Mitigation of Ionospheric Errors by Penalised Least Squares Technique for Hight Precision Medium Distances GPS Positioning

Ionospheric errors are a major systematic error source for high precision medium distance GPS positioning. Ionospheric errors impair the precision and reliability of both the ambiguity resolution and estimated site coordinates. In this paper, the vector semiparametric and penalised least squares technique is used for mitigation of ionospheric errors to improve the ambiguity resolution and site coordinates. The main point of the vector semiparametric and penalised least squares technique is that ionospheric errors are modelled as systematic error functions which vary smoothly with time (experiments have shown that ionospheric errors change smoothly with time from epoch to epoch). Ionospheric systematic error functions, ambiguities and site coordinates are estimated simultaneously. As a result, ambiguities and site coordinates are estimated with a better reliability and accuracy than with conventional least squares, but over much shorter time spans. Two medium distance GPS baselines (72 km and 86 km) have been processed to demonstrate the potential of this technique. INTRODUCTION Since GPS signals propagate from the GPS satellite to the GPS receiver through the atmosphere rather than through a vacuum, their speed and direction of propagation can vary with time. The ionosphere is the atmospheric layer encountered by the GPS signal which is charged with a high number of free electrons that refract the GPS signal. As the ionosphere is a dispersive medium, the resulting delay depends on the signal frequency. Another important fact is that the ionosphere advances carrier phase signals and delays code signals. Ionosphere errors are a major error source for high precision medium distance positioning, which not only can hamper the reliable resolution of ambiguities, but also can degrade the accuracy of estimated baseline solutions. Several techniques have been developed for the mitigation of ionospheric errors to improve ambiguity resolution and the determination of site coordinates. One such technique is the ionosphere-free linear combination of L1 and L2 observations, which uses the frequency-dependent characteristics of ionospheric propagation to eliminate the delay (e.g. Seeber, 1993; Leick, 1995; Hofmann-Wellenhof et al.,1997). However, the ionosphere-free linear combination only eliminates first order ionospheric errors. Residual ionospheric errors can have a magnitude of several centimetres (Brunner and Gu, 1991; Bassiri and Hajj, 1993). In addition, residual tropospheric errors, in particular the wet component, still exist in the combined data. Furthermore, the ionosphere-free linear combination can seriously amplify any multipath errors that may be present in the raw GPS data. An expensive and complex, but more reliable alternative for ionospheric error reduction is to compute ionosphere error corrections using a reference station network (e.g. Wanninger, 1997; Han and Rizos, 1997; Raquet, 1998; Odijk et al., 2000; Schaer et al., 2000). Prerequisites for application of a reference station network are that the accurate coordinates of the reference stations themselves must to be known, and that the inter-station ambiguities of the network must be reliably determined. In this paper, a vector semiparametric and penalised least squares technique is used for mitigation of ionosphere errors. The double-differenced dual frequency carrier phase and pseudorange are the basic observables for this study. SEMIPARAMETRIC AND PENALISED LEAST SQUARES TECHNIQUE A semiparametric model is a model in which estimated variables are divided into two parts, namely, the parametric part and the nonparametric part. The latter is described by a set of smoothing functions varying with time. Usually, the parametric part is of interest to the users. In the case of GPS, the parametric part can be the parameters of the carrier phase ambiguities and site coordinates. The nonparametric part can represent a combination of any error functions that smoothly vary with time. A vector semiparametric model can be expressed as n i t g N x A y i i i i i ... , 1 , ) ( = + + = ε (1) j i E N j i i i ≠ = ∑ , 0 ) ( ), , 0 ( ~ ε ε ε (2) where n is the number of epochs; m i i y R ∈ , ε are the random errors and the observations, respectively, at ith epoch; m is the number of the observations per epoch; q m i N × R ∈ is the incidence matrix (Green and Silverman, 1994) consisting of the coefficients of the ionosphere error functions (Rothacher and Mervart, 1996); q i t g R ∈ ) ( are the ionospheric error functions; q is the number of the error functions; i t is the time index; p m i p A x × R ∈ R ∈ , are the estimated parameters including the carrier phase ambiguities and site coordinates, and the design matrix, respectively; p is the number of the estimated parameters; i ∑ denotes the random observation error covariance matrix. Equation (1) contains n m× observations and p n q + × unknowns. Two cases can occur for such an equation system. First, the number of the unknowns is larger than the number of the observations. In this case, equation (1) cannot be solved by using the traditional least squares technique. Second, even if the number of the unknowns is less than the number of the observations, equation (1) cannot provide a stable solution when the traditional least squares technique is used due to the number of unknowns. In order to obtain reliable resolution, a penalising term is often added. To achieve this purpose a regularisation procedure can be used (Moritz, 1989). Here, a natural cubic spline (Green and Silverman, 1994) is used for the regularisation. The merit of this regularisation procedure is that the estimated error functions vary smoothly with time. This is the case when comparing the ionosphere error change with the random observation errors even during severe ionospheric disturbances (Rothacher and Mervart, 1996; Wanninger, 1997). The regularised least squares technique is called the penalised least square technique, which can be expressed as min )) ( ( )) ( ( )) ( (