2003 Moussa HP On morphometric properties of basins, scale effects and hydrologic response (1) (original) (raw)
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On morphometric properties of basins, scale effects and hydrological response
Hydrological Processes, 2003
One of the important problems in hydrology is the quantitative description of river system structure and the identification of relationships between geomorphological properties and hydrological response. Digital elevation models (DEMs) generally are used to delineate the basin's limits and to extract the channel network considering pixels draining an area greater than a threshold area S. In this paper, new catchment shape descriptors, the geometric characteristics of an equivalent ellipse that has the same centre of gravity, the same principal inertia axes, the same area and the same ratio of minimal inertia moment to maximal inertia moment as the basin, are proposed. They are applied in order to compare and classify the structure of seven basins located in southern France. These descriptors were correlated to hydrological properties of the basins' responses such as the lag time and the maximum amplitude of a geomorphological unit hydrograph calculated at the basin outlet by routing an impulse function through the channel network using the diffusive wave model. Then, we analysed the effects of the threshold area S on the topological structure of the channel network and on the evolution of the source catchment's shape. Simple models based on empirical relationships between the threshold S and the morphometric properties were established and new catchment shape indexes, independent of the observation scale S, were defined. This methodology is useful for geomorphologists dealing with the shape of source basins and for hydrologists dealing with the problem of scale effects on basin topology and on relationships between the basin morphometric properties and the hydrological response. Copyright © 2002 John Wiley & Sons, Ltd.
Fractal relation of mainstream length to catchment area in river networks
Water Resources Research, 1991
Hjelmfelt 1988) using eight rivers in Missouri. The fractal dimension of river length, d, is derived here from the Horton's laws of network composition. This results in a simple function of stream length and stream area ratios, that is, d --max (1, 2 log Roe/log RA). Three case studies are reported showing this estimate to be coherent with measurements of d obtained from map analysis. The scaling properties of the network as a whole are also investigated, showing the fractal dimension of fiver network, D, to depend upon bifurcation and stream area ratios according to D --min (2, 2 log R B/log R A). These results provide a linkage between quantitative analysis of drainage network composition and scaling properties of river networks.
Influence of Valley Type on the Scaling Properties of River Planforms
Water Resources Research, 1996
Scaling properties of 44 individual river planforms from the Cascade and Olympic Mountains of Washington State were defined using the divider method. Analysis of the standardized residuals for least squares linear regression of Richardson plots reveals systematic deviations from simple self-similarity that correlate with the geomorphological context defined by valley type. A single fractal dimension describes rivers flowing through bedrock valleys. Those flowing in inherited glacial valleys exhibit two distinct fractal dimensions, with a larger fractal dimension at small scales. Rivers flowing in alluvial valleys are also described by two fractal dimensions, but with a larger dimension at large scales. We further find that the wavelength of the largest meander defines an upper limit to the scaling domain characterized by fractal geometry. These results relate scaling properties of river planforms to the geomorphological processes governing valley floor morphology. GoodchiM, 1980; Mark and Aronson, 1984; GoodchiM and Mark, 1987; Klinkenberg, 1992], few studies relate D to processes governing these forms [Woronow, 1981; Phillips, 1993]. Indeed, the relatively narrow range of fractal dimensions describing a variety of natural patterns suggests the futility of searching for ties with physical processes [Turcotte, 1992]. Some geomorphic features, however, exhibit scale-dependent variations in D, motivating examination of physical causes of such variations [Church and Mark, 1980; GoodchiM, 1980; Dutton, 1981; Lam and Quattrochi, 1992; Beauvais et al., 1994]. Efforts to evaluate potential connections between geomorphological process and scaling properties of river planforms are complicated by the many ways to calculate D. Some workers calculate the fractal dimension of river planforms from the relation between mainstream length and basin area [Hack, 1957]: L -= I3A • (1) where oe is the length of the river planform,/3 is a constant of proportionality, andA is the drainage area. Mandelbrot [1977], and later Church and Mark [1980] and Hjelmfelt [1988], interpreted the exponent a as being half the fractal dimension of the river planform (i.e., a -D/2). The relatively small range of a for most drainage basins (see data compiled by Montgomery and Dietrich [1992]) implies that D defined in this manner is equal to approximately 1.2 for rivers in general [Mandelbrot, 1983; Tarboron et al., 1988; Turcotte , 1992]. This implies that the D derived from length-area relations is not useful for examining differences among rivers. Furthermore, Robert and showed the unreliability of using a to infer the fractal dimension of rivers due to the effect of cartographic generalization.
Nature Communications
Since the 1950s river networks have been intensely researched in geosciences and hydrology. This led to the definition of scaling laws that described the organisation of landscapes under fluvial incision and were later explored by statistical physics and fractal mathematics. The emblematic Hack's Law proposes a power-law relationship between watershed area and main stream length. Though extensively documented, a wide range of values is still reported for Hack's parameters. Some authors associate this dispersion to local geologic and climatic conditions. Here based on the analysis of large sets of river basins in various climatic and geological settings, we confirm the geometric similarity of river networks. We demonstrate that basin shape is mostly related to Hack's coefficient and not to the exponent, independently of external forcing such as lithology and pluviometry.
International Journal of Geosciences, 2013
Complex nonlinear dynamic systems are ubiquitous in the landscapes and phenomena studied by earth sciences in general and by geomorphology in particular. Many natural landscape features have an aspect such as fractals. In the many geomorphologic phenomena such as river networks and coast lines this is visible. In recent years fractal geometry offers as an option for modeling river geometry and physical processes of rivers. The fractal dimension is a statistical quantity that indicates how a fractal scales with the shrink, the space occupied. This theory has the mathematical basis but also applied in geomorphology and also shown great success. Physical behavior of many natural processes as well as using fractal geometry is predictable relations. Behavior of complex natural phenomena as rivers has always been of interest to researchers. In this study using data basic maps, drainage networks map and Digital Elevation Model of the ground was prepared. Then applying the rules Horton-Strahler river network, fractal dimensions were calculated to examine the relationship between fractal dimension and some rivers geomorphic features were investigated. Results showed fractal dimension of watersheds have meaningful relations with factors such as shape form, area, bifurcation ratio and length ratio in the watersheds.
River network fractal geometry and its computer simulation
Water Resources Research, 1993
The hierarchical ordinal and statistical models of river networks are proposed. Their investigation has been carried out on the basis of river networks computer simulation as well as on empirical data analysis. The simulated river networks display self-similar behavior on small scales (the fractal dimension D-• 1.52 and Hurst's exponent H = 1.0) and self-affine behavior on large scales (the lacunary dimension D• • 1.71, H • 0.58). Similar behavior is also qualitatively characteristic for natural river networks (for catchment areas from 142 to 63,700 km 2 we obtained D • • 1.87 and H •-0.73). Thus in both cases one finds a region of scales with self-affine behavior (H < 1) and with Do < 2. Proceeding from fractal properties of the river networks, the theoretical basis of scaling relationships L-• A t• and 3?-A *, widely used in hydrology, are given (L, 37, and A denote the main river length, the total length of the river network, and catchment area, respectively); fi = 1/(1 + H) and, = Do/2. Nn as well as those of Panov [1948] and Schumm [1956]: An+l An = A iR,•-! R A =• (3) An and Zavoianu' s [ 1985] Pn+l Pn = PIR•-1 Rp = Pn (4) where L n, N n, A n, and P n denote the length, the number, the catchment area, and its perimeter for n-order streams, respectively. The following relationships for estimating the
Describing scales of features in river channels using fractal geometry concepts
Regulated Rivers: Research & Management, 2000
Quantitative description of spatial patterns is often at the heart of ecological research in aquatic systems, particularly for investigations of how biota respond to physical habitat. A common first step for approximating a river channel is tessellation, or the discretization of the channel into cells of approximately uniform size, and assigning each cell a representative value for velocity or other characteristics. More innovative methods may use the fractal dimension to characterize patterns of features in spatially complex geological structures, such as channel bed forms. Unfortunately, these methods lose information because they either force continuous data into a grid framework or assume that complexity is constant over a range of scales. The current understanding of aquatic processes would improve if information about the scale of channel features could be preserved throughout the analysis instead of being discarded in the first step because simplifying assumptions were used. New methods are presented that characterize complex spatial data sets with minimal use of assumptions or simplifying approximations. The new methods identify dominant features in a set of coordinate data, locate the positions of such features in the cross section, describe how kinetic energy is distributed in these features, and quantify how features of different scales relate to one another. The effectiveness of this technique on mathematical constructs having known characteristics is demonstrated. The methods are then used to describe a Missouri River cross section before and after river regulation to illustrate how the methods can be used to quantify changes in physical habitat patterns that may not be apparent using other methods. Improved description of complex shapes in aquatic environments may lead to increased understanding of aquatic processes in general, and in particular, the way aquatic organisms relate to physical habitat.
Fractal analyses of tree-like channel networks from digital elevation model data
Journal of Hydrology, 1996
Digital elevation models (DEMs) are generally used to automatically map the channel network and to delineate subbasins. The most common approach to extract a channel network from DEMs consists of specifying a threshold area S which is the minimum area required to drain to a point for a channel to form. This threshold area is usually specified arbitrarily, although it is recognized that different threshold areas will result in substantially different channel networks for the same basin. In this paper, we study the effect of S (that is also the scale of observation) on the morphometric properties (external and internal links, length of drainage paths, mainstream length) and s~ling properties (such as Horton's and Strahier's laws, and fractal dimension). Three basins, located in southern France, were extensively studied. The results indicate that morphometric properties vary considerably with S, and thus values reported without their associated S should be used in hydrologic analysis with caution. Then, the fractal geometry is used to take into account the dependence of measured values on observation scales, which is not possible with classical hydrological indexes. The use of fractals allows, first, to point out self-similarity in the structure of channel networks and then to quantify the tree-like organization. Hew catchment shape indexes, independent of the observation scale S, are defined. These indexes are useful for comparing catchments and for measuring the irregularity level of the channel network.
CHAPTER 1. Lithologic control on the scaling properties of the first order streams of drainage networks: A monofractal analysis ABSTRACT 1.1. INTRODUCTION 1.2. STUDY AREA OVERVIEW 1.3. MATERIALS AND METHODS 1.3.1. Data set 1.3.2. Box-counting method 1.4. RESULTS 1.4.1. Quantitative framework of fluvial geomorphology 1.4.2. Fractal dimension of the drainage systems 1.4.3. Fractal dimension of the first-order streams 1.4.4. Intercepts 1.4.5. Lacunarity 1.5. DISCUSSION 1.5.1. Suitability of the assumed scaling model 1.5.2. Properties of fluvial geomorphology 1.5.3. Fractal scaling parameters 1.6. CONCLUSIONS CHAPTER 2. Identification of bedrock lithology using fractal dimensions of drainage networks extracted from medium resolution LiDAR digital terrain models ABSTRACT 2.1. INTRODUCTION 2.2. MATERIALS AND METHODS 2.2.1. Study area 2.2.2. Data set 2.2.3. Method of drainage network extraction 2.2.4. Geomorphological approach to the lithological map units (LMUs) 2.2.5. Fractal analysis: Box-counting method 2.2.6. Statistical analysis 2.3. RESULTS Tesis doctoral de JOAQUÍN CÁMARA GAJATE _____________________________________________________________________________ II 2.3.1. Morphological properties of the lithological map units 2.3.2. Drainage density 2.3.3. Fractal box-counting dimension 2.3.4. LMUs results grouped by type of rock 2.3.5. Classification model of LMUs in different types of rock 2.4. DISCUSSION 2.5. CONCLUSIONS CHAPTER 3. Quantifying the relationship between drainage networks at hillslope scale and particle size distribution at pedon scale
On the multifractal characterization of river basins
Geomorphology, 1992
ljjasz-Vasquez, E.J., Rodriguez-Iturbe, I. and Bras, R.L., 1992. On the multifractal characterization of river basins. In: J.D. Phillips and W.H. Renwick (Editors), Geomorphic Systems. Geomorphology, 5: 297-310.