The geometry of the earth - Physical Geodesy - page 199

4. For network points other than the common points, transform the co-
ordinates ( X, Y, Z ) GPS into ( X, Y, Z ) LS via Eq. (5-41) using the trans-
formation parameters determined in the previous step.

5. Transform the Cartesian coordinates ( X, Y, Z ) LS computed in the pre-
vious step into ellipsoidal coordinates ( ϕ, λ, h ) LS , e.g., by the iterative
procedure given in (5-28) through (5-34).

6. Map the ellipsoidal surface coordinates ( ϕ, λ ) LS computed in the pre-
vious step into plane coordinates ( y, x ) LS by the appropriate mapping
formulas.

The advantage of the three-dimensional approach is that no a priori infor-
mation is required for the seven parameters of the similarity transformation.
The disadvantage of the method is that for the common points ellipsoidal
heights (and, thus, geoidal heights) are required. However, as reported by
Schmitt et al. (1991), incorrect heights of the common points often have
a negligible effect on the plane coordinates ( y, x ). For example, incorrect
heights may cause a tilt of a 20 km

×

20 km network by an amount of 5 m
in space; however, the effect on the plane coordinates is only approximately
1 mm.
For large areas, the height problem can be solved by adopting approx-
imate ellipsoidal heights for the common points and performing a three-
dimensional ane transformation instead of the similarity transformation.

5.7.4 Differential formulas for other datum transformations

Now we consider simplified cases. Suppose that the geocenter does not co-
incide with the center of the reference ellipsoid, but that the geocentric axes
and the ellipsoidal axes are parallel . Such a parallel shift is also called a
translation (Fig. 5.8). Assume a rectangular coordinate system XY Z whose
origin is the geocenter, the axes being directed as usual. Let the coordinates
of the center of the ellipsoid with respect to this system be x 0 ,y 0 ,z 0 ,as
stated previously. Then Eqs. (5-27) must obviously be modified so that they
become

X = x 0 +( N + h )cos ϕ cos λ,

Y = y 0 +( N + h )cos ϕ sin λ,
Z = z 0 + b 2

(5-53)

a 2 N + h sin ϕ.

These equations form the starting point for various important differential
formulas of coordinate transformation.