Mixed-domain migration theory

Downward continuation is the process in which we recursively extrapolate in depth the wavefield recorded at the surface. Mathematically, this operation amounts to a phase shift applied to the wavefield Claerbout (1985)

$$U_{z+\Delta z}= \mathcal TU_z,$$

where U_z is the wavefield at depth z, and $U_{z+\Delta z}$ is the extrapolated wavefield at depth $z+\Delta z$. The downward continuation operator at depth z is  

$$\mathcal T= e^{i k_z\Delta z},$$ (1)

with the vertical wavenumber, k_z, given by the one-way wave equation, also known as the single square root (SSR) equation

$$k_z= \sqrt{\omega^2 {s}^2 - \left\vert \bf k_m\right\vert^2},$$

where $\omega$ is the temporal frequency, s is the laterally variable slowness of the medium, and $\bf k_m$ is the horizontal wavenumber.

Since downward continuation by phase shift can be applied for slowness models that only vary with depth, we need to split the operator $\mathcal T$ into two parts: a constant slowness continuation operator applied in the $\omega-{\bf k}$ domain, which accounts for the propagation in depth, and a screen operator applied in the $\omega-{\bf x}$ domain, which accounts for the wavefield perturbations due to the lateral slowness variations. In essence, we approximate the vertical wavenumber k_z with its constant slowness counterpart k_zo, corrected by a term describing the spatial variability of the slowness function (Figure ).

 

screen Figure 1 A sketch of mixed-domain migration. The wavefield at depth z is downward continued to depth $z+\Delta z$ trough a variable-slowness screen.