Multiscale Total Variation Regularization Method For Visco Elastic Seismic Waveform Inversion

Due to the wide application and enormous potential economic benefits, seismic waveform inversion has always been the hotspots and the focus of research and theory. So, developing a practical and robust numerical waveform inversion method is of great practical significance and theoratical value.In this paper, we first introduce the background and current situation of seismic waveform inversion. As the existing model cannot simulate the real geological parameter very well, the advance of visco-elastic wave equation is discussed here. After introducing the foundamental theory of visco-elastic medium, the wave equation of Kelvin medium and its discrete form are derived. Then we give a numerical simulation of the forward problem and present the snapshots of wave field when homogeneous and heterogeneous parameters are selected respectively. Therefore, the accuracy of the forward simulation is verified.With the forward simulation of visco-elastic equation, we obtain the formulation of the least squares functional. To deal with the inherent ill-poseness of the inverse problem and the nonlinearity of the visco-elastic wave equation, we introduce the Total Variation(TV) regularization, which is a better choice when block-like parameter contained sharp edges or piecewise smooth segments is to be reconstructed. For the specialty of the visco-elastic wave equation, the corresponding least squares target functional is formulated. We then give the matrix form of discrete TV penalty term.To determinate the minimum of the TV regualrization functional, we linearize the original functional step by step and derive the Gauss-Newton method and its discrete iterative form. As Multi-grid method can effectively avoid being trapped in local extremums, it is utilized to solve the nonlinear problem above. We select double-grid method, nine-point continuation operator and nine-point restriction operator and give the multiscale TV regularization method for visco-elastic waveform inversion. Then we carry out two different numerical examples in which hierarchy and porous parameters are to be reconstructed. The updating process of the parameter and the final result are presented. From the results obtained, we can see that the desired parts of the parameter are inversed opporapriatly, which enable the method to be applied practically and to reconstruct the layer parameter well.