基于Kriging的地形高程插值

将地形高程作为区域化变量,根据普通Kriging法由散乱的高程点进行地形高程插值,并采用Matlab软件开发专门的程序,实现研究区高程插值计算与结果可视化分析。以广州市南沙区10 km2范围内的200个高程点数据为例,分别运用球面模型、指数模型和高斯理论变差函数模型进行10 m×10 m格网插值,借助Matlab可视化分析插值结果及其精度,表明采用指数模型效果最好。     

小区域癌症数据典型空间插值方法比较研究

癌症已成为危害全球居民健康的重大民生问题,选取合适的空间插值方法分析小区域癌症数据的空间特征可对区域性癌症防控工作的有效开展提供依据。本研究以湖南省苏仙区2012和2016年以村为单位的肺癌死亡率数据为研究对象,以平均误差和均方根误差为评价指标,对反距离加权（IDW）、普通克里金（OK）、趋势面分析（TSA）、多元线性回归（MLR）与协同克里金（CK）五种典型空间插值方法进行精度效果对比及参数优选,并结合不同插值方法的优缺点,确定癌症数据的最优插值方法。结果表明：插值精度方面,CK法的均方根误差最小、插值精度最高,OK、IDW（幂值=1）和MLR次之,TSA（阶数=5）最低;插值效果方面,五种插值方法的实测值和预测值均显著相关,除CK外,其它四种方法均对死亡率低估程度较大,CK和OK插值结果的空间分布效果更好。同时考虑空间因素和影响因子的CK方法是小区域苏仙区2012年、2016年肺癌死亡率最优插值方法,应用该方法可对区域性癌症防控工作的有效开展提供最优的技术支撑。本论文的研究思路也可为小区域癌症数据空间插值方法及参数优选提供参考。     

Spatial interpolation using MLP–RBFN hybrid networks

It is easy for a multi-layered perception (MLP) to fit a stratified spatial interpolation pattern whose form is close to open surface; while it is easy for a radial basis function network (RBFN) to fit a pocket (radial) spatial interpolation pattern whose form is close to closed surface. However, in the real world, the spatial interpolation pattern may consist of stratified and pocket patterns. Neither MLP nor RBFN can fit the pattern easily. To combine their advantages to fit the complex hybrid spatial interpolation patterns, in this article we propose a novel neural network, MLP–RBFN hybrid network (MRHN), whose hidden layer contains sigmoid and Gaussian units at the same time. Although there are two kinds of processing units in MRHN, in this study we used the principle of minimizing the error sum of squares to derive the supervised learning rules for all the network parameters. This research took rainfall distribution in Taiwan as a case study. The results show that (1) the prediction error of the testing dataset outside the training dataset demonstrated that MRHN was the most accurate among the three networks, RBFN was the next best, and MLP was the worst; (2) the MLP model seriously underestimated the values of high observed rainfall; (3) over-learning may be a serious shortcoming of using RBFN in spatial interpolation applications; (4) MRHN may have better generalization learning capacity than RBFN in spatial interpolation applications.     

土壤粒径空间预测方法研究综述

土壤粒径（砂粒、粉粒和黏粒）是各种陆表过程和生态系统服务评估等模型的关键参数。作为一种土壤成分数据,土壤粒径的空间预测方法有和为1（或100%）等特殊要求,其空间分布精度受预测方法影响较大。本文针对土壤粒径相较于其他土壤属性的特殊性,提出了土壤粒径空间预测方法框架,综述了土壤粒径数据变换、空间插值和精度验证等系列方法,总结了提升土壤粒径空间预测精度的各种途径,包括通过有效的数据变换改善数据分布、结合数据分布特点选择合适的预测方法、结合辅助变量提升制图精度和分布合理性、使用混合模型提升插值精度、使用多成分联合模拟模型提升预测的系统性等。最后,提出了今后土壤粒径空间预测方法研究的未来方向,包括从考虑数据变换原理和机制角度改善数据分布、发展多成分联合模拟模型和高精度曲面建模方法,以及引入土壤粒径函数曲线并与随机模拟结合等。     

Propagation of vertical and horizontal source data errors into a TIN with linear interpolation

Digital elevation models (DEMs) have been widely used for a range of applications and form the basis of many GIS-related tasks. An essential aspect of a DEM is its accuracy, which depends on a variety of factors, such as source data quality, interpolation methods, data sampling density and the surface topographical characteristics. In recent years, point measurements acquired directly from land surveying such as differential global positioning system and light detection and ranging have become increasingly popular. These topographical data points can be used as the source data for the creation of DEMs at a local or regional scale. The errors in point measurements can be estimated in some cases. The focus of this article is on how the errors in the source data propagate into DEMs. The interpolation method considered is a triangulated irregular network (TIN) with linear interpolation. Both horizontal and vertical errors in source data points are considered in this study. An analytical method is derived for the error propagation into any particular point of interest within a TIN model. The solution is validated using Monte Carlo simulations and survey data obtained from a terrestrial laser scanner.     

土壤厚度的空间插值方法比较——以青海三江源地区为例

利用青海三江源地区533个土壤剖面中的厚度数据, 在GIS技术的支持下, 采用确定性内插(反距离加权、全局多项式、局部多项式和径向基函数)和地统计内插(普通克里格、简单克里格、泛克里格和协同克里格)两类共八种插值方法对研究区土壤厚度的空间分布进行了预测, 并综合比较了各种插值方法的预测误差、统计特征值和插值结果分布图。结果表明:(1)在地统计内插方法中, 普通克里格方法(一阶)插值效果比普通克里格方法(二阶)要好;在普通克里格方法(一阶)的半方差函数模型中, 球状模型的插值效果优于指数模型和高斯模型;普通克里格方法在四种地统计内插方法中预测误差最小、预测结果准确性最好。(2)确定性内插方法中, 反距离加权(指数为1)法的误差较小, 并且对区域与局部趋势的反映效果最好。(3)从预测误差大小和对区域总体及局部趋势的综合反映效果来看, 有异向性的球状模型普通克里格(一阶)插值方法预测结果最能准确反映青海三江源地区土壤厚度的空间分布。     

Reconstructing pre-erosion topography using spatial interpolation techniques:A validation-based approach

Understanding the topographic context preceding the development of erosive landforms is of major relevance in geomorphic research, as topography is an important factor on both water and mass movement-related erosion, and knowledge of the original surface is a condition for quantifying the volume of eroded material. Although any reconstruction implies assuming that the resulting surface reflects the original topography, past works have been dominated by linear interpolation methods, incapable of generating curved surfaces in areas with no data or values outside the range of variation of inputs. In spite of these limitations, impossibility of validation has led to the assumption of surface representativity never being challenged. In this paper, a validation-based method is applied in order to define the optimal interpolation technique for reconstructing pre-erosion topography in a given study area. In spite of the absence of the original surface, different techniques can be nonetheless evaluated by quantifying their capacity to reproduce known topography in unincised locations within the same geomorphic contexts of existing erosive landforms. A linear method (Triangulated Irregular Network, TIN) and 23 parameterizations of three distinct Spline interpolation techniques were compared using 50 test areas in a context of research on large gully dynamics in the South of Portugal. Results show that almost all Spline methods produced smaller errors than the TIN, and that the latter produced a mean absolute error 61.4% higher than the best Spline method, clearly establishing both the better adjustment of Splines to the geomorphic context considered and the limitations of linear approaches. The proposed method can easily be applied to different interpolation techniques and topographic contexts, enabling better calculations of eroded volumes and denudation rates as well as the investigation of controls by antecedent topographic form over erosive processes.     

中国中部六省预期寿命时序加密估算研究

年龄组死亡率是利用年龄分组人口数据计算预期寿命的关键参数,而非采样年份的统计年鉴中年龄分组死亡率缺失导致无法计算预期寿命。针对该问题,本文将人口普查数据与统计年鉴数据融合,首次提出一种基于拉格朗日插值的中国省级预期寿命时间序列加强密集度（时序加密）的算法,以解决非采样（即未进行人口普查或1%人口抽样调查）年份省级预期寿命的估算问题。以中国中部六省为例,在所选取年份省级预期寿命估算实验中,绝对精度表明年龄分组人口比例线性插值计算的精度明显高于人口比例抛物线插值和直接插值算法的精度,故为推荐算法。本研究为高时间分辨率下省级预期寿命值的获取提供了一个新的可行思路,为分省较精确地进行预期寿命趋势分析奠定基础。     

基于分布式感知深度神经网络的高分辨率PM2.5值估算

细颗粒物(PM2.5)个体暴露水平是健康效应研究中的关键问题,然而历史数据缺失和地面监测点覆盖范围小阻碍了相关研究.基于美国国家航空航天局遥感数据提供的气溶胶光学厚度(AOD),融合地面监测、气象等多源数据进行建模来估算近地面PM2.5浓度,所得结果的空间覆盖范围广、时间连续性强、方法成本低.本文基于2018年京津冀鲁...     

薄盘光滑样条插值中三种协变量方法的降水量插值精度比较

在薄盘光滑样条插值中,高相关协变量的选取决定了插值结果的精确性。以2001-2009年全国728个气象站点日降水为数据源,提取年降水量数据,在分析多年平均降水量与两协变量高程（DEM）和距海岸线距离（DCL）的空间相关性基础上,利用ANUSPLIN软件,比较不同协变量下降水量插值结果精度在全国尺度以及区域尺度上的差异。以DEM、DCL及DEM-DCL分别为协变量对降水量数据进行空间插值发现：①在全国尺度上,DEM法的平均绝对误差（MAE）为47.79,略低于DEM-DCL法（48.90）,但显著低于DCL法（55.54）;且DEM法的平均相对误差和均方根误差也明显低于其它两种方法。②在区域尺度上,除西藏地区外的其他7个区域,3种方法的插值误差与全国尺度上相一致。西藏地区降水插值结果以DCL法的精度最高,而DEM法则较差。研究建议除在西藏地区的降水量插值研究中采用DCL法,在全国其他大部分区域采用DEM法。     

Surface modeling of DEMs based on a sequential adjustment method

A sequential adjustment (SA) method is employed to decrease the computational cost of high-accuracy surface modeling (HASM), and the SA of HASM (HASM-SA) is being developed. A mathematical surface was used to comparatively analyze the computing speed of SA and the classical iterative solvers provided by MATLAB 7.7 for solving the system of linear equations of HASM. Results indicate that SA is much faster than the classical iterative solvers. The computing time of HASM-SA is determined by not only the total number of grid cells but also the number of sampling points in the computational domain. A real-world example of surface modeling of digital elevation models (DEMs) with various resolutions shows that HASM-SA is averagely more accurate and much faster than the commonly used interpolation methods, such as inverse distance weighting (IDW), kriging, and three versions of spline, namely regularized spline (RSpline), thin-plate spline (TPS), and ANUDEM in terms of root mean square error (RMSE), mean absolute error (MAE), and mean error (ME). In particular, the ME of HASM-SA at different spatial resolutions is averagely smaller than those of IDW, kriging, RSpline, TPS, and ANUDEM by 85%, 83%, 83%, 53%, and 19%, respectively. The high speed and high accuracy of HASM-SA can be due to the absence of matrix inversion computation, combined with the perfect fundamental theorem of HASM.     

Evaluating representation and scale error in the maximal covering location problem using GIS and intelligent areal interpolation

A common problem in location-allocation modeling is the error associated with the representation and scale of demand. Numerous researchers have investigated aggregation errors associated with using different scaled data, and more recently, error associated with the geographic representation of model objects has also been studied. For covering problems, the validity of using polygon centroid representations of demand has been questioned by researchers, but the alternative has been to assume that demand is uniformly distributed within areal units. The spatial heterogeneity of demand within areal units thus has been modeled using one of two extremes – demand is completely concentrated at one location or demand is uniformly distributed. This article proposes using intelligent areal interpolation and geographic information systems to model the spatial heterogeneity of demand within spatial units when solving the maximal covering location problem. The results are compared against representations that assume demand is either concentrated at centroids or uniformly distributed. Using measures of scale and representation error, preliminary results from the test study indicate that for smaller scale data, representation has a substantial impact on model error whereas at larger scales, model error is not that different for the alternative representations of the distribution of demand within areal units.     

Climate spatial variability and data resolution in a semi-arid watershed, south-eastern Arizona

In the 10,000 km² San Pedro River watershed area in south-eastern Arizona, high-resolution spatial patterns of long-term precipitation and temperature were better reproduced by kriging climate data with elevation as external drift (KED) than by multiple linear regression on station location and elevation as judged by the spatial distribution of interpolation error. Mean errors were similar overall, and interpolation accuracy for both methods increased with increasing correlation between climate variables and elevation. Uncertainty in station locations had negligible effect on mean estimation error, although error for individual stations varied as much as 27%. Our future ability to examine spatial aspects of climate change at high spatial resolution will be severely limited by continuing closures of climate stations in this part of the United States.     

不同时间尺度、季节的气温数据空间平稳特征及其对插值结果的影响

不同时间尺度、季节的气温数据表现出不同的空间平稳特征。为探讨分析空间平稳性对气温插值的影响规律,采用趋势线法对气温数据进行空间平稳性探索,并对比分析不同空间平稳性条件下,普通线性回归、普通克里格、回归克里格的气温插值精度及插值结果的空间分布特点。结果显示：冬季日均、月均气温与年均气温呈现空间非平稳,插值精度随时间序列的增长而提高,随着气温数据逐渐趋于稳定,精度提高的幅度逐渐下降;夏季日均、月均气温呈现空间平稳,随时间序列的增长,插值精度的提高并不显著;夏季日均气温各插值方法的插值精度普遍高于冬季日均气温。与普通克里格相比,回归克里格能有效提高空间非平稳数据的插值精度。时间序列的增长削弱了不同插值算法之间的插值精度差异和插值结果空间分布差异。     

Book reviews

A digital elevation model (DEM), which is used to represent a terrain surface, is normally constructed by applying an interpolation method on given sample elevation points. Interpolation methods can be classified into two classes: linear methods, which have a low time cost and are suitable for terrains where there is little change in elevation, and nonlinear methods, which normally consume comparatively more time and are more suitable for terrains where there are frequent changes in elevation. A hybrid interpolation method, which involves both a linear method and a nonlinear method of interpolation, is proposed in this paper. The proposed method aims to integrate the advantages of both linear and nonlinear interpolation methods for the refinement of regular grid DEM. Here, the bilinear is identified as the linear method, and the bi‐cubic is taken to be the nonlinear interpolation method. The hybrid method is an integration of a linear model and nonlinear interpolation model with a parameter that defines the weights for each of the models. The parameter is dependent on the complexity of the terrain, for which a DEM is to be interpolated. The experimental results in this study demonstrate that the hybrid method is effective for interpolating DEMs for various types of terrain.     

薄板光顺样条插值与中国气候空间模拟

利用720个气象台网的长期平均气象数据拟合具有三维地理空间的气候曲面,并与1km空间分辨率的数字高程模型相结合,对气候变量的规则栅格进行插值估计。对各月平均最低温度、平均最高温度和降水量的插值结果构成了基础数字气候空间,以满足地理信息系统的数据分析需求。插值过程提供的误差统计表明插值的温度误差普遍小于0.6度,降水误差范围在8%~15%,明显优于其它插值方法。样条法利用线性模型反映地形对气候的影响,并提供了简便的误差诊断程序,具有良好的实用性。     

喀斯特地区春季土壤水分空间插值方法对比

以杨眉河小流域为研究区,通过土壤水分采样,选取辅助变量,采用普通克里金、协同克里金、回归克里金3种地统计学方法对土壤水分数据进行空间插值。结果表明：1）回归克里金对研究区土壤水分估算误差最小,其次为协克里金,普通克里金的误差最大;2）普通克里金生成的土壤水分表面最为平滑,而回归克里金最大程度反映了研究区实际的土壤水分空间变化;3）对于协同克里金,以湿度指数（WI）样点数据作为辅助变量的估算误差小于将WI栅格数据作为辅助变量的估算误差。总之,在可获得有效辅助变量的条件下,回归克里金对研究区土壤水分估算的效果优于协同克里金与普通克里金。     

Influence of survey strategy and interpolation model on DEM quality

Accurate characterisation of morphology is critical to many studies in the field of geomorphology, particularly those dealing with changes over time. Digital elevation models (DEMs) are commonly used to represent morphology in three dimensions. The quality of the DEM is largely a function of the accuracy of individual survey points, field survey strategy, and the method of interpolation. Recommendations concerning field survey strategy and appropriate methods of interpolation are currently lacking. Furthermore, the majority of studies to date consider error to be uniform across a surface. This study quantifies survey strategy and interpolation error for a gravel bar on the River Nent, Blagill, Cumbria, UK. Five sampling strategies were compared: (i) cross section; (ii) bar outline only; (iii) bar and chute outline; (iv) bar and chute outline with spot heights; and (v) aerial LiDAR equivalent, derived from degraded terrestrial laser scan (TLS) data. Digital Elevation Models were then produced using five different common interpolation algorithms. Each resultant DEM was differentiated from a terrestrial laser scan of the gravel bar surface in order to define the spatial distribution of vertical and volumetric error. Overall triangulation with linear interpolation (TIN) or point kriging appeared to provide the best interpolators for the bar surface. Lowest error on average was found for the simulated aerial LiDAR survey strategy, regardless of interpolation technique. However, comparably low errors were also found for the bar-chute-spot sampling strategy when TINs or point kriging was used as the interpolator. The magnitude of the errors between survey strategy exceeded those found between interpolation technique for a specific survey strategy. Strong relationships between local surface topographic variation (as defined by the standard deviation of vertical elevations in a 0.2-m diameter moving window), and DEM errors were also found, with much greater errors found at slope breaks such as bank edges. A series of curves are presented that demonstrate these relationships for each interpolation and survey strategy. The simulated aerial LiDAR data set displayed the lowest errors across the flatter surfaces; however, sharp slope breaks are better modelled by the morphologically based survey strategy. The curves presented have general application to spatially distributed data of river beds and may be applied to standard deviation grids to predict spatial error within a surface, depending upon sampling strategy and interpolation algorithm.     

Estimation of precipitation fields from 1-minute rain gauge time series – comparison of spatial and spatio-temporal interpolation methods

Accurate estimations of spatio-temporal fields at unsampled locations are important in a number of applications. Often, spatio-temporal fields are advected, which means the change in field values over time at a particular point in space stems to a large extent from motion of a more or less constant spatial field. For such dynamic fields, interpolation methods including information on the motion behaviour of the field are promising extensions of solely spatial (snapshot) and symmetric spatio-temporal methods. In this paper, the performance of different deterministic and geostatistical interpolation methods is compared for precipitation estimation from 1-minute time series of spatially distributed rain gauges. The focus is on spatio-temporal methods that include information on the motion behaviour of the rainfield, estimated from weather radar using optical flow. The different interpolation methods are introduced and evaluated using rain gauge measurements of a 15-day period and cross-validation. The results show that including information on the motion behaviour significantly improves interpolation quality in terms of RMSE.     

Why GPS makes distances bigger than they are

Global navigation satellite systems such as the Global Positioning System (GPS) is one of the most important sensors for movement analysis. GPS is widely used to record the trajectories of vehicles, animals and human beings. However, all GPS movement data are affected by both measurement and interpolation errors. In this article we show that measurement error causes a systematic bias in distances recorded with a GPS; the distance between two points recorded with a GPS is – on average – bigger than the true distance between these points. This systematic ‘overestimation of distance’ becomes relevant if the influence of interpolation error can be neglected, which in practice is the case for movement sampled at high frequencies. We provide a mathematical explanation of this phenomenon and illustrate that it functionally depends on the autocorrelation of GPS measurement error (C). We argue that C can be interpreted as a quality measure for movement data recorded with a GPS. If there is a strong autocorrelation between any two consecutive position estimates, they have very similar error. This error cancels out when average speed, distance or direction is calculated along the trajectory. Based on our theoretical findings we introduce a novel approach to determine C in real-world GPS movement data sampled at high frequencies. We apply our approach to pedestrian trajectories and car trajectories. We found that the measurement error in the data was strongly spatially and temporally autocorrelated and give a quality estimate of the data. Most importantly, our findings are not limited to GPS alone. The systematic bias and its implications are bound to occur in any movement data collected with absolute positioning if interpolation error can be neglected.