On Resolution Matrices

Abstract

Solution appraisal, which has been realized on the basis of projections from the true medium to the solution, is an essential procedure in practical studies, especially in computer tomography. The projection operator in a linear problem or its linear approximation in a nonlinear problem is the resolution matrix for the solution (or model). Practical applications of a resolution matrix can be used to quantitatively retrieve the resolvability of the medium, the constrainability of the solution parameters, and the relationship between the solution and the factors in the study system. A given row vector of the matrix for a solution parameter can be used to quantify the resolvability, deviation from expectation, and difference between that solution parameter and its neighbor from the main-diagonal element, row-vector sum, and difference between neighboring elements in the row vector, respectively. The resolution length of a solution parameter should be estimated from the row vector, although it may be unreliable when the vector is unstable (e.g., due to errors). Comparatively, the resolution lengths that are estimated from the column vectors of the observation-constrained parameters are reliable in this instance. Previous studies have generally employed either the direct resolution matrix or the hybrid resolution matrix as the model resolution matrix. The direct resolution matrix and hybrid resolution matrix in an inversion with damping (or general Tikhonov regularization) are Gramian (e.g., symmetric). The hybrid resolution matrix in an inversion using zero-row-sum regularization matrices (e.g., higher-order Tikhonov regularizations) is one-row-sum but is not a stochastic matrix. When the two resolution matrices appear in iterative nonlinear inversions, they are not a projection of the solution, but rather the gradient of the projection or a projection of the solution improvement immediately after a given iteration. Regardless, their resultant resolution lengths in iterative nonlinear inversions of surface-wave dispersion remain similar to those from the projection of the solution. The solution is influenced by various factors in the study, but the direct resolution matrix is derived only from the observation matrix, whereas the hybrid resolution matrix is derived from the observation and regularization matrices. The limitations imply that the appropriateness using the two resolution matrices may be questionable in practical applications. Here we propose a new complete resolution matrix to overcome the limitations, in which all of the factors (e.g., errors) in linear or nonlinear (inverse or non-inverse) studies can be incorporated. Insights on all of the above are essential for ensuring a reliable and appropriate application of the resolution matrix to appraise the model/solution and understand the relationship between the solution and all of the factors in the study system, which is also important for improving the system.