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Wave extrapolation

The basis for wavefield extrapolation is an operator, W(k), that marches the wavefield q, at depth z, down to depth z+1.  
q_{z+1}=W \; q_{z}.\end{displaymath} (16)
Ideally, $W({\bf k})$, will have the form of the phase-shift operator Gazdag (1978),  
W({\bf k})=e^{i \sqrt{\alpha^2-\vert{\bf k}\vert^2}}.\end{displaymath} (17)

Due to lateral velocity variations, and the desire to avoid spatial Fourier transforms, approximations to $W({\bf k})$ are often applied in the $(\omega-{\bf x})$ domain. Typically $W({\bf k})$ is split into a `thin-lens' term that propagates the wave vertically, and a `diffraction' term that models more complex wave phenomena. In the $(\omega-{\bf x})$ 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 1: Comparison of the mathematical form of various wavefield extrapolators
Gazdag: $W({\bf k}) = e^{i \sqrt{\alpha^2-\vert{\bf k}\vert^2}}$
Implicit: $W({\bf k}) = e^{i\alpha} \;
\frac{A({\bf k})}{B({\bf k})}$
Implicit with helical factorization: $W({\bf k}) = e^{i\alpha} \; 
\frac{U_A({\bf k}) L_A({\bf k})}{U_B({\bf k})L_B({\bf k})}$
Explicit: $W({\bf k}) = e^{i\alpha} \; C({\bf k})$
Helmholtz factorization: $W({\bf k}) = \frac{1}{L({\bf k})}$

Implicit extrapolation approximates $W({\bf k})$ with a rational form, consisting of a convolutional filter, and an inverse filter. The traditional Crank-Nicolson implicit formulation ensures the pair of convolutional operators, $A({\bf k})$ and $B({\bf k})$, are complex conjugates, and so the resulting extrapolator is unitary. Implicit methods apply an extrapolator of the form  
W({\bf k})=e^{i\alpha} \;\frac{A({\bf k})}{B({\bf k})}.\end{displaymath} (18)

Although implicit extrapolation is often the method of choice in 2-D, unfortunately the cost of the matrix inversion means traditional implicit extrapolation is rarely possible in 3-D. Helical boundary conditions facilitate 3-D implicit methods by providing a way to decompose the filters into an upper and lower triangular pair, which can be easily inverted Rickett et al. (1998). The extrapolator in equation (18), therefore, becomes
W({\bf k})=e^{i\alpha} \; 
\frac{U_A({\bf k}) L_A({\bf k})}{U_B({\bf k})L_B({\bf k})}.\end{displaymath} (19)

Most practical 3-D extrapolation is done with an explicit operator, using McClellan transforms. This approach amounts to approximating $W({\bf k})$ by with a simple convolutional filter, $C({\bf k})$.Explicit extrapolators, therefore, have the form
W({\bf k})=e^{i\alpha} \; C({\bf k}).\end{displaymath} (20)

In contrast to these methods, the minimum-phase factorization of the Helmholtz equation provides appears to provide a recursive depth extrapolator of the different form  
W({\bf k})=\frac{1}{L({\bf k})},\end{displaymath} (21)
where $L({\bf k})$ is a minimum-phase filter.

The inverse of the Helmholtz factorization, therefore, is all poles and no zeros. However, the apparent contradiction that we are approximating the unitary operator in equation (17) with the minimum-phase extrapolator in equation (21) is resolved by examining the impulse response of the operator $\frac{1}{L({\bf
k})}$ shown in Figure 2. The inverse of the factor is indeed minimum-phase if you consider the response at depths z and z+1. However, we are only interested in the response at depth step z+1, i.e. the second bump in the lower panel of Figure 2, which is symmetric and tapers to zero away from the location of the impulse. We are not concerned by the first bump on the lower panel of Figure 2, as this corresponds to the response of the filter at depth step z.

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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