- Article
- Open Access

# Differential ionosphere modelling for single-reference long-baseline GPS kinematic positioning

- H. Dekkiche
^{1}Email author, - S. Kahlouche
^{1}and - H. Abbas
^{1}

**62**:620120915

https://doi.org/10.5047/eps.2010.11.004

© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB 2010

**Received:**3 March 2009**Accepted:**16 November 2010**Published:**3 February 2011

## Abstract

The ionospheric effect is considered to be one of the most important error sources limiting the quality of GPS kinematic positioning. Over longer distances, differential ionospheric residuals become larger and may affect the ambiguity resolution process. We present here a Kalman-filter-based GPS ionosphere model for long-baseline kinematic applications. This observational model includes the differential ionosphere as an additional unknown factor with position coordinates and ambiguities, while the temporal correlations of the state vector are specified in the dynamic model. The temporal behaviour of ionospheric residuals is determined by the analysis of their autocorrelation function. This newly developed method was applied on a set of data collected by a roving receiver located offshore of Oran (Algeria). The results show that for baselines of about 80 km, the root mean square is at the level of a few centimetres. For tests of baselines of about 51 km, the comparison between short- and longbaseline solutions revealed that mean differences of a few millimetres and 2 cm are obtained for the horizontal coordinates and vertical component, respectively, and the standard deviation (σ) of differences on the scale of a few centimetres.

## Key words

- GPS kinematic positioning
- long baseline
- Kalman filter
- differential ionosphere
- temporal-correlation

## 1. Introduction

The development of the GPS kinematic technique has enabled real-time accurate positioning of a mobile platform to be performed. For such accurate GPS kinematic positioning, however, it is necessary to determine the integer number of carrier phase cycles, which is called “integer ambiguities”. Over short baselines, the double-difference (DD) technique can be applied to cancel out most of the correlated errors. Another option—and a common practice—is simply to disregard their effects. However, in the case of long-baseline kinematic positioning, the ionospheric effect become increasingly more decorrelated, and it may affect the ambiguity resolution process, or even make it impossible to compute (Vollath *et al.*, 2000; Wielgosz *et al.*, 2005). Therefore, reducing the differential ionospheric effect is one of the most important steps towards improving ambiguity resolution and to achieve accurate medium- and long-range kinematic positioning (Odijk, 2000; Vollath *et al.*, 2000; Kashani *et al.*, 2005; Wielgosz *et al.*, 2005). In recent years, many approaches have been developed to enable high-accuracy GPS kinematic positioning over longer distances (Wübbena *et al.*, 1996; Han, 1997; Raquet, 1998; Wanninger, 1999; Lachapelle *et al.*, 2000; Odijk *et al.*, 2000; Hernandez-Pajares *et al.*, 2000; Cannon *et al.*, 2001; Rizos, 2002; Hu *et al.*, 2003; Chen *et al.*, 2004; Wielgosz *et al.*, 2004, 2005). All of these investigations involve the use of multi-reference stations. However, a network of GPS reference stations is not always available and its implementation is costly. In addition, in certain cases (e.g., marine long-baseline applications), interpolated corrections are not reliable because the rover receiver is usually outside the network coverage area. A new approach based on a single-reference-station mode has been published by Kim and Langley (2007) that nullifies the effect of the differential ionosphere in an ambiguity search process; this method provides a number of interesting results.

In this paper, GPS kinematic positioning is implemented in a single-reference-station mode. Our approach is based on the use of the Kalman filtering method to estimate the differential ionospheric delay as a state with unknown position coordinates and ambiguities. To model the ionospheric delays as a state, an additional parameter for each satellite must be included in the observation equations that involves a singularity of the equation system (i.e., the number of unknown parameters becomes greater than the number of observations). To overcome this problem, the redundancy is increased by using dual-frequency carrier phases (L1 and L_{2}) and code pseudo-range (P_{1}, P_{2} and C/A) measurements.

The Kalman filtering method is based on the use of two models: the observation model (for updating or correction) and the underlying dynamic model (for prediction). The observation model describes the relationship between the observations and unknown parameters, while the dynamic model allows the user to predict the state parameters from a previous time step. The dynamic model describes the time-dependent relationship between successive values of the same state. The ionospheric effect is usually considered to be a time-dependent signal that continuously changes in time as the electron density varies with time and location in the ionosphere (Coco, 1991; Klobuchar, 1991; Ming, 1999). To characterise the temporal behaviour of the ionosphere, we have calculated and analysed the autocorrelation of phase DD ionospheric residuals and thereby obtained the dynamic model. The developed method was applied, with success, on a set of data collected by a roving receiver offshore Oran (Algeria).

## 2. Methodology

### 2.1 GPS observation equations

*r*and

*m*(subscript) receivers, and satellites

*i*and

*j*(superscript) are given as follows: In Eq. (1), ϕ

_{1}, ϕ

_{2}and

*P*

_{ 1 },

*P*

_{ 2 }are the phase ranges and pseudo-ranges measured at the L

_{1}and L

_{2}frequencies, respectively. The term ρ is the geometric distance between the two satellite antennas and the two receiver antennas, while

*I/f*

^{2}stands for the first-order ionosphere refraction. The wavelengths of the L

_{1}and L

_{2}phases are γ

_{1}≈ 19 cm and γ

_{2}≈ 24 cm, respectively. The tropospheric refraction is

*T*, and the measurement noise is characterised by the error terms ε and

*e.*The integer ambiguities associated with the L

_{1}and L

_{2}phases are denoted as

*N*

_{ 1 }and

*N*

_{2}, respectively. Multipath and receiver antenna phase centre variations are ignored in Eq. (1). One can combine the geometric distance p and the tropospheric delay

*T*to obtain the ideal pseudorange as: In this study, the Goad and Goodman model is used to estimate tropospheric delay. Actually, several tropospheric models may be applied, of which are those of Essen and Froome (1951), Hopfield (1969), Saastamoinen (1972), Goad and Goodman (1974), and Chao (1974). When the elevation angle is >20°, different models give very similar estimates of the tropospheric refraction (Ho, 1990; Rothacher, 1992; Solheim, 1993).

*Z*is the observed-minus-computed vector for the four measurements. The error vector is

*V*, and

*H*is known as the design matrix. The state vector used in this study is given, for one epoch, as: where ; represent the corrections made to the approximate receiver position. The term stands for the L

_{1}DD ionospheric residual. N

_{1}and N

_{1}— N

_{2}are the DD ambiguity integers of the L1 and L1 — L2.

### 2.2 Ionosphere modelling

In order to stochastically analyse the ionospheric residuals, we collected and used test data from a base-receiver situated in the harbour of Oran and data from a rover antenna embedded on a 9-m-long boat located offshore Oran. Data were recorded by two Ashtech-Z12 GPS receivers during the morning (0800—1000 hours Local Time) of June 17, 2002, with a sampling rate of 01 s. The analysis is based on the calculation of the autocorrelation function of geometry-free DD measurements (DD ionospheric residuals) for the selected receiver-satellite pairs. All the DD pairs are based on the reference-rover baseline, which varies from 100 m to 80 km, and two sets of satellite pairs (PRN 02-03, and PRN 02-31) were chosen for this analysis.

#### 2.2.1 DD ionospheric residuals:

The correlated nature of the ionosphere allows the major part of its effect to be eliminated by forming the DD phase observables. Nevertheless, when baselines are longer than few tens of kilometres, residual errors still remain.

*I*may be obtained by forming the so called geometry-free linear combination of L1 and L2 phase measurements (Xinhua, 1992).

#### 2.2.2 Analysis of the autocorrelation of DD ionospheric residuals:

In order to model the differential ionosphere, its autocorrelation function is analysed. From the results of this analysis, it is possible to model the ionospheric residuals as a first-order Gauss-Markov process.

*I*represents the DD ionospheric residuals. where

*E*[ ] denotes the expectation operator, and is the Dirac delta function.

*t*. The autocorrelation function of a first-order Gauss-Markov process is given by Zhang (1999):

### 2.3 Kalman filter

*X*, which is grouped as follows: The prediction (—) of the state vector at the next epoch

*n*+1 is derived from the update (+) of epoch

*n*as: The parameter

*Xp*in Eq. (11) contains the three-dimensional position coordinates, the velocity, and the acceleration of the moving receiver. is the residual DD ionosphere. The remaining parameters refer to L

_{1}and wide-lane (L1—L2) ambiguities. Each group is treated independently, i.e., there are no correlations between the groups. Therefore, the state transition matrix ϕ and the process noise covariance matrix

*W*are in block diagonal forms: Position, velocity and acceleration vectors

*Xp*are expressed in the local topocentric northing, easting and height coordinate system. We ignore any inter-correlation between each component, thus treating each of these as independent of each other. Also, similar to the majority of vehicle motion, the modelling of position and velocity in each dimension (second order system) is considered.

*x*is the position, v is the velocity, ϕ

*t*is the observation period, and The symbol

*q*

_{ p }denotes variance of the process noise.

*X*

_{ p }, the transition matrix ϕ

_{ p }and the process noise matrix

*Wp*for a typical dynamic system, in the northing, easting, and height coordinate system, are in which

*(x, y, z)*are the vector components of northing, easting, and height, respectively, in the geodetic horizon plane at the fixed station, (v

_{x}, v

_{y}, v

_{z}) are the respective velocity components, and

*(qx qy qz)*are the variance of the process noise in the same components. All other empty entries are zero.

*n*double-differences are: with

*q*

_{ i }being the variance of the residual ionosphere process noise for the correlation time .

The DD integer ambiguities are simply modeled as constants. The states are transformed into L1 integer ambiguities and wide-lane integer ambiguities to take advantage of the longer wide-lane wavelength (86 cm).

### 2.4 DD ambiguity resolution

To achieve a GPS kinematic position on the centimetre scale, it is necessary to resolve the integer ambiguities in the carrier phase measurements. Float ambiguities are estimated as part of the state vector with a Kalman filtering algorithm. The DD integer ambiguities are modelled as constants and are treated as independent from each other.

After an initial transition period and when the estimated integer ambiguities variances reach a predetermined criteria, an integer ambiguity search is launched using estimated values as initial guesses. Ambiguities will be fixed to integers when the ambiguity search criteria is met.

To take advantage of the longer wide-lane wavelength (≈86 cm), we first fix the wide-lane integer ambiguities, following which it is relatively easy to determine the L1 integer ambiguities.

## 3. Test Results and Analysis

Ionosphere models (processed cases).

Case | Ionosphere model |
---|---|

1 | N/A |

2 | White noise |

3 | Gauss-Markov |

4 | Random walk |

In the Kalman filtering method, the dynamic model should be correctly specified. In our study, horizontal coordinates are modelled as an integrated velocity, while the average vertical component is modelled as a constant. For differential ionosphere modelling, the autocorrelation analysis of the DD ionospheric residuals, performed previously, shows that ionospheric residuals can be modelled as a firstorder Gauss-Markov random process.

Several alternatives have been considered in modelling the correlation time of the ionospheric residuals, including the white noise model (WN), Gauss-Markov process, and the random walk model.

To quantify the position error induced by bad modelling of the differential ionosphere, the Gauss-Markov model (with 100 s of correlation time) is taken as a reference because residuals are minimum. Rover position coordinates are calculated for the four previous cases. Figure 7 shows the ship’s trajectory; rover positions are determined a 1-s intervals and are expressed in a local topocentric coordinate system for which the reference station is the origin. The observed baselines vary from some hundreds of metres to 80 km.

*(x*) and the height

*(z)*component, respectively. Differences are calculated between the third case and the other remaining cases described in Table 1. The non-modelling of ionospheric residuals introduces a maximal error of 28 cm in the

*x*component and 25 cm in the

*z*component. At the beginning of observations, differences of about 60 cm correspond to the initial transition period, where ambiguities are not yet resolved.

For distances of about 40 km, the random walk model gives similar results to those obtained for the Gauss-Markov model (with a correlation time of 100 s). This may be discerned from Figs. 8 and 9 where the differences in *x* and *z* fluctuate around zero, until the time tag 32.75 (which corresponds to a baseline of about 40 km).

To validate our approach, we compared the long- and short-baseline solutions. The short-baseline solution is used as reference since the all of the systematic errors, those on the ionospheric effect in particular, are correlated and therefore eliminated by differencing.

## 4. Conclusion

In the study reported here, we have used the single-reference GPS kinematic positioning method to accurately determine the trajectory of a ship in the water offshore of Oran (Algeria).

In kinematic positioning, the ionospheric effect is the main source of error affecting the integer ambiguity resolution process and, consequently, the position accuracy. The ionospheric effect becomes increasingly decorrelated as the distance separating the base and the rover receivers increases.

Summary statistics for the difference between short and long baseline solutions.

Mean (m) | Std (m) | |
---|---|---|

Easting | 0.003 | 0.010 |

Northing | −0.001 | 0.012 |

Up | 0.0233 | 0.021 |

We investigated only the temporal correlation of the ionospheric residuals and did not take into account the spatial correlation, except for determining the parameters used for the initialisation of the model; the stochastic parameters of the model, the initial uncertainty and perturbation are calculated with respect to the baseline length.

## Declarations

### Acknowledgements

The authors greatly appreciate constructive and useful comments from reviewers.

## Authors’ Affiliations

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