双相介质波动方程孔隙率反演的同伦方法

HOMOTOPY METHOD FOR INVERSING THE POROSITY OF 1-D WAVE EQUATION IN POROUS MEDIA

从材料响应的理论合成应与实际测量数据相拟合这一出发点,将双相介质波动方程参数反演问题转化为非线性算子方程的零点求解问题,从而应用一种大范围收敛的同伦方法来求解非线性算子方程,并把这种方法用于Simon(1984)给出的具有解析解的一维双相介质模型的数值模拟,最后的数值结果表明,给出的算法是十分有效的.

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.