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## Kinematic imaging of pegleg multiples in a laterally-homogeneous earth

In a 1-D Earth'' (horizontally-stratified, v(z) medium), the normal-moveout (NMO) equation Taner and Koehler (1969) describes the prestack traveltime curve of a primary reflection at relatively short source-receiver offset: (16)
Applied as an offset-dependent vertical time shift to a CMP gather, the NMO equation flattens an arbitrary primary to its zero-offset traveltime , where (half) offset is denoted x and the root-mean-square (RMS) velocity, , is defined in a laterally-homogeneous earth as: (17)
The earth is parameterized by layers of time thickness , with constant interval velocity vi in each layer.

Analogously, a modified NMO equation can image pegleg multiples in a 1-D Earth, as motivated graphically by Figure . From the figure, we see that kinematically, a first-order pegleg can be conceptualized as a pseudo-primary'' with the same offset, but with an additional two-way zero-offset traveltime to the multiple generator, . In equation form, let us extend this intuition to the general case of a -order pegleg to write an NMO equation for peglegs: (18) is the effective RMS velocity of the pseudo-primary shown in Figure . To derive an expression for , we modify the definition of RMS velocity, equation ( ), to reflect a -order pegleg multiple's additional travel between the surface and multiple generator: (19)
Analogously, is the number of assumed layers between the earth's surface and the multiple generator. Notice that the two terms inside the parentheses of equation ( ) are simply the definition of RMS velocity at and , respectively. We can substitute equation ( ) accordingly to derive the final expression for : (20)
Wang (2003) derives a similar series of expressions.

 schem Figure 3 Pegleg multiples S201G'' and S102G'' have the same traveltimes as pseudo-primary'' with the same offset and an extra zero-offset traveltime .       Next: Amplitude corrections for pegleg Up: Particular Implementation of LSJIMP Previous: Particular Implementation of LSJIMP
Stanford Exploration Project
5/30/2004