Double-difference Ionospheric Delay Research Articles

Assessment of Quad-Frequency Long-Baseline Positioning with BeiDou-3 and Galileo Observations

Quad-frequency signals have thus far been available for all satellites of BeiDou-3 and Galileo systems. The major benefit of quad-frequency signals is that more extra-wide-lane (EWL) combinations can be formed with quad-frequency than with triple- or dual-frequency, of which the ambiguities can be fixed instantaneously in medium and long baselines. In this paper, the long-baseline positioning algorithm based on optimal triple-frequency EWL/wide-lane (WL) combinations of BeiDou-3 and Galileo is proposed. First, the theoretical precision of multi-frequency combinations of BeiDou-3 and Galileo is studied, and EWL/WL combinations with a small noise amplitude factor and a small ionospheric scalar factor are selected. Then, geometry-free methods are used to estimate the a priori precision of EWL/second EWL/WL signals for different combination schemes. Second, the double-differenced (DD) geometry-based function models of quad-frequency configurations and three different triple-frequency configurations are given, and the DD ionospheric delays are estimated as unknown parameters. In the end, the real BeiDou-3 and Galileo data are used to evaluate the positioning preference. The results show that, when using fixed EWL observations to constrain WL ambiguities, the proposed triple-frequency EWL/WL signals composed of (B1I,B3I,B2a) of BeiDou-3 and (E1,E5a,E6) of Galileo can achieve the same precision as the quad-frequency signals. Therefore, the method proposed in this article can realize long-baseline instantaneous decimeter-level positioning while reducing the dimension of matrix and improving calculation efficiency.

EFFECT OF SPATIAL CORRELATION ON THE PERFORMANCES OF MODERNIZED GPS AND GALILEO IN RELATIVE POSITIONING

In the context of processing GNSS (Global Navigation Satellite System) data, it is known that the estimation of the ionospheric delays decreases the strength of the observation model and makes significant the time required to fix the ambiguities namely in case of long baselines. However, considering the double-differenced (DD) ionospheric delays as stochastic quantities, the redundancy in this case increases and leads to the reduction of time of fixing the ambiguities. The approach developed in the present paper makes two considerations: 1) the DD ionospheric delays are assumed as stochastic quantities and, 2) the spatial correlation of errors is accounted for based on a simple model of correlation. A simulation is made and aims to study the effect of these two mentioned considerations on the performances of the three multifrequency GNSSs; modernized GPS, Galileo and BDS which are not yet in full capability. For each GNSS, dual-frequency combinations of frequencies as well as triple-frequency combination are investigated in the simulation. The performances studied include: the time to fix the ambiguities with high success rate and the precision of coordinates in static relative positioning with varying baseline length. A method is developed to derive what we call the spatial correlation model which approximately gives the covariance between the individual errors belonging to two stations. Furthermore, the stochastic models that follow from accounting and neglecting the spatial correlation are developed. The LAMBDA (Least-squares Ambiguity Decorrelation Adjustment) method is implemented for ambiguity decorrelation. The results show that the time to fix the ambiguities caused by accounting the spatial correlation is less than the time of fix without the spatial correlation. Also, a slight superiority of Galileo in terms of performances is seen compared to the other GNSS. For all the dualfrequency combinations investigated, when processing a baseline length of 500 km with accounted spatial correlation, the time needed to successfully fix the ambiguities lies between 5 and 9 min, whereas it becomes only between 2.5 and 3 min for all the triple-frequency combinations, this is with a sampling time of 5 s. In addition, for all different combinations, the coordinates precision is less than 8 mm even for 500 km. We think that these high performances result from: 1) the precise codes of future GNSS signals, 2) the high redundancy in the observations equation and, 3) taking into account the spatial correlation in the definition of the stochastic model.

Extra-Wide Lane Ambiguity Resolution and Validation for a Single Epoch Based on the Triple-Frequency BeiDou Navigation Satellite System

The ambiguity resolution (AR) and validation of the global navigation satellite system (GNSS) have been challenging tasks for some decades. Considering the reliability problem of extra-wide-lane (EWL) ambiguity in the triple-carrier ambiguity resolution (TCAR), a method for validating the reliability of the EWL ambiguity using a single epoch was proposed for the BeiDou Navigation Satellite System (BDS). For the initial EWL ambiguity, obtained using a rounding estimator with a geometry-free (GF) model, the double-difference ionospheric delay was first estimated to construct a relative positioning model with an initial fixed ambiguity. Second, based on the theory of gross error detection and the AR characteristics of EWL, the second-best ambiguity candidate was constructed. Finally, among the two sets of ambiguities, the one with the smaller posterior variance was taken as the reliable ambiguity. The study showed that, for a single epoch, when only one or two satellites had incorrect ambiguities, the AR success rate after ambiguity validation and correction could reach 100% for medium baselines. For long baselines, due to the increase of atmospheric error, the validation was affected to some extent. However, the AR success rates for two long baselines increased from 96.82% and 98.44% to 98.80% and 99.67%, respectively.

Accurate ionospheric delay model for real-time GPS-based positioning of LEO satellites using horizontal VTEC gradient estimation

Ionospheric delays compensation is a mandatory step for precise absolute and relative positioning of Low Earth Orbit Satellites (LEO) by GPS measurements. The most frequently used ionosphere model for real-time GPS-based navigation in LEO is an isotropic model proposed by Lear, which uses the Vertical Total Electron Content (VTEC) above the receiver and a mapping function for TEC evaluation along a given ray path. Based on significant assessed results available for ground-based GPS receivers, we propose the use of a different model relying on the thin shell assumption and a bilinear horizontal variation of the VTEC as a function of latitude and longitude in the shell. It is expected that this model is capable of better describing horizontal gradients in the ionosphere, thus improving ionospheric delay estimation, especially in intense ionospheric conditions. This model is referred to as Linear Thin Shell (LTS). LTS performance in estimating undifferenced and double-differenced ionospheric delays is checked by comparing measured and predicted delays computed using flight data from the GRACE mission. Results show that the LTS always outperforms the isotropic model, especially in case of high solar activity. Moreover, the LTS model provides a higher performance uniformity over a wide range of ionospheric delays, thus ensuring good performance in different conditions. The results obtained demonstrate that the LTS model improves the ionosphere delays estimation accuracy by 20 and 40% for undifferenced and double-differenced delays, respectively. This suggests the LTS model can effectively contribute to improving precision in LEO positioning applications.

MSTIDs impact on GNSS observations and its mitigation in rapid static positioning at medium baselines

<p>Total electron content (TEC) fluctuation in the ionosphere is one of the main unresolved problems degrading ambiguity resolution and thus affect the reliability of satellite positioning. With regard to conditions prevailing at mid-latitudes, the impact of the upper atmospheric layers is primarily associated with the occurrence of medium-scale traveling ionospheric disturbances (MSTIDs). This study contains the combined analysis involving MSTIDs patterns observed in raw phase data and their impact on rapid static positioning. The first part is aimed at discrepancies in wavelike MSTIDs signatures detected in time series of high elevated measurements. It was demonstrated that the differences in slant ionospheric delays at medium baselines often exceed 0.5 TEC units and more importantly, the detected MSTIDs patterns were highly mutable. The next section presents the impact of MSTIDs on double differenced ionospheric delays and performance of rapid static positioning. The positioning was executed using two approaches. The first one adopted as a benchmark strategy was the geometry-based relative positioning model with weighted ionosphere and troposphere parametrization. The second approach was the improved version of the former. It used the rate of TEC corrections, which allowed the mitigation of MSTIDs through the reduction of epoch-wise ionospheric delays to one parameter for each double-differenced arc. The comparison of results for both strategies indicated significant improvement of positioning performance after the application of proposed algorithm.</p>

Effect of stochastic ionospheric delay modeling for GPS ambiguity resolution

Abstract The network-derived ionospheric delay can improve the fast and accurate determination of the long baseline in both the rapid-static and kinematic Global Positioning System (GPS) positioning mode. In this study, an interpolation of the undifferenced (UD) ionospheric delays is performed on a satellite-by-satellite and epoch-by-epoch basis, respectively, using the least-squares collocation (LSC) to provide not only ionospheric delays but also their variances. The developed method retains the simplicity of the two-dimensional (2-D) model, but it does not introduce errors due to the thin-shell assumption made in the single-layered model. Our method also provides flexibility in forming the predicted double-differenced (DD) ionospheric delays. Faster and more reliable positioning solutions can be obtained when the developed method is used to predict DD ionospheric delays. The numerical test applying the method to the Ohio Continuously Operating Reference Station network shows a 23% improvement in mean time-to-fix with the network-derived ionospheric delays.

The impact of the ionospheric correction latency on long-baseline instantaneous kinematic GPS positioning

The primary objective of this paper is to estimate the influence of the double-difference (DD) ionospheric corrections latency on the instantaneous (one-epoch) ambiguity resolution (AR) in longrange RTK under typical ionospheric conditions. The key to the success in integer AR rests mainly in the mitigation of the atmospheric errors, i.e., the ionospheric and tropospheric delays. Between these two, the former has the greatest influence on the AR, since both ambiguities and ionospheric delay are frequency-dependent. Instantaneous RTK is presently one of the most challenging topics in precise GPS applications. The research presented here addresses this topic through the development and testing of a multiple reference station approach implemented in the MPGPS™ (Multi Purpose GPS Processing Software) software. Atmospheric corrections are used in order to obtain a high quality RTK position over long distances. In our approach, DD ionospheric correction prediction derived from the previous correctly resolved epoch is applied. Yet, at the beginning of the session, a short initialization period is still required in order to produce the initial prediction. After the initialization the method is based on single epoch solution. This method assures a high success rate of the instantaneous AR for long baselines (over 100 km). Since the previous-epoch ionospheric delay is used, and instantaneous mode is applied in the algorithm, the proposed method is robust against cycle slips and data gaps, and still capable of producing centimetre-level RTK positions. The RTK solution was simulated in the post-processing mode. Namely, different DD ionospheric delay correction latencies were simulated in 10 s increments and sent to the (simulated) rover in order to test the AR performance. The AR results were compared and analyzed, and the performance of the RTK positioning was assessed based on the static true solution. Several hours of GPS data, collected by the State of Israel permanently tracking network, were processed. The analyses show that about 90 s latency may exist while the instantaneous ambiguities could still be resolved correctly. The numerical tests presented in this study show the centimetre-level positioning results for mobile receiver.

Toward Instantaneous Network-Based Real-Time Kinematic GPS over 100 km Distance

ABSTRACT: Instantaneous (single-epoch) ambiguity resolution may not always be possible because of the increasing noise and lack of sufficient redundancy to verify the integer selection in long-range real-time kinematic (RTK) GPS positioning. The algorithm proposed here is based on a single-baseline solution aided by double-difference (DD) ionospheric delay from the previous epoch (up to 60 s latency in DD ionospheric delay can be accepted), which supports the ambiguity resolution and provides a quality instantaneous kinematic position over long distances. However, obtaining the initial DD ionospheric delay requires atmospheric corrections provided by the reference network and some accumulation at the beginning of the session—i.e., on-the-fly (OTF) initialization is needed. Afterwards, instantaneous RTK positioning is assured. In this paper, a feasibility test of instantaneous real-time rover positioning is presented, and the effect of the latency of the DD ionospheric corrections on the instantaneous ambiguity resolution in long-range RTK is investigated. The positioning accuracy achieved in the tests, expressed as the differences between the resultant coordinates and the known “true” coordinates, is at millimeter-level accuracy for the horizontal components and centimeter-level for the vertical one for baselines over 100 km long.