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Article(id=1322506571238719741, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322506501562937604, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2024.12.604, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1736352000000, receivedDateStr=2025-01-09, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793239737567, onlineDateStr=2026-10-29, pubDate=1765728000000, pubDateStr=2025-12-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793239737567, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793239737567, creator=13701087609, updateTime=1793239737567, updator=13701087609, issue=Issue{id=1322506501562937604, tenantId=1146029695717560320, journalId=1281212831689347082, year='2025', volume='45', issue='12', pageStart='1211', pageEnd='1316', issueExtLink='null', onlineDate='null', pubDate='1765728000000', pubDateStr='2025-12-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793239720954, creator='13701087609', updateTime=1793239720954, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=1215, endPage=1220, ext={EN=ArticleExt(id=1322506571435852031, articleId=1322506571238719741, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=GNSS-IR Soil Moisture Inversion Accounting for Residual SNR Unmodeled Errors, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

To address the issue of deviations in amplitude extraction caused by unmodeled errors in global navigation satellite system interferometric reflectometry(GNSS-IR) soil moisture inversion, this study proposes a method that utilizes a semi-parametric model to introduce nonparametric vectors compensating for unmodeled errors, thereby optimizing amplitude extraction. The soil moisture inversion results based on parametric models are compared with those derived from semi-parametric models. The findings demonstrate that the amplitude extracted by the semi-parametric model achieves higher accuracy in both single satellite and multi-satellite fusion scenarios. The proposed approach effectively improves inversion precision by mitigating unmodeled errors through semi-parametric compensation.

, authors=Yan ZHOU1, Qian CHEN1, Xilong KOU1, Haigang PANG1, Bo SUN1, *, authorsList=Yan ZHOU, Qian CHEN, Xilong KOU, Haigang PANG, Bo SUN, authorCompany=null, correspAuthors=Bo SUN, authorNote=

ZHOU Yan, PhD, lecturer, majors in GNSS data processing and applications, E-mail:

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针对全球导航卫星系统干涉反射(GNSS-IR)反演土壤湿度中，未建模误差导致振幅提取结果存在一定偏差的问题，提出一种利用半参数模型引入非参数向量补偿未建模误差以优化振幅提取的方法。将参数模型与半参数模型提取振幅的土壤湿度反演结果进行对比。结果表明，相较于参数模型，半参数模型提取的振幅在单星及多星融合上进行土壤湿度反演精度更优，利用半参数模型补偿未建模误差能够提高反演结果的精确性。

, authors=周岩1, 陈倩1, 寇希龙1, 逄海港1, 孙波1, *, authorsList=周岩, 陈倩, 寇希龙, 逄海港, 孙波, authorCompany=null, correspAuthors=孙波, authorNote=

周岩，博士，讲师，主要研究方向为GNSS数据处理及应用，E-mail：

, correspAuthorsNote=

孙波，博士，讲师，主要研究方向为GNSS反射信号遥感、人工智能，E-mail：

SUN Bo, PhD, lecturer, majors in GNSS reflectometry remote sensing, artificial intelligence, E-mail:

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ZHOU Yan, PhD, lecturer, majors in GNSS data processing and applications, E-mail:

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周岩，博士，讲师，主要研究方向为GNSS数据处理及应用，E-mail：

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周岩，博士，讲师，主要研究方向为GNSS数据处理及应用，E-mail：

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Journal of Shanghai Jiaotong University, 2020, 54 (6): 569- 576, articleTitle=null, refAbstract=null)], funds=[Fund(id=1322506580097089861, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506571238719741, awardId=ZR2021MD082, language=EN, fundingSource=Natural Science Foundation of Shandong Province(ZR2021MD082), fundOrder=null, country=null), Fund(id=1322506580172587334, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506571238719741, awardId=ZR2021MD082, language=CN, fundingSource=山东省自然科学基金(ZR2021MD082), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1322506574841626893, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506571238719741, xref=null, ext=[AuthorCompanyExt(id=1322506574854209806, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506571238719741, companyId=1322506574841626893, language=EN, country=null, province=null, city=null, postcode=null, 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Scheme settings

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方案一元线性回归(单星)多元线性回归(多星)
方案设置方案1-1方案1-2方案2-1方案2-2
方案释义半参数模型参数模型半参数模型参数模型
), ArticleFig(id=1322506579627327808, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506571238719741, language=CN, label=表1, caption=

方案设置

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方案一元线性回归(单星)多元线性回归(多星)
方案设置方案1-1方案1-2方案2-1方案2-2
方案释义半参数模型参数模型半参数模型参数模型
), ArticleFig(id=1322506579702825281, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506571238719741, language=EN, label=Tab. 2, caption=

The precision statistics of single satellite inversion using two methods

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方法精度卫星
PRN12PRN14PRN25PRN28
半参数模型 R2 0.643 4 0.922 5 0.352 2 0.725 8
RMSE/cm3·cm-3 0.017 1 0.005 5 0.015 8 0.010 3
MAE/cm3·cm-3 0.009 6 0.003 7 0.012 3 0.009 0
参数模型R2 0.538 5 0.915 4 0.165 7 0.674 5
RMSE/cm3·cm-3 0.013 3 0.005 7 0.017 9 0.011 2
MAE/cm3·cm-3 0.011 4 0.003 8 0.015 6 0.009 9
), ArticleFig(id=1322506579795099970, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506571238719741, language=CN, label=表2, caption=

单颗卫星2种方法反演精度统计

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方法精度卫星
PRN12PRN14PRN25PRN28
半参数模型 R2 0.643 4 0.922 5 0.352 2 0.725 8
RMSE/cm3·cm-3 0.017 1 0.005 5 0.015 8 0.010 3
MAE/cm3·cm-3 0.009 6 0.003 7 0.012 3 0.009 0
参数模型R2 0.538 5 0.915 4 0.165 7 0.674 5
RMSE/cm3·cm-3 0.013 3 0.005 7 0.017 9 0.011 2
MAE/cm3·cm-3 0.011 4 0.003 8 0.015 6 0.009 9
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The mean accuracy statistics of multi-satellite inversion

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卫星个数 使用模型 评价指标
R2 RMSE/cm3·cm-3 MAE/cm3·cm-3
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参数 0.573 5 0.012 0 0.010 2
双星 半参数 0.859 6 0.007 1 0.005 5
参数 0.776 7 0.008 9 0.007 0
3星 半参数 0.914 7 0.006 1 0.004 2
参数 0.864 4 0.006 9 0.005 1
4星 半参数 0.943 4 0.004 7 0.003 4
参数 0.922 4 0.005 5 0.003 5
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多卫星反演精度均值统计

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卫星个数 使用模型 评价指标
R2 RMSE/cm3·cm-3 MAE/cm3·cm-3
单星 半参数 0.661 0 0.010 8 0.008 6
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参数 0.776 7 0.008 9 0.007 0
3星 半参数 0.914 7 0.006 1 0.004 2
参数 0.864 4 0.006 9 0.005 1
4星 半参数 0.943 4 0.004 7 0.003 4
参数 0.922 4 0.005 5 0.003 5
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顾及残余SNR未建模误差的GNSS-IR土壤湿度反演
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周岩 1 , 陈倩 1 , 寇希龙 1 , 逄海港 1 , 孙波 1, *
大地测量与地球动力学 | 现代大地测量 2025,45(12): 1215-1220
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大地测量与地球动力学 |现代大地测量 2025 , 45 (12) : 1215 -1220
顾及残余SNR未建模误差的GNSS-IR土壤湿度反演
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周岩1 , 陈倩1, 寇希龙1, 逄海港1, 孙波1, *
作者信息
  • 1 山东农业大学信息科学与工程学院, 山东省泰安市岱宗大街61号, 271018
通讯作者:

孙波，博士，讲师，主要研究方向为GNSS反射信号遥感、人工智能，E-mail：

SUN Bo, PhD, lecturer, majors in GNSS reflectometry remote sensing, artificial intelligence, E-mail:

作者简介:

周岩，博士，讲师，主要研究方向为GNSS数据处理及应用，E-mail：

GNSS-IR Soil Moisture Inversion Accounting for Residual SNR Unmodeled Errors
Yan ZHOU1 , Qian CHEN1, Xilong KOU1, Haigang PANG1, Bo SUN1, *
Affiliations
  • 1 College of Information Science and Engineering, Shandong Agricultural University, 61 Daizong Street, Taian 271018, China

    • ZHOU Yan, PhD, lecturer, majors in GNSS data processing and applications, E-mail:

出版时间: 2025-12-15 doi: 10.14075/j.jgg.2024.12.604
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摘要
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针对全球导航卫星系统干涉反射(GNSS-IR)反演土壤湿度中，未建模误差导致振幅提取结果存在一定偏差的问题，提出一种利用半参数模型引入非参数向量补偿未建模误差以优化振幅提取的方法。将参数模型与半参数模型提取振幅的土壤湿度反演结果进行对比。结果表明，相较于参数模型，半参数模型提取的振幅在单星及多星融合上进行土壤湿度反演精度更优，利用半参数模型补偿未建模误差能够提高反演结果的精确性。

GNSS-IR  /  土壤湿度  /  半参数模型  /  多星融合  /  反演精度

To address the issue of deviations in amplitude extraction caused by unmodeled errors in global navigation satellite system interferometric reflectometry(GNSS-IR) soil moisture inversion, this study proposes a method that utilizes a semi-parametric model to introduce nonparametric vectors compensating for unmodeled errors, thereby optimizing amplitude extraction. The soil moisture inversion results based on parametric models are compared with those derived from semi-parametric models. The findings demonstrate that the amplitude extracted by the semi-parametric model achieves higher accuracy in both single satellite and multi-satellite fusion scenarios. The proposed approach effectively improves inversion precision by mitigating unmodeled errors through semi-parametric compensation.

GNSS-IR  /  soil moisture  /  semi-parametric model  /  multi-satellite fusion  /  inversion accuracy
周岩, 陈倩, 寇希龙, 逄海港, 孙波. 顾及残余SNR未建模误差的GNSS-IR土壤湿度反演. 大地测量与地球动力学, 2025 , 45 (12) : 1215 -1220 . DOI: 10.14075/j.jgg.2024.12.604
Yan ZHOU, Qian CHEN, Xilong KOU, Haigang PANG, Bo SUN. GNSS-IR Soil Moisture Inversion Accounting for Residual SNR Unmodeled Errors[J]. Journal of Geodesy and Geodynamics, 2025 , 45 (12) : 1215 -1220 . DOI: 10.14075/j.jgg.2024.12.604
正文
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土壤湿度是量化陆地和大气能量交换的关键参数，准确获取高精度、实时的土壤湿度监测和预报数据，对指导农业生产具有重要意义。传统的土壤湿度测量方法，如烘干称重法，难以满足大范围的实时监测需求[1]。GNSS-IR作为一种新型遥感技术，可通过非接触方式监测土壤湿度，并支持多站点布置，可为大范围区域提供土壤湿度数据。国内外学者已在GNSS-IR领域展开了多学科、多角度的研究。Larson等[2]发现反射信号影响信噪比数据的收集，且产生的振幅变化与土壤含水量相关。Rodriguez-Alvarez等[3]提出利用直接信号与经过地面散射后的反射信号之间的干涉信号功率进行土壤湿度探测。Chew等[4]发现多路径信号的相位、振幅受土壤湿度的影响，相位与表层土壤水分呈线性相关。敖敏思等[5]利用GNSS观测数据与实测土壤湿度数据进行对比分析，发现反射信号的相位能够反映土壤湿度的变化。汉牟田等[6]考虑天线增益、土壤介电常数和噪声的影响，提出利用GNSS干涉信号振幅进行土壤湿度反演的方法。Roussel等[7]分析了2°~70°卫星高度角SNR数据中提取出的观测量，发现由低高度角中提取的特征参量与土壤湿度间相关性更高。张双成等[8]利用多卫星赋权融合解译土壤湿度变化的算法进行土壤湿度反演，实验表明，多卫星融合的反演精度较单颗卫星有较大提升。郭斐等[9]提出了一种多特征融合的GNSS-IR土壤湿度反演方法，证明了多特征融合后的精度较单一特征反演有很大提升。
综合已有研究表明，在使用GNSS-IR技术进行土壤湿度反演时，需要利用最小二乘法拟合残余SNR振荡项的时间序列得到振幅。但由于使用谐波函数模型拟合残余SNR振荡项未充分考虑卫星性能和多路径环境差异等因素的影响，在提取精度上有所欠缺。因此，本文引入半参数模型进行振幅提取，并研究单星、多星数据融合的反演算法，最后通过实验验证在单星或多星条件下，半参数模型可以有效提高土壤湿度反演的精度。
1 GNSS-IR基本理论
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在GNSS测量中，测站周围物体的反射信号会对卫星的直射信号产生干扰，造成观测值与真实值的偏差而影响测量精度，GNSS-IR技术就是利用从卫星发射的直射信号与地面反射回接收机的反射信号产生的干涉现象来进行地表反射面物理特征的提取。
GNSS-IR技术的核心在于利用接收到的SNR数据来分析GNSS的直射信号和反射信号的变化[10]。由于GNSS卫星的运动，直射、反射信号的干涉特征参量会随时间不断变化，接收机记录这些变化信息，并将其表现为SNR的变化。因此，研究SNR的变化可以揭示干涉特征参量的变化，从而反演土壤湿度。
在地基单天线系统中，由于平台固定和天线架设较低，直射信号与反射信号的频率相近，因此这2种信号在接收天线处叠加时会形成相对稳定的干涉信号。干涉信号SNR可表示为[11]：
$\mathrm{SNR}^{2}=A_{d}^{2}+A_{m}^{2}+2 A_{d} A_{m} \cos \varphi$
式中，Ad和Am为直射信号和多路径信号的振幅；φ为2个信号之间的相位差。
用低阶多项式对SNR原序列进行拟合以去除趋势项，残余SNR振荡项可表示为：
$\mathrm{SNR}_{m}=A \cos \left(\frac{4 \pi h}{\lambda} \sin E+\varphi\right) $
式中，A为振幅；h为天线相位中心到地面的高度；λ为GNSS信号波长；E为卫星高度角；φ为相移。
由$f=\frac{2 h}{\lambda}、x=\sin E$，式(2)可转化为：
$\mathrm{SNR}_{m}=A \cos (2 \pi f x+\varphi)$
已有研究表明，卫星反射信号干涉特征参量(包括振幅、相位和频率)和土壤湿度之间存在较强的相关性，能够用于土壤湿度反演。
2 半参数模型基本理论
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2.1 半参数模型
在实际问题中，影响观测值的因素很多，仅使用单一参数模型不能很好地解决问题。因此，结合参数模型与非参数模型发展出了一种统计模型，即半参数模型，它既包含了参数部分又包含了非参数部分，可表示为：
$\boldsymbol{L}=\boldsymbol{B} \boldsymbol{X}+\boldsymbol{s}+\boldsymbol{\varDelta}$
式中，$\boldsymbol{L}$ 为观测值；$\boldsymbol{B}$ 为列满秩设计矩阵；$\boldsymbol{X}$ 为待求参数向量；$\boldsymbol{s}$ 为非参数分量，用来拟合未建模误差；$\boldsymbol{\varDelta}$ 为观测误差。
由式(4)可得误差方程式为：
$\boldsymbol{V}=\boldsymbol{B} \hat{\boldsymbol{X}}+\hat{\boldsymbol{s}}-\boldsymbol{L} $
式中，$\hat{\boldsymbol{X}}$ 为待求参数估值；$\hat{\boldsymbol{s}}$为非参数分量估值。
进一步根据补偿最小二乘原则[12]可得：
$\boldsymbol{V}^{\mathrm{T}} \boldsymbol{P} \boldsymbol{V}+\alpha \hat{\boldsymbol{s}}^{\mathrm{T}} \boldsymbol{R} \hat{\boldsymbol{s}}=\min$
式中，P为正定矩阵；α为平滑因子；R为正则化矩阵，在极小化过程中对V和s起平衡作用。
确定正则化矩阵R与平滑因子α非常重要，由于SNR值是观测序列，相邻时刻模型误差相差不大，因此使用时间序列法来确定正则化矩阵：
$\boldsymbol{R}=\boldsymbol{T}^{\mathrm{T}} \boldsymbol{T} $
$\begin{gather*}\boldsymbol{T}=\left[\begin{array}{cccc}-1 & 1 & 0 & \cdots \\0 & -1 & \cdots & \cdots \\\cdots & \cdots & \cdots & 1 \\0 & \cdots & \cdots & -1\end{array}\right]\end{gather*}$
使用L曲线法确定最佳平滑因子[13]。在补偿最小二乘原则中，设噪音范数为$\boldsymbol{V}^{\mathrm{T}} \boldsymbol{P V}$，信号范数为$\hat{\boldsymbol{s}}^{\mathrm{T}} \boldsymbol{R} \hat{\boldsymbol{s}}, L$曲线是以信号范数作为横轴，噪音范数作为纵轴拟合而成的曲线，拐点处的值对应的α值即为最佳平滑因子。求最佳平滑因子有2条标准：一是离原点距离最小时的α值；二是曲线曲率最大时的α值[14]。
基于补偿最小二乘原则，根据拉格朗日极值法求解非参数分量s的估计值和待求参数X的估计值：
$\begin{align*}& \left\{\begin{array}{l} \hat{\boldsymbol{s}}=(\alpha \boldsymbol{R}+\boldsymbol{M})^{-1} \boldsymbol{M} \boldsymbol{L} \\\boldsymbol{M}=\boldsymbol{P}-\boldsymbol{P} \boldsymbol{B}\left(\boldsymbol{B}^{\mathrm{T}} \boldsymbol{P} \boldsymbol{B}\right)^{-1} \boldsymbol{B}^{\mathrm{T}} \boldsymbol{P}\end{array}\right. \end{align*}$
$ \hat{\boldsymbol{X}}=\left(\boldsymbol{B}^{\mathrm{T}} \boldsymbol{P} \boldsymbol{B}\right)^{-1} \boldsymbol{B}^{\mathrm{T}} \boldsymbol{P}(\boldsymbol{L}- \hat{\boldsymbol{s}})$
2.2 半参数模型求解特征参量
传统方法一般利用谐波函数模型拟合残余SNR振荡项，通过最小二乘法进行振幅特征参量的提取。然而在GNSS测站收集信号时，观测数据容易受到外界环境的影响，并且这些影响因素众多，它们之间的函数关系复杂。如果仅采用参数回归的方法来建立模型，那么所建立的参数模型会与实际情况存在差异，从而影响最终的估计结果。因此，将半参数模型引入残余SNR振荡项数据处理中，改进常用的参数模型，使用非参数对受卫星性能和多路径环境差异等因素的影响而产生的未建模误差进行补偿，以得到性能更佳的参数。
使用半参数模型提取干涉特征参量的操作流程如下：
1) 将残余SNR振荡项转化为半参数模型表示。将式(3)根据半参数模型式(4)改为：
$L(x)=A \cos (2 \pi f x+\varphi)+s(x)+\varDelta(x) $
将式(11)展开：
$\begin{gather*}L(x)=A \cos (2 \pi f x) \cos \varphi- \\A \sin (2 \pi f x) \sin \varphi+s(x)+\varDelta(x) \end{gather*}$
令$B_{1}=\cos (2 \pi f x)、B_{2}=-\sin (2 \pi f x)、X_{1}= A \cos \varphi、X_{2}=A \sin \varphi$，得出：
$L(x)=B_{1} X_{1}+B_{2} X_{2}+s(x)+\varDelta(x) $
2) 获取频率与高度角。通过Lomb-Scargle谱分析获取SNR振荡项频率，获得对应2°~30°高度角的正弦值。
3) 确定拟合未建模误差。通过时间序列法来确定正则化矩阵，使用L曲线法确定最佳平滑因子，根据补偿最小二乘原则，求解未建模误差s(x)。
4) 求解振幅。通过式(10)得到X1与X2的值，根据式(14)求出振幅：
$A= \pm \sqrt{X_{1}^{2}+X_{2}^{2}} $
3 实际数据处理结果
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3.1 数据来源
本文所用数据为作者团队与法国图卢兹第三大学团队于2014-02-04—03-21在法国图卢兹市拉马斯克雷(Lamasquere)小镇的一处农场采集的GPS数据。实验场地平坦开阔，实验期间地表近似裸土，实验场地如图 1所示。
GPS信号采集设备是徕卡公司的GR25型接收机和AR10基准站天线，天线架设高度为1.69~1.70 m。土壤湿度采集设备为2枚ML3型FDR土壤湿度传感器，传感器埋置深度为2 cm和5 cm，测量点距离约为2 m，采样时间间隔为2 min。考虑到电磁波在土壤中的穿透能力，选取0~5 cm深度的土壤湿度数据作为参照。
3.2 数据处理
按照土壤湿度反演流程(图 2)对GPS系统接收数据进行处理，具体步骤如下：
1) 数据预处理。对RINEX数据文件进行预处理，考虑到高度角太高或太低都会影响反射信号的采集，选取接收功率较强的2°~30°高度角的GPS Ll频段的SNR信号，使用低阶多项式去除SNR趋势项。
2) 特征提取。通过Lomb-Scargle谱分析获取SNR振荡项频率，分别通过参数模型与半参数模型获取SNR振荡项振幅与相位。
3) 建模分析。经验证，在振幅、相位和频率3种干涉特征参量中，振幅特征参量与土壤湿度的相关性最高，故选取振幅作为输入数据，土壤湿度实测值作为输出数据，建立线性回归模型。
综合考虑实验场地的经纬度、高度角、天线架设位置和方位角的限制，选用PRN12、PRN14、PRN25、PRN28卫星进行实验研究，并以决定系数(R2)、均方根误差(RMSE)、平均绝对误差(MAE)作为性能评价指标，对半参数模型辅助下的土壤湿度反演与参数模型辅助下的土壤湿度反演进行对比。
为验证在单星和多星条件下半参数模型提取干涉特征参量进行土壤湿度反演的可行性，设置4种方案进行对比分析，如表 1所示。
方案1-1将半参数模型提取的振幅作为输入参量，使用一元线性回归模型($y=b_{0}+b_{1} x+\varDelta$)进行单颗卫星土壤湿度反演；方案1-2将参数模型提取的振幅作为输入参量进行单颗卫星土壤湿度反演；方案2-1是将半参数模型提取的振幅使用2~4颗卫星的数据作为输入参量，使用多元线性回归模型$\left(y=b_{0}+b_{1} x_{1}+b_{2} x_{2}+\cdots+b_{n} x_{n}+\right. \varDelta)$ 进行土壤湿度反演；方案2-2是将参数模型提取的振幅使用2~4颗卫星的数据作为输入参量进行土壤湿度反演。
3.3 结果分析
首先按照方案1-1和方案1-2，对4颗卫星使用半参数模型与参数模型进行单星土壤湿度反演，图 3为半参数模型与参数模型的土壤湿度线性回归分析图。从图 3中可以看出，PRN14显示出最高的相关性，而PRN25由于受到植被的影响，其相关性相对较低。对比图 3(a)和图 3(b)可以发现，相较于参数模型，采用半参数模型提取的振幅进行一元线性回归时具有更高的相关性。
在半参数模型与参数模型条件下，PRN14和PRN28的反演值与土壤湿度真实值的绝对误差如图 4所示，可以看出，半参数模型条件下的整体误差值更小。这是因为半参数模型相对于参数模型拟合了未建模误差，从而提高了提取精度。
从图 3、4可以看出，半参数模型辅助下的线性拟合程度相对于传统的参数模型辅助下的线性拟合程度更优。结合表 2可以看出，相对于参数模型，利用半参数模型提取的振幅特征干涉参量在进行土壤湿度反演时精度有所提升，且半参数模型方法对相关性较差的卫星数据提升明显。
图 5为方案2-1与方案2-2中部分组合土壤湿度反演值与土壤湿度实测值的拟合图。可以看出，在双星、3星和4星融合情况下，使用半参数模型提取的振幅进行多元线性回归的反演精度优于使用参数模型的反演结果。
为了进一步验证半参数模型的可行性，分别使用半参数模型与参数模型进行土壤湿度反演，将土壤湿度反演结果与土壤湿度实测值进行对比，如图 6所示。从图 6可以看出，相对于参数模型，半参数模型提取的振幅进行土壤湿度反演的值更接近于真实值。此外，4星融合反演比单星、双星融合、3星融合的反演精度更高。
结合图 6与表 3可以看出，在单星及多星条件下，通过半参数模型提取的振幅进行土壤湿度反演的精度均优于参数模型。此外，随着卫星融合数量的增加，土壤湿度反演精度也在提高。综上，由于多星轨道分布的多样性，通过利用不同卫星之间的性能差异可以有效提高土壤湿度反演精度。
4 结语
收起
针对传统的参数模型在拟合残余信噪比振荡项提取振幅时由于未建模误差的影响，导致提取的振幅有偏差这一问题，研究了一种基于半参数模型的干涉特征参量提取方法，并结合实验结果对其在土壤湿度反演方面的性能进行分析。实验结果表明：
1) 引入非参数向量表示未建模误差，可以改善振幅提取精度，从而提升土壤湿度反演性能，在卫星配置相同时，采用半参数模型较参数模型结果更优。
2) 由于多星轨道分布的多样性，不同卫星的轨道位置和观测角度不同，接收到地面反射的信号也不同，因此所用算法相同时，融合多卫星比单颗卫星反演结果更优。
3) 本文所采用的半参数方法对于其他类型的GNSS-IR应用也有一定的参考价值，例如水面测高等。
综上所述，采用半参数模型提取振幅进行多星融合的土壤湿度反演是可行的。此外，由于卫星性能不一，半参数模型在小样本或数据丢失较多卫星的反演方法中的应用有待进一步研究。
基金
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  • 山东省自然科学基金(ZR2021MD082)
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2025年第45卷第12期
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doi: 10.14075/j.jgg.2024.12.604
  • 接收时间：2025-01-09
  • 首发时间：2026-10-29
  • 出版时间：2025-12-15
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  • 收稿日期：2025-01-09
基金
Natural Science Foundation of Shandong Province(ZR2021MD082)
山东省自然科学基金(ZR2021MD082)
作者信息
    1 山东农业大学信息科学与工程学院, 山东省泰安市岱宗大街61号, 271018

通讯作者:

孙波，博士，讲师，主要研究方向为GNSS反射信号遥感、人工智能，E-mail：

SUN Bo, PhD, lecturer, majors in GNSS reflectometry remote sensing, artificial intelligence, E-mail:

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2种不同金属材料的力学参数
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Number of
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Number of
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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