Geology Reference
In-Depth Information
Fig. 5.24
Polar
coordinates for the
computation of integral
(
5.94
)
§
x
x
0
;y
y
0
;
z
the upward continuation integral is performed on
a finite but sufficiently large rectangle below the
point (
x
,
y
,
z
0
-
z
). An alternative approach to the
upward continuation is based on the Fourier do-
main representation, which allows to determine
V
on any plane, given the values of the function on
a reference level surface. We note that the integral
(
5.88
) is a form of two-dimensional convolution
of
V
by a harmonic function §:
1
2
@
@
z
1
r
D
(5.92)
where
r
is the distance of the point (
x
,
y
,
z
0
-
z
)
from points on the plane
z
0
D
z
0
(Fig.
5.23
). The
Fourier transform of (
5.92
)gives:
1
r
1
2
@
@
z
F
F
Τ
D
(5.93)
To calculate the Fourier transform of 1/
r
,let
us move the observation point to (0,0,0). In this
instance, the Fourier transform is performed over
the plane
z
D
z
, with
r
D
[
x
2
C
Z
C
Z
V
x
0
;y
0
;
z
0
V.x;y;
z
0
z
/
D
1
1
C
y
2
C
z
2
]
1/2
.
§
x
x
0
;y
y
0
;
z
dx
0
dy
0
Then,
(5.89)
1
r
C
Z
C
Z
where
§
x
x
0
;y
y
0
;
z
1
C
z
2
e
ik¡
dxdy
(5.94)
F
D
p
x
2
C
y
2
1
1
z
D
2
h
.x
x
0
/
2
C
z
2
i
3=2
(5.90)
This integral can be evaluated more easily
passing to polar coordinates (¡,™,¥), as shown in
Fig.
5.24
.
We h ave :
C
.y
y
0
/
2
Therefore, we can perform a Fourier transfor-
mation of both sides of (
5.89
) and apply the con-
volution property. If V is the upward continuation
of
V
, then:
1
r
C
Z
Z
2
¡
C
z
2
e
ik¡cos.¥/
d¡d™
(5.95)
F
D
p
¡
2
0
0
F
V
D
F
F
ŒV
Τ
(5.91)
Making the substitution ™
0
D
™ - ¥ and taking
into account that exp(-
ik
¡cos™
0
) is periodic with
period , we obtain:
To determine the Fourier transform of
§
,itis
useful to rewrite (
5.90
) as follows:
