Weighting GPS Dual Frequency Observations: Bearing the Cross of Cross-Correlation P.J.G. Teunissen, N.F. Jonkman, C.C.J.M. Tiberius Department of Mathematical Geodesy and Positioning Delft University of Technology The Netherlands e-mail: mgp@geo.tudelft.nl ABSTRACT A proper choice of the observation weight matrix is of importance for both adjusting and testing GPS data. Our understanding of the noise characteristics of GPS observations, on which the weight matrix should be based, is however still underdeveloped. This makes it difficult to draw up an appropriate weight matrix. The first and foremost purpose of this contribution is therefore to draw attention to the need to improve upon our rudimentary knowledge of the GPS stochastic model. To this end, results will be presented of a relatively simple casestudy in which the possible presence of cross-correlation between observables is considered. With these results we hope to spur further discussion and research on this important topic. 1. INTRODUCTION GPS data are usually processed with algorithms based on the least-squares principle. In order to apply the least-squares principle one needs to specify both the observation equations and the observation weights. The observation equations link the GPS observables, like pseudo ranges and carrier phases, to the unknown parameters, such as, for example, baseline coordinates, carrier phase ambiguities and atmospheric delays. The observation weights, which are collected in a weight matrix, allow one to specify by how much the individual observations should contribute to the overall solution. For instance, it is sensible to give lower weights to the noisier observations and higher weights to the less noisy observations. The choice of weights is optimal when the weight matrix equals the inverse of the variance-covariance (vc-) matrix of the observations. In that case the balance between the relative weights is such that the best possible precision is obtained in the computed solution. The GPS observation equations are (sufficiently) known and well documented. However, the same can not be said of the vc-matrix of the GPS observations. In the many GPS-textbooks available, one will usually find only a few comments, if any, on the vc-matrix of the GPS observations. Also advertisement or data sheets for GPS receivers are usually vague in their specifications of the precision characteristics of the data outputted by the receiver. Due to this lack of information in the public domain, most of us are probably inclined to start with the simplest weight matrix possible, a scaled unit matrix for instance per observation type (pseudo ranges and carrier phases). Such a choice may however be an oversimplification that fails to do justice to the more complicated noise characteristics of the data. A proper choice of the vc-matrix is of relevance for all subsequent stages of data processing. The least-squares solution for instance, will loose its property of ‘minimum variance’ when a misspecified vc-matrix is used. In addition, the detection power of the statistical tests, employed for model validation and quality control (e.g. outliers and cycle-slips), will become smaller when the noise characteristics are not properly taken into account. And finally, the a posteriori quality description of the computed results will also be affected when mispecified or oversimplified vc-matrices are used. At present research into the stochastic model of GPS observables is still in its infancy. Only a few studies have been reported in the literature. Examples are [Euler and Goad, 1991], [Jin and de Jong, 1996], [Gerdan, 1995] and [Gianniou, 1996], who studied the elevation dependence of the observation variances, and [Jonkman, 1998] and [Tiberius, 1998], who considered time-correlation and cross-correlation of the pseudo ranges and carrier phases as well. A systematic study of the stochastic model is of course far from trivial. Not only do the noise characteristics depend on the mechanization of the measurement process, and therefore on the make and type of the receiver used, but the residual terms which are not captured by the observation equations, such as environmental effects, will also have their influence. Despite these difficulties though, we believe that the time has come to put more effort into the stochastic