To take full advantages of Galileo and BDS-3 penta-frequency signals, a long-baseline RTK positioning method based on Galileo and BDS-3 penta-frequency ionosphere-reduced (IR) combinations is proposed. First, the high-quality signals with low-noise and weak-ionospheric delay characteristics of Galileo and BDS-3 are analyzed. Second, the multi-frequency extrawide-lane (EWL)/ wide-lane (WL) combinations with long-wavelengths are constructed. Third, the IR-EWL combinations are calculated by geometry-free (GF) method, then the resolved IREWL combinations are used to constrain the IR-WL, of which the ambiguities can be obtained in a single epoch. There is no need to consider the influence of ionospheric parameters in the third step because the ionospheric delay factors of IR-EWL/WL combinations are close to 0. Compared with the estimated ionosphere model, the proposed method can improve the availability of positioning and reduce the number of parameters by half and the required operation time is greatly reduced. Therefore, it reduces the dimension of parameter estimation and is suitable for the use of multi-frequency and multi-system real-time RTK. The results using real data show that stepwise fixed model of the IR-EWL/WL combinations can realize long-baseline instantaneous decimeter-level positioning.
BDS-3
2022 Copyright for this paper by its authors. accuracy of ambiguity resolution are improved compared with the traditional dual-frequency ionosphere-free combination [6]. Based on the new frequency signals of Galileo and BDS-3, we studied the selection criteria for the combination of ionosphere-reduced observations suitable for long baselines. The optimal ionosphere-reduced combinations of Galileo and BDS-3 penta-frequency signals are analyzed. The estimated ionosphere model is used to compare the positioning performance with the IR model, and the decimeter-level positioning of Galileo and BDS-3 penta-frequency minimum number of parameters to be estimated for long-baseline RTK positioning is realized.
The purpose of this article is to study multi-frequency combination signals of Galileo and BDS-3 with ionospheric delay factor close to 0 and low combined observation noise. With the comparation of IR model and estimated ionosphere model, positioning performance and positioning efficiency of IR model is studied. In the following section, we define the conditions to be met by IR-EWL/WL combinations, and provide the geometric model required by the algorithm. Then experiments were conducted using a set of long baseline data. Finally, the experimental results are analyzed and summarized.
Ignore satellite systems, the observation equation of the basic pseudo-range and carrier-phase observations is: Pi = +iI1 + T + P i = −iI1 + T + iNi + where, the symbol " " represents DD operation; Pi and i represent pseudo-range and carrier observations, respectively; is the geometric distance between the satellite and the receiver; I1 is the first-order ionospheric delay at the first frequency; i is the first order ionospheric scale factor; T indicates tropospheric delay; P and are observation noise of pseudo-range and carrier-phase respectively; N indicates integer ambiguity; is the carrier wavelength.
Correspondingly, the DD observation equation of the penta-frequency signals after basic observation equation linear combination is:
(i, j,k,m,n) = −(i, j,k,m,n)I1 + T + (i, j,k,m,n)N(i, j,k,m,n) + (i, j,k,m,n) where, each parameter is expressed as: (i, j,k,m,n) = i f1 1 + j f2 2 + k f3 3 + m f4 4 + n f5
i f1 + j f2 + k f3 + m f4 + n f5 (i, j,k,m,n) = f12 (i f1 + j f 2 + k f3 + m f4 + n f5 )
i f1 + j f2 + k f3 + m f4 + n f5 (i, j,k,m,n) = (i, j,k,m,n) = i f1 + j f2 + k f3 + m f4 + n f5 (if1)2 + ( jf2 )2 + (kf3 )2 + (mf4 )2 + (nf5 )2
c
f(i, j,k,m,n)
(
The combination of penta-frequency EWL combination can construct infinite combined observations according to different coefficient values, but most signals do not have the characteristics of low-noise and weak-ionospheric scale factor. Referring to literature [6], this paper selects the IREWL/WL combinations for long-baseline positioning, and makes the following constraints on the characteristics of combined observations:
(
c
The ionospheric delay corresponding to 5 m has less than 0.1 cycles on ambiguity resolution. If the impact on ranging is less than 5 cm, it is required. In fact, the ionospheric residuals for the 100-500 km baseline double-differences are less than 1.5m [5].
(
(
Based on the above three conditions, take [-10, 10] as the search interval of combination coefficient,
the characteristics of combinations meeting the above conditions are shown in Table 1. The sequence
of corresponding Galileo and BDS-3 signal types in the Table 1 is: E1/E5a/E5b/E5/E6 and
B1C/B1I/B3I/B2a/B2b. It can be seen from the table that there are five EWL/WL combinations of
BDS3 and four EWL/WL combinations of Galileo that meet the conditions, of which BDS-3 (
(i, j,k,m,n) / cycle m−1
Equation (
P[a,b,c,d,e] − (i, j,k,m,n) N(i, j,k,m,n) =
(i, j,k,m,n) P[a,b,c,d,e] = aP1 + bP2 + cP3 + dP4 + eP5 (i, j,k,m,n) -0.0005 0.0027 -0.0029 -0.0060 0.0023 where, • represents the rounding operator, P[a,b,c,d,e] is the linear combination form of DD observations of pseudo-range combination, and the combination coefficient of pseudo-range a,b,c, d,e is any real number. As shown in equation (10-12), considering that the sum of pseudo-range coefficients is 1, the sum of ionospheric scale factor of combined pseudo-range observations and IR is 0, and the combined noise is the smallest, the optimal pseudo-range coefficient can be calculated by the minimum norm method, as shown in Table 2. It should be noted that only up to five significant figures are displayed in the table.
a + b + c + d + e = 1
f 2 f 2 f 2 f 2 (i, j,k,m,n) + a + b f122 + c f132 + d f142 + e f152 = 0
a2 + b2 + c2 + d 2 + e2 = min N(i,j,k,m,n) = (if1)2 + ( jf2 )2 + (kf3 )2 + (mf4 )2 + (nf5 )2
2 f(i, j,k,m,n)
2 + (a2 + b2 + c2 + d 2 + e2 ) 2P (i, j,k,m,n) where, and P represent the DD noise of non-combined carrier observation value and pseudorange observation respectively, and the values in this paper are 0.5 m and 5 mm respectively.
The optimal pseudo-range coefficient combination of each IR combination is brought into respectively, and the rounding success rate of ambiguities of IR-WL Ps is calculated by equation (14) [8].
Ps (−0.5 x ) = −00.5.5 12 exp − ( x2−2 )2 dx GF = Kijkmn,abdceI (i, j,k,m,n) , Kijkmn,abdce = (i, j,k,m,n) + (a,b,c,d,e) can be regarded as 0, while Kijkmn,abdce = 0 .
It can be seen from Table 2 that BDS-3 (
variance covariance matrix of parameter estimation, which reflects the accuracy of parameter estimation, and its diagonal element is the variance of estimated parameters. The IR-EWL combinations is affected by the unmodeled atmospheric residual and orbit error when GB model used. Specifically, in equation (14) can be calculated by equation (18): GB = orb + T −(i, j,k,m.n)I
(i, j,k,m.n)
Referring to literature [5], it is assumed that under the conditions of medium and long-baseline, the tropospheric residuals are 10 cm and 15 cm respectively, and the first-order ionosphere residuals are 80 cm and 100 cm respectively. The corresponding ambiguity accuracy and success rate used GB model are shown in Table 3.
BDS-3
(
Galileo
(
Therefore, the IR-EWL combinations can be obtained directly by rounding using GF model.
In this paper, the IR-EWL combinations of BDS-3 or Galileo is calculated by GF method. As a comparison, with reference to equation (19), which estimates ionospheric parameters with low-noise EWL combinations: vvvPPP123 = BBB 132IIIsss 000 ioxn − lllPPP132 (19) vvvEPPW54L BBB −54IIEssWL Is 00EWL Is N EWL lllEPP54WL
After fixing EWL, the calculation of WL ambiguity is selected according to its combination characteristics. Generally, the fixed EWL combinations are used to restrict the ambiguity of WL combinations, as shown in equation (20).
vE'WL = B vWL B − EWL Is −WL Is
x 0 ion − l E'WL WL Is NWL lWL
The WL is constrained with IR-EWL by GB model, and ionosphere can be ignored because it is very little. Its estimation equation is: vE'WL = B 0 x l E'WL
− WL Is NWL lWL vWL B
It can be seen that using the IR-EWL combination to restrict the IR-WL combinations does not need to estimate the ionospheric parameters, and the dimension of parameter estimation can be reduced. 2.4.
Based on equation (21), calculate the DD float ambiguities of the remaining four IR-WL combinations
in Table 1 after ambiguities of BDS-3 IR-EWL combination (
It can be seen that the accuracy of WL combination Galileo (
In this paper, a group of 189.4 km long-baseline of IGS station are used for the experiment. The data comes from TIT2 and FFMJ stations of BKG data center. The observation date is UTC time, October 1, 2021 (24 hours), day of year is 274, and the sampling interval is 30 s. During the calculation, the cutoff angle of the satellite in the calculation is set to 15°.
The number of BDS-3 and Galileo satellites with five frequencies and their RDOP in this period are shown in Figure 1. The number of common view satellites of the two stations fluctuates in the range of 8-14 mostly. In the 2279th epoch, rdop increased sharply due to the small number of visible satellites. Figure 2 shows the sky plots of BDS-3 and Galileo various satellites in the experiment.
Figure 3 and Figure 4 shows the fraction bias of EWL combinations ambiguities using estimated ionosphere model and IR-GF model, respectively. Different colors correspond to different satellite pairs. It can be seen that the estimated ionosphere model can be all within 0.25 cycles and can be reliably rounded and fixed.
However, either BDS-3 or Galileo, the accuracy of IR-EWL ambiguities calculating by GIF method is poor because of greater pseudo-range noise, which affects the ambiguity accuracy. Therefore, the influence of the ionosphere can be properly ignored to reduce the noise of pseudo-range observations. This paper only analyzes the case without pseudorange noise reduction. (b) Figure 3: The fractions of EWL ambiguity of IR-GF model: BDS-3 (a) and Galileo(b) (d) Figure 4: The fractions of EWL ambiguity of estimated ionosphere model: BDS-3 (c) and
Galileo(d)
The true values of IR-EWL ambiguities are all obtained by multi-epoch filtering. It can be seen from the Table 5 that although the accuracy of IR-EWL ambiguity calculated by GIF model is not high, the rounding reaches 99.04% (BDS-3) and 97.89% (Galileo). In the experiment, the threshold of decimal deviation is set as 0.3 when rounding EWL.
Pseudo-range Coefficients b c d
e 1.169 -0.123 -0.741 -0.519
1.916 -0.679 -0.351 -0.512 0.326 2.420 Noise Factor 0.5 0.96 2.11
EWL Fractions/cycle 0.5 99.04 97.89 0.4 97.06 94.64 0.3 91.02 86.77
% 0.2 75.58 70.07
The WL ambiguities are calculated by IR-GB model and estimated ionosphere model and the suboptimal/optimal ambiguity variance ratio (Ratio value) is shown in Figure 5 and Figure 6. In the figure, the values corresponding to the red lines in the upper and lower figures are 10 and 2.5 respectively. It can be seen that ratio value of IR-GB model is greater than that of estimated ionosphere model. The reason is that IR-GB model does not need to estimate the ionospheric delay, so the strength of the parameter estimation model is greater.
The threshold for LAMBDA estimation of WL ambiguities is 0.2 cycles. If the number of WL ambiguity float solutions satisfying the condition is less than 4, the positioning result of this epoch is considered invalid. After the WL ambiguities are fixed, the observation equations are brought back to obtain the coordinate solution under the fixed solution. The positioning error of the corresponding solution coordinates in the East (E), North (N) and Up (U) directions is shown in Figure 7 and Figure 8.
Finally, the positioning accuracy statistics of the two methods are shown in Table 6. It can be seen that the accuracy of the IR-GB model and that of the estimated ionosphere model is basically the same. However, the IR does not need to estimate the ionospheric delay term, so it can achieve higher ambiguity calculation efficiency. Especially in the multi-level step-by-step resolution of multi-system ambiguity, it will be more obvious. Figure 9 and Table 7 shows the operation time of these two models. It can be seen that the operation time of the IR model is significantly lower than that of estimated ionosphere model, which is consistent with the analysis.
In this paper, a step-by-step method for fixing the ambiguities of IR-EWL/WL combinations is proposed. First, the IR-EWL combinations are calculated by GF method, then the fixed IR-EWL combinations are used to constrain the IR-WL combinations, of which the ambiguities can be obtained in a single epoch. The proposed IR model does not need to estimate the ionospheric delay, so the strength of the parameter estimation model is greater.
The main difference between the ionosphere-reduced model and estimated ionospheric model is the accuracy of the WL float ambiguities. Once the EWL of the IR model is successfully fixed, a positioning performance comparable to that of the estimated ionosphere model can be obtained.
The experiment shows that the positioning performance of the IR model is comparable to that of estimated ionosphere model (0.168/0.191/0.375m vs. 0.159/0.206/0.408m), and positioning performence is more effective (99.5% vs. 98.4%) and the computation time is shorter (3.7ms vs. 13.7ms).
The performance of the ratio depends on the strength of the model. The ionosphere-reduced model is essentially an ionospheric-fixed model, which reduces the number of parameters to be estimated to improve the model and obtain a higher-precision float solutions.
IGS MGEX is gratefully acknowledged for providing Galileo and BDS-3 data. This research is supported by the National Key R&D Program of China (No. 2021YFC3000502); the National Natural Science Foundation of China (No. 41904022); and the Foundation of Laboratory of Science and Technology on Marine Navigation and Control, China State Shipbuilding Corporation (No. 2021010104). 6. References