schem
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.
Figure 1 |

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.

Figure 2

Brown (2002a) derived an extension to the conventional NMO equation which images pegleg multiples at the zero-offset traveltime of the target reflector:

(1) |

(2) |

(3) |

(4) |

7/8/2003