## Sobolev scalar products
in the construction of velocity models:
Application to model Hess and to SEG/EAGE Salt Model

**Petr Bulant**
### Summary

The minimization of the Sobolev norm during linearized inversion
of given data
enables to control the model parameters unresolved by the data being fitted.
Even if a reasonably looking model can be obtained without minimizing
the Sobolev norm, it may be too rough for some computational methods.
We may construct models optimally smooth for given computational
methods by minimizing the corresponding Sobolev norm during the inversion.

Probably the smoothest models are required by the ray methods.
The efficiency of ray tracing can be evaluated in terms of
the ``average Lyapunov exponent'' for the model.
The ``average Lyapunov exponent'' may be approximated by
the square root of the corresponding Sobolev norm of the model,
which allows models
most suited for ray tracing to be constructed.
### Keywords

Model specification, smoothing, inversion,
ray methods,
Sobolev norm, Lyapunov exponent.

### Whole paper

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### Extended version

Bulant, P.:
Sobolev scalar products in the construction of velocity models -
application to model Hess, to SEG/EAGE Salt Model, and to model Pluto 1.5.
In: Seismic Waves in Complex 3-D Structures, Report 11,
pp. 133-159, Dep. Geophys., Charles Univ., Prague, 2001.

*Pure and appl. Geophys.*, **159** (2002), 1487-1506.

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