## Optimization of the structural Gabor functions
in a homogeneous velocity model
for a zero-offset surface seismic reflection survey

**Ludek Klimes**
### Summary

We consider the algorithm of true
linearized inversion of the complete set of seismograms recorded
for all shots during seismic reflection survey,
based on the
correspondence between the structural Gabor functions
and the corresponding scattered Gaussian packets.
The first step of this linearized inversion consists
in specifying the finite set of structural Gabor functions
to which the wanted deviations of the material parameters
from the given velocity model will be decomposed.
If possible, the shape of the Gabor functions should be optimized,
and the Gabor functions should form a frame.
We present a simple attempt to specify
the finite set of structural Gabor functions
for this inversion.

The shape of the Gabor functions
and the space-wavenumber lattice of their central points
are optimized analytically,
but for a homogeneous velocity model only.
The derived set of structural Gabor functions
may be applied also to slightly heterogeneous models
in which the shape of Gabor functions is not optimum
but is still probably better than
the shape of Gabor functions specified by chance.
The envelopes of the structural Gabor functions are chosen
isotropic, but are allowed to be complex-valued.
The shape of Gabor functions is optimized
for a zero-offset surface seismic reflection survey only,
but is close to optimum
for a narrow-offset surface seismic reflection survey.

### Keywords

Gabor function, Gabor transform,
frame bounds, discretization error,
phase-space metric tensor,
elastic waves, Gaussian packets, wavefield inversion.

### Whole paper

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In: Seismic Waves in Complex 3-D Structures, Report 18,
pp. 115-127, Dep. Geophys., Charles Univ., Prague, 2008.

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