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Stability analysis

The starting point for wavefield extrapolation algorithms is an equation that governs the evolution of the wavefield in depth,  
\frac{dq}{dz}=-Rq.\end{displaymath} (50)
The Crank-Nicolson finite-difference scheme makes a numerical approximation of the depth derivative, by equating the two sides of equation ([*]) at a point midway between depth steps n and n+1:  
\frac{1}{\Delta z} \left( {\bf q}_{n+1} - {\bf q}_{n} \right...
 ...{1}{2} \, {\bf R}\, \left( {\bf q}_{n+1} + {\bf q}_{n} \right).\end{displaymath} (51)
Rearranging terms gives the implicit system,  
\left( {\bf I} + \frac{\Delta z}{2} {\bf R} \right) {\bf q}_...
 ...( {\bf I} - \frac{\Delta z}{2} {\bf R} \right) {\bf q}_{n}. \\ \end{displaymath} (52)

An extrapolator in equation ([*]) will be stable if $q' \, q$ either decreases or remains constant with depth. Therefore for stable extrapolation,
\frac{d}{dz}(q' \, q) & \leq & 0 \nonumber \\  
q' \, \frac{dq}...{dq'}{dz} \, q & \leq & 0 \nonumber \\ q' (R + R') q & \geq & 0.\end{eqnarray}
This implies that if R+R' is symmetric non-negative definite, then extrapolation with equation ([*]) will be unconditionally stable. With a similar proof, Godfrey et al. (1979) showed the same condition applies to the extrapolation matrix ${\bf R}$ under Crank-Nicolson extrapolation with equation ([*]). If ${\bf R}+{\bf R}'$ is symmetric non-negative definite, then stable extrapolation is guaranteed.

While the helical factorization begins with equation ([*]), we really solve the implicit system,
{\bf L}\,{\bf L}^T \; {\bf q}_{n+1} & = &
\left( {\bf L}\,{\bf ...
 ...}, \\ 
{\bf A} \; {\bf q}_{n+1} & = &
{\bf A}^\ast \; {\bf q}_{n},\end{eqnarray} (54)
where ${\bf A} = {\bf L}\,{\bf L}^T 
\approx {\bf I} + \frac{\Delta z}{2} {\bf R}$.To relate this to equation ([*]), we can premultiply that equation by an invertible matrix ${\bf B}$ to provide a slightly more general Crank-Nicolson extrapolation system with the same stability requirements:  
{\bf B} \, 
\left( {\bf I} + \frac{\Delta z}{2} {\bf R} \rig...
 ...left( {\bf I} - \frac{\Delta z}{2} {\bf R} \right) {\bf q}_{n}.\end{displaymath} (56)

Equating equations ([*]) and ([*]) produces a formula for ${\bf R}$ in terms of the helical factorization, ${\bf A}$:
2 \, {\bf B} & = &
{\bf A} + {\bf A}^\ast \\ \Delta z \, {\bf B...
 ...+ {\bf A}^\ast
{\bf A} - {\bf A}^\ast
\right).\end{eqnarray} (57)

Bulletproof stability requires ${\bf R}+{\bf R}'$ to be symmetric non-negative definite. For the helical factorization this matrix is given by,  
\frac{\Delta z}{2} \left( {\bf R}+{\bf R}' \right) =
 ...\bf A}^\ast
{\bf A} + {\bf A}^\ast
\right)^{-1}.\end{displaymath} (60)

In the constant velocity case, the matrix ${\bf A}$ represents a stationary filtering operation. Therefore the composite matrices, ${\bf A} + {\bf A}^\ast$ and ${\bf A} - {\bf A}^\ast$ commute with each other. Under this scenario, the matrix ${\bf R}+{\bf R}'$ becomes the zero matrix, which clearly satisfies the non-negative definite criterion required for stability. Constant velocity extrapolation with equation ([*]) is therefore unconditionally stable.

Unfortunately, however, if the velocity varies laterally, ${\bf A}$and ${\bf A}^\ast$ no longer commute with each other, and so the composite matrices ${\bf A} + {\bf A}^\ast$ and ${\bf A} - {\bf A}^\ast$ do not commute either. Consequently stable extrapolation cannot be guaranteed. Furthermore, there are no obvious steps we can take to ensure that equation ([*]) remains non-negative definite in areas of strong lateral velocity variations. In practice, equation ([*]) does indeed encounter stability problems in some areas. Section [*] illustrates this problem with some examples.

Essentially the problem revolves around the fact that I factor ${\bf I}+ \Delta z/2 \; {\bf R}$ and ${\bf I}- \Delta z/2 \; {\bf R}$, rather than ${\bf R}$ itself. We can ensure our factorization is symmetric non-negative definite, but not the extrapolator itself.

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