SEISMIC RECIPROCITY

Geophysics ◽  
1959 ◽  
Vol 24 (4) ◽  
pp. 681-691 ◽  
Author(s):  
Leon Knopoff ◽  
Anthony F. Gangi
Keyword(s):  
Scalar Product ◽  
Point Sources ◽  
Simple Experiment ◽  
Reciprocity Relations ◽  
Green's Tensor ◽  
Vector Case ◽  
Green’S Tensor

The reciprocity relationship describing the relations among the fields resulting from the interchange of point sources and receivers may be extended to the seismic case. Seismic reciprocity can be described either in terms of the scalar product of the vectors representing the excitation of the source and the field at the receiver, or in terms of a Green’s tensor describing these two quantities. Theoretical reciprocity relations give no information concerning reciprocity in the cases for which the scalar product vanishes. A simple experiment in the vector case demonstrates that reciprocity is not obtained when the scalar product of the two vectors vanishes.

Computation of Green’s tensor integrals for three‐dimensional electromagnetic problems using fast Hankel transforms

Geophysics ◽  
1984 ◽  
Vol 49 (10) ◽  
pp. 1754-1759 ◽  
Author(s):  
Walter L. Anderson
Keyword(s):  
Half Space ◽  
Three Dimensional ◽  
Hankel Transform ◽  
Electromagnetic Modeling ◽  
Electromagnetic Problems ◽  
Hankel Transforms ◽  
Homogeneous Half Space ◽  
Green's Tensor ◽  
Green’S Tensor ◽  
The Many

A new method is presented that rapidly evaluates the many Green’s tensor integrals encountered in three‐dimensional electromagnetic modeling using an integral equation. Application of a fast Hankel transform (FHT) algorithm (Anderson, 1982) is the basis for the new solution, where efficient and accurate computation of Hankel transforms are obtained by related and lagged convolutions (linear digital filtering). The FHT algorithm is briefly reviewed and compared to earlier convolution algorithms written by the author. The homogeneous and layered half‐space cases for the Green’s tensor integrals are presented in a form so that the FHT can be easily applied in practice. Computer timing runs comparing the FHT to conventional direct convolution methods are discussed, where the FHT’s performance was about 6 times faster for a homogeneous half‐space, and about 108 times faster for a five‐layer half‐space. Subsequent interpolation after the FHT is called is required to compute specific values of the tensor integrals at selected transform arguments; however, due to the relatively small lagged convolution interval used (same as the digital filter’s), a simple and fast interpolation is sufficient (e.g., by cubic splines).

Reply by the author to R. Ignetik

Geophysics ◽  
1989 ◽  
Vol 54 (11) ◽  
pp. 1501-1502
Author(s):  
Art P. Raiche
Keyword(s):  
Loop Modeling ◽  
Magnetic Dipoles ◽  
Layered Earth ◽  
Starting Point ◽  
Green's Tensor ◽  
Green’S Tensor ◽  
Receiver Point

The failure of Ignetik to recognize the logic underlying my approach to polygonal‐loop modeling demonstrates a need to present some points more clearly. The starting point was that I wanted to represent the transmitter as an array of vertical magnetic dipoles rather than horizontal linear dipoles. The reason for this was to minimize the computation needed for downhole receivers and multiple transmitter loops. The layered‐earth Green’s tensor elements for horizontal linear dipoles for a general “receiver” point are very complicated for the N‐layer case. Those for the vertical dipole are very simple and require much less computation.

scholarly journals A Green's tensor approach to the modeling of nanostructure replication and characterization

Radio Science ◽  
2003 ◽  
Vol 38 (2) ◽  
pp. n/a-n/a ◽  
Author(s):  
Michael Paulus ◽  
Olivier J. F. Martin
Keyword(s):  
Green's Tensor ◽  
Green’S Tensor
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