The effective characterization of topographic surfaces is a central tenet of geomorphology. Differences in land surface properties reveal variations in structural controls and the nature and efficacy of Earth‐shaping processes. In this paper, we employ the Hölder exponents, α, characterizing the local scaling behavior of topography and commonly used in the study of the (multi)fractal properties of landscapes and show that the joint probability distribution of the area of the terrain with a given elevation and α contains a wealth of information on topographic structure. The conditional distributions of the hypsometric integrals as a function of α, that is, I_hyp|α, are shown to capture this structure. A multivariate analysis reveals three metrics that summarize these conditional distributions: Strahler's original hypsometric integral, the standard deviation of the I_hyp|α, and the nature of any trend of the I_hyp|α against α. An analysis of five digital elevation models (DEMs) from different regions of the United States shows that only one is truly described by the hypsometric integral (Mettman Ridge from central Oregon). In the other cases, the new metrics clearly discriminate between instances where topographic roughness is more clearly a function of elevation, as captured by the conditional variables. In a final example, we artificially sharpen the ridges and valleys of one DEM to show that while the hypsometric integral and standard deviation of I_hyp|α are invariant to the change, the trend of I_hyp|α against α captures the changes in topography.

中文翻译：

---

Hölder条件下的测湿法：对经典的地形特征描述方法的改进

地形表面的有效表征是地貌学的中心原则。土地表面特性的差异揭示了结构控制以及造地球过程的性质和功效的变化。在本文中，我们采用了Hölder指数α来表征地形的局部缩放行为，该指数通常用于研究景观的（多）分形特性，并表明在给定条件下地形区域的联合概率分布高程和α包含了大量有关地形结构的信息。催眠积分的条件分布是α的函数，即I _{h y p |。α}显示为捕获此结构。多元分析揭示了总结这些条件分布的三个指标：Strahler的原始测压积分，I _{h y p |}的标准偏差。_α以及I _{h y p |}的任何趋势的性质。_α对α。对来自美国不同地区的五个数字高程模型（DEM）进行的分析显示，实测积分（俄勒冈州中部的梅特曼里奇）仅能真正描述一个模型。在其他情况下，新指标可以清楚地区分地形粗糙度是由条件变量捕获的海拔高度函数的实例。在最后一个示例中，我们人为地锐化了一个DEM的波谷和波谷，以证明当I _{h y p | α}是不变的，I _{h y p |}的趋势。_α对α捕获地形变化。