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Traveltime Equation for Pegleg Multiples in a Non-Flat Earth

 
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Figure 1
NMO for multiples in a 1-D earth. Pegleg multiples ``S201G'' and ``S102G'' have the same traveltimes as ``pseudo-primary'' with the same offset and an extra zero-offset traveltime $\tau^*$, given the velocity and time-thickness of the top layer.

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A first-order pegleg multiple consists of two unique arrivals: the event with a multiple bounce over the source, and the event with a bounce over the receiver. Figure [*] shows that in a flat earth, both ``legs'' of a pegleg (denoted S102G and S201G) arrive simultaneously; when the reflector geometry varies with position, they generally do not. In some cases, pegleg multiples are actually observed to ``split'', though humans rarely observe the phenomenon unambiguously in field data, unless the reflector geometry is uniformly dipping over a large distance (see Morley (1982), for a good example).

The practical non-observation of split peglegs notwithstanding, geologic heterogeneity is a first-order effect in the accurate modeling of their kinematic and amplitude behavior. Mild variations in reflector depth over a cable length can introduce significant destructive interference between the legs of a pegleg multiple at far offsets - interference impossible to predict with a 1-D theory.

Levin and Shah (1977) deduced analytic kinematic moveout equations for 2-D pegleg multiples arising from two dipping layers, and Ross et al. (1999) extended the work to 3-D. Both approaches assume constant velocity and locally planar reflectors - assumptions which, depending on local geology, may be unrealistic in practice. In this section, I present ``HEMNO'' (short for Heterogeneous Earth Multiple NMO Operator), a simplified moveout equation based upon a more practically realizable conceptual model. In Appendix A, I show that the HEMNO equation is equivalent to Levin and Shah's in the limit of small dip angle.

 
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schem-pegleg-dip2
Figure 2
Raypath for single-CMP approximate imaging operator. Panel a): True raypath in constant-velocity earth. The zero-offset traveltimes to the seabed and subsea reflector are $\tau^*(y_0)$ and $\tau(y_0)$, respectively. Panel b): Assumed reflection points under flat-earth assumption. xp is defined in equation (4). Panel c): The appoximation. Stretch legs of raypath vertically to match measured $\tau^*(y_0-x_p/2)$ and $\tau(y_0 + (x-x_p)/2)$. Panel d): Final step. Connect legs of raypath. The solid line that connects the reassembled raypath is the final result.


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Brown (2002a) derived an extension to the conventional NMO equation which images pegleg multiples at the zero-offset traveltime of the target reflector:  
 \begin{displaymath}
t = \sqrt{ (\tau+j\tau^{*})^2 + x^2/V_{eff}^2 }.\end{displaymath} (1)
$j\tau^{*}$ is the two-way traveltime of a $j^{\mbox{th}}$-order pegleg in the multiple-generating layer. $V_{eff}(\tau)$ is the effective RMS velocity of the equivalent primary shown in Figure [*].  
 \begin{displaymath}
V_{eff}(\tau) = \left( j \tau^{*} V^{*}(\tau^*) + \tau V(\tau) \right)/
 \left( j \tau^{*} + \tau \right)\end{displaymath} (2)
Figure [*] graphically illustrates HEMNO in a constant-velocity earth. The solid line in panel d) is the final result. It has the equation of a hyperbola with zero-offset traveltime $\tau^*(y_0-x_p/2) + 
\tau(y_0 + (x-x_p)/2)$ and the same offset, x:  
 \begin{displaymath}
t^2 = \left[\tau(y_0-(x-x_p)/2) + \tau^*(y_0-x_p/2)\right]^2 + \frac{x^2}{V_{eff}^2}\end{displaymath} (3)
The following expression for xp, the width of the pegleg's primary leg in an assumed 1-D earth, is derived in Brown (2003a).  
 \begin{displaymath}
x_p^2 = \frac{x^2 \tau^2 V^4}{(\tau+j\tau^*)^2 V_{eff}^4 + x^2(V_{eff}^2-V^2)}\end{displaymath} (4)
Although equation (3) was derived by seemingly ad hoc means, I show in Appendix A that the HEMNO raypath is a first-order approximation of Levin and Shah's 1977 true raypath. The Veff shown in equation (3) is modified relative to that in equation (2); $\tau^*(y_0-x_p/2)$ is substituted for $\tau^*$ and $\tau(y_0-(x-x_p)/2)$ is substituted for $\tau$.


 
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Next: Practical implementation of the Up: Brown: Imaging operator for Previous: Introduction
Stanford Exploration Project
7/8/2003