Abstract This paper concerns the random source problems for the time-harmonic acoustic and elastic wave equations in two and three dimensions. The goal is to determine the compactly supported external force from the radiated wave field measured in a domain away from the source region. The source is assumed to be a microlocally isotropic generalized Gaussian random function such that its covariance operator is a classical pseudo-differential operator. Given such a distributional source, the direct problem is shown to have a unique solution by using an integral equation approach and the Sobolev embedding theorem. For the inverse problem, we demonstrate that the amplitude of the scattering field averaged over the frequency band, obtained from a single realization of the random source, determines uniquely the principle symbol of the covariance operator. The analysis employs asymptotic expansions of the Green functions and microlocal analysis of the Fourier integral operators associated with the Helmholtz and Navier equations.

中文翻译：

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时谐声波和弹性波的逆随机源问题

摘要 本文研究二维和三维时谐声波和弹性波方程的随机源问题。目标是从远离源区的域中测量的辐射波场确定紧密支撑的外力。假设源是一个微局部各向同性广义高斯随机函数，这样它的协方差算子是一个经典的伪微分算子。给定这样的分布源，通过使用积分方程方法和 Sobolev 嵌入定理，直接问题被证明具有唯一的解决方案。对于逆问题，我们证明了从随机源的单个实现中获得的在频带上平均的散射场幅度，唯一确定协方差算子的主要符号。该分析采用格林函数的渐近展开和与亥姆霍兹和纳维方程相关的傅立叶积分算子的微局部分析。