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Implemetation

The PS-AMO operator described above works on a regular sampled cube, our data is recorded on an irregular mesh. Following the methodology described in Clapp (2006) we first map our data to a regular 5-D mesh. For this problem we chose nearest neighbor interpolation, designated by the operator $\bf L'$. For migraion efficiency, and because the five-dimensional space is sparsely populated, we want to reduce the dimensionality of our dataset. A common goal, and the one we chose to implement, was to create common azimuth volume orriented along the inline direction. As a result we want to eliminate the hy axis.

Our PS-AMO operator (diagramed in Figure [*]) allows to transform between various vector offsets. We use it to transform data from $h_y\neq 0$ to hy=0. We can think of it in terms of an operator $\bf Z'$ which is a sumation over hy. We can allow for some mixing between hx by expanding our sumation to form hx=a hy=0, by summing over all hy and  
 \begin{displaymath} \sum_{a - \Delta h_x}^{a+\Delta h_x}, \end{displaymath} (8)
where $\Delta h_x$ is small.

 
flow
Figure 1 Diagram flow for the implementation of the PS-AMO operator. 		flow
view

We can combine these two operators to estimate a 4-D model ($\bf m$) from a 5-D irregular dataset ($\bf d$) through,  
 \begin{displaymath} \bf m= \bf Z' \bf L' \bf d . \end{displaymath} (9)
Equation 9 amounts to just running the adjoint of the inversion implied by,  
 \begin{displaymath} Q(\bf m)= \vert\vert\bf d- \bf L\bf Z\vert\vert^2 . \end{displaymath} (10)
The adjoint solution is not ideal. The irregularity of our data can lead to artifical amplitude artifacts. A solution to this problem is to approximate the Hessian implied by equation 10 with a diagonal matrix based on a reference model Rickett (2001),
\begin{displaymath} \bf m= \bf H \bf Z' \bf L', \end{displaymath} (11)
where  
 \begin{displaymath} \bf H = \rm diag \left[ \frac{ \bf Z'\bf L'\bf L\bf Zm_{{\rm ref}}}{{\bf m}_{{\rm ref}}} \right]. \end{displaymath} (12)
We set our reference model $\bf m_{{\rm ref}}=1$.

next up previous print clean
Next: Parallel implementation Up: Rosales and Clapp: PS-AMO Previous: f-k log-stretch PS-AMO
Stanford Exploration Project
4/5/2006