Geoscience Reference
In-Depth Information
the surface element becomes
=
dx dy ,
R
2
dσ
(6-47)
and we further have
r
=
R
+
H,
(6-48)
.
=2
RH.
r
2
R
2
=(
r
+
R
)(
r
−
−
R
)
Thus, (6-44) becomes the plane formula
∞
∞
∞
∞
H
2
π
T
l
3
H
2
π
T
(
x
2
+
y
2
+
H
2
)
3
/
2
T
P
dx dy
=
dx dy .
=
−∞
−∞
−∞
−∞
(6-49)
This important formula is called the
“upward continuation integral”
.Itper-
forms the computation of the value of the harmonic function
T
at a point
above the
xy
-plane from the values of
T
given on the plane, that is, the up-
ward continuation of a harmonic function. Both
T
and its partial derivatives,
∂T/∂x, ∂T/∂y, ∂T/∂z
, are harmonic, because if
∂
2
T
∂x
2
+
∂
2
T
∂y
2
+
∂
2
T
∂z
2
=0
,
(6-50)
then we also have
∂
2
∂z
2
=0
.
(6-51)
Thus, the upward continuation integral (6-49), which applies for any har-
monic function, may also be applied to
∂T/∂x, ∂T/∂y
,and
∂T/∂z
.
As
T
is the disturbing potential, its partial derivatives are the compo-
nents of the gravity disturbance:
∂x
2
∂T
+
∂y
2
∂T
+
∂z
2
∂T
=
∂
2
T
∂x
2
∂
2
∂
2
+
∂
2
T
∂y
2
+
∂
2
T
∂
∂x
∂x
∂x
∂x
∂T
∂x
∂T
∂y
∂T
∂z
=
δg
ϕ
,
=
δg
λ
,
=
δg
r
.
(6-52)
We are not writing
δg
x
,δg
y
,δg
z
because we wish to reserve this notation
for the components in the geocentric global coordinate system, which should
not be confused with the local system introduced in this section. As usual,
r, ϕ, λ
denote geocentric spherical coordinates (see Sect. 6.3) corresponding
to the spherical approximation.
Thus, we have in addition to (6-49)
∞
δg
r
l
3
H
2
π
δg
r
=
dx dy ,
(6-53)
−∞
