A simple method for determining the spatial resolution of a general inverse problem

Abstract

Summary The resolution matrix of an inverse problem defines a linear relationship in which each solution parameter is derived from the weighted averages of nearby true-model parameters, and the resolution matrix elements are the weights. Resolution matrices are not only widely used to measure the solution obtainability or the inversion perfectness from the data based on the degree to which the matrix approximates the identity matrix, but also to extract spatial-resolution or resolution-length information. Resolution matrices presented in previous spatial-resolution analysis studies can be divided into three classes: direct resolution matrix, regularized/stabilized resolution matrix and hybrid resolution matrix. The direct resolution matrix can yield resolution-length information only for ill-posed inverse problems. The regularized resolution matrix cannot give any spatial-resolution information. The hybrid resolution matrix can provide resolution-length information; however, this depends on the regularization contribution to the inversion. The computation of the matrices needs matrix operation, however, this is often a difficult problem for very large inverse problems. Here, a new class of resolution matrices, generated using a Gaussian approximation (called the statistical resolution matrices), is proposed whereby the direct determination of the matrix is accomplished via a simple one-parameter non-linear inversion performed based on limited pairs of random synthetic models and their inverse solutions. Tests showed that a statistical resolution matrix could not only measure the resolution obtainable from the data, but also provided reasonable spatial/temporal resolution or resolution-length information. The estimates were restricted to forward/inversion processes and were independent of the degree of inverse skill used in the solution inversion; therefore, the original inversion codes did not need to be modified. The absence of a requirement for matrix operations during the estimation process indicated that this approach is particularly suitable for very large linear/linearized inverse problems. The estimation of statistical resolution matrices is useful for both direction-dependent and direction-independent resolution estimations. Interestingly, even a random synthetic input model without specific checkers provided an inverse output solution that yielded a checkerboard pattern that gave not only indicative resolution-length information but also information on the direction dependence of the resolution.