ION GNSS 2012
Session E6: Precise Point Positioning 2

Title: Satellite Bias Determination with Global Station Network and Best-Integer Equivariant Estimation
Author(s): Z. Wen, P. Henkel and C. Gunther, Technische Universitaet Muenchen, Germany
Date/Time: Friday, September 21, 2012, 2:12 p.m.
Room: 209/210 (NCC)

For Precise Point Positioning, the tropospheric delay is in general modeled as a mapping function times a zenith delay. The Global mapping function (GMP), the Niell mapping function (NMP) and the Vienna mapping function (VMP) are widely used and assumed to be unbiased, i.e. errors of these mapping functions are typically not considered although they degrade ambiguity resolution especially for low elevation satellites.
In this paper, the statistical distribution of tropospheric mapping function errors is determined, i.e. the GMP, VMP and NMP are compared to the mapping function based on tropospheric slant and zenith delays from raytracing with radiosonde measurements. The latter ones were taken from the atmospheric sounding maps as provided by the University of Wyoming [1]. These maps include vertical temperature, pressure and humidity information every few hundred meters for one up to ten radiosondes per state in the US and per country in Europe. The statistics of the tropospheric mapping function are then included in the precise point positioning.
A Best-Integer Equivariant (BIE) estimator [2] is used to estimate the receiver position, tropospheric zenith delay and integer ambiguities with an ionosphere-free linear combination. The BIE estimator is a Minimum Mean Square Error (MMSE) estimator, which is always better than both the float and fixed integer solution, and can be efficiently implemented by a small set of marginal distributions for the most likely integer candidates [2],[3]. The BIE estimators represent a class of integer estimators that only impose the integer-remove-restore property, i.e. which means that shifting the float solution by an integer amount always shifts the fixed solution by the same amount. Classical inter least-squares estimators (e.g. the LAMBDA method) additionally require that the set union of all pull-in regions does not leave any gaps and that there is no overlapping of pull-in regions. As the BIE estimator does not require these conditions, it also includes the classical integer-least squares estimation and integer aperture estimation as special cases. The BIE estimator achieves significantly more accurate position and tropospheric zenith delay estimates than the classical fixed integer least-squares solution, which considers the integer estimates as deterministic while they are actually stochastic.
In this paper, we use the BIE estimator also for the generation of undifferenced phase bias corrections. As the estimation of one individual phase bias per satellite and per receiver and of one ambiguity per satellite-receiver link is not feasible, a parameter mapping is applied as proposed by Ge et al [4]. This mapping splits the RK undifferenced ambiguities (for R receivers and K satellites) into two subsets: a subset of (R-1)(K-1) double differenced integer ambiguities and a second subset of RK-(R-1)(K-1)=R+K-1 real-valued ambiguities, which can also be interpreted as R receiver phase biases and K-1 satellite phase biases [4]. The estimation of undifferenced phase and code biases is performed by a dual stage Kalman filter. In the first stage, the orbital error, receiver and satellite clock offsets, and tropospheric delay are mapped to a single geometry term, which is estimated jointly with the ionospheric delay, double differenced integer ambiguities and undifferenced phase biases. The mapping of the non-dispersive terms to a single geometry term enables a fast integer ambiguity fixing. In the second stage, these a posteriori geometry estimates are refined, i.e. satellite orbit corrections, receiver clock offsets, satellite code biases (clock corrections) and tropospheric zenith delays are estimated by another Kalman filter. As the first Kalman filter has introduced some temporal correlations, a generalized Kalman filter for colored measurement noise has to be used in the second stage. A pseudo-time differencing of measurements and a subsequent decorrelation of measurements and process states is performed to whiten the measurements [5].
Measurement analysis has shown that the proposed dual stage Kalman filtering with an MMSE BIE estimation of carrier phase integer ambiguities substantially improves the convergence speed of precise point positioning.

[1] Atmospheric sounding maps: Vertical pressure, temperature and humidity profiles based on radiosonde measurements, University of Wyoming: http://weather.uwyo.edu/upperair/europe.html, called on March 1, 2012.
[2] Sandra Verhagen, The GNSS integer ambiguities: estimation and validation, PhD Thesis, TU Delft, 2004.
[3] P. Teunissen, GNSS Best Integer Equivariant Estimation, Proc. of Intern. Assoc. of Geodesy (IAG) Symp., 2005, Volume 128, Symposium G04, pp. 422-427, 2005.
[4] X. Zou, W.-M. Tang, M. Ge, J. Liu and H. Cai, A New Network RTK based on Transparent Reference Selection in Absolute Positioning Mode, Journal of Surveying Engineering, submitted, 2010.
[5] Bryson and Henrikson, A. E. Bryson, Jr. and L. J. Henrikson, Estimation using sampled-data containing sequentially correlated noise. J. of Spacecraft and Rockets, 5(6), 662-665, 1968.

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