The data consists of irregular traces in a 5-D space (). The AMO operator acts on regularly sampled () cubes, so we map from the irregular data space to the regular model space using a simple linear interpolation operator . Figure shows two cube views from the five dimensional space the data is mapped into. Notice the sparseness of the data in these cubes. In standard marine acquisition, a single cross-line offset is acquired for each midpoint. The standard multi-streamer acquisition results in variation of the cross-line offset that is filled as we scan over .
Figure 1 The location of the input traces for a simple synthetic. The left panel is a constant offset cube (fixed hx and hy). The right panel is a single midpoint (fixed and ).
For common azimuth migration, we want all of our data to reside at hy=0. As a result, we need to use AMO to transform from the hy that the data was recorded at to hy=0. The operator is a sum over the () cubes that have been transformed to hy=0. Figure shows two cube views of the result of applying to the small synthetic. In this case we still have significant holes along . I will discuss why I created these holes later in the section.
Finally, we need to add in our regularization term. Generally, after NMO, our data should be smooth as a function of offset. We can think of adding a derivative operator along the offset axis. We can improve this estimate even further by applying a derivative on cubes that have been transformed to the same offset using AMO .We can write our fitting goals as