A methodology is presented to understand the role of the statistical self-similar topology of real river networks on scaling, or power law, in peak flows for rainfall-runoff events. We created Monte Carlo generated sets of ensembles of 1000 random self-similar networks (RSNs) with geometrically distributed interior and exterior generators having parameters ip/isubi/sub and ip/isube/sub, respectively. The parameter values were chosen to replicate the observed topology of real river networks. We calculated flow hydrographs in each of these networks by numerically solving the link-based mass and momentum conservation equation under the assumption of constant flow velocity. From these simulated RSNs and hydrographs, the scaling exponents β and φ characterizing power laws with respect to drainage area, and corresponding to the width functions and flow hydrographs respectively, were estimated. We found that, in general, φ β, which supports a similar finding first reported for simulations in the river network of the Walnut Gulch basin, Arizona. Theoretical estimation of β and φ in RSNs is a complex open problem. Therefore, using results for a simpler problem associated with the expected width function and expected hydrograph for an ensemble of RSNs, we give heuristic arguments for theoretical derivations of the scaling exponents βsupi(E)/i/sup and φsupi(E)/i/sup that depend on the Horton ratios for stream lengths and areas. These ratios in turn have a known dependence on the parameters of the geometric distributions of RSN generators. Good agreement was found between the analytically conjectured values of βsupi(E)/i/sup and φsupi(E)/i/sup and the values estimated by the simulated ensembles of RSNs and hydrographs. The independence of the scaling exponents φsupi(E)/i/sup and φ with respect to the value of flow velocity and runoff intensity implies an interesting connection between unit hydrograph theory and flow dynamics. Our results provide a reference framework to study scaling exponents under more complex scenarios of flow dynamics and runoff generation processes using ensembles of RSNs.
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机译：提出了一种方法，以了解实际河流网络的统计自相似拓扑在降雨径流事件的峰值流量中对水垢或功率定律的作用。我们创建了由Monte Carlo生成的1000个随机自相似网络（RSN）的集合，这些集合具有几何分布的内部和外部生成器，其参数分别为 p i 和 p e 。选择参数值来复制实际河网的观测拓扑。我们在恒定流速的假设下，通过对基于链接的质量和动量守恒方程进行数值求解，从而在每个网络中计算了流量水文图。根据这些模拟的RSN和水位图，比例指数β和φ等于1。估算了关于流域面积的幂律特征，并分别对应于宽度函数和流量水位图。我们发现，通常，＆phi; >＆beta ;，支持类似的发现，该发现首次报道用于亚利桑那州Walnut Gulch盆地的河网模拟。 β和φ的理论估计RSN中的问题是一个复杂的开放问题。因此，使用针对与预期宽度函数相关的更简单问题的结果以及针对一组RSN的预期水位图的结果，我们给出了缩放比例指数＆beta; （E） < / sup>和＆ （E） ，它们取决于河流长度和面积的霍顿比率。这些比率又取决于RSN发生器的几何分布参数。在＆beta; （E） 的解析推测值之间找到了很好的一致性，由RSN和水位图的模拟合奏估算的值。标度指数＆ （E） 和＆phi;的独立性。关于流速和径流强度的值意味着单位水文理论与流动力学之间的有趣联系。我们的结果提供了一个参考框架，以研究使用RSN集成在更复杂的流动动力学和径流生成过程中的比例指数。
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