Amplitude Radiation Pattern of a Virtual Source - Estimation and Correction

In the Virtual Source (VS) method we cross-correlate recordings at two receiver locations to create data as if one of these receivers is a Virtual Source and the other is a receiver. We study the amplitude radiation pattern of Virtual Sources. This pattern can be estimated by autocorrelation of the spatial Fourier transform of the downgoing wave field that is used for the VS creation in the special case of a laterally invariant medium. The generated VS data can be improved by deconvolution with the estimated amplitude radiation pattern in the FK-domain. The methodology is tested on a 1D elastic model, where it is shown that almost perfect amplitude retrieval is possible within a limited aperture of VS radiation. In general heterogeneous media the spatial Fourier transforms are not laterally invariant and the VS amplitude radiation pattern has to be estimated in a different way. We argue that Wigner distribution functions can serve this purpose. We show how these functions can be used for diagnosis of the spatial distributions of the wave fields that are used for VS creation, with a VS synthetic data example generated by the Peace River 2D elastic model.