Introduction

Moveout analysis is an important component of many data processing steps. The most basic application is Normal Moveout (NMO), where curvature is related to velocity. Radon based multiple removal techniques take advantage of the differing moveout of primaries and multiples Hampson (1986). Migration velocity analysis often measures curvature in common reflection point gathers in either the offset domain or angle domain.

Moveout analysis involves testing potential trajectories by applying a series of moveout functions. A new domain is created where one (or more) of the axes is now a moveout parameter. The creation of this domain can be setup as an adjoint operation, inverse problem (Guitton and Symes (1999); Lumley et al. (1994), or in terms of semblance analysis. Often we want to choose a single parameter at each time (or depth) that accurately represents the moveout at the time (or depth). Unfortunately this a non-linear problem. Toldi (1985) and Symes and Carazzone (1991) discuss ways of linearizing the problem. The problem becomes more complicated if we wish to describe moveout by more than a single parameter. The volume formed by scanning over multiple moveout parameters results in very large model spaces. Previous authors have suggested sparse inversion techniques Alvarez (2006), or successive scanning.

Another approach to the problem is to use dip information to gain moveout information. () suggested applying a rough NMO correction then estimating the median of the implied $v_{{\rm rms}}$ from the dip information at a given zero offset traveltime $\tau$. Guitton et al. (2004) built more directly on the flattening work of Lomask and Claerbout (2002); Lomask and Guitton (2006); Lomask (2006). Guitton et al. (2004) $\tau$-based tomography problem Clapp (2001) based on the time-shifts calculated from flattening the data. The advantage of this formulation is that picking becomes unnecessary. The problem with these approaches, when applied to moveout analysis, is the non-linear nature of flattening can easily lead to unrealistic local minima and may not converge to a satisfactory result.

In this paper I also take advantage of the power of flattening while attempting to avoid its pitfalls by limiting the model space. The first approach is to set up an inverse problem from the time shifts needed to flatten a series of Common Reflection Point (CRP). I first invert for a single parameter at each depth, and then two parameters. In the second approach I set up a non-linear inverse problem that relates dips directly to velocity. Both techniques show promise, but additional work is needed.