HOMOTOPY METHOD FOR INVERSING THE POROSITY OF 1-D WAVE EQUATION IN POROUS MEDIA
双相介质波动方程孔隙率反演的同伦方法

According to that the computed and measured response should be fitted, the porosity parameter inversion problem in porous media is reduced to solve a problem of nonlinear equations' zero with the homotopy method in this paper.The essence of traditional optimum methods such as gradient method, perturbation method or time-convolution regularization iterative method is based on Newton iterative method with local convergence. The homotopy method is a newly developed powerful device for solving nonlinear problems. It is introduced to improve on the convergent state of Newton iterative method. The basic idea of the homotopy method is to construct a homotopy map with a homotopy parameter, then tracking the homotopy path with the homotopy parameter as the variable numerically to yield the solution that is needed.Because of the complexity of the wave equations in porous media, the associated dynamic problems are often solved by numerical methods. Up to now, a few analytical solutions for these initial-boundary value problems have been obtained. The analytical solution with high theoretical value obtained by Simon (1984) using Laplace integral transform and inverse technique for the transient response of the one-dimensional wave equation in porous media is used to inverse the porosity parameter with the homotopy method in this paper. The Euler predictor-Newton corrector algorithm is taken to tracking the homotopy path. In the end, the inversion results are compared with those computed by the time-convolution iterative method. The numerical results show that the porosity inversion by using the homotopy method is very effective. It is convergent for any arbitrary initial values of porosity very well. It is proved that the homotopy method is a widely convergent method. It not only suits to solve highly nonlinear inverse problems but also suits to get the initial parameter value that is very difficult to obtain.