非均匀介质散射问题的体积分方程数值解法

将非均匀介质视为某一均匀背景介质中的扰动，可建立用均匀背景介质格林函数作基本解的体积分方程．给出了配置法求解体积分方程的数值方法，首先解得扰动域内各点以速度扰动为权的波场函数，然后回代计算得到观测面上各接收点的散射波场．与边界元法和Ｂｏｒｎ近似法计算结果比较表明该方法具有很高的精度，可得到穿过非

VOLUME INTEGRAL EQUATION METHOD FOR SCATTERING PROBLEMS OF INHOMOGENEOUS MEDIA

It is an effective technique to treat inhomogeneous media as perturbations on the back ground of homogeneous media and to treat the perturbations as equivalent source. Two forms of corresponding integral equations are: 1) boundary integral equation with both boundary integral items for boundary displacement and volume integral items for displacement in wavefield; 2) volume integral equation including integral only for inhomogeneous bounded region. Much work has been done in this field. Interpreting inhomogeneous media as a perturbation of the reference medium one can derive a volume integral equation based on the Green's function of the homogeneous reference media. In the present paper, the collocation method is employed to solve the volume integral equation for the velocity weighted wavefield within the scattering region, which includes transmission wave, reflection wave, diffraction wave, head wave and multiscattered wave. Using Fredholm theorem, we can see that the integral equation for the velocity weighted wavefield has unique solution. Then the observed wavefield can be computed through iterated substitutions of the integral equation. Two kinds of techniques for solving the matrix form of the equation are Gaussian elimination and iterative algorithm. The former is more rigorous, but only suitable for smaller scale problems, while the latter is faster and suitable for larger scale problems. In our numerical examples, the volume integral equation method is applied to seismic modeling, with a comparison with those given by the BE method and the approach using the Born approximation, which shows that the volume integral equation is a numerical method with high accuracy for inhomogeneous media.