Geology Reference
In-Depth Information
be a real continuous non-periodic function of
variable
x
, such that
f
(
x
)
!
0as
x
!˙1
.
The Fourier transform of
f
is a complex func-
tion of a real parameter
k
, defined as follows:
[
f
]ofareal
function has the basic properties listed below:
1.
Symmetry
.Re(
F
) is a symmetric function,
Im(
F
) is antisymmetric:
The Fourier transformation
F
C
Z
Re .F .-k//
D
Re .F.k//
I
Im.F .-k//
f.x/e
ikx
dx
F.k/
D
(5.71)
D
-Im .F.k//
(5.77)
1
2.
Linearity
. For any pair of constants
a
and
b
,
if
f
and
g
are real continuous non-periodic
functions of
x
, then:
The variable
k
in (
5.71
) is called
wavenumber
and has units of m
1
. It is related to the
wave-
2
œ
F
Œaf
C
bg
D
a
F
Œf
C
b
F
Œg (5.78)
k
D
(5.72)
3.
Scaling
. For an arbitrary constant
a
¤
0, if
g
(
x
)
D
f
(
ax
)and
G
D
F
Being a complex function, on the basis of
Euler's formula
F
can be written in the form:
[
g
], then:
F
k
a
1
j
a
j
G.k/
D
(5.79)
F.k/
Dj
F.k/
j
e
i‚.k/
(5.73)
where the functions
j
F
(
k
)
j
and‚(
k
) are called re-
spectively
amplitude
and
phase
. The key feature
of Fourier transforms is that always exists an
inverse Fourier transform
that allows to go back
from the Fourier wavenumber domain to the
space domain:
4.
Shifting
.If
g
(
x
)
D
f
(
x
-
x
0
) is a shift of function
f
, then its transform adds a linear phase factor
to the Fourier transform of
f
, leaving the am-
plitude spectrum unaffected:
G.k/
D
F.k/e
ikx
0
(5.80)
C
Z
5.
Convolution
.Let
f
and
g
two real functions
with Fourier transforms
F
and
G
, respectively.
The following integral function is called
con-
volution
of
f
and
g
:
1
2
F.k/e
ikx
dk
f.x/
D
(5.74)
1
Therefore, the basic idea in Fourier-domain
modelling is to simplify complex operations by
application of the Fourier transform, then going
back to the space domain through an inverse
transformation. In the case of multivariate func-
tions, the Fourier transform defines a complex
function of a wavevector
k
, thereby, the transfor-
mation and its inverse assume the form:
C
Z
h.x/
D
f.Ÿ/g.x
Ÿ/dŸ
f
g
1
(5.81)
Then, the convolution property states that:
H.k/
D
F.k/G.k/
(5.82)
C
Z
C
Z
C
Z
f.r/e
ikr
dxdyd
z
(5.75)
F.k/
D
6.
Derivative
. The operation of differentiation
in the space domain is transformed into a
multiplication in the wavenumber domain:
1
1
1
C
Z
C
Z
C
Z
1
.2 /
3
F.k/e
ikr
dk
x
dk
y
dk
z
f.r/
D
d
n
f
dx
n
D
.ik/
n
F.k/
1
1
1
F
(5.83)
(5.76)
