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Wave extrapolation

The basis for wavefield extrapolation is an operator, R, that marches the wavefield q, at depth z, down to depth $z+\Delta z$.  
q_{z+\Delta z}=R \; q_{z}.\end{displaymath} (28)
In constant velocity, R will be a function of horizontal wavenumber, ${\bf k_x}=(k_x \hspace{0.35cm} k_y)^T$, and ideally, $R({\bf k_x})$will have the form of the phase-shift operator Gazdag (1978),  
R({\bf k_x})=e^{i \sqrt{\frac{\omega^2}{v^2}-\vert{\bf k_x}\vert^2}}.\end{displaymath} (29)

Due to lateral velocity variations, and the desire to avoid spatial Fourier transforms, approximations to R are often applied in the $(\omega,x)$ domain. Typically R is split into a `thin-lens' term that propagates the wave vertically, and a `diffraction' term that models more complex wave phenomena. In the $(\omega,x)$ domain, the thin-lens term can be applied as a simple phase-shift, while the diffraction term is approximated by a small finite-difference filter. The method of extrapolation determines the nature of the finite-difference filter. The mathematical forms of different extrapolators are summarized in Table 1, and discussed below.

Table 3.1: Comparison of the mathematical form of various wavefield extrapolators.
Gazdag: $R({\bf k_x}) = e^{i \sqrt{\omega^2/v^2-\vert{\bf k_x}\vert^2}}$
Implicit: $R({\bf k_x}) = e^{i \omega/v} \;
\frac{A({\bf k_x})}{B({\bf k_x})}$
Explicit: $R({\bf k_x}) = e^{i \omega/v} \; C({\bf k_x})$
Helmholtz factorization: $R({\bf k_x}) \simeq \frac{1}{L({\bf k})}$

Implicit extrapolation (discussed in more detail in following chapters) approximates $R({\bf k_x})$ with a rational form, consisting of a convolutional filter, and an inverse filter,
R({\bf k_x})=e^{\frac{i \omega}{v}} \;\frac{A({\bf k_x})}{B({\bf k_x})}.\end{displaymath} (30)
In constant velocity, the traditional Crank-Nicolson implicit formulation ensures the pair of convolutional operators, A and B, are complex conjugates, and so the resulting extrapolator is unitary.

Practical 3-D extrapolation is often done with an explicit operator using McClellan transforms. This approach amounts to approximating $R({\bf k_x})$ by with a simple convolutional filter, $C({\bf k_x})$.Explicit extrapolators, therefore, have the form
R({\bf k_x})=e^{\frac{i \omega}{v}} \; C({\bf k_x}).\end{displaymath} (31)

In contrast to these methods, the minimum-phase factorization of the Helmholtz operator provides a recursive depth extrapolator of a different form:  
R({\bf k_x}) \simeq \frac{1}{L({\bf k})},\end{displaymath} (32)
where $L({\bf k})$ is a minimum-phase filter. Because L is a function of ${\bf k}=(k_x \; k_y \; k_z)^T$,rather than ${\bf k_x}$,extrapolation with the Helmholtz factorization does not fit exactly with equation ([*]). In practice the wavefield needs to be zero-padded in depth before extrapolation, and so equation ([*]) is not written as an equality.

The apparent contradiction that we are approximating the unitary (delay) operator in equation ([*]) with the minimum-phase extrapolator in equation ([*]) is resolved by examining the impulse response of the operator $\frac{1}{L({\bf k})}$ shown in Figure [*]. The impulse response consists of two `bumps'. The first bump is the response of the impulse at the same depth step as the impulse. Because it looks like a delta function, it leaves that depth step essentially unchanged. The second bump, on the other hand, is the response to the impulse at the following depth step -- it describes the wave propagation in depth. When taken together, the first and second bumps are indeed minimum phase; however, the second bump controls wave propagation in depth and is almost pure delay.

Figure 2
Amplitude of impulse response of polynomial division with minimum-phase factorization of the Helmholtz equation. The top panel shows the location of the impulse. The bottom panel shows the impulse response. Helical boundary conditions mean the second bump in the impulse response corresponds to energy propagating to the next depth step.
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