Estimating multiple physical parameters in subsurface using full-waveform inversion (FWI) methods suffers from parameter crosstalk challenge. The inherent ambiguities among different physical parameters make the inverse problems much more non-linear. The parameter resolution is highly inﬂuenced by the parameterization selected for describing the subsurface elastic and anisotropic medium. Quantify the parameter resolution for determining the optimal parameterization in multi-parameter FWI becomes essential for reducing the parameter crosstalk difﬁculty. In recent years, researchers devote signiﬁcant effort for evaluating the resolving abilities of different parameterizations for elastic and anisotropic full-waveform inversion based on the analytic solutions of the Fréchect derivative waveﬁelds (so called "scattering" or "radiation" patterns). However, these studies may not be able to evaluate the resolving abilities of different parameter classes completely because of the inherent defects of the scattering patterns. The goal of this research is to develop new strategies for quantifying the parameter resolution for multi-parameter elastic and anisotropic full-waveform inversion. We ﬁnd that the multi-parameter Hessian, the secondorder derivative of the misﬁt function, provides direct and complete measurements of the inter-parameter trade-off. The investigations based on scattering patterns can be interpreted as an asymptotic approximation of the multi-parameter Gauss-Newton Hessian. With the block-diagonal approximation of the multi-parameter Gauss-Newton Hessian, we are able to assess the parameter resolution by taking the geometrical spreading and complex model into consideration. Furthermore, with the adjoint-state technique, we are able to calculate one column of the multi-parameter Hessian (multi-parameter point spread function), deﬁned as parameter resolution kernel in this research, with which we can evaluate the inter-parameter mapping of different parameters at adjacent positions locally by considering ﬁnite-frequency effects. With the help of random probing technique, we can infer the characteristics of the multi-parameter Hessian within the whole model at affordable computation cost for large-scale inverse problems. Thus, with the multi-parameter Hessian, the parameter resolution can be assessed more completely.
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