Spatially bandlimited sources

Previous: Reciprocity in approximations
Up: Wave equations and reciprocity
Next: What does it mean for seismic data ?
Previous Page: Reciprocity in approximations
Next Page: What does it mean for seismic data ?

Spatially bandlimited sources

The previous example was for a spatially impulsive source activated on the finite-difference grid. The spatial frequency content of the source thus generates frequencies up to the spatial Nyquist frequency. It is remarkable that even for the highest frequencies (where the finite-difference approximation becomes less accurate and dispersive) numerical reciprocity holds. However, from a physical point of view, sources on a finite-difference grid are usually not introduced as a spatial impulse but in a bandlimited manner in order to reduce the spatial frequency content such that the difference operators are able to approximate spatial derivatives accurately. If reciprocity has to be maintained, then a spatially bandlimited receiver has to be used. In this way the duality of the experiment is maintained. Let denote the staggered-grid finite-difference operator that propagates the entire wavefield, and an operator that injects sources at certain locations in the medium; denotes a related operator that extracts the wavefield at the receiver points. For point sources and receivers these two operators just consist of -functions at the source and receiver locations. For a source function , the recorded data are given by

which in matrix form might appear schematically, like this

Spatially bandlimited receivers and sources can be implemented using appropriate weight functions in the projection operators and . Commonly used weights are multi-dimensional Gaussian weights. denotes the vector of impulsive sources on the gridded model, while is the staggered-grid finite-difference modeling operator. Injection and extraction operators and maintain reciprocity if are transposes .

Figures 6 and 7 show the above example with bandlimited source, but using point receivers, hence not reciprocal. The Gaussian weight is of the general form and extends over four gridpoint halfwidth. The data traces match remarkably quite well, but deviations in the waveform are noticable. If now also the receiver is spatially bandlimited in the same way as the source, reciprocity is again restored and the waveforms match exactly. Figure 6: [ IMAGE ]

Figure 7: [ IMAGE ]

In many cases of data processing, imaging, inversion or optimization, reciprocity arguments are invoked. If such arguments are used, numerical implementations of operators should be designed to be reciprocal. I showed the staggered-grid finite-difference wave equation operator as one such example that when implemented conventionally is not quite reciprocal. However by symmeterizing the kernel, complete reciprocity can be obtained.