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The kind of optimization problems that we usually face in exploration
geophysics are non-linear, high-dimensional, with a complex search space that
may be riddled with many local minima or maxima.
Usually our first line of attack is linearization of the problem around some given
smooth, ``easily'' computed initial model. Seismic tomography is a good
example of the success of this approach. There are many cases, however, where
linearization is impractical or undesirable and the full non-linear problem
must be solved. Broadly speaking, there are two ways to attack these kind of
problems: deterministic search methods, for example non-linear conjugate
gradient, quasi-Newton methods or Levenberg-Marquardt method Gill and Murray (1981), and
stochastic search methods such as Montecarlo, simulated annealing and genetic
algorithms Davis (1987); Goldberg (1989a). Deterministic methods are attractive because
they are natural extensions of familiar linear methods and because, in certain
applications, they can be made to run extremely fast. The downside is the need
to compute first and/or second order derivatives of the cost function and the
dependence of the solution (and sometimes even the convergence of the method
itself) on a
suitable starting point. In other words, deterministic methods are extremely
efficient at locating the bottom of the valley, provided they start the search
somewhere inside the valley. This is a serious shortcoming since in many
problems locating the valley that contains the minimum may be a problem as
difficult as locating the minimum itself. We could say that deterministic
methods are poor at ``exploration'' (locating the best valleys) but are very
good at ``exploitation'' (given the valley, locating its floor).
Stochastic methods, on the other hand, perform a much more exhaustive search
of the model space but are not as good at exploiting the early results of the
search. It appears that a hybrid method combining the strengths of both
techniques would be the best choice. The situation is not so clear cut,
however, because it may be difficult to identify the most promising valleys
and it may also be difficult to compute the derivatives of the cost function.
In geophysics such a hybrid approach between a genetic algorithm and conjugate
gradient was used to solve the problem of estimating velocities from
refraction seismic data, although the results were not conclusively better
than those obtained by the genetic algorithm alone Boschetti (1995).

An interesting alternative is the use of the so-called micro-genetic
algorithms Krishnakumar (1989) which aim at improving the relatively poor
exploitation characteristic of the genetic algorithms without affecting their
strong exploration capabilities. In this paper I compare the results of
applying both a standard and a micro-genetic algorithm to the problem of matching
a seismic trace. This is part of the more interesting problem of inverting a
zero-offset trace for interval velocities addressed in a companion paper in
this report ().
Figure shows the input sub-sampled sonic log and the
corresponding reference seismic trace obtained by a simple computation of the
normal incidence reflection coefficients assuming no multiples, no absorption
and no noise.

**input
**

Figure 1 Left, sub-sampled sonic log used to
generate the synthetic seismic trace on the right

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Stanford Exploration Project

11/11/2002