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.
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Footnotes and References
G. L. Fitzpatrick, Acoustic Imaging and Holography, Vol. 1, No. 1, Crane-Russak, New York 1978.
The functions ψ1 and ψ2 are normalized with respect to amplitude variations. That is, in practice ψ1 = A1e p-iø, ψ2 = A2e-iø’ where A1 and A2 are not equal in general. Normalizing ψ1, Thus we use only phase factors in (8 and 9) and the actual holograms which produce ψ1, ψ2 are normalized before being combined so as to insure that ψ1, ψ2 are “normalized”
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.
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 =-cosΔφo and β =-sinΔφo. 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 meanin
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© 1980 Plenum Press, New York
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Fitzpatrick, G., Morgan, T., Hilterman, F., Wang, K., Haider, A. (1980). Holographic Enhancement of Boundaries in Seismic Applications. In: Wang, K.Y. (eds) Acoustical Imaging. Acoustical Imaging, vol 9. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-3755-3_46
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DOI: https://doi.org/10.1007/978-1-4684-3755-3_46
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