**Geophysical Journal International**

# Velocity shift in two-dimensional anisotropic random media using the Rytov method

## SUMMARY

When high-frequency waves propagate through a randomly inhomogeneous medium, the apparent wave velocity is larger than the spatial average of the velocity distribution. The difference between the two velocities is referred to as the velocity shift. The present study formulates the velocity shift in 2-D anisotropic random media using the Rytov method. Anisotropic random media in (*x*, *z*) coordinates are characterized by autocorrelation functions (ACFs) with a long correlation distance along the *x*-axis and a short correlation distance along the *z*-axis, where the velocity structure varies smoothly along the *x*-axis and rapidly along the *z*-axis. Note that the spatial average values of the velocity distribution along the *x*-axis and the *z*-axis are the same. The formulation gives the velocity shift for general types of ACF. An analytic solution is obtained for the case of a Gaussian ACF. To examine the reliability of the Rytov method, the velocity shift is estimated from numerical simulations of wave propagation using Ricker wavelets with dominant frequencies 80 and 40 Hz. The random media are realized with a spatial average velocity of 2700 m s^{−1} and an exponential ACF with 5 per cent rms fractional velocity fluctuation, a correlation distance of 80 m along the *x*-axis and 40 m along the *z*-axis. Numerical simulations show that waves apparently propagate faster with increasing travel distance, frequency and the angle of incidence measured from the *z*-axis to the global ray direction. For example, in the case of the 80 Hz Ricker wavelet at a distance of 520 m, the values of the velocity shift are about 0.9 and 0.3 per cent along the *x*-axis and the *z*-axis, respectively. The Rytov method quantitatively explains these general tendencies except for short travel distances along the *x*-axis. The discrepancy at short travel distances could be due to the small-angle scattering approximation and the long travel distance approximation employed in the Rytov method. Observations of *P*-wave velocity anisotropy have usually been interpreted in terms of preferred orientations of cracks and minerals in past studies. However this study indicates that wave scattering due to anisotropic random media can provide an alternative explanation for those observations.