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Adjoint Implementation

To map the irregular recorded seismic data onto the regular mesh is a far from trivial. A common approach in industry is to think of the problems in the same way we approach Kirchhoff migration, namely to loop over data space and spread into our regular model space. The spreading operation can be governed by something like AMO Biondi et al. (1998), which maps data from one offset vector to another. If we think of the AMO operator $\bf T$ as mapping from the regular model space $\bf m$to the irregular data space $\bf d$, our estimation procedure becomes,  
\bf m= \bf \bf T' \bf d
.\end{displaymath} (1)

The wavenumber domain AMO operator works on a regular sampled cube, so the problem is more complicated. We first must map the data to a regular sampled space by applying the interpolation operator $\bf L'$.The regular sampled cube $\bf s$ is now a full five dimensional volume ($t,{\rm cmpx},{\rm cmpy},\rm hx,\rm hy$). We can produce the model $\bf m$ at a given (hx,hy) by summing nearby cubes (t, cmpx, cmpy) that have transformed to our desired (hx,hy) through AMO. To write this in a mathematical form we need to make some definitions. We will define ihx and ihy as the offset indicies of the expanded space $\bf s$.These indicies correspond to the half-offset hx and hy. The output space, $\bf m$, is defined as a coralary ihx' and ihy` which also correspond to hx and hy. The notation $\bf m (ihx',ihy')$ correspond to the 3-D subcube (t,cmpx,cmpy) at the given ihx' and ihy'. Finally $\bf T_{{\bf x_1} \Rightarrow {\bf x_2}}$ refers to transforming the cube through AMO from the offset vector ${\bf x_1}$to ${\bf x_2}$, nx and ny is the region in sampling of $\bf s$ that we wish to sum over; and ${\rm dhx}$ and ${\rm dhy}$ is the sampling of the cube in $\rm hx$ and $\rm hy$ respectively. We obtain  
\bf m(ihx',ihy') = 
\sum_{iy = -ny}^{ny} 
\sum_{ix = -nx}^{n...
 ...hy} ) 
\Rightarrow (\rm hx,\rm hy) 
{\bf s}(ix+ihx ,iy+ihy) .\end{displaymath} (2)
In we to write our regularization problem in the form of equation (2), $\bf S$ is a spraying operation where the columns of the matrix are defined by equation (1). We then obtain or model by applying  
\bf m= \bf S' \bf L' d
.\end{displaymath} (3)
The formulation suffers from all of the usual problems associated with applying an adjoint operation. We are spraying into a regular mesh, but the data is not regular. Areas with higher concentration of data traces will tend to map to artificially higher amplitudes in the model space. In the Kirchoff formulation we can do some division by hit count to help minimize this effect. Because we are operating in the wave number domain we can't normalize by something as simple as hit count. We can accomplish something similar by following the approach of Claerbout and Nichols (1994) and Rickett (2001). We approximate the Hessian of the least squares solution,
\bf m= \left( \bf S \bf L' \bf L\bf S \right)^{-1} \bf S' \bf L' \bf d,\end{displaymath} (4)
by the diagonal operator $\bf W$.We form $\bf W$ by  
{\bf W^{-1}} = {\rm diag} \left[ \left( \bf S' \bf L' \bf L\bf S {\bf 1} +\alpha \right)\right]
,\end{displaymath} (5)
where $\bf 1$ is a vector of 1s, $\alpha$ is a stabilization term, and ${\rm diag}[]$ map the vector to the diagonal of the matrix. We scale our adjoint solution by $\bf W$ obtaining  
\bf m= \bf W \bf S' \bf L' d
.\end{displaymath} (6)