Medium-Range Kinematic Positioning Constraint by Ionospheric Pseudo-Observation with Elevation-Dependent Weight Yihe Li and Yunzhong Shen
Abstract The performance of RTK (real time kinematic positioning) is much degraded with increasing distance between receivers due to the presence of distance-dependent GPS errors, notably ionosphere and troposphere refraction. In this paper, we investigate the methodology for medium-range kinematic positioning constraint by ionosphere pseudo-observation with elevation-dependent weight. The ionospheric delays of double differenced (DD) observations are treated as pseudo-observations having a priori values and respective weights. This means that the ionospheric delays are modeled as the unknown parameters with prior stochastic information which is expressed as the form of ionospheric pseudoobservations. The weight of ionospheric pseudo-observation is determined according to satellite elevation. Moreover, the relative variation constraint to the DD ionospheric delay between consecutive epochs is also taken into account. A field experiment is conducted to verify and demonstrate the proposed method. The dual frequency GPS data were collected with sampling interval of 1 s in two CORS stations spacing 94.6 km. The results demonstrate that the presented method can provide high-quality DD ionospheric delay estimates instantaneously, which can improve the speed and reliability of the Ambiguity Resolution (AR). Furthermore, medium-range (up to 100 km) RTK solution precision achieves centimeters in all components. The precision of the horizontal component is less than 2 cm, horizontal component is about 3 cm.
Y. Li (&) Y. Shen Department of Surveying and Geo-informatics Engineering, Tongji University, 20092 Shanghai, China e-mail:
[email protected] Y. Li Center for Spatial Information Science and Sustainable Development, Tongji University, 20092 Shanghai, China
J. Sun et al. (eds.), China Satellite Navigation Conference (CSNC) 2012 Proceedings, Lecture Notes in Electrical Engineering 159, DOI: 10.1007/978-3-642-29187-6_38, Ó Springer-Verlag GmbH Berlin Heidelberg 2012
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Keywords GNSS pseudo-observation
Y. Li and Y. Shen
Medium-range kinematic Elevation-dependent weight
positioning
Ionospheric
1 Introduction In recent years, GPS users have more and more interest in long-range high precision RTK (real time kinematic) for a wide range of precise applications. The success of precise GPS positioning over long baselines depends on the ability of fast or even instantaneously resolving the integer phase ambiguities. Unfortunately, the ambiguity resolution (AR) process is often disrupted dramatically in the presence of severe systematic biases, especially the ionospheric delay and the tropospheric delay [1]. Regarding the tropospheric delay, a priori correction of the troposphere can be obtained from several tropospheric correction models, e.g., Hopfield, Saastamoinen, and UNB3. Despite the differences among the various models, when the elevation angle of satellites is beyond 20°, different models give very similar estimates. After the correction with a tropospheric model, 99% of tropospheric error is corrected, the residual part, namely wet part can be parameterized and absorbed by RZTD (relative zenith tropospheric delay) plus mapping function or even neglected with mediumrange baseline [2]. The ionospheric delay is the major error source for GNSS precise positioning [3, 4]. To reliably resolve the ambiguities, these errors have to be kept as small as possible. In conventional long-distance AR algorithm, the ionospheric delays can be basically eliminated by using the so-called ionosphere-free combination of two radio frequencies [1]. However, the ionosphere-free combination enlarges the observation noise and causes long convergence time for AR process. Another way to overcome this problem is to include additional ionosphere parameters, which is called ionosphere-float model [5, 6]. This method has been actually equivalent to the ionosphere-free model. Both above two ways of AR decrease the model strength by eliminating ionospheric delays or setting ionospheric parameters so that fast AR is impossible to be realized. In order to achieve fast or instantaneous AR, the ionosphere-weighted model has been popularly applied, in which the double-differenced ionospheric delays are treated stochastically instead of deterministically ([2, 5, 7–10]). Its popularity stems from improving the model strength through adding prior stochastic information of ionospheric delays in term of zerovalued ionospheric pseudo-observations and stochastic model the ionospheric pseudo-observations which assume DD ionospheric delays are epoch-wisely independent and the priori standard deviations of all DD ionospheric delays are same. However, some features, such as dependency on the elevation angle or time-correlation of the DD ionospheric delays, have not been taken into account [11]. Thus, the ionospheric pseudo-observations and their stochastic models might not represent the behaviour of the true ionosphere error process. An adaptive scheme had to be used here to track the ionospheric conditions. Considering the ionosphere delay is elevation dependent, the noise of the ionospheric pseudo-observation is certainly
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elevation dependent. Therefore, the weight of ionospheric pseudo-observation is determined according to satellite elevation. Moreover, the relative variation of the DD ionosphere delay between consecutive epochs is also taken into account in term of pseudo-observations. In the following sections, a method for fast AR in presence of ionospheric delays is developed by using an ionosphere-weighted model along with elevation-dependent weighting scheme and temporal correlation constraint for the ionospheric pseudoobservations. Subsequently, the performance of this method will be tested and its capability for fast AR and high accuracy user positioning is demonstrated.
2 The Observation Model with Wide-Lane (WL) Ambiguity Constraints The original DD observables of carrier-phases and pseudo-ranges with (m ? 1) satellites can be expressed as UijAB;1;k ¼ qijAB;k þ T ijAB;k gijAB;k þ k1 N ijAB;1 þ eU1 UijAB;2;k ¼ qijAB;k þ T ijAB;k
f12 ij g þ k2 N ijAB;2 þ eU2 f22 AB;k
ð1Þ
PijAB;1;k ¼ qijAB;k þ T ijAB;k þ gijAB;k þ eP1 PijAB;2;k ¼ qijAB;k þ T ijAB;k þ
f12 ij g þ eP 2 f22 AB;k
where, the subscripts ‘‘A’’ and ‘‘B’’ indicate the terms associated to the A and B receivers; the subscripts ‘‘1’’ and ‘‘2’’ indicate the terms associated to the L1 and L2 frequencies, respectively; the subscripts ‘‘i’’ and ‘‘j’’ indicate the terms associated to the i and j satellites; the subscripts ‘‘k’’ indicates the terms associated to the kth epoch. U; P indicate the DD carrier-phases and pseudoranges observables vectors, respectively; q is the actual DD satellite-to receiver distance; T is DD residual tropospheric delay corrected by UNB3 standard tropospheric model; g is DD ionospheric delay vector at L1 frequancy; N is DD integer ambiguity vectors. e is the random noise of normal distribution with zeros mean, and f and k are frequency and wavelength of carriers, respectively. Due to the long wavelength (0.86 m), the WL ambiguity is much easier to be determined than L1 or L2 alone. We firstly resolve the WL ambiguity parameter, the float WL ambiguity solved is ^ ij N AB;W ¼
k¼n . X UijAB;W;k PijAB;N;k kw k¼1
k¼n X f1 UijAB;1;k f2 UijAB;2;k f1 PijAB;1;k þ f2 PijAB;2;k ¼ f1 f2 f1 þ f2 k¼1
, kw
! ð2Þ
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where, the subscripts ‘‘N’’ and ‘‘W’’ indicate the terms associated to the narrow-lane and wide-lane frequencies; kw ¼ c=ðf1 f2 Þ; c is the speed of light. With increasing observables are accumulated, the precision of WL ambiguity float solution can achieve below 0.3 cycle, finally WL ambiguities are successfully fixed. After that, N ijAB;2 can be alternated with N ijAB;1 NijAB;W as a constraint to improve AR model strength. The tropospheric delays T can be obtained from UNB3 standard tropospheric model using standard atmosphere parameters and the residual component is neglected in the medium-range positioning [2]. Since the following derivations are based on DD observables, thus the subscripts corresponding to receivers and satellites are omitted. The observation equation can be expressed as vk ¼ Ak xk þ Bk zk lk Rk ¼ r20 P1 k
ð3Þ
where, vk is the residual vector; xk are a set of instantaneous parameters including coordinates parameters drk and ionosphere parameter gk with its the design matrix Ak ¼½ Acoord;k Aiono;k ; zk is accumulated parameters with its the design matrix Bk ¼ k1 I m . In this paper, an elevation-dependent weighting function is specified for determining variance of original observations [12]. ( h [ 30 r20 2 2 r ð hÞ ¼ ð4Þ r0 h 30 2 sin h where, h is satellite elevation angle.
3 Ionosphere Pseudo-Observation with Elevation-Dependent Weight Due to adding ionospheric parameters to model, The model (3) is too weak, thus the prior stochastic information of DD ionospheric delays is treated as pseudoobservations to improve the model strength gk ¼ g0k þ egk ;
R0gk ¼ r2g0 P1 gk
ð5Þ
where, g0k is the prior ionospheric biases with its variance–covariance matrix R0gk ; Generally, g0k ¼ 0, while rg0 stands for the priori standard deviations corresponding to the largest DD residual ionosphere delay in current epoch, which can be determined according to the baseline length and local time. If ionospheric delay is relatively fierce, a small rg0 may result in the float solutions with considerable biases despite it is helpful to improve AR model strength. Conversely, if rg0 is too large, it will casue that ionospheric-constrained model lose effectiveness on improving AR.
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When GPS signals transmit through ionosphere, ionospheric delays mainly depend on the total electron content (TEC) along the propagation path. If the ionosphere is assumed as single layer model, the TEC in slant propagation path can be mapped to vertical direction, the DD ionospheric delays are expressed as gk ¼
40:28 f2
STEC¼
40:28 VTEC f 2 sinðhÞ
ð6Þ
where, STEC and VTEC stand for slant TEC and vertical TEC, respectively. From (6) we can find out DD ionospheric delays increase when satellite’s elevation decreases. Besides, from Fig. 2, we can find out the values of DD ionospheric delays become larger when the elevation of satellite is lower. This reveals DD ionospheric delays are elevation dependent. Considering the ionosphere delay is elevation dependent, the noise of the ionosphere pseudo-observation is certainly elevation dependent, thus the assumption that the priori standard deviations of all DD ionospheric delays are same might give DD ionospheric delays with higher satellite elevation with large r2g0 , which directly undermines the efficiency of AR process. Similar to variance of original observations, the elevation-dependent weighting function is also employed to determine the variance of ionospheric pseudo-observations 8 > sin hmin r2g0 > > h [ h0 < sin h0 ð7Þ r2g ðhÞ¼ > sin hmin r2g0 > > : h h0 sin h where, h0 is the threshold of satellite’s elevation. Here, one thing should be mentioned that (7) is slightly different with (4), because r2g0 in (7) is defined as the variance component of the prior ionospheric biases corresponding to the pseudoobservation with minimum elevation, while r20 in (4) is the corresponding to the observation with elevation which is larger than h0 . Since the variance component of prior ionospheric biases corresponding to the pseudo-observation with minimum elevation is relatively easy to be determined according to the length of baseline and local time, we choose variance component of prior ionospheric biases corresponding to the pseudo-observation with minimum elevation as unit variance component. Therefore, (4) and (7) are intrinsically consistent but only different in format.
4 Relative Variation Constraint to Ionospheric Delays Besides dependency on the elevation angle, DD ionospheric delays are also time correlated. From Fig. 4, we can find out the variation of DD ionospheric delays between consecutive epochs is regular, for most is within 5 mm, which can be taken as a constraint to enhance the model strength. The relative variation constraint is set up as
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gk ¼ gk1 þ wgk
Rwgk ¼ r20 P1 w
ð8Þ
where, wk is relative variation vector with its variance–covariance matrix Rwgk ¼ diagð r1wgk . . . rwj gk Þ; the subscripts ‘‘j’’ indicates the terms associated to the jth DD ionospheric delay; while Rwgk should be determined according to the dispersion of time-differencing ionosphere delay. The variance component corresponding to the jth relative variation of DD ionosperic delays can be computed as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , un j 1 uX j j 2 j t r ¼ g g ðn j 1Þ ð9Þ wg k
kþ1
k
k¼1
where, n j is the number of epochs corresponding to the jth relative variation of DD ionospheric delay.
5 Fast GPS Ambiguity Resolution Estimation Collecting and rewriting the observation equations (3), the ionospheric pseudoobservation equation (5) and relative variation equations (8) with respect to the ionospheric delays as error equation type, we set r20 ¼ r2U as variance component of unit weight, thus r2g0 ¼ k r20 , k is a variance ratio between r2g0 and r20 . vk ¼ Ak xk þ Bk zk lk Rk ¼ r20 P1 k
ð10Þ
~ 0gk ¼ r2g0 P1 ~vk ¼ gk wR gk
ð11Þ 1
vk ¼ gk wR0 gk ¼ Rwgk þ Rgk1 ¼ r20 P0 gk
ð12Þ
where, ~vk and vk are residual vector of ionospheric pseudo-observation equation ~ ¼ g0k ¼0 and w¼gk1 are constant vector. and relative variation equations; w 0 R gk is the variance–covariance matrix of relative variation. The total observation equation can be expressed as k xk þ B k zk lk vk ¼ A . . T . . k ¼ A Ak 0..I m ..0 0..I m ..0 ; Bk ¼ ð Bk 0 0 P where, T B lk ¼ lk w Pg ~ w ; Pk ¼ @
ð13Þ 0 ÞT 1
k
P0 g k
C A
The LS criterion corresponding to (3), (5) and (8) is expressed as,
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T
X ¼ vTk Pk vk þ ~vTk Pgk ~vk þ vk P0 gk vk ¼ min The normal equation is derived as T k P Tk PB lk A k A k ^ xk A ATk P ¼ lk A k BT PB k ^zk BTk P BT P k
ð14Þ
ð15Þ
k
In order to simplifying the expression, epoch-wised parameter vector ^xk is firstly eliminated, its LS solution is expressed as 2 1 ^zk ¼ N 1 zjx uzjx ; R^zk ¼ r0 N zjx
Nxx where, Nzx
T A k N xz ¼ AkT P A k N zz Bk P
ð16Þ
T Tk PB k P k lk ux A A k ; uz ¼ BT P BTk PB k lk
1 1 N zjx ¼ N zz Nzx N 1 xx N xz ; N xjz ¼ N xx N xz N zz N zx ; uzjx ¼ uz N zx N xx ux ; uxjz 1 ¼ ux N xz Nzz uz
If integer ambiguities keep unchanged for consecutive n epochs, thus accumulated LS solution is as !1 !1 n k¼n n X X X ^z ¼ ^20 Nzjx uzjx ; R^zk ¼ r N zjx ð17Þ k¼1
k¼1
k¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T vTk Pk vk þ ~vTk Pgk ~vk þ vk P0 gk vk ^20 ¼ r n
ð18Þ
After the float ambiguity vector and covariance matrix are obtained, LAMBDA [13] is used to fix the ambiguities to their integer values. Ratio test is used for the integer selection validation.
6 Experiment Design The static field experiment is carried out with the data from CORS (continuously operating reference station) system in USA, the collecting time span is from March 1st 0 h 0 min 1 s, 2009 to March 1st 3 h 15 min 0 s, 2009. In Fig. 1, both GPS stations, i.e. GLVT and NSCH, are mounted with TRIMBLE 4700 dual-frequency receiver with the sampling frequency of 1 Hz. The baseline between two stations is 94.6 km and the number of visible satellites in both stations varies from 5 to 9 during the whole test process. The kinematic positioning mode is applied to process with the baseline, namely GLVT is regarded as reference station while NSCH is rover station, of which the initial coordinates are computed by using pseudo-range single point positioning (SPP).
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Fig. 1 DD ionospheric delays over 94 km baseline
0.2 0.15 0.1 0.05 0
−0.05 −0.1 −0.15 −0.2 10
20
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°
Fig. 2 DD ionospheric delays with elevation change
Figure 1 presents the actual DD ionospheric delays over the baselines connecting the user receiver with the reference stations. Most of DD ionospheric delays are within 5 cm, but some of them indeed exceed 15 cm. In order to avoid the float solutions with considerable biases, we must give rg0 with a large value, for example 20 cm. From Fig. 2, we can find out that there is an increasing trend of DD ionospheric delays with the decrease of satellite’ elevation, this motivates us to give ionospheric pseudo-observations corresponding to high elevation with a smaller rg0 , which will improve AR process without resulting in biases.
Time−differenced DD iono delays [mm]
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10 8 6 4 2 0 −2 −4 −6 −8 −10
0
0.5
1
1.5
2
2.5
3
Elapsed time (hour)
Fig. 3 Time-differencing ionospheric delays over 94.6 km baseline
Figure 3 presents the actual time-differenced ionospheric delays over the baselines. DD ionospheric delays are stable with time, which have small changes between consecutive epochs. Figure 4 shows the variance component corresponding to the relative variation of DD ionospheric delays, most of them do not exceed 2 mm, some with low satellite elevation exceed 5 mm. In order to test the capability for fast AR and high-accuracy positioning of the presented methods, we first computes the mean, and standard deviation (repeatability) of the float ambiguities. The data set was divided into 195 1-min sessions. The sessions were processed independently, yielding hence 195 solutions. Three scenarios are analyzed and compared. Scenario 1: Ionospheric pseudo-observation with the same priori standard deviations for all DD ionospheric delays. Scenario 2: Ionospheric pseudo-observation with elevation-dependent weight. The standard deviation of DD ionospheric delays which correspond to satellite’s elevation below than 30° is set to 15 cm. Scenario 3: Ionospheric pseudo-observation with elevation-dependent weight and relative variation constraint. The standard deviation of DD ionospheric delays correspond to the lowest satellite’s elevation is set to 20 cm. Besides, the variance component corresponding to the relative variation of DD ionospheric delays is determined by (9). Figures 5, 6 show the mean, standard deviation of those 12 pairs of DD float ambiguity biases, respectively. The reference satellite PRN is No.14. The result shows the improvements of the method by using Ionospheric pseudoobservation with elevation-dependent weight and relative variation constraint in terms of float ambiguity biases. The means of float ambiguity solutions with Scenario 2 are closer to 0 than their counterparts of Scenario 1. While the means of float ambiguity solutions with Scenario 3 are even much closer to 0 compared to their counterparts of Scenario 2. Besides, the standard deviations of float ambiguity solutions with Scenario 3 are smaller than those with Scenarios 1 amd 2, which
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16 18 20 22 Satellite PRN
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Fig. 4 Time-differencing ionospheric delays over 94.6 km baseline 0.5 Scenario 1 Scenario 2 Scenario 3
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Mean (cycle)
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-0.2
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Fig. 5 Mean of 12 pairs of float ambiguity biases
indicate smaller biases of float ambiguity and significantly improved estimated precision in ambiguity. Then we compare the values of Ratio test which is used for the integer selection validation with three scenarios. Generally, the larger ratio value means the higher reliability that selected integer ambiguity set is correct. Thus, the result of ratio
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Scenario 1 Scenario 2 Scenario 3
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Fig. 6 STD of 12 pairs of float ambiguity biases
value is able to indicate the capability for fast AR with three above scenarios. Figure 7 shows ratio values of 195 1-min session with three scenarios. If scenario 3 is employed, there are 127 sessions among 195 sessions with the ratio value larger than 2, which are more than 120 sessions corresponding to scenarios 2 and 110 sessions with scenarios 1. The success rate of ambiguities is defined as the ratio between the number of sessions in which ambiguities passed the discrimination test and the number of total sessions. The success rates for three scenarios are 56.4, 61.5 and 65.5%. Moreover, the mean ratio values of 195 sessions with three scenarios are 4.24, 4.94 and 5.33, respectively. The above statistical results indicate the ambiguities are easier to be fixed if scenario 3 is employed. Due to the Ionospheric pseudo-observation with elevation-dependent weight, the success rate of passing ambiguity validation is improved compared to that with scenario 1. As we know, the ambiguities passed the discrimination test are not always the real integer ambiguities. Therefore we further compare the correct rate of ambiguities passed the discrimination test. The Table 1 shows the correct rates of ambiguities passed the discrimination test with two statistical modes. The ‘‘1-min’’ mode means the ambiguities are resolved with 1-min session data. The ambiguities which passed the discrimination test are selected and validated with the real integer ambiguities. In the instantaneous mode, we do not wait 1 min but instantaneously resolved the ambiguities, if they pass the discrimination test, we pick them up and validate with real integer ambiguities. As a result, the correct rates for all of three scenarios are 100% with 1-min mode. This means that the
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Y. Li and Y. Shen 45 Scenarios 1 Scenarios 2 Scenarios 3
40 35
Ratio value
30 25 20 15 10 5 0
0
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100
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No. of sessions
Fig. 7 Ratios of ambiguity validation with three scenarios
Table 1 The correct rates of ambiguities with three scenarios
Mode
Scenario 1 (%) Scenario 2 (%) Scenario 3 (%)
1-min 100 Instantaneous 99.5
100 99.8
100 100
fixed ambiguities with all of three scenarios are robust after enough observational accumulation. In the instantaneous AR mode, the scenario 3 achieves higher correct rate which amounts to 100 compared to 99.5 and 99.8% when scenario 1 and scenario 2 are applied, respectively. Now, we analyze the RTK solutions. The plots of horizontal positional errors as well as the vertical positional errors are shown in Fig. 8. Since systematic errors such as ionospheric and tropospheric delays are parameterized or eliminated, if the ambiguities are incorrectly fixed, they will lead to significant positional biases to the real position. As a result, we can totally validate whether the integer ambiguities are resolved correctly or not from positioning results. From Fig. 8, the time-series plot of positional errors show that the positional error in north and east component is smaller than 5 cm for most epochs, and smaller for than 10 cm in height although all of them are relatively larger at the a short span between 2 and 2.5 h due to the severe observational situation. The statistics of RTK solutions are listed in Table 2. Regarding to the mean of positional error, the RTK’s north, east and height are 0.68, -0.27 and -1.52 cm, respectively. Furthermore, the results about the standard deviations are also satisfactory. The standard deviations in north, east and up components are 1.06, 1.20 and 2.78 cm. The above positional results are enough to testify the correct of fixed ambiguity. However, both of the mean and standard deviation in up component are larger than north and east show that systematic errors still exist.
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(cm)
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dn
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Epoch (hour)
Fig. 8 The positional errors in north, east and height component [cm]
Table 2 Statistics for RTK solutions [cm]
Statistic (cm)
North
East
Up
Mean STD
0.68 1.06
-0.27 1.20
-1.52 2.78
7 Concluding Remarks This paper presents an investigation on fast or instantaneous AR constraint by ionospheric pseudo-observation with elevation-dependent weight and mediumrange kinematic positioning. The results demonstrate that the presented method can provide high-quality DD ionospheric delay estimates instantaneously, which can improve the speed and reliability of the ambiguity resolution (AR). If the ionosphere pseudo-observation with elevation dependent weight and relative constraint of ionosphere delays are added the rate amounts to 65.5% with 1-min observational accumulation. Besides, the correct rates of the ambiguities sets passed the discrimination test are 100% with both 1-min and instantaneous modes if scenario 3 is employed. Furthermore, medium-range (94.6 km) RTK solution precision achieves centimeters in all components.
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Acknowledgments This work is supported by the National Natural Science Funds of China (grant 40874016, 41074018) and partially supported by Kwang-Hua Fund for College of Civil Engineering, Tongji University.
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