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Improvement of Data Precision and Spatial Resolution of cGNSS-R Altimetry Using Improved Device With External Atomic Clock Lifeng Bao, Nazi Wang, and Fan Gao
Abstract—The applications of Global Navigation Satellite System Reflectometry (GNSS-R) altimetry will mainly rely on the precision of this remote-sensing system. Comparing with nadir measurement of satellite altimetry and water gauges, GNSS-R altimetry makes off-nadir measurements with high spatial and temporal resolution come true. To enhance the data precision and spatial resolution, an improved cGNSS-R altimetry system, including one uplooking geodetic GNSS receiver, one downward left-handed circularly polarized GNSS-R receiver, and one tamed atomic clock, was adopted. The objective of the introduced atomic clock is to provide an accurate and synchronous time frequency signal for both receivers, and receiver clock errors can be removed in the regular single difference geodetic processing. Therefore, the unknown parameters in the computation of cGNSS-R were reduced to only one, caused by the phase subtraction between two receivers. By this proposed method, water level height can be derived from one GNSS satellite’s signals, and the spatial resolution will be improved greatly due to more observations. An experimental water level measurement over a smooth inland lake is presented with GNSS-R carrier-phase processing. Then, GNSS-R-derived local water level height is compared with in situ observations. A high degree of agreement is found in these two independent data, and the standard deviation of the derived water level height is better than 1 cm at 1-Hz sample. Index Terms—Atomic clock, cGNSS-R, water level monitoring.
I. I NTRODUCTION
T
HE potential for Global Navigation Satellite System (GNSS) reflected signals to be used as a remote-sensing device was first envisioned by Martin-Neira [1]. The delay of the GNSS reflected signal with respect to the rough surface could provide information on the differential paths between direct and reflected signals, so the delay measurements associated with the properties of the reflecting surface can be used to produce the surface parameters and to determine the surface characteristics. Manuscript received October 21, 2015; revised November 30, 2015; accepted December 1, 2015. Date of publication December 23, 2015; date of current version January 19, 2016. This work was supported in part by the National Natural Science Foundation of China under Grants 41321063, 41274050, and 41374021 and by the Key Program of Chinese Academy of Sciences under Grant Y309451046. L. Bao is with the State Key Laboratory of Geodesy and Earth’s Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China (e-mail:
[email protected]). N. Wang and F. Gao are with the Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China, and also with University of Chinese Academy of Sciences, Beijing 100049, China (e-mail: wnz@whigg. ac.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2015.2506186
The versatility and availability of GNSS Reflectometry (GNSS-R) signals result in many new applications. Generally, two classes of GNSS-R applications have rapidly emerged in the community: surface altimetry, which aims at retrieving the surface height like water, ice, and snow heights [2]–[6], and surface reflectometry, which focuses on the surface attribution like sea roughness, wind, and soil moisture [8]–[13]. A number of experiments and missions using GNSS reflected signals from the ocean and land surface have been tested and applied. GNSSR has been a powerful and potentially disruptive technology for remote sensing, profited from its wide coverage, its being passive, its being precise, its long term, its being all-weather, and its being multipurpose. It should be noted that the GNSS reflected signals will provide height measurements if they could be received and decoded from any surfaces, including ocean, ice, and land surface [14]. For ocean altimetry, to date, tide gauge (for coastal area) and satellite altimeter systems (for deep ocean) have demonstrated their ability to measure the water/ocean surface topography with an accuracy of 1 cm. At coastal areas, the satellite altimeter has worse accuracy. The main reason in developing high-accuracy GNSS-R altimetry is to measure the ocean surface with finer altimetry spatial and temporal resolutions for better understanding of the global ocean surface characteristics and its change. The precision and data coverage of water level measurement will also affect the estimation of sea level rise and research on ocean mesoscale processes. The technology for measuring water level at a coastal site is well established via water gauges. However, even an accurate water gauge measures the relative water level, not an absolute water level. Effects such as glacial isostatic adjustment, coseismic and postseismic deformation, and land subsidence make it difficult to use water gauges either to measure water level directly or to calibrate altimeters [15]. Furthermore, most of the water gauges are located along the coastline, and it only provides the point-value of regional sea surface and is hard to indicate the true global sea surface. The conventional satellite radar technique measures sea surface height at high spatial resolution along its ground track, but the cross-track distance is usually quite large (e.g., about 158 km for typical T/P and Jason-1/2). The temporal resolution is also lower with the 10-day repeat orbit of T/P and subsequent Jason-1/2, and future wide-swath altimeters, even if its spatial coverage has been improved. Although the concept of GNSS altimetry is simple, the crucial requirement in GNSS-R is the strength of the signal-to-noise
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ratio because the reflected signal is significantly weaker than the direct signal. Only a professionally designed GNSS-R receiver can be used to process the left-handed circularly polarized (LHCP) GNSS signal after reflection. Due to the receivers used in the GNSS-R technology, GNSS-R can be divided into two kinds [16]: one is iGNSS-R from interferometric technology, correlating the direct signal and the reflected signals from the same GNSS satellite to get the time delay between the two signals [5], [17], [18]. At present, the precision of the reflector heights from iGNSS-R can be 1 cm; the other kind is cGNSSR by conventional geodetic receivers. The local water level is derived from the geometry path differences of the direct and reflected GNSS signals [19] and [20]. This letter presents the first step on improving retrieval reflector heights with improved cGNSS-R device and external atomic clock. A 1-cm precision of water level heights has been demonstrated by the comparison with in situ observations. Potential applications of this improved cGNSS-R altimetry system were also discussed. II. E FFECTS OF R ECEIVER C LOCK E RROR ON A LTIMETRY P RECISION The receiver clock is usually a quartz, and its precision is about 0.1 × 10−7 ∼ 0.2 × 10−7 s [21], which corresponds to a 3–6-m error in the range. The precision of the reflector height derived from the carrier phase will also be affected by the clock error. Receiver clock error has been one important item in the retrieval of the reflector height in cGNSS-R altimetry, it is regarded as an unknown parameter in the single difference phase data processing, but it can be eliminated in double difference phase data processing. Generally, the receiver clock errors from two receivers in the cGNSS-R altimetry system are not the same, and the drift of the clock error also differs with each other. Following common cGNSS-R retrieval steps, after the ambiguity is determined by the suggested method [6], there are two unknown parameters that need to be resolved: one is the reflector height, and the other is the receiver clock error in every epoch. To show the impact of the receiver clock error on the reflector height, we present one example from Trimble GNSS R8 geodetic receiver in Fig. 1. The clock error has been calculated in precise point positioning with the direct signals. It is clear that the receiver clock jumps exited in the phase measurement. Kim and Zhang have discussed the influence of the receiver clock jumps on the precision of the carrier phase [22]–[24]. To reduce the influence due to the clock error, simultaneous observations from at least two GNSS satellites are required to derive the reflector height, but the availability of GNSS-R signals will decrease. In fact, once the receiver clock error is corrected/removed, the reflector height could be derived from one GNSS observation. The more reflector heights will certainly improve the spatial resolution of cGNSS-R altimetry. III. I MPROVED GNSS-R A LTIMETRY D EVICE AND A LGORITHM The aim of this ground-based experiment is to evaluate the precision of the improved GNSS-R receiver over an inland
Fig. 1. Example of clock jumps from one Trimble GNSS R8 receiver.
smooth water surface. Results are shown for the experiment conducted at the Qinghe arch bridge located at approximately (30◦ 33 N, 114◦ 26 W), over the East Lake of Wuhan, China, on July 02, 2015. It is a closed pond in the south of the bridge, and the closest distance is about 800 m to the opposite bank. The field experiment lasted for about 3 h, decided by a portable power supply. In our improved GNSS-R altimetry system, the zenithpointing antenna (Leica AR10 with GM10 receiver) is used to receive the direct signal, which is right-handed circularly polarized and the same as the transmitted signal from GNSS satellites, and the nadir-pointing antenna (Leica AR20 with GR10 receiver) is optimized to receive the reflected signal, which becomes primarily LHCP after reflection. Both receivers for direct and reflected signals are synchronized by one tamed rubidium atomic clock, which means that the time/frequency from the clock has been adjusted by the GNSS system. The accuracy of short stability could be 2.0 × 10−11 s at 1-Hz sampling frequency. Both geodetic-quality carrier-phase GNSS receivers were operated at 1 sample/s. Here, a special AR20 antenna is designed to only receive the left-handed GNSS signal, which has been optimized with commercial tracking loops. We can process those reflected signals as common direct signals without any technical details of the instrument. The antennae were mounted on the same vertical axis and fixed on a steel frame out of the bridge of about 1.5 m. The GNSS-R antenna was installed under the arch and about 5.9 m above the water surface. By adjusting the location of GNSS-R altimetry at the bridge, we kept empty as much as possible over outside the bridge to avoid multipath influence. The basic principle in GNSS-R altimetry is that the reflected signal will arrive later than the direct one, since it will travel a longer path to the receiver. Since it is a ground-based GNSS-R experiment and the distance between the GNSS-R antenna and the water surface is only several meters, the impact due to the Earth’s curvature will be neglected. The experimental setup is then equivalent to having a “virtual” nadir-pointing antenna placed at the mirror image point of the real downward antenna with respect to the water surface. The geometry algorithm of the GNSS-R altimetry is shown in Fig. 2.
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Fig. 2. Geometry of cGNSS-R altimetry. The signals transmitted by the GNSS satellite are received through their direct path by the zenith-pointing antenna but also after being reflected by the water surface by a suitable nadir-pointing reflectometry antenna. The relative delay between the two receivers provides an observable GNSS-R altimetry.
We perform combined carrier-phase measurements from different carrier frequencies (L1&L2). The delay path length ρr − ρd , which is the difference of the direct path length ρd and reflected path length ρr , will be used to estimate h, the height of the reflectometry antenna above the water surface. In practice, the phase centers of two receivers could not be located in the same position. Given a known vertical distance d between phase centers of two receivers, h can be derived from the following: ρ1r (ti ) − ρ1d (ti ) = λ1 L11r (ti ) − λ1 L11d (ti ) − cδtrd (ti ) + λ1 (Nr1 − Nd1 ) = (2h + d) sin(θ).
receivers have been synchronized by one accurate rubidium atomic clock, so the difference of the clock errors from the receivers and satellite can be removed in the delay path length. Then, besides the true delay path length, there is only an unknown difference of integer ambiguity between two receivers in the observation. It will be a short baseline relative positioning, and we employ a classical geodetic kinematic algorithm to estimate the difference of the integer ambiguity. After Nr1 − Nd1 is determined using the method in [6], the reflector height h can be carried out.
(1)
Here, θ is the complementary angle of reflection, where and ρ1d (ti ) are for the ρr and ρd , respectively, from satellite 1 to the antennae at epoch ti ; λ1 is for the wavelength of GPS L1 ; L11r (ti ) is for the Li measurement of the reflected receiver from satellite 1; L11d (ti ) is for the L1 measurement of the direct receiver from satellite 1; c is the speed of light, and δtrd (ti ) is for the difference of two receivers’ clock offsets, whose value is zero because of the same external receiver clock. Nr1 is the ambiguity of satellite 1, and the receiver recorded the reflected signal. Nd1 is the ambiguity of satellite 1 and direct signal receiver. Nr1 − Nd1 will be a constant over an arc if there is no cycle slip. Nr1 − Nd1 can be determined on the data of the entire observation arc [25]. After Nr1 − Nd1 is determined, h still remains unknown. In the case that the signals in the zenith-point antenna and nadir-pointing antenna come from one same GNSS satellite, taking into consideration that the direct path and reflected path are close to each other, the atmospheric propagation error in both paths could be assumed as the same. Moreover, two GNSS ρ1r (ti )
Fig. 3. Estimated distance between the nadir-pointing antenna and the smooth water surface from GNSS-R altimetry with L1 (top) and L2 carrier phases (bottom).
IV. R ESULTS AND VALIDATION The heights of the GNSS-R antenna above the water surface were retrieved following the method in Section III, over the entire data collection period. The carrier phases at L1 and L2 were taken as basic measurements. First, the direction paralleled to the bridge has been rejected. Then, all directions (in azimuth) lying on the same side of the bridge as the antenna were considered good, while the elevation was restricted to be greater than 10◦ . Moreover, an additional water level measurement from an accurate electronic total station, which is 5.85 m, will be employed as an independent observation to assess the precision of GNSS-R altimetry measurement from the aforementioned algorithm. The results from the L1 and L2 carrier-phase measurements are shown in Fig. 3. The mean and standard deviation of the height from GNSS-R from the L1 and L2 carrier phases are listed in Table I. For both frequencies, the mean of the residual is less than 2 cm, and the precision of each profile is better than 1 cm. Considering that it is a calm/smooth water surface, which make the L1 and L2
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 13, NO. 2, FEBRUARY 2016
TABLE I S TATISTICS OF R EFLECTOR H EIGHTS F ROM L1 AND L2 C ARRIER P HASES AND I TS D IFFERENCE F ROM IN SITU M EASUREMENTS
TABLE II S TATISTICS OF R EFLECTOR H EIGHTS F ROM P REVIOUS GNSS-R E XPERIMENT W ITHOUT ATOMIC C LOCK AND I TS D IFFERENCE F ROM IN SITU M EASUREMENTS
Fig. 4. Geographic location of reflector heights during one whole day (July 2, 2015) in imperfect conditions, given the truncated elevations of the GNSS satellites as 10◦ and the inland lake as wide enough. The sparsity in the north direction is caused by the spatial distribution of the GPS satellite constellation.
signals remain coherent long enough to achieve direct tracking, a formulation similar to L1 and L2 was found for the height of the nadir-pointing antenna. However, the L2-based solution is noisier than the L1 result because of the lower power of the emitted L2 signal compared with that of the L1 signal, and the codeless technique used internally by the Turbo-Rogue receiver for the L2 carrier-phase extraction introduces noise into the system. Although it is a calm inland lake, there is still a small wave, caused by wind or other factors, on the water surface. The amplitude in the observed height from GNSS-R altimetry coincides with the field scene. To prove the improvement of the retrieved water level height due to the introduced atomic clock, we also present the comparison of the reflector heights from GNSS-R altimetry without external atomic clock in Table II. In that experiment, a simultaneous observation at two GNSS receivers was required, and the reflector height was derived by (1) at those epochs with simultaneous observations. Obviously, the valid data observations will be less than the observations with atomic clock. A short-period, about 25 min, reflector height series from the pair of PRN16 and PRN32 at 1-Hz sampling shows that the root-mean-square (rms) agreement with in situ measurements is about 2–3 cm. From the aforementioned analysis, this proposed cGNSSR altimetry system will enhance the spatial resolution of the reflector heights because no simultaneous observations are required. When an external atomic clock is used, the valid observations of reflector heights will depend on the sampling rate. If the inland lake is wide enough and given the truncated elevations of GNSS satellites as 10◦ , in one whole day (e.g., July 2, 2015), there are 687 593 reflector heights that will be valid (Fig. 4), which will provide a dense water surface measurement.
V. C ONCLUSION AND D ISCUSSION GNSS-R altimetry could be used as a water gauge due to its high temporal and spatial resolution, once the data precision is met. The receiver clock error is one of the main errors on the retrieval computation of water level height. To reduce the influence of the receiver clock errors in cGNSS-R altimetry, we have proposed one improved cGNSS-R system, including one uplooking geodetic GNSS receiver, one downward LHCP GNSS-R receiver, and one tamed atomic clock. A groundbased field experiment using a GNSS-R antenna to sense the inland water level under a calm/smooth condition has been successfully conducted on the Qinghe arch bridge over East Lake, Wuhan, China, and a 1-cm precision water measurement was achieved by processing the GNSS L1 and L2 carrier-phases in 1-s snapshots. The results from this letter imply that GNSS-R is applicable for Earth’s science, which requires a high precision of surface measurement. Different from the point-value from the traditional water gauge observation, the GNSS-R altimetry provides a quantity of off-nadir measurements above the surface with a high precision. For obtaining more validation data, it was suggested to install two or more GNSS-R altimetry systems at the same time. When the multi-GNSS-R altimetry systems were installed at the right location, decided by the sharpness of the water surface and the GNSS constellation, the distribution of reflection points will be a complex cross. The difference at the crossover points between/among the results from multiGNSS-R altimetry systems can be used to assess the precision of GNSS-R altimetry. It is also useful in determining the systemic bias in the GNSS-R altimetry. The advantage of GNSS-R altimetry over monostatic radar altimeters is that the receiver could produce about dozens of simultaneous measurements, distributed over its surrounding
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area. Thus, with an optimized spaceborne orbit, the global ocean sea surface could be measured in a short repeat period with almost seamless coverage. GNSS-R is poised to potentially make a significant new contribution to Earth scientific studies involving the ocean altimetry, sea surface change, and mass fluxes. Development in the GNSS-R instrument has pushed the application of GNSS-R, especially in altimetry. Researchers can focus on the algorithms and detailed applications without concerning the complex signal processing and Doppler delay mapping. Those mature algorithms to the common GNSS receiver will also be valid to the new GNSS-R receiver. In this GNSS-R altimetry data processing, the delays due to atmospheric propagation in direct and reflected signals are regarded as the same. In fact, there is an extra influence due to the delay path between the direct and reflected signals. This extra influence can be used to retrieve the water vapor content between the antenna and the water surface. Furthermore, this 1-cm precision GNSS-R altimetry has proved the feasibility of mesoscale mapping, spatial scale of approximately 50–200 km (e.g. eddies, fronts, warm and cold rings, and jets). These shortscale variations tend to occur over time periods of weeks to months, currently dominate the global climate modeling errors, and require finer altimetric spatial and temporal resolutions than that offered by a traditional altimeter. The expected amplitudes for mesoscale eddies and meanders ant mid- and high latitudes are in the range 5–30 cm, down to a few centimeters for highlatitude fronts. Assuming a required sensitivity equal to 20% of the signal, mesoscale ocean altimetry calls for a precision between 1 and 5 cm [2]. Lofgren [19] has presented the significant ocean tidal signals at fortnightly, diurnal, semidiurnal, and quarter-diurnal periods based on three months of local sea level derived from a GNSS-based water gauge. It also has been proved, at least in the simulation, that GNSS-R altimetry with the low-Earth orbiter carrying a high-gain antenna will improve the accuracy limitations of the global ocean tide models from the water gauge observation and radar altimeters. Extending this concept to use satellite signals of other GNSSs will improve the temporal resolution of the geodetic water gauge and allow a more comprehensive comparison. The future of the GNSS-R altimetry is promising based on the enhanced capabilities that GNSS systems will offer in the coming decades. Due to its prominent ability of measuring the sea surface and its change with a high spatial and temporal resolution, GNSS-R altimetry will greatly contribute to many fields of Earth science. Our upcoming work will focus on the practical application of GNSS-R altimetry with more frequencies, and we are planning to install our GNSS-R altimetry at one oil platform in the South China Sea for improving the local ocean tide model. ACKNOWLEDGMENT The authors would like to thank Z. Li for the assistance with the experiments and Dr. R. Yang for the thoughtful suggestion and discussion on the carrier-phase processing.
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R EFERENCES [1] M. Martín-Neira, “A Passive Reflectometry and Interferometry System (PARIS): Application to ocean altimetry,” Ecol. Soc. Amer. J., vol. 17, pp. 331–355, Nov. 1993. [2] M. Martín-Neira, M. Caparrini, J. Font-Rosselló, S. Lannelongue, and C. S. Vallmitjana, “The PARIS concept: An experimental demonstration of sea surface altimetry using GPS reflected signals,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 1, pp. 142–149, Jan. 2001. [3] S. T. Lowe et al., “5-cm-precision aircraft ocean altimetry using GPS reflections,” Geophys. Res. Lett., vol. 29, no. 10, pp. 1–4, May 2002. [4] S. T. Lowe et al., “First spaceborne observation of an Earth-reflected GPS signal,” Radio Sci., vol. 37, no. 1, pp. 1–28, Jan. 2002. [5] K. M. Larson, J. S. Löfgren, and R. Haas, “Coastal sea level measurements using a single geodetic GPS receiver,” Adv. Space Res., vol. 51, no. 8, pp. 1301–1310, Apr. 2013. [6] J. S. Löfgren, R. Hass, and J. M. Johansson, “Monitoring coastal sea level using reflected GNSS signals,” Adv. Space Res., vol. 47, pp. 213–220, Jan. 2011. [7] F. Fabra et al., “Monitoring sea-ice and dry snow with GNSS reflections,” in Proc. IEEE IGARSS, Jul. 2010, pp. 3837–3840. [8] A. Komjathy, J. Maslanik, V. U. Zavorotny, P. Axelrad, and S. J. Katzberg, “Sea ice remote sensing using surface reflected GPS Signals,” in Proc. IEEE IGARSS, 2000, vol. 7, pp. 2855–2857. [9] A. Rius et al., “Altimetry with GNSS-R interferometry: First proof of concept experiment,” GPS Solutions, vol. 16, pp. 231–241, Apr. 2012. [10] M. P. Clarizia et al., “Analysis of GNSS-R delay-Doppler maps from the UK-DMC satellite over the ocean,” Geophys. Res. Lett., vol. 36, no. 2, Jan. 2009, Art. ID L02608. [11] O. Germain et al., “The eddy experiment: GNSS-R speculometry for directional sea-roughness retrieval from low altitude aircraft,” Geophys. Res. Lett., vol. 31, no. 21, Apr. 2004, Art. ID L21307. [12] D. S. Masters, “Surface remote sensing applications of GNSS bistatic radar: Soil moisture and aircraft altimetry,” Ph.D. dissertation, Dept. Aeorsp. Eng. Sci., Univ. Colorado, Denver, CO, USA, 2004. [13] J. L. Garrison et al., “Estimation of sea surface roughness effects in microwave radiometric measurements of salinity using reflected global navigation satellite system signals,” IEEE Geosci. Remote Sens. Lett., vol. 8, no. 6, pp. 1170–1174, Nov. 2011. [14] C. K. Shum et al., “Prospects of Global Navigation Satellite System (GNSS) reflectometry for geodynamic studies,” Adv. Space Res., vol. 47, no. 10, pp. 1814–1822, May 2011. [15] K. M. Larson and F. G. Nievinski, “GPS snow sensing: Results from the EarthScope plate boundary observatory,” GPS Solutions, vol. 17, no. 1, pp. 41–52, Mar. 2012. [16] A. Camps et al., “Optimization and performance analysis of interferometric GNSS-R altimeters: Application to the PARIS IoD mission,” IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens., vol. 7, no. 5, pp. 1436–1451, May 2014. [17] B. Wilmhoff, F. Lalezari, V. Zavorotny, and E. Walsh, “GPS ocean altimetry from aircraft using the P(Y) code signal,” in Proc. IEEE IGARSS, Jul. 2007, pp. 5093–5096. [18] C. Wagner and J. Kloko˘cník, “The value of ocean reflections of GPS signals to enhance satellite altimetry: Data distribution and error analysis,” J. Geodesy, vol. 77, no. 3/4, pp. 128–138, 2003. [19] J. S. Löfgren, R. Haas, H. G. Scherneck, and M. S. Bos, “Three months of local sea level derived from reflected GNSS signals,” Radio Sci., vol. 46, no. 6, Dec. 2011, Art. ID RS0C05. [20] J. S. Löfgren and H. Rüdiger, “Sea level measurements using multifrequency GPS and GLONASS observations,” EURASIP J. Adv. Signal Process., vol. 1, pp. 1–13, 2014. [21] Z. Li and J. Huang, GPS Surveying and Data Processing. Wuhan, China: IEEE, Feb. 2014. [22] C. Zhang and X. Jia, “The influence and detection method of receiver clock jumps on GPS positioning,” Bull. Surv. Mapping, vol. 12, pp. 7–9, 2009. [23] D. Kim and R. B. Langley, “Instantaneous real-time cycle-slip correction of dual frequency GPS data,” in Proc. Int. Symp. Kinematic Syst. Geodesy, Geomatics Navigat., Banff, AB, Canada, Jun. 2001, pp. 5–8. [24] D. Kim and R. B. Langley, “Quality Control Techniques and Issues in GPS Applications: Stochastic Modeling and Reliability Test,” presented at the Int. Symposium GPS/GNSS, Jeju, Korea, 2001. [25] B. Hoffmann-Wellenhof, H. Lichtenegger, and E. Walse, GNSS— Global Navigation Satellite Systems GPS, GLONASS, Galileo and More. Shenzhen, China: IEEE, Aug. 2009.
