The migration operator

The migration operator and its adjoint (L) that are used in this inversion scheme are linear operators. For the 2-D case, we choose to use the downward continuation migration operator introduced by Prucha et al. (1999). This 2-D downward continuation migration operator takes an input of seismic data with the dimensions of common midpoint (CMPX), offset (h_x), and frequency ($\omega$). Its output is a model (image) with the dimensions of depth (z), common reflection point (CRPX), and offset ray parameter (p_hx), which is related to the reflection angle for a given subsurface point. This downward continuation migration operator can be formulated as a 3-D process by adding the crossline common midpoint (CMPY) and crossline offset (h_y) to the input, but that would be a very computationally expensive process. Fortunately, to reduce costs in 3-D, we could also use a Common Azimuth Migration (CAM) operator Biondi and Palacharla (1996). For this, we add the CMPY dimension, but not the crossline offset.