Background field: Forward operator

Migration by downward continuation, in post-stack or prestack, is done in two steps: the first step is to downward continue the data (D) measured at the surface, and the second is to apply the imaging condition, that is, to extract the wavefield at time t=0, or the image (R) at the moment the reflectors explode Claerbout (1985).

1.

Downward continuation

The first step of migration consists of downward continuation of the wavefield measured at the surface (a.k.a. the data), which is done by the recursive application of the equation

 

u₀^z+1 = T₀^z u₀^z (1)

initialized by the wavefield at the surface  

$$u_0^{1} = f \cdot d$$ (2)

where

• $u_0^{z}(\omega_{\!})$ is the wavefield $u_0^{\!}(\omega_{\!})$ at depth z,
• $u_0^{1}(\omega_{\!})$ is the wavefield $u_0^{\!}(\omega_{\!})$ at the surface (z=0),
• $T_0^{z}(\omega_{\!},s_0)$ is the downward continuation operator at depth z,
• $\d(\omega_{\!})$ is the data, i.e., the wavefield at the surface, and
• $f(\omega_{\!})$ is a frequency-dependent scale factor for the data.

2.

Imaging

The second step of the migration by downward continuation is imaging. According to the exploding reflector concept, the image is found by selecting the wavefield at time t=0, or equivalently, by summing over the frequencies $\omega_{\!}$: 

$$\i_0^{z}=\sum_1^{N_{\omega_{\!}}} u_0^{z}(\omega_{\!})$$ (3)

where

• $\i_0^{z}$ is the image (reflectivity) corresponding to a given depth level z.