A generalized Rytov approximation for accurate calculation of phase variation in strong perturbation media

Abstract

SUMMARY In the case of long-range propagation of forward scattering, due to the accumulation of phase changes caused by the velocity perturbations, the validity of the Born approximation will be violated. In contrast, the phase-change accumulation can be handled by the Rytov approximation, which has been widely used for long-distance propagation with only forward scattering or small-angle scattering involved. However, the weak scattering assumption (i.e. small velocity perturbation) in the Rytov approximation limits its scope of application. To address this problem, we analyse the integral kernel of the Rytov transform using the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation and we demonstrate that the integral kernel is a function of velocity perturbation and scattering angle. By applying a small scattering angle approximation, we show that the phase variation has a linear relationship with the slowness perturbation, no matter how strong the magnitude of perturbation is. Therefore, the new integral equation is then referred to as the generalized Rytov approximation (GRA) because it overcomes the weak scattering assumption of the Rytov approximation. To show the limitations of the Rytov approximation and the advantages of the proposed GRA method, first we design a two-layer model and we analytically calculate the errors introduced by the small scattering angle assumption using plane wave incidence. We show that the phase (traveltime) variations predicted by the GRA are always more accurate than the Rytov approximation. Particularly, the GRA produces accurate phase variations for the normal incident plane wave regardless of the magnitude of velocity perturbation. Numerical examples using Gaussian anomaly models demonstrate that the scattering angle has a crucial impact on the accuracy of the GRA. If the small scattering angle assumption holds, the GRA can produce an accurate phase approximation even if the velocity perturbation is very strong. On the contrary, both the first-order Rytov approximation and the GRA fail to get satisfying results when the scattering angle is large enough. The proposed GRA method has the potential to be used for traveltime modelling and inversion for large-scale strong perturbation media.