GPS Solutions

, Volume 21, Issue 3, pp 949–961 | Cite as

Particle filter-based estimation of inter-system phase bias for real-time integer ambiguity resolution

  • Yumiao Tian
  • Maorong Ge
  • Frank Neitzel
  • Jianjun Zhu
Original Article

Abstract

Although double-differenced (DD) observations between satellites from different systems can be used in multi-GNSS relative positioning, the inter-system DD ambiguities cannot be fixed to integer because of the existence of the inter-system bias (ISB). Obviously, they can also be fixed as integer along with intra-system DD ambiguities if the associated ISBs are well known. It is critical to fix such inter-system DD ambiguities especially when only a few satellites of each system are observed. In most of the existing approaches, the ISB is derived from the fractional part of the inter-system ambiguities after the intra-system DD ambiguities are successfully fixed. In this case, it usually needs observations over long times depending on the number of observed satellites from each system. We present a new method by means of particle filter to estimate ISBs in real time without any a priori information based on the fact that the accuracy of a given ISB value can be qualified by the related fixing RATIO. In this particle filter-based method, the ISB parameter is represented by a set of samples, i.e., particles, and the weight of each sample is determined by the designed likelihood function related to the corresponding RATIO, so that the true bias value can be estimated successfully. Experimental validations with the IGS multi-GNSS experiment data show that this method can be carried out epoch by epoch to provide precise ISB in real time. Although there are only one, two, or at most three Galileo satellites observed, the successfully fixing rate increases from 75.5% for GPS only to 81.2%. In the experiment with five GPS satellites and one Galileo satellites, the first successfully fixing time is reduced to half of that without fixing the inter-system DD ambiguities.

Keywords

Ambiguity fixing Multi-GNSS integration Phase inter-system bias Particle filter 

Introduction

Two global navigation satellite systems (GNSS) are currently operational with their full constellations, namely GPS and GLONASS, while the European Galileo and the Chinese BeiDou system (BDS) are rapidly advancing toward their operational services. Besides those four global systems, the Japanese Quasi-Zenith Satellite System (QZSS) and the Indian Regional Navigation Satellite System (IRNSS) are also being developed. Those multi-GNSS constellations largely increase the number of available satellites in orbits and enhance positioning performance in terms of accuracy, reliability, and availability as demonstrated by Force and Miller (2013), Odolinski et al. (2014), and Li et al. (2015).

In order to employ multi-GNSS observations in positioning, the problem of the differences in space and time reference frame can be solved through multi-GNSS precise orbit and clock determination (Li et al. 2015; Dach et al. 2009) or can be eliminated by forming differenced observations between nearby stations. However, the inter-system bias (ISB) in both range and phase observations must be handled properly. The range ISB can be estimated along with the coordinate parameters (Odijk and Teunissen 2013a), while the phase ISB can be absorbed by ambiguity parameters without any negative consequence to the real-valued solution. In this case, although double-differenced (DD) ambiguities among each individual system, named intra-system DD ambiguities, are still integer natural, the ones between satellites of different systems referred to as inter-system DD ambiguities are not. In most of the studies, for each system, one satellite is selected as reference to define DD ambiguities within each individual system for fixing (Julien et al. 2003; Odijk et al. 2014; Odolinski et al. 2014; Ineichen et al. 2008). Obviously, inter-system DD ambiguities cannot be fixed due to the existence of phase ISBs. However, they can also be resolved to integer, if ISBs are well known or can be precisely estimated. Actually, the integer part of ISB does not affect the results because it lumps into the integer ambiguities, whereas the fractional part of ISB, labeled F-ISB, plays the decisive role to recover the integer feature of the inter-system DD ambiguities. Fixing the inter-system DD ambiguities is very critical for positioning in severe environments, such as urban areas where signals are easily blocked by high buildings and trees. In this case, the number of observed satellites of each single system can be very low. For example, for a four-system constellation, assuming two satellites of each system to be observed, there are eight satellites with only four DD ambiguities of natural integer and three inter-system DD ambiguities which cannot be fixed without known ISBs. Hence, scientists started to investigate the ISB characteristics and attempted to resolve inter-system ambiguities.

GPS L1 and Galileo E1 are two typical frequency bands with the same frequency and therefore have been focused on in such studies to avoid the complexity caused by the difference in wavelength (Odijk and Teunissen 2013a, b; Paziewski and Wielgosz 2015).

One straightforward way to handle the ISB problem is to introduce the ISB parameter to the DD observation model directly. However, this will make the corresponding linear equation system rank defect and cannot be solved. Odijk and Teunissen (2013b) suggested that the reference satellites for GPS and Galileo are selected, separately. Then, the ISB is derived from the estimate of the inter-system DD ambiguity of the two reference satellites. This method usually needs observations over long times when satellites from each system are few, i.e., the inter-system DD ambiguity is crucial in ambiguity fixing. Paziewski and Wielgosz (2015) proposed another approach, assigning the F-ISB parameter an a priori value with a priori sigma of half of the phase cycle to constrain the ISB, so that the rank deficiency is removed. Apparently, the results of the ambiguity resolution depend on the actual accuracy of this a priori value.

Similar to the ISB problem, the inter-frequency bias (IFB) was a crucial issue in the GLONASS ambiguity resolution. Tian et al. (2015) proposed a new approach to estimate the GLONASS IFB based on the relationship between IFB accuracy and the fixed RATIO by means of particle filtering to avoid estimating IFB with the other parameters simultaneously. Since IFB and ISB have similar characteristics according to the ISB analysis by Odijk et al. (2014), we investigate the possibility of employing a similar approach to rapidly estimate ISBs in multi-GNSS data processing.

We propose a new approach based on particle filter to estimate F-ISBs for fixing all DD ambiguities including the inter-system ones for multi-GNSS data processing. The approach is validated with a number of short baselines in the IGS Multi-GNSS Experiment (MGEX). The next section will introduce the basic mathematical models of multi-GNSS data processing, followed by the investigation of the characteristics of F-ISBs. Then, the new approach and the related algorithm will be described in detail. Finally, the experiment and the validation results are presented and discussed before the conclusions are drawn.

Multi-GNSS observation model

According to Teunissen and Kleusberg (1996), the undifferenced (UD) model of GPS pseudorange and carrier phase observations can be expressed as
$$P_{a}^{i,G} = \rho_{a}^{i,G} - \left( {\delta t_{a} - \delta t^{i,G} } \right)c + d_{a}^{G} - d^{i,G} + I_{a}^{i,G} + T_{a}^{i,G} + \varepsilon_{a}^{i,G}$$
(1a)
$$\lambda^{i,G} \varPhi_{a}^{i,G} = \rho_{a}^{i,G} - \left( {\delta t_{a} - \delta t^{i,G} } \right)c + \mu_{a}^{G} - \mu^{i,G} + \lambda^{i,G} B_{a}^{i,G} - I_{a}^{i,G} + T_{a}^{i,G} + \xi_{a}^{i,G}$$
(1b)
where P is the pseudorange observation and Φ is carrier phase observation, indices i, a, and G refer to the satellite, receiver, and GPS, respectively, ρ is the distance between satellite and receiver, \(\delta t_{a}\) and \(\delta t^{i,G}\) are the clock offsets for receiver and satellite, c is the light speed in vacuum, \(d_{a}^{G}\) and \(d^{i,G}\) are the hardware delay for receiver and satellite on pseudorange, respectively, \(\mu_{a}^{G}\) and \(\mu^{i,G}\) are the hardware delay for receiver and satellite on carrier phase, respectively, \(I_{a}^{i,G}\) is the ionospheric delay, \(T_{a}^{i,G}\) is the tropospheric delay, \(\varepsilon_{a}^{i,G}\) and \(\xi_{a}^{i,G}\) are the measurement noise for pseudorange and carrier phase observations, respectively.
Due to the time difference of Galileo with respect to GPS, the parameter \(\delta t_{\text{sys}}^{\text{GE}}\) has to be introduced to transmit Galileo system time to GPS time (Hahn and Powers 2005). According to Odijk and Teunissen (2013b), the UD models of Galileo in the same reference frame can be written as
$$P_{a}^{j,E} = \rho_{a}^{j,E} - \left( {\delta t_{a} - \delta t^{j,E} - \delta t_{sys}^{GE} } \right)c + d_{a}^{j,E} - d^{j,E} + I_{a}^{j,E} + T_{a}^{j,E} + \varepsilon_{a}^{i,E}$$
(2a)
$$\lambda^{j,E} \varPhi_{a}^{j,E} = \rho_{a}^{j,E} - \left( {\delta t_{a} - \delta t^{j,E} - \delta t_{\text{sys}}^{\text{GE}} } \right)c + \mu_{a}^{j,E} - \mu^{j,E} + \lambda^{j,E} B_{a}^{j,E} + k^{j} \bar{\gamma }_{a} - I_{a}^{j,E} + T_{a}^{j,E} + \xi_{a}^{i,E}$$
(2b)
Afterward, the SD observations between station a and b can be formed, where the satellite clock offsets are eliminated. Furthermore, the ionospheric and tropospheric delays are largely reduced and can be neglected in the SD observations if the inter-station distance is small, for example within 15 km. Then, the SD models for GPS and Galileo are
$$P_{ab}^{i,G} = \rho_{ab}^{i,G} - \delta t_{ab} c + d_{ab}^{G} + \varepsilon_{ab}^{i,G}$$
(3a)
$$\lambda^{i,G} \varPhi_{ab}^{i,G} = \rho_{ab}^{i,G} - \delta t_{ab} c + \mu_{ab}^{G} + \lambda^{i,G} B_{ab}^{i,G} + \xi_{ab}^{i,G}$$
(3b)
$$P_{ab}^{j,E} = \rho_{ab}^{j,E} - \delta t_{ab} c + d_{ab}^{E} + \varepsilon_{ab}^{i,E}$$
(4a)
$$\lambda^{j,E} \varPhi_{ab}^{j,E} = \rho_{ab}^{j,E} - \delta t_{ab} c + \mu_{ab}^{E} + \lambda^{j,E} B_{ab}^{j,E} + \xi_{ab}^{i,E}$$
(4b)
The DD models within GPS L1 and Galileo E1 observations are
$$P_{ab}^{ij,GG} = \rho_{ab}^{ij,GG} + \varepsilon_{ab}^{ij,GG}$$
(5a)
$$\lambda^{G} \varPhi_{ab}^{ij,GG} = \rho_{ab}^{ij,GG} + \lambda^{G} B_{ab}^{ij,GG} + \xi_{ab}^{ij,GG}$$
(5b)
and
$$P_{ab}^{ij,EE} = \rho_{ab}^{ij,EE} + \varepsilon_{ab}^{ij,EE}$$
(6a)
$$\lambda^{E} \varPhi_{ab}^{{ij,{\text{EE}}}} = \rho_{ab}^{{ij,{\text{EE}}}} + \lambda^{E} B_{ab}^{{ij,{\text{EE}}}} + \xi_{ab}^{{ij,{\text{EE}}}}$$
(6b)
respectively.
The DD observations between GPS and Galileo satellites can also be built in a similar way. But the hardware delay cannot be eliminated and remains as ISB parameter \(d_{ab}^{GE}\) and \(\mu_{ab}^{GE}\). The corresponding models can be expressed as
$$P_{ab}^{ij,GE} = \rho_{ab}^{ij,GE} + d_{ab}^{GE} + \varepsilon _{ab}^{ij,GE}$$
(7a)
$$\lambda \varPhi_{ab}^{ij,GE} = \rho_{ab}^{ij,GE} + \mu_{ab}^{GE} + \lambda B_{ab}^{ij,GE} + \xi_{ab}^{ij,GE}$$
(7b)
The estimation can be carried out by means of Kalman filter or least square (LSQ) adjustment. For LSQ estimation, after linearization, the models (5a), (6a), and (7a) can be contributed together to the normal equation (NEQ), written as
$$\left[ {\begin{array}{*{20}c} {\varvec{N}_{{\varvec{xx}}} } & {\varvec{N}_{{\varvec{xb}}} } & {\varvec{N}_{{\varvec{xy}}} } \\ {} & {\varvec{N}_{{\varvec{bb}}} } & {\varvec{N}_{{\varvec{by}}} } \\ {\text{sym}} & {} & {\varvec{N}_{{\varvec{yy}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} \varvec{x} \\ \varvec{b} \\ \varvec{y} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\varvec{W}_{\varvec{x}} } \\ {\varvec{W}_{\varvec{b}} } \\ {\varvec{W}_{\varvec{y}} } \\ \end{array} } \right]$$
(8)
where vector x is composed by unknown coordinate’s parameters, b contains unknown DD ambiguity parameters, and y includes unknown ISB parameters.
Because of the additional ISB parameter, the system (8) is rank deficient and cannot be solved directly (Odijk and Teunissen 2013b; Paziewski and Wielgosz 2015). Even though some approaches have been proposed to deal with the problem, in general, the ISB cannot be estimated explicitly from (8). However, if the ISB parameter is given, the full-rank NEQ can be obtained based on observations of either a single epoch or accumulated multi-epochs and the float DD ambiguities and the corresponding variance–covariance (VC) matrix are derived as
$$\left[ {\begin{array}{*{20}c} {\hat{\varvec{x}}} \\ {\hat{\varvec{b}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\varvec{Q}_{{\varvec{xx}}} } & {\varvec{Q}_{{\varvec{xb}}} } \\ {\varvec{Q}_{{\varvec{bx}}} } & {\varvec{Q}_{{\varvec{bb}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varvec{W}_{\varvec{x}} - \varvec{N}_{{\varvec{xy}}} \varvec{y}} \\ {\varvec{W}_{\varvec{b}} - \varvec{N}_{{\varvec{by}}} \varvec{y}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\hat{\varvec{x}}} \\ {\hat{\varvec{b}}} \\ \end{array} } \right]_{{\varvec{y} = 0}} - \left[ {\begin{array}{*{20}c} {\varvec{Q}_{{\varvec{xx}}} } & {\varvec{Q}_{{\varvec{xb}}} } \\ {\varvec{Q}_{{\varvec{bx}}} } & {\varvec{Q}_{{\varvec{bb}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varvec{N}_{{\varvec{xy}}} \varvec{y}} \\ {\varvec{N}_{{\varvec{by}}} \varvec{y}} \\ \end{array} } \right]$$
(9)
Afterward, integer ambiguity can usually be fixed according to their integer nature as demonstrated in Blewitt (1989), Dong and Bock (1989), and Ge et al. (2008) and the integer ambiguity candidate can be derived by the LAMBDA method presented in Teunissen (1995). Then, if the RATIO value proposed by Euler and Schaffrin (1991) is larger than a certain threshold, for example 3, the integer ambiguity candidate can be accepted. The relationship between RATIO and F-ISB will be investigated in the next section.

Relationship between RATIO and F-ISB

The relationship between F-ISB parameter and RATIO values is investigated numerically by processing and analyzing experimental data. Additionally, for the F-ISB estimation, there could be two converged points within an approximated distance of one cycle, because the integer part of ISB does not affect the integer feature of the ambiguities. A typical case is that the F-ISB value is close to half a L1 wavelength, for example 0.095 m, and the F-ISB is initialized within [−0.1, …, 0.1] m, and then, values close to the two endpoints are good candidates of the F-ISB. Although both values are valid as correct values, they must be shifted together before applying particle filter. This problem will be referred to as ISB half-cycle problem and studied in this section.

Experimental data

As mentioned above, the ISBs are present at both pseudorange and carrier phase observations. However, the pseudorange ISB has rather small effect to the carrier phase ISB, because ranges are significantly down-weighted in the estimation. Moreover, the pseudorange ISB can also be estimated in advance using pseudorange measurements only and treated as known in this estimation. Therefore, the unknown parameters include just the station coordinate, DD ambiguities, and phase ISBs.

The baseline employed for this study on ISB characteristics is composed of two MGEX stations TLSG and TLSE in France. TLSG is equipped with a SEPT POLARX4TR receiver and a TRIMBLE 59800-00 antenna, while TLSE is equipped with a TRIMBLE NETR9 receiver and a TRM59800.00 antenna. The data used are collected on the day of year (DOY) 001 of 2015 with epoch interval of 30 s. The GPS and Galileo satellite sky plots for TLSE are shown in the top panel and bottom panel of Fig. 1, respectively. The number of satellites for the baseline is presented in Fig. 2.
Fig. 1

Satellite sky plots of GPS (top) and Galileo (bottom) for station TLSE on DOY 001 of 2015

Fig. 2

Numbers of GPS and Galileo satellites for baseline TLSG–TLSE on DOY 001 of 2015, with an elevation mask of 10°

The baseline composed of two MGEX stations GOP6 and GOP7 in the Czech Republic is selected to demonstrate the half-cycle problem. The two stations employ LEICA GRX1200 and JAVAD TRE_G3TH DELTA receivers, respectively, but use antennas of the same type, i.e., a LEIAR25.R4 LEIT.

Relationship between ISB and fixing RATIO

After the NEQ (8) is constructed with single-epoch observations including GPS L1 and Galileo E1, as well as GPS L2 for assistance, the F-ISB candidates within a pre-defined searching region for GPS L1 and Galileo E1 integration are introduced one by one as known F-ISB to obtain the NEQ (9) for ambiguity resolution using the LAMBDA method. The associated fixing RATIOs are derived for further investigations.

First, the interval [−0.20, …, 0.20] m, which is about [−1.0, …, 1.0] cycle for GPS L1 and Galileo E1 phase observations, is evenly sampled with a sample interval of 1 mm. This results in a total number of 400 F-ISB candidates. For the epochs with Galileo observations, RATIO values corresponding to those pre-defined F-ISB samples are calculated. The RATIO values of the first calculated epoch are presented in the top panel of Fig. 3, while those for all the epochs are depicted in the bottom panel. Two peaks in the top panel and two ridges in the bottom panel with relatively large RATIO values can be observed. The major reason is that the integer part of ISB can be fully absorbed by the related DD ambiguity and only the F-ISB affects the RAITO values
Fig. 3

Relationship between F-ISB and fixing RATIO for the first epoch with only one Galileo satellite (top) and that for all epochs (bottom)

The previous interval is then expanded into [−1.0, …, 1.0] m to investigate samples over a wider region. Thus, about 2000 samples are generated with the same sampling interval of 1 mm. In this case, most of the F-ISB values have an absolute value larger than one wavelength. The 3D RATIO map is presented in the top panel of Fig. 4, and the mean values of all the epochs are shown in the bottom panel, where the periodic characteristic can be clearly observed.
Fig. 4

Three-dimensional RATIO map of GPS and Galileo integration (top), as well as the mean values along epoch time (bottom)

Figures 3 and 4 show that there are many ISB samples which can provide a fixed solution with significantly large RATIO value, for example larger than 3. From this test, with any ISB within [−3, …, 3] cm around a peak, a fixed solution can be achieved. Certainly, most of them are biased and thus result in a contaminated fixed solution. In order to investigate the accuracy of the fixed solutions, the positions of the fixed solutions are calculated for all the F-ISB candidates and are compared with the GPS only fixed solution. As a typical example, the differences of all the fixed solutions at epoch 1406 are shown in Fig. 5. The fixed solutions with a peak RATIO are indicated in red, while the others are indicated in blue. For F-ISBs without a fixed solution, the differences are not shown. These F-ISB samples are about one-third of the total number depending on the RATIO threshold used. It is clear that the position differences show a periodic performance with respect to the wavelength. In other words, the ISB values with the same fractional part have similar biases in the fixed solution with respect to the wavelength of about 19 cm. The solutions corresponding to the peaks of the RATIO values overlap very well with the GPS only solutions, which indicate that the corresponding ISB values are most likely close to the true value.
Fig. 5

Impact of ISB biases in the estimated station positions compared to the ground true values from GPS L1 solutions. The result is from Epoch 1406 and only ISBs with fixed solutions are shown

Selecting the F-ISB values corresponding to the local maximum RATIO is a choice to determine the F-ISB value. However, it is difficult to judge the reliability of the estimated IFB. To demonstrate this, the local maximum RATIOs in Fig. 3 are presented in Fig. 6. Although with large maximum RATIO values, the F-ISBs have a variety reaching as large as 4 cm. The bias in F-ISB will affect the ambiguity fixing and the fixed solutions. The particle filter can estimate the results more precisely with the standard deviation (STD) indicating the precision and thus will be employed in this study to estimate the F-ISB.
Fig. 6

Local maximum RATIO and the corresponding F-ISB

F-ISB half-cycle problem

Instead of ISB, F-ISB is employed to recover the integer nature of inter-system ambiguities which is usually restricted over the interval [−0.5, …, 0.5] cycles. In the F-ISB estimation using particle filter presented in the next section, all samples over the interval will be introduced as known value to derive the corresponding RATIO as its quality index. In the case that the true value of an ISB is very close to 0.5 cycles, both −0.5 and 0.5 are equivalent optimal solutions. However, current particle filter cannot handle such problem properly as the particles are split into two groups and cannot converge as the filter proceeds, and thus could not provide a satisfying ISB estimate. This is referred to as the F-ISB half-cycle problem. Here an example of such problem with the data from baseline GOP6-GOP7 is presented. The RATIO value at each epoch reaches to two peaks at −0.095 m/0.5 cycles and 0.095 m/0.5 cycles, as shown in Fig. 7. Since no a priori information about the expected ISB value is available, we cannot precisely narrow the searching interval to avoid such problem.
Fig. 7

Three-dimensional RATIO map for baseline GOP6-GOP7, the F-ISB is 0.095 m which is very close to 0.5 cycles. There are two RATIO peaks within the initial interval [−0.1, …, 0.1] m

This problem was also encountered while estimating the uncalibrated phase delay for the integer ambiguity resolution of precise point positioning (Ge et al. 2008). Here, an approach based on cluster analysis in data mining is proposed to classify all particles into clusters. As the filtering proceeds, the distance between the centroids of clusters becomes close to one cycle. Then, the clusters can be detected and shifted together to a single cluster for F-ISB estimation.

The centroids need to be calculated more than one time in the K-means algorithm, which is one of the traditional but widely used clustering algorithms (Tan et al. 2006). The computation procedure can be refined as follows. First, the two particles with the largest distance are selected as the first point of each cluster. Then, all particles are sorted to the closest cluster, and the centroids of the clusters are calculated. Finally, if the distance between two centroids is close to one wavelength, the particles in one cluster are transformed to another cluster by shifting one cycle. This procedure is carried out just after the update step in the estimation procedure in the next section.

Estimation procedure

From the investigation of the previous section, a pre-defined value of F-ISB which is closer to the true value leads to relatively larger RATIO. This indicates that RATIO can be used as a quality index of a given F-ISB value. At each epoch, after the float solution is derived using LSQ adjustment, for all defined F-ISB samples scattered over the true F-ISB value, the corresponding RATIO can be calculated for each sample through ambiguity resolution using the solution in forms of (9). Then, the F-ISB value can be determined based on these samples and their RATIO values. This idea can be well realized by the particle filter (Gordon et al. 1993) in a similar way as for the IFB rate estimation (Tian et al. 2015). It should be pointed out the particle filter is conducted at each epoch based on the LSQ result for integer ambiguity resolution.

Particle filter

Usually, Kalman filter is applied to linear Gaussian models to derive the optimal solution (Doucet et al. 2000), whereas a particle filter is more suitable for nonlinear and/or non-Gaussian problems, although both are based on the Bayesian estimation. In the latter case, the probability density function (PDF) of the model parameters can be represented by a number of samples instead of analytic functions (Doucet et al. 2000, 2001; Gustafsson 2010; Haug 2012). In this section, we focus only on the practical particle filter algorithm by Gordon et al. (1993), Doucet et al. (2001), Gustafsson et al. (2002) and Haug (2012).

The prediction model and the measurement model for the problem to be solved are expressed as
$$x\left(k \right) = f\left({x\left({k - 1} \right),\epsilon \left(k \right)} \right) = f_{k - 1} \left({x_{k - 1}} \right) + v_{k}$$
(10a)
$$\varvec{y}\left( k \right) = h\left( {\varvec{x}\left( k \right), \varvec{e}\left( k \right)} \right) = h_{k} \left( {\varvec{x}_{k} } \right) +\varvec{\omega}_{k}$$
(10b)
where yk is the measurement state at epoch k, x is the state vector including unknown parameters, \(f_{k - 1} ()\) is the prediction function and \(h_{k} ()\) is the measurement function, and \(\epsilon_{k}\) and ek are the process noise and the measurement noise, respectively. We assume that the noise can be separated from the function and expressed by vk and wk, respectively.

At the beginning, an initial interval of the unknown parameter, within which the true value may be located, is sampled with an equivalent step size and N particles are generated. Each particle \(\varvec{x}_{0}^{i}\) is then assigned an initial weight \(w_{0}^{i} = \frac{1}{N}\). So the initial collection of particles is \(\left\{ {\varvec{x}_{0}^{i} , w_{0}^{i} } \right\}_{i = 1}^{N}\), where i is the number of particles. Particle filter is then described with three main steps: update, resampling, and prediction.

In the update step, the likelihood function of the observation \(p\left( {\varvec{y}_{k} |\varvec{x}_{k}^{i} } \right)\) is derived and the weight of each particle is updated according to \(p\left( {\varvec{y}_{k} |\varvec{x}_{k}^{i} } \right)\) by
$$\bar{w}_{k}^{i} = w_{k - 1}^{i} p\left( {\varvec{y}_{k} |\varvec{x}_{k}^{i} } \right)$$
(11)
The weights are then normalized by \(\hat{w}_{k}^{i} = \bar{w}_{k}^{i} /\sum\nolimits_{j = 1}^{N} {\bar{w}_{k}^{j} }\). Afterward, the parameters in the state vector and its variance can be derived by
$$\hat{\varvec{x}}_{k} \approx \mathop \sum \limits_{i = 1}^{N} \hat{w}_{k}^{i} \varvec{x}_{k}^{i}$$
(12a)
$$\hat{p}_{k} \approx \mathop \sum \limits_{i = 1}^{N} \left( {\varvec{x}_{k}^{i} - \hat{\varvec{x}}_{k} } \right)\left( {\varvec{x}_{k}^{i} - \hat{\varvec{x}}_{k} } \right)^{T} \hat{w}_{k}^{i}$$
(12b)
After the update, particles with small weight will contribute very little to the estimated parameter in (12a). If fewer particles have large weights, the particle set cannot well represent the PDF of the parameters. Therefore, the resampling step is necessary to generate a new set of particles.

The resampling step can be implemented by different methods, one of which is so-called stratified resampling method by Kitagawa (1996). First, the series \(\left\{ {u_{i} = \frac{{\left( {i - 1} \right) + \tilde{u}_{i} }}{N}} \right\}_{i = 1}^{N}\) with \(\tilde{u}_{i} \sim\,U\left( {0,1} \right)\) and accumulated weights collection \(\left\{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{w}_{k}^{j} = \sum\nolimits_{h = 1}^{j} {\hat{w}_{k}^{h} } } \right\}_{j = 1}^{N}\) are calculated, where \(U\left( {0,1} \right)\) is the standard uniform distribution. Second, from \(i = 1, \ldots , N\), for each ui, find the first element in the collection \(\left\{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{w}_{k}^{j} } \right\}_{j = 1}^{N}\) which is larger than ui. Then, put \(\varvec{x}_{k}^{j}\) in a new collection or duplicate \(\varvec{x}_{k}^{j}\) if it has been in the new collection. At last, N particles with assigned weight \(w_{k}^{j} = \frac{1}{N}\) are generated. The resampling step is time-consuming, and therefore, a test is usually employed to judge whether the resampling step is needed. The test is realized by comparing the parameter \(N_{eff} = \frac{1}{{\mathop \sum \nolimits_{i = 1}^{N} \left( {w_{k}^{i} } \right)^{2} }}\) with a threshold value Nth, which is usually set as \(\frac{2}{3}N\). Afterward, the prediction step moves the resampled particles to the next epoch with model (10b). Those three steps are then repeated in the next epoch.

Basic F-ISB estimate procedure

The F-ISB is supposed to be stable at least within a short time. Thus, the value is set as equal to that of the last epoch plus process noise in the prediction model. The introduced noise can also help to increase the diversity of the particles. Therefore, the prediction model employed here is
$$x_{abk}^{i} = x_{abk - 1}^{i} + v_{k}$$
(13)
In the particle filter, the likelihood function of the observations is used to update the weight of the parameters with (11) for calculating their estimates with (12a). However, the statistic information of carrier phase measurements does not directly provide obvious information about the ISB parameter. For each given F-ISB, integer ambiguity resolution can be carried out to obtain the best integer ambiguity candidate. The closer the F-ISB is to the true value, the better the determined integer ambiguity candidate will be. Therefore, the likelihood functions of the ISB particles can be given by the quality of the integer ambiguity candidate instead of the measurements.
The RATIO index indicates the closeness of the float solution to its nearest integer vector (Verhagen and Teunissen 2013) and thus is usually utilized to judge the quality of the integer ambiguity candidates in practice. The larger the RATIO is, the better the results will be. Based on the investigation of the relationship between RATIO and F-ISB, the likelihood function is built according to the RATIO values here. As the RATIO value varies from epoch to epoch, a normalized RATIO value
$$p\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{b}_{k} |x_{k}^{i} } \right) = \frac{{{\text{RATIO}}_{i} }}{{\mathop \sum \nolimits_{i = 1}^{N} {\text{RATIO}}_{i} }}$$
(14)
is selected (Tian et al. 2015).
In conclusion, the complete approach to estimate the ISB is as follows:
  1. 1.

    Process the carrier phase and pseudorange observations to get the (8) with F-ISB parameter.

     
  2. 2.

    For the first epoch, initial particles are derived by sampling [−0.1, 0.1 m] with an equal step size and each particle is assigned the same weight of 1/N. For the other epochs, the particles are prepared at the step of resampling of the previous epoch.

     
  3. 3.

    For each particle, its value is inserted into the (9) to derive the associated float ambiguity and VC matrix. Then, the LAMBDA method is employed to obtain the corresponding RATIO value.

     
  4. 4.

    Detect and solve the half-cycle problem by cluster analysis where two clusters are shifted together if their centroid distance approximately equals the wavelength.

     
  5. 5.

    Normalize the RATIO values and then update the weights according to the likelihood function (14). Calculate the estimated F-ISB and the corresponding variance with (12b). If the STD is smaller than a threshold, the filtering can be considered converging and the estimated F-ISB can be used as corrections.

     
  6. 6.

    Resample the particles if Neff is smaller than Nth.

     
  7. 7.

    Predict the particles for the next epoch according to the prediction model (13).

     
Compared with the procedure proposed in Tian et al. (2015), besides the differences in GNSS data processing and parameterization, the cluster analysis is applied for solving the F-ISB half-cycle problem. This whole procedure for the data processing is depicted by a flowchart in Fig. 8.
Fig. 8

Flowchart of the data processing for F-ISB estimation

Experimental validation

The estimation procedure is implemented to process the data for the experimental validation. All the test baselines involved in this study are from the IGS network. The baseline TLSG–TLSE is processed for validating the convergence of the new approach and the efficiency of fixing all DD ambiguities including inter-system ones. Then, the baseline GOP6–GOP7 is processed to confirm that the cluster analysis can properly solve the half-cycle problem. In order to evaluate the long-term stability of ISBs, short baselines in the MGEX network with valid data are processed and the estimated ISBs are investigated.

Results of baseline TLSG–TLSE

The data of baseline TLSG–TLSE on DOY 001 of 2015 are processed using the presented approach with the initial ISB samples over [−0.1, …, 0.1] m whose width approximately equals one wavelength of the L1 phase. The STD of the noise in (13) is set as 1 mm, and the number of particles is 200. The estimated F-ISBs of all the epochs are presented in Fig. 9 with their corresponding STD. In view of the number of satellites shown in Fig. 2, it is clear that the STD of the estimated ISB decreases along with the increased number of Galileo satellites. The mean of the estimated F-ISB values is about 0.040 m.
Fig. 9

Estimated F-ISB values and STD of baseline TLSG–TLSE on the DOY 001 of 2015 with GPS L1 and Galileo E1 data

The convergence time of this method is similar to that of the IFB estimation in Tian et al. (2015), where both convergence time and computation time for each epoch have been thoroughly analyzed. The convergence time is 3.5 min, i.e., 7 epochs, with only one Galileo satellite using a convergence threshold of 6 mm. When more Galileo satellites are available or their elevation angle increases, the convergence time will be shorter. The computation time for each epoch is around 1 s.

The single-epoch positioning results using GPS L1 and Galileo E1 observations with the estimated F-ISB of 0.040 m fixed are shown in the top panel of Fig. 10. As shown in Fig. 2, only a maximum of three Galileo satellites can be tracked from time to time. The fixed rate of the whole day is improved from 75.5% for GPS only to 81.2%. The corresponding RATIO values are presented in the bottom panel of Fig. 10, from which it is clear that larger RATIO values can be derived when the Galileo satellites are involved.
Fig. 10

TLSG–TLSE baseline solutions (top) and the RATIO values (bottom) on the DOY 001 of 2015 with GPS L1 and Galileo E1 data in single-epoch approach

The inter-system DD ambiguity has integer nature after F-ISB correction has been applied and therefore can be fixed along with the other DD ambiguities. Here, an experiment is designed to demonstrate the advantage of involving inter-system DD ambiguity by comparing the first fixing time with and without fixing the inter-system ambiguity. If there are a large number of GPS satellites, the fixing is already achievable just with GPS data of a few epochs and the advantage of fixing the inter-system ambiguity could hardly be shown. We choose a constellation with only five GPS satellites and one Galileo satellite. Data from 1:30:00 to 8:30:00, during which Galileo E12 is observed, are selected and divided into 15 sessions with a length of half an hour. Two processing cases are carried out including integer ambiguity resolution with and without the inter-system DD ambiguity. The data lengths of the first successfully fixing of the two cases are plotted in Fig. 11.
Fig. 11

Observation time needed for the first successful ambiguity fixing for the processing cases with and without inter-system DD ambiguity

The averaged first fixing times without and with the inter-system ambiguity are 11.2 min and 5.1 min, respectively. Fixing the inter-system ambiguity needs only about half of the observing time of fixing only intra-system ones in order to get a fixed solution.

ISB half-cycle problem

The data with the half-cycle problem as shown in Fig. 7 are processed with and without the cluster analysis procedure. The estimated F-ISBs and their STDs without cluster analysis are shown in the top and bottom panels of Fig. 12, respectively, while that with the cluster analysis procedure are shown in the top and bottom panels of Fig. 13, respectively. It is obvious that the procedure solves the problem very well. Its computation time can be ignored compared to the whole processing time.
Fig. 12

Estimated F-ISB (top) and the corresponding STD (bottom) without cluster analysis for baseline GOP6-GOP7 on DOY 001 of 2015

Fig. 13

Estimated F-ISB (top) and the corresponding STD (bottom) with cluster analysis for baseline GOP6-GOP7 on DOY 001 of 2015

Analysis of long-term data

In order to investigate the temporal stability of ISBs, almost all the short baselines in MGEX are selected and long-term data of these baselines are processed using this new approach to obtain the F-ISBs between GPS L1 and Galileo E1. At first, we selected three days, DOY 001, 120 and 181 of 2015 to give a snapshot of the F-ISB change. If there are no data on the specified day, data of the nearest day within one week are taken. In case of significant changes among the three daily F-ISB estimates, more data of the related baseline will be processed for further investigation.

There are in total 18 baselines associated with 27 stations. The F-ISB results are presented in Table 1 with the corresponding baseline lengths. Since the F-ISB value may depend on the receiver type and firmware (Odijk and Teunissen 2013a), they are listed in Table 2 for all the involved stations.
Table 1

F-ISB estimation results of short baselines in MGEX

Baseline

Length (m)

F-ISB (m)

DOY 001

DOY 120

DOY 181

DUND-OUS2

6888

 

−0.038

−0.042

GOP6–GOP7

0

−0.094

−0.094

−0.092

HARB–HRAG

2249

−0.040

−0.087

−0.086

KIR8–KIRU

4469

 

−0.039

−0.040

OHI2–OHI3

3

0.095

0.095

0.094

RGDG–RIO2

49

−0.040

−0.087

−0.087

SIN0–SIN1

0

0.040

0.040

0.040

TLSE–TLSG

1266

−0.040

−0.087

−0.087

UNB3–UNBD

19

−0.035

−0.035

−0.035

UNB3–UNBN

0

0.055

0.055

0.055

UNBD–UNBN

19

0.090

−0.100

0.090

UNX2–UNX3

0

0.000

0.000

0.000

WTZ3–WTZR

69

−0.095

0.095

0.095

WTZ3–WTZZ

69

0.000

0.000

0.000

WTZR–WTZZ

0

0.095

0.095

−0.095

ZIM2–ZIM3

0

 

0.000

0.000

ZIM2–ZIMJ

8

 

−0.039

−0.036

ZIM3–ZIMJ

8

−0.038

−0.039

−0.036

Table 2

Receiver type and firmware series for each station in the short baselines of MGEX

Station name

Receiver type

Receiver firmware

DOY 001

DOY 120

DOY 181

DUND

Trimble NetR9

 

4.81

4.81

GOP6

LEICA GRX1200 + GNSS

8.71/6.112

8.71/6.112

8.71/6.112

GOP7

JAVAD TRE_G3TH DELTA

3.5.1

3.5.1

3.5.1

HARB

TRIMBLE NETR9

4.85

4.85

5.01

HRAG

JAVAD TRE_G2T DELTA

3.6.1

3.6.1

3.6.1

KIR8

TRIMBLE NETR9

 

4.85

5.01

KIRU

SEPT POLARX4

 

2.5.2-esa3

2.5.2-esa3

OHI2

JAVAD TRE_G3TH DELTA

3.5.3

3.6.1

3.6.1

OHI3

LEICA GR25

3.11.1639/6.403

3.11.1639/6.403

3.11.1639/6.403

OUS2

JAVAD TRE_G3TH DELTA

 

3.5.7

3.5.7

RGDG

TRIMBLE NETR9

4.85

4.85

5.01

RIO2

JAVAD TRE_G3TH DELTA

3.4.7

3.4.7

3.4.7

SIN0

JAVAD TRE_G3TH DELTA

3.4.7

3.6.1

3.6.1

SIN1

TRIMBLE NETR9

4.80

4.80

4.80

TLSE

TRIMBLE NETR9

4.85

4.85

5.01

TLSG

SEPT POLARX4TR

2.5.2

2.5.2

2.9.0

UNB3

TRIMBLE NETR9

4.85

4.85

5.01

UNBD

JAVAD TRE_G2T DELTA

3.6.1

3.6.1

3.6.1

UNBN

NOV OEM60510RN0

000

000

000

UNX2

JAVAD TRE_G3TH DELTA

3.4.7

3.6.1

3.6.1

UNX3

SEPT ASTERX3

2.3.4

2.3.4

2.3.4

WTZ3

JAVAD TRE_G3TH DELTA

3.4.14

3.6.1

3.6.1

WTZR

LEICA GR25

3.11.1639/6.403

3.11.1639/6.403

3.11.1639/6.403

WTZZ

JAVAD TRE_G3TH DELTA

3.6.0

3.6.1

3.6.2

ZIM2

TRIMBLE

NETR5 4.85

NETR9 4.85

NETR9 5.01

ZIM3

TRIMBLE NETR9

4.85

4.93

5.01

ZIMJ

JAVAD TRE_G3TH DELTA

3.4.9

3.4.9

3.4.9

From Table 1, most of the baselines have nonzero F-ISB because different types of receivers are used with one exception. Baseline UNX2-UNX3 has zero F-ISB, even the receivers are from different manufacturers: JAVAD and SEPT.

Comparing F-ISBs of the different days, the F-ISBs change hardly with time for all but three baselines. The three baselines HARB-HRAG, RGDG-RIO2 and TLSE-TLSG have a jump of −47 mm from the first selected DOY 001 to the second one, DOY 120. However, there was no documented change in either hardware or firmware.

For further investigation, data over a longer time for the three baselines are processed. The estimated F-ISB time series are plotted in Fig. 14. The results show that the F-ISB is very stable except for the jumps.
Fig. 14

F-ISB of data over long time

From the above numerical study on long-term ISB characteristics, we find that F-ISBs are usually nonzero if different types of receivers are employed, but they are very stable in time and can be estimated as correction values. However, there are unreasonably rapid changes, which need further investigation. Since the ISB of a receiver actually includes delays caused by its hardware and firmware, as well as the initial carrier phase, any variation of these factors can lead to ISB changes. This means that the presented approach is required in order to carry out real time or in situ calibration.

Conclusion

Although DD observations between any two satellites can be employed in the multi-GNSS data processing, the inter-system DD ambiguities do not have integer nature due to the existence of phase ISBs and thus cannot be fixed to integer. In severe environments where only few satellites of each system are tracked, fixing inter-system DD ambiguities will significantly enhance the availability, reliability, and accuracy of multi-GNSS positioning. Because the phase ISB lumps with the inter-system DD ambiguity parameters, it is difficult to be estimated using a short set of data without any a priori information, especially in the above-mentioned observing environments.

In this contribution, we demonstrated that the inter- and intra-system DD ambiguities can be fixed together if the ISB is accurately known, and thus, the fixing RATIO can be used to quality the given ISB values. Based on this fact, a new approach is proposed to estimate F-ISBs by means of particle filter using the fixing RATIO to judge the quality of particles. Since the integer part of ISB is absorbed by the corresponding ambiguity, the procedure is not able to distinguish two F-ISB candidates with a distance of about 1 cycle. Thus, a method of cluster analysis is introduced to deal with of the half-cycle problem.

Experimental validations show that the improved approach can estimate and track the F-ISB accurately. With the method presented, the ISB between GPS L1 and Galileo E1 phase observations can be estimated precisely. Although only a maximum of three Galileo satellites was available, the fixing rate increases from 75.5 to 81.2% by fixing the inter-system DD ambiguity.

It is also demonstrated that fixing inter-system DD ambiguities together with intra-system ambiguities can significantly reduce the observing time needed for the successfully ambiguity fixing with a designed constellation of five GPS satellites and one Galileo satellite. This advantage is crucial for GNSS applications in severe environments.

We also calculated F-ISB values of short baselines in MEGX and presented the results. Apparently, the update of firmware does not affect the F-ISB value, and the F-ISB values have the characteristic of long-term stability. However, unexpected large jumps are also detected and need further investigation. Therefore, we should be careful when regarding the F-ISB value between a certain receiver types as a fixed value.

Notes

Acknowledgements

The first author is financially supported by the China Scholarship Council (CSC) for his study at the Technische Universität Berlin and the German Research Centre for Geosciences (GFZ). This research is also partly supported by the Collaborative Innovation Center of Geospatial Technology of China.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Yumiao Tian
    • 1
  • Maorong Ge
    • 2
  • Frank Neitzel
    • 1
  • Jianjun Zhu
    • 3
  1. 1.Technische Universität BerlinBerlinGermany
  2. 2.German Research Centre for GeosciencesPotsdamGermany
  3. 3.Central South UniversityChangshaChina

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