Geoscience Reference
In-Depth Information
where
N
c
is the undulation of the cogeoid, the immediate result of Stokes'
formula, and
δN
is the indirect effect. By differentiating
N
in a horizontal
direction, we get the deflection component along this direction:
∂N
c
∂s
−
∂N
∂s
∂
(
δN
)
∂s
ε
=
−
=
−
.
(8-8)
This means that we must add to the immediate result of Vening Meinesz'
formula,
∂N
c
/∂s
, a term representing the horizontal derivative of
δN
(see
also Sect. 3.7).
To repeat, the main purpose is to obtain a simple boundary surface.
The geoid approximated by an ellipsoid or even a sphere is a much easier
boundary surface than the physical surface of the earth, to which we turn
now.
−
8.3
Geodetic boundary-value problems
It is, however, quite easy to understand the general principles. In space we
have the well-known fact that the gravity vector
g
and the gravity potential
(geopotential)
W
are related by
∂W
∂x
,
∂W
∂y
∂W
∂z
g
=grad
W
≡
,
,
(8-9)
which shows that the force
g
is the gradient vector of the potential.
Let
S
be the earth's topographic surface and let
W
and
g
be the geopo-
tential and the gravity vector on this surface. Then there exists a relation
g
=
f
(
S, W
)
,
(8-10)
the gravity vector
g
on
S
is a function of the surface
S
and the geopotential
W
on it. This can be seen in the following way. Let the surface
S
and the
geopotential
W
on
S
be given. The gravitational potential
V
is obtained
by subtracting the potential of the centrifugal force Φ, which is simple and
perfectly known (Sect. 2.1):
V
=
W −
Φ
.
(8-11)
The potential
V
outside the earth is a solution of Laplace's equation ∆
V
=0
and consequently
harmonic
(Sect. 1.3). Thus, knowing
V on S
, we can ob-
tain
V
outside
S
by solving Dirichlet's boundary-value problem, the first
boundary-value problem of potential theory, which is practically always
uniquely solvable (Sect. 1.12) at least if
V
is suciently smooth on
S
.After
