参考波速线性变化时的声波方程逆散射反演

声波方程的逆散射反演乃是求解双曲型偏微分方程系数项反问题的一种解析方法,一般利用Born近似把这一非线性反问题线性化,并给出了恒参考波速介质中反问题解的解析表达式.由于Born近似假定波速扰动为一级无穷小,因此,在大多数情况下,恒参考波速介质模型的反问题的解无法得以应用.本文研究介质参考波速沿某个方向线性变化时的声散射理论,导出了声波方程逆散射问题解的解析表达式,从而既可使Born近似的假定在大多数情况下能得以满足,又可利用快速Fourier变换快速实现介质波速扰动的反演成象.

INVERSION OF THE ACOUSTIC WAVE EQUATION BY INVERSE SCATTERING IN A MEDIUM OF LINEAR REFERENCE VELOCITY

The inverse scattering of the acoustic wave equation is an analytic method used to solve the inverse problem of determining coefficient terms in a hyperbolic partial differential equation. In the method, the Born approximation has been employed to linearize the inverse problem and an analytic solution has been obtained when the reference velocity is assumed to be constant. Because the assumption of small velocity disturbances in the Born approximation can hardly be satisfied in most cases, it is difficult to apply the solution of the constant reference velocity model to estimating the velocity disturbances. Considering a more realistic medium of a linear reference velocity, we first show the Green function and the expression of acoustic scatter waves, then obtain the analytic solution of the inverse scattering problem, resulting in a new procedure for inversion of the velocity disturbances. The inversion procedure must enjoy both high accuracy, due to satisfying the assumption of small velocity disturbances, and high computation speed by making it possible to employ the FFT algorithm.