空间激光链路时频传递算法及仿真分析

陈俊平; 王彬; 孟文东; 张晶宇

doi:10.13203/j.whugis20230165

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空间激光链路时频传递算法及仿真分析

学术论文 | 更新时间：2023-06-29

- 空间激光链路时频传递算法及仿真分析
- Simulation Analysis of Time and Frequency Transfer Algorithm for Space Laser Link
- 武汉大学学报（信息科学版）
- 作者机构：
  
  1.中国科学院上海天文台，上海，200030
  2.中国科学院大学，北京，100049
  3.上海市空间导航与定位技术重点实验室，上海，200030
- 作者简介：
  
  陈俊平，博士，研究员，主要从事卫星导航定位等方面的研究。junping@shao.ac.cn
  
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  王彬，博士，副研究员。binw@shao.ac.cn
  
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- 基金信息：
  
  国家自然科学基金(41904034);上海市优秀学术带头人计划(20XD1404500)
- DOI：10.13203/j.whugis20230165
  中图分类号： P228
- 纸质出版日期：2023-07-05，
  
  收稿日期：2023-05-05，
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引用本文
陈俊平,王彬,孟文东，等.空间激光链路时频传递算法及仿真分析[J].武汉大学学报（信息科学版）,2023,48(7):1082-1088.

CHEN Junping,WANG Bin,MENG Wendong,et al.Simulation Analysis of Time and Frequency Transfer Algorithm for Space Laser Link[J].Geomatics and Information Science of Wuhan University,2023,48(7):1082-1088.
陈俊平,王彬,孟文东，等.空间激光链路时频传递算法及仿真分析[J].武汉大学学报（信息科学版）,2023,48(7):1082-1088. DOI：10.13203/j.whugis20230165

CHEN Junping,WANG Bin,MENG Wendong,et al.Simulation Analysis of Time and Frequency Transfer Algorithm for Space Laser Link[J].Geomatics and Information Science of Wuhan University,2023,48(7):1082-1088. DOI：10.13203/j.whugis20230165

空间激光链路时频传递算法及仿真分析 Simulation Analysis of Time and Frequency Transfer Algorithm for Space Laser Link 1 2 3 first-author 陈俊平，博士，研究员，主要从事卫星导航定位等方面的研究。junping@shao.ac.cn 陈俊平，博士，研究员，主要从事卫星导航定位等方面的研究。junping@shao.ac.cn junping@shao.ac.cn 1 2 3 corresp 王彬，博士，副研究员。binw@shao.ac.cn 王彬，博士，副研究员。binw@shao.ac.cn binw@shao.ac.cn 1 2 1 中国科学院上海天文台，上海，200030 中国科学院上海天文台，上海，200030 Shanghai Astronomical Observatory,Chinese Academy of Sciences, Shanghai 200030, China Shanghai Astronomical Observatory,Chinese Academy of Sciences, Shanghai 200030, China 中国科学院大学，北京，100049 中国科学院大学，北京，100049 University of Chinese Academy of Sciences,Beijing 100049, China University of Chinese Academy of Sciences,Beijing 100049, China 上海市空间导航与定位技术重点实验室，上海，200030 上海市空间导航与定位技术重点实验室，上海，200030 Shanghai Key Laboratory of Space Navigation and Positioning Techniques,Shanghai Astronomical Observatory,Chinese Academy of Sciences, Shanghai 200030, China Shanghai Key Laboratory of Space Navigation and Positioning Techniques,Shanghai Astronomical Observatory,Chinese Academy of Sciences, Shanghai 200030, China 利用空间高精度时频系统提供的超高精度时间频率信号，可以开展一系列空地时频传递和基础物理实验，支撑相对论及相关理论的高精度检验。针对空间激光链路时频传递算法进行了理论推导及仿真分析。首先，从星地激光双向时间比对理论公式出发，对比分析了X型星地双向时间比对与Lambda型星地双向时间比对的优缺点。其次，针对空间站轨道高度讨论了相对论效应对坐标时与原时转换的影响。最后，讨论了空地时频比对数据不连续性对空间站高精度原子钟稳定性评估的影响。结果表明：(1)基于激光测距方式的Lambda型双向时间比对可以抵消一阶多普勒对上下行距离项差异的影响，且Sagnac项影响的增大可忽略；(2)星地时间比对中坐标时和原时转换的相对论项对空间站位置的速度精度提出了高要求，为实现1×10 -18 量级的频率偏差比对精度，空间站地心距精度要求为1 dm，速度精度要求为0.1 mm/s；(3)受空间站对地可见性影响，仅利用国内测站无法通过星地链路进行中短期原子钟稳定性的评估。 Objectives 2 With the ultra-high precision time and frequency signals provided by space high precision time and frequency system, series of space-to-ground time and frequency transfer and fundamental physics experiments can be carried out to support high precision validation of relativity and related theories. Methods 2 First, theoretical derivation and simulation analysis are conducted on the time and frequency transfer algorithm of space laser links. Theoretical formula of space-to-ground laser two way time transfer are reviewed, and the advantages and disadvantages of X-type and Lambda-type approaches are compared and analyzed. Then, the impact of relativistic effects on the conversion between proper time and coordinate time is discussed for the specific orbital height of China space station. Finally, the impact of space-to-ground time and frequency transfer discontinuity on the stability evaluation of high-precision atomic clocks onboard space stations is discussed. Results 2 The results show that: (1) Lambda-type two-way time transfer based on satellite laser ranging can make up the distance difference of first-order Doppler effect on the uplink and downlink, and increase of Sagnac effect can be ignored. (2) The relativistic effect of the conversion between proper time and coordinate time for satellite ground time transfer makes high demands on the velocity accuracy of space station positions, in order to achieve 1×10 －18 frequency transfer accuracy, geocentric distance accuracy of 1 dm and speed accuracy of 0.1 mm/s of space station is required; (3) Influenced by the visibility of space stations to the ground, it is not possible to evaluate the medium term stability of atomic clocks with only the observation data of domestic stations satellite for space-to-ground links. Conclusions 2 High precision navigation solution is important to ensure high-precision time and frequency transfer between space and ground. How to optimize navigation algorithms and use various payloads onboard to achieve high-precision position and velocity solution is a key issue that needs to be studied. 中国空间站激光双向时频传递广义相对论高精度时频信号 China space station two-way laser time and frequency transfer general relativity high precision time and frequency signal 光钟的频率稳定度和准确度均已优于1×10 －18 水平［ 1 1 - 2 2 1-2 ］，有利于国际单位制秒长的重新定义［ 3 3 - 4 4 3-4 ］。光钟不仅可用于相对论大地测量［ 5 5 ］，也可用于空间任务［ 6 6 - 7 7 6-7 ］开展基础物理学以及天文学研究。空间原子钟组计划（atomic clock ensemble in space，ACES）是欧洲空间局和法国空间局负责实施的基于国际空间站微重力环境下的空间微波原子钟实验验证计划，核心载荷是一台激光冷原子铯钟和一台空间主动型氢钟，利用双向微波链路预期实现100 ps的空地时间同步，以及1×10 －16 水平的频率比对准确度，此外利用激光链路预期实现50 ps准确度、单弧段4 ps精度的空地原子钟比对［ 8 8 ］。空间光钟（space optical clock，SOC）计划是ACES计划的后续，将在空间站上搭载光钟，其频率准确度优于5×10 －17 ，频率稳定度优于1×10 －15 τ －0.5［ 9 9 ］，其中τ表示平滑时间。中国空间站梦天实验舱于2022-10-31发射升空，其中高精度时频实验柜科学实验系统包括冷原子铷钟、主动氢原子钟和锶原子光钟等不同特性的原子钟，在空间微重力环境下时频基准实现精度优于地面时频基准1~2个量级［ 10 10 - 11 11 10-11 ］。中国空间站原子钟组能连续产生秒稳定度优于2×10 －15 、日稳定度和不确定度为8×10 －18 的高精度时频信号，稳定度可用谱密度分别为1.28×10 －35 、8.00×10 －30 的调频白噪声和调频闪变噪声近似［ 12 12 ］，地面概念验证模型稳定度符合2.6×10 －15 τ －0.5 变化［ 7 7 ］。中国空间站高精度时频实验柜科学实验系统与ACES计划均采用微波双向和激光双向两种链路实现空地原子钟时频比对，不同之处在于ACES上行链路采用Ku波段，两条下行链路采用S波段和Ku波段计算电离层大气时延，而中国空间站下行链路包含3路信号，两路信号在Ku波段，一路在频率更高的Ka波段，在电离层延迟修正以及天线相位中心修正方面更具优势［ 10 10 ， 12 12 ］。中国空间站星地激光时频传递时间稳定度指标优于0.1 ps/300 s、1 ps/1 d，采用10 kHz高重频测量提升激光时频比对的短期稳定性［ 11 11 ］，采用温漂补偿实现高精度单光子激光时间传递［ 13 13 ］。利用空间高精度时频系统提供的超高精度时间频率信号，可以开展一系列空地时频传递和基础物理实验，如引力红移测量、精细结构常数变化探测等，支撑相对论及相关理论的高精度检验［ 14 14 ］。科学目标的实现离不开空地高精度时频比对链路，高精度的时频比对数据是众多基础物理实验的关键。ACES观测模型时间比对精度需求为1 ps，相对频率偏差准确度需求为1×10 －16 ，除了要考虑相对论效应外，还应顾及非惯性力（如大气阻力、太阳辐射压等）以及空间站动态环境（如姿态变化）的影响［ 8 8 ］。ps量级的时间比对要求需要在广义相对论框架下建立卫星坐标时计时与时间比对模型［ 15 15 ］，讨论卫星轨道摄动对时间比对的影响［ 16 16 ］。轨道误差是空间站共视法时间比对的主要误差源，其影响在几百ps量级，通过选择特定的星站几何关系可将百ps量级的站间时间比对误差降低到几十ps ［ 17 17 ］。 ACES计划以及中国空间站时频比对系统在微波和激光时频传递链路方面做了很多分析工作［ 8 8 ， 18 18 - 20 20 18-20 ］。但针对性能指标更高的空间激光时频比对实验，需要针对实际情况开展进一步的分析与设计，比如频率不确定度1×10 －18 量级的单向Λ构型链路的相对论分析［ 21 21 ］、特定的微波时频链路系统体制［ 10 10 ］，以及高测量频率（10 kHz）的卫星激光双向时频比对［ 11 11 ］等。为此，本文针对空间激光链路时频传递算法开展进行了理论推导及仿真分析。 1　星地激光双向时间比对 1 星地激光双向时间比对原理如图1 图1 所示，地面站在本地时刻 <math id="M1"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079206&type= 2.28600001 3.72533321 向卫星发射激光脉冲，该脉冲在卫星时刻 <math id="M2"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079246&type= 2.28600001 3.72533321 到达卫星后被反射，地面站在 <math id="M3"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079226&type= 2.28600001 3.72533321 时刻接收到回波信号，综合卫星下传的 <math id="M4"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079246&type= 2.28600001 3.72533321 时刻信息，从而解算得到星地钟差［ 22 22 ］。假定地面时 T A 与卫星时 T B 之间的时差为Δ T ，本地时刻 <math id="M5"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079206&type= 2.28600001 3.72533321 与地面秒脉冲之间的时间间隔为 <math id="M6"><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078693&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078691&type= 4.23333359 3.72533321 ，卫星时刻 <math id="M7"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079246&type= 2.28600001 3.72533321 与卫星秒脉冲之间的时间间隔为 <math id="M8"><msub><mrow><mi>T</mi></mrow><mrow><mi>S</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078685&type= 3.89466691 3.72533321 ，上行信号光行时用 <math id="M9"><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078699&type= 6.68866634 3.72533321 表示，下行信号光行时用 <math id="M10"><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">w</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078703&type= 9.14400005 3.72533321 表示，则 <math id="M11"><mi mathvariant="normal">Δ</mi><mi>T</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078706&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078719&type= 34.29000092 3.72533321 。 1.1　星地单向时间比对 2 一阶后牛顿假设下，坐标时在渐进平直空间坐标系下（当地心距 <math id="M12"><mi>r</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078726&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078725&type= 9.90600014 2.79399991 时，空间度规分量 <math id="M13"><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078716&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078714&type= 11.43000031 4.57200003 ， <math 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mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mrow><mrow><mn mathvariant="normal">2</mn><mi>U</mi></mrow><mo>/</mo><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></mrow></mfenced><mfenced separators="|"><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi><mi mathvariant="normal">d</mi><msup><mrow><mi>ϕ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078736&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078734&type= 60.11333084 12.10733318 式中， <math id="M16"><mi mathvariant="normal">d</mi><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078754&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078737&type= 3.13266683 2.79399991 为相对论线元； <math id="M17"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079415&type= 1.26999998 2.79399991 为坐标时； <math id="M18"><mi>r</mi><mi mathvariant="normal">、</mi><mi>θ</mi><mi mathvariant="normal">、</mi><mi>ϕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078742&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078739&type= 9.05933285 3.64066648 分别为球坐标直径、余纬和经度； <math id="M19"><mi>U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078762&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078744&type= 3.30200005 2.79399991 为地球引力位以及其他天体潮汐位； <math id="M20"><msub><mrow><mi>L</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>=</mo><mrow><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>g</mi></mrow></msub></mrow><mo>/</mo><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078747&type= 16.93333435 4.40266657 ； <math id="M21"><msub><mrow><mi>U</mi></mrow><mrow><mi>g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078769&type= 4.23333359 3.72533321 为大地水准面上引力位； c 为光速。旋转坐标系（ <math id="M22"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078771&type= 2.53999996 2.79399991 为旋转速度）下的信号传播坐标时 <math id="M23"><msub><mrow><mi>T</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078785&type= 3.30200005 3.72533321 为： <math display="block" id="M24"><msub><mrow><mi>T</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msubsup><mo>∫</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></msubsup><mrow><mfrac><mrow><mi mathvariant="normal">d</mi><mi>u</mi></mrow><mrow><mi>c</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>g</mi></mrow></msub><mi mathvariant="normal">d</mi><mi>u</mi></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>ω</mi><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi><mi mathvariant="normal">d</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>u</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mfrac><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi><mi mathvariant="normal">d</mi><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>U</mi><mi mathvariant="normal">d</mi><mi>u</mi></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>O</mi><mfenced separators="|"><mrow><msup><mrow><mi>c</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078796&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078794&type= 139.02265930 11.34533310 式中， <math id="M25"><mi mathvariant="normal">d</mi><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078802&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078798&type= 3.72533321 2.79399991 为传播路径上的线长，沿信号发射点 <math id="M26"><mi>a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079447&type= 1.86266661 2.79399991 到信号接收点 <math id="M27"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078810&type= 1.86266661 2.79399991 的信号传播路径进行积分； <math id="M28"><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078813&type= 3.13266683 4.57200003 为旋转坐标系下经度；忽略 c －4 以上的项 <math id="M29"><mi>O</mi><mfenced separators="|"><mrow><msup><mrow><mi>c</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078825&type= 11.17599964 4.48733330 。式中等号右边前4项（即 <math id="M30"><msubsup><mo>∫</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></msubsup><mrow><mfrac><mrow><mi mathvariant="normal">d</mi><mi>u</mi></mrow><mrow><mi>c</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>g</mi></mrow></msub><mi mathvariant="normal">d</mi><mi>u</mi></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>ω</mi><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi><mi mathvariant="normal">d</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>u</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mfrac><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi><mi mathvariant="normal">d</mi><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078820&type= 100.66867065 11.34533310 ）为几何项，第5项（ <math id="M31"><msubsup><mo>∫</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></msubsup><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mi>U</mi><mi mathvariant="normal">d</mi><mi>u</mi></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078831&type= 12.86933327 8.21266651 ）为引力位项。几何项可通过非旋转框架下信号发射位置和接收位置间距离的计算得到，忽略的地球引力场以及大气折射影响小于1 ps。假定信号发射时刻 <math id="M32"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078834&type= 2.28600001 3.72533321 卫星位置为 <math id="M33"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>a</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078843&type= 9.14400005 3.97933316 ，地面测站位置为 <math id="M34"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078837&type= 9.14400005 3.97933316 ，信号接收时刻 <math id="M35"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078849&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078856&type= 2.28600001 3.72533321 地面测站位置为 <math id="M36"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078859&type= 9.14400005 3.97933316 ，此时信号传播时间可表示为： <math display="block" id="M37"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>T</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>T</mi><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>g</mi></mrow></msub></mrow></mfenced><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>/</mo><mi>c</mi><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></mfenced><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>b</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>/</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">2</mn><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>+</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>a</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>+</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>+</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>a</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>-</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078862&type= 71.79733276 24.04533195 式中， <math id="M38"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078884&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078870&type= 2.87866688 2.79399991 为几何时延； <math id="M39"><msub><mrow><mi>T</mi></mrow><mrow><mi>g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078886&type= 3.72533321 3.72533321 为引力时延； <math id="M40"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>a</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078872&type= 31.58066559 3.97933316 ； <math id="M41"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078876&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078894&type= 3.89466691 3.72533321 为向量 <math id="M42"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078878&type= 3.97933316 3.72533321 的模长； <math id="M43"><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078881&type= 33.18933105 3.97933316 ； <math id="M44"><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078903&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078914&type= 3.97933316 3.72533321 为旋转框架下地面测站的运动速度； <math id="M45"><mi mathvariant="bold-italic">ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078905&type= 2.62466669 2.79399991 为地球自转向量； <math id="M46"><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><mfenced separators="|"><mrow><mi mathvariant="bold-italic">ω</mi><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi><mi>b</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>r</mi><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078907&type= 59.77466202 4.74133301 ； <math id="M47"><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi>r</mi><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078923&type= 3.89466691 3.72533321 为旋转框架下地面测站的运动加速度； <math id="M48"><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078929&type= 7.61999989 3.72533321 为地球引力常数。当信号接收测站相对地球静止时， <math id="M49"><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi><mi>b</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078927&type= 10.66800022 3.72533321 ，式（3）式（3）中第2项 <math id="M50"><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></mfenced><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078936&type= 17.52599907 3.97933316 ）为Sagnac改正项： <math display="block" id="M51"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mn mathvariant="normal">2</mn><mi>ω</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078948&type= 32.17333603 4.40266657 当 <math id="M52"><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi><mi>b</mi></mrow></msub><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078952&type= 10.66800022 3.97933316 ，有： <math display="block" id="M53"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mn mathvariant="normal">2</mn><mi>ω</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>a</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi><mi>b</mi></mrow></msub><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078956&type= 73.82933807 4.74133301 式中，等号后第1项为Sagnac改正；等号后第2项为多普勒改正项； <math id="M54"><msub><mrow><mi>A</mi></mrow><mrow><mi>E</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078966&type= 4.57200003 3.72533321 为信号传播路径点的地心向径在传播过程中扫出的面积在赤道面内的投影。 <math id="M55"><msub><mrow><mi>A</mi></mrow><mrow><mi>E</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078966&type= 4.57200003 3.72533321 具有方向性，如果信号为东方向传播，则 <math id="M56"><msub><mrow><mi>A</mi></mrow><mrow><mi>E</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078966&type= 4.57200003 3.72533321 为正，否则为负。式（3）式（3）中的第3行为Shaprio改正。 1.2　星地双向时间比对 2 1.2.1　X型星地双向时间比对 3 信号分别在 <math id="M57"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079206&type= 2.28600001 3.72533321 和 <math id="M58"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45078990&type= 10.83733368 3.80999994 时刻从测站 <math id="M59"><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079273&type= 1.52400005 2.79399991 和卫星 <math id="M60"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079291&type= 1.26999998 2.79399991 发出，测站和卫星位置分别为 <math id="M61"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079014&type= 8.89000034 3.97933316 和 <math id="M62"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079022&type= 17.35666656 4.06400013 ，并于 <math id="M63"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079246&type= 2.28600001 3.72533321 和 <math id="M64"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079226&type= 2.28600001 3.72533321 时刻分别被卫星和测站接收，测站和卫星位置分别为 <math id="M65"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079040&type= 8.80533314 3.97933316 和 <math id="M66"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079245&type= 8.89000034 3.97933316 （见图2 图2 ）。假定测站 <math id="M67"><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079273&type= 1.52400005 2.79399991 和卫星 <math id="M68"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079291&type= 1.26999998 2.79399991 的原子钟都进行了改正，利用钟面时即可得到坐标时间隔： <math display="block" id="M69"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>t</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079079&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079062&type= 29.71800041 9.90600014 为了进行同步，须求取 <math id="M70"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079298&type= 3.72533321 2.79399991 。光行时方程为： <math display="block" id="M71"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079074&type= 27.77066803 9.05933285 求解 <math id="M72"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079298&type= 3.72533321 2.79399991 可得： <math display="block" id="M73"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></mfenced><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>δ</mi></mtd></mtr><mtr><mtd><mi>δ</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>/</mo><mn mathvariant="normal">2</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079099&type= 33.27399826 10.58333302 改正 <math id="M74"><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079115&type= 1.94733346 2.79399991 来自于时间同步框架下卫星和测站的运动以及传播时间 <math id="M75"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079334&type= 3.72533321 3.72533321 到 <math id="M76"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079124&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079107&type= 3.72533321 3.72533321 的引力时延。讨论几何项 <math id="M77"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079320&type= 12.27666664 3.72533321 ，由式（3）式（3）可得： <math display="block" id="M78"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub></mrow></mfenced><mfenced separators="|"><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>/</mo><mi>c</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></mfenced><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>O</mi><mfenced separators="|"><mrow><mfenced separators="|"><mrow><mi>v</mi><mo>/</mo><mi>c</mi></mrow></mfenced><mfenced separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>/</mo><mi>c</mi></mrow></mfenced><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>/</mo><mi>c</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079143&type= 111.08267212 5.24933338 其中， <math display="block" id="M79"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">Δ</mi><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>=</mo><mo>-</mo><mfenced separators="|"><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">ω</mi><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mi mathvariant="bold-italic">ω</mi><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079152&type= 54.77933884 26.58533096 式中， <math id="M80"><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079135&type= 2.96333337 3.72533321 为卫星对地速度。式（10）式（10）中的第一行与 <math id="M81"><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079157&type= 2.96333337 3.72533321 以及 <math id="M82"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079298&type= 3.72533321 2.79399991 有关。如果 <math id="M83"><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>≈</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079167&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079165&type= 10.32933331 2.79399991 ，该项的影响取决于残余速度项的影响。该项可通过某种方式消去，即引入某些特定时延的 <math id="M84"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079298&type= 3.72533321 2.79399991 ，使得其能与式（10）式（10）第2行公式等号右边第3项（ <math id="M85"><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079188&type= 19.38866615 3.97933316 ）相消，达到 <math id="M86"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>≈</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079201&type= 15.49400043 3.72533321 的效果。 1.2.2　Lambda型星地双向时间比对 3 信号在 <math id="M87"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079206&type= 2.28600001 3.72533321 时刻从测站 <math id="M88"><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079273&type= 1.52400005 2.79399991 发出，测站位置为 <math id="M89"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079228&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079197&type= 8.89000034 3.97933316 ， <math id="M90"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079246&type= 2.28600001 3.72533321 时刻被卫星接收后反射，卫星位置为 <math id="M91"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079224&type= 8.80533314 3.97933316 ，回波信号在 <math id="M92"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079226&type= 2.28600001 3.72533321 时刻被测站 <math id="M93"><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079273&type= 1.52400005 2.79399991 接收，信号接收时刻测站位置为 <math id="M94"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079245&type= 8.89000034 3.97933316 （见图3 图3 ）。与X型时间传递的不同之处在于：Lambda型星地时间传递不存在两个具有同时性的事件假设，卫星 s 的坐标时 <math id="M95"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079246&type= 2.28600001 3.72533321 实际上为 <math id="M96"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079266&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079250&type= 10.83733368 3.80999994 ，需要先求得单向的信号传播时间 <math id="M97"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079334&type= 3.72533321 3.72533321 ，然后才能求得星地同步所需的 <math id="M98"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079298&type= 3.72533321 2.79399991 。假定测站 <math id="M99"><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079273&type= 1.52400005 2.79399991 和卫星 <math id="M100"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079291&type= 1.26999998 2.79399991 的钟都进行了改正，利用钟面时即可得到坐标时时间间隔： <math display="block" id="M101"><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079283&type= 16.25600052 3.72533321 光行时方程为： <math display="block" id="M102"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079309&type= 19.21933365 8.97466660 求解 <math id="M103"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079298&type= 3.72533321 2.79399991 可得： <math display="block" id="M104"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>δ</mi></mtd></mtr><mtr><mtd><mi>δ</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>/</mo><mn mathvariant="normal">2</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>δ</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079314&type= 60.11333084 19.81200027 <math id="M105"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079320&type= 12.27666664 3.72533321 的表达式不变，同式（9）式（9），其中， <math display="block" id="M106"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">Δ</mi><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>=</mo><mo>-</mo><mfenced separators="|"><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">ω</mi><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mi mathvariant="bold-italic">ω</mi><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079322&type= 54.77933884 26.58533096 将 <math id="M107"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079334&type= 3.72533321 3.72533321 用一阶近似替换，可得： <math display="block" id="M108"><mfrac><mrow><mi mathvariant="normal">Δ</mi><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079329&type= 39.45466614 8.55133343 且 <math display="block" id="M109"><mfrac><mrow><mi mathvariant="normal">Δ</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">ω</mi><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi mathvariant="bold-italic">ω</mi><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced><mfrac><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079348&type= 96.85866547 8.55133343 式（15）式（15）为一阶多普勒效应，与式（16）式（16）中最后一项形成相消效果。剩余部分与地球自转以及卫星对地速度有关。与X型星地双向时间比对相比，Lambda型星地时间比对通过调整下行信号的发射时刻，抵消了距离不对称性与一阶多普勒影响，削弱了多普勒的总体影响，但增大了Sagnac的影响。对于空间站卫星，假定轨道高度为370 km，星对地速度假定为7.3 km/s，对于赤道测站而言，式（16）式（16）中 <math id="M110"><mi mathvariant="bold-italic">ω</mi><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfenced><mfrac><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079337&type= 24.29933357 8.21266651 项的最大值在0.05 ps量级，该影响非常小；而 <math id="M111"><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">ω</mi><mo>⋅</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></mfenced></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079342&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079340&type= 21.59000015 8.55133343 项的最大值约为70 ns量级，采用已知地面测站坐标、卫星轨道代入，1 m的轨道误差带来的修正影响小于0.01 ps量级。 2　星地激光双向时间比对中的相对论效应 1 地心地固系下时间同步的前提为：卫星和地面的原子钟都已进行了引力位和速度改正，使其运行频率与坐标时定义频率保持一致。主要讨论卫星钟运行速率与坐标时进行同步所需的速度项（狭义相对论）和引力位（广义相对论）项改正。相对参考于坐标时的原子钟频率（包含所有大于1× <math id="M112"><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">18</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079355&type= 7.61999989 3.04800010 项影响）［ 23 23 ］为： <math display="block" id="M113"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">v</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">G</mi></mrow></msub></mrow></mfrac><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">w</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><msup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>+</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>T</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo>+</mo><mi mathvariant="bold-italic">w</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>T</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>T</mi><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfenced></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">T</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">G</mi></mrow></msub></mrow></mfrac><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>g</mi></mrow></msub></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079357&type= 114.13066101 20.57399940 式中， <math id="M114"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079359&type= 4.57200003 3.72533321 为卫星原时； <math id="M115"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">G</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079368&type= 7.11199999 3.72533321 为地心坐标时； <math id="M116"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079370&type= 5.67266655 3.72533321 为大地水准面上原时； <math id="M117"><mi mathvariant="bold-italic">w</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079384&type= 3.13266683 2.79399991 为卫星三维位置坐标向量 <math id="M118"><mo stretchy="false">(</mo><msup><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079386&type= 25.06133461 3.80999994 ； <math id="M119"><mfenced separators="|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">G</mi></mrow></msub><mo>,</mo><msup><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079393&type= 15.91733265 4.48733330 构成地心框架时空坐标；下标E表示地球质心； <math id="M120"><msub><mrow><mi>U</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">w</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079392&type= 11.09133339 3.97933316 和 <math id="M121"><msub><mrow><mi>U</mi></mrow><mrow><mi>T</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079401&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079400&type= 10.41399956 3.97933316 分别为地球牛顿引力位和外部天体的牛顿引力位。将外部天体的影响看做潮汐位影响 <math id="M122"><mi mathvariant="normal">Δ</mi><mi>V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079412&type= 5.67266655 2.87866688 ，利用开普勒定律以及轨道根数对式（17）式（17）进行积分［ 24 24 - 25 25 24-25 ］，可得卫星坐标时 <math id="M123"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079415&type= 1.26999998 2.79399991 ： <math display="block" id="M124"><mtable columnalign="left"><mtr><mtd><mi>t</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">v</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">l</mi></mrow></msup><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">v</mi></mrow></msub><mo>-</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfrac><mi>t</mi><mo>+</mo><mo>∫</mo><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn><mi>a</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>R</mi></mrow><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi></mrow><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow></mfrac></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079421&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079419&type= 72.05133057 14.81666565 其中， <math id="M125"><mi mathvariant="normal">Δ</mi><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">l</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079422&type= 5.92666674 3.04800010 为相对论效应， <math id="M126"><msub><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mrow><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow><mo>/</mo><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>g</mi></mrow></msub></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079440&type= 22.43666649 4.14866638 为地球平均半径； <math id="M127"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079445&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079430&type= 2.79399991 2.79399991 为地球非球形效应影响； <math id="M128"><mi>a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079447&type= 1.86266661 2.79399991 为卫星半长轴。利用轨道动力学特性以及轨道根数 <math id="M129"><mi>a</mi><mi mathvariant="normal">、</mi><mi>e</mi><mi mathvariant="normal">、</mi><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079453&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079435&type= 9.73666668 3.04800010 ，对式（18）式（18）进行积分可得： <math display="block" id="M130"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">Δ</mi><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">l</mi></mrow></msup><mo>=</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi></mrow></msub><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>∫</mo><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mi>R</mi></mrow><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi></mrow><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow></mfrac></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn><mi>a</mi></mrow></mfrac></mrow></mfenced><mi>t</mi><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mroot><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mi>a</mi></mrow><mrow/></mroot><mi>e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>E</mi></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>∫</mo><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mi>R</mi></mrow><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi></mrow><mrow><mi>G</mi><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow></mfrac></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079457&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45079455&type= 63.16133118 36.83000183 式（19）式（19）为原时与坐标时转换公式，可知相对论的影响表现为频差和时差两个方面，频差受轨道半长轴、地球非球形效应以及外部天体潮汐位的影响，时差受轨道位置的影响。假定轨道高度为370 km，偏心率为4.5×10 －4 ，时差的影响在500 ps量级。忽略地球非球形效应以及外部天体潮汐位影响时，频差的影响在2.9×10 －10 量级。为分析式（17）式（17）中地球引力位、天体潮汐位以及卫星速度对相对论修正性能的影响，需要精确的地球重力场以及其他天体重力场模型信息，以及地心惯性系下卫星位置和速度。对以上修正效果进行模拟分析，实验中地球重力场模型采用EGM2008模型，太阳及其他行星的引力位采用一阶近似，卫星位置和速度信息的采样间隔为300 s，忽略该时间内引力位和速度的时变性，分析相对论改正。图4 图4 为利用中国空间站两行轨道提供的卫星位置、速度信息计算的相对论改正。图4 图4 （a）为广义和狭义相对论改正时序图，图4 图4 （b）为对应时段的空间站地心距变化以及速度变化。从图4 图4 中可以看出，广义和狭义相对论引起的相对频率偏差影响分别为6.570×10 －10 和3.285×10 －10 量级，变化范围分别为1.100×10 －12 和1.000×10 －12 ，对应的地心距在11.00 km范围内变化，速度在11.00 m/s范围内变化。由此根据式（17）式（17）可得当地心距在0.1 m范围内变化（或者存在0.1 m误差）时，对应广义相对论频率偏差误差9.7×10 －18 量级，速度在0.1 mm/s范围内变化（或者存在0.1 mm/s误差）时，对应狭义相对论频率偏差误差8.5×10 －18 量级。 3　空间站对地可见性对稳定性分析的影响 1 空间站是空间激光比对系统的载体，在与地面进行双向激光时频比对时，其可见情况将对指标评估产生影响。中国空间站的轨道周期约为90 min，具有较大的对地运行速度，单一测站单弧段高度角10°以上的可视时长在5 min左右。即使利用全国测站观测数据，同一轨道周期高度角10°以上的可视时长在10 min左右。由此会存在大量的数据间断，如果利用内插方法对数据进行补充，会导致稳定性方差存在偏差。通常利用有限观测值组合计算扩展的阿伦方差［ 26 26 ］。采用仿真方法分析空间站对地可见性对在轨原子钟稳定性结果的影响。中国空间站原子钟稳定度可用谱密度分别为1.28×10 －35 、8.00×10 －30 的调频白噪声和调频闪变噪声近似［ 12 12 ］。图5 图5 中蓝色线为中国空间站原子钟稳定性仿真数据，秒稳为2×10 －15 ，万秒稳为1×10 －17 量级，天稳为2×10 －18 量级。黑色线为欧洲SOC计划原子钟预期稳定性曲线，服从1×10 －15 τ －0.5 曲线变化。利用空间站两行轨道根数计算中国境内上海、三亚、拉萨、昆明、喀什、西安、北京、长春、成都等激光测站对空间站可视时间段（截止高度角10°），依据间断数据重新计算稳定性方差，并与仿真数据进行比较。图5 图5 中红色线为利用间断数据计算的原子钟稳定性，可以看出，受数据间断影响，500~40 000 s之间原子钟的稳定性不可评估，并且受数据间断导致数据数目稀少，使得长期（如40 000 s）稳定度估值不可靠。 4　结语 1 随着中国空间站等空间卫星系统的建设，利用空间高精度时频系统提供的超高精度时间频率信号能够开展一系列空地时频传递和基础物理实验。本文讨论了空间激光链路双向时间比对的算法并进行了仿真验证，结果表明： 1）基于激光测距方式的Lambda型双向时间比对可以抵消一阶多普勒对上下行距离项差异的影响，且Sagnac项影响的增大可忽略； 2）星地时间比对中坐标时和原时转换的相对论项对空间站位置速度精度提出了高要求，空间站地心距在0.1 m范围内变化时，广义相对论频率偏差在9.7×10 －18 范围内变化，速度在0.1 mm/s范围变化时，狭义相对论频率偏差在8.5×10 －18 范围内变化； 3）受空间站对地可见性影响，仅利用国内测站不可评估中短期原子钟的稳定性，中长期稳定性估值也会因为数据缺失而变得不可靠。空间高精度导航位置速度解是确保高精度空地时频传递的关键，如何对现有导航算法进行优化，利用搭载的各类载荷实现高精度位置速度的求解是需要重点研究的问题。参考文献 McGrew W F ， Zhang X ， Fasano R J ， et al . 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A High-Precision Relativistic Model for Two-Way Frequency Transfer of Λ-Configuration in GCRS 程然，韩文标，何克亮 . 一种GCRS下Λ型双向频率传输的高精度相对论模型［J］. 天文学报， 2019 ， 60 （ 3 ）： 80 - 95 . 一种GCRS下Λ型双向频率传输的高精度相对论模型 Wang Yuanming ， Yang Fumin ， Huang Peicheng ， et al . Principle Prototype and Ground Simulation Comparison Experiment of Satellite Ground Laser Time Comparison ［J］. Science in China（Series G：Physics， Mechanics and Astronomy）， 2008 （ 2 ）： 217 - 224 . Principle Prototype and Ground Simulation Comparison Experiment of Satellite Ground Laser Time Comparison 王元明，杨福民，黄佩诚，等 . 星地激光时间比对原理样机及地面模拟比对试验［J］. 中国科学（G辑：物理学力学天文学）， 2008 （ 2 ）： 217 - 224 . 星地激光时间比对原理样机及地面模拟比对试验 Wolf P ， Petit G . Relativistic Theory for Clock Syntonization and Realization of Geocentric Coordinates Times ［J］. Astronomy and Astrophysics ， 1995 （ 304 ）： 653 - 661 . Relativistic Theory for Clock Syntonization and Realization of Geocentric Coordinates Times Kouba J . Improved Relativistic Transformations in GPS ［J］. 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Application of the Dynamic Allan Variance for the Characterization of Space Clock Behavior 10.13203/j.whugis20230165 P228 A 1671-8860（2023）07-1082-07 陈俊平,王彬,孟文东，等.空间激光链路时频传递算法及仿真分析[J].武汉大学学报（信息科学版）,2023,48(7):1082-1088.DOI: 10.13203/j.whugis20230165 http://dx.doi.org/10.13203/j.whugis20230165 陈俊平,王彬,孟文东，等.空间激光链路时频传递算法及仿真分析[J].武汉大学学报（信息科学版）,2023,48(7):1082-1088.DOI:10.13203/j.whugis20230165 CHEN Junping,WANG Bin,MENG Wendong,et al.Simulation Analysis of Time and Frequency Transfer Algorithm for Space Laser Link[J].Geomatics and Information Science of Wuhan University,2023,48(7):1082-1088.DOI: 10.13203/j.whugis20230165 http://dx.doi.org/10.13203/j.whugis20230165 CHEN Junping,WANG Bin,MENG Wendong,et al.Simulation Analysis of Time and Frequency Transfer Algorithm for Space Laser Link[J].Geomatics and Information Science of Wuhan University,2023,48(7):1082-1088.DOI:10.13203/j.whugis20230165 国家自然科学基金（41904034）；上海市优秀学术带头人计划（20XD1404500）。 2023-05-05 2023-07-05 2023-06-29 39072449 武汉大学学报（信息科学版） 7 48

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Simulation Analysis of Time and Frequency Transfer Algorithm for Space Laser Link

DOI：10.13203/j.whugis20230165