An inverse random source problem for the one-dimensional Helmholtz equation with attenuation
An inverse random source problem for the one-dimensional Helmholtz equation with attenuation
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DOI：
10.1088/1361-6420/abcd43
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发表时间：
2020-09
期刊：
影响因子：
2.1
通讯作者：
Peijun Li;Xu Wang
中科院分区：
文献类型：
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作者：
Peijun Li;Xu Wang
研究一维带衰减的随机亥姆霍兹方程的逆随机源问题。假设源为微局部各向同性高斯随机场，其协方差算子为经典伪微分算子。所考虑的随机源等同于广义分数高斯随机场，其中包括粗糙场，甚至可能比白噪声更粗糙，因此应被解释为分布。在分布意义上建立了直接源问题的适定性。证明了随机源的微相关强度，即协方差算子主符号中的强度，是由开放测量集中的波场唯一决定的。通过白噪声模型的数值实验验证了该方法的有效性。
This paper is concerned with an inverse random source problem for the one-dimensional stochastic Helmholtz equation with attenuation. The source is assumed to be a microlocally isotropic Gaussian random field with its covariance operator being a classical pseudo-differential operator. The random sources under consideration are equivalent to the generalized fractional Gaussian random fields which include rough fields and can be even rougher than the white noise, and hence should be interpreted as distributions. The well-posedness of the direct source problem is established in the distribution sense. The micro-correlation strength of the random source, which appears to be the strength in the principal symbol of the covariance operator, is proved to be uniquely determined by the wave field in an open measurement set. Numerical experiments are presented for the white noise model to demonstrate the validity and effectiveness of the proposed method.