Improving the Accuracy of Static GPS Positioning with a New Stochastic Modelling Procedure

For high precision static GPS positioning applications, carrier phase measurements have to be processed. It is well known that there are two important aspects to the optimal processing of GPS measurements: the definition of the functional model, and the associated stochastic model. These two models must be correctly defined in order to achieve high reliability in the positioning results. The functional model is nowadays sufficiently known, however the definition of the stochastic model still remains a challenging research topic. Previous studies have shown that the GPS measurements have a heteroscedastic, spaceand time-correlated error structure. Therefore, a realistic stochastic modelling procedure should take all of these error features into account. In this paper, a new stochastic modelling procedure is introduced. This procedure also takes into account the temporal correlations in the GPS measurements. To demonstrate its performance, both simulated and real data sets for short to medium length baselines have been analysed. The results indicate that the accuracy of GPS results can be improved to the millimetre level. INTRODUCTION GPS carrier phase measurements are extensively used for all high precision static and kinematic positioning applications. The least-squares estimation technique is usually employed in the data processing step, and basically requires the definition of two models: (a) the functional model, and (b) the stochastic model. The functional model describes the mathematical relationship between the GPS observations and the unknown parameters, while the stochastic model describes the statistical characteristics of the GPS observations (see, eg., Leick, 1995; Rizos 1997; and other texts). The stochastic model is therefore dependent on the selection of the functional model. A double-differencing technique is commonly used for constructing the functional model as it can eliminate many of the troublesome GPS biases, such as the atmospheric biases, the receiver and satellite clock biases, and so on. However, some unmodelled biases still remain in the GPS observables, even after such data differencing. Many researchers have emphasised the importance of the stochastic model, especially for high accuracy applications, for example, Barnes et al. (1998), Cross et al. (1994), Han (1997), Teunissen (1997), Wang (1998), Wang et. al. (2001) for both the static and kinematic positioning applications. In principle it is possible to further improve the accuracy and reliability of GPS results through an enhancement of the stochastic model. Previous studies have shown that GPS measurements have a heteroscedastic, spaceand timecorrelated error structure (eg., Wang 1998; Wang et al., 1998a). The challenge is to find a way to realistically incorporate such information into the stochastic model. This paper deals only with the static positioning case. Several stochastic modelling techniques have recently been proposed to accommodate the heteroscedastic behaviour of GPS observations. Some are based on the signal-to-noise (SNR) ratio model (eg., Barnes et al., 1998; Brunner et al., 1999; Hartinger & Brunner, 1998; Lau & Mok, 1999; Talbot, 1988), others use a satellite elevation dependent approach (eg., Euler & Goad, 1991; Gerdan, 1995; Han, 1997; Jin, 1996; Rizos et al., 1997). Both of these techniques are equally applicable to static and kinematic GPS positioning. The more rigorous Minimum Norm Quadratic Unbiased Estimation (MINQUE) procedure has been suggested by Wang et al. (1998a), but its applicability is generally restricted to the static positioning case. A comparative study of two GPS quality indicators, namely the SNR and the satellite elevation, has been reported in Satirapod & Wang (2000). This study revealed that in some cases both the use of SNR and satellite elevation information failed to reflect the true data quality. In such cases it is necessary to consider a more rigorous statistical method such as the MINQUE technique for estimating the stochastic model of GPS observations. All these methods, however, do not take into account temporal correlations. An exponential function was empirically derived by ElRabbany (1994) in an attempt to model the temporal correlations in GPS observations. Han & Rizos (1995) and Howind et al. (1999) investigated the effect of temporal correlations on the accuracy, and the repeatability, of static receiver coordinates. These investigations assumed that all one-way GPS measurements have the same variance and same temporal correlations. It has been shown that different satellites have different temporal correlation coefficients (Wang, 1998), hence it is not appropriate to make such assumptions. An iterative stochastic assessment procedure, which takes into account all the error features, has been proposed by Wang et al. (2001). In the following sections, a brief description of the iterative stochastic modelling procedure is given. A new procedure is then proposed to deal with long observation period data sets, and in order to reduce the computational load. The effectiveness of the new procedure is tested using both real data and simulated data sets for short to medium length baselines. ITERATIVE STOCHASTIC MODELLING PROCEDURE The basic procedure requires the double-differenced (DD) carrier phase measurements to be transformed into a set of new measurements using estimated temporal correlation coefficients. The transformed measurements are free of temporal correlations and thus have a block diagonal variance-covariance matrix (Wang, 1998). Consequently, the immense memory usage and computational load for the inversion of a fully populated variance-covariance matrix can be avoided, and the variance-covariance matrix for the transformed measurements can be estimated using a rigorous statistical method such as the MINQUE method (Rao, 1971). An iterative process is performed until sufficient accuracy is achieved, as illustrated in Figure 1. A detailed description of the iterative stochastic modelling procedure can be found in Wang et al. (2001). Figure 1. Flowchart of the iterative stochastic modelling procedure. This procedure is suitable for short observation periods as it assumes that the temporal correlation coefficients and the variance of GPS measurements are constant for the whole observation period. Initial experiments based on this procedure have demonstrated encouraging results in the case of short observation periods and for short baselines (Wang et al., 2001). In order to deal with long observation periods, the following issues need to be addressed: 1) The assumption that the temporal correlation coefficients and the variance of GPS measurements are constant for the whole observation period is not realistic. 2) In practice, an observation period of several hours may be expected for some applications. Thus the memory usage and computational load can be unbearable when the standard MINQUE technique is applied. Establish the functional and the default stochastic models stochastimodelstochasticdelmod el GPS data reduction Compute the residuals of the original measurements Estimate temporal correlation coefficients Transform the measurements Estimate the unknown parameters for both functional and stochastic models using the transformed measurements Is the accuracy sufficient?