Definition of new equivalent indices of Horton-Strahler ratios for the derivation of the Geomorphological Instantaneous Unit Hydrograph

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1 Click Here for Full Article Definition of new equivalent indices of Horton-Strahler ratios for the derivation of the Geomorphological Instantaneous Unit Hydrograph Roger Moussa 1 WATER RESOURCES RESEARCH, VOL. 45, W09406, doi: /2008wr007330, 2009 Received 3 August 2008; revised 31 May 2009; accepted 10 June 2009; published 10 September [1] The Horton-Strahler concept has been important in river basin geomorphology; it describes scaling properties and identifies GIUH but has also attracted criticisms, because it depends on the threshold area S, which is used to extract the channel network from digital elevation models (DEM), on the position of the outlet and on the few number of ratios used to identify the scaling laws. To overcome these limitations, this paper proposes new indices independently of S and which have similar properties than the Horton-Strahler ratios. Applications were conducted on seven French basins. The methodology consists on analyzing the morphometric properties, such as the drained area, the number of sources, and the length of the channel network as a function of S. The results show that Horton-Strahler ratios vary considerably with S. Particularly, while the W-order length L W is a staircase function of S, results show that the channel network is articulated around two main nodes independent of S, which are descriptors of L W. Then, we compared the properties of Horton-Strahler laws to those obtained from self-similarity analysis, which enables to define new indices independent of S. These indices are calculated automatically from DEM, have similar properties as the Horton-Strahler ratios, and are used to calculate the GIUH independently of S. The four new descriptors R Be,R Le,R Ae, and L e, which have similar properties as R B,R L,R A, and L W respectively, are equivalent to those of Horton-Strahler s laws and can be used as indicators of hydrological similarity for catchment comparison and regionalization. Citation: Moussa, R. (2009), Definition of new equivalent indices of Horton-Strahler ratios for the derivation of the Geomorphological Instantaneous Unit Hydrograph, Water Resour. Res., 45, W09406, doi: /2008wr Introduction [2] The measurement and the quantitative expression of river networks has been the subject of interest to both geomorphologists and hydrologists since the mid-twentieth century. The quantitative study of channel networks architecture began with Horton s [1932, 1945] concept of classifying channels by order. Later on, Strahler [1952, 1957, 1964] revised Horton s scheme to avoid some ambiguities, and proposed a modified version of Horton s ordering scheme. Horton and Strahler classifications slightly differ. In the Horton classification, a channel of any order extends headward to the place the most distant tip ends, near the basin divide [Leopold et al., 1964], while Strahler restricted the designation of order to stream segments. Classification by segments is easier in that one can start at the upstream end and assign order numbers without any required change. Because of the ease of application, most workers use Strahler system rather than the original Horton scheme. However, disadvantages of the Strahler system, seldom acknowledged, arise from two sources. First, the delineation of a channel as a blue line on a topographic 1 INRA, UMR LISAH, 2 Place Pierre Viala, F Montpellier, France. Copyright 2009 by the American Geophysical Union /09/2008WR007330$09.00 W09406 map is an incomplete picture of the occurrence of channels on the ground. If this delineation is improved by adding definite channels to those shown as printed blue lines, then the Strahler ordering invariably leads to an unrealistically short length of the highest stream order. Second, the data on the number and length derived from the Strahler ordering system are sensitive to the inclusion or omission of ephemeral tributaries. The addition or omission of a minor channel can materially change the order designation of segments downstream. The Horton scheme is rather insensitive to such a minor change in the mapped network. However, in most common application in hydrogeomorphology, Strahler s method is generally preferred because of its simplicity [Smart, 1967, 1968; Eagleson, 1970], and is commonly noted Horton-Strahler ordering scheme as suggested by Rodríguez-Iturbe and Rinaldo [1997, p. 5]. [3] Horton-Strahler s ratios of stream ordering (the bifurcation ratio R B, the length ratio R L, and the area ratio R A ) are often used as indicators of hydrological similarity for catchment comparison and regionalization under the assumption that if catchment attributes are identical, one would expect the surface hydrologic response to be similar [Blöschl and Sivapalan, 1995; Aryal et al., 2002; Blöschl, 2005]. As mentioned by Kirchner [1993, 1994a, 1994b], most of these laws don t describe the details of the processes involved, because the stream order just provides a simple objective, and though not always consistent, measure 1of24

2 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 of size or scale. However, Horton-Strahler s laws were extensively used in geomorphological applications to classify river systems [e.g., Raff et al., 2003; Reis, 2006], to establish relations with the fractal nature of channel network as detailed by Rodríguez-Iturbe and Rinaldo [1997] [e.g., Beer and Borgas, 1993; La Barbera and Roth, 1994; Rodríguez-Iturbe et al., 1994; Tarboton, 1996], and to characterize scale properties [Claps et al., 1996; Peckham and Gupta, 1999; Veitzer and Gupta, 2000; Dodds, 2000; Dodds and Rothman, 1999, 2000, 2001a, 2001b, 2001c]. Following the pioneering work of Rodríguez-Iturbe and Valdés [1979], they were incorporated into the theoretical scheme of the Geomorphological Instantaneous Unit Hydrograph (GIUH) and largely used to link the hydrological response to the geomorphological features [Rodríguez- Iturbe and Valdés, 1979; Gupta et al., 1980; Rosso, 1984; Gupta et al., 1986; Gupta and Mesa, 1988; Bras and Rodríguez-Iturbe, 1989; Jin, 1992; Rinaldo et al., 1995; Rodríguez-Iturbe and Rinaldo, 1997; Gupta and Waymire, 1998; Saco and Kumar, 2002a, 2002b, 2004; Bhunya et al., 2003, 2004, 2007, 2008; Rodriguez et al., 2005; Kumar et al., 2007; Singh et al., 2007; Lee et al., 2008], to estimate peak discharge using GIUH [Sorman, 1995], to establish statistical scaling laws of mean annual discharge [De Vries et al., 1994] and peak flows [Gupta et al., 1996; Mantilla et al., 2006]. Horton-Strahler s ordering scheme were also applied in other domains, such as coding binary trees [Devroye and Kruszewski, 1994; Kruszewski, 1999; Zaliapin et al., 2006], or establishing hierarchical structures of cities [Chen and Zhou, 2008]. [4] However, since the sixties, Horton-Strahler ordering scheme has attracted criticisms because it is related to the map scale [Scheidegger, 1968a, 1968b; Smart, 1967, 1968, 1972]. Moreover, there is no agreement between geographers on how blue lines found on maps should be interpreted [Wharton, 1994; Hancock and Evans, 2006], because there are so many errors on the mapping of first-order streams, and because of the temporal evolution of the channel network sources as streams can move up and down valleys respectively during flood and drought periods [Dunne and Black, 1970]. Also, Horton-Strahler ordering is not always consistent when calculated automatically from digital elevation models (DEM), due especially to the difficulties in defining first-order streams [Kennedy, 1978, 1992; Helmlinger et al., 1993; Snell and Sivapalan, 1994; Puente and Castillo, 1996], and due to the extreme sensitivity of Horton-Strahler s laws to the few number of ratios used in the calculation, and the dependence on the position of the outlet. [5] Indeed, an important aspect of any channel network extraction algorithm from DEM is to decide where to begin the channels [Montgomery and Dietrich, 1988]. The earliest, simplest and commonly method used for specifying drainage directions in grid-based digital elevation models (DEMs) is the D8 approach which assign a pointer from each cell to one of its eight neighbors, in the direction of the steepest downward slope [O Callaghan and Mark, 1984; Marks et al., 1984]. The D8 approach has disadvantages arising from the discretization of flow into one of eight possible directions, separated by 45 [Fairfield and Leymarie, 1991; Quinn et al., 1991; Costa-Cabral and Burges, 1994; Tarboton, 1997]. To overcome these problems different alternative methods have been proposed in the literature. Fairfield and Leymarie [1991] assign a random flow direction to one of the downslope neighbors. Multiple flow direction methods have also been suggested [Freeman, 1991; Quinn et al., 1991]. The D8 method has also been improved by the D1 multiple flow direction method [Tarboton, 1997], the D8-LAD (Least angular deviation) and D8-LTD (Least transversal deviation) method [Orlandini et al., 2003]. All these methods mitigate some disadvantages of the D8 method but introduce new disadvantages as expressed by Tarboton [1997] and Orlandini et al. [2003]. In particular, multiple drainage directions produce numerical dispersion of area from a DEM cell to all neighboring cells with a lower elevation, which may be inconsistent with the physical definition of upstream drainage area. The D1 method constitutes a reasonable compromise between the simplicity of the D8 method and the sophistication introduced by randomly multiple flow direction methods; however, a certain degree of dispersion is maintained by the D1 method. In this respect, nondispersive methods using a single drainage direction (such as D8, D8-LAD and D8-LTD) appear preferable, because they are consistent with the physical definition of upstream drainage area, and this appears an essential requirement for the quantification of upstream releases (e.g., runoff, sediments, or pollutants from nonpoint sources) at a given section of the channel network. When using nondispersive methods, various approaches were developed to identify channel initiation. The most common one, consists of specifying a threshold area S, usually assumed constant, which is the minimum area required to drain to a point for a channel to form [e.g., Band, 1986; Jenson and Domingue, 1988; Tarboton et al., 1988, 1991; Montgomery and Dietrich, 1989]. The channel initiation problem has also been addressed in the literature by considering thresholds on the sp e quantity where s is the upstream specific drained area (upslope contributing area per unit contour width), p is the local slope and e is an exponent [Montgomery and Dietrich, 1988; Dietrich and Dunne, 1993; Istanbulluoglu et al., 2002], or on the normalized divergence [Howard, 1994]. The threshold (i.e., S, sp e, etc.) for channel initiation represents a scale of observation of the basin; it is recognized that different thresholds will result in substantially different channel networks for the same basin [e.g., Helmlinger et al., 1993; Snell and Sivapalan, 1994; Moussa and Bocquillon, 1996] and consequently the order of individual streams, and the characteristics of Horton-Strahler laws, depend on the threshold (i.e., for the threshold area S, see Kennedy [1978, 1992], Puente and Castillo [1996], and Da Ros and Borga [1997]). Another problem in the calculation of Horton-Strahler s laws concerns the basin order, because consideration of the whole basin as a mature catchment of order W is often misleading. What often happens is that the basin outlet is little downstream of a junction of two subbasins of order W 1. In these cases the basin is not really representative of a fully developed network of order W, and this affects the value of W-length and W-area order, and consequently the estimates of Horton-Strahler ratios [Claps and Oliveto, 1996]. The last problem concerns the determination of the ratios R B,R L and R A, because the average configurations of the natural basins analyzed in literature seldom have Horton orders greater than 5 [Claps 2of24

3 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 Figure 1. The seven catchments (Tech, Têt, Agly, Aude, Orb, Hérault, and Vidourle) used for applications, located in southern France and having their outlets in the Mediterranean Sea (the coordinates of the left bottom corner are: latitude N and longitude: E). and Oliveto, 1996]; this leads to just a few numbers of ratios with discontinuities while the drainage network is a continuous set. [6] Consequently, the morphometric properties and the GIUH are strongly affected by threshold (i.e., S, sp e, etc.) selection to the point that reporting these values without reference to the threshold area used is meaningless. The analysis of Horton-Starhler properties as a function of the threshold enable to quantify the effect of uncertainty in the selection of the channel initiation scale on morphometric properties and GIUH derivation. Since DEM are more and more used in hydrologic applications, efforts should be directed to resolve the problem of channel initiation, which can be done by defining new morphometric descriptors independent of the threshold value. To overcome these limitations, this paper proposes new indices, independent of the value of the threshold, that have similar properties than the Horton-Strahler ratios, and which can be used for the calculation of the GIUH. The paper is structured into four sections: (1) Properties of the Horton-Strahler ordering scheme; (2) Self-similarity properties of channel networks; (3) Definition of new morphometric indices verifying Horton- Strahler laws properties on the basis of self-similarity properties; (4) Application case for hydrological modeling of flood events. 2. Properties of Horton-Strahler Ordering Scheme [7] Seven catchments located in southern France and having their outlets in the Mediterranean Sea, were studied: 3of24 the Tech, the Têt, the Agly, the Aude, the Orb, the Hérault and the Vidourle (Figure 1). The DEM data are in the form of the French National Geographic Institute (IGN, Institut Géographique National) blocks with a grid resolution of 50 m. Altitudes range between 0 (all outlets are at the sea level) and 2832 m, and the catchment areas S 0 range between 738 (Vidourle) and 5346 km 2 (Aude) Horton-Strahler Laws [8] Figure 2a shows an example of a channel network. Sources are points furthest upstream, and a point at which two upstream channels join to form one downstream channel is called a junction or node. Exterior links are the segments of channel between a source and a node, and interior links are the segments of channel between two successive nodes or a node and the outlet. Each link has certain properties: length along the stream and the contributing area, which is the total area draining through the link measured at the downstream. Streams are defined as segments of the channel network, which are composed of continuous links of the same order. Horton-Strahler classification can be defined in terms of a recursive rule as follows. [9] The iterative procedure starts with the channel network of Figure 2a (step = 0). The first step (Figure 2b; step = 1) consists in pruning away all exterior streams and their downstream nodes in Figure 2a, and classify them as order w = 1. This leads to a new, more structurally coarse tree that has its own set of exterior streams (Figure 2b). At the second step = 2 (Figure 2c), we prune away all external streams, and all streams pruned are referred as order w = 2.

4 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 Figure 2. Horton-Strahler ordering scheme. (a) Basic network. (b) Created by, removing all source streams from the network in Figure 2a; these are denoted as first-order streams. The new source streams in the pruned network of Figure 2b are labeled as second-order streams and are themselves removed to give Figure 2c, a third-order stream. If we continue to prune in this way, the tree will disappear entirely after a finite number of iterations. We will refer to the streams that get pruned in the w th iteration as the order w. Finally, the order of the basin is the highest stream order W. The quantitative expressions of Horton-Strahler s laws are summarized below: [10] 1. Law of stream numbers [11] 2. Law of stream lengths [12] 3. Law of stream areas N w 1 N w ¼ R B ð1þ L w 1 L w ¼ 1 R L ð2þ A w 1 A w ¼ 1 R A where N w is the number of streams of order w, L w is the mean length of streams of order w, and A w is the mean area contributing to streams of order w and its tributary (1 w W). The values of the parameters R B,R L and R A are generally determined by plotting, N w,l w and A w versus w on semilog diagram and determining the best fit straight lines by least squares analysis; the slopes of the lines are then log R B,logR L and, respectively. The equations of the straight lines adjusted on semilog diagram for N w (w), L w (w) and A w (w) are ð3þ where N*, L* and are parameters to be adjusted. They correspond respectively to the values of the adjusted equations (4), (5) and (6) for w = W. In the case of ideal Hortonian channel network, the Horton-Strahler laws (equations (1), (2) and (3)) are verified for all streams order w, and the measured values of N w,l w and A w fits on the straight lines of equations (4), (5) and (6); consequently N* = 1, L* = L W (L W being the measured value of the W-order stream) and = S 0. [13] Shreve [1966] showed that in a topologically random population of networks with a given number of sources, the most probable network order W is that which makes R B closest to 4. In the literature, values of R B of natural channel networks typically range between 2.5 and 5.0, values of R L between 1.5 and 3.5, and values of R A between 3.0 and 6.0 [Smart, 1968; Rodríguez-Iturbe and Rinaldo, 1997]. The Horton-Strahler ratios were used to characterize fractal properties of channel networks [La Barbera and Rosso, 1989; Tarboton et al., 1988] and to calculate the GIUH [Rodríguez-Iturbe and Valdés, 1979; Rosso, 1984; Rodríguez- Iturbe and Rinaldo, 1997] GIUHðÞ¼ t t a 1 e t k k kgðþ a R A and k ¼ 0:70 R B R L with a ¼ 3:29 R 0:78 B R 0:07 0:48 L W v where v is the velocity of the flow and G(x) is the Gamma function of argument x. Consequently, the GIUH can be expressed as a function of four parameters derived from Horton-Strahler s laws: R B,R L,R A and L W. R A L ð7þ N w ¼ N* R W w B ð4þ 1 W w L w ¼ L* ð5þ R L 1 W w A w ¼ ð6þ R A 4of Properties of Horton-Strahler Laws as a Function of the Method Used to Extract the Channel Network From DEM [14] The flow directions are identified from DEM using the TraPhyC-BV code [Moussa and Bocquillon, 1994] using nondispersive methods; three methods are available D8, D8-LAD and D8-LTD. Two data sets are generated, the distance from each pixel to the outlet and the upstream drained area on each pixel. Then, the channel network is extracted by identifying pixels draining more than a spec-

5 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 ified upstream threshold area S [Band, 1986; Tarboton et al., 1988; Montgomery and Dietrich, 1989] or by considering a threshold on the sp e quantity [Montgomery and Dietrich, 1988; Dietrich and Dunne, 1993; Istanbulluoglu et al., 2002]. Tarboton et al. [1992] and Montgomery and Foufoula-Georgiou [1993] had discussed in detail the identification of the threshold area S which ranges between 0.5 and 5 km 2 for catchments located in southern France as given by Moussa [1991]. Montgomery and Dietrich [1992] and Dietrich and Dunne [1993] showed that for data from channel heads observed in the Tennessee valley, the exponent e varies between 1 and 2 and that all the channel heads observed are captured between two topographic threshold lines with sp e values of 25 m < sp e < 200 m. This reveals a factor of eight to ten variation in the contributing area sizes required for channel initiation which can be due to both spatial and temporal variation in the hydrologic and erosional processes governing channel initiation [Montgomery and Dietrich, 1992]. As the two methods D8-LAD and D8- LTD gave very comparable results, this section compares the drainage network extracted from DEM using D8 and D8-LAD methods, and then the channel network obtained using a threshold S or sp e. Then, we analyze the properties of Horton-Strahler ordering scheme function of the threshold value. [15] Figure 3 shows for the Hérault catchment the channel network topology obtained when using the D8 method for three values of S (0.5, 1 and 2 km 2 ). Figure 3 shows also the measured values (points) and the adjusted laws (dashed lines) N w (w), L w (w) and A w (w) of equations (4), (5) and (6) for the three values of S (NB: the position of the point M on Figure 3 will be explained later in this section). For each of the three values of S, the six parameters (N*, L*,, R B, R L and R A ) of the three equations (equations (4), (5) and (6)) were adjusted using the least squares method (Table 1). Results show that all six parameters depend on the value of S. As the adjusted lines in Figure 3 doesn t necessary fit within the measured points, the value of N* is not equal to 1, the value of L* differ also from the measured value L W (equal to 103.1, 47.5 and km respectively for the three values of S) and the value differ from the catchment area S 0 (2617 km 2 ). [16] Table 1 compares the results obtained using the D8 and D8-LTD methods to define the drainage direction, and then the extraction of the channel network obtained using a threshold S or sp e. We observe that both D8 and D8-LTD methods gave different topologies of the channel network, and consequently different values of the six parameters (N*, L*,, R B,R L and R A ). The value of the six parameters differs also function of the value of the threshold used to extract the channel network. These results show that the Horton-Strahler ratios depend on the method used to define the drainage direction and the threshold used to extract the channel network from DEM. This paper doesn t aim to improve the methods of channel network extraction from DEM, neither to compare nondispersive and dispersive methods, but aims to analyze the sensitivity of the Horton- Strahler ratios (R B,R L,R A ) function of the method used to extract the channel network. In the following, we use the D8 method and a constant threshold area S to extract the channel network from DEM, which remain the most simplest and the most commonly used algorithms. The same methodology can be applied to D8-LAD, D8-LTD and other threshold criteria. [17] Figures 4, 5 and 6 show for the studied basins, the relationships between the threshold area S for various values of 0.5 km 2 <S<5km 2, and the morphometric properties that enter into the calculation of Horton-Strahler ratios. Figure 4 shows that when S increases: (1) the number of links decreases (Figure 4a) and consequently, the total length of the channel network (T) and the drainage density (T/S 0 ) also decrease because the channel network is shortest; (2) the basin order decreases (Figure 4b) as a staircase function (i.e., for the Hérault, W = 6 for 0.5 < S < 1.83 km 2 and W =5for2<S<5km 2 ); (3) the mean length of the first order streams (L 1 ) generally increases (Figure 4c); however this is not always the case as for the Tech and the Vidourle for4<s<5km 2. [18] Figure 5 shows that the W-order stream length (noted L W and represented by black points on Figure 5) takes only fixed values, and is a discontinuous function of S (NB: the equivalent parameter L e on Figure 5 will be discussed later in section 4.5). No simple relationships can be established between L W and S. In the case of the Hérault in Figure 3, L W is equal to the path flow OM (O being the outlet and M the internal node), and is equal to km for S = 0.5 km 2, to km for S = 1 km 2, and once again to km for S=2km 2. We observe that the location of M on the channel network changes drastically when S varies slightly between 0.5 and 1 km 2. Consequently, this result shows the extreme sensitivity of L W on the uncertainty on the value of S and on the position of source nodes. This result is also observed for all seven catchments when S varies between 0.5 and 5 km 2. Figure 5 shows that L W takes a limited number of values for each catchment: only four values for the Tech (L W = 22.68, 50.03, and km), three values for the Têt (L W = 76.50, and km), two values for the Agly (L W = and km), three values for the Aude (L W = 27.48, and km), four values for the Orb (L W = 25.72, 60.93, and km), two values for the Hérault (L W = and km), and three values for the Vidourle (L W = 45.05, and km). Results show also that the value of the constant L* of equation (5) (solid line on Figure 5) differs from L W but fluctuates around values of the same order of L W for the Aude, Orb and Hérault, inferior than L W for the Tech, Têt and Vidourle, and superior than L W for the Agly. These results show that both L W and L* vary enormously as a function of S and are very sensitive to the value of S. While L W is a discontinuous function of S, L* is a continuous function of S. A small uncertainty on S can lead to an extreme variability on L W of more than 300% (i.e., for the Aude, when S varies from 0.6 to 0.7 km 2,L W varies from 27.5 to km; for the Orb, when S varies from 1.3 to 1.6 km 2,L W varies from 25.7 to 92.8 km). Similar results are also observed for L*. As L W enters in the calculation of R L (equation (5)) and the GIUH (equation (7)), both the value of R L and the shape of the GIUH, are expected to be very sensitive to the value of S. [19] Figure 6 shows the relationships between S and the remaining five parameters of Horton-Strahler equations N*,, R B,R L and R A (NB: The equivalent parameter R Be,R Le and R Ae on Figure 6 will be discussed later in sections 4.3 and 4.4). The value of N* fluctuates around a value equal to 1 (Figure 6a) and the value of generally fluctuates 5of24

6 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 Figure 3. Example of the channel network of the Hérault extracted using the D8 method for three values of the drained area S (0.5, 1, and 2 km 2 ) and position of the point M on the channel network, such as OM = L W. The three Horton-Strahler s laws N(w), L(w), and A(w), adjusted in a semilog plot, are shown for each value of S. 6of24

7 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 Table 1. Main Characteristics of the Horton-Strahler Laws Adjusted on the Hérault Catchment for Two Methods (D8 and D8-LTD) and Three Different Values of the Threshold Area S = 0.5, 1, and 2 km 2 or the Threshold sp e = 50, 100, and 200 m a Method Threshold L W (km) N* L* (km) (km 2 ) R B R L R A D8 S = 0.5 km D8 S = 1.0 km D8 S = 2.0 km D8-LTD S = 0.5 km D8-LTD S = 1.0 km D8-LTD S = 2.0 km D8 sp e = 50 m D8 sp e = 100 m D8 sp e = 200 m D8-LTD sp e = 50 m D8-LTD sp e = 100 m D8-LTD sp e = 200 m a Wherein S is the upstream drained area in km 2, s is the specific upstream catchment area (upslope contributing area per unit contour width), p is the local slope, and e is an exponent considered herein equal to 1. Figure 3 shows the example of the D8 method; S = 0.5, 1, and 2 km 2. values of the four parameters R B,R L,R A and L W can be compensated when calculating the GIUH. We observe that for the Têt, Agly and the Vidourle, all calculated GIUH are very similar for all ranges of S, because for both catchments the parameters R B,R L,R A and L W (Figures 5 and 6) vary very slightly with S in comparison to the remaining five catchments. However, for the Tech, Aude, Orb and Hérault, we observe for each catchment two groups of GIUH (Figure 7): the first group corresponds to a short time of occurrence of peak flow and a high value of the peak flow, and the second corresponds to a long time of occurrence of peak flow and a low value of the peak flow. The first group corresponds in general to the cases where L W is small (Figure 5) as for the examples of the Tech for S = 4 km 2 and L W = 22.7 km, of the Aude for S = 0.6 km 2 and L W = 27.5 km, of the Orb for S=1or1.3km 2 and L W = 25.7 km, and for the Hérault for 0.7 < S < 1.9 or 3.5 < S < 5 km 2 and L W = 47.5 km. In the opposite, the second group corresponds to the cases with high values of L W. We observe also a common trend in the shape of the hydrograph (i.e., peakflow, time of occurrence of peakflow) as a function of the threshold area S. For the range of S when the basin order is constant (i.e., for the Hérault, W = 6 for 0.5 < S < 1.83 km 2 or W =5for2<S<5km 2 ), around a value equal to the catchment area S 0 (Figure 6b). For all seven basins, and for all ranges of 0.5 < S < 5 km 2, the values of R B ranges mainly between 3.5 and 4.5, R L ranges mainly between 2 and 3, and R A ranges mainly between 3.5 and 7. These values are of the same order of those given in the literature [e.g., Smart, 1968; Rodríguez- Iturbe and Rinaldo, 1997]. However, there is no clear relationship between S and R B,R L, and R A (respectively Figures 6c, 6d and 6e). We observe that for the range of S where the basin order is constant (i.e., for the Hérault, W =6 for 0.5 < S < 1.83 km 2 ), all three variables R B,R L,R A decrease when S increases (in the example of the Hérault, we observe also similar results for W =5for2<S<5km 2 ). In the opposite, when the basin order decreases (i.e., for the Hérault, when S increases from 1.83 to 2 km 2, the basin order W decreases from 6 to 5), the three variables R B,R L and R A increase. These relationships between the threshold area S and the bifurcation, area and length ratios of Figure 6 are similar to those obtained by Snell and Sivapalan [1994, Figure 8, p. 2318] on the Conjurunup catchment in Australia. We observe similar range of values of R B,R L and R A and similar fluctuation as a function of S, and the basin order W. As for the results obtained for L W and L* above, all five parameters N*,, R B,R L and R A vary enormously as a function of S and are very sensitive to the value of S (e.g., R B varies between 4 and 6 for the Tech, R L varies between 2 and 4 for the Tech, Têt and Orb, and R A varies between 4 and 6 for the Tech, Têt and Orb) GIUH as a Function of the Threshold Area S [20] The results presented above show that N*, L*,, R B,R L,R A and L W depend on the value of the threshold S. The variation of the GIUH (calculated from equation (7) using as parameters R B,R L,R A and L W ) as a function of S was then investigated. Figure 7 depicts the computed GIUH for the seven studied basins with different values of 0.5 km 2 < S<5km 2 (NB: The GIUH e (bold line) on Figure 7 will be discussed later in section 4.5). Note that the variability of the 7of24 Figure 4. Relationships between the threshold area S and the number of links, the basin order (W), and the mean length of first-order streams (L 1 ) for the seven catchments.

8 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 Figure 5. Relationships between the threshold area S, the length of W-order stream L W (points), and the parameter L* (solid line) for the seven catchments. The values of the two indices of the self-similarity properties (OE = m 1 and OI 1 ), and the value of the calculated equivalent distance L* e are also indicated. and when the value of L W remains constant, we observe that when S increases, the peakflow decreases and the lag time increases. However, as the value of L W may vary as a function of S, on the interval of S when W remains constant (Figure 5), no general relationship can be established between the shape of the GIUH and S. These results stress the strong influence of the threshold area selection S on the GIUH showing the high variability that sometimes occurs in the passage from a threshold area value to immediately successive area considered. Again, this effect is mainly related to the lack of completeness of channel networks and the strong variability of R B,R L,R A and L W function of S. The next section aims to analyze the self-similarity properties of the channel network, and then to compare these properties to those of Horton-Strahler laws in order to deduce new descriptors independent of S, and equivalent to R B,R L,R A and L W. 3. Self-Similarity Properties of Channel Networks [21] Herein, we use the D8 method, and the channel network data set is derived from the flow accumulation data set by selecting only the pixels with values greater than a threshold S (0.5 < S < 5 km 2 in Southern France). This technique can be extrapolated for higher values of S in order to analyze the morphometric properties of the channel network as a function of the upstream drained area on each pixel. The aim of this section is to analyze self-similarity properties of the channel network for various values of S < S 0. Two morphometric characteristics are analyzed at each threshold S: the total number of source nodes n(s) and the total length of channel network T(S). Note that the same approach can be applied if another method than D8 is applied, or another threshold used (i.e., sp e ) Effects of the Threshold Area S on the Topology of the Channel Network [22] Figure 8 shows for the Hérault, the evolution of the channel network topology for various values of S < S 0. For S=S 0 (Figure 8a), the channel network doesn t exist and there are neither internal nor external nodes. When S decreases and when S 1 <S<S 0 (Figure 8b), the channel network is represented by only one reach, and has only one source node (E). We define the threshold S 1 such as the total number of source nodes varies from n = 1 (for S = S 1 + e, 8of24 the term e being a small number; Figure 8b) to n = 2 (for S = S 1 ; Figure 8c). For S = S 1 the first internal node (I 1 )ofthe channel network appears, and the basin has only one source node in E. S 1 is defined as S 1 ¼ maxðs for 0 S S 0 Þ such as nðs 1 Þ ¼ 2 ð8þ [23] The position of the nodes E and I 1, characterized respectively by the path flows OE and OI 1, and the area S 1 are descriptors of the channel network independent of S. [24] For the studied basins, Figure 5 compares the values of L W (which depend on the value of S), to the two new indices OE and OI 1. Let L W,max and L W,min be respectively the maximum and the minimum value of L W for 0.5 < S < 5km 2 on Figure 5. We observe that OE is equal to L W,max for the Tech (L W,max = OE = km) and the Orb (L W,max = OE = km). However, OE is slightly superior to L W,max for the Têt (OE = km and L W,max = km), the Agly (OE = km and L W,max = km), the Hérault (OE = km and L W,max = km), and the Vidourle (OE = km and L W,max = km); OE is slightly inferior to L W,max for the Aude (OE = km and L W,max = km). In all cases, the value of OE is very close to L W,max with an error (OE L W,max )/OE less than 3% for five of the seven basins, and less than 14% for the two remaining basins. We observe also that OI 1 is equal to L W,min for the Tech (OI 1 =L W,min = 22.7 km), Têt (OI 1 = L W,min = 76.5 km), Agly (OI 1 =L W,min = 32.0 km), Hérault (OI 1 =L W,min = 47.6 km) and Vidourle (OI 1 =L W,min = 45.1 km). However, OI 1 is superior than L W,min for the Aude (OI 1 = 84.5 km and L W,min =27.5km)andOrb(OI 1 = 60.9 km and L W,min =25.7km). [25] Figure 9 synthesizes these results and shows the correlations between OE and L W,max, and between OI 1 and L W,min. The descriptor OE is of the same order as L W,max for all seven catchments, and OI 1 is of the same order as L W,min for five catchments. However, L W and L* are highly correlated and are of the same order for a given catchment (Figure 5), with L W = L* for ideal Hortonian channel networks. While L W and L* varies drastically as a function of the threshold S, the two new indices OE and OI 1 are descriptors of the length of the W-order channel (L W and L*) independently of the threshold area S.

9 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 Figure 6. Relationships between the threshold area S and (1) the parameter N* (solid line) to be compared to the value of 1 for ideal Hortonian network, (2) the parameter (solid line) to be compared to the catchment area S 0 for ideal Hortonian network, and the Horton-Starhler s parameters R B,R L, and R A (solid lines) to be compared, respectively, to the equivalent parameters R Be,R Le, and R Ae Relationships n(s) and T(S) [26] When S decreases (S = 256, 32 and 4 km 2 ; Figures 8d, 8e and 8f), the number of source nodes increases and the total length of the channel network increases. The points on Figure 10 show for the Hérault basin, the calculated variables (n and T) from the DEM represented as a function of S/S 0. We observe a linear relationships in a log-log diagram of n(s) and T(S) for S < S 1. As such, we define a simple empirical relation n(s) where a and l are two parameters to be adjusted; they describe the distribution of external nodes. We can define a similar relationship for T(S). However, in order to reduce the number of parameters, Moussa [2008a, 2008b] established a simple relationship T(S) for S < S 1 TS ð Þ ¼ m 1 þ bs 1 2 S aþ bs 2 0 ð10þ S 1 ns ð Þ ¼ l S a ð9þ S 0 where b is a parameter to be adjusted, and m 1 = OE and S 1 are basin characteristics calculated from the DEM. Equation (9) 9of24

10 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 Figure 7. The GIUHs (dotted lines) calculated from equation (7) (for v = 1 m s 1 ) for various values of the parameters R B,R L,R A, and L W corresponding to various values of the threshold area 0.5 < S < 5 km 2 and the GIUH e (bold line) calculated for the equivalent parameters R Be,R Le,R Ae, and L* e (from Table 4). 10 of 24

11 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 Figure 8. Example of the channel network of the Hérault with pixels draining an area superior than a threshold S (for 4 km 2 S S 0 ) and position of the external node E and the internal node I 1 for the particular case when S = S 1. verifies the continuity for T(S 1 )=m 1. Note that b can be negative if S aþ 1 2: T < m 1 [27] Table 2 gives the values of S 1,s 1 =S 1 /S 0,m 1 = OE, and OI 1 calculated from the DEM, and the calculated values of a, l and b of the empirical equations (9) and (10) using S 1 the least squares method. Figure 10 (dashed lines) shows the adequacy of the simulated curves n(s) and T(S) for S < S 1 for the Hérault with a correlation coefficient R 2 = For all seven catchments, the adequacy of the simulated curves n(s) and T(S) for S < S 1 gives R 2 >0.97[Moussa, 2008a, 2008b]. The parameter a ranges between 0.91 and 1, the parameter l ranges between 0.25 and 0.42, and the parameter b ranges between 0.39 and All five parameters (S 1,m 1, l, a and b) of equations (9) and (10) are catchment Figure 9. Relationships between OE and L W,max = max (L W for 0.5 < S < 5 km 2 ) and between OI 1 and L W,min = min (L W for 0.5 < S < 5 km 2 ) for the seven catchments. 11 of 24

12 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 Figure 10. Relationships between the total number of source nodes n, the total length of the channel network T, and the threshold area S for the Hérault catchment (solid line calculated from DEM; dotted line simulated). shape descriptors independent of the scale of observation S, and can be automatically calculated from DEM. These descriptors allow the definition of simple mathematical relationships between the total number of extremities or source basins, the total length of the channel network, and the threshold area S. 4. New Descriptors Independent of S and Verifying Horton-Strahler Laws Properties [28] To overcome the dependence of Horton-Strahler law descriptors N*, L*,, R B,R L,R A and L W as a function of S, this section aims to define new morphometric descriptors independent of S, on the basis of the self-similarity properties established in the previous section. For the calculation of a GIUH independent of S, we define new equivalent indices noted R Be,R Le,R Ae and L e (the index e refers to equivalent ), independent of S, and which have similar morphometric properties respectively as R B,R L,R A and L W. In this section, we first analyze the scaling properties of Horton-Strahler s laws, then we compare the Horton-Strahler s properties to those obtained from the self-similarity approach, and finally we calculate the equivalent descriptors by analogy between the two approaches Scaling Properties of Horton-Strahler Laws [29] The Horton-Strahler ordering scheme allows decomposing the channel network into a discrete set of scales. For example, the channel network represented in Figures 2 and 11 has an order W = 3, and each step corresponds to a scale of observation or level of details of the channel network. In this section, the index notation HS refers to Horton-Strahler laws hypotheses. As for the previous section 3.1, we characterize each scale of observation of Figures 2 and 11 by the value of the threshold S = S HS,the number of sources n HS (S HS ) and the total length of the channel network T HS (S HS ). [30] First, for step = 0 (Figure 11), let A 0 be the value of the threshold area S used to extract the channel network from the DEM. The scale of observation is S HS =A 0 and the total number of source basins is n HS (S HS )=n HS (A 0 )=N 1 = 12 of 24 n(a 0 ), where N 1 is the number of Horton-Strahler 1st-order streams and n(a 0 ) the value of the function n(s) in equation (9) for S = A 0. Appendix A gives the total length of the channel network T HS ða 0 Þ ¼ XW k¼1 2 1 R W 3 B 6 R N k L k ¼ 4 L 1 R B R L 7 5N*L* ð11þ For S HS =A 0, we have T HS (A 0 )=T(A 0 ). If R B /R L 1, as is most often the case in river channel networks, the series diverges; for large W we get T HS (S HS )=T HS (A 0 ) / R B W 1 R L. [31] Second, for step = 1 (Figure 11), all first order streams are pruned, and consequently the channel network observed corresponds to pixels draining an area superior than S HS 2A 1 (Figure 11) where A 1 is the mean area drained by 1st-order Horton-Strahler streams. The value of the threshold S HS is estimated as 2A 1 because at step = 1, all source nodes correspond to upstream nodes of 2nd-order Horton-Strahler streams which generally drains two 1st-order streams. Consequently, the number of sources n HS (S HS )= n HS (A 1 )=N 2, where N 2 is the number of Horton-Strahler 2nd-order streams, and the total length of channel network T HS (S HS )= PW N k L k. k¼2 Table 2. Main Characteristics of the Studied Channel Networks, Calculated From DEM (S 0,S 1,s 1,m 1 = OE and OI 1 ), and Adjusted Parameters (l, a, and b) of the Laws n(s) and T(S) Basin S 0 (km 2 ) S 1 (km 2 ) s 1 =S 1 /S 0 m 1 =OE (km) OI 1 (km) l a b Tech Têt Agly Aude Orb Hérault Vidourle

13 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 Figure 11. Channel network observed at different scales of observation using Horton-Strahler ordering scheme (S HS =A 0,2A 1,2A 2,...,2A W 1 ). [32] Generally, for a step w, all 1st, 2nd,... and w th order are pruned; the channel network is observed for pixels draining an area superior to S HS with (Appendix A) 1 W w S HS 2A w ¼ 2 for 1 w W 1 ð12þ R A The value of the threshold S HS is approximated to 2A w because at a step w, all source nodes correspond to upstream nodes of (w + 1)-order Horton-Strahler streams which generally drains two w-order streams. The total number of source nodes n HS (S HS ) is equal to the number of streams of Horton-Strahler order w + 1 n HS ðs HS Þ ¼ n HS ð2a w Þ ¼ N wþ1 ð13þ The total length of the channel network T HS (S HS ) is equal to the total lengths of streams of order (w + 1), (w + 2),..., (W 1) and W T HS ðs HS Þ ¼ T HS ð2a w Þ ¼ XW N k L k k¼wþ1 ð14þ [33] Note that the two functions n HS (S HS ) and T HS (S HS ) are discontinuous functions, and are defined for S = A 0 and for S = 2A w (with 1 w W 1). To express Horton- Strahler s laws independently of the channel order w, the two functions n HS (S HS ) and T HS (S HS ) are expressed as a function of the average area draining w-order stream (S HS ). Appendix A gives the relationships n HS (S HS ) and T HS (S HS ) with S HS =2A w n HS ðs HS Þ ¼ N* 1 R B 2 log R B S HS log R B ð15þ 13 of 24 T HS ðs HS " Þ ¼ N*L* 1 1 log R L log R B # S log R L log R B log R HS A 1 RB 2 R L for 1 w W 1 ð16þ For the particular case when w = 0, T HS (A 0 ) corresponds to the total length of the channel network (equation (11)). [34] Horton-Strahler s laws (1), (2) and (3) or the equivalent relations (15) and (16) expressed with the total number of source points n HS (S HS ) and the total length of channel network T HS (S HS ) at a given w-order scale of observation, are scaling laws that relate properties at small scale (low stream order) to properties at large scale (high stream order). They characterize scaling at scales larger than the basic and may apply down to infinitesimally small scale if the notion of a lower bound basic scale is rejected. The Horton- Strahler analysis may be regarded as an attempt to give a quantitative description of drainage basin composition in terms of six parameters: R B,R L,R A, N*, L* and. Once these parameters are known, approximate values for other geomorphic parameters, such as the total number of source basins n HS (S HS ) and the total stream length T HS (S HS ) can easily be obtained Comparison Between the Horton-Strahler and the Self-Similarity Approaches [35] Two approaches to describe the channel network were compared. The first one is based on the Horton- Strahler ordering scheme (section 4.1) and the second based on self-similarity properties (section 3.2). Table 3 summarizes and compares the major hypotheses and the analytical relationships between the number of source nodes n HS (S HS ) for the first approach and n(s) for the second approach, and between the total length of the channel network T HS (S HS ) for the first approach and T(S) for the second approach. The

14 W09406 MOUSSA: EQUIVALENT INDICES OF HORTON-STRAHLER RATIOS W09406 first approach is based on equations (15) and (16) as a function of six parameters R B,R L,R A, N*, L* and while the second approach is based on equations (9) and (10) as a function of also six parameters a, l,b,m 1,S 1 and S 0. Under the hypotheses, that the thresholds areas S HS and S of the two approaches are equivalent, that the value of is close to the catchment area S 0, and that the value of N* is close to 1, the two functions n HS (S HS ) and T HS (S HS ) have similar mathematical expression respectively as n(s) and T(S). The differences between the two approaches are that n HS (S HS ) and T HS (S HS ) are discontinuous functions defined only for W values of S HS =A 0,2A 1,2A 2,... 2A W 1, while n(s) and T(S) are continuous functions defined for all values of S<S 1. Note also that the six parameters R B,R L,R A, N*, L* and of the first approach depend on the value of the threshold S used to extract the channel network, and are calculated using very few number of ratios (in general 4 to 7), while the six parameters a, l,b,m 1,S 1 and S 0 of the second approach are calculated independently of S using a large number of points Definition of New Equivalent Descriptors R Be,R Le, R Ae, and L e Independent of S [36] By analogy, we can compare the terms of equations (9) and (15), and the terms of equations (10) and (16). Let R Be,R Le,R Ae and L e be respectively the values of R B,R L, R A and L* in equations (15) and (16) such as equation (9) is equal to equations (15) and (10) equal to equation (16) (see Table 3). Under the assumption that S = S HS, we have when comparing the exponents of S in equations (9) and (15) log R Be e ¼ a ) R Be ¼ R a Ae ð17þ [37] When comparing the exponents of S in equations (10) and (16), we have [40] When comparing the multiplicative coefficient of S in equations (11) and (16), we have m 1 þ bs 0:5 0 S aþ0:5 1 ¼ N*L e* 1 log R Le log R Be e R Be R 1 2 Le ð21þ [41] The analytical expressions of n(s) and n HS (S HS )are equivalent without any conditions, while the equivalence between T(S) and T HS (S HS ) is conditioned by the equality of the constant ( bs ) of equation (10) and the constant N*L* of equation (16). The term ( bs R B 0 ) of RL equation (10) ranges between 1.49 and 0.40 S with a mean value of 0.68 S (Table 2) and the term N*L* in 1 R B RL equation (16) can be approximated to ( L*) under the assumption that N* 1 and R B 2R L as shown in Figure 6. The two approximated terms ( 0.68 S ) and ( L*) are of the same order. Moreover, when S and S HS tends to A 0 (or to zero) in equations (10) and (15) respectively, both ( 0.68 S ) and ( L*) can be neglected in comparison to the other terms in equations (10) and (15) respectively. Consequently, the equality between equations (10) and (15) needs only the equality of both the exponents and the multiplicative coefficients of S. [42] From the analysis above, and by identifying the exponents and the multiplicative coefficients of the variable S such as equation (9) is equal to equation (15), and equation (10) equal to equation (16), we had established four relations (17), (19), (20) and (21). However, the Horton-Strahler approach has six parameters (R B,R L,R A, N*, L* and ) and the self-similarity approach has also six parameters (a, l, b,m 1,S 1 and S 0 ). In order to calculate the equivalent parameters, we need to add two further conditions. We choose the two hypotheses that are generally verified on Figure 6: the first assumption is that is equal to the catchment area S 0, and the second assumption is that N* = 1. By combining equations (20) and (17), we have a þ 0:5 ¼ log R Le log R Be e ð18þ R Be ¼ 2a l ð22þ [38] By combining equations (17) and (18), we obtain [43] By substituting equation (22) into equations (17) and (19), we obtain log R Le ¼ 0:5 ) R Le ¼ R 0:5 Ae e ð19þ 1 R Ae ¼ 2a a 1 and R Le ¼ 2a 2a l l ð23þ [39] The condition log R Le /e = 0.5 established in equation (19) verifies the relations between the three parameters R A,R B and R L in an ideal Hortonian channel network: R B 2, R S >R B, and R A =R L 2 [Beer and Borgas, 1993]. When comparing the multiplicative coefficient of S in equations (10) and (15), we have [44] Substituting R Be,R Ae and R Le obtained in equations (22) and (23), into equation (21) gives L e ¼ m 1 þ bs 0:5 S a 0:5 1 R Be R Le 0 2S 0 R Le ð24þ l S a ¼ N* log R 1 Be e 0 R Be 2 ð20þ [45] The equivalent ratios R Be,R Ae and R Le are related to the exponent a and the coefficient l of the empirical law on source numbers n(s) (equation (10)) as shown in equations (22) and (23), while the equivalent length L e is related to the parameters a, l,b,m 1 and S 1. Equations (22), (23) and (24) 14 of 24