Adaptive Kalman filter based on variance component estimation for the prediction of ionospheric delay in aiding the cycle slip repair of GNSS triple-frequency signals

In order to incorporate the time smoothness of ionospheric delay to aid the cycle slip detection, an adaptive Kalman filter is developed based on variance component estimation. The correlations between measurements at neighboring epochs are fully considered in developing a filtering algorithm for colored measurement noise. Within this filtering framework, epoch-differenced ionospheric delays are predicted. Using this prediction, the potential cycle slips are repaired for triple-frequency signals of global navigation satellite systems. Cycle slips are repaired in a stepwise manner; i.e., for two extra wide lane combinations firstly and then for the third frequency. In the estimation for the third frequency, a stochastic model is followed in which the correlations between the ionospheric delay prediction errors and the errors in the epoch-differenced phase measurements are considered. The implementing details of the proposed method are tabulated. A real BeiDou Navigation Satellite System data set is used to check the performance of the proposed method. Most cycle slips, no matter trivial or nontrivial, can be estimated in float values with satisfactorily high accuracy and their integer values can hence be correctly obtained by simple rounding. To be more specific, all manually introduced nontrivial cycle slips are correctly repaired.     

Higher order ionospheric effects in precise GNSS positioning

With the increasing number of precise navigation and positioning applications using Global Navigation Satellite Systems (GNSS) such as the Global Positioning System (GPS), higher order ionospheric effects and their correction become more and more important. Whereas the first-order error can be completely eliminated by a linear combination of dual- frequency measurements, the second- and third-order residual effects remain uncorrected in this approach. To quantify the second-order residual effect, a simple formula has been derived for GNSS users in Germany. Our proposed correction algorithm reduces the second-order effects to a residual error of fractions of 1 mm up to 2 mm at a vertical total electron content level of 10¹⁸ electrons/m² (100 TECU), depending on satellite azimuth and elevation angles. The correction formula can be implemented in real-time applications as it does not require the knowledge of the geomagnetic field or the electron density distribution in the ionosphere along the signal path. It is expected that the correction will enable more accurate positioning using the line-of-sight carrier-phase measurements.     

Mitigation of higher order ionospheric effects on GNSS users in Europe

Current dual-frequency GPS measurements can only eliminate the first-order ionospheric term and may cause a higher-order range bias of several centimeters. This research investigates the second-order ionospheric effect for GNSS users in Europe. In comparison to previous studies, the electron density profiles of the ionosphere/plasmasphere are modeled as the sum of three Chapman layers describing electron densities of the ionospheric F₂, F₁ and E layers and a superposed exponential decay function describing the plasmasphere. The International Geomagnetic Reference Field model is used to calculate the geomagnetic field vectors at numerous points along the incoming ray paths. Based on extended simulation studies, we derive a correction formula to compute the average value of the longitudinal component of the earth’s magnetic field along the line-of-sight as a function of geographic latitude and longitude, and geometrical parameters such as elevation and azimuth angles. Using our correction formula in conjunction with the total electron content (TEC) along the line-of-sight, the second-order ionospheric term can be corrected to the millimeter level for a vertical TEC level of 10¹⁸ electrons/m².     

Carrier phase bias estimation of geometry-free linear combination of GNSS signals for ionospheric TEC modeling

The ionosphere can be modeled and studied using multi-frequency GNSS signals and their geometry-free linear combination. Therefore, a number of GNSS-derived ionospheric models have been developed and applied in a broad range of applications. However, due to the complexity of estimating the carrier phase ambiguities, most of these models are based on low-accuracy carrier phase smoothed pseudorange data. This, in turn, critically limits their accuracy and applicability. Therefore, we present a new methodology of estimating the phase bias of the scaled L1 and L2 carrier phase difference which is a function of the ambiguities, the ionospheric delay, and hardware delays. This methodology is suitable for ionospheric modeling at regional and continental scales. In addition, we present its evaluation under varying ionospheric conditions. The test results show that the carrier phase bias of geometry-free linear combination can be estimated with a very high accuracy, which consequently allows for calculating ionospheric TEC with the uncertainty lower than 1.0 TECU. This high accuracy makes the resulting ionosphere model suitable for improving GNSS positioning for high-precision applications in geosciences.     

基于GNSS的电离层模型研究进展

在利用全球导航卫星系统GNSS进行精密定位和导航时,电离层延迟误差是影响其精度和准确度的主要误差源之一,故对电离层模型研究至关重要。本文将电离层模型分成了经典电离层模型和现代电离层模型,并对经典电离层模型进行了比较,重点介绍了目前全世界电离层模型的研究热点、存在问题及研究方向。     

层析模型在GNSS探测电离层中的研究进展

在利用全球导航卫星系统GNSS(GPS、GLONASS、Galileo等卫星定位系统)进行精密定位和导航时,电离层延迟误差是影响其精度和准确度的主要误差源之一,故对电离层模型研究至关重要。本文介绍了传统电离层模型的缺点,重点阐述了不同的电离层层析方法,同时分析了现在电离层层析方法所存在的主要问题,最后介绍了目前全世界电离层层析模型的研究展望。     

中国大陆导航连续站周期信号空间规律研究

为了进一步分析我国GNSS周期信号的空间分布规律,文章基于我国大陆及周边IGS站与中国地壳运动观测网络基准站的坐标时间序列,给出定量标准剔除质量差的测站;采用分段线性拟合法提取并扣除时间序列的趋势项;采用"拉依达准则"剔除时间序列中的奇异值;采用4参数三角函数拟合出时间序列的周期与幅度;分析东向、北向、垂向3个方向的年周期、半年周期的振幅与最大值出现月份在我国的空间分布情况。分析结果表明:3方向年周期的振幅都是半年周期振幅的两倍以上,3个方向不同周期振幅的空间分布规律差异明显。     

几种附电离层约束GNSS单频PPP性能评估

针对单频精密单点定位(PPP)两种常用的定位模型:非组合模型和附加电离层约束模型,同时综合考虑电离层约束模型三种不同约束策略(常数约束,时空约束,逐步松弛),对比分析了其使用GPS单系统及GPS+BDS双系统观测值的定位收敛时间,定位精度及其优缺点.实验结果表明:使用GPS单系统,附加不同电离层约束对单频PPP收敛时间...     

鞍山区域电离层模型构建及GNSS应用研究

针对电离层延迟误差目前是GNSS导航定位精度最重要误差源的现状,通过GNSS参考站或跟踪站实测数据计算电子总含量值,建立区域电离层模型,监测区域电离层变化,进而找到削弱或消除电离层延迟误差影响方法。利用曲面拟合实现建模,在模型的建立过程中通过对不同的模型阶数进行设置,对比不同情况下的模型精度,从而确定特定区域内最佳数据采样间隔及阶数设置,并在最佳阶数设置情况下,比较了预报不同时段的精度,进而对延迟量预报问题进行探讨,得出一些有益结论。可以通过该模型单独解算流动站站点的实时电离层延迟信息,这对多基站CORS的站间距离选择和单基站CORS基准站和流动站之间距离设计,尤其对提高单频接收机以及GIS产品用户的定位精度和差分模型的覆盖范围都具有实际参考意义。     

Evaluation of three ionospheric delay computation methods for ground-based GNSS receivers

GNSS observables for ionospheric estimation are commonly based on carrier-to-code leveling (CCL) and precise point positioning (PPP) methods. The CCL method is a geometry-free method which uses carrier phase to level pseudorange observation for decreasing multipath error and observation noise. However, the ionospheric observable based on the CCL has been proven to be affected by leveling errors. The leveling errors are caused by pseudorange multipath and intraday variation of receiver DCB. To obtain more accurate ionospheric observable, the PPP method takes advantage of precise satellite-to-ground range for retrieving slant total electron content and is less affected by the leveling errors. Previous studies have only proven that the ionospheric observables extracted by the two methods are affected by the leveling errors. The influence on ionospheric observable by the pseudorange inter-receiver satellite bias (IRSB) of the receiver has not been taken into consideration. Also, the magnitude of the differences between the ionospheric observables extracted by the two methods has also not been given. In this work, three methods, namely, the CCL, the conventional ionospheric-free PPP method which uses the ionospheric-free Hatch–Melbourne–Wubbena (HMW) function, and the University of Calgary (UOFC) PPP method, are selected to analyze and compare the differences of ionospheric observables and the global ionospheric maps, using a large number of measured data from international GNSS service global stations. Experimental results show that the accuracy of ionospheric observables obtained by the three methods is not only related to the leveling error, but also pseudorange IRSB. The IRSB of the receiver exerts a major effect on the ionospheric observables obtained by the CCL method and a minor effect on the ionospheric observables obtained by the HMW and UOFC methods. The accuracies in the latter case are similar and superior to those obtained by the CCL. The differences of the ionospheric observables obtained by the CCL and UOFC methods, or the CCL and HMW methods, are at decimeter level, whereas the difference of the ionospheric observables obtained by the UOFC and HMW methods is at centimeter level. The UOFC method presented the highest single-frequency pseudorange positioning accuracy using estimated global ionospheric products, followed by the HMW and the CCL methods which presented the lowest positioning accuracy.     

台风“尼伯特”的电离层扰动分析

针对台风运动过程中大气层与电离层的耦合效应问题,该文采用总电子含量与电子密度数据分析了台风"尼伯特"达到最大风速时刻的电离层变化情况。结果显示在台风达到最大时刻的当天,风眼处电离层电子含量显著降低,在台风附近的IGS测站处也有明显的电子含量负异常出现。电子含量异常空间分布图显示最大异常区域位于风眼的南侧,表明台风边缘处的电离层异常幅度要大于风眼处的异常幅度。电子密度三维模型显示在250～350km高度上,风眼附近有显著电子密度减弱现象,尤其在300km高度上,异常幅度达到-6～-8×105 el/cm3。诸多证据表明此次电子含量负异常与台风"尼伯特"造成的大气层-电离层耦合效应有关,这可为以后的台风-电离层扰动研究提供一定的参考。     

Mitigating the impact of ionospheric cycle slips in GNSS observations

Processing of data from global navigation satellite systems (GNSS), such as GPS, GLONASS and Galileo, can be considerably impeded by disturbances in the ionosphere. Cycle-slip detection and correction thus becomes a crucial component of robust software. Still, dealing with ionospheric cycle slips is not trivial due to scintillation effects in both the phase and the amplitude of the signals. In this contribution, a geometry-based approach with rigorous handling of the ionosphere is presented. A detailed analysis of the cycle-slip correction process is also tackled by examining its dependence on phase and code noise, non-dispersive effects and, of course, the ionosphere. The importance of stochastic modeling in validating the integer cycle-slip candidates is emphasized and illustrated through simulations. By examining the relationship between ionospheric bias and ionospheric constraint, it is shown that there is a limit in the magnitude of ionospheric delay variation that can be handled by the cycle-slip correction process. Those concepts are applied to GNSS data collected by stations in northern Canada, and show that enhanced cycle-slip detection can lead to decimeter-level improvements in the accuracy of kinematic PPP solutions with a 30-s sampling interval. Cycle-slip correction associated with ionospheric delay variations exceeding 50 cm is also demonstrated, although there are risks with such a procedure and these are pointed out.     

Different mechanisms of the ionospheric influence on GPS occultation signals

A local mechanism for strong ionospheric effects on radio occultation (RO) global positioning satellite system (GPS) signals is described. Peculiar zones centered at the critical points (the tangent points) in the ionosphere, where the gradient of the electron density is perpendicular to the RO ray trajectory, strongly influence the amplitude and phase of RO signals. It follows from the analytical model of local ionospheric effects that the positions of the critical points depend on the RO geometry and the structure of the ionospheric disturbances. Centers of strong ionospheric influence on RO signals can exist, for example, in the sporadic E-layers, which are inclined by 3–6° relative to the local horizontal direction. Also, intense F2 layer irregularities can contribute to the RO signal variations. A classification of the ionospheric influence on the GPS RO signals is introduced using the amplitude data, which indicates different mechanisms (local, diffraction, etc.) for radio waves propagation. The existence of regular mechanisms (e.g., local mechanism) indicates a potential for separating the regular and random parts in the ionospheric influence on the RO signals.     

Contribution of ionospheric irregularities to the error of dual-frequency GNSS positioning

This paper investigates the third-order residual range error in the dual-frequency correction of ionospheric effects on satellite navigation. We solve the two-point trajectory problem using the perturbation method to derive second-approximation formulas for the phase path of the wave propagating through an inhomogeneous ionosphere. It is shown that these formulas are consistent with the results derived from applying perturbation theory directly to the eikonal equation. The resulting expression for the phase path is used in calculating the residual range error of dual-frequency global positioning system (GPS) observations, in view of second- and third-order terms. The third-order correction includes not only the quadratic correction of the refractive index but also the correction for ray bending in an inhomogeneous ionosphere. Our calculations took into consideration that the ionosphere has regular large-scale irregularities, as well as smaller-scale random irregularities. Numerical examples show that geomagnetic field effects, which constitute a second-order correction, typically exceed the effects of the quadratic correction and the regular ionospheric inhomogeneity. The contribution from random irregularities can compare with or exceed that made by the second-order correction. Therefore, random ionospheric irregularities can make a significant (sometimes dominant) contribution to the residual range error.     

Study on the cross-correlation of GNSS signals and typical approximations

In global navigation satellite system (GNSS) receivers, the first signal processing stage is the acquisition, which consists of detecting the received GNSS signals and determining the associated code delay and Doppler frequency by means of correlations with a code and carrier replicas. These codes, as part of the GNSS signal, were chosen to have very good correlation properties without considering the effect of a potential received Doppler frequency. In the literature, it is often admitted that the maximum GPS L1 C/A code cross-correlation is about ?24 dB. We show that this maximum can be as high as ?19.2 dB when considering a Doppler frequency in a typical range of [?5, 5] kHz. We also show the positive impact of the coherent integration time on the cross-correlation and that even a satellite with Doppler outside the frequency search space of a receiver impacts the cross-correlation. In addition, the expression of the correlation is often provided in the continuous time domain, while its implementation is typically made in the discrete domain. It is then legitimate to ask the validity of this approximation. Therefore, the purpose of this research is twofold: First, we discuss typical approximations and evaluate their regions of validity, and second, we provide characteristic values such as maximums and quantiles of the auto- and cross-correlation of the GPS L1 C/A and Galileo E1 OS codes in the presence of Doppler, for frequency ranges up to 50 kHz and for different integration times.     

Performance evaluation of MSK and OFDM modulations for future GNSS signals

The objective of this work is to investigate the performances of orthogonal frequency division multiplexing (OFDM) and minimum frequency shift keying (MSK) modulations as potential future global navigation satellite systems (GNSS) signal modulation schemes. MSK is used in global system for mobile communications because of its spectral efficiency, while OFDM is used in WLAN and digital video broadcast-terrestrial because of its multipath mitigation capability. These advantages of MSK and OFDM modulations render them as promising modulation candidates for future GNSS signals to offer enhanced performances in challenging environments. Gabor bandwidth and multipath error envelopes of these two modulations were computed and compared with those of the current global positioning system (GPS), Galileo, and Beidou signal modulations. The results show that OFDM modulation demonstrated promises as a viable future GNSS modulation, especially for signals that require pre-filtering bandwidths larger than 2 MHz, while MSK modulation is more desirable for pre-filtering bandwidth below 2 MHz where it exhibits the largest Gabor bandwidth.     

ERTK: extra-wide-lane RTK of triple-frequency GNSS signals

Triple-frequency signals have thus far been available for all satellites of BeiDou and Galileo systems and for some GPS satellites. The main benefit of triple-frequency signals is their formation of extra-wide-lane (EWL) combinations whose ambiguities can be instantaneously fixed for several 10–100 km baselines. Yet, this benefit has not been fully exploited and only used as a constraint for narrow-lane (NL) ambiguity resolution (AR) in most previous studies. In this study, we comprehensively investigate the real-time kinematic (RTK) capabilities of EWL observations, also referred to as EWL RTK (ERTK). We begin by mathematically expressing the ease of EWL AR and the difficulty of NL AR, respectively, using a numerical demonstration. We then present the mathematical models for ERTK including the ionosphere-ignored, ionosphere-float and ionosphere-smoothed types. The experiments are conducted using a four-station network of real triple-frequency BeiDou data with baseline lengths from 33 to 75 km. The results show that the ionosphere-ignored ERTK achieves real-time solutions with a horizontal accuracy of about 10 cm. Although the ionosphere-float ERTK solutions are very noisy, they can be quickly improved at the centimetre level by further applying the ionosphere-smoothed model. Note that such accurate results are very promising and already satisfy many applications without complicated NL AR. To the best of our knowledge, this is the first comprehensive study to make full use of EWL observations of triple-frequency signals on RTK.     

改进CODE GIM的GNSS广播电离层模型评估方法

针对如何有效地对各类电离层模型在建模实现、模型精度、模型时效性等方面进行综合评估问题,该文提出一种基于改进CODE模型(CODE+模型)全球电离层图(GIM)的预报电离层精度评估方法,通过增加我国陆态网监测站数据,提高我国及周边地区的电离层建模精度,弥补了CODE GIM在我国及周边地区因观测数据少而精度受限的不足。通过试验分析表明,改进CODE GIM能够满足精度要求,且更好地与我国实际电离层情况相吻合;并以此为基准评估GNSS广播电离层精度,对比分析了GPS、BDS、Galileo电离层模型的精度,得到一些初步结论。