A number of researchers have cast velocity analysis as an *
optimization* problem: that is, they propose an objective function to
be minimized or maximized at a correct velocity model
Al Yahya (1989); Cao et al. (1990); Clément and Chavent (1993); Fowler (1986); Kolb et al. (1986); Martinez and McMechan (1991); Sen and Stoffa (1991); Sevink and Herman (1993); Toldi (1985).
Extremization of the objective is then an automatic process, to be
accomplished through numerical optimization algorithms. The most
widely investigated objectives - variants of stack power or RMS data
fit error (``output least squares'') - are velocity dependent
quadratic forms in the data. These functions are believed to be
multimodal and very ill-conditioned Gauthier et al. (1986); Scales et al. (1991).
The presumed existence of many local minima appears to mandate global
search methods such as simulated annealing. These usually require
orders of magnitude more function evaluations than do gradient-based
methods which find local minima. The computational cost of global
search methods renders them unsuitable for industrial scale velocity
estimation.

The mechanism underlying these features of stack power and similar
objectives is *asymptotic instability*: none of these functions
have limiting shapes as source and data bandwidth become
infinite. Besides accounting for multimodality, saturation, ill
conditioning, and other undesirable mathematical properties,
asymptotic instability of stack power, output least squares, and
similar objectives also inhibits analysis of local and global features
*via* of high frequency asymptotics.

These observations lead to the question: do there exist velocity dependent
quadratic forms in the data, extremized by the correct velocity with
model-consistent data, which also have stable high frequency asymptotics?
The answer is ``yes'', and the nature of such forms is specified
completely as part of the answer: they express semblance through
comparison of *neighboring traces*. In contrast, stack power and
output least squares objectives measure semblance (explicitly or
implicitly) by comparing of traces with widely
differing offset and/or midpoint, and this fact accounts for the
asymptotic instability of these functions. In the ideal limit of
continuous sampling, traces to be compared should be infinitely near,
so I have used the phrase *differential semblance* to describe
these asymptotically stable forms. For mathematical details
of the connection between differential semblance and stable asymptotics
in the context of the simpler but similar *plane wave detection*
problem, see Kim and Symes (1998), also Claerbout (1992), pp. 93 ff.

The asymptotic stability of differential semblance opens up the possibility of analysing its global shape by asymptotic methods. This paper presents such an analysis for a simple special case applicable to field data, based on the convolutional model of CMPs for layered acoustic Earth response. The layered medium assumption leads to simple explicit expressions for all quantities figuring in this approach to velocity estimation. The analysis shows that, for noise-free (model consistent) data in the continuous sampling limit and velocities limited to natural admissible sets, the length of the gradient bounds the objective, up to an error which vanishes in the high frequency limit. Therefore every stationary point is asymptotically a global minimum of the differential semblance objective function in this case. That is, differential semblance does not suffer from the local minima which plague other optimization formulations of velocity inversion.

Simple estimates bound the effect of noise. Numerical experiments have shown that random noise has virtually no effect on the location of DS stationary points, whereas strong coherent noise, such as multiple reflections, has a maximal effect. In any case the influence of noise is bounded, i.e. the differential semblance velocity estimate ``degrades gracefully'' as noise of any sort is added to the data.

These conclusions - stable asymptotics, unimodality, bounded influence of noise, significance of coherent noise - conform to the results of many numerical experiments with field and synthetic data. The present paper concerns only analysis: details of computational implementation and results appear elsewhere Araya et al. (1996); Chauris et al. (1998a,b); Gockenbach and Symes (1997); Symes (1997, 1998).

The ubiquitous presence of various ``multis'' - multiple reflection,
multiple refraction (transmission caustics), multiple wave modes, and
of course multidimensional geometry - necessarily limits the practical
importance of this or any other technique for velocity estimation (or
imaging) based on primaries only layered acoustic modeling. Note that
the differential semblance concept is *not* in any way limited to
layered medium models or 1D velocity functions, any more than stack
power is limited to NMO-based velocity analysis. The abstract Chauris et al. (1998b) and earlier work of this author Kern and Symes (1994); Symes (1993) present examples of multidimensional velocity estimation by differential semblance optimization.

I begin with an abstract definition of differential semblance. After defining the convolutional model for layered acoustics, I discuss various types of error inherent in this approximation, the construction of mutes, and natural admissible model sets. This groundwork supports an asymptotic analysis of the differential semblance objective, which reveals that in the case of noise free data it is essentially a data-weighted mean square error in RMS square slowness. This observation leads directly to the main result, and to some convenient estimates of the influence of noise in general data. It also suggests an approach to optimal parametrization of velocity profiles.

4/20/1999