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Snell Resampling Removes AVO/Attenuation Differences

If we are modeling seabed pegleg multiples, Figure [*] illustrates the fact that (ignoring the seabed reflection) in a v(z) medium, there exists a single offset xp such that a pegleg with offset x and primary with offset xp are physically invariant with respect to AVO behavior and anelastic attenuation (water is assumed perfectly elastic). Ottolini (1982) introduced the concept of ``Snell Traces'' - a resampling of multi-offset reflection data along curves of constant time dip, or ``stepout''. I adopt a similar line of reasoning to infer xp as a function of x.

Figure 1
A primary and pegleg multiple with the same emergence angle ($\theta$) and midpoint (y). Note different offsets (x and xp) and a shift ($\Delta y$) in reflection point.


Since the pegleg multiple and primary in Figure [*] have the same emergence angle, $\theta$, the stepout of the two events is the same at x and xp. This fact is the basis of the derivation in Appendix A, which obtains the following result for xp as a function of x.
x_p^2 &=& \frac{x^2 \tau^2 V^4}{(\tau+j\tau^*)^2 V_{eff}^4 + x^...
 ...{ j \tau^{*} v^{*}(\tau^{*}) + \tau V(\tau) }{ j \tau^{*} + \tau }\end{eqnarray} (1)
$j\tau^{*}$ is the two-way traveltime of a $j^{\mbox{th}}$-order pegleg multiple in the top layer. Equation (1) defines a time-variable compression of the offset axis. In constant velocity, Veff=V, and equation (1) reduces to the radial trace resampling used by Taner (1980) for long-period deconvolution of peglegs. Figure [*] demonstrates the Snell resampling on the first- and second-order pegleg multiples of a synthetic dataset.

Graphically (Figure [*]), we infer that the shift in midpoint, $\Delta y$, of the reflection points of the primary and pegleg is  
\Delta y = \left(x-x_p\right)/2.\end{displaymath} (3)
As a function of time, $\Delta y$ decreases asymptotically to zero from a maximum of x/4 at the seabed. The deeper the reflector, the smaller $\Delta y$ becomes.

Oftentimes, non-seabed pegleg multiples (e.g. top of salt) are strong enough to merit modeling. In a v(z) earth, the results derived in this section are equally valid, with one exception: attenuation. In this case, the effects of attenuation are encoded - in a possibly non-linear way - in the effective reflection coefficient that we estimate in a subsequent section.

Figure 2
Snell resampling demonstration. A synthetic dataset with flattened primaries and resampled first- and second-order peglegs. The transformation warps the vertical black lines horizontally, according to equation (1). Notice that the raw data has five unrecorded near offset traces and two dead traces at medium offsets. Snell resampling spreads information from the multiples into these no-data zones. The multiples provide a direct, additional constraint on the amplitude of the primaries where no data is recorded.

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