Integer aperture bootstrapping: a new GNSS ambiguity estimator with controllable fail-rate

In this contribution, we introduce a new bootstrap-based method for Global Navigation Satellite System (GNSS) carrier-phase ambiguity resolution. Integer bootstrapping is known to be one of the simplest methods for integer ambiguity estimation with close-to-optimal performance. Its outcome is easy to compute due to the absence of an integer search, and its performance is close to optimal if the decorrelating Z-transformation of the LAMBDA method is used. Moreover, the bootstrapped estimator is presently the only integer estimator for which an exact and easy-to-compute expression of its fail-rate can be given. A possible disadvantage is, however, that the user has only a limited control over the fail-rate. Once the underlying mathematical model is given, the user has no freedom left in changing the value of the fail-rate. Here, we present an ambiguity estimator for which the user is given additional freedom. For this purpose, use is made of the class of integer aperture estimators as introduced in Teunissen (2003). This class is larger than the class of integer estimators. Integer aperture estimators are of a hybrid nature and can have integer outcomes as well as non-integer outcomes. The new estimator is referred to as integer aperture bootstrapping. This new estimator has all the advantages known from integer bootstrapping with the additional advantage that its fail-rate can be controlled by the user. This is made possible by giving the user the freedom over the aperture of the pull-in region. We also give an exact and easy-to-compute expression for its controllable fail-rate.     

An optimality property of the integer least-squares estimator

A probabilistic justification is given for using the integer least-squares (LS) estimator. The class of admissible integer estimators is introduced and classical adjustment theory is extended by proving that the integer LS estimator is best in the sense of maximizing the probability of correct integer estimation. For global positioning system ambiguity resolution, this implies that the success rate of any other integer estimator of the carrier phase ambiguities will be smaller than or at the most equal to the ambiguity success rate of the integer LS estimator. The success rates of any one of these estimators may therefore be used to provide lower bounds for the LS success rate. This is particularly useful in case of the bootstrapped estimator. Received: 11 January 1999 / Accepted: 9 July 1999     

一个新的GNSS模糊度估计类

介绍了一类新的GNSS模糊度估计。因为该类遵循移去一恢复原理，称之为整数等变估计类。本文将说明整数等变估计类较整数估计类和线性无偏估计类的范围要大，同时给出一个相当有用的整数等变估计类的表达式。这个表达式揭示了整数等变估计类的结构，并显示该表达式如何在浮点解的基础上实现整数等变估计。最后还提出最优整数估计。     

Influence of ambiguity precision on the success rate of GNSS integer ambiguity bootstrapping

In this contribution, we study the dependence of the bootstrapped success rate on the precision of the GNSS carrier phase ambiguities. Integer bootstrapping is, because of its ease of computation, a popular method for resolving the integer ambiguities. The method is however known to be suboptimal, because it only takes part of the information from the ambiguity variance matrix into account. This raises the question in what way the bootstrapped success rate is sensitive to changes in precision of the ambiguities. We consider two different cases. (1) The effect of improving the ambiguity precision, and (2) the effect of using an approximate ambiguity variance matrix. As a by-product, we also prove that integer bootstrapping is optimal within the restricted class of sequential integer estimators.     

GNSS双频整周关系约束模糊度算法研究

根据GNSS双频载波相位观测值间的特定关系,提出了一种基于双频整周关系约束的模糊度解算方法(FirCAR)。该方法在局部整数范围内可将载波相位的等效波长增长,以利于整周模糊度的快速解算。不同长度的基线数据实验证明了该算法的正确性和有效性,并分析了卫星截止高度角对解算结果的影响。     

Theory of integer equivariant estimation with application to GNSS

Carrier phase ambiguity resolution is the key to high-precision global navigation satellite system (GNSS) positioning and navigation. It applies to a great variety of current and future models of GPS, modernized GPS and Galileo. The so-called fixed baseline estimator is known to be superior to its float counterpart in the sense that its probability of being close to the unknown but true baseline is larger than that of the float baseline, provided that the ambiguity success rate is sufficiently close to its maximum value of one. Although this is a strong result, the necessary condition on the success rate does not make it hold for all measurement scenarios. It is discussed whether or not it is possible to take advantage of the integer nature of the ambiguities so as to come up with a baseline estimator that is always superior to both its float and its fixed counterparts. It is shown that this is indeed possible, be it that the result comes at the price of having to use a weaker performance criterion. The main result of this work is a Gauss–Markov-like theorem which introduces a new minimum variance unbiased estimator that is always superior to the well-known best linear unbiased (BLU) estimator of the Gauss–Markov theorem. This result is made possible by introducing a new class of estimators. This class of integer equivariant estimators obeys the integer remove–restore principle and is shown to be larger than the class of integer estimators as well as larger than the class of linear unbiased estimators. The minimum variance unbiased estimator within this larger class is referred to as the best integer equivariant (BIE) estimator. The theory presented applies to any model of observation equations having both integer and real-valued parameters, as well as for any probability density function the data might have. AcknowledgementsThis contribution was finalized during the authors stay, as a Tan Chin Tuan Professor, at the Nanyang Technological Universitys GPS Centre (GPSC) in Singapore. The hospitality of the GPSCs director Prof Law Choi Look and his colleagues is greatly appreciated.     

On the probability density function of the GNSS ambiguity residuals

Integer GNSS ambiguity resolution involves estimation and validation of the unknown integer carrier phase ambiguities. A problem then is that the classical theory of linear estimation does not apply to the integer GPS model, and hence rigorous validation is not possible when use is made of the classical results. As with the classical theory, a first step for being able to validate the integer GPS model is to make use of the residuals and their probabilistic properties. The residuals quantify the inconsistency between data and model, while their probabilistic properties can be used to measure the significance of the inconsistency. Existing validation methods are often based on incorrect assumptions with respect to the probabilistic properties of the parameters involved. In this contribution we will present and evaluate the joint probability density function (PDF) of the multivariate integer GPS carrier phase ambiguity residuals. The residuals and their properties depend on the integer estimation principle used. Since it is known that the integer least-squares estimator is the optimal choice from the class of admissible integer estimators, we will only focus on the PDF of the ambiguity residuals for this estimator. Unfortunately the PDF cannot be evaluated exactly. It will therefore be shown how to obtain a good approximation. The evaluation will be completed by some examples.     

Testing of a new single-frequency GNSS carrier phase attitude determination method: land,ship and aircraft experiments

Global navigation satellite system (GNSS) ambiguity resolution is the process of resolving the unknown cycle ambiguities of the carrier phase data as integers. The sole purpose of ambiguity resolution is to use the integer ambiguity constraints as a means of improving significantly on the precision of the remaining GNSS model parameters. In this contribution, we consider the problem of ambiguity resolution for GNSS attitude determination. We analyse the performance of a new ambiguity resolution method for GNSS attitude determination. As it will be shown, this method provides a numerically efficient, highly reliable and robust solution of the nonlinearly constrained integer least-squares GNSS compass estimators. The analyses have been done by means of a unique set of extensive experimental tests, using simulated as well as actual GNSS data and using receivers of different manufacturers and type as well as different platforms. The executed field tests cover two static land experiments, one in the Netherlands and one in Australia, and two dynamic experiments, a low-dynamics vessel experiment and high-dynamics aircraft experiment. In our analyses, we focus on stand-alone, unaided, single-frequency, single-epoch attitude determination, as this is the most challenging case of GNSS compass processing.     

Precise GRACE baseline determination using GPS

Precision relative navigation is an essential aspect of spacecraft formation flying missions, both from an operational and a scientific point of view. When using GPS as a relative distance sensor, dual-frequency receivers are required for high accuracy at large inter-satellite separations. This allows for a correction of the relative ionospheric path delay and enables double difference integer ambiguity resolution. Although kinematic relative positioning techniques demonstrate promising results for hardware-in-the-loop simulations, they were found to lack an adequate robustness in real-world applications. To overcome this limitation, an extended Kalman Filter modeling the relative spacecraft dynamics has been developed. The filter processes single difference GPS pseudorange and carrier phase observations to estimate the relative position and velocity along with empirical accelerations and carrier phase ambiguities. In parallel, double difference carrier phase ambiguities are resolved on both frequencies using the least square ambiguity decorrelation adjustment (LAMBDA) method in order to fully exploit the inherent measurement accuracy. The combination of reduced dynamic filtering with the LAMBDA method results in smooth relative position estimates as well as fast and reliable ambiguity resolution. The proposed method has been validated with data from the gravity recovery and climate experiment (GRACE) mission. For an 11-day data arc, the resulting solution matches the GRACE K-Band Ranging System measurements with an accuracy of 1 mm, whereby 83% of the double difference ambiguities are resolved.     

基于格论的GNSS模糊度解算

快速、准确地解算整周模糊度是实现GNSS载波相位实时高精度定位的关键,由于模糊度之间的强相关,基于整数最小二乘估计准则时,需要较长的时间才能搜索出最优的整周模糊度向量。为了提高模糊度的搜索效率,本文在扼要介绍格论的理论框架基础上,引入基于格论的模糊度解算方法,通过格基规约来降低模糊度之间的相关性,从而快速搜索出最优的整数模糊度向量。与此同时,将GNSS领域的主要降相关方法统一到格论框架下,探讨了并建议采用Bootstrapping成功率作为格基规约的性能指标之一。最后实验分析了三频多系统长基线相对定位情况下,不同格基规约可获得的性能。     

一种机载GNSS高精度定位算法

提出部分模糊度固定的加权电离层模型提高大范围全球卫星导航系统(GNSS)航空定位的精度、可靠性及连续性.该方法的主要思路包括:自适应调整大气扰动等误差影响以实现短基线与长基线两类解算模式之间的灵活切换;施加虚拟电离层观测约束信息,提高基线动态定位的浮点解精度;采用部分模糊度固定方法有效挖掘若干模糊度参数的整周约束.试验表明,提出的方法可提高模糊固定效率与定位精度,克服传统方法有效观测信息利用率不足、定位精度较差、可靠性不高以及连续性较差的问题.实验结果表明,部分模糊度固定算法可在2 min内固定95%以上宽巷模糊度解算与80%以上窄巷模糊度,约20 min后可固定所有模糊度.     

一种GPS整周模糊度单历元解算方法

仅利用单历元的载波相位观测值进行整周模糊度解算,观测方程秩亏,给单历元模糊度解算带来很大困难.因此,本文提出一种单历元确定GPS整周模糊度的方法.利用单历元测码伪距观测值和双频载波相位观测值组成双差观测方程,根据方差矩阵对宽巷模糊度进行分组,采用基于LABMDA方法的逐步解算方法来确定双差相位观测值的宽巷模糊度.确定宽...     

网络RTK参考站间模糊度动态解算的卡尔曼滤波算法研究

提出一种适用于参考站网络的站间模糊度解算方法,该方法使用CA码与相位的电离层无关组合解算宽巷模糊度,利用多路径效应的周期性削弱CA码多路径效应。在宽巷模糊度得到固定后,利用卡尔曼滤波对L1模糊度进行估计,并使用模糊度失相关搜索算法,动态地确定模糊度。这种方法已经应用在自主开发的网络RTK软件上,并以四川GPS综合服务网络SIGN(Sichuan Integrated GPS Network)作为试验网络,进行了试验和分析。     

ADOP in closed form for a hierarchy of multi-frequency single-baseline GNSS models

Successful carrier phase ambiguity resolution is the key to high-precision positioning with Global Navigation Satellite Systems (GNSS). The ambiguity dilution of precision (ADOP) is a well-known scalar measure which can be used to infer the strength of the GNSS model for carrier phase ambiguity resolution. In this contribution we present analytical closed-form expressions for the ADOP. This will be done for a whole class of different multi- frequency single baseline models. These models include the geometry-fixed, the geometry-free and the geometry-based models, respectively. And within the class of geometry-based models, we discriminate between short and long observation time spans, and between stationary and moving receivers. The easy-to-use ADOP expressions can be applied to infer the contribution of various GNSS model factors. They comprise, for instance, the type, the number and the precision of the GNSS observations, the number and selection of frequencies, the presence of atmospheric disturbances, the length of the observation time span and the length of the baseline.     

Efficiency of carrier-phase integer ambiguity resolution for precise GPS positioning in noisy environments

Precise GPS positioning relies on tracking the carrier-phase. The fractional part of carrier-phase can be measured directly using a standard phase-locked loop, but the integer part is ambiguous and the ambiguity must be resolved based on sequential carrier-phase measurements to ensure the required positioning precision. In the presence of large phase-measurement noise, as can be expected in a jamming environment for example, the amount of data required to resolve the integer ambiguity can be large, which requires a long time for any generic integer parameter estimation algorithm to converge. A key question of interest in significant applications of GPS where fast and accurate positioning is desired is then how the convergence time depends on the noise amplitude. Here we address this question by investigating integer least-sqaures estimation algorithms. Our theoretical derivation and numerical experiments indicate that the convergence time increases linearly with the noise variance, suggesting a less stringent requirement for the convergence time than intuitively expected, even in a jamming environment where the phase noise amplitude is large. This finding can be useful for practical design of GPS-based systems in a jamming environment, for which the ambiguity resolution time for precise positioning may be critical.     

北斗信号体制下三频CIR法模糊度解算方法研究

未来的GNSS系统将广泛使用三个频点的载波信号,极大地推动了利用载波对用户位置和速度等信息进行高精度解算方法的研究。利用载波进行高精度解算的关键是高成功率的确定载波相位整周模糊度。CIR算法基于整数引导估计准则确定载波相位模糊度整数解,算法实现简单,模糊度解算成功率较高,具有很大的实际应用价值。然而目前CIR法在实际应用中主要针对GPS信号体制,在北斗信号体制下是否具有良好的性能还有待验证。本文将CIR法应用于北斗信号体制下三频载波整周模糊度的解算,利用仿真数据对算法进行验证,并将解算结果与GPS信号体制下解算结果进行对比,验证CIR法在北斗信号体制下的可行性,分析不同信号体制下CIR法的解算性能和适用范围。     

组合导航卫星系统定位技术进展

多星座导航系统的组合共用是近年来卫星导航领域的发展方向之一。综述了基于组合导航卫星系统定位技术的最新研究进展,从多源信息融合理论的角度阐述了卫星导航系统组合定位的思路,同时在多星座导航系统组合的新背景下分析了载波相位模糊度解算这一关键问题的发展,探讨了系统组合共用实施中有待于进一步研究解决的新问题。     

长距离网络RTK参考站间双差模糊度快速解算算法

提出了一种长距离网络RTK参考站间双差整周模糊度快速解算方法,该方法利用双频载波相位模糊度间的线性关系确定宽巷模糊度,然后选取双频载波相位的备选模糊度组合,通过计算参考站间对流层误差和轨道误差等非色散误差,对双频载波相位整周模糊度进行搜索。实验结果表明,此方法能够快速、可靠地解算长距离参考站间的双差整周模糊度。     

下三角Cholesky分解的整数高斯变换算法

针对全球导航卫星系统(GNSS)载波相位测量中,基于整数最小二乘估计准则解算整周模糊度问题。目前以LAMBDA降相关算法和Lenstra-Lenstra-Lovász(LLL)为代表的规约算法应用最为广泛。由于不同算法采用的模糊度方差-协方差阵的分解方式不同,导致难以合理地进行不同算法性能的比较。该文通过分析LAMBDA算法的降相关特点,从理论上推出基于下三角Cholesky分解多维情形下的整数高斯变换的降相关条件及相应公式,并与分解方式不同的LAMBDA和LLL算法作了对比。实验结果表明,降相关采用的分解方式将会直接影响计算复杂度和解算性能,因此该文推导的整数高斯变换算法便于今后基于下三角Cholesky分解的降相关算法间的合理比较。     

差分GPS载波相位整周模糊度快速解算方法

本文提出了一种整周模糊度的快速求解方法,将差分GPS的测量值分配到主要测量值集合和次要测量值集合中,用主要集合中的相位测量值限定简约搜索空间,而次要集合中的相位测量值用来验证候选集合。利用已知的基线长度的约束条件,对搜索空间进行了简约,提高了求解整周模糊度的速度,同时,通过Cholesky分解提高搜索效率。