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Solving the equation

Equations (6) and (7) usually contain an enormous number of variables and, as will be shown later, are extremely ill-conditioned. However, good results may be obtained in a small number of iterations if a conjugate gradient method is used. To solve the equation
\begin{displaymath}
\bold L \bold u = \bold d\end{displaymath} (8)
we can minimize the function
\begin{displaymath}
\bold f = {\Vert \bold L \bold u - \bold d \Vert }^2 ,\end{displaymath} (9)
or for stochastic inversion,
\begin{displaymath}
\bold f = {{\Vert \bold L \bold u - \bold d \Vert }^2 \over \sigma_d^2 }
+ {{\Vert \bold u \Vert}^2 \over \sigma_u^2},\end{displaymath} (10)
where $\sigma_d^2$ and $\sigma_u^2$ are the variances of the data and the model spaces respectively.

When a velocity analysis panel is obtained by the solution of equation (6) or equation (7), an inversion is easily done by application of the operator $\bold P$ or $\bold H^{\bold T}$ on the velocity analysis panel $\bold u$.

We can also consider solving the equation  
 \begin{displaymath}
\bold H \bold d = \bold u,\end{displaymath} (11)
where $\bold u$ was obtained by the application of $\bold H$ on (t,x)-space, and  
 \begin{displaymath}
\bold P^{\bold T} \bold d = \bold u,\end{displaymath} (12)
where $\bold u$ was obtained by the application of $\bold P^{\bold T}$ on (t,x)-space.


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Next: COMPUTATION OF ADJOINT OPERATORS Up: VELOCITY ANALYSIS Previous: VELOCITY ANALYSIS
Stanford Exploration Project
1/13/1998