EVALUATION OF L-CURVE AND Θ-CURVE APPROACHES FOR THE SELECTION OF REGULARIZATION PARAMETERS IN ANISOTROPIC TRAVELTIME TOMOGRAPHY

摘要

Estimation of anisotropic models from traveltime tomography is an ill-posed problem, even in the case of full coverage. To overcome this data incompleteness, additional information must be included to regularize the inverse problem. We use derivative operators for regularization. This introduces smoothness constraints in the solution and it is known in the literature by several names, specially as Tikhonov regularization. When a first derivative matrix is used, the order is called the first. Then, second order regularization is when the matrix is formed by second order differences, and order zero means that the regularization matrix is the identity. Regularization has an input parameter with a crucial role, known as the regularization parameter. In this work we propose a new criterion to select the optimum regularization parameter, λ,which we call the Θ-curve and compare its performance with the L-curve criterion, which was reintroduced in the literature of inverse problems by Hansen (1992a). We applied both criteria in crosshole traveltime tomography. In this tomography approach, the 2D velocity field is estimated based on the measured values of traveltime between sources and receivers. In numerical experiments with synthetic data and in a real data set application, the proposed criterion was effective and more robust than the L-curve approach.