4.1. INTRODUCTION

This part of the Polish National Report on Geodesy is the quadrennial report of works on theory, evaluation and methodology performed in Poland in a period from 1995 to 1998. It summarises investigations such as deformation analysis, least squares theory and evaluation, research on GPS, etc. These activities were conducted mainly at the following research centres listed in an alphabetic order:

• Department of Geodesy, University of Agriculture in Cracow;
• Department of Geodesy and Photogrammetry, Agricultural University in Wroclaw;
• Department of Mining Geodesy and Environmental Engineering, Academy of Mining and Metallurgy in Cracow;
• Department of Planetary Geodesy, Space Research Centre, Polish Academy of Sciences in Warsaw;
• Institute of Geodesy, Olsztyn University of Agriculture and Technology;
• Institute of Geodesy and Geodetic Astronomy, Warsaw University of Technology.

2. DEFORMATION ANALYSIS

 

 

New procedures for modelling and processing kinematic and quasi-kinematic networks in application to deformation analysis were presented (Kadaj and Plewako, 1996). An examination of the problem led to a definition of kinematic vector fields as 4D deformation models for investigated objects.

A generalised algorithm for direct determining displacement components by the method of differences was presented (Beluch, 1997). It consisted also of a critical analysis of previous presentations of the method.

Electronic measurements allow for station positioning by means of the so-called free station technology. Equations for determination a covariance matrix for two separate cases of intersection were derived (Beluch, 1998). The obtained average errors differ from their expectations resulting from the geometry of the intersection.

Two kinematic models of adjustment of a vertical control network in areas of intensive vertical movements were investigated (Beluch and Plewako, 1995). In order to overcome the singularity of the system the authors introduced the Meissl’s condition of free adjustment, as well as an additional pseudo-observation equation. It was shown that an improper assumption of the point motion might cause a significant blunder in the results.

A generalised model of displacements and deformations was given (Czaja, 1996). Minimisation of the quadratic form is executed for random deviations, as well as for the constraints preserving continuity of deformations. An extended theoretical basis of the problem was given (Czaja, 1997).

3. LEAST SQUARES – THEORY AND EVALUATION

 

 

An algorithm for processing observations with the use of a generalised matrix inverse was presented (Pietruszka, 1995). It allowed a free choice of an optimal co-ordinate system. The analysis of accuracy was based on the matrix eigenvalues.

General concept of the adjustment method using Edgewood series was presented (Dumalski and Wiśniewski, 1995). The series approximates a certain class of probability distributions and the proposed method can then replace the set of adjustment methods derived with application of specific probabilistic model observation errors.

Definitions of the reduced vectors of the second, third and fourth order moments were given (Wiśniewski, 1995, 1996a). The vectors concerned mutually independent random variables. Also, the propagation laws of the defined vectors were given. The major goal of the study was to present an estimation method of the moment vectors after the LS adjustment. The estimation was carried out under the assumption that the preliminary estimations of the third and fourth order moments were known. Implementation of the proposed method would facilitate practical applications of such adjustment methods in which knowledge of values of the higher order moments (besides of the variances) would be required.

Relationship between the variance estimate and the matrix of excess coefficients was given (Wiśniewski, 1996b). It made it possible to assess the effectiveness of the analysed estimate with respect to the excess values of the set of observations to be adjusted. There were also derived the relationships to determine critical and admissible values of the excess. Moreover, some optimisation problems connected with the determination of the variance in the class of the ?-locally minimum variance quadratic unbiased invariant estimators were investigated (Wi?niewski, 1998).

The method of estimation according to Henderson was applied to determine local variances (Duchnowski, 1996). The main purpose of the study was to estimate local variances in 3D networks.

The alternative forms of commonly used adjustment algorithms were elaborated (Oszczak, 1996) taking advantage of the introduced by author symbols of generalised Kalman weighting matrices.

The sequential estimation algorithm has a compact and simple form and can be used both for batch and sequential processing procedure of GPS observations.

Matrix expressions for minimum semi-norm least-squares solutions and the related inverse were investigated (Prószyński and Sosnowski, 1995). The expressions had initially been derived for the needs of data analysis in geodetic monitoring of relative movements in engineering structures. The mathematical models used there were linearised rank–deficient systems solved by least squares method. The solutions are often required to be of minimum l² – norm for a specified sub-vector of unknowns.

The initial approach was later expanded into a more general problem of seeking the minimum N – semi-norm M – least squares solutions (with the matrices N and M positive semi-definite) and determining an inverse to generate those solutions.

The derivations were based on the Moor–Penrose inverse, providing a clearly interpretable structure of the matrix expressions as well as enabling a prior analysis to detect areas of non-unique solutions. With the auxiliary assumption of minimum l² – norm for the full solution vectors a unique minimum N – semi-norm M – least squares inverse was defined.

An algorithm for calculating the rms of a function C of two sets of variables: (1) the set of s unknowns x calculated from the system of observation equations with known rms m, and (2) the set of r known variables w, for which some of rms were also known was derived (Skórczyński, 1997a). The idea of the solution consisted in the assumption that all w variables are observations. Hence r additional observation equations  with appropriate weights were obtained. Then the inverse of the (regular) matrix (s+r)×(s+r) of parametric normal equations allows for calculating the rms of C.

The exact practical formula for the rms of a linear misclosure of a traverse with initial azimuth calculated from co-ordinates was derived (Skórczyński, 1997b). It was used to show substantial influence of the misclosure’s azimuth onto misclosure’s rms.

The robustness potential of the least-squares estimation with geodetic illustration was investigated (Prószyński, 1997). The structure of the projection operator transforming the vector of standardised observations onto the vector of standardised residuals was analysed. On this basis the properties of the model responses to observational disturbances (i.e. blunders) were derived. A final outcome of the research was: (1) suggesting characteristic quantities of the robustness of a given model and linking them to the local measures of internal reliability, and (2) determining the internal reliability levels satisfying the specified requirements for the robustness (i.e. possibility of detecting at least one of the k observational disturbances). The theory and a numerical example showed that a system properly designed, with respect to a given level of internal reliability, could provide accordingly high level of robustness.

A comparative analysis of several approaches to reliability measures for correlated observations was given (Prószyński, 1998). The approaches were compared on the basis of numerical examples used in recent publications. An attempt was made to explain the discrepancies in the results. The conclusions contained possible directives for future investigation of the problem.

A perspective modelling of the adjustment and maintenance of the Polish levelling system was studied (Pachelski, 1998). The Polish national levelling system consists of 1^st and 2^nd class precise levelling networks and amounts to over 43 thousand benchmarks. The international EUVN programme assumes, among others, its integration with the ETRS'89 system, the last one realised through GPS measurements on the EUREF network.

The above task was considered to be solved by a simultaneous (“single batch”) adjustment of both classes of the network linked (constrained) to a number of GPS reference points, as well as by its possible further systematic maintenance through repeated GPS, precise levelling and gravity measurements. That caused certain numerical problems reflected in the evident ill conditioning and large dimensions of matrices, cumulating round-off errors, possible instability of the solution, and others. To overcome the problems it was suggested to use the so-called sequential adjustment algorithm applying the sparse matrix technique. Its basic properties consisted in high numerical stability (no matrix inversion performed), possible on-line control of the covariance matrix in the course of the one-by-one processing of observation equations, and possible augmenting (as well as reducing) the number of unknowns.

The study gave a basic idea of the sequential algorithm for this task. It also presented results of its tests on a simulated levelling system, including its performance in the course of updating the system and estimating possible inaccuracies due to the used sparse matrix technique. Positive results indicated possible application of the approach also in other types of geodetic control systems.
 
 

4. RESEARCH ON GPS

A concept of single epoch ambiguity and slip resolution applied to undifferenced GPS phases was presented (Pachelski, 1996). According to it an observation equation of the GPS carrier phase contains a bias  independent of epoch and specific for a station-satellite pair, in which  is an integer ambiguity and ?^s(t₀), ?_r(t₀) are transmitter and receiver initial phases. Through sequential processing of phases we update estimates in each epoch, provided specific minimal configurations of satellites, stations and already processed epochs are satisfied. All second differences of the phases with respect to a given reference satellite and reference receiver, , should be then integers on each L1 and L2 band. These conditions can be solved for all ’s (thus implying new-values) about current estimates of the ’s as soon as the integer values are found by means of a proper search procedure.

Cycle slips come into view as outliers of observations caused by rapid changes of particular  values. In that case a new observation sequence is created, for which new ?-parameters are estimated, and then consequently constrained for ambiguities.

A series of works concerning integration of the total station and GPS measurements in real time were published (Osada, 1996, 1998; Osada and Trojanowicz, 1998; Osada et al., 1997: Bosy, 1996; Bosy and Osada, 1997).

Problems of the GPS levelling, updating and optimal designing were investigated (Osada, 1998). Height and local gravity models in GPS network were discussed (Osada and Trojanowicz, 1996; Osada, 1997; Osada and Bosy. 1995).

A strategy of GPS data evaluation for local geodynamic polygons was presented (Bosy and Kontny, 1998).
 
 

References