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Constraints are critical in non-linear problems to limit the number of possible
solutions and to increase the chances of finding a solution that satisfies
all aspects of the problem. An obvious constraint to be imposed on the inversion
is that the computed velocities be within a reasonable interval. In this case
I chose 2200 < *V* <3200 in m/s. This constraint, however, is not enough because
we need to account for the velocity-depth trend present in the sonic log.
It is a well-known fact that in general wave propagation velocities increase with
depth due to several factors such as compaction, cementation, closing
of small fractures due to overburden, etc. There may be effects such as overpressure
that locally change this behavior, but in average we expect a trend of velocities
increasing with depth. When inverting seismic data for interval velocities, it is
disappointing to find that this general trend is not recovered. The reason is that
the information in the velocity trend would correspond to very low frequencies
(lower than, say, 5 Hz) and these low frequencies are not recorded in the standard
seismic method. Furthermore, with land data we usually attenuate or eliminate
low frequencies because of the detrimental effect of ground-roll on the seismic
data. Since this trend is not resolvable by the direct inversion of the seismic
trace, it is necessary to impose a second constraint to honor it. This constraint
will be discussed in more detail below.

** Next:** Results
** Up:** Velocity Inversion
** Previous:** Convergence Criteria
Stanford Exploration Project

11/11/2002