弱横向非均匀介质中的动态反问题
Dynamic inverse problem in a weakly laterally inhomogeneous medium
AI总结：
针对弱横向非均匀半空间声波传播反问题，通过小参数展开将问题降维为一系列一维反问题，并给出零阶和一阶近似下的求解算法，用于数值重建速度和界面。
摘要：
在声学近似下，考虑波在占据半空间的弱横向非均匀介质中传播的反问题。该半空间由上层和半无限底层组成，两者之间有一个界面。弱横向非均匀性的假设意味着波传播速度和界面形状对水平坐标$x=(x_1,x_2)$的依赖较弱，而对垂直坐标$z$的依赖较强，从而引入小参数$\e <<1$。将速度展开为$\e$的幂级数，我们得到一维反问题的递归系统。我们提供了求解零阶和一阶近似问题的算法。在零阶近似中，相应的一维反问题被简化为非线性Volterra型积分方程组。在一阶近似中，相应的一维反问题被简化为耦合的线性Volterra积分方程组。这些方程用于数值重建两层中的速度和界面，精度达到$O(\e^2)$。
英文摘要：
An inverse problem of wave propagation into a weakly laterally inhomogeneous medium occupying a half-space is considered in the acoustic approximation. The half-space consists of an upper layer and a semi-infinite bottom separated with an interface. An assumption of a weak lateral inhomogeneity means that the velocity of wave propagation and the shape of the interface depend weakly on the horizontal coordinates, $x=(x_1,x_2)$, in comparison with the strong dependence on the vertical coordinate, $z$, giving rise to a small parameter $\e <<1$. Expanding the velocity in power series with respect to $\e$, we obtain a recurrent system of 1D inverse problems. We provide algorithms to solve these problems for the zero and first-order approximations. In the zero-order approximation, the corresponding 1D inverse problem is reduced to a system of non-linear Volterra-type integral equations. In the first-order approximation, the corresponding 1D inverse problem is reduced to a system of coupled linear Volterra integral equations. These equations are used for the numerical reconstruction of the velocity in both layers and the interface up to $O(\e^2)$.
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