The L-curve Method for the Regularization Parameter Search in Cross-Well Attenuation Geophysical Tomography

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 have the first order. Then, second order regularization is when the matrix is formed by second order differences, and zeroth order means that the regularization matrix is the identity. There is a crucial problem in regularization, which is the selection of the regularization parameter lambda. We used the L-curve as a tool for the lambda selection, which was reintroduced in the literature of inverse problems by Hansen in 1992. The application was in geophysical attenuation tomography in the cross-hole acquisition geometry. The goal is to obtain the 2-D attenuation distribution from the measured values of the amplitudes. We present several simulation results with synthetic data, for the first order regularization. We validate the necessity of some kind of regularization, as well as the feasibility of both parameter selection approaches. The input data used for the inversion procedure are the M measures amplitudes between M1 sources and M2 receivers, being M = M1 x M2. The inversion output are the N slowness values which describe the 2-D medium. We present two synthetic models that simulate different geological structures like mixed structural traps with potential hydrocarbon reservoirs. The geological models have a rectangular shape with 20 layers and 40 columns, resulting thus into 800 blocks. Each block has a constant velocity or slowness (velocity reciprocal), but the velocity may differ between two neighboring blocks. For the acquisition we considered one borehole with 30 sources and another one with 30 receivers, in such a way that we have 900 rays connecting sources and receivers, or 900 equations. Thus, the problem is said to be overdetermined since there are more equations (900) than unknowns (800). In order to evaluate the robustness we added to the amplitude data different levels of Gaussian noise. We employed three regularization orders: zero, first and second. For each order we generated L-curves. The reconstructed models, for the three regularization orders, have good concordance with the true model, allowing us to validate the L-curve approach. The results were also good when Gaussian noise was added to the data.