Geoscience Reference
In-Depth Information
∞
δg
ϕ
l
3
H
2
π
δg
ϕ
=
dx dy ,
−∞
(6-54)
δg
λ
l
3
∞
H
2
π
δg
λ
=
dx dy .
−∞
On the left-hand side of these equations, the components of
δ
g
refer to the
elevated point
P
; in the integral on the right-hand side, they are taken at
sea level and are to be computed from the expressions
∆
g
+
2
γ
0
R
N
,
δg
r
=
−
δg
=
−
(6-55)
δg
ϕ
=
−
γ
0
ξ,
(6-56)
δg
λ
=
−
γ
0
η,
which follow from (2-264) together with (6-42) and (6-43) applied to sea
level. The symbols
R
and
γ
0
denote, as usual, a mean earth radius and a
mean value of gravity on the earth's surface.
Hence, we may compute
T
and
δ
g
by means of the upward continuation
integral if the geoidal undulations
N
and the deflection components
ξ
and
η
at the earth's surface are given.
The plane approximation is su
cient except for very high altitudes (e.g.,
>
250 km). Otherwise, we must use the spherical formula (6-44) for
T
.For
the radial component
δg
r
, formula (6-44) may also be applied with
T
re-
placed by
rδg
,since
rδg
and
r
∆
g
are harmonic as we know from Sect. 2.14.
The corresponding spherical formulas for the upward continuation of the hor-
izontal components
δg
ϕ
and
δg
λ
arenotknown.Thereasonwhythesame
formula, the upward continuation integral, applies for
T
and the components
of
δ
g
in the planar case only is that the derivatives of
T
are harmonic only
when referred to a Cartesian coordinate system.
6.5
Additional considerations
Reference surface
The preceding formulas for the disturbing potential
T
and the gravity dis-
turbance vector
δ
g
are rigorously valid if the reference surface is a sphere.
In practice, the gravity anomalies are referred to an ellipsoid. The above
formulas for
T
and
δ
g
are also valid for an ellipsoidal reference surface if a
relative error of the order of the flattening
f
.
=0
.
3% is neglected, that is, as
a spherical approximation. The reader is reminded that
this does not mean
