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GPS/GLONASS Carrier Phase Elevation-Dependent Stochastic Modelling Estimation and Its Application in Bridge Monitoring

: The Global Positioning System (GPS) based monitoring technology has been recognised as an essential tool in the long-span bridge health monitoring throughout the world in recent years. However, the high observation noise is still a big problem that limits the high precision displacement extraction and vibration response detection. To solve this problem, GPS double-difference model and many other specific function models have been developed to eliminate systematic errors e.g. unmodeled atmospheric delays, multipath effect and hardware delays. However, relatively less attention has been given to the noise reduction in the deformation monitoring area. In this paper, we first proposed a new carrier phase elevation-dependent precision estimation method with Geometry-Free (GF) and Melbourne-Wübbena (MW) linear combinations, which is appropriate to regardless of Code Division Multiple Access (CDMA) system (GPS) or Frequency Division Multiple Access (FDMA) system (GLONASS). Then, the method is used to estimate the receiver internal noise and the realistic GNSS stochastic model with a group of zero-baselines and short-baselines (served for the GNSS and Earth Observation for Structural Health Monitoring of Bridges (GeoSHM) project), and to demonstrate their impacts on the positioning. At last, the contribution of integration of GPS and GLONASS is introduced to see the performance of noise reduction with multi-GNSS. The results show that the higher level receiver internal noise in cost effective receivers has less influences on the short-baseline data processing. The high noise effects introduced by the low elevation satellite and the geometry variation caused by rising and dropping satellites, can be reduced by 10%-20% with the refined carrier phase


Introduction
Global Navigation Satellite System (GNSS) is now gradually recognized as an essential tool However, the large background noise commonly exists in the GNSS observation. Hence, the low-frequency component (primarily under 0.05 Hz) of the displacement time series will contain bias, and, in the high-frequency component, the vibration amplitude of the response under the ambient excitation will contain false distortion signal or be covered by noise (Górski 2017). This is still a big problem that prevents the GNSS based monitoring technology to extract displacements in high accuracy and to detect correct vibration response. However, the noise level control method in deformation monitoring is still rarely studied.
The GNSS observation noise mainly comes from receiver internal noise, multipath effects and residual atmosphere delays, etc. In this study, we firstly proposed a new single differential (SD) geometry-free method to estimate the carrier phase elevation-dependent precisions of GPS and GLONASS with MW and GF combinations. Then, based on a testing platform served for the GeoSHM project, a project sponsored by the European Space Agency, the internal noise of two receiver brands and a realistic stochastic elevation based model was built with a group of zero baselines and short baselines. Thirdly, the effect of the realistic GNSS elevation-dependent weighting and integration of GPS and GLONASS on positioning are numerically analysed. Finally, a real-life bridge monitoring data set is applied to demonstrate the performance of the realistic GNSS elevation-dependent stochastic model and the multi-GNSS application in the false distortion signal elimination.
2 Elevation-dependent precision estimation with SD geometry-free method

The SD Geometry-free functional model
As pseudorange is easily contaminated by multipath and hardware delays, biases in the pseudorange are significant (Yu et al., 2017, Chu and Yang 2018, Chu et al., 2016. Hence, only carrier phase observations are usually used in our monitoring work and only the elevation-dependent precision of carrier phase is estimated in this study. As proposed by Li et al. (2016), SD geometryfree stochastic model estimation has the advantages of only one satellite involving in a SD observation and no mathematical correlation between satellites, which is suitable for the stochastic modelling estimation. On ultrashort (shorter than 10 m) or zero baselines, the systematic errors can be assumed to be completely eliminated in a SD observation. In this way, only the pure random errors remain, which can help us to estimate the precision of the satellite-specific variances (the stochastic model).
Then the single-epoch, single-frequency between-receiver SD geometry-free GPS or GLONASS phase observation equation can be read where the subscripts j is the frequency number ( is the SD integer ambiguity vector of the j-th frequency, which are expressed in cycles. Note that in this study, the receivers we use are with the same hardware configuration (i.e., same manufacturer, receiver type, firmware version, and antenna type) for GLONASS data. Thus, the inter-frequency bias (IFB) will be out of the consideration [44].
In this study, the zero baseline and ultrashort baseline are used, and the antennas are set on the known positions. In theory,   equal zeros in the zero-baseline or obtained by the known short- f and 2 f are frequencies of GPS or GLONASS dual-frequency observation;  6 where   and P  are the undifferenced standard deviation in theory.
Based on Eq. (3), the float solution of SD ambiguities in the two frequencies can be solved as:  (6) and the accuracy of SD ambiguities are With Eq. (4) and Eq. (7), the accuracy of estimated SD ambiguities can be obtained. To fixed the ambiguities estimated in Eq. (6), in this study, a small search produce is applied and the search space is decided by the accuracy of the float SD ambiguities, as 11 221 1 12 is the candidates and "   " means rounding up to the nearest integer.
"round" indicates fixing to its nearest integer. Generally, without significant disturbed delays and multipath effects, we assume the standard deviation of undifferenced carrier phase and code measurements are 0.5 cm and 1 m.
After fixing the SD ambiguities, the Eq. (1) can be rewritten as The benefit of this method is that it not only can be used in CDMA systems, including GPS, BeiDou and Galileo, but also available for FDMA system (GLONASS).

Carrier Phase Elevation-Dependent Precision Estimation
In this study, we only consider the effect of elevation-dependent precision on structural    are overall better than L2 in both receivers, especially in elevations lower than 40°. From the perspective of receivers, LEICA is slightly better than PANDA receivers on L1. However, on L2, the residuals from the PANDA receiver pairs are extremely larger than the LEICA ones. That means for two receiver pairs, the noise in L1 is smaller than L2, and the internal noise of the PANDA receiver is higher than the LEICA receiver. In contrast, for GLONASS system, the noise is lower for L2 than that of L1, and the elevation-dependent characteristic is not that obvious as GPS.  Table 2.
It is clearly shown that, from Fig 4, the two models can fit the overall elevation-dependent precisions very well and they show a high agreement with each other. For L1 observations, the precision is higher than 1 mm for all data set. Except for GLONASS L2 observations whose precision is also better than 1 mm, the internal noise is larger than 2 mm for GPS at elevation 10°.
The lowest precision is shown in L2 frequency of the PANDA receivers with larger than 4 mm for 10° elevation. However, the precisions tend to be constant to 0.5 mm for the elevation upper than 30° for all observations.

GPS/GLONASS carrier phase elevation-dependent precision estimation with short baselines
In this section, SHM7 with LEICA GM30 receiver served as the reference station and four  As it is shown, the carrier phase residuals for all data are overall elevation-dependent. The mean precision is slightly better for L1 observations than L2. Even though the high level of internal noise, the highest precision are both found in PANDA results for the two frequencies. For the residual changes with elevation, when the elevation is lower than 30°, the maximum residual can even reach up to 4 cm, mostly within 2.5 cm in GPS. GLONASS is slightly lower, within 2 cm and 2.5 cm for L1 and L2 observations respectively. For the elevation higher than 30°, the residual time series tend to converge within 4 mm with the increasing of elevation angles until to 90°. It demonstrates that the value of 3 mm usually used as the prior precision of GNSS measurements is unrealistic. Fig 6 gives Table 3 Fitting parameters estimated of two predefined elevation-dependent models.  Table 3. It can be seen that Model A has a better fitting performance than Model B. However, the fitted curves and parameters estimated are quite similar for the two frequencies and two receiver brands. Although the receiver noise level is high for the PANDA receiver, they give almost the same result with the LEICA receiver. Based on this experiment, we know that the higher internal noise of receivers in millimeter level will not influence the baseline solution too much. The cost effective receivers can also give the same performance compared with the high quality equipment. To further analyse the effect of the refined stochastic model in positioning, we processed the data of the short-baseline formed by the LEICA receiver pair and the PANDA receiver pair with the home-made GNSS real-time data processing software named GNSSDEM. The GNSSDEM software now supports dual-and triple-frequency data processing with GPS, BeiDou, GLONASS and integrated GPS, BeiDou and GLONASS using the double difference (DD) method. Table 4 GPS and GLONASS data processing models and strategies used in GNSSDEM.

Options Processing Strategy
Ephemeris GPS and GLONASS broadcast ephemeris The paper also gives the standard deviations (STD) of the time series, which can be calculated with: (19) where  is the average bias; n is the number of epochs. The statistical precision of time series are shown in Table 5. We can see that the time series with the refined model is higher than the ones with the empirical model, with the precision improvement from 10% to 20%, and the four baselines show the same STD values with a same model. This also confirmed that the PANDA receiver with a higher internal noise level can have the same positioning performance with the high quality ones. Then, we process the integrated GPS and GLONASS data for SHM7-SHM5 with the refined model in Table 3. The STD values for the baselines solutions are also listed in Table 5. Compared with the GPS only empirical model solutions, the precision improves up to 30% to 40%.   One can found that, since the data quality is good enough for this session, the ASR is higher than 98% for GPS-only data regardless of empirical and refined model. However, the ASR of refined model is slightly better than the empirical model with higher than 99%. After combined with GLONASS data, the ASR can achieve to 100% for both of the empirical and refined model. From the ratio value of every epoch, however, we can noticed that the refined model is larger than the empirical model most of the time. Therefore, the refined stochastic model may give a more realistic weight for the observations, which will be beneficial for the ambiguity resolution.

Performance of the refined stochastic model and multi-GNSS used on the real-life bridge monitoring
In   Table 3  10 For comparing purpose, we processed the GNSS data of SHM4 in the four schemes listed in Table 6. We can see that the posteriori residuals of G17 is extremely larger than other satellites. Therefore, we can conclude that the signal between 16:30 to 16:40 are large noise caused by G17.
Note that this may cause two problems. Firstly, the large noise can cover the small amplitude vibration signals. Secondly, it could be recognised as a signal to be extracted from the displacements.
In order to reduce the false signal, we rise up the cutoff elevation to 15° in scheme (b). We can see that the signal between 16:30 and 16:40 is excluded. However, the noise is still large between 16:45 to 17:00. From Fig 14, it indicates that this is because the dropping of G27. The PDOP value is changing and the geometry strength is becoming weak.

Conclusion
In this study, we first proposed a GPS and GLONASS carrier phase elevation-dependent precision modelling method. Then, based on a testing platform served for the GeoSHM project with a group of zero-baselines and short-baselines, we analyse the impacts of receiver internal noise and refined GNSS stochastic model on the positioning. At last, the contribution of integration of GPS and GLONASS is also introduced. After the experimental analysis, some useful conclusions can be summarised as follows: -The precision estimated for GPS and GLONASS measurements is overall elevation-dependent for both zero-baseline and short-baseline. In the zero-baseline experiment, the internal noise features are accounting for the systems, receiver brands and frequencies. For GPS observations, the precision of L1 is better than L2. The cost effective receiver PANDA has a high level internal noise than LEICA. For GLONASS, the elevation-dependent characteristic is not obvious compared with GPS. The precision of GLONASS L2 observation is slightly better than L1.
-In the short-baseline experiment, however, we got almost the same precisions for GPS observations in L1 and L2 frequencies. The PANDA receivers with high internal noise can have the same performance with the LEICA ones. For GLONASS, the precision of L2 is worse than L1, and they show a lower mean precision compared with GPS. However, at low elevation region, they show a little bit higher precision than GPS. After fitting observation precisions of GPS and GLONASS by elevation-dependent functions, we use them in the DD data processing, and compare with the empirical model solutions. We found that the refined model can improve the precision of the baseline time series from 10% to 20%. If GLONASS data were joint, the precision can further improve by 30% to 40%. From further analysis with box plots, we found that the extreme data points and the values within the 25th and 75th percentiles are all trend to be smaller after applying the refined stochastic model and using multi-GNSS observations.
-By using the refined model estimated with the short-baseline data into the real-life bridge monitoring data processing, the large noise caused by the observations of the low elevation satellites and rising and dropping satellites can be reduced. The integration of GPS and GLONASS with refined model tend to completely eliminate the false distortion signals. The ambiguity resolution rate is also improved with the refined stochastic model, since the random errors in the GPS/GLONASS measurements are reasonably described.
Finally, the paper suggests that, for the long-term bridge deformation monitoring, the receiver internal noise, the realistic stochastic model can be evaluated and estimated in advance. Multi-GNSS data could be applied to achieve high precision of displacements and reliable vibration parameters.