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 systems
(“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 longitude 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 deficiencies 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 squares, for example, P. J. Young (1994), (ii) ridge type regression (Tychonov regularization), for example, A. K. Saleh
and B. M. G. Kibria (1993), and (iii) truncated or total least squares, for example, R. D. Fierro and
J. R. Bunch (1994). Indeed, in the analysis of the curvilinear datum transformation Jacobi matrix
for regional geodetic networks, we identified a spectral condition number (the ratio of the largest
and smallest eigenvalue) of the order of 10
9 . For some reasons, we have accordingly chosen partial
least squares in analyzing the datum parameters from horizontal control, exclusively. Our results are
presented in Section 21-1.
Section 21-2.
In contrast, Section 21-2 is devoted to the synthesis of ellipsoidal longitude and ellipsoidal latitude
known in the local reference system and to be transformed into the global reference system or,
equivalently into Gauss–Krueger coordinates or UTM coordinates from local to global reference. A
real data example is given.
Section 21-3.
Finally, Section 21-3 three reviews the error propagation in the analysis and synthesis of a curvilinear
datum transformation.
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 systems
(“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 longitude 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 deficiencies 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 squares, for example, P. J. Young (1994), (ii) ridge type regression (Tychonov regularization), for example, A. K. Saleh
and B. M. G. Kibria (1993), and (iii) truncated or total least squares, for example, R. D. Fierro and
J. R. Bunch (1994). Indeed, in the analysis of the curvilinear datum transformation Jacobi matrix
for regional geodetic networks, we identified a spectral condition number (the ratio of the largest
and smallest eigenvalue) of the order of 10
9 . For some reasons, we have accordingly chosen partial
least squares in analyzing the datum parameters from horizontal control, exclusively. Our results are
presented in Section 21-1.
Section 21-2.
In contrast, Section 21-2 is devoted to the synthesis of ellipsoidal longitude and ellipsoidal latitude
known in the local reference system and to be transformed into the global reference system or,
equivalently into Gauss–Krueger coordinates or UTM coordinates from local to global reference. A
real data example is given.
Section 21-3.
Finally, Section 21-3 three reviews the error propagation in the analysis and synthesis of a curvilinear
datum transformation.