The inverse scattering imaging condition for anisotropic reverse time migration

The inverse scattering imaging condition (ISIC) is widely used in reverse time migration (RTM), which naturally eliminates the low-wavenumber noise by combining a time-derivative imaging condition with a spatial-derivative imaging condition. However, when applying ISIC to anisotropic RTM, we observe that the low-wavenumber noise is not entirely eliminated. This phenomenon is the leakage of low-wavenumber noise caused by anisotropy. In anisotropic media, due to the deviation of the vertical velocity from the phase velocity, the spatial-derivative imaging condition does not generate correct weights to cancel the low-wavenumber noise obtained by using the time-derivative imaging condition. By adding an anisotropic correction term to compensate for this deviation, we obtain the anisotropic ISIC, which can fully eliminate the low-wavenumber noise. In addition, we construct the connection between the anisotropic ISIC and the scattering function. The scattering function describes the power of the media to generate the scattering field, and it encompasses the effect of model perturbations and their scattering patterns. Under the small perturbation assumption and weak anisotropy approximation, we theoretically find that the ISIC reverses the scattering process and recovers the scattering function from the scattering field. Through three numerical examples, we determine the effectiveness of our method in suppressing the low-wavenumber noise for anisotropic RTM. The anisotropic ISIC also has potential applications in inversion-based imaging algorithms.