Comparative appraisal of linear inverse models constructed via distinctive parameterizations (comparing distinctly inverted models)

Abstract

Most geophysical inverse problems deal with models of the continuous Earth structure. The classical Backus‐Gilbert's theory demonstrates that the resolvable model variation is the truth viewed through the resolution kernel that is woven by the data kernels. The actual numerical resolution amounts to the inversion of the Gram matrix formed by the inner products among data kernels. However, due to the usually sizable amount of data constraints and/or imperfection of the forward theory, the practical implementations are usually tackled through certain a priori finite parameterizations based on rather arbitrary choices of bases such as spatial voxels, splines, spherical harmonics, or spherical wavelets. The cross assessment on the consistency among inverse models parameterized or regularized differently has long been downplayed. It is shown in this study that straightforward conversions among different model representations also enable the direct conversions of the Gram matrices. This leads to significant flexibility in formulating the forward data rule in one representation and then carrying out the actual inversion in an alternate domain. Furthermore, it is also fairly easy to convert both the model covariance and resolution matrices across different representations. These conversions thus enable direct assessments across inverse models obtained via different parameterizations and different regularization schemes. An example utilizing preliminary results of an experiment of ambient noise tomography of a plate boundary region of complex tectonics for the northeast coast and offshore area of Taiwan is shown to demonstrate such comparisons.