  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.
