The paper presents a suggestion of modification of Otrebski' s theorem for some special structures of geodetic networks. The modification leads to forming up the conditional equation system with unknowns. A new parameter as a global criterion of evaluation of the quality of the networks characterized by an inhomogenous observation system, has been introduced as well.

The paper presents two alternative proposals for processing kinematic modular networks. The first method employs the idea of multi-group transformation which may be reduced to setting up a system of conditional equations with unknowns. The kinematic parameters (point motion velocities) are in this case determined after the observations are adjusted, together with point coordinates. The other proposal is based on the classic idea of the parametric method. The theoretical relationships for functional models of the network adjustment for each of the methods have been provided. The practical conditions have been presented for the application of the proposed models (methods) in constructing detailed computational algorithms. The modular network technology may be an appropriate method of geodetic determination of displacements, especially in difficult terrain conditions (slopes, trees, unfavourable exposition to satellite signals).

The aim of the work is to develop and test an algorithm of adjustment of geodetic observations, resistant to gross errors (method of robust estimations), with the use of the damping function, proposed by the author. Detailed formulae of the damping function as a component of the objective function in a modified classic least squares method were derived. The selection criteria for the controlling parameters of the damping functions have also been provided. The effectiveness of the algorithm has been verified with two numerical examples. The results have been analysed with reference to the methods of resistant compensation, which apply other damping functions, e.g. Hampel's function.

The results of analysis of geometrical structure of modular networks are discussd in the paper. The criteria of technical correctness of such construction were determined. The algebraic relationship between the network components, e.g. station number, tie points, number of measurements, was analysed. The determination conditions for a single module and for a surface network have been introduced considering the existence of elementary modules that are not internally determined. A comparative test for modular and classical models of network was performed using a computer program. The results illustrate positioning accuracy achievable with use of modular networks. The conclusions presented might be helpful when designing surveying networks.

Mathematical torrnulacs for rigorous adjustment of surveying modular networks are presented in the paper. The method is based on the idea of multigroup transformation. The solution leads to general problem of non-I i near adjusting of conditional equations with unknowns by the least square method. Il is a modification or the classical way of approximate adjustment of the modular networks.

The geodesy literature seems to offer comprehensive insight into the planar Helmert transformation with Hausbrandt corrections. Specialist literature is mainly devoted to the issues of 3D transformation. The determination of the sought values, coordinates in the target system, requires two stages of computations. The classical approach yields ‘new’ coordinates of reference points in the target system that should not be changed in principle. This requires the Hausbrandt corrections. The paper proposes to simplify the two-stage process of planar transformation by assigning adjustment corrections to reference point coordinates in the source system. This approach requires solving the Helmert transformation by adjusting conditioned observations with unknowns. This yielded transformation results consistent with the classical method. The proposed algorithm avoided the issue of correcting the official coordinates of the control network and using additional (post-transformation) corrections for the transformed points. The proposed algorithm for solving the plane Helmert transformation for �� > 2 reference points simplifies the computing stages compared to the classical approach. The assignment of adjustment corrections to coordinates of reference points in the source system helps avoid correcting coordinates of the reference points in the target system and additional corrections for transformed points. The main goal of any data adjustment process with the use of the least squares method is (by definition) obtaining unambiguous quantities that would strictly meet the mathematical relationships between them. Therefore, this work aims to show such a transformation adjusting procedure, so that no additional computational activities related to the correction of already aligned results are necessary.