Evaluation of Tomographic Inverse Models Resolved from Various Travel- time Theories and Parameterizations

Whether different forward theories or parameterization methods employed in seismic tomographic imaging lead to the improvement of the resulting Earth structures has been a focus of attention in the seismological community. Recent advance in tomographic theory has gone beyond classical ray theory and incorporated the 3-D sensitivity kernels of frequency-dependent travel-time data into probing the mantle velocity heterogeneity with unprecedented resolution. On the other hand, the conception of multi-scale parameterization has been introduced to deal with naturally uneven data distribution and spatially-varying model resolution for the tomographic inverse problems. The multi-resolution model automatically built through the wavelet decomposition and synthesis results in the non-stationary spatial resolution and data-adaptive resolvable scales. Because the Gram matrix of Frechét derivatives that relates observed data to seismic velocity variations is usually too large to be practically inverted by singular value decomposition (SVD), the iterative LSQR algorithm is instead employed in the inversion which inhibits the direct calculation of resolution matrix to assess the model performance. Recently, with the increasing computing power, we are now able to calculate the SVD of the Gram matrix more efficiently using the parallel PROPACK solver. In this study, we compute the ground-truth pseudo-spectral seismograms in random media with certain heterogeneity strengths and scale lengths. The finite-frequency travel-time residuals measured from waveform cross correlation are then used to invert for the implanted random structure based on different forward theory and model parameterization. For each inversion approach, the trade-off between model covariance and model spread is utilized to determine the optimal solution, showing that the multi-scale model yields a much lower model covariance and remains better spectral resolution for longer-wavelength velocity structures than the simple grid one. The spreadness and geometry of the resulting resolution matrices reveal that both the 3-D finite-frequency kernel and multi-scale parameterization tend to broaden and smooth the structures having less smearing toward the non-crossing ray directions. Furthermore, the comparison of the misfits between the resolved and initial random model among all the optimal solutions indicates that the models obtained with finite-frequency theory have better fits to the true model because wavefront healing effect is properly taken into account in modeling cross-correlation travel-time residuals.