The generalized one‐dimensional synthetic seismogram

The normal‐incidence synthetic seismogram for an elastic and horizontally stratified medium has been thoroughly studied for a relatively restricted number of source and receiver locations. Most existing treatments are concerned with the special case in which the source as well as the receiver are situated at the surface; few attempts have dealt with completely arbitrary source and receiver geometries. Here we examine arbitrary geometries with the aid of the layer matrix approach, in which upgoing and downgoing wave motion at each interface is expressed in terms of z-transform polynomials. Such an approach brings to light a number of physically important relations that the model satisfies. For example, the synthetic seismograms generally have the familiar autoregressive‐moving average (ARMA) structure for the surface‐source, surface‐receiver case. For particular combinations of reflection coefficients, however, the seismograms reduce to purely autoregressive (AR) representations. In all cases, we work out the delay properties that the respective autoregressive and moving average components must obey. The present solutions are easily reduced to a useful form for practical computation. One application of particular current interest is the simulation of vertical seismic profiling (VSP) surveys, where we have extended the theoretical treatment to include expressions for the derivatives of the seismograms with respect to the reflection coefficients. The resulting time series, which we call Jacobograms, are indicative of the sensitivity of the seismogram to the various reflection coefficients and are thus diagnostic of the model’s behavior.