Acoustical Imaging pp 795-816 | Cite as

# Holographic Enhancement of Boundaries in Seismic Applications

Chapter

## Abstract

There are many different ways of forming images from seismic wave data recorded on the earth’s surface. One way is to record or produce a hologram at this surface and by an appropriate algorithm or analog procedure reconstruct the hologram to produce an image. This procedure will be employed here when we verify theoretical ideas; however, any method that properly “migrates” seismic waves back into the earth will suffice to take advantage of the techniques we are about to describe.

## Keywords

Seismic Wave Image Space Quarter Wave Frequency Pair Contrast Measure
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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## Footnotes and References

- 1.G. L. Fitzpatrick, Acoustic Imaging and Holography, Vol. 1, No. 1, Crane-Russak, New York 1978.Google Scholar
- 2.The functions ψ
_{1}and ψ_{2}are normalized with respect to amplitude variations. That is, in practice ψ_{1}= A_{1e}^{p-iø}, ψ_{2}= A_{2}e_{-iø}’ where A_{1}and A_{2}are not equal in general. Normalizing ψ_{1}, Thus we use only phase factors in (8 and 9) and the actual holograms which produce ψ_{1}, ψ_{2}are normalized before being combined so as to insure that ψ_{1}, ψ_{2}are “normalized”Google Scholar - 3.It was erroneously stated in Ref. 1 that a contrast of-1 was of no interest. We see here that in searching for a high contrast image (C = +1) we need first to find the C = −1 image. Even though the C = −1 is by itself usually not a good image (sometimes adjacent boundaries with different properties can make a C = −1 image reasonably “good”) it is needed to find the “best” C = +1 image.Google Scholar
- 4.Clearly, the foregoing procedure could have great value in analyzing an image since it would aid in finding the spatial extent of many uniform boundary conditions present. However, such a technique would not work if for any choice of Z there always appeared a minimum in the functional I for a given boundary. That this is not the case can be “proved” as follows: We know that the functional has a minimum only where α =-cosΔφ
_{o}and β =-sinΔφ_{o}. For any other choice of Z, Z ≠ Z, I does not possess a minimum. When we combine this fact with the reasonable assumption that there are at least a limited number of boundaries, i.e a limited number of Z’s characterizing these boundaries, any Z that is not equal to one of these will not produce a minimum in I anywhere in the image space. Hence the appearance of a minimum in I is a unique signal that a “proper” physically relevant Z has been found. These points will be clarified when we examine real data and the degree of uniqueness can be given a more tangible meaninGoogle Scholar

## Copyright information

© Plenum Press, New York 1980