Theory

Estimating a regularly sampled common-azimuth volume $\bf m$ from our irregular input data $\bf d$ can be set up as a least squares inversion problem. The data consists of irregular traces in a 5-D space ($t,{\rm cmp}_x,{\rm cmp}_y,hx,hy$). I want to apply a wave-number domain AMO operator Biondi and Vlad (2001) that works on regular cube, so the first step is to map the data onto a regular mesh. The operator $\bf L$ performs linear interpolation between the irregular dataspace and a regular 5-D mesh.

We want to create a common azimuth volume, specifically the zero azimuth volume, so we want to create a dataset where hy=0. AMO provides a way to translate between different offsets. We can create an operator $\bf Z$ that sums over hy volumes that have been translated to hy=0 using AMO. Unlike Clapp (2005a), we will also sum over a small range of hx. This additional summation allows for additional mixing of information in the five-dimensional space. We can define transforming from hx₁,hy₁ to hx₂,hy₂ using AMO as Z_{hx1,hy1,hx2,hy2}. We can define our hx sampling interval as d_hx, the number of samples as n_hx, and the first location as o_hx. Similarly our sampling hy is defined by n_hy,o_hy, and d_hy. Given $\Delta hx$ samples we wish to sum over we can write an equation relating our domain m and range d through

$$m(ix) = \sum_{ix'=ix-\Delta x}^{ix+\Delta x} \sum_{iy=0}^{ny... ...} + o_{hx}, iy d_{hy} +o_{hy},ix d_{hx} +o_{hx},0} (d(ix',iy)).$$ (1)

Where m(ix) and d(ix',iy) are 3-D cubes ($\bf t,{\rm cmp}_x,{\rm cmp}_y$).

Finally, we need to add in our regularization term. Generally, after NMO, our data should be smooth as a function of offset. By applying a derivative operator along the offset axis we can emphasize this smoothness and help fill in acquisition holes caused by data irregularity. We can improve this estimate even further by applying a derivative on cubes that have been transformed to the same offset using AMO $\bf D_h$.We can write our objective function as

$$Q(\bf m)= \vert\vert \bf d- \bf L\bf Z\bf m\vert\vert^2 + \epsilon^2 \vert\vert\bf D_h\bf m\vert\vert^2$$ (2)

where $\epsilon$ controls the importance of consistency along the offset axis. We can speed up the convergence of this problem by preconditioning the model with the inverse of our regularization operator. In this case, we replace taking the derivative of AMO cubes with performing causal integration of AMO cubes $\bf C_h$. Our new fitting goals then become  

$$Q(\bf p)= \vert\vert \bf d- \bf L\bf Z\bf C_h\bf p\vert\vert^2 + \epsilon^2 \vert\vert\bf p\vert\vert^2$$ (3)

where $\bf m= \bf C_h\bf p$.