DEM地表坡向变率的向量几何计算法

作为计算坡向变率的数据基础,坡向矩阵具有方向性,以标量的方式计算带有方向属性的数据,将带来计算方式的误区及计算结果的偏差。本文以数学高斯曲面和不同黄土地貌样区5 m分辨率DEM数据为基础,针对坡向数据具有方向性的特点,设计基于数学向量的坡向变率计算方法。首先针对坡向数据进行极坐标转换,形成坡向矩阵的向量几何表达;然后以该坡向向量数据为基础来计算坡向变率;最后将本文方法的计算结果与传统标量方法的计算结果展开对比分析。试验结果显示,本文方法的坡向变率计算有效地避免了正北方向产生的极大偏差以及坡向差超过180°时的不准确现象,同时其他大部分区域也得出更为合理准确的坡向变率计算结果。在不同分辨率DEM下,本文方法能得到较为稳定的结果。本文所提出的基于向量几何的坡向变率计算方法可为精准数字地形分析提供参考,也是借鉴数学向量几何的方法解决数字地形分析问题的重要实践。

Vector geometry based method for the extraction of slope of aspect by using DEMs

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.