Datum Problems - Map Projections: Cartographic Information Systems - page 507

21
Datum Problems
Analysis versus synthesis, Cartesian approach versus curvilinear approach. Local reference
system versus global reference system. Datum parameters, collinearities, error propagation.
Partial least squares, ridge type regression (Tychonov regularization), truncated or total least
squares. Gauss-Krueger coordinates and UTM coordinates. Stochastic pseudo-observations
and variance-covariance matrix, dispersion matrix.
The evolutionary process of (2+1)-dimensional geodesy separating horizontal control and vertical
control towards three-dimensional geodesy, namely enforced by satellite global positioning sys-
tems (“global problem solver”: GPS), confronts us with the problem of curvilinear geodetic datum
transformations of the following type. In a local two-dimensional geodetic network ellipsoidal lon-
gitude and ellipsoidal latitude (equivalent: Gauss-Krueger coordinates, UTM coordinates) are
available in a local geodetic reference system. From a global three-dimensional geodetic network,
namely for a few identical points, ellipsoidal longitude, ellipsoidal latitude, and ellipsoidal height
are known in a global geodetic reference system. Here, we aim at the analysis of datum parameters
(seven parameter global conformal group C 7 (3): translation, rotation, scale). We set up curvilinear
linearized pseudo-observational equations for given ellipsoidal longitude and ellipsoidal latitude,
with incremental parameters of translation (three parameters), rotation (three parameters), and
scale (one parameter), and with incremental form parameters of the ellipsoid-of-revolution (two
parameters: semi-major axis, squared eccentricity). In particular, we investigate the rank deficien-
cies in the curvilinear datum transformation Jacobi matrix (collinearities). A strict collinearity
between the incremental datum parameter t z and the incremental semi-major axis δA has been
identified. For geodetic networks of regional extension, we experienced also configurations close
to a collinearity.
Section 21-1.
A regression system close to a collinearity (near linear dependence) experiences damaging
effects on the ordinary least squares estimator, as small changes in the Jacobi matrix or in
the vector of pseudo-observations may result in unproportionally large changes in the solution.
Three methods of overcoming the problem of collinearity are currently used: (i) partial least