General Calibration Methodology for a Combined Horton-SCS Infiltration

Nat. Hazards Earth Syst. Sci., 8, 1317–1327, 2008 www.nat-hazards-earth-syst-sci.net/8/1317/2008/ Natural Hazards © Author(s) 2008. This work is distributed under and Earth the Creative Commons Attribution 3.0 License. System Sciences

General calibration methodology for a combined Horton-SCS infiltration scheme in flash flood modeling

S. Gabellani1,2, F. Silvestro1, R. Rudari1,2, and G. Boni1,2 1CIMA Research Foundation, University Campus, Armando Magliotto, 2. 17100 Savona, Italy 2DIST, Dipartimento di Informatica Sistemistica e Telematica, University of Genoa, Genoa, Italy

Abstract. Flood forecasting undergoes a constant evolution, 1 Introduction becoming more and more demanding about the models used for hydrologic simulations. The advantages of developing When dealing with flash flood forecasting, there is always the distributed or semi-distributed models have currently been need to combine different requirements that, in many cases, made clear. Now the importance of using continuous dis- force tradeoffs on the completeness of the processes schema- tributed modeling emerges. A proper schematization of the tization used in models. A model that is to be implemented infiltration process is vital to these types of models. Many in a flash flood forecasting chain is required to be reliable popular infiltration schemes, reliable and easy to implement, and portable, because, due to their limited sizes, the major- are too simplistic for the development of continuous hydro- ity of the catchments affected by this particular hazard are logic models. On the other hand, the unavailability of de- un-gauged. Catchments affected by this particular hazard are tailed and descriptive information on soil properties often ungauged, so that a large ensemble of model runs is usually limits the implementation of complete infiltration schemes. needed to deliver a usable forecast product (Siccardi et al., In this work, a combination between the Soil Conservation 2005; Franz et al., 2005). From this, the need for simple Service Curve Number method (SCS-CN) and a method de- schematization has led to the diffuse utilization of hydrolog- rived from Horton equation is proposed in order to overcome ical models working at event scale, using simplified schemes the inherent limits of the two schemes. The SCS-CN method for rainfall abstraction. Although event models have been is easily applicable on large areas, but has structural limi- successfully employed in this field, nowadays their value, tations. The Horton-like methods present parameters that, which is based on their simplicity and robustness, is more though measurable to a point, are difficult to achieve a reli- and more confined to off-line applications such as risk as- able estimate at catchment scale. The objective of this work sessment, hydraulic design, etc. Real-time forecasting prob- is to overcome these limits by proposing a calibration proce- lems now call for continuous, spatially distributed models, dure which maintains the large applicability of the SCS-CN because of their ability in estimating initial conditions (i.e., method as well as the continuous description of the infiltra- soil moisture distribution) in combination with distributed tion process given by the Horton’s equation suitably modi- forcing (e.g., rainfall, temperature, radiation) provided by fied. The estimation of the parameters of the modified Horton real-time measurements (e.g. meteorological radars, satel- method is carried out using a formal analogy with the SCS- lites) and meteorological models (Reed and Zhang, 2007). CN method under specific conditions. Some applications, at The common belief for flash floods is that the the influ- catchment scale within a distributed model, are presented. ence on peak discharge of initial moisture conditions is poor, flaws when the performance of a forecasting chain is evalu- ated during a long period and not on the basis of a few case studies that produced extreme discharge values. The added value of a forecasting chain resides in its ability to dealing with ambiguous cases, when false or missed alarms can oc- cur. In these conditions, a good representation of the initial state, together with a proper infiltration scheme, can substan- Correspondence to: S. Gabellani tially improve the forecasting chain skill, enabling the model ([email protected]) to discriminate events that produce different runoff volumes

Published by Copernicus Publications on behalf of the European Geosciences Union. 1318 S. Gabellani et al.: Calibration methodology for a combined Horton-SCS infiltration scheme starting from similar volumes and intensities of rainfall. model (Giannoni et al., 2000, 2003, 2005). The capability The need to develop continuous and distributed modeling of the calibrated modified Horton’s infiltration model to cor- frameworks has generated many infiltration models able to rectly represent the rainfall-runoff separation and the porta- run continuously in time starting from schematization origi- bility of the results, are discussed, presenting some case stud- nally intended for event modeling. Many of these efforts con- ies. siders the modifications of the SCS-CN schematization (e.g., Mishra and Singh, 2003), which is one of the most widely used because of its applicability and reliability (e.g., Ponce 2 The Soil Conservation Service Curve Number (SCS- and Hawkins, 1996). This method was developed within the CN) method and its limitations empirical framework, giving it world-wide credibility, and The SCS-CN method was developed in the 1954 and doc- asks for a deep understanding of the mathematical consis- umented in the United States Department of Agriculture tency of the formula when a modification or addition is pro- (1954). Since then, this method has been widely used and posed. Recently, it has been proven that it is impossible to modified (McCuen, 1982; Mishra and Singh, 2003). The introduce a continuous schematization of the soil moisture method is simple, useful for ungauged watersheds, and ac- accounting without rewriting the method consistently (e.g., counts for most runoff producing watershed characteristics as Michel et al., 2005). In doing this, it is the opinion of the au- soil type, land use, surface condition, and antecedent mois- thors that one of the major added values of the method, which ture condition. It is founded on the hypothesis that the ratio resides in the tremendous amount of experimental work done between potential runoff volume (P −I ), where I is the ini- since it was developed, tends to be lost. It is hardly accept- a a tial abstraction, and runoff volume R equals the ratio between able to modify the structure of an empirically driven method potential maximum retention S and infiltration volume F: without going through an extensive calibration which takes into account the proposed modifications. It would then be more straight forward to use an infiltration model with less (P − I ) S a = (1) empirical implications, conceptually suitable to run in a con- R F tinuous framework. The combination between this hypothesis and a simplified Many choices are possible and several applications are mass balance equation gives: present in the literature that could be of help in such a choice. Given the operational focus of this work, a suitable choice is 2 a method which is not over-parameterized and does not re- (P − Ia) quire the acquisition of soil or land use information with de- R = for P > Ia (2) R + S − Ia tails normally not available outside experimental catchments. Simple schematization of this type, based on physically in- The potential maximum retention of the soil is determined terpretable parameters, start from the Horton equation (e.g., by selecting a Curve Number parameter (CN): Bauer, 1974; Aron, 1990; Diskin and Nazimov, 1994). The limit of these models is usually the difficulty of setting pa- 1000 rameter values based on physical characteristics of the catch- S = 25.4( − 10) [mm] (3) CN ment which are representative for basin scale simulations. This often forces the hydrologist to consider them as only pa- The Curve Number is a function of the soils type, land rameters to be calibrated on available rainfall/discharge time cover and the so called Antecedent Moisture Condition series, thus limiting the portability of the model itself. (AMC). Three levels of AMC are considered: AMC-I dry The aim of this paper is to formalize a simple conceptual soil (but not to the wilting point), AMC-II average case and schematization based on a modification of the Horton’s in- AMC-III saturated soil. The initial abstraction term Ia com- filtration equation and propose a calibration methodology of bines the short term losses – surface storage, interception its parameters on the basis of some formal analogies with the and infiltration prior to runoff – and it is linearly related to SCS-CN method. The final result is a general relation be- the potential maximum retention S. The results of empir- tween SCS-CN and modified parameters of Horton’s meth- ical studies give Ia=0.2S. Hydrologists have shown that ods that increases the portability of the latter exploiting the the SCS-CN formulation is nominally consistent with both proved applicability and reliability of the SCS-CN method. infiltration-excess (Hjelmfelt, 1980a,b) and saturation-excess The proposed methodology leads to an infiltration method or Varied Source Area – VSA – hydrology (Steenhuis et al., which behaves as the SCS-CN method in the range of events 1995). The traditional SCS-CN method is most commonly where the latter ought to perform at its best with no additional used after the assumption that infiltration excess is the pri- calibration, and requires little calibration for events whose mary runoff mechanism. The strength of the method results intensity and duration is out of this range. from an exhaustive field investigation carried out in several The calibration is performed by implementing the two in- USA watersheds and it is empirically corrected to match the filtration schemes in the same semi-distributed hydrological outputs of the watersheds used for this study. The SCS-CN

Nat. Hazards Earth Syst. Sci., 8, 1317–1327, 2008 www.nat-hazards-earth-syst-sci.net/8/1317/2008/ S. Gabellani et al.: Calibration methodology for a combined Horton-SCS infiltration scheme 1319 method is, in fact, empirically calibrated in order to satisfy, rate representation of the catchment initial state, hampering for the calibration events, at catchment scale, Eq. (1)(United the performance of the model. All these issues limit the ap- States Department of Agriculture, 1954). plicability of the SCS-CN method for operational flood fore- The impressive sample size used in the calibration gives casting and leads us to move towards another scheme for the reliability and, as proved by different authors (Singh et al., infiltration process simulation. 2002), good method application when used to evaluate the cumulative runoff on durations comparable to the ones used for its calibration, (i.e., 24 h, in most cases the curve num- 3 The modified Horton’s equation ber was developed using daily rainfall-runoff records corre- Horton (1933) proposed an exponential decay equation to de- sponding to the maximal annual flow derived from gauged scribe the variation in time of the infiltration capacity of the watersheds (United States Department of Agriculture, 1954; soil during a rainfall event as: Mishra and Singh, 2003). Recent applications have also shown the possibility of using the SCS-CN method in a distributed way and on time scales finer than the event scale, − − f (t − t ) = f + (f − f )e k(t t0) (4) so that each cell of a distributed model is described by a 0 1 0 1 CN value and the runoff/infiltration separation is performed f (t−t0)=infiltration rate at time t−t0 from the begin- at cell scale over a time step (Grove et al., 1998; Moglen, ning of the rainfall event [LT −1], 2000). This is not surprising, due to the general scalability of −1 the continuity equation which is at the base of the method, al- f0=initial infiltration rate at t0 [LT ], though a careful recalibration would be needed (Grove et al., −1 1998; Michel et al., 2005). The SCS-CN method does not f1=infiltration rate for (t−t0)→∞ [LT ], consider the infiltration to lower soil layers, therefore, set- t =beginning time of the rainfall event [T ], ting the soil residual infiltration capacity to zero. Gener- 0 ally, this assumption is physically unacceptable, but holds k= exponential decay coefficient [T −1]. when daily or shorter rainfall annual maximum precipitation events are studied and and limited sizes of catchments are of The main restrictions to the application, at catchment a concern. From this, comes the necessity of re-initializing scale, of the Horton’s equation in its original form are: the the moisture conditions of the watershed for long rainfall difficulty of taking into account rainfall with intensities lower events. The revision of the antecedent moisture condition than f0, the impossibility to describe the effect of (even varies considerably with the morphology and climatology of short) dry periods inside the rainfall event, and the difficulty the site where the hydrologic model is applied. For an ex- of obtaining reliable estimates for its parameters, namely ample, in Northwestern Apennines the antecedent moisture f0, f1 and k. Several authors suggested modifications of conditions should be revised mostly after about 36–40 h, this the Horton’s equation to estimate infiltration for intermit- comes from the decade-long experience of the authors in ap- tent storm events including rainfall intensities lower than f0. plying this method operationally in in the area. So the SCS- Bauer (1974) accounted for dry periods through the recovery CN method, in its classical configuration, is neither suitable of soil infiltration capacity by coupling the Horton’s equation for continuous simulations nor to represent long events with with a drainage equation into lower soil layers. Following complex rainfall temporal distribution. Another intrinsic lim- Bauer’s formulation, Aron (1990) proposed a modification of itation of the method is that it refers to cumulated rainfall the Horton’s equation that makes the infiltration rate a func- and not to instantaneous intensity inputs. This, for equal tion of cumulative antecedent infiltration estimating the wa- volumes, leads to a constant runoff coefficient independent ter storage capacity of the soil from the potential maximum from the temporal distribution of rainfall. This characteris- retention S given by the SCS-CN. A similar approach is pro- tic, though the method has been used successfully to repro- posed in the SWAT model (Arnold et al., 1993). Aron (1990) duce flood events, fails theoretically in capturing complex smoothed the SCS-CN infiltration process by linearly filter- temporal distribution of runoff when infiltration-excess is the ing the infiltration increments. He computed the cumulative dominant runoff-producing mechanism. With the increasing infiltration F according to SCS-CN and defined the potential knowledge of rainfall spatial and temporal distribution pro- infiltration rate, f , proportional to the available soil moisture vided by modern sensors, this limitation becomes more and storage capacity, given by S−F . more evident. The modification of Horton’s equations, proposed here, Initial soil moisture conditions are crucial in determin- starts from the original formulation of (Bauer, 1974) and ing the performance of a hydrologic model in an operational (Diskin and Nazimov, 1994). It accounts for: (i) initial soil flood-forecasting chain. In the SCS-CN method the AMC moisture conditions linking them to initial infiltration capac- condition is discretized by only three different values. On ity f0; (ii) intermittent and low-intensity rainfall (namely one hand, this simplicity enhances the applicability of the lower than f0). The terms are properly managed through a method, but, on the other hand, it does not allow for an accu- mass balance equation applied to the “root zone” (RZ). RZ www.nat-hazards-earth-syst-sci.net/8/1317/2008/ Nat. Hazards Earth Syst. Sci., 8, 1317–1327, 2008 41320 S. S.Gabellani Gabellani et al.: et Calibration al.: Calibration methodology methodology for a combined for a combined Horton-SCS Horton-SCS infiltration scheme infiltration scheme

−1 where dpEquations(t) is the percolation 7 can be applied rate to deep to numerical layers [LT models], assuming −1 p(t) isthat the rainfallp(t) can rate be [LT considered] and f1 is a the constant percolationpi on rate time intervals −1 at t→∞i ·[∆LTt, (i].+ The 1) mass· ∆t, balance i ∈ N1 equation. Integrating takes then it over the∆t gives: form:   if p ≤ g : if p(t) ≤g(t)i: i h i   f1   piVmax − V ·∆t piVmax   + e maxV (t) V (ti) − p(t) − dp(t) f=1 p(t) − f1 f1 dV (t)  Vmax V=(ti+1) = (7) (8) dt if pi:>· gi : ¸ if p(t) > g(t)   − f0 ∆t − f0 ∆t  V 1− e Vmax + V (t )e Vmax g(t) − d(t)max= f 1 − V (t) i  p 0 Vmax

Equationwhere (7)p cani and beg appliedi are the to numerical values of modelsp(t) and assumingg(t) assumed con- that p(t)stantcan inside be considered the time a interval constantip·i∆ont, time(i + intervals 1) · ∆t. The tuning i·1t:(i+1)·1t, i∈N 1. Integrating it over 1t gives: of the 3 model parameters, f0, Vmax, f1 is described in the next section.  ≤ : if pi gi Fig.Fig. 1. 1.AA scheme scheme of the the modified modified Horton Horton method method for the for parameters the parameters   − f1 · h i g(t) p r d  pi Vmax V 1t pi Vmax calibration;calibration;g(t)isis the the filter, filter, pisis the the rainfall, rainfall,isr theis runoff, the runoff,is thed is the 4 Combining + e SCS-CNmax V (tandi) −modified Horton’s equation:  f1 f1 infiltration and dp is the percolation to lower strata.  infiltration and dp is the percolation to lower strata. V (ti+1) = a methodology for parameters calibration(8) if pi > gi :   f f One of the most− appealing0 1t characteristics− 0 1t of the SCS-CN Vmax 1 − e Vmax + V (ti)e Vmax valuesoil layerf0 for is dry modelled soil condition as a linear and reservoir the value of totalf1 capacityfor saturated method and the reason of its wide application in catchment soilVmax condition. The state (Diskin of the and reservoir Nazimov, is quantified 1994). by the state modeling is the simplicity in estimating its parameters. Thus, variable V (t−t ) (0 f0 − (f0 − f1) : expression of f0, it is necessary to introduce strong hypothe- = g(t) − dp(t) Vmax (5) based methods. The easiest way to link Horton’s equation dt  V (t) f0 − (f0 − f1) and SCS-CNses which is to hamper assume analytical the use of equality the method between in a their continuous-like Vmax The input and output terms are modeled as follows: outputsframework under the same (see rainfall appendix and soil A for moisture the analytical conditions. details). It is (e.g., Brasnecessary, 1990; toMishra change and perspective Singh, 2003). in The order assump- to overcome such tions thatlimitations. are needed to equate the two models lead the final  V (t) if Vp(t)(t)≤ f0 − (f0 − f1) : formulation of the Horton’s equation to fall back in many of dp(t) =f1 Vmax (6b) SCS − CN parameters were calibrated using daily r(t)Vmax the limitations of the SCS-CN method (e.g., Pitt et al., 1999). g(t) = (6a) rainfall-runoff records corresponding to maximum annual − − V (t) : The equations yield a null percolation rate which is not phys- if p(t) > f0 (f0 f1) V flows derived from gauged watersheds in the U.S.A. for  max −1 ically appealing. Moreover, in order to obtain an analytical where d(t) is the percolationV (t) rate to deep layers [LT ], p f0 − (f0 − f1) which information on their soils, land cover, and hydro- Vmax−1 expression of f0, it is necessary to introduce strong hypothe- p(t) is the rainfall rate [LT ] and f1 is the percolation rate logic conditions were available (United States Department of −1 ses which hamper the use of the method in a continuous-like at t → ∞ [LT ]. The mass balance equation takes then the Agriculture, 1954). Because of the empirical nature of the V (t) 1Usually 1t is the step on which rainfall is measured by rain- form: = method, the CN values would need, if not a complete ”re- dp(t) f1 (6b) gauges or radar (typically ranges from 10 min to 1 h). Vmax calibration”, at least an adjustment on historical events to  match the observed discharge in the catchments which might Nat. Hazardsif Earthp(t) Syst.≤ g( Sci.,t): 8, 1317–1327, 2008 www.nat-hazards-earth-syst-sci.net/8/1317/2008/  belong to different morphology contexts. This problem is of-  V (t) dV (t) p(t) − dp(t) = p(t) − f1 ten overlooked by hydrologists who rely on literature values. = Vmax (7) dt if p(t) > g(t):  ³ ´ 1Usually ∆t is the step on which rainfall is measured by rain-  V (t) g(t) − dp(t) = f0 1 − Vmax gauges or radar (typically ranges from 10 minutes to 1 hour) S. Gabellani et al.: Calibration methodology for a combined Horton-SCSS. Gabellani infiltration et al.: scheme Calibration methodology for a 1321 combined Horton-SCS infiltration scheme 5 framework (see Appendix A for the analytical details). It is tics of maximum flows, the SCS-CN method gives good re- necessary to change perspective in order to overcome such sults for such durations. Such limited temporal extension jus- limitations. tifies also the assumption of negligible cumulate losses and, SCS-CN parameters were calibrated using daily rainfall- therefore, of null residual infiltration rate. Therefore, the to- runoff records corresponding to maximum annual flows derived from gauged watersheds in the USA for which infor- tal duration of the design events N has been set equal to 12 mation on their soils, land cover, and hydrologic conditions hours, and the cumulated rainfall P have been fixed by the were available (United States Department of Agriculture, annual maximum values recorded from 125 stations in Lig- 1954). Because of the empirical nature of the method, the uria Region from 1951 to 2001. CN values would need, if not a complete “re-calibration”, at The fact that the CN method gives identical results in least an adjustment on historical events to match the observed terms of final cumulated runoff regardless of the hyetograph discharge in the catchments which might belong to different shape leads to an ill-posed mathematical problem. Further morphology contexts. This problem is often overlooked by hydrologists who rely on literature values. Although the au- assumptions are therefore needed in order to define a proper thors feel that this is a key problem in the application of the distribution of the rainfall intensities Int. Several different SCS-CN method, it is out of the scope of the present work Fig.Fig. 2. 2. SchemeScheme of the the modified modified Horton Horton method method for the for parameters the parameters shapes are possible (linear increasing, Chicago Hyetograph, calibration; g(t) is the filter, p is the rainfall, r is the runoff, d is the to discuss it in detail. It is only necessary to notice that the calibration; g(t) is the filter, p is the rainfall, r is the runoff, d is the etc.) which give slightly different values of f0. In order to infiltration. CN values used in the presented test-case have been carefully infiltration. avoid an arbitrary choice we have searched for the more ap- tested for the study area (Liguria Region, see below) on the propriate event’s shape by comparing the runoff produced by basis of the maximum annual flows recorded there, follow- of the design events N has been set equal to 12 h, and the cu- the two methods not only on the total duration N of the event, ing techniques similar to the original SCS-CN work (Boni Althoughmulated rainfall the authorsP have feel been that fixed this by is the a key annual problem maximum in the ap- et al., 2007). Their work proved that CN values give a reli- but also on smaller durations. The shape of the hyetograph plicationvalues recorded of the SCS-CN from 125method, stations in it Liguriais out of Region the scope from of the able description of the hydrological behavior of the soils in that optimizes such a comparison has been sought by mini- 1951 to 2001. the region when the class of events that causes the maximum present work to discuss it in detail. It is only necessary to no- mizing the following objective function: The fact that the CN method gives identical results in terms annual flows is of concern. It would, therefore, be reasonable tice that the CN values used in the presented test-case have of final cumulated runoff regardless of the hyetograph shape s to force the modified Horton’s method to give the same re- been carefully tested for the study area (Liguria Region, see P leads to an ill-posed mathematical problem. Further assump- N (r − r )2 sults for the same class of events. In order to properly select i=1 Hi CNi below)tions are on therefore the basis needed of the in order maximum to define annual a proper flows distribu- recorded RMSE = (9) the events to be used for extensive calibration of the modified N there,tion of following the rainfall techniques intensities I similar. Several to different the original shapes areSCS-CN Horton’s model parameters, it is necessary to identify un- nt workpossible (Boni (linear et al., increasing, 2007). ChicagoTheir work Hyetograph, proved etc.) that whichCN values where N is the duration of the event in hours, r and ambiguously the characteristics of the intense events in the Hi givegive a slightly reliable different description values of of thef . hydrological In order to avoid behavior an ar- of the r are the runoffs obtained by the modified Horton method region. Specifically the characteristics influencing, at most, 0 CNi bitrary choice, we have searched for the more appropriate 2 the runoff production: the total duration, the range of rainfall soils in the region when the class of events that causes the and the SCS-CN method, respectively . The procedure has event’s shape by comparing the runoff produced by the two intensities spanned and their temporal distribution. maximum annual flows is of concern. It would be therefore been repeated for each CN. The unknowns of the minimiza- methods not only on the total duration N of the event, but According to the SCS-CN method, only for this calibration reasonable to force the modified Horton’s method to give the tion problem are: the duration N, the intensity I of the also on smaller durations. The shape of the hyetograph that nt step, the final infiltration rate f is set to null. This limit is 1 sameoptimizes results such for a the comparison same class has beenof events. sought In by order minimizing to properly N − 1 showers, the cumulated rainfall P and the initial in- justified by the fact that percolation cumulates on these spe- selectthe following the events objective to be function: used for extensive calibration of the filtration rate f0. In total there are N + 2 unknowns. The cific kind of events and short durations is negligible if com- modified Horton’s model parameters it is necessary to iden- system of equations is constituted by the N + 2 derivatives pared to runoff and infiltration; this hypothesis, not feasible tify unambiguously the characteristics of the intense events for real catchments, will be relaxed further on in the paper. s of equation 9. However we have reduced the unknowns hav- PN 2 in the region. Specifically(rH − r the) characteristics influencing at ing fixed the values of cumulated rainfall P and of the total Following Aron (1990), the maximum soil capacity Vmax is RMSE = i=1 i CNi (9) set equally to the potential maximum retention S in average most the runoff production:N the total duration, the range of duration of the event N. conditions (AMC-II): rainfall intensities spanned and their temporal distribution. The variables of function 9 are hence the N − 1 intensities where N is the duration of the event in hours, r and r Figure 2 illustrates the scheme adopted for this step of the According to the SCS-CN method, only forHi this calibrationCNi of the showers, I , and the initial infiltration rate, f . The are the runoffs obtained by the modified Horton method and nt 0 calibration. Under the formulated assumptions, f0 is the only step,the SCS-CN the final method, infiltration respectively rate f12.is The set procedure null. This has constrain been is minimum research procedure is based on the Conjugate Gra- parameter of the modified Horton method which needs cali- justified by the fact that percolation cumulates on this spe- dient Method and starts from a random distribution of hourly bration. repeated for each CN. The unknowns of the minimization Referring to the analysis of historical series and ME- cificproblem kind are: of events the duration and shortN, the durations intensity isInt negligibleof the N− if1 com- showers. The procedure is iterated starting from different TEOSAT satellite images (Deidda et al., 1999; G.N.D.C.I., paredshowers, to runoff the cumulated and infiltration; rainfall P and this the hypothesis, initial infiltration not feasible initial distributions of the single showers. By looking at the 1992; Boni et al., 2007) a typical duration of the intense for real catchments, will be relaxed further on in the paper. hyetographs obtained by the minimization of the equation 9 2RMSE has been adopted to quantify the error in terms of the events in Liguria region is about 12 h. As already mentioned, Followingunit of the variable.Aron (1990), It is know the that, maximum like variance, soil meancapacity squaredVmax is (see figure 3) one can notice that in the first part of the event because of the extensive calibration on regional statistics of seterror equal has the to disadvantage the potential of weighting maximum outliers. retention This isS ain result average the values of hourly showers are apparently random because maximum flows, the SCS-CN method gives good results for conditionsof the squaring (AMC of each− term,II): which effectively weights large errors the cumulated rainfall is less than the initial abstraction Ia in such durations. Such limited temporal extension justifies also moreFigure heavily 2 illustrates than small ones. the schemeHowever using adopted mean for absolute this errorsstep of the the SCS-CN method, and the runoff is null. When the cumu- the assumption of negligible cumulate losses and, therefore, (MAE) leads to similar results indicating that significant outliers are of null residual infiltration rate. Therefore, the total duration calibration.not generated Under within the the comparison. formulated assumptions, f0 is the only lated rainfall reaches Ia the runoff production begins and the parameter of the modified Horton method which needs cali- 2 bration. RMSE has been adopted to quantify the error in terms of the www.nat-hazards-earth-syst-sci.net/8/1317/2008/ Nat. Hazards Earth Syst. Sci., 8, 1317–1327, 2008 unit of the variable. It is know that, like variance, mean squared Referring to the analysis of historical series and ME- error has the disadvantage of weighting outliers. This is a result TEOSAT satellite images (Deidda et al., 1999; G.N.D.C.I., of the squaring of each term, which effectively weights large errors 1992; Boni et al., 2007) a typical duration of the intense more heavily than small ones. However using mean absolute errors events in Liguria region is about 12 hours. As already men- (MAE) leads to similar results indicating that significant outliers are tioned, because of the extensive calibration on regional statis- not generated within the comparison. S. Gabellani et al.: Calibration methodology for a combined Horton-SCS infiltration scheme 7 1322 S. Gabellani et al.: Calibration methodology for a combined Horton-SCS infiltration scheme

Fig. 3. Examples of hyetographs obtained by searching rainfall events that minimize the differences between the runoff obtained by the Fig.modified 3. ExamplesHorton method of hyetographs and the SCS-CN obtained method. by searchingVmax is rainfallset equal event to S(AMC-II). that minimize the differences between the runoff obtained by the modified Horton method and the SCS-CN method. Vmax is set equal to S(AMC-II). rate f0. In total, there are N+2 unknowns. The system of Once the cumulated rainfall exceeds Ia, the SCS-CN Tableequations 2. Nash-Sutcliffe is constituted coefficient by the N (NS),+2 derivativesroot mean square of Eq. error (9). (RMSE)method and mean always absolute produces error runoff, (MAE) whilefor the the considered runoff produced events in the caseHowever, of use weof the havemodified reducedHorton the unknowns method and by in fixing the case the of valuesSCS-CN method.by the modified Horton method is a function of the rainfall in- of cumulated rainfall P and of the total duration of the event tensity (if p(t)

Nat. Hazards Earth Syst. Sci., 8, 1317–1327, 2008 www.nat-hazards-earth-syst-sci.net/8/1317/2008/ 8 S. Gabellani et al.: Calibration methodology for a combined Horton-SCS inﬁltration scheme

tensity. It could or not entirely inﬁltrate depending on temporal distribution of P . The same problem affects the initial condition V (ti−1), so that it is necessary to make an assumption to integrate the equations from ti−1 to ti. dP Moreover we have to bind dt (t) to be always grater than g(t). In this way an analytical equation of the methods is possible. Equating 11 and 13, assuming Vmax = S and starting from V (ti−1) = Ia = 0.2Vmax, which assumes dP dt (t) < g(t) in the period before the cumulates reaches Ia, the equation 14 is obtained:

if pi ≤ gi : · ¸ piVmax − f1 ·∆t piVmax V (ti+1) = + e Vmax V (ti) − (12) f1 f1

Fig. 4. An example of hyetograph used in the assessment of the if pi > gi : · ¸ initial infiltration capacity f0. 8 S. Gabellani et al.: Calibration methodology for a combined Horton-SCS infiltration scheme − f0 ∆t − f0 ∆t S. Gabellani et al.: Calibration methodology for a combined Horton-SCS infiltration scheme 1323 V (ti+1) = Vmax 1 − e Vmax + V (ti)e Vmax (13)

tensity. It could or not entirely infiltrate depending on· tem- ¸ − f0 ∆t − f0 ∆t poral distribution of P . Vmax 1 − e Vmax + 0.2Vmaxe Vmax = The same problem affects the initial condition V (ti−1), so that it is necessary to make an assumption to integrate the (P − 0.2Vmax) = Vmax (14) equations from ti−1 to ti. (P + 0.8Vmax) dP Moreover we have to bind dt (t) to be always graterand solving than for f0 g(t). In this way an analytical equation of the methods is if P > 0.2Vmax : possible. Equating 11 and 13, assuming Vmax = S and Vmax (Vmax) starting from V (ti−1) = Ia = 0.2Vmax, which assumes f0 = − ln (15) dP 0.8∆t (P + 0.8Vmax) dt (t) < g(t) in the period before the cumulates reaches Ia, the equation 14 is obtained: On the other hand the modified Horton method switches between the two equations which describe the evolution of if pi ≤ gi : V (t) (equations 8) depending on the temporal distribution · ¸ dP piVmax − f1 ·∆t piVmaxof P , therefore on the values of the rainfall intensities dt (t). V (ti+1) = + e Vmax V (ti) − The switch(12) in the equations is difficult to manage analytically Fig. 4. An example of hyetograph used in the assessment of the Fig. 5. Trend of the initial infiltrationf1 rate f0 as a function of the f1 Fig. 5. Trend of the initial infiltration rate f0 as a function of the and can be tackled numerically as proposed in the paper. initial infiltration capacity f0. CN(AMC-II). Standard deviation error bars are plotted. Fig. 4. An example of hyetograph used in the assessment of the if pi > gi : CN(AMC-II). Standard deviation· error bars are plotted.¸ The hypothesis of equating runoff or infiltration respec- initial infiltration capacity f0. − f0 ∆t − f0 tively∆t operated in the numerical approach and in the mathe- V (t ) = V 1 − e Vmax + V (t )e Vmax (13) been originally conceived for catchment scale application). Table 1. Parametersi+1 f0 andmaxVmax of the modified Horton method i matical comparison are identical under the assumption P > The modified Horton method, during the calibration process, as resulting from the calibration; cv is the coefficient of variation of Appendix: Analytic comparison 0.2Vmax. ∆t and P in the analytic comparison are the cu- f f and σf is the standard deviation of f . adapts the parameter 0 trying to reproduce the effects on the 0 0 · ¸0 mulated and duration on the whole event while in the numer- runoff estimation caused by the described issues. − f0 ∆t − f0 ∆t The analyticalVmax derivation1 − e follows.Vmax + 0.2Vmaxe Vmax = ical framework the differences between the two methods are With this calibration procedure, from the information CNAMC−II f0 [mm/h] σf0 [mm/h] cv Vmax [mm] of CN maps, the spatial distribution of modified Horton Infiltration by SCS-CN is given by: computed on each single step. 40 70 33.6(P 0.38− 0.2V 375 ) method’s parameters have been obtained. 45 60 27.1 0.36 306max As can be easily deduced from equation 15 the value (P − I=) Vmax (14) 50F 51= 23.4a (P 0.38S+ 0.8V 250max) (11) of f0 results to be strongly dependent on cumulated rain- 55 47 24.4 0.39 205 (P + S − Ia) fall P and on which duration 4t it is occurred, indepen- 60and solving 43 for f 22.5 0.38 167 5 Catchment applications 0 dently of its temporal distribution. Knowing this latter the Cumulated65 infiltration 41 by modified 20.5 0.36Horton 135 is given by In order to evaluate the modified Horton method behavior in 70if P > 390.2Vmax 16.6: 0.30 107 15 could be used step by step always under the assumption equations75 8. To carry 36 out the analytical 11.8 0.21 comparison 83 we have dP hydrological simulations, this infiltration scheme has been that dt > g(t), but in this case the evolution of V (t) needs to make some80 significant 30 hypotheses 6.9Vmax due 0.12 to the different 63(Vmax) for- implemented in a semi-distributed rainfall-runoff model, 85 22f0 = 3.8− 0.08ln 44 to be introduced(15) (V (ti−1) 6= 0). DRiFt – Discharge River Forecast – (Giannoni et al., 2005). malization of the two methods. 0.8∆t (P + 0.8Vmax) 90 14 2.3 0.08 28 Acknowledgements. This work was supported by Italian Civil Pro- Several “multi-peaks” events, for which the SCS-CN method First of95 all the expression 7 of cumulated 1.2 0.07 infiltration 13 F of the On the other hand the modified Horton methodtection switches Department within the PROSCENIO Research Project. usually failed, have been considered. DRiFt with the modi- SCS-CN method is valid when P > Ia with Ia = 0.2Vmax. Prof. Fabio Castelli is deeply acknowledged for the fruitful dis- fied Horton infiltration scheme uses the distribution of the This portionbetween of rainfall the ( twoI ) in equations the case of whichmodified describeHorton the evolution of a cussions on the matter. The authors thanks the referees for their soil capacity Vmax and the initial infiltration f0 maps as pre- method is processedV (t) (equations by the equations 8) depending and its on analytical the temporal ex- distribution viously determined from the Curve Numbers for the differ- of P , therefore on the values of the rainfall intensitiescarefuldP revision(t). that helped improving the quality of this paper. pressionthe most. depends The initial on the moisture temporal conditions distribution have been of rainfall con- in- dt ent catchments (see Table 1). The final infiltration rate f1 has The switch in the equations is difficult to manage analytically been re-introduced using the scheme of Fig. 1 as a percentage sistently chosen with the cumulated rainfall of the five days Fig. 5. Trend of the initial infiltration rate f0 as a function of theantecedentand the can event be for tackled both modified numericallyHorton and as SCS-CN proposed in the paper. of the infiltration rate for dry soil f0: CN(AMC-II). Standard deviation error bars are plotted. methods. FiguresThe6 hypothesis, 7, 8 and 9 show of equating some validation runoff events or infiltration respec- that comparetively the operated performances in of the both numerical methods with approach the ob- and in the mathe- served hydrographs. In Table 2, the values of Nash-Sutcliffe f1 = cf f0 (10) coefficientmatical (NS), root comparison mean square are error identical (RMSE) and under mean the assumption P > Appendix: Analytic comparison absolute error0.2V (MAE). are∆t reported.and P in the analytic comparison are the cu- where cf is a calibration parameter that belongs to interval max [0,1] and represents a physically meaningful simplification. Figuremulated6 reports and the duration observed hydrographon the whole for event the 6– while in the numer- The analyticalIn this way, derivation the spatial distribution follows. if f1 is fixed by the dis- 13 Novemberical framework 1997 event at the the differences Bisagno creek between closed at the two methods are 2 Infiltrationtribution of byf0SCS-CNand the parameteris givencf by:is a calibration param- Gavette (89computed km ) and the on simulations each single obtained step. with the mod- eter at catchment scale. Just one event per each catchment ified Horton method and with the SCS-CN method. When (not shown) has been used to provide calibration for cf , cho- using the SCS-CNAs can method be easilyit is not deducedpossible to simulatefrom equation the 15 the value (P − I ) sen in a range whereF = the residual infiltrationa S capacity counts (11)whole eventof butf0 itresults is necessary to be to break strongly it into three dependent parts and on cumulated rain- (P + S − Ia) fall P and on which duration 4t it is occurred, indepen- www.nat-hazards-earth-syst-sci.net/8/1317/2008/dently Nat. Hazards of its Earth temporal Syst. Sci., distribution. 8, 1317–1327,Knowing 2008 this latter the Cumulated infiltration by modified Horton is given by 15 could be used step by step always under the assumption equations 8. To carry out the analytical comparison we have that dP > g(t), but in this case the evolution of V (t) needs to make some significant hypotheses due to the different for- dt to be introduced (V (t ) 6= 0). malization of the two methods. i−1 First of all the expression of cumulated infiltration F of the Acknowledgements. This work was supported by Italian Civil Pro- tection Department within the PROSCENIO Research Project. SCS-CN method is valid when P > Ia with Ia = 0.2Vmax. This portion of rainfall (I ) in the case of modified Horton Prof. Fabio Castelli is deeply acknowledged for the fruitful dis- a cussions on the matter. The authors thanks the referees for their method is processed by the equations and its analytical ex- careful revision that helped improving the quality of this paper. pression depends on the temporal distribution of rainfall in- 1324 S. Gabellani et al.: Calibration methodology for a combined Horton-SCS infiltration scheme

Table 2. Nash-Sutcliffe coefﬁcient (NS), root mean square error (RMSE) and mean absolute error (MAE) for the considered events in the case of use of the modiﬁed Horton method and in the case of SCS-CN method.

Horton Mod. SCS method Date Basin NS RMSE MAE NS RMSE MAE 6 Nov 1997 Bisagno 0.87 11.8 6.8 0.55 22.0 10.9 S. Gabellani et al.: Calibration methodology25 Nov 2002 for Bisagno a combined 0.71 Horton-SCS 29.7 7.6 inﬁltration 0.48 34.4 scheme 10.8 9 2 Aug 1965 Magra 0.90 77.5 37.5 0.96 80.6 40.9 2 Dec 1966 Magra 0.91 115.8 20.2 0.85 147.8 26.0

2 Fig. 6. 6-13Fig. November 6. 6–13 November 1997 event, 1997 event,Bisagno Bisagno river river closed closed at at Gavette Gavette (89 (89 kmkm).2). Comparison Comparison between between the simulations the simulations performed performed with the with the DRiFt model using the modiﬁed Horton method and the SCS-CN method. The event has been broken into three parts for simulation using DRiFt modelSCS-CN using method. the modiﬁed Horton method and the SCS − CN method. The event has been break in three parts for simulation using SCS − CN method.

re-initialize the model3, while the modified Horton method 6 Conclusions allows for a continuous simulation of the event, improving the performance in reproducing the last peak. Figures 7, 8 and 9 illustrate some events in which the performances of A procedure that maps the CN values onto the initial infiltra- the two model are comparable. tion rate f0 of a modified Horton method is proposed. The procedure allows for an easy calibration of the main parameter of the modified Horton method at catchment scale exploiting the advantages offered by the SCS-CN method. The two methods are bounded to give the same performance in terms of cumulated run-off in a specific range of rainfall intensities and durations where the hypothesis of the SCS-CN 3The antecedent moisture conditions for the SCS-CN method method do not represent a limitation. A further calibration are coherently given with the rainfall observed in the five days pre- step is due for f1 on events where the residual infiltration ceding the event. rate is important. The f1 spatial distribution simply shifts the

Nat. Hazards Earth Syst. Sci., 8, 1317–1327, 2008 www.nat-hazards-earth-syst-sci.net/8/1317/2008/

Fig. 7. 25-27 November 2002 event, Bisagno river closed at La Presa (34 km2). Comparison between the simulations performed with the DRiFt model using the modiﬁed Horton method and the SCS − CN method. S. Gabellani et al.: Calibration methodology for a combined Horton-SCS inﬁltration scheme 9

Fig. 6. 6-13 November 1997 event, Bisagno river closed at Gavette (89 km2). Comparison between the simulations performed with the DRiFt model using the modiﬁed Horton method and the SCS − CN method. The event has been break in three parts for simulation using SCS − CN method.

S. Gabellani et al.: Calibration methodology for a combined Horton-SCS inﬁltration scheme 1325

10 S. Gabellani et al.: Calibration methodology for a combined Horton-SCS inﬁltration scheme

Fig. 7. 25–27 November 2002 event, Bisagno river closed at La Presa (34 km2). Comparison between the simulations performed with the 2 Fig. 7. 25-27DRiFt November model using 2002 the modiﬁed event, BisagnoHorton method river and closed the SCS-CN at La Presa method. (34 km ). Comparison between the simulations performed with the DRiFt model using the modiﬁed Horton method and the SCS − CN method.

Fig. 8. 2–5 August 1965 event, Magra river closed at Calamazza (936 km2). Comparison between the simulations performed with the DRiFt 2 Fig. 8. 2-5model August using 1965 the event,modiﬁed MagraHorton river method closed and the at CalamazzaSCS-CN method. (936 km ). Comparison between the simulations performed with the DRiFt model using the modiﬁed Horton method and the SCS − CN method.

www.nat-hazards-earth-syst-sci.net/8/1317/2008/ Nat. Hazards Earth Syst. Sci., 8, 1317–1327, 2008

Fig. 9. 2-3 December 1966 event, Magra river closed at Calamazza (936 km2). Comparison between the simulations performed with the DRiFt model using the modiﬁed Horton method and the SCS − CN method. 10 S. Gabellani et al.: Calibration methodology for a combined Horton-SCS inﬁltration scheme

Fig. 8. 2-5 August 1965 event, Magra river closed at Calamazza (936 km2). Comparison between the simulations performed with the DRiFt model using the modiﬁed Horton method and the SCS − CN method.

1326 S. Gabellani et al.: Calibration methodology for a combined Horton-SCS inﬁltration scheme

2 Fig. 9. 2–3 December 1966 event, Magra river closed at Calamazza (936 km 2). Comparison between the simulations performed with the Fig. 9. DRiFt2-3 December model using 1966 the modiﬁed event, MagraHorton river method closed and the at CalamazzaSCS-CN method. (936 km ). Comparison between the simulations performed with the DRiFt model using the modiﬁed Horton method and the SCS − CN method.

f0 spatial distribution at catchment scale. The results within pression depends on the temporal distribution of rainfall in- a simple distributed hydrological model show how the cali- tensity. It could, or not entirely, inﬁltrate depending on tem- brated Horton method outperforms the SCS-CN method on poral distribution of P . long, multi-peak events. The structure of modiﬁed Horton The same problem affects the initial condition V (ti−1), so method is suitable for implementation in continuous models that it is necessary to make an assumption to integrate the without schematization straining and, within the proposed equations from ti−1 to ti. dP framework, maintains simplicity in the calibration phase of Moreover, we have to bind dt (t) to be always greater than its main parameters. g(t). In this way, an analytical equation of the methods is possible. Equations A1 and A3, assuming Vmax=S and start- = = dP ing from V (ti−1) Ia 0.2Vmax, which assumes dt (t)

Inﬁltration by SCS-CN is given by: if pi>gi :

f f − 0 1t − 0 1t = − Vmax + Vmax (P − I ) V (ti+1) Vmax 1 e V (ti)e (A3) F = a S (A1) (P + S − Ia)

f f Cumulated infiltration by modified Horton is given by − 0 1t − 0 1t Vmax 1 − e Vmax + 0.2Vmaxe Vmax = equations 8. To carry out the analytical comparison we have to make some significant hypotheses due to the different for- (P − 0.2Vmax) = Vmax (A4) malization of the two methods. (P + 0.8Vmax) First of all, the expression of cumulated infiltration F of and solving for f0 if P >0.2Vmax : the SCS-CN method is valid when P >Ia with Ia=0.2Vmax. This portion of rainfall (Ia), in the case of modified Horton V (V ) = − max max method, is processed by the equations and its analytical ex- f0 ln (A5) 0.81t (P + 0.8Vmax)

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On the other hand, the modified Horton method switches Giannoni, F., Roth, G., and Rudari, R.: Can the behaviour of dif- between the two equations which describe the evolution of ferent basins be described by the same model’s parameter set? V (t) (Eq. 8) depending on the temporal distribution of P , A geomorphologic framework, Phys. Chem. Earth, 28, 289–295, dP 2003. therefore, on the values of the rainfall intensities dt (t). The switch in the equations is difficult to manage analytically and Giannoni, F., Roth, G., and Rudari, R.: A procedure for drainage can be tackled numerically as proposed in the paper. network identification from geomorphology and its application to the hydrologic response, Adv. Water Resour., 28, 567–581, The hypothesis of equating runoff or infiltration respec- 2005. tively operated in the numerical approach and in the math- CNR-G.N.D.C.I.: Linea 3 Rapporto di evento: Savona – ematical comparison are identical under the assumption 22 September, Genova – 27 September 1992, 1994. P >0.2Vmax. 1t and P in the analytic comparison are the Grove, M., Harborand, J., and Engel, B.: Composite vs distributed cumulated and the duration on the whole event while in the curve numbers: Effects on estimates of storm runoff depths, J. numerical framework the differences between the two meth- Am. Water Resour. Assoc., 5(34), 1015–1033, 1998. ods are computed on each single step. Hjelmfelt, A.: Curve number Procedure as infiltration Method, J. As can be easily deduced from Eq. A5, that the value of Hydr. Div. ASCE, 106, 1107–1110, 1980a. Hjelmfelt, A.: An empirical investigation of the curve number tech- f results are strongly dependent on cumulated rainfall P 0 nique, J. Hydr. Div. ASCE, 106, 1471–1476, 1980b. 4 and on which duration t it is occurred, independently of Horton, R.: The role of infiltration in the hydrological cycle, Trans., its temporal distribution. Knowing this, the Eq. A5 could be Am. Geophisical Union, 14, 446–460, 1933. dP used step by step always under the assumption that dt >g(t), McCuen, R.: A Guide to Hydrologic Analysis using SCS Methods, but in this case the evolution of V (t) needs to be introduced Prentice-Hall, Englewood Cliffs, NY, 1982. (V (ti−1)6=0). Michel, C., Andrassian, V., and Perrin, C.: Conservation Ser- vice Curve Number method: How to mend a wrong soil mois- Acknowledgements. This work was supported by Italian Civil ture accounting procedure?, Water Resour. Res., 41, W02011, Protection Department within the PROSCENIO Research Project. doi:10.1029/2004WR003191, 2005. Fabio Castelli is deeply acknowledged for the fruitful discussions Mishra, S. and Singh, V.: Soil Conservation Service Curve Number on the matter. The authors thank the referees for their careful (SCS-CN) Methodology, Kluwer Academic Publisher, 2003. revision that helped improve the quality of this paper. Moglen, G.: Effect of orientation of spatially distributed curve numbers in runoff calculations, J. Am. Water Resour. Ass., 6(36), Edited by: F. Guzzetti 1391–1400, 2000. Reviewed by: T. Moramarco and another anonymous referee Pitt, R., Lilburn, M., Nix, S., Durrans, S., Burian, S., Voorhees, J., and Martinson, J.: Guidance Manual for Integrated Wet Weather Flow (WWF) Collection and Treatment Systems for Newly Ur- banized Areas (New WWF Systems), US Environmental Protec- References tion Agency, 1999. Ponce, V. and Hawkins, R.: Runoff Curve Number: Has It Reached Arnold, J., Allen, P., and Bernhardt, G.: A comprehensive surface maturity?, J. Hydrol. Eng., 1(1), 11–19, 1996. groundwater flow model, J. Hydrol, 142, 47–69, 1993. Reed, S. J. and Zhang, Z.: A distributed hydrologic model and Aron, G.: Adaptation of Horton and SCS infiltration equations to threshold frequency-based method for flash flood forecasting at complex storms, J. Irrig. and Drainage Eng., 118, 275–284, 1990. ungauged locations, J. Hydrol., 337, 402–420, 2007. Bauer, S.: A modified Horton equation during intermittent rainfall, Risse, L., Liu, B., and Nearing, M.: Using Curve Number to Deter- Hydrol. Sci. Bull., 19, 219–229, 1974. mine Baseline Values of Green-Ampt Effective Hydraulic Con- Boni, G., Ferraris, L., Giannoni, F., Roth, G., and Rudari, R.: Flood ductivities, Water Resour. Bull., 31, 147–158, 1995. probability analysis for un-gauged watersheds by means of a sim- Siccardi, F., Boni, G., Ferraris, L., and Rudari, R.: A hydromete- ple distributed hydrologic model, Adv. Water Resour., 30, 2135– orological approach for probabilistic flood forecast, J. Geophys. 2144, 2007. Res., 110, D05101, doi:10.1029/2004JD005314, 2005. Bras, R.: Hydrology, Addison-Wesley Publishing Co., Reading, Singh, P., Frevert, D. K., and Meyer, S.: Mathematical Models of MA, 1990. Small Watershed Hydrology and Applications, Water Resources Deidda, R., Benzi, R., and Siccardi, F.: Multifractal modeling Publications, Highlands Ranch, CO., 2002. of anomalous scaling laws in rainfall, Water Resour. Res., 35, Steenhuis, T., Winchell, M., Rossing, J., Zollweg, J. A., and Wal- 1853–1867, 1999. ter, M.: SCS Runoff Equation Revisited for Variable-Source Diskin, M. H. and Nazimov, N.: Linear riservoir with feedback reg- Runoff Areas, Journal of Irrigation and Drainage Engineering, ulated inlet as a model for the infiltration process, J. Hydrol., 172, May/June, 1995. 313–330, 1994. United States Department of Agriculture, S. C. S., National En- Franz, K., Ajami, N., Schaake, J., and Buizza, R.: Hydrologic geneering Handbook, Section 4, US Department of Agriculture, Ensemble Prediction Experiment Focuses on Reliable Forecasts, Washington, DC, 1954. Eos. Trans. AGU, 86(25), 239, 2005. Giannoni, F., Roth, G., and Rudari, R.: A Semi – Distributed Rain- fall – Runoff Model Based on a Geomorphologic Approach, Phys. Chem. Earth, 25, 665–671, 2000. www.nat-hazards-earth-syst-sci.net/8/1317/2008/ Nat. Hazards Earth Syst. Sci., 8, 1317–1327, 2008