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 h_y 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 h_y=0. We can think of it in terms of an operator $\bf Z'$ which is a sumation over h_y. We can allow for some mixing between h_x by expanding our sumation to form h_x=a h_y=0, by summing over all h_y and  

$$\sum_{a - \Delta h_x}^{a+\Delta h_x},$$ (8)

where $\Delta h_x$ is small.

 

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

We can combine these two operators to estimate a 4-D model ($\bf m$) from a 5-D irregular dataset ($\bf d$) through,  

$$\bf m= \bf Z' \bf L' \bf d .$$ (9)

Equation 9 amounts to just running the adjoint of the inversion implied by,  

$$Q(\bf m)= \vert\vert\bf d- \bf L\bf Z\vert\vert^2 .$$ (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),

$$\bf m= \bf H \bf Z' \bf L',$$ (11)

where  

$$\bf H = \rm diag \left[ \frac{ \bf Z'\bf L'\bf L\bf Zm_{{\rm ref}}}{{\bf m}_{{\rm ref}}} \right].$$ (12)

We set our reference model $\bf m_{{\rm ref}}=1$.