remote sensing Article
Site-Specific Unmodeled Error Mitigation for GNSS Positioning in Urban Environments Using a Real-Time Adaptive Weighting Model Zhetao Zhang 1,2 1 2
*
ID
, Bofeng Li 1, *
ID
, Yunzhong Shen 1 , Yang Gao 2 and Miaomiao Wang 1
College of Surveying and GeoInformatics, Tongji University, Shanghai 200092, China;
[email protected] (Z.Z.);
[email protected] (Y.S.);
[email protected] (M.W.) Department of Geomatics Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada;
[email protected] Correspondence:
[email protected]; Tel.: +86-136-3634-4065
Received: 25 June 2018; Accepted: 20 July 2018; Published: 22 July 2018
Abstract: In Global Navigation Satellite System (GNSS) positioning, observation precisions are frequently impacted by the site-specific unmodeled errors, especially for the code observations that are widely used by smart phones and vehicles in urban environments. The site-specific unmodeled errors mainly refer to the multipath and other space loss caused by the signal propagation (e.g., non-line-of-sight reception). As usual, the observation precisions are estimated by the weighting function in a stochastic model. Only once the realistic weighting function is applied can we obtain the precise positioning results. Unfortunately, the existing weighting schemes do not fully take these site-specific unmodeled effects into account. Specifically, the traditional weighting models indirectly and partly reflect, or even simply ignore, these unmodeled effects. In this paper, we propose a real-time adaptive weighting model to mitigate the site-specific unmodeled errors of code observations. This unmodeled-error-weighted model takes full advantages of satellite elevation angle and carrier-to-noise power density ratio (C/N0). In detail, elevation is taken as a fundamental part of the proposed model, then C/N0 is applied to estimate the precision of site-specific unmodeled errors. The principle of the second part is that the measured C/N0 will deviate from the nominal values when the signal distortions are severe. Specifically, the template functions of C/N0 and its precision, which can estimate the nominal values, are applied to adaptively adjust the precision of site-specific unmodeled errors. The proposed method is tested in single-point positioning (SPP) and code real-time differenced (RTD) positioning by static and kinematic datasets. Results indicate that the adaptive model is superior to the equal-weight, elevation and C/N0 models. Compared with these traditional approaches, the accuracy of SPP and RTD solutions are improved by 35.1% and 17.6% on average in the dense high-rise building group, as well as 11.4% and 11.9% on average in the urban-forested area. This demonstrates the benefit to code-based positioning brought by a real-time adaptive weighting model as it can mitigate the impacts of site-specific unmodeled errors and improve the positioning accuracy. Keywords: Global Navigation Satellite System (GNSS) positioning; code weighting; site-specific unmodeled error; urban environment
1. Introduction Currently, Global Navigation Satellite System (GNSS) positioning techniques based on code observations including single-point positioning (SPP) and code real-time differenced (RTD) positioning have been widely used due to their easy-to-implement and low-cost advantages, particularly in smart
Remote Sens. 2018, 10, 1157; doi:10.3390/rs10071157
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phones and vehicles [1,2]. However, the signals of these devices could be distorted by the buildings and trees especially in urban canyon areas [3]. Consequently, the code precisions will be contaminated by the site-specific unmodeled effects. Here, the site-specific unmodeled effects refer to the signal distortions that are not always considered in the mathematical model [4,5], such as multipath and other space loss caused by the signal propagation like non-line-of-sight (NLOS) reception. As usual, the observation precisions are captured by the weighting function in a stochastic model [6,7]. Hence, the site-specific unmodeled errors should be better considered by a realistic weighting function which determines the contribution of each observation to the least-squares (LS) solution. One can obtain the high-precision and high-reliability positioning results only if a realistic weighting function is applied. That is, any inappropriate weighting model cannot obtain the minimum variance estimators [8,9], and the power of the statistical tests, such as outlier and cycle slip detections, will also be reduced [10,11]. Therefore, much research has been focused on how to estimate the observation precisions accurately with a weighting function. In earlier times, the assumption of homoscedasticity was used. That is, the variances of observations are equal in a stochastic model [12]. Apparently, this assumption cannot meet the reality most of the time. To improve the positioning accuracy, two main indicators are then applied to reflect the heteroscedasticity of observations. The first one is the satellite elevation angle since the measurements with higher elevations are more precise [13]. In positioning programs, the most widely used model is based on the cosecant function [14–17]. The second one is the carrier-to-noise power density ratio (C/N0) which is the ratio of the signal power to noise power in a 1-Hz bandwidth [18,19]. The principle of this method is that the C/N0 and GNSS measurements are recorded by the same tracking loops, so the signal qualities C/N0 are highly consistent with the precision of GNSS measurements [20]. It was firstly considered by Talbot [21], and the SIGMA series models [22–24] are frequently applied. Although the elevation and C/N0 can both reflect the observation precisions to some extent, they have different characteristics. The elevation model can reflect better the receiver noise and the atmospheric delays. It is proven that the receiver noise is a function of elevation by the zero-baseline method [25–27]. Additionally, the atmospheric delays have the longer propagation path in the case of low elevations [28,29]. Hence, the precision of the atmospheric correction models are also elevation-dependent. On the contrary, the C/N0 model is much more sensitive to the site-specific errors, especially the reflective and diffractive multipath [18,30,31]. It is found that the observations with low C/N0 values usually suffer the multipath or even the NLOS reception [19,32,33], hence, the observation precisions become low. Unfortunately, most existing weighting models use the elevation only or C/N0 only as an indicator, and seldom take full advantages of the above two indicators. In addition, it is found that C/N0 will deviate from the nominal values when the signal distortions are severe [31,33,34], whereas the traditional C/N0 models do not fully consider this important property and just indirectly and partly reflect the severity of site-specific unmodeled errors. Since the conventional weighting models cannot reflect the reality especially in urban environments, where the site-specific unmodeled errors are easily significant, a priori weighting model considering these unmodeled effects is usually needed. In this research, we propose a real-time adaptive weighting model for the code observations considering the site-specific unmodeled errors. By taking full advantages of the complementarity of elevation and C/N0, a combination of these two indicators is applied into the proposed model, which consists of two parts. The first one is the precision of receiver noise and atmospheric delay described by the elevation, and the second one is the precision of site-specific unmodeled error described by the C/N0. Unlike the traditional C/N0 models, the template functions of C/N0 and its precision, which can estimate the nominal values, are applied to adaptively estimate the precisions of site-specific unmodeled errors. The principle of the second part is that the measured C/N0 will deviate from the nominal values when the signal distortions are severe. It is worth noting that different from the phase observations which have a time-lag problem between multipath and C/N0 [35,36], the signal distortions of
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code observations can be directly reflected by the C/N0 [20,31]. Therefore, the proposed model is especially suitable for the undifferenced code observations. Actually, this model can be regarded as an unmodeled-error-weighted model, which can be degenerated to the unmodeled-error-ignored or unmodeled-error-float models. Specifically, when the precision of these site-specific unmodeled errors is set to zero in the unmodeled-error-weighted model, the solutions are the same as those of the unmodeled-error-ignored model. On the other hand, the solutions of the proposed model become equivalent with those of the unmodeled-error-float model if the precision of these site-specific unmodeled errors is close to infinity. To evaluate the performance of the proposed method, static and kinematic experiments with distinct impacts of site-specific unmodeled errors were carried out, where both the SPP and RTD solutions are tested. 2. Methodology Firstly, this section describes the functional and stochastic models of SPP and RTD, and the unreality of traditional weighting schemes is emphasized. Secondly, a real-time adaptive weighting model is proposed, which can be used to mitigate the site-specific unmodeled errors adaptively. 2.1. GNSS SPP and RTD Mathematical Models The GNSS observation equation of code observations for the receiver k and satellite p on frequency i reads [37,38]: p p p p p p Pk,i = ρk + cδtk − cδt p + ξ k,i − ξ p,i + Ik,i + Tk + Mk,i + ε k,i , (1) p
where ρk is the receiver-to-satellite range; c is the speed of light in a vacuum; δtk and δt p are the receiver and satellite clock errors, respectively; ξ k, i and ξ p,i are the code hardware delays with respect to receiver p p p and satellite, respectively; Ik, i and Tk are the ionospheric and tropospheric delays, respectively; Mk, i is p
the site-specific unmodeled error (mainly referring to the multipath and NLOS reception); and ε k, i is the receiver noise. It is worth noting that the time group delay and the intersignal correction are assumed to be corrected by the ephemeris file, and the antenna code center offsets of the receiver and satellite are also corrected. Hence, they are not listed in the observation equation. When using the SPP, the satellite position and clock errors can be computed by the broadcast ephemerides. The ionospheric delay can be corrected by the Klobuchar model where the coefficients are given in the broadcast ephemerides [39]. The tropospheric delay can be corrected by the Hopfield model using the standard meteorological data [40]. Finally, the receiver position and clock errors can be estimated by the LS adjustment. However, it is noted that the correction model cannot fully eliminate the atmospheric delays, and the site-specific unmodeled errors are always existent in the functional model. In case of RTD, the functional model of the double differenced (DD) code observations reads: pq
pq
pq
pq
pq
pq
Pkm,i = ρkm + Ikm,i + Tkm + Mkm,i + ε km,i ,
(2)
where k and m are the reference and rover stations, respectively; and p and q are the reference and common satellites, respectively. It can be found that the RTD is more precise than the SPP since the receiver and satellite clock errors as well as their hardware delays are all eliminated. The ionospheric and tropospheric delays can be further removed by the DD technique to a great extent. The residual atmospheric delays can even be ignored if the baseline length is not too long (e.g., less than 10 km). However, the multipath effects are usually ignored even they are significant. For the stochastic modeling, as mentioned above, the two most popular indicators are elevation and C/N0. The elevation model σele often uses the sine functions as [16,17,26]: 2 σele = a/(b + sin θ )2 ,
(3)
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where θ is the elevation in degree; a and b are the parameters to be estimated. As usual, the elevation model can be simplified as [14,15]: 2 = a/ sin2 θ, (4) σele The C/N0 model σC/N0 can be expressed as [24]: 2 = c + d·10− σC/N0
C/N0 10
,
(5)
where c and d are also the parameters to be estimated. Similarly, the more commonly used simplified model is [21–23]: C/N0 2 σC/N0 = d·10− 10 , (6) If the terms on the right side of Equations (1) or (2) can all be parameterized and/or modeled perfectly, the observation precisions can be calculated: σP2 = σε2 ,
(7)
where σP and σε are the standard deviations (STDs) of the code observations and receiver noise, respectively. However, due to the existence of these site-specific unmodeled errors in code positioning, and the limited precisions of these atmospheric correction models, the observation precisions are often influenced significantly, thus indicating the unreality of Equation (7) at this time. The precisions then can be expressed in a more precise form: 2 σP2 = σU + σ2A + σ2ε ,
(8)
where σU and σA are the STDs of the site-specific unmodeled errors and atmospheric delays after being corrected by the correction models. It indicates that a more reliable and accurate weighting model should consider all these error sources, especially the site-specific unmodeled effects. It is worth noting that if the part of ionospheric delays is significant, the σA can also be treated by an ionosphere-weighted model [41]. 2.2. A Real-Time Adaptive Weighting Model To describe the observation precisions more precisely, especially when the site-specific unmodeled errors are significant, a real-time adaptive weighting model is proposed. By taking full advantages of the complementarity of C/N0 and elevation, these two indicators are combined in the proposed model. Specifically, the C/N0 will be stable when the observation environment is ideal, then the nominal values for a certain elevation, i.e., C/N0nom (θ ). can be estimated by a so-called template function. The template function is determined by the reference data observed in a low-multipath environment. Note that unless the signal strength is changed, the template function needs to be determined for only once in advance since it is highly stable for a giving receiver and satellite on one frequency [42]. Additionally, the template function is a function of elevation since the C/N0 values are elevation-dependent mainly due to the antenna gain. Therefore, the C/N0 template function is: C/N0nom (θ ) = f 1 (θ ),
(9)
where the function f 1 usually denotes the third-order polynomial. Based on the reference data observed in a low-multipath environment, the STDs of the corresponding nominal C/N0, i.e., STDC/N0nom (θ ), can also be estimated by another template function of elevation: STDC/N0nom (θ ) = f 2 (θ ), where the function f 2 also usually denotes the third-order polynomial.
(10)
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In order to determine the highly reliable template functions, the modeling procedure is generally as follows. Firstly, the C/N0 patterns from different satellites need to be checked whether they are significantly different. Only when the C/N0 of different satellites share the same pattern, can we use the same template function. Secondly, the raw C/N0 observations can be removed if the values exceed double or triple STD ranges in each elevation interval. In the end, the template functions of C/N0 and its precision can be determined by using Equations (9) and (10), respectively. Based on the above template functions, since the measured C/N0 will fluctuate around the nominal values when the site-specific unmodeled errors are severe, the precisions of code multipath and other space loss caused by the signal propagation (e.g., NLOS reception) can be described according to Equations (5) and (6): 2 σU
− C/N0 nom (θ ) 2 − C/N010 2 , 10 − 10 = σC/N0 − σC/N0nom = d· 10
(11)
Here the absolute symbol ‘|.|’ is used because the C/N0 may also be larger than the nominal values when these unmodeled effects are significant. On the other hand, since the receiver noise and atmospheric delays can be estimated better by the elevation, the elevation model is set as the foundation part of the adaptive model as follows: σ2A + σ2ε = a/ sin2 θ,
(12)
According to Equations (8), (11), and (12), the fundamental mathematical expression of the priori adaptive model σada can be given by: C/N0nom (θ ) C/N0 2 , 10 σada = a/ sin2 θ + d· 10− 10 − 10−
(13)
However, Equation (13) cannot be applied directly because there are two unknown coefficients a and d, which cannot be estimated easily by fitting the precisions with the LS criterion mainly due to the unpredictable of site-specific unmodeled errors. To solve this problem, the above fundamental mathematical model needs to be refined. In this research, based on Equations (4) and (6), when the elevation is 90 degrees and the measurement environment is ideal, the observation precision can be regarded as the only result of receiver noise to a great extent [27,43]. It leads to the basic relations: σP2 = σ2ε = a/ sin2 (90◦ ) = d·10− which can be slightly rearranged to: d = a·10
C/N0nom (90◦ ) 10
C/N0nom (90◦ ) 10
,
(14)
,
(15)
Finally, the thresholds based on the template functions of the C/N0 and its precision are used to make the adaptive model more stable in real applications. That is, if the difference between the measured C/N0 and the nominal C/N0 reaches or exceeds the given threshold, the unmodeled error mitigation will be applied. In conclusion, the practical mathematical expression of the priori adaptive model is given by: 2 σada =
a/ sin2 θ
− C/N0 C/N0nom (θ ) − 2 10 a/ sin θ + a · γ · 10 10 − 10 C/N0nom (90◦ )
|C/N0 − C/N0nom (θ )| < T |C/N0 − C/N0nom (θ )| ≥ T
,
(16)
10 where γ = 10 , and the threshold T = β·STDC/N0nom (θ ) with a user-defined scale factor β. Because the nominal C/N0 for a certain elevation can be treated independent and normally distributed, the scale factor can be usually set to 2 since an approximately 95% confidence interval can be obtained. That is, the threshold T is applied to test the significance of these site-specific
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unmodeled errors. If the differences between the C/N0 and the nominal C/N0 are not greater than the threshold, the site-specific unmodeled errors are not significant. Hence, if there is not much need to mitigate these site-specific unmodeled effects for such cases, then the σU is equal to zero. In fact, this unmodeled-error-weighted model is the most general method for unmodeled error mitigation. If the unmodeled error is insignificant, the template functions can be used to decrease the σU . When we set σU → 0 , the solutions of the proposed model equal the ones of the unmodeled-error-ignored model. On the other hand, if the unmodeled error is significant, the template functions can also be used to increase the σU . When the σU → ∞ , the solutions of the proposed model become equivalent with the ones of the unmodeled-error-float model. 3. Data and Experiments In this study, we tested a Trimble R10 with the antenna embedded in the receiver. It is manufactured by the Trimble company in Sunnyvale, California, USA. Firstly, the template functions of Remote Sens. 2018, 10, x FOR PEER REVIEW 6 of 18 C/N0 and its precision are determined. As aforementioned, the template functions can be determined of C/N0 precision are receiver determined. aforementioned, the template functions be was in advance forand onlyitsonce. The test wasAs placed on a permanent reference station, can which determined in advance for only once. The test receiver on a permanent reference station, on the rooftop of the Engineering building (ENB) at was the placed University of Calgary. In this situation, which was ondual the rooftop of the Engineering building at the University Calgary. In this of twenty-four hour frequency C/N0 observations on(ENB) frequencies C1 and P2ofwith the sampling situation, twenty-four hour dual frequency C/N0 observations on frequencies C1 and P2 with the 1 s were obtained from 00:00:00 to 23:59:59 on 24 November 2017 (GPS time). This ideal environment sampling of 1 s were obtained from 00:00:00 to 23:59:59 on 24 November 2017 (GPS time). This ideal can help us to obtain the reliable reference nominal C/N0, then all the available Global Positioning environment can help us to obtain the reliable reference nominal C/N0, then all the available Global System (GPS) satellites with 0 to 90 degrees can be tracked, as can be seen from the daily skyplot of Positioning System (GPS) satellites with 0 to 90 degrees can be tracked, as can be seen from the daily visible satellites in Figure 1. in Figure 1. skyplot of visible satellites
Figure Twenty-four hour skyplot of visible satellites observed from observed the permanent reference station. Figure 1. 1.Twenty-four hour skyplot of GPS visible GPS satellites from the permanent reference station. Secondly, in order to validate the effectiveness of the proposed method, two static and one kinematic experiments were carried out. To evaluate the performance of the proposed method, two Secondly, in order to validate are the used. effectiveness method, two static and types of code-based positioning The first of onethe is proposed the SPP, where the ionospheric and one tropospheric delays were the Klobuchar andthe Hopfield models, respectively. The second kinematic experiments werecorrected carriedby out. To evaluate performance of the proposed method, one is of thecode-based RTD, wherepositioning the ionosphere-fixed model applied. The and two types are used. The [41] firstand onethe is Hopfield the SPP, model wherewere the ionospheric reference station is UCAL (−1,641,945.119 m, −3,664,803.809 m, and 4,940,009.252 m, in WGS84 tropospheric delays were corrected by the Klobuchar and Hopfield models, respectively. The second coordinate system) which is also located at the University of Calgary from the International GNSS one is the RTD, where the ionosphere-fixed model [41] and the Hopfield model were applied. Service [44]. Four types of weighting models are applied into the SPP and RTD for comparison: (A) The reference station is UCAL (−1,641,945.119 m, −3,664,803.809 m, and 4,940,009.252 m, in WGS84 the equal-weight model (EQUM) which assumes the variances of all the observations are equal; (B) coordinate system) which is also located at the University of Calgary from the International GNSS the elevation model (ELEM) using Equation (4); (C) the C/N0 model (CN0M) using Equation (6); (D) Service Four adaptive types of weighting models areEquation applied(16). intoFor the these SPP and for comparison: the[44]. proposed model (ADAM) using four RTD weighting models, to (A) the equal-weight model (EQUM) assumes the variances all the are equal; (B) the obtain the comparable results,which the cut-off elevations are the of same, andobservations the scale factor 𝛽 of the elevation model using Equation (4); (C) the C/N0 model (CN0M) using Equation (6); (D) the ADAM is set(ELEM) to 2. The first static experiment was carried out as dataset no. 1. We recorded 1-Hz GPS C1 and P2 data for consecutive 1 h from 19:00:00 to 19:59:59 on 25 November 2017 (GPS time). The coordinates of the test station are precisely known by applying the real-time kinematic positioning (RTK). When using the relative positioning, the baseline length is approximately 133 m. Figure 2 shows the environment of the test receiver placed nearby the blocks of ENB. It can be seen that the observation environment is like in a dense high-rise building group that could offer a variety of reflective multipath and NLOS reception, where the signals could be reflected and blocked by the ENB.
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proposed adaptive model (ADAM) using Equation (16). For these four weighting models, to obtain the comparable results, the cut-off elevations are the same, and the scale factor β of the ADAM is set to 2. The first static experiment was carried out as dataset no. 1. We recorded 1-Hz GPS C1 and P2 data for consecutive 1 h from 19:00:00 to 19:59:59 on 25 November 2017 (GPS time). The coordinates of the test station are precisely known by applying the real-time kinematic positioning (RTK). When using the relative positioning, the baseline length is approximately 133 m. Figure 2 shows the environment of the test receiver placed nearby the blocks of ENB. It can be seen that the observation environment is like in a dense high-rise building group that could offer a variety of reflective multipath and NLOS reception, where the signals could be reflected and blocked by the ENB. Remote Sens. 2018, 10, x FOR PEER REVIEW 7 of 18 Remote Sens. 2018, 10, x FOR PEER REVIEW
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Figure 2. Observation environment of dataset no. 1 near buildings.
Figure 2. Observation environment of dataset no. 1 near buildings. The second static experiment was carried out as dataset no. 2. One-hour GPS C1 and P2 Figure 2. Observation environment of dataset no. 1 near buildings.
observations with the sampling was interval of 1 sout were 21:30:00 to 22:29:59 The second static experiment carried as collected dataset from no. 2. One-hour GPS on C127and P2 November 2018 (GPS time). The coordinates of the test station are precisely known by applying the observations with the sampling intervalwas of 1carried s wereout collected fromno. 21:30:00 to 22:29:59 on 27 The second static experiment as dataset 2. One-hour GPS C1 andNovember P2 RTK, and the baseline length is approximately 396 m when using the RTK and RTD. Figure 3 2018 (GPS time). with The the coordinates of the test precisely applying observations sampling interval of 1station s were are collected from known 21:30:00 by to 22:29:59 on the 27 RTK, illustrates the test receiver deployed nearby the trees on the campus of University of Calgary. It is November (GPSistime). The coordinates test station known applying the and the baseline2018 length approximately 396 of mthe when using are theprecisely RTK and RTD.byFigure 3 illustrates clear that the observation environment is like in a heavy urban-forested area where a variety of RTK, and the baseline length is approximately 396 m when using the RTK and RTD. Figure 3 the testdiffractive receiver multipath, deployedand nearby trees on the occur, campus of University even the NLOS reception, caused by the trees.of Calgary. It is clear that the illustrates the test receiver deployed nearby the trees on the campus of University of Calgary. It is observation environment is like in a heavy urban-forested area where a variety of diffractive multipath, clear that the observation environment is like in a heavy urban-forested area where a variety of and even NLOS reception, occur, caused by the trees. diffractive multipath, and even NLOS reception, occur, caused by the trees.
Figure 3. Observation environment of dataset no. 2 near trees.
A kinematic experiment was then carried out as dataset no. 3. 1-Hz GPS C1 data was recorded Figure 3. Observation environment of dataset no. 2 near trees. in approximately Figure consecutive 40 min from 19:58:33 toof20:38:23 November 3. Observation environment dataseton no.262 near trees. 2018 (GPS time). When using the RTD, the longest length of this kinematic baseline is less than 1 km. This kinematic A kinematic experiment was then carried out as dataset no. 3. 1-Hz GPS C1 data was recorded experiment was deployed on the main road at the University of Calgary, as shown in Figure 4. It can in approximately consecutive 40 min from 19:58:33 to 20:38:23 on 26 November 2018 (GPS time). be found that the observation environment could offer different situations during the test. When using the RTD, the longest length of this kinematic baseline is less than 1 km. This kinematic Specifically, it is noted that the site-specific unmodeled errors will be significant when the test experiment was deployed on the main road at the University of Calgary, as shown in Figure 4. It can receiver is located between the Energy Environment Experiential Learning (EEEL) and Earth Science be found that the observation environment could offer different situations during the test. (ES) buildings, as shown in the left panel (a) of Figure 4 marked with yellow rectangle. These Specifically, it is noted that the site-specific unmodeled errors will be significant when the test
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A kinematic experiment was then carried out as dataset no. 3. 1-Hz GPS C1 data was recorded in approximately consecutive 40 min from 19:58:33 to 20:38:23 on 26 November 2018 (GPS time). When using the RTD, the longest length of this kinematic baseline is less than 1 km. This kinematic experiment was deployed on the main road at the University of Calgary, as shown in Figure 4. It can be found that the observation environment could offer different situations during the test. Specifically, it is noted that the site-specific unmodeled errors will be significant when the test receiver is located between the Energy Environment Experiential Learning (EEEL) and Earth Science (ES) buildings, Remote Sens. 2018, 10, x FOR PEER REVIEW 8 of 18 as shown in the left panel (a) of Figure 4 marked with yellow rectangle. These unmodeled effects can be further confirmed bybe thefurther specificconfirmed environments of specific EEEL and ES buildingsof(see the and rightES panel (b) of unmodeled effects can by the environments EEEL buildings Figure 4). (see the right panel (b) of Figure 4).
Figure 4. (a) Trajectory of dataset no. 3 represented as a blue line on the main road at the University Figure 4. (a) Trajectory of dataset no. 3 represented as a blue line on the main road at the University of Calgary, where the EEEL and ES buildings are marked with yellow rectangles. The green and red of Calgary, where the EEEL and ES buildings are marked with yellow rectangles. The green and red points denote the starting point and finishing point, respectively; and (b) the test environment points denote the starting point and finishing point, respectively; and (b) the test environment between between the EEEL and ES buildings. the EEEL and ES buildings.
4. Results and Discussion 4. Results and Discussion In this section, the template functions are analyzed firstly. After that, the results of the two static In this section, the template functions are analyzed firstly. After that, the results of the two static and one kinematic experiments are discussed, where the proposed model ADAM is compared with and one kinematic experiments are discussed, where the proposed model ADAM is compared with the other three traditional models EQUM, ELEM, and CN0M. the other three traditional models EQUM, ELEM, and CN0M. 4.1. Template Template Functions Functions 4.1. As aforementioned, aforementioned, the test receiver receiver need need to to be be As the highly highly reliable reliable template template functions functions of of the the test determined. Firstly, the means and STDs of C/N0 values of each satellite (except for G04 due to outage) determined. Firstly, the means and STDs of C/N0 values of each satellite (except for G04 due to from elevations of 35 to of 8035 degrees are compared. The reason that most influences can be outage) from elevations to 80 degrees are compared. The is reason is thatexternal most external influences reduced above 35-degree elevations [27,43] and several satellites do not have the measurements can be reduced above 35-degree elevations [27,43] and several satellites do not have the measurements above 80-degree 80-degree elevations. elevations. Figure Figure 55 illustrates illustrates the the mean meanC/N0 C/N0 on as well well as as the the one-time one-time above on C1 C1 and and P2, P2, as STD ranges ranges of of each each satellite. STD satellite. It It can can be be seen seen that that the the means means and and STDs STDs are are rather rather similar similar with with each each other, other, thus indicating that the same template functions can be used among all the GPS satellites. thus indicating that the same template functions can be used among all the GPS satellites. Secondly, Secondly, the raw raw C/N0 C/N0 observations in each each one-degree one-degree the observations are are removed removed if if the the values values exceed exceed double double STD STD ranges ranges in interval. Figure and P2P2 (C/N02). It isItclear thatthat the interval. Figure 66 shows shows the the refined refined reference referenceC/N0 C/N0ofofC1 C1(C/N01) (C/N01) and (C/N02). is clear reference measurements from different GPS satellites are elevation-dependent and consistent with the reference measurements from different GPS satellites are elevation-dependent and consistent with each other, other, which which can can be be fit fit by by the the third-order third-orderpolynomials. polynomials. each
from elevations of 35 to 80 degrees are compared. The reason is that most external influences can be reduced above 35-degree elevations [27,43] and several satellites do not have the measurements above 80-degree elevations. Figure 5 illustrates the mean C/N0 on C1 and P2, as well as the one-time STD ranges of each satellite. It can be seen that the means and STDs are rather similar with each other, thus indicating that the same template functions can be used among all the GPS satellites. Secondly, the raw C/N0 observations are removed if the values exceed double STD ranges in each one-degree Remote Sens. 2018, 10,interval. 1157 Figure 6 shows the refined reference C/N0 of C1 (C/N01) and P2 (C/N02). It is clear that the reference measurements from different GPS satellites are elevation-dependent and consistent with each other, which can be fit by the third-order polynomials.
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Figure 5. Mean C/N0 values of each satellite, where the error bars are the one-time STD ranges for each satellite. (a) C/N0 C1 (C/N01); and (b)where C/N0 of P2 (C/N02). Figure 5. Mean C/N0 values of ofeach satellite, the error bars are the one-time STD ranges for each satellite. (a) C/N0 of C1 (C/N01); and (b) C/N0 of P2 (C/N02). Remote Sens. 2018, 10, x FOR PEER REVIEW
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Figure 6. Refined reference C/N0 of C1 (C/N01) and P2 (C/N02) as a function of elevation.
Figure 6. Refined reference C/N0 of C1 (C/N01) and P2 (C/N02) as a function of elevation. In the end, the template functions of C/N0 on two frequencies are determined by using Equation (9): C/N01
𝜃 = 37.79 + 0.2611𝜃 − 6.044 × 10 𝜃 − 7.707 × 10 𝜃 ,
(17)
In the end, the template functions of C/N0 on two frequencies are determined by using (18) 𝜃 = 18.12 + 0.5213𝜃 − 4.302 × 10 𝜃 + 1.140 × 10 𝜃 , C/N02 Equation (9): The template functions of C/N0 precision on these two frequencies can also be determined by the STDs of every 1-degree elevation interval by using Equation (10):
−4 2 −6 3 C/N01nomSTD + 0.2611θ −+6.044 −×7.707 θ , (θ )/ = 37.79 𝜃 = 2.672 − 0.06788𝜃 1.106 ×× 1010 𝜃 −θ 5.317 10 𝜃 ,× 10 (19) STD
/
𝜃 = 3.310 − 0.07618𝜃 + 9.216 × 10 𝜃 − 3.438 × 10 𝜃 ,
(17)
(20)
C/N02nom (θ ) = 18.12 + 0.5213θ − 4.302 × 10−3 θ 2 + 1.140 × 10−5 θ 3 ,
(18)
4.2. Static Experiment Near Buildings
Figure 6. Refined reference C/N0 of C1 (C/N01) and P2 (C/N02) as a function of elevation. The template functions of C/N0 precision on these two frequencies can also be determined by the The results of the first static experiment are shown here. In dataset no. 1, the averages of satellite numbers positional dilution of precision (PDOP) values areare 6.5 and 4.2, respectively, as shown In theand end, the template functions of on twoEquation frequencies determined by using Equation (9): STDs of every 1-degree elevation interval byC/N0 using (10):
in Figure 7. It can be used to further confirm the poor observation environment of dataset no. 1. C/N01 𝜃 = 37.79 + 0.2611𝜃 − 6.044 × 10 𝜃 − 7.707 × 10 𝜃 , (17)
−3 2 −6 3 STDC/N01nom C/N02 − 0.06788θ 1.106 θ ×− (θ ) = 2.672 𝜃 = 18.12 + 0.5213𝜃 −+ 4.302 × 10 ×𝜃 10 + 1.140 105.317 𝜃 , × 10 (18)θ ,
(19)
The template functions of C/N0 precision on these two frequencies can also be determined by the STDs of every elevation interval by using Equation STD − 0.07618θ + 9.216 ×(10): 10−4 θ 2 − 3.438 × 10−6 θ 3 , (θ1-degree ) = 3.310
(20)
C/N02nom
STD
/
𝜃 = 2.672 − 0.06788𝜃 + 1.106 × 10 𝜃 − 5.317 × 10 𝜃 ,
4.2. Static Experiment Near Buildings 𝜃 STD /
= 3.310 − 0.07618𝜃 + 9.216 × 10 𝜃 − 3.438 × 10 𝜃 ,
(19) (20)
The results of static experiment are shown here. In dataset no. 1, the averages of satellite 4.2.the Staticfirst Experiment Near Buildings numbers and positional dilution of precision (PDOP) are 6.5 4.2, respectively, as shown in The results of the first static experiment are shownvalues here. In dataset no. 1,and the averages of satellite numbers and positional dilution of precision (PDOP) values are 6.5 and 4.2, respectively, as shown Figure 7. It can be used to further confirm the poor observation environment of dataset no. 1. in Figure 7. It can be used to further confirm the poor observation environment of dataset no. 1. Figure 7. (a) Satellite numbers; and (b) PDOP values. The data are calculated from the C1 observations of dataset no. 1.
The positioning errors (relative to the known precise coordinates) of the single-frequency SPP results by using the EQUM, ELEM, CN0M, and ADAM are illustrated in Figure 8. The E, N, and U denote the east, north, and up directions, respectively. It can be clearly seen that the CN0M and ADAM perform much better than the other two models, especially for the first around 3000 epochs. One may conclude that the C/N0-dependent models (CN0M and ADAM) are necessary in challenging environments. Compared with the results of CN0M and ADAM, the ADAM is more
Figure 7. (a) Satellite numbers; and (b) PDOP values. The data are calculated from the C1 observations
Figure 7. (a) Satellite numbers; and (b) PDOP values. The data are calculated from the C1 observations of dataset no. 1. of dataset no. 1. The positioning errors (relative to the known precise coordinates) of the single-frequency SPP results by using the EQUM, ELEM, CN0M, and ADAM are illustrated in Figure 8. The E, N, and U denote the east, north, and up directions, respectively. It can be clearly seen that the CN0M and ADAM perform much better than the other two models, especially for the first around 3000 epochs. One may conclude that the C/N0-dependent models (CN0M and ADAM) are necessary in challenging environments. Compared with the results of CN0M and ADAM, the ADAM is more
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The positioning errors (relative to the known precise coordinates) of the single-frequency SPP results by using the EQUM, ELEM, CN0M, and ADAM are illustrated in Figure 8. The E, N, and U denote the east, north, and up directions, respectively. It can be clearly seen that the CN0M and ADAM perform much better than the other two models, especially for the first around 3000 epochs. One may conclude that the C/N0-dependent models (CN0M and ADAM) are necessary in challenging Remote Sens. 2018, Compared 10, x FOR PEER REVIEW 10 of 18 environments. with the results of CN0M and ADAM, the ADAM is more precise. It indicates that, unlike the ADAM, the CN0M only indirectly and partly reflect the precisions of site-specific precise. It indicates that, unlike the ADAM, the CN0M only indirectly and partly reflect the precisions unmodeled errors. That is, the template functions in ADAM are, indeed, effective. Table 1 presents the of site-specific unmodeled errors. That is, the template functions in ADAM are, indeed, effective. corresponding statistical results of SPP and RTD using the four types of weighting models. It can be Table 1 presents the corresponding statistical results of SPP and RTD using the four types of confirmed that, compared with the SPP results of EQUM, ELEM, and CN0M, the three-dimensional weighting models. It can be confirmed that, compared with the SPP results of EQUM, ELEM, and root mean square (3D RMS) values of ADAM are improved by 50.0%, 41.4%, and 13.9%, respectively. CN0M, the three-dimensional root mean square (3D RMS) values of ADAM are improved by 50.0%, The dual-frequency RTD results by using the four types of weighting models are illustrated in Figure 9. 41.4%, and 13.9%, respectively. The dual-frequency RTD results by using the four types of weighting Similarly, it can be clearly seen that the proposed method is the most precise. According to the Table 1, models are illustrated in Figure 9. Similarly, it can be clearly seen that the proposed method is the the improvement percentages of the ADAM compared with the EQUM, ELEM, and CN0M are 27.8%, most precise. According to the Table 1, the improvement percentages of the ADAM compared with 19.6%, and 5.3%, respectively. the EQUM, ELEM, and CN0M are 27.8%, 19.6%, and 5.3%, respectively.
Figure 8. 8. Positioning Positioningdifferences differencesbetween betweenthe theSPP SPPsolutions solutionsand andthe theprecise precisecoordinates coordinates dataset Figure ofof dataset no.no. 1. 1. (a) Results of the equal-weight model (EQUM); results of the elevation model (ELEM); results (a) Results of the equal-weight model (EQUM); (b)(b) results of the elevation model (ELEM); (c) (c) results of of the C/N0 model (CN0M); and results adaptive model (ADAM). the C/N0 model (CN0M); and (d)(d) results of of thethe adaptive model (ADAM). Table1.1.Statistical Statisticalresults resultsof ofSPP SPPand andRTD RTDusing usingthe theequal-weight equal-weightmodel model(EQUM), (EQUM), elevation elevation model model Table (ELEM), C/N0 model (CN0M), and adaptive model (ADAM). The RMS values of E, N, and U (ELEM), C/N0 model (CN0M), and adaptive model (ADAM). The RMS values of E, N, and U directions, directions, and 3D RMS werefrom calculated and 3D RMS values werevalues calculated datasetfrom no. 1dataset (m). no. 1 (m).
EQUM EQUM E 11.73 11.73 N E 22.44 N 22.44 U 36.03 U 36.03 3D 3D 44.04 44.04
SPP SPP ELEM CN0M ELEM CN0M 11.35 11.79 11.35 11.79 18.43 12.37 18.43 12.37 30.74 18.99 30.74 18.99 37.59 25.55 37.59 25.55
ADAM ADAM 11.15 11.15 10.74 10.74 15.65 15.65 22.01 22.01
EQUM EQUM 4.36 4.36 8.69 8.69 14.92 14.92 17.81 17.81
RTD CN0M ADAM CN0M ADAM 3.83 3.59 3.83 3.595.91 6.00 6.00 5.91 11.57 10.84 11.57 10.84 13.58 12.86 13.58 12.86
RTD
ELEM ELEM 4.19 4.19 7.63 7.63 13.43 13.43 16.00 16.00
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Figure 9. Positioning differences between RTD solutions and precise coordinatesofofdataset datasetno. 1. Figure 9. Positioning differences between thethe RTD solutions and thethe precise coordinates no. 1. (a) Results of the equal-weight model (EQUM); (b) results of the elevation model (ELEM); (c) (a) Results of the equal-weight model (EQUM); (b) results of the elevation model (ELEM); (c) results of results of the C/N0 model (CN0M); and (d) results of the adaptive model (ADAM). Figuremodel 9. Positioning between solutions and(ADAM). the precise coordinates of dataset the C/N0 (CN0M);differences and (d) results of the the RTD adaptive model no. 1. (a) Results of the equal-weight model (EQUM); (b) results of the elevation model (ELEM); (c) The reason why the ADAM is better than any other traditional methods is that the ADAM can results of the C/N0 model (CN0M); and (d) results of the adaptive model (ADAM).
The reason why the ADAM is better than any 10 other traditional methodsand is that the ADAM can mitigate these unmodeled effects greatly. Figure illustrates the elevations dual-frequency C/N0 of all the used satellites denoted with different colors. It can be seen that the C/N0 are mitigate these unmodeled effects greatly. Figure 10 illustrates the elevations and dual-frequency C/N0 The reason why the ADAM is better than any other traditional methods is that the patterns ADAM can significantly different from the reference values in Figure 6. The reason is that several GPS signals of allmitigate the used satellites denoted with different colors. It can be seen that the C/N0 patterns these unmodeled effects greatly. Figure 10 illustrates the elevations and dual-frequency are suffer the multipath and even the NLOS reception the blocks ofreason ENB.that In itpatterns canGPS alsoare be C/N0 of all the usedfrom satellites with different colors. It can seen the C/N0 significantly different the denoted reference values in due Figure 6. Thebe is addition, that several signals found that the low elevations do not necessarily mean the low C/N0. It is reasonable since the different values in Figure 6. The reason is that In several GPS signals suffersignificantly the multipath and from eventhe thereference NLOS reception due the blocks of ENB. addition, it can also observations with high cannot ensure that they are free from In the multipath andalso NLOS suffer the multipath and elevations even the reception due the blocks of ENB. it can be the be found that the low elevations doNLOS not necessarily mean the low C/N0. addition, It is reasonable since reception. Actually, this is why the positioning results of the ELEM and CN0M are significantly found that the low elevations do not necessarily mean the low C/N0. It is reasonable since the observations highother elevations cannot ensure that they are free from the multipath and NLOS different with withwith each in this situation. observations high elevations cannot ensure that they are free from the multipath and NLOS reception. Actually, this is why the positioning results of the ELEM and CN0M are significantly reception. Actually, this is why the positioning results of the ELEM and CN0M are significantly different with each other in this situation. different with each other in this situation.
Figure 10. (a) Elevations of used satellites; (b) C/N0 of C1 (CN01) of used satellites; and (c) C/N0 of P2 (CN02) of used satellites. Each color denotes one satellite calculated from dataset no. 1. Figure 10. (a) Elevations of used satellites; (b) C/N0 of C1 (CN01) of used satellites; and (c) C/N0 of Figure 10. (a) Elevations of used satellites; (b) C/N0 of C1 (CN01) of used satellites; and (c) C/N0 of P2 (CN02) of used satellites. Each color denotes one satellite calculated from dataset no. 1. P2 (CN02) of used satellites. Each color denotes one satellite calculated from dataset no. 1.
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The results of the second static experiment are then discussed. In dataset no. 2, the averages The resultsand of thePDOP second static experiment are then In dataset no.as 2, the averages of satellite numbers values are 11.7 and discussed. 1.5, respectively, shown inofFigure 11. satellite numbers and PDOP values are 11.7 and 1.5, respectively, as shown in Figure 11. Therefore, Remote Sens. 2018,with 10, x FOR PEER REVIEW of 18 Therefore,compared compared dataset no. 1, the proposed method can be evaluated in a 12 relatively good with dataset no. 1, the proposed method can be evaluated in a relatively good observation observation condition. condition. 4.3. Static Experiment Near Trees The results of the second static experiment are then discussed. In dataset no. 2, the averages of satellite numbers and PDOP values are 11.7 and 1.5, respectively, as shown in Figure 11. Therefore, compared with dataset no. 1, the proposed method can be evaluated in a relatively good observation condition.
Figure 11. (a) Satellite numbers; and (b) PDOP values. The data are calculated from the C1
Figure 11. observations (a) Satelliteofnumbers; and (b) PDOP values. The data are calculated from the C1 observations dataset no. 2. of dataset no. 2. The positioning errors of the single-frequency SPP results by using the EQUM, ELEM, CN0M, and ADAM are presented in Figure 12. It can be easily seen that the ADAM is still the most precise. The positioning errors of the SPP results using EQUM, ELEM, CN0M, Figure 11. (a)accuracy Satellite numbers; (b) CN0M PDOP values. The than databy are calculated from the ADAM, C1 Specifically, the of single-frequency EQUMand and are lower those of the ELEM and observations of U dataset no. 2. It12. and ADAM are presented in Figure It canthat be the easily seenisthat the ADAMindicator is still the most precise. particularly in the direction. indicates elevation an indispensable in some cases. Hence, it is necessary to use the elevation as the foundation part of the proposed model. Table Specifically, theThe accuracy of EQUM and CN0M are lower than those of ELEM and ADAM, particularly positioning errors of the single-frequency SPP results by using the EQUM, ELEM, CN0M, 2 gives the corresponding statistical results of SPP and RTD using the four types of weighting models. in the U direction. Itare indicates that the elevation an indispensable indicator some cases. Hence, and ADAM presented in Figure 12. It canELEM, beis easily that the still thein precise. Compared with the SPP results of EQUM, andseen CN0M, the ADAM 3D RMSis values ofmost ADAM are Specifically, the accuracy of EQUM and CN0M are lower than those of ELEM and ADAM, it is necessary to use the elevation as the foundation part of the proposed model. Table 2 gives the improved by 18.8%, 6.1%, and 9.4%, respectively. The dual-frequency RTD results by using the four particularly in the U direction. It indicates that the elevation is an indispensable indicator in some corresponding and RTD using types of weighting types statistical of weightingresults modelsof areSPP all shown in Figure 13. the Oncefour again, the proposed methodmodels. is still theCompared cases.precise. Hence, it isshown necessary to use2,the as thepercentages foundation part ofADAM the proposed model. Table most As in Table theelevation improvement of the compared with theimproved with the SPP results of EQUM, ELEM, and CN0M, the 3D RMS values of ADAM are 2 gives the corresponding SPP and RTD using the four types of weighting models. EQUM, ELEM and CN0Mstatistical are 16.8%,results 5.6%, of and 13.2%, respectively. by 18.8%, Compared 6.1%, andwith 9.4%, The dual-frequency results by using the four the respectively. SPP results of EQUM, ELEM, and CN0M,RTD the 3D RMS values of ADAM are types of by 18.8%, 6.1%, and 9.4%, respectively. The dual-frequency RTDmethod results byisusing weighting improved models are all shown in Figure 13. Once again, the proposed still the thefour most precise. of weighting models are all shown in Figure 13. OnceADAM again, the proposed method is still the As shown types in Table 2, the improvement percentages of the compared with the EQUM, ELEM most precise. As shown in Table 2, the improvement percentages of the ADAM compared with the and CN0MEQUM, are 16.8%, 5.6%, and 13.2%, respectively. ELEM and CN0M are 16.8%, 5.6%, and 13.2%, respectively.
Figure 12. Positioning differences between the SPP solutions and the precise coordinates of dataset no. 2. (a) Results of the equal-weight model (EQUM); (b) results of the elevation model (ELEM); (c) results of the C/N0 model (CN0M); and (d) results of the adaptive model (ADAM).
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Figure 12. Positioning differences between the SPP solutions and the precise coordinates of dataset Remoteno. Sens. 10, 1157of the equal-weight model (EQUM); (b) results of the elevation model (ELEM); (c) 13 of 18 2.2018, (a) Results
results of the C/N0 model (CN0M); and (d) results of the adaptive model (ADAM). Table 2. Statistical results of SPP and RTD using the equal-weight model (EQUM), elevation model Table 2. Statistical results of SPP and RTD using the equal-weight model (EQUM), elevation model (ELEM), C/N0 model (CN0M), and adaptive model (ADAM). The RMS values of E, N, and U directions, (ELEM), C/N0 model (CN0M), and adaptive model (ADAM). The RMS values of E, N, and U and 3D RMS values were calculated from dataset no. 2 (m). directions, and 3D RMS values were calculated from dataset no. 2 (m).
E E N N U U 3D 3D
SPP SPP EQUM EQUM ELEM ELEM CN0M CN0M 3.203.20 3.183.18 3.02 3.02 3.69 3.973.97 3.643.64 3.69 4.36 5.085.08 3.933.93 4.36 7.20 6.23 6.46 7.20 6.23 6.46
RTD RTD
ADAM ADAM 3.01 3.01 3.49 3.49 3.60 3.60 5.85 5.85
EQUM EQUM
ELEM ELEM
0.41 0.41 0.65 0.65 2.32 2.32 2.44
0.43 0.43 0.62 0.62 2.01 2.01 2.15
2.44
2.15
CN0M ADAM ADAM CN0M 0.38 0.38 0.58 0.58 2.23 2.23 2.34 2.34
0.380.38 0.620.62 1.901.90 2.03
2.03
Figure 13. 13. Positioning Positioning differences differences between between the the RTD RTD solutions solutions and and the the precise precise coordinates coordinates of of dataset dataset Figure no. 2. 2. (a) results of of thethe elevation model (ELEM); (c) no. (a) Results Resultsof ofthe theequal-weight equal-weightmodel model(EQUM); (EQUM);(b)(b) results elevation model (ELEM); results of the C/N0 model (CN0M); and (d) results of the adaptive model (ADAM). (c) results of the C/N0 model (CN0M); and (d) results of the adaptive model (ADAM).
Taking a closer look at the SPP results of the ELEM and ADAM, the ADAM can also eliminate Taking a closer look at the SPP results of the ELEM and ADAM, the ADAM can also eliminate the the spikes around 2350 s, whereas the ELEM cannot. One may conclude that the ADAM can adjust spikes around 2350 s, whereas the ELEM cannot. One may conclude that the ADAM can adjust the the precisions of the site-specific unmodeled errors adaptively, since the site-specific unmodeled precisions of the site-specific unmodeled errors adaptively, since the site-specific unmodeled errors errors may become significant around 2350 s. It can be confirmed by the elevations and C/N0 of the may become significant around 2350 s. It can be confirmed by the elevations and C/N0 of the satellite satellite denoted with the cyan color in Figure 14, which illustrates the elevations and dual-frequency denoted with the cyan color in Figure 14, which illustrates the elevations and dual-frequency C/N0 of C/N0 of all the used satellites. Specifically, the satellite denoted by the cyan color has a minimum all the used satellites. Specifically, the satellite denoted by the cyan color has a minimum C/N0 around C/N0 around 2350 s, whereas its elevations are not a minimum. In conclusion, the other advantage 2350 s, whereas its elevations are not a minimum. In conclusion, the other advantage of the proposed of the proposed method is that it can adaptively mitigate the spikes of the positioning results. method is that it can adaptively mitigate the spikes of the positioning results.
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Figure 14. (a) Elevations of used satellites; (b) C/N0 of C1 (CN01) of used satellites; and (c) C/N0 of
Figure 14. (a) Elevations of used satellites; (b) C/N0 of C1 (CN01) of used satellites; and (c) C/N0 of P2 (CN02) of used satellites. Each color denotes one satellite calculated from dataset no. 2. P2 (CN02) of used satellites. Each color denotes one satellite calculated from dataset no. 2. Figure 14. (a) Elevations of used satellites; (b) C/N0 of C1 (CN01) of used satellites; and (c) C/N0 of
4.4. Kinematic P2 (CN02)Experiment of used satellites. Each color denotes one satellite calculated from dataset no. 2.
4.4. Kinematic Experiment
In the end, we present the results of the kinematic experiment. Figure 15 illustrates the satellite 4.4. Kinematic Experiment numbers andwe PDOP values no. 3.experiment. It is clear thatFigure between 150 and In the end, present thecalculated results offrom thedataset kinematic 15around illustrates the 500 satellite In the end, we present the results of the kinematic experiment. Figure 15 illustrates the satellite s, the satellite numbers become fewer and the PDOP values become larger. As aforementioned, this500 s, numbers and PDOP values calculated from dataset no. 3. It is clear that between around 150 and numbers and values calculated from dataset no. and 3. It ES is clear that between around 150 andGPS 500 is because the PDOP test receiver located between the EEEL buildings at this and several the satellite numbers becomeisfewer and the PDOP values become larger. Astime aforementioned, this is s, the satellite numbers It become fewer and thethe PDOP valuesand become larger. As aforementioned, signals are obstructed. also indicates that multipath NLOS reception inevitably existthis in because the test receiver is located between the EEEL and ES buildings at this time and several GPS is because test receiver is located between the EEEL ESthe buildings this time and several GPS some otherthe GPS signals at this time. The SPP results by and using EQUM,at ELEM, CN0M, and ADAM signals are obstructed. It also also indicates thatthat the the multipath and NLOS reception inevitably exist signals are obstructed. indicates multipath reception existininsome are illustrated in Figure It 16. Since this experiment was carriedand out NLOS in Calgary, the X inevitably and Y coordinates otherin GPS signals atsignals this time. The SPP byCompared using thewith EQUM, ELEM, CN0M, and ADAM some GPS atsystem this time. The SPP results by using the EQUM, ELEM, CN0M, and ADAM the other WGS84 coordinate are allresults negative. the other models, the ADAM is the are illustrated in Figure 16. Since thisthe experiment was carried out Calgary, the X and and coordinates are illustrated in Figure 16. Since this carried outinin Calgary, the X YYcoordinates most precise, especially when testexperiment receiver is was between EEEL and ES buildings. It can be further in the WGS84 coordinate system areofare alleach negative. theother other models, the ADAM in the WGS84 system all negative. Compared with the models, theusing ADAM theis the confirmed by coordinate the small panels subplotCompared from Figurewith 16. The RTD results by theisfour most precise, especially when the test receiver is between EEEL and ES buildings. It can be further types of weighting models are illustrated in Figure 17. Once again, the proposed method is the most most precise, especially when the test receiver is between EEEL and ES buildings. It can be further confirmed byinthe small panels each subplot from Figure 16. RTD results by using four precise, Figure 16. Actually, the reason from is that the observations contaminated the siteconfirmed bylike the small panels ofofeach subplot Figure 16.The The RTD results byby using the four types of weighting models are illustrated in Figure 17. Once again, the proposed method is the most specific unmodeled errors are adaptively down-weighted by the ADAM. In conclusion, types of weighting models are illustrated in Figure 17. Once again, the proposed method is theitmost precise, like inthat Figure 16. Actually, the reason is that thesite-specific observations contaminated bytothe sitedemonstrates theActually, proposed method canismitigate unmodeled errors a great precise, like in Figure 16. the reason that thethe observations contaminated by the site-specific specific unmodeled errors are adaptively down-weighted by the ADAM. In conclusion, it extent in kinematic modes regardless of SPP or RTD solutions, and has a great potential in urban unmodeled errors are adaptively down-weighted by the ADAM. In conclusion, it demonstrates that demonstrates that the proposed method can mitigate the site-specific unmodeled errors to a great canyon areas. the proposed method can mitigate the site-specific unmodeled errors great extent in in urban kinematic extent in kinematic modes regardless of SPP or RTD solutions, and hasto aa great potential modes regardless canyon areas. of SPP or RTD solutions, and has a great potential in urban canyon areas.
Figure 15. (a) Satellite numbers; and (b) PDOP values. The data are calculated from dataset no. 3. Figure 15. (a) Satellite numbers; and (b) PDOP values. The data are calculated from dataset no. 3.
Figure 15. (a) Satellite numbers; and (b) PDOP values. The data are calculated from dataset no. 3.
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Figure 16. SPP results (blue lines) of dataset no. 3, where the horizontal and vertical coordinates
Figure 16. SPP results (blue lines) of dataset no. 3, where the horizontal and vertical coordinates denote Figure the 16. X SPP (blue lines) of dataset no. 3,system whererespectively, the horizontal vertical coordinates denote andresults Y directions in WGS84 coordinate andand the detailed results (red the X and Y directions in WGS84 coordinate system respectively, and the detailed results (red points) in denote the X andwhen Y directions WGS84iscoordinate systemEEEL respectively, the detailed results points) in meter the testin receiver located between and ES and buildings. (a) Results of(red the meter when the test when receiver is located between EEEL and ES buildings. (a) Results of Results the equal-weight points) in meter the test(b) receiver between EEEL and ES(c) buildings. of the equal-weight model (EQUM); resultsisoflocated the elevation model (ELEM); results of(a) the C/N0 model model (EQUM); (b)results results of the elevation model (ELEM); (c) results of the C/N0 model (CN0M); equal-weight model (EQUM); resultsmodel of the elevation (CN0M); and (d) of the (b) adaptive (ADAM). model (ELEM); (c) results of the C/N0 model and (CN0M); (d) results of the adaptive model (ADAM). and (d) results of the adaptive model (ADAM).
Figure 17. RTD results (blue lines) of dataset no. 3, where the horizontal and vertical coordinates Figure the 17. X RTD (blue in lines) of dataset no. 3, whererespectively, the horizontal vertical coordinates denote andresults Y directions WGS84 coordinate system andand the detailed results (red Figure 17. RTD results (blue lines) of dataset no. 3, where the horizontal and vertical coordinates denote denote the X andwhen Y directions WGS84iscoordinate systemEEEL respectively, the detailed results points) in meter the testin receiver located between and ES and buildings. (a) Results of(red the the X and Y directions in WGS84 coordinate system respectively, and the detailed results (red points) in points) in meter when the test(b) receiver between EEEL and ES(c) buildings. of the equal-weight model (EQUM); resultsisoflocated the elevation model (ELEM); results of(a) theResults C/N0 model meter when the test receiver is located between EEEL and ES buildings. (a) Results of the equal-weight equal-weight model (EQUM); resultsmodel of the elevation (CN0M); and (d) results of the (b) adaptive (ADAM). model (ELEM); (c) results of the C/N0 model model (EQUM); (b) results of the elevation model (ELEM); (c) results of the C/N0 model (CN0M); (CN0M); and (d) results of the adaptive model (ADAM).
and (d) results of the adaptive model (ADAM).
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5. Conclusions In this paper, we developed a new real-time adaptive weighting model which can mitigate the site-specific unmodeled errors of undifferenced code observations. The proposed method combines the elevation and C/N0, where the precision of the receiver noise and atmospheric delay are described by the elevation, and the precision of site-specific unmodeled error is described by the C/N0. Unlike the traditional C/N0 models, the template functions of C/N0 and its precision are applied to adaptively estimate the precisions of site-specific unmodeled errors. The effectiveness of the adaptive weighting model is validated by the static and kinematic experiments. For the static experiment near buildings, the 3D RMS of the proposed model are improved by 35.1% and 17.6% on average compared with the other traditional models. It can be found that the observations with high elevations cannot ensure large C/N0 values in difficult environments. Additionally, the results indicate that the traditional C/N0-based model only indirectly and partly reflect the precisions of site-specific unmodeled errors. However, by taking full advantage of the template functions that can estimate the nominal values, the proposed method can mitigate these unmodeled effects to a great extent. For the static experiment near trees, 11.4% and 11.9% of improvements are obtained by using the proposed model compared with the other traditional models. The results indicate that it is necessary to use the elevation as the foundation part of the proposed model. In addition, the other advantage of the proposed model is that it can adaptively mitigate the spikes of the positioning results. For the kinematic experiment, the proposed model is also effective since the observations contaminated by the site-specific unmodeled errors can be down-weighted adaptively. In conclusion, the SPP and RTD results show that the adaptive model is superior to the conventional equal-weight, elevation and C/N0 models especially in challenging environments, such as dense high-rise building group and urban-forested area. It is worth noting that the effectiveness of the proposed weighting model relies on the precisions of template functions of used receiver. Specifically, when the precisions of template functions are high, the proposed weighting model can mitigate the site-specific unmodeled effects even if they are not very severe. On the contrary, when the precisions of template functions are low, the proposed weighting model is not very sensitive to these unmodeled effects even if they are severe. At last, in GNSS precise positioning, this new model is an important supplementary for the theory of GNSS unmodeled error processing and can be easily used to extend the elevation-based weighting model in GNSS positioning programs. Author Contributions: Z.Z. and B.L. conceived the initial idea, designed the experiment, performed the analysis, and wrote the manuscript; Y.S. and Y.G. provided guidance; and M.W. helped polish the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (41574023, 41622401, and 41731069), the Scientific and Technological Innovation Plan from Shanghai Science and Technology Committee (17511109501, 17DZ1100802, and 17DZ1100902), the National Key Research and Development Program of China (2016YFB0501802), and the Fundamental Research Funds for the Central Universities. Acknowledgments: The first author acknowledges the support of the China Scholarship Council (CSC) for his visiting Ph.D. studies at the University of Calgary. Conflicts of Interest: The authors declare no conflict of interest.
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