Seismic interferometry by cross-convolution

Seismic interferometry by cross-correlating recordings from transient or noise sources is often used to retrieve the Green's function between two receivers. However, in order to do this, a configuration is required where the sources surround both receivers. Interferometry by cross-correlation fails to work when such a configuration of sources cannot be achieved. Seismic interferometry by cross-convolution can be applied instead when only one of the receivers is surrounded by the source distribution. In this work, it is investigated whether the convolution approach can be used to retrieve the Green's function using recordings from transient sources or noise recordings. First, it is shown that the reciprocity theorem leads to a Green's function representation. This representation can be applied to retrieve the Green's function between two receivers using the convolution of the corresponding recordings. When the sources are transient, the convolutions have to be corrected for the source time function before integration over the source positions. When the source time function is known, the convolution approach is suitable for the retrieval of the Green's function from earthquake recordings. For uncorrelated noise sources, the convolution of the noise recordings provides no information about the Green's function between the receivers. Therefore, the cross-convolution approach cannot be used to retrieve the Green's function from noise recordings