Virtual reflector multidimensional deconvolution: Inversion issues for convolutive-type interferometry

We examine the multidimensional deconvolution (MDD) approach for virtual-reflector (VR) signal representation by cross-convolution. Assuming wavefield separation at receivers, the VR signal can be synthesized by cross-convolution of inward and outward wavefields generated from a multiplicity of transient sources. Under suitable conditions, this virtual signal is representable as the multidimensional composition of (1) the outward wavefield from the redatumed virtual sources at receivers and (2) of the so-called point-spread function (PSF) for VRs. Multidimensional inversion of the PSF provides the solution to get deblurred signals and recover the Green's function either of transmitted wavefields or reflectivity. This approach is similar to the MDD by backward seismic interferometry of cross-correlation type. The forward approach by cross-convolution poses the issue to use suitable projections and representations by functions with convex trends in space and time. This work discusses the main differences for illumination and stability in the cross-convolution and cross-correlation approaches, providing, under appropriate coverage conditions, equivalent and robust inversion results.