The seismic example was designed to mimic the passive seismic experiment and reflects the injection of a few simple but very important assumptions about the nature of the subsurface noise-field. These assumptions immediately move us away from the truly random ambient noise experiment. Strictly, I am assuming that the length of any particular subsurface source is fairly short in duration i.e. less than a few seconds) and randomly distributed throughout the recording time of the experiment. Cross-talk between sources and their reflections about the subsurface will be introduced under these assumptions if the subsurface sources are not completely uncorrelated. Some degree of correlation will arise if they are not separated in time by at least the two-way traveltime to the deepest reflector and if the sources have correlable waveforms.
Previously, I have manufactured synthetic passive data by convolving transmission wavefields from individual sources with different, long, random traces. The signal to noise ratio of the result can in this case be shown to improve with the square root of the number of time samples in the random source function and the square root of the number of transmission wavefields, that is number of sources, used. There are however important physical implications to modeling in this manner. Specifically, it implies many sources exploding simultaneously with infinite duration and perfect distribution of energy during this entire period. A more reasonable experiment instead is one where the sources have durations of less than a few seconds, and are randomly distributed during the course of the recording interval.
The synthetic examples presented here are generated with random source time functions no longer than 3 seconds and a random bulk time shift. This facilitates the exploitation of this subsampling strategy to its limit. Under these assumptions, increasing the signal-to-noise ratio of the image will require recording more subsurface sources or allowing the sources to ring for longer than 3 seconds. In either case, the signal to noise ratio will increase with the square root of the quantity considered.
Figure explores the use of Fourier subsampling on the migration results of the synthetic passive data. Three data volumes were created and imaged without first cross-correlating the traces Artman et al. (2004). All three panels used 280 subsurface energy sources. The first panel was modeled with a total experimental duration of 8 seconds. The second and third panels assumed the recording time was 260 seconds. The first and second panel directly migrated all the data from the entire recording duration. They are identical to within machine precision. The third panel subsampled the 260 second data set in the Fourier domain to a level that will only support 4 seconds of an inverse Fourier transform. This image is identical to the others.