Wave-equation Datuming for laterally variable velocity

For laterally variant media, Gazdag and Sguazzero (1984) propose to downward extrapolate the wavefield one depth interval at a time with several velocities. We can apply the same idea to upward propagate the wavefield with several velocities. To obtain a single upward propagated wavefield at each space point the value of the resulting wavefield is interpolated between the two wavefields with the closest velocities. Linear interpolation between two values P₁ and P₂ is an operation of the type

$$\left[ \begin{array} {c} \tilde{P} \\ \end{array}\right] ... ...] \left[ \begin{array} {c} P_1 \\ P_2 \\ \end{array}\right]$$

where the conjugate transpose operator is spreading a value with two weights w₁ and w₂

$$\left[ \begin{array} {c} \tilde{P}_1 \\ \tilde{P}_2 \\ \... ...ay}\right] \left[ \begin{array} {c} P \\ \end{array}\right].$$

The PSPI upward datuming can be written as  

$$\displaystyle{ \left[ \begin{array} {ccc} A&B&C \end{arra... ... \left[ \begin{array} {c} P(x,z=0,{\omega})\end{array}\right] }$$ (9)

where the matrices U_i represent the operator for upward continuation of the wavefield to the depth level i. The matrix U_i can be further decomposed into the sequence  

$$\displaystyle{ \left[ \begin{array} {c} FT^* \\ \end{arra... ...left[ \begin{array} {c} w_1 \\ w_2 \\ \end{array}\right] }.$$ (10)

The matrix

$$\left[ \begin{array} {c} w_1 \\ w_2 \\ \end{array}\right]$$

contains two diagonal matrices w₁ and w₂ which have constant coefficients along the diagonal. Multiplication by this matrix, which is the transpose to linear interpolation, splits the data with two different weights. The matrix

$$\left[ \begin{array} {cc} W_i1 & \\ & W_i2 \\ \end{array}\right]$$

contains two diagonal matrices, which perform the upward extrapolation with different velocities. Each element of the diagonal is a complex exponential as in equation (7). And finally, the matrix

$$\left[ \begin{array} {cc} I & I \\ \end{array}\right]$$

contains two identity matrices which add the two wavefields.

The conjugate transpose algorithm is found by transposing each matrix and reversing the multiplication order:  

$$\displaystyle{ \left[ \begin{array} {c} U^*_1\\ U^*_2 U^*_1... ...ft[ \begin{array} {c} P(x,z_{dat},{\omega})\end{array}\right] }$$ (11)

The matrices U^*_i represent the operator for downward continuation of the wavefield to the depth level i. The matrix U^*_i can be further decomposed into the sequence  

$$\displaystyle{ \left[ \begin{array} {cc} w_1 & w_2 \\ \en... ...\right] \left[ \begin{array} {c} FT \\ \end{array}\right] }.$$ (12)

The physical interpretation of equation (12) is that the wavefield, after Fourier transformation, is split and downward continued with two different velocities. The two wavefields are independently inverse Fourier transformed and then interpolated. This sequence is then repeated for each depth level.

The Split-Step algorithm is very similar to the PSPI algorithm, with the difference being that a single average velocity is used and an extra phase-shift correction is applied. Although the up-datuming sequence is exactly the same as in equation (9), the values of U_i are different then in equation (10). The matrix U_i is decomposed in the sequence  

$$\displaystyle{ \left[ \begin{array} {c} FT^* \\ \end{arra... ...right] \left[ \begin{array} {c} S_i \\ \end{array}\right] }.$$ (13)

The only new matrix here is the matrix S_i which has the form  

$$S_i= \left[ \begin{array} {cccccc} e^{i({1 \over {v(x_1,z_i... ...)}}-{1 \over {v_{med}}})\omega \Delta z} \\ \end{array}\right]$$ (14)

and is responsible for a laterally varying phase-shift correction.

The conjugate transpose downward-datuming has the form given in equation (11) except that the values of U_i are found by transposing the matrices in equation (13)  

$$\displaystyle{ \left[ \begin{array} {c} S^*_i \\ \end{arr... ...\right] \left[ \begin{array} {c} FT \\ \end{array}\right] }.$$ (15)