Geoscience Reference
In-Depth Information
Inserting this into the above integrals, we can easily evaluate them. Perform-
ing the integration with respect to
α
first and noting that
2
π
0
dα
=2
π,
2
π
0
sin
αdα
=
2
π
0
cos
αdα
=
2
π
0
sin
α
cos
αdα
=0
,
(2-429)
2
π
0
sin
2
αdα
=
2
π
0
cos
2
αdα
=
π,
we find
s
0
∆
g
P
+
s
2
4
ds ,
1
γ
0
N
i
=
(
g
xx
+
g
yy
)+
···
(2-430)
0
s
0
1
2
γ
0
ξ
i
=
−
(
g
x
+
···
)
ds ,
0
(2-431)
s
0
1
2
γ
0
η
i
=
−
(
g
y
+
···
)
ds ,
0
∂
∆
g
∂H
s
0
=
1
4
(
g
xx
+
g
yy
+
···
)
ds .
(2-432)
s
=0
i
We now perform the integration over
s
, retaining only the lowest nonvanish-
ing terms. The result is
s
0
γ
0
∆
g
P
,
N
i
=
(2-433)
s
0
2
γ
0
g
x
,
s
0
2
γ
0
g
y
,
ξ
i
=
−
i
=
−
(2-434)
∂
∆
g
∂H
=
s
0
4
(
g
xx
+
g
yy
)
.
(2-435)
i
We see that the effect of the innermost circular zone on Stokes' formula
depends, to a first approximation, on the value of ∆
g
at
P
; the effect on
Vening Meinesz' formula depends on the first horizontal derivatives of ∆
g
;
and the effect on the vertical gradient depends on the second horizontal
derivatives.
Note that the contribution of the innermost zone to the total deflection
of the vertical has the same direction as the line of steepest inclination of
the “gravity anomaly surface”, because the plane vector
ϑ
=[
ξ
i
,η
i
]
(2-436)
is proportional to the horizontal gradient of ∆
g
,
grad ∆
g
=[
g
x
,g
y
]
.
(2-437)
