An Algorithm of Long Distance Undifference Network RTK with BDS-3/BDS-2 Multi-frequency Data

ZHU Huizhong; ZHANG Jie; ZHAO Qile; XU Aigong

doi:10.13203/j.whugis20220797

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An Algorithm of Long Distance Undifference Network RTK with BDS-3/BDS-2 Multi-frequency Data

Updated：2023-06-29

- An Algorithm of Long Distance Undifference Network RTK with BDS-3/BDS-2 Multi-frequency Data
- Geomatics and Information Science of Wuhan University
- Affiliations：
  
  1.School of Geomatics,Liaoning Technical University,Fuxin 123000,China
  2.GNSS Research Center,Wuhan University,Wuhan 430079,China
- Author bio：
- Funds：
- DOI：10.13203/j.whugis20220797
  CLC： P228
- Published：05 July 2023，
  
  Received：02 April 2023，
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祝会忠,张杰,赵齐乐，等.BDS-3/BDS-2多频数据的长距离非差网络RTK算法[J].武汉大学学报（信息科学版）,2023,48(7):1146-1159.

ZHU Huizhong,ZHANG Jie,ZHAO Qile,et al.An Algorithm of Long Distance Undifference Network RTK with BDS-3/BDS-2 Multi-frequency Data[J].Geomatics and Information Science of Wuhan University,2023,48(7):1146-1159.
祝会忠,张杰,赵齐乐，等.BDS-3/BDS-2多频数据的长距离非差网络RTK算法[J].武汉大学学报（信息科学版）,2023,48(7):1146-1159. DOI：10.13203/j.whugis20220797

ZHU Huizhong,ZHANG Jie,ZHAO Qile,et al.An Algorithm of Long Distance Undifference Network RTK with BDS-3/BDS-2 Multi-frequency Data[J].Geomatics and Information Science of Wuhan University,2023,48(7):1146-1159. DOI：10.13203/j.whugis20220797

BDS-3/BDS-2多频数据的长距离非差网络RTK算法 An Algorithm of Long Distance Undifference Network RTK with BDS-3/BDS-2 Multi-frequency Data 1 first-author 祝会忠，教授，博士，主要研究方向为GNSS高精度定位算法。zhuhuizhong@whu.edu.cn 祝会忠，教授，博士，主要研究方向为GNSS高精度定位算法。zhuhuizhong@whu.edu.cn zhuhuizhong@whu.edu.cn 1 2 1 辽宁工程技术大学测绘与地理科学学院，辽宁阜新，123000 辽宁工程技术大学测绘与地理科学学院，辽宁阜新，123000 School of Geomatics,Liaoning Technical University,Fuxin 123000,China School of Geomatics,Liaoning Technical University,Fuxin 123000,China 武汉大学卫星导航定位技术研究中心，湖北武汉，430079 武汉大学卫星导航定位技术研究中心，湖北武汉，430079 GNSS Research Center,Wuhan University,Wuhan 430079,China GNSS Research Center,Wuhan University,Wuhan 430079,China 北斗三号卫星导航系统(BeiDou-3 satellite navigation system，BDS-3)已正式建成并提供服务，网络实时动态定位（real time kinematic，RTK）算法是提高北斗卫星导航系统（BeiDou satellite navigation system，BDS）定位精度的主要手段。提出了一种基于BDS多频多系统观测数据的长距离非差网络RTK算法。首先，通过顾及大气误差参数的多频载波相位整周模糊度解算模型，确定长距离BDS参考站的多频载波相位整周模糊度。然后，利用准确确定的参考站间载波相位整周模糊度，通过线性变化得到各参考站的非差整周模糊度，并计算各参考站的非差观测误差。考虑长距离测站间观测误差空间相关性减弱的特点，根据误差特性的不同分离出参考站网分类的非差误差改正数。最后，采用反距离加权法计算流动站的分类非差误差改正数，流动站通过非差误差改正数进行观测误差的改正和高精度定位。使用长距离连续运行参考站网的多频实测数据进行实验，结果表明：BDS-3的定位精度相对于北斗二号卫星导航系统 (BeiDou-2 satellite navigation system，BDS-2)在东、北、天各方向分别有15.7%、28%、11.9%的提高，定位性能与GPS相当甚至更优，多频数据的模糊度固定成功率和收敛时间较GPS双频数据有一定优势，BDS-3和BDS-2双系统定位结果优于单系统，二者均能实现流动站厘米级定位。该方法可避免双差观测值的相关性影响，空间误差计算效率高，流动站的定位更加灵活，充分发挥了BDS多频观测数据的优势。 Objectives 2 BeiDou-3 satellite navigation system (BDS-3) has been officially completed and provides services, network real time kinematic(RTK) is the main means to improve positioning accuracy, we present a long-distance undifference network RTK method based on BDS multi frequency and multi system observation data. Methods 2 First, the ambiguity resolution model considering atmospheric error parameters is used to determine the integer ambiguity of multi-frequency carrier phase of long-distance BeiDou satellite navigation system (BDS) reference station.Then,the double difference integer ambiguity between reference stations is converted into non‑difference integer ambiguity of each reference station through linear change,and calculate the undifference observation error of each reference station.Considering the weakening of spatial correlation of observation errors between long-distance stations.According to the difference of error characteristics,obtain the correction number of classified undifference error of reference station network.Finally, by inverse distance weighted interpolation method, we calculate the error correction of mobile station, high precision positioning for users. Results 2 Select the measured data of long-distance continuous operation reference station network for experiment, the results show that: Compared with BeiDou-2 satellite navigation system (BDS-2), BDS-3 increases by 15.7%, 28% and 11.9% in E, N and U directions respectively. Compared with GPS, the positioning performance is equivalent or even better. The ambiguity fixing success rate and convergence time of multi-frequency data are better than GPS dual-frequency data. Dual system positioning results are better than single system, both of which can achieve centimeter level positioning of mobile stations. Conclusions 2 This method can avoid the correlation effect of double difference observations, high efficiency of spatial error calculation,mobile station positioning is more flexible,shows the advantages of BDS multi frequency observation data. BDS 长距离参考站网分类误差非差误差改正数网络RTK BDS long-distance reference station network classification error undifference error correction network RTK 2020-07-31，中国自主建设的北斗三号卫星导航系统（BeiDou-3 satellite navigation system，BDS-3）正式建成并投入使用，开始为全球用户提供定位、导航和授时服务，这宣告了中国北斗卫星导航系统（BeiDou navigation satellite system，BDS）建设“三步走”计划的顺利完成［ 1 1 ］。北斗二号卫星导航系统（BeiDou-2 satellite navigation system，BDS-2）目前也有在轨工作卫星，卫星工作状态良好，包括3颗中圆地球轨道（medium Earth orbit，MEO）卫星、5颗地球同步轨道（geostationary Earth orbit，GEO）卫星、7颗倾斜地球同步轨道（inclined geostationary orbit，IGSO）卫星，并且全星座播发三频信号。相对于BDS-2，BDS-3在覆盖能力、空间信号精度、空间信号可用性、空间信号连续性等方面均有显著提升［ 2 2 ］。BDS-3还提供包括国际搜救、短报文通信、星基增强等服务，大幅度提升了BDS的应用场景和全球影响力。BDS-3星座由24颗MEO、3颗GEO、3颗IGSO卫星组成，频率是在BDS-2的基础上新增了B1c、B2a、B2b频率，播发B1I（1 561.098 MHz）、B2b（1 207.14 MHz）、B3I（1 268.52 MHz）、B1c（1 575.42 MHz）、B2a（1 176.45 MHz）5个频率信号。观测值的增加提高了自身系统的数据冗余度和定位性能，更多的频率为定位解算过程中模糊度的固定提供了保障，定位精度也能得到明显提高。特别是在一些观测条件恶劣的情况下，更多的频率能获取到更好的定位结果。同时，还提高了与其他全球卫星导航系统的兼容性和互操作性，BDS-3的B1I、B3I信号与BDS-2兼容，B1c、B2a与GPS的L1/L5、伽利略卫星导航系统（Galileo satellite navigation system，Galileo）的E1、E5a信号兼容，为实现BDS-3与其他系统的组合定位提供了有利条件。文献［ 3 3 ］对BDS-3的信号质量进行分析，发现BDS-3卫星信号的多路径噪声水平优于BDS-2卫星且未发现与高度角相关的系统偏差，B1c受多路径及噪声的影响更为显著。文献［ 4 4 ］对BDS-3观测信息的高度角相关随机模型进行研究，结果表明BDS-3的伪距和载波相位观测值精度均与高度角相关，并且观测类型不同，相关程度也不同。文献［ 5 5 ］则是对建成后的BDS-3与Galileo重叠频率间的系统间偏差进行研究，并对两个系统组合的实时动态定位（real time kinematic，RTK）性能进行分析，结果表明双系统相较于单系统在模糊度固定成功率和定位精度上都有明显提升。这些为BDS-3的研究提供了重要参考。网络RTK算法的参考站间距离一般为几十千米或上百千米，电离层延迟、对流层延迟、卫星轨道误差的相关性低，观测值的残余误差大于载波相位观测值的半个波长，即使已知参考站坐标也很难将整周模糊度与误差分离，特别是在长距离的情况下，如何准确固定参考站整周模糊度是长距离网络RTK的核心问题之一［ 6 6 ］。网络RTK算法还包括区域误差模型的建立和用户载波相位整周模糊度的解算与定位。文献［ 7 7 ］研究了网络RTK相较于传统RTK的优势。文献［ 8 8 ］提出了利用电离层约束模型实现网络RTK宽巷、窄巷模糊度依次固定的整周模糊度解算方法。文献［ 9 9 ］在顾及大气因素和卫星高度角的前提下提出了一种网络RTK参考站模糊度快速解算方法。文献［ 10 10 ］实现了顾及GEO卫星实际大气延迟变化约束和整周模糊度约束的长距离BDS三频载波相位整周模糊度解算方法。文献［ 11 11 ］研究了BDS参考站三频载波相位整周模糊度单历元确定方法。文献［ 12 12 ］提出了网络RTK的非差算法，求解区域非差改正数。文献［ 13 13 ］研究了BDS-2三频数据频间组合的网络RTK算法。文献［ 14 14 ］研究了在加入载波频间约束的条件下BDS-2三频数据非差非组合网络RTK解算方法。文献［ 15 15 ］研究了BDS长距离网络RTK算法。文献［ 16 16 ］研究了BDS-2和GPS双系统在加入大气约束条件下的非差网络RTK算法。文献［ 17 17 ］研究了一种长距离GPS/BDS双系统网络RTK算法，利用宽巷、窄巷模糊度逐级固定方法实现了BDS-2/GPS双系统网络RTK。文献［ 18 18 ］为了满足稀疏参考站的高精度定位，提出了一种基于虚拟大气约束的网络RTK算法。这些方法主要是以BDS-2和GPS两个系统为主进行研究，关于BDS-3系统长距离网络RTK算法的研究较少。长距离网络RTK可以扩大参考站的覆盖范围，提高系统的作业效率。但是长距离参考站间距离较大，电离层延迟、对流层延迟等误差的空间相关性较小，整周模糊度与残余误差的分离和高精度区域误差改正模型的建立比较困难。非差网络RTK算法可避免双差观测值间的相关性，非差误差改正数能够以单颗卫星的形式生成，不受基准星变化的限制。本文提出了一种BDS-3/BDS-2多频多系统长距离非差网络RTK算法，充分利用BDS全星座提供三频观测数据的优势，首先基于大气延迟约束模型计算参考站网的多频载波相位整周模糊度，并将其转换为非差整周模糊度；然后计算参考站的分类非差误差改正数，利用反距离加权算法内插得到流动站的分类非差误差改正数，并进行流动站BDS-3/BDS-2单系统、双系统组合的多频载波相位整周模糊度解算和位置参数计算；最后使用实测连续运行参考站（continuously operating reference stations，CORS）网数据进行BDS-3/BDS-2长距离非差网络RTK算法验证和实验分析。该算法在流动端与目前最前沿的PPP-RTK技术类似，但是不需要引入精密产品以及相位未校准硬件延迟等产品，只需要引入广播星历便可实现流动站厘米级定位，BDS-3单系统多频情况下能够在2 min左右实现收敛，并且只需要给流动站播发非差误差改正信息，对信号播发的要求也不高。 1　长距离BDS非差网络RTK算法 1 1.1　参考站网多频载波相位整周模糊度解算 2 由于长距离网络RTK参考站间距离较长（100 km以上），电离层延迟和对流层延迟等误差对双差观测值的影响远大于0.5周，即使在使用多频观测数据和参考站坐标已知的情况下，整周模糊度与误差也难以分离。本文采用的载波相位非差观测方程可表示为［ 15 15 ］： <math display="block" 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076782&type= 1.60866666 2.79399991 为电离层误差； <math id="M12"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076793&type= 2.87866688 2.79399991 为对流层误差； <math id="M13"><msubsup><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076795&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076807&type= 4.31799984 4.06400013 为整周模糊度； <math id="M14"><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076812&type= 2.70933342 3.72533321 为接收机端相位硬件延迟； <math id="M15"><msup><mrow><mi>σ</mi></mrow><mrow><mi>s</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076817&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076814&type= 2.70933342 3.04800010 为卫星端相位硬件延迟； <math id="M16"><msub><mrow><mi>ε</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076821&type= 2.20133328 3.72533321 为载波观测噪声；上标 <math id="M17"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076838&type= 1.26999998 2.79399991 代表卫星编号；下标 <math id="M18"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 代表频率；下标 <math id="M19"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076859&type= 1.69333339 2.79399991 代表接收机。伪距线性化后的非差观测方程可表示为： <math display="block" id="M20"><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>H</mi><mo>⋅</mo><mi>X</mi><mo>+</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>r</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>+</mo><mi>c</mi><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>-</mo><msup><mrow><mi>d</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></mfenced><mo>+</mo><msup><mrow><mi>O</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>+</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076864&type= 90.42400360 4.48733330 式中， <math id="M21"><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076851&type= 2.62466669 2.79399991 为伪距观测值； <math id="M22"><msub><mrow><mi>d</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076856&type= 3.13266683 3.72533321 为接收机端伪距硬件延迟； <math id="M23"><msup><mrow><mi>d</mi></mrow><mrow><mi>s</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076868&type= 3.13266683 3.04800010 为卫星端伪距硬件延迟； <math id="M24"><msub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076871&type= 2.70933342 3.72533321 为伪距观测噪声。以 A 、 B 、 C 3个参考站同步观测基准星 q 和卫星 p 的第 <math id="M25"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 个频率载波相位观测值为例，参考站坐标精确已知，则进行参考站载波相位整周模糊度解算的双差观测方程为： <math display="block" id="M26"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mi>R</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>B</mi><mi>C</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mi>R</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mi>C</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>C</mi><mi>A</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>C</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mi>R</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mi>A</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076880&type= 99.31400299 15.66333294 其中， <math display="block" id="M27"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>p</mi></mrow></msubsup><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>q</mi></mrow></msubsup><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msubsup><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msubsup><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076892&type= 65.27799988 16.17133331 。式中， <math id="M28"><mi mathvariant="normal">Δ</mi><mo>∇</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076895&type= 5.33400011 3.13266683 为双差运算符；下标为参考站；上标为卫星； <math id="M29"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076898&type= 2.45533323 2.79399991 为常数项； <math id="M30"><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076905&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076903&type= 3.89466691 2.87866688 为对流层投影函数； <math id="M31"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076918&type= 2.79399991 2.79399991 为两个参考站的相对天顶对流层延迟误差；下标 <math id="M32"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 表示第 <math id="M33"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 个频率。观测方程中的主要误差源为包含对流层在内的非色散性误差和电离层在内的色散性误差［ 8 8 ］。使用全球投影函数和天顶对流层延迟误差表示参考站观测值中的对流层延迟误差［ 19 19 - 20 20 19-20 ］。两个相邻参考站的同颗卫星投影函数值较接近，将其天顶对流层延迟误差合并为相对天顶对流层延迟误差参数。虽然双差观测方程消除了卫星和接收机钟差、轨道误差、卫星端和接收机端的硬件延迟、观测噪声等大部分误差，但是对于长距离RTK而言，如果不采用合理的措施对大气延迟进行处理，那么对双差模糊度的固定速度和正确性都会造成很大影响。为了防止由于每个历元都估计大气延迟参数导致方程秩亏问题，同时要考虑大气延迟的时变性，本文对电离层参数和对流层参数采用历元间随机游走模型约束的方式进行处理，对流层延迟谱密度取 <math id="M34"><mn mathvariant="normal">1</mn><mtext> </mtext><mi mathvariant="normal">c</mi><mi mathvariant="normal">m</mi><mo>/</mo><mroot><mrow><mi mathvariant="normal">h</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076915&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076913&type= 14.73200035 4.14866638 ，电离层延迟谱密度取 <math id="M35"><mn mathvariant="normal">1</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mo>/</mo><mroot><mrow><mi mathvariant="normal">h</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076926&type= 13.20800018 4.14866638 ［ 21 21 ］。参数解算时方程中包含每颗卫星前后两个历元的电离层延迟误差参数，以及前后两个历元的对流层延迟误差参数。以BDS为例，假设两参考站间共视卫星各 s +1颗，则多频多系统组成法方程为： <math display="block" id="M36"><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><msubsup><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msubsup><mo>⋯</mo><msubsup><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msubsup><mtext> </mtext><msubsup><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msubsup><mo>⋯</mo><msubsup><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msubsup></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msup><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msup><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mtable><mtr><mtd><msub><mrow><mi>R</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>R</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi>I</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msubsup></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr><mtd><msubsup><mrow><mi>I</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msubsup></mtd></mtr><mtr><mtd><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr><mtr><mtd><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msup></mtd></mtr></mtable></mtd></mtr></mtable></mrow></mfenced><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076931&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076929&type= 102.87000275 28.61733437 <math display="block" id="M37"><mfenced open="[" close="]" separators="|"><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msubsup><mo>⋯</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msubsup><mtext> </mtext><msubsup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msubsup><mo>⋯</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076937&type= 31.91933441 4.48733330 <math display="block" id="M38"><msup><mrow/><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076944&type= 1.69333339 3.38666677 其中， <math id="M39"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>S</mi></mrow></msubsup><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mtable><mtr><mtd><msubsup><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd/></mtr></mtable><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi>s</mi><mo>×</mo><mn mathvariant="normal">1</mn></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076950&type= 22.52133369 17.18733406 ； <math id="M40"><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>S</mi></mrow></msup><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>M</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>×</mo><mi>s</mi><mo stretchy="false">)</mo><mo>×</mo><mn mathvariant="normal">1</mn></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076957&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076966&type= 22.60600090 18.20333290 ; <math id="M41"><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>S</mi></mrow></msup><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">b</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mrow><mrow><msubsup><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mo>/</mo><mrow><msubsup><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mrow><mo stretchy="false">)</mo><mi mathvariant="bold-italic">b</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mrow><mrow><msubsup><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mo>/</mo><mrow><msubsup><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mrow><mo stretchy="false">)</mo><mi mathvariant="bold-italic">b</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>×</mo><mi>s</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">[</mo><mn mathvariant="normal">2</mn><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076971&type= 33.02000046 25.06133461 ； <math id="M42"><msup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>S</mi></mrow></msup><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>S</mi></mrow></msubsup><mi mathvariant="bold-italic">b</mi></mtd><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋱</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>S</mi></mrow></msubsup><mi mathvariant="bold-italic">b</mi></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>×</mo><mi>s</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">[</mo><mi>i</mi><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076962&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076976&type= 33.02000046 16.93333435 ； <math id="M43"><msubsup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>S</mi></mrow></msubsup><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>S</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>S</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">2</mn><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>×</mo><mn mathvariant="normal">1</mn></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076980&type= 26.41600037 11.34533310 ； <math id="M44"><msup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>S</mi></mrow></msup><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mo stretchy="false">[</mo><mi>i</mi><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>×</mo><mn mathvariant="normal">1</mn></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076984&type= 23.36800003 18.11866570 ； <math id="M45"><msubsup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>S</mi></mrow></msubsup><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mtable><mtr><mtd><msubsup><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr></mtable></mtd></mtr><mtr><mtd><msubsup><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi>s</mi><mo>×</mo><mn mathvariant="normal">1</mn></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076990&type= 19.64266586 17.86466599 ； <math id="M46"><mi mathvariant="bold-italic">b</mi><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋱</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi>s</mi><mo>×</mo><mfenced separators="|"><mrow><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45076999&type= 52.66266251 27.00866890 。式中，上标 <math id="M47"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077005&type= 3.72533321 3.72533321 、 <math id="M48"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077012&type= 3.72533321 3.72533321 分别表示BDS-2、BDS-3两系统卫星标识； <math id="M49"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077029&type= 2.20133328 2.79399991 表示不同系统； <math id="M50"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi mathvariant="normal">*</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077017&type= 5.24933338 3.72533321 表示卫星 <math id="M51"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077044&type= 3.38666677 3.72533321 与 <math id="M52"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077043&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077041&type= 3.30200005 3.72533321 作星间差分， <math id="M53"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077044&type= 3.38666677 3.72533321 表示不同系统的基准星； <math id="M54"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 表示不同频率； <math id="M55"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>S</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077051&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077049&type= 4.82600021 4.06400013 为对应残差向量； <math id="M56"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077068&type= 2.53999996 4.57200003 表示不同载波相位观测值对应的频率值； <math id="M57"><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077072&type= 2.53999996 3.72533321 表示不同频率载波相位观测值对应的波长； <math id="M58"><mi mathvariant="normal">Δ</mi><mi mathvariant="bold-italic">N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077076&type= 5.67266655 2.96333337 为站间单差整周模糊度矢量； <math id="M59"><msub><mrow><mi>R</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077084&type= 3.55599999 3.72533321 、 <math id="M60"><msub><mrow><mi>R</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077088&type= 7.11199999 3.72533321 为两个参考站的相对天顶对流层延迟量； <math id="M61"><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077090&type= 2.37066650 3.72533321 、 <math id="M62"><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077092&type= 5.92666674 3.72533321 为非差电离层延迟量； <math id="M63"><mi mathvariant="bold-italic">L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077102&type= 2.53999996 2.79399991 为站星间双差载波观测方程的常数项； <math id="M64"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077097&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077095&type= 1.26999998 2.79399991 、 <math id="M65"><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077116&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077114&type= 7.78933382 2.96333337 表示历元。因为未知数个数过多，方程性质减弱，加入大气历元间约束，将两历元间电离层、对流层双差差值均值设置为0，作为历元间约束条件，以增强方程性质，提高方程的解算速度。约束条件方程为： <math display="block" id="M66"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">o</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">o</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mo>-</mo><mn mathvariant="normal">1</mn></mtd><mtd><munder><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></munder></mtd><mtd><munder><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></munder></mtd><mtd><munder><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></munder></mtd><mtd><munder><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></munder></mtd><mtd><munder><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mo stretchy="false">[</mo><mn mathvariant="normal">2</mn><mi>i</mi><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></munder></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi mathvariant="bold-italic">E</mi></mtd><mtd><mo>-</mo><mi mathvariant="bold-italic">E</mi></mtd><mtd><munder><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></munder></mtd><mtd><munder><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></munder></mtd><mtd><munder><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">[</mo><mn mathvariant="normal">2</mn><mi>i</mi><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></munder></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><munder><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></munder></mtd><mtd><munder><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></munder></mtd><mtd><mi mathvariant="bold-italic">E</mi></mtd><mtd><mo>-</mo><mi mathvariant="bold-italic">E</mi></mtd><mtd><munder><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">[</mo><mn mathvariant="normal">2</mn><mi>i</mi><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></munder></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mtable><mtr><mtd><msub><mrow><mi>R</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>R</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi>I</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msubsup></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr><mtd><msubsup><mrow><mi>I</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msubsup></mtd></mtr><mtr><mtd><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr><mtr><mtd><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msup></mtd></mtr></mtable></mtd></mtr></mtable></mrow></mfenced><mo>-</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077104&type= 140.46200562 28.61733437 式中， <math id="M67"><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077106&type= 6.43466663 3.80999994 为对流层参数的历元间变化量； <math id="M68"><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077109&type= 6.60400009 3.80999994 为电离层参数的历元间变化量； <math id="M69"><mi mathvariant="bold-italic">E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077123&type= 2.87866688 2.79399991 为单位矩阵。由于电离层延迟误差与频率成反比，以及对流层延迟误差使用的是相对天顶对流层延迟量，B1、B2、B3频率采用的是同一个电离层和对流层参数，只对B1频率进行历元间约束便可以加强多频观测数据的直接约束性，因此BDS的多频数据对电离层参数的约束和解算有利。需要注意的是，B1、B3分别表示BDS-2与BDS-3的重叠频率B1I、B3I，B2则分别表示BDS-2的B2b和BDS-3的B2a。当然，B1也可表示BDS-3的B1c频率。联立观测方程和约束条件，采用参数消去法对法方程进行消参处理，利用最小二乘算法进行未知数求解，采用最小二乘模糊度降相关平差法（least square ambiguity decorrelation adjustment，LAMBDA）进行模糊度固定，最终使用Ratio>3的双差整周模糊度代数和为零这一闭合条件对整周模糊度结果进行检验，以提高整周模糊度固定结果的可靠性。Ratio值为在采用搜索算法确定整周模糊度参数的整数值时，产生次最小的单位权方差与最小的单位权方差的比值，一般Ratio值越高，说明模糊度固定解的可靠性越高，采用Ratio值进行检验时，阈值可以取1.5、2、3。 1.2　分类非差误差改正数的计算 2 参考站网多频载波相位的双差整周模糊度确定后，以BDS系统第 <math id="M70"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 个频率双差整周模糊度为例，得到卫星 p 、 q 、 k （ q 为基准星）在参考站 A 、 B 、 C 上的非差整周模糊度和对应的双差整周模糊度关系为： <math display="block" id="M71"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077145&type= 57.91199875 14.56266594 <math display="block" id="M72"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi><mi>q</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi><mi>q</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi><mi>q</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077147&type= 57.91199875 14.56266594 式（6）式（6）、式（7）式（7）中各只有两个线性独立的双差整周模糊度。为分离出单个测站上单颗卫星的非差整周模糊度，定义参考站非差基准整周模糊度、卫星非差基准整周模糊度并分离双差整周模糊度。选择参考站 A 和卫星 q 作为非差基准模糊度，可以设定为常数，该整数值不会影响用户站的定位［ 22 22 ］。将与参考站和基准星相关的非差整周模糊度取值为0，则可得到其他卫星的模糊度： <math display="block" id="M73"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo></mtd></mtr><mtr><mtd><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo></mtd></mtr><mtr><mtd><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup></mtd></mtr></mtable></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077152&type= 61.80666733 24.12999916 <math display="block" id="M74"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi><mi>q</mi></mrow></msubsup><mo>=</mo></mtd></mtr><mtr><mtd><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi><mi>q</mi></mrow></msubsup></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi><mi>q</mi></mrow></msubsup><mo>=</mo></mtd></mtr><mtr><mtd><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi><mi>q</mi></mrow></msubsup><mo>-</mo><mi mathvariant="normal">Δ</mi><mo>∇</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi><mi>q</mi></mrow></msubsup></mtd></mtr></mtable></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077156&type= 61.80666733 24.12999916 式中， <math id="M75"><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077158&type= 6.18066692 4.06400013 、 <math id="M76"><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077136&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077134&type= 6.18066692 4.06400013 、 <math id="M77"><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077140&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077138&type= 6.18066692 4.06400013 、 <math id="M78"><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077162&type= 6.01133299 4.06400013 和 <math id="M79"><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077142&type= 6.01133299 4.06400013 为非差基准模糊度； <math id="M80"><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077178&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077176&type= 7.53533268 4.06400013 、 <math id="M81"><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077167&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077180&type= 7.36600018 4.06400013 、 <math id="M82"><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi><mi>q</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077169&type= 7.53533268 4.06400013 和 <math id="M83"><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>k</mi><mi>q</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077188&type= 7.36600018 4.06400013 为已知的双差整周模糊度，通过上述过程便可由双差整周模糊度转换为非差整周模糊度。确定了每颗卫星对应测站的非差整周模糊度后，就可以计算得出每个参考站对应卫星的非差误差改正数，使用反距离加权内插算法建立起区域高精度的非差误差改正模型，利用参考站网的非差误差改正数计算用户站的非差误差改正数，并对用户站误差进行改正。 A 、 B 、 C 3个参考站对应卫星 p 的非差误差改正数的计算公式为： <math display="block" id="M84"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>F</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>φ</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>N</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><mi>c</mi><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>-</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfenced><mo>+</mo><msup><mrow><mi>O</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>-</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>-</mo><msup><mrow><mi>σ</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi>F</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>φ</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>N</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><mi>c</mi><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>-</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfenced><mo>+</mo><msup><mrow><mi>O</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>-</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>-</mo><msup><mrow><mi>σ</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi>F</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>φ</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><mi>c</mi><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>-</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfenced><mo>+</mo><msup><mrow><mi>O</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>-</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>-</mo><msup><mrow><mi>σ</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077192&type= 121.49666595 18.62666702 式中， <math id="M85"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077195&type= 5.84200001 3.97933316 、 <math id="M86"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077198&type= 5.58799982 3.97933316 、 <math id="M87"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077182&type= 5.67266655 3.97933316 分别为各参考站上卫星 p 的非差误差改正数。从式（10）式（10）中可以看出，非差误差改正数中主要包含了卫星和接收机钟差、卫星和接收机端硬件延迟、电离层延迟误差、对流层延迟误差、轨道误差等。网络RTK中可以将综合非差误差改正数直接用于流动站的误差改正，但是对于长距离情况，综合误差的处理方式得出的改正信息并不准确，需要按照误差性质对非差误差改正信息进行分类处理。分类区域误差内插方法的示意图如图1 图1 所示， U 为流动站， I 为电离层穿刺点。在参考站和流动站所在的地球表面上对非差非色散性误差进行内插计算，在高度为中心电离层高度的内插平面上进行流动站非差电离层延迟误差的内插计算［ 17 17 ］。只需要计算出参考站网中各分类误差，便可以按照反距离加权内插算法得到流动站相应的分类误差改正值为： <math display="block" id="M88"><msubsup><mrow><mi>S</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>d</mi></mrow><mrow><mi>U</mi><mi>A</mi></mrow></msub></mtd><mtd><msub><mrow><mi>d</mi></mrow><mrow><mi>U</mi><mi>B</mi></mrow></msub></mtd><mtd><msub><mrow><mi>d</mi></mrow><mrow><mi>U</mi><mi>C</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>S</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi>S</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi>S</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077220&type= 45.80466461 13.54666710 式中， <math id="M89"><msubsup><mrow><mi>S</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077224&type= 5.50333309 3.97933316 为流动站 p 卫星第 <math id="M90"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 个频率观测值的分类误差； <math id="M91"><msub><mrow><mi>d</mi></mrow><mrow><mi>U</mi><mi>A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077232&type= 5.41866684 3.72533321 、 <math id="M92"><msub><mrow><mi>d</mi></mrow><mrow><mi>U</mi><mi>B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077215&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077212&type= 5.24933338 3.72533321 、 <math id="M93"><msub><mrow><mi>d</mi></mrow><mrow><mi>U</mi><mi>C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077239&type= 5.33400011 3.72533321 分别为 A 、 B 、 C 3个参考站的加权系数； <math id="M94"><msubsup><mrow><mi>S</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077216&type= 5.41866684 3.97933316 、 <math id="M95"><msubsup><mrow><mi>S</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077244&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077251&type= 5.16466665 3.97933316 、 <math id="M96"><msubsup><mrow><mi>S</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077254&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077247&type= 5.24933338 3.97933316 为各参考站 p 卫星第 <math id="M97"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 个频率观测值的分类误差。每个频率观测值的分类误差改正数生成方式与第 <math id="M98"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 个频率误差改正数的生成过程相同。非色散性误差与电离层延迟误差内插方法一致，只是进行内插的平面不同，电离层延迟误差的加权系数根据穿刺点位置获得。 1.3　流动站多频载波相位整周模糊度解算及定位 2 以流动站 U 观测 p 、 q 卫星（ q 为基准星）第 <math id="M99"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 个频率观测数据为例，流动站第 <math id="M100"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 个频率载波相位非差观测方程为： <math display="block" id="M101"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>=</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>⋅</mo><mi>X</mi><mo>+</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>N</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>-</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>T</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo><mi>c</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>U</mi></mrow></msub><mo>-</mo><mi>c</mi><msup><mrow><mi>t</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>U</mi></mrow></msub><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>+</mo><msubsup><mrow><mi>ε</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>⋅</mo><mi>X</mi><mo>+</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>N</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>O</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>-</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>T</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><mi>c</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>U</mi></mrow></msub><mo>-</mo><mi>c</mi><msup><mrow><mi>t</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>U</mi></mrow></msub><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><msubsup><mrow><mi>ε</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077275&type= 66.37866211 21.33600044 通过非差误差改正数进行流动站误差改正之后，能够消除观测值中的卫星钟差及卫星硬件延迟，基本消除或大大削弱了对流层延迟误差、电离层延迟误差和卫星轨道误差等误差，得到误差方程： <math display="block" id="M102"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>=</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>⋅</mo><mi>X</mi><mo>+</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>N</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>t</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>U</mi></mrow></msub><mo>-</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>U</mi></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>⋅</mo><mi>X</mi><mo>+</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>N</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>t</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>U</mi></mrow></msub><mo>-</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>U</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077285&type= 60.02866364 20.91266632 式中， <math id="M103"><msubsup><mrow><mi>W</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077286&type= 7.19666624 3.97933316 、 <math id="M104"><msubsup><mrow><mi>W</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077298&type= 7.19666624 3.97933316 表示通过多个参考站获得的流动站第 <math id="M105"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077311&type= 1.10066664 2.79399991 个频率的分类非差误差改正数之和； <math id="M106"><msub><mrow><mi>t</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>U</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077304&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077314&type= 5.24933338 3.72533321 表示残余接收机钟差； <math id="M107"><msub><mrow><mi>σ</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>U</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077318&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077305&type= 5.84200001 3.72533321 表示残余接收机端的载波相位硬件延迟；下标 R ， U 表示流动站与参考站做站间差分。为了消除这部分残余误差，利用式（13）式（13）进行星间差可得： <math display="block" id="M108"><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>V</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><mo>-</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mo>⋅</mo><mi>X</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mo stretchy="false">{</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>-</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mo stretchy="false">(</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077323&type= 67.14067078 15.83266735 其中， <math id="M109"><mtext> </mtext><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>h</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><mo>-</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>q</mi></mrow></msup></mtd></mtr><mtr><mtd><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>-</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077338&type= 70.01933289 15.83266735 。可得流动站载波相位模糊度和位置参数解算的误差方程为： <math display="block" id="M110"><msubsup><mrow><mi>V</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup><mo>⋅</mo><mi>X</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>p</mi><mi>q</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077329&type= 59.85933304 4.65666676 与参考站相对应地，设BDS在流动站观测 s +1颗卫星（基准星序号为1），则多频多系统误差方程的矩阵形式为： <math display="block" id="M111"><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>U</mi><mo>,</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msubsup><mtext> </mtext></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>U</mi><mo>,</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msup></mtd><mtd><msup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">X</mi></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="normal">B</mi><mn mathvariant="normal">3</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077331&type= 61.72200012 31.24200058 其中， <math id="M112"><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><mi>S</mi></mrow></msubsup><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>V</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mrow></msubsup></mtd><mtd><mo>⋯</mo></mtd><mtd><msubsup><mrow><mi>V</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077347&type= 42.67200089 5.24933338 ； <math id="M113"><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>S</mi></mrow></msup><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mrow></msubsup></mtd><mtd><msubsup><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mrow></msubsup></mtd><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd><mtd><msubsup><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr></mtable></mtd><mtd><mtable columnalign="left"><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr></mtable></mtd><mtd><mtable columnalign="left"><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr></mtable></mtd></mtr><mtr><mtd><msubsup><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd><mtd><msubsup><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>×</mo><mi>s</mi><mo stretchy="false">)</mo><mo>×</mo><mn mathvariant="normal">3</mn></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077353&type= 42.33333206 39.87799835 ； <math id="M114"><msup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>S</mi></mrow></msup><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>S</mi></mrow></msubsup><mi mathvariant="bold-italic">b</mi></mtd><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋱</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>⋯</mo></mtd><mtd><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>S</mi></mrow></msubsup><mi mathvariant="bold-italic">b</mi></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>×</mo><mi>s</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>i</mi><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077368&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077356&type= 33.02000046 16.93333435 ； <math id="M115"><mi mathvariant="bold-italic">X</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077373&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077371&type= 12.78466606 14.39333248 ； <math id="M116"><msup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>S</mi></mrow></msup><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,1</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,1</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr></mtable></mtd></mtr><mtr><mtd><msubsup><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mo stretchy="false">[</mo><mi>i</mi><mo>×</mo><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>×</mo><mn mathvariant="normal">1</mn></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077376&type= 23.36800003 39.87799835 ； <math id="M117"><msup><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>S</mi></mrow></msup><mo>=</mo><munder><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>L</mi></mrow><mrow><mi>U</mi><mo>,</mo><mi>i</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>×</mo><mi>s</mi><mo stretchy="false">)</mo><mo>×</mo><mn mathvariant="normal">1</mn></mrow></munder></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077382&type= 19.64266586 30.14133072 。式中， <math id="M118"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077393&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077387&type= 1.26999998 2.79399991 、 <math id="M119"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077394&type= 2.87866688 2.79399991 和 <math id="M120"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077399&type= 1.86266661 2.79399991 分别为位置参数的系数； <math id="M121"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077403&type= 2.20133328 2.79399991 、 <math id="M122"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077408&type= 2.11666679 3.72533321 和 <math id="M123"><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45077414&type= 1.77800000 2.79399991 分别表示WGS-84坐标系下的位置参数。流动站进行非差误差改正后，与参考站相比未知参数除了新增的3个位置参数，就只剩下整周模糊度参数。首先通过参数消去的最小二乘方法估计得到流动站多频数据模糊度的浮点解和协方差矩阵，再将其代入LAMBDA算法中进行单个频率上的模糊度搜索和固定，最后将整周模糊度代入观测方程进行流动站位置解算。 2　算例与分析 1 本文采用自编的长距离非差网络RTK算法程序对长基线观测数据分别进行BDS-2、BDS-3、BDS-2+BDS-3双系统多频非差网络RTK解算，并与GPS网络RTK结果进行对比分析。实验使用中国某省CORS网2022年年积日第140天观测数据，采样间隔为1 s，观测时长为24 h，卫星截止高度角设置为15°。观测数据有 A 、 B 、 C 、 D 4个参考站和两个流动站 U 1 、 U 2 ，参考站间距如表1 表1 所示，其中基线长 AB 为167 km， BC 为105 km， CA 为161 km， AD 为133 km， DC 为178 km，两流动站距离 U 1 U 2 为69 km。本文将参考站分为参考站网1（ A 、 D 、 C ）和参考站网2（ A 、 B 、 C ）两组分别对 U 1 、 U 2 两个流动站进行定位解算，测站分布如图2 图2 所示。基线长度/km 基线长度km AB 167 AD 133 BC 105 DC 178 CA 161 U 1 U 2 69 两个流动站位于1个区域，卫星可见数差异不大，以流动站 U 1 的卫星可见数为例，如图3 图3 所示。BDS-2和BDS-3的卫星数都在8~9颗左右，但BDS-2卫星数较多，GPS的卫星数在7颗左右，均能满足解算要求。根据流动站 U 1 与 U 2 的观测数据以及广播星历计算出各个系统的位置精度强弱度（position dilution of precision，PDOP）时间序列，如图4 图4 所示，BDS与GPS的PDOP值统计信息如表2 表2 所示。一般地，PDOP值小于3为比较理想状态［ 23 23 ］，由表2 表2 可见BDS-2和BDS-3的PDOP值都小于3且平均值小于2；GPS的PDOP值小于其他两个系统，BDS-3优于BDS-2；BDS-2的PDOP值为1~2.3，BDS-3的PDOP值为0.77~1.89，GPS的PDOP值为0.59~1.05，平均值分别约为1.59、1.13、0.72。各个系统的PDOP值均小于3，满足定位时需要的理想状态。定位系统流动站最大值最小值平均值 2 BDS-2 U 1 2.215 7 1.063 8 1.613 8 U 2 2.215 5 1.000 3 1.575 1 2 BDS-3 U 1 1.882 2 0.867 9 1.187 7 U 2 1.496 7 0.770 9 1.068 9 2 GPS U 1 1.045 4 0.602 6 0.743 4 U 2 0.847 6 0.591 0 0.693 2 1）参考站多频载波相位整周模糊度的解算。参考站网多频载波相位整周模糊度的准确解算是网络RTK算法的核心问题，也是本文双差模糊度转换为非差模糊度从而计算综合非差误差改正数的关键。本文在模糊度固定前需要得到模糊度浮点解，利用模糊度搜索法以实现模糊度的快速准确固定。以长度最长的参考站网1中基线 DC 为例，载波相位整周模糊度解算时各系统部分卫星双差模糊度浮点解的收敛过程见图5 图5 。由于对残余大气延迟误差进行估计分离，从各个卫星双差模糊度浮点解的收敛过程不难看出，模糊度浮点解一段时间后能稳定在准确的整周模糊度附近，使用LAMBDA算法搜索固定解时可迅速提高获得整周模糊度准确解的效率，计算量小且固定成功率更高。整周模糊度搜索解算的Ratio值如图6 图6 所示，部分历元Ratio值出现跳变是因为基准卫星发生变化，变换基准卫星时，模糊度已经固定，可对当前弧段观测值进行基准卫星的变换，继续使用，并进行随后历元的双差载波相位整周模糊度解算。将闭合参考站间的双差模糊度和为0作为模糊度的约束条件之一，附加到模糊度解算的观测方程中，提高模糊度搜索的效率和成功率。对两参考站网观测数据BDS-2、BDS-3多频和GPS双频载波相位整周模糊度固定成功率进行统计（其中，GPS系统双频观测值的解算作为比较对象），统计结果如表 3 3 ~ 5 5 所示。从表3~5中能够得出两个参考站网的模糊度固定成功率都很高，参考站网2略优于参考站网1。BDS-3的模糊度固定率最优，BDS-2与GPS相当，BDS-3的模糊度固定率优于BDS-2，这主要是因为BDS-3在系统性能上有一定的提升［ 2 2 ］。BDS-2、BDS-3多频载波相位整周模糊度固定成功率都超过了98%，特别是BDS-3的成功率超过了99%。这体现了多频数据在模糊度固定成功率方面较双频数据更有优势。卫星号参考站网观测历元数固定历元数未固定历元数固定成功率/% 2 C01 1 86 400 85 146 1 254 98.55 2 86 400 85 274 1 126 98.69 2 C03 1 86 400 85 144 1 256 98.55 2 86 400 85 274 1 126 98.70 2 C08 1 32 143 31 664 479 98.51 2 32 143 31 779 364 98.87 2 C11 1 24 583 24 583 0 100 2 24 583 24 582 1 99.99 2 C13 1 37 354 36 783 571 98.47 2 44 599 43 922 677 98.48 卫星号参考站网观测历元数固定历元数未固定历元数固定成功率/% 2 C24 1 26 327 26 315 12 99.95 2 26 327 26 318 9 99.97 2 C27 1 18 070 17 916 154 99.15 2 18 070 18 001 69 99.62 2 C32 1 19 452 19 452 0 100 2 19 452 19 452 0 100 2 C41 1 18 278 18 278 0 100 2 18 278 18 278 0 100 2 C45 1 12 211 12 211 0 100 2 12 211 12 211 0 100 卫星号参考站网观测历元数固定历元数未固定历元数固定成功率/% 2 G02 1 18 660 18 433 227 98.78 2 18 660 18 467 193 98.97 2 G06 1 17 977 17 664 313 98.26 2 17 977 17 707 270 98.50 2 G12 1 23 808 23 406 402 98.31 2 23 808 23 444 364 98.47 2 G21 1 11 681 11 530 151 98.71 2 11 681 11 562 119 98.98 2 G31 1 14 226 14 006 220 98.45 2 19 106 18 902 204 98.93 2）非差误差改正数的计算。首先将准确确定的双差整周模糊度进行转换得到非差整周模糊度，然后获得每个参考站对应的非差综合误差改正数，以C01、C39、G20卫星在 A 、 B 、 C 、 D 4个参考站上第一个频率连续可见的时段内的非差综合误差改正数为例（见图 7 7 ~ 9 9 ）。可以看出，每个卫星的非差误差改正数呈线性变化，并且同一颗卫星在4个参考站上的变化趋势相近，符合其变化规律。对于长距离网络RTK，大气延迟误差相关性减弱会降低定位结果精度。为了保证由反距离加权内插算法获得的流动站非差改正数的精度和可靠性，对综合非差误差改正数进行分类处理，选取每颗卫星其中两个频率的综合非差误差改正数，分别进行无几何距离组合和无电离层组合计算获得对应的第一频率非差电离层延迟误差和非色散性误差。BDS-2中C01（GEO卫星）、C10（IGSO卫星）、C14（MEO卫星），BDS-3中C21（MEO卫星）、C39（IGSO卫星）、C42（MEO卫星），GPS中G01、G20、G30（MEO卫星）在4个参考站上对应的误差计算结果如图 10 10 ~ 12 12 所示。由于不同类型卫星所处轨道不同，所以卫星可见时段也不相同。从图 10 10 ~ 12 12 中可以看出非差电离层延迟误差前后历元间变化比较平缓，同一颗卫星4个参考站的非差电离层延迟误差变化趋势基本一致。对于非色散性误差，历元之间的非色散性误差变化较小，符合正常情况下对流层变化的规律，也能够反映出载波相位整周模糊度固定是正确的。非色散性误差本身比较稳定，因此同一颗卫星4个参考站的非色散性误差在变化趋势方面也基本一致。 3）定位精度。将非差误差改正数代入流动站观测方程，准确解算出流动站载波相位整周模糊度和流动站的准确位置。流动站初次实现厘米级定位的收敛时间统计结果见表6 表6 。 U 1 、 U 2 两个流动站各个系统收敛时间差异不大， U 2 的收敛时间整体较短。但是能够明显看出BDS-2的收敛时间最长，而BDS-3的收敛时间最短，GPS次之，这主要是得益于亚太地区能够观测到的BDS卫星较GPS更多，BDS-3的多频观测数据有利于整周模糊度的快速确定，相较于GPS双频数据，收敛时间缩短了1~2 min左右，并且BDS-3较BDS-2在各个方面都有显著提升［ 2 2 ］。同时给出了参考站网载波相位整周模糊度解算耗时结果（见表7 表7 ），其结果与收敛时间相对应，依然是参考站网2优于参考站网1，BDS-3的载波相位整周模糊度解算耗时最少。定位系统流动站收敛时间/s 2 BDS-2 U 1 424 U 2 415 2 BDS-3 U 1 106 U 2 67 2 GPS U 1 171 U 2 125 定位系统参考站网模糊度初始化耗时/s 每小时数据平均耗时/s 2 BDS-2 1 400 412 2 359 364 2 BDS-3 1 202 215 2 178 188 2 GPS 1 303 297 2 286 288 流动站 U 1 、 U 2 观测数据GPS、BDS-3、BDS-2以及BDS-2+BDS-3的定位结果偏差如图 13 13 、 14 14 所示。从定位误差时间序列和统计信息可知，流动站GPS、BDS观测值的系统误差基本被消除。结合上文给出的卫星数情况，虽然BDS-2的卫星数最多，但其定位结果并不是最好的，这主要是由于BDS-2观测到的卫星中GEO卫星有4~5颗，GEO卫星运行速度较慢，星座空间几何构型较差，不利于高采样率观测数据的连续解算。相同情况下BDS-3和GPS系统可见卫星数分别为7~8颗，其中BDS-3的定位偏差总体最小且较稳定，GPS定位偏差优于BDS-2。相对于BDS-2，BDS-3卫星星座更加合理，系统性能更优，所以BDS-3虽然可见卫星数少于BDS-2，但是定位精度明显高于BDS-2，GPS网络RTK结果与BDS-3相当。由于卫星数量的增加，网络RTK数据解算精度得到了提升，可以看出BDS-2+BDS-3的精度超过包括BDS-3在内的任意单系统数据解算精度，BDS-2在加入BDS-3后，东（east， E）、北（north， N）、天（up， U） 3个方向精度分别平均升了26.4%、39.5%、29.5%。同时，相较于GPS单系统，BDS-2+BDS-3多频数据的定位结果在E、N、U 3个方向分别平均提升了18.1%、26.8%、25.7%。为了方便分析，对各解算方式定位偏差值计算均方根（root mean square，RMS）并进行统计，结果如表8 表8 所示。由表8 表8 可以看出，除BDS-2以外，其他解算方式RMS均在2 cm之内，而且BDS-3网络RTK精度明显高于BDS-2，与GPS定位效果相近。双系统定位由于卫星数增加，多频率冗余观测值增加，进而定位精度明显提高。可以看出参考站网2的定位结果各系统间的差异与参考站网1相似，但其定位结果精度更高。通过单系统以及组合系统解算结果精度、卫星可见数、PDOP值、模糊度固定成功率等方面可以看出BDS-3相较于BDS-2性能取得了很大的提升，其网络RTK算法在定位精度、稳定性等方面都有着显著提高，已经达到与GPS相当甚至更优的精度，并且在多频模式下，模糊度固定成功率、定位结果收敛时间、定位精度均有一定提升。 2 方向 2 流动站 4 RMS BDS-2 BDS-3 GPS BDS-2+BDS-3 2 E U 1 0.013 0.011 0.012 0.009 U 2 0.011 0.009 0.009 0.008 2 N U 1 0.017 0.012 0.014 0.010 U 2 0.011 0.007 0.008 0.006 2 U U 1 0.021 0.018 0.019 0.014 U 2 0.018 0.015 0.016 0.012 为了对解算结果进行检验，图15 图15 给出了两组观测数据BDS-2、BDS-3、GPS中的部分卫星定位结果残差值，空白部分是由于这个时段未能观测到此颗卫星。残差值是通过最终的流动站定位结果以及中间过程中估计的误差参数代入观测方程中得到的。BDS-2、BDS-3两个系统的部分卫星观测时长比GPS的部分卫星更长，C01作为GEO卫星在全时段都能观测，但其残差值在定位前期有明显的波动，分布不均匀，这与GEO卫星的轨道精度较低有关，虽然GEO卫星对地面的覆盖范围广，但是卫星轨道精度也较低。BDS-3、GPS的卫星观测时长不长，但其残差值总体趋于0附近，符合残差值的变化规律。3个系统的残差值由于每颗卫星观测时长不同而有所差异，但是都能保持在2 cm以内，BDS-3、GPS更加稳定。 4　结语 1 本文研究了一种利用多频观测数据的长距离BDS-3/BDS-2非差网络RTK方法。使用CORS网数据进行BDS-2、BDS-3、GPS、BDS-2+BDS-3 4种解算方式的定位解算和精度分析。实验结果表明，采用本文的方法可以实现参考站整周模糊度的准确快速固定，建立非差区域误差改正模型为流动站提供准确的分类非差误差改正数。BDS-3在卫星数、PDOP值、模糊度固定成功率、稳定性、收敛时间、定位精度等方面总体优于其他两个系统，以GPS双频数据为参考，BDS-3多频数据在模糊度固定成功率和收敛时间方面有一定优势，BDS-3/BDS-2单系统、双系统多频数据定位结果与坐标已知值在E、N、U坐标分量方向的偏差均达到厘米级水平，能实现厘米级实时动态定位。参考文献 Yang Yuanxi ， Yang Cheng ， Ren Xia . 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Functions of Retired GEO Communication Satellites in Improving the PDOP Value of CAPS 10.13203/j.whugis20220797 P228 A 1671-8860（2023）07-1146-14 祝会忠,张杰,赵齐乐，等.BDS-3/BDS-2多频数据的长距离非差网络RTK算法[J].武汉大学学报（信息科学版）,2023,48(7):1146-1159.DOI: 10.13203/j.whugis20220797 http://dx.doi.org/10.13203/j.whugis20220797 祝会忠,张杰,赵齐乐，等.BDS-3/BDS-2多频数据的长距离非差网络RTK算法[J].武汉大学学报（信息科学版）,2023,48(7):1146-1159.DOI:10.13203/j.whugis20220797 ZHU Huizhong,ZHANG Jie,ZHAO Qile,et al.An Algorithm of Long Distance Undifference Network RTK with BDS-3/BDS-2 Multi-frequency Data[J].Geomatics and Information Science of Wuhan University,2023,48(7):1146-1159.DOI: 10.13203/j.whugis20220797 http://dx.doi.org/10.13203/j.whugis20220797 ZHU Huizhong,ZHANG Jie,ZHAO Qile,et al.An Algorithm of Long Distance Undifference Network RTK with BDS-3/BDS-2 Multi-frequency Data[J].Geomatics and Information Science of Wuhan University,2023,48(7):1146-1159.DOI:10.13203/j.whugis20220797 国家自然科学基金（42030109，42074012）；卫星导航系统与装备技术国家重点实验室开放基金（CEPNT-2018KF-13）；辽宁省“兴辽英才计划”（XLYC2002101，XLYC2008034，XLYC2002098）。 2023-04-02 2023-07-05 2023-06-29 39072098 武汉大学学报（信息科学版） 7 48

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An Algorithm of Long Distance Undifference Network RTK with BDS-3/BDS-2 Multi-frequency Data

DOI：10.13203/j.whugis20220797