## Sobolev scalar products in the construction of velocity models

**Petr Bulant** ** , **
**Ludek Klimes** ** & **
**Karel Zacek**
### 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
optimum for ray tracing to be constructed.

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### Related full papers

Bulant, P.:
Smoothing 2-D model HESS for Kirchhoff migrations.
In: Seismic Waves in Complex 3-D Structures, Report 10,
pp. 75-82, Dep. Geophys., Charles Univ., Prague, 2000.

Klimes, L.:
Sobolev scalar products in the construction of velocity models.
In: Seismic Waves in Complex 3-D Structures, Report 10,
pp. 15-40, Dep. Geophys., Charles Univ., Prague, 2000.

Bulant, P.:
Sobolev scalar products in the construction of velocity models -
application to model Hess and to SEG/EAGE Salt Model.
*Pure and appl. Geophys.*, **159** (2002), 1487-1506.

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.

Zacek, K:
Smoothing the Marmousi model.
In: Seismic Waves in Complex 3-D Structures, Report 10,
pp. 41-62, Dep. Geophys., Charles Univ., Prague, 2000.

Zacek, K:
Smoothing the Marmousi model.
In: Seismic Waves in Complex 3-D Structures, Report 11,
pp. 161-179, Dep. Geophys., Charles Univ., Prague, 2001.

Zacek, K.:
Smoothing the Marmousi model.
*Pure appl. Geophys.*, **159** (2002), 1507-1526.

Extended Abstracts of 63rd EAGE Conference (Amsterdam),
EAGE, Houten, 2001, O024.

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