On the Realistic Stochastic Model of GPS Observables: Implementation and Performance

Abstract

ABSTRACT:High-precision GPS positioning requires a realistic stochastic model of observables. A realistic GPS stochastic model of observables should take into account different variances for different observation types, correlations among different observables, the satellite elevation dependence of observables precision, and the temporal correlation of observables. Least-squares variance component estimation (LS-VCE) is applied to GPS observables using the geometry-based observation model (GBOM). To model the satellite elevation dependent of GPS observables precision, an exponential model depending on the elevation angles of the satellites are also employed. Temporal correlation of the GPS observables is modelled by using a first-order autoregressive noise model. An important step in the high-precision GPS positioning is double difference integer ambiguity resolution (IAR). The fraction or percentage of success among a number of integer ambiguity fixing is called the success rate. A realistic estimation of the GNSS observables covariance matrix plays an important role in the IAR. We consider the ambiguity resolution success rate for two cases, namely a nominal and a realistic stochastic model of the GPS observables using two GPS data sets collected by the Trimble R8 receiver. The results confirm that applying a more realistic stochastic model can significantly improve the IAR success rate on individual frequencies, either on L1 or on L2. An improvement of 20% was achieved to the empirical success rate results. The results also indicate that introducing the realistic stochastic model leads to a larger standard deviation for the baseline components by a factor of about 2.6 on the data sets considered.* Corresponding author INTRODCTIONGPS data processing is usually performed by the least squares adjustment method for which both the functional and stochastic models must be correctly specified. The functional model, describing the mathematical relation between the observations and the unknown parameters, is usually well known either in the relative positioning or in the single/precise point positioning (Seeber, 2003;Hofmann-Wellenhof et al., 2008;Leick, 2004; Teunissen and Kleusberg, 1998;Rizos, 1997). However, the stochastic model, expressing the statistical characteristics of the GPS observations by means of a covariance matrix, is still a challenging problem. Misspecification in the stochastic model leads to unreliable and suboptimal estimates. The realistic stochastic model should thus be utilized aiming at obtaining reliable least squares estimates.There are a few common stochastic models for GPS observables: (1) Equal-weight stochastic model in which identical variances for each measurement type are chosen (i.e. the same variance for the code and the same variance for the phase observations). This structure ignores the correlations among different observables, (2) Satellite elevation-angle dependent model in which the observations weighting procedure is performed using trigonometric or exponential functions. T...

Positioning accuracy improvement in high‐speed GPS receivers using sequential extended Kalman filter

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Improvement of Multi-GNSS Precision and Success Rate Using Realistic Stochastic Model of Observations

“…For more information, refer to [32,33]. The reliability of the resolved ambiguities on the stochastic model was investigated in [34] by comparing the IAR success rate when a nominal and realistic stochastic model for the GPS observables is considered. It was shown that when a realistic stochastic model was used, the IAR success rate on individual frequencies improved by 20%, whether on L1 or L2.…”

“…Moreover, the weight matrix of the observables is taken into account to affect the estimated baseline components and their uncertainties. As a result, the estimated baseline uncertainties were evaluated in [34] using a nominal and realistic stochastic model of the GPS observables. The findings demonstrate that employing realistic stochastic model results in realistic baseline component uncertainty.…”

A New Variance-Covariance Matrix for Improving Positioning Accuracy in High-Speed GPS Receivers

“…The stochastic model represents the statistical characteristics of GPS observables using a variance-covariance matrix, and its misspecification leads to unreliable and non-optimal estimates. The stochastic models for GPS observations can be divided into three general categories: (1) equal-weight models in which the identical variances are selected, (2) elevation-based models in which the weighting of observations is performed, using exponential or trigonometric functions, and (3) SNR-based models in which the weighting of observations depends on the SNR values [11].…”

“…So far, various methods have been proposed to determine the VCM. In these methods, the VCM matrix is either defined as diagonal, which results in less accurate positioning, or is defined as fully populated, which results in an excessive increase in the computational load in each iteration of the positioning algorithm [11,[13][14][15].…”