Acoustical Imaging pp 513-540 | Cite as

# Acoustic Imaging by Wave Field Extrapolation Part I: Theoretical Considerations

Chapter

## Abstract

In this paper it is shown that the focussing problem can be approached from the theory of inverse filtering. First a forward model is derived with the aid of the acoustic wave equation:
Each row of complex-valued data matrix P(z
for i=1,2...,N.

$$ Q\left( {{z_O}} \right) = \sum\limits_{i = O}^N {W\left( {{z_i},{z_{i + 1}}} \right)R\left( {{z_{i + 1}}} \right)W\left( {{z_{i + 1}},{z_i}} \right)} ;P\left( {{z_O}} \right) = S\left( {{z_O}} \right)Q\left( {{z_O}} \right)D\left( {{z_O}} \right). $$

(1)

_{o}) contains the response of one source (or one source array) at a given position in acquisition plane z_{o}. Transducer matrices S(z_{o}) and D(z_{o}) define the acquisition geometry and the transducer configurations. Propagation matrices W(z_{i},z_{i+1}) and W(z_{i+1},z_{i}) are defined by the wave equation and quantify the propagation effects in layer (z_{i}, z_{i+1}) for downward and upward travelling waves respectively. Scattering matrix R determines the acoustic reflectivity properties of the medium. In the*forward*problem (‘modeling’) P(z_{o}) is computed for a given set of R(z_{i}). In the*inverse*problem (‘focussing’) R(z_{i}) is computed for a given P(z_{o}). The inverse operator, being derived from physical model (1), simulates a suite of new unfocussed images for fictitious recording planes*inside*the medium of investigation:$$ P\left( {{z_{{i + 1}}}} \right) = {W^{{ - 1}}}\left( {{z_{{i + 1}}},{z_{i}}} \right)P\left( {{z_{i}}} \right){W^{{ - 1}}}\left( {{z_{i}},{z_{{i + 1}}}} \right) $$

(2)

The foussed image at depth level z_{i+1} is obtained by selecting from the unfocussed image P(z_{i+1} ) the data around zero travel time.

It is shown that the propsed method is most suitable for media with variable propagation velocities and variable absorption

The practical aspects and an illustrative example of this focusing technique will be given in part II of this paper

## Keywords

Depth Level Inverse Filter Acoustic Wave Equation Lateral Velocity Variation Recording Plane
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- French, W.S., 1975,
*Computer migration of oblique seismic reflectionprofiles*: Geophysics, vol. 40, p. 961–980.Google Scholar - Claerbout, J.F., 1976,
*Fundamentals of geophysical data processing*: New York, McGraw-Hill.Google Scholar - Berkhout, A.J., 1977,
*Least-squares inverse filtering and wavelet deconvolution*: Geophysics, vol. 42, p. 1369–1383.Google Scholar - Schneider, W.A., 1978,
*Integral formulation in two and threedimensions*: Geophysics, vol. 43, p. 49–76.Google Scholar - Berkhout, A.J., 1980.,
*Seismic migration - imaging of acoustic energy by wave field extrapolation-*: Amsterdam/New York, Elsevier North Holland Publ. Comp.Google Scholar

## Copyright information

© Plenum Press, New York 1982