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):  

  $$x^2 \Rightarrow x_1^2 + x_2^2,$$ (31)

  where x₁²+x₂² is the squared norm of the offset vector $[x_1 \; x_2]^T$. 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 x_p, the width of the primary leg of a pegleg multiple [equation ()]. Like the offset vector in 3-D, x_p also becomes a vector quantity:  

  $$x_p \Rightarrow \left[\begin{array} {c} x_{p,1} \ x_{p,... ...x_2^2) (V_{\rm eff}^2-V_{\rm rms}^2)} } \end{array}\right].$$ (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 x_p,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 x_p derived in section [equation ()]. The zero-offset traveltimes to multiple generator and reflector and the effective velocity are measured at midpoints y_m and y_p, defined specifically for the first-order S102G pegleg in equation (), can be rewritten:  

  $$y_m \Rightarrow \left[\begin{array} {c} y_0 - x_{p,1}/2 \... ...- x_{p,1})/2 \ y_0 + (x_2 - x_{p,2})/2 \end{array}\right].$$ (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 ().