Advanced Geometric Geodesy - GS760
Autumn Quarter 2000
Distributed: 29 September 2000
Date Due: 9 October 2000
Problem Set 1
1. The Cartesian coordinates of 12 points are observed in two coordinate systems. Their
observed values are given in files FILE11.DAT and FILE12.DAT on the “geodesy” server under
directory ‘pub/jekeli/data’. Each file contains three rows of 12 numbers; the first row contains
all x-coordinates, the second all y-coordinates, and third row all z-coordinates of the 12 points.
The standard deviations for all coordinates within a file are, respectively, 1 = 0.2 m and 2 =
0.5 m. Perform a least-squares adjustment to determine the transformation parameters
(translation, rotation, scale) between the two systems. Also determine the covariance matrix of
the adjusted parameters and discuss the correlations (hint: map the points). Do the same for files
FILE21.DAT and FILE22.DAT.
2. a) Verify Rapp’s equ. (2.51) and derive

x
T
 = R ( 0, 0)  y

z
b) Show that Rapp’s equ. (2.90) is the same as (2.87); and provide a correct equ. (2.98).
3. The differential changes in geodetic coordinates due to Cartesian coordinate changes is:
d
dx
-1 RT(,) dy
=
D
d
dz
dh
where D and R are defined in the class notes (and da=0 and df=0). Suppose the coordinate
system is rotated by (x,y,z). Show that the changes in geodetic coordinates are given by
equations (2.126), (2.127), and (2.128) (ibid); and show that these are equivalent to Vincenty's
(1985) equations (2.10), (2.11), and (2.12).
4. Suppose the conventional terrestrial system rotates by x=1", y=-0.5", and z=0.7". The
geodetic datum is based on the International Ellipsoid and its origin point is to remain fixed at
0=35° , 0=-100° . Before rotation, at a point P with geodetic coordinates (=40° , =-80° ), the
deflections of the vertical are  = 0.5" and  = -0.1". The vertical angle at P to an observed point
is 10° . Assume an azimuth of 30° to the observed point.
a) Compute the change in astronomic azimuth at P due to this rotation. In the formula use
first astronomic coordinates and then geodetic coordinates and compare your results by
computing the difference.
b) Compute the change in the geodetic azimuth of the observed point at P due to the
rotation.
c) Compute the change in astronomic coordinates at P due to this rotation.