NETWORK ADJUSTMENTS - GPS Satellite Surveying - page 307

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For example, consider the case that coordinates of some of the stations are known
in the “local datum” (u) and that (u) does not coincide with (x), i.e., the coordinate
system of the GPS vectors. Let it be further required that if the adjusted coordinates
should be expressed in (u), then the following model

1 a = f 1 ( x a ) (8.23)

2 a = f 2 (s, η , ξ , α , x a ) (8.24)

m ight be applicable. The model (8.23) pertains to the terrestrial observations, denoted
he re as the 1 set. This model is discussed in Chapter 2. In adjustment notation the

pa rameters x a denote station coordinates in the geodetic system (u). The observa-
tio ns for Model (8.24) are the Cartesian coordinate differences between stations as
ob tained from GPS carrier phase processing. The additional parameters in (8.24) are
th e scale correction s and three rotation angles. The rotation angles are small as they
re late the nearly aligned geodetic coordinate systems (u) and (x). Because GPS yields
th e coordinate differences, there is no need to include the translation parameter t .If
th e coordinate systems (u) and (x) coincide, then the estimate of the rotation angles
sh ould statistically be zero. Even if

[30

Lin
—
0.7
——
No
PgE

1 in (8.23) does not contain observations at all,
so me of the station coordinates in the (u) system can still be treated as observed
pa rameters and thus allow the estimation of the scale and rotation parameters. This
is a simple way to implement the GPS vector observations into the existing network.
The mathematical model (8.24) follows directly from the transformation expres-
sio n (8.14). Applying this expression to the coordinate differences for stations k and
m yields

[30

( 1 + s) M ( λ 0 ,ϕ 0 , η , ξ , α ) ( u k − u m ) − ( x k − x m ) = o (8.25)

The coordinate differences

x km = x k − x m (8.26)

represent the observed GPS vector between stations k and m . Thus the mathematical
model (8.24) can be written as

x km = ( 1 + s) M ( λ 0 ,ϕ 0 , η , ξ , α ) ( u k − u m ) (8.27)

Af ter substituting (8.16) into (8.27), we readily obtain the partial derivatives of
th e design matrix. Table 8.1 lists the partial derivatives with respect to the station

TABLE 8.1 Design Submatrix for Stations Occupied with Receivers

Pa rameterization Station m Station k

( u )
(ϕ,

( 1

+ s ) M −

( 1

+ s ) M

,h)
(n, e, u)

λ ( 1 + s) MJ (ϕ m , λ m ) − ( 1 + s) MJ (ϕ k , λ k )

λ m ) H − 1 (ϕ m ) λ k ) H − 1 (ϕ k )

( 1 + s) MJ (ϕ m , − ( 1 + s) MJ (ϕ k ,