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

- 1.8k Downloads

## 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

*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.

*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

*is composed by unknown coordinate’s parameters,*

**x***contains unknown DD ambiguity parameters, and*

**b***includes unknown ISB parameters.*

**y**## 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 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.

### F-ISB half-cycle problem

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).

**y**_{k}is the measurement state at epoch

*k*,

*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*

**x**

**e**_{k}are the process noise and the measurement noise, respectively. We assume that the noise can be separated from the function and expressed by

**v**_{k}and

**w**_{k}, 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.

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 *u*_{i}, 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 *u*_{i}. 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 *N*th, 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

- 1.
Process the carrier phase and pseudorange observations to get the (8) with F-ISB parameter.

- 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.
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.
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.
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.
Resample the particles if

*N*_{eff}is smaller than*N*th. - 7.
Predict the particles for the next epoch according to the prediction model (13).

## 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 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 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

### 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.

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 |

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.

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.

### References

- Blewitt G (1989) Carrier phase ambiguity resolution for the global positioning system applied to geodetic baselines up to 2000 km. J Geophys Res 94(B8):10187–10203CrossRefGoogle Scholar
- Dach R, Brockmann E, Schaer S, Beutler G, Meindl M, Prange L, Ostini L (2009) GNSS processing at CODE: status report. J Geodesy 83(3):353–365CrossRefGoogle Scholar
- Dong D, Bock Y (1989) Global positioning system network analysis with phase ambiguity resolution applied to crustal deformation studies in California. J Geophys Res 94(B4):3949–3966CrossRefGoogle Scholar
- Doucet A, Godsill S, Andrieu C (2000) On sequential Monte Carlo sampling methods for Bayesian filtering. Stat Comput 10(3):197–208CrossRefGoogle Scholar
- Doucet A, Freitas N, Gordon N (2001) Sequential Monte Carlo methods in practice. Springer, New YorkCrossRefGoogle Scholar
- Euler HJ, Schaffrin B (1991) On a measure of discernibility between different ambiguity solutions in the static-kinematic GPS-mode. In: Proceedings of the international symposium on kinematic systems in Geodesy, surveying, and remote sensing, Banff, Canada, September, 1990. Springer, New York, pp 285–295Google Scholar
- Force D, Miller J (2013) Combined global navigation satellite systems in the space service volume. In: Proceedings of the ION international technical meeting, San Diego, USAGoogle Scholar
- Ge M, Gendt G, Rothacher M, Shi C, Liu J (2008) Resolution of GPS carrier phase ambiguities in precise point positioning (PPP) with daily observations. J Geodesy 82(7):389–399CrossRefGoogle Scholar
- Gordon J, Salmond J, Smith F (1993) Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc Radar Signal Process 140(2):107–113CrossRefGoogle Scholar
- Gustafsson F (2010) Particle filter theory and practice with positioning applications. IEEE Aerosp Electron Syst Mag 25(7):53–82CrossRefGoogle Scholar
- Gustafsson F, Gunnarsson F, Bergman N, Forssell U, Jansson J, Karlsson R, Nordlund P (2002) Particle filters for positioning, navigation, and tracking. IEEE Trans Signal Process 50(2):425–437CrossRefGoogle Scholar
- Hahn J, Powers E (2005) Implementation of the GPS to Galileo time offset (GGTO). In: Proceedings of 2005 joint IEEE international frequency control symposium and precise time and time interval (PPTI) systems & applications meeting, Vancouver, Canada, pp 33–37Google Scholar
- Haug A (2012) Bayesian estimation and tracking: a practical guide. Wiley, New JerseyCrossRefGoogle Scholar
- Ineichen D, Brockmann E, Schaer S (2008) Processing combined GPS/GLONASS data at swisstopo’s local analysis center. In: Proceedings of EUREF symposium, Brussels, BelgiumGoogle Scholar
- Julien O, Alves P, Cannon E, Zhang W (2003) A tightly coupled GPS/GALILEO combination for improved ambiguity resolution. In: Proceedings of the European navigation conference, Graz, AustriaGoogle Scholar
- Kitagawa G (1996) Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J Comput Graph Stat 5(1):1–25Google Scholar
- Li X, Ge M, Dai X, Ren X, Mathias F, Jens W (2015) Accuracy and reliability of multi-GNSS real-time precise positioning: GPS, GLONASS, BeiDou, and Galileo. J Geodesy 89(6):607–635CrossRefGoogle Scholar
- Odijk D, Teunissen P (2013a) Characterization of between-receiver GPS-Galileo inter-system biases and their effect on mixed ambiguity resolution. GPS Solut 17(4):521–533CrossRefGoogle Scholar
- Odijk D, Teunissen P (2013b) Estimation of differential inter-system biases between the overlapping frequencies of GPS, Galileo, BeiDou and QZSS. In: Proceedings of the 4th international colloquium scientific and fundamental aspects of the Galileo program, Prague, Czech RepublicGoogle Scholar
- Odijk D, Teunissen P, Khodabandeh A (2014) Galileo IOV RTK positioning: standalone and combined with GPS. Surv Rev 46(337):267–277CrossRefGoogle Scholar
- Odolinski R, Teunissen P, Odijk D (2014) Combined BDS, Galileo, QZSS and GPS single-frequency RTK. GPS Solut 19(1):151–163CrossRefGoogle Scholar
- Paziewski J, Wielgosz P (2015) Accounting for Galileo-GPS inter-system biases in precise satellite positioning. J Geodesy 89(1):81–93CrossRefGoogle Scholar
- Tan P, Steinbach M, Kumar V (2006) Introduction to data mining. Pearson Addison Wesley, BostonGoogle Scholar
- Teunissen P (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geodesy 70(1):65–82CrossRefGoogle Scholar
- Teunissen P, Kleusberg A (1996) GPS observation equations and positioning concepts. In: Kleusberg A, Teunissen P (eds) GPS for Geodesy. Springer, Berlin, pp 175–218CrossRefGoogle Scholar
- Tian Y, Ge M, Neitzel F (2015) Particle filter-based estimation of inter-frequency phase bias for real-time GLONASS integer ambiguity resolution. J Geodesy 89(11):1145–1158CrossRefGoogle Scholar
- Verhagen S, Teunissen P (2013) The ratio test for future GNSS ambiguity resolution. GPS Solut 17:535–548CrossRefGoogle Scholar