﻿ DEM地表坡向变率的向量几何计算法
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DEM地表坡向变率的向量几何计算法

1. 南京师范大学地理科学学院, 江苏 南京 210023;
2. 南京师范大学虚拟地理环境教育部重点实验室, 江苏 南京 210023;
3. 江苏省地理信息资源开发与利用协同创新中心, 江苏 南京 210023

Vector geometry based method for the extraction of slope of aspect by using DEMs
HU Guanghui1,2,3, XIONG Liyang1,2,3, TANG Guoan1,2,3
1. School of Geography, Nanjing Normal University, Nanjing 210023, China;
2. Key Laboratory of Virtual Geographic Environment(Nanjing Normal University), Ministry of Education, Nanjing 210023, China;
3. Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing 210023, China
Abstract: Aspect matrix, as the source data for calculating the slope of aspect (SOA), has the characteristic of directionality. Thus, misunderstanding and error would be produced if SOA was calculated by scalar method because its source data has the directional property. On a basis of the mathematical Gaussian surface and 5 m resolution DEM data of different loess landform sample areas, the mathematical vector method is proposed to calculate SOA with a full consideration of directional property of the aspect matrix. In this method, the original aspect matrix has been transformed into the polar coordinate system, and the vector geometric representation of aspect matrix could be achieved. Then the SOA is calculated on a basis of this vector transformed aspect data. In the end, a comparative analysis is conducted among the proposed method and traditional scalar methods. The results show that SOA calculated by the proposed method could effectively avoid the extreme error in the north direction and the inaccuracy when the aspect difference exceeds 180°. Meanwhile, a more reasonable SOA result could be achieved in the other main areas, and a more stable result can be obtained by using our method in different resolution DEMs. The proposed vector geometry method could help to provide a reference for accurate digital terrain analysis, and it is also an important practice to solve problem in DTA by mathematical vector geometry.
Key words: DEM    slope of aspect    aspect    vector geometry method

20世纪以来，数字高程模型(DEM)与数字地形分析(DTA)的提出与应用，为传统的地学分析方法带来革命性的变化[9-13]。现阶段，基于DEM数据，一阶地形因子能够较为方便地计算[14]。但是，坡向变率为二阶地形因子，其计算方法更为复杂[15]。起初，坡向变率的计算是参考坡度的计算方法(对高程数值矩阵求偏导)，也就是对坡向数值矩阵求取偏导数，即计算坡向的坡度。据此，随着对坡向变化理解的深入，前人依次提出直接法、正反DEM法、六分法实现了标量条件的坡向变率求解，并取得了广泛的应用[16-19]。可见，前人对坡向变率的计算作了初步的研究，取得了一定的研究成果。但是，坡向变率的计算是一个基于坡向数据基础二阶地形因子的提取[20-21]。该坡向数值矩阵与原始的高程数值矩阵存在本质的不同，即高程数值在很大程度上被看成是一个没有方向特征的常规数学标量，标量值只有大小关系，而无须考虑方向属性。此时对其进行一阶求偏导可直接进行高程求差，即一阶地形因子。而相对二阶地形因子坡向变率而言，坡向数值矩阵的每个像元都代表着其独有的坡面指向，这种指向代表着其特有的方向性[22]。因此，坡向数值矩阵不是一个标量矩阵，简单数学标量的代数运算方式用在坡向数值矩阵上不可避免地违背了数学运算机制。

 图 1 数学视角下两种不同作差运算 Fig. 1 Two different operations from perspective of mathematics

1 研究方法 1.1 坡向变率计算基础

(1)

1.2 坡向矩阵向量化表达

(1) 坡向在坡向极坐标系下的向量表达。坡向向量可以认为处在一个以正北方向为起始方向，顺时针为旋转方向的特殊极坐标系下。本文称之为坡向极坐标系，如图 2(a)所示。坡向值应当首先在坡向极坐标系下表达为向量。其表示为

 图 2 两种不同的极坐标系 Fig. 2 Two different polar coordinate systems

(2)

(2) 坡向在普通极坐标系下的向量表达。如图 2(b)所示，普通极坐标系是较为常用的一种坐标系，并且可以与平面直角坐标系进行快速转换。坡向极坐标转为普通极坐标的方法如下

(3)

(4)

(3) 坡向在平面直角坐标系下的向量表达。平面直角坐标系更适合于向量之间的运算，因此需要将极坐标系下的坡向向量转换到平面直角坐标系下进行表达。转换公式如下

(5)

(4) 向量在平面直角坐标系下的坐标表示。分别取与X轴、Y轴方向相同的两个单位向量ij作为基底向量。任作一个向量a，由平面向量基本定理可知，有且只有一对实数xy，可以使得等式a=xi+yj成立，于是将(x, y)叫作向量a的直角坐标表示，记作a=(x, y)，如图 3所示。

 图 3 平面直角坐标系下向量表示特征 Fig. 3 Vector representation method of plane Cartesian coordinate system

1.3 坡向变率向量计算

(1) 已知a=(x1, y1), b=(x2, y2)，则a+b=(x1+x1, y2+y2)。

(2) 已知a=(x1, y1), b=(x2, y2)，则a-b=(x1-x2, y2-y2)。

(3) 已知a=(x, y)和实数λ, 则λa=(λx, λy)。

(4) 已知a=(x, y), 则a的模长是

 图 4 基于向量几何法的地表坡向变率提取算法流程 Fig. 4 Flow chart of slope of aspect extraction algorithm based on vector geometry

2 试验样区与数据

(6)

(7)

A=3，B=10，C=1/3，m=500，n=500，标准差为1.326 0，均值为0.624 7，格网分辨率为5×5构建地形曲面如图 5(a)所示。针对高斯数学曲面分别求取偏导fxfy，再根据式(7)求得坡向结果作为原始坡向数值矩阵。将此坡向数值矩阵按照2.2节向量化后的坡向矢量场图如图 5(b)所示，图 5(c)(d)(e)为局部放大图。

 图 5 模拟高斯曲面及曲面坡向 Fig. 5 Simulated Gaussian surface and its aspect

 图 6 试验样区位置及其地貌晕眩 Fig. 6 Locations of study areas and the hill shade in sample areas

 样区 高程/m 平均坡度/(°) 地貌类型 绥德 814~1188 29.28 黄土峁 吴起 1329~1721 28.42 黄土墚 宜君 768~1158 19.23 黄土塬

3 试验结果与分析 3.1 高斯曲面坡向变率计算结果与分析

 图 7 不同方法基于高斯曲面的坡向变率计算结果s Fig. 7 SOA results with different methods by using Gaussian surface

 图 8 4个样区不同方法坡向变率计算结果频率分布 Fig. 8 Probability distribution of SOA results with different methods in the four sample areas

3.2 不同样区坡向变率计算结果与分析

 图 9 宜君样区坡向变率不同方法的计算结果 Fig. 9 SOA results by different methods in Yijun area

 图 10 吴起样区坡向变率不同方法的计算结果 Fig. 10 SOA results by different methods in Wuqi area

 图 11 绥德样区坡向变率不同方法的计算结果 Fig. 11 SOA results by different methods in Suide area

 样区 统计量 [0, 5) [5, 10) [10, 15) [15, 20) [20, 25) [25, 30) [30, 35) [35, 40) [40, 45) [45, 50) [50, 55) 宜君 P(d)/(%) 46.15 23.01 11.51 6.44 3.99 2.52 1.83 1.74 1.80 0.90 0.000 08 E 2.156 7.204 12.23 17.31 22.28 27.33 32.41 37.55 42.69 46.55 50.03 吴起 P(d)/(%) 36.20 24.87 13.05 8.07 5.52 3.71 2.70 2.38 2.25 1.15 0.001 3 E 2.674 7.212 12.27 17.35 22.31 27.34 32.40 37.51 42.63 46.70 50.11 绥德 P(d)/(%) 31.06 26.82 14.68 8.56 5.19 3.07 2.41 2.85 3.45 1.82 0.001 E 2.810 7.263 12.25 17.35 22.23 27.34 32.48 37.63 42.82 46.57 50.11

3.3 不同分辨率向量法坡向变率计算结果分析

 图 12 基于不同分辨率的三样区DEM数据的向量法计算结果频率分布 Fig. 12 Probability distribution of SOA results based different resolutions using vector methods in the three sample areas

 图 13 基于不同分辨率的三样区DEM数据向量法计算结果统计 Fig. 13 Statistic result based on different resolution DEMs using vector method in the three sample areas

4 结论

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http://dx.doi.org/10.11947/j.AGCS.2019.20180447

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#### 文章信息

HU Guanghui, XIONG Liyang, TANG Guoan
DEM地表坡向变率的向量几何计算法
Vector geometry based method for the extraction of slope of aspect by using DEMs

Acta Geodaetica et Cartographica Sinica, 2019, 48(11): 1404-1414
http://dx.doi.org/10.11947/j.AGCS.2019.20180447