Global Positioning System Reference
In-Depth Information
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the truncation error increases rapidly. The basic shape of this figure and the values do
not change significantly for different latitudes of the center. Because the ellipsoid is
a figure of rotation, the longitude of the center does not matter.
The GML solution satisfies typical geodetic applications. In the unlikely case that
they are not sufficient because long lines are involved, one can always replace them
with other solutions that are valid for long geodesics.
9.1.3 Network Adjustment on the Ellipsoid
Th
e geodesic azimuths, geodesic distances, and the angles between geodesics form a
ne
twork of stations on the ellipsoidal surface that can be adjusted using standard least-
sq
uares techniques. The ellipsoidal network contains no explicit height information.
Th
e height information was used during the transition of the 3D geodetic observ-
ab
les to the geodesic observables on the ellipsoid. Conceptually, this is expressed by
{
[32
,
α
, s
. The geodetic height
h
is no
lo
nger a parameter, and geodesic observables do not include quantities that directly
co
rrespond to the geodetic vertical angle, the geodetic height difference
ϕ,
λ
,h
} → {
ϕ,
λ}
and
{α
,
δ
,
β
,s,
∆
h,
∆
N
} →{
δ
}
Lin
—
1
——
Nor
*PgE
∆
h
,orthe
ge
oid undulation difference
N
.
Least-squares techniques are discussed in detail in Chapter 4. For discussion in
th
is section, we use the observation equation model
∆
v
=
Ax
+
(
0
−
b
)
(9.4)
In
the familiar adjustment notation the symbol
v
denotes the residuals,
A
is the design
m
atrix, and
x
represents the corrections to the approximate parameters
x
0
. The symbol
b
denotes the observations, in this case the geodesic observables, and
[32
0
represents
th
e observables as computed from the approximate parameters
x
0
=
···
···
T
ϕ
i,
0
λ
i,
0
(9.5)
us
ing the GML functions. If we further use the (2-1-3) subscript notation to denote
th
e angle measured at station 1 from station 2 to station 3 in a clockwise sense, then
th
e geodesic observables can be expressed as
α
12
,b
= α
12
,b
+ ∆α
12
(9.6)
δ
213
,b
= δ
213
,b
+ ∆α
13
− ∆α
12
(9.7)
s
12
=
s (s
12
,R,h
1
,h
2
)
(9.8)
In order to make the interpretation of the coordinate (parameter) shifts easier, it is
advantageous to reparameterize the parameters to northing
(dn
i
=
M
i
dϕ
i
)
and east-
ing
(de
i
=
λ
i
)
. Using the partial derivatives in Table B.3, the observation
equations for the geodesic observables become
N
i
cos
ϕ
i
d
