GNSS Solutions
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A precise observation should have GNSS a higher weight and thus contribute more to the computed parameters than an imprecise one. Proper observation weighting is only possible if the vari- Solutions: ance-covariance matrix (VCM) of the observations is known, and in fact LS Weighting How important is estimation and Kalman filtering yield GNSS observation the most precise results only if the cor- GNSS rect VCM is used (advanced approach- weighting? es with less stringent requirements are beyond the scope of this column). Observations Knowledge of the VCM is even n the early days of GPS, most more important in view of statistical and Variations receivers tracked only as many failure detection and identification; of GNSS/INS satellites as were required to com- inappropriate weights may cause outli- Ipute a position. This meant that ers to remain undetected and truly observation weighting was not needed accurate observations to be rejected, Integration and not even possible when process- thus inverting the desired benefit of ing on the epoch-by-epoch level. quality control into a considerable loss “GNSS Solutions” is a Soon, though, receivers were capable of accuracy. Both redundant observa- regular column featuring of tracking all satellites “in view,” and tions and proper observation weighting questions and answers instead of the four minimum pseu- are essential for obtaining a precise and dorange observations required for a reliable estimate. about technical aspects of three-dimensional position, five, six, Proper observation weighting, as GNSS. Readers are invited or more pseudoranges could be avail- it turns out, is not a trivial task with to send questions to the able at each epoch. GNSS observation GNSS observations. The reason is columnists, Professor redundancy will increase further as that the variance must incorporate all GLONASS and Galileo approach their unmodeled effects and thus depends Gérard Lachapelle and Dr. full constellations. on factors such as tracking loop char- Mark Petovello, Department Inevitably, redundant observations acteristics, receiver and antenna hard- of Geomatics Engineering, are inconsistent. At first sight, this ware, signal strength, receiver dynam- University of Calgary, might seem a nuisance best avoided ics, multipath effects, atmospheric by selecting a suitable non-redundant propagation effects, and so forth, most who will find experts to subset of the observations to compute of which can hardly be controlled or answer them. Their e-mail position and receiver clock bias — for determined accurately. The practical addresses can be found with example, the one yielding minimum solution is to use a simple variance their biographies at the GDOP. In reality, however, GDOP model that comes close enough to real- tells nothing about the actual errors ity so that LS estimation or Kalman conclusion of the column of the observations, and the chosen filtering yield nearly optimum results subset may produce a larger position and reliability checking works well. error than other subsets would. It is The simplest variance model far better to exploit the inconsisten- assigns identical variance to all observ- cies using statistical methodsGPS | GALILEO such as | GLONASSables of the same type produced by least-squares (LS) estimation, Kal- the same receiver, for example, 1m2 for man filtering, and hypothesis testing. C/A pseudorange, 1 Hz2 for L1 Dop- This increases the positioning preci- pler, and 0.012 cyc2 for L1 carrier phase. sion, allows checking for failures, and Several studies have shown that this is reduces the probability of undetected not a suitable variance model, unless gross errors. the receiver is operated only in clear- However, exploiting the inconsis- sky environment under line-of-sight tencies requires that the relative preci- conditions, and only high elevation sat- sion of each observation with respect ellites are used. Elevation or C/N0 (car- to the other observations be known. rier-to-noise-density ratio) dependent 26 InsideGNSS JANUARY/FEBRUARY 2007 www.insidegnss FIGURE 1 Normal probability plot of standardized pseudorange errors from low-multipath dataset (left) and high-multipath data set (right) FIGURE 2 Error of estimated receiver positions in a low-multipath environment (left) and high-multipath environment (right; note different scale of axes). Percentages of solutions with more than four satellites are shown in brackets in the legend. variance models usually perform sig- high-multipath data set collected in a receiver and antenna types, and were nificantly better, with the C/N -based dense urban environment (right). determined in advance. 0 GPS | GALILEO | GLONASS models being more widely applicable. The coordinates of both locations If the observations are normally This can be shown both in the observa- were precisely known beforehand; so, distributed and their variance is tion domain (Figure 1) and in the coor- ‘pseudorange errors’ could be esti- matched by any of the variance mod- dinate domain (Figure 2). mated. These errors were then stan- els, the corresponding standardized Figure 1 presents a so-called nor- dardized using three different variance errors should lie exactly on the blue mal probability plot of pseudorange models: (i) identical variances σ2 (ID), straight line in Figure 1. Actually, all 2 errors obtained from two different (ii) elevation dependent variances σ 0 three models represent the majority of static GPS data sets: a three-hour /sin2 E (ELV), and (iii) SIGMA-ε vari- the low-multipath data set well (left), low-multipath data set collected on a ances C2 • 10-C/N0/10 (EPS). The model with a slight advantage shown for EPS mountain slope (left) and a five-hour parameters σ, σ0, and C depend on the (see vertical axis range over which the www.insidegnss.com JANUARY/FEBRUARY 2007 InsideGNSS 27 GNSS SOLUTIONS plotted data coincide approximately ance model is also crucial for success- Wieser, A., and M. Gaggl and H. Hartinger, with the straight line). However, both ful ambiguity resolution. “Improved positioning accuracy with high-sen- ID and ELV fail completely in the high- The above examples highlight the sitivity GNSS receivers and SNR aided integrity monitoring of pseudo-range observations,” in multipath environment (right), while fact that proper GNSS observation Proceedings ION GNSS 2005, 18th Int. Technical EPS still represents more than 90 per- weighting is in fact very important in Meeting of the Satellite Division, Sept. 13–16, cent of the data. order to obtain the most precise and Long Beach, CA: 1545 – 1554, (2005) Figure 2 shows that, indeed, the reliable position, velocity, and time Wieser, A., “Robust and fuzzy techniques precision of the computed coordinates solutions that can be computed from a for parameter estimation and quality assess- depends on the chosen variance model. given set of redundant observations. ment in GPS,” Ph.D. dissertation, Graz University of Technology, Shaker Verlag, Aachen, ISBN The plot on the left contains the epoch- ANDREAs wiESER by-epoch results obtained from the 3826598075 (2002) low-multipath data set at the mountain Dr. Andreas Wieser is a university assistant slope. The identical raw observations with the Institut für Ingenieurgeodäsie und were processed separately using each of Messsysteme at the Graz University of Technol- the three variance models. Obviously, ogy. He has worked on GNSS weight models and “What is the EPS yields the highest precision and quality control as part of his Ph.D. dissertation, the fewest large errors. and he has been actively involved in parameter difference between estimation and GNSS research since 1998. The plot on the right contains the corresponding position solutions Editor’s Note For details of observation ‘loose’, ‘tight’, obtained for the high-multipath site, weighting models, see the following references: ‘ultra-tight’ and and again EPS yields the best results: Collins, J.P., and R.B Langley, “Possible Weighting Schemes for GPS Carrier Phase Obser- 95% of the position errors are within ‘deep’ integration vations in the Presence of Multipath,” Geodetic 38 meters, as opposed to 60 meters Research Laboratory, University of New Bruns- strategies for INS when using the other variance models. wick, Canada, Report to the United States Army Furthermore, fewer observations are Corps of Engineers Topographic Engineering and GNSS?” rejected as potential outliers by the Center, <http://gauss.gge.unb.ca/papers.pdf/ quality control kernel when using EPS, acereport99.pdf>, (1999) and thus the percentage of epochs with Euler, H.J., and C. C. Goad, “On optimal he terms loose, tight, ultra-tight, a controlled solution based on more filtering of GPS dual frequency observations and deep are used to describe than four pseudorange observations without using orbit information,” Bulletin Géo- the way in which information (see numbers in legend) is higher. désique 65: 130–143 (1991) from an inertial navigation Hartinger, H., and F. K. Brunner, “Variances T Similar patterns can also be found with system (INS) and a GNSS receiver of GPS Phase Observations: the SIGMA- Model,” Doppler processing and with carrier are fused in an integrated navigation GPS Solutions 2/4: 35–43, (1999) phase processing, where a proper vari- system. More than a decade ago, R. L. GPS | GALILEO | GLONASS 28 InsideGNSS JANUARY/FEBRUARY 2007 www.insidegnss Greenspan in his seminal work on INS/GPS integration (see citation in Further Readings section at the end of this Solu- tion.) described the “loose” and “tight” integration architec- tures in the way they were understood at the time. Over the years, however, a slight departure from these definitions has occurred. Unfortunately, this has led to some confusion especially because — as a review of recent literature on the subject indicates — an alternate consensus has emerged regarding the terms used to describe the vari- ous INS/GNSS architectures. Another source of confusion is the fact that some INS/GNSS architectures contain ele- ments of the various fusion schemes such that they cannot be described simply as loose, tight, or deep.