F-19. Fourier transform pairs. The time functions on the left are Fourier transforms of the frequency functions on the right and vice-versa. Many more transform pairs could be shown. The above are all even functions and hence have zero phase. Transforms for real odd functions are imaginary, i.e., they have a phase shift of +π/2. Transforms of functions that are neither odd nor even involve variations of phase with frequency. Note
f=1/
t.
Formulas that convert a time function g(t) (waveform, seismic record trace, etc.) into its frequency-domain representation G(f) and vice versa. G(f) and g(t) constitute a Fourier-transform pair; see Figure F-19 below. An example is
The inverse transform is
Finding G(f) from g(t) is called Fourier analysis and finding g(t) from G(f) is called Fourier synthesis. G(f) is the complex spectrum, the real part being the cosine transform and the imaginary part the sine transform whenever g(t) is real. Another expression for G(f) is
where the functions A(f) and γ(f) are real. They are, respectively, the amplitude spectrum and the phase spectrum of g(t):
γ(f) is in the first or second quadrant if the imaginary part is positive, in the first or fourth quadrant if the real part is positive. A record trace h(t) that extends only from 0 to T may be assumed to be repeated indefinitely and so expanded in a Fourier series of period T:
where
and
The same rules for quadrants apply to γn as expressed for γ(f); a0 is the zero-frequency component (or dc shift). The frequency spectrum is discrete if the function is periodic. If ht is a sampled time series sampled at intervals of time t2, then we can stop summing when n>2T/t2 (see sampling theorem). In this case an and bn can be expressed as sums:
and
Also see phase response and fast Fourier transform.
F-20. Equivalence of Fourier transform operations. Doing the time operation is equivalent to doing the frequency operation on the transform of the data. Note:
g(
t)↔
G(
f) and
h(
t)↔
H(
f).
Operations in one domain have equivalent operations in the transform domain (see Figure F-20). Computations can sometimes be carried out more economically in one domain than the other and Fourier transforms provide a means of accomplishing this. The Fourier-transform relations can be generalized for more than one dimension (see Figure F-21). For example,
and
The 1/4π factor is sometimes distributed between the two integrals; where calculations involve an arbitrary scaling factor, the 1/4π factor may be dropped entirely.
Fourier transforms are discussed in Sheriff and Geldart (1995, 277, 532–533)[1]. Theorems relating to Fourier transforms are shown in Figure F-22.
F-22. Fourier transform theorems.
Notations and sign conventions
The notations and sign conventions used above are common in the electrical engineering world. Many exploration geophysicists may be familiar, however, with the conventions
used by physicists and mathematicians. These differ subtly from those used above. Many geophysicists whose education may in those fields, or who draw
from the scientific results from those fields prefer conventions different from those of electrical engineers. It is, therefore, important to include a short
discussion of these conventions.
It is more common in the world of mathematics and theoretical physics for the convention of
rather than
.
Electrical engineers prefer to use the letter
because the letter
is reserved for current. Mathematicians and mathematical
physicists prefer using the angular frequency
where the units are radians per time, rather than cycles per time. This means
that there may be a factor of
discrepancy between computations using the differing conventions. Finally, there may be different sign
conventions on the exponent of the exponentials in the Fourier transform definitions. Because signals recorded in the space-time domain are causal, meaning
that there are no arrivals before time
the forward temporal transform integration will start at
rather than
.
1D transforms
Putting all of these together, we obtain a common notational convention for the forward Fourier transform in time as
and the inverse Fourier transform
The 1D spatial forward Fourier transform differs from the temporal transform in that the integration is on infinite limits and the sign of the exponent in
the exponential is negative
and the inverse spatial Fourier transform is similarly different
Transforms in
spatial dimensions
In
dimensions, the spatial transforms become
and the inverse spatial Fourier transform is similarly different
Here we have used the conventions that
and
Transforms in 1 temporal and 3 spatial dimensions
In 3 dimensions of space and 1 dimension of time, as is encountered in problems dealing with the wave equation, we have the forward Fourier transform from
and the corresponding inverse Fourier transform
Here,
and
References
- ↑ Sheriff, R. E. and Geldart, L. P., 1995, Exploration Seismology, 2nd Ed., Cambridge Univ. Press.