THE ARNO MODEL

By Prof. Ezio TODINIDepartment of Earth and Geo-Environmental Sciences

University of Bologna, Bologna, Italy

ABSTRACT: The paper describes the ARNO model, acontinuous time semi-distributed rainfall-runoff model, widelyused in the last two decades as the basic tool for severalapplications such as water master planning, analysis of extremefloods, realisation of real-time flood forecasting systems and lastbut not least to represent the soil component in GeneralCirculation Models (GCM). The ARNO model is based upon aparametric description of the main hydrological processes takingplace at the catchment scale, such as infiltration, evapo-transpiration, drainage, percolation, routing, etc., and canrepresent a large catchment as a tree shaped cascade of sub-catchments. The paper will also briefly discuss the parametercalibration and present the application of the ARNO model to theTiber river in Italy.

1. INTRODUCTIONMost discrete time rainfall-runoff models operable over longuninterrupted periods can be classified as Explicit Soil MoistureAccounting models, ESMA, and of these models probably themost famous is the Stanford Watershed Model (Crawford andLinsley, 1966). In a typical ESMA-type model the soil moisturestorage is modelled as three reservoirs: upper zone, lower zone,and active or deep groundwater storage, and the overland flowoccurs from impervious as well as from non-impervious areas.The channel flow routing in ESMA type models is either aMuskingum routing, a kinematic wave, or more frequently anarbitrary linear transfer function. The more complex ESMAmodels usually make provision for multiple sub-watershed inputswithin the channel routing phase. Various other mechanisms areprovided by most ESMA type models that tend to make theoverall model highly non-linear. Unfortunately the ESMA type

models have been over-parameterised and as reported by theWMO inter-comparison of conceptual models (1975) and byFranchini and Pacciani (1991) the large number of parameters isnot directly related to improved performances or to betterphysical representations.

In the late seventies Zhao (1977) showed how the formationof runoff, in regions not suffering from prolonged dryness, washighly related to the dynamics of saturated surfaces as a functionof the precipitation volumes, and he developed the Xinanjiangmodel, which was widely and successfully used in China;similar results were also found by Moore and Clarke (1981) whodeveloped the Probability Distributed Soil Capacity model.Following the work of Zhao, Todini (1988) found that a majorimprovement could be obtained by allowing the soil moisture tobe depleted not only by evapotranspiration, as in the originalXinanjiang, but also by drainage into the river network andpercolation into the water table.

This gave rise to a new conceptual model, which wasdeveloped by Todini (1988) within the frame of the hydrologicalstudy of the river Arno on behalf of the Tuscany RegionalGovernment in Italy, from which the name of ARNO model.

2. DESCRIPTION OF THE ARNO MODELThe ARNO model is a tentative of setting up a comprehensiveframework for describing most of the processes lumped atcatchment or sub-catchment scale. The processes represented inthe ARNO model are the following:

- Soil moisture balance- Drainage- Percolation- Groundwater- Evapo-transpiration- Snow accumulation and melting- Parabolic overland runoff routing- Parabolic in stream routing

Precipitation Temperature

ModuleEvapotranspiration

Snow Accumulation and Melting Module Snow accumulation

Rainfall + Snowmelt

Surface WaterModule

Surface Runoff

Infiltration

Percolation Soil Water Balance Module

Drainage Flow Drainage Module Groundwater Module

Groundwater Flow

Discharge

Channel NetworkModule

Fig. 1. A flowchart representing the ARNO model

Figure 1 shows a flow chart representing the data, theprocesses and the modules used in the ARNO model. Thesemodules are described in the sequel, together with theirparameterisation and the relevant assumptions made, while thefull derivation of the equations reported in this paper can befound in Todini (1996).

2.1. Soil Moisture Balance ModuleAs pointed out by Dooge (1973) and also shown by Todini andWallis (1977) the major problem in representing the rainfallrunoff process lies in the strongly non-linear behaviour and feed-back due to the soil moisture storage, which controls the runoffproduction. This is why the different rainfall runoff are qualifiedand distinguished on the basis of the soil moisture balancerepresentation they adopted. The soil moisture balance module ofthe ARNO model derives from the Xinanjiang model developedby Zhao (1977, 1984), who expressed the spatial distribution of

the soil moisture capacity in the form of a probability distributionfunction, similar to that advocated by Moore and Clarke (1981)and Moore (1985). Successively, in order to account moreeffectively for soil depletion due to drainage, this model schemewas modified by Todini (1988), who originated the ARNOmodel within the frame of the hydrological study of the riverArno.

The basic assumptions expressed in the soil moisturebalance module of the ARNO model are:

- the precipitation input to the soil is considered uniform overthe catchment (or sub-catchment) area;

- the catchment is composed of an infinite number ofelementary areas (each with a different soil moisturecapacity and a different soil moisture content) for each ofwhich the continuity of mass can be written and simulatedover time;

- all the precipitation falling over the soil infiltrates unless thesoil is either impervious or it has already reached saturation;the proportion of elementary areas which are saturated isdescribed by a spatial distribution function;

- the spatial distribution function describes the dynamics ofcontributing areas which generate surface runoff;

- the total runoff is the spatial integral of the infinitesimalcontributions deriving from the different elementary areas;

- the soil moisture storage is depleted by the evapo-transpiration, by lateral sub-surface flow (drainage) towardsthe drainage network and by the percolation to deeper layers;

- both drainage and percolation are expressed by simpleempirical expressions.

A sub-basin of given surface area ST (excluding the surfaceextent of water bodies such as reservoirs or lakes) is in generalformed by a mixture of pervious and less pervious soils, theresponse to precipitation of which will be substantially different.For this reason the total area ST is divided into the impervious

area SI and the pervious area SP :

PIT SSS += (1)

In order to derive the expressions needed for the continuousupdating of the soil moisture balance, let us first deal with theamount of precipitation that falls over the pervious area. Giventhat from equation (1) the entire pervious area is:

ITp SSS −= (2)

if one denotes by (S - SI) the generic surface area at saturation, x,defined as:

IT

I

SSSS

x−−

= (3)

will indicate the proportion of pervious area at saturation. Zhao(1977) demonstrated that the following relation holds reasonablywell between the area at saturation and the local proportion ofmaximum soil moisture content w/wm, where w is the elementaryarea soil moisture at saturation and wm is the maximum possiblesoil moisture in any elementary area of the catchment.

b

mwwx

−−= 11 (4)

This can be inverted to give the cumulative distribution forthe elementary area soil moisture at saturation, shown in Figure 2and defined as:

( )

−−= bxww m

1

11 (5)

In the ARNO model, an interception component (Rutter etal., 1971, 1975) is not explicitly included. Nevertheless in orderto allow for a substantially larger evapotranspiration when thecanopies are wet (without obviously explicitly accounting for thedisappearance of the stomatal resistance (Shuttleworth, 1979)),the following succession of operations is followed.

Fig. 2. Cumulative distribution for the elementary area soilmoisture at saturation

If the precipitation P is larger than the potentialevapotranspiration ETp, the actual evapotranspiration, for thereasons expressed above, is assumed to coincide with thepotential, i.e.:

ETa= ETp (6)

and so an "effective" meteorological input Me , defined as thedifference between precipitation and potential evapo-transpiration, becomes:

0>−=−= ape ETPETPM (7)

With reference to Figure 3, the surface runoff R generated bythe entire catchment is obtained as the sum of two terms, the firstone is the product of the meteorological effective input and thepercentage of impervious area while the second one is theaverage runoff produced by the pervious area, which is obtainedby integrating the soil moisture capacity curve, which gives:

( )∫+−+=

wM

wT

ITe

T

I edx

SSSM

SSR ξξ (8)

if me wwM <+ , or

( )[ ]

−−

−+= ∫

mw

we

T

ITe

T

I dxMS

SSM

SS

R ξξ1 (9)

if me wwM ≥+ .

Equations (8) and (9), can also be expressed in terms of thecatchment average soil moisture content W and that at saturationWm and after integration they become

( ) ( )

+−

−−−

−+=

+

+

1

11

11

b

b

mWbeM

mWW

mWWmWS

SSMR

T

ITe (10)

if ( )1

1

110+

−+<<

b

mme W

WWbM , or

( )WWS

SSMR m

T

ITe −

−−= (11)

if ( )1

1

11+

−+≥

b

mme W

WWbM .

Fig. 3. Runoff R generated by an effective meteorologicalinput Me>0.

If the precipitation P is smaller than the potentialevapotranspiration ETp the effective meteorological input Me,becomes negative:

0<−= pe ETPM (12)

which implies that the runoff R is zero. The actualevapotranspiration is then computed as the precipitation P plus aquantity which depends upon Me reduced by the average degreeof saturation of the soil, which gives:

( ) ( )1

1

11

111

111

+

+

−−

+

−−

+

−−+=

b

b

m

mm

T

ITpa

WW

b

WW

bWW

SSS

PETPET (13)

These equations, which represent the average surface runoffproduced in the sub-catchment, must be associated with anequation of state in order to update the mean water content in thesoil. This equation takes the form:

IDRETPWW attt −−−−+=+∆ (14)

where all the quantities represent averages over the sub-basin andare expressed in mm:

Wt+∆t is the soil moisture content at time t+∆t;Wt is the soil moisture content at time t .P is the area precipitation between t and t+∆t;ETa is the evapotranspiration loss between t and t+∆t;R is the total runoff between t and t+∆t;D is the loss through drainage between t and t+∆t;I is the percolation loss between t and t+∆t;

The non-linear response of the unsaturated soil toprecipitation, represented by the shape of the distribution curvegiven by equation (5), is strongly affected by the horizontaldrainage and vertical percolation losses. The drainage loss D isan important quantity to be reproduced in a hydrological model,since on the one hand it affects the soil moisture storage and onthe other hand it controls the hydrograph recession. Theoreticalconsiderations on the unsaturated zone flow as well asapplication results have suggested in time various types ofempirical relationships to express the drainage loss D as a non-linear function of the soil moisture. The most recently adoptedexpression in the ARNO model is the following, which althoughwritten as a lumped equation valid at catchment scale, lookssimilar to the one, valid at a point, which is due to Brooks andCorey (1964):

1c

ms W

WDD

= (15)

where

mWW is the percentage saturation;

c1 is an exponent related to the soil types;

Ds is the drainage at saturation;

The percolation loss I, which feeds the groundwater module,is represented with a similar expression, although generally withdifferent parameters. which will control the base flow in the

model, varies less significantly over time if compared with theother terms; nevertheless a non-linear behaviour is also assumedas follows:

2c

ms W

WII

= (16)

c2 = exponent related to the soil types;

Is = infiltration at saturation;

The total runoff per unit area produced by the precipitation Pis finally expressed as:

Rtot = R + D + B (17)

where:

B is the base flow generated by the presence of a groundwatertable fed by the percolation, and computed by thegroundwater module.

2.2. The Evapotranspiration ModuleIn the ARNO model, a second forcing factor is provided byevapotranspiration, which can be either computed externally anddirectly introduced as an input to the model or estimatedinternally as a function of the air temperature. The internalevapotranspiration estimator uses in fact a simplified techniquewhich has proven more than accurate for modelling the rainfallrunoff process, where evapotranspiration plays a major role notreally in terms of its instantaneous impact, but in terms of itscumulative temporal effect on the soil moisture volumedepletion; this reduces the need for an extremely accurateexpression, provided that its integral effect be well preserved.Although acknowledging the fact that the Penman-Monteithequation is the most rigorous theoretical description for thiscomponent, for a practical utilisation a simplified approach isgenerally necessary because in most countries the requiredhistorical 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 forecastingapplications. In the ARNO model, the effects of the vapourpressure and wind speed are explicitly ignored andevapotranspiration is calculated starting from of a simplifiedequation known as the radiation method (Doorembos et al.,1984):

ET0d = CvWtaRs = CvWta 0. 25 + 0.50nN

Ra (18)

where:

ET0d is the reference evapotranspiration, i.e. evapo-transpirationin soil saturation conditions caused by a reference crop(mm/day) ;

Cv is an adjustment factor obtainable from tables as a functionof the mean wind speed;

Wta is a compensation factor that depends on the temperatureand altitude;

Rs is the short wave radiation measured or expressed as afunction of Ra in equivalent evaporation (mm/day);

Ra is the extraterrestrial radiation expressed in equivalentevaporation (mm/day);

n/N is the ratio of actual hours of sunshine to maximum hoursof sunshine (values measured or estimated from meanmonthly values).

Hence the calculation of Rs requires both knowledge of Ra,obtainable from tables as a function of latitude, and knowledgeof actual n/N values, which may not be available. In the absenceof the measured short wave radiation values Rs or of the actualnumber of sunshine hours otherwise needed to calculate Rs as afunction of Ra (see equation (18)), an empirical equation wasdeveloped that relates the reference potential evapotranspirationET0d, computed on a monthly basis using one of the availablesimplified expressions such as for instance the one due toThornthwaite and Mather (1955), to the compensation factor Wta,the mean recorded temperature of the month T and the maximum

number of hours of sunshine N. The developed relationship islinear in temperature (and hence additive), and permits thedissaggregation of the monthly results on a daily or even on anhourly basis, while most other empirical equations are ill-suitedfor time intervals shorter than one month.

The relation used, which is structurally similar to theradiation method formula in which the air temperature is taken asan index of radiation, is:

ET0 = α +β NWtaTm (19)

where:ET0 is the reference evapotranspiration for a specified time step

∆t (in mm/∆t);

α,β are regression coefficients to be estimated for each sub-basin;

Tm is the area mean air temperature averaged over ∆t;

N is the monthly mean of the maximum number of dailyhours of sunshine (tabulated as a function of latitude).

Wta for a given sub-basin can be either obtained from tables orapproximated by a fitted parabola:

CTBTAWta ++= 2(20)

where:

A, B, C are coefficients to be estimated;T is the long term mean monthly sub-basin temperature (oC).

2.3. The Snowmelt ModuleAgain, for reasons of limited data availability, in the ARNOmodel the snowmelt module is driven by a radiation estimatebased upon the air temperature measurements; in practice theinputs to the module are the precipitation, the temperature, andthe same radiation approximation which is used in theevapotranspiration module.

Given the role that altitude may play in combination with thethermal gradient, the sub-catchment area is subdivided into anumber of equi-elevation zones (snow-bands) according to thehypsometric curve, and for each zone simplified mass and energybudgets are continuously updated. For each snowband thefollowing steps, similar to those adopted in SHE (Abbott et al.,1986), are followed:

- Estimation of radiation at the average elevation of thesnowband;

- Decision whether the precipitation is solid or liquid;- Estimation of the water mass and energy budgets based on

the hypothesis of zero snowmelt;- Comparison of the total available energy with that sustained

as ice by the total available mass at 273 °K;- Computation of the snowmelt produced by the excess

energy; and updating the water mass and energy budgets.

2.3.1. Estimation of radiation at the average elevation of thesnowbandThe estimation of the radiation is performed by re-converting thelatent heat (which has already been computed as the referenceevapotranspiration ET0 ) back into radiation, by means of aconversion factor Cer (Kcal Kg-1) which can be found in anythermodynamics textbook as:

( )0 695.05.606 TTCer −−= (21)

where T0 is the temperature of fusion of snow (273 °K).

In addition, in order to account for albedo which plays anextremely important role in snowmelt, it is necessary to apply anefficiency factor which will be assumed approximately as η = .6for clear sky and η = .8 for overcast conditions: this leads to thefollowing estimate for the driving radiation term:

( )[ ] 00695.05.606 ETTTRad −−=η (22)

Given the lack of information concerning the status of thesky when simulating with historical data, for practical purposes it

is generally assumed in the ARNO model that the sky is clear ifthere is no precipitation and overcast if precipitation is beingmeasured.

2.3.2. Decision whether the precipitation is solid or liquidInformation concerning the status of precipitation (solid orliquid) is rarely available as a continuous record; therefore it isnecesary to define a mechanism, mainly based upon the airtemperature measurements and the historical precipitation. If oneplots the frequency of the usually scattered observations withwhich precipitation was observed to be liquid or snow as afunction of the air temperature, a Gaussian distribution isgenerally obtained, with a mean value Ts which very seldom willcoincide with T0. For this reason, the following rules are adopted:

- Precipitation is taken as liquid if the air temperature T >Ts;- Precipitation is taken as snow otherwise.

The value of Ts (which generally ranges between 271 and275 °K) must be derived, as previously mentioned, by plottingthe frequency of the status of historically recorded precipitationas a function of the air temperature.

2.3.3. Estimation of the water and energy budgets on thehypothesis of zero snowmeltThe water equivalent mass is estimated with the following simplemass balance equation, where all quantities are expressed in mmof water:

Zt +∆t∗ = Zt + P (23)

The water equivalent at the end of the time step is designatedwith a star because it is a tentative value which does not yetaccount for the eventual snowmelt.

Similarly to the mass, the energy is estimated in thefollowing way, by computing the increase (or decrease) of totalenergy E:

- if the precipitation is zero:

Et+∆t∗ = Et + Rad (24)

-if the precipitation is non-zero and the precipitation is in thesolid phase ( snTT ≤ )

PTCRadEE sgttt 0++=∗+∆ (25)

-if the precipitation is non-zero and the precipitation is in theliquid phase (T > Tsn )

( )[ ]PTTCCTCRadEE salfsittt 00 −++++=∗+∆ (26)

Again the total energy at the end of a time step is designatedwith a star to denote a tentative value; in the previous equationsCsi is the specific heat of ice (0.5 Kcal °K-1 Kg-1), Clf the latentheat of fusion of water (79.6 Kcal Kg-1) and Csa the specific heatof water (1 Kcal °K-1 Kg-1).

2.3.4. Estimation of snowmelt and updating of mass andenergy budgetsIf the total available energy is smaller or equal to that required tomaintain the total mass in the solid phase at the temperature T0i.e; CsiZt+∆t

∗ T0 ≥ Et+ ∆t∗ ,it means that the available energy is not

sufficient to melt part of the accumulated water, and therefore:

=

=

=

∗++

∗++

tttt

tttt

sm

EE

ZZ

R

∆∆

∆∆

0 (27)

where Rsm is the snowmelt expressed in mm. If the totalavailable energy is larger than that required to maintain the totalmass in the solid phase at the temperature T0 , it means that partof the accumulated water will melt, and therefore the followingenergy balance equation holds:

( ) ( ) smlfsittsmttsi RCTCETRZC +−=− ∗+

∗+ 00 ∆∆ (28)

from which the snowmelt and the mass and energy statevariables can be computed as:

( )

+−=

−=

−=

∗++

∗++

∗+

∗+

smlfsitttt

smtttt

lf

ttsittsm

RCTCEE

RZZ

CZTCE

R

0

0

∆∆

∆∆

∆∆

(29)

2.4. The Groundwater ModuleThe groundwater module represents the overall response of itsstorage by means of a cascade of linear reservoirs, characterisedby two parameters: the number of reservoirs n and their timeconstant K , a model which is well known in hydrology as theNash model (Nash, 1958). For practical reasons, instead of usingthe Gamma distribution function to express the impulse response,a numerical procedure has been adopted. The expression to beused can easily be derived from the assumption of a cascade oflinear reservoirs where the volume of the ith reservoir isproportional to its outflow, i.e. St = K Bt

i ; thus, the continuityequation written for the generic ith reservoir becomes:

it

it

it BB

tB

K −= −1

∂∂

(30)

which, after discretization in time with a finite difference schemecentred at time t+∆t/2, becomes:

( )22

11 it

itt

it

itti

ti

ttBBBB

BBt

K +−

+=− +

−−+

+∆∆

∆∆ (31)

and the required expression is then obtained by making Bt+ ∆ti

explicit:

( )11

222 −−

++ ++

++−= i

ti

ttit

itt BB

tKtB

tKtKB ∆∆ ∆

∆∆∆

(32)

where:

Bi is the outflow from the ith reservoir

B0=I is the percolation given by the soil moisture balance

equation

Bn=B is the resulting base flow

Equation (32) is then recursively solved n times at each timestep.

2.5. The Parabolic Routing ModuleThe routing in the ARNO model is performed by assuming forthe catchment the "open book" shape, as shown in Figure 4 anddividing the routing problem into a sequence of three separatelinear routings. The first one relates to the routing to the mainchannel of the total runoff, given by equation (17) and assumedto be uniformly produced over the hillslopes on a unit widthstrip; the second one relates to the routing in the main channel ofthe distributed inflow (per unit width strip) produced by thehillslopes; the third one, which is used to link together severalsub-catchments, relates to the routing of upstream inflows to theexiting section of the downstream sub-catchment. The three stepsare illustrated in Figure 4 and described in the sequel.

Channel routingwith distributed runoff

inflow

Hillslope routingwith distributed runoff

inflow

Channel routing with concentrated upstream inflow

Fig. 4. Schematic representation of the three steps in routingThe hillslope routing and channel routing of distributed

inflows are both performed using a distributed input linearparabolic model, while channel routing of upstream inflows to adownstream sub-basin is performed by means of a concentratedinput parabolic model. The parameters of the two linearparabolic transfer functions can be estimated as a function of theslope, the length and the roughness of the drainage system.

2.5.1. Routing of upstream inflowsThe propagation of inflows from upstream is carried out bymeans of a linear model consisting of a parabolic unithydrograph deriving from the analytical integration of theunsteady flow equations when the inertia effects are ignored andthe coefficients are linearised around mean outflow values. Inthis case the following differential equation is obtained:

xQC

xQD

tQ

∂∂

∂∂

∂∂

−=2

2

(33)

where D and C are the diffusivity and the convectivitycoefficient respectively. The discrete time solution of equation(33) for mean values of the coefficients C and D is (Todini andBossi, 1986):

( ) ( ) ( ) ( )[ ]ttIFtIFttIFt

tttU x ∆∆∆

∆∆ −+−+=+ 21,2 (34)

where

( ) ( ) ( ) ϑττϑϑϑ

∆ ddudFtIFt

x

t

∫ ∫∫ ==0 00

(35)

with u∆x , the impulse response relevant to equation (35), whichcan be written as:

( )( )

ττ∆

∆τπ

∆τ DCx

x eD

xu 43

2

4

−−

= (36)

Integrating equation (35), after substitution from equation(36) one obtains:

( ) ( ) ( )

+−++

−−−=DtCtxNexCt

DtCtxNxCt

CtIF D

xC

221 ∆∆∆∆

∆(37)

where N(*) is the value of the Standard Normal probabilitydistribution in (*), and ∆x is the length of the channel reach.

2.5.2. Diffuse lateral inflow routingThe runoff generated in each sub-basin moves first along theslopes and then reaches the drainage network as diffuse inflowthat, for simplicity, may be considered uniformly distributed. Inthis case the following differential equation applies:

CqtQ

xQC

xQD −=−−

∂∂

∂∂

∂∂

2

2

(38)

where q is the lateral inflow per unit length. The discrete timesolution of equation (38) for mean values of the coefficients Cand D is (Franchini and Todini, 1989):

( ) ( ) ( ) ( )[ ]ttIGtIGttIGt

tttLU x ∆∆∆

∆∆ −+−+=+ 21,2 (39)

where:

( ) ( )[ ]tIIFtCtIG −= 2

2(40)

with

( ) ( ) ( ) ( ) ξϑττξϑϑξξξ ϑ

∆

ξ

ddduddFdIFtIIFt

x

tt

∫ ∫ ∫∫ ∫∫ ===0 0 00 00

(41)

Integration of equation (41), after substitution of u∆x fromequation (36), yields:

( )

( )

+

+

+−−

−−

−+

+

+−+

−−

+−=

−−

DtCtx

DxC

DxC

eDtC

xC

eDtCtxN

DtCtxN

Cxt

CxC

eDtCtxN

DtCtxN

CxttCtIG

42

2

2

222

2

22

222

2

222

∆

∆

∆

π∆

∆∆∆∆

∆∆∆

(42)

The results provided by equation (38) when substitued forIG(t) given by equation (42), differ from the ones obtained byNaden (1992), in that they represent the response of a discretetime system to a time discretised input (see Todini and Bossi,1986).

3. THE ARNO MODEL PARAMETERSAll the ESMA type models require calibration of several ofparameters. Some of these parameters relate to the mainprocesses, such as the soil moisture dynamical balance, and playan important role in the model performances in that they maymodify the output by an order of magnitude. Other parameters,such as for instance the routing parameters, will only affect thetiming of the response or may produce errors of the order of 10-20%. This is why the comparison among the different ESMAtype models, is generally based upon the soil componentparameters.

In the ARNO model, the following parameters are soilcomponent parameters that require accurate calibration by meansof historical data and verification using split sample testingtechniques:

Wm represents the average volume in mm of soil moisturestorage. This parameter can be derived as a function of thesoil types and its value ranges in general from 50 to 350mm;

b represents a shape factor for the soil moisture vs saturatedareas curve; its value, which ranges between .01 and .5, is

somehow related to the soil types and their spatialvariability, but must be estimated from the data.

Ds is the maximum drainage (in mm per time unit) that shouldbe expected when the soil is completely saturated; it is afunction of many factors including the geo-morphologicalstructure of the catchment, the soil types, the space andtime scales of the problem, and must be estimated from thedata.

c1 exponent (which depends on soil types with values ranginggenerally between 1.5 and 3) used to represent drainagewhen saturation is not reached which value is estimatedfrom historical data.

Is is the maximum percolation (in mm per time unit) thatshould be expected when the soil is completely saturated;similarly to Ds it depends upon several items and must beestimated from the data.

c2 exponent (which depends on soil types with values ranginggenerally between 1 and 2) used to represent percolationwhen saturation is not reached which value is estimatedfrom historical data.

Other parameters appearing in the equations, such as ST, SPand SI, do not require estimation from historical data, but can bedirectly evaluated on the basis of geo-morphological, soil andland use maps.

The ARNO model is therefore essentially based upon 6parameters, a small number when compared to more classicalESMA type models (generally more than 20 parameters), but stillthe double of the parameters required in the TOPMODEL(Beven and Kirkby, 1979). Moreover, although related to theproblem physics, not all the ARNO model parameters can bedirectly estimated or derived from maps, which is the majormodel limitation when wishing to use the model on ungaugedcatchments.

As mentioned earlier, the estimation of the parametersrelevant to the routing model components (hillslopes and

channels) are not critical. A convectivity C and diffusivity Dcoefficient must be provided for each response function whichvalue may be related to the dimensions, slopes and lengths of theindividual sub-basins. In general very small values for D areassumed for the hillslopes, given the marked kinematic nature ofthe phenomenon; generally, D increases with the size of the riverand with the inverse of the bottom slope. Table 1 gives anindication of the orders of magnitude of C and D.

C (m/s) D(m2/s)

Hillslopes 1-2 1-100

Brooks 1-3 100-1000

Rivers 1 - 3 1000-10000

Large rivers 1 - 3 10000-100000

Table 1 -Approximate initial guess for parameters C and D.

4. APPLICATION OF THE ARNO MODELSince its derivation, the ARNO model, given its simplicity androbustness, was used in various parts of the world as the basictool for the development of several real-time operational floodforecasting systems. Moreover, the soil moisture balancecomponent was successfully used as an alternative to the Manabe(1969) soil bucket in the several GCMs, such as the ECHAM2model (Dümenil and Todini, 1992, Roeckner et al., 1992), theUK Meteorological Office (UKMO) model (Rowntree and Lean,1994), and the LMD model (Polcher et al., 1995) while modifiedversions of the ARNO model have also been developed such asfor instance the VIC model by Wood et al. (1992) who extendedthe ARNO model concepts to multiple soil layers or the ADMmodel (Franchini and Galeati, 1997) a simplified version of theARNO.

Fig. 5. The river Tiber catchment and the sub-basinsFrom the application point o view, several rivers have been

successfully modelled by means of the ARNO model: theFuchun river in China, the Danube river in Germany, and therivers Arno, Tiber, Fortore, Ofanto, Adda, Ticino, Oglio, Agno-Guà, Tronto in Italy. In order to illustrate the ARNO modelperformances, only its application to the case of the real timeflood forecasting system of the river Tiber will be described.

The Tiber, the third largest river in Italy, is 403 km longwith a catchment area of over 17,000 km2 originating fromMount Fumaiolo at an altitude of 1260 m a.s.l. and debauchinginto the Thyrrenean sea near Rome, the capital of Italy.

Fig. 6. The schematic representation of river Tibercatchment and its sub-basins in the ARNO model

Figure 5 shows a map of the Tiber basin which was dividedin 30 sub-basins in order to set up the ARNO model. Thissubdivision is required for three main reasons. The first onerelates to the need of preserving homogeneity in the geo-morphological and meteo-climatical characteristics in thedifferent sub-units over which the processes are lumped (whichgenerally requires sub-units of a few hundred square kilometresin size); the second one relates to the availability of rating curvesfor the calibration of the different model components; the thirdone is imposed by the sections of interest, namely the river crosssections for which the computations or the forecasts are needed.Figure 6 shows the interconnections of the different sub-unitsused in the ARNO model for the development of the singlerainfall-runoff models and for their routing to the river mouth.

Continuous records of hydro-meteorological data from June1993 to May 1994 were available at 47 rain gauge stations, 18thermometric stations and 22 level gauge stations and used forthe calibration of the ARNO model.

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Fig. 7. Calibration results for a flood period at Rome Ripettastation

Figure 7 shows the calibration results obtained in the floodperiod at the ending section of Rome Ripetta where 12 hoursadvance forecasts are needed in order to close a number ofsluices thus avoiding the waters to flow into the city streets. Thefigure also shows that the model also captured the marked effectof the upstream reservoir of Corbara, which is located in themiddle part of the Tiber, and which affects particularly the lowflows. A model for the reservoir had to be implemented and usedas an additional module in the ARNO model. The quality of thecalibration was considered as successful with over 95% ofexplained variance. The calibrated parameters were then used forthe development of a real time flood forecasting system based onEFFORTS (European Flood Forecasting Operational Real TimeSystem) (ET&P, 1992) which includes the ARNO model

together with other models such as the hydraulic flood routingmodel PAB (Todini and Bossi, 1986) and the Kalman filter basederror model MISP (Todini, 1978).

Fig. 8. Simulated real time 12 hours in advance forecastscompared to the measured discharges on the Tiber at Rome

Ripetta.

Fig. 9. Operational real time 12 hours in advance forecastscompared to the measured discharges on the Tiber at Rome

Ripetta.The quality of the real time forecasting results is illustrated

in Figure 8 comparing the measured values with the successionof 12 hours in advance simulated real time flood forecasts atRome Ripetta station during the flood period in the calibrationrecord.

The system was then operationally installed in October 1994and figure 9 compares the measured values with the successionof 12 hours in advance, this time operational, real time floodforecasts at Rome Ripetta station during November 1994,showing practically the same pattern of reliable 12 hours inadvance flood forecasts.

5. CONCLUSIONSThe ARNO model described in this paper has proven to be areliable and flexible tool for rainfall-runoff modelling atcatchment or at sub-catchment scale as shown by the numeroussuccessful applications on large and complex rivers as well as onsmall and flash flood prone brooks. The model parameterisationis a compromise between the complexity and space and timevariability of phenomena to be reproduced and the simplicityoffered by its lumped formulation.

The limitations of the approach lie mainly in the lack ofdirect interpretation of some of its parameters that do not allowfor the extension of model calibration to ungauged catchments.On the other hand, when long records of rainfall and runoff areavailable, the model can be successfully calibrated and used bothfor simulation and for real time flood forecasting.

In order to overcome the above mentioned limitations whilepreserving the ARNO model contributing areas dynamics,further studies have indicated the possibility of deriving a newgeneration of distributed models such as for instance theTOPKAPI (Todini and Ciarapica, 2001), which parameters arephysically based and calibration is possible on the basis of digitalelevation, soil and land use maps; but showing the interestingcharacteristics of collapsing into a model similar to the ARNOmodel, when lumped at catchment or at sub-catchment scales.

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