Jacobian matrix for the inversion of P- and S-wave velocities and its accurate computation method

Abstract

The optimization inversion method based on derivatives is an important inversion technique in seismic data processing, where the key problem is how to compute the Jacobian matrix. The computation precision of the Jacobian matrix directly influences the success of the optimization inversion method. Currently, all AVO (Amplitude Versus Offset) inversion techniques are based on approximate expressions of Zoeppritz equations to obtain derivatives. As a result, the computation precision and application range of these AVO inversions are restricted undesirably. In order to improve the computation precision and to extend the application range of AVO inversions, the partial derivative equation (Jacobian matrix equation (JME) for the P- and S-wave velocities inversion) is established with Zoeppritz equations, and the derivatives of each matrix entry with respect to P- and S-wave velocities are derived. By solving the JME, we obtain the partial derivatives of the seismic wave reflection coefficients (RCs) with respect to P- and S-wave velocities, respectively, which are then used to invert for P- and S-wave velocities. To better understand the behavior of the new method, we plot partial derivatives of the seismic wave reflection coefficients, analyze the characteristics of these curves, and present new understandings for the derivatives acquired from in-depth analysis. Because only a linear system of equations is solved in our method, the computation of Jacobian matrix is not only of high precision but also is fast and efficient. Finally, the theoretical foundation is established so that we can further study inversion problems involving layered structures (including those with large incident angle) and can further improve computational speed and precision.