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Journal of Geo-information Science >

2018 , Vol. 20 >Issue 11: 1579 - 1591

DOI: https://doi.org/10.12082/dqxxkx.2018.180289

Research on Surface Deformation Based on GLM-PSO-coKriging Model

  • NIU Teng , 1 ,
  • YUE Depeng , 1 ,
  • LI Qian 2 ,
  • YU Qiang 1 ,
  • YU Jiaxin 1 ,
  • FANG Minzhe 1
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  • 1. Beijing Key Laboratory of Precision Forestry, Beijing Forestry University, Beijing 100083, China
  • 2. Forestry bureau of Qinhuangdao, Qinhuangdao 066004, China
*Corresponding author: YUE Depeng, E-mail：

Received date: 2018-06-15

  Request revised date: 2018-09-11

  Online published: 2018-11-20

Supported by

Special Fund for Basic Research Business Expenses of Central-level Public Welfare Research Institutes（CAFYBB2017MB026）

National Science and Technology Support Program of China, No.2012BAD16B00.

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《地球信息科学学报》编辑部 所有
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Abstract

Taking the Chengguan District of Lanzhou City as a research area in the slope disaster-prone area, the surface deformation rate of surface deformation points is extracted by PS-InSAR technology, and the deformation rate can effectively reflect the distribution and uplifting of geological disasters in the spatial range. Based on the coKriging interpolation, combined with the generalized linear model (GLM) and the particle swarm optimization (PSO) algorithm, the coKrigong interpolation is optimized by fitting the linear model to construct the PSO-GLM-coKriging interpolation model to the surface deformation rate. The main variables, DEM, geotechnical porosity and NDVI fitting parameters were covariates, and spatial interpolation simulations were performed. Compared with the co-Kriging model and the GLM-co-Kriging model, the PSO-GLM-coKriging interpolation model has higher precision and better simulation effect, eliminating the complexity of multi-dimensional generation and improving the small-scale range. Interpolation effect, the error of the three models is 1.25mm/year, 0.70mm/year, 0.47mm/year. By comparison, the PSO-GLM-coKriging interpolation model has higher simulation accuracy and better simulation results. In the overall range, the interpolation results of the three models are similar in spatial distribution, in line with the actual situation of the surface. Therefore, the interpolation simulation of the blank area of ​​the deformation point is carried out by the PSO-GLM interpolation model to fill the gap that the PS-InSAR technology can not extract the surface information at the non-deformation point, and the ground subsidence and uplift with sudden degeneration and sudden landslides will be completed. Geological disasters have been effectively combined, and the coupling relationship between geological disasters with high uncertainty and the monitoring of surface deformation can be established, which provides certain data and theoretical support for the planning and construction of urbanization in Chengguan District.

Key words： PS-InSAR; geological disaster; surface deformation rate; co-Kriging; PSO-GLM-coKriging interpolation model

Cite this article

NIU Teng , YUE Depeng , LI Qian , YU Qiang , YU Jiaxin , FANG Minzhe . Research on Surface Deformation Based on GLM-PSO-coKriging Model[J]. Journal of Geo-information Science, 2018 , 20(11) : 1579 -1591 . DOI: 10.12082/dqxxkx.2018.180289

1 引言

PS-InSAR技术是目前国内外用来监测地表形变最为前沿的方法。众多学者利用PS-InSAR技术进行轨道交通,断裂带,建筑载荷,冻土等地表形变相关检测[1,2,3,4,5,6]。在大尺度下,PS-InSAR技术能够充分展现其在地表形变检测的优越性,通过多幅雷达影像与地面高程的干涉处理,能够去除地形误差和时间失相干,有效地监测多年间的地表形变速率。但在较小的尺度,形变点分布不均匀的情况下,地表形变的监测就充满了不确定性,如：形变点分布不均,离散程度较高,受多种地面因子的影响过大,各维度的变化量不同等多种多样的问题。为了解决这些问题,有学者提出用插值法模拟地表形变。但熊思婷等[7]也证明普通的插值方法对地表形变进行插值的结果误差很大,因此,改进地表形变插值模型是很有必要的。
兰州市城关区属于黄河河谷型城市,地表山体不稳定,斜坡居多,且易发生滑坡、泥石流和山体崩塌等地质灾害,由于地表形变的缓变性,可以通过相关的监测手段,来有效的预测和防护地质灾害的发生[8,9]。本文利用协克里金插值模型[10],结合地面高程、植被覆盖度和岩土疏松度3个协变量拟合主变量地表形变速率,解决形变点分布不均和离散度较高的问题。在函数拟合的过程中利用广义线性模型[11],将离散型数据和连续性数据拟合为一个线性模型,并加入粒子群算法对广义线性模型进行优化[12,13],进一步对模型进行优化,提高预测精度。通过GLM-PSO-coKriging插值模型对城关区的地表形变结果进行模拟,以期为城关区的地表形变的监测与地质灾害的预测提供理论与方法支持。

2 材料与方法

2.1 研究区概况

兰州是中国西北地区的第二大城市群中心,市区南北方向群山环抱,东西方向黄河穿城而过。城关区作为兰州市的政治、经济、文化、科研、交通中心,在兰州市范围内占据很重要的地位,城关区地理位置介于北纬35°58′~36°09′,东经103°46′~103°59′之间,位于兰州盆地东部,地形地貌复杂多样。城关区内有黄河穿城而过,城区内宽度约在0.5~0.8 km之间,黄河北岸主要是为丘陵沟壑区,海拔1600~2032 m,其中西北部由庙儿岔、大破沟至黄河沿岸一带有城区呈扇形分布。黄河南岸为山地区,海拔在1900~2171 m之间,其上为大面积黄土覆盖,厚度约200 m左右,山坡坡度较陡,一般15~25°左右,从城关区最西端至最东端,黄河南岸的城区呈纺锤型分布,城区最宽处从黄河沿岸到将军山山脚有 4 km的距离。由于黄土斜坡所特有的地形地貌、地层岩性、气象水文、植被覆盖和强烈的人类活动,导致城关区范围内地质灾害频发,如滑坡、泥石流、地面塌陷、崩塌等。地质灾害主要分布在黄河北岸的沿河沟壑地带和黄河南岸的城区与山地接触边缘,地质灾害主要沿这两条地表形变带分布。而不同的地质灾害的构成因素和易发程度与附近的地面抬升或沉降有较强的相关性。

2.2 数据来源与处理

地表形变点的形变速率是反映地表形变的重要体现。地表形变速率对于景观格局的适应性评价和规划有着重要作用,地表形变反映的是一个地区的土壤疏松性和地块活跃性。利用PS-InSAR技术提取研究区范围内的形变点和形变速率。对获取41幅欧空局2010-2016年的Envisat SAR影像数据,利用Linux系统平台下的STAMPS/MIT软件进行PS-InSAR处理,选取2014年5月17日的影像为主影像,与剩余40幅影像形成40个干涉对。将外部引入的DEM与干涉图进行差分干涉处理,对得到的差分干涉图提取PS点。在具备PS点的基础上,用DEM进行相位纠正和相位解缠,进而得到PS点的形变速率图[14,15,16,17,18,19,20,21,22,23]。
兰州市城关区范围内共提取PS点8336个,研究区范围内形变点密度为37.89 PS/km²,远大于有效探测要求的4 PS/km²。由于PS点位过于密集,大部分PS点的形变速率过小,为降低插值中整体区域的误差,按照国际范例,选取地表形变大值点范围内形变速率大于2 mm/year的PS点,共108个,占形变速率大值点的40%,基本覆盖所有形变大值点所在的区域,形变速率小于2 mm/year的PS点54个,如图1,共162个形变点进行空间插值分析,形变速率在-6 ~ 5 mm/year之间,有效PS点基本覆盖研究区所有范围。
Fig. 1 Study area and PS point distribution map

图1 研究区及PS点位分布图

在地表形变PS点提取的基础上,发现地表形变点受很多因素的影响,如：地面高程（DEM）,植被覆盖指数（NDVI）,气候及降水,土壤岩性,人类利用等等。在这些环境因子的共同作用下,地表形变点呈一种不规则的状态分布,影响因子在区域范围内不同方向分布的不均匀的,在定量描述地表形变速率时需要通过构建一种更加有效的模型进行模拟分析。其中DEM数据来源于日本地球遥感观测数据中心下载的30 m分辨率的GEDM数据[24];土壤岩性数据来源于1:20万地质图,根据地质图进行矢量化得到研究区的岩性数据,岩性代表岩石的风化程度,决定了固体松散物的形成,岩性可通过不同的质地和强度进行分级,区分出不同的岩土疏松度;植被覆盖指数（NDVI）来源于兰州市城关区的Landsat 8影像计算结果,经过波段计算后得到同样为30 m分辨率栅格数据。

2.3 co-Kriging模型

协同克里金插值（co-Kriging）是在区域范围内进行插值的主变量与多个属性变量之间存在很强的空间相关性[25],相对于普通克里金插值方法,协克里金插值是将区域的空间估计值的参考变量在一个主变量的基础上增加了多个区域化协同变量的方法,通过模型的拟合,分析主变量与协同变量之间的相关性[26],更好地估算整个区域范围内的因变量分布情况[27,28,29,30,31,32,33]。
本文将地表形变点的形变速率v作为协同克里金插值主变量,选取多个影响因子作为插值方法的协同变量,因在协同克里金插值的模型里,不仅考虑了主变量的自相关性,还将具有交叉相关性的协变量的因素考虑在内[34]。本文将与地表形变速率具有强相关性的地面高程数据/归一化植被指数和岩土疏松度作为协变量进行参考分析。
协同区域化的协变量k所对应的影响因子数据{Zk（x）,k=1,2,…,k},其中k0为主变量地表形变速率。在各地表形变点上形变速率 Z k 0 ( x i ) （i=1,2,3,…,n）的数学期望存在并为一个定值 m k 0 。
ZVk0对中心点x0在待估测区域V（x0）上区域化变量地表形变速率k0估测的平均值：
Z Vk 0 = 1 V k 0 ∫ V k 0 Z k 0 x dx （1）
Zαk是在承载Vαk上的平均值：
Z αk = 1 V αk ∫ V k 0 Z k x dx （2）
整个区域范围内的地表形变速率估测值 Z V k 0 * 是k个协同区域变化量的有效数值的线性组合。 Z V k 0 * 是 Z V k 0 的无偏估计量,是在无偏和估计方差最小的基础上,求解μαk。
Z V k 0 * = ∑ k = 1 K ∑ αk nk μ αk Z αk （3）
式中： Z V k 0 * 为主变量地表形变速率的估测值;μαk为协克里金插值各因素权系数;Zαk为协同化区域变量
协同化区域变量在地表形变点{x1,x2,x3, …,xn}的范围内符合二阶平稳假设,二阶平稳假设是指区域化变量Z（x）的任一n维分布函数不因空间点x发生位移而改变,具有数学期望存在且平稳,方差和协方差存在且平稳的性质。在本文中协同化区域变量为DEM,岩土疏松度和NDVI,分别用ui(i=1, 2, …, n),vj(j=1, 2, …, m),ps(q=1, 2, …, r)来表示：
Z V k 0 * = ∑ i = 1 n a i u i + ∑ j = 1 m b j v j + ∑ q = 1 r c s p s （4）
式中：ai,bj,cs分别为DEM、岩土疏松度和NDVI的权重系数。

2.4 GLM-co-Kriging模型

传统的协同克里金插值模型对于多种协变量和主变量进行插值分析时,没有考虑到多种协变量与主变量之间的定量关系,仅仅用主变量和协变量进行简单的插值,无法有效反映不同变量对于插值点的作用程度,应该通过一定的模型拟合,赋给不同协变量相应的权重,定量地反映协变量DEM,NDVI和岩土疏松度与主变量地表形变之间的相互关系。需要通过一种模型拟合,在不影响协变量与地表形变相互关系的基础上,更加具体地反映底边形变速率的空间分布。
广义线性模型是常规正态线性模型的推广和延伸,它适用于连续性状和离散形状数据的统计分析[35]。相对于典型的线性模型,广义线性模型没有正态性的要求;相对于非线性模型,它没有明确随机误差分布[36]。本文中的地表形变速率受DEM,NDVI和岩土疏松度的影响,其中DEM和NDVI是连续性数据,而岩土疏松度根据不同岩石的硬度、脆性和弹性,不同土壤颗粒的松散度不同,分成多个级别,属于离散型数据[37,38]。GLM-co-Kriging模型虽然在插值过程中增加了模型拟合过程,但是量化了协变量与主变量的相互关系,实际上将插值中的模拟误差转移到模型拟合中,在相互关系不变的情况下,双变量的协克里金插值过程误差范围会相应减小,3个协变量的权也会相应减小。
广义线性模型主要由3种要素构成。
随机成分,指的是响应分布类型或者误差结构
f Y y , θ , ϕ = exp yθ - b θ a ϕ + c y , ϕ （5）
因变量服从比正态分布更一般的概率分布,即指数分布。因变量的每个观测值相互独立且服从指数型分布族中的一个分布,指数型分布族包括许多常见分布,如正态分布,泊松分布,逆高斯分布,二项分布,伽马分布等[39,40]。
线性预测：
y x = f T x β （6）
即解释变量的线性组合,表示为μ=β1x1+…βpxp。
连接函数,随机成分和系统成分之间连接函数
y x = g μ x （7）
随机成分：
f Y y , θ , ϕ = exp yθ - b θ a ϕ + c y , ϕ （8）
似然函数：
L = log f = yθ - b θ a θ + c y , ϕ （9）
式中： a ϕ , b θ , c y , ϕ 均为已知函数。θ为典范参数（Canonical Parameter）,φ为离差参数（Dispersion Parameter）,广义线性模型的方差为一个依赖于均值的函数。将Y地表形变速率,θ对应的DEM和NDVI, ϕ 对应的岩土疏松度代入不同的回归模型进行拟合分析。

2.5 GLM- PSO-coKriging模型

粒子群算法（PSO）源于鸟群捕食行为的研究,是通过无数的迭代从随机位置寻找最优解的过程,再利用适应度来评价解的品质,之后在当前最优解的基础上寻找全局最优解[41,42,43]。
将粒子群算法与广义线性模型相结合,改进为GLM-PSO-coKriging模型,在广义线性模型的基础上,通过粒子群算法优化参数拟合过程：
v D = a × f NDVI + b × g DEM + c × q L i + d （10）
式中：vD表示地表形变速率;NDVI表示归一化植被指数;DEM表示地表高程;Li表示岩性;a、b、c分别表示这3类指标在线性拟合中所占的比例,d为拟合系数。
在拟合这些系数的过程中,通过粒子群算法降低广义线性模型拟合的误差,以训练样本的输出值与实际的均方差作为适应度函数寻找最优解。实现过程如下：
（1） 设置粒子群计算参数;
（2） 设置适应度函数;
（3） 将第i组学习样本影响因子（DEMi,NDVIi,Lii）代入GLM拟合函数（转换成二次函数）,计算适应度值;
（4） 比较粒子所经过的所有位置的适应度值,确定其最优位置;
（5） 比较所有粒子在其最优位置的适应度值,确定整个种群的最优位置;根据各粒子自身位置及最优粒子位置调整粒子的速度和位置;
（6） 达到迭代终止条件,得到函数拟合最优解vDi;
（7） 将最优解与地表形变实际值vi进行对比,通过最优解vDi与实际值vi的差值重新拟合模型参数a,b,c,d。
GLM-PSO模型流程图如图2。
Fig. 2 Particle swarm optimization roadmap

图2 粒子群优化技术路线图

相较于原始的co-Kriging模型,GLM-PSO-coKriging模型在增加协同变量的基础上,利用广义线性模型,将协变量的组合进行优化,突出了主变量地表形变速率的主导影响,直接将主变量与协变量建立联系,通过协变量组合参数,尽量消除主变量的偶然误差。在此基础上,利用粒子群优化算法实现函数拟合最优解,使组合参数的误差降到最低。

3 结果与分析

3.1 基于co-Kriging模型的插值分析

PS-InSAR提取的地表形变数据,高精度形变点无法覆盖所有区域,因此,通过改进模型后的空间插值分析来确定未知点的地表形变速率。克里金插值是最基础的插值分析方法,现实空间上的地表形变速率不仅要通过空间距离的权重进行数值内插研究,更要充分考虑各种影响因子对于的地表形变速率的影响,对比地形地貌、气象水文、地层岩性、地质构造、人类活动等各种环境影响因子,选取其中空间相关性最大的DEM,NDVI和岩土疏松度,如图3所示。DEM主要反应地面高程信息,随着海拔的升高,地表形变速率的绝对值是降低的;相对应NDVI反应的是植被覆盖指数,植被覆盖是地表形变的一个重要影响因素,植被覆盖度越高,产生较大地面形变的概率越小;岩土疏松度越大,地表形变速率是升高的。利用协克里金插值,确定地表形变点的形变速率为主变量,将在1485~2161 m范围取值的地面高程DEM,主要区间在0.25-0.5的NDVI和量化指标的岩土疏松度作为协变量,共同模拟形变点空白地区的形变速率[44,45,46]。
Fig. 3 Covariate regional distribution map

图3 协变量区域分布图

从162个地表形变点中提取135个形变点作为样本点,剩余27个点作为检验样本。在协克里金插值中选择不同插值方式和参数应用,确定最好的插值效果图像。在ArcGIS中分别选择普通克里金插值（Ordinary）,简单克里金插值（Simple）,泛克里金插值（Universal）,和析取克里金插值（Disjunctive）进行插值比较,普通克里金插值多用于单个变量的无偏估计,泛克里金插值需要知道整个插值空间的整体变化趋势,析取克里金插值是非线性的插值模型,基于研究地表形变及其影响因子的耦合关系分析,确定简单克里金插值效果最好。通过分布直方图和正态QQ图的检验,主变量地表形变速率和 3个协变量都满足正态分布。选择不同协方差函数进行函数拟合精度分析,在环形模型（Circular）,球面模型（Spherical）,三球模型（Tetraspherical）,五球模型（Pentaspherical）,指数模型（Exponential）,高斯模型（Gaussian）,有理二次模型（Rational Quandratic）,空穴模型（Hole Effect）,K-贝塞尔模型（K-Bessal）,J-贝塞尔模型（J-Bessal）和稳定模型（Stable）11个函数模型的对比中,在对协方差模型的精度验证中,选择最优模型的标准是通过标准平均值（Mean Standardized）,均方根预测误差（Root Mean Square）,平均标准误差（Average Standard Error）和标准均方根预测误差（Root Mean Square Standardized）4个指标来构建。当标准平均值趋近于0,均方根预测误差最小,平均标准误差趋近于均方根预测误差,标准均方根预测误差趋近于1时,说明此协方差模型是最优模型。由此判断,在11个模型中稳定模型的效果最好[47,48,49]。
对地表形变速率构建协方差模型,经过拟合后的稳定协方差函数为：
Var（地表形变速率）-Var（DEM）
Model：-0.32671×Stable（0.056957,2）,如图4（a）。
Fig. 4 Correlation analysis of deformation rate and covariate

图4 形变速率与协变量相关性分析

Var（地表形变速率）-Var（岩土疏松度）
Model：0.36996×Nugget+0.35143×Stable（0.056957,2）,如图4（b）。
Var（地表形变速率）-Var（NDVI）
Model：0.18513×Nugget+0.82464×Stable（0.056957,2）,如图4（c）。
其中,Nugget为块金值,反映最小抽样尺度以下变量的变异性及测量误差;Stable为稳定模型协方差数学方程。
将地表形变速率作为主变量,DEM,NDVI和岩土疏松度作为协变量的稳定模型协克里金插值的模拟结果如图5所示。将模拟结果与27个检验样本叠加,提取对应形变点的协克里金插值的预测值,与形变点实际的形变速率进行对比。
Fig. 5 Training sample distribution map

图5 训练样本分布图

27个检验样本形变速率的对比如表1所示：在整体的分布上,大部分形变点的地表形变速率预测值的绝对值相对于实际值都偏小,在协克里金插值的过程中,通过稳定协方差函数拟合3个影响因子的过程中,增加了插值的复杂度,3个影响因子的维度多样性制约了插值过程中某个方向上较大值的出现;在较小的尺度上,地面沉降较大的形变点分布在黄河南岸主城区和山地区的连接带,城关区西北部丘陵区,少部分分散于黄河北岸丘陵区,其中黄河北岸的大值点误差相对较大,南岸城区山地区连接带误差相对较小;地面抬升的形变点主要分布在黄河沿岸,少部分分布在黄河北岸丘陵区,地面抬升的形变点的形变速率预测值整体比实际值小2mm/year;绝对值较小的地面形变点相对来说误差范围很小,综合各种协变量的协克里金插值,插值范围在-2 ~2 mm/year之间。
Tab. 1 Model comparison and evaluation

表1 模型对比评价

OBJECTID 地表形变速率（实际） 地表形变速率（co-Kriging） 地表形变速率（GLM-co-Kriging） 地表形变速率（PSO-GLM-coKriging）
1 -5.6 -1.658 598 -3.746 961 -4.618 687
2 -2.3 -1.296 831 -2.134 503 -2.317 376
3 -2.4 -1.273 715 -2.093 304 -2.275 046
4 -2.7 -1.250 287 -2.051 728 -2.231 728
5 -4.3 -1.814 903 -2.879 605 -3.130 972
6 -2 -1.555 483 -1.663 416 -1.798 855
7 -2.1 -2.037 225 -2.296 454 -2.394 602
8 2.1 0.824 218 1.614 265 1.708 631
9 2 0.391 450 1.074 233 1.137 565
10 -3.1 -2.075 240 -2.734 641 -2.734 221
11 3.7 2.059 876 2.232 309 3.111 206
12 2.1 0.533 751 1.116 056 2.006 622
13 -3.5 -2.120 437 -2.331 069 -3.405 210
14 -3.6 -1.580 881 -2.870 344 -2.769 015
15 -2.2 -1.026 174 -2.185 163 -2.781 297
16 2.4 0.467 513 1.093 025 1.102 090
17 2.2 0.231 962 1.412 381 1.466 303
18 -2.1 -0.169 692 -1.212 495 -1.578 251
19 2.3 0.923 981 1.146 161 2.017 496
20 0 -0.836 368 -0.316 787 -0.462 244
21 0 -0.947 541 -0.434 894 -0.480 970
22 -1.9 -1.393 763 -1.317 613 -1.512 581
23 1.6 1.222 541 1.248 370 1.357 471
24 0.5 0.966 701 1.020 924 0.841 816
25 0.7 0.844 806 0.919 411 0.647 823
26 -1.3 -0.317 138 -0.447 999 -0.653 810
27 -0.5 -0.635 594 0.018 314 -0.247 485

3.2 基于GLM-co-Kriging模型的插值分析

通过训练样本对协克里金插值的结果验证,发现简单的协克里金插值在综合多个协变量后的结果预测值出现聚集效应,90%的预测值在 -2~2 mm/year的范围内。基于此,在协克里金插值的基础上,通过加入广义线性模型,进一步明确地面形变点形变速率的主变量效应,突出形变速率的主导性。广义线性模型对DEM,岩土疏松度和NDVI 3个协变量的拟合的优势体现在广义线性模型可以将离散变量岩土疏松度和连续变量DEM,NDVI放到同一个线性模型中进行函数拟合。再将拟合后的函数公式用来对整个研究区范围内的协变量影响因子计算,并将这个函数拟合值作为一个协变量代入到协克里金插值的运算中。
通过数理统计软件SPSS对样本点的线性属性进行检测（图6）,在SPSS的回归分析检验下,对地表形变速率取绝对值,将地面抬升和地面塌陷的形变速率统一于一个函数体系中,整体用地表形变量表示,DEM,NDVI和岩土疏松度在研究区的空间范围内与地面形变速率是呈近似线性相关分布的：DEM在城关区与地表形变速率的绝对值是呈负相关的分布,城关区属于河谷型城区,黄河对于河谷平原的侵蚀,人类的各种开发利用,疏松的岩土质地和低植被覆盖度等各种因素导致随着DEM的增加,地表形变速率呈现出近似线性减小的趋势;NDVI也与地表形变速率呈现出线性负相关的关系;岩土疏松度与地表形变速率呈现出近似正相关的关系,在岩土疏松度较小,土壤质地较为疏松,岩石硬度较低的几类岩土地类中,随着岩土疏松度的增加,地表形变速率增大,而当岩土疏松度增大到某个量级时,地表形变速率趋于平稳,不会发生太大变化。
Fig. 6 Covariates and linear variables of main variables

图6 协变量与主变量线性分析

通过统计软件SPSS对162个PS点形变数据进行广义线性模型拟合,选择地表形变速率的绝对值为因变量,DEM,NDVI和岩土疏松度为协变量,模型构建项中选择因子模型,参数估计方法中选择混合模型的极大似然估计,通过函数拟合,得到广义线性模型公式：
V = 0.017 F - 0.003 H - 18.833 P + 13.585 （11）
式中：V代表研究区范围内各点的广义线性模型的拟合值;F代表对应点岩土疏松度;H代表地面高程DEM;P代表植被覆盖指数NDVI。
利用广义线性模型拟合出来的函数式,计算135个样本点的地表形变速率协变量拟合值,将其作为样本点的协变量,消除之前协克里金插值协变量过多引发的误差,通过协克里金插值得到城关区研究区范围内的GLM-co-Kriging模型图像。将 co-Kriging模型和GLM-co-Kriging模型的插值图像进行对比,插值的整体趋势是类似的,西北向东南地面形变速率逐渐减小再逐渐增大,由地面抬升到地面沉降。从总体来看,GLM-co-Kriging模型的插值的取值区间更大,插值结果的渐变性也更为明显,灵敏度也高于co-Kriging模型。co-Kriging模型的插值结果的渐变较为迟钝,而实际情况是临近各个形变点的形变速率数值差值和形变方向差距可能会很大,而GLM-co-Kriging模型在这方面差值效果较好,陆家沟马家沟村一带有一个地表抬升向地表塌陷的突变,徐家山国家公园和黄河改道一带也有一个地表抬升向塌陷的突变。整个研究区插值空间,尤其是这些区域,形变速率为0或者趋近于0的范围很少,更贴近实际情况。分析2种模型插值情况的具体差异,其中GLM-co-Kriging模型生成的图像在西南山区五泉山公园和兰山公园附近的形变速率趋近于0,而co-Kriging模型拟合出地面抬升;GLM-co-Kriging模型相对于co-Kriging模型在研究区在西北部到东北部福儿沟-马家沟村-台湾沟一线地面沉降的值也较大。
通过27个检验样本点对GLM-co-Kriging模 型和co-Kriging模型进行量化对比和精度验证,如图7。对误差进行统计分析,co-Kriging模型中误差大于1 mm/year的检验样本有17个点,GLM-co-Kriging模型中只有5个检验样本点的误差大于 1 mm/year,对2种模型的形变速率误差的绝对值取均值,co-Kriging模型的误差为1.25 mm/year,GLM-co-Kriging模型的误差为0.7 mm/year。在实际形变速率偏大的区域,co-Kriging模型的插值精度远小于GLM-co-Kriging模型。
Fig. 7 Comparison of error between co-Kriging model and GLM-co-Kriging model

图7 co-Kriging模型与GLM-co-Kriging模型误差对比图

3.3 基于PSO-GLM-coKriging模型的插值分析

由GLM-co-Kriging模型的插值结果与实际地表形变速率对比分析可知,这种模型还存在着一定的误差,在地表形变速率较大的形变点,依旧存在较大的误差。为了降低这种误差,将粒子群优化算法加入模型中,通过无数次的迭代运算,优化广义线性模型的拟合参数,通过统计软件MATLAB的粒子群优化算法相应工具箱,对其中一部分代码进行修正之后,在MATLAB R2014 a中设置迭代次数为2000,输入135个学习样本点和广义线性模型对应的联系函数,对其进行迭代运算。通过PSO-GLM模型拟合的插值函数为：
V = 0.0121 F - 0.0035 H - 18.754 P + 14.585 （12）
将PSO-GLM-coKriging模型拟合的插值函数导入到ArcGIS中,再次对135个地表形变样本点计算预测值,并将预测值作为协变量和主变量地表形变速率进行协克里金插值。对插值结果图像进行分析,在整体趋势上,地面形变速率的绝对值进一步增大,与实际的地面形变进一步趋同,同时地表形变速率大于0,即地面抬升的形变点更加集中。相较于GLM-co-Kriging模型,PSO-GLM-coKriging模型的插值大值区域在减小,也就是在某些范围内的插值变化很明显,反应迟钝现象大大减少,在插值结果中体现为在城关区西北部新城区附近石门沟村和徐家山国家森林公园一带PSO-GLM-coKriging模型的插值结果比周围地面抬升地区的形变速率有一段突增,效果图中颜色的对比很明显;而在GLM-co-Kriging模型中,地面抬升的整个范围内插值的渐变效果较为平缓,同时在地面沉降地区,也具备这个特点,地面沉降的大值点集中分布在研究区的中东部主城区的东部,因此可以看出PSO-GLM-coKriging模型对于地表形变空间分布引起的变化更为灵敏。而且,PSO-GLM-coKriging插值模型在小范围地区的插值效果相较于GLM-co-Kriging模型更能突显细节,从研究区西北部到东北部的王家庄-陡道沟-大红沟-桑院子村一线地面抬升带并不是一个连续的形变带,在北部陡道沟大浪村沟和东北部大牛圈附近地表形变值趋近于0。福儿沟-马家沟村-台湾沟的地面沉降带也更加明显。因此,在较小的尺度下,PSO-GLM-coKriging插值模型也能体现出其特有的优越性。
将PSO-GLM-coKriging插值模型的数据提取到27个检验样本点,在小尺度上通过检验样本量化对比3个模型的预测效果。用折线图来展示3个误差的范围和变化趋势,由图8、图9和表1可以明显看出,co-Kriging模型,GLM-co-Kriging模型和PSO-GLM-coKriging插值模型的预测精度是逐步提高的,3个模型的平均误差分别为1.25、0.70、0.47 mm/year,尤其是在地表形变较大的地区PSO-GLM-coKriging插值模型的预测精度远超于其他 2个模型,预测的误差大部分在1 mm/year之内。其中误差较大的形变点1,形变点5,形变点9和形变点16主要分布在研究区中部的主城区内,因主城区内地表形变量受新增建筑等多种人为因素的影响,所以预测误差在这部分范围内较大[55]。
Fig. 8 Prediction map of surface deformation rate in Chengguan District

图8 城关区地表形变速率预测图

Fig. 9 Model error comparison chart

图9 模型误差对比图

通过PSO-GLM-coKriging插值模型对城关区研究区地表形变的模拟分析,能够较为精确的构建地面高程,岩土疏松度,植被覆盖度与地表形变速率的联系,并将样本点的地表形变速率作为主变量,3个影响因子拟合的地表形变速率预测值为协变量通过协克里金插值构建整个城关区空间范围内地表形变的预测模型。通过这个模型,填补了西北部和北部丘陵九州台墩洼山大破沟地区和狼舌头和刘家坪地区,南部山地白家湾皋兰山乡红沟村地区形变点较少,无法通过PS-InSAR技术提取地表形变速率的不足。
通过PSO-GLM插值模型的预测,对城关区城区的城市化规划建设提出相应的参考依据。通过地表形变PSO-GLM插值模型提取的地表形变数值来评价和分析相应地区的发展规划,区分不同类型的地区,并对该地区的城市化提出相关的建议。通过地表形变的量化分析,有效规划对旧城区,棚户区和城中村的改造,对黄河北岸新城区的开发,对建城区的生态修复和休闲旅游观光产业的升级。
PSO-GLM-coKriging插值模型对地表形变和地质灾害建立了相关联系,对具有缓变性的地面沉降和抬升与突发性的山体滑坡等地质灾害进行了有效的结合。结合近年来兰州市城关区的地质灾害发生情况,其中,地质灾害中的地面坍塌和崩塌在城关区内发生频率较小,主要分布在城关区东部老城建城区附近,属于地面沉降的集中区;另外3种地质灾害,泥石流灾害在城关区主要分布在建城区与山区的交界地带,南北两条,带状分布,较为均匀,泥石流的地表形变主要发生在地表抬升或沉降2 mm/year的范围内;滑坡灾害主要分布在城关区的西部,地表形变以地面抬升为主;不稳定斜坡在研究区范围内较为分散,没有统一的规律来体现这种灾害的分布情况。将具有高度不确定性的地质灾害与可监测的地表形变建立相关的耦合关系,通过对地表形变监测来预测各类斜坡灾害的发生情况是未来的发展方向之一,并具有重要意义。但插值方法毕竟是理论化的一种分析手段,还需要结合更多的环境影响因子合实地情况进行综合监测才能达到最好的预测效果。

4 结论

（1）利用PS-InSAR技术提取地表形变点的形变速率描述兰州市城关区的各种地质灾害的发生情况,之后通过收集地表形变速率信息就可以对地质灾害进行有效的分析和预测。而在小尺度范围内,PS-InSAR技术下的形变点分布不均匀,因此提取的形变速率在空间范围内的部分也很不均匀。利用协克里金插值的方法对研究区范围内地表形变速率进行模拟,结合DEM,岩土疏松度和NDVI这3个协变量,弥补了普通克里金插值主变量地表形变速率单一,样本点少引起模拟精度不高等缺陷。
（2）在简单的协克里金插值的基础上对模型进行改进,加入广义线性模型,可以将离散型数据岩土疏松度,连续型数据DEM和NDVI拟合到同一个公式中,在普通协克里金插值中,过多的协变量制约了多维度的最大值出现,广义线性模型拟合后的公式降低的协变量的复杂度。在细节方面,通过粒子群优化算法不断对拟合系数进行迭代,使形变速率模拟值在较小的尺度上也能具备很高的精度。对比3种模型,co-Kriging模型、GLM-co-Kriging模型和PSO-GLM-coKriging插值模型的平均误差分别为1.25、0.70、0.47 mm/year。但利用改进的PSO-GLM-coKriging插值模型相较于实际的地表形变还是存在一定的误差,要达到最好的预测效果,还需要结合更多的环境因素和实地的综合监测。
（3）通过对比可知,PSO-GLM-coKriging插值模型具备更高的模拟精度和更好的模拟效果。在整体范围内,3种模型的插值结果在空间分布上是类似的,符合地表实际情况。通过PSO-GLM插值模型对形变点空白区的插值模拟,可以对地质灾害进行初步的预测,对城关区城镇化的规划建设提供一定的数据和理论支撑。

The authors have declared that no competing interests exist.

References

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DOI

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DOI

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DOI

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DOI

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DOI

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Fokker P A, Wassing B B T, Leijen F J V, et al. Application of an ensemble smoother with multiple data assimilation to the Bergermeer gas field, using PS-InSAR[J]. Geomechanics for Energy & the Environment, 2016,5:16-28.61Satellite radar data were used to constrain subsurface model parameters.61Ascending and descending line-of-sight measurements were used without decomposition.61A multiple data assimilation ensemble smoother was applied.

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Qin X, Yang M, Liao M, et al.Exploring temporal-spatial characteristics of Shanghai road networks settlement with multi-temporal PSInSAR technique[J]. Geomatics & Information Science of Wuhan University, 2017,42(2):170-177.The continuous monitoring of the stability of urban road networks is not only of utmost significance to mitigate the financial and human losses,but also beneficial to the sustainable development of economy and society.For the long-distance and long-span real time monitoring requirements,this article introduced PSInSAR(persistent scatterer synthetic aperture radar interferometry)technology to the deformation monitoring and warning of urban road networks.We processed 26 time series TerraSAR-X images in Shanghai and focused on the temporal-spatial analysis of the subsidence along the road networks.In the space,the overall land subsiding characteristics of road network was first demonstrated,and then the detailed subsidence detail and the subsidence driving force of the typical road.In the temporal time,we discussed the temperature variation associated with temporal displacements of asphalt pavement and verified the accuracy of the results.The results show the settlement of Shanghai road network is mainly distributed in Pudong district,which is associated with the density of roads.The rapid urbanization construction has become a main cause of urban road network subsidence.The temporal deformation shows some time correlation with the temperature variation.The verify results showing fairly consistent agreement.

DOI

[18]
Chen F, Wu Y, Zhang Y, et al.Surface motion and structural instability monitoring of Ming Dynasty city walls by two-step Tomo-PSInSAR approach in Nanjing city, China[J]. Remote Sensing, 2017,9(4):371.Spaceborne Multi-Temporal Synthetic Aperture Radar (SAR) Interferometry (MT-InSAR) has been a valuable tool in mapping motion phenomena in different scenarios. Recently, the capabilities of MT-InSAR for risk monitoring and preventive analysis of heritage sites have increasingly been exploited. Considering the limitations of conventional MT-InSAR techniques, in this study a two-step Tomography-based Persistent Scatterers (PS) Interferometry (Tomo-PSInSAR) approach is proposed for monitoring ground deformation and structural instabilities over the Ancient City Walls (Ming Dynasty) in Nanjing city, China. For the purpose of this study we utilized 26 Stripmap acquisitions from TerraSAR-X and TanDEM-X missions, spanning from May 2013 to February 2015. As a first step, regional-scale surface deformation rates on single PSs were derived (ranging from 40 to +5 mm/year) and used for identifying deformation hotspots as well as for the investigation of a potential correlation between urbanization and the occurrence of surface subsidence. As a second step, structural instability parameters of ancient walls (linear motion rates, non-linear motions and material thermodynamics) were estimated by an extended four-dimensional Tomo-PSInSAR model. The model applies a two-tier network strategy; that is, the detection of most reliable single PSs in the first-tier Delaunay triangulation network followed by the detection of remaining single PSs and double PSs on the second-tier local star network referring to single SPs extracted in the first-tier network. Consequently, a preliminary phase calibration relevant to the Atmospheric Phase Screen (APS) is not needed. Motion heterogeneities in the spatial domain, either caused by thermal kinetics or displacement trends, were also considered. This study underlines the potential of the proposed Tomo-PSInSAR solution for the monitoring and conservation of cultural heritage sites. The proposed approach offers a quantitative indicator to local authorities and planners for assessing potential damages as well as for the design of remediation activities.

DOI

[19]
Bayer B, Schmidt D, Simoni A.The influence of external digital elevation models on PS-InSAR and SBAS results: Implications for the analysis of deformation signals caused by slow moving landslides in the Northern Apennines (Italy)[J]. IEEE Transactions on Geoscience & Remote Sensing, 2017,99:1-14.Advanced interferometric synthetic aperture radar (InSAR) postprocessing, like persistent scatterer InSAR (PS-InSAR), offers the possibility to investigate slow moving landslides, where standard interferometry is problematic. These advanced algorithms involve the analysis of a series of SAR acquisitions in both time and space. One input that requires particular attention for landslide applications is the external digital elevation model (DEM) that is used to correct the interferograms for the topographic phase term. When multiple elevation data sets are available for a given study area, it is difficult to decide which one should be used. In this paper, we test the sensitivity of PS-InSAR/Small Baseline Subset (SBAS) results to different DEMs. The study area is located in the Northern Apennines of Italy, where chaotic clay shales and fine-grained flysch host slow-moving earth flows and ancient rock slides. C-band (Envisat) and X-band (Cosmo-SkyMed) data are processed with different DEMs. We describe a simple framework to statistically analyze the influence of these models on the final PS-InSAR/SBAS results. We find that individual interferograms do not vary much depending on the DEM, while the results from PS-InSAR and SBAS analysis do vary. This is likely caused by the way the DEM error is estimated. We find also that the quality of the DEM is more important than the resolution and that X-band InSAR data are more sensitive to the choice of the DEM than C-band. The significance of the results is discussed with reference to two landslide areas.

DOI

[20]
张永红,张继贤,龚文瑜,等.基于SAR干涉点目标分析技术的城市地表形变监测[J].测绘学报,2009,38(6):482-487.通过深入研究干涉点目标的相位模型，提出基于空间搜索的邻近点目标干涉相位差解缠方法，用以计算点目标的地形残差和线性形变，以及分离点目标大气延迟相位和非线性形变相位的时空域滤波方法，解决干涉点目标分析中的关键问题。最后，以苏州地区地表沉降监测为应用试验，利用形成的SAR干涉点目标形变信息提取技术，获取苏州市区1992--2002年间的地表沉降信息。研究结果与已有文献记录保持比较好的一致性，证明SAR干涉点目标技术完全可以发展成为应用于城市地表形变监测的实用化技术。

DOI

[ Zhang Y H, Zhang J X, Gong W Y, et al.Monitoring urban subsidence based on SAR interferometric point target analysis[J]. Acta Geodaetica et Cartographica Sinica, 2009,38(6):482-487. ]

[21]
陈强,罗容,杨莹辉,等.利用SAR影像配准偏移量提取地表形变的方法与误差分析[J].测绘学报,2015,44(3):301-308.单一的InSAR观测技术可提取地表沿雷达视线方向(LOS)上的一维位移,而利用SAR影像配准过程中的同名像素偏移量可提取地表沿雷达方位向(近南北向)与距离向(近东西向)的二维形变场,与LOS方向的一维形变形成优势互补.本文在分析SAR影像配准偏移量提取地表形变场方法的基础上,推导建立了雷达方位向与距离向形变提取的误差模型,探讨了该方法提取地表形变的主要误差源.以Bam地震ASAR影像和玉树地震PALSAR影像为数据源,分别开展同震形变场的提取与误差分析试验,结果表明,基于SAR像素配准偏移量提取同震形变场的精度主要受匹配窗口尺寸与过采样因子影响,形变提取误差随匹配计算窗口的增大而减小,形变提取精度随过采样因子的增大有适量提高,地形起伏效应在高差较大的SAR影像区域中表现较为显著.

DOI

[ Chen Q, Luo R, Yang Y H, et al.Method and accuracy of extracting surface deformation field form SAR image coregistration[J]. Acta Geodaetica et Cartographica Sinica, 2015,44(3):301-308. ]

[22]
杨艳. 北京地面沉降InSAR监测效果分析[J].上海国土资源,2013,34(4):21-24.在分析北京地面沉降灾害现状、现有监测方法和技术手段基础上，对比InSAR监测与传统分层动态监测、高精度水准测量、GPS测量等在区域地面沉降监测时效性、精确性和经济性等之间的差别，并以北京地铁13号线地面沉降InSAR监测为例，分析其优势与不足，为优化北京地面沉降监测资源、全面开展线性工程地面沉降灾害监测提供参考，以助于推进首都减灾防灾工作进程。

DOI

[ Yang Y.Effectiveness of InSAR monitoring of land subsidence in Beijing[J]. Shanghai Land & Resources, 2013,34(4):21-24. ]

[23]
刘志敏,李永生,张景发,等.基于SBAS-InSAR的长治矿区地表形变监测[J].国土资源遥感, 2014,26(3):37-42.小基线集InSAR（SBAS-InSAR）时序分析方法能够较好地克服InSAR时空失相干限制，抑制地形和大气影响，增加时间采样率，在监测地表形变随时间演化方面取得了较好的应用。为了有效监测山西省长治矿区地表形变，利用DInSAR方法监测开采矿区的快速大形变，得到形变区30 d的最大沉降量为11 cm；利用SBAS方法监测矿区边缘微小缓慢形变，得到2003年7月-2010年7月期间区内地面沉降的空间展布及时间序列相对形变量。对于矿区周围相干性保持较好的居民区，SBAS方法监测结果表明其整体形变表现为沉降趋势，沉降面积较大，沉降速率为5~15 mm/a，最大累计沉降为90 mm。矿区因开采时间、开采方式、采储量以及地形等因素的不同而呈现出不同的沉降结果。

DOI

[Liu Z M,Li Y S,Zhang J F, et al. An analysis of surface deformation in the Changzhi mining area using small baseline In SAR[J]. Remote Sensing for Land and Resources, 2014,26(3):37-42. ]

[24]
薛亚婷. 基于雷达干涉测量技术的不同环境影响因子下兰州市区斜坡灾害识别及敏感性分析研究[D].兰州:兰州大学,2015.

[ Xue Y T.Slope hazards identification and susceptibility analysis in Lanzhou urban based on PS-InSAR[D]. Lanzhou: Lanzhou University, 2015. ]

[25]
卢月明,王亮,仇阿根,等.一种基于主成分分析的协同克里金插值方法[J]. 测绘通报, 2017(11):51-57.针对协同克里金插值方法在插值时,辅助变量较多造成计算复杂度增加,而辅助变量较少引起插值精度降低这一问题,提出了一种基于主成分分析的协同克里金插值方法(PCA-CoKriging)。该方法首先使用主成分分析对插值相关变量进行将维,得到较少几个综合指标,然后里利用这几个综合指标作为辅助变量进行协同克里金插值。为验证该方法的有效性和数据分布对该方法的影响,本文选取了2016年北京市范围内4个季节中PM2.5浓度满足正态分布效果不同的4组数据,分别使用PCA-CoKriging和普通克里金插值方法、常规协同克里金插值方法,进行了插值试验。结果表明,本文方法与普通克里金插值方法、常规协同克里金插值法在4组试验中的平均绝对误差分别为4.91、6.04、5.61,平均均方根误差分别为6.65、8.76、7.57。综合比较,本文方法比常规协同克里金插值的平均绝对误差与均方根误差分别提升了10.73%、12.56%,比普通克里金插值法的平均绝对误差与均方根误差分别提升了18.71%、24.09%。

[ Lu Y M, Wang L, Chou A G, et al.A CoKriging interpolation method based on principal component analysis[J]. Bulletin of Surveying and Mapping, 2017(11):51-57. ]

[26]
翟进乾.克里金(kriging)插值方法在煤层分布检测中的应用研究[D].太原:太原理工大学,2008.

[ Zhai J Q.Applied reseach of kriging interpolation method in distributed detection of coal seam[D]. Taiyuan: Taiyuan University of Technology, 2008. ]

[27]
Cellmer R, Zrobek S.The cokriging method in the process of developing land value maps[C]. Baltic Geodetic Congress. 2017:364-368.

[28]
Adhikary S K, Muttil N, Yilmaz A G.Cokriging for enhanced spatial interpolation of rainfall in two Australian catchments[J]. Hydrological Processes, 2017,31(12):2143-2161.Abstract Rainfall data in continuous space provide an essential input for most hydrological and water resources planning studies. Spatial distribution of rainfall is usually estimated using ground-based point rainfall data from sparsely positioned rain-gauge stations in a rain-gauge network. Kriging has become a widely used interpolation method to estimate the spatial distribution of climate variables including rainfall. The objective of this study is to evaluate three geostatistical (ordinary kriging (OK), ordinary cokriging (OCK), kriging with an external drift (KED)) and two deterministic (inverse distance weighting, radial basis function) interpolation methods for enhanced spatial interpolation of monthly rainfall in the Middle Yarra River catchment and the Ovens River catchment in Victoria, Australia. Historical rainfall records from existing rain-gauge stations of the catchments during 1980-2012 period are used for the analysis. A digital elevation model of each catchment is used as the supplementary information in addition to rainfall for the OCK and KED methods. The prediction performance of the adopted interpolation methods is assessed through cross-validation. Results indicate that the geostatistical methods outperform the deterministic methods for spatial interpolation of rainfall. Results also indicate that among the geostatistical methods, the OCK method is found to be the best interpolator for estimating spatial rainfall distribution in both the catchments with the lowest prediction error between the observed and estimated monthly rainfall. Thus, this study demonstrates that the use of elevation as an auxiliary variable in addition to rainfall data in the geostatistical framework can significantly enhance the estimation of rainfall over a catchment.

DOI

[29]
Tan J, Li A, Lei G.Contrast on anusplin and cokriging meteorological spatial interpolation in Southeastern Margin of Qinghai-Xizang Plateau[J]. Plateau Meteorology, 2016,23(5):124-129.Meteorological data is the essential data of ecological,resources,environment,global change and other research areas.However,the meteorological in Mountain are rare and complex than in plain.And acquiring precise spatial grid meteorological data has been a difficult task in this area.Anusplin and Cokriging interpolation method are one of the most common method that considering terrain'impacts during the Meteorological Spatial Interpolation.To ensure which one is more suit for complex area,we take the most complex mountain area(the Southeastern margin of the Qinghai-Xizang Plateau)as study area to compare the two methods.Based on Anusplin and Cokriging meteorological spatial interpolation method separately,combined with the terrain data and 96 meteorological stations in the southeastern margin of the Qinghai-Xizang Plateau region,250 m resolution average temperature and total precipitation interpolated surfaces in 2010 was obtained.With the cross-validation method comparison method,the interpolation accuracy of Anusplin and Cokriging was compared,and the spatial distribution of errors was analyzed.Applying relevant information,accurate of the two methods in local area where the result of interpolation are quite different was qualitatively analyzed.Through this,the method which is more suit for this area is sought out and the applicability of Anusplin in this area was assessed.The results showed that,Anusplin interpolation outperformed Cokriging.In the comparing of mean square error(RMSE)of the interpolation of temperature and precipitation,Anusplin temperature is only 0.82 and Cokriging is 1.45 ,the RMSE of precipitation of the two methodes are consistent,but Anusplin are superior to Cokriging in the highly heterogeneous area.Therefore Anusplin can achieve better results than Cokriging,indicating that Anusplin is suit for the interpolation in Southeastern Margin of Qinghai-Xizang Plateau.

[30]
Xu H, Russell B, Innanen K A.Determination of reservoir thickness and distribution using improved rescaled cokriging[C]. Seg Technical Program Expanded, 2016:2967-2971.

[31]
Sun W.Comparison of a cokriging method with a Bayesian alternative[J]. Environmetrics, 2015,9(4):445-457.

[32]
Wang Q, Atkinson P M, Shi W.Indicator cokriging-based subpixel mapping without prior spatial structure information[J]. IEEE Transactions on Geoscience & Remote Sensing, 2014,53(1):309-323.Indicator cokriging (ICK) has been shown to be an effective subpixel mapping (SPM) algorithm. It is noniterative and involves few parameters. The original ICK-based SPM method, however, requires the semivariogram of land cover classes from prior information, usually in the form of fine spatial resolution training images. In reality, training images are not always available, or laborious work is needed to acquire them. This paper aims to seek spatial structure information for ICK when such prior land cover information is not obtainable. Specifically, the fine spatial resolution semivariogram of each class is estimated by the deconvolution process, taking the coarse spatial resolution semivariogram extracted from the class proportion image as input. The obtained fine spatial resolution semivariogram is then used to estimate class occurrence probability at each subpixel with the ICK method. Experiments demonstrated the feasibility of the proposed ICK with the deconvolution approach. It obtains comparable SPM accuracy to ICK that requires semivariogram estimated from fine spatial resolution training images. The proposed method extends ICK to cases where the prior spatial structure information is unavailable.

DOI

[33]
Gratiet L L, Cannamela C.Cokriging-based sequential design strategies using fast cross-validation techniques for multi-fidelity computer codes[J]. Technometrics, 2015,57(3):418-427.Cokriging-based surrogate models have become popular in recent decades to approximate a computer code output from a few simulations using both coarse and more complex versions of the code. In practical applications, it is common to sequentially add new simulations to obtain more accurate approximations. We propose a method of cokriging-based sequential design, which combines both the error evaluation provided by the cokriging model and the observed errors of a leave-one-out cross-validation procedure. This method is proposed in two versions, the first one selects points one at a time. The second one allows us to parallelize the simulations and to add several design points at a time. The main advantage of the suggested strategies is that at a new design point they choose which code versions should be simulated (i.e., the complex code or one of its fast approximations). A multifidelity application is used to illustrate the efficiency of the proposed approaches. In this example, the accurate code is a two-dimensional finite element model and the less accurate one is a one-dimensional approximation of the system. This article has supplementary material online.

DOI

[34]
徐成东. 基于线性加权回归模型的降水量空间插值方法研究[D].郑州:河南大学,2008.

[ Xu C D.Research on the spatial interpolation for precipitation data using weighted Linear Regression Model[D]. Zhengzhou: Henan University, 2008. ]

[35]
马彦辉. 基于GLM的非正态响应稳健设计研究[D].天津:天津大学,2008.

[ Ma Y H.Study on Non-Normal response robust design based on GLM[D]. Tianjin: Tianjin University, 2008. ]

[36]
赵林城,尹长明.广义线性模型中极大拟似然估计的强相合性[J].中国科学,2005,35(3):312-317.假定在一个具有一般联系函数的广义线性模型中,q×1响应变量yi是可观测的,p×q协变量Zi是固定设计的,以λ-n记∑ni=1 ZiZli的最小特征根.在λ-n≥nα(对某个α＞0),supi≥1 E‖yi‖r＜∞(对某个r＞1/α)和其他正则条件下,证明了以概率为1,当n充分大时,未知回归参数向量的拟似然方程有一个解,它收敛于参数真值.这一结果是对文献中的相应结果的实质性改进.

DOI

[ Zhao L C, Yin C L.Strong consistency of maximal quasi-likelihood estimators in generalized linear models[J]. Chinese Science, 2005,35(3):312-317. ]

[37]
Peter H E, Kevin E S.GLM estimation of trade gravity models with fixed effects[J]. Empirical Economics, 2016,50(1):137-175.Many empirical gravity models are now based on generalized linear models (GLM), of which the poisson pseudo-maximum likelihood estimator is a prominent example and the most frequently used estimator....

DOI

[38]
Bosquet C, Boulhol H.Applying the GLM variance assumption to overcome the scale-dependence of the negative Binomial QGPML Estimator[J]. Econometric Reviews, 2014,33(7):772-784.Recently, various studies have used the Poisson Pseudo-Maximal Likehood (PML) to estimate gravity specifications of trade flows and non-count data models more generally. Some papers also report results based on the Negative Binomial Quasi-Generalised Pseudo-Maximum Likelihood (NB QGPML) estimator, which encompasses the Poisson assumption as a special case. This note shows that the NB QGPML estimators that have been used so far are unappealing when applied to a continuous dependent variable which unit choice is arbitrary, because estimates artificially depend on that choice. A new NB QGPML estimator is introduced to overcome this shortcoming.

DOI

[39]
闫莉,陈夏.缺失数据下广义线性模型的经验似然推断[J].统计与信息论坛,2013,28(2):14-17.

[ Yan L, Chen X.Numerical experiment on the analytic hierarchy process[J]. Statistics & Information Forum, 2013,28(2):14-17. ]

[40]
苏凯,岳德鹏,YANG Di,等.基于改进力导向模型的生态节点布局优化[J].农业机械学报, 2017,48(11):215-221.在西北干旱半干旱生态脆弱区,构建生态网络可以连接破碎的生境斑块,提高景观之间的连通性。而生态节点的布局优化能够降低能量损耗,增加稳定性,对维持区域生态环境安全稳定具有重要意义。以生态脆弱区典型县域磴口县为研究区,在现有生态网络基础上,改进了H＆V算法的力导向（force-directed）模型,通过优化生态节点的布局对生态网络进行优化。研究结果表明,在磴口县选取的局部研究区内,与H＆V算法相比,改进force-directed模型优化的生态节点布局覆盖率达到90.79%,提升了4.08个百分点;平均聚类系数升高至0.071,是未改进H＆V算法的1.4倍;分布均匀度降低至2.629,比未改进H＆V算法降低了0.629。通过模型优化使得网络结构清晰、生态节点布局均匀,节点覆盖率更高,表明优化后生态网络结构更为稳定。

DOI

[Su K, Yue D P, Yang D, et al. Layout optimization of ecological nodes based on improved force-directed model[J]. Transactions of the Chinese Society for Agricultural Machinery, 2017,48(11):215-221. ]

[41]
Chuang L Y, Chang H W, Tu C J, et al.Improved binary PSO for feature selection using gene expression data[J]. Computational Biology & Chemistry, 2008,32(1):29-38.Gene expression profiles, which represent the state of a cell at a molecular level, have great potential as a medical diagnosis tool. Compared to the number of genes involved, available training data sets generally have a fairly small sample size in cancer type classification. These training data limitations constitute a challenge to certain classification methodologies. A reliable selection method for genes relevant for sample classification is needed in order to speed up the processing rate, decrease the predictive error rate, and to avoid incomprehensibility due to the large number of genes investigated. Improved binary particle swarm optimization (IBPSO) is used in this study to implement feature selection, and the -nearest neighbor (-NN) method serves as an evaluator of the IBPSO for gene expression data classification problems. Experimental results show that this method effectively simplifies feature selection and reduces the total number of features needed. The classification accuracy obtained by the proposed method has the highest classification accuracy in nine of the 11 gene expression data test problems, and is comparative to the classification accuracy of the two other test problems, as compared to the best results previously published.

DOI PMID

[42]
Niknam T, Amiri B.An efficient hybrid approach based on PSO, ACO and -means for cluster analysis[J]. Applied Soft Computing, 2010,10(1):183-197.Clustering is a popular data analysis and data mining technique. A popular technique for clustering is based on k-means such that the data is partitioned into K clusters. However, the k-means algorithm highly depends on the initial state and converges to local optimum solution. This paper presents a new hybrid evolutionary algorithm to solve nonlinear partitional clustering problem. The proposed hybrid evolutionary algorithm is the combination of FAPSO (fuzzy adaptive particle swarm optimization), ACO (ant colony optimization) and k-means algorithms, called FAPSO-ACO–K, which can find better cluster partition. The performance of the proposed algorithm is evaluated through several benchmark data sets. The simulation results show that the performance of the proposed algorithm is better than other algorithms such as PSO, ACO, simulated annealing (SA), combination of PSO and SA (PSO–SA), combination of ACO and SA (ACO–SA), combination of PSO and ACO (PSO–ACO), genetic algorithm (GA), Tabu search (TS), honey bee mating optimization (HBMO) and k-means for partitional clustering problem.

DOI

[43]
Sudheer C, Maheswaran R, Panigrahi B K, et al.A hybrid SVM-PSO model for forecasting monthly streamflow[J]. Neural Computing & Applications, 2014,24(6):1381-1389.

[44]
张好,徐涵秋,李乐,等.成都市热岛效应与城市空间发展关系分析[J].地球信息科学学报, 2014,16(1):70-78.利用Landsat卫星影像反演成都市中心城区1992、2001和2009年的地表温度，建筑用地和植被等信息，计算其城市热岛比例指数（URI），对成都市中心城区热岛效应与城市空间发展关系进行了分析。结果表明，在1992-2009年期间成都市主城区范围从91.24km<sup>2</sup>扩展到403.8km<sup>2</sup>。成都市建成区的大面积扩展导致了城市热岛空间分布发生迁移，从单中心聚集分布转变为多中心环状分布。回归分析说明，建筑用地和植被都是影响地表温度的重要因素，其中建筑用地与地表温度呈指数型正相关关系，而植被与地表温度呈负相关关系。总的看来，成都市中心城区在这17年间的热岛效应有了明显的缓解，城市热岛比例指数从0.72下降到0.33。城市植被覆盖率的增加和合理的规划对缓解城市热岛效应起到了积极的作用。

DOI

[Zhang H, Xu H Q, Li L, et al. Analysis of the relationship between urban heat island effect and urban expansion in Chengdu, China[J]. Journal of Geo-information Science, 2014,16(1):70-78. ]

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DOI

[ Zhang Z Q, Jia D X, Deng S H, et al.Quantitative research of urban spatial morphology: A case study of the main urban zone of Chongqing[J]. Journal of Geo-information Science, 2013,15(2):297-306. ]

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[ Jia Y, Deng S W, Yao X M, et al.Kriging interpolation algorithm based on constraint particle swarm optimization[J]. Journal of Chengdu University of Technology, 2015(1):104-109. ]

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