基于IGGIII模型的多模多频GNSS-MR潮位反演融合研究

基于IGGIII模型的多模多频GNSS-MR潮位反演融合研究 Sea level combined retrievals using multi-GNSS multipath reflectometry based on the IGGIII scheme first-author 陈殊，1997年生，男，硕士研究生，研究方向为GNSS空间环境分析。E-mail：cshu@hhu.edu.cn 陈殊，1997年生，男，硕士研究生，研究方向为GNSS空间环境分析。E-mail：cshu@hhu.edu.cn cshu@hhu.edu.cn corresp 何秀凤，1962年生，女，教授，研究方向为卫星定位导航，海洋监测，卫星遥感技术。E-mail：xfhe@hhu.edu.cn 何秀凤，1962年生，女，教授，研究方向为卫星定位导航，海洋监测，卫星遥感技术。E-mail：xfhe@hhu.edu.cn xfhe@hhu.edu.cn 河海大学 地球科学与工程学院,南京 211100 河海大学 地球科学与工程学院,南京 211100 School of Earth Science and Engineering,Hohai University,Nanjing 211100,China School of Earth Science and Engineering,Hohai University,Nanjing 211100,China 潮位是保证沿海安全、监测海洋气候、维持高程基准的重要参数。近年来基于地基Global Navigation Satellite Systems （GNSS）反射信号的遥感方法被证实可以用于潮位监测。相较于传统的潮位测量方法，GNSS-multipath reflectometry （GNSS-MR）技术有成本低、连续跟踪、全天候等优势；但是目前技术的精度不高、时间分辨率较低。通过获取更多GNSS卫星系统的观测数据可以提高潮位监测结果的时间分辨率，本文利用GPS、GLONASS、Galileo和BeiDou的观测数据，采用基于IGGIII模型的稳健回归方法对四系统的潮位反演数据进行融合研究。测站选取BRST站和HKQT站，这两个测站均可接收四系统数据；实验结果表明，利用多模多频GNSS-MR进行潮位反演，二个测站的反演精度分别优于13cm和 8cm，相比于单系统单频精度有40%-70%的提升，而且能够大大提高时间分辨率。 Sea level is an important parameter to ensure coastal safety, monitor marine climate and maintain elevation datum. In recent years, the remote sensing method using ground-based GNSS reflection signal can be used for sea level monitoring. Compared with the traditional sea level measurement method, GNSS multipath reflectometry (GNSS-MR) technology has the advantages of low cost, continuous tracking and can observe all-weather and all day. However, GNSS-MR technology is limited by two problems: low accuracy and low time resolution. The time resolution of retrievals can be improved by acquiring more observation data from more satellite systems. In this paper, a robust regression strategy based on the IGGIII scheme is proposed to address the two limitations. This method uses the SNR data of GPS, GLONASS, Galileo and Beidou, the Lomb–Scargle periodogram (LSP) method in the classical tide level inversion principle is used to obtain the sea level estimates of each frequency band from quad-constellation, then a specific time window is established, and the state transition equation set is established in each time window considering the sea surface dynamic change and tropospheric delay. Finally，the sea level time series is solved by a robust estimation model. In order to prove the feasibility and effectiveness of this method, BRST station in France and HKQT station in Hong Kong were chosen to validate the performance of the proposed method. The root-mean-square errors (RMSEs) between sea level combined retrievals of multi-GNSS signals and the tide gauge records are calculated. The RMSE of BRST station is 12.43cm, which is about 40% - 60% higher than the results of single signal of each system. The RMSE of HKQT station is 7.09cm, which is about 72% higher than the results of the four systems. Both BRST station and HKQT station can formulate a 10-min sea level time series, which greatly improves the time resolution of sea level retrievals compared with single signal retrievals. Comparing the inversion results of the two stations, it can be concluded that using robust regression strategy based on the IGGIII scheme can lead to a clear increase in precision and achieve a higher temporal sampling because of the more frequent GNSS retrievals and a better retrieval combination strategy. The estimated value of sea level has good agreement with the data of tide gauge records, and can clearly describe the fluctuation of sea level. In essence, it is a method of quality control and optimal valuation for GNSS-MR that is theoretically suitable for different geographical environments. GNSS-MR 潮位反演 多模多频 稳健回归 IGGIII GNSS-MR sea level estimation multiple system robust regression IGGIII 1　引言 1 沿海地区是世界上经济活动和城市化最集中的地区，目前世界上有一半人口生活在离海60公里以内地区，而且沿海地区的人口增长速度正在加快。然而由于风暴潮和海平面上升等自然因素，沿海地区将面临越来越多威胁，所以对海面变化进行实时监测并研究其变化规律具有重要意义（ Melet等，2018 Melet等，2018 ）。随着GNSS技术的不断发展与完善，GNSS信号不仅可以用于定位、导航和授时，利用GNSS反射信号还可以监测潮位变化（ Jin等，2014 Jin等，2014 ）。 自 Martin-Neira （1993） Martin-Neira （1993） 提出用GNSS反射信号反演海潮变化以来，GNSS-MR技术逐渐成熟，成为沿海海平面测高的另一种可行方案（ Geremia-Nievinski等，2020 Geremia-Nievinski等，2020 ）。但是，GNSS-MR潮位反演技术在当前的实际应用依然存在精度不够高和分辨率低的两大问题（ Larson等，2017 Larson等，2017 ）。反演结果的单点精度因信号类型而异，质量好的信号反演的精度更高，采样率由卫星轨迹数量、signal-to-noise ratio （SNR）数据质量以及环境的几何形状决定。多模多频GNSS可以观测到更多的卫星、获取质量更好的信号和更为开阔的信号感知区域，这为GNSS-MR反演潮位提供了更多的反演值从而提高精度和时间分辨率（ Larson等，2017 Larson等，2017 ； Tabibi等，2017 Tabibi等，2017 ）。 Löfgren等（2014） Löfgren等（2014） 分别使用了GPS和GLONASS的L1和L2频段的信号进行潮位反演，证实了GLONASS系统也可以用于潮位反演。 Roussel等（2015） Roussel等（2015） 基于最小二乘方法，对GPS和GLONASS系统反演的潮位值进行融合。 Strandberg等（2016） Strandberg等（2016） 分别用L1、L2C和L2P频段的SNR数据进行潮位反演，并对信号的质量进行分析。 金双根等（2017） 金双根等（2017） 第一次用BeiDou系统L2、L6、L7信号估计海面变化。王笑蕾等（2019）用稳健估计的方法对四个系统（GPS、GLONASS、Galileo和BeiDou）潮位反演结果进行融合，验证了多模多频GNSS-MR可以有效地提高潮位反演的精度和时间分辨率。 何秀凤等（2020） 何秀凤等（2020） 利用多模多频GNSS信号反演了沿海台风风暴潮，证实了在极端天气情况下，GNSS-MR技术依然可以探测到潮位的异常变化。 目前，多模多频GNSS-MR融合算法主要使用“bisquare”定权方法（王笑蕾等，2019），并将定权中涉及的常系数确定为1；这种定权方法对于偏差大于1的观测值无法很好的处理。然而，由于GNSS-MR潮位反演中存在各类误差，反演观测值经常大于1 m。考虑到IGGIII定权模型（ Yang等，2002 Yang等，2002 ） 可以对不同大小的偏差进行更细致更合理的处理，文章提出基于IGGIII模型的稳健估计方法来进行多模多频GNSS-MR融合，并验证多模多频GNSS-MR融合方法在不同测站环境用于潮位监测的可行性。 2　方法与原理 1 2.1　GNSS-MR经典海面反演原理 2 多路径效应主要与反射面的结构和电介质参数有关，当GNSS测站位于海边时，接收机在海域方向会接收到卫星直射信号与海水表面反射信号相干的合成信号，这种相干性表现在卫星观测文件SNR的变化上（ 金双根等，2017 金双根等，2017 ）。SNR可以表示为（ Larson等，2008 Larson等，2008 ） <math display="block" id="M1"><mi>S</mi><mi>N</mi><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>A</mi></mrow><mrow><mi>d</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626021&type= 49.27600098 4.82600021 式中， <math id="M2"><msub><mrow><mi>A</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626023&type= 3.21733332 3.72533321 和 <math id="M3"><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626025&type= 3.72533321 3.72533321 分别表示为直射信号和反射信号的振幅； <math id="M4"><mi>ψ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626027&type= 2.37066650 3.38666677 是直射信号和反射信号之间的相位偏差。 直射信号与反射信号的相位差为 <math display="block" id="M5"><mi>ψ</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>π</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mi>D</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>π</mi><mi>h</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626030&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626029&type= 36.15266418 7.70466709 式中， <math id="M6"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626031&type= 2.53999996 2.62466669 为反射信号与直射信号的路径差， <math id="M7"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626033&type= 2.28600001 2.62466669 为信号波长， <math id="M8"><mi>e</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626036&type= 1.43933344 2.62466669 为卫星高度角， <math id="M9"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626106&type= 1.86266661 2.62466669 为天线相位中心到反射面的垂直距离。 假设反射面时静止的，根据 公式（2） 公式（2） 有 <math display="block" id="M10"><mn mathvariant="normal">2</mn><mi>π</mi><mi>f</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>ψ</mi></mrow><mrow><mi>d</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mn mathvariant="normal">4</mn><mi>π</mi><mfrac><mrow><mover accent="true"><mi>h</mi><mo>¯</mo></mover></mrow><mrow><mi>λ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626041&type= 35.13666534 8.97466660 式中， <math id="M11"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626059&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626058&type= 1.10066664 3.47133350 是SNR中受干涉影响部分的信号频率， <math id="M12"><mover accent="true"><mi>h</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626046&type= 1.86266661 3.13266683 是静态假设下得到的反射高度，（3）中给出了 <math id="M13"><mover accent="true"><mi>h</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626238&type= 1.86266661 3.13266683 与 <math id="M14"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626059&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626058&type= 1.10066664 3.47133350 的关系，即 <math id="M15"><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><mi>λ</mi><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626052&type= 10.24466610 7.61999989 。对于 <math id="M16"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626059&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626058&type= 1.10066664 3.47133350 的提取，经典方法是先用低阶多项式来去除SNR序列中的直射信号，再用LSP（Lomb–Scargle periodogram）分析方法（ Lomb等，1976 Lomb等，1976 ； Scargle等，1982 Scargle等，1982 ）从去除直射信号的残差序列 <math id="M17"><msub><mrow><mi>δ</mi></mrow><mrow><mi>S</mi><mi>N</mi><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626108&type= 5.24933338 3.72533321 中求出 <math id="M18"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626059&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626058&type= 1.10066664 3.47133350 。 2.2　GNSS-MR潮位监测误差改正 2 在GNSS-MR经典海面反演原理中，有效高度 <math id="M19"><mover accent="true"><mi>h</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626060&type= 1.86266661 3.13266683 的求解原理是基于海面静止的静态假设下。顾及到海面的动态变化， Larson等（2013） Larson等（2013） 首次提出对海面动态变化引起的高度误差做改正。设 <math id="M20"><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626074&type= 2.28600001 4.40266657 和 <math id="M21"><mover><mrow><mi>e</mi></mrow><mrow><mo>·</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626065&type= 1.86266661 4.40266657 分别为海面高度和卫星高度角随时间的变化率，结合（2）和（3）式可得 <math display="block" id="M22"><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow></mfrac><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover><mo>+</mo><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626069&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626068&type= 25.56933212 9.48266602 为了求解动态变化的高度 <math id="M23"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626106&type= 1.86266661 2.62466669 ，首先是得到 <math id="M24"><mover accent="true"><mi>h</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626072&type= 1.86266661 3.13266683 的时间序列，再由此来估算 <math id="M25"><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626074&type= 2.28600001 4.40266657 ，从时间序列 <math id="M26"><mover accent="true"><mi>h</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626146&type= 1.86266661 3.13266683 中扣除 <math id="M27"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow></mfrac><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626080&type= 23.53733444 9.48266602 ，即可得到改正后的时间序列 <math id="M28"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626106&type= 1.86266661 2.62466669 。 卫星信号在经过对流层区域时会发生折射效应，这种对流层延迟会使求得的垂直反射距离出现偏差（ Williams等，2017 Williams等，2017 ）。在GNSS反射信号反演海面高的研究中， Anderson（2000） Anderson（2000） 首先提出了大气折射引起的无线信号几何弯曲可能会影响反演结果精度。 Santamaría-Gómez等（2017） Santamaría-Gómez等（2017） 在GPS-MR海平面估计中使用了基于大气压力和温度的经验天文折射模型来消除对流层延迟。 Williams等（2017） Williams等（2017） 使用GPT2W（Global Temperature and Pressure）和VMF1（Vienna mapping functions）映射函数，来消除对流层延迟的影响。对流层延迟 <math id="M29"><msub><mrow><mi>τ</mi></mrow><mrow><mi>T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626094&type= 3.04800010 3.72533321 计算公式为 <math display="block" id="M30"><msub><mrow><mi>τ</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>Z</mi></mrow></msubsup><mo>⋅</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>h</mi></mrow></msub><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>Z</mi></mrow></msubsup><mo>⋅</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>w</mi></mrow></msub><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626086&type= 51.64666748 4.91066647 式中， <math id="M31"><mi mathvariant="normal">Δ</mi><msup><mrow><mi>τ</mi></mrow><mrow><mi>Z</mi></mrow></msup><mo>=</mo><msup><mrow><mi>τ</mi></mrow><mrow><mi>Z</mi></mrow></msup><mo stretchy="false">(</mo><mo>-</mo><mi>H</mi><mo stretchy="false">)</mo><mo>-</mo><msup><mrow><mi>τ</mi></mrow><mrow><mi>Z</mi></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626088&type= 34.29000092 3.80999994 是天线和地面位置的天顶延迟差， <math id="M32"><msub><mrow><mi>m</mi></mrow><mrow><mi>h</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626091&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626090&type= 3.89466691 3.72533321 和 <math id="M33"><msub><mrow><mi>m</mi></mrow><mrow><mi>w</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626092&type= 4.06400013 3.72533321 分别为干延迟映射函数和湿延迟映射函数。于是由对流层延迟 <math id="M34"><msub><mrow><mi>τ</mi></mrow><mrow><mi>T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626094&type= 3.04800010 3.72533321 引起的反演高度偏差 <math id="M35"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626256&type= 5.58799982 3.89466691 为 <math display="block" id="M36"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mo>∂</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mrow><mo>∂</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626099&type= 26.07733345 8.80533314 结合（4）式和（6）式，静态假设下垂直反演距离 <math id="M37"><mover accent="true"><mi>h</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626102&type= 1.86266661 3.13266683 和实际情况下垂直距离的关系有 <math display="block" id="M38"><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow></mfrac><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover><mo>+</mo><mi>h</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626105&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626104&type= 35.47533035 9.48266602 这样经过海面动态改正和对流层延迟改正后即可得到GNSS接收机到海面实际的垂直距离 <math id="M39"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626106&type= 1.86266661 2.62466669 ，再用接收机的大地高减去垂直距离便是真实的潮位值。 2.3　多模多频GNSS-MR潮位反演值融合方法 2 GNSS系统中各个频段的SNR数据都可以按照GNSS-MR经典海面反演理论及误差改正方法对潮位进行监测。在整个技术流程中，首先需要确定高度角和方位角区间，获得来自海面的反射信号，再对截取的残差序列 <math id="M40"><msub><mrow><mi>δ</mi></mrow><mrow><mi>S</mi><mi>N</mi><mi>R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626108&type= 5.24933338 3.72533321 用LSP分析方法求得有效高度 <math id="M41"><mover accent="true"><mi>h</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626110&type= 1.86266661 3.13266683 。在LSP分析过程中，需要进行质量控制，一般选取峰值与背景噪声的比值（peak-to-noise ratio）大于3的反演值（ Larson等，2013 Larson等，2013 ）。最后从海面动态变化和对流层延迟两个方面对垂直高度进行修正。对于多模多频GNSS-MR潮位反演值，本文采用基于状态方程和稳健估计的方法，考虑海面动态变化以及对流层延迟对数据进行融合（ Wang等，2019 Wang等，2019 ），实现了每10分钟获得一个潮位值。在建立方程之前，首先需要设定一个滑动窗口，窗口的宽度需要容纳足够多的潮位反演点以满足建立方程求解。设窗口长度为 <math id="M42"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626112&type= 2.11666679 2.62466669 ，窗口的滑动步长为 <math id="M43"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></mfrac><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626115&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626114&type= 5.16466665 7.70466709 ，第 <math id="M44"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626185&type= 1.10066664 2.62466669 个窗口的时间 <math id="M45"><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></mfrac><mi>h</mi><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mi>h</mi><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>h</mi><mo>,</mo><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mi>h</mi><mo>,</mo><mfrac><mrow><mn mathvariant="normal">5</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></mfrac><mi>h</mi><mo>,</mo><mn mathvariant="normal">1</mn><mi>h</mi><mo>,</mo><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626119&type= 57.31933212 8.80533314 ，则在第 <math id="M46"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626185&type= 1.10066664 2.62466669 个窗口内可建立状态方程 <math display="block" id="M47"><msub><mrow><mover accent="true"><mi>h</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub></mrow></mfrac><msub><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626124&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626123&type= 97.02799988 10.75266743 式中， <math id="M48"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626125&type= 0.84666669 3.47133350 表示信号类型， <math id="M49"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626128&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626127&type= 1.10066664 2.62466669 表示窗口内反演值的序号，其中 <math id="M50"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo><</mo><mi>T</mi><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626130&type= 19.72733307 5.16466665 。则第 <math id="M51"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626185&type= 1.10066664 2.62466669 个窗口内所有的反演值可以用方程组表示 <math display="block" id="M52"><mfenced separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>h</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><msub><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>h</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><msub><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>h</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><msub><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626135&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626134&type= 63.58466721 72.30532837 可以简化为矩阵方程 <math display="block" id="M53"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">H</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626136&type= 34.29000092 5.41866684 式中， <math id="M54"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">H</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>h</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626138&type= 45.55066681 13.97000027 ， <math id="M55"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626141&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626140&type= 40.30133438 20.48933220 ， <math id="M56"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfenced close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mfenced open="" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626142&type= 16.51000023 5.16466665 ， <math id="M57"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626144&type= 13.80066681 11.76866722 。 如果不同信号的反演值 <math id="M58"><mover accent="true"><mi>h</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626146&type= 1.86266661 3.13266683 具有相同的精度，那么根据最小二乘原理，矩阵方程（10）的解为 <math display="block" id="M59"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>T</mi></mrow></msubsup><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">H</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626148&type= 27.68600082 5.84200001 但是不同系统和频段信号的SNR不同，这导致了在LSP分析过程中每个SNR弧段的主导频率存在差异，进而反演精度也不相同（ Tabibi等，2017 Tabibi等，2017 ），所以不同的信号应对应不同的权重，本文采用基于IGGIII模型的稳健估计方法进行自动赋权。 稳健估计的基本原理为：首先通过最小二乘法得到反演值的估计残差；根据残差确定的各反演值新的权因子，进行下一轮最小二乘解算；反复迭代，直到前后两次估计值的变化值小于限差（ 方兴等，2018 方兴等，2018 ）。 根据 公式（11） 公式（11） 进行第一次迭代 <math display="block" id="M60"><msup><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>T</mi></mrow></msubsup><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">H</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626150&type= 30.22599983 5.84200001 第一次迭代的残差为 <math display="block" id="M61"><msup><mrow><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">H</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626152&type= 26.50066566 4.91066647 在接下来的迭代过程中，精度较低的反演值将被重新定权，逐渐降低其在求解垂直距离 <math id="M62"><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626192&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626191&type= 2.62466669 3.72533321 过程中的作用，以便达到提高精度的目的。潮位反演值主体服从正态分布，可疑和显著异常反演值只是占少部分。在稳健估计中，应充分利用主体可靠信息，所以本文采用IGGIII模型来自动赋权， 图1 图1 为BRST站反演数据迭代过程中残差 <math id="M63"><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626157&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626156&type= 2.28600001 3.72533321 所对应的权重，从图中可以看出权重大致分为三段，即正常段，保持原有的权不变；可疑段，降权，使其小于原始权；淘汰段，权值为0，淘汰掉异常值，这不但能够削弱粗差对结果的影响还可以充分的保留精确数据，避免了数据严重失真（ Yang等，2002 Yang等，2002 ）。IGGIII模型表示为 <math display="block" id="M64"><msup><mrow><msub><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msup><mrow><msub><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mtable><mtr><mtd/><mtd/><mtd><mtable><mtr><mtd/><mtd/><mtd><mtable><mtr><mtd><mtable><mtr><mtd/><mtd/></mtr></mtable></mtd><mtd/></mtr></mtable></mtd></mtr></mtable></mtd><mtd><mfenced open="|" close="|" separators="|"><mrow><msup><mrow><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced><mo>≤</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><msup><mrow><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mfenced open="|" close="|" separators="|"><mrow><msup><mrow><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><msub><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo 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separators="|"><mrow><msup><mrow><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced><mo>></mo></mtd></mtr></mtable><msub><mrow><mi>k</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=22626159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626158&type= 77.47000122 28.44799995 式中， <math id="M65"><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mover accent="true"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mo>̂</mo></mover><mroot><mrow><mo 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626164&type= 2.96333337 4.57200003 是 <math id="M68"><mover accent="true"><mi mathvariant="bold-italic">P</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626167&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626166&type= 2.37066650 3.13266683 中的元素， <math id="M69"><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>v</mi><mi>v</mi></mrow></msub><mo>=</mo><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">P</mi><mo>¯</mo></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><mi mathvariant="bold-italic">A</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>T</mi></mrow></msup><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">P</mi><mo>¯</mo></mover><mi mathvariant="bold-italic">A</mi><mo 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mathvariant="bold-italic">X</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mfenced><mo><</mo><mi mathvariant="bold-italic">ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626184&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626183&type= 28.36333466 5.41866684 时，迭代结束，得出第 <math id="M76"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626185&type= 1.10066664 2.62466669 个窗口内经过海面动态改正和对流层延迟改正的垂直高度 <math id="M77"><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626192&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626191&type= 2.62466669 3.72533321 和 <math id="M78"><msub><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626189&type= 3.38666677 5.41866684 ，在用接收机的大地高减去 <math id="M79"><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626192&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626191&type= 2.62466669 3.72533321 即可得到潮位高。 3　GNSS站点和数据分析 1 本文选取BRST和HKQT两个测站来验证多模多频GNSS-MR技术用于潮位变化监测的有效性，这两个测站均可接收来自四系统的卫星观测数据。 3.1　站点介绍 2 BRST站（48.4°N，4.5°W）位于法国西海岸的Brest海港岸边，周围海平面日涨落幅在7m左右（ Santamaría-Gómez等，2015 Santamaría-Gómez等，2015 ）。 图2（a） 图2（a） 即为BRST站点的测站环境，BRST站配备了Trimble NETR9大地测量型接收机，提供了GPS、GLONASS、Galileo、BeiDou和SBAS的卫星观测数据，采样间隔为30s。为了验证多模多频GNSS-MR技术监测潮位的精度，在实验中利用距离BRST站292m的Brest验潮站实测数据进行对比分析。 由于利用GNSS-MR技术监测潮位变化仅需获取来自海面的卫星反射信号，本文以5°，10°，20°，30°的高度角来绘制第一菲涅尔反射区（ Roesler等，2018 Roesler等，2018 ），BRST站海面信号反射区域如 图2（b） 图2（b） 所示。为了获取来自海面的反射信号，本文设置两个方位角和高度角区间来剔除陆地的干扰信号。在方位角为130°–165°的区域，高度角区间为5°–20°的反射信号被使用；在方位角为165°–330°的区域，高度角区间为12°–25°的反射信号被使用（ Wang等，2019 Wang等，2019 ）。 1,2; HKQT站（22.3°N，114.1°E）位于香港北边海岸，安装了Trimble NETR5大地测量型接收机，提供GPS、GLONASS、Galileo、BeiDou、QZSS和SBAS的卫星观测数据，采样间隔有1s、5s和30s。HKQT测站环境及海面反射信号如 图3 图3 所示，该站点来自海面反射信号区域的方位角为-60°–105°，有效高度角为4°–9°。距离HKQT站2m的Quarry Bay实测潮位数据可以用于实验进行对比分析。 1,2; 3.2　数据分析 2 以BRST站为例，本文选取BRST站2020年年积日为197-253的SNR数据进行实验，其中信噪比数据GPS有S1C、S2W、S2X、S5X四种，GLONASS有S1C、S1P、S2C、S2P四种，Galileo有S1X、S5X、S7X、S8X、S6X五种，BeiDou有S1X、S5X、S2I、S7I、S6I五种。 实验期间内来自各个系统每天可用的SNR弧段数统计结果如 图4 图4 所示，从图中可以看出GPS和GLONASS系统实验期间可用的SNR弧段数波动较小，信号质量较为稳定，每天使用的弧段数在110左右。Galileo和BeiDou系统可用的SNR弧段数波动较大，信号质量不稳定，但是由于记录的信噪比类型都有5种，平均每天可用的弧段数在120左右。为了进一步分析各系统不同波段的SNR信号质量及其反演能力，以2020年年积日为227的四个系统数据进行分析。 图 5 图 5 为四个系统各信号SNR序列和对应的LSP 分析结果，其中BeiDou系统由于数据记录不完整，只接收到S1X、S5X、S2I、S6I四种信号。 1;2;3;4;5;6;7;8; 由 图 5 图 5 左边的四系统SNR序列可知，在低高度角情况下，GPS中S2W的能量明显低于其他三种信号，SNR的最大差值在10db左右，GLONAS、Galileo和BeiDou的不同载波信号SNR差值较小，整体趋势较为一致。 图 5 图 5 右边为四系统SNR序列对应的LSP分析结果，其中波峰对应的横坐标即为垂直反射距离，可以看出四个系统不同信号反演结果差值在0.08m左右。GPS中S2W的SNR序列能量最低，在LSP分析中整体振幅也是最低；S2W和S2X的LSP分析结果有明显的次波峰，信号质量较差，而S5X在LSP分析中只有一个较大波峰，信号质量最好。GLONASS和Beidou中信号的SNR序列有较好的一致性，其LSP分析结果振幅值比较稳定，都只有一个明显的波峰。Galileo中除了S1X在LSP中振幅值最低且最大波峰不明显，S5X、S7X、S8X、S6X四种信号LSP结果都没有虚假高峰，能够较好的得到反演结果。四系统不同信号的LSP表现出不同的结果，每个信号对应于不同的峰值，这表明同一系统不同类型的SNR会导致精度水平不一样的结果。 4　单模单频GNSS-MR潮位反演序列 1 以BRST测站为例，对去除趋势项的SNR序列进行LSP分析，可以获得每种信号的垂直反射距离 <math id="M80"><mover accent="true"><mi>h</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626238&type= 1.86266661 3.13266683 的时间序列。在设定LSP分析中峰值噪声比大于3后，四个系统潮位有效反演值如 图6 图6 所示。 1;2;3;4; 由 图6 图6 可知，四个系统每个信号通过GNSS-MR获取的潮位反演结果都能够描述潮位变化的趋势，但是这些结果与验潮站潮位测量值存在一定偏差。为了进一步评估各个信号的反演效果，同时对海面动态变化误差 <math id="M81"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626254&type= 5.67266655 4.65666676 和对流层延迟误差 <math id="M82"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626256&type= 5.58799982 3.89466691 进行改正，BRST测站每个信号反演序列 <math id="M83"><mover accent="true"><mi>h</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626258&type= 1.86266661 3.13266683 、 <math id="M84"><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626260&type= 11.93799973 4.99533367 和 <math id="M85"><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626262&type= 22.26733208 6.09600019 的均方根误差统计结果如 表1 表1 所示，经过两次误差改正后的精度提高百分比如 图7 图7 所示。 2 系统 2 信号 2 日均反演值数量 3 均方根误差（m） <math id="M86"><mover accent="true"><mi>h</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626264&type= 1.43933344 2.45533323 <math id="M87"><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mstyle displaystyle="true"><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mstyle></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626266&type= 9.31333351 4.14866638 <math id="M88"><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mstyle displaystyle="true"><mover><mrow><mi>h</mi></mrow><mrow><mo>·</mo></mrow></mover></mstyle></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>h</mi></mrow><mrow><mi>T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=22626268&type= 17.27199936 4.99533367 GPS S1C 38.5333 0.4640 0.3006 0.2758 S2W 28.5333 0.4669 0.2955 0.2652 S2X 25.9667 0.4730 0.3045 0.2716 S5X 19.4667 0.4517 0.2867 0.2502 GLONASS S1C 25.0333 0.4636 0.2382 0.2045 S1P 25.2000 0.4748 0.2491 0.2068 S2C 24.9333 0.4460 0.2308 0.2040 S2P 25.9333 0.4487 0.2272 0.2008 Galileo S1X 20.5333 0.4633 0.2617 0.2400 S5X 25.8667 0.5070 0.2675 0.2106 S7X 25.1000 0.4974 0.2733 0.2272 S8X 24.0333 0.5322 0.2670 0.2254 S6X 25.1333 0.4900 0.2593 0.2316 BeiDou S1X 23.9667 0.4752 0.2731 0.2488 S5X 28.0333 0.4994 0.2686 0.2360 S2I 26.4667 0.4681 0.3009 0.2749 S7I 3.8333 0.4746 0.3523 0.3202 S6I 31.2333 0.4900 0.2590 0.2268 结合 表1 表1 和 图7 图7 可知通过海面动态改正和对流层延迟改正，四个系统的反演精度大约提高了50%，GPS中S5X反演结果优于S1C、S2W和S2X，精度最高；GLONASS中S2C和S2P反演精度要优于S1C和S1P；Galileo的S5X反演精度最高，S8X、S7X和S6X较次之，S1X精度最低；BeiDou系统整体反演精度要低于其他三个系统，其中S7I反演精度最低，S6I反演精度最高。 对于BRST测站而言，经过误差改正后精度最好的SNR数据分别是GPS的S5X、GLONASS的S2P、Galileo的S5X和BeiDou的S61。分析可知随着信号的频率下降，反演的精度会随之提高，这可能是因为频率会影响天线增益模式和随机表面粗糙度，从而影响信号的质量（ Nievinski等，2014 Nievinski等，2014 ）。 5　GNSS-MR多模多频联合潮位反演 1 5.1　BRST站联合反演结果 2 在BRST测站中，结合 表1 表1 和 图6 图6 可以看出单系统单频的反演结果精度低且时间分辨率差，而单系统多频又会有反演结果在某一时间段堆叠的情况。为了解决上述问题，本文在窗口长度为2h，滑动步长为10min的时间窗口中建立状态方程，在稳健回归模型下实现四系统反演结果融合。滑动窗口长度的选择非常关键：长度过短，窗口中数据量过少，不足以进行可靠的抗差估计；窗口过长，则潮位的非线性运动造成的融合反演误差越大。文章从精度和数据损失率（损失数据量/窗口数量）两个方面分别比较0.5h、1h、1.5h、2h、2.5h、3h窗口潮位反演差异（如 表2 表2 所示），最终确定最优窗口长度为2h。 窗口长度 均方根误差（m） 数据损失率（%） 0.5h 0.2980 74.91 1h 0.1983 19.37 1.5h 0.1387 4.21 2h 0.1243 0.20 2.5h 0.1278 0.05 3h 0.1377 0 图8（a） 图8（a） 为四系统联合反演的结果，图中黑色实线表示验潮站实测数据，蓝色的点为联合反演结果，可以看出联合反演的结果较好的对应了验潮站的实测数据，同时实现了10min一个点的均匀采样，满足日常潮位监测的要求。改正后精度为12.43cm，相较于各系统单信号的反演结果，精度大约提高了40%-60%。 图8（b） 图8（b） 为反演潮位与实测潮位的残差分布，散点大都在-0.5m~0.5m范围内，在涨潮和落潮时残差波动较大。 图8（c） 图8（c） 是反演潮位与实测潮位的线性回归模型，相关系数为99.56%。 1;2;3; 5.2　HKQT站联合反演结果 2 为了进一步验证基于稳健回归模型融合算法的有效性，本文选取HKQT站2020年年积日为203-232期间 GPS、GLONASS、Galileo、BeiDou系统的SNR数据进行实验。其中信噪比数据GPS有S1C、S2W、S2X、S5X四种，GLONASS有S1C、S1P、S2C、S2P四种，Galileo有S1X、S5X、S7X、S8X四种，BeiDou有S2I、S7I两种。经过海面动态改正和对流层延迟改正后，四系统整体的潮位反演精度约为25.42cm。 在HKQT站中，同样从精度和数据损失率两个方面分别比较不同窗口潮位反演差异（如 表3 表3 所示），在保证精度和数据利用率的情况下最后选择2h的时间窗口，滑动步长为10min，然后在每个窗口中进行四系统反演结果融合。 图9（a） 图9（a） 为四系统联合反演的结果，图中黑色实线表示验潮站实测数据，红色的点为联合反演结果，联合反演的结果可以清晰的反映潮位的起伏变化。改正后精度为7.09cm，相较于四系统整体的反演结果，精度提高了约72%。 图9 图9 （b）为反演潮位与实测潮位的残差分布，散点大都集中在0m左右，整体表现更为稳定。 图9（c） 图9（c） 是反演潮位与实测潮位的线性回归模型，相关系数为99.06%。与BRST站的联合反演结果相比，基于稳健回归模型的数据融合算法在HKQT站即潮位起伏较小的区域表现更好，精度提高更多。虽然BRST站获取的信号种类多于HKQT站，但是由于Galileo和BeiDou系统存在个别信号质量低而影响了整体反演精度。 窗口长度 均方根误差（m） 数据损失率（%） 0.5h 0.1012 46.36 1h 0.0698 18.37 1.5h 0.0690 12.69 2h 0.0709 6.87 2.5h 0.0772 3.31 3h 0.0856 1.37 1;2;3; 6　结论 1 针对GNSS-MR技术在实际用于潮位监测中存在精度不高和时间分辨率低两大问题，本文提出一种基于IGGIII模型的稳健回归方法，在综合考虑海面动态变化和对流层延迟的情况下对多模多频GNSS-MR反演结果进行融合，有效改善了潮位反演结果的精度和时间分辨率。本文主要得出如下结论： （1）在潮位反演中多模多频数据相较于单系统单频数据可以大大提高反演结果的时间分辨率，但即使经过海面动态改正和对流层延迟改正，反演结果的精度依旧没有得到有效提高。在稳健回归模型下考虑海面动态变化误差和对流延迟，利用GNSS-MR技术进行潮位反演，BRST站和HKQT站反演结果RMSE分别为12.43cm和7.09cm，与传统方法相比精度提高约40%-70%。 （2）传统方法获取潮位反演值由于SNR数据采样时间不规律，潮位值也无法实现均匀采样。在基于稳健回归模型的数据融合算法中可以在特定的时间窗口内建立方程求解潮位值，BRST站和HKQT站均可实现10min的均匀采样，这大大提高了GNSS-MR技术用于潮位监测的时间分辨率，满足了潮位监测需求。 利用这种基于IGGIII模型的稳健回归方法对潮位反演值进行融合目前也存在一定局限性：（1）IGGIII模型也无法完全去除异常值对结果的影响，系统内存在的低质量信号会影响整体反演精度。（2）某一个时间点可能存在卫星数量少或者信号质量差的情况，这会导致这一时间段状态方程无法求解从而导致潮位数据缺失。（3）反演潮位时获得的潮位值是根据一段时间的SNR序列获得，一般将其与SNR序列的平均时刻对应；而验潮站测量结果也为短时间的平均值，因此反演值和观测值之间会存在时间尺度上的表示性误差。对于此类误差，需要在未来进一步研究。 参考文献（References） Anderson , Kenneth D . 2000 . 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Robust estimator for correlated observations based on bifactor equivalent weights 10.11834/jrs.20211227 陈殊，何秀凤，王笑蕾，宋敏峰.XXXX.基于IGGIII模型的多模多频GNSS-MR潮位反演融合研究.遥感学报，XX（XX）： 1-14 陈殊，何秀凤，王笑蕾，宋敏峰.XXXX.基于IGGIII模型的多模多频GNSS-MR潮位反演融合研究.遥感学报，XX（XX）： 1-14 CHEN Shu，He Xiufeng，WANG Xiaolei，SONG Minfeng. XXXX. Sea level combined retrievals using multi-GNSS multipath reflectometry based on the IGGIII scheme. National Remote Sensing Bulletin， XX（XX）：1-14 CHEN Shu，He Xiufeng，WANG Xiaolei，SONG Minfeng. XXXX. Sea level combined retrievals using multi-GNSS multipath reflectometry based on the IGGIII scheme. National Remote Sensing Bulletin， XX（XX）：1-14 国家自然科学基金重点项目（编号：41830110）；国家重点研究发展计划项目（编号：2018YFC1503603） Supported by Supported by National Natural Science Foundation of China (No. 41830110); National Key Research and Development Project of China (No. 2018YFC1503603) 2021-04-18 2021-12-08 遥感学报