Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 34

Fourier transform of f(t). The function f(t) in equation (15) is called the Fourier transform
(inverse transform) of F(ω) and the pair may be represented by a notation f(t)↔F(ω). F(ω) is
called the amplitude of the time domain field, f(t).
So far we have used as variables t and ω, representing time and angular frequency,
respectively. Mathematics will, of course, be the same if we change the names of these
variables. In describing the spatial variations of a wave, it is more natural to use either r or x,
y, and z to represent distances. In a function of time, the period T is the time interval after
which the function repeats itself. In a function of distance, the corresponding quantity is
called the wavelength λ, which is the increase in distance that the function will repeat itself.
Thus, if t is replaced by r, then the angular frequency ω, which is equal to 2 π/T, should be
replaced by a quantity equal to 2 π/λ, which is known as the wave number, k. In three
dimensions, we can define a Fourier transform pair as

=

( ) / ∭ () . ( ) / ∭ ( ) .

and ()=

Where and are radius and wave number vectors respectively. If the wave number, is
along the z-axis of the coordinate space, then . = kr cosθ and d 3 r = r 2 sinθdθdrd ϕ and so
the Fourier transform of f(r) becomes, =

∞

( )

as ϕ runs from 0 to 2π, r

runs from 0 to ∞ and θ runs from 0 to π.

Again, how to split 1 / (2 π ) 3 between the Fourier transform and its inverse is somewhat
arbitrary. Here we split them equally to conform to most standard expressions.
There are number of useful theorems that connect the Fourier transform pair in equations
(14) and (15) and these can be found in standard texts on the subject. We shall only mention
a few ones that are easily provable as follows:
a.

The Convolution Theorem - If f 1 (t)↔F 1 (ω) and f 2 (t)↔F 2 (ω) are Fourier pairs, then

(). ()↔

()∗ () and ()∗ ()↔ (). ()
This a very important theorem which has a wide field application. It simply says that
the spectrum of a product of two time functions is the convolution of their individual
spectra. The theorem can be extended to any number of functions as

[ ()∗ ()∗ ()… ()] .
Note that in general a convolution is expressed as

[ (). (). ()… ()]↔

( ) ∗ ( ) = ( ) ( − )

and

()∗ ()= () (−)

.

The mathematical operation of convolution consists of the following steps:
1. Take the mirror image of f 2 ( τ ) about the coordinate axis to create f 2 ( −τ ) from f 2 (τ ) .

2. Shift f 2 ( −τ ) by an amount t to get f 2 ( t − τ ) . If t is positive, the shift is to the right, if it

is negative, to the left.