Simultaneous inversion for 3-D crustal structure and hypocentres including direct, refracted and reflected phases. I: Development, validation and optimal regularization of the method

SUMMARY A new technique for the simultaneous inversion for three-dimensional (3-D) seismic structure and hypocentres (SSH) is developed and validated. The seismic forward problem is solved with a new non-linear 3-D ray tracing procedure that includes direct (as first or later arrivals), refracted and reflected phases from various crustal discontinuities. The non-linear inversion problem is formulated in terms of a modified, iterative Levenberg-Marquardt (LM) technique. It is demonstrated that this method, in spite of its computational inexpediencies, has various properties that make it best suited for the solution of both the general non-linear inverse problem (near-quadratic convergence, due to the approximate restoration of the full Hessian of the objective function) and its linearized subproblems [where LM is equivalent to classical damped least squares (ridge regression)]. Therefore LM is able to play a dual role for the optimal regularization (OR) of both the non-linear and the linear ill-posed SSH problem. The SSH method is validated through simulated inversions of local traveltimes from the Rhine Graben region. The results demonstrate that the usual trade-off between focal depths and seismic velocities can be reduced strongly through the combined use of all crustal phases. A new non-linear statistical F-test shows that the Newton-like LM method gives a quasi-linear ‘appearance’ of the SSH problem: i.e. convergence is essentially obtained already in the first iteration. This is further confirmed from the results of the OR performed within both the non-linear and the linear model space employing the regularization techniques (RT’s) of Tikhonov, the ridge regression, the stochastic inverse, the method of Backus & Gilbert, and a new method (Backus subjective). Some of these RTs rely on various forms of a priori model information so that they can, in principal, be derived from a general Bayesian formulation. However, the results of the OR illustrate the limitations of this approach, which is attributed to difficulties in specifying the appropriate covariance matrices in model and data space and discrepancies between the theoretical optimal model and its numerical realization. Therefore, those ORTs have been found the most useful, that rely on some form of graphical trade-off curves to systematically explore the model and data space. Based on their theoretical foundations and practical performances, the ORTs of Backus (subjective) and of Tikhonov are being chosen as the reference for the other RTs. Using the ORT of Backus, the minimum of the true error (MSE) is computed, after an a priori upper bound on the model has been selected. This allows one to estimate the statistical bias of the model which is proved to be formally equivalent to the resolution error, as epitomized in the Backus & Gilbert formalism. The Tikhonov ORT, which is used in its original form (the objective one) and as a new version (the subjective approach), allows to constrain best the optimal solution from both the model- and data viewpoints. All of the other Bayesian ORT’s and that of Backus & Gilbert have not been found satisfactory, because they are not able to delimit the optimal model space appropriately.