Vertical Seismic Profiling: The One-dimensional Forward and Inverse Problems

This study develops two one-dimensional (1-D) inversion procedures to estimate the seismic parameters (velocity, density, attenuation, upgoing and downgoing wavefields) from VSP data. A Marquardt-Levenberg traveltime inversion using 1-D ray tracing is designed and applied to synthetic and field data from the Gulf Coast, Sulphur Springs, Texas and Colorado. The errors in the data and parameters and their relationship to one another are considered. Optimal velocity thickness (40ft-150ft) in the inversion depends partly on the observation spacing and data noise. The travel time inversion is found to provide a stable and accurate 1-D estimate of the velocity section. The VSP velocities are found to be consistently several percent smaller than the sonic velocities. Both P and SH velocities in the Gulf Coast survey are used to estimate gas saturation in a thin sand. The lower bound on the gas saturation is about 10%.

Comparing the VSP traveltimes to the integrated sonic travel times from surveys conducted in the Anadarko Basin, Texas, the above field data and literature uncovers a discrepancy between the seismic and sonic traveltimes. The seismic traveltimes are from 2.0 to 7.0 ms/100ft longer than their corresponding sonic traveltimes. Wave equation synthetic data and field results indicate that this discrepancy may well be explained by wave propagation effects. Velocity dispersion associated with attenuation (nearly-constant Q) appears to cause the most significant time delays while short-path multiples have a smaller but observable effect. Equations to predict these effects are developed.

To allow usage of the full VSP waveform in constraining the seismic parameters, two wave equation based inversions are devised. The weighted damped least-squares inversion is used to simultaneously estimate the velocity, attenuation and upgoing and downgoing waves in a group of four vertically-adjacent VSP seismograms. This process is repeated for the entire set of VSP seismograms. Results from both synthetic and field data show very good parameter estimation. Especially useful is the extracted upgoing wave; it may be used to pinpoint the depth of its generation and to estimate the underlying impedance mismatch. The separated downgoing wave provides a source signature as well as constraint on the velocity and attenuation of the medium.

Impedances are also included as independent parameters in the forward model of stochastic inversion using the four trace group. In areas of large reflection coefficients, impedance contrasts are reasonably estimated. Basically, the algorithm finds the impedances which fit the reflection coefficient and are also closest to the initial guesses. A good first guess is critical. Density may be computed from these impedances. In the field example, the densities have been well estimated near a strong impedance contrast.

Several related theoretical results are developed. Analysis for the extension of the wave equation inversion of the elastic and dipping interface cases is outlined. A procedure for the simultaneous inversion of the complete VSP data set is devised. The lateral resolution (Fresnel zone) is calculated for the VSP geometry and wavelength.

The procedures developed and the results found in this work provide a coherent and reasonably complete analysis of the 1-D vertical seismic profile.