INTRODUCTIONMy Least-squares Joint Imaging of Multiples and Primaries (LSJIMP) algorithm (Brown, 2003b) separates pegleg multiples and primaries. LSJIMP computes separate images of the peglegs and primaries, and then uses the mutual consistency of the images to discriminate against unwanted noise types in each image. The images must be consistent in two respects: kinematics and amplitudes. A companion paper (Brown, 2003a) paper addresses the amplitude issue. In this paper, I address the kinematics. Kinematically, the events must be correctly positioned in time and flat with offset. To this end, I derive new time imaging operator, "HEMNO" (Heterogeneous Earth Multiple NMO Operator).To correctly image primary or multiple reflections in a heterogeneous earth, migration reigns supreme, although its computational cost may be nontrivial over complex geology, in 3-D, or when a "true amplitude" image is required. My previously presented (Brown, 2002a) joint imaging technique used normal moveout (NMO), because of NMO's speed and its amplitude predictability, despite its inability to account for nonflat geology. HEMNO correctly accounts for moderate subsurface structure but retains the speed and amplitude advantages of NMO. I show that the 2-D and 3-D pegleg moveout equations for two dipping planes derived by Levin and Shah (1977) and Ross et al. (1999), respectively, reduce to HEMNO in the small dip angle limit.When used in conjunction with LSJIMP, I show that HEMNO produces improved results on the moderately complex Mississippi Canyon 2-D multiples test dataset, relative to a flatearth NMO operator. Nevertheless, the real value of HEMNO lies in 3-D. Guitton (2003) demonstrates that the "Delft method" (Verschuur et al., 1992) (plus advanced multiple subtraction technology), almost perfectly separates surface-related multiples from 2-D data with complex 2-D structure. However, in real-world 3-D situations, acquisition and computational constraints diminish the method's applicability. I demonstrate that in a simple, yet realistic, 3-D synthetic example that HEMNO can accurately image pegleg multiples from a seabed that dips in both directions. In the future, HEMNO should prove a useful advance in 3-D multiple separation.
S201G S102GA 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 1 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 d...