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# Modifications to the 2-D Theory

In this section I enumerate the necessary modifications to my particular LSJIMP implementation, presented earlier in section , to move from 2-D data to 3-D data. The narrow azimuth geometry illustrated in this chapter considerably simplifies this move, but for completeness, I nontheless discuss both the narrow azimuth and general 3-D implementations.
• Regularization operators: In section I introduced the three LSJIMP regularization operators. The first (differencing between images, section ) and third (crosstalk penalty weights, section ) extend to 3-D with no modification. However, in the full 3-D case, the second operator, which differences across offset (section ), must operate along both inline and crossline offset axes. Thus the corresponding model residual [equation ()] becomes a vector quantity, as it is nothing more than a finite difference spatial gradient. As mentioned earlier, in the narrow azimuth case, the crossline offset axis is ignored, in which case the operator is the same as in the 2-D case.
• 1-D Imaging of multiples: The 1-D imaging operator for multiples, derived in section changes in 3-D. For the full 3-D case, the NMO equation for both primaries and multiples changes. In equations () and (), the squared inline offset (x2) changes to the sum of the squared inline (x12) and crossline offsets (x22):    (31)
where x12+x22 is the squared norm of the offset vector . Equation () applies to the full 3-D and narrow azimuth cases alike. The difference is in the implementation: in the full 3-D case, a computer program loops over the x2 axis, but not in the narrow azimuth case, where the crossline offset at a given CMP location must be pre-defined and passed as an input parameter.
• Amplitude correction operators: An important quantity for my implementation of LSJIMP was xp, the width of the primary leg of a pegleg multiple [equation ()]. Like the offset vector in 3-D, xp also becomes a vector quantity:    (32)
As noted earlier in this chapter, with narrow azimuth data it makes the most sense not to do Snell Resampling in the crossline direction. Still, the crossline offset of the reduced'' CMP gather may still be nonzero, and will affect the value of xp,1.

The differential geometric spreading correction derived in section remains unchanged, with the exception of substituting equation () for squared offset in equations () and ().

The estimation of a multiple generator's reflection coefficient in 3-D remains similar to the 2-D case, although the model is a function of two varibles, CMPx and CMPy, and the data may (full 3-D) or may not (narrow azimuth) be a function of crossline offset.

• HEMNO: HEMNO is strongly dependent on the quantity xp derived in section [equation ()]. The zero-offset traveltimes to multiple generator and reflector and the effective velocity are measured at midpoints ym and yp, defined specifically for the first-order S102G pegleg in equation (), can be rewritten:    (33)
An accurate 3-D dip estimate is also required. Event tracking in 3-D is just as straightforward as in 2-D The only other change required to HEMNO is to change squared inline offset in equation () to the squared norm of the offset vector, equation ().

Next: 3-D Results \label>chapter:results3d> Up: 3-D Theory \label>chapter:theory3d> Previous: LSJIMP and wide tow
Stanford Exploration Project
5/30/2004