Water for the Future: Hydrology in Perspective (Proceedings of the Rome Symposium, April 1987). IAHSPubl. no. 164, 1987.

The role of quantitative geomorphology in the hydrologicai response of river networks

K. ANDAH, Water Resources Research and Documentation Centre, Villa La Colombella, 06080 Colombella, Perugia, Italy R. ROSSO Hydraulic Institute, Politecnico di Milano, Italy A. C, TARAMASSO Hydraulic Institute, University of Genoa, Italy

ABSTRACT Although many drainage network schemes based on Horton's stream ordering have been developed and used in many studies, there has not emerged any one model that answers to the complex influences of the controlling variables of climate, hydrology, vegetation cover and soil characteristics. Analysis of various schemes shows that these schemes tend to put more emphasis on individual physical processes than to address the actual phenomenon involving the mutual interactions between the hydrologicai and geomorphological characteristics and the drainage network structure. An analysis of the physical milieu defining the drainage network structure has been carried out showing the interconnections between the various ordering schemes. This approach paves the way for understanding the hydrologicai response of the network structure at the basin scale.

Le rôle de la géomorphologie quantitative dans la réponse hydrologique des réseaux de rivières RESUME Bien que beaucoup de projets de réseaux de drainage basés sur la méthode de l'ordre d'un cours d'eau (Horton) aient été mis au point et utilisés pour beaucoup d'études, on n'a vu apparaître jusqu'ici aucun modèle particulier qui réponde aux influences complexes des variables déterminantes du climat, de l'hydrologie, de la couverture végétale et des caractéristiques des sol. L'analyse de différents projets montre qu'ils tendent à souligner avec insistance les processus physiques individuels plutôt que se mettre à examiner le phénomène réel qui comprend les réactions réciproques entre les caractéristiques hydrologiques et gêomorphologiques et la structure du réseau de drainage. On a effectué une analyse du milieu physique qui définit la structure du réseau de drainage. Elle démontre les communications réciproques entre les plans d'ordre différents. Cette démarche ouvre la voie a la compréhension de la réponse hydrologique à la structure du réseau de drainage à l'échelle du bassin.

93

94 K.Andah et al.

NOTATION

An mean area of basin of order n D drainage density H n mean drop, or fall, of stream of order n L n mean lengths of streams of order n an = ^n/^n m link order n order of basin or stream segment N n number of streams of order n fi highest order of a drainage network Q n mean discharge of stream of order n RA basin area ratio defined as AJJ+I/AJJ Rb bifurcation ratio defined as Nn/Nn+i RL stream length ratio defined as Ln+x/Ln

RQ discharge ratio defined as Qn+l/Qn Rw slope ratio defined as S n/S n + 1

Sn channel of stream slope defined as DNJJ/LJJ V coefficient of variation of mean annual flood (daily) discharge V 0 coefficient of variation of mean annual discharge

n 0 = Qm/Qo Hi = Vm/Vo

INTRODUCTION

This paper attempts to review critically some of the basic concepts underlying the interaction of quantitative geomorphology with hydrology at the basin scale, introducing some of the unresolved problems posed by recent advances in this area of activity. The ordering schemes often used have passed through the traditional concepts of river networks (Horton, 1945) through systems development (Strahler, 1952; Rzhanitsin, 1960) to structural or link ordering (e.g. Shreve, 1967). Basically these schemes should reflect the changes in the physical characteristics controlling the natural flow processes as the network develops. Although, the elementary basin of first order is defined identically in all the schemes, the relative weight of mutual interactions between the morphometric, hydraulic and hydrological processes in the river network finds different expressions with each ordering scheme. The Horton-Strahler approach represents a rather morphological concept of form as opposed to the more process oriented scheme of Shreve. Analysis of the various geomorphological forms of the IUH (Rosso, 1984; Rod­riguez, 1984) reveals the dependence of the hydrological response on the Hortonian characteristic ratios, which in turn vary with the ordering scheme. Due to the erratic nature of upland streams, and the absence of spatial scale in the various assessment schemes, presupposing a general homogeneity in the river basin may not be valid in nature. Presently, the problems posed by the interaction between planform and altitude remain unresolved both in form-form and form-process analysis (see e.g. Gupta et al., 1986), especially necessary in sediment transport if a unit sedimentograph is possible. Even though the various stages of the network development are supposed to evolve under diverse physical mechanisms (Abrahams &

The role of quantitative geomorphology 95

Miller, 1982), most attempts at probability distribution frequency analysis on network parameters, lump network links together into external and internal ones. Finally some preliminary results of analysis on the river networks of Ghana (Andah, 1983) and Italy are presented to illustrate some future perspectives and the impact of these results on energy distribution in stream networks.

THE STREAM ORDER AND THE DRAINAGE NETWORK STRUCTURE

In order to establish the basic characteristics of streams and channels, follow their changes from small streams to larger rivers (i.e. from streams of lower orders to those of higher orders) and the mutual interaction between them, it is necessary to sufficiently and clearly define the basic concepts underlying the river network structure. These principles controlling the formation and the behaviour of drainage network structure can be summarized as follows:

(a) The river network is the product of the development of a physico-geographical process, and at the same time, a factor influen­cing the further development of this process.

(b) Delineation of the basic phases and types of the water regime conforms with and is defined by their origin.

(c) There exists a basis for territorial-geographic links in the formation of surface runoff with climatic factors.

The basis of the quantitative description of the form and process of the drainage basin network and their interrelationships was first introduced by Horton (1945) when he proposed empirical relations between landform characteristics and the controlling variables of climate, hydrology, vegetation and soil properties. These relation­ships were expressed through the hierarchical structure of the stream network, referred to as "stream ordering". The stream order expres­ses in quantitative terms the level of magnitude in the drainage network hierarchy. Among other explanations of the physical signifi­cance of stream order Rzhanitsin (1960) showed graphically that there are points of leap-change in the values of drainage area per unit length of channel, as a function of the number of channels and he characterized them as points of change in stream order. Furthermore, Yang (1971), considering the drainage basin analogous to a thermo­dynamic system, followed the distribution of entropy along the river network and concluded that for any drainage basin in dynamic equilibrium, the ratio of the average fall between any two different stream orders in the same basin is unity.

These observations show that the stream order is not an artificial classification or subdivision of the network as it may seem, but rather it reflects the ratio between soil erodibility and the eroding force, in the form of runoff or discharge, exerted on the soil surface. The change in stream order therefore reflects a sudden change in the physical characteristics relating the form and the process taking place in the basin as the network develops. Any given stream order can be characterized by a system of morphometrical, hydrographical and hydrological parameters (Rzhanitsin, 1960; Andah, 1983). From the foregoing analysis, it must be emphasized that the

traditional delineation of a river network into a main river, into which flows an uncountable number of tributaries must give way to a

96 K.Andah et al.

systems appraisal of the drainage network. It is in this context that Horton drew an almost erroneous conclusion to his brilliant work on stream ordering. Even though Strahler (1952) modified Horton's conclusion, the traditional concept on river network still persists (Schumm, 1956; Shreve, 1974; Jarvis and Sham, 1981). In nature, the river network structure is a vivid example of a system structure defined by its boundaries, mass and energy balance. From a systems point of view, in a river network there cannot be a main river with tributaries; every river as it flows from its source to its mouth changes its physical characteristics.

COMPARATIVE ANALYSIS OF STREAM ORDERING SCHEMES

There are presently various schemes of stream ordering, attributed mainly to Horton (1945), Strahler (1952), Rzhanitsin (1960), Shreve (1964), and Scheidegger (1966). The basic principle of stream ordering introduced by Horton states that: "if two streams of the same order, Vn, merge, the resultant stream has an order higher by unity", i.e.

V , = 2 * V (1) n+1 n

As stated elsewhere, Horton concluded by extending the highest order of the network upstream through the path of least deviation, thereby assigning the resultant main channel with the same highest order. This, in physical terms, means that the main channel from the source to the mouth maintains the same physical characteristics. Strahler (1952) modified this conclusion of Horton by assigning the highest order to only the point of change of the order and hence for the first time introduced a new understanding of the development process of the network hierarchy. The ordering scheme introduced by Shreve (1967) is fundamentally different from the Horton-Strahler scheme. It is basically process-oriented and generally referred to as magnitude or link ordering, in which stream magnitudes are equal to the number of external links or sources. For example, if a stream of first order flows into a stream of fifth order, the resultant stream attains a higher order of six. Rzhanitsin (1960) proposed an ordering scheme in which both commutative and associative effects of merging streams are taken into consideration. This scheme differs from that of Scheidegger (1965) which actually negates the sequence of the merging streams and thereby contradicts systems understanding of river network development. In the case of Rzhanit­sin (1960), for example, if two second-order streams flow consecu­tively into a third order stream, the water content of the resultant stream increases to a value equal to the merging of two third-order streams and the resultant order should therefore increase to four, i.e.

V = v + 2 * V „ (2) n+1 n n-1 K '

Generally most of the ordering schemes can be represented symbolically as below:

The role of quantitative geomorphology 97

V = 2 * V n+1 n

V _ = V + 2 * V „ n+1 n n-1

V = V + V + 2 * V 0 (3) n+1 n n - 1 n - 2

V ., = V + V + V „ + . . . V n+1 n n - 1 n - 2 1

Analyses of the above expressions indicates that their differen­ces are basically due to the level of emphasis placed on the physical milieu of the drainage basin. At the same time, all the schemes tend to an elementary stream referred to as a stream of first order or external link, defined as an unbranched stream into which no other stream flows. Comparative analysis of the Horton-Strahler scheme with that of Shreve reveals two extreme physical processes, mutual in essence, namely geomorphological and hydrological respec­tively. The purely geomorphological approach underplays the role of cumulative or associative effects of streams of lower order in the hydrological characteristics and can therefore be referred to as channel or valley ordering. On the other hand, the purely hydrolo­gical approach does not consider the mutual interactions between the morphometrical and hydrological characteristics of the channels and hence physically signifies an ordering of streamflow amounts. The physical significance of the Rzhanitsin (1960) proposal can be seen as an attempt at considering the cumulative effect of streams of lag orders on the mutual interaction between the channel and streamflow processes defining the natural flow regime. The obvious problem inherent in this approach is the determination of the truncation level of lag effect, such that the cumulative effect of streams of lower order could produce a change in the physical characteristics of the natural flow process equal to the merger of two streams of the same order. This approach can be referred to as hydrogeomorphologi-cal since it reflects more on the basic principle underlying the structural and hydrological formation of river network.

BASIC LAWS OF DRAINAGE NETWORK COMPOSITION

The most important contribution of Horton to the development of quantitative analysis of drainage network was his very important formulation of the network structure that there is an orderly development of the geometrical qualities of a drainage system.

Horton introduced three basic laws of drainage network composi­tion, i.e. the change in stream numbers, mean stream lengths and mean channel slope are in geometric progression with stream orders.

(a) The Law of stream numbers: N = N /R n _ 1

n n b

98 K.Andah et al.

_ -J

(b) The Law of stream lengths: L = L R n 1 ii

(c) The Law of stream slopes: S = S /R n 1 w

Schumm (1956) further added corresponding laws for drainage areas :

A = A .R n- X

n 1 A

Furthermore Rzhanitsin (1960), Hirsch (1962), Stall & Fok (1967) found similar relationship between discharge and stream order:

Qn = V RQ n - 1

Many other forms or combinations of drainage basin characteristic have also been found to respond to the basic geometrical relation­ship (e.g. Rzhanitsin, 1960; Fok, 1971; Andah, 1983).

BASIN SCALE

There have been many attempts toward quantitative analysis of non-homogeneities inherent in the spatial processes occurring in the drainage basin due to climatic, soil, topographical and other conditions influencing the natural flow process. The presence of these heterogeneities in the runoff formation from overland flow through the hillslopes up to the basin response rules out the feasibility of a single channel equation to describe the streamflow processes at the outlet of the basin. It is in this context that efforts have been made to describe runoff formation at different scales, namely hydrodynamic, hillslope and the more practical and physically significant, the basin scale (e.g. Kirkby, 1978, 1985; Rodriguez-Iturbe & Valdes, 1979; Gupta & Waymire, 1983). The main characteristic which signifies the spatial distribution of the channel network in a basin is the drainage density, D. The basic principle is that amongst the assemblage of subnetworks within a drainage basin, the spatial scale at which the value of D is invariant, is the basin scale. Due to the uncertainties involved in topographical maps and also map scales, Gupta et al. (1986) intro­duced a new form of the drainage density Dm(h) using a link ordering with respect to altitude. Accordingly, the drainage number density is defined as the ratio:

D (h) = N(h)/A(h) (4) m

where N(h) and A(h) denote the number of links in an interval around an altitude h and the basin area enclosed within the interval defined by order m respectively. A basin at the basin scale in this context is defined to be homogeneous if the drainage number density does not vary with altitude, i.e.

The role of quantitative geomorphology 99

D (h) E D m m

It must be noted that the concept of basin scale is not tied to the size of a basin, but rather to the size of an area within which a basin possesses an identifiable channel network which governs runoff generation from a system of hillslopes. A basin scale, therefore, is a spatial scale within the drainage basin at which the collective description of runoff generation from the system of hillslopes and its transmission to the basin outlet through the channel network, can admit simple descriptions by averaging out complex physical processes involved at the hydrodynamic and hill-slope scales.

Empirically Andah (1983), Andah & Rosso (1984), in an analysis of the river network of Ghana confirmed the above conclusions. This work involved a water resources assessment of river networks of Ghana, with areas ranging from 800 to 397 000 km , using ordered topographical maps at the scale of 1:250 000.

The results to be later discussed in more detail, indicated a change in the regression of stream length on order at the sixth order, creating two distinct values of the stream length ratio RL. More interestingly, the Volta River basin of maximum order of fifteen also had only one point of change at the sixth order. Preliminary results of an on-going work by the authors involving the Fontanabuona basin in Italy at the scale of 1:25 000 shows a similar change, but at the fifth order.

Furthermore, Coffman et al. (1971) carried out an analysis of ordered thresholds from the relationship between the number of links and the number of segments of natural drainage network in the form of a theoretical general equation:

• Y = 2X - (2n - 1) (5)

where Y is the number of links, X is the number of segments and n is the Strahler stream order. They found out that all networks up to the fourth order fall within their respective threshold fields with those above the fifth order not conforming to their respective threshold boundaries. These observations amongst others (e.g. Schumm, 1956) offer the possibility of delineating within the drainage basin two distinct geomorphological homogeneities, namely hillslope, and channel dominated processes. Much more analytical work is needed to confirm such preliminary results which could introduce a physically based concept into the search for a basin scale.

HYDR0L0GICAL RESPONSE VERSUS THE GEOMORPHOLOGICAL QUANTITATIVE PROPERTIES OF A RIVER BASIN

General

The basic principle underlying geomorphological approaches to water resources assessment of drainage networks can be stated as follows: "if the river network is the final link of a definite physico-

100 K.Andah et al.

geographical process of mutual interactions of climatical, hydrolo-gical and geomorphological factors, and definite laws can be observed within its structure, then sufficiently strong relation­ships must also exist inbetween the morphometrical elements of the channel, the hydraulic and hydrological characteristic of the flow, the character and the structure of the drainage network". The early works of Horton (1924, 1932) in their original formulation of basin morphometry were partly carried out with a view to deriving a hydrological method by which discharge events could be predicted for ungauged rivers.

The most widely used approach is the multiple regression tech­nique, in which regression equations are developed between basin form variables and some selected basin process indices (Leopold & Maddock, 1953; Benson, 1964; Trainer, 1969; Orsborn, 1974; Caroni et al., 1985). For example, Morisawa (1959) suggested a general functional relationship between streamflow Q and basin form parameters such as total stream length, basin area, first order stream frequency, stream gradient, circularity and relief ratio in the following form:

Q = f(A, El, 1 f, f, f) c n

and actually developed corresponding empirical solutions. Other indices of discharge (e.g. lag time) have also been related to basin morphometrical characteristics with some success (e.g. Kennedy & Watt, 1967; Quimpo, 1983; Mimikou, 1984). The enormous work produced in this sphere cannot be reported here since this paper intends to put more emphasis on the Hortonian stream-order approach to drainage network assessment. There are two wide approaches involved in the Hortonian techniques, namely, empirical regression analysis of actual observed data with stream order, and process-response analysis based on the geomorphological structure of the drainage basin.

Hortonian regression analysis

In this approach, attempts are made towards establishing a system of drainage basin characteristics namely hydrographical, morphomet­rical and hydrological for each stream order. A successful result can create a basis for predicting flow at ungauged sites, on the basis of either regional or single-basin data transfer through the use of a single index - the stream order.

Regional analysis

Some results of an earlier work by the authors (Andah, 1983; Andah et al., 1984; Andah & Rosso, 1984) are briefly reported here to show that in spite of uncertainties involved in the quality of observed data, stream ordering techniques could provide some reliable predictions of streamflow. Topographical maps of all river network of Ghana at the scale of 1:250 000 were used in establishing the stream orders, taking into consideration the cumulative effects of

The role of quantitative geomorphology 101

lower orders of merging streams up to lag two (Rzhanitsin, 1960). On the basis of drainage patterns with respect to basin relief, river networks of Ghana were grouped into three types, characterized by the transition from hilly areas to flat plains (Andah, 1983). As regards the hydrological regime, the river basins fall into two main types, referred to as "Northern" and "Southern" rivers. The northern rivers, lying in the northern savannah zone, are charac­terized by unimodal distribution of annual runoff. The southern rivers, lying in more humid and hilly areas and covered mainly by forest, experience two runoff peaks.

The main hydrographical characteristics considered were river lengths, stream lengths, stream drainage areas and the stream area/ length ratio. From Fig.l, it can be observed that the relationships of stream lengths and stream drainage area/length ratio with stream order change at the sixth order, except for streams in flat plains (type III) in the case of area length ratio. The respective charac­teristic ratios are reported in Table 1. These observations show that the hydrographical characteristics are dependent on the physico-

II ill IV V VI VII VIII IX X X! XII XIII XIV STREAM ORDER

FIG.l Variation of hydrographical characteristics with stream order for relief type I, II and III and Fontana-buona.

102 K.Andah et al.

TABLE 1 Hydrographical characteristics of rivers of Ghana

Relief type Characteristic ratios: R- R*

an

1st order values: L (km) 1 (km) a (km)

I II III

1.58 1.35-1.58 1.73-1.33 1.58 1.26-1.90 1.78-1.29 1.57 1.15-1.90 1.33

3.76 2.66 0.40 8.91 6.30 1.58

13.70 10.70 10.96

*The first value is associated with stream orders from 1 to 6 and the second from 7 to 14.

geographical conditions of the drainage basin, especially on basin relief.

The hydrological characteristics considered are the mean annual discharge, Q0, the mean annual flood (daily) discharge, and their respective coefficients of variation V0, and Vm. The relationship of discharge characteristics with stream orders obtained for the two water regime types are shown in Fig.2, and Table 2. In the case of VQ and Vm, there was a larger scatter in their relationships due to uneveness in observation periods, and as such, more work is being carried out. Since generally there are fewer gauging stations on upland streams, the behaviour of the hydrological characteristics for streams of lower order could not be actually established but for

Q m^s

1ooo

100

CC < I o

Q m 3 s"'

VI VII VIII IX

a.NORTHERN RIVERS

XI XII Kill XIV

STREAM ORDER n II IV V VI VII VIII IX

b.SOUTHERN RIVERS

x xi XII xm Xiv STREAM ORDER n

FIG.2 Variation of characteristic discharges with stream order (Ghana).

The role of quantitative geomorphology 103

TABLE 2 Hydrological characteristics of rivers of Ghana

Water regime type

Northern Southern

Characteristic ra RQO

2.13 2.06

Qm DO

1.85 0.87 1.80 0.87

tios :

R -, r\l

1.03 0.97

1st order values:

Qo

0.06 0.08

Qm

2.00 1.50

\

30.0 19.0

n2

0.84 1.48

the purpose of handy predictions could be extrapolated from graphical relationships. The combination of the hydrographical and hydrologi­cal characteristics obtained through a single network index (order) could be used to predict runoff for any project site, especially for prefeasibility studies.

When compared with traditional regression analysis of hydrologi­cal variables versus basin area, altitude, mainstream length and slope, and similar global catchment parameters which are used as independent variables, one can observe that regional analysis based upon stream ordering seems to provide better results as far as the reliability of the estimates are concerned. In fact, the power equations which are usually fitted by the first method can provide unbiased and minimum variance estimates if the range of the gauged rivers under examination is considered. But highly biased estimates can be sometimes found when the power equations are used to extra­polate ungauged rivers. The estimates obtained by Hortonian regression analysis are usually affected by larger variance within the range of the gauged rivers, but they generally provide much more robust predictions within the extrapolated range.

Single basin analysis

This is a brief account of preliminary results of on-going research in a basin in Italy being carried out by the authors. The Fontana-buona River basin of drainage area 364 km2 lies in the northeastern part of Italy. It is characterized by mountainous terrain of maximum altitude 1500 m a.m.s.l. with a mean value of 545 m. Three main basin characteristics are considered, namely morphological, hydrographical and hydrological. Stream orders were established by the schemes of Strahler and Rzhanitsin (with lag-one cumulative effect), using topographical maps at the scale of 25 000. The highest basin orders obtained were six and ten for the Strahler and Rzhanitsin schemes respectively.

The hydrographical and morphological characteristics analysed include stream lengths, drainage areas, stream fall and stream slopes (gradient) and also their various interrelations. The relationships of the mean values of stream lengths and stream slopes with order changed at the fourth and fifth orders with respect to the Strahler and Rzhanitsin schemes. Further analysis with the Strahler scheme was discontinued since only one gauging station differed in order out of the existing four stations and would therefore not offer any appreciable relationship of discharge with stream order. The relationship and characteristic ratios reported

104 K.Andah et al.

in Fig.l and Table 1, respectively, refer to the Rzhanitsin scheme. It is interesting to observe that mean stream falls did not show any clear correlation with stream order. However, the mean stream falls seem to conform more with the postulation of Yang (1971) than the mathematically derived law of stream relief proposed by Fok (1971). Much more work is being carried out in search of a more conclusive analysis on this most important aspect of the interaction between the vertical and planimetrical scales of the drainage network.

The hydrological analysis include the mean annual discharge, various flow states of the flow duration curve (10, 91, 192, 274 and 350 days) and the absolute daily and instantaneous maximum flow values. From Fig.3, it can be seen that apart from Q350 (not shown) all the other mean flow states fall into the same relationship with stream order, showing a constant mean discharge ratio of 2.75. The corresponding discharge ratios for the absolute values of maximum flows, both daily and instantaneous are 4.17 and 5.0 respectively. The values of discharge ratios reported here (Ghana and Fontana-buona) seem to indicate their tendency to increase from higher time averaged values to instantaneous rather than the flow states.

STREAM ORDER

0 MEAN ANNUAL FLOW

o 10 DAY FLOW

0 91 DAY FLOW

a 192 DAY FLOW

e 7Jl\ DAY FLOW

FIG.3 Variation of mean annual flow with stream order (Fontanabuona).

The results of work being done to evolve a system of drainage basin characteristics that could give some guide to exploitable energy survey in river basins, if not absolute, at least in relative terms, for run-of-river plants are too preliminary to be reported

The role of quantitative geomorphology 105

here. Suffice it to say that the characteristics being analysed involve the stream area-length ratio as a benefit-cost index, the product of discharge with mean stream slope and mean stream fall. The general problem encountered in the form-process interaction can be found in the absence of hydrometrical stations especially in the upland sub-basins, making it difficult to proceed with strict phenomenological analysis of drainage basins.

Geomorphological synthesis of hydrological response at the basin scale

The channel network within a drainage basin forms as a result of runoff generation and its subsequent transmission to the basin outlet. Recent developments in analytical models of drainage patterns (e.g. Cordova et al., 1982; Roth et al., 1985) show that the process of network formation over a loose boundary is governed by the climatic input, i.e. the rainfall intensity, for a given tectonic and soil constraints. Although these studies of process-response characteristics are in their early stage of development, a close connection between climatic input and stream network as represented by its quantitative geomorphological description and runoff transportation at the outlet, has been observed both in nature and by controlled experiments (Roth & Rosso, 1986). This unique interconnection suggests that network geometry can provide the basic knowledge to analyse how and why a given network forms as well as the mode of response of a particular network to rainfall input.

The pioneering works of Rodriguez-Iturbe & Valdes (1979) and Gupta et al. (1980) provided firm arguments to support the identifi­cation of the instantaneous unit hydrograph (IUH), i.e. the basin response function to a unit input of rainfall excess, with the probability density function (pdf) of the basin holding time. Such an identification can also be extended to the concept of instan­taneous unit response functions (IRF) to relax the linear assumption (see e.g. Wang et al., 1981; Rodriguez-Iturbe et al., 1982; Caroni et al., 1986). The importance of this identification lies in the proper theoretical setting it provides for a physically based analysis of the density of the basin holding time. In this context, the basin scale concept is approached from the random character of the holding time and of the interaction of various processes taking place at lower scales. Even though the equations governing the transport of water over land and in saturated and unsaturated soils are best understood at the hydrodynamical scale, the characteristics describing the network structure such as order ratios or link parameters are more meaningful at the larger basin scale and hence runoff produced by the basin should be represented at this scale.

Geomorphological models of the hydrological response The first model linking the IUH and network geometry, based on Strahler's scheme of network patterns was proposed by Rodriguez-Iturbe & Valdes (1979) within a mechanical statistical framework. Due to the simple characterisation of the drainage network, simple expressions of the IUH could be achieved. Moreover, Rosso (1984) has demonstrated that

106 K.Andah et al.

this model is equivalent to the Nash model (1959), the parameters of which are found to depend on the characteristic order ratios of the basin. Accordingly, the shape of the IUH can be determined from only catchment geomorphology, while the time scale of the IUH is the product of catchment geomorphology and streamflow dynamics. This latter aspect has recently been developed by Agnese et al. (1986) who derived a simple and straightforward approach to analyse the time scale of the IUH at the basin scale. Applications of this model, generally referred to as the geomorphological IUH, to dif­ferent climatic and geomorphological conditions have been extensively carried out by various authors and the results generally are quite satisfactory for practical purposes, especially if one takes into account the uncertainties involved in parameter estimation of more complex models (Caroni & Rosso, 1986). Other areas where this model has found success in its applications are in frequency analysis (e.g. Hebson & Wood, 1982; Diaz-Granados et al., 1984) and in storm pollutant analysis (e.g. Marano & Rinaldo, 1986).

Recent developments initiated by Gupta et al. (1980) have been directed towards linking the IUH with the basin structure, in order to provide a closer description of the process of runoff generation and flow than the one achievable by means of the Horton-Strahler scheme of network geometry. Although the manner of treating network geometry in these models subsequently required modifications (e.g. Gupta & Waymire, 1983), such approach has been shown to provide a flexible and meaningful representation of the hydrological response of a basin to an excess of rainfall over it. As compared with the first approach, the formulations which can be derived by the link structure relation to runoff generation and flow lead to more complex expressions. Nevertheless, this approach could allow a deeper insight of the process, since network and hillslope response functions can be identified (e.g. Gupta et al., 1986; Mesa & Mifflin, 1986) as well as the averaging properties of channel networks can be directly linked with the associated properties of flow hydrodynamics (e.g. Troutman & Karlinger, 1986).

Gravity and network struct ure in basin runoff The briefly discussed forms of the geomorphological models have been based on only network geometry. One can observe that gravity and for that matter altitude plays a relevant role in the generation and flow process and hence should be considered in modelling the hydrological response. A first attempt has been made by Gupta et al. (1986), who observed that a structural regularity among the runoff producing areas of the hillslopes does exist. In which case, the third dimension and more specifically, the drops in elevation among the channel links of the network should also be connected through some unifying principles. These principles can permit the postulation of energy considerations, dependent on the amounts of runoff produced by the hillslopes which are considered to be intimately linked with the drainage network. Under this viewpoint, Gupta et al. (1986) postulated that "in the absence of geologic controls, the networks that form in nature have statistically independent link heights having an exponential density function". As a result, the total runoff generated over a long time span by any sub-basin associated with each link of a basin has a gamma distribution with parameters

The role of quantitative geomorphology 107

X/y and 2*(link magnitude) - 1, where X is the rate of exponential link heights and y is the runoff volume per link per unit elevation.

It can be observed that this result does not hold if X varies with altitude and as such the pdf of link heights play a dominant role in the assessment of basin runoff. In other words, a functional relationship of À with altitude would lead to a non-homogenous process of link development in the third dimension, while this process could be homogenous in the plane. Randomization of X, moreover, will lead to a double stochastic process of three-dimen­sional link structure, the effect of which will introduce a further fluctuation in the runoff production of a basin.

As further observed by Gupta et al. (1986), "in searching for structural regularities between fluctuations in potential energy, kinetic energy and losses in channel morphology, it will be neces­sary to quantify the channel flow paths parameterised by altitudes". Only after a suitable relationship between altitudes and flow paths is established and empirically tested, will it be possible to compute flows from a basin as well as to begin to address the fundamentally important problem of hydrological similarity among basins. Moreover the transformation of potential energy to kinetic energy appears to be tied to the interrelationship between the basin time-scale and spatial-scale.

FINAL REMARKS

Since the introduction of quantitative geomorphology into river network analysis by Horton some forty years ago, there has still not emerged a basic understanding of the controlling geometry defining the network structure (see e.g. Strahler, 1952; Rzhanitsin, 1960; Shreve, 1967; Scheidegger, 1965; Rodriguez-Iturbe & Valdes, 1979; Gupta et al., 1980, 1986 etc.). As shown by expression (3), the main problem involved in the search for an objective river network structure in nature is due to the contradiction between the superim­position of analytical models on the river network system and the actual physical processes generating and controlling the drainage basin. Even though the Horton-Strahler ordering scheme has been found on fundamental grounds to be inadequate to describe the river network geometry, the most preferred Shreve's scheme cannot also be adequate since it reduces the network structure to a simple collec­ting and transporting system of matter through the channels. A river network geometry in nature must of necessity be able to describe the various processes interconnecting climatic, morpholo­gical and hydrological variables. It is therefore necessary that analytical models must be geared towards the network structure in nature and not vice versa. In this case an attempt at bridging the gap between the rich store of empirical and analytical results already achieved would be in the right direction.

The best prospect for the application of the Hortonian analysis towards drainage network analysis and assessment lies with the fast advancing computer oriented Digital Terrain Models (see e.g. Clarke et al., 1982; Carrara, 1986) which would reduce the amount of work involved in the establishment of stream orders and measurements of various geomorphological characteristics. It is precisely due to

108 K.Andah et al.

the tediousness of this approach that has limited most researchers to small subcatchments and also to schemes that are more mechanical in their use at the expense of the more objective form-process-response reality of the drainage network.

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