ABSTRACT Elevation gradients are primary components of slope and aspect. Significant concerns remain when computing gradients if noise (perturbing non-DEM data) is present. There is still a need to find ways to balance accuracy of the gradient and stability to noise for specific types of DEM. In this study, six algorithms are compared using four DEMs and analyzed for stability to base level DEM noise and added random noise. Theoretical stability and accuracy of the formulae are analyzed using harmonic (frequency or spatial scale) response. The results provide a basis to determine the most appropriate algorithm for different situations. They show that: (1) the set (Evans-Young (EY), Sharpnack (Sp), Sobel (Sb)) has a better stability to noise ratio than the set (Zevenbergen (Z), Florinsky (F), Horn (H)). EY has the smoothest surface and the highest stability to noise ratio. If stability is the primary measure in mid-frequencies, EY is a good choice. (2) Sb is good because of its accuracy in mid to high frequencies. Out to the highest frequencies, Sb is the best. (3) F has potential but should not be used with very high-frequency noise. (4) H and Z should not be used when there is substantial noise.

中文翻译：

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高程梯度算法中的不确定性和错误

摘要 高程梯度是坡度和坡向的主要组成部分。如果存在噪声（干扰非 DEM 数据），则在计算梯度时仍然存在重大问题。对于特定类型的 DEM，仍然需要找到平衡梯度精度和噪声稳定性的方法。在这项研究中，六种算法使用四个 DEM 进行了比较，并分析了对基本级 DEM 噪声和添加的随机噪声的稳定性。使用谐波（频率或空间尺度）响应分析公式的理论稳定性和准确性。结果为确定最适合不同情况的算法提供了基础。他们表明：（1）集合（Evans-Young（EY），Sharpnack（Sp），Sobel（Sb））比集合（Zevenbergen（Z），Florinsky（F），Horn（）具有更好的稳定性噪声比H））。EY 拥有最光滑的表面和最高的稳定性噪声比。如果稳定性是中频的主要衡量标准，安永是一个不错的选择。(2) Sb 是好的，因为它在中高频上的精度很高。对于最高频率，Sb 是最好的。(3) F 具有潜力，但不应与非常高频的噪声一起使用。(4) 当有大量噪音时，不应使用 H 和 Z。