Estimation of phase velocities and Q‐factors from zero‐offset, vertical seismic profile data

Abstract

In seismic processing, we face the problem of characterizing the effect of anelastic attenuation on the seismic signal during propagation. To solve this problem, approximate physical properties of the porous fluid‐filled rock are needed. We give a strategy for estimating effective parameters governing absorption and dispersion of waves in viscoelastic media by inverting zero‐offset vertical seismic profiling (VSP) data acquired in a medium with plane horizontal layers. The VSP data are filtered such that all energy except for the direct downgoing wave and the primary reflected wave from each interface is zeroed. This procedure requires a model with layer thicknesses greater than a minimum limit. A stack of thin layers must be replaced by a layer with average physical properties and with thickness exceeding the required minimum. The model parameter vector is partitioned into two vectors. The first contains the frequency‐dependent complex propagation velocity in each layer, and is evaluated over the frequency band where the signal‐to‐noise ratio is acceptable. In the second vector, the geophone‐to‐formation coupling factors, which are assumed to be frequency‐independent, are gathered. It is straightforward to determine both the frequency‐dependent phase velocities and the frequency‐dependent quality (Q‐) factors from the frequency‐dependent propagation velocities. We assume that layer boundaries and layer densities can be obtained from well logs. We give the equations for a simplified forwardmodeling scheme and the equations for the solution of the nonlinear inverse problem. The algorithm is applied to both synthetic and real data. Inversion of synthetic data shows that the phase velocities can be satisfactorily estimated, and that Q‐factors below approximately 50 are well‐resolved, even for large errors in the geophone‐to‐formation coupling factors. The estimated phase velocities from the real data behave fairly stable as a function of frequency. The results for the quality factors are less conclusive, but the low Q‐factors may be of correct size.