Research On The Forward And Inversion Multiscale Method For Porous Medium Wave Equations

In the research of seismic wave propagation in underground rock,the theory of porous medium describes the underground formation more accurately. As the coefficients of porous medium wave equations contain much related parameter information of underground formation, the direction and inversion of these coefficients can provide a lot of geological information and more accurate seismic data for practice. But wave equation inverse problem faces with much essential difficulties, such as nonlinearity and ill-posedness of operator equations. In addition, a huge amount of computation and workload of numerical solution will be generated in practical applications. Therefore, it is necessary to the deeply study the wave equation direct problem, inverse problem, and their numerical algorithm. It has very broad application prospects.In this paper,aiming to the characteristics of the wave equation of the porous medium as well as the current problems encountered in the numerical inversion, the mathematical model of the porous medium wave equations is selected based on the Biot theory and related algorithm is constructed. After the model is discreted with finite difference method, the regularization processing is used for the nonlinear problem. Due to discontinuous of the problem and good regularization effect of total variation regularization for discontinuous function, the ill-posed problem is solved by the total variation regularization. Then the conjugate gradient iteration method is adapted to form the single scale the total variation regularization-conjugate gradient method. Generally speaking, wave equation inverse problem has heavy calculation. In order to reduce calculation and improve efficiency, the multiscale the total variation regularization-conjugate gradient method is constructed on the base of the thought of multigrid.The feasibility, efficiency and stability of the algorithm, we carry out numerical inversion of porous medium wave equation porosity parameters on a three-layer medium model and single anomaly body model. The numerical simulation results robustly indicated feasibility and effect of the algorithm, which solve some difficult in inversion to some extent.