降相关对模糊度解算中搜索效率的影响分析

首先理论分析了条件数、正交缺陷度、S(A)等降相关评价指标所表示的几何意义,然后采用LAMBDA算法、LLL规约算法和Seysen规约算法通过模拟和实际数据对模糊度的搜索效果和不同评价指标之间的关系进行了深入计算分析。进一步验证得出"降低模糊度方差分量间的相关性实现最大程度地压缩椭球可以提高搜索效率"的观点是片面的,并通过结果分析表明提高搜索效率的本质在于尽可能地促使基向量按照一定方向排序。     

三种GNSS模糊度解算方法性能的对比分析

探讨了三种GNSS模糊度解算方法,对三种模糊度解算方法的规约时间、搜索时间、总体时间和条件数进行了比对分析。通过多次不同维度的模拟数据和实测数据实验,验证了排序QR分解算法在模糊度解算过程中总体最优,原因在于排序QR分解算法在降相关过程中对条件方差进行了升序排列,用较少的降相关过程获得了一个更有利于搜索的降相关矩阵;LLL算法的规约性能不稳定,取决于数据结构;LAMBDA算法在本文模拟实验中总体性能相对较差,与降相关矩阵的分解方式有关,实测实验中搜索性能较差的原因在于条件方差的排序不稳定。     

误差椭圆搜索法EES固定GNSS整周模糊度

LAMBDA方法和改进的LAMBDA方法都使用整数矩阵进行降相关,使得变换后的模糊度方差阵更加对角化,但有时变换后的方差阵的对角线元素的数值量级相差很大,使得搜索空间有些扁长,为了避免这些情况的发生,本文提出一种更加理想的新的EES(error ellipse search,误差椭圆搜索)方法来进行整周模糊度的固定。由于实数矩阵可以使降相关达到各种理想的状态,在降相关方面比整数矩阵更有优势,因此EES方法利用实数矩阵进行降相关,搜索每两个模糊度之间的最佳误差椭圆,使得变换后的方差阵对角线元素趋于同一量级,搜索空间更加接近于球形。通过实验表明,随着基线长度的增加,EES方法固定模糊度的成功率远远高于LAMBDA方法,缩短了固定整周模糊度需要的历元数,是一种切实可行有效的固定整周模糊度的方法。     

一种基于对角线预排序的模糊度降相关方法

通过分析基于升序排列和下三角乔列斯基分解的降相关算法的优缺点,提出了一种模糊度降相关新方法。该方法是基于对角线预排序和上三角乔列斯基分解的降相关算法,不仅保证每次乔列斯基分解的降相关程度最高,而且使降相关后的条件方差大致降序排列。在分析当前常用的降相关效果评价指标的基础上,选取条件数和等价相关系数作为新方法降相关效果的评价指标。应用实测数据进行降相关计算得出,与基于升序排列和下三角乔列斯基分解的降相关算法相比,新方法降相关程度更高,迭代次数更少,可以提高整周模糊度解算过程中条件搜索的效率。     

基于LAMBDA整周模糊度解算方法中的整数Z变换算法

LAMBDA算法是目前公认求解整周模糊度效果最好的方法,该算法主要包括模糊度去相关处理(Z变换)和整周模糊度搜索,其中Z变换对高度相关的整周模糊度进行降相关处理是LAMBDA算法的核心内容。本文分析了Z变换中迭代法和联合去相关法两种算法的基本原理,并通过实例对两种算法进行了性能评价,实验分析表明两种算法去相关水平相当,迭代次数无明显差异,但矩阵维数越大,去相关效果越有所下降。总体而言,联合去相关效果略高于迭代法。     

整周模糊度去相关的两种实现方法

整周模糊度搜索一直是GPS快速精确定位的关键问题。短时间的观测会导致观测方程和整周模糊度方差、协方差矩阵的高相关性，因而急剧增大整周模糊度的搜索空间，对整周模糊度未知数方差、协方差矩阵进行去相关性处理，可以有效地压缩搜索空间。本文对整周模糊度去相关的迭代法和联合变换法从原理上进行了阐述，并结合实际算倒进行了分析和比较。     

Householder正交变换在模糊度降相关算法中的应用

在GNSS模糊度解算的过程中,由于模糊度之间存在相关性,为减少搜索时间需要对模糊度的协方差矩阵进行降相关处理。降相关算法的优劣将直接影响到模糊度搜索的效率。本文基于Householder正交变换提出了一种新的降相关算法,并利用随机模拟数据和北斗实测数据,从谱条件数、平均相关系数和规约时间3个方面将Householder算法与目前较为流行的LLL算法以及逆整数Cholesky算法进行了对比。通过实验分析得出,Householder算法能够明显改善降相关处理的效果。但是该算法仍存在规约时间较长的不足,需要进一步完善。     

基于ICSO的DGPS整周模糊度的求解方法

针对差分全球定位系统(DGPS)模糊度解算过程中效率低,搜索慢的问题,对鸡群优化算法(CSO)进行适应性改进,并将改进后的鸡群优化算法(ICSO)应用到整周模糊度的快速解算中,利用卡尔曼滤波求出双差模糊度的浮点解和协方差矩阵,采用Lenstra-Lenstra-Lovasz (LLL)降相关算法对模糊度的浮点解和方差协方差矩阵进行降相关处理,以降低模糊度各分量之间的相关性,在基线长度固定的情况下,利用ICSO搜索整周模糊度的最优解. 采用经典算例进行仿真,仿真结果表明,与已有文献相比在整周模糊度的解算过程中改进的鸡群优化算法能有效提高搜索速度和求解成功率.      

一种单频单历元BDS／GPS组合整周模糊度解算方法

针对单频单历元组合载波相位差分技术(RTK)定位过程中存在的秩亏及模糊度解算病态等问题,提出了一种模糊度降相关的新方法。该方法引入伪距观测值进行辅助解算。首先采用经验分权法对伪距与载波相位观测值分配权重,并通过加权最小二乘法获得整周模糊度浮点解及协方差。然后通过对整周模糊度浮点解的方差-协方差矩阵进行降序排列和剔除病态模糊度。最后利用修正后的浮点解迭代搜索模糊度的整数解。试验结果表明而且可以起到良好的模糊度降相关的效果定位。      

改进的GPS模糊度降相关LLL算法

模糊度降相关技术可以有效提高模糊度求解的效率及成功率,LLL(A.K.Lenstra,H.W.Lenstra,L.Lovasz)算法是新出现的模糊度降相关方法。详细分析LLL算法,针对该算法中存在的缺陷,提出逆整数乔勒斯基、整数高斯算法和升序调整矩阵辅助的改进LLL算法。利用谱条件数及平均相关系数为准则,以300个随机模拟的对称正定矩阵作为模糊度方差-协方差矩阵,对LLL算法和改进的LLL算法进行仿真计算。比较与分析结果表明,改进LLL算法模糊度降相关处理更加彻底,能有效地加速整周模糊度搜索及成功解算。     

Orthogonality defect and reduced search-space size for solving integer least-squares problems

In the context of ambiguity resolution (AR) of global navigation satellite systems (GNSS), decorrelation among entries of an ambiguity vector, integer ambiguity search, and ambiguity validations are three standard procedures for solving integer least-squares problems. This paper contributes to AR issues from three aspects. Firstly, the orthogonality defect is introduced as a new measure of the performance of ambiguity decorrelation methods and compared with the decorrelation number and with the condition number, which are currently used as the judging criterion to measure the correlation of ambiguity variance–covariance matrix. Numerically, the orthogonality defect demonstrates slightly better performance as a measure of the correlation between decorrelation impact and computational efficiency than the condition number measure. Secondly, the paper examines the relationship of the decorrelation number, the condition number, the orthogonality defect, and the size of the ambiguity search space with the ambiguity search candidates and search nodes. The size of the ambiguity search space can be properly estimated if the ambiguity matrix is decorrelated well, which is shown to be a significant parameter in the ambiguity search progress. Thirdly, a new ambiguity resolution scheme is proposed to improve ambiguity search efficiency through the control of the size of the ambiguity search space. The new AR scheme combines the LAMBDA search and validation procedures together, which results in a much smaller size of the search space and higher computational efficiency while retaining the same AR validation outcomes. In fact, the new scheme can deal with the case there are only one candidate, while the existing search methods require at least two candidates. If there are more than one candidate, the new scheme turns to the usual ratio-test procedure. Experimental results indicate that this combined method can indeed improve ambiguity search efficiency for both the single constellation and dual constellations, respectively, showing the potential for processing high-dimension integer parameters in multi-GNSS environment.     

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。     

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。     

A semivariance-based index for spatially correlated errors in rapid differential GNSS applications

In this work, a regional network of permanent Global Navigation Satellite Systems (GNSS) receivers is used to estimate the decorrelation of the spatially correlated errors in differential GNSS positioning. Emphasis is laid on the dispersive errors (i.e. mainly the ionosphere). A new index, based on variance as function of station separation (semivariance) is proposed and compared to the existing I95 index. This study uses data from the 29–30th October 2003, a period with severe ionospheric activity. The proposed index is shown to give realistic predictions of differential measurement accuracy, and has potential for further development towards use in RTK-networks.     

大型桥梁桥塔监测的GNSS相对对流层延迟估计

针对大型桥梁桥塔与基站高程差异较大,残余对流层延迟成为影响全球卫星导航系统(GNSS)监测成功率与精度的主要因素之一。该文基于随机过程理论,对桥梁监测GNSS残余对流层湿延迟进行参数估计,有效地提高了桥梁塔顶监测GNSS模糊度固定率。通过采用对流层经验模型改正对流层干延迟,将基准站和塔顶观测站对流层湿延迟组成相对对流层湿延迟,并联合位置参数和模糊度参数建立双差卡尔曼模型,最后利用最小二乘模糊度降低相关平差法(LAMBDA)对双差模糊度进行固定,并估计位置参数与相对对流层延迟参数。实验结果表明,该方法可以有效估计相对对流层延迟,有效提高GNSS模糊度固定率。     

The least-squares ambiguity decorrelation adjustment: its performance on short GPS baselines and short observation spans

The least-squares ambiguity decorrelation adjustment is a method for fast GPS double-difference (DD) integer ambiguity estimation. The performance of the method will be discussed, and although it is stressed that the method is generally applicable, attention is restricted to short-baseline applications in the present contribution. With reference to the size and shape of the ambiguity search space, the volume of the search space will be introduced as a measure for the number of candidate grid points, and the signature of the spectrum of conditional variances will be used to identify the difficulty one has in computing the integer DD ambiguities. It is shown that the search for the integer least-squares ambiguities performs poorly when it takes place in the space of original DD ambiguities. This poor performance is explained by means of the discontinuity in the spectrum of conditional variances. It is shown that through a decorrelation of the ambiguities, transformed ambiguities are obtained which generally have a flat and lower spectrum, thereby enabling a fast and efficient search. It is also shown how the high precision and low correlation of the transformed ambiguities can be used to scale the search space so as to avoid an abundance of unnecessary candidate grid points. Numerical results are presented on the spectra of conditional variances and on the statistics of both the original and transformed ambiguities. Apart from presenting numerical results which can typically be achieved, the contribution also emphasizes and explains the impact on the method's performance of different measurement scenarios, such as satellite redundancy, single vs dual-frequency data, the inclusion of code data and the length of the observation time span. Received: 31 October 1995 / Accepted: 21 March 1997     

Fast integer least-squares estimation for GNSS high-dimensional ambiguity resolution using lattice theory

GNSS ambiguity resolution is the key issue in the high-precision relative geodetic positioning and navigation applications. It is a problem of integer programming plus integer quality evaluation. Different integer search estimation methods have been proposed for the integer solution of ambiguity resolution. Slow rate of convergence is the main obstacle to the existing methods where tens of ambiguities are involved. Herein, integer search estimation for the GNSS ambiguity resolution based on the lattice theory is proposed. It is mathematically shown that the closest lattice point problem is the same as the integer least-squares (ILS) estimation problem and that the lattice reduction speeds up searching process. We have implemented three integer search strategies: Agrell, Eriksson, Vardy, Zeger (AEVZ), modification of Schnorr–Euchner enumeration (M-SE) and modification of Viterbo-Boutros enumeration (M-VB). The methods have been numerically implemented in several simulated examples under different scenarios and over 100 independent runs. The decorrelation process (or unimodular transformations) has been first used to transform the original ILS problem to a new one in all simulations. We have then applied different search algorithms to the transformed ILS problem. The numerical simulations have shown that AEVZ, M-SE, and M-VB are about 320, 120 and 50 times faster than LAMBDA, respectively, for a search space of dimension 40. This number could change to about 350, 160 and 60 for dimension 45. The AEVZ is shown to be faster than MLAMBDA by a factor of 5. Similar conclusions could be made using the application of the proposed algorithms to the real GPS data.     

Integer least-squares theory for the GNSS compass

Global navigation satellite system (GNSS) carrier phase integer ambiguity resolution is the key to high-precision positioning and attitude determination. In this contribution, we develop new integer least-squares (ILS) theory for the GNSS compass model, together with efficient integer search strategies. It extends current unconstrained ILS theory to the nonlinearly constrained case, an extension that is particularly suited for precise attitude determination. As opposed to current practice, our method does proper justice to the a priori given information. The nonlinear baseline constraint is fully integrated into the ambiguity objective function, thereby receiving a proper weighting in its minimization and providing guidance for the integer search. Different search strategies are developed to compute exact and approximate solutions of the nonlinear constrained ILS problem. Their applicability depends on the strength of the GNSS model and on the length of the baseline. Two of the presented search strategies, a global and a local one, are based on the use of an ellipsoidal search space. This has the advantage that standard methods can be applied. The global ellipsoidal search strategy is applicable to GNSS models of sufficient strength, while the local ellipsoidal search strategy is applicable to models for which the baseline lengths are not too small. We also develop search strategies for the most challenging case, namely when the curvature of the non-ellipsoidal ambiguity search space needs to be taken into account. Two such strategies are presented, an approximate one and a rigorous, somewhat more complex, one. The approximate one is applicable when the fixed baseline variance matrix is close to diagonal. Both methods make use of a search and shrink strategy. The rigorous solution is efficiently obtained by means of a search and shrink strategy that uses non-quadratic, but easy-to-evaluate, bounding functions of the ambiguity objective function. The theory presented is generally valid and it is not restricted to any particular GNSS or combination of GNSSs. Its general applicability also applies to the measurement scenarios (e.g. single-epoch vs. multi-epoch, or single-frequency vs. multi-frequency). In particular it is applicable to the most challenging case of unaided, single frequency, single epoch GNSS attitude determination. The success rate performance of the different methods is also illustrated.     

An alternative approach to calculate the posterior probability of GNSS integer ambiguity resolution

When precise positioning is carried out via GNSS carrier phases, it is important to make use of the property that every ambiguity should be an integer. With the known float solution, any integer vector, which has the same degree of freedom as the ambiguity vector, is the ambiguity vector in probability. For both integer aperture estimation and integer equivariant estimation, it is of great significance to know the posterior probabilities. However, to calculate the posterior probability, we have to face the thorny problem that the equation involves an infinite number of integer vectors. In this paper, using the float solution of ambiguity and its variance matrix, a new approach to rapidly and accurately calculate the posterior probability is proposed. The proposed approach consists of four steps. First, the ambiguity vector is transformed via decorrelation. Second, the range of the adopted integer of every component is directly obtained via formulas, and a finite number of integer vectors are obtained via combination. Third, using the integer vectors, the principal value of posterior probability and the correction factor are worked out. Finally, the posterior probability of every integer vector and its error upper bound can be obtained. In the paper, the detailed process to calculate the posterior probability and the derivations of the formulas are presented. The theory and numerical examples indicate that the proposed approach has the advantages of small amount of computations, high calculation accuracy and strong adaptability.