## Sensitivity of seismic waves to the structure

**Ludek Klimes**
### Summary

We study how the perturbations of a generally heterogeneous
isotropic or anisotropic structure manifest themselves
in the wavefield, and which perturbations can be detected
within a limited aperture and a limited frequency band.
A short-duration broad-band incident wavefield
with a smooth frequency spectrum is considered.
Infinitesimally small perturbations of elastic moduli and density
are decomposed into Gabor functions.
The wavefield scattered by the perturbations is then composed of
waves scattered by individual Gabor functions.
The scattered waves are estimated using
the first-order Born approximation with
the paraxial ray approximation.

For each incident wave,
each Gabor function generates at most 5 scattered waves,
propagating in specific directions.
A Gabor function corresponding to a low wavenumber
may generate a single broad-band unconverted wave
scattered in forward or narrow-angle directions.
A Gabor function corresponding to a high wavenumber
usually generates 0 to 5 narrow-band Gaussian packets
scattered in wide angles,
but may also occasionally generate
a narrow-band P to S or S to P converted Gaussian packet
scattered in a forward direction,
or a broad-band S to P
(and even S to S in a strongly anisotropic background)
converted wave scattered in wide angles.

For a particular source, each wave scattered by a Gabor function
at a given spatial location is sensitive to just a single linear combination
of 22 values of the elastic moduli and density
corresponding to the Gabor function.
This information about the Gabor function is lost
if the scattered wave does not fall into the aperture
covered by the receivers
and into the legible frequency band.

The derived approximate solution of the forward scattering problem
may be utilized in designing the migration algorithm based on true
linearized inversion of the complete set of seismograms recorded
for all shots.

### Keywords

Elastic waves, elastic moduli, perturbation,
Born approximation, paraxial ray approximation,
inversion, seismic anisotropy, heterogeneous media.

### Whole paper

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### Revised version

Klimes, L.:
Sensitivity of seismic waves to structure.
*Stud. geophys. geod.*, **56** (2012), 483-520.

In: Seismic Waves in Complex 3-D Structures, Report 17,
pp. 27-61, Dep. Geophys., Charles Univ., Prague, 2007.

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