Source wavefield downward extrapolation

The recursion in equations (2) and (3) can be also written in matrix form as  

$$( I- W^{+}) \, \, P^+= F,$$ (13)

$$\begin{eqnarray} \left[ \begin{array} {cccccc} {1} & 0 & 0 &...& 0 & 0 \\ {-w^+... ...ay} {ccccc} f\\ 0\\ 0\\ ...\\ 0 \end{array} \right], \nonumber \end{eqnarray}$$

where

• W⁺ is a lower bidiagonal matrix containing the downward continuation operator for all depth levels,
• P⁺ is a column vector containing the source wavefield at all depth levels, and
• F is a column vector containing the source signature.

Equation (13) represents the downward 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:  

$$\left( {\mathcal I} - {\mathcal W}^+ \right){\mathcal P}^+={\mathcal F},$$ (14)

$$\begin{eqnarray} \left[ \begin{array} {cccc} { I- W^+(\omega_1)} & 0 &...& 0 \\ ... ..._2)\\ ...\\ F(\omega_{N_\omega}) \end{array} \right], \nonumber \end{eqnarray}$$

where

• $\left( {\mathcal I} - {\mathcal W}^+ \right)$ is a lower bidiagonal matrix containing the downward continuation operators for all the frequencies in the data,
• ${\mathcal P}^+$ is a column vector containing the wavefield data for all the frequencies, and
• ${\mathcal F}$ is a column vector containing the source signature for all the frequencies.

Equation (14) represents the downward 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:  

$$\left( {\bf I} - {\bf W}^{+} \right){\bf P}^+={\bf f},$$ (15)

$$\begin{eqnarray} \left[ \begin{array} {cccc} { {\mathcal I- \mathcal W}^+(s_1)} ... ...2)\\ ...\\ {\mathcal F}(s_{N_s}) \end{array} \right], \nonumber \end{eqnarray}$$

where

• $\left( {\bf I} - {\bf W}^{+} \right)$ is a lower bidiagonal matrix containing the downward continuation operators for all the shots in the data,
• ${\bf P}^+$ is a column vector containing the wavefield data for all the shots, and
• ${ \bf f}$ is a column vector containing the source signature for all the shots.

B