Area-to-point Kriging with inequality-type data

In practical applications of area-to-point spatial interpolation, inequality constraints, such as non-negativity or more general constraints on the maximum and/or minimum attribute value, should be taken into account. The geostatistical framework proposed in this paper deals with the spatial interpolation problem of downscaling areal data under such constraints, while: (1) explicitly accounting for support differences between sample data and unknown values, (2) guaranteeing coherent (mass-preserving) predictions, and (3) providing a measure of reliability (uncertainty) for the resulting predictions. The formal equivalence between Kriging and spline interpolation allows solving constrained area-to-point interpolation problems via quadratic programming (QP) algorithms, after accounting for the support differences between various constraints involved in the problem formulation. In addition, if inequality constraints are enforced on the entire set of points discretizing the study domain, the numerical algorithms for QP problems are applied only to selected locations where the corresponding predictions violate such constraints. The application of the proposed method of area-to-point spatial interpolation with inequality constraints in one and two dimension is demonstrated using realistically simulated data.     

Data-enriched interpolation for temporally consistent population compositions

This research evaluates the performance of areal interpolation coupled with dasymetric refinement to estimate different demographic attributes, namely population sub-groups based on race, age structure and urban residence, within consistent census tract boundaries from 1990 to 2010 in Massachusetts. The creation of such consistent estimates facilitates the study of the nuanced micro-scale evolution of different aspects of population, which is impossible using temporally incompatible small-area census geographies from different points in time. Various unexplored ancillary variables, including the Global Human Settlement Layer (GHSL), the National Land-Cover Database (NLCD), parcels, building footprints and the proprietary ZTRAX® dataset are utilized for dasymetric refinement prior to areal interpolation to examine their effectiveness in improving the accuracy of multi-temporal population estimates. Different areal interpolation methods including Areal Weighting (AW), Target Density Weighting (TDW), Expectation Maximization (EM) and its data-extended approach are coupled with different dasymetric refinement scenarios based on these ancillary variables. The resulting consistent small area estimates of white and black subpopulations, people of age 18–65 and urban population show that dasymetrically refined areal interpolation is particularly effective when the analysis spans a longer time period (1990–2010 instead of 2000–2010) and the enumerated population is sufficiently large (e.g., counts of white vs. black). The results also demonstrate that current census-defined urban areas overestimate the spatial distribution of urban population and dasymetrically refined areal interpolation improves estimates of urban population. Refined TDW using building footprints or the ZTRAX® dataset outperforms all other methods. The implementation of areal interpolation enriched by dasymetric refinement represents a promising strategy to create more reliable multi-temporal and consistent estimates of different population subgroups and thus demographic compositions. This methodological foundation has the potential to advance micro-scale modeling of various subpopulations, particularly urban population to inform studies of urbanization and population change over time as well as future population projections.     

Singly- and Doubly-Constrained Methods of Areal Interpolation for Vector-based GIS

Traditionally, areal interpolation has referred to techniques for transferring attribute values from one partitioning of space to a different partition of space but this is only one of several situations that create the need for estimating unknown data values for areal units. This paper presents a categorization of four areal interpolation problems that includes the "missing" data problem, the traditional "alternative geography" problem, the overlay of a choropelth and an area-class data layer, and the overlay of two choropleth data layers and demonstrates the relationship between the last three problems and general spatial interaction modelling. The "alternative geography" and overlay of choropleth and area-class data layers mirrors a singly constrained spatial interaction model while the overlay of two choropleth layers is analogous to a doubly constrained interaction model. Iterative proportional fitting techniques with and without ancillary data are developed to solve these three classes of problems.     

Accuracy of areal interpolation: A comparison of alternative methods

This paper discusses the accuracy of spatial data estimated by areal interpolation, a process of transferring data from one zonal system to another. A stochastic model is proposed which represents areal interpolations in diverse geographic situations. The model is used to examine the relationship between estimation accuracy and the spatial distribution of estimation error from a theoretical viewpoint. The analysis shows that the uniformity in error distribution improves the accuracy of areal interpolation. Four areal interpolation methods are then assessed through numerical examinations. From this it is found that the accuracy of simple interpolation methods heavily depends on the appropriateness of their hypothetical distributions, whereas the accuracy of intelligent methods depends on the fitness of the range of supplementary data for that of true distribution. Received: 19 February 1999/Accepted 17 September 1999     

Interpolating Spatial Interaction Data1

Spatial interpolation has been widely used to improve the spatial granularity of data, or to mediate between inconsistent zoning schemes of spatial data. Traditional areal interpolation methods translate values of source zones to those of target zones. These methods have difficulty in dealing with flow data, as each instance is associated with a pair of zones. This study develops a new concept, flow line interpolation, to fill the abovementioned gap. We also develop a first flow line interpolation method to estimate commuting flow data between spatial units in a target zoning scheme based on such data in a source zoning scheme. Three models (i.e., areal‐weighted, intelligent, and gravity‐type flow line interpolation) are presented. To test the estimation accuracy and the application potential of these models, a case study of Fulton County in Georgia is conducted. The results reveal that both the areal‐weighted and intelligent models are very promising flow line interpolation methods.     

Inferring Spatial Scale Change in an Isopleth Map

This study reunites areal interpolation with the isopleth mapping process to construct an inferred larger scale isopleth map. Intelligent areal interpolation is used to construct two types of population density surfaces that are used as inputs for pycnophylactic interpolation of an isopleth surface. One is a target zone population density surface (TZPDS) and the other is a control zone population density surface (CZPDS). Results suggest that an inferred isopleth map with remote sensing control data is a better surface depiction than an isopleth map without any control data, and the quality of such an isopleth map is further improved by enhancing the remote sensing data with residential parcel information. A CZPDS-derived intelligent isopleth map also has more peaks and variations in population distribution patterns than does a TZPDS-derived one due to the larger scale of the control data.     

Multivariate Analysis of Incomplete Mapped Data

Classical multivariate analyses are based on matrix algebra and enable the analysis of a table containing measurements of a set of variables for a set of sites. Incomplete mapped data consist of measurements of a set of variables recorded for the same geographical region but for different zonal systems and with only a partial sampling of this zone. This kind of data cannot be analysed with usual multivariate methods because there is no common system of sites for all variables. We propose a new approach using GIS technology and NIPALS, an iterative multivariate method, to analyse the spatial patterns of this kind of data. Moreover, an extension of our method is that it can be used for areal interpolation purposes. We illustrate the method in analysing data concerning the distribution of roe deer weights over several years in a reserve.     

An Implementation of the Robinson Map Projection Based on Cubic Splines

The Robinson world map projection has been in existence since 1963. Mapping equations for the projection are based on table interpolation. Cubic splines with the choice of appropriate boundary conditions are shown to possess favorable characteristics as interpolative functions. The evaluation of cubic splines for given latitudes provides a basis for efficient forward mapping equations. Inverse mapping equations are based on an efficient inversion of the cubic spline functions utilizing Newton's method. Derivatives of the cubic spline functions lead to formulas for scale factors along projected meridians and parallels and for areal scale factors. Details for a computer implementation of the mapping equations and the computation of scale factors are given.     

滑动式广义延拓插值法在GLONASS钟差插值中的应用

在对卫星钟差数据进行插值处理时,所采用插值算法的精度,直接影响到卫星钟差插值结果的精度,继而影响了卫星导航定位的精度. 因此在对卫星钟差数据进行插值时,应选择适宜的插值方法. 将Lagrange插值法和切比雪夫多项式拟合法进行滑动,利用这两种传统的插值方法和滑动式广义延拓插值法,分别对历元间隔为5 min的GLONASS卫星钟差数据插值到历元间隔为30 s的钟差数据,再与历元间隔为30 s的精密钟差数据进行对比,分析三种插值方法在GLONASS卫星钟差数据中的应用效果. 结果表明:利用这三种插值方法对GLONASS钟差数据进行插值时,插值精度均能满足要求,且滑动式广义延拓插值法的插值精度最高.      

Instructions for Contributors to Cartography and Geographic Information Science

A number of areal interpolation methods have been developed to estimate population for overlapping, discontinuous, or fragmented areas. Previous studies examined the relative accuracy of various methods; this research advances those endeavors by comparing the effectiveness of seven different methods using a national random sample of census block groups and blocks. As the results show, the areal interpolation methods produce good population estimates for nested census blocks except in areas of heterogeneous land use or unusual contexts. In addition, estimation conducted in areas with small populations or low population density was vulnerable to high percentage error. Amongst the different methods, road network allocation and statistical regression (with area and roads as predictors) produced the best population estimates for the sample blocks.     

Data matching of building polygons at multiple map scales improved by contextual information and relaxation

The aim of matching spatial data at different map scales is to find corresponding objects at different levels of detail (LODs) that represent the same real-world phenomena. This is a prerequisite for integrating, evaluating and updating spatial data collected and maintained at various scales. However, matching spatial data is not straightforward due to the ambiguities caused by problems like many-to-many correspondence, non-systematic displacement and different LODs between data sets. This paper proposes an approach to matching areal objects (e.g. buildings) based on relaxation labeling techniques widely applied in pattern recognition and computer vision. The underlying idea is to utilize contextual information (quantified by compatibility coefficient) in an iterative process, where the ambiguities are reduced until a consistent matching is achieved. This paper describes (1) a domain-specific extension to previous relaxation schemes and (2) a new compatibility coefficient that exploits relative relationships between areal object pairs in spatial data. Our approach were validated through extensive experiments using building data sets at 1:10k and 1:50k as an example. Our contextual approach showed superior performance against a non-contextual approach in general and especially in ambiguous situations. The proposed approach can also be applied to matching other areal features and/or for a different scale range.     

A Comparative Analysis of Areal Interpolation Methods

Over the years many approaches to areal interpolation have been developed and utilized. They range from the simple 2-D areal weighing method which uses only the spatial Z variable being processed, to more sophisticated approaches which use auxiliary variable(s) to provide more specificity to the results. In the research reported here, four promising approaches are implemented and comparatively tested. These approaches have widely varying conceptual foundations, and different auxiliary variables, if used. The areal weighing reflects many earlier methods which assumes uniform distributions of the spatial Z variable, and does not use any auxiliary variable. Tobler's pycnophylactic method uses a volumetric preservation approach, which assumes spatial Z variable is heterogeneously distributed, but does not use any auxiliary variable. The traditional dasymetric method of Wright is used with remote sensing spectral data of land use. Xie's road network hierarchically weighted interpolation uses the road network as the auxiliary variable, and assumes that population density is related to the class of the road, and to the density of the road network. The research design implemented here uses Census population distributions at different levels in the hierarchy as the source and target variables analyzed. The source zone population is taken at the Census Tract level, and the target zones are specified at the Census Block Group level in the hierarchy. The first two tests use only the Census population Z data, and no auxiliary variables, whereas the next uses remotely sensed land use data as the auxiliary data variable, and the fourth test utilizes the road network hierarchy as the auxiliary variable. The paper reports on the findings from these tests, and then interprets them in a spatial setting in terms of accuracy and effectiveness. This research points to the network method as the most accurate of the areal interpolation methods tested in this research.     

若干平滑和插值方法对图像空间分辨力估算的影响

图像空间分辨力是图像质量评估中的一个重要指标,而对图像空间分辨力的估算通常采用调制传递函数（ＭＩＦ）解算。由于图像噪声的存在和采样数据的离散性,平滑和插值是必不可少的两个环节,因此,平滑和插值算法的好坏直接影响最后的评估结果。采用了两种最常用的平滑算法和两种插值算法,并按８种实验方案对两幅典型的图像进行了实验对比研究,获得了这两种算法对图像空间分辨力评估的影响机理,从而遴选出了较好的平滑和插值算法     

基于广义延拓逼近法的QZSS卫星钟差内插精度分析

卫星钟差数据插值是高精度定位数据处理中的重要环节,其插值结果直接影响定位精度,但常用的插值或拟合方法具有不同缺点.本文尝试将广义延拓逼近法应用于准天顶卫星系统(QZSS)卫星钟差数据的处理中,介绍了Lagrange插值法、切比雪夫拟合法和广义延拓逼近法的原理,以及滑动式与非滑动式的区别;然后使用QZSS钟差数据探讨三种方法的参数(组)取值与插值结果精度的关系;最后比较三种方法在各自最优参数(组)取值情况下对QZSS卫星钟差的插值精度.结果表明:选取合理的参数组合,广义延拓逼近法完全适用于QZSS卫星钟差的插值,且插值精度明显高于其他两种方法.     

人口统计数据的空间分布化研究

分析了传统的人口空间分布密度衰减函数－指数型和Gauss型，指出了其应用的局限性，对于有两个中心以上的城市，提出了将人口统计数据空间分布化的思路和方法。     

利用内切圆内插等高线的算法

等高线内插在地图自动综合、地图数字化、三维地形重建等过程中都具有重要意义。许多等高线内插算法在等高线急剧变化以及闭合等高线处存在问题。在分析已有等高线内插算法优缺点的基础上,提出了一种等高线内插算法。该算法以等高线上的节点为圆心,建立与相邻等高线之间的内切圆来探测相邻等高线之间的空间关系,并获取等高线间的辅助线,进而内插出等高线,一方面弥补了已有等高线内插方法中的问题,另一方面有效提高了等高线内插的速度和质量。通过与其他内插算法之间的实验对比分析,验证了本方法的科学性和先进性。     

插值条件下格网DEM坡度计算模型的噪声误差分析

针对目前DEM（Digital Elevation Model, DEM）数字地形分析的精度评价多数不考虑DEM误差的空间自相关性或仅仅采用经验的自相关性模型问题,本文从DEM插值入手,从理论上推导了插值条件下格网DEM邻域窗口内坡度噪声误差的空间自相关性模型以及坡度精度模型,并选取典型的插值方法和坡度差分算法,从实验角度分析了在顾及和不顾及空间自相关性两种情况下的格网DEM坡度计算模型的噪声精度,实验结果表明：坡度精度受DEM噪声误差的空间自相关性影响较大,并与DEM插值方法和坡度计算模型中的差分算法有关。     

无人机机载相机曝光时刻摄影中心三维坐标高精度插值算法研究

随着机载定位定向系统（POS）的不断完善,无人机技术已成为高精度摄影测量的重要途径．由于POS系统中全球卫星导航系统（GNSS）接收机的采样频率有限,需要对机载相机曝光点时刻摄影中心三维坐标进行插值计算．本文对常用的四种插值算法进行了研究,并通过实测数据对比不同插值算法的优劣,结果表明,当GNSS原始数据采样频率高于1 Hz时,三次样条插值算法误差均方根误差优于5 cm,更适用于无人机曝光时刻摄影中心三维坐标的高精度插值．      

一种面向CPU/GPU异构环境的协同并行空间插值算法

CPU/GPU异构混合系统是一种新型高性能计算平台,但现有并行空间插值算法仅依赖CPU或GPU进行加速,迫切需要研究协同并行空间插值算法以充分利用异构计算资源,进一步提升插值效率。以薄板样条函数插值为例,提出一种CPU/GPU协同并行插值算法以加速海量激光雷达（light detector & ranger,LiDAR）点云生成数字高程模型（DEM）。通过插值任务的分解与抽象封装以屏蔽底层硬件执行模式的差异性,同时在多级协同并行框架基础上设计了Greedy-SET动态调度策略,策略顾及底层硬件能力的差异性,以实现异构并行资源的充分利用和良好负载均衡。实验表明,协同并行插值算法在高性能工作站上取得19.6倍的加速比,相比单一CPU或GPU并行算法,其效率提升分别达到54%和44%,实现了高效的协同并行处理。     

基于小波的图像插值方法

本文在分析已有插值基函数和小波函数、小波变换的基础上提出一种应用小波作为插值基函数进行图像插值的方法,给出了该方法在一般图像插值方面在多幅图像的信息揉合和累积方面的应用,并分析了一般图像插值方法在图像之间存在辐照度差异时的可行性和在同名点之间存在相对空间信息差异时的操作方法。最后,采用４幅ＰＯＬＤＥＲ多角度遥感图像对本文所述方法进行了验证。实验表明,本文所述方法是可以较好地进行图像插值的。