Forward operator: Migration

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

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:  

$$\fbox {$ u_0^{z+1}(\omega_{\!}) = T_0^{z}(\omega_{\!},s_0)u_0^{z}(\omega_{\!}) $}$$ (10)

initialized by the wavefield at the surface, as follows:  

$$u_0^{1}(\omega_{\!}) = f(\omega_{\!}) \d(\omega_{\!})$$ (11)

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.

The recursion in Equations (A-1) and (A-2) can be also rewritten in matrix form as  

$$\fbox {$ [\bf {I}-\bf {T_0}]\bf{U_0}=\bf {D} $}$$ (12)

$$\begin{eqnarray} \left[ \matrix { {1} & 0 & 0 &...& 0 & 0 \cr {-T_0^{1}} & {1} &... ...left[ \matrix{ f~d \cr 0 \cr 0 \cr...\cr 0 \cr } \right]\nonumber \end{eqnarray}$$

where

• $\bf {T_0}$ is a square matrix containing the downward continuation operator for all depth levels,
• $\bf{U_0}$ is a column vector containing the wavefield at all depth levels, and
• $\bf {D}$ is a column vector containing the scaled data.

Equation (A-3) represents the downward continuation recursion written for a given frequency. We can write a similar relationship for each of the frequencies in the analyzed data, and group them all in the matrix relationship  

$$\fbox {$ (\mathcal I-\mathcal T_0) \mathcal U_0= \mathcal D $}$$ (13)

$$\begin{eqnarray} \left( \matrix { {\bf {I}-\bf {T_0}{(\omega_{1},s_0)}} & 0 &...... ......\cr \bf {D}{(\omega_{N_{\omega_{\!}}})} \cr } \right)\nonumber \end{eqnarray}$$

where

• $\mathcal T_0$ is a diagonal matrix containing the downward continuation operators for all the frequencies in the data,
• $\mathcal U_0$ is a column vector containing the wavefield data for all the frequencies, and
• $\mathcal D$ is a column vector containing the scaled data at all frequencies.

It follows from Equation (A-4) that the background wavefield ($\mathcal U_0$) can be computed as a function of the measured data ($\mathcal D$), as follows:  

$$\mathcal U_0= (\mathcal I-\mathcal T_0)^{-1} \mathcal D$$ (14)

2.

Imaging

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

$$\fbox {$ \i_0^{z}=\sum_1^{N_{\omega_{\!}}} u_0^{z}(\omega_{\!}) $}$$ (15)

where

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

We can write the Equation (A-6) in matrix form as  

$$\fbox {$ \mathcal R_0= \mathcal H\mathcal U_0 $}$$ (16)

$$\begin{eqnarray} \left( \matrix{ \i_0^{1} \cr \i_0^{2} \cr ...\cr \i_0^{N_z} \cr... .....\cr \bf{U_0}{(\omega_{N_{\omega_{\!}}})} \cr } \right)\nonumber \end{eqnarray}$$

where

• $\mathcal H$ is an operator performing the summation over frequency for every depth level z, and
• $\mathcal R_0$ is a column vector containing the image at every depth level.

Therefore, the image ($\mathcal R_0$), corresponding to the background velocity field, can be computed from the measured data ($\mathcal D$) using the summation ($\mathcal H$) and the downward continuation operators ($\mathcal T_0$) as  

$$\fbox {$ \mathcal R_0= \mathcal H(\mathcal I- \mathcal T_0)^{-1} \mathcal D $}$$ (17)