P-wave and S-wave separation in thin beds

Compressional (P) and shear (S) waves can be separated in vector vertical-seismic profile recordings through the use of multichannel multidimensional filters. These filters have impulse responses that are spatially and temporally infinite, and their application to recordings of finite spatial extent, such as those made in thin beds, results in a truncation error. The use of constrained shift-varying filters helps to minimize this error. The constrained filters are derived under the assumption that most of the energy in the recording is concentrated around a few apparent (vertical) velocities. Implementation of these shift-varying filters requires little additional computation beyond that required by simpler shift-invariant filters. The reduction of truncation error accomplished by this processing scheme is demonstrated with several synthetic data sets