Acoustic Imaging by Wave Field Extrapolation Part I: Theoretical Considerations

  • A. J. Berkhout
  • J. Ridder
  • L. F. v.d. Wal
Part of the Acoustical Imaging book series (ACIM, volume 10)


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:
$$ Q\left( {{z_O}} \right) = \sum\limits_{i = O}^N {W\left( {{z_i},{z_{i + 1}}} \right)R\left( {{z_{i + 1}}} \right)W\left( {{z_{i + 1}},{z_i}} \right)} ;P\left( {{z_O}} \right) = S\left( {{z_O}} \right)Q\left( {{z_O}} \right)D\left( {{z_O}} \right). $$
Each row of complex-valued data matrix P(zo) contains the response of one source (or one source array) at a given position in acquisition plane zo. Transducer matrices S(zo) and D(zo) define the acquisition geometry and the transducer configurations. Propagation matrices W(zi,zi+1) and W(zi+1,zi) are defined by the wave equation and quantify the propagation effects in layer (zi, zi+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(zo) is computed for a given set of R(zi). In the inverse problem (‘focussing’) R(zi) is computed for a given P(zo). 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:
$$ P\left( {{z_{{i + 1}}}} \right) = {W^{{ - 1}}}\left( {{z_{{i + 1}}},{z_{i}}} \right)P\left( {{z_{i}}} \right){W^{{ - 1}}}\left( {{z_{i}},{z_{{i + 1}}}} \right) $$
for i=1,2...,N.

The foussed image at depth level zi+1 is obtained by selecting from the unfocussed image P(zi+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


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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Copyright information

© Plenum Press, New York 1982

Authors and Affiliations

  • A. J. Berkhout
    • 1
  • J. Ridder
    • 1
  • L. F. v.d. Wal
    • 1
  1. 1.Department of Applied Physics Group of AcousticsDelft University of TechnologyDelftThe Netherlands

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