|Imaging in shot-geophone space|
The principle of reciprocity says that the same seismogram should be recorded if the locations of the source and geophone are exchanged. A physical reason for the validity of reciprocity is that no matter how complicated a geometrical arrangement, the speed of sound along a ray is the same in either direction.
Mathematically, the reciprocity principle arises because symmetric matrices arise. The final result is that very complicated electromechanical systems mixing elastic and electromagnetic waves generally fulfill the reciprocal principle. To break the reciprocal principle, you need something like a windy atmosphere so that sound going upwind has a different velocity than sound going downwind.
Anyway, since the impulse-response matrix is symmetric, elements across the matrix diagonal are equal to one another. Each element of any pair is a response to an impulsive source. The opposite element of the pair refers to an experiment where the source and receiver have had their locations interchanged.
A tricky thing about the reciprocity principle is the way antenna patterns must be handled. For example, a single vertical geophone has a natural antenna pattern. It cannot see horizontally propagating pressure waves nor vertically propagating shear waves. For reciprocity to be applicable, antenna patterns must not be interchanged when source and receiver are interchanged. The antenna pattern must be regarded as attached to the medium.
I searched our data library for split-spread land data that would illustrate reciprocity under field conditions. The constant-offset section in Figure 9.7 was recorded by vertical vibrators into vertical geophones.
Figure 7. Constant-offset section from the Central Valley of California. (Chevron)
The survey was not intended to be a test of reciprocity, so there likely was a slight lateral offset of the source line from the receiver line. Likewise the sender and receiver arrays (clusters) may have a slightly different geometry. The earth dips in Figure 9.7 happen to be quite small although lateral velocity variation is known to be a problem in this area.
In Figure 9.8, three seismograms were plotted on top of their reciprocals.
Figure 8. Overlain reciprocal seismograms.
Each constant time slice in Figure 9.9 shows the reciprocity of many seismogram pairs.
Figure 9. Constant time slices after NMO at 1 second and 2.5 seconds.
In the laboratory, reciprocity can be established to within the accuracy of measurement. This can be excellent. (See White's example in FGDP). In the field, the validity of reciprocity will be dependent on the degree that the required conditions are fulfilled. A marine air gun should be reciprocal to a hydrophone. A land-surface weight-drop source should be reciprocal to a vertical geophone. But a buried explosive shot need not be reciprocal to a surface vertical geophone because the radiation patterns are different and the positions are slightly different. Under varying field conditions Fenati and Rocca found that small positioning errors in the placement of sources and receivers can easily create discrepancies much larger than the apparent reciprocity discrepancy.
Geometrical complexity within the earth does not diminish the applicability of the principle of linearity. Likewise, geometrical complexity does not reduce the applicability of reciprocity. Reciprocity does not apply to sound waves in the presence of wind. Sound goes slower upwind than downwind. But this effect of wind is much less than the mundane irregularities of field work. Just the weakening of echoes with time leaves noises that are not reciprocal. Henceforth we will presume that reciprocity is generally applicable to the analysis of reflection seismic data.
|Imaging in shot-geophone space|