Velocity-Depth Ambiguity in the Imaging of Multiples  

As shown in Figure , the HEMNO approach for imaging pegleg multiples that I introduced in section assumes that the reflection points of multiples do not move in midpoint from their ``1-D Earth'' position. This assumption is violated in the presence of nonzero reflector dip or lateral velocity variation. Reflector dip affects the kinematic properties of multiples in two ways, which were quantified by Levin (1971) in a seminal paper. First, dip always leads to a negative shift in the zero-offset traveltime of multiples relative to the 1-D case. Second, multiples from dipping reflectors always have a higher apparent velocity than those from flat reflectors.

Prestack migration methods naturally and automatically unravel the mystery of dip to correct seismic data for the effects of nonzero offset. However, HEMNO is a more mechanical operation. To image a multiple, it requires an estimate of zero-offset traveltime and the multiple's stacking velocity. Unfortunately, these quantities are inherently coupled. The goal of HEMNO is to ``best'' align a multiple and its primary as a function of offset. A small perturbation in $\tau^*$ may better align the multiple and primary at zero offset, but will change the multiple's apparent velocity and possibly worsen alignment at far offsets. Conversely, if a multiple is nonflat after imaging, a small velocity perturbation may improve far-offset alignment but will not change near-offset alignment.

In this thesis I take a pragmatic view of the velocity-depth ambiguity problem. Pure multiples do not split. If a primary is flat after imaging, but its pure multiple is not, then any residual moveout in the multiple is due to dip and/or lateral velocity variation. I use a two-step process to handle the nonlinear coupling of velocity and reflector position:

1.

Compute perturbation in $\tau^*$ by aligning near-offset stacks of primary and its pure multiple with a cross-correlation approach Rickett and Lumley (2001).

2.

Compute perturbation in multiple velocity by performing residual stacking velocity analysis for the (pure) multiple event of interest. The velocity perturbation is applied for all $\tau$ in $V_{\rm eff}$ [equation ()].