Estimating and correcting the amplitude radiation pattern of a virtual source

Abstract

In the virtual source (VS) method we crosscorrelate seismic recordings at two receivers to create a new data set as if one of these receivers were a virtual source and the other a receiver. We focus on the amplitudes and kinematics of VS data, generated by an array of active sources at the surface and recorded by an array of receivers in a borehole. The quality of the VS data depends on the radiation pattern of the virtual source, which in turn is controlled by the spatial aperture of the surface source distribution. Theory suggests that when the receivers are surrounded by multi-component sources completely filling a closed surface, then the virtual source has an isotropic radiation pattern and VS data possess true amplitudes. In practical applications, limited sourceaperture and deployment of a single source type create an anisotropic radiation pattern of the virtual source, leading to distorted amplitudes. This pattern can be estimated by autocorrelating the spatial Fourier transform of the downgoing wavefield in the special case of a laterally invariant medium. The VS data can be improved by deconvolving the VS data with the estimated amplitude radiation pattern in the frequency-wavenumber domain. This operation alters the amplitude spectrum but not the phase of the data. We can also steer the virtual source by assigning it a new desired amplitude radiation pattern, given sufficient illumination exists in the desired directions. Alternatively, time-gating the downgoing wavefield before crosscorrelation, already common practice in implementing the VS method, can improve the radiation characteristics of a virtual source.