Johnson S A, Tracy M L
Ultrason Imaging. 1983 Oct;5(4):361-75. doi: 10.1177/016173468300500406.
A new method for solving the inverse scattering problem for the scalar, inhomogeneous, exact, Helmholtz wave equation is presented. No perturbation approximations are used and the method is applicable even for many cases where weak to moderate attenuation and moderate to strong refraction of incident fields occur. The ill-posed nature of the inverse scattering problem for a single monochromatic source is known. However, the use of multiple sources, the collection of redundant (i.e., overdetermined) data, and the constraining of the fields and complex refractive index to be spatially band limited constitutes a new problem. The cases we have tested by computer simulation indicate that the new problem is well posed, a unique solution, and is stable with noisy data. The method is an application of the well-known method of moments with sinc basis and delta testing functions to discretize the problem. The inverse scattering solution may be obtained by solving the resulting set of simultaneous, quadratic, multivariate equations. Several algorithms for solving these equations are given.
提出了一种求解标量、非均匀、精确亥姆霍兹波动方程逆散射问题的新方法。该方法不使用微扰近似，甚至适用于许多入射场存在弱到中等衰减以及中等到强折射的情况。单个单色源的逆散射问题具有不适定性是已知的。然而，使用多个源、收集冗余（即超定）数据以及将场和复折射率限制在空间带限内构成了一个新问题。我们通过计算机模拟测试的案例表明，这个新问题是适定的、有唯一解的，并且对噪声数据是稳定的。该方法是将著名的具有 sinc 基和 delta 测试函数的矩量法应用于离散该问题。通过求解所得的联立二次多元方程组可以得到逆散射解。给出了几种求解这些方程的算法。
