Particle filter-based estimation of inter-frequency

Particle filter-based estimation of interfrequency phase bias for real-time GLONASS integer ambiguity resolution Yumiao Tian, Maorong Ge & Frank Neitzel

Journal of Geodesy Continuation of Bulletin Géodésique and manuscripta geodaetica ISSN 0949-7714 J Geod DOI 10.1007/s00190-015-0841-1

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Author's personal copy J Geod DOI 10.1007/s00190-015-0841-1

ORIGINAL ARTICLE

Particle filter-based estimation of inter-frequency phase bias for real-time GLONASS integer ambiguity resolution Yumiao Tian1 · Maorong Ge2 · Frank Neitzel1

Received: 6 May 2015 / Accepted: 13 July 2015 © Springer-Verlag Berlin Heidelberg 2015

Abstract GLONASS could hardly reach the positioning performance of GPS, especially for fast and real-time precise positioning. One of the reasons is the phase inter-frequency bias (IFB) at the receiver end prevents its integer ambiguity resolution. A number of studies were carried out to achieve the integer ambiguity resolution for GLONASS. Based on some of the revealed IFB characteristics, for instance IFB is a linear function of the received carrier frequency and L1 and L2 have the same IFB in unit of length, most of recent methods recommend estimating the IFB rate together with ambiguities. However, since the two sets of parameters are highly correlated, as demonstrated in previous studies, observations over several hours up to 1 day are needed even with simultaneous GPS observations to obtain a reasonable solution. Obviously, these approaches cannot be applied for real-time positioning. Actually, it can be demonstrated that GLONASS ambiguity resolution should also be available even for a single epoch if the IFB rate is precisely known. In addition, the closer the IFB rate value is to its true value, the larger the fixing RATIO will be. Based on this fact, in this paper, a new approach is developed to estimate the IFB rate by means of particle filtering with the likelihood function derived from RATIO. This approach is evaluated with several sets of experimental data. For both static and kinematic cases, the results show that IFB rates could be estimated precisely just with GLONASS data of a few epochs depending on the baseline length. The time cost with a normal PC can be controlled around 1 s and can be further reduced. With the estimated IFB rate, integer ambiguity resolution is available

B

Yumiao Tian [email protected]

1

Technische Universität Berlin, 10623 Berlin, Germany

2

German Research Centre for Geosciences, 14473 Potsdam, Germany

immediately and as a consequence, the positioning accuracy is improved significantly to the level of GPS fixed solution. Thus the new approach enables real-time precise applications of GLONASS. Keywords GLONASS · Integer ambiguity resolution · Phase inter-frequency bias · Particle filter · Real-time applications

1 Introduction Although GLONASS has been running with its full constellation since December 2011 and is today still the only one completed GNSS system along with GPS, its usage lags behind GPS, especially in rapid and real-time precise positioning, due to the lack of approaches for efficient integer ambiguity resolution. One of the reasons is the existence of the inter-frequency bias (IFB) of phase observations, which is caused by the analog hardware delay and the digital signal processing (Wanninger and Wallstab-Freitag 2007; Sleewagen et al. 2012). Since all GPS satellites have the unique frequency for each carrier band, such delays are similar for the same type of observations and can be eliminated by forming differences between satellites. Unfortunately, GLONASS employs frequency division multiple access (FDMA) technology, i.e., different frequencies for satellites, thus, this bias is different from satellite to satellite (Pratt et al. 1998; Wanninger and Wallstab-Freitag 2007; Zinoviev et al. 2009; Sleewagen et al. 2012). The existence of phase IFB prevents the GLONASS integer ambiguity resolution, because it could not be eliminated in the double-differenced (DD) observations/ambiguities. It is worth to point out that range observations have also such bias as well, but in this study IFB

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Author's personal copy Y. Tian et al.

always means receiver phase IFB, as we focus on the integer ambiguity resolution. To fix the integer ambiguity for GLONASS, a number of studies were carried out to investigate the IFB characteristics. Although it is concluded that receivers of the same manufacturer have similar bias in principle, there are exceptions (Wanninger 2012). In practice, we cannot always employ devices from the same manufacturer, as the number of receiver manufacturers is increasing. Moreover, the antenna and cable, as well as the restart of receivers can also contribute to the IFB (Wanninger and Wallstab-Freitag 2007). Therefore, we should not assume that IFB can always be eliminated in the differenced observations/ambiguities. Pratt et al. (1998) and Wanninger and Wallstab-Freitag (2007) showed that GLONASS receiver IFB is nearly linearly correlated to the frequency number, and can be, therefore, represented by a constant offset and IFB rate with respect to the frequency number. It was also shown that the IFBs in L1 and L2 are similar in unit of length as well. These characteristics are always utilized in the IFB modeling and estimating (Wanninger 2012; Al-Shaery et al. 2013). Based on the linear relationship between IFB and the signal frequency number, several approaches were developed to estimate the IFB rate. Wanninger and Wallstab-Freitag (2007) and Wanninger (2012) employed single-differenced (SD) GPS and GLOANSS observations between two stations to determine the GLONASS IFBs. However, this method needs an a priori value of the IFB rate with certain accuracy, so that at least one of the ambiguities can be fixed. Afterwards, the remaining ambiguities are estimated along with the IFB rate parameter. It was not clearly addressed how to obtain such an initial IFB rate value. Zhang et al. (2011) ignored the differences in wavelengths and estimated DD-ambiguities using 1-day GPS and GLONASS data. Then ambiguities are fixed by simply rounding the float estimate to the nearest integer. Afterwards, the IFBs are derived according to the fixed ambiguities. Al-Shaery et al. (2013) presented a method similar to the aforesaid one but not ignoring the differences in wavelengths by estimating SD-ambiguities and then mapping them to DD-ambiguities for fixing. IFB rate was estimated along with SD float ambiguities. The method was applied to a zero baseline with 23-hour GPS and GLONASS data. The results have shown that the approach works well, but there is still the need for a new approach for the fast estimation of IFB rates. Besides, Sleewagen et al. (2012) demonstrated that the dominant linear correlation between IFB and the channel number is caused in the digital signal process of the receivers according to the linear relationship. Banville et al. (2013) also proposed an approach to fix GLONASS ambiguity without any external IFB calibration. However, this approach requires that two GLONASS satellites with adjacent frequency numbers are observed simultaneously. Furthermore, in the demonstration,

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the fixing rate is only 70 % in the case of using GLNOASS only. In general, almost all the current approaches try to estimate the ambiguities and the IFB rate simultaneously. Unfortunately, due to the high correlation between the two sets of parameters, the estimation is rather weak even with a long data set including simultaneous GPS observations. Consequently, none of these methods can provide a fast or real-time solution of IFB rate for GLONASS integer ambiguity resolution without a priori IFB or IFB rate information. Assuming that the IFB rate is exactly known, it is easy to understand that for zero- or short baselines employed in the above-mentioned studies, the integer ambiguity resolution could be carried out very reliably, i.e., with a significant large RATIO. It is obvious that statistically the closer the IFB rate is to its true value, the larger the RATIO will be. As IFB rate is usually within an interval of [−0.10, . . ., 0.10] in unit of meters per frequency number (m/FN) (Wanninger 2012), a limited number of samples of IFB rate uniformly distributed can be defined over the interval. After introducing the samples one by one into the processing for integer ambiguity resolution, in principle the best estimate of IFB rate can be found out according to the resulting RATIO values. The rigorous estimation can fortunately be realized using the particle filter which was developed exactly for providing solution to such kind of estimation problem (Gordon et al. 1993; Doucet et al. 2000; Gustafsson et al. 2002). Therefore, instead of estimating IFB rate and ambiguities simultaneously, a new approach is developed in this paper to find out the IFB rate estimation which can bring the best performance of integer ambiguity resolution for observations over all epochs. The estimation is realized by means of particle filtering with likelihood function derived from RATIO. Experimental validations show that this approach can provide a very precise IFB rate estimation just with GLONASS data of a few epochs, of course, depending on the inter-station distance. As soon as IFB rate has converged, GLONASS integer ambiguity resolution is available and the position accuracy can be significantly improved. Hence, the new approach can be applied to real-time applications without any a priori IFB information. This article is organized as follows: Sect. 2 presents the GLONASS data process model. Section 3 analyses the relationship between RATIO and IFB. The particle filter procedure and the new approach are presented in Sects. 4 and 5. Section 6 presents experiment results and conclusions are given in Sect. 7.

2 GLONASS observation model The observation equations for GLONASS pseudo-range and carrier-phase observations can be expressed as (Al-Shaery et al. 2013)

Author's personal copy Particle filter-based estimation of inter-frequency…

Pai = ρai − c(dt i − dta ) − Iai + Tai + δai + ξai , λin ϕai

=

ρai

+ λin Bai

+ γai

− c(dt − dta ) − i

Iai

+

(1a) Tai

+ εai . (1b)

with P, ϕ the pseudo-range and the carrier-phase measurement, respectively; λ the wavelength; i and a the index for satellites and receivers, respectively; ρ the geometric distance between the satellite and receiver antenna; c the speed of light in vacuum; dt i and dta the satellite and receiver clock offset, respectively; I and T the ionospheric and tropospheric delays; δ the IFB for pseudo-range; B the float carrier-phase ambiguity; n = 1, 2, the frequency bands of L1 and L2; γ the IFB for carrier phase; ξ and ε the measurement noise for range and phase, respectively. The satellite clock offset can be eliminated by forming SD observations between two receivers and the ionospheric and tropospheric delays are reduced to a negligible level for short baselines within about 15 km, whereas the IFBs may still exist because they are receiver-type dependent and can be different even for receivers from the same manufacturer or even of the same type. Therefore, we obtain the SD observation equations for GLONASS as i i i i i = ρab − cdtab + δab + ξab , Pab i λin ϕab

=

i ρab

i + λin Bab

i + γab

(2a)

i − cdtab

i + εab ,

(2b)

where b is the other receiver of the baseline. Furthermore, for another satellite j, the equations of the DD-observations between two satellites and receivers can be derived from (2) as ij

ij

ij

ij

Pab = ρab + δab + ξab , j

j

ij

(3a) j

j

ij

ij

i i λin ϕab − λn ϕab = ρab + λin Bab − λn Bab + γab + εab . (3b)

Besides the IFBs, the major difference of the GLONASS DD-observation equations to that of GPS is the different wavelength of the SD-ambiguities. As a consequence, we cannot directly estimate the DD-ambiguities with integer feature. In several studies, the phase observations are expressed in the unit of cycles instead of length as in the equations above, so that DD-ambiguities with integer feature are directly available. However, in this case, receiver clock biases cannot be eliminated due to the different wavelength and must be estimated with certain accuracy (Pratt et al. 1998; Leick 1998; Han et al. 1999). In fact, SD-ambiguities in (3) can be estimated directly with a proper constraint on one or several of them, then the estimated SD-ambiguities and their covariance matrix can be transformed to that of DD-ambiguities for integer ambiguity resolution (Leick 1998; Wang 2000; Alber et al. 2000; Takasu and Yasuda 2009). Although these two types of methods

have similar performance from the comparative studies by Li and Wang (2011) and Al-Shaery et al. (2012), the method by Takasu and Yasuda (2009) as summarized from Eq. (1) to (3) is employed in this study. The IFBs for L1 and L2 have the similar value in units of length (Wanninger 2012), so they are modeled with the same parameter in this paper. Although the IFBs for pseudoranges are not the same as those of carrier phases, ranges are significantly down-weighted with respect to phases due to their much larger noise. Hereby, the differences between IFBs for pseudo-ranges and carrier phases can be ignored without noticeable bad effects on the solution, so that they can be represented by the same IFB parameter. According to the linear relationship between IFBs and the frequency numbers, the IFB for each DD-observation can be expressed by the frequency numbers and the IFB rate as ij

γab = (k i − k j ) γab ,

(4)

ij

where γab is the IFB for the DD-observation; γab is the differenced IFB rate between the two receivers; k is satellite frequency numbers . Inserting (4) into (3) yields ij

ij

ij

Pab = ρab + (k i − k j ) γab + ξab , i λin ϕab

j j − λn ϕab

=

ij ρab

(5a)

j j − λn Bab ij ) γab + εab .

i + λin Bab i j

+ (k − k

(5b)

From the observations (5), the station coordinates which are involved implicitly in the geometric distance, ambiguity parameters and the IFB rate can be estimated simultaneously. After linearization, the equation system (5) can be written as v = Ax + Db + C y + l,

P,

(6)

where vector x contains the unknown station coordinates, b the unknown SD-ambiguities, y the unknown IFB rate and P is the weight matrix of the observations. To compute the least-squares solution for the unknown parameters, the normal equations (NEQ) ⎤⎡ ⎤ ⎡ T ⎤ A Pl AT P A AT P D AT P C x ⎣ DT P D DT P C ⎦ ⎣ b ⎦ = ⎣ DT P l ⎦ . y s ym CT PC CT Pl ⎡

(7)

are formed. For further computations, the notation ⎡

⎤⎡ ⎤ ⎡ ⎤ N x x N xb N x y x Wx ⎣ N bb N by ⎦ ⎣ b ⎦ = ⎣ W b ⎦ s ym N yy Wy y

(8)

for the NEQ is used.

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As demonstrated in several publications, model (8) needs rather long time to converge, even with the aid of GPS observations. That makes the current strategies not applicable for in-situ calibration for IFBs and real-time applications. Once the IFB rate γab , i.e., y is known, the solution with float SD-ambiguities can be written as −1 xˆ N x x N xb Wx − Nxy y = N bb W b − N by y bˆ Wx − Nxy y Q x x Q xb . = Q bb W b − N by y

(9)

Afterwards, the SD-ambiguities and their covariance matrix in (9) can be transformed into that of DD-ambiguities. Since the transformed DD-ambiguities have the integer nature, they can be fixed to integer as usually done for GPS (Blewitt 1989; Dong and Bock 1989; Wang et al. 2001; Ge et al. 2005). Then the LAMBDA method is employed to fix the DD ambiguity (Teunissen 1995) where the performance of the fixing is normally judged by the statistical parameter RATIO defined as (Teunissen 2004) bˆ − bˇ Q bˆ σˇ 2 . RATIO = 2 = σˇ ˇ 2 Qˆ bˆ − b b 2

(10)

where σˇ 2 and σˇ 2 are the squared norm of the ambiguity residuals for the best integer ambiguity candidate bˇ and the second best one bˇ respectively. Vector bˆ contains the float ambiguities and Q bˆ is the associated variance–covariance (VC) matrix. The larger the RATIO is, the better the fixing performance will be. In practice, a threshold around 3 is selected for making the decision whether the fixing could be accepted or not. The estimation discussed above can be carried out either epoch-by-epoch independently or for accumulated epochs in case of epoch-wise fixing is not feasible. The performance of the ambiguity resolution depends on the baseline length, as more unmodeled errors could occur over longer baselines. In general, similar to the case of using GPS observations, singleepoch ambiguity resolution is achievable for baselines of a few kilometers using L1 and L2 observations.

3 Relationship between RATIO and IFB rate To investigate the relationship between IFB rates and the performance of integer ambiguity resolution, data from three baselines are employed in the following numerical analysis. The first baseline is a zero baseline using the same type of receivers and antennas, i.e., Trimble NetR9 and TRM55971.00, respectively, with a sampling rate of 5 s. The

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data were collected on 1st July 2014, from 9:10:35 AM to 12:20:00 PM. The second one contains two co-located IGS stations KOSG and KOS1 in Holland, with a baseline length of about 814 m. The data were collected on 17th February 2014, 24 h, with a sampling rate of 30 s, where KOSG is equipped with LEICA GRX1200GGPRO receiver and AOAD/M_B antenna, while KOS1 is equipped with SEPT POLARX4 receiver and LEIAR25.R3 antenna. The third one is a kinematic baseline composed by a reference (REF6) and a rover station (AIR5) and the data of 1 Hz sampling rate were taken near Munich in Germany on 6th June 2012, from 4:21:05 AM to 05:05:27 AM, with a maximum inter-station distance of about 1 km. Both stations are equipped with JAVAD DELTA G3T receivers, but with different antennas ACCG5ANT_42AT1 and LEIAS10, respectively. As shown in the previous research by Wanninger (2012), it is reasonable to assume that the largest γab is less than 0.10 m/FN. Therefore, γab is assumed to be within [−0.10, . . ., 0.10] in unit of m/FN. This interval is evenly sampled with a step size of 1 mm, so that there are 200 samples. Then each sample value is used as exactly known γab to obtain the solution of (9) from (8) with IFB rate as unknown. Then, the LAMBDA method is applied to obtain the RATIO value of the integer ambiguity resolution. The processing is carried out for all the epochs over the three above-mentioned baselines using L1 and L2 as independent observations. The three-dimensional RATIO maps for all IFB rate samples over all the epochs are shown in the three sub-plots in Fig. 1 for the three baselines, respectively. The RATIO results for the first epoch corresponding to the three baselines are presented in Fig. 2 for a clear vision of their epoch behavior. Theoretically, the highest RATIO value at each epoch should correspond to the true value of IFB rate. For both the static cases (Fig. 1a, b) and kinematic data (Fig. 1c), there is a clearly detectable peak series in a straight line with remarkable high RATIO values. The peak series line with almost the same IFB rate for all epochs also shows its stability. The high RATIO values of the zero baseline are centralized to the straight line and much higher than that of another static baseline. For the kinematic baseline, although the RATIO values are lower than that of the others, they are definitely strong enough for making the positive fixing decision except from a few outliers. In general, if the IFB rate is given with certain accuracy, the ambiguities can usually be fixed to integer epoch-by-epoch for both static and kinematic baselines. This can also be seen from the distribution of RATIOs for the first epoch of the three baselines shown in Fig. 2. It is also noticed that there are points very close to the straight line with rather lower RATIOs, especially for the baseline KOSG–KOS1 and the kinematic baseline, whereas there are also points far away from the straight line with a rather high


Fig. 1 Three-dimensional RATIO distribution along with epochs and IFB rate samples for the zero baseline (a), KOSG–KOS1 (b) and REF6–AIR5 (c) (The part corresponding to RATIOs which are larger than 50 are not depicted)

Fig. 2 RATIO values of the first epoch corresponding to different IFB rate samples for the zero baseline (a), KOSG–KOG1 (b) and REF6–AIR5 (c)

Fig. 3 The distribution of the points with RATIO larger than the threshold of 3 for the zero baseline (a), KOSG–KOS1 (b) and REF6–AIR5 (c)

RATIO. This is probably mainly caused by the inaccurate handling of station-specified errors. For clarity, Fig. 3 shows the distribution of the points with RATIO larger than 3, which can be approximately considered as the threshold for the acceptance of the corresponding ambiguity candidates. In each sub-plot, there is a very narrow stripe with a width smaller than ±4 mm/FN around a line with constant IFB rate for all epochs. This means only when

the given IFB rate is within the above-mentioned width, the corresponding ambiguities could be fixed correctly to integer. For some existing methods, where an a priori IFB rate value is required (Wanninger 2012), it might be rather difficult to provide such an accurate value for reliable initial ambiguity resolution. From the narrow stripe of the zero baseline, it seems that the IFB rate could be obtained by taking the mean of the time

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Fig. 4 The RATIO plot at the epoch 112 (a), 323 (b) and 800 (c) for baseline REF6–AIR5

Fig. 5 Statistics of the relationship of the maximum RATIO and the IFB rate over all epochs for the three baselines, the zero baseline (a), KOSG–KOS1 (b), and REF6–AIR5 (c)

series. However, there are numerous points scattered beyond and even far away from the stripe. The boundary of the stripe changes along with the epochs, particularly for the KOSG– KOS1 and the kinematic baseline. This means the highest RATIO could lead to wrong IFB rates. Figure 4 shows the RATIO distribution of three typical epochs: Fig. 4a is the worst case where all RATIOs are very low and the highest RATIO is related to a fully biased IFB rate; in Fig. 4b there are two peaks both close to the correct IFB rate value; and Fig. 4c is the usual case with a single peak corresponding to the correct value. To obtain a statistical interpretation of their relationship, the maximum RATIO and the corresponding IFB rate for all epochs are depicted in Fig. 5 for the three baselines, where each point in the figures corresponds to one epoch. In Fig. 5a, IFB rates with the maximum RATIO are almost the same, while in Fig. 5b, c, some maximum RATIO values correspond to wrong IFB rate values. Obviously, selecting the IFB rate value corresponding to the maximum RATIO is not always reliable. Although there might be a number of methods to estimate the correct IFB rate based on the relationship between RATIO and IFB rate samples shown in Fig. 1, we

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are going to develop a new method based on particle filtering to obtain a more reliable solution in real time.

4 Particle filter Particle filter (Gordon et al. 1993) is also known as the sequential Monte Carlo method, which is in fact a recursive Bayesian filter implemented by Monte Carlo simulation. It has a good performance for even nonlinear and nonGaussian problem (Arulampalam et al. 2002). In the Monte Carlo method, the state vector is represented by a set of discrete samples with corresponding weights to approximate the probability density function (PDF) of the estimates. If the number of samples is large enough, the state vector can be accurately estimated based on the samples. The approximation error in Monte Carlo method can be controlled under a given value within a certain probability. The basic idea of the particle filter is to improve the representation of the samples by changing their weights and then their values according to their weights. Finally, a set of samples close to the true values with proper weight is achieved, so that the weighted


mean can be taken as the best estimate of the state vector. The theoretical details can be found in the work by Doucet et al. (2001), Dimov (2008), Gustafsson (2010) and Haug (2012). Here, we focus on the practical algorithm of the bootstrap particle filter. Let the state equations and the observations equation of the estimation of the filter problem be x(k) = f (x(k − 1), (k)),

(11a)

y(k) = h(x(k), e(k)),

(11b)

where x(k) is the state vector, y(k) the observation vector, (k) and e(k) are system and observation noise, respectively. The algorithm of particle filter starts from generating an N initial set of particles x0i , w0i i=1 with N being the total number of particles. Each particle is expressed by its value x0i and initial weight w0i . These particles are uniformly distributed (or any other distribution based on a priori information) and scattered over the regions where state vector could be. For epoch k, the value of each particle is judged, for example, by its agreement with the observations at the epoch. Then, the weight wki for each particle xki can be updated according to the conditional likelihood of the observation p(yk |xki ) as

j

with the element of the ordered series u i . If wˇ k is smaller than j j u i , the particle xk will be deleted and check the next wˇ k . As j soon as one wˇ k equal or larger than u i is found, the particle j xk is duplicated once. Then come to u i+1 , repeat the step for u i with i = 2. . .N . Finally, N particles are copied and are j assigned with the same weight of wk = N1 . After that, with the resampled particles, the values for the next epoch k + 1 are predicted and drawn by (11a), so that the particles are ready for the next epoch.

5 The new approach Assume that the GLONASS observations at a single epoch are processed using the observation equations (5) and that the NEQ of (8) is generated at each epoch, instead of solving the NEQ with both unknown parameters include ambiguities and IFB rate, the particle filter in Sect. 4 is implemented to estimate the IFB rate. The state variable is the IFB rate γab which is actually a constant. The state equation is extended as i + γ , γabi = γab k−1 k

i w¯ ki = wk−1 p(yk |xki ).

(12)

As soon as the weights for all particles are updated, with the

j normalized weights wˆ ki = w¯ ki / Nj=1 w¯ k , the estimated value of state vector and its variance are given by xˆk ≈

N

wˆ ki xki ,

(13a)

i=1

pˆ k ≈

N (xki − xˆk )(xki − xˆk )T wˆ ki .

(13b)

i=1

Now that the normalized weights are not the same, the particles with too small weights are removed, whereas those with larger weights are represented by more particles around and with an average weight to keep the number of total particles the same. This procedure is called resampling and is important to the particle filter. One of the resampling methods is the stratified resampling method by Kitagawa (1996). The principle is that on average each particle should have a weight around 1/N , particles with too small weight are removed and that with large weight are duplicated. The stratified resampling method implemented in the algorithm is as follows. u˜ i N }i=1 with u˜ i ∼ First, an ordered series {u i = (i−1)+ N U (0, 1) standard uniform distribution and the accumulated

j j weights with wˇ k = h=1 wˆ kh are generated, where h is an integer number. Then, the accumulated weights are compared

(14)

where γ is assumed to be normal distributed noise. The key issue here is the likelihood function which is usually p(yk |xk ) derived from observations to update the particle weight according to (12). However, p(yk |xk ) does not tell anything about the quality of the IFB rate, since IFB rate and ambiguities are correlated. In other words, for any IFB rate we have the same observation residuals in the adjustment. According to the relationship between the IFB rate and the corresponding RATIO in Sect. 3, the RATIO can be used as the probability of xki to judge its quality. Approximately, the PDF for ambiguities to be fixed to the right integers under a given IFB rate can be expressed as p( bˇ k |xki ) ∝ RATIOi .

(15)

where ∝ denotes direct proportionality. Because RATIO for all the particles at each epoch are usually of very much different quantity, the normalized RATIO value RATIOi p( bˇ k |xki ) = N i=1 RATIOi

(16)

is selected. It must be pointed out that (16) is an empirical expression, although its efficiency is validated in the following experimental evaluation. Based on the above definition, the particle filter for estimating the IFB rate can be carried out as follows:

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1. Process the GLONASS observations at current epoch using the observation equations (5) and generate the NEQ of (9). Of course accumulated NEQ over several epochs can also be used if single-epoch ambiguity resolution does not perform well. 2. For the first epoch, an initial set of particles with a certain number of elements must be generated, let us say N . These particles should be uniformly distrib{x0i , w0i }i=1 uted over the interval [−0.10, . . ., 0.10] m/FN and the weights of all particle are 1/N . The total number of the particles N is 200 for this study. As described in Sect. 4, the larger the number of particles is, the more accurate the results will be. But a larger number of particles also increases the computational burden. For other epochs k = 2, 3, . . . the particles are already prepared in the processing of the previous epoch. 3. For each of the particles, a solution of (9) is derived by inserting the IFB rate value of this particle into the NEQ in step 1. Then integer ambiguity resolution is undertaken using the LAMBDA method and fixing RATIO is obtained. At the end of this step, we have the RATIOs for all the particles. 4. Update the weight of each particle using (12) with the empirical PDF p( bˇ k |xki ) of (16). Then normalize the weights and calculate the estimated IFB rate and its standard deviation (STD) by (13) as well. 5. Resample the particles as described in Sect. 4 and transmit each particle to next epoch by (14), then the particle set for the next epoch is ready. 6. Repeat the steps 1–5 for the epoch k + 1. If the STD of the estimated value is smaller than the threshold value, the procedure can be ended and IFB rate is set as known parameter. The flowchart of this procedure is presented in Fig. 6. This algorithm can be applied for precise IFB rate calibration, for example using long data set and even without known stations’ coordinates. It can also be run for fast and even real-time calibration. In this case, the particle procedure can be stopped, as soon as the estimated IFB rate has converged, for example its STD is smaller than a threshold value, and then the IFB rate value can be fixed for deriving precise positioning with ambiguity-fixing. Certainly, a procedure to monitor its possible changes should be involved in the data processing as part of the quality control.

6 Experimental validation The performance of the new approach presented in Sect. 5 is investigated in this section using the same three data sets described in Sect. 3. It must be pointed out that in all the processing for IFB rate estimation or GLONASS ambigu-

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Fig. 6 Flowchart of the procedure for IFB estimation, where stdth is the STD threshold value

ity resolution, only dual-frequency GLONASS data alone are employed and GPS data are processed independently for comparison. In the following, three major results of the experimental validation are presented and analyzed in details to confirm whether the new approach could be applied to realtime applications. The first part is to evaluate the convergence and accuracy of the IFB rate estimation. Then, the performance of the baseline processing with integer ambiguity resolution is investigated with IFB rate estimation procedure or with estimated IFB rate. The last part is to analyze the time consumption of the new approach. 6.1 IFB rate estimation For the three baselines, the estimated IFB rates of all the epochs are drawn in Fig. 7. For the zero baseline (Fig. 7a), the mean value after convergence is −0.0017 mm/FN, with STD of 0.16 mm/FN. For baseline KOSG–KOS1 (Fig. 7b), the mean value after convergence is −24.9 mm/FN, with STD of 0.36 mm/FN. For the kinematic baseline REF6–AIR5 (Fig. 7c), the estimated bias is −0.043 mm/FN with STD of 0.65 mm/FN. Figure 8 shows the convergence process of the IFB rate and its STD for the baselines. It is clear that the estimated IFB rate converges quickly and it needs at maximum about three minutes to become stable. By the way, it could be even faster if a better initial value is available. However, there are obviously some fluctuations of about few mm/FN for baseline KOSG-KOG1 and the kinematic base-


Fig. 7 The estimated IFB rates using the particle filter for the zero baseline (a), KOSG–KOS1 (b) and REF6–AIR5 (c)

Fig. 8 The convergence of the estimated IFB rate (solid line) and STD (dash lines) for zero baseline (a), KOSG–KOS1 (b) and baseline REF6–AIR5 (c). The sampling rate is 5, 30, 1 s, respectively

Fig. 9 The convergence procedures of the estimated IFB rate versus the number of epochs for the zero baseline (a), KOSG–KOS1 (b) and REF6–AIR5 (c). The star symbols denote the converged point. The sampling rate is 5, 30, and 1 s, respectively

line as shown in Fig. 7b, c, respectively. This is probably caused by inaccurate modeling and improper quality controls, as KOGS-KOG1 are equipped with different type of antennas and REF6–AIR5 is in kinematic mode. To further investigate the convergence time for the IFB rate parameter, the data of the three baselines are processed in short sessions. The first session starts at the data begin-

ning and then moves forwards with a step size of 20 epochs, so there are about 110–150 sessions for each baseline. The processing of each session keeps going until the IFB rate is converged with STD smaller than a threshold which is set to 2 mm/FN for all baselines in this study. Figure 9a–c shows the convergence process of the IFB rate of all the sessions for the three baselines, respectively.

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Each line shows the IFB rate estimates for a single session and ends at the epoch meeting the convergence criteria. For clarity, the end point is marked with a star symbol. The statistics of convergence times are presented in Table 1. From the statistics, for the zero baseline and the kinematic baseline, the new approach could obtain a converged IFB rate with 10 epochs of data collected with 30 s sampling interval. However, for the KOSG–KOS1 baseline with a sampling rate of 30 s four minutes of data is needed. There is a big difference in the averaged time needed for the convergence but the numbers of needed epochs are closer to each other. This can be clearly seen from Fig. 10 showing the distribution of the number of epochs needed. Therefore, one possibility is that a Table 1 The statistics of the convergence time for the IFB rate estimated by the new approach for the three baselines

Baseline

Sampling rate (s)

certain number of epochs are necessary for the particle filter to obtain a converged IFB rate. Another reason could be that KOSG–KOS1 is the only baseline with different receivers, so that it needs longer time for convergence. Since KOSG and KOS1 do not provide 1 s high rate data, another two collocated IGS stations STR1 and STR2 in Australia with different type of receivers are chosen for validation. The data processing is carried out in the same way as for the previous three baselines, but using sample rates of 1, 5 and 30 s. The results shown in Fig. 11 confirm that a certain number of epochs are needed for the new approach, so for baselines with different receivers IFB rate could also be precisely estimated using 1 Hz data collected within 30 s.

# Total epochs

#Sessions/ removed

Time(s)/epochs for convergence Max

Min

Mean

Zero Baseline

5

2270

113/0

40/8

15/3

24/5

KOSG–KOS1

30

2880

144/1

630/21

120/4

240/8

1

2663

133/6

29/29

6/6

11/11

REF6–AIR5

Fig. 10 The statistics of the time/epochs needed for the convergence of IFB rate for the zero baseline (a), KOSG–KOS1 (b) and REF6–AIR5 (c). The sampling rate is 5, 30 and 1 s, respectively

Fig. 11 The statistics of the epochs needed for the convergence of IFB rate for baseline STR1–STR2 with different type of receivers. The sampling rate is 1 s (a), 5 s (b) and 30 s (c), respectively

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Author's personal copy Particle filter-based estimation of inter-frequency… Fig. 12 Three-dimensional RATIO distribution (left) and the estimated IFB rates (right) for baseline KOSG-APEL with a length of 11 km

Fig. 13 Position differences with respect to the ground truth for GLONASS float solution (red) and fixed solution (blue) with the initial calibration of IFB rate in kinematic mode. The right panel is a snapshot

of the first 30 epochs (15 min) of the left one to show the process of the IFB rate estimation and the first ambiguity-fixing afterwards

Anyway, the quick convergence and the stable estimation provide a chance for a field and real-time calibration of the IFBs for instantaneous ambiguity resolution without any a priori information. For longer baseline, this approach still works but the performance degrades due to the effects of inaccurately modelled error sources. Baseline KOSG-APEL of 11 km is also tested with data collected on 17th December 2013. Both stations are equipped with LEICA GR25 receiver, but with antennas LEIAR25.R4 LEIT and AOAD/M_B NONE, respectively. The three-dimensional RATIO distribution and the estimated IFB rate are presented in Fig. 12. It is clear that the approach works well but with a lower RATIO compared to the shorter baselines .

tion and at the seventh epoch the estimated IFB rate reaches the converged status with STD of 0.78 mm/FN. Therefore, the particle filter is stopped as described in the procedure in Sect. 5. The remaining data are processed with the estimated IFB rate as known and ambiguities are fixed with LAMBDA method at each epoch. The results show that the ambiguity resolution has a success rate of 98.6 %. The baseline components in the results are presented in Fig. 13, together with the float solution with respect to the averaged GLONASS fixed solutions. The left of Fig 13 shows the whole time series of the position difference, while the right one shows the result of the first 15 min where the effect of the integer ambiguity resolution is clearly visible. The position time series of the GLONASS fixed solution is also compared with that of the GPS one in Fig. 14 with the averaged GPS fixed solutions as reference. The differences in East, North and Up directions are 1.3, 1.0, 1.7 mm with STD of 2.3, 2.8, 5.5 mm, respectively. This indicates that GLONASS has the same performance as GPS for real-time kinematic positioning using the new method.

6.2 Kinematic solution with ambiguity resolution The data from baseline KOSG–KOS1 are employed for the kinematic positioning with IFB rate calibration to demonstrate the position accuracy of the fixed solution using the new method. The processing started with IFB rate estima-

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Fig. 14 The comparison of GLONASS fixed solution and GPS fixed solution

Fig. 16 The computational time for a single epoch including the particle filter for baseline KOSG–KOS1. The upper thin dash line and the thick solid line close to the bottom are for the search interval of [−0.10, . . ., 0.10] and [−0.04, . . ., 0] m/FN, respectively. The number of satellites is also plotted

soon as the ambiguities are fixed, the position STD reaches an accuracy of about 2, 2 and 5 mm in East, North and Up directions. There are very few epochs without fixed solution with a very large bias. For the single-epoch float solution, the position accuracy is about several decimeters for each component. Obviously, the new approach can fix the integer ambiguities on single-epoch base and position results are largely improved by reliable integer ambiguity resolution. 6.4 Computational efficiency Fig. 15 Comparison of the GLONASS single-epoch solution with (blue) and without (red) integer ambiguity resolution for baseline KOSG–KOS1

6.3 Single-epoch ambiguity resolution As a further step of the experiment, the GLONASS data of the baseline KOSG–KOS1 are processed epoch-by-epoch independently with integer ambiguity resolution by fixing the IFB rate to the estimate in Sect. 6.2, i.e., only data of the single epoch are used without any information from the previous epochs. For this baseline as the IFB rate is significantly nonzero, the ambiguity resolution is rarely available if IFB rate is not estimated and corrected. The related solution is actually single-epoch float solution. The impact of the GLONASS ambiguity resolution can be demonstrated by comparing the aforementioned single-epoch fixed solution and the float solution. Figure 15 shows the position differences with respect to the static result of the fixed solution (blue dots) and float solution (red dots).The success rate of the single-epoch ambiguity resolution is about 97.9 %. As

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From the algorithm of the new approach, at each epoch, about 200 particles must be tested for ambiguity resolution. Therefore, the computational time at each epoch is of course a critical concern, especially for real-time applications. To give an estimation of the computation efficiency, the computational time at each epoch is recorded in a personal computer with a processor of 2.8 GHz and plotted in Fig. 16. The computation time is somehow correlated with the number of satellites involved at the epoch. Generally, it could be completed within 1 s for most of the epochs. The computational time could be reduced significantly if a better initial IFB rate is available, as fewer particles are needed. For example, it takes about 0.17 s if the searching is within [−0.04, . . ., 0] m/FN.

7 Conclusion Due to the existence of the IFB of the phase observations, GLONASS integer ambiguity resolution is still very difficult for relative positioning using different type of receivers, especially for real-time applications. The major reason is that


almost all the recent methods estimate the IFB rate together with the ambiguities which are highly correlated with each other and the situation cannot be improved very much using simultaneous GPS data. Hence, rather long data sets of several hours up to 1 day should be employed in order to get a converged IFB rate. During this process, GLONASS ambiguity resolution is hardly available, which is a critical obstacle for fast and real-time precise positioning. In this study, we demonstrated that integer ambiguity resolution is very reliable even only using a few epochs of GLONASS observations if the IFB rate is precisely known. Moreover, the closer the IFB rate is to the true value, the better the fixing performance will be. Therefore, RATIO of the fixing can be used as a reliable index to qualify a given IFB rate. Furthermore, the variation of IFB rates is set within [−0.10, . . ., 0.10] m/FN. Based on these prerequisites, a new method is developed to estimate IFB rate by means of particle filter. The new method is evaluated with three data sets for one zero baseline and one kinematic experiment with an interstation distance up to 1 km, and one short baseline of 814 m with different type of receivers. The statistical result shows that the IFB rate could be estimated precisely for integer ambiguity resolution using 1 Hz GLNOASS data collected within 30 s. As soon as the IFB rate is estimated, the fixed solution is available immediately and its positioning accuracy is similar to that of GPS for both static and kinematic positioning. The computational time needed for the estimation is also investigated. In general, the computational time is slightly correlated with the number of satellites involved in the data processing but depends on the accuracy of the initial value of IFB rate. On average, it takes 0.2 and 0.8 s for obtaining a converged IFB rate for an initial value within [−0.04, . . ., 0] and [−0.10, . . ., 0.10] m/FN, respectively. From the outcome of the experimental evaluation, the new method enables for the first time the quick estimation of IFB rate using only GLONASS data of few epochs and without a priori value. Therefore, the method can not only be used for calibration of the receiver IFB rate in advance, but also can be realized GLONASS fast and real-time precise positioning using any type of receivers no matter its IFB rate is calibrated or not. Acknowledgments The first author is financially supported by the China Scholarship Council (CSC) for his study at the Technische Universität Berlin and the German Research Center of Geosciences (GFZ).

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