The article discusses the features of constructing predictive mathematical models for studying the deformation process of displacements of the crest of the dam of the Sayano-Shushenskaya hydroelectric power station. Studies of the most successfully designed predictive mathematical models for various combinations of influencing factors, including the results of field observations and the calculated values of temperature components of the displacements, were performed. Based on the constructed models, the forecasting of the controlled points of the dam body was carried out for the stages of its operation that were different in time.

Assessment of the operational state of a hydraulic structure and its technical safety should be carried out by comparing the obtained quantitative and qualitative diagnostic indicators with their criteria values. For this purpose, predictive mathematical models of the structure's behavior should be developed, which are recommended to be calibrated according to field observations. The article considers features of constructing predictive mathematical models for studying deformation process of displacements of the dam crest of the Sayano-Shushenskaya hydroelectric power plant. For various combinations of input influencing factors, including the results of field observations and calculated values of component displacements, the most successfully designed predictive mathematical models were studied, on the basis of which the dam body points were forecasted for stages of its operation in different times. The advantages of using the created forecast models for various temperature conditions of the structure (medium, warm and cold temperatures of year) are presented.

The article presents the results of the initial data preparation for the construction of a predictive mathematical model in the study of the deformation process of a high-rise building. The possibilities of forming statistically homogeneous groups of realizations using graphs constructed in the software product Surfer and analytical calculations using the coefficients of variation, calculation of Mahalanobis distances and Hotteling criterion when grouping statistically uniform observation cycles have been shown. The forecast of possible precipitations with the use of the constructed predictive model was performed.

The article discusses the possibility of using mathematical modeling to assess the state of the main technological equipment of the chamber for receiving and launching cleaning and diagnostic tools (PPC) to predict possible changes in the position of equipment elements from the corresponding design values. Based on the completed planned cycles of geodetic observations, based on the measurement results, calculations were made to build predictive mathematical models for the subsequent study of the deformation processes of the main technological equipment of the chamber for receiving and launching cleaning and diagnostic tools. The mechanisms of sequential processing of the measurement results for the sample belonging to the law of normal distribution by an approximate method and fur-ther data preparation by forming statistically homogeneous groups of process implementations in order to build predictive mathematical models are shown. According to the prediction results, deformation marks were identified, for which the process of deformation proceeds with a higher rate of upsetting in comparison with other marks. Recommendations for further geodetic observations are proposed.

Owing to the widespread use of GNSS technologies in geodetic practice, the problem arises of transition from rectangular spatial coordinates of points to spatial geodetic coordinates, which are necessary for the transition to flat rectangular coordinates in the Gauss-Kruger projection. The authors proposed five algorithms for converting rectangular coordinates of points in the plane of the meridian ellipse into geodetic heights and latitudes. The first two algorithms are geometrically related to the intersection point of the ellipse with the normal passing through the point at which the rectangular spatial coordinates were obtained. The formulas of the other three algorithms are based on the geometric relationships of the point of intersection of the meridian ellipse with the straight line connecting the point with the center of curvature of the meridian. As a result of the experiments, deviations of the calculated latitudes and heights from the reference values of the given grid of geodetic coordinates were obtained. The formulas were tested not only for points under and on the earth's surface, but also outside the earth at different heights up to an altitude of 20,000 km.