Issues of Different Estimation Models for Epoch-by-Epoch Double-Difference GPS Observation Equations: A Comparative Study Hendy F. Suhandri 1 , Eugenio Realini 2 1 Institute of Navigation, University of Stuttgart, Germany email: suhandri@nav.uni-stuttgart.de 2 Research Institute for Sustainable Humanosphere, Kyoto University, Japan email: eugenio realini@rish.kyoto-u.ac.jp Abstract —In short-timespan processing of GPS ob- servations the combination of code and carrier phase observations has the shortcoming that the normal matrix is often ill-conditioned and giving unstable computation. The regularized least-squares (RLS) and the iterative least- squares (ILS) methods are often proposed as alternatives to the conventional least-squares (CLS) method. The RLS are claimed to give better reliability of GPS ambiguity solving for short-period observations [1], [6], [18], and to improve the quality of the normal equation, also reducing the condition number of the normal matrix. The regular- ization parameter is determined by minimizing the trace of the mean square error matrix MSEα{ ˆ x}. However, the regularization induces a biased estimation and its benefits are difficult to be confirmed [12]. On the other hand, the ILS do not improve the estimate results and their stochastic properties, apart from stabilizing the normal matrix and reducing its condition number: this, however, depends on the given initial estimate vector. In this work we investigate the performance of RLS and ILS as compared to CLS when applied to single-frequency epoch-by-epoch processing with ambiguity solving by LAMBDA method. Results show that, while the investigated methods do not produce significant differences in terms of estimated baseline precision, improvements are instead observed in the condition number of the normal matrix, with ILS producing the best results when using estimated initial values from CLS. On the other hand, the RLS method fails to improve the condition number for epoch-by-epoch strategy. All methods also give practically equal reliability for ambiguity resolution, where the evaluation is taken in terms of success rate. BIOGRAPHIES Hendy F. Suhandri holds a master’s degree in 2008 from the Geoengine program at the University of Stuttgart, Germany. Since 2009 he has been as a research fellow at the Institute of Navigation, Stuttgart. His research focuses on stochastic modelling of GNSS observation, ambiguity resolution, and kinematic attitude determination. Eugenio Realini received his master’s degree in Envi- ronmental Engineering in 2005 and his Ph.D. in Geodesy and Geomatics in 2009, both from Politecnico di Milano (Italy). His current research interests include precise po- sitioning by using low-cost single-frequency GNSS re- ceivers and the analysis of the troposphere by GNSS observations. He is co-founder and developer of the open source positioning software goGPS (http://www.gogps- project.org). I. I NTRODUCTION High precision GPS positioning requires the use of carrier phase measurements. Once integer phase am- biguities are correctly solved, the carrier phase mea- surement acts as if it were a high precision code. For this reason the integer ambiguity resolution of phase observations is a crucial issue for obtaining precise posi- tioning. Precise and reliable resolution of integer ambigu- ities depends on the receiver-satellites geometry, on the stochastic properties of observations and on the chosen estimation strategies. Any change in those components will also affect the availability and reliability of the solved integer values. When performing epoch-by-epoch observation pro- cessing, using only carrier phase observations leads to rank deficiency. This problem can be solved either by jointly processing multiple epochs (batch processing), or by adding code pseudoranges to the epoch-by-epoch solution. The former method has the shortcoming that the normal equation is often ill-conditioned when dealing with short-timespan observations, due to slowly varying satellite geometry. The latter method, on the other hand, increases the number of observations without adding further unknowns as ambiguity terms, but it may still be ill-conditioned in case of poor satellite geometry, and the added code pseudoranges introduce significant noise. The epoch-by-epoch solution provides also the funda- mental benefit of being applicable in real-time. In this work we investigate two methods for mitigating the ill-conditioned problem of the normal equation: the first involves the use of a Tykhonov-Phillips regularization parameter, which is determined by minimizing the trace of the mean square error matrix MSE α { ˆ x} [1]–[6], [15]; the second is based on the iteration of the normal equation [12], [16], [17], [19]. Both methods are reported to improve the normal equation [6], [12], but from our point of view