Seismic scattering and velocity dispersion due to heterogeneous lithology

Abstract

Seismic velocity obtained from different measurements, such as Vertical Seismic Profiling (VSP) and sonic logging, is often found to vary as a function of scale or frequency. This variation is believed to be due to intrinsic attenuation and intrabed multiples. The latter effect results from scattering from heterogeneous formations. In this study, we investigate the relationship between seismic scattering and the resulting velocity dispersion. Our approach is based on the numerical simulation of one-dimensional (1-D) wave propagation through heterogeneous media. These media are characterized by a heterogeneity scale length and a distribution that is fractal at small wavelengths. By simulating wave propagation in a broad frequency range, seismic wave attenuation due to scattering can be determined as a function of frequency for the 1-D case. We found that for the fractal distribution, the attenuation (expressed as Q-l) is linearly proportional to frequency at low frequencies (Rayleigh scattering in 1-D) and is inversely proportional to frequency at high frequencies (diffusion scattering). The transition between the two scattering regimes depends on the scale length of the medium. Therefore, Q-’ can be analytically expressed in terms of the rms impedance fluctuation and heterogeneity scale length of the medium. By requiring that the seismic wave propagation through the scattering medium be causal, we can apply the Kramers-Kronig integrals to derive a velocity dispersion relation from the scattering attenuation measured from the 1-D numerical simulation. This relation predicts that low-frequency waves (e.g. VSP seismics) are delayed with respect to high-frequency (logging) waves. Numerical experiments were performed to verify this dispersion relation, and the results are in good agreement with the theory. This relation will be useful in the calibration of sonic log with VSP or surface seismics. For example, the delay of VSP travel t.imes with respect to the integrated sonic times can be corrected for scattering effects by measuring the scale length and rms fluctuation of the medium from sonic logs and using this relation.