Forward modelling formulas for least-squares reverse-time migration

Reverse-time migration can be formulated in a least-squares inversion framework. This is referred to as least-squares reverse-time migration, which attempts to find an optimal model of the reflectors that fits the observed data in a least-squares sense. Based on different representations of the model, different formulas of the forward modelling for least-squares reverse-time migration can be derived. In this paper, we derive two different formulas. One formula is to recover the impedance-perturbation-related images based on Born approximation. The other is to invert the reflectivity-related images based on Kirchhoff approximation. The theoretical analysis unveils there is an iω difference between the two formulas. Consequently, the seismic image using the two formulas has different shape/phase: the one based on Born approximation produces anti-symmetric images; the other based on Kirchhoff approximation gives symmetric images. Two numerical examples demonstrate the similarities and differences between the two formulas.