Phase-shift datuming with lateral velocity variation

For laterally-variant media, phase-shift datuming can be performed by using an adaptation of Gazdag and Sguazzero's (1984) phase shift plus interpolation (PSPI) method. In this method, the wavefield is extrapolated at multiple velocities and a single upward-propagated wavefield is obtained by interpolation. Linear interpolation of two wavefields 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 adjoint operator spreads 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 operator for the geometry of Figure  can be written as  

$$\displaystyle{ \left[ \begin{array} {ccc} U_3 U_2 U_1 & U_3... ...] \left[ \begin{array} {c} P(x,z_s,\omega)\end{array}\right], }$$ (18)

where the matrices U_i represent the extrapolation operators for laterally-variant velocity. The matrix U_i can be further decomposed into the sequence  

$$\displaystyle{ \left[ \begin{array} {cc} w_1 & w_2 \\ \en... ...}\right] \left[ \begin{array} {c} F \\ \end{array}\right] }.$$ (19)

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

The adjoint algorithm is found by transposing each matrix and reversing the multiplication order, as follows:  

$$\displaystyle{ \left[ \begin{array} {ccc} A & B & C \end... ... \begin{array} {c} P(x,z_{\rm dat},\omega)\end{array}\right]. }$$ (20)

The matrices U^*_i represent the operator for downward continuation of the wavefield to the depth level i. The matrix U^*_i is obtained by taking the adjoint of equation  ():  

$$\displaystyle{ \left[ \begin{array} {c} F^* \\ \end{array... ...left[ \begin{array} {c} w_1 \\ w_2 \\ \end{array}\right] }.$$ (21)

Popovici (1992) shows that the Split-Step formulation is similar, and that the only difference is in the U_i matrices.