地球信息科学学报 2018 , 20 (1): 108-118 https://doi.org/10.12082/dqxxkx.2018.170319

遥感科学与应用技术

地表组合粗糙度不确定性引起的SAR反演土壤水分的不确定性分析

陈鲁皖, 韩玲^*^,, 王文娟, 秦小宝

长安大学地质工程与测绘学院,西安 710064

Uncertainty Analysis of SAR-retrieved Soil Moisture Induced by Uncertainty of Soil Surface Combined Roughness

CHEN Luwan, HAN Ling^*^,, WANG Wenjuan, QIN Xiaobao

College of Geology Engineering and Geomatics, Chang’an University, Xi'an 710064, China

通讯作者: *通讯作者：韩玲（1964- ）,女,博士,教授,主要从事遥感地质领域的研究。E-mail: hanling@chd.edu.cn

收稿日期: 2017-07-12

修回日期: 2017-11-6

网络出版日期: 2018-01-20

基金资助: 国家重大高分专项军事测绘专业处理与服务系统地理空间信息融合分系统（GFZX04040202-07）;中央高校基本科研业务费专业资金项目（310826175031）

作者简介:

作者简介：陈鲁皖（1980- ）,男,博士生,讲师,主要从事微波土壤水分反演研究。E-mail: 368848532@qq.com

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摘要

地表粗糙度的不确定性是引起SAR土壤水分反演结果不确定性的主要因素,现有研究大多着重于研究单个粗糙度参数（主要是相关长度）的不确定性,直接研究地表组合粗糙度不确定性的较少。本文使用偏度、峰度和四分位距3个指标来量化不确定性,通过在组合粗糙度中加入不同量级高斯噪声进行随机扰动的方法,研究组合粗糙度不确定性在反演过程中的传递,并对反演土壤水分的不确定性进行定量分析。进一步研究反演土壤水分的均方根误差对组合粗糙度不同比例误差范围的响应特征,得到满足反演精度要求的组合粗糙度误差控制范围。样区的实验分析结果表明：组合粗糙度高斯噪声标准差在0-0.045之间时,峰度取值从-0.1984到1.2501,偏度取值从0.0191到0.6791,四分位距取值从0.0018到0.0167,3个量化指标都随组合粗糙度高斯噪声量级的增大而增大,土壤水分反演值有集中在众数附近的趋势,土壤水分低估倾向比高估倾向更明显;本文提出的组合粗糙度误差控制范围可满足反演精度要求,误差控制范围与入射角负相关。

关键词： 不确定性量化 ; 组合粗糙度 ; 土壤水分 ; 峰度 ; 偏度 ; 四分位距 ; 误差控制范围

Abstract

Soil moisture is a key factor in the energy and water balance of the earth's surface, and it also plays an important role in the ecological environment. Soil moisture inversions based on Synthetic Aperture Radar （SAR） have shown promising progress but do not easily meet expected application requirements because a number of inversion algorithms cannot quantify the uncertainty of soil moisture inversions. Uncertainty of surface roughness is the main factor that causes uncertainty of SAR-retrieved soil moisture. Most of the existing studies focused on the uncertainty of single roughness parameter (correlation length), and seldom directly studied the uncertainty of surface combined roughness. The uncertainty was usually estimated by probability distribution of model parameter values in existing studies. Then, the probability distribution was propagated through the inversion process. Finally, the probability distribution of soil moisture inversion was obtained. The uncertainty was quantified by using skewness, kurtosis, and interquartile range in this paper. First of all, the range and distribution of the measured soil moisture data and roughness data in sampling area were counted and analyzed. Input values and scope of the AIEM model parameters were obtained. Then, effective correlation length was calculated by using the LUT (look up tables) method based on the measured soil moisture data and backscattering coefficients, and the effective combined roughness was obtained. The nonlinear relationship between the effective combined roughness and backscattering coefficients was constructed. By adding different levels of Gaussian noise to surface combined roughness, the uncertainty propagating of surface combined roughness in the process of retrieved soil moisture was studied, and the uncertainty of soil moisture retrieval was quantitatively analyzed. For each Gauss noise level, 1000 effective combined roughness sampling values with noise were obtained. By using the nonlinear relationship between the effective combined roughness and the backscattering coefficient, the backscattering coefficients corresponding to the sampling value of each effective combined roughness were derived. The soil moisture was obtained by using the empirical equation of soil moisture inversion. The skewness, kurtosis and interquartile range of the effective combined roughness and the inversion results were calculated. By using the AIEM （Advanced Integrated Equation Model） model and the limited range of input parameters, a large number of simulated data were obtained. The effective combined roughness of the simulation was introduced into different proportion error according to the initial value, and the soil moisture was obtained by the empirical equation of soil moisture inversion. Furthermore, according to the response characteristics between RMSE （Root Mean Square Error） of retrieved soil moisture and the error range of combined roughness, the error control range that meets the inversion accuracy requirement was obtained. The experimental results of sample area show that kurtosis range is -0.1984 to 1.2501, the deviation range is 0.0191 to 0.6791, and interquartile range is 0.0018 to 0.0167 when gaussian noise standard deviation range of composite roughness is 0 to 0.045. Also, these three quantitative indexes increase with the increase of combined roughness gaussian noise. Soil moisture inversion values tend to be concentrated near mode, and the tendency to underestimate soil moisture is more obvious than the overestimation tendency. The error range of combined roughness should be controlled within a certain range of the initial value to meet the inversion accuracy requirement, and it is negatively related to the incident angle. The error control range is suitable for bare soil with low surface roughness and low sparse vegetation coverage area.

Keywords： uncertainty quantification ; surface combined roughness ; soil moisture ; skewness ; kurtosis ; quartile ; error control range

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本文引用格式导出 EndNote Ris Bibtex

陈鲁皖, 韩玲, 王文娟, 秦小宝. 地表组合粗糙度不确定性引起的SAR反演土壤水分的不确定性分析[J]. 地球信息科学学报, 2018, 20(1): 108-118 https://doi.org/10.12082/dqxxkx.2018.170319

CHEN Luwan, HAN Ling, WANG Wenjuan, QIN Xiaobao. Uncertainty Analysis of SAR-retrieved Soil Moisture Induced by Uncertainty of Soil Surface Combined Roughness[J]. Journal of Geo-information Science, 2018, 20(1): 108-118 https://doi.org/10.12082/dqxxkx.2018.170319

1 引言

土壤水分对地球表层的能量和水分平衡具有重要作用,土壤水分的时空分布与动态变化对陆地-大气间热量平衡、大气环流产生显著影响,土壤水分还是农业、林业、气象学、水文学和生态学等领域研究的重要参数^[1]。利用空间分辨率较高的微波遥感反演土壤水分,可以实现对土壤水分动态、实时、大范围和较高精度的估算。

土壤水分反演过程中的不确定性分析是当前研究的热点,Keysera等^[2]提出了一种基于线性回归的粗糙度参数化方法,并使用概率分布对其不确定性进行估计,通过反演模型进一步传播不确定性,得到反演土壤水分的概率分布。FernA´ndez-GA´lvez^[3]认为大多数土壤水分反演方法依赖于土壤水分与介电特性之间的关系,比较了包含这种关系的9种反演模型,并对土壤水分和介电特性之间的不确定性引起的误差进行量化。Ma等^[4]提出了一种基于贝叶斯定理和马尔可夫链蒙特卡罗方法的土壤水分概率反演（PI）算法,能够定量描述土壤水分反演的不确定性,通过最大似然估计得到高精度的土壤水分估计。这些研究主要集中在粗糙度（如相关长度）、土壤介电常数等参数的不确定性分析上,而对于反演中常用的地表组合粗糙度的不确定性分析的研究较少。根据Verhoest等^[5]的研究,利用SAR进行土壤水分反演,地表粗糙度参数的不确定性是导致土壤水分反演误差的重要来源。地表粗糙度的不确定性包括测量过程中的不确定性和反演过程中的不确定性。而地表粗糙度参数是难以准确测量的,测量结果取决于测量设备的剖面长度^[6]和测量技术,即可以从相同的土壤表面得到不同的粗糙度参数值。研究表明,仅依赖单个传统的粗糙度参数（均方根或相关长度）很难描述粗糙度效应,需要同时考虑多个粗糙度参数^[7],如地表组合粗糙度,可以参与构建反演经验方程得到较好的地表参数反演结果。

本文使用Rahman的LUT（Look-Up Tables）方法^[8]计算有效相关长度（l_mod）来代替实际测量值,可避免引入粗糙度参数测量过程中的不确定性,并使用 $\begin{matrix} Z_{s} = \sqrt{\frac{S^{2}}{l}} \end{matrix}$ 形式^[9]得到地表有效组合粗糙度Z_s。在粗糙度的不确定性是土壤含水量不确定性的主要来源的假设下,主要解决2个问题：① 有效Z_s的不确定性如何影响反演土壤水分（M_v）的不确定性?② 有效Z_s的误差控制在多大范围才能得到符合精度要求的M_v反演结果?目前在现有土壤水分反演不确定性研究中,通常通过偏度、峰度、四分位距3个模型参数指标来量化数据的不确定性,得到反演土壤水分值分布的不对称程度和方向,土壤水分分布相对于正态分布而言是更陡峭还是平缓,以及土壤水分分布的离散程度;根据不同程度的高斯噪声对Z_s的取值进行随机扰动,得到大量Z_s采样值,并构建Z_s与后向散射系数（ $\begin{matrix} σ_{pq}^{0} \end{matrix}$ ）之间的非线性回归关系,利用该关系计算得到对应的 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ ;通过 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 、M_v和Z_s构成的反演经验方程反演M_v,分析Z_s的不确定性在反演过程中的传递,并使用偏度、峰度和四分位距来分析反演的M_v。为满足一定的M_v反演精度要求,需要控制有效Z_s的误差范围。通过统计裸土、草地、农田、芦苇地和盐碱地等不同类型的地表参数,限定AIEM（Advanced Integrated Equation Model）模型的输入参数范围,得到模拟数据集合,然后对Z_s加入不同比例的误差,通过反演经验方程得到M_v反演结果,分析M_v反演值的RMSE,最终得到满足反演精度的Z_s误差范围。

2 研究区与数据源

研究区位于甘肃省黑河中游的临泽样地和黑河上游的青海省阿柔样地,其中临泽样地较为平坦,选择了2个360 m×360 m的样方D、E,样点间距为60 m,D样方地表类型是苜蓿,E样方地表类型是大麦地。临泽地区气候类型为大陆性荒漠草原气候,年蒸发量1830.4 mm,平均降水量118.4 mm,年平均气温7.7 $\begin{matrix} ℃ \end{matrix}$ 。阿柔样地位于八宝河阶地上,地势平坦,植被覆盖为草地,选择了2个90 m×90 m的样方,样点间距为30 m。阿柔地区年平均气温1 ℃,年降水量约在420 mm之间,属典型的高原大陆性气候。

地面实验数据来源于中国科学院寒区旱区科学数据中心所提供的“黑河综合遥感联合试验”。为定量研究土壤水分反演的不确定性,选取了与2008年7月11日ASAR影像及同期的98组地面样方观测数据,观测数据包括M_v、均方根高度S和相关长度l。为验证本文得到的不同入射角下Z_s误差范围的有效性,选取2007年10月17日、18日和2008年5月24日、7月5日、7月11日共5幅ASAR影像及同期的305组地面样方观测数据。

选用欧空局的ENVISAT-1卫星上ASAR传感器获取的SAR影像作为土壤含水量反演的数据源。选取了包含临泽地区的2008年5月24日、2008年7月11日和包含阿柔地区的2007年10月17日、18日、2008年7月5日一共5幅ASAR数据,入射波段为C波段（f=5.331GHz）,经度范围（99°28′~100°43′E）,纬度范围（38°02′~39°48′N）,地面分辨率为12.5 m×12.5 m,工作模式为Alternating Polarization,极化方式为VV和VH。图1（a）、（b）分别为临泽研究区的TM5、4、3假彩色合成影像和ASAR的VV极化影像。

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图1 黑河流域临泽地区遥感影像

Fig. 1 Images of Lin Ze in basin of Heihe

3 研究方法

本文着重于定量分析Z_s的不确定性如何影响M_v反演结果的不确定性,并得到满足反演精度要求的Z_s误差范围。由于实验的样区地形较为平坦,各样区相距较近,入射角变化很小,局部雷达入射角对后向散射系数的影响可以忽略,为简便起见,将入射角进行归一化。图2为SAR土壤水分反演不确定性分析流程图。

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图2 SAR土壤水分反演不确定性分析流程图

Fig. 2 Flowchart of uncertainty analysis for SAR-retrieved soil moisture

为了实现不确定性的定量分析,通过以下几个步骤进行研究：

（1）对临泽样区的实测土壤水分数据和粗糙度数据的取值范围和分布进行统计分析,得到AIEM模型部分输入参数的取值和限定范围;

（2）利用临泽样区的实测土壤水分数据和同期ASAR影像得到的 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ ,多次运行AIEM模型计算得到l_mod,从而得到有效Z_s参数值,并构建有效Z_s和 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 之间的非线性关系;

（3）根据Z_s的均值和标准方差产生不同量级的高斯噪声对Z_s的取值进行随机扰动,每个量级得到1000个带噪声的Z_s采样值,计算Z_s的偏度、峰度和四分位距;通过Z_s和 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 之间的非线性关系,反推出每个Z_s采样值对应的 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 。

（4）通过土壤水分反演经验方程得到M_v,计算反演结果的偏度、峰度和四分位距,量化Z_s的不确定性引起的反演M_v的不确定性。

（5）通过使用AIEM模型和限定范围的输入参数,获取到大量模拟数据,对模拟的Z_s按照初始值的不同比例引入误差,并通过土壤水分反演经验方程得到M_v,计算反演M_v的RMSE,得到满足反演精度要求的Z_s误差范围。

3.1 AIEM模型

AIEM模型（Advanced Integrated Equation Model）即高级积分方程模型,是对积分方程模型（Integrated Equation Model,IEM）的改进,是计算随机粗糙面单次电磁散射问题的近似方法^[10],该模型的优势在于它适合各种粗糙程度的地表后向散射的模拟,且模型精度较高,是目前使用最广泛的裸露土壤表面散射模型。它从电场和磁场所满足的积分方程出发,得到了表面上切向电场和磁场的表达式。

AIEM模型单次散射表达式为^[11,12]：

（1）

$\begin{matrix} I_{pq}^{n} = {(k_{ss} + k_{s})}^{n} f_{pq} \exp (- s^{2} k_{s} k_{ss}) + \\ \frac{{(k_{ss})}^{n} F_{pq} (- k_{x}, - k_{y}) + {(k_{ss})}^{n} F_{pq} (- k_{sx}, - k_{sy})}{2} \end{matrix}$ （2）

式中：pq表示极化方式;k1是波数;S是地表均方根高度;Wⁿ（k_sx-k_x, k_sy-k_y）是地表相关函数的n阶傅里叶变换;k_z=kcosθ_i;k_sz=kcosθ_s;k_x=ksinθ_icosφ; k_sx=ksinθ_scosφ_s;k_y=ksinθ_isinφ;k_sy=ksinθ_ssinφ_s;θ_i是入射角,φ是入射方位角;θ_s是散射角;φ_s是散射方位角;f_pq和F_pq分别是与菲涅尔反射率相关的函数^[13]。

AIEM模型的输入参数包括传感器参数和地表参数。其中需输入的传感器参数为频率f和入射角θ,地表参数包括M_v、S和l、表面自相关函数、地表土壤有效温度st和土壤质地（砂土比例sv和黏土比例cv）。

由于不同样区,地表参数的取值不尽相同,为研究方便,对AIEM模型的部分输入参数的取值进行统一标定。选择的ASAR影像获取时间为2008年7月11日上午11时26分（北京时间）,使用针式温度计获得各样点0~5 cm的平均土壤温度,地表土壤有效温度取平均土壤温度的均值st=22.23 ℃;土壤质地参数取值根据“黑河流域HWSD土壤质地数据集”中临泽地区砂土和黏土比例的平均值来确定,sv=24%,cv=32%;雷达入射角归一化为 θ=33.5°。Davidson等^[6]的实验结果表明,地表粗糙度较小时的光滑地表,其自相关函数为指数函数时能取得很好的模拟结果,而地表粗糙度较大的地表的自相关函数接近高斯自相关函数。临泽地区D、E样区地表类型为牧草地和农田,地形平坦地表粗糙度较小,所以自相关函数应为指数函数。

3.2 有效相关长度的计算

研究显示,在对粗糙度的采样中,l的测量误差远远大于S的误差,较短的廓线长度和较大的采样间距会造成l的严重低估^[14]。研究发现,AIEM模拟的 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 与雷达实际 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 之间的误差,主要由l的不准确造成的。因此,Su等^[15]引入了有效粗糙度参数的思想,利用 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 和M_v实测值来校准后向散射模型的粗糙度参数,使用校准后的粗糙度参数代替土壤粗糙度原始测量值。目前使用较多的有效相关长度反演方法有查找表（look-up tables,LUT）法^[8]、线性回归模型^[16]、最小二乘法^[17]和迭代方法^[18]。

本文使用Rahman的LUT方法来计算有效相关长度。主要步骤如下^[19]：首先利用AIEM模型模拟不同粗糙度和不同土壤水分下的后向散射系数值,将模拟的后向散射系数值存储于表格中,即LUT表;然后利用SAR影像中获得的后向散射系数和成本函数在表格中查找,查找规则是根据不同极化方式的后向散射系数实测值和AIEM模型模拟值查找符合成本函数最小值的后向散射系数记录,LUT表中该后向散射系数记录对应的相关长度数值即为有效相关长度。成本函数如下：

$\begin{matrix} σ_{\min}^{0} = \min (\sqrt{(σ_{VV, M}^{0} - σ_{VV, AIEM}^{0})^{2} + (σ_{VH, M}^{0} - σ_{VH, AIEM}^{0})^{2}}) \end{matrix}$ （3）

式中： $\begin{matrix} σ_{VV, M}^{0} \end{matrix}$ 和 $\begin{matrix} σ_{VH, M}^{0} \end{matrix}$ 为SAR影像中实测的后向散射系数; $\begin{matrix} σ_{VV, AIEM}^{0} \end{matrix}$ 和 $\begin{matrix} σ_{VH, AIEM}^{0} \end{matrix}$ 为AIEM模拟的后向散射系数。

Lievens等^[16]实验了大量不同的（S,l）组合,发现当固定S的取值时,通过AIEM模型反演得到校准的l,此时得到的（S,l）组合反演M_v效果较好,并发现S取不同值时,得到的有效相关长度精度不同,试验发现S=1 cm,l∈（1 cm,120 cm）时,有效相关长度的反演精度较高。

为获取到适合研究区地表类型的S取值,首先利用AIEM模型对后向散射情况模拟,土壤水分输入实测采样点的M_v,粗糙度参数的取值范围通过对样地采样数据的统计分析获得。为了尽可能精确地反映自然地表的粗糙度,采用美国NASA和农业部的合作实验“Washita '92”的数据^[20]和中国科学院寒区旱区科学数据中心所提供的“黑河综合遥感联合试验（WATER '2008）”中的299组实测数据,测量的地表类型有裸土地表、草地、农田、芦苇地、盐碱地等。通过分析S实测数值的分布情况,发现均方根高度的范围S∈(0.2 cm, 2.5 cm)时,在所有实测数据中所占比例达到98%。因此取S∈(0.2 , 2.5 ) cm,步长为0.1 cm,相关长度的范围为l∈（1 ,120） cm,步长为1 cm,建立 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 模拟数据集,通过LUT法结合成本函数查找有效相关长度值,反复试验后确定S为 1.4 cm。即在本文研究区内,S=1.4 cm,l∈（1 ,120 ）cm时,有效相关长度的反演精度较高。

3.3 地表组合粗糙度的不确定性分析

为减少反演过程中的参数,在很多文献中地表粗糙度使用S和l的组合形式Z_s来表示。在土壤水分的反演研究中现有的Z_s形式有多种^[22],对“黑河综合遥感联合试验”中样区的实测粗糙度数据进行统计分析,表1为样区中不同形式的Z_s统计分析表。

表1 样区不同形式Z_s统计

Tab. 1 Statistics of different Z_s in the experimental area

Z_s	均值	最大值	最小值	偏度
Z_s=S³/l²	0.0075	0.0274	0.0014	1.7085
Z_s=S²/l	0.0961	0.196	0.0436	0.8530
$\begin{matrix} Z_{s} = \sqrt{\frac{S^{2}}{l}} \end{matrix}$	0.3049	0.4427	0.2087	0.4010
Z_s=S³/l	0.1346	0.2744	0.0610	0.8530

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从表1可看出, $\begin{matrix} Z_{s} = \sqrt{\frac{S^{2}}{l}} \end{matrix}$ 形式数值分布范围较大,分布较为均匀,而且与另外3种形式组合粗糙度相比, $\begin{matrix} Z_{s} = \sqrt{\frac{S^{2}}{l}} \end{matrix}$ 参数值的偏度较小,取值的分布更接近于正态分布,所以选择 $\begin{matrix} Z_{s} = \sqrt{\frac{S^{2}}{l}} \end{matrix}$ 的形式来表示组合粗糙度。

为了定量描述Z_s的不确定性,根据不同程度的高斯噪声对Z_s参数值进行随机扰动,即以各样区采样点的Z_s均值为期望,标准差从小到大设置8个不同量级：0.01、0.015、0.02、0.025、0.03、0.035、0.04、0.045。每个量级生成1000个加入高斯噪声的Z_s集合。

利用临泽D、E样区中的实测数据得到各采样点有效相关长度l_mod,然后根据 $\begin{matrix} Z_{s} = \sqrt{\frac{S^{2}}{l}} \end{matrix}$ 计算各采样点的有效Z_s。结合同期ASAR影像中采样点的 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 可以构建如下非线性经验方程：

$\begin{matrix} σ_{pq}^{0} = aln Z_{s} + b \end{matrix}$ （4）

式中： $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 为极化后向散射系数;a、b为方程的经验系数。将带噪声的Z_s代入式（4）中得到模拟的 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 集合。

使用偏度、峰度和四分位距^[4]这些指标来定量描述Z_s以及反演M_v的不确定性。这3个指标都是用来描述数据的概率分布的形状,偏度能够判定数据分布的不对称程度以及方向,峰度能够判定数据分布相对于正态分布而言是更陡峭还是平缓,四分位距常用于描述数据分布的离散程度。

偏度SK计算如下：

$\begin{matrix} SK = E [{(\frac{X - μ}{σ})}^{3}] \end{matrix}$ （5）

峰度K计算如下：

$\begin{matrix} K = E [{(\frac{X - μ}{σ})}^{4}] \end{matrix}$ （6）

式中：X为随机变量;μ为均值;E为数学期望;o为标准差。

四分位距IQR计算前需要先对数据进行从小到大的排序,然后按照如下公式计算：

$\begin{matrix} IQR = Q_{3} - Q_{1} \end{matrix}$ （7）

$\begin{matrix} Q_{1} 的位置 = \frac{n + 1}{4} \end{matrix}$ （8）

$\begin{matrix} Q_{3} 的位置 = \frac{3 (n + 1)}{4} \end{matrix}$ （9）

式中：Q₁是下四分位数;Q₃是上四分位数;n是项数。

3.4 土壤水分反演经验方程

裸土和低矮植被覆盖地表的 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 与Z_s、M_v之间的关系可以表示为^[23]：

$\begin{matrix} σ_{pq}^{0} = Aln M_{v} + Bln Z_{s} + C \end{matrix}$ （10）

通过样地的实测数据结合同期ASAR影像中的 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 进行多元线性回归分析即可求得其中的经验系数A、B、C。将带高斯噪声的Z_s和通过式（4）获取的模拟 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 代入式（10）,可得土壤含水量M_v。

3.5 误差控制

由于地表粗糙度的不确定性是导致土壤水分反演误差的重要来源^[5],为满足一定的土壤水分反演精度要求,需要把Z_s的误差控制在一定范围内。首先统计裸土、草地、农田、芦苇地和盐碱地等不同类型的地表参数,限定AIEM模型的输入参数范围,可得到模拟数据集合（包括Z_s、M_v和对应的 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ ）。然后在Z_s中加入不同比例的误差,通过反演经验方程式（10）得到新的反演结果M_v′,通过分析新的反演值M_v′的RMSE,得到满足反演精度的Z_s误差范围。由于反演经验方程的不确定性不是本文研究的范畴,所以为了避免式（10）的不确定性,反演值M_v′的RMSE比对的对象并不是初始的M_v,而是未加入误差的Z_s通过式（10）反演得到的土壤水分值。

4 分析结果

4.1 组合粗糙度不确定性量化

选取了临泽地区的2个样区D、E,其中D样区土壤含水量较高,M_v∈(31.5%, 49.9%),而E样区中土壤含水量较低,M_v∈(13.4%, 23.3%)。利用临泽D、E样区中98个采样点的土壤水分实测数据,将S固定为1.4 cm,结合同期ASAR影像中的后向散射数据,使用Rahman的LUT方法反演得到采样点相应的有效相关长度,以此代替原始实测的相关长度,然后根据 $\begin{matrix} Z_{s} = \sqrt{\frac{S^{2}}{l}} \end{matrix}$ 计算各采样点的有效组合粗糙度。

以各样区采样点的组合粗糙度中值为期望,标准差从小到大设置8个不同量级：0.01、0.015、0.02、0.025、0.03、0.035、0.04、0.045,每个量级生成1000个加入高斯噪声的组合粗糙度集合。图3为样区E加入量级为0.03高斯噪声后得到的Z_s的直方图。

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图3 带高斯噪声（Std=0.03）的Z_s直方图

Fig. 3 Histogram of Z_s with gauss noise （Std=0.03）

分析Z_s的采样值,使用偏度、峰度和四分位距量化Z_s参数的不确定性。表2为各样区Z_s采样值的不确定性量化统计。

表2 带不同量级高斯噪声Z_s的不确定性量化统计

Tab. 2 Statistics of Z_s uncertainty quantification with different magnitude of gaussian noise

样区	噪声量级（标准差）	中值/cm	峰度	偏度	四分位距
样区D	0.01	0.2601	-0.0021	-0.0525	0.0140
	0.015	0.2598	-0.0687	-0.0076	0.0203
	0.02	0.2597	-0.2262	0.0813	0.0276
	0.025	0.2594	-0.0580	0.0703	0.0350
	0.03	0.2597	-0.1405	-0.0238	0.0393
	0.035	0.2604	0.0249	-0.1222	0.0466
	0.04	0.2619	0.0917	0.0269	0.0525
	0.045	0.2594	0.0085	0.0374	0.0604
样区E	0.01	0.2866	-0.2475	-0.0902	0.0144
	0.015	0.2854	0.1039	0.1754	0.0201
	0.02	0.2858	0.0518	0.0238	0.0270
	0.025	0.2868	0.2010	0.0502	0.0337
	0.03	0.2852	-0.0823	0.0317	0.0409
	0.035	0.2847	0.1743	-0.0084	0.0450
	0.04	0.2849	0.0206	0.0548	0.0528
	0.045	0.2870	-0.0515	-0.0309	0.0622

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从表2可看出,加入不同量级高斯噪声进行随机扰动后的Z_s数值的分布基本上符合正态分布。

利用样区D中采样点的有效Z_s,结合同期ASAR影像中采样点的 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ 可以构建如下非线性经验方程：

$\begin{matrix} σ_{vv}^{0} = 13.512 \ln Z_{s} + 7.0243 \end{matrix}$ （11）

式中：R²=0.98, $\begin{matrix} σ_{vv}^{0} \end{matrix}$ 为VV极化后向散射系数。

同样,样区E中采样点的Z_s也可构建如下经验方程：

$\begin{matrix} σ_{vv}^{0} = 12.319 \ln Z_{s} + 3.4816 \end{matrix}$ （12）

式中：R²=0.97,将各样区中加入不同量级高斯噪声的Z_s采样值分别代入式（11）、（12）中,得到Z_s对应的后向散射系数 $\begin{matrix} σ_{vv}^{0} \end{matrix}$ ,然后利用Z_s、 $\begin{matrix} σ_{vv}^{0} \end{matrix}$ 来反演土壤水分。

4.2 土壤含水量不确定性量化

构建临泽样区D的土壤含水量反演经验方程,分析各样区有效Z_s、实测土壤含水量,并结合同期ASAR影像中得到的后向散射系数,得到适用于样区D的土壤水分反演经验方程：

$\begin{matrix} σ_{vv}^{0} = 3.10693 \ln M_{v} + 14.08189 \ln Z_{s} + 10.71639 \end{matrix}$ （13）

式中：R²=0.99,同样,样区E也可构建如下土壤水分反演经验方程：

$\begin{matrix} σ_{vv}^{0} = 2.70720 \ln M_{v} + 12.81281 \ln Z_{s} + 8.55339 \end{matrix}$ （14）

式中：R²=0.98,把加入不同量级高斯噪声的Z_s采样值、对应的 $\begin{matrix} σ_{vv}^{0} \end{matrix}$ ,分别代入到式（13）、（14）中,得到反演的土壤水分。图4为土壤水分反演结果的直方图。

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图4 土壤水分反演结果的直方图

Fig. 4 Histogram of retrieved soil moisture content

分析土壤水分反演结果,使用偏度、峰度和四分位距量化土壤水分反演结果的不确定性。表3为土壤水分反演结果的不确定性量化统计。

表3 反演土壤水分不确定性量化统计

Tab. 3 Statistics of uncertainty quantification of soil moisture retrieval

样区	噪声量级（标准差）	中值/cm	峰度	偏度	四分位距	样区	噪声量级（标准差）	中值/cm	峰度	偏度	四分位距
样区D	0.010	0.3901	0.1157	0.1926	0.0039	样区E	0.010	0.1929	-0.1984	0.2034	0.0018
	0.015	0.3902	-0.0191	0.2068	0.0056		0.015	0.1931	0.0971	0.0191	0.0025
	0.020	0.3902	-0.1261	0.1677	0.0076		0.020	0.1930	0.1954	0.2410	0.0033
	0.025	0.3903	0.2613	0.2849	0.0096		0.025	0.1929	0.4067	0.3084	0.0041
	0.030	0.3902	0.1655	0.4135	0.0109		0.030	0.1931	0.0766	0.3403	0.0051
	0.035	0.3900	0.8019	0.6583	0.0128		0.035	0.1932	0.6807	0.5092	0.0056
	0.040	0.3896	0.8234	0.5938	0.0145		0.040	0.1931	0.5589	0.4892	0.0066
	0.045	0.3903	1.2501	0.6791	0.0167		0.045	0.1929	0.4850	0.6059	0.0077

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4.3 对比分析

图5为样区D、E中各采样点的组合粗糙度在不同量级的随机扰动下,土壤水分反演结果的不确定性量化指标响应折线,其中图5（a）、（d）为峰度比较,图5（b）、（e）为偏度比较,图5（c）、（f）为四分位距比较。

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图5 参数不确定性量化指标响应折线

Fig. 5 Response line of parameter uncertainty quantification index

从表3和图5可看出,由于Z_s采样值在不同量级的随机扰动下都符合正态分布,所以随着Z_s加入的高斯噪声量级的增大,其偏度和峰度并无明显变化,取值都趋近于0。而反演土壤水分的峰度随高斯噪声量级的增大而增大,在高斯噪声量级0<Std<0.045范围内,-0.1984<K<1.2501,表明反演土壤水分的分布相对于正态分布约束较差,高斯噪声量级的越大,土壤水分反演值集中在众数附近的趋势越大。反演土壤水分的偏度随高斯噪声量级的增大而增大,在高斯噪声量级0<Std<0.045范围内,0.0191<SK<0.6791,土壤水分分布明显呈右偏态,说明大部分土壤水分反演值都在均值左侧,土壤水分反演值的低估倾向比高估倾向更明显。这种非正态分布是由于M_v、Z_s和 $\begin{matrix} σ_{vv}^{0} \end{matrix}$ 之间的非线性关系引起的,符合Ma C^[4]和verhoest等^[5]的观察结果。反演土壤水分的四分位距IQR随着量级的增大而增大,与Z_s的IQR变化趋势一致,表明土壤水分反演值的离散程度受Z_s离散程度的影响。对比样区D与E的量化指标,发现样区D的偏度要大于样区E的,说明土壤水分含量越低其数值分布越趋向于正态分布;样区D的IQR要高于样区E的,说明样区D中土壤水分分布的离散程度与样区E相比较大。

4.4 误差控制范围

利用AIEM模型和限定的输入参数范围得到后向散射系数模拟数据集,AIEM模型输入参数做如下限定：S=1.4 cm,表面相关长度l∈(10 cm, 40 cm)（统计采样点有效相关长度得到）,步长为1 cm, M_v∈(5%,50%),步长为1%,st=22.23 ℃;sv=24%, cv=32%;雷达入射角分别取18.5°,22.5°,28.5°,33.5°,37°,41°,44°共7个值,可得到模拟数据集合（包括Z_s、M_v和对应的 $\begin{matrix} σ_{pq}^{0} \end{matrix}$ ）。利用式（10）结合模拟数据得到如下7个不同入射角下土壤水分反演经验公式：

$\begin{matrix} σ_{vv}^{0} = 3.17011 \ln M_{v} + 6.96978 \ln Z_{s} + 9.99514 θ = 18 . 5^{°} \end{matrix}$ （15）

$\begin{matrix} σ_{vv}^{0} = 3.17 134 \ln M_{v} + 9.46531 \ln Z_{s} + 10.63559 θ = 22.5 \end{matrix}$ （16）

$\begin{matrix} σ_{vv}^{0} = 3.17 362 \ln M_{v} + 12.08064 \ln Z_{s} + 10.67288 θ = 28.5 \end{matrix}$ （17）

$\begin{matrix} σ_{vv}^{0} = 2.96032 \ln M_{v} + 13.56833 \ln Z_{s} + 9.96122 θ = 33.5 \end{matrix}$ （18）

$\begin{matrix} σ_{vv}^{0} = 3.17775 \ln M_{v} + 14.35597 \ln Z_{s} + 9.70449 θ = 37^{°} \end{matrix}$ （19）

$\begin{matrix} σ_{vv}^{0} = 3.18050 \ln M_{v} + 15.06279 \ln Z_{s} + 9.03248 θ = 41^{°} \end{matrix}$ （20）

$\begin{matrix} σ_{vv}^{0} = 3.18345 \ln M_{v} + 15.48714 \ln Z_{s} + 8.47769 θ = 44^{°} \end{matrix}$ （21）

按照模拟数据中Z_s初始值的不同比例设置误差量级,在Z_s中加入不同量级的误差：±1%、±2%、 ±3%、±4%、±5%、±6%,然后将带有不同量级误差的Z_s代入到不同入射角下反演经验方程（式（15）-（21））中得到新的反演土壤水分。通过分析新的土壤水分反演值的RMSE,得到满足反演精度的Z_s误差范围。图6为不同入射角下不同误差量级Z_s时土壤水分反演结果的RMSE响应折线。

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图6 不同θ下不同误差量级Z_s时M_v的RMSE响应折线

Fig. 6 Response line of RMSE with different error magnitude and different radar incident angle

从图6可看出,所有RMSE响应折线都与入射角正相关,即入射角越小,不同误差量级的Z_s反演土壤水分的RMSE也越小,Z_s的误差对反演结果的影响也越小。以SMOS土壤水分产品的精度RMSE=0.04 cm³/cm³为标准,当θ<18.5°时,把Z_s的误差控制在初始值的（-5%,6%）之内就可以满足精度要求;对于样区D、E,θ=33.5°,Z_s的误差控制在初始值的（-2%,3%）之内可满足精度要求。从表4可看出,Z_s误差控制范围与入射角的大小负相关。

表4 不同θ时Z_s的误差控制范围

Tab. 4 Error control range of Z_s at different incidence angles

入射角θ/°	Z_s误差控制范围
18.5	（-5%,6%）
22.5	（-4%,4%）
28.5	（-3%,3%）
33.5	（-2%,3%）
37	（-2%,3%）
41	（-2%,3%）
44	（-2%,2%）

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4.5 试验验证

为了验证本文得到的不同θ时Z_s误差控制范围的有效性,本文选取了包含临泽地区的2008年5月24日、2008年7月11日和包含阿柔地区的2007年10月17日、18日、2008年7月5日共5幅ASAR数据,入射角θ分别进行归一化,利用临泽样区中采样点的土壤水分实测数据,将均方根高度固定为1.4 cm,结合同期ASAR影像中的后向散射数据,利用LUT法反演得到采样点相应的有效相关长度和土壤水分。使用表4中Z_s误差控制范围为Z_s加入干扰噪声,然后利用LUT法反演得到带噪声的土壤水分,计算ASAR影像中样区采样点土壤水分反演值的RMSE,见表5。

表5 反演土壤水分RMSE

Tab. 5 RMSE of the inversed soil moisture

ASAR获取时间	样区	归一化入射角/°	$\begin{matrix} Z_{s} \end{matrix}$ 误差控制范围/%	RMSE
2008-05-24	临泽	18.5	5	0.029
2007-10-17	阿柔	22.5	4	0.036
2008-07-11	临泽	33.5	3	0.037
2007-10-18	阿柔	41	3	0.039
2008-07-05	阿柔	44	2	0.032

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表5的结果表明,通过实测数据验证,本文得到的不同θ时Z_s误差控制范围在试验研究区内是有效的。

5 结论与讨论

本文使用ENVISAT ASAR影像结合实测土壤水分数据,反演计算得到有效组合粗糙度,使用对Z_s加入不同量级的高斯噪声进行随机扰动的方法,来对Z_s的不确定性以及反演土壤水分的不确定性进行定量分析,使用峰度、偏度和四分位距这3个指标来量化不确定性,并得到满足反演精度要求的Z_s误差控制范围。通过样区D、E进行实验,对比分析不确定性指标的响应折线和土壤水分的RMSE响应折线,发现反演土壤水分的峰度、偏度和四分位距都随Z_s高斯噪声量级的增大而增大,并选择临泽样区和阿柔样区不同入射角的ASAR影像和同步实测数据对Z_s误差控制范围进行验证,得到以下结论：

（1）在高斯噪声0<Std<0.045时,-0.1984<K<1.2501,说明高斯噪声量级越大,土壤水分反演值集中在众数附近的趋势越大;0.0191<SK<0.6791,土壤水分分布呈右偏态,土壤水分反演值的低估倾向比高估倾向更明显,这种非正态分布是由于M_v、Z_s和 $\begin{matrix} σ_{vv}^{0} \end{matrix}$ 之间的非线性关系引起的;反演土壤水分的IQR与Z_s的IQR变化趋势一致,表明土壤水分反演值的离散程度受Z_s离散程度的影响;

（2）通过样区D与E的量化指标进行对比,发现土壤水分含量越低其数值分布越趋向正态分布;

（3）入射角越小,Z_s的误差对反演结果的影响也越小,Z_s误差控制范围与入射角的大小负相关;通过不同入射角的ASAR影像及同步实测数据的验证,本文得到的Z_s误差控制范围在临泽样区和阿柔样区内均可满足精度要求。

本文在对Z_s的不确定性以及反演土壤水分的不确定性进行定量分析时,所采用的实测数据来自于甘肃省临泽样地的有限数据,进行试验验证时采用的数据也都来自于地势较为平坦的临泽样地和阿柔样地。而地表的异质性对土壤水分反演结果有显著影响,需要综合考虑土壤质地、植被覆盖、地表温度和粗糙度等多种因素的不确定性,所以考虑到临泽样地和阿柔样地的地表类型,本文得到的结论适用于地表粗糙度较小的裸土和低矮稀疏植被覆盖区域。

致谢：感谢中国科学院寒区旱区科学数据中心（http://westdc.westgis.ac.cn）所提供的“黑河综合遥感联合试验”Envisat ASAR数据和同步野外观测数据。

The authors have declared that no competing interests exist.

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[1]	Engman E T, Chauhan N. Status of microwave soil moisture measurements with remote sensing [J]. Remote Sensing of Environment, 1995,51(1):189-198. https://doi.org/10.1016/0034-4257(94)00074-W URL [本文引用: 1] 摘要 Active and passive zaicrowave remote sensing techniques have demonstrated their potential for measurements of soil moisture. However, the soil moisture response from them is coupled to vegetation and surface roughness effects, and therefore the interaction among all three needs to be understood. This paper reviews the progress made in the measurement of soil moisture and the factors such as vegetation and surface roughness that affect these measurements. The active techniques, particularly those employing synthetic aperture radar (SAR), provide opportunities for soil moisture studies over a large area, and varbus aircraft and space missions have been carried out to achieve them. Still, there are unresolved questions about deriving soil moisture from these missions, and research is underway to develop algorithms so that soil moisture information can be obtained on a local as well as a global basis.
[2]	Keyser E D, Vernieuwe H, Lievens H, et al. Assessment of SAR-retrieved soil moisture uncertainty induced by uncertainty on modeled soil surface roughness [J]. International Journal of Applied Earth Observations & Geoinformation, 2012,18(1):176-182. https://doi.org/10.1016/j.jag.2012.01.017 URL [本文引用: 1] 摘要 The Integral Equation Model (IEM) is frequently used to retrieve moisture content of bare soils from synthetic aperture radar (SAR) images. This physically-based backscatter model requires surface roughness parameters, generally obtained by in situ measurements, which unfortunately often result in inaccurately retrieved soil moisture contents. Furthermore, when the retrieved soil moisture contents need to be used in data assimilation applications, it is important to also assess the retrieval uncertainty. Therefore, in this paper a regression-based method is developed that allows for the parameterization of roughness and that provides an estimation of its uncertainty by means of a probability distribution. By further propagating this distribution through the inversion of the IEM, a probability distribution of soil moisture content is obtained. It was found that 70% of the thus obtained distributions are skewed and non-normal. Furthermore, it is shown that their interquartile range varies depending on soil moisture conditions. Comparison of soil moisture measurements with the retrieved median values of the soil moisture histograms results in a root mean square error (RMSE) of approximately 3.5vol%.
[3]	FernÁndez-GÁlvez J. Errors in soil moisture content estimates induced by uncertainties in the effective soil dielectric constant [J]. International Journal of Remote Sensing, 2008,29(11):3317-3323. https://doi.org/10.1080/01431160701469115 URL [本文引用: 1] 摘要 Soil moisture content (SMC) is an important variable in the hydrological cycle. Sensors have been developed to enable continuous monitoring of SMC (remote sensing from satellite and devices for in situ measurements), many of which are based on measurement of the soil dielectric properties from which SMC is inferred. Most SMC estimation techniques depend on the relationship between SMC and its dielectric properties, but the lack of a universal effective soil dielectric model introduces an additional source of error on SMC estimates. This work compares nine models of this relationship, and quantifies errors due to uncertainties in the relation between water content and dielectric properties. Errors of around 4% are reported with maximum figures at relatively low water content. Inconsistency between these models and the inconsistency in the effect of soil texture on SMC retrieval from satellite ased sensors is also evaluated.
[4]	Ma C, Li X, Notarnicola C, et al. Uncertainty quantification of soil moisture estimations based on a bayesian probabilistic inversion [J]. IEEE Transactions on Geoscience & Remote Sensing, 2015,99:1-14. https://doi.org/10.1109/TGRS.2017.2664078 URL [本文引用: 3] 摘要 Soil moisture (SM) inversions based on active microwave remote sensing have shown promising progress but do not easily meet expected application requirements because a number of inversion algorithms can only produce point estimates of SM and cannot quantify the uncertainty of SM inversions. Although previous studies have reported Bayesian maximum posterior estimations that are capable of retrieving SM within a probabilistic framework, they have primarily focused on the optimal estimators of SM and have typically ignored the uncertainty of SM inversions. This paper presents an SM probabilistic inversion (PI) algorithm based on Bayes' theorem and the Markov Chain Monte Carlo technique and capable of revealing the uncertainty of SM inversions and obtaining highly accurate SM estimates via maximum likelihood estimations (MLEs). The algorithm is implemented based on the advanced integral equation model, water cloud model simulations, and dual-polarized TerraSAR-X observations. The ground SM and vegetation water content (VWC) measurements from the Heihe watershed allied telemetry experimental research experiments are applied for validation. The results show that: 1) uncertainties in SM inversions, defined with respect to the measures of dispersion of SM posterior probability distribution, are approximately 0.1-0.12 m /m and 2) an acceptable inversion accuracy is obtained via MLEs, which present an SM Root Mean Square Error (RMSE) of 0.045 and 0.047 m /m for bare and vegetated soils, respectively, and a VWC RMSE of 0.45 kg/m虏. The presented PI can quantify the uncertainty in SM inversions; therefore, it should be useful for improving active microwave remote sensing estimations of SM.
[5]	Verhoest N E C, Lievens H, Wagner W, et al. On the soil roughness parameterization problem in soil moisture retrieval of bare surfaces from synthetic aperture radar [J]. Sensors, 2008,8(7):4213. https://doi.org/10.3390/s8074213 URL PMID: 3697171 [本文引用: 3] 摘要 Synthetic Aperture Radar has shown its large potential for retrieving soil moisture maps at regional scales. However, since the backscattered signal is determined by several surface characteristics, the retrieval of soil moisture is an ill-posed problem when using single configuration imagery. Unless accurate surface roughness parameter values are available, retrieving soil moisture from radar backscatter usually provides inaccurate estimates. The characterization of soil roughness is not fully understood, and a large range of roughness parameter values can be obtained for the same surface when different measurement methodologies are used. In this paper, a literature review is made that summarizes the problems encountered when parameterizing soil roughness as well as the reported impact of the errors made on the retrieved soil moisture. A number of suggestions were made for resolving issues in roughness parameterization and studying the impact of these roughness problems on the soil moisture retrieval accuracy and scale.
[6]	Davidson M, Toan T L, Mattia F, et al. On the characterization of agricultural soil roughness for radar remote sensing studies [J]. IEEE Transactions on Geoscience & Remote Sensing, 2000,38(2):630-640. https://doi.org/10.1109/36.841993 URL [本文引用: 2] 摘要 Abstract The surface roughness parameters commonly used as inputs to electromagnetic surface scattering models (SPM, PO, GO, and IEM) are the root mean square (RMS) height s, and autocorrelation length l. However, soil moisture retrieval studies based on these models have yielded inconsistent results, not so much because of the failure of the models themselves, but because of the complexity of natural surfaces and the difficulty in estimating appropriate input roughness parameters. In this paper, the authors address the issue of soil roughness characterization in the case of agricultural fields having different tillage (roughness) states by making use of an extensive multisite database of surface profiles collected using a novel laser profiler capable of recording profiles up to 25 m long. Using this dataset, the range of RMS height and correlation values associated with each agricultural roughness state is estimated, and the dependence of these estimates on profile length is investigated. The results show that at spatial scales equivalent to those of the SAR resolution cell, agricultural surface roughness characteristics are well described by the superposition of a single scale process related to the tillage state with a multiscale random fractal process related to field topography
[7]	Shi J, Chen K S, Li Q, et al. A parameterized surface reflectivity model and estimation of bare-surface soil moisture with L-band radiometer [J]. IEEE Transactions on Geoscience & Remote Sensing, 2010,40(12):2674-2686. https://doi.org/10.1109/TGRS.2002.807003 URL [本文引用: 1] 摘要 Soil moisture is an important parameter for hydrological and climatic investigations. Future satellite missions with L-band passive microwave radiometers will significantly increase the capability of monitoring earth's soil moisture globally. Understanding the effects of surface roughness on microwave emission and developing quantitative bare-surface soil moisture retrieval algorithms is one of the essential components in many applications of geophysical properties in the complex earth terrain by microwave remote sensing. In this study, we explore the use of the integral equation model (IEM) for modeling microwave emission. This model was validated using a three-dimensional Monte Carlo model. The results indicate that the IEM model can be used to simulate the surface emission quite well for a wide range of surface roughness conditions with high confidence. Several important characteristics of the effects of surface roughness on radiometer emission signals at L-band 1.4 GHz that have not been adequately addressed in the current semiempirical surface effective reflectivity models are demonstrated by using IEM-simulated data. Using an IEM-simulated database for a wide range of surface soil moisture and roughness properties, we developed a parameterized surface effective reflectivity model with three typically used correlation functions and an inversion model that puts different weights on the polarization measurements to minimize surface roughness effects and to estimate the surface dielectric properties directly from dual-polarization measurements. The inversion technique was validated with four years (1979-1982) of ground microwave radiometer experiment data over several bare-surface test sites at Beltsville, MD. The accuracies in random-mean-square error are within or about 3% for incidence aneles from 20 to 50 .
[8]	Rahman M M, Moran M S, Thoma D P, et al. A derivation of roughness correlation length for parameterizing radar backscatter models [J]. International Journal of Remote Sensing, 2007,28(18):3995-4012. https://doi.org/10.1080/01431160601075533 URL [本文引用: 2] 摘要 Surface roughness is a key parameter of radar backscatter models designed to retrieve surface soil moisture (θS) information from radar images. This work offers a theory‐based approach for estimating a key roughness parameter, termed the roughness correlation length (L c). The L c is the length in centimetres from a point on the ground to a short distance for which the heights of a rough surface are correlated with each other. The approach is based on the relation between L c and h RMS as theorized by the Integral Equation Model (IEM). The h RMS is another roughness parameter, which is the root mean squared height variation of a rough surface. The relation is calibrated for a given site based on the radar backscatter of the site under dry soil conditions. When this relation is supplemented with the site specific measurements of h RMS, it is possible to produce estimates of L c. The approach was validated with several radar images of the Walnut Gulch Experimental Watershed in southeast Arizona, USA. Results showed that the IEM performed well in reproducing satellite‐based radar backscatter when this new derivation of L c was used as input. This was a substantial improvement over the use of field measurements of L c. This new approach also has advantages over empirical formulations for the estimation of L c because it does not require field measurements of θS for iterative calibration and it accounts for the very complex relation between L c and h RMS found in heterogeneous landscapes. Finally, this new approach opens up the possibility of determining both roughness parameters without ancillary data based on the radar backscatter difference measured for two different incident angles.
[9]	余凡,李海涛,张承明,等. 利用双极化微波遥感数据反演土壤水分的新方法 [J].武汉大学学报·信息科学版,2014,39(2):225-228. https://doi.org/10.13203/j.whugis20120527 [本文引用: 1] 摘要提出了一种基于微波双极化数据的土壤水分反演经验模型,该模型引入了新的综合粗糙度参数Rs=√S2/L来描述地表粗糙状况,将两个粗糙度参数均方根高度S和相关长度L合二为一,因而模型的未知量仅为Rs与法向菲涅尔反射系数Γ0.基于AIEM模型数值模拟,建立了后向散射系数与Rs、Γ 0的经验关系,并利用两个极化的微波数据同时反演得到粗糙度参数Rs和Γ0,进而得到地表土壤水分.实测数据表明,该模型反演的土壤水分与地袁实测值相关性较高(R2 =0.681,RMS=0.043),在土壤水分反演方面具有较大的潜力. [ Yu F, Li H T, Zhang C M, et al. A new approach for surface soil moisture retrieval using two-polarized microwave remote sensing data [J]. Geomatics and Information Science of Wuhan University, 2014,39(2):225-228. ] https://doi.org/10.13203/j.whugis20120527 [本文引用: 1] 摘要提出了一种基于微波双极化数据的土壤水分反演经验模型,该模型引入了新的综合粗糙度参数Rs=√S2/L来描述地表粗糙状况,将两个粗糙度参数均方根高度S和相关长度L合二为一,因而模型的未知量仅为Rs与法向菲涅尔反射系数Γ0.基于AIEM模型数值模拟,建立了后向散射系数与Rs、Γ 0的经验关系,并利用两个极化的微波数据同时反演得到粗糙度参数Rs和Γ0,进而得到地表土壤水分.实测数据表明,该模型反演的土壤水分与地袁实测值相关性较高(R2 =0.681,RMS=0.043),在土壤水分反演方面具有较大的潜力.
[10]	郭立新,王蕊,吴振森.随机粗糙面散射的基本理论与方法[M].北京:科学出版社,2010:>:32-39. [本文引用: 1] [ Guo L X, Wang R. The basic theory and method of scattering from random rough surface[M]. Beijing: The Science Publishing Company, 2010:32-39. ] [本文引用: 1]
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[12]	T D Wu, K S Chen, J C Shi, et al. A transition model for the reflection coefficient in surface scattering [J]. IEEE Transactions on Geoscience and Remote Sensing, 2001,39(4):2040-2050. https://doi.org/10.1109/36.951094 URL [本文引用: 1] 摘要 In the development of wave scattering models for randomly dielectric rough surfaces, it is usually assumed that the Fresnel reflection coefficients could be approximately evaluated at either the incident angle or the specular angle. However, these two considerations are only applicable to their respective regions of validity. A common question to ask is what are the conditions under which we would choose one or the other of these two approximations? Since these approximations are basically roughness-dependent, how can we handle the in-between cases where neither is appropriate? In this paper, a physical-based transition function that naturally connects these two approximations is proposed. The like-polarized backscattering coefficients are evaluated with the model and are compared with those calculated with a moment method simulation for both Gaussian and non-Gaussian correlated surfaces. It is found that the proposed transition function provides an excellent prediction for the backscattering coefficient in the frequency and angle trends
[13]	Wu T D, Chen K S. A reappraisal of the validity of the IEM model for backscattering from rough surface [J]. IEEE Transactions on Geoscience and Remote Sensing, 2004,42(4):743-753. https://doi.org/10.1109/TGRS.2003.815405 URL [本文引用: 1] 摘要 An integral equation method (IEM) surface scattering model was examined in terms of its applicability to laboratory measurement and numerical simulations. New expressions for both single scattering and multiple scattering were obtained by rederiving the scattering coefficient to keep all the phase terms in the spectral representation of the Green's function. After quite intricate mathematical manipulations, a fairly compact form is obtained for the scattering coefficients. In addition, the Fresnel reflection coefficients used in the model were replaced by a transition function that takes surface roughness and permittivity into account. The results of comparisons with both the numerical simulations and measurements for the backscattering case indicate that the IEM is improved, becoming more accurate and practical to use.
[14]	Oh Y, Hong J Y. Effect of surface profile length on the backscattering coefficients of bare surfaces [J]. IEEE Transactions on Geoscience & Remote Sensing, 2007,45(3):632-638. https://doi.org/10.1109/TGRS.2006.888137 URL [本文引用: 1] 摘要 The root mean square (rms) height s and autocorrelation length l are commonly used as the surface roughness input parameters to surface scattering models. Whereas it is well known that the surface roughness parameters of a natural soil surface are underestimated with a short surface profile, it is not clear how much the underestimated surface parameters affect the backscattering coefficients of the surface for various incidence angles and polarizations. In this paper, the backscattering coefficients of simulated and measured surface profiles are computed using the integral equation method and analyzed to answer this question. A 4000lmacr-long rough surface is generated numerically, where lmacr is the true correlation length of the surface, and the backscattering coefficients of the surface are computed and analyzed for various conditions. The rms error of the backscattering coefficient at a medium range of incidence angles is less than 1.5 dB for vv-polarization and 0.5 dB for hh-polarization if the profile length is larger than 5lmacr for a surface with ks=1.0, kl=10.0, and epsiv r=(10.0,2.0). Similar results are obtained from numerous simulations with various roughness conditions and various wavelengths. It is also shown that the rms error of the backscattering coefficients between 5- and 1-m-long measured surface profiles is 1.7 dB for vv-polarization and 0.5 dB for hh-polarization at a medium range of incidence angle (15deglesthetasles70deg), whereas the surface roughness parameters are significantly reduced from 2.4 to 1.5 cm for the rms height s and from 35.1 to 10.0 cm for the autocorrelation length l
[15]	Su, Z, Troch, P A, De Troch, F P. Remote sensing of soil moisture using EMAC/ESAR data[C]// Geoscience and Remote Sensing Symposium, 1996. IGARSS'96. 'Remote Sensing for a Sustainable Future.', International. IEEE, 1997:1303-1305. [本文引用: 1]
[16]	Lievens H, Verhoest N E C, Keyser E D, et al. Effective roughness modelling as a tool for soil moisture retrieval from C- and L-band SAR [J]. Hydrology & Earth System Sciences Discussions, 2010,7(4):151-162. [本文引用: 2]
[17]	N. Baghdadi, N. Holah, M. Zribi. Soil moisture estimation using multi-incidence and multi-polarization ASAR data [J]. International Journal of Remote Sensing, 2006,27(10):1907-1920. https://doi.org/10.1080/01431160500239032 URL [本文引用: 1] 摘要 The potential of Advanced Synthetic Aperture Radar (ASAR) for the retrieval of surface soil moisture over bare soils was evaluated for several ASAR acquisition configurations: (1) one date/single channel (one incidence and one polarization); (2) one date/two channels (one incidence and two polarizations); (3) two dates/two channels (two incidences and one polarization); and (4) two dates/four channels (two incidences and two polarizations). The retrieval of soil moisture from backscattering measurements is discussed, using empirical inversion approaches. When compared with the results obtained with a single polarization (HH or HV), the use of two polarizations (HH and HV) does not enable a significant improvement in estimating soil moisture. For the best estimates of soil moisture, ASAR data should be acquired at both low and high incidence angles. ASAR proves to be a good remote sensing tool for measuring surface soil moisture, with accuracy for the retrieved soil moisture that can reach 3.5% (RMSE).
[18]	Alvarez-Mozos J, Casali J, Gonzalez-Audicana M, et al. Assessment of the operational applicability of RADARSAT-1 data for surface soil moisture estimation [J]. IEEE Transactions on Geoscience & Remote Sensing, 2006,44(4):913-924. https://doi.org/10.1109/TGRS.2005.862248 URL [本文引用: 1] 摘要 The present paper focuses on the ability of currently available RADARSAT-1 data to estimate surface soil moisture over an agricultural catchment using the theoretical integral equation model (IEM). Five RADARSAT-1 scenes acquired over Navarre (north of Spain) between February 27, 2003 and April 2, 2003 have been processed. Soil moisture was measured at different fields within the catchment. Roughness measurements were collected in order to obtain representative roughness parameters for the different tillage classes. The influence of the cereal crop that covered most of the fields was taken into account using the semiempirical water cloud model. The IEM was run in forward and inverse mode using vegetation corrected RADARSAT-1 data and surface roughness observations. Results showed a great dispersion between IEM simulations and observations at the field scale, leading to inaccurate estimations. As the surface correlation length is the most difficult parameter to measure, different approaches for its estimation have been tested. This analysis revealed that the spatial variability in the surface roughness parameters seems to be the reason for the dispersion observed rather than a deficient measurement of the correlation length. At the catchment scale, IEM simulations were in good agreement with observations. The error values obtained in the inverse simulations were in the range of in situ soil moisture measuring methods (0.04 cm/sup 3//spl middot/cm/sup -3/). Taking into account the small size of the catchment studied, these results are encouraging from a hydrological point of view.
[19]	孔金玲,李菁菁,甄珮珮,等. 微波与光学遥感协同反演旱区地表土壤水分研究 [J].地球信息科学学报,2016,18(6):857-863. https://doi.org/10.3724/SP.J.1047.2016.00857 URL Magsci [本文引用: 1] 摘要 <p>土壤水分是水文循环中的关键因素,尤其对旱区的生态环境具有十分重要的意义。微波遥感是反演土壤水分的有效手段,而植被是影响土壤水分反演精度的重要因素。因此,对土壤水分的反演需要考虑植被的影响。本文以内蒙古乌审旗为研究区,利用Radarsat-2雷达数据与TM光学数据,对旱区稀疏植被覆盖地表土壤水分反演进行研究。利用TM数据,分别选取NDVI和NDWI指数对植被含水量进行反演,通过水云模型消除植被层对土壤后向散射系数的影响;在此基础上,根据研究区地表植被特性,提出一种基于AIEM 模型的反演土壤水分的改进算法,反演了不同粗糙度参数、不同极化（VV极化和HH极化）条件下的研究区土壤水分。反演结果与野外实测数据的对比结果表明,本文提出的基于地表植被特性的土壤水分改进算法,具有更好的适应性;土壤水分反演模式<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="Mml2-1560-8999-18-6-857"><mml:mtable frame="none" columnlines="none" rowlines="none"><mml:mtr><mml:mtd><mml:maligngroup/><mml:mrow><mml:mi>M</mml:mi><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">σvv</mml:mi><mml:mfenced separators="\|"><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi mathvariant="italic">lh</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></inline-formula>（VV极化方式下采用NDVI去除植被影响的反演模式）更适合于旱区考虑稀疏植被覆盖影响的地表土壤水分的反演。</p> [ Kong J L, Li J J, Zhen P P, et al. Inversion of soil moisture in arid area based on microwave and optical remote sensing data [J]. Journal of Geo-information Science, 2016,18(6):857-863. ] https://doi.org/10.3724/SP.J.1047.2016.00857 URL Magsci [本文引用: 1] 摘要 <p>土壤水分是水文循环中的关键因素,尤其对旱区的生态环境具有十分重要的意义。微波遥感是反演土壤水分的有效手段,而植被是影响土壤水分反演精度的重要因素。因此,对土壤水分的反演需要考虑植被的影响。本文以内蒙古乌审旗为研究区,利用Radarsat-2雷达数据与TM光学数据,对旱区稀疏植被覆盖地表土壤水分反演进行研究。利用TM数据,分别选取NDVI和NDWI指数对植被含水量进行反演,通过水云模型消除植被层对土壤后向散射系数的影响;在此基础上,根据研究区地表植被特性,提出一种基于AIEM 模型的反演土壤水分的改进算法,反演了不同粗糙度参数、不同极化（VV极化和HH极化）条件下的研究区土壤水分。反演结果与野外实测数据的对比结果表明,本文提出的基于地表植被特性的土壤水分改进算法,具有更好的适应性;土壤水分反演模式<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="Mml2-1560-8999-18-6-857"><mml:mtable frame="none" columnlines="none" rowlines="none"><mml:mtr><mml:mtd><mml:maligngroup/><mml:mrow><mml:mi>M</mml:mi><mml:msubsup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">σvv</mml:mi><mml:mfenced separators="\|"><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi mathvariant="italic">lh</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></inline-formula>（VV极化方式下采用NDVI去除植被影响的反演模式）更适合于旱区考虑稀疏植被覆盖影响的地表土壤水分的反演。</p>
[20]	USDA ARS Grazinglands Research Lab.Washita '92[EB/OL]. . URL [本文引用: 1]
[21]	Zribi M, Baghdadi N, Guerin C. Analysis of surface roughness heterogeneity and scattering behavior for radar measurements [J]. IEEE Transactions on Geoscience & Remote Sensing, 2006,44(9):2438-2444. https://doi.org/10.1109/TGRS.2006.873742 URL 摘要 The use of a theoretical backscatter model to analyze medium to low spatial resolution microwave data is still very complicated, particularly because of the difficulty in defining a unique roughness parameter, capable of adequately representing heterogeneous terrain. In this paper, an approach is proposed for roughness analysis and the modeling of backscattering, under conditions of surface heterogeneity. This paper is based on the use of a semiempirical backscattering model, defined with a single roughness parameter Zs=s2/l (s being the root mean square surface height and l the correlation length). The proposed backscattering model has been validated with integral equation model simulations, for high radar incidence angles, and within its domain of roughness validity. A range of experimental measurements was used to validate the model expressions. The effective low spatial resolution roughness, inferred from signals backscattered from a surface of heterogeneous roughness, is defined for different roughness classes
[22]	甄珮珮. 基于粗糙度参数的风沙滩地区土壤水分微波遥感反演模型研究 [D].西安:长安大学,2016. [本文引用: 1] [ Zhen P P. Research on soil moisture retrieval using microwave remote sensing data based on roughness parameter in Blown-sand region [D]. Xi'an: Chang'an University, 2016. ] [本文引用: 1]
[23]	任鑫. 多极化、多角度SAR土壤水分反演算法研究 [D].北京:中国科学院大学,2004. [本文引用: 1] [ Ren X. A surface moisture inversion technique using multi-polarization and multi-angle radar images [D]. Beijing: Chinese Academy of Sciences, 2004. ] [本文引用: 1]

Status of microwave soil moisture measurements with remote sensing

1995

... 土壤水分对地球表层的能量和水分平衡具有重要作用,土壤水分的时空分布与动态变化对陆地-大气间热量平衡、大气环流产生显著影响,土壤水分还是农业、林业、气象学、水文学和生态学等领域研究的重要参数^[1].利用空间分辨率较高的微波遥感反演土壤水分,可以实现对土壤水分动态、实时、大范围和较高精度的估算. ...

Assessment of SAR-retrieved soil moisture uncertainty induced by uncertainty on modeled soil surface roughness

2012

... 土壤水分反演过程中的不确定性分析是当前研究的热点,Keysera等^[2]提出了一种基于线性回归的粗糙度参数化方法,并使用概率分布对其不确定性进行估计,通过反演模型进一步传播不确定性,得到反演土壤水分的概率分布.FernA´ndez-GA´lvez^[3]认为大多数土壤水分反演方法依赖于土壤水分与介电特性之间的关系,比较了包含这种关系的9种反演模型,并对土壤水分和介电特性之间的不确定性引起的误差进行量化.Ma等^[4]提出了一种基于贝叶斯定理和马尔可夫链蒙特卡罗方法的土壤水分概率反演（PI）算法,能够定量描述土壤水分反演的不确定性,通过最大似然估计得到高精度的土壤水分估计.这些研究主要集中在粗糙度（如相关长度）、土壤介电常数等参数的不确定性分析上,而对于反演中常用的地表组合粗糙度的不确定性分析的研究较少.根据Verhoest等^[5]的研究,利用SAR进行土壤水分反演,地表粗糙度参数的不确定性是导致土壤水分反演误差的重要来源.地表粗糙度的不确定性包括测量过程中的不确定性和反演过程中的不确定性.而地表粗糙度参数是难以准确测量的,测量结果取决于测量设备的剖面长度^[6]和测量技术,即可以从相同的土壤表面得到不同的粗糙度参数值.研究表明,仅依赖单个传统的粗糙度参数（均方根或相关长度）很难描述粗糙度效应,需要同时考虑多个粗糙度参数^[7],如地表组合粗糙度,可以参与构建反演经验方程得到较好的地表参数反演结果. ...

Errors in soil moisture content estimates induced by uncertainties in the effective soil dielectric constant

2008

Uncertainty quantification of soil moisture estimations based on a bayesian probabilistic inversion

2015

... 使用偏度、峰度和四分位距^[4]这些指标来定量描述Z_s以及反演M_v的不确定性.这3个指标都是用来描述数据的概率分布的形状,偏度能够判定数据分布的不对称程度以及方向,峰度能够判定数据分布相对于正态分布而言是更陡峭还是平缓,四分位距常用于描述数据分布的离散程度. ...

... 从表3和图5可看出,由于Z_s采样值在不同量级的随机扰动下都符合正态分布,所以随着Z_s加入的高斯噪声量级的增大,其偏度和峰度并无明显变化,取值都趋近于0.而反演土壤水分的峰度随高斯噪声量级的增大而增大,在高斯噪声量级0<Std<0.045范围内,-0.1984<K<1.2501,表明反演土壤水分的分布相对于正态分布约束较差,高斯噪声量级的越大,土壤水分反演值集中在众数附近的趋势越大.反演土壤水分的偏度随高斯噪声量级的增大而增大,在高斯噪声量级0<Std<0.045范围内,0.0191<SK<0.6791,土壤水分分布明显呈右偏态,说明大部分土壤水分反演值都在均值左侧,土壤水分反演值的低估倾向比高估倾向更明显.这种非正态分布是由于M_v、Z_s和

\begin{matrix} σ_{vv}^{0} \end{matrix}

之间的非线性关系引起的,符合Ma C^[4]和verhoest等^[5]的观察结果.反演土壤水分的四分位距IQR随着量级的增大而增大,与Z_s的IQR变化趋势一致,表明土壤水分反演值的离散程度受Z_s离散程度的影响.对比样区D与E的量化指标,发现样区D的偏度要大于样区E的,说明土壤水分含量越低其数值分布越趋向于正态分布;样区D的IQR要高于样区E的,说明样区D中土壤水分分布的离散程度与样区E相比较大. ...

On the soil roughness parameterization problem in soil moisture retrieval of bare surfaces from synthetic aperture radar

2008

... 由于地表粗糙度的不确定性是导致土壤水分反演误差的重要来源^[5],为满足一定的土壤水分反演精度要求,需要把Z_s的误差控制在一定范围内.首先统计裸土、草地、农田、芦苇地和盐碱地等不同类型的地表参数,限定AIEM模型的输入参数范围,可得到模拟数据集合（包括Z_s、M_v和对应的

\begin{matrix} σ_{pq}^{0} \end{matrix}

）.然后在Z_s中加入不同比例的误差,通过反演经验方程式（10）得到新的反演结果M_v′,通过分析新的反演值M_v′的RMSE,得到满足反演精度的Z_s误差范围.由于反演经验方程的不确定性不是本文研究的范畴,所以为了避免式（10）的不确定性,反演值M_v′的RMSE比对的对象并不是初始的M_v,而是未加入误差的Z_s通过式（10）反演得到的土壤水分值. ...

\begin{matrix} σ_{vv}^{0} \end{matrix}

On the characterization of agricultural soil roughness for radar remote sensing studies

2000

... 由于不同样区,地表参数的取值不尽相同,为研究方便,对AIEM模型的部分输入参数的取值进行统一标定.选择的ASAR影像获取时间为2008年7月11日上午11时26分（北京时间）,使用针式温度计获得各样点0~5 cm的平均土壤温度,地表土壤有效温度取平均土壤温度的均值st=22.23 ℃;土壤质地参数取值根据“黑河流域HWSD土壤质地数据集”中临泽地区砂土和黏土比例的平均值来确定,sv=24%,cv=32%;雷达入射角归一化为 θ=33.5°.Davidson等^[6]的实验结果表明,地表粗糙度较小时的光滑地表,其自相关函数为指数函数时能取得很好的模拟结果,而地表粗糙度较大的地表的自相关函数接近高斯自相关函数.临泽地区D、E样区地表类型为牧草地和农田,地形平坦地表粗糙度较小,所以自相关函数应为指数函数. ...

A parameterized surface reflectivity model and estimation of bare-surface soil moisture with L-band radiometer

2010

A derivation of roughness correlation length for parameterizing radar backscatter models

2007

... 本文使用Rahman的LUT（Look-Up Tables）方法^[8]计算有效相关长度（l_mod）来代替实际测量值,可避免引入粗糙度参数测量过程中的不确定性,并使用

\begin{matrix} Z_{s} = \sqrt{\frac{S^{2}}{l}} \end{matrix}

形式^[9]得到地表有效组合粗糙度Z_s.在粗糙度的不确定性是土壤含水量不确定性的主要来源的假设下,主要解决2个问题：① 有效Z_s的不确定性如何影响反演土壤水分（M_v）的不确定性?② 有效Z_s的误差控制在多大范围才能得到符合精度要求的M_v反演结果?目前在现有土壤水分反演不确定性研究中,通常通过偏度、峰度、四分位距3个模型参数指标来量化数据的不确定性,得到反演土壤水分值分布的不对称程度和方向,土壤水分分布相对于正态分布而言是更陡峭还是平缓,以及土壤水分分布的离散程度;根据不同程度的高斯噪声对Z_s的取值进行随机扰动,得到大量Z_s采样值,并构建Z_s与后向散射系数（

\begin{matrix} σ_{pq}^{0} \end{matrix}

）之间的非线性回归关系,利用该关系计算得到对应的

\begin{matrix} σ_{pq}^{0} \end{matrix}

;通过

\begin{matrix} σ_{pq}^{0} \end{matrix}

、M_v和Z_s构成的反演经验方程反演M_v,分析Z_s的不确定性在反演过程中的传递,并使用偏度、峰度和四分位距来分析反演的M_v.为满足一定的M_v反演精度要求,需要控制有效Z_s的误差范围.通过统计裸土、草地、农田、芦苇地和盐碱地等不同类型的地表参数,限定AIEM（Advanced Integrated Equation Model）模型的输入参数范围,得到模拟数据集合,然后对Z_s加入不同比例的误差,通过反演经验方程得到M_v反演结果,分析M_v反演值的RMSE,最终得到满足反演精度的Z_s误差范围. ...

... 研究显示,在对粗糙度的采样中,l的测量误差远远大于S的误差,较短的廓线长度和较大的采样间距会造成l的严重低估^[14].研究发现,AIEM模拟的

\begin{matrix} σ_{pq}^{0} \end{matrix}

与雷达实际

\begin{matrix} σ_{pq}^{0} \end{matrix}

之间的误差,主要由l的不准确造成的.因此,Su等^[15]引入了有效粗糙度参数的思想,利用

\begin{matrix} σ_{pq}^{0} \end{matrix}

和M_v实测值来校准后向散射模型的粗糙度参数,使用校准后的粗糙度参数代替土壤粗糙度原始测量值.目前使用较多的有效相关长度反演方法有查找表（look-up tables,LUT）法^[8]、线性回归模型^[16]、最小二乘法^[17]和迭代方法^[18]. ...

利用双极化微波遥感数据反演土壤水分的新方法

2014

... 本文使用Rahman的LUT（Look-Up Tables）方法^[8]计算有效相关长度（l_mod）来代替实际测量值,可避免引入粗糙度参数测量过程中的不确定性,并使用

\begin{matrix} Z_{s} = \sqrt{\frac{S^{2}}{l}} \end{matrix}

\begin{matrix} σ_{pq}^{0} \end{matrix}

）之间的非线性回归关系,利用该关系计算得到对应的

\begin{matrix} σ_{pq}^{0} \end{matrix}

;通过

\begin{matrix} σ_{pq}^{0} \end{matrix}

利用双极化微波遥感数据反演土壤水分的新方法

2014

... 本文使用Rahman的LUT（Look-Up Tables）方法^[8]计算有效相关长度（l_mod）来代替实际测量值,可避免引入粗糙度参数测量过程中的不确定性,并使用

\begin{matrix} Z_{s} = \sqrt{\frac{S^{2}}{l}} \end{matrix}

\begin{matrix} σ_{pq}^{0} \end{matrix}

）之间的非线性回归关系,利用该关系计算得到对应的

\begin{matrix} σ_{pq}^{0} \end{matrix}

;通过

\begin{matrix} σ_{pq}^{0} \end{matrix}

2010

... AIEM模型（Advanced Integrated Equation Model）即高级积分方程模型,是对积分方程模型（Integrated Equation Model,IEM）的改进,是计算随机粗糙面单次电磁散射问题的近似方法^[10],该模型的优势在于它适合各种粗糙程度的地表后向散射的模拟,且模型精度较高,是目前使用最广泛的裸露土壤表面散射模型.它从电场和磁场所满足的积分方程出发,得到了表面上切向电场和磁场的表达式. ...

2010

Backscattering from a randomly rough dielectric surface

1992

... AIEM模型单次散射表达式为^[11,12]： ...

A transition model for the reflection coefficient in surface scattering

2001

... AIEM模型单次散射表达式为^[11,12]： ...

A reappraisal of the validity of the IEM model for backscattering from rough surface

2004

... 式中：pq表示极化方式;k1是波数;S是地表均方根高度;Wⁿ（k_sx-k_x, k_sy-k_y）是地表相关函数的n阶傅里叶变换;k_z=kcosθ_i;k_sz=kcosθ_s;k_x=ksinθ_icosφ; k_sx=ksinθ_scosφ_s;k_y=ksinθ_isinφ;k_sy=ksinθ_ssinφ_s;θ_i是入射角,φ是入射方位角;θ_s是散射角;φ_s是散射方位角;f_pq和F_pq分别是与菲涅尔反射率相关的函数^[13]. ...

Effect of surface profile length on the backscattering coefficients of bare surfaces

2007

... 研究显示,在对粗糙度的采样中,l的测量误差远远大于S的误差,较短的廓线长度和较大的采样间距会造成l的严重低估^[14].研究发现,AIEM模拟的

\begin{matrix} σ_{pq}^{0} \end{matrix}

与雷达实际

\begin{matrix} σ_{pq}^{0} \end{matrix}

之间的误差,主要由l的不准确造成的.因此,Su等^[15]引入了有效粗糙度参数的思想,利用

\begin{matrix} σ_{pq}^{0} \end{matrix}

De Troch, F P. Remote sensing of soil moisture using EMAC/ESAR data[C]// Geoscience and Remote Sensing Symposium, 1996.

1997

... 研究显示,在对粗糙度的采样中,l的测量误差远远大于S的误差,较短的廓线长度和较大的采样间距会造成l的严重低估^[14].研究发现,AIEM模拟的

\begin{matrix} σ_{pq}^{0} \end{matrix}

与雷达实际

\begin{matrix} σ_{pq}^{0} \end{matrix}

之间的误差,主要由l的不准确造成的.因此,Su等^[15]引入了有效粗糙度参数的思想,利用

\begin{matrix} σ_{pq}^{0} \end{matrix}

Verhoest N E C, Keyser E D, et al. Effective roughness modelling as a tool for soil moisture retrieval from C- and L-band SAR

2010

... 研究显示,在对粗糙度的采样中,l的测量误差远远大于S的误差,较短的廓线长度和较大的采样间距会造成l的严重低估^[14].研究发现,AIEM模拟的

\begin{matrix} σ_{pq}^{0} \end{matrix}

与雷达实际

\begin{matrix} σ_{pq}^{0} \end{matrix}

之间的误差,主要由l的不准确造成的.因此,Su等^[15]引入了有效粗糙度参数的思想,利用

\begin{matrix} σ_{pq}^{0} \end{matrix}

... Lievens等^[16]实验了大量不同的（S,l）组合,发现当固定S的取值时,通过AIEM模型反演得到校准的l,此时得到的（S,l）组合反演M_v效果较好,并发现S取不同值时,得到的有效相关长度精度不同,试验发现S=1 cm,l∈（1 cm,120 cm）时,有效相关长度的反演精度较高. ...

Soil moisture estimation using multi-incidence and multi-polarization ASAR data

2006

... 研究显示,在对粗糙度的采样中,l的测量误差远远大于S的误差,较短的廓线长度和较大的采样间距会造成l的严重低估^[14].研究发现,AIEM模拟的

\begin{matrix} σ_{pq}^{0} \end{matrix}

与雷达实际

\begin{matrix} σ_{pq}^{0} \end{matrix}

之间的误差,主要由l的不准确造成的.因此,Su等^[15]引入了有效粗糙度参数的思想,利用

\begin{matrix} σ_{pq}^{0} \end{matrix}

Assessment of the operational applicability of RADARSAT-1 data for surface soil moisture estimation

2006

... 研究显示,在对粗糙度的采样中,l的测量误差远远大于S的误差,较短的廓线长度和较大的采样间距会造成l的严重低估^[14].研究发现,AIEM模拟的

\begin{matrix} σ_{pq}^{0} \end{matrix}

与雷达实际

\begin{matrix} σ_{pq}^{0} \end{matrix}

之间的误差,主要由l的不准确造成的.因此,Su等^[15]引入了有效粗糙度参数的思想,利用

\begin{matrix} σ_{pq}^{0} \end{matrix}

微波与光学遥感协同反演旱区地表土壤水分研究

2016

... 本文使用Rahman的LUT方法来计算有效相关长度.主要步骤如下^[19]：首先利用AIEM模型模拟不同粗糙度和不同土壤水分下的后向散射系数值,将模拟的后向散射系数值存储于表格中,即LUT表;然后利用SAR影像中获得的后向散射系数和成本函数在表格中查找,查找规则是根据不同极化方式的后向散射系数实测值和AIEM模型模拟值查找符合成本函数最小值的后向散射系数记录,LUT表中该后向散射系数记录对应的相关长度数值即为有效相关长度.成本函数如下： ...

微波与光学遥感协同反演旱区地表土壤水分研究

2016

USDA ARS Grazinglands Research Lab.Washita '92[EB/OL].

... 为获取到适合研究区地表类型的S取值,首先利用AIEM模型对后向散射情况模拟,土壤水分输入实测采样点的M_v,粗糙度参数的取值范围通过对样地采样数据的统计分析获得.为了尽可能精确地反映自然地表的粗糙度,采用美国NASA和农业部的合作实验“Washita '92”的数据^[20]和中国科学院寒区旱区科学数据中心所提供的“黑河综合遥感联合试验（WATER '2008）”中的299组实测数据,测量的地表类型有裸土地表、草地、农田、芦苇地、盐碱地等.通过分析S实测数值的分布情况,发现均方根高度的范围S∈(0.2 cm, 2.5 cm)时,在所有实测数据中所占比例达到98%.因此取S∈(0.2 , 2.5 ) cm,步长为0.1 cm,相关长度的范围为l∈（1 ,120） cm,步长为1 cm,建立

\begin{matrix} σ_{pq}^{0} \end{matrix}

模拟数据集,通过LUT法结合成本函数查找有效相关长度值,反复试验后确定S为 1.4 cm.即在本文研究区内,S=1.4 cm,l∈（1 ,120 ）cm时,有效相关长度的反演精度较高. ...

Analysis of surface roughness heterogeneity and scattering behavior for radar measurements

2006

基于粗糙度参数的风沙滩地区土壤水分微波遥感反演模型研究

2016

... 为减少反演过程中的参数,在很多文献中地表粗糙度使用S和l的组合形式Z_s来表示.在土壤水分的反演研究中现有的Z_s形式有多种^[22],对“黑河综合遥感联合试验”中样区的实测粗糙度数据进行统计分析,表1为样区中不同形式的Z_s统计分析表. ...

基于粗糙度参数的风沙滩地区土壤水分微波遥感反演模型研究

2016

多极化、多角度SAR土壤水分反演算法研究

2004

... 裸土和低矮植被覆盖地表的

\begin{matrix} σ_{pq}^{0} \end{matrix}

与Z_s、M_v之间的关系可以表示为^[23]： ...

多极化、多角度SAR土壤水分反演算法研究

2004

... 裸土和低矮植被覆盖地表的

\begin{matrix} σ_{pq}^{0} \end{matrix}

与Z_s、M_v之间的关系可以表示为^[23]： ...

〈

〉

地表组合粗糙度不确定性引起的SAR反演土壤水分的不确定性分析

Uncertainty Analysis of SAR-retrieved Soil Moisture Induced by Uncertainty of Soil Surface Combined Roughness

1 引言

2 研究区与数据源

3 研究方法

3.1 AIEM模型

3.2 有效相关长度的计算

3.3 地表组合粗糙度的不确定性分析

3.4 土壤水分反演经验方程

3.5 误差控制

4 分析结果

4.1 组合粗糙度不确定性量化

4.2 土壤含水量不确定性量化

4.3 对比分析

4.4 误差控制范围

4.5 试验验证

5 结论与讨论

参考文献 文献选项 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

参考文献

原文顺序

文献年度倒序

文中引用次数倒序

被引期刊影响因子