Burkhard Schaffrin

Modifying Cadzow's algorithm to generate the optimal TLS-solution for the structured EIV-Model of a similarity transformation

Modifying Cadzows algorithm to generate the optimal TLS-solution for the structured EIV-Model of a similarity transformation In 2005, Felus and Schaffrin discussed the problem of a Structured Errors-in-Variables (EIV) Model in the context of a parameter adjustment for a classical similarity transformation. Their proposal, however, to perform a Total Least-Squares (TLS) adjustment, followed by a Cadzow step to imprint the proper structure, would not always guarantee the identity of this solution with the optimal Structured TLS solution, particularly in view of the residuals. Here, an attempt will be made to modify the Cadzow step in order to generate the optimal solution with the desired structure as it would, for instance, also result from a traditional LS-adjustment within an iteratively linearized Gauss-Helmert Model (GHM). Incidentally, this solution coincides with the (properly) Weighted TLS solution which does not need a Cadzow step.

An alternative approach to robust collocation

The now classical collocation method in geodesy has been derived byH. Moritz (1970; 1973) within an appropriate Mixed Linear Model. According toB. Schaffrin (1985; 1986) even a generalized form of the collocation solution can be proved to represent a combined estimation/prediction procedure of typeBLUUE (Best Linear Uniformly Unbiased Estimation) for the fixed parameters, and of type inhomBLIP (Best inhomogeneously LInear Prediction) for the random effects with not necessarily zero expectation. Moreover, “robust collocation” has been introduced by means of homBLUP (Best homogeneously Linear weakly Unbiased Prediction) for the random effects together with a suitableLUUE for the fixed parameters. Here we present anequivalence theorem which states that the robust collocation solution in theoriginal Mixed Linear Model can identically be derived as traditionalLESS (LEast Squares Solution) in amodified Mixed Linear Model without using artifacts like “pseudo-observations”. This allows us a nice interpretation of “robust collocation” as an adjustment technique in the presence of “weak prior information”.

On The Errors-In-Variables Model With Singular Dispersion Matrices

Abstract While the Errors-In-Variables (EIV) Model has been treated as a special case of the nonlinear Gauss- Helmert Model (GHM) for more than a century, it was only in 1980 that Golub and Van Loan showed how the Total Least-Squares (TLS) solution can be obtained from a certain minimum eigenvalue problem, assuming a particular relationship between the diagonal dispersion matrices for the observations involved in both the data vector and the data matrix. More general, but always nonsingular, dispersion matrices to generate the “properly weighted” TLS solution were only recently introduced by Schaffrin and Wieser, Fang, and Mahboub, among others. Here, the case of singular dispersion matrices is investigated, and algorithms are presented under a rank condition that indicates the existence of a unique TLS solution, thereby adding a new method to the existing literature on TLS adjustment. In contrast to more general “measurement error models,” the restriction to the EIV-Model still allows the derivation of (nonlinear) closed formulas for the weighted TLS solution. The practicality will be evidenced by an example from geodetic science, namely the over-determined similarity transformation between different coordinate estimates for a set of identical points.

Empirical Affine Reference Frame Transformations by Weighted Multivariate TLS Adjustment

In order to determine the transformation parameters between two reference frames empirically, a sufficient number of point coordinates (or possibly higher dimensional features such as, e.g., straight lines or conics) need to be observed in both systems. A proper adjustment of the observed data must take the different variances and covariances into account.

On the Gauss-Helmert model with a singular dispersion matrix where B Q is of smaller rank than B

The case of a singular dispersion matrix within the Gauss-Helmert Model has been considered before, usually assuming a sufficiently small rank deficiency in order to guarantee a unique solution for both the residual vector as well as the estimated parameter vector of type Best Linear Uniformly Unbiased Estimate (BLUUE). In this contribution the emphasis is shifted towards establishing necessary and sufficient conditions for a unique residual vector, along with a unique estimate of type Best Linear Uniformly Minimum Bias Estimate (BLUMBE) for the parameter vector. Should uniformly unbiased estimates exist, the BLUMBE obviously becomes the BLUUE.

The planar trisection problem and the impact of curvature on non-linear least-squares estimation

Abstract The planar trisection problems is an idealized version of a routine task in geodesy, namely the determination of one points 2D-coordinates by measuring the distances to three known points, all points lying in the same plane. In this case we can easily visualize the non-linear least-squares estimation process as the orthogonal projection of the observation vector in the 3D-Euclidean space onto the respective 2D-Riemannian manifold. We compute the principle curvatures and discuss these results in view of a lemma by A. Pazman (1984b) that ensures the solution of the “normal equations” to become the sought “least-squares estimate” under certain conditions.

Biased Kriging on The Sphere

It is well known among geodesists that homogeneous-isotropic processes on the sphere which are both, Gaussian and ergodic, do not exist. Thus we have to give up one of the two properties as soon as we consider global phenomena such as the (residual) topography, the disturbing gravity field, etc. In case of a non-Gaussian process we are facing problems in deriving the proper distribution functions for our test statistics, beside the fact that such a process is not completely defined by its first two moments. On the other hand, a non-ergodic process does not allow us to equivalently express “expectation”, and “covariance”, as spatial integral over any of its realizations. Hence our usual methods to derive the covariance function (or the semi-variogram) fail to produce consistent - or even unbiased - estimates.

Adjusting the Errors-In-Variables Model: Linearized Least-Squares vs. Nonlinear Total Least-Squares

It has long been known that the Errors-In-Variables (EIV) Model is a special case of the nonlinear Gauss–Helmert Model (GHM) and can, therefore, be adjusted by standard least-squares techniques in iteratively linearized GH-Models, which is the approach by Helmert (Adjustment Computations Based on the Least-Squares Principle (in German), 1907) and – later – by Deming (Phil Mag 11:146–158, 1931; Phil Mag 17:804–829, 1934).

NEW AFFINE TRANSFORMATION PARAMETERS FOR THE HORIZONTAL NETWORK OF SEOUL/KOREA BY MULTIVARIATE TLS-ADJUSTMENT

Abstract The new approach of multivariate Total Least-Squares (TLS) adjustment has been applied to 33 stations of the horizontal network of Seoul/Korea in order to estimate best-fitting affine transformation parameters. The results are presented along with variances and covariances (in first order approximation), and comparisons are drawn with the standard Least-Squares (LS) approach. Although the estimates from TLS do not show practical differences with respect to LS, the fit does yield significant improvements. Furthermore, the estimates could be substantially different when the data quality drops considerably. No weights are taken into consideration at this time.

Biharmonic Spline Wavelets versus Generalized Multi-quadrics for Continuous Surface Representations

In some respect, the continuous representation of surfaces such as the geoid, the topography, or other spatial phenomena, is superior to discrete forms like the TIN (Triangular Irregular Network) or a raster DEM (Digital Elevation Model) as long as these surfaces exhibit a certain degree of local smoothness. In this contribution, we shall concentrate on the special study of biharmonic spline wavelets and of generalized multi-quadrics, with the emphasis on increased efficiency while maintaining the local approximation quality up to the desired resolution.