Maximum depth is crucial for many lake processes and biota, but attempts to explain its variation have achieved little predictive power. In this paper, we describe the probability distribution of maximum depths based on recent developments in the theory of fractal Brownian motions. The theoretical distribution is right-tailed and adequately captures variations in maximum depth in a dataset of 8164 lakes (maximum depths 0.1–135 m) from the northeastern United States. Maximum depth increases with surface area, but with substantial random variation—the 95% prediction interval spans more than an order of magnitude for lakes with any specific surface area. Our results explain the observed variability in lake maximum depths, capture the link between topographic characteristics and lake bathymetry, and provide a means to upscale maximum depth-dependent processes, which we illustrate by upscaling the diffusive flux of methane from northern lakes to the atmosphere.

中文翻译：

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湖面面积与最大深度关系的理论

最大深度对许多湖泊过程和生物群至关重要，但解释其变化的尝试几乎没有取得预测能力。在本文中，我们根据分形布朗运动理论的最新发展描述了最大深度的概率分布。理论分布是右尾的，并且充分捕捉了美国东北部 8164 个湖泊（最大深度 0.1–135 m）的数据集中最大深度的变化。最大深度随着表面积的增加而增加，但随机变化很大——对于具有任何特定表面积的湖泊，95% 的预测区间跨度超过一个数量级。我们的结果解释了观察到的湖泊最大深度的变化，捕捉了地形特征与湖泊测深之间的联系，