more difficult than to solve an inverse problem. Visual inspection of the restoration of a synthetic structure widely applied in tomographic studies can give indicative information on spatial resolution distribution, however, resolution matrix estimation can give quantitative information of spatial resolution length for a general inverse problem. Resolution matrices obtained by matrix operation may be divided into three classes: direct resolution matrix, regularized resolution matrix and hybrid resolution matrix. Each matrix can give part of the information on the inversion, and then the simultaneous implementation of all three resolution matrices in a single study can potentially provide a complete understanding on the resolution length information. An (2012) proposed a new class of resolution matrices generated by a simple one-parameter nonlinear inversion performed based on limited pairs of random synthetic models and their inverse solutions. The estimates were directly retrieved from synthetic models and their inverse solutions, and then it can include the information on the whole inversion procedure; The independence on the degree of inverse skill used and the absence of a requirement for matrix operations indicated that this approach is particularly suitable for very large linear/linearized inverse problems. Inversion examples even for 3D inversion problem demonstrated that reasonable resolution lengths can be determined from statistic spatial resolution calculation.