L- and Θ-curve approaches for the selection of regularization parameter in geophysical diffraction tomography

Abstract

Since inverse problems are usually ill-posed it is necessary to use some method to reduce their deficiencies. The method that we choose is the regularization by derivative matrices. 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. There is a crucial problem in regularization, which is the selection of the regularization parameter λ. We used the L-curve as a tool for the selection of λ, and we propose a new extension, which we call the Θ-curve. The tool was applied in geophysical diffraction tomography in two acquisition geometries: cross-hole and vertical seismic profile (VSP), where the goal is to obtain the 2-D velocity distribution from the measured values of the scattered acoustic field. We present several simulation results with synthetic data, for the three regularization orders mentioned above. We validate the necessity of some kind of regularization, as well as the feasibility of both parameter selection approaches.