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Next: Extension to 3-D Up: Brown: Pegleg amplitudes Previous: Estimation of Seabed Reflection

Applying the Seabed Reflection Coefficient in Practice

Figure [*] illustrates that a first-order pegleg multiple consists of two unique arrivals, each with the same traveltime in a v(z) medium. If the material properties of the water bottom and target reflector do not vary across midpoint, then the two arrivals also have the same strength. In this idealized case, the expression for total pegleg multiple amplitude derived by Backus (1959),  
 \begin{displaymath}
r_{tot} = r_p \cdot (j+1)r^j,\end{displaymath} (7)
holds, where j is the order of the multiple, r is the seabed reflection coefficient, and rp is the target reflector reflection coefficient. In real data, sedimentary bedding layers always exhibit at least some ``texture'', or variation of reflection strength with midpoint Claerbout (1985). If r(y) and rp(y) are space-variable reflection coefficients of the seabed and subsea reflector, respectively, then the total strength of the first-order pegleg is  
 \begin{displaymath}
r_{tot} = r(y_{m,1}) \cdot r_p(y_{p,1}) \ + \ r(y_{m,2}) \cdot r_p(y_{p,2}).\end{displaymath} (8)
In practice, a tractable compromise between the simplistic Backus model of pegleg amplitude [equation (7)] and the complicated model of equation (8) is justified. Let us assume that the total reflection strength of a first-order pegleg can be modeled as  
 \begin{displaymath}
r_{tot} = \left[ r(y_{m,1}) + r(y_{m,2}) \right] \ r_p(y).\end{displaymath} (9)
In other words, we ignore variations in reflection strength of the target reflector, but not the seabed. Notice from Figure [*] that the two pegleg ``splits'' impinge upon the seafloor at yp,1 and yp,2, and that the primary reflection occurs at y. Therefore, if rp varies linearly in the neighborhood of y, then the average of rp(yp,1) and rp(yp,2) is rp(y), and we can safely ignore the variation in reflection strength of the target. Equation (3) shows that $\Delta y_p$ is always less or equal to than |ym,1-ym,2|, making local linearity in reflection strength more likely.

 
schem-pegleg
Figure 5
In a 1-D earth, both pegleg multiple events shown here arrive at the same time. At fixed offset, the multiple legs of the two events impinge on the multiple-generating layer at ym,1 and ym,2, and on the target reflector at yp,1 and yp,2. $\Delta y_p$, which we ignore in equation (9), goes asmptotically to zero as $\tau$ increases.
schem-pegleg
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next up previous print clean
Next: Extension to 3-D Up: Brown: Pegleg amplitudes Previous: Estimation of Seabed Reflection
Stanford Exploration Project
7/8/2003