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Lateral velocity variation

When the velocity varies laterally, the matrix M in the extrapolation operator at equation (7) has coefficients which varies along the diagonal, but we can still derive a symmetric matrix (Godfrey et al., 1979). The symmetry property, which will guarantee the unconditional stability, can be obtained by putting the velocity term on both sides as follows

\begin{displaymath}
{\partial \over \partial z} P = -{1\over \sqrt{v}}[-\omega^2...
 ...tial x}v^2{\partial \over \partial x}]^{1/2}{1\over \sqrt{v}} P\end{displaymath} (20)

Now the small block matrices which are located along the diagonal in split matrices Me and Mo will have a symmetric form

\begin{displaymath}
E =\Delta z 
\left( 
\begin{array}
{cc}
a&c\\ c&b\end{array} 
\right),\end{displaymath}

where

\begin{displaymath}
a=-{i\over 2}({\omega \over v(x)} -{v^2(x)+v^2(x-1) \over 2 v(x)\omega \Delta x^2}),\end{displaymath}

\begin{displaymath}
b=-{i\over 2}({\omega \over v(x+1)} -{v^2(x+1)+v^2(x) \over 2 v(x+1)\omega \Delta x^2}),\end{displaymath}

and

\begin{displaymath}
c=-i{v(x)^2 \over \sqrt{v(x+1)v(x)}2\omega \Delta x^2}.\end{displaymath}

The eigenvalues of $\exp(E)$ are given by

\begin{displaymath}
\lambda=\exp\left({a + b \pm \sqrt{(a-b)^2+4c^2} \over 2}\right)\end{displaymath}

and lie on the unit circle since a and b are imaginary, and $(a-b)^2 + 4c^2 \leq 0$.It follows that the matrix norms $\Vert\exp(\Delta z M_e)\Vert = 1 $and $\Vert\exp(\Delta z M_o)\Vert = 1 $.To prove the unconditional stability of the algorithm we need only show that $\Vert B\Vert\leq 1$. This follows immediately since $\Vert B\Vert=\Vert\exp(\Delta z M_e)
\exp(\Delta z M_o)\Vert \leq\Vert\exp(\Delta z M_e)\Vert\Vert\exp(\Delta z M_o)\Vert$, each of which is unity according to the preceding argument.


previous up next print clean
Next: WIDE-ANGLE DEPTH MIGRATION Up: 15-DEGREE DEPTH MIGRATION Previous: Accuracy
Stanford Exploration Project
12/18/1997