Analysis of the Relationship between Vegetation and Radar Interferometric Coherence
植被与雷达干涉相干性的关系分析

Abstract  抽象

To effectively reduce the impact of vegetation cover on surface settlement monitoring, the relationship between normalized difference vegetation index (NDVI) and coherence coefficient was established. It provides a way to estimate coherence coefficient by NDVI. In the research, a new method is tried to make the time range coincident between NDVI results and coherence coefficient results. Using the coherence coefficient results and the NDVI results of each interference image pair in the study area, the mathematical relationship between NDVI and the coherent coefficient was established based on statistical analysis of the fitting results of the exponential model, logarithmic model, and linear model. Four indicators were selected to evaluate the fitting results, including root mean square error, determinant coefficient, prediction interval coverage probability, and prediction interval normalized average width. The fitting effect of the exponential model was better than that of the logarithmic model and linear model. The mean of error was −0.041 in study area ROI1 and −0.126 in study area ROI2.The standard deviation of error was 0.165 in study area ROI1 and 0.140 in study area ROI2. The fitting results are consistent with the coherence coefficient results. The research method used the NDVI results to estimate the InSAR coherence coefficient. This provides an easy and efficient way to indirectly evaluate the interferometric coherence and a basis in InSAR data processing. The results can provide pre-estimation of coherence information in Ningxia by optical images.
为有效降低植被覆盖对地表沉降监测的影响，建立了归一化差值植被指数 （NDVI） 与相干系数的关系。它提供了一种通过 NDVI 估计相干系数的方法。在研究中，尝试了一种新方法使 NDVI 结果和相干系数结果之间的时间范围重合。利用研究区各干涉图像对的相干系数结果和 NDVI 结果，在对指数模型、对数模型和线性模型的拟合结果进行统计分析的基础上，建立了 NDVI 与相干系数的数学关系。选取均方根误差、行列式系数、预测区间覆盖概率和预测区间归一化平均宽度 4 个指标对拟合结果进行评价。指数模型的拟合效果优于对数模型和线性模型。研究区域 ROI1 的误差平均值为 -0.041，研究区域 ROI2 的误差平均值为 -0.126。研究区 ROI1 的误差标准差为 0.165，研究区 ROI2 的误差标准差为 0.140。拟合结果与相干系数结果一致。该研究方法使用 NDVI 结果估计 InSAR 相干系数。这为间接评估干涉相干性和 InSAR 数据处理的基础提供了一种简单有效的方法。结果可为宁夏地区光学图像相干信息提供预估计。

1. Introduction  1. 引言

Interferometric synthetic aperture radar (InSAR) is a useful technique in generating a digital elevation model and monitoring surface deformation [1,2,3,4]. It is based on the phase difference of the echoes received by a satellite or aircraft. It has the advantage of acquiring data over a site during day or night time under all weather conditions and not impeded by cloud cover.
干涉合成孔径雷达 （InSAR） 是生成数字高程模型和监测表面变形的一种有用技术 [ 1， 2， 3， 4]。它基于卫星或飞机接收到的回波的相位差。它的优势在于，在所有天气条件下，白天或黑夜都会在站点上采集数据，并且不受云层覆盖的阻碍。

Interferometric coherence is one of the most important parameters for measuring the quality of interferograms in data processing of InSAR. It is predominantly evaluated using the coherence coefficient. The main factors affecting the interferometric coherence are Doppler decorrelation, thermal noise decorrelation, spatial decorrelation, and temporal decorrelation. Doppler decorrelation refers to a decorrelation phenomenon due to the inconsistency of the Doppler center frequency for twice imaging, which can be suppressed by azimuth filtering and can be further reduced by estimating the Doppler function [5]. The thermal noise decorrelation generates in the radar system when transmitting, receiving electromagnetic wave signals, and recording ground echo information [2]. Radar systems mainly rely on hardware settings to reduce the impact of system thermal noise. Spatial decorrelation refers to a decorrelation phenomenon caused by the spatial baseline of two radar images being too long and the side looking angles observed on the same targets for two images being different, resulting in considerable changes in phase [6]. The critical baseline can be used to evaluate the spatial decorrelation. With the baseline increasing, the correlation decreases. When the baseline increases to critical baseline, the coherence coefficient is 0. To reduce the effect of spatial decorrelation, it is necessary to set a short spatial baseline. Temporal decorrelation refers to a decorrelation phenomenon caused by large changes in the scattering characteristics of the ground objects when imaging twice [2]. To reduce the effect of temporal decorrelation, it is necessary to set a short temporal baseline. Additionally, long wavelength has a good effect on temporal correlation [7].
干涉相干性是 InSAR 数据处理中衡量干涉图质量的最重要参数之一。它主要使用相干系数进行评估。影响干涉相干性的主要因素是多普勒去相关、热噪声去相关、空间去相关和时间去相关。多普勒去相关是指由于两次成像的多普勒中心频率不一致而产生的去相关现象，这种现象可以通过方位滤波来抑制，也可以通过估计多普勒函数来进一步降低 [ 5]。雷达系统中在发射、接收电磁波信号和记录地面回波信息时会产生热噪声去相关 [ 2]。雷达系统主要依靠硬件设置来减少系统热噪声的影响。空间去相关是指由于两幅雷达图像的空间基线过长，并且两幅图像在同一目标上观测到的侧视角度不同，导致相位发生较大变化而引起的去相关现象 [ 6]。临界基线可用于评估空间去相关。随着基线的增加，相关性降低。当基线增加到临界基线时，相干系数为 0。要减少空间去相关的影响，必须设置一个较短的空间基线。时间去相关是指在成像两次时，由于地面物体的散射特性发生较大变化而引起的去相关现象 [ 2]。为了减少时间去相关的影响，有必要设置一个较短的时间基线。此外，长波长对时间相关性有很好的影响 [ 7]。

The growth of surface vegetation is one of the main factors causing the decorrelation of radar images. It is difficult to reduce its impact through post-processing. The normalized difference vegetation index (NDVI) can be obtained through calculations based on optical imagery and it is often used to monitor vegetation cover and growth. The relationship between NDVI and the coherence coefficient of radar images can effectively reflect the influence of vegetation cover on the interferometric coherence. There has been research conducted on it [8,9,10,11,12,13,14,15]. Bai et al. (2020) built a liner relationship between interferometric coherence and NDVI and found that the months with low NDVI are suitable for deformation monitoring [10]. Chen et al. (2021) used Landsat and ALOS-1 data to establish the relationship between NDVI and coherence coefficient in Meitanba in Hunan Province [14]. Liu et al. (2021) studied the relationship between the coherence coefficient and NDVI in southern China using linear and power function models and determined that it has a negative correlation relationship, considering the accuracy of the power function model is higher than that of the linear function model [15]. The relationship can be used in simple estimating of coherence coefficient.
地表植被的生长是导致雷达图像脱切的主要因素之一。很难通过后处理来减少其影响。归一化差值植被指数 （NDVI） 可以通过基于光学影像的计算获得，通常用于监测植被覆盖和生长。NDVI 与雷达图像相干系数的关系可以有效反映植被覆盖对干涉相干性的影响。已经对它进行了研究 [ 8， 9， 10， 11， 12， 13， 14， 15]。Bai et al. （2020） 在干涉相干性和 NDVI 之间建立了线性关系，发现 NDVI 低的月份适合进行变形监测 [ 10]。Chen et al. （2021） 使用 Landsat 和 ALOS-1 数据建立了湖南省眉丹坝 NDVI 与相干系数之间的关系 [ 14]。Liu et al. （2021） 使用线性和幂函数模型研究了南方地区相干系数与 NDVI 之间的关系，并确定其具有负相关关系，考虑到幂函数模型的准确性高于线性函数模型 [ 15]。该关系可用于简单估计相干系数。

The function between NDVI and coherence coefficient can be effectively used to extract the decoherent area covered by vegetation in the radar image with accessible optical images. It may provide an easy way to get the decoherent area in SAR images and provide a basis in InSAR data processing. In this study, we got the Sentinel-1 and Sentinel-2 images acquired in Ningxia Hui Autonomous Region, China. We established three models to establish the relationship between NDVI and the coherent coefficient and selected four indicators to evaluate the fitting results. The results can provide pre-estimation of coherence information in Ningxia by optical images.
利用 NDVI 与相干系数之间的函数，可以有效地提取雷达图像中植被覆盖的退相干区域，并提供可访问的光学图像。该方法为获取 SAR 图像中的退相干面积提供了一种简便的方法，为 InSAR 数据处理提供了依据。在本研究中，我们获得了在中国宁夏回族自治区采集的 Sentinel-1 和 Sentinel-2 图像。我们建立了 3 个模型来建立 NDVI 与相干系数之间的关系，并选择了 4 个指标来评价拟合结果。结果可为宁夏地区光学图像相干信息提供预估计。

This paper is organized as follows. In the next section, the study area and the data information of Sentinel-1 and Sentinel-2 are described in detail. The critical method for establishing the model is introduced in Section 3. A new method is tried to make the time range of NDVI results and coherence coefficient results coincident. Section 4 is dedicated to the presentation of the obtained results. Finally, conclusions are addressed.
本文的组织结构如下。在下一节中，将详细介绍 Sentinel-1 和 Sentinel-2 的研究区域以及数据信息。第 3 节介绍了建立模型的关键方法。尝试了一种新的方法，使 NDVI 结果和相干系数结果的时间范围重合。第 4 节专门介绍获得的结果。最后，得出结论。

2. Overview of the Study Area and Data
2. 研究区域和数据概述

2.1. Overview of the Study Area
2.1. 研究区域概述

Located in the eastern part of Ningxia, the study area is in Yanchi County and is represented by red rectangles in Figure 1. ROI1 is used for model building and model accuracy validation. ROI2 is used for model feasibility analysis. The study area has a mid-temperate semi-arid continental monsoon climate. The climate is dry and hot, and the winter is severely cold and the summer is extremely hot with changeable conditions [16]. There are high diurnal fluctuations in temperature and the average annual temperature is approximately 9 °C, with the average annual precipitation being 205.2 mm [16]. The terrain of the study area is high in the south and low in the north, with a gentle terrain in the north.
研究区位于宁夏东部，位于盐池县，图 1 中用红色矩形表示。ROI1 用于模型构建和模型准确性验证。ROI2 用于模型可行性分析。研究区属中温带半干旱大陆性季风气候。气候干燥炎热，冬季严寒，夏季极热，条件多变 [ 16]。温度昼夜波动很大，年平均气温约为 9 °C，年平均降水量为 205.2 mm [ 16]。研究区地形南高北低，北平缓地形。

Figure 1. (a) is China map and (b) is the location of the study area and remote sensing image coverage. Two red angles represent the study areas. A purple rectangle represents Sentinel-1 image coverage. A blue rectangle represents Sentinel-2 image coverage. And basemap bases on SRTM 30 m digital elevation model (DEM) data. Note: Figure (a) is based on the standard map with the review number GS(2019) 1673 downloaded from the standard map service website of the National Administration of Surveying, Mapping and Geographic Information, and the basemap has not been modified.
图 1.（a） 是中国地图，（b） 是研究区域的位置和遥感影像覆盖范围。两个红色角表示研究区域。紫色矩形表示 Sentinel-1 影像覆盖范围。蓝色矩形表示 Sentinel-2 影像覆盖范围。底图基于 SRTM 30 m 数字高程模型 （DEM） 数据。注：图（a）基于从国家测绘地理信息局标准地图服务网站下载的审核编号为 GS（2019） 1673 的标准地图，底图未作修改。

2.2. Data  2.2. 数据

2.2.1. Radar Data  2.2.1. 雷达数据

Sentinel-l is a C-band radar satellite with a wavelength of approximately 5.6 cm, which has advantages of all-weather and day-and-night observation, a stable revisit time, and convenient access. Sentinel-1 has a 12-day revisit cycle and can be downloaded from the Earth Data website (https://asf.alaska.edu/ (accessed on 6 December 2021)). The time range of SAR images is from September 2017 to August 2018. The information of acquisition date is shown in Table A1. During the time range, study area ROI1 was stable without obvious deformation. It can reduce the impact of deformation on the model. Single look complex images were used in the research. The data scan mode is IW mode, with an ascending track mode. The Sentinel-1 image coverage is represented by a purple rectangle in Figure 1.
Sentinel-l 是一颗波长约为 5.6 cm 的 C 波段雷达卫星，具有全天候、昼夜观测、重访时间稳定、接入方便等优点。Sentinel-1 的重访周期为 12 天，可从 Earth Data 网站下载（https://asf.alaska.edu/（于 2021 年 12 月 6 日访问））。SAR 图像的时间范围为 2017 年 9 月至 2018 年 8 月。采集日期信息如表 A1 所示。在时间范围内，研究区域 ROI1 稳定，无明显变形。它可以减少变形对模型的影响。研究中使用了单视复杂图像。数据扫描模式为 IW 模式，具有升序跟踪模式。Sentinel-1 影像覆盖范围在图 1 中由紫色矩形表示。

2.2.2. Optical Image Data
2.2.2. 光学图像数据

Sentinel-2 is a high-resolution multispectral imaging satellite, which consists of two satellites, Sentinel-2A and Sentinel-2B. Under cloudless conditions, the single-satellite revisit cycle is 10 days and the double-satellite revisit cycle can reach 5 days. Sentinel-2 has 12 bands and three resolutions, with detailed information shown in Table 1. The red band and the near-infrared band are used for NDVI calculation, and the resolution is 10 m. The Sentinel-2 image coverage is represented by a blue rectangle in Figure 1. The Sentinel-2 data includes two product levels: the L1C product and the L2A product. The L2A product can be obtained using the L1C product and atmospheric correction processing. The L2A product was used in the research and the cloud coverage was less than 10%. The information of optical image data used in the research is shown in Table A2.
Sentinel-2 是一颗高分辨率多光谱成像卫星，由 Sentinel-2A 和 Sentinel-2B 两颗卫星组成。在无云条件下，单星重访周期为 10 d，双星重访周期可达 5 d。Sentinel-2 有 12 个波段和 3 个分辨率，详细信息如表 1 所示。红波段和近红外波段用于 NDVI 计算，分辨率为 10 m。Sentinel-2 影像覆盖范围在图 1 中由蓝色矩形表示。Sentinel-2 数据包括两个产品级别：L1C 产品和 L2A 产品。L2A 产品可以使用 L1C 产品和大气校正处理获得。研究中使用了 L2A 产品，云覆盖率不到 10%。研究中使用的光学图像数据信息如表 A2 所示。

Table 1. Band information of Sentinel-2.

Sensor	Band Number	Band Name	Sentinel-2A		Sentinel-2B		Resolution (meters)
			Central Wavelength (nm)	Bandwidth (nm)	Central Wavelength (nm)	Bandwidth (nm)
MSI	4	Red	664.5	30	665	30	10
MSI	8	NIR	835.1	115	833	115	10

3. Modeling Processing  3. 建模处理

3.1. Preprocessing  3.1. 预处理

Critical steps in preprocessing include obtaining the coherence coefficient of the radar images and the NDVI from optical images of the study area, resampling the NDVI results, selecting images, and using the maximum value composite (MVC) algorithm to calculate the NDVI.
预处理的关键步骤包括从研究区域的光学图像中获取雷达图像和 NDVI 的相干系数、对 NDVI 结果进行重采样、选择图像以及使用最大值合成 （MVC） 算法计算 NDVI。

3.1.1. Coherence Coefficient Calculation
3.1.1. 相干系数计算

Interferometric coherence is one of the key parameters for measuring the quality of interferograms in the data processing of InSAR and is mainly evaluated using the coherence coefficient. The coherence coefficient value is between [0,1]$\left[0,1\right]$. The higher the value, the better the interferometric coherence. The coherence coefficient calculation equation is shown in (1) [17,18].
干涉相干性是 InSAR 数据处理中衡量干涉图质量的关键参数之一，主要使用相干系数进行评价。相干系数值介于 之间 [0,1]$\left[0,1\right]$ 。值越高，干涉相干性越好。相干系数计算方程如 （1） [ 17， 18] 所示。

𝛾=𝐸[𝑣1𝑣∗2]𝐸[𝑣1𝑣∗1]−−−−−−−√𝐸[𝑣2𝑣∗2]−−−−−−−√$$\gamma =\frac{E\left[v_1{v}_2^{\ast }\right]}{\sqrt{E\left[v_1{v}_1^{\ast }\right]}\sqrt{E\left[v_2{v}_2^{\ast }\right]}}$$

(1)

where 𝑣1$v_1$ and 𝑣2$v_2$ represents the reference and secondary images, ∗$\ast$ represents the conjugate complex number. The tool to calculate coherence coefficient is GMTSAR, an open source InSAR processing system [5,19], and the spatial resolution of the coherence coefficient results is 25 m. The DEM data is SRTM 30 m digital elevation model (DEM) data.
其中 𝑣1$v_1$ 和 𝑣2$v_2$ 表示参考图像和辅助图像， ∗$\ast$ 表示共轭复数。计算相干系数的工具是 GMTSAR，一个开源的 InSAR 处理系统 [ 5， 19]，相干系数结果的空间分辨率为 25 m。DEM 数据是 SRTM 30 m 数字高程模型 （DEM） 数据。

The information about interferometric pairs is shown in Table A3, and the coherence coefficient results are shown in Figure A1. The temporal baseline threshold for the interferometric images was 24 days, and the vertical baseline threshold was 100 m. A short perpendicular baseline means the high baseline correlation, which represents the high spatial correlation [20]. The equation for calculating baseline correlation is shown in (2) [21].
有关干涉对的信息如表 A3 所示，相干系数结果如图 A1 所示。干涉图像的时间基线阈值为 24 d，垂直基线阈值为 100 m。较短的垂直基线意味着高基线相关性，这代表了高空间相关性 [ 20]。计算基线相关性的方程式如 （2） [ 21] 所示。

𝛾𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒=1−𝐵⊥𝐵⊥,𝑐𝑟𝑖𝑡$${\gamma }_{baseline}=1-\frac{B_{\perp }}{B_{\perp ,crit}}$$

(2)

where 𝐵⊥$B_{\perp }$ represents perpendicular baseline and 𝐵⊥,𝑐𝑟𝑖𝑡$B_{\perp ,crit}$ represents critical perpendicular baseline. The equation for calculating critical baseline is shown in (3) [21].
其中 𝐵⊥$B_{\perp }$ ，表示垂直基线， 𝐵⊥,𝑐𝑟𝑖𝑡$B_{\perp ,crit}$ 表示临界垂直基线。计算临界基线的方程式如 （3） [ 21] 所示。

𝐵⊥,𝑐𝑟𝑖𝑡=𝜆𝑅𝑡𝑎𝑛(𝜃−𝛼)2𝑑𝑅$$B_{\perp ,crit}=\frac{\lambda Rtan\left(\theta -\alpha \right)}{2d_R}$$

(3)

where 𝜆$\lambda$ is the wavelength, 𝑅$R$ is the slant range, 𝜃$\theta$ is the incident angle, 𝛼$\alpha$ is the slope, 𝑑𝑅$d_R$ is the range resolution. Set Sentinel-1 as an example. 𝑅$R$ is 90 km, 𝜃$\theta$ is 44 degrees. We can get the critical perpendicular baseline is about 5 km by using (3) in the flat range. When the perpendicular baseline is 100 m, baseline correlation is about 0.98.
其中 𝜆$\lambda$ 是波长， 𝑅$R$ 是倾斜范围， 𝜃$\theta$ 是入射角， 𝛼$\alpha$ 是斜率， 𝑑𝑅$d_R$ 是距离分辨率。以 Sentinel-1 为例。 𝑅$R$ 是 90 公里， 𝜃$\theta$ 是 44 度。我们可以通过在平坦范围内使用 （3） 得到临界垂直基线约为 5 公里。当垂直基线为 100 m 时，基线相关性约为 0.98。

3.1.2. NDVI Calculation  3.1.2. NDVI 计算

The NDVI value is between [−1,1] and a negative value indicates that the surface is covered with water, ice, or snow. Zero indicates bare soil, rock, or buildings. Positive values indicate the presence of vegetation, and the NDVI value increases as vegetation cover increases. The NDVI calculation equation is shown in (4) and the NDVI results are shown in Figure A2.
NDVI 值介于 [−1,1] 之间，负值表示表面被水、冰或雪覆盖。零表示裸露的土壤、岩石或建筑物。正值表示存在植被，NDVI 值随着植被覆盖率的增加而增加。NDVI 计算方程如图 （4） 所示，NDVI 结果如图 A2 所示。

𝑁𝐷𝑉𝐼=𝜌𝑁𝐼𝑅−𝜌𝑅𝐸𝐷𝜌𝑁𝐼𝑅+𝜌𝑅𝐸𝐷$$NDVI=\frac{{\rho }_{NIR}-{\rho }_{RED}}{{\rho }_{NIR}+{\rho }_{RED}}$$

(4)

where 𝜌𝑁𝐼𝑅${\rho }_{NIR}$ is the reflectance value of the near-infrared band and 𝜌𝑅𝐸𝐷${\rho }_{RED}$ is the reflectivity value of the red band. The tool to calculate NDVI is SNAP, an open-source data processing software provided by the European Space Agency (ESA).
其中 𝜌𝑁𝐼𝑅${\rho }_{NIR}$ 是近红外波段的反射率值， 𝜌𝑅𝐸𝐷${\rho }_{RED}$ 是红色波段的反射率值。计算 NDVI 的工具是 SNAP，这是欧洲航天局 （ESA） 提供的开源数据处理软件。

3.1.3. Resampling, Selecting Images, and Calculating the NDVI Using the MVC Algorithm
3.1.3. 使用 MVC 算法重新采样、选择图像和计算 NDVI

The resolution of the coherent coefficient images is equal to interferometric images. Because it is lower than the resolution of the NDVI images, the resolution of the NDVI images are resampled to the resolution of the coherent coefficient images so that there are same numbers of grids in the coherent coefficient and NDVI images.
相干系数图像的分辨率等于干涉图像。由于它低于 NDVI 影像的分辨率，因此 NDVI 影像的分辨率将重采样为相干系数影像的分辨率，以便相干系数和 NDVI 影像中的格网数量相同。

In ideal conditions, the SAR data have a 12-day revisit cycle, and optical data have a 5-day revisit cycle. The imaging time being inconsistent between the NDVI, and the interferometric pairs is the main problem. Chen et al. (2021) solved this problem by establishing the relationship between the time baselines and the coherence coefficient, and introducing the time variable 𝑡$t$ into the function of the NDVI and the coherence coefficient [14]. Liu et al. (2021) solved the inconsistency imaging problem between the NDVI and the interferometric pair based on the monthly NDVI dataset from 2016–2017 and by using the average coherence coefficient of the interferometric pairs for the months [15]. In this research, the MVC algorithm was used to calculate the NDVI images in the time range between the imaging time of the main and auxiliary images, so that the interferometric pair can correspond with the NDVI image. MVC refers to obtaining the maximum value of each grid in the NDVI images over a certain period, which is relatively easy to achieve. It can effectively reduce the influence of cloud, atmosphere, solar altitude angle, and other factors on remote sensing images and is widely used in the production of NDVI. The MVC results are shown in Figure A3.
在理想条件下，SAR 数据的重访周期为 12 天，光学数据的重访周期为 5 天。NDVI 和干涉对之间的成像时间不一致是主要问题。Chen et al. （2021） 通过建立时间基线和相干系数之间的关系，并将时间变量 𝑡$t$ 引入 NDVI 和相干系数的函数中来解决这个问题 [ 14]。Liu et al. （2021） 基于 2016-2017 年的月度 NDVI 数据集，并使用干涉测量对在月份的平均相干系数 [ 15] 解决了 NDVI 和干涉对之间的不一致成像问题。本研究采用 MVC 算法计算主、副像成像时间之间时间范围内的 NDVI 图像，使干涉对与 NDVI 图像相对应。MVC 是指获取 NDVI 图像中每个网格在一定时间内的最大值，相对容易实现。能有效降低云、大气、太阳高度角等因素对遥感影像的影响，广泛应用于 NDVI 的制作中。MVC 结果如图 A3 所示。

3.2. Data Sampling  3.2. 数据采样

The images are sampled according to main parameter: the sample window size is 40×40$40\times 40$ pixels. Although the NDVI images were processed using the MVC method, there is a considerable number of grids containing information from the cloud, with water also having a considerable influence on the experiment. Because compared with the images of coherence coefficient and the NDVI in water, some values of the coherence coefficient and the NDVI are smaller than 0.2, so the grid need be masked. The mean value and the correlation coefficients R are calculated for sampling and selecting. If the correlation coefficient was greater than 0.6, the mean value of the coherence coefficient and the NDVI are fitted. The correlation coefficient calculation formula is shown in (5).
根据 main 参数对图像进行采样：样本窗口大小为 40×40$40\times 40$ 像素。尽管 NDVI 图像是使用 MVC 方法处理的，但有相当数量的网格包含来自云的信息，其中水也对实验有相当大的影响。因为与水中相干系数和 NDVI 的图像相比，相干系数和 NDVI 的某些值小于 0.2，因此需要对网格进行遮罩。计算平均值和相关系数 R 以进行抽样和选择。如果相关系数大于 0.6，则拟合相干系数和 NDVI 的平均值。相关系数计算公式如 （5） 所示。

𝑅=∑𝑁𝑖=1(𝑥𝑖−𝑥̲)(𝑦𝑖−𝑦̲)∑𝑁𝑖=1(𝑥𝑖−𝑥̲)2−∑𝑁𝑖=1(𝑦𝑖−𝑦̲)2−−−−−−−−−−−−−−−−−−−−−−−−−√$$R=\frac{{\sum }_{i=1}^N\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}{\sqrt{{\sum }_{i=1}^N{\left(x_i-\overline{x}\right)}^2-{\sum }_{i=1}^N{\left(y_i-\overline{y}\right)}^2}}$$

(5)

where 𝑥𝑖$x_i$ and 𝑦𝑖$y_i$ represent the value of each raster element, and 𝑥̲$\overline{x}$ and 𝑦̲$\overline{y}$ represent the mean value of the raster element being sampled.
其中 𝑥𝑖$x_i$ and 𝑦𝑖$y_i$ 表示每个栅格元素的值， 𝑥̲$\overline{x}$ 而 and 𝑦̲$\overline{y}$ 表示正在采样的栅格元素的平均值。

3.3. Data Fitting Results Evaluation
3.3. 数据拟合结果评估

The fitting functions are evaluated using the root mean square error (RMSE) and determined coefficient (𝑅2$R^2$). Under the condition of a 99% confidence level, the prediction intervals coverage probability (PICP) and the prediction intervals normalized averaged width (PINAW) are used to evaluate the fitting functions.
拟合函数使用均方根误差 （RMSE） 和确定的系数 （ 𝑅2$R^2$ ） 进行评估。在 99% 置信水平的条件下，预测区间覆盖率概率 （PICP） 和预测区间归一化平均宽度 （PINAW） 用于评估拟合函数。

𝑅𝑀𝑆𝐸$RMSE$ is the deviation between the true value and the predicted value, which is calculated as shown in (6). The smaller the RMSE value, the smaller the gap between the true and predicted values.
𝑅𝑀𝑆𝐸$RMSE$ 是真实值与预测值之间的偏差，计算方式如 （6） 所示。RMSE 值越小，真实值和预测值之间的差距就越小。

𝑅𝑀𝑆𝐸=1𝑁∑𝑖=1𝑁(𝑦(𝑖)𝑟𝑒𝑎𝑙−𝑦̂(𝑖)𝑝𝑟𝑒𝑑𝑖𝑐𝑡)2−−−−−−−−−−−−−−−−−−⎷$$RMSE=\sqrt{\frac{1}{N}\sum _{i=1}^N{\left(y_{real}^{\left(i\right)}-{\widehat{y}}_{predict}^{\left(i\right)}\right)}^2}$$

(6)

where 𝑦(𝑖)𝑟𝑒𝑎𝑙$y_{real}^{\left(i\right)}$ is the true value and 𝑦̂(𝑖)𝑝𝑟𝑒𝑑𝑖𝑐𝑡${\widehat{y}}_{predict}^{\left(i\right)}$ is the predictive value calculated by fitting function.
其中 𝑦(𝑖)𝑟𝑒𝑎𝑙$y_{real}^{\left(i\right)}$ 是 true 值， 𝑦̂(𝑖)𝑝𝑟𝑒𝑑𝑖𝑐𝑡${\widehat{y}}_{predict}^{\left(i\right)}$ 是 Fitting 函数计算的预测值。

𝑅2$R^2$ is used to determine the fitting degree of the regression equation, which is calculated as shown in (7). The value field is [0,1]. The larger the 𝑅2$R^2$ value, the better the degree of the model fitting.
𝑅2$R^2$ 用于确定回归方程的拟合度，其计算方式如 （7） 所示。值字段为 [0,1]。 𝑅2$R^2$ 值越大，模型拟合的程度越好。

𝑅2=∑𝑁𝑖=1(𝑦̂(𝑖)𝑝𝑟𝑒𝑑𝑖𝑐𝑡−𝑦̲𝑟𝑒𝑎𝑙)2∑𝑁𝑖=1(𝑦(𝑖)𝑟𝑒𝑎𝑙−𝑦̲𝑟𝑒𝑎𝑙)2=1−∑𝑁𝑖=1(𝑦(𝑖)𝑟𝑒𝑎𝑙−𝑦̂(𝑖)𝑝𝑟𝑒𝑑𝑖𝑐𝑡)2∑𝑁𝑖=1(𝑦(𝑖)𝑟𝑒𝑎𝑙−𝑦̲𝑟𝑒𝑎𝑙)2$$R^2=\frac{{\sum }_{i=1}^N{\left({\widehat{y}}_{predict}^{\left(i\right)}-{\overline{y}}_{real}\right)}^2}{{\sum }_{i=1}^N{\left(y_{real}^{\left(i\right)}-{\overline{y}}_{real}\right)}^2}=1-\frac{{\sum }_{i=1}^N{\left(y_{real}^{\left(i\right)}-{\widehat{y}}_{predict}^{\left(i\right)}\right)}^2}{{\sum }_{i=1}^N{\left(y_{real}^{\left(i\right)}-{\overline{y}}_{real}\right)}^2}$$

(7)

where 𝑦̂(𝑖)𝑝𝑟𝑒𝑑𝑖𝑐𝑡${\widehat{y}}_{predict}^{\left(i\right)}$ is the predictive value calculated by fitting function, 𝑦(𝑖)𝑟𝑒𝑎𝑙$y_{real}^{\left(i\right)}$ is the true value, and 𝑦̲𝑟𝑒𝑎𝑙${\overline{y}}_{real}$ is the average value.
其中 𝑦̂(𝑖)𝑝𝑟𝑒𝑑𝑖𝑐𝑡${\widehat{y}}_{predict}^{\left(i\right)}$ 是 Fitting 函数计算的预测值， 𝑦(𝑖)𝑟𝑒𝑎𝑙$y_{real}^{\left(i\right)}$ 是 true 值， 𝑦̲𝑟𝑒𝑎𝑙${\overline{y}}_{real}$ 是平均值。

𝑇𝑆𝑆=∑𝑖=1𝑁(𝑦̂(𝑖)𝑝𝑟𝑒𝑑𝑖𝑐𝑡−𝑦̲𝑟𝑒𝑎𝑙)2$$TSS={\displaystyle \sum }_{i=1}^N{\left({\widehat{y}}_{predict}^{\left(i\right)}-{\overline{y}}_{real}\right)}^2$$

(8)

𝑅𝑆𝑆=∑𝑖=1𝑁(𝑦(𝑖)𝑟𝑒𝑎𝑙−𝑦̂(𝑖)𝑝𝑟𝑒𝑑𝑖𝑐𝑡)2$$RSS={\displaystyle \sum }_{i=1}^N{\left(y_{real}^{\left(i\right)}-{\widehat{y}}_{predict}^{\left(i\right)}\right)}^2$$

(9)

𝐸𝑆𝑆=∑𝑖=1𝑁(𝑦(𝑖)𝑟𝑒𝑎𝑙−𝑦̲𝑟𝑒𝑎𝑙)2$$ESS={\displaystyle \sum }_{i=1}^N{\left(y_{real}^{\left(i\right)}-{\overline{y}}_{real}\right)}^2$$

(10)

The relationship among the three is:
这三者之间的关系是：

𝑇𝑆𝑆=𝑅𝑆𝑆+𝐸𝑆𝑆$$TSS=RSS+ESS$$

(11)

where 𝑇𝑆𝑆$TSS$ is the total square sum, 𝑅𝑆𝑆$RSS$ is the residual square sum, and 𝐸𝑆𝑆$ESS$ is the explained square sum.
其中 𝑇𝑆𝑆$TSS$ 是总平方和， 𝑅𝑆𝑆$RSS$ 是残差平方和， 𝐸𝑆𝑆$ESS$ 是解释的平方和。

PICP represents the actual frequency that observations fall within the prediction interval and is used to evaluate the reliability of the prediction interval, which is calculated as shown in (12) [22]. The higher the PICP value, the better the prediction result and the more reliable the model.
PICP 表示观测值落在预测区间内的实际频率，用于评估预测区间的可靠性，其计算方式如 （12） [ 22] 所示。PICP 值越高，预测结果越好，模型越可靠。

𝑃𝐼𝐶𝑃=1𝑁∑𝑖=1𝑁𝜀𝑖,𝜀𝑖={1,𝑦𝑖∈[𝐿𝑖,𝑈𝑖]0,𝑦𝑖∉[𝐿𝑖,𝑈𝑖]$$PICP=\frac{1}{N}{\displaystyle \sum }_{i=1}^N{\epsilon }_i,{\epsilon }_i=\left\{\begin{array}{c}1,y_i\in \left[L_i,U_i\right]\\ 0,y_i\notin \left[L_i,U_i\right]\end{array}\right.$$

(12)

where 𝑁$N$ is the sample size, 𝜀𝑖${\epsilon }_i$ is a Boolean variable that represents the relationship between the prediction interval and the observed value, 𝑦𝑖$y_i$ is the sample truth value, and 𝐿𝑖$L_i$ and 𝑈𝑖$U_i$ are the smallest and biggest bounds of the prediction interval, respectively.
其中 𝑁$N$ 是样本量， 𝜀𝑖${\epsilon }_i$ 是表示预测区间和观察值之间关系的布尔变量， 𝑦𝑖$y_i$ 是样本真值， 𝐿𝑖$L_i$ 和 𝑈𝑖$U_i$ 分别是预测区间的最小和最大边界。

The formula for calculating the 𝑃𝐼𝑁𝐴𝑊$PINAW$ is shown in (13). A smaller 𝑃𝐼𝑁𝐴𝑊$PINAW$ value means a more sensitive prediction interval and more reliable model [22].
计算公式 𝑃𝐼𝑁𝐴𝑊$PINAW$ 如 （13） 所示。较小的 𝑃𝐼𝑁𝐴𝑊$PINAW$ 值意味着更敏感的预测区间和更可靠的模型 [ 22]。

𝑃𝐼𝑁𝐴𝑊=1𝑁𝑅∑𝑖=1𝑁(𝑈𝑖−𝐿𝑖)$$PINAW=\frac{1}{NR}{\displaystyle \sum }_{i=1}^N\left(U_i-L_i\right)$$

(13)

where 𝑁$N$ is the sample size, 𝑅$R$ is the range of the sample target variable, and 𝐿𝑖$L_i$ and 𝑈𝑖$U_i$ represent the smallest and biggest bounds of the prediction interval, respectively.
其中 𝑁$N$ 是样本大小， 𝑅$R$ 是样本目标变量的范围， 𝐿𝑖$L_i$ 和 𝑈𝑖$U_i$ 分别表示预测区间的最小和最大边界。

3.4. The Method of Model Accuracy Validation
3.4. 模型准确性验证方法

Comparing the calculated coherence coefficient with the real coherence coefficient, the error can be obtained to evaluate the precision of the fitting function. The error calculation equation is shown in (14):
将计算出的相干系数与实际的相干系数进行比较，可以获得误差来评估拟合函数的精度。误差计算方程如 （14） 所示：

𝐸𝑟𝑟𝑜𝑟=𝛾𝑡𝑟𝑢𝑒−𝛾𝑝𝑟𝑒𝑑𝑖𝑐𝑡$$Error={\gamma }_{true}-{\gamma }_{predict}$$

(14)

where 𝛾𝑡𝑟𝑢𝑒${\gamma }_{true}$ represents the true coherence coefficient, and 𝛾𝑝𝑟𝑒𝑑𝑖𝑐𝑡${\gamma }_{predict}$ represents the predicted coherence coefficient calculated by 𝛾1${\gamma }_1$ and the NDVI image.
其中 𝛾𝑡𝑟𝑢𝑒${\gamma }_{true}$ 表示真正的相干系数， 𝛾𝑝𝑟𝑒𝑑𝑖𝑐𝑡${\gamma }_{predict}$ 表示由 𝛾1${\gamma }_1$ 和 NDVI 影像计算的预测相干系数。

4. Results and Discussion
4. 结果与讨论

4.1. Data Fitting Results
4.1. 数据拟合结果

The results from sampling are shown in Figure 2a. The range of the NDVI is [0.084717,0.709237]$\left[0.084717,0.709237\right]$ and the range of the coherence coefficient is [0.175292,0.88656]$\left[0.175292,0.88656\right]$. The NDVI and the coherence coefficient show a negative relationship. With the increase in NDVI, the interferometric coherence decreases. The exponential model, linear model, and logarithmic model are used to fit and the fitting results are shown in (15), (16), and (17), and four evaluation indicators in Section 3.3 are used to evaluate the fitting functions.
采样结果如图 2a 所示。NDVI 的范围是 [0.084717,0.709237]$\left[0.084717,0.709237\right]$ ，相干系数的范围是 [0.175292,0.88656]$\left[0.175292,0.88656\right]$ 。NDVI 和相干系数呈负相关。随着 NDVI 的增加，干涉相干性降低。采用指数模型、线性模型和对数模型进行拟合，拟合结果如（15）、（16）、（17）所示，3.3 节中的四个评价指标用于评价拟合函数。

𝛾1=0.9259e−3.982∗NDVI+0.1753,0.029≤𝑁𝐷𝑉𝐼≤1$${\gamma }_1=0.9259{\mathrm{e}}^{-3.982\ast \mathrm{NDVI}}+0.1753,0.029\le NDVI\le 1$$

(15)

𝛾2=−1.226𝑁𝐷𝑉𝐼+0.8673,0≤𝑁𝐷𝑉𝐼≤0.7074$${\gamma }_2=-1.226NDVI+0.8673,0\le NDVI\le 0.7074$$

(16)

𝛾3=−0.315𝑙𝑛(𝑁𝐷𝑉𝐼)+0.08049,0.054<𝑁𝐷𝑉𝐼≤1$${\gamma }_3=-0.315ln\left(NDVI\right)+0.08049,0.054<NDVI\le 1$$

(17)

Figure 2. Fitting result of exponential model 𝛾1${\gamma }_1$ (a) is the fitted curve graph, the red line is the fitted function curve, and the blue break line is the prediction interval curve. (b) is the residual graph.
图 2.指数模型 𝛾1${\gamma }_1$ （a） 的拟合结果是拟合曲线图，红线是拟合函数曲线，蓝色折断线是预测区间曲线。（b） 是残差图。

As shown in Figure 2a, almost all the points are inside the prediction interval. Figure 2b is a residual distribution map, showing that that the residuals are roughly distributed [−0.2,0.2]$\left[-0.2,0.2\right]$ and the residual distribution is relatively uniform. According to the statistical results, 𝑅𝑀𝑆𝐸$RMSE$ is 0.06272 and 𝑅2$R^2$ is 0.8767. Under the condition of a 99% confidence level, 𝑃𝐼𝐶𝑃$PICP$ is 99.52% and 𝑃𝐼𝑁𝐴𝑊$PINAW$ is 0.3249.
如图 2a 所示，几乎所有的点都在预测区间内。图 2b 是残差分布图，显示残差分布大致， [−0.2,0.2]$\left[-0.2,0.2\right]$ 残差分布相对均匀。根据统计结果， 𝑅𝑀𝑆𝐸$RMSE$ 分别为 0.06272 和 𝑅2$R^2$ 0.8767。在 99% 置信水平的情况下， 𝑃𝐼𝐶𝑃$PICP$ 为 99.52% 和 𝑃𝐼𝑁𝐴𝑊$PINAW$ 0.3249。

As shown in Figure 3a, almost all the points are inside the prediction interval. Figure 3b is a residual distribution plot, showing that the residuals are roughly distributed [−0.3,0.3] and there is an anomaly in the residuals, which becomes larger when the NDVI value is higher than 0.6. According to the statistical results, 𝑅𝑀𝑆𝐸$RMSE$ is 0.07551 and 𝑅2$R^2$ is 0.8207. Under the condition of a 99% confidence level, 𝑃𝐼𝐶𝑃$PICP$ is 99.36% and 𝑃𝐼𝑁𝐴𝑊$PINAW$ is 0.3908.
如图 3a 所示，几乎所有的点都在预测区间内。图 3b 是一个残差分布图，显示残差大致分布 [−0.3,0.3]，残差中存在异常，当 NDVI 值高于 0.6 时，残差会变得更大。根据统计结果， 𝑅𝑀𝑆𝐸$RMSE$ 分别为 0.07551 和 𝑅2$R^2$ 0.8207。在 99% 置信水平的条件下， 𝑃𝐼𝐶𝑃$PICP$ 为 99.36% 和 𝑃𝐼𝑁𝐴𝑊$PINAW$ 0.3908。

Figure 3. Fitting result of linear model 𝛾2${\gamma }_2$ (a) is the fitted curve graph, the red line is the fitted function curve, and the blue break line is the prediction interval curve. (b) is the residual graph.
图 3.线性模型 𝛾2${\gamma }_2$ （a） 的拟合结果是拟合曲线图，红线是拟合函数曲线，蓝色折断线是预测区间曲线。（b） 是残差图。

As shown in Figure 4a, almost all the points are inside the prediction interval. Figure 4b is a residual distribution plot, showing that the residuals are roughly distributed [−0.2,0.2] and the residual distribution is relatively uniform. According to the statistical results, 𝑅𝑀𝑆𝐸$RMSE$ is 0.06407 and 𝑅2$R^2$ is 0.8711. Under the condition of a 99% confidence level, 𝑃𝐼𝐶𝑃$PICP$ is 99.36% and 𝑃𝐼𝑁𝐴𝑊$PINAW$ is 0.3316.
如图 4a 所示，几乎所有的点都在预测区间内。图 4b 是一个残差分布图，显示残差大致分布 [−0.2,0.2]，残差分布相对均匀。根据统计结果， 𝑅𝑀𝑆𝐸$RMSE$ 分别为 0.06407 和 𝑅2$R^2$ 0.8711。在 99% 置信水平的情况下， 𝑃𝐼𝐶𝑃$PICP$ 为 99.36% 和 𝑃𝐼𝑁𝐴𝑊$PINAW$ 0.3316。

Figure 4. Fitting result of logarithmic model 𝛾3${\gamma }_3$ (a) is the fitted curve graph, the red line is the fitted function curve, and the blue break line is the prediction interval curve. (b) is the residual graph.
图 4.对数模型 𝛾3${\gamma }_3$ （a） 的拟合结果是拟合曲线图，红线是拟合函数曲线，蓝色折断线是预测区间曲线。（b） 是残差图。

The results of the evaluation are shown in Table 2. Comparing the 𝑅𝑀𝑆𝐸$RMSE$, it can be considered that the gap between the true value and the predicted value of 𝛾1${\gamma }_1$ is less those of than 𝛾2${\gamma }_2$ and 𝛾3${\gamma }_3$. Comparing 𝑅2$R^2$, it can be seen that the model fits of 𝛾1${\gamma }_1$ is better than those of 𝛾2${\gamma }_2$ and 𝛾3${\gamma }_3$. Under the condition of a 99% confidence level, comparing 𝑃𝐼𝐶𝑃$PICP$ and 𝑃𝐼𝑁𝐴𝑊$PINAW$, it can be seen that model reliability and prediction interval sensitivity of 𝛾1${\gamma }_1$ is higher than those of the others. In summary, the 𝛾1${\gamma }_1$ evaluation results are better than 𝛾2${\gamma }_2$ and 𝛾3${\gamma }_3$. Therefore, it is more reliable to choose 𝛾1${\gamma }_1$ as the fitting function.
评估结果如表 2 所示。比较 𝑅𝑀𝑆𝐸$RMSE$ ，可以认为 的真实值和预测值之间的差距 𝛾1${\gamma }_1$ 小于 𝛾2${\gamma }_2$ 和 的 𝛾3${\gamma }_3$ 差距。比较 𝑅2$R^2$ ，可以看出 的 𝛾1${\gamma }_1$ 模型拟合值优于 𝛾2${\gamma }_2$ 和 𝛾3${\gamma }_3$ 的模型拟合值。在 99% 置信水平的条件下，比较 𝑃𝐼𝐶𝑃$PICP$ 和 𝑃𝐼𝑁𝐴𝑊$PINAW$ 可以看出，模型可靠性和预测区间敏感性 高于 𝛾1${\gamma }_1$ 其他模型。综上所述， 𝛾1${\gamma }_1$ 评估结果优于 𝛾2${\gamma }_2$ 和 𝛾3${\gamma }_3$ 。因此，选择 𝛾1${\gamma }_1$ 作为拟合函数更可靠。

Table 2. Evaluation results.

Fitting Curve	𝑹𝑴𝑺𝑬$\mathit{R}\mathit{M}\mathit{S}\mathit{E}$	𝑹𝟐${\mathit{R}}^2$	𝑷𝑰𝑪𝑷/%$\mathit{P}\mathit{I}\mathit{C}\mathit{P}/\%$	𝑷𝑰𝑵𝑨𝑾$\mathit{P}\mathit{I}\mathit{N}\mathit{A}\mathit{W}$
𝛾1${\gamma }_1$	0.06272	0.8767	99.52	0.3249
𝛾2${\gamma }_2$	0.07551	0.8207	99.36	0.3908
𝛾3${\gamma }_3$	0.06407	0.8711	99.36	0.3316

4.2. Model Accuracy Validation
4.2. 模型准确性验证

One of interferometric pairs is selected to verify the precision of the fitting function in ROI1. The reference imaging time is 15 August 2019, and the secondary imaging time is 27 August 2019. The results are shown in Figure 5. Figure 5a shows the ROI1 NDVI result in the study area. Figure 5b shows the ROI1 coherence coefficient result in the study area. Figure 5c shows the error between the true values and the prediction values of the coherence coefficient. The overall fitting effect of the study area is good, and the larger error values are concentrated in the south of the study area. Figure 5d represents the error distribution, and the error is approximately followed by a normal distribution. The mean of error is −0.041, and the standard deviation of error is 0.165, with the error mainly being concentrated [−0.3,0.3]$\left[-0.3,0.3\right]$. Figure 5e is the slope distribution in ROI1.
选择干涉对中的一个来验证 ROI1 中拟合函数的精度。参考成像时间为 2019 年 8 月 15 日，二次成像时间为 2019 年 8 月 27 日。结果如图 5 所示。图 5a 显示了研究区域的 ROI1 NDVI 结果。图 5b 显示了研究区域中的 ROI1 相干系数结果。图 5c 显示了相干系数的真实值和预测值之间的误差。研究区整体拟合效果较好，较大的误差值集中在研究区南部。图 5d 表示误差分布，误差大约后跟正态分布。误差均值为 −0.041，误差标准差为 0.165，误差主要集中 [−0.3,0.3]$\left[-0.3,0.3\right]$ 。图 5e 是 ROI1 中的斜率分布。

Figure 5. Fit function accuracy verification plot of interferometric pair 20190815–20190827 in ROI1, (a) is the NDVI result; (b) is the coherence coefficient map; (c) is the error result; (d) is the error distribution; (e) is the slope of ROI1.
图 5.ROI1 中干涉对 20190815–20190827 的拟合函数精度验证图，（a） 为 NDVI 结果;（b） 是相干系数映射;（c） 是错误结果;（d） 是误差分布;（e） 是 ROI1 的斜率。

It can be seen from Figure 5c that the error in the northern part of the study area is [−0.2,0.2]$\left[-0.2,0.2\right]$. However, the estimated values in some areas are higher than the true values. In the southern part of the study area, there are larger error values. The estimated values for this area are predominantly lower than the true values. From Figure 5c,e, the large error values and high slope degrees have roughly the same spatial distribution. The high relief maybe the main reason for larger error values [14].
从图 5c 中可以看出，研究区域北部的误差为 [−0.2,0.2]$\left[-0.2,0.2\right]$ 。但是，某些区域的估计值高于真实值。在研究区域的南部，存在较大的误差值。此区域的估计值主要低于真实值。从图 5c，e 中可以看出，较大的误差值和高坡度具有大致相同的空间分布。高浮雕可能是较大误差值的主要原因 [ 14]。

Then, another interferometric pair is selected, which has a temporal baseline of 24 days. The reference imaging time is 3 August 2019, and the secondary imaging time is 27 August 2019. The results are shown in Figure 6. Figure 6a shows the ROI1 NDVI result in the study area. Figure 6b shows the ROI1 coherence coefficient result in the study area. Figure 6c shows the error between the true values and the prediction values of the coherence coefficient. The overall fitting effect of the study area is good, and the larger error values are concentrated in the south of the study area. Figure 6d represents the error distribution and the error is approximately followed by a normal distribution. The mean of error is −0.069, and the standard deviation of error is 0.170, with the error mainly being concentrated [−0.3,0.3]$\left[-0.3,0.3\right]$. Figure 6e is the slope distribution in ROI1. The results shown in Figure 6 are consistent with the results shown in Figure 5. The mean and standard deviation of error show a small increase, and the large error values and high slope degrees show the same spatial distribution as Figure 5.
然后，选择另一个干涉测量对，其时间基线为 24 天。参考成像时间为 2019 年 8 月 3 日，二次成像时间为 2019 年 8 月 27 日。结果如图 6 所示。图 6a 显示了研究区域的 ROI1 NDVI 结果。图 6b 显示了研究区域中的 ROI1 相干系数结果。图 6c 显示了相干系数的真实值和预测值之间的误差。研究区整体拟合效果较好，较大的误差值集中在研究区南部。图 6d 表示误差分布，误差大约后跟正态分布。误差均值为 −0.069，误差标准差为 0.170，误差主要集中 [−0.3,0.3]$\left[-0.3,0.3\right]$ 。图 6e 是 ROI1 中的斜率分布。图 6 所示的结果与图 5 所示的结果一致。误差的平均值和标准差显示小幅增加，大误差值和高坡度数显示与图 5 相同的空间分布。

Figure 6. Fit function accuracy verification plot of interferometric pair 20190803–20190827 in ROI1, (a) is the NDVI result; (b) is the coherence coefficient map; (c) is the error result; (d) is the error distribution; (e) is the slope of ROI1.
图 6.ROI1 中干涉对 20190803–20190827 的拟合函数精度验证图，（a） 是 NDVI 结果;（b） 是相干系数映射;（c） 是错误结果;（d） 是误差分布;（e） 是 ROI1 的斜率。

The other study area, ROI2, is used to verify model feasibility. The results are shown in Figure 7. Figure 7a shows the NDVI result of ROI2 and Figure 7b shows the prediction result of coherence coefficient in ROI2 by Figure 7a and fitting model. Figure 7c shows the true value of coherence coefficient in ROI2. Figure 7d shows the error between the true values and the prediction values of the coherence coefficient, and the overall fitting effect of the study area is good. Figure 7e shows the error distribution, and the error is approximately followed by the normal distribution. The mean of error is −0.126, and the standard deviation of error is 0.140, with the error being mainly concentrated [−0.4,0.2]$\left[-0.4,0.2\right]$. Figure 7f is the map of DEM, and Figure 7g is the map of slope. Figure 7h is the map of land use type distribution in 2018.
另一个研究区域 ROI2 用于验证模型的可行性。结果如图 7 所示。图 7a 显示了 ROI2 的 NDVI 结果，图 7b 显示了图 7a 和拟合模型对 ROI2 中相干系数的预测结果。图 7c 显示了 ROI2 中相干系数的真实值。图 7d 显示了相干系数的真实值与预测值之间的误差，研究区域的整体拟合效果良好。图 7e 显示了误差分布，误差大致后跟正态分布。误差均值为 −0.126，误差标准差为 0.140，误差主要集中 [−0.4,0.2]$\left[-0.4,0.2\right]$ 。图 7f 是 DEM 的地图，图 7g 是斜率的地图。图 7h 是 2018 年土地利用类型分布图。

Figure 7. Fit function accuracy verification plot of interferometric pair 20190803–20190827 in ROI2, (a) is the NDVI result; (b) is the prediction result of coherence coefficient; (c) is the coherence coefficient map; (d) is the error result; (e) is the error distribution; (f) is the map of DEM; (g) is the map of slope; (h) is the map of land use type distribution in 2018 [23], and T represents the land use type. Among them, 1 represents cultivated land, 2 represents woodland, 3 represents grassland, 4 represents waters, 5 represents urban and rural, industrial and mining, and residential land, construction land, and 6 represents unused land. Note: (f) is provided by Data Center for Resources and Environmental Sciences, Chinese Academy of Sciences (RESDC) (http://www.resdc.cn(accessed on 20 July 2021)).
图 7.ROI2 中干涉对 20190803–20190827 的拟合函数精度验证图，（a） 是 NDVI 结果;（b） 是相干系数的预测结果;（c） 是相干系数图;（d） 是错误结果;（e） 是误差分布;（f） 是 DEM 的地图;（g） 是坡度地图;（h） 为 2018 年土地利用类型分布图 [ 23]，T 表示土地利用类型。其中，1 表示耕地，2 表示林地，3 表示草地，4 表示水域，5 表示城乡、工矿、居住用地、建设用地，6 表示未利用地。注：（f） 由中国科学院资源与环境科学数据中心 （ http://www.resdc.cn（于 2021 年 7 月 20 日浏览）） 提供）。

Compared with Figure 7c, the prediction result of coherence coefficient in Figure 7b is consistent. The high NDVI value region in Figure 7a is the region with low coherence coefficient in Figure 7b,c. It can be seen from Figure 7d that the error is [−0.3,0.3], which is similar to the result showed in Figure 7e. Compared with the same interferometric pair result in ROI1, the absolute value of the mean error became larger and the standard deviation became smaller, with both following the normal distribution. Figure 7f,g shows ROI2 is flat area, and it is inconsistent between error distribution and slope degree. Figure 7d,h show that the larger error area distribution is generally in areas of urban and rural, industrial and mining and residential land, construction land, which are mostly misestimated. However, the results from ROI2 show that 𝛾1${\gamma }_1$ fits well for most of the area. Therefore, the model is effective.
与图 7c 相比，图 7b 中相干系数的预测结果是一致的。图 7a 中的高 NDVI 值区域是图 7b，c 中低相干系数的区域。从图 7d 中可以看出，误差为 [−0.3,0.3]，这与图 7e 中显示的结果相似。与相同的干涉对结果 ROI1 相比，平均误差的绝对值变大，标准差变小，均服从正态分布。图 7f、g 显示 ROI2 是平坦区域，误差分布和斜率程度不一致。图 7d、h 显示较大的误差区域分布一般在城市和农村、工矿和住宅用地、建设用地等区域，这些区域大多被误估。然而，ROI2 的结果表明，这 𝛾1${\gamma }_1$ 非常适合大部分地区。因此，该模型是有效的。

In this paper, an empirical model for NDVI and the coherence coefficient is obtained for study area ROI1 and the model accuracy is verified in study areas ROI1 and ROI2. Research shows that the model is reliable. The research result is useful in DInSAR data processing of deformation monitoring:
本文获得了研究区 ROI1 的 NDVI 和相干系数的经验模型，并在研究区 ROI1 和 ROI2 中验证了模型的准确性。研究表明，该模型是可靠的。研究结果可用于形变监测的 DInSAR 数据处理：

(1) The NDVI value can be used as a standard for evaluating the usability in DInSAR. It would be good with lower NDVI and bad in higher NDVI area. It is not useful in area when NDVI higher than 0.4. In this condition, longer band SAR data or technology is better.
（1） NDVI 值可以作为评价 DInSAR 可用性的标准。NDVI 较低是好事，而 NDVI 较高地区则是坏事。当 NDVI 高于 0.4 时，它没有用。在这种情况下，较长的波段 SAR 数据或技术更好。

(2) It can be used to reduce the impact of vegetation in deformation monitor. The deformation got in DInSAR technology should be masked in higher NDVI to improve the deformation result reliable.
（2） 可用于减少植被对变形监测的影响。采用 DInSAR 技术得到的变形应在较高的 NDVI 中进行掩盖，以提高变形结果的可靠性。

(3) It also can used as a basis to evaluate the precision of InSAR monitoring results. The DInSAR deformation results and precision with different NDVI area should be discussed separately. It is an important basis to calculate the error of DInSAR deformation results.
（3） 也可以作为评价 InSAR 监测结果精度的依据。不同 NDVI 面积下的 DInSAR 变形结果和精度应单独讨论。它是计算 DInSAR 形变结果误差的重要依据。

In the future, the different types of land cover in the correlation between NDVI and interferometric coherence should be considered [24]. Also, introducing a time parameter to the model maybe a good idea [14]. In addition, the model connected among multi-band, multi-polarization, and multi-vegetation indexes can be built [9].
未来，应考虑 NDVI 和干涉相干性相关性中不同类型的土地覆盖 [ 24]。此外，在模型中引入 time 参数可能是一个好主意 [ 14]。此外，还可以构建多波段、多极化和多植被指数之间的连接模型 [ 9]。

5. Conclusions  5. 结论

The relationship between NDVI and interferometric coherence in Ningxia was studied using optical images and C-band SAR images and an empirical model between the NDVI and the coherence coefficient was constructed through fitting analysis. The main conclusions were as follows:
利用光学图像和 C 波段 SAR 图像研究了宁夏 NDVI 与干涉相干的关系，并通过拟合分析构建了 NDVI 与相干系数之间的经验模型。主要结论如下：

(1) In the research, a new way is used to make the time range of NDVI results and coherence coefficient results coincident. The good fitting results show that it may be a good way to study the relationship between NDVI and interferometric coherence.
（1）在研究中，采用了一种新的方法使 NDVI 结果和相干系数结果的时间范围重合。良好的拟合结果表明，这可能是研究 NDVI 与干涉相干性之间关系的好方法。

(2) Three statistical models between the coherence coefficient and the NDVI were constructed using the exponential model, logarithmic model, and linear model. The RMSE, determinant coefficient, PICP, and PINAW, under the condition of a 99% confidence level, are calculated, respectively. Results show that the fitting effect of the exponential model is better than the other models. Based on the exponential function, the calculation accuracy was analyzed in study areas ROI1 and ROI2. Results showed that the error of the calculation conformed to the normal distribution. The mean of error was −0.041 in ROI1 and −0.126 in ROI2. The standard deviation of error was 0.165 in ROI1 and 0.140 in ROI2, which showed good fitting results using the exponential function.
（2） 使用指数模型、对数模型和线性模型构建了相干系数和 NDVI 之间的三个统计模型。在 99% 置信水平的条件下，分别计算 RMSE 、行列式系数、PICP 和 PINAW。结果表明，指数模型的拟合效果优于其他模型。基于指数函数，分析研究区域 ROI1 和 ROI2 的计算精度。结果表明，计算误差符合正态分布。ROI1 的误差均值为 -0.041，ROI2 的误差均值为 -0.126。误差标准差在 ROI1 中为 0.165，在 ROI2 中为 0.140，使用指数函数显示出良好的拟合结果。

(3) The result shows that it is possible to evaluate the coherence coefficient values using optical remote sensing data. It can be the complementary content to the study of the relationship between NDVI and coherence and can guide the selection of incoherent areas in radar images due to excessive vegetation coverage, which can provide the basis for DInSAR data processing of deformation monitoring. The results can provide pre-estimation of coherence information in Ningxia by optical images.
（3） 结果表明，可以使用光学遥感数据评估相干系数值。它可以作为研究 NDVI 与相干性关系的补充内容，可以指导雷达图像中由于植被覆盖过大而出现的非相干区域的选择，可为 DInSAR 的变形监测数据处理提供依据。结果可为宁夏地区光学图像相干信息提供预估计。