Fehr E, Kadau D, Araújo N A M, Andrade J S, Herrmann H J
IfB, ETH Zürich, CH-8093 Zürich, Switzerland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Sep;84(3 Pt 2):036116. doi: 10.1103/PhysRevE.84.036116. Epub 2011 Sep 27.
We study the morphology of watersheds in two and three dimensional systems subjected to different degrees of spatial correlations. The response of these objects to small, local perturbations is also investigated with extensive numerical simulations. We find the fractal dimension of the watersheds to generally decrease with the Hurst exponent, which quantifies the degree of spatial correlations. Moreover, in two dimensions, our results match the range of fractal dimensions 1.10≤d(f)≤1.15 observed for natural landscapes. We report that the watershed is strongly affected by local perturbations. For perturbed two and three dimensional systems, we observe a power-law scaling behavior for the distribution of areas (volumes) enclosed by the original and the displaced watershed and for the distribution of distances between outlets. Finite-size effects are analyzed and the resulting scaling exponents are shown to depend significantly on the Hurst exponent. The intrinsic relation between watershed and invasion percolation, as well as relations between exponents conjectured in previous studies with two dimensional systems, are now confirmed by our results in three dimensions.
我们研究了二维和三维系统中不同程度空间相关性下流域的形态。还通过广泛的数值模拟研究了这些对象对微小局部扰动的响应。我们发现流域的分形维数通常随赫斯特指数降低，赫斯特指数量化了空间相关性的程度。此外，在二维中，我们的结果与自然景观观测到的分形维数范围1.10≤d(f)≤1.15相匹配。我们报告称流域受到局部扰动的强烈影响。对于受扰动的二维和三维系统，我们观察到原始流域和位移后流域所包围区域（体积）的分布以及出水口之间距离的分布呈现幂律缩放行为。分析了有限尺寸效应，结果表明所得缩放指数显著依赖于赫斯特指数。我们在三维中的结果证实了流域与入侵渗流之间的内在关系，以及先前二维系统研究中推测的指数之间的关系。
