Receiver wavefield extrapolation

The recursion in equation (6) can be also written in matrix form as  

$$(I-W^{-})\, \, P^-=P^+\, \, {\bf r},$$ (16)

$$\begin{eqnarray} \left[ \begin{array} {cccccc} {1} & {-{w}^{-}(z_0,z_i)} & 0 &.... ...c} 0\\ p^-_r(z_i)\\ 0\\ ...\\ 0 \end{array} \right], \nonumber \end{eqnarray}$$ (17)

where

• W^- is a upper bidiagonal matrix containing the upward continuation operator for all depth levels,
• P^- is a column vector containing the receiver wavefield at all depth levels,
• P⁺ is a diagonal square matrix containing the source wavefield at all depth levels, and
• ${\bf r}$ is the reflectivity at all depth levels.

Equation (16) represents the upward continuation recursion written for a given frequency. We can write a similar relationship for each of the frequencies in the data, and group them all in a matrix relationship:  

$$({\mathcal I}-{\mathcal W}^{-})\, \, {\mathcal P}^-={\mathcal P}^+\, \Sigma^t_\omega\, {\bf r},$$ (18)

where

• $\left( {\mathcal I} - {\mathcal W}^- \right)$ is a upper bidiagonal matrix containing the upward continuation operators for all the frequencies in the data,

  $$\begin{eqnarray} ({\mathcal I}-{\mathcal W}^{-})=\left[ \begin{array} {cccc} { I... .....& { I- W^-(\omega_{N_{\omega}})} \end{array} \right], \nonumber \end{eqnarray}$$

• ${\mathcal P}^-$ is a column vector containing the receiver wavefield for all the frequencies,

  $$\begin{eqnarray} {\mathcal P}^-=\left[ \begin{array} {cccc} P^-(\omega_1)\\ P^... ...)\\ ...\\ P^-(\omega_{N_\omega}) \end{array} \right], \nonumber \end{eqnarray}$$

• ${\mathcal P}^+$ is a diagonal square matrix containing the source wavefield for all the frequencies,

  $$\begin{eqnarray} {\mathcal P}^+=\left[ \begin{array} {cccc} { P^+(\omega_1)} & 0... ...& 0 &...& {P^+(\omega_{N_\omega})} \end{array} \right], \nonumber \end{eqnarray}$$

• and $\Sigma_\omega$ is the sum over frequency matrix with dimensions $N_Z \times (N_Z\times N_\omega)$ (the transpose of the spreading over frequencies),

  $$\begin{eqnarray} \Sigma_\omega = \left[ \left\vert \begin{array} {cccc} {1} & 0 ... ..... \\ 0 & 0 &...& {1} \end{array} \right\vert\right]. \nonumber \end{eqnarray}$$

Equation (18) represents the upward continuation recursion written for a given shot position. We can write a similar relationship for each of the shot positions in the data, and group them all in a matrix relationship:  

$$({\bf I}-{\bf W}^{-})\, \, {\bf P}^-={\bf P}^+\, \Sigma^t_{\omega s} \, {\bf r},$$ (19)

where

• $\left( {\bf I} - {\bf W}^{-} \right)$ is a upper bidiagonal matrix containing the upward continuation operators for all the shots in the data,

  $$\begin{eqnarray} \left( {\bf I} - {\bf W}^{-} \right)=\left[ \begin{array} {cccc... ...thcal I- \mathcal W}^{-}(s_{N_s})} \end{array} \right], \nonumber \end{eqnarray}$$

• ${\bf P}^-$ is a column vector containing the receiver wavefield for all the shots,

  $$\begin{eqnarray} {\bf P}^-=\left[ \begin{array} {cccc} {\mathcal P}^-(s_1)\\ {... ...\\ ...\\ {\mathcal P}^-(s_{N_s}) \end{array} \right], \nonumber \end{eqnarray}$$

• ${\bf P}^+$ is a diagonal square matrix containing the source wavefield for all the shots,

  $$\begin{eqnarray} {\bf P}^+=\left[ \begin{array} {cccc} { {\mathcal P}^+(s_1)} & ... ...& 0 &...& {{\mathcal P}^+(s_{N_s})} \end{array}\right], \nonumber \end{eqnarray}$$

• and $\Sigma_{\omega s}$ is the sum over the frequency matrix with dimensions $N_Z \times (N_Z\times N_\omega \times N_s)$ (the transpose of the spreading over frequencies),

  $$\begin{eqnarray} \Sigma_{\omega s} = \left[ \left\vert \begin{array} {cccc} {1} ... ..... \\ 0 & 0 &...& {1} \end{array} \right\vert\right]. \nonumber \end{eqnarray}$$