Geoscience Reference
In-Depth Information
Linearization
The linearization applies equally well for the Molodensky problem and the
GPS problem. The geometry is familiar (Fig. 8.2).
We recall the surface Σ, the telluroid , which is defined by the condition
U ( Q )= W ( P ) .
(8-28)
We note that (8-28) is the surface equivalent to the classical relation for sea
level (Fig. 8.3)
U ( Q 0 )= W ( P 0 ) .
(8-29)
Equation (8-28) would apply with
W ( P 0 )= W 0 = constant
(8-30)
if S were an equipotential surface, the geoid, which is the case only over the
oceans with the usual simplifying assumption that the surface of the ocean
is an equipotential surface not changing with time (Fig. 8.3).
Molodensky's theory does not use the geoid directly but the physical
earth's surface. We repeat once more that this is Molodensky's epochal idea
which radically changed the course of physical geodesy since 1945.
We shall, however, use the fictitious case of S being an equipotential sur-
face, but only as a first (or zero-order) assumption in a perturbation approach
for the real earth's surface (Molodensky series). This first approximation is
the spherical case to be considered in the next section.
Now we consider the linearization in more detail. The ellipsoidal height
h is directly determined by GPS. It may be decomposed into
h = H + ζ.
(8-31)
Here, H is the normal height and ζ is the height anomaly, whose definitions
are seen from Fig. 8.2. In the GPS case we do know the earth's surface S
directly, but the telluroid Σ and the height anomalies ζ are still required for
formulating the boundary condition, just as the knowledge of the geoid does
not make superfluous the reference ellipsoid.
P 0
geoid
N
ellipsoid
Q 0
Fig. 8.3. Geoid and ellipsoid
Search WWH ::




Custom Search