Based in Mahone Bay, nova scotia, agile libre is an independent publisher of technical books about the subsurface.  

The magic of Fourier

If I were to respond to the question ‘Which single mathematical idea had the greatest contribution to the modern world of technology?’ I would reply, with­out any hesitation, ‘The Fourier transform’. No one can deny the great role that the Fourier transform has played in telecommunications, the electronics industry, and signal-processing methods over the past century. Therefore, it is no surprise that the Fourier transform also has an indispensable part to play in geophysical data analysis. While the presence of Fourier can be felt in every discipline of geophysics, here are some highlights from exploration seismology:

Data acquisition.  Almost all of the electronic and mechanical devices used in field data acquisition are designed and tested using the Fourier analysis. Also the collected signals have to be checked by Fourier analysis to ensure acquisition of a proper bandwidth of data. Bandwidth has a direct relationship with the strength and resolution of a signal; it’s a good measure of a signal’s capacity to obtain information from desired depths of the subsurface. The Fourier trans­form also plays a prominent part in data compression and storage.

Noise elimination.  The collected seismic data are often contaminated with random or coherent noises. In the traditional time-space coordinates that the seismic data are recorded in, one can not easily distinguish between noise and signal. However, by transforming the data to frequency and wave-number domains using the Fourier transform a significant portion of noise is separated from the signal and can be eliminated easily.

Interpolation.  Seismic records contain specific features of simplicity such as a stable wavelet and spatial continuity of events. This simplicity has even more interesting consequences in the Fourier domain. Seismic data composed of linear events will only have a few non-zero wave-number coefficients at each frequency. This sparsity, as it is called, is used in most of the seismic data in­terpolation methods to produce a regular spatial grid from irregular spatial sampling scenarios. Also, a simple relationship between the low frequencies and high frequencies of the seismic records helps to overcome the spatial aliasing issues by reducing spatial sampling intervals.

Time-frequency analysis.  Seismic traces can exhibit non-stationary proper­ ties. In other words, the frequency content of the data varies at different arrival times. Likewise, spatially curved seismic events have non-stationary wave-number content. In order to analyse these signals, time-frequency analysis is required. Several transforms such as Wavelet, Gabor, Stockwell, and Curvelet transforms have been introduced to perform these analyses. In essence, all of these transforms are windowed inverse Fourier transforms applied on the Fou­rier representation of the original data. These Fourier-based transforms can be used for interpolation and noise elimination of very complex seismic records.

Migration and imaging.  Migration of seismic records is a necessary process­ing step to obtain the true depth and dip information of seismic reflections. The Fourier transform is a very common mathematical tool for solving wave-equation problems. Many of the wave-equation migration techniques deploy the multidimensional fast Fourier transforms inside a least-squares fitting algorithm for imaging the subsurface. The Fourier-based wave-equation migration algo­rithms avoid the common pitfalls of ray-tracing migration algorithms known as ray-crossing.

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