Technical Article • July/August 2016
First ResultsGLONASS CDMA L3 Ambiguity ResolutionResearchers in Australia present their first results of GLONASS CDMA L3 ambiguity resolution. Based on the observations from two GLONASS satellites that were collected at Curtin University, this article assesses the performance of geometryfree and geometryfixed L3 ambiguity resolution methods and compares the outcomes with their GPS L5 counterparts.
Share via: Slashdot Technorati Twitter Facebook In February 2011, Russia launched the first satellite of the GLONASSK1 series, i.e., SVN (space vehicle number) 801 (R26), which in addition to the legacy frequency division multiple access (FDMA) signals, for the first time was enabled to transmit code division multiple access (CDMA) signals on the GLONASS L3 frequency (1202.025 MHz). Later in 2014, the GLONASS program added SVNs 802 (R17) of series K1 and 755 (R21) of series M, and in 2016, SVN 751 of series M, with the capability of transmitting CDMA L3 signals to the constellation. The GLONASS FDMA doubledifferenced (DD) ambiguity resolution is known to be hampered by the inherent interfrequency biases. Several calibration procedures have been proposed to deal with this impediment. With GLONASSCDMA however, the standard methods of integer ambiguity resolution can be applied to resolve the integer DD ambiguities. The goal of this article is to provide a first assessment of this L3 ambiguity resolution performance.
Measurement Setup Figure 2 shows their observed carriertonoise densities (C/N_{0}).As their C/N_{0} graphs show a similar signature, their signals are expected to have similar noise characteristics. Figure 1 also shows the skyplot of the mentioned satellite pairs at Perth. For both the GLONASS and the GPS satellites, we used the broadcast ephemeris data. Table 2 provides further information on the dataset that we used.
Model of Observations Equation (1) (see inset photo, above right, for all equations) in which p and φ are the DD code and phase observable, respectively, ρ the DD receiversatellite range and a the DD integer ambiguity in cycles. The ambiguity a is linked to the DD phase observable through the signal wavelength λ. With the elevationdependent weighting function wθ_{s} (s = 1, 2) for the s_{th} satellite with elevation angle θ_{s}, respectively, the final weight becomes Equation (1a) Here wθ_{s} is taken as Equation (2) where θ_{s} is in degrees. The zenithreferenced standard deviations of the undifferenced code and phase observables are denoted as σ_{p} and σ_{φ}. In our analysis we considered two different models. These are arranged in ascending order of strength as: 1. Geometryfree model (GFr): This is the model as formulated in (1). As it is parametrized in ρ, it is free from the receiversatellite geometry. The singleepoch DD ambiguity is then estimated as Equation (3) 2. Geometryfixed model (GFi): In this model, the information on receiver position, from e.g. surveying, and satellite position, from navigation file, is available and thus ρ is assumed known. The singleepoch DD ambiguity is then estimated as Equation (4) Note that although the observations of only two satellites are used, both the geometryfree and geometryfixed models are instantaneously solvable, i.e., based on data of only a single epoch. See the article by P. J. G. Teunissen (1997) listed in the Additional Resources section near the end of this article for a more detailed discussion of these models.
Ambiguity Resolution In Figure 3, the time series of â − a and ă − a are shown for the receiver pair CUT3CUCC, for both the GLONASS satellite pair R21R26 (left column) and the GPS satellite pair G10G26 (right column). While the geometryfixed results of the two signals are comparable, the GPS L5 geometryfree ambiguity resolution outperforms that of the GLONASS L3, which can be explained by means of the satellites’ elevations: the higher the elevation, the lower the noise level, thus the better the ambiguity resolution performance (cf. 2, 3 and 4). The bottom set of graphs in Figure 3 also illustrates the elevation time series of the chosen satellite pair (in blue) in addition to the geometryfixed DD ambiguities. Here we can see that the elevations of the GPS satellite pair is higher than those of the GLONASS satellite pair. Also, the low elevation of R26 at the end of the period and the low elevation of G10 at the beginning of the period describe the larger fluctuations of, respectively, the GLONASS DD ambiguities and the GPS DD ambiguities at those time instants. For both the geometryfree and the geometryfixed scenario, we computed the formal and empirical ambiguity successrates, defined as the probability of correct integer estimation. The formal ambiguity successrate can be computed, as discussed in the article by P. J. G. Teunissen, (1998) cited in Additional Resources, as Equation (5) being the standard normal probability density function (PDF). For the computation of the formal successrate, the ambiguity standard deviation was taken as the squareroot of an average of the formal variances, i.e., as Equation (6)
Table 3 lists the empirical and formal successrates for both GLONASS and GPS corresponding with Figure 3. Based on these results, the empirical values are consistent with their formal counterparts. Moreover, as the model gets stronger from oneepoch geometryfree to geometryfixed, the ambiguity resolution successrates experience a significant improvement. In case of the oneepoch geometryfree model, σ_{â} is governed by the code precision σ_{p}. Including the observations of k epochs, the corresponding σ_{â} of kepoch geometryfree model is improved by almost √ k Switching from geometryfree to geometryfixed model, σ_{â} is then governed by the phase precision σ_{φ} which is much better than the code precision. For the geometryfree model to achieve a successrate of more than 0.999, 40 epochs of observation in the case of GLONASS L3 and 10 epochs in the case of GPS L5 are required. To further confirm the consistency between the data and models, we compare, for both the geometryfree and the geometryfixed model, the formal PDF with the histogram of the estimated DD ambiguity. Normalizing the estimated DD ambiguity by means of the elevation weighting function results in a new quantity, i.e., √ w (â  a) which, assuming the data to be normally distributed, has a central normal distribution with the standard deviation of √ w σ_{â}. Depending on whether the underlying model is geometryfree or geometryfixed, the value of √ w σ_{â} can be obtained from (3) or (4), respectively. Figure 4 displays the histograms of the normalized DD ambiguity √ w (â  a), for geometryfree and geometryfixed model. The corresponding formal distribution is also shown by the red curve. It demonstrates the consistency between the empirical and formal distributions.
Conclusion For our analyses, we made use of the GLONASS L3 signal transmitted by the satellite pair R21R26 and of the GPS L5 signal from the satellite pair G10G26. The carriertonoise densities of both signals were shown to have similar signatures. The integer ambiguity resolution performance in the framework of geometryfree and geometryfixed observational model was demonstrated. As the model gets stronger from geometryfree to geometryfixed model, the ambiguity resolution improves significantly. Our empirical results (in the form of successrates and normalized ambiguity PDF) showed a good agreement with their formal counterparts, thereby showing the consistency between data and models. The ambiguity resolution of GPS L5 was better than that of the GLONASS L3, which was attributed to the higher elevation of the GPS satellites w.r.t the GLONASS satellites during the considered period.
Additional Resources ManufacturersThe GPS/GLONASS receivers used to observe satellite signals were JAVAD TRE_G3TH_8 receivers from Javad GNSS, San Jose, California USA. They were connected with TRM59800.00 SCIS antennas from Trimble Navigation Ltd., Sunnyvale, California USA.Copyright © 2017 Gibbons Media & Research LLC, all rights reserved. 
