The geometry of the earth - Physical Geodesy - page 207

the common adjustment of GPS observations and terrestrial data is inves-
tigated. The problem encountered here is that GPS data refer to the three-
dimensional geocentric Cartesian system WGS 84, whereas terrestrial data
refer to the individual local level (tangent plane) systems at each measure-
ment point referenced to plumb lines. Furthermore, terrestrial data are tra-
ditionally separated into position and height, where the position refers to an
ellipsoid and the (orthometric) height to the geoid.
For a joint adjustment, a common coordinate system is required to which
all observations are transformed. In principle, any arbitrary system may be
introduced as common reference. One possibility is to use two-dimensional
(plane) coordinates in the local system as proposed by Daxinger and Stir-
ling (1995). Here, a three-dimensional coordinate system is chosen. The ori-
gin of the coordinate system is the center of the ellipsoid adopted for the
local system, the Z -axis coincides with the semiminor axis of the ellipsoid,
the X -axis is obtained by the intersection of the ellipsoidal Greenwich merid-
ian plane and the ellipsoidal equatorial plane, and the Y -axis completes the
right-handed system. Position vectors referred to this system are denoted by
X LS , where LS indicates the reference to the local system.
After the decision on the common coordinate system, the terrestrial mea-
surements referring to the individual local level systems at the observing
sites must be represented in this common coordinate system. Similarly, GPS
baseline vectors regarded as measurement quantities are to be transformed
to this system.

5.10.2 Representation of measurement quantities

Distances
The spatial distance s ij as function of the local level coordinates is given in
(5-69). If n ij ,e ij ,u ij , the components of x ij , are substituted by (5-65), the
relation

s ij = n 2

ij + e 2 ij + u 2 ij

(5-70)

= ( X j − X i ) 2 +( Y j − Y i ) 2 +( Z j − Z i ) 2

is obtained, where (5-64) has also been taken into account, namely, the
fact that n i , e i , u i are unit vectors. Obviously, the second expression arises
immediately from the Pythagorean theorem. Differentiation of (5-70) yields

ds ij = X ij
s ij

dX i )+ Y ij
s ij

dY i )+ Z ij
s ij

( dX j − ( dY j − ( dZ j − dZ i ) ,

(5-71)