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تعداد صفحات این فایل: ۳۶ صفحه

بخشی از مقاله انگلیسیعنوان انگلیسی:Extending generalized Horton laws to test embedding algorithms for topologic river networks~~en~~

Abstract

River networks in the landscape can be described as topologic rooted trees embedded in a three-dimensional surface. We examine the problem of embedding topologic binary rooted trees (BRTs) by investigating two space-filling embedding procedures: Top-Down, previously developed in the context of random selfsimilar networks (RSNs), and Bottom-Up, a new procedure developed here. We extend the concept of generalized Horton laws to interior sub catchments and create a new set of scaling laws that are used to test the embedding algorithms. We compare the two embedding strategies with respect to the scaling properties of the distribution of accumulated areas A and network magnitude M for complete order streams . The Bottom-Up procedure preserves the equality of distributions Aw/E[Aw]=dMw/E[Mw] ; a feature observed in real basins. The Top-Down embedded networks fail to preserve this equality because of strong correlations of tile areas in the final tessellation. We conclude that the presence or absence of this equality is a useful test to diagnose river network models that describe the topology/geometry of natural drainage systems. We present some examples of applying the embedding algorithms to self similar trees (SSTs) and to RSNs. Finally, a technique is presented to map the resulting tiled region into a three-dimensional surface that corresponds to a landscape drained by the chosen network. Our results are a significant first step toward the goal of creating realistic embedded topologic trees, which are also required for the study of peak flow scaling in river networks in the presence of spatially variable rainfall and flood-generating processes.

۱ Introduction

The study of river network topology is an active area of research in geomorphology (Meakin et al., 1991; Maritan et al., 1996; Dodds and Rothman, 2000; Molnar, 2005). Several mathematical models have been introduced in the literature to describe, using simple principles, the complex network topology of river networks (Tokunaga, 1966; Scheidegger, 1967; Shreve, 1967; Veitzer and Gupta, 2000). These topologic models have helped put their mathematical foundations on a firm footing, and many of them have been successful in explaining major geomorphic features observed in natural river networks. Topologic river network models, by definition, only describe how nodes of the network are connected with each other. These models do not provide a description of the spatial embedding of the topology in the three-dimensional landscape of a river basin. As a result, they are unsuitable to study the interaction between spatially correlated rainfall and runoff generation processes and its impact on the transport of flows through a river network. Some network models, such as optimal channel networks (Rigon et al., 1993; Maritan et al., 1996; Rinaldo et al., 2006), Gibbsian networks (Troutman and Karlinger, 1994, 1998), and networks generated by random walks (Leopold and Langbein, 1962; Meakin et al., 1991) include explicitly the spatial geometry of networks by embedding them on a twodimensional lattice. However, two disadvantages of lattice models relative to topologic models are, first, that generation of lattice models is typically more computationally demanding and, second, that it is more difficult to obtain analytic results for lattice models. Self-similar river network models play a fundamental role in understanding observed scaling in the magnitudes of peak flows (Gupta et al., 2010). For example, Gupta et al. (1996), Menabde and Sivapalan (2001), and Troutman and Over (2001) considered idealized self-similar networks embedded in a two-dimensional space to understand how the interaction between multifractal rainfall and self-similar river networks determines the magnitude and scaling characteristics of peak flows. Gupta et al. (1996) assumed that rainfall follows spatially correlated beta random cascade that is deposited on a Peano channel network. Likewise, Menabde and Sivapalan (2001) assumed that rainfall follows a log-normal cascade on a Mandelbrot–Viscek network. Troutman and Over (2001) made more general assumptions regarding spatial variability of rainfall and self-similar network structure to study scaling in peak flows. In all these studies, however, the spatial cascade structure of rainfall was assumed to be aligned with the topology of the network. Mantilla (2007) conducted simulation of scaling in peak flows on RSNs. In order to generalize this approach to spatially correlated rainfall embedding realistic1 topological networks in a three-dimensional space is necessary. The embedding process is schematically illustrated in Fig. 1 for a very simple network topology. We begin with, first, a network for which it is assumed that only the connectivity structure (a topologic tree) is known (Fig. 1a) and, second, a two-dimensional region with a fixed shape (Fig. 1b) into which the network will be embedded. The goal is to develop an algorithm, which yields the embedded network, as shown in Fig. 1c. The embedding process, as defined in this paper, must therefore fulfill two minimal requirements: (i) it must be space filling, which means that every point in the given region has to be assigned to, or drain into, a unique link of the binary rooted tree (BRT); the process is known as tiling, and (ii) the topologic structure of the network needs to be preserved. These two minimal requirements of the embedding process do not uniquely define an algorithm for obtaining the embedded network, so clearly any proposed procedure needs to be further tailored so as to preserve, to the extent possible, important geometrical properties of real drainage basins. Examination of Fig. 1c makes clear that the distribution of the size of areas assigned to the individual links, and the resulting size of accumulated areas of sub basins is a fundamental characteristic of embedded networks. It is the focus here. A key contribution of this paper is to apply generalized Horton’s laws (Peckham and Gupta, 1999; Veitzer and Gupta, 2000; Troutman, 2005) to obtain a sensitive measure of how well the area distribution of embedded networks conforms with behavior observed in real networks. Generalized Horton’s laws are based on the idea of statistical selfsimilarity (SSS) of the distribution of basin variables defined for streams of different Strahler orders (Strahler, 1957). Strahler ordering of streams is determined by the following rules: first-order streams are those with no upstream inflows and the stream immediately downstream from the junction of two streams of order 1 and 2 is 1+ 1 if 1=2 and the maximum between 1 and 2 if 12 The two basin variables of interest here are drainage area, A, and network magnitude, M, defined as the number of first-order streams in the network. A comparison of the statistical distribution of these two variables provides a test of embedding algorithms. In Section 2 we provide some background on the problem of embedding BRTs and also present data from real networks. In Section 3, the concept of basin decomposition into hillslopes is used to introduce an extension of Horton’s laws (Horton, 1945; Strahler, 1957). It is used to test the properties of the tessellations generated by our embedding strategies. In Section 4 we present two embedding strategies. First, in Section 4.1 we present the embedding procedure that Veitzer (1999) developed that we call the top-down embedding (TDE) algorithm. We explain the deficiencies of this algorithm with respect to the generalized Horton’s laws. In Section 4.2 the bottom-up embedding (BUE) algorithm is described, and advantages of using this approach are explained. Some examples of the BUE are presented using self similar trees (SSTs) and RSNs. In Section 5 we develop a method to generate three-dimensional landscapes using the tiled region obtained by the BUE. Finally, in Section 6 our conclusions are presented and areas of future research are highlighted.

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