Geoscience Reference
In-Depth Information
dinates are related to rectangular coordinates
x, y, z
by the equations
x
=
r
cos
ϕ
cos
λ,
y
=
r
cos
ϕ
sin
λ,
z
=
r
sin
ϕ
(6-18)
or inversely by
r
=
x
2
+
y
2
+
z
2
,
ϕ
=tan
−
1
z
x
2
+
y
2
,
λ
=tan
−
1
y
(6-19)
x
.
Now it is convenient to start with the components
δg
r
,δg
ϕ
,δg
λ
of the
gravity disturbance vector
δ
g
, Eq. (6-3), in the spherical coordinates
r, ϕ, λ
.
In analogy to (2-377), we have
δg
r
=
∂T
ϕ
=
1
r
∂T
∂ ϕ
,
1
r
cos
ϕ
∂T
∂λ
.
∂r
,
λ
=
(6-20)
Since we are dealing with the relatively small quantities of the disturbing
field, a spherical approximation may be sucient (Sect. 2.13), as it was in
the case of Stokes' formula.
The disturbing potential
T
may be expressed in terms of the free-air
anomalies at the earth's surface by the formula of Pizzetti, Eqs. (2-302) and
(2-303),
∆
gS
(
r, ψ
)
dσ ,
=
T
(
r, ϕ, λ
)=
R
4
π
T
P
(6-21)
σ
where
S
(
r, ψ
) is the extended Stokes function,
cos
ψ
5+3ln
r
,
R
2
r
2
S
(
r, ψ
)=
2
R
l
+
R
3
Rl
−
R
cos
ψ
+
l
2
r
r
−
r
2
−
(6-22)
and
l
=
r
2
+
R
2
−
2
Rr
cos
ψ.
(6-23)
According to (6-20), we must differentiate (6-21) with respect to
r, ϕ
,and
λ
. Here we note that the integral on the right-hand side of (6-21) depends
on
r, ϕ, λ
only through the function
S
(
r, ψ
). Thus, ∆
g
being constant with
