Adaptive Homotopy Regularization Methods For Direction And Inversion Of Wave Equation

In field of seismic exploration,it is very valuable to research the propagation of waves in the underground medium and it is more practical to research the forward and inverse problems of wave equation.The forward modeling of the wave equation is an important part of the inversion and the results are used to simulate geological conditions.The inversion of wave equation has a widespread applications in practice,although it is inherently difficult which non-adaptive and non-linear.In this paper,a two-dimensional wave equation is taken as an example to study the forward and inverse of the wave equation in this limited area.This paper applies the alternating direction implicit difference scheme and explicit scheme to discretize the two-dimensional wave equation and analyzes its stability.To test the effect of different locations of the source on the propagation of waves in various underground media,MATLAB numerical simulation is performed on the two constructed difference scheme while different conditions.The simulation results indicate that the alternate direction implicit scheme is a step-by-step method,which both reduces the calculation amount and overcomes the limitation of the explicit scheme on the step size,while maintaining accuracy.In this paper,aiming to the ill-posedness and non-linear nature of the inverse problem of wave equation,the widely convergent homotopy method and regularization method are combined to constructing of minimum optimization equation.The selection of regularization parameters are important key for the adaption of regularization method.The linear relationship between homotopy parameters and regularization parameters is constructed in order to avoid the uncertainty of regularization parameters selected by experience.Then an adaptive homotopy regularization inversion method is formed by combining this technique with the Regularization-Gauss-Newton method.Finally,MATLAB numerical simulation was performed.The paper adopts the alternating direction implicit difference scheme instead of the original direct discrete scheme to perform forward modeling of the wave equation,which overcomes the difficulty of the large calculation amount of the original method.Meanwhile,a new inversion method is formed by constructing a linear relationship between regularization parameters and homotopy parameters,and combining the Regularization-Gauss-Newton method,which overcomes the dependence of the inversion algorithm on the initial value as well as ensures the effect of the inversion.