坡度作为最基本的地形因子,是构建地学分析模型的基础数据,基于格网DEM的坡度计算模型的精度分析一直是地学分析的重要研究内容。本文通过分析计算坡度的 差分模型误差来源及特征,提出了一种新的模型——5节点二阶差分模型。其基本思想是顾及多距离邻近点高程信息,采用5×5分析窗口,建立单倍格网和两倍格网的差分计算线性组合模型。并通过数学分析的理论推导证明该模型可以显著提高计算精度。实验选取典型数学曲面并对其采用不同的分辨率(1m和5m)进行离散化精度验证。计算结果表明,与现有的差分模型相比,5节点二阶差分模型可以显著提高坡度的精度,对应于两种分辨率的DEM,该模型计算坡度的精度可以分别提高7×10⁴倍、3×10³倍以上,表明该模型对于高分辨率DEM数据更为适用。本研究不仅丰富了数字地形分析的方法体系,为相关地学模型分析提供更精确的坡度数据,而且为将来改进地形曲率因子等计算模型提供了一定的参 考和借鉴。

Slope is the most basic terrain parameter to construct geo-science models. The accuracy analysis of the computing model is one of the most important issues in the geo-science. A lot of studies focused on error analysis of the computing model over the past few decades. The source of the error and the relationship between the error and the other parameters were studied. However, few studies were aimed to put forward a new model to decrease the error, which is more important for high resolution DEMs. By analyzing the error source of the slope computing model, this paper put forward a new model called 5 Node Second-order Finite Difference (5N-2FD). The characteristic of this model is that it is able to consider the multi-distance neighbor node: firstly it builds two difference models using different grid sizes, and then combines the two models into one with different weights. It is proved that it can improve the accuracy of the results remarkably through mathematical analysis. In this paper, a mathematical surface is selected to verify the accuracy of the new model. In order to study the influence of the resolution to the accuracy of the slope, this paper generated DEMs from the mathematical surface with different resolution, one is 1m, and the other is 5m. The results show that the new model can significantly improve the accuracy of the result compared with the common models. This study enriched the method system of digital terrain analysis, and provided slope data of high accuracy for many geo-science models. In addition, besides slope, there are many terrain parameters which are calculated through finite difference, such as aspect and various kinds of curvature, and the methods of this paper could afford some useful references in improving the accuracy of such terrain parameters.