Reciprocity in approximations

Previous: Reciprocity and self-adjointness of operators
Up: Wave equations and reciprocity
Next: Spatially bandlimited sources
Previous Page: Reciprocity and self-adjointness of operators
Next Page: Spatially bandlimited sources

Reciprocity in approximations

As shown in the previous section, a reciprocity relation holds for a large variety of wave propagation problems. Reciprocity in a wave propagation problem may be defined as a symmetry property of the wavefield, due to symmetry of the Green's function. An important question raised by Ravazy and Lenoachca [(Razavy and Lenoachca,1986)] was whether an approximation to the original problem still preserves the original reciprocity relationships. Even if such an approximation is more accurate, it might implicitly result in a Green's kernel that is no longer symmetric and thus violates spatial reciprocity. They indeed found when investigating the scalar wave equation, that some analytical high frequency approximations and some numerical finite-difference approximations destroyed the reciprocity relationships of the original problem. It is therefore very important to verify that if reciprocity arguments are used to derive a data processing operation, the resulting algorithm and its numerical implementation should maintain reciprocity.

Figure 2 shows a combination of two homogeneous elastic media in which a pair of source/receiver locations are marked. The two materials have different stiffnesses and densities. At both locations, a source with identical time history is activated and at both locations the wavefield is recorded. Source and receiver activate and register both and components. Figure 3 shows a plot of the four components of received wavefields. Across a row the receiver component is the same, while across the column the source component is the same. In each quadrant two seismograms are overlaid, one at location (1) the other at location (2). The diagonal plots show identical source and receiver components and the seismograms match perfectly. The off-diagonal plots clearly show nonidentical seismograms; the source component is different from the receiver component. Compare this to Figure 4, where the off-diagonal components show a perfect match. In contrast to Figure 3, reciprocal components are selected. All seismograms are now reciprocal and match perfectly. Thus the anisotropic elastic wave equation operator is symmetrically implemented using high order finite-differences on a staggered grid. Approximating the continuous wave equation has not broken the original symmetry. Figure 2: [ IMAGE ]

Figure 3: [ IMAGE ]

Figure 4: [ IMAGE ]

Figure 5: [ IMAGE ]