格林函数反演中的互相关核与时间反演声学
Crosscorrelation kernel in the Green function retrieval and time reversal acoustics
摘要：
利用严格的积分方程数学理论，揭示了互相关波场反演格林函数的互相关结构。先前通过将在由相同源产生的两个位置处记录的波场互相关并且然后对边界上的所有源上的这种互相关求和来提取格林函数的实践是不充分的，并且在恢复格林函数中的低频内容方面具有限制。为了恢复精确的格林函数，我们需要互相关的波场观察到的两个位置产生的不同源，分别。互相关结构在离散情况下可以被视为矩阵，在连续情况下可以被视为积分算子。以前的格林函数检索方法对应于单位矩阵乘以常数。如果矩阵是对角的，则由于相同的源，波场互相关仍然在波场内，但是对于不同的源应该应用不同的加权。我们已经推导出解析的互相关内核的两个重要的情况下，平面边界和圆形边界，这两个内核是对称的差分内核，对应于卷积运算。对于高频波或远场源，核减少到已知的结果，即狄拉克δ分布（或单位矩阵）。对于一般几何形状的其他边界，可以使用边界元法等数值方案来求解核矩阵。
英文摘要：
Crosscorrelation structures in the Green function retrieval by crosscorrelating wavefields are revealed using rigorous mathematical theory on integral equations. The previous practice on extracting the Green function by crosscorrelating the wavefields recorded at two locations produced by the same source and then summing such crosscorrelations over all sources on a boundary is inadequate and has limitations in recovering the low frequency content in the Green function. To recover the exact Green function, we need crosscorrelate the wavefields observed at the two locations generated by different sources, respectively. The crosscorrelation structure can be viewed as a matrix in the discrete case or an integral operator in the continuous case. The previous Green function retrieval method corresponds to the identity matrix multiplying a constant. If the matrix is diagonal, the wavefield crosscorrelation is still within the wavefield due to the same source but different weighting should be applied for different sources. We have derived analytically the crosscorrelation kernels for two important cases, the plane boundary and the circular boundary, and both kernels are symmetrical difference kernels, which correspond to convolutional operations. For high frequency waves or far-field sources, the kernels reduce to the know result which is the Dirac delta distribution (or an identity matrix). For other boundaries of general geometric shapes, numerical schemes like the boundary element method can be used to solve for the kernel matrix.
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