Singular-value decomposition analysis of source illumination in seismic interferometry by multidimensional deconvolution

We have developed a method to analytically evaluate the relationship between the source-receiver configuration and the retrievedwavefieldinseismicinterferometryperformedbymultidimensional deconvolution (MDD). The MDD method retrieves thewavefieldwiththedesiredsource-receiverconfigurationfrom the observed wavefield without source information. We used a singular-value decomposition (SVD) approach to solve the inverse problem of MDD. By introducing SVD into MDD, we obtained quantities that revealed the characteristics of the MDD inverse problem and interpreted the effect of the initial sourcereceiverconfigurationforasurveydesign.Wenumericallysimulated the wavefield with a 2D model and investigated the rank of the incident field matrix of the MDD inverse problem. With a source array of identical length, a sparse and a dense source distribution resulted in an incident field matrix of the same rank and retrievedthesamewavefield.Therefore,theoptimumsourcedistribution can be determined by analyzing the rank of the incident field matrix of the inverse problem. In addition, the introduction ofscatterersintothemodelimprovedthesourceilluminationand effectively increased the rank, enabling MDD to retrieve a better wavefield. We found that the ambiguity of thewavefield inferred from the model resolution matrix was a good measure of the amount of illumination of each receiver by the sources. We used the field data recorded at the two boreholes from the surface sourcestosupportourresultsofthenumericalmodeling.Weevaluated the rank of incident field matrix with the dense and sparse source distribution. We discovered that these two distributions resulted in an incident field matrix of almost the same rank and retrieved almost the samewavefield as the numerical modeling.Thisiscrucialinformationfordesigningseismicexperiments using the MDD-based approach.