3_arno_joh_1996 by Enrique Ortiz

Journal of

Hydrology ELSEVIER

Journal of Hydrology 175 (1996) 339-382

The ARNO rainfall-runoff

model

E. Todini Institute for Hydraulic Construction, University of Bologna, Via/e Risorgimento 2,40136 Bologna, Italy Received 2 March 1995; accepted 30 March 1995

Abstract

This paper describes in detail a semi-distributed conceptual rainfall-runoff model known as the ARNO model, which is now in widespread use both in land-surface-atmosphere process research and as an operational flood forecasting tool on several catchments in different parts of the world. The model, which derives its name from its first application to the Arno River, incorporates the concepts of a spatial probability distribution of soil moisture capacity and of dynamically varying saturated contributing areas. The ARNO model is characterized by two main components: the tirst and most important component represents the soil moisture balance, and the second describes the transfer of runoff to the outlet of the basin. The relevance of the soil component emerges from the highly nonlinear mechanism with which the soil moisture content and its distribution controls the dynamically varying size of the saturated areas mainly responsible for a direct conversion of rainfall into runoff. The second component describes the way in which runoff is routed and transferred along the hillslopes to the drainage channels and along the channel network to the outlet of the basin. Additional components, such as the evapotranspiration, snowmelt and groundwater modules, are also described. A discussion on the advantages of the model, calibration requirements and techniques is also presented, together with the physical interpretation of model parameters. Finally, after describing the original calibration of the ARNO model on the Arno basin, and a comparison with several conceptual models, recent applications of the ARNO model, as part of a real-time flood forecasting system, as a tool for investigating land use changes and as an interesting approach to the evaluation of land-surface-atmosphere interactions at general circulation model (GCM) scale, are illustrated.

1. Introduction The literature contains many works that summarize the level of understanding of the complex physics governing the transformation of rainfall into runoff (Dunne, 0022-1694/96/$15.00 0 1996 - Elsevier Science B.V. All rights reserved SSDI 0022-1694(95)02853-6

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1978; Freeze, 1980). Many efforts have been made to schematize the whole process so as to develop mathematical models (Dooge, 1957, 1973; Amorocho and Hart, 1964; Freeze and Harlan, 1969; Todini, 1989). These range from the simple lumped calculation of design discharge to the distributed representation of the various processes based on the conservation of mass, energy and momentum (Bathurst, 1986; Abbott et al., 1986a,b; Beven et al., 1987; Binley et al., 1989). Taken together, such models form the broad category of distributed differential models (Todini, 1986); they are frequently referred to as ‘physically based models’ to highlight the fact that their respective parameters are (or should be) reflected in the field measurements (Beven, 1989). Given their nature, such models are appropriate for studying the effects of land use changes, soil erosion, surface groundwater interactions, etc., but are less suitable for conventional rainfall-runoff applications at catchment scale. Another category of models, which was developed mainly for operational purposes, is that of the complex ‘conceptual models’. From the early 1960s a large number of these models were described, from the Stanford Watershed Model IV (Crawford and Linsley, 1966) to the SSARR (Rockwood et al., 1972), the Sacramento (Burnash et al., 1973) and the Tank (Sugawara et al., 1976), which represented in different ways the response mechanisms of the various phenomena, but mostly by means of non-linear reservoirs and thresholds, directly connected or linked either by means of linear transfer functions or by linear or non-linear hydrologic or hydraulic type routing. The underlying reason for developing these models was to represent the hydrologic cycle by linking together process components which described physical concepts, on the presumption that the model parameters would also bear physical meaning, so that they could be assigned values without reference to the observed data. In other words, it was assumed that most of their parameters (such as storage coefficients, roughness coefficients or thresholds present in the various sub-components) could be defined from the physiographic characteristics of the basins. In reality, the parameters needed to be estimated by minimizing objective functions *(e.g. the sum of squared deviations), which generally led to groups of unrealistic parameters incorporating both data measurement errors and the errors presemin the structure of the model itself; in addition, parameter observability conditions could not always be guaranteed (Sorooshian and Gupta, 1983). Although they are in widespread use throughout the world, it is now understood that the basic failure of these models to represent catchment response with a small number of parameters is essentially due to their inability to reproduce the dynamic variation of the saturated areas within the catchment (Beven et al., 1983). Indeed, in recent years, a general consensus has been reached on the fact that it is this dynamic variation, a function of the accumulation and horizontal movement of water in the upper soil layers (see Todini (1995) and Franchini et al. (1996)), which is mainly responsible for the highly non-linear nature of catchment response to storm ,events. Most of the conceptual modellers tried to compensate for the inadequacy of their models by adding more and more process components as well as parameters, but they failed to reproduce the actual phenomena and reduced the models to extremely

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complex black boxes with an exceedingly high number of parameters (frequently larger than 20) to be estimated from the available records. This emerged clearly from the WMO intercomparison of conceptual models (World Meteorological Organization (WMO), 1975) where the results of all the different models did not appear to be significantly better than those produced by the constrained linear systems (CLS) model, a simple piecewise linear black-box model (Natale and Todini, 1977). More recently, newly developed conceptual models have regarded the soil moisture replenishment, depletion and redistribution mechanism as directly responsible for the dynamic variation of the areas contributing to direct runoff. From this concept, a number of models originated which use a probability distribution of the soil moisture content, as described by Zhao (1977) and Moore and Clarke (1981) or the distribution of a topographic index, as in TOPMODEL (Beven and Kirkby, 1979; Beven et al., 1984). The advantage of these models lies in their capability to reproduce catchment response with a smaller number of physically meaningful parameters than the more traditional lumped conceptual models (see Franchini and Pacciani, 1991). The ARNO model falls in this class of models, as it derives its soil moisture accounting module directly from the distribution function approach of Zhao (1977) augmented by the introduction of drainage and percolation losses in the soil moisture balance. The major advantage of the ARNO model is the fact that it is entirely driven by the total catchment soil moisture storage, which is functionally related, by means of simple analytical expressions, to the dynamic contributing areas, and to the drainage and the percolation amounts. The soil moisture module of the ARNO model has been extensively used in hydrological practice; in particular, it has become the kernel of a real-time operational flood forecasting system developed on behalf of the Commission of the European Communities (the European Flood Forecasting Operational Real-Time System (EFFORTS); ET&P, 1992) which is already operational on several rivers in many countries: the Fuchun in China; the Danube in Germany; the PO, the Arno, the Tiber, the Adda and the Oglio in Italy. Recently, it has been tested in joint meteorologicalhydrological experiments, such as the Spatial Variability of Land Surface Processes (SLAPS) project (Dooge et al., 1994) and by a number of meteorologists (Rowntree and Lean, 1994; Polcher et al., 1995) and, given the simplicity of its formulation, it was successfully included in the Hamburg climate model (Dtimenil and Todini, 1992). In addition to the crucial soil moisture component, the ARNO model includes evapotranspiration, snowmelt and groundwater components, all of which are parsimonious descriptions in line with the overall philosophy. Moreover, once a satisfactory description of the so-called â&#x20AC;&#x2DC;runoff production functionâ&#x20AC;&#x2122; has been achieved, the runoff contributions of the area units considered must then be transferred downstream, and aggregated as they move along the slopes and the drainage network. A wide choice of runoff transfer or routing methods is available, and one may find the folllowing in the literature: numerical integration of the non-linear parabolic flow equations as in SHE (Abbott et al., 1986b); numerical integration of the non-linear kinematic flow equation (Kibler and Woolhiser, 1970); methods based on the Muskingum-Cunge algorithm (Cunge, 1969; Zhao, 1977); methods based on a

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gamma distribution (Nash, 1958; Kalinin and Miljukov, 1958); methods based on the derivation of single transfer functions (Nalbantis et al., 1988a,b) or multiple transfer functions (Natale and Todini, 1977); derivation of the geomorphological unit hydrograph (GIUH) (Rodriguez Iturbe and Valdes, 1979; Rodriguez Iturbe et al., 1982; Rosso, 1983, 1984); methods based on a linear parabolic representation leading to an inverse Gaussian distribution (Todini, 1988a; Franchini and Pacciani, 1991; Naden, 1992). However, backed by numerous experimental results (e.g. Franchini and Pacciani, 1991) and by various authoritative opinions (Cordoba and Rodriguez-Iturbe, 1983), it may safely be stated that the routing component, which is essential for linking together the various area units, is not the real kernel of the problem, and, whatever the representation used, the results will not be very different, except perhaps for more or less physical significance to be attributed to the parameters. In the ARNO model, the linear parabolic approach has been successfully used with parameter values that can be established according to physical reasoning, without the need for extensive trial and error or highly non-linear optimization procedures.

2. Basic concepts in the ARNO model The ARNO model is a semi-distributed conceptual model in integral form (Todini, 1988b), which is based on the schematic representation of catchment hydrology shown in Fig. 1. The catchment is divided into a series of sub-basins to each of which the rainfall-runoff model is applied; this division takes place according to

LATERAL OR lKFLOWS

IJTFIDW DLSCHARGIZS Fig. 1. Schematic

representation

of processes

within a catchment

unit.

E. Todini / Journal of Hydrology 175 (1996) 339-382

Fig. 2. Example of a catchment sub-division.

the natural sub-basin boundaries so that the sub-basin closing sections coincide with the cross-sections of interest along the river and its tributaries (Fig. 2). These sections are chosen according to the presence of hydrometric measuring stations, either because they are of interest for flood forecasting or for reasons connected with the morphology of the basin. The sub-catchments can then be represented as a tree (Fig. 3) which defines the order in which the computations are to be carried out. The following operations are performed on each sub-basin for each time interval: calculation of the evapotranspiration and runoff production for the ith generic subbasin; transfer of runoff, by means of a simplified hydraulic routing model, along the slopes and in the channel reach inside the ith sub-basin to the closing section; transfer of the input hydrograph from the upstream closing section to that of the sub-basin

w G

Fig. 3. Schematic tree representing the catchment of Fig. 2.

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immediately downstream using the same form of routing model; summation at the downstream closing section of the input hydrograph transferred from the upstream sub-basin with the hydrograph produced by the ith sub-basin. The main physical phenomena represented in the ARNO model are the following (Fig. 1): water balance in the soil, on the basis of present soil moisture content, rainfall, runoff, evapotranspiration, drainage and percolation; water losses through evapotranspiration, evaluated on the basis of the air temperature data and the soil moisture content; snow accumulation and/or melt, based upon the energy balance of

p(t) e(t) r(t)

precipitation evapo-transpiration runoff inflow discharges

outflow discharges snowmelt runoff interflow percolation baseflow

QMi N :: I B

Fig. 4. Processes and quantities represented in the ARNO model.

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175 (1996) 339-382

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the snow cover (when present) as a function of the air temperature and precipitation data; groundwater flow represented by means of a multiple linear reservoir-type model; overland and channel flow routing represented with linear parabolic models. Accordingly, as mentioned above, several modules have been developed to represent the individual aspects of the overall rainfall-runoff process, as follows: soil moisture balance module; evapotranspiration module; snowmelt module; groundwater module; parabolic transfer module. Fig. 4 shows, for each subcatchment, a schematic flow diagram connecting the different modules considered in the ARNO model. Geomorphological data such as average catchment elevation, catchment surface area and length of stream, as well as parameters including celerity and diffusivity, thermal gradient, and parameters characterizing the spatial distribution of soil moisture storage must be provided to apply the model to a catchment. Rainfall and air temperature inputs are provided to the model as area averages by means of weights generally based upon Thiessen polygons. In the case of temperature, the calculation of the average temperature over a sub-basin takes into account the effects of a thermal gradient with elevation. Once the runoff is obtained from the soil moisture balance, the routing of runoff on the hillslopes of the catchment is simulated by applying a distributed inflow linear

Schematic representation of channel routing with distributed lateral inflow Fig. 5. Hillslope and channel routing schemes, where q(t) is the lateral distributed inflow discharge, Q(r) is the outflow discharge and r(t) is the distributed runoff inflow.

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parabolic model to an â&#x20AC;&#x2DC;open bookâ&#x20AC;&#x2122; representation of the hillslope elements (Fig. 5). The channel routing is also performed by means of a distributed input linear parabolic model, and the contribution from upstream sub-catchments is routed downstream by means of a linear concentrated-input parabolic model (Fig. 5). The sub-catchment outflows are then routed downstream according to the tree scheme (Fig. 3) which represents the overall catchment.

3. The soil moisture balance module The soil moisture balance module of the ARNO model derives from the Xinanjiang model developed by Zhao (1977, 1984), who expressed the spatial distribution of the soil moisture capacity in the form of a probability distribution function, similar to that advocated by Moore and Clarke (1981) and Moore (1985). Successively, to account more effectively for soil depletion owing to drainage, the original Xinanjiang model scheme was modified by Todini (1988a), who originated the ARNO model within the frame of the hydrological study of the river Arno. The basic assumptions expressed in the soil moisture balance module of the ARNO model are as follows: (1) the precipitation input to the soil is considered uniform over the catchment (or sub-catchment) area; (2) the catchment is composed of an infinite number of elementary areas (each with a different soil moisture capacity and a different soil moisture content), for each of which the continuity of mass can be written and simulated over time; (3) all the precipitation falling over the soil infiltrates unless the soil is either impervious or it has already reached saturation; (4) the proportion of elementary areas which are saturated is described by a spatial distribution function; (5) the spatial distribution function describes the dynamics of contributing areas which generate surface runoff, (6) the total runoff is the spatial integral of the infinitesimal contributions deriving from the different elementary areas; (7) the soil moisture storage is depleted by the evapotranspiration as well as by lateral sub-surface flow (drainage) towards the drainage network and the percolation to deeper layers; (8) both drainage and percolation are expressed by simple empirical expressions. A sub-basin of given surface area ST (excluding the surface extent of water bodies such as reservoirs or lakes) is in general formed by a mixture of pervious and less pervious terrains, the response to precipitation of which will be substantially different. For this reason, the total area ST is divided into the impervious area S, and the pervious area SP: ST = s, + sp

(1)

To derive the expressions needed for the continuous updating of the soil moisture balance, let us first deal with the amount of precipitation that falls over the pervious area. Given that, from Eq. (1) the entire pervious area is s, = S* - s,

(2)

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341

if one denotes by (S - S,) the generic surface area at saturation, then x, defined as s - s, x=sT.

(3)

will indicate the percentage of pervious area at saturation. Zhao (1977) demonstrated that the following relation holds reasonably well between the area at saturation and the local proportion of maximum soil moisture content W/W,, where w is the elementary area soil moisture at saturation and w, is the maximum possible soil moisture in any elementary area of the catchment: b

(4) This is similar to defining the cumulative distribution for the elementary area soil moisture at saturation, shown in Fig. 6 and defined as w = Wm[l- (1 -X)$

(5) In the ARNO model, an interception component (Rutter et al., 1971, 1975) is not explicitly included. Nevertheless, to allow for a substantially larger evapotranspiration when the canopies are wet (without obviously explicitly accounting for the disappearance of the stomata1 resistance (Shuttleworth, 1979)), the following succession of operations is followed. If the precipitation P is larger than the potential evapotranspiration ETp, the actual evapotranspiration, for the reasons expressed above, is assumed to coincide with the potential, i.e. ET, = ETP

(6)

and so an â&#x20AC;&#x2DC;effectiveâ&#x20AC;&#x2122; meteorological input M,, defined as the difference between precipitation and potential evapotranspiration, becomes M,=P-ET,=P-ET,>0

Fig. 6. Cumulative distribution for the elementary area soil moisture at saturation.

(7)

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With reference to Fig. 7, the surface runoff R generated by the entire catchment is obtained as the sum of two terms; the first one is the product of the meteorological effective input and the percentage of impervious area, and the second one is the average runoff produced by the pervious area, which is obtained by integrating the soil moisture capacity curve, which gives M,+W sT

R=zM,+-

40 d5

if M,+w<

(8)

(9)

if M,+waw,

Eqs. (8) and (9) can also be expressed in terms of the catchment average soil moisture content ( W)and that at saturation ( W,), and after integration they become

1I bfl

R=M,+

- (b +:

forO<M,<(b+l)W,,, R=M,+

V(W,-

forM,Z(b+l)w,,,

Fig. 7. Runoff R generated by an effective meteorological input M, > 0.

E. Todini / Journal of Hydrology 175 (19%)

339-382

If the precipitation P is smaller than the potential evapotranspiration effective meteorological input it4, becomes negative: M,=P-ET,<0

349

ET,, the

(11)

which implies that the runoff R is zero. The actual evapotranspiration is then computed as the precipitation P plus a quantity which depends upon A& reduced by the average degree of saturation of the soil (see Appendix A), which gives (1+Ek)-(1-E)” ET, = P + (ETp - P) (STs; ‘I)

(12) (1+$

- (l-E)&

These equations, which represent the average surface runoff produced in the subcatchment, must be associated with an equation of state to update the mean water content in the soil. This equation takes the form W(t + At) = W(t) + P(t, t + At) - ET,(t, t + At) - R(t, t + At) - D(t, t + At) - Z(t, t + At)

(13)

where P(t, t + At) is the area precipitation between t and t + At; ET,(t, t + At) is the loss through evapotranspiration between t and t + At; R(t, t + At) is the total runoff between t and t + At; D(t, t + At) is the loss through drainage between t and t + At; Z(t, t + At) is the percolation loss to groundwater between t and t + At; W(t,+ At) is the soil moisture content at time t + At; W( t)‘is the soil moisture content at time t. All the quantities representing averages over the sub-basin are expressed in millimetres. The non-linear response of the unsaturated soil to precipitation, represented by the shape of the distribution curve given by Eq. (5), is strongly affected by the horizontal drainage and vertical percolation losses. The drainage loss D is an important quantity to be reproduced in a hydrological model, because on the one hand it affects the soil moisture storage and on the other hand it controls the hydrograph recession. Experience derived from applications suggested the use of the following empirical relationship to express also the drainage loss D as a non-linear function of the soil moisture content: D = D,,,$D = Dmin5

m m

for W < W, (14) + (Dmax-Dtin)(EIFJ

for W>WW,

where c is the exponent of the variation law, Dti,, and D,,, are drainage parameters at saturation and W, is .moisture content threshold value. The percolation loss Z, which feeds the groundwater module, which will control the base flow in the model, varies less significantly over time if compared with the other

E. Todini / Journal of Hydrology 175 (19%)

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339-382

terms; nevertheless, a non-linear behaviour is also assumed as follows: Z= 0 Z=U(W-

for W < Wi Wi)

for Wa Wi

(15)

where Wi represents the moisture content threshold value below which the percolation may be considered negligible, and a is an empirical coefficient. The total runoff per unit area produced by the precipitation P is finally expressed as Rtot = R i- D + B

(16)

where B is the base flow generated by the presence of a groundwater table fed by the percolation, and computed by the groundwater module.

4. The evapotranspiration module Although the Penman-Monteith equation is the most rigorous theoretical description for this component, in practice many simplifications are necessary because in most countries the required historical data for its estimation are not extensively available, and, in addition, apart from a few meteorological stations, almost nowhere are real-time data available for flood forecasting applications. Moreover, it should be clearly understood that evapotranspiration plays a major role in the rainfall-runoff process not in terms of its instantaneous impact, but in terms of its cumulative temporal effect on the soil moisture volume depletion; this reduces the need for an extremely accurate expression, provided that its integral effect be well preserved. In the ARNO model, the effects of the vapour pressure and wind speed are explicitly ignored and evapotranspiration is calculated starting from a simplified equation known as the radiation method (Doorembos et al., 1984): ETod = C, W,R, = C, W, (0.25 + 0.50 ;;> R,

(17)

where ETod is the reference evapotranspiration, i.e. evapotranspiration in soil saturation conditions caused by a reference crop (mm day-‘); C, is an adjustment factor obtainable from tables as a function of the mean wind speed; W,, is a compensation factor that depends on the temperature and altitude; R, is the short-wave radiation measured or expressed as a function of R, in equivalent evaporation (mm day-‘); R, is the extraterrestrial radiation expressed in equivalent evaporation (mm day-‘); n/N is the ratio of actual hours of sunshine to maximum hours of sunshine (values measured or estimated from mean monthly values as described in Appendix B). Hence the calculation of R, requires both knowledge of R,, obtainable from tables as a function of latitude, and knowledge of actual n/N values, which may not be available. In the absence of the measured short-wave radiation values R, or of the actual number of sunshine hours otherwise needed to calculate R, as a function of R, (see Eq. (17)), an empirical equation was developed that relates the reference potential evapotranspiration ETod,computed on a monthly basis using one of the available simplified expressions, to the compensation factor W,, the mean recorded

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175 (1996) 339-382

351

temperature of the month T and the maximum number of hours of sunshine N. The developed relationship is linear in temperature (and hence additive), and permits the dissaggregation of the monthly results on a daily or even on an hourly basis, whereas most other empirical equations are ill-suited for time intervals shorter than 1 month. The relation used, which is structurally similar to the radiation method formula in which the air temperature is taken as an index of radiation, is ET0 = cx + PNW,,T,,,

(18)

where ET0 is the reference evapotranspiration for a specified time step At (in mm (At)-‘); cxand ,L3are regression coefficients to be estimated for each sub-basin; T,,, is the area mean air temperature averaged over At; N is the monthly mean of the maximum number of daily hours of sunshine (tabulated as a function of latitude). W,, for a given sub-basin can be either obtained from tables or approximated by a fitted parabola: W,,=AT2+BT+C

(19)

where A, B and C are coefficients to be estimated; T is the long-term mean monthly sub-basin temperature (“C). Further details on the estimation of the evapotranspration parameters are given in Appendix B.

5. The snowmelt module

Again, for reasons of limited data availability, the snowmelt module is driven by a radiation estimate based upon the air temperature measurements; in practice, the inputs to the module are the precipitation, the temperature, and the same radiation approximation which is used in the evapotranspiration module. Given the role that altitude may play in combination with the thermal gradient, the sub-catchment area is subdivided into a number of equi-elevation zones (snow-bands) according to the hypsometric curve, and for each zone simplified mass and energy budgets are continuously updated. For each snowband the following steps, similar to those adopted in SHE (Abbott et al., 1986a,b), are followed: (1) estimation of radiation at the average elevation of the snowband; (2) decision on whether precipitation is solid or liquid; (3) estimation of the water mass budget based on the hypothesis of zero snowmelt; (4) estimation of the energy budget based on the hypothesis of zero snowmelt; (5) comparison of the total available energy with that sustained as ice by the total available mass at 273°K; (6) computation of the snowmelt produced by the excess energy; (7) updating of the water mass budget; (8) updating of the energy budget. 5.1. Estimation

of radiation at the average elevation of the snowband

The estimation of the radiation is performed by re-converting the latent heat (which has already been computed as the reference evapotranspiration ETo) back into radiation, by means of a conversion factor C,, (kcal kg-‘), which can be found in

352

E. Todini 1 Journal of Hydrology 175 (1996) 339-382

any thermodynamics

textbook as

C,, = 606.5 - 0.695(T - T,,)

(20)

where TOis the temperature of fusion of snow (273째K). In addition, to account for albedo, which plays an extremely important role in snowmelt, it is necessary to apply an efficiency factor, which will be assumed approximately as r] = 0.6 for clear sky and n = 0.8 for overcast conditions; this leads to the following estimate for the driving radiation term: Rad = ~[606.5 - 0.695( T - TO)]ETO

(21)

Given the lack of information concerning the status of the sky when simulating with historical data, for practical purposes it is generally assumed in the ARNO model that the sky is clear if there is no precipitation and overcast if precipitation is being measured. 5.2. Decision on whether the precipitation is solid or liquid Information concerning the status of precipitation (solid or liquid) is rarely available as a continuous record; therefore it is necesary to define a mechanism, mainly based upon the air temperature measurements and the historical precipitation. If one plots the frequency of the usually scattered observations with which precipitation was observed to be liquid or snow as a function of the air temperature, a Gaussian distribution is generally obtained, with a mean value T, which very seldom will coincide with TO. For this reason, the following rules are adopted: precipitation is taken as liquid if the air temperature T > T,; precipitation is taken as snow otherwise. The value of T, (which generally ranges between 271 and 275째K) must be derived, as mentioned above, by plotting the frequency of the status of historically recorded precipitation as a function of the air temperature. 5.3. Estimating the water mass The water equivalent mass is estimated with the following simple mass balance equation, where all quantities are expressed in millimetres of water:

Zr;& = 2, + P

(22)

The water equivalent at the end of the time step is designated with a star because it is a tentative value which does not yet account for the eventual snowmelt. 5.4. Estimating the energy Similarly to the mass, the energy is estimated in the following way, by computing the increase (or decrease) of total energy E: if the precipitation is zero, E&at = El + Rad

(23)

E. Todini / Journal of Hydrology 175 (1996) 339-382

if the precipitation E&

353

is non-zero and the precipitation is in the solid phase (T < T,),

= Et + Rad + CSgToP

(24)

if the precipitation is non-zero and the precipitation is in the liquid phase (T > T,), E*t+~t = El + Rad + [CsiTo + Cy + C,,(T - To)]f’

(25)

Again, the total energy at the end of a time step is designated with a star to denote a tentative value; in the previous equations CSi is the specific heat of ice (0.5 kcal “K-’ kg-‘), C, is the heat of fusion of water (79.6 kcal kg-‘) and C,, is the specific heat of water (1 kcal “K-’ kg-‘). 5.5. Estimation of snowmelt and updating of mass and energy state variables If the total available energy is smaller than or equal to that required to maintain the total mass in the solid phase at the temperature To, i.e. CsiZ:+AtTo 2 ,!?:+A~,it means that the available energy is not sufficient to melt part of the accumulated water, and therefore R,

&+At

= z:+At

E t+At

(26)

E,*,A~

where R, is the snowmelt expressed in millimetres. If the total available energy is larger than that required to maintain the total mass in the solid phase at the temperature To, it means that part of the accumulated water will melt, and therefore the following energy balance equation holds: Csi(Z,*,At

- Km) To = G+A~

- (~J~To + C/f&n

(27)

from which the snowmelt and the mass and energy state variables can be computed as E’ t+At - C~iT&+A,

&,, =

C/f

Z r+At - zt+At

- &m

E t+Ar

G+A~

(CsiTo

(28) + Clf)&m

6. The groundwater module The groundwater module represents the overall response of its storage by means of a cascade of linear reservoirs, characterized by two parameters: the number of reservoirs n and their time constant k, a model which is well known in hydrology as the Nash model (Nash, 1958). For practical reasons, instead of using the gamma

E. Todini / Journal of Hydrology 175 (1996) 339-382

354

distribution function to express the impulse response, a numerical procedure has been adopted. The expression to be used can easily be derived from the assumption of a cascade of linear reservoirs where the volume of the ith reservoir is proportional to its outflow, i.e. S, = kBf; thus, the continuity equation written for the generic ith reservoir becomes

which, after discretization t + At/2, becomes

;(BaAr_ Bf)

in time with a finite difference scheme centred at time

&i-,

+ Bf-1

‘+A;+B’

(30)

and the required expression is then obtained by making B:+A~explicit: 2k - At Bi + B’t+b =2k+At f &

(B&A, + Bf’)

(31)

where B’ is the outflow from the ith reservoir; B” = Z is the percolation given by the soil moisture balance equation; B” = B is the resulting base flow. Eq. (31) is then recursively solved n times at each time step.

7. The parabolic routing module As outlined in Section 2, the hillslope routing and channel routing of distributed inflows are both performed using a distributed inflow linear parabolic model, whereas channel routing of upstream inflows to a sub-basin is performed by means of a concentrated input parabolic model. The parameters of the two linear parabolic transfer functions can be estimated as a function of the slope, the length of the drainage system and the roughness. 7.1. Routing of upstream inflow The propagation of inflows from upstream is carried out by means of a linear model consisting of a parabolic unit hydrograph deriving from the analytical integration of the unsteady flow equations when the inertia effects are ignored and the coefficients are linearized around mean outflow values. In this case the following differential equation is obtained:

ae=.a’e_,ae at

ax2

(32)

where D and C are the diffusivity and the convectivity coefficient, respectively. The discrete time solution of Eq. (23) for mean values of the coefficients C and D is (Todini

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of Hydrology 175 (1996) 339-382

355

and Bossi, 1986)

VA&,

t + At) =

-&[ZF(t + At) -

2ZF(t) + ZF(t -

At)]

(33)

where IF(t) = jF(O)dO=

jlr&)didS

(34)

with uax, the impulse response relevant to Eq. (34), which can be written as (35) Integrating Eq. (34), after substitution from Eq. (35) one obtains

ZF(t)=i{(Ct-Ax)iV[-Giz]

+(Ct+Ax)exp(y)

(36) where N(*) is the value of the standard normal probability distribution in (*), and Ax is the length of the channel reach. 7.2. D&se

lateral inflow routing

The runoff generated in each sub-basin moves first along the slopes and then reaches the drainage network as diffuse inflow, which, for simplicity, may be considered uniformly distributed. In this case, the following differential equation applies:

-DaZQ

ax2

cdQ __-= aQ dx at

--(A

(37)

where q is the lateral inflow per unit length. The discrete time solution of Eq. (37) for mean values of the coefficients C and D is (Franchini and Todini, 1989) m&t,

t + At) = -$[ZG(t

+ At) - 2ZG(t) + ZG(t - At)]

(38)

where ZG(t) = ; [t2 - ZZF(t)]

(39)

with ZZF(t) = jZF(E)d[= 0

jjP(O)dt9d< 00

= jjj~&)drdRdf 000

(40)

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356

Integration of Eq. (40), after substitution of uAx from Eq. (35), yields &&lZ)

++$C2$ezp(%F)]}

-+$)t$wo)]}

(41)

The results provided by Eq. (38) when substituted for IG(t) given by Eq. (41), differ from those obtained by Naden (1992), in that they represent the response of a discrete time system to a time discretized input (see Todini and Bossi, 1986). As mentioned above, the parameters of each of the models are naturally composed of the two convectivity (C)and diffusivity (D) coefficients for each response function which are related to the dimensions, slopes and lengths of the individual sub-basins. In general, very small values for D are assumed for the hillslopes, given the marked kinematic nature of the phenomenon; generally, D increases with the size of the river and with the inverse of the bottom slope. Table 1 gives an indication of the orders of magnitude of C and D.

8. Model calibration requirements To adapt the ARNO model to a specific basin, the following data are needed: (1) an orographic map of the basin, at an appropriate â&#x20AC;&#x2DC;manageableâ&#x20AC;&#x2122; size (1:25 OOO1:lOO000) depending on the actual size of the catchment showing the hydrographic network. (2) Continuous historical records of precipitation, temperature and river levels at several measurement stations within the catchment, sampled at the appropriate time steps (1 h for sub-catchment sizes of 200-300 km2 or 3 h for sub-catchments of the order of 1000-2000 km2 as an upper limit). The length, of these records must be sufficient to include periods both of dry and wet soil moisture conditions. When the model is used for simulating historical data, it is advisable to use at least 2 or 3 years of data as a calibration period. Nevertheless, when the model is part of an Table 1 Approximate initial guess for parameters C and D

Hillslopes Brooks Rivers Large rivers

l-2 l-3 l-3 l-3

l-100 100-1000 looO-loo00 10000-100000

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on-line real-time flood forecasting system, 1 year of data may be suIlicient to start, and additional calibrations can be performed after 1 or 2 years of real-time operations. This is generally advisable because the historical observation network rarely coincides with the real-time observation network. (3) Historical records, not necessarily continuous in time, of monthly average temperature, for at least 10 or 20 years, should be acquired for estimating the longterm potential evapotranspiration as described in Appendix B. Whenever possible, data should be collected for the same stations as referred to in (2), although a sub-set can eventually be used. (4) Rating curves for all the hydrometric stations where the discharge is to be simulated. (5) Geographical coordinates and elevation above mean sea-level, for all the measurement stations. (6) Whenever possible, a soil map and a land use map (a coarse scale would be enough) should be used to obtain an impression of the order of magnitude for the various sub-catchments’ soil moisture capacity parameter values W,,,. The calibration phase starts by resealing the measured temperatures at the average sub-catchment elevation, by means of a thermal gradient, and then proceeding to the estimation of the potential ’ evapotranspiration using the technique described in Section 4. After dividing the basin into a number of sub-catchment units following the river geomorphology, a connection tree can be identified and its topological characteristics used to generate the correct succession of operations and routing from the uppermost sub-basins to the downstream closing section. For each subcatchment, one has then to determine the hypsographic curve from the available orographic information, and all the precipitation and temperature data must be averaged in space (for instance, according to Thiessen polygons), over each of the sub-catchments. After these preliminary activities, initial values are assigned to the model parameters. From the geometrical characteristics of all the sub-catchments, the average length of slopes can be determined, on the assumption of the ‘open book’ schematization, as L, = A/2L, with L the length of the main channel in the specific sub-catchment. An initial guess for the values of the parameters C and D can be roughly given as C M 2 and D x l/So, where S0 is the average terrain slope or the average bed slope, depending on whether the hillslopes or the river are considered. When all the routing parameters have been assigned, all the necessary transfer functions are evaluated and are then provided to the ARNO model in the form of unit graphs. An initial guess for all the other model parameters must also be provided. Initially, in the absence of soil and vegetation maps, the same value is used for all the sub-catchments; it is not easy to give good starting values, but one could start by considering that the catchment average soil water storage W,,, generally ranges between 50 and 300 mm and the other soil moisture curve parameter 6, which expressses the degree of homogeneity of soil characteristics, should generally be taken between 0.1 and 0.01. The initial soil moisture condition W is not of great relevance, because a wrong assumption for this will be evident after the first calibration run. Unfortunately, there are no fixed rules for initializing the drainage curve

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parameter values, except that c should be taken between one and two, but after one or two runs one will be able to reproduce the recession of hydrographs by adjusting the initial guess for the drainage parameter values. Finally, after a few calibration trials, both the long-term water balance and the shape of the baseflow will be easily matched by acting on the percolation and on the groundwater model parameter values. Calibration is not performed automatically; the reason is that, in most situations, the available information does not allow for resolving the parameter values, unless additional knowledge of the system as well as hydrological understanding is introduced in the estimation phase. For instance, in the case of two upstream subcatchments, both contribuing to the outflow in a downstream section which is to be used for calibration, it is very unlikely that an automatic procedure can correctly estimate the differences between the parameter values for the two sub-catchments. In this case, one has to find one or more precipitation events when precipitation was occurring on one of the two and use that event (or those events) for adjusting the parameter values of the specific sub-catchment. Calibration proceeds from the upstream sub-catchments to the downstream ones. The first step requires the assessment of the routing parameters. This is done by routing individually, to the downstream closing section, the flood waves produced by the upstream ungauged sub-catchments (i.e. without combining their effects) and by comparing the results with the observed discharges. It is relatively easy to see how the different flood waves add together and to identify those with substantially wrong parameters in terms of travel time or subsidence. After this first assessment of the routing parameters, attention is focused on the calibration of the two sets of most important model parameters. The first set refers to the shape parameter b of the soil moisture storage curve and the maximum soil moisture storage W,,,, and the second set relates to the parameters of the drainage curve. Although interdependence exists among the parameters, it is generally not difficult to reach, after a few trial-and-error iterations, a good combination of parameters, given that their effects are easily detectable. A very large value for the soil moisture curve shape parameter (i.e. b > 1) will produce peaky runoff responses even when the soil is mostly dry, and a large value for W,,, can reduce the outflow by a factor of 100%. Furthermore, the drainage parameters are mainly associated with the recession curves of the observed hydrographs, and again, even if some interaction with the soil moisture storage exists, it is not too difficult to reproduce the observed effects. With respect to the estimation of the other parameters relevant to the percolation and the groundwater storage, they tend to be easily identified on long recession periods and by matching the overall mass balance. Needless to say, the most important aspect in calibrating a continuous time rainfall-runoff model must be its capacity to reproduce the â&#x20AC;&#x2DC;entireâ&#x20AC;&#x2122; record, including minor flood events and long dry spells, and not just a few large flood events. This will give a better guarantee of good operational performance, because the model learns the different ways of reacting to rainfall events as a function of its initial soil moisture conditions. The effect of the initial conditions on runoff production can only be disregarded in urban or in mostly impervious catchments. In fact, if one compares the amount of available soil moisture

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359

storage (50-300 mm) with maximum rainfall intensities (in most cases smaller than 100 mm h-‘) one can determine the different responses obtained by 100 mm of rainfall over a dry soil or over an entirely saturated soil. After calibration of the upstream sub-catchments, one can proceed downstream, following the river network connection tree, until it reaches the downstream closing section. Calibration is repeated, checking that the water balance is matched between observed and computed discharges, until a reasonable value for the explained variance is obtained. In general, if the available hydrological data are of good quality as well as representative of the spatial variation, this reasonable agreement ranges between 85% and 90% of explained variance for upstream small sub-catchments and between 90% and 98% for the downstream larger ones, with very small biases.

1’

“,

. R

N ‘43 ;

“irr(“’ j CANA

Fig. 8. The Amo river catchment.

360

E. Todini / Journal of Hydrology I75 (1996) 339-382

Fig. 9. The Arno river catchment

closed at Nave di Rosano

and its division

into sub-catchments.

E. Todini / Journal of Hydrology I75 (1996) 339-382

361

9. Examples of application of the ARNO model The ARNO model has been applied to a number of river basins both as a catchment model and as the basis for a real-time flood forecasting system. The calibration results, although strongly affected by the quality of the available data, are generally more than adequate for the simulation of the catchment responses to rainfall, with explained variances ranging from 90% to 98% without the need for an automatic parameter estimation procedure based upon the minimization of a quadratic function of residuals. As mentioned above, this is done on purpose to preserve the physical meaning of the parameters (Todini, 1988a).

SIEVE

Fig. 10. Schematic representation of the Amo river sub-catchments in the ARNO model.

362

E. Todini/ Journal of Hydrology 175 (1996) 339-382

2500 ---------

Obaawed

discharges

Computed

discharges

2000

1500

1000

500

0 1

Fig. 11. Amo at Nave di Rosano. Calibration: December 1959.

Fig. 12. Amo at Nave di Rosano. Validation: January 1960.

363

E. Todini / Journal of Hydrology I75 (1996) 339-382

1000

--___-500

-l-7-7--T-l--r--7--r-1-1-l-T-

Fig. 13. Amo at Nave di Rosano. Validation: February 1960.

2500r----------.----

-.-----

2000

1500

Observed discharger G,qut.d discharges

-~.-_.-.___

--_-_-

1000

Fig. 14. Amo at Nave di Rosano. Validation: March 1960.

364

E. Todini / Journal of Hydrology 175 (1996) 339-382

9.1. The ARNO application

The ARNO model was originally developed for the Arno river in Tuscany (Fig. 8) closed at Nave di Rosano, a few kilometres upstream of Florence. The catchment area, of approximately 4000 km*, was divided into a number of sub-catchments as shown in Figs. 9 and 10. The calibration of the model was performed using the few continuous records of rainfall, temperature and water levels (sampled at 1 h intervals) which were available in computer compatible form at the time, and rating curves for five stations. The calibration period considered was the month of December 1959 (Fig. 1l), when a succession of medium-sized floods was observed. Three successive months (January, February and March 1960) were then used as the validation period (Figs. 12-14). Although the calibration and the validation periods were limited, according to the previously stated requirements, the results were more than adequate, presumably as a consequence of the exceptional quality of the data. The statistical analysis of the results for the calibration and validation periods is summarized in Table 2, where EV stands for explained variance and DC for determination coefficient (see Franchini et al. (1996) for definitions). The quality of calibration can easily be seen, both from Table 2 and the graphical comparison of simulated and observed flows in Fig. 11. The statistics in Table 2 and the graphical comparisons in Figs. 12-14 for the validation period indicate a comparable level of agreement to that obtained for the calibration period. Moreover, an additional validation test shows that the validity of the calibration remained unchanged after 20 years. Fig. 15 demonstrates good agreement between simulated and observed flows, and also shows the reconstruction of the flows during a flooding period in 1981, when the recording level station was not operational for a number of hours (steady discharge measures during the latest flood event). After calibration on the Arno river, the model was compared with other existing conceptual models by Franchini and Pacciani (1991) and Franchini et al. (1996) on the Sieve river, a tributary of the Arno (note that, for the comparison, the Sieve catchment was not divided into sub-basins, which explains a small difference in the Table 2 Statistical analysis of ARNO model residuals Catchment

Amo model Calibration

Amo-Nave di Rosanoâ&#x20AC;&#x2122; Amo-Nave di Rosanob Sieve-Fomacina Amo-Subbiano Chiana-P. Ferrovia

Validation

0.922 0.943 0.886 0.901 0.891

0.922 0.943 0.878 0.885 0.886

0.932 0.949 0.866 0.901 0.822

0,922 0.939 0.865 0.898 0.805

â&#x20AC;&#x2122; Rainfall over the entire catchment. b Rainfall over Valdamo, Ambra and Sieve only.

.----_--- ---__--r 365

E. Todini / Journal of Hydrology 175 (1996) 339-382

2500

--------

---

2000

1500

---.---

0bsarve.d

discharges

Computad

disqhargas

-----.---__----

I- __--_-_----_

------A..----___

r.--._--

1000

_____,_(_

..-..-

leve.1km@e

stuck

500

-iv--

0 i

I 3

111-T-T-m 9

Fig. 15. Amo at Nave di Rosano. Split sample test: December 1981.

ARNO model results when compared with Table 2). From Table 3, TOPMODEL (Franchini et al., 1996) and the ARNO model clearly perform best, and, although TOPMODEL performs better in the calibration period, the ARNO model shows a smaller degradation in performance in the validation period, indicating good predictive stability.

Table 3 Statistical analysis of different model residuals Catchment

Sieve Fomacina results Calibration

ARNO TOPMODEL Xinanjiang Stanford IV Sacramento Tank APIC SSARR

Validation

0.888 0.914 0.880 0.843 0.836 0.875 0.776 0.867

0.880 0.912 0.840 0.830 0.821 0.856 0.751 0.829

0.853 0.852 0.822 0.845 0.835 0.847 0.820 0.834

0.851 0.846 0.821 0.844 0.833 0.845 0.779 0.824

366

E. Todini / Journal of Hydrology 175 (1996) 339-382

9.2. Real-time forecasting applications

Following its derivation and application to the Arno river data, the model has become the basic component of the European Flood Forecasting Operational Real-Time System (EFFORTS), a fully automatic on-line computer package for real-time flood forecasting developed under a CEC-funded R&D project for the study and the implementation of a real-time forecasting scheme for the Fuchun river (ET&P, 1992), whose catchment lies in the Zhejiang province in mainland China, SW of Nanjing and Shanghai. The application of the ARNO model can be considered fairly successful, given that, although the number of raingauges (23) was limited when compared with the size of the cat&n-rent area to be modelled (18230 km2), in, the calibration phase the model explained 96% of the total observed runoff variance, and Fig. 16 also shows the good quality of the results. A real validation period is not available, as, in practical applications, all the available data are generally used to improve calibration; nevertheless, the following comments were reported by Professor Liu Gu Chong (personal communication, 1994), the Chinese Project Leader, after the second year of on-line running: â&#x20AC;&#x153;From a flood forecasting point of view, the system had experienced several floodings occurred during the reported period. Undoubtedly the flood forecasting softwares, EFFORTS and HFS, were put into use for those floodings and satisfactory forecasts were obtained. For example, for the flooding of June 19, 1993 a forthcoming discharge of 11 100 cubic meters per second at Lanxi station was forecast, compared with an actually

_____

Fig. 16. Fuchun at Lan Xi. Calibration period: 24 April 1992-1 August 1992.

obwrvad compiled

E. Todini / Journal of Hydrology 175 (1996) 339-382

367

measured value of 11200 cubic meters per second. Such a forecast not only helped the action of flood mitigation measures within the basin, but increased the power production by regulating the Fuchun hydropower plant.â&#x20AC;? It should be noted that both the EFFORTS package and the HFS (Hydrological Forecasting System), developed under sub-contract by the Laboratoire dâ&#x20AC;&#x2122;Hydraulique de France, operate with the ARNO model. In 1993, the ARNO model was applied to three rivers in Italy: the upper Tiber in Central Italy, and the Adda and the Oglio rivers in Northern Italy. The application to the Tiber was performed on behalf of the Italian National Electric Company (ENEL), to control better the reservoir of Corbara which closes a catchment area of approximately 5250 km*. In Fig. 17 the results of the model are compared with data for the station of Ponte Nuovo (4179 km*), where the explained variance is 90%. The relatively low value of the variance is mainly associated with the lack of spatial coverage of the rain gauges; the number has recently been increased and a new calibration is under way. Application of the model to the Adda and Oglio rivers was also aimed at controlling the levels of the Lake Como and of Lake Iseo, respectively. In this case, not only were very few flow data available for calibration, but also all the natural low flows were considerably modified by the large number of interconnected upstream reservoirs, which made it impossible to provide a meaningful measure of the agreement between observed and computed values. Nevertheless, although no information is available for the Iseo river, the reported

Fig. 17. Tevere at Ponte Nuovo. Calibration period: 1 April 1991-31 December 1991.

368

E. Todini / Journal of Hydrology I75 (1996) 339-382

operational performances of the model on the Adda river are rather impressive. If one considers that the operational range of Lake Como is 1.3 m, it is possible to appreciate the following performances described by Eng. Luigi Bertoli (personal communication, 1994), the Technical Director of the Adda Consortium, who is in charge of the regulation of the Lake Como gates: â&#x20AC;&#x153;In the only available flood, good forecasts were obtained particularly for the Lake Como levels; for instance the level (in cm) in Malgrate above the hydrographic zero was: Time of forecast issue: Sept. 13, 1994, 01 p.m.; present level +53. 12 hours in advance forecast (Sept. 14,01 a.m.): Forecast +69, Observed +68. Time of forecast issue: Sept. 14, 1994, 03 p.m.; present level +82. 12 hours in advance forecast (Sept. 15,03 a.m.): Forecast +106, Observed +104.â&#x20AC;? In 1994 the ARNO model was also applied to the upper Danube (4037 km*) on behalf of the Baden-Wurttemberg flood forecasting centre. The model performances are adequate in this case, with 94% of explained variance, and the results for the measurement station of Berg are shown in Fig. 18. Again, in 1994, a new calibration was performed on the Arno river using the data collected by the new real-time telemetering data acquisition system recently installed by the Tuscany Regional Government. The new calibration substantially improved the previous one, with an explained variance of 96%. Fig. 19 shows the entire calibration record, and Fig. 20 enlarges the rainy season to give a better appreciation of the results. One can observe that most of the errors produced by the model are a consequence of the upstream reservoir operations, which are evident in the medium or low flow periods and

Fig. 18. Donau at Berg. Calibration period: 1 March 1993-31 December 1993.

E. Todini / Journal of Hydrology 175 (1996) 339-382

369

Fig. 19. Amo at Nave di Rosano. Calibration period: 1 January 1992-31 December 1992.

â&#x20AC;&#x2DC;.

obsmed --

compihd

Fig. 20. Amo at Nave di Rosano. Calibration period: 1 September 1992-31 December 1992.

370

E. Todini/ Journal of Hydrology I75 (1996) 339-382

disappear during flood conditions; the model could not account operations because of the lack of knowledge of the operating rule. At present, several other operational applications of the ARNO way in Italy, such as the model of the Reno river, the model of the the extension of the Tiber model from Corbara to the river mouth, warning system developed for the City of Rome.

for the reservoir model are under Ticino river, and as part of a flood

9.3. Land use change application Within the framework of the NERC-ESRC Land Use Programme (NELUP) of research being undertaken at Newcastle University, the ARNO model has been fully integrated into a decision support system (DSS) designed to assess the impacts of future land use changes on the hydrology, ecology and economics of large river basin systems. Its role within the DSS is to screen rapidly alternative future land use change scenarios for significant shifts in hydrological behaviour; significant changes are then explored in more detail using the distributed flow and transport modelling system SHETRAN. Alternative futures can include both a redistribution of land cover types and new land and water management strategies. As a test of the ARNO model, and the DSS as a whole, the system has been applied to two river basins in the UK: the Tyne River basin in NE England and the Cam River basin in East Anglia. The two basins are hydrologically very different. The Tyne basin, which covers an area of 2000 km2, is primarily an upland catchment possessing little groundwater and a rapid surface response to rainfall and snow melt events. Water management is restricted to controls at a small number of reservoirs. The smaller Cam basin covers an area of approximately 800 km2 and includes both groundwater and surface water dominated sub-catchments. Water management and agricultural activity are both intensive. For the adequate representation of both catchments, the ARNO model has been adapted to include more complete representations of groundwater storage and movement, land cover distribution, and water abstractions and effluent returns. It has been very successfully applied to the surface water dominated sub-cat&rents, and tests on the groundwater-dominated sub-catchments have shown acceptable but not complete agreement between the simulated and observed runoff for a range of meteorological conditions including dry, wet and average years. For further details, the reader is referred to Adams et al. (1995). 9.4. Land-surface-atmosphere

applications

Interesting research results have been obtained with the ARNO model in representing the land-surface-atmosphere exchanges within the framework of global climate change studies. A noticeable improvement in the behaviour of the ECHAM general circulation model (GCM) was obtained by substituting the ARNO soil moisture balance component (Diimenil and Todini, 1992) for the Manabe bucket model (Manabe, 1969). In this case, the availability of a major rivers data bank (the Global Runoff Data Centre-GRDC) provided the opportunity to compare not only the

E. Todini 1 Journal of Hydrology 175 (1996) 339-382

371

precipitation, evapotranspiration and temperature fields produced by the ECHAMGCM with those available from climatological atlases, but also the predicted monthly river discharges with those observed, a step in the calibration of GCMs which is now considered of the utmost importance. Successively, the ARNO model was compared with a number of different land surface-atmosphere schemes used in GCMs, within the frame of the EC-funded research project SLAPS (Soil-Land-Atmosphere Process Simulation) and, while preserving the overall meteorological exchanges, it was found to produce more realistic outflow discharges than the schemes generally used by meteorologists (Polcher et al., 1995); this effect was also confirmed in a comparison of the ARNO scheme with the UK Meteorological Office soilatmosphere column model on the basis of the River Thames data (Rowntree and Lean, 1994).

10. Conclusions and future perspectives As a result of the increased collaboration between meteorologists, hydrologists and soil scientists, several studies are now under way aimed at developing a new parameterization of soil-atmosphere exchanges. Several international programmes, such as the Global Energy and Water Cycle Experiment (GEWEX) or the International Geosphere-Biosphere Programme (IGBP) with its Core Project â&#x20AC;&#x2DC;Biospheric Aspects of the Hydrological Cycleâ&#x20AC;&#x2122; (BAHC), deal with the problem. The possibility of using topographical information to parameterize as well as to evaluate the lumped model parameter values, a concept originally introduced by Beven and Kirkby (1979) with TOPMODEL, is attracting the interest of scientists, given the practical impossibility of extending to the required scales the distributed differential models such as the SHE or the IHDM model. Although improvements in the rainfall-runoff process representation can be sought mostly through lumping in space at the different scales the model parameters, as a function of the distributed topographic, pedologic and topologic information, nevertheless, the TOPMODEL approach still retains a number of physical inconsistencies (Franchini et al., 1996), which may prevent its generalized use. From a detailed analysis of the effects of lumping in space and a comparison of the physical behaviour of both the ARNO model and TOPMODEL, Todini (1995) recently proposed the TOPKAPI approach, which transforms the rainfall-runoff process into two non-linear reservoir differential equations. Both derive from the integration in space of the non-linear kinematic wave model; the first represents the drainage in the soil and the second represents the overland flow on saturated or impervious soils. The parameter values of the model are shown to be scale dependent and obtainable from digital elevation maps and soil maps in terms of slopes, soil permeabilities and topology (Todini, 1995). In the mean time, when choosing among the recently available models, and in particular between the ARNO model and TOPMODEL, in spite of the enthusiasm of TOPMODEL proponents for its immediate matching with geographical information systems as well as with rasterized radar images, which may give the illusion of operating in distributed system form, one has to recognize that the

372

E. Todini / Journal of Hydrology 175 (1996) 339-382

ARNO model allows for a more comprehensive approach to rainfall-runoff modelling. In addition, the integration of. the ARNO model components into a standard and well-documented package not only made possible the application of the model to a wide variety of medium and large size catchments, but also allowed for its operational use. Operational packages must in fact not only perform well, but also be simple, well documented and require extremely simplified calibration operations, possibly in line with those traditionally applied by hydrologists. Last but not least, the ARNO model, which has shown high flexibility by achieving fairly good calibrations in a variety of different climate and geomorphological conditions, has shown a noticeable stability of results, in that the drop in performance between calibration and validation periods is always very small. Until the time comes, and it could be tomorrow, that a new scheme is demonstrated to improve either the physical interpretation of the phenomena or to simplify and improve the calibration requirements, it is the belief of the author that the ARNO model constitutes a robust operational tool also providing, in an extremely simplified form, a reasonable interpretation of the overall rainfall-runoff process at the catchment scale.

Appendix A. Derivation of the formulae for the soil moisture balance module in the ARNO model A sub-basin of given surface area ST (excluding the surface extent of water bodies such as reservoirs or lakes) is, in general, formed by a mixture of pervious and less pervious areas, the response to precipitation of which will be substantially different. For this reason, the total area ST is divided into impervious area S1 and pervious area s,: ST = s, i- s,

(Al)

s - s,

x=gq

(AZ) then x, defined

(A3)

will indicate the percentage of pervious area at saturation. Zhao (1977) demonstrated that the following relationholds reasonably well between the area at saturation and the local proportion of maximum soil moisture content w/w,, where w is the elementary area soil moisture at saturation and w, is the maximum possible soil

E. Todini / Journal of Hydrology I75 (1996) 339-382

373

moisture content in any elementary area of the catchment: b

(A4) This is similar to defining a cumulative distribution saturation, as shown in Fig. 6, defined as

for the local soil moisture at

w = Wm[l- (1 - x)“]

(A5)

Following the description in Section 3, an analysis was first carried out for the case where the precipitation P is larger than the potential evapotranspiration ET,, and therefore the actual evapotranspiration ET, is assumed to coincide with the potential, i.e. ET, = ETP

646)

In this case, the ‘effective’ meteorological input M,, defined as the difference between precipitation and potential evapotranspiration, is larger than zero: M,=P-ET,=P-ET,>0

047)

With reference to Fig. 7, the surface runoff generated by the entire catchment is obtained as the sum of two terms: the first is the product of the effective rainfall times the percentage of impervious area, and the second is the average runoff produced by the pervious area, which is obtained by integrating the soil moisture capacity curve, to give h4,+tV

40 dt

if M,+w<w,

648)

or 649) Substituting x from Eq. (A4) into Eq. (A8) gives

(Al01

E. Todini / Journal of Hydrology 175 (1996) 339-382

374

Eq. (AlO) can be integrated by substitution as follows:

,=1-L

WOl’

C=wm(l -q),

dc= -w,dn

q=l_W

c=o,

Wi?l i

t =

p,l_M,.tw

to give ~-~w/Y?l)

R=ikf,+-w - SI ST

nbdn = 44, +

ST '-(‘+fe+w%n EM

+sT-% STb+l

[(l

--K)b+l-(l

-y)b+‘]

(Al 1)

The catchment average amount of soil moisture content can then be evaluated by integrating Eq. (A5), to give W=jw(<)dt+(l-x)w=

w”j,l

- (1 -#]d5+(1

-.x)w,,,[l - (1 -x)“]

=w,+w,/(l-#dc+w,-w,x-w,(l-x)?

=wm[l-(l:#-t(l-&d[]

(Al2)

Integration of Eq. (A12) can also be performed by substitution as follows: <=1-q,

rl=l-J, c=o,

d‘$=-dn

r]=l

{ E = x7 7/=1-x to give

1-(1-x)9-

= w,

j&+&(l-x)y]

=5[1-(1-x)Y]

[

(Al3)

E. Todini / Journal of Hydrology I75 (1996) 339-382

If all the pervious area has reached saturation, x = 1, Eq. (A13) becomes

w,=wâ&#x20AC;?

b+l

which can be rewritten as W, = (b+ l)Wm

(Al5)

Substitution of Eq. (A15) in Eq. (A13) gives w=

W,[l -(l

-x)Y]

(Al(j)

Therefore the percentage of pervious surface area that has reached saturation as well as the percentage of unsaturated area become respectively

(A17)

Substituting for x from Eq. (A4) into the first equation of (A17) gives the relationship between the average and the elementary area soil moisture storage quantities: b+l

Eq. (A18) can also be written as (A19) from which one can obtain

and finally, bearing in mind Eq. (Al 5) one finds the value of the elementary area soil moisture at saturation, corresponding to a specific percentage x of the pervious surface, with x written in terms of the catchment average amount of soil moisture content as in Eq. (A17): w=(b+l)W,,,[l-

(l-E)&]

6421)

Returning to Eq. (Al 1) and substituting for w and w, from Eqs. (A21) and (Al5),

E. Todini / Journal

376

of Hydrology I75 (19%) 339-382

respectively, one obtains sT

R-M,+-

-I Wi?l

(1-F > m

=M,+

b+l

M,+(b+l)wJl-

cl-aâ&#x20AC;?]

(b+ l)W,

(Wm- W)-

(b+l)W,,,-Me--(b+l)W,+(b+l)W, X (b+

1)Wm

(A221 Eq. (A22), after simple algebraic manipulations, allows for estimating the runoff when M, + w < w,, or in terms of the catchment average quantities when

M,<(b+l)W,,,

(A23)

as R=M,+ 6424)

In the other case, when M, + w 2 w,, or in terms of the average quantities when 6425)

the integral appearing in Eq. (A9) is easily evaluated in terms of the catchment average quantities, given that the integral appearing in Eq. (A9) can be written as W,,, - W, which gives R= sâ&#x20AC;&#x2122;M,+ ST

VIM,-(W,-

W)]

W6)

E. Todini 1 JournaL of Hydrology 175 (19%)

339-382

377

Eq. (A26) can be finally rewritten as

R=M,-qqw,-

(A271

An analysis is now carried out of the case where the precipitation P is smaller than the potential evapotranspiration ETP and the effective meteorological input M,, becomes negative, i.e. M,=P-ET,<0

(A281

ET, is now implying that the runoff R is zero. The actual evapotranspiration computed as the precipitation P plus a quantity which depends upon M,, which is taken to be unchanged for the surface area at saturation, whereas, for the remaining unsaturated surface area, it is reduced by a factor which depends upon the ratio between the actual moisture content and the total: ET,=P+M,

[

s,-s -

s - s, ST

w-w,

wm-ws

SW,-SW,-SIW,+SIW,+STW-STW,-SW+SW,

=P+M,

sT(wm=P+M

ws)

(S-S,)W,+(ST-S)W-(ST-SI)W, e sT(wm

- w,)

= P+M,ST =p+M~T-SIXW,+(~-~)W-w~ e

(~29) wl?l-ws

Substituting for x from Eq. (A15) into Eq. (A29), one obtains s

_,[1-(1-~)fi]w,.+(1+)~w-w~

ET, = P+MeL

= P+M,-

sT;;sI,l

_ (1

K)';;i

(A30)

With reference to Fig. 7, the soil moisture content W, of the unsaturated area (1 - x), can be computed as w,=

w-w(l-x)=

w-(b+1)w~[1-(1-x)~](l-.x)

= w-(b+1)W,(1-x)+(b+l)W,(1-x)~

(A311

378

E. Todini / Journal of Hydrology 175 (19%)

339-382

and by substituting into Eq. (A30) for x, from Eq. (A4), one obtains

= W~[l+b(l-~)-(b+l)(l-$-)m]

6432)

The desired expression for the actual evapotranspiration when the precipitation is smaller than the potential evapotranspiration is finally found by substituting for W, from Eq. (A32) into Eq. (A30), to give (1 -E)â&#x20AC;&#x153;(l

-$K)

ET,=P+M,p STs,s1

â&#x20AC;&#x2122; -(b+

l)(l

_?-)â&#x20AC;&#x2DC;-b(1

=p+

-g)

1 (A33)

Appendix B. Estimation of parameters for the evapotranspiration module in the ARNO model For each of the sub-basins, the estimation of the evapotranspiration expression (Eq. (18)) is based upon the following steps: (1) computation of the long-term monthly average temperatures at the measurement stations; (2) computation of the thermal gradient; (3) computation of the spatially averaged, long-term monthly mean temperature for each sub-basin; (4) estimation of the average monthly potential evapotranspiration according to Thornthwaite; (5) computation of the regression coefficients cx and a for each sub-basin. Bl. Computation of long-term monthly temperature averages Long-term monthly temperature averages are calculated at the available temperature measurement stations, for the calendar months over a long period of time (a minimum of 10 years is required), and constitute the basis for the estimation of the evapotranspiration.

379

E. Todini / Journal of Hydrology I75 (1996) 339-382

B2. Computation of the thermal gradient

The influence of the temperature gradient may be significant if the stations are located at different heights or if the catchment shows large topographical variation. The temperature gradient varies according to place and time of the year. For the purpose of the ARNO model an average value of the gradient is used and estimated by means of long-term averages. If this is not available a table such as the ‘US Standard Atmosphere’ or ‘NACA Standard Atmosphere’ can be used (see Hess, 1959, p. 85), which is based on the following assumptions: (1) the air temperature at sea-level is 15°C and the pressure is 1013.25 hPa; (2) the air is dry and follows the perfect gas law; (3) the gravity acceleration is assumed constant and equal to 980.665 cm se2. B3. Computation of the long-term monthly temperature for a sub-basin

All the long-term mean temperatures computed for the various temperature stations are reduced to a common level (for instance, mean sea-level) by means of the thermal gradient described above. By using Thiessen polygons, a spatial average temperature is computed for each sub-basin, which is then reduced to the mean elevation of the sub-catchment, again using the thermal gradient. B4. Estimation of the average monthly potential evapotranspiration according Thornthwaite

As was mentioned in Section 4, evapotranspiration plays a major role in the rainfall-runoff process not really in terms of its instantaneous value, but in terms of its accumulated effect over time on the soil moisture volume depletion. Therefore it is extremely important that the long-term mean values be accurately reproduced, even if the expression used, which is generally different from the Penman-Monteith equation owing to the lack of data, may not be very precise on a short time interval such as 1 or 3 h. The long-term values of the average daily reference potential evapotranspiration to be used for the calibration of the evapotranspiration parameters in the ARNO model (although any alternative formula could be used) are generally computed for a given sub-catchment according to Thornthwaite and Mather (1955), by means of the following formula:

44 W Edi) = 1630~

[

T(i)

(Bl)

where Eth(i) is the average monthly potential evapotranspiration (mm day-‘) in month i, T(i) is long-term average temperature (“C) in month i, a(i) = [n(i)/30][N(i)/12] is a sunshine percentage index, n(i) is number of days in month i, N(i) is mean daily duration of maximum possible sunshine hours (Doorembos et al., 1984), kl is the thermal index defined as Cjz1[T(i)/5]1.514, and k2 = 0.49239 + (1792 x 10-5)b - (771 x 10-7)b2 + (675 x 10-9)b3.

E. Todini / Journal of Hydrology 175 (1996) 339-382

380

B5. Computation of the regression coejkients a and b for each sub-basin The expression actually used in the ARNO model for the estimation of the reference evapotranspiration at each time step is a linear expression in T, in such a way that El,,(i) can be disaggregated to short time steps without losing the long-term balance. It has been found that the radiation formula presented in Section 4, rewritten in a more convenient parametric form, closely approaches the Thornthwaite estimates, i.e. one can write l&(i) = (Y + ,L?N(i)W,,(i)‘T(i)

032)

where a and p are two model parameters. W,, is a compensation factor dependent upon the monthly average temperature and elevation above sea-level, which can be determined by means of a table given by Doorembos et al. (1984) that can be well approximated by means of the following parabola: W,,(i) = AT(i)2 + BT(i) + C

(B3)

where T(i) is the monthly long-term temperature (in “C) and A, B and C are three fitting parameters. Eq. (B2) very closely approximates Eq. (Bl), generally providing a determination coefficient of R2 = 0.999 or larger. Given the linearity of the equation with respect to temperature, once the parameters a and /3 have been estimated, the following expression is finally used at a more appropriate time step for the rainfall-runoff model: ET0 = h +,&V(i) W,(i)T

034)

where ET,, is the reference evapotranspiration for a specified time step (mm per time step), & and fi are the estimated values for the two model parameters, and T is the average temperature over a specified time step (in “C).

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