Transient effects at a velocity discontinuity in up/down filtering

The up/down convolutional velocity filter is commonly used in vertical seismic profiling to separate an upgoing from a downgoing wave field. In the neighborhood of a velocity discontinuity one expects to see transient effects. For a simple medium containing a single velocity discontinuity it is possible to derive an explicit formula for these transients. This analysis predicts that the transients decay at a rate inversely proportional to the distance from the velocity discontinuity. Despite the slow decay rate, these transients are found to be quite insignificant