The Fourier transform is of central importance in virtually all aspects of petroleum seismology. It is a method of analysis that detects periodicity in signals, a mathematical tool for solving partial differential equations, and a fertile domain for implementing processing algorithms. It is the mechanism for understanding the properties of waves including bandwidth, amplitude spectrum, phase spectrum, and stability with respect to inversion.

It is said of the ancient mathematician Euclid (circa 300 BCE) author of The Elements of Geometry: that Ptolemy once asked him if there was in geometry any shorter way than that of the Elements, and he replied that there was no royal road to geometry.

In a distant echo of that thought, there is no non-mathematical shortcut to understanding the Fourier transform and its uses. For a brief, but intensely detailed, account of the Fourier transform see Press et al. ^[149].

A.1 Definitions

Consider a time function, g( t), which is the measured output of some kind of experiment. An example would be pressure measured during the passage of a sound wave. This function is said to be in the time domain and the Fourier transform (FT) casts it into the frequency domain. Since the FT moves us from the physical domain (time) to the alternate domain (frequency), it is termed the forward transform. The definition of the forward FT is

where

i

=

? ?1, and ? is angular frequency.

The notation g( ?)...