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 , given the velocity and time-thickness of the top layer.
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.
Brown (2002a) derived an extension to the conventional NMO equation which images pegleg multiples at the zero-offset traveltime of the target reflector: