Computation of the third‐order partial derivatives from a digital elevation model

Loci of extreme curvature of the topographic surface may be defined by the derivation function (T) depending on the first‐, second‐, and third‐order partial derivatives of elevation. The loci may partially describe ridge and thalweg lines. The first‐ and second‐order partial derivatives are commonly calculated from a digital elevation model (DEM) by fitting the second‐order polynomial to a 3×3 window. This approach cannot be used to compute the third‐order partial derivatives and T. We deduced formulae to estimate the first‐, second‐, and third‐order partial derivatives from a DEM fitting the third‐order polynomial to a 5×5 window. The polynomial is approximated to elevation values of the window. This leads to a local denoising that may enhance calculations. Under the same grid size of a DEM and root mean square error (RMSE) of elevation, calculation of the second‐order partial derivatives by the method developed results in significantly lower RMSE of the derivatives than that using the second‐order polynomial and the 3×3 window. An RMSE expression for the derivation function is deduced. The method proposed can be applied to derive any local topographic variable, such as slope gradient, aspect, curvatures, and T. Treatment of a DEM by the method developed demonstrated that T mapping may not substitute regional logistic algorithms to detect ridge/thalweg networks. However, the third‐order partial derivatives of elevation can be used in digital terrain analysis, particularly, in landform classifications.     

Evaluation on the accuracy of digital elevation models

1 IntroductionDigital elevation model (DEM) is digital representation of relief. It is one of the most important components in the database of GIS. At present, DEM is playing a key role in the field of survey and mapping, remote sensing and almost all the terrain related geographical analyses. DEM can be grouped into regular grids (raster) and triangulated irregular networks (TIN). Both have their advantages and disadvantages in application. It is generally believed that grid DEM will …     

Evaluation on the accuracy of digital elevation models

There is a growing interest in investigating the accuracy of digital elevation model (DEM). However people usually have an unbalanced view on DEM errors. They emphasize DEM sampling errors, but ignore the impact of DEM resolution and terrain roughness on the accuracy of terrain representation. This research puts forward the concept of DEM terrain representation error (Et) and then investigates the generation, factors, measurement and simulation of DEM terrain representation errors. A multi-resolution and multi-relief comparative approach is used as the major methodology in this research. The experiment reveals a quantitative relationship between the error and the variation of resolution and terrain roughness at a global level. Root mean square error (RMS Et) is regressed against surface profile curvature (V) and DEM resolution (R) at 10 resolution levels. It is found that the RMS Et may be expressed as RMS Et = (0.0061 × V+ 0.0052) × R - 0.022 × V + 0.2415. This result may be very useful in forecasting DEM accuracy, as well as in determining the DEM resolution related to the accuracy requirement of particular application.     

Geomorphometry on the surface of a triaxial ellipsoid: towards the solution of the problem

Existing algorithms of geomorphometry can be applied to digital elevation models (DEMs) given with plane square grids or spheroidal equal angular grids on the surface of an ellipsoid of revolution or a sphere. Computations on spheroidal equal angular grids are trivial for modelling of the Earth, Mars, the Moon, Venus, and Mercury. This is because: (a) forms of these celestial bodies can be described by an ellipsoid of revolution or a sphere and (b) for these surfaces, there are well-developed theory and algorithms to solve the inverse geodetic problem as well as to determine spheroidal trapezoidal areas. It is advisable to apply a triaxial ellipsoid for describing the forms of small moons and asteroids. However, there are no geomorphometric algorithms intended for such a surface. In this article, first, we formulate the problem of geomorphometric modelling on a triaxial ellipsoid surface. Then, we recall definitions and formulae for coordinate systems of a triaxial ellipsoid and their transformation. Next, we present analytical and computational solutions, which provide the basis for geomorphometric modelling on the surface of a triaxial ellipsoid. The Jacobi solution for the inverse geodetic problem has a fundamental mathematical character. The Bespalov solutions for determination of the length of meridian/parallel arcs and the spheroidal trapezoidal areas are computationally efficient. Finally, we describe easy-to-code algorithms for derivation of local and non-local morphometric variables from DEMs based on a spheroidal equal angular grid of a triaxial ellipsoid.     

数字地形分析的理论、方法与应用

数字高程模型自20世纪50年代首次被提出以来,就以其简洁的数据组织方式、对地形的直观表达、简单高效的地形因子解译方法而显示了其在地学领域应用的巨大潜力。本文在全面检索和分析前人相关研究成果的基础上,首先将数字地形分析 (Digital Terrain Analysis,简称DTA) 的各种分析方法总结为四类,即坡面地形因子分析、特征地形要素分析、地形统计分析以及基于DEM的地学模型分析,并简要总结了每种分析方法的主要内容;其次,探讨了数字地形分析的精度及尺度问题,指出尺度效应、最适宜尺度选择以及尺度转换是DEM地形分析中的基本尺度问题;然后,论文介绍了DTA在地貌、水文、土壤、地质灾害、农业等领域的应用现状;最后,提出了数字地形分析发展方向,指出数字地形分析的概念框架亟待完善、模型构建精度有待提高、地形分析方法需要扩展,尤其是实现DEM模型嵌入的一体化分析方法,以求得到与现实世界地理环境更为接近的模拟。     

Multi-scale landform characterization

One fundamental objective in geomorphometry is to extract signatures of geomorphologic processes on different spatial scales from digital terrain models (DTMs) and to describe the complexity of landforms as the synthesis of those individual imprints. We present an approach for characterizing land surfaces on multiple, spatially varying local scales. We approximate terrain surfaces locally to calculate surface derivatives at different window sizes. Local scale behaviour diagrams are used to define dominant scale ranges and multiple curvatures for each surface point. Multi-scale landform analysis leads to improved models of surface derivatives and new landform classifications, applicable in geomorphology, soil science and hydrology.     

基于不同算法等高线曲率的提取与分析

等高线曲率是一个重要的地形属性,反应了地形表面在水平方向的凹凸性,表达了地表物质运动的发散和汇合模式。基于安塞县县南沟小流域的矢量等高线数据和DEM,分别利用圆拟合算法（相邻三点法、间隔三点法和最小二乘法）和曲面拟合模型（E模型、S模型和Z模型）提取等高线曲率,通过对实地地形的对比分析,结果表明：（1） 在矢量等高线数据的计算结果中,三点拟合法计算结果相比最小二乘法结果差异大,对等高线曲率空间格局分布描述更准确;（2） 最小二乘法计算的结果频数分布集中程度最高,两种三点拟合法计算结果频数曲线差别微小;（3） 在栅格数字高程模型的计算结果中,基于S模型计算结果在空间格局上较E模型和Z模型的结果差别大,基于E模型的计算结果对等高线曲率描述效果更好。结果能准确说明采用不同算法计算等高线曲率的差别,对在实际数字地形分析中有重要的意义,可为等高线曲率计算提供重要参考。     

Simulation on slope uncertainty derived from DEMs at different resolution levels： a case study in the Loess Plateau

Slope is one of the crucial terrain variables in spatial analysis and land use planning, especially in the Loess Plateau area of China which is suffering from serious soil erosion. DEM based slope extracting method has been widely accepted and applied in practice. However slope accuracy derived from this method usually does not match with its popularity. A quantitative simulation to slope data uncertainty is important not only theoretically but also necessarily to applications. This paper focuses on how resolution and terrain complexity impact on the accuracy of mean slope extracted from DEMs of different resolutions in the Loess Plateau of China. Six typical geomorphologic areas are selected as test areas, representing different terrain types from smooth to rough. Their DEMs are produced from digitizing contours of 1:10,000 scale topographic maps. Field survey results show that 5 m should be the most suitable grid size for representing slope in the Loess Plateau area. Comparative and math-simulation methodology was employed for data processing and analysis. A linear correlativity between mean slope and DEM resolution was found at all test areas, but their regression coefficients related closely with the terrain complexity of the test areas. If taking stream channel density to represent terrain complexity, mean slope error could be regressed against DEM resolution (X) and stream channel density (S) at 8 resolution levels and expressed as(0.0015S2 0.031S-0.0325)X-0.0045S2-0.155S 0.1625, with a R2 value of over 0.98. Practical tests also show an effective result of this model in applications. The new development methodology applied in this study should be helpful to similar researches in spatial data uncertainty investigation.     

不同分辨率DEM提取地面坡度的不确定性模拟:以在黄土高原的试验为例

Slope is one of the crucial terrain variables in spatial analysis and land use planning,especially in the Loess Plateau area of China which is suffering from serious soil erosion. DEM based slope extracting method has been widely accepted and applied in practice. However slope accuracy derived from this method usually does not match with its popularity. A quantitative simulation to slope data uncertainty is important not only theoretically but also necessarily to applications. This paper focuses on how resolution and terrain complexity impact on the accuracy of mean slope extracted from DEMs of different resolutions in the Loess Plateau of China. Six typical geomorphologic areas are selected as test areas, representing different terrain types from smooth to rough. Their DEMs are produced from digitizing contours of 1:10,000 scale topographic maps. Field survey results show that 5 m should be the most suitable grid size for representing slope in the Loess Plateau area. Comparative and math-simulation methodology was employed for data processing and analysis. A linear correlativity between mean slope and DEM resolution was found at all test areas,but their regression coefficients related closely with the terrain complexity of the test areas. If taking stream channel density to represent terrain complexity, mean slope error could be regressed against DEM resolution (X) and stream channel density (S) at 8 resolution levels and expressed as (0.0015S2+0.031S-0.0325)X-0.0045S2-0.155S+0.1625, with a R2 value of over 0.98. Practical tests also show an effective result of this model in applications. The new development methodology applied in this study should be helpful to similar researches in spatial data uncertainty investigation.     

Simulation on slope uncertainty derived from DEMs at different resolution levels: a case study in the Loess Plateau

Slope is one of the crucial terrain variables in spatial analysis and land use planning, especially in the Loess Plateau area of China which is suffering from serious soil erosion. DEM based slope extracting method has been widely accepted and applied in practice. However slope accuracy derived from this method usually does not match with its popularity. A quantitative simulation to slope data uncertainty is important not only theoretically but also necessarily to applications. This paper focuses on how resolution and terrain complexity impact on the accuracy of mean slope extracted from DEMs of different resolutions in the Loess Plateau of China. Six typical geomorphologic areas are selected as test areas, representing different terrain types from smooth to rough. Their DEMs are produced from digitizing contours of 1:10,000 scale topographic maps. Field survey results show that 5 m should be the most suitable grid size for representing slope in the Loess Plateau area. Comparative and math-simulation methodology was employed for data processing and analysis. A linear correlativity between mean slope and DEM resolution was found at all test areas, but their regression coefficients related closely with the terrain complexity of the test areas. If taking stream channel density to represent terrain complexity, mean slope error could be regressed against DEM resolution (X) and stream channel density (S) at 8 resolution levels and expressed as (0.0015S2+0.031S-0.0325)X-0.0045S2-0.155S+0.1625, with a R2 value of over 0.98. Practical tests also show an effective result of this model in applications. The new development methodology applied in this study should be helpful to similar researches in spatial data uncertainty investigation.     

Sensitivity analysis of a decision tree classification to input data errors using a general Monte Carlo error sensitivity model

We analysed the sensitivity of a decision tree derived forest type mapping to simulated data errors in input digital elevation model (DEM), geology and remotely sensed (Landsat Thematic Mapper) variables. We used a stochastic Monte Carlo simulation model coupled with a one‐at‐a‐time approach. The DEM error was assumed to be spatially autocorrelated with its magnitude being a percentage of the elevation value. The error of categorical geology data was assumed to be positional and limited to boundary areas. The Landsat data error was assumed to be spatially random following a Gaussian distribution. Each layer was perturbed using its error model with increasing levels of error, and the effect on the forest type mapping was assessed. The results of the three sensitivity analyses were markedly different, with the classification being most sensitive to the DEM error, than to the Landsat data errors, but with only a limited sensitivity to the geology data error used. A linear increase in error resulted in non‐linear increases in effect for the DEM and Landsat errors, while it was linear for geology. As an example, a DEM error of as small as ±2% reduced the overall test accuracy by more than 2%. More importantly, the same uncertainty level has caused nearly 10% of the study area to change its initial class assignment at each perturbation, on average. A spatial assessment of the sensitivities indicates that most of the pixel changes occurred within those forest classes expected to be more sensitive to data error. In addition to characterising the effect of errors on forest type mapping using decision trees, this study has demonstrated the generality of employing Monte Carlo analysis for the sensitivity and uncertainty analysis of categorical outputs that have distinctive characteristics from that of numerical outputs.     

An alternative approach to transverse and profile terrain curvature

Terrain curvature is one of the most important parameters of land surface topography. Well-established methods used in its measurement compute an index of plan or profile curvature for every single cell of a digital elevation model (DEM). The interpretation of these outputs may be delicate, especially when selected locations have to be analyzed. Furthermore, they involve a high level of simplification, contrasting with the complex and multiscalar nature of the surface curvature itself. In this paper, we present a new method to assess vertical transverse and profile curvature combining real-scale visualization and the possibility to measure these two terrain derivatives over a large range of scales. To this purpose, we implemented a GIS tool that extracts longitudinal and transverse elevation profiles from a high-resolution DEM. The performance of our approach was compared with some of the most commonly used methods (ArcMap, Redlands, CA, USA; ArcSIE, Landserf) by analyzing the terrain curvature around charcoal production sites in southern Switzerland. The different methods produced comparable results. While conventional methods quickly summarize terrain curvature in the form of a matrix of values, they involve a loss of information. The advantage of the new method lies in the possibility to measure and visualize the shape and size of the curvature, and to obtain a realistic representation of the average curvature for different subsets of spatial points. Moreover, the new method makes it possible to control the conditions in which the index of curvature is calculated.     

Exploring the sensitivity of coastal inundation modelling to DEM vertical error

As sea level is projected to rise throughout the twenty-first century due to climate change, there is a need to ensure that sea level rise (SLR) models accurately and defensibly represent future flood inundation levels to allow for effective coastal zone management. Digital elevation models (DEMs) are integral to SLR modelling, but are subject to error, including in their vertical resolution. Error in DEMs leads to uncertainty in the output of SLR inundation models, which if not considered, may result in poor coastal management decisions. However, DEM error is not usually described in detail by DEM suppliers; commonly only the RMSE is reported. This research explores the impact of stated vertical error in delineating zones of inundation in two locations along the Devon, United Kingdom, coastline (Exe and Otter Estuaries). We explore the consequences of needing to make assumptions about the distribution of error in the absence of detailed error data using a 1 m, publically available composite DEM with a maximum RMSE of 0.15 m, typical of recent LiDAR-derived DEMs. We compare uncertainty using two methods (i) the NOAA inundation uncertainty mapping method which assumes a normal distribution of error and (ii) a hydrologically correct bathtub method where the DEM is uniformly perturbed between the upper and lower bounds of a 95% linear error in 500 Monte Carlo Simulations (HBM+MCS). The NOAA method produced a broader zone of uncertainty (an increase of 134.9% on the HBM+MCS method), which is particularly evident in the flatter topography of the upper estuaries. The HBM+MCS method generates a narrower band of uncertainty for these flatter areas, but very similar extents where shorelines are steeper. The differences in inundation extents produced by the methods relate to a number of underpinning assumptions, and particularly, how the stated RMSE is interpreted and used to represent error in a practical sense. Unlike the NOAA method, the HBM+MCS model is computationally intensive, depending on the areas under consideration and the number of iterations. We therefore used the HBM+ MCS method to derive a regression relationship between elevation and inundation probability for the Exe Estuary. We then apply this to the adjacent Otter Estuary and show that it can defensibly reproduce zones of inundation uncertainty, avoiding the computationally intensive step of the HBM+MCS. The equation-derived zone of uncertainty was 112.1% larger than the HBM+MCS method, compared to the NOAA method which produced an uncertain area 423.9% larger. Each approach has advantages and disadvantages and requires value judgements to be made. Their use underscores the need for transparency in assumptions and communications of outputs. We urge DEM publishers to move beyond provision of a generalised RMSE and provide more detailed estimates of spatial error and complete metadata, including locations of ground control points and associated land cover.     

Chinese progress in geomorphometry

Geomorphometry, the science of digital terrain analysis (DTA), is an important focus of research in both geomorphology and geographical information science (GIS). Given that 70% of China is mountainous, geomorphological research is popular among Chinese scholars, and the development of GIS over the last 30 years has led to significant advances in geomorphometric research. In this paper, we review Chinese progress in geomorphometry based on the published literature. There are three major areas of progress: digital terrain modelling methods, DTA methods, and applications of digital terrain models (DTMs). First, traditional vector- and raster-based terrain modelling methods, including the assessment of uncertainty, have received widespread attention. New terrain modelling methods such as unified raster and vector, high-fidelity, and real-time dynamic geographical scene modelling have also attracted research attention and are now a major focus of digital terrain modelling research. Second, in addition to the popular DTA methods based on topographical derivatives, geomorphological features, and hydrological factors extracted from DTMs, DTA methods have been extended to include analyses of the structure of underlying strata, ocean surface features and even socioeconomic spatial structures. Third, DTMs have been applied to fields including global climate change, analysis of various typical regions, lunar surface and other related fields. Clearly, Chinese scholars have made significant progress in geomorphometry. Chinese scholars have had the greatest international impact in areas including high-fidelity digital terrain modelling and DTM-based regional geomorphological analysis, particularly in the Loess Plateau and the Tibetan Plateau regions.     

面向地貌学本源的数字地形分析研究进展与展望

数字地形分析(DTA)是地理信息科学(GIS)研究的热点.但是,当前基于数字高程模型(DEM)的数字地形分析在地貌学研究中存在重形态轻机理、重现象轻过程、重地上轻地下等问题,急需从单一的地貌形态分析,迈向面向成因、过程与机理等地貌学本源问题的研究转变.据此,本文系统梳理了面向地貌学本源的数字地形分析的相关研究现状,并从...     

A universal spectral analytical method for digital terrain modeling

There are three major mathematical problems in digital terrain analysis: (1) interpolation of digital elevation models (DEMs); (2) DEM generalization and denoising; and (3) computation of morphometric variables through calculating partial derivatives of elevation. Traditionally, these three problems are solved separately by means of procedures implemented in different methods and algorithms. In this article, we present a universal spectral analytical method based on high-order orthogonal expansions using the Chebyshev polynomials of the first kind with the subsequent Fejér summation. The method is intended for the processing of regularly spaced DEMs within a single framework including DEM global approximation, denoising, generalization, as well as calculating the partial derivatives of elevation and local morphometric variables.

The method is exemplified by a portion of the Great Rift Valley and central Kenyan highlands. A DEM of this territory (the matrix 480 × 481 with a grid spacing of 30″) was extracted from the global DEM SRTM30_PLUS. We evaluated various sets of expansion coefficients (up to 7000) to approximate and reconstruct DEMs with and without the Fejér summation. Digital models of horizontal and vertical curvatures were computed using the first and second partial derivatives of elevation derived from the reconstructed DEMs. To evaluate the approximation accuracy, digital models of residuals (differences between the reconstructed DEMs and the initial one) were calculated. The test results demonstrated that the method is characterized by a good performance (i.e., a distinct monotonic convergence of the approximation) and a high speed of data processing. The method can become an effective alternative to common techniques of DEM processing.     

Uncertainty modeling and analysis of surface area calculation based on a regular grid digital elevation model (DEM)

In the field of digital terrain analysis (DTA), the principle and method of uncertainty in surface area calculation (SAC) have not been deeply developed and need to be further studied. This paper considers the uncertainty of data sources from the digital elevation model (DEM) and SAC in DTA to perform the following investigations: (a) truncation error (TE) modeling and analysis, (b) modeling and analysis of SAC propagation error (PE) by using Monte-Carlo simulation techniques and spatial autocorrelation error to simulate DEM uncertainty. The simulation experiments show that (a) without the introduction of the DEM error, higher DEM resolution and lower terrain complexity lead to smaller TE and absolute error (AE); (b) with the introduction of the DEM error, the DEM resolution and terrain complexity influence the AE and standard deviation (SD) of the SAC, but the trends by which the two values change may be not consistent; and (c) the spatial distribution of the introduced random error determines the size and degree of the deviation between the calculated result and the true value of the surface area. This study provides insights regarding the principle and method of uncertainty in SACs in geographic information science (GIScience) and provides guidance to quantify SAC uncertainty.     

基于球面DQG的矢量与地形数据无缝集成

现有的全球大规模空间数据可视化系统主要侧重于影像和地形数据的综合表达,针对矢量与地形的集成可视化能力相对较弱。该文以球面退化四叉树格网(Degenerate Quad-tree Grids,DQG)为基础,通过DQG格网的三角化过程构建了地表DEM模型,并提出了从矢量线对象到地形格网表面的映射方法。采用GTOPO30数据集和国界矢量数据进行了相关实验,结果表明:该方法能实现矢量数据与多分辨率DEM的无缝集成,并能有效地避免矢量对象"悬浮"和"入地"等现象。     

Derivation of topographic variables from a digital elevation model given by a spheroidal trapezoidal grid

Digital elevation models (DEMs) given by spheroidal trapezoidal grids are more appropriate for large regional, sub-continental, continental and global geological and soil studies than square-spaced DEMs. Here we develop a method for derivation of topographic variables, specifically horizontal (k) and vertical h (k) landsurface curvatures, from spheroidal trapezoidal-spaced DEMs. First, we v derive equations for calculation of partial derivatives of elevation with DEMs of this sort. Second, we produce formulae for estimation of the method accuracy in terms of root mean square errors of partial derivatives of elevation, as well as k h and k (m and m respectively). We design the method for the case that the v kh k v Earth's shape can be ignored, that is, for DEM grid sizes of no more than 225 km. We test the method by the example of fault recognition using a DEM of a part of Central Eurasia. A comparative analysis of test results and factual geological data demonstrates that the method actually works in regions marked by complicated topographic and tectonic conditions. Upon increasing DEM grid size, one can produce generalised maps of k and k. Spatial distributions of m and m h v kh k v depend directly on the distribution of elevation RMSE. Areas with high values of m are marked by low values of m, and vice versa, areas with high values kh k v of m are marked by low values of m. Data on m and m should be utilised k v kh kh k v to control and improve applications of k and k to geological studies. The method h v developed opens up new avenues for carrying out some 'conventional' raster operations directly on geographical co-ordinates.     

地形复杂度对坡度坡向的影响

采用三阶不带权差分算法，研究了地形复杂度与坡度坡向的关系，澄清了目前关于坡度坡向误差空间分布的矛盾观点，并分别在凹向椭球和高斯合成曲面数学模型曲面DEM上对其进行验证。通过研究得出：①坡度、坡向误差与坡度值正相关；②坡度坡向误差主要分布在平坦地区；③坡向误差较坡度误差对DEM高程数据误差敏感，较小的DEM误差引起较大的坡向误差。