# Evaluation of statistical models of precise point positioning based on satellites elevation angles

Document Type : Research Article

Authors

1 Assistant Professor, Department of Surveying and Geomatics Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

2 Professor, Department of Surveying and Geomatics Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

3 Ph.D. Student Department of Surveying and Geomatics Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

Abstract

Due to the importance of using an accurate stochastic model in estimation of the coordinates of points using the GNSS observation method, this study investigates the role of the elevation angle-dependent stochastic model. The least-squares estimation method is usually used in processing GNSS observations. This method requires the use of two essential models, one is functional model and the other is stochastic model. The functional model illustrates the relationship between observations and unknown parameters. The stochastic model presents the covariance matrix and the statistical properties (expectation) and dispersion of errors in observation, which expresses the accuracy and the correlation between the types of observations. A precise and detailed stochastic model for observations, expresses the receiver's internal noise, residual errors, and the correlation between the variables. Moreover, by choosing a suitable stochastic model, we can provide the necessary preconditions for solving the reliable phase ambiguity and precise positioning. In this study, we investigate and compare 9 stochastic models based on the satellite elevation angle. These 9 models are expressed as equations of four families of trigonometric functions , , improved trigonometric functions, and exponential functions. To do this, we use observations of a single point in two time epoch where simulated displacement was applied to it very precisely by the device. First, by using precise point positioning method, the horizontal coordinates of the point in two epochs were estimated by using 9 stochastic models. According to the accomplished comparison, we present the closest estimated value to the simulated real value of the stochastic models which are trigonometric functions , improved trigonometric functions and exponential functions respectively. Among these four models, The results of exponential function is closest to the simulated real value. Online services are then used to process point-of-view observations, according to which the two OPUS and AUSPOS services are most closest to the simulated real observations. Then the estimation of the accuracy of the horizontal components is examined by means of 9 presented stochastic models. According to the presented results, the highest accuracy and least error are related to the use of stochastic model  (improved trigonometric functions). Then, two stochastic models and matrix weights  and  (exponential functions) showed the highest accuracy. Using these three models with the highest accuracy, the average accuracy obtained for the East component is between 0.03 mm and 2.8 mm and for the North component is between 0.04 mm and 3.1 mm. In the next section, due to the accuracy obtained for the horizontal coordinate components in all epochs (sampling interval of 5 s) using these three stochastic models, fewer epochs are required to reach the level of accuracy of Dosimeter, Centimeter and Millimeter. In such a way that, for desired point, 277 epochs for the East component and 405 epochs for the North component are required to reach the millimeter precision level. Finally, considering the 5cm convergence condition for the horizontal components East and North, due to the three models used, this convergence condition is achievable after 5 minutes and 50 seconds for the East component and after 8 minutes and 5 seconds for the component North.

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