Inversion of P and SV waves from multicomponent offset vertical seismic profiles

Downgoing waves in multicomponent VSP experiments are used to obtain seismic P- and S-wave velocities as a function of depth and angle of incidence. If P and SV waveforms do not overlap in time at the depth of interest, local velocities of the medium are obtained by separate analysis of these events. The apparent velocity of the event (P or SV) is computed from the moveout across several neighboring depth locations. The angle of incidence of the same event is computed from the particle-motion hodogram within an appropriately chosen time window. Then, the local medium velocity (P wave or S wave depending on the chosen event) is given by the apparent velocity multiplied by the cosine of the angle of incidence.Layer interfaces with reasonably sharp velocity contrasts are efficient P-wave to SV-wave converters, even at moderate angles of incidence. In offset VSP experiments, converted SV waves are generated with varying strengths at practically all depths. Consequently, the converted SV waveforms partially overlap with the direct P waveforms, making the separate event analysis difficult and inaccurate. These overlapping waveforms can be handled properly by modeling the data in a given time window as a superposition of several events. In particular, the downgoing data at each depth level are modeled as a superposition of a P wave and an SV wave, with local P and S velocities, angles of incidence, and waveforms as model parameters. These parameters are then estimated by minimizing the squared error between the observed data and the model-generated data. The unknown waveforms are eliminated from the minimization problem, leaving only four nonlinear parameters (velocities and angles) for estimation. Once these four parameters are found, least-squares estimates of waveforms are obtained by evaluating a simple expression.