A continuous simulation model for design ...

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A continuous simulation model for design-hydrograph estimation in small and ungauged watersheds S. Grimaldi

a b c

d

, A. Petroselli & F. Serinaldi

e f

a

Dipartimento per l'Innovazione nei Sistemi Biologici, Agroalimentari e Forestali (DIBAF), University of Tuscia, Via San Camillo De Lellis snc, I-01100, Viterbo, Italy b

Honors Center of Italian Universities (H2CU), Sapienza University of Rome, Via Eudossiana 18, I-00184, Roma, Italy c

Department of Mechanical and Aerospace Engineering, Polytechnic Institute of New York University, Six MetroTech Center, Brooklyn, New York, 11201, USA d

Dipartimento di Scienze e Tecnologie per l'Agricoltura, le Foreste, la Natura e l'Energia (DAFNE), University of Tuscia, Via San Camillo De Lellis snc, I-01100, Viterbo, Italy e

School of Civil Engineering and Geosciences, Newcastle University, Newcastle Upon Tyne, NE1 7RU, UK f

Willis Research Network, 51 Lime St., London, EC3M 7DQ, UK Version of record first published: 01 Aug 2012. To cite this article: S. Grimaldi, A. Petroselli & F. Serinaldi (2012): A continuous simulation model for design-hydrograph estimation in small and ungauged watersheds, Hydrological Sciences Journal, 57:6, 1035-1051 To link to this article: http://dx.doi.org/10.1080/02626667.2012.702214

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Hydrological Sciences Journal – Journal des Sciences Hydrologiques, 57(6) 2012

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A continuous simulation model for design-hydrograph estimation in small and ungauged watersheds S. Grimaldi1,2,3 , A. Petroselli4 and F. Serinaldi5,6 1

Dipartimento per l’Innovazione nei Sistemi Biologici, Agroalimentari e Forestali (DIBAF), University of Tuscia, Via San Camillo De Lellis snc, I-01100 Viterbo, Italy [email protected]

Downloaded by [Universita Studi la Sapienza] at 06:45 05 November 2012

2

Honors Center of Italian Universities (H2CU), Sapienza University of Rome, Via Eudossiana 18, I-00184 Roma, Italy

3

Department of Mechanical and Aerospace Engineering, Polytechnic Institute of New York University, Six MetroTech Center, Brooklyn, New York 11201, USA

4

Dipartimento di Scienze e Tecnologie per l’Agricoltura, le Foreste, la Natura e l’Energia (DAFNE), University of Tuscia, Via San Camillo De Lellis snc, I-01100 Viterbo, Italy [email protected] 5

School of Civil Engineering and Geosciences, Newcastle University, Newcastle Upon Tyne, NE1 7RU, UK [email protected] 6

Willis Research Network, 51 Lime St., London, EC3M 7DQ, UK

Received 3 October 2011, accepted 2 April 2012, open for discussion until 1 February 2013 Editor D. Koutsoyiannis Citation Grimaldi, S., Petroselli, A., and Serinaldi, F., 2012. A continuous simulation model for design-hydrograph estimation in small and ungauged watersheds. Hydrological Sciences Journal, 57 (6), 1035–1051.

Abstract The estimation of design hydrographs for small and ungauged watersheds is a key topic in hydrology. In this context, event-based procedures, which transform design storms deduced from intensity–duration– frequency curves by lumped rainfall–runoff models, are commonly applied. This study introduces a continuous simulation model that involves a two-stage rainfall generator, a geomorphological rainfall–runoff model and a flood frequency analysis applied to simulated runoff time series. The resulting design hydrograph with an assigned return period preserves peak and volume information. The case study results indicate that the continuous model is able to return a range of flood scenarios corresponding to a wide range of possible watershed physical conditions. Moreover, the rainfall–runoff model applied using empirical calibration without observations appears to be as accurate as other models based on regionalized information. Key words synthetic design hydrograph; copula-based rainfall generator; scale universal multi-fractal model; rainfall downscaling; WFIUH model; continuous models; ungauged basins

Un modèle de simulation continu pour l’estimation d’hydrogrammes de projet sur des petits bassins versants non jaugés

Résumé L’estimation d’hydrogrammes de projet sur des petits bassins versants non jaugés est un sujet clé en hydrologie. Dans ce contexte, on applique couramment des procédures événementielles, dans lesquelles des modèles pluie–débit globaux transforment des pluies de projet déduites de courbes intensité–durée–fréquence. Cette étude présente un modèle de simulation continu qui associe un générateur de pluie en deux étapes, une modèle pluie–débit géomorphologique et une analyse fréquentielle des crues appliquée à la série des débits simulés. L’hydrogramme de projet résultant pour la période de retour attribuée préserve l’information sur le volume et le pic. Les résultats de l’étude de cas indiquent que le modèle continu est en mesure de produire une gamme de scénarios de crue correspondant à un large éventail de conditions physiques possibles des bassins. En outre, le modèle pluie–débit appliqué en utilisant un calage empirique sans observations semble être aussi fiable que d’autres modèles basés sur une information régionalisée. Mots clefs hydrogramme de projet synthétique; générateur de pluie basé sur des copules; modèle multifractal universel d’échelle; descente d’échelle des pluies; modèle WFIUH; modèles continus; bassins non jaugés

ISSN 0262-6667 print/ISSN 2150-3435 online © 2012 IAHS Press http://dx.doi.org/10.1080/02626667.2012.702214 http://www.tandfonline.com

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1 INTRODUCTION Producing synthetic design hydrographs (SDHs) is a crucial objective of hydrological analyses because they represent the input of many hydraulic infrastructure designs and flood-risk mapping procedures (PAI 2006, EU 2007, FEMA 2009). Several methods for estimating SDHs are described in the literature, for which the appropriate approach is strongly dependent on watershed characteristics and data availability. In this study, the specific case of a small and ungauged basin is investigated. Here, the term “small” refers to watersheds with a drainage area of less than 150–200 km2 , for which it is reasonable to assume linear behaviour and to apply instantaneous unit hydrograph (IUH) theory (Dooge 1973). Since small basins are usually ungauged, defining appropriate models is a critical task for hydrologists. Here, the term “ungauged” is related to watersheds that lack discharge observations, while it is assumed that common digital elevation models (DEMs) with a standard resolution, soil-use digital support (i.e. CORINE 2000) and rainfall raingauge data are accessible. Under these conditions, two main categories of methods are available for SDH estimation: eventbased models and continuous models. Event-based schemes (Soczyñska et al. 1997, Hsieh et al. 2006, Alfieri et al. 2008) involve defining a design hyetograph with an assigned return time using intensity–duration–frequency (IDF) curves and applying a lumped rainfall–runoff model to obtain the design hydrograph. The event-based procedure is widely used, because IDF curves are easily estimated through the extreme value observations recorded in annual reports of hydrological offices, or by using regional IDF parameter maps when available. However, this approach is associated with some drawbacks and inaccuracies that are difficult to quantify. The hyetograph shape, the critical rainfall duration concept, the pre-event soil-moisture conditions, and the similar return time assumed for the design storm and design hydrograph could affect the results (Rahman et al. 2002, Hoes and Nelen 2005, Alfieri et al. 2008, Verhoest et al. 2010). These drawbacks are relevant, considering that, in addition to the peak flow, the hydrograph volume and duration are also crucial for risk-mapping procedures. Indeed, the available event-based methods could provide rough approximations in defining these additional SDH properties (Grimaldi et al. 2005). Continuous schemes (Boughton and Droop 2003, Faulkner and Wass 2005, Haberlandt et al.

2008, McMillan and Brasington 2008, Moretti and Montanari 2008, Blazkova and Beven 2009, Calver et al. 2009, Viviroli et al. 2009) consist of generating a long synthetic rainfall time series and using it as an input for a continuous rainfall–runoff model. By selecting the maximum annual hydrographs from the obtained synthetic runoff time series, it is possible to determine the design hydrograph with an assigned return time through flood frequency analysis combined with a SDH procedure (Pramanik et al. 2010, Serinaldi and Grimaldi 2011). As an example of the performance of continuous frameworks, in the following we briefly recall the main findings of four studies that discuss the uncertainty of the resulting peak flow with high return period. A comparison among event-based procedures and continuous simulation methods is presented by Calver et al. (2009), who considered 107 watersheds in Great Britain with drainage areas varying from 10 to 1200 km2 . The model application was developed with simple regionalization approaches using observed rainfall time series only. The mean and standard deviation of the absolute percentage errors estimated for the predicted and observed 50-year returntime flood-peak values were 39.8 and 43.6, respectively, for the event-based approach and 29.0 and 36.0, respectively, for the continuous approach (Calver et al. 2009, p. 28: Table 1). A spatially-distributed rainfall–runoff model was applied by Moretti and Montanari (2008) in a continuous framework simulating rainfall and temperature time series in a watershed with an area of 1294 km2 . A simulation of 100 years of hourly river flows allowed comparison of the annual peak flow frequency distributions of observed and simulated data. At a 50-year empirical return time, the peak flow exhibited an approximately 40% difference from the observed data (Moretti and Montanari 2008, p. 1149: Figure 6). The continuous approach was developed by Haberlandt et al. (2008) using a 100-year synthetic rainfall time series, which was simulated using the alternating renewal model and the HEC-HMS model calibrated with the observed data. The empirical probability-distribution functions show that the 50-year return time peak flow related to ten 100 year simulations could differ up to 50% (Haberlandt et al. 2008, p. 1364: Figure 10(a)). Finally, Viviroli et al. (2009) applied the continuous distributed PREVAH model to a large number of gauged and ungauged watersheds (with drainage areas varying from 10 to 2000 km2 ) to evaluate the performance of several regionalization techniques. The maximum deviations from the observed 100-year

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A continuous simulation model for design-hydrograph estimation in small and ungauged watersheds

peak flow values varied from –75 to 200% (and more in some cases) when the regionalization approach was changed for model calibration and using local standard methods (Viviroli et al. 2009, p. 215: Figure 5). All of the reports described above refer to regionalization methods that are particularly promising, as described in the recent literature (Merz and Blöschl 2004, 2005, Shu and Ouarda 2007, Castiglioni et al. 2010). However, in this paper, we address a critical situation in which regionalization methods are not applicable and, thus, only the use of empirical formulas is assumed to be possible for parameter calibration. This situation is common in small and ungauged basin hydrological analyses, and examples of continuous models devoted to this particular case are not available in literature. Moreover, the aforementioned studies focus on the model performance in terms of peak flow, whereas volume and duration of design floods are not usually taken into account. In this study, we propose a continuous empirical procedure (named COSMO4SUB: Continuous Simulation Model for Small and Ungauged Basins) that allows the definition of a SDH using minimal information from rainfall data and digital terrain support. The proposed procedure includes several modules. First, a daily rainfall simulation model is coupled to a disaggregation method to simulate synthetic rainfall time series at a 15-min resolution. Consequently, a rainfall excess scheme is applied followed by a geomorphological rainfall–runoff model, and a flood frequency analysis on the simulated runoff time series is performed for SDH estimation. The paper is organized as follows: an overview of the model is presented in Section 2, in which the various modules and related parameters are explained. The observed data used for the model analysis are described in Section 3, and a case study is illustrated in detail in Section 4. A sensitivity analysis of the rainfall excess and rainfall–runoff model parameters is developed in Section 5 to investigate their effect on the variability of the model output. 2 MODEL DESCRIPTION 2.1 Overview As noted in the Introduction, the aim of the COSMO4SUB framework is to provide a continuous runoff simulation useful for SDH estimation in a desired watershed outlet. As we deal with modelling of small and ungauged watersheds, the input data are

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limited to the watershed DEM at standard resolution (20–30 m), the soil-use maps or similar sources, and daily and sub-daily raingauge observations. The first step of the procedure is the simulation of rainfall time series at a sub-daily resolution. Here, a two-stage rainfall model is applied, wherein synthetic daily rainfall series are simulated by a single-site copula-based generator (Serinaldi 2009a) and disaggregated by the continuous-in-scale universal multi-fractal model (Schertzer and Lovejoy 1987). The parameters related to this step (six per month for the daily rainfall simulator and three for the disaggregation model) are calibrated using the rainfall observations. Given the rainfall scenario, the second step is the analysis of rainfall excess. The net rainfall has been defined by the Soil Conservation Service-Curve Number (SCS-CN) model (USDA-SCS 1986, Chow et al. 1988, Tramblay et al. 2012), which is one of the most widely-applied in the literature. Referring to the aforementioned literature for a description of this method, it is worth recalling that the SCS-CN is an event-based approach and, here, is continuously implemented on the entire simulated rainfall time series allowing the Curve Number (CN) parameter to vary according to the antecedent moisture condition (AMC). Therefore, the SCS-CN parameters (λ and CN) must be complemented with an additional parameter, referred to as the separation time, Ts , which is the no-rain time interval that separates two sequential storms that can be considered independent in terms of initial infiltration conditions. Thus, Ts is used to define the instant at which the cumulative gross and excess precipitation should be reset to zero. The effect and role of this parameter are discussed in detail in the sensitivity analysis in Section 5. Therefore, the rainfall-excess step requires three parameters, which must be empirically set up in the case of a lack of runoff observations. Given the rainfall-excess time series, a geomorphological rainfall–runoff model is applied to obtain the runoff scenario. Here, an advanced version of the well-known width-function instantaneous unit hydrograph (WFIUH) is considered (Grimaldi et al. 2012) that allows the definition of the watershed IUH using one parameter (WFIUH-1par), which, in the case of a lack of runoff observations, can be empirically quantified through the concentration time (Tc ) which is widely-used in practical applications related to ungauged basins. Given an appropriate IUH, a continuous convolution is carried out and the runoff scenario is obtained.

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In this step, it is appropriate to round off the rainfall and runoff data. Indeed the convolution provides a multitude of negligible runoff values, which does not allow easy and automatic flood-event selection, resulting in an overestimation of hydrograph volume and duration. To reduce this noise effect, the excess rainfall data are rounded off to 0.01 mm and the runoff data to 0.01 m3 /s. Given the synthetic runoff time series, the SDH is estimated following a multivariate flood frequency approach (Serinaldi and Grimaldi 2011), which identifies the design hydrograph with the assigned return time considering peak and volume information. In the following sub-sections, more details are provided concerning the rainfall simulation procedure, the WFIUH-1par model approach, and the SDH method, which were only recently described in the literature. 2.2 Rainfall simulation As previously mentioned, the rainfall simulation was performed using a two-stage model. Because subdaily rainfall sequences are usually shorter than the daily rainfall series provided by raingauge networks, the idea behind the approach applied in this study was to use a daily rainfall generator calibrated on a long daily series and a disaggregation scheme for which the parameters are estimated based on shorter, fine-scale rainfall records. The at-site daily generator is based on the simulation from the discretecontinuous conditional distribution of the rainfall, Xt , at the generic day, t, given the value Xt−1 on the previous day, which is deduced from a copula-based discrete-continuous bivariate distribution (Serinaldi, 2008, 2009a, 2009b, Villarini et al. 2008):  p00 + p01 #(xt )  "Xt |Xt−1 (xt |Xt−1 = 0 ) =    p00 + p01    "Xt |Xt−1 (xt |Xt−1 = xt−1 , Xt−1 > 0 )      p + p11 ∂C(u,v)  ∂u  = 01 p01 + p11

(1)

where p00 , p01 , and p11 , are the probabilities of the pairs (Xt−1 = 0, Xt = 0), (Xt−1 = 0, Xt > 0) and (Xt−1 > 0, Xt > 0), respectively; Ψ is the marginal distribution of positive rainfall; ∂C(u,v)/∂u = P(Xt < xt |Xt−1 = xt−1 , Xt−1 > 0, Xt > 0); C is a copula; and u = Ψ (xt−1 ) and v = Ψ (xt ). Further details on the basic hypotheses of the model, the inference procedure and the simulation algorithm can be found in Serinaldi (2009a).

The synthetic daily series are, therefore, disaggregated by the continuous-in-scale universal multifractal (CUM) model introduced by Schertzer and Lovejoy (1987). This class of models represents the so-called “second generation” of multiplicative random cascade models based on Lévy stable random variables (e.g. Schertzer and Lovejoy 1987, 1997, Lovejoy and Schertzer 1995, Lovejoy et al. 2008). The CUM model is physically-based and relies on the expected scaling properties of the rainfall process. Denoting K(q) as the moment scaling exponent and q as the moment order (e.g. Lovejoy et al. 2008), the theoretical K(q) of the CUM process is a threeparameter function:  C 1  (qα − q) α − 1 K (q) − qH =  C1 q log(q)

for α " = 1

(2)

for α = 1

where α is the index of stability of the Lévy stable distribution; C 1 is the co-dimension of the mean singularity, and describes the sparseness of the mean of process (e.g. Schertzer and Lovejoy 1987, Tessier et al. 1993); and H is the “non-conservation parameter”, since H = 0 is a quantitative statement of ensemble average conservation across the scales (e.g. Lovejoy and Schertzer 1995, Lovejoy et al. 2008). Referring to the aforementioned literature, in this case, the CUM simulation, which is based on fractional integrations, is performed using the algorithms described by Wilson et al. (1991) and Lovejoy and Schertzer (2010a, 2010b). 2.3 The WFIUH-1par rainfall–runoff model and DEM pre-processing We start from the unit hydrograph definition (Sherman 1932): Qt = Aw

!t 0

IUH(t − τ )p(τ )dτ

(3)

where Aw is the watershed area, p is the excess rainfall intensity, t is time, and τ is the starting time of the storm. Here, the IUH(t), which represents the traveltime probability density function, is defined using the geomorphological WFIUH. The WFIUH formulation included in the proposed framework (Mesa and Mifflin 1986, Naden 1992, Rodríguez-Iturbe and Rinaldo 1997, Grimaldi

A continuous simulation model for design-hydrograph estimation in small and ungauged watersheds

et al. 2010, Grimaldi et al. 2012) consists of defining the IUH(t) using the watershed flow-time (FT) distribution:

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FTi =

Lci Lhi + uc uh

for i = 1, . . . , n

(4)

where FTi is the flow time related to the path of cell i, Lci is the path length in the channel, uc is the channel velocity, Lhi and uh are the same parameters defined for hillslope areas, and n is the total number of watershed DEM cells. The selected DEM pre-processing methods used for the implementation of equation (4), including the removal of pit and flat areas, the definition of flow direction, automatic blue-line extraction, flowpath and hillslope-area identification, and hillslopevelocity estimation, are summarized in the following steps:

(a) Correction of pit and flat areas in the watershed DEM using the physically-based erosion model for pit removal, the PEM4PIT method (Grimaldi et al. 2004, 2007, Santini et al. 2009). (b) Definition of flow direction using the single-flow method, D8-LTD (Orlandini et al. 2003). (c) Extraction of drainage network using the drop-analysis approach (Tarboton et al. 1991, Tarboton and Ames 2001). (d) Definition of runoff path from each cell to the watershed outlet using a mixed flow direction approach: single-flow D8-LTD in the channel and multi-flow D∞ in hillslope areas (Nardi et al. 2008). (e) Estimation of hillslope velocity (uh ) by applying the NRCS formula cell by cell (NRCS 1997, Grimaldi et al. 2010).

Following equation (4), each flow path length (L), estimated through the DEM pre-processing procedure, is divided by the related velocities. For hillslope cells, Lh is divided by the uh obtained in step (e), while for channel cells, Lc is divided by the unknown constant value representing the single WFIUH-1par calibration parameter. In cases lacking runoff observations, this constant is selected so that the maximum FT abscissa is equal to the basin concentration time, Tc .

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2.4 Definition of SDH based on GSTSP distribution The general concept of the approach consists of representing a flood hydrograph using a distribution function with finite support (Yue et al. 2002). Because the support and area underlying this distribution are equal to one, the base of the hydrograph is divided by its duration (D), and each ordinate Q(t) is multiplied by the ratio D/V, where V is the flood volume. The resulting signal can be considered as a density function with support [0,1], and a distribution can then be fitted on it. To return to the original space, the base and ordinates of the fitted distribution can be multiplied by D and V/D, respectively. It should be emphasized that, using this approach, only two hydrograph attributes (among Qp , V and D) can be arbitrarily assigned when the density is back-transformed into the final SDH. The ordinates Q(t) of the SDH are linked to density ordinates, f (t), by the relationship Q(t) = f (t)V/D. It follows that Qp = f (tp )V/D, where tp is the peak time. Therefore, if Qp and V are fixed, D cannot be arbitrarily chosen. The 3-parameter generalized standard two-sided power (GSTSP) distribution (Kotz and van Dorp 2004) is considered, because it allows the modelling of highly leptokurtic shapes appropriate for small watershed floods (Serinaldi and Grimaldi 2011):  % &n1 −1 t n1 , n3      θ n3 + (1 − θ )n1 θ     for 0 ≤ t < θ % & (5) f (t; θ , n1 , n3 ) = n1 , n3 1 − t n3 −1      θ n3 + (1 − θ )n1 1 − θ     for 0 ≤ t < 1

where t is time and θ , n1 and n3 are the distribution parameters. In other words, θ is the point of switch between the two branches of the GSTSP density function and coincides with the mode, whereas n1 and n3 are the power exponents of the left-side and rightside power distributions, respectively, and control the shape of the distribution. The procedure applied here is summarized in the following steps:

(a) Definition of the driving variable; that is, the variable on which the univariate frequency analysis is performed to assign the return time. Here, the peak discharge Qp,T with a return period T is chosen.

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(b) Hydrograph selection: given the synthetic runoff time series, flood events are identified. Each event is characterized by peak, volume and duration values. The maximum annual hydrograph is selected by choosing the event with the maximum annual peak. (c) Definition of the dimensionless hydrographs from each hydrograph selected from the simulated runoff, dividing the base by its duration (D) and multiplying each ordinate Q(t) by the ratio D/V. (d) Estimation of GSTSP parameters. For each selected dimensionless hydrograph, the three GSTSP parameters (θ , n1 , n3 ) are estimated using the formulas provided by Nadarajah (2007). (e) Definition of the GSTSP hydrograph related to the driving variable, with computation of the parameters of the GSTSP hydrograph referring to the driving variable: θ Qp,T = E[θ |Qp,T ], n1 ,Qp,T = E[n1 |Qp,T ], n3 ,Qp,T = E[n3 |Qp,T ]. (f) Definition of the non-driving variables, computing V Qp,T = E[V|Qp,T ] and DQp,T = f (tp ,Qp,T ) V Qp ,T /Qp,T , where tp ,Qp,T = θ Qp,T . (g) Definition of the SDH, with the multiplication of the hydrograph bases, defined in step (d), by DQp,T and the ordinates by V Qp,T /DQp,T . 3 DATA AND MATERIALS For the application of the case study and sensitivity analysis, the Wattenbach River basin, a small and gauged watershed (area 71.6 km2 ) located in the central eastern Alps in Austria, is selected. Its main characteristics, including DEM source (ASTER GDEM 2009), outlet coordinates and general morphometric parameters, are listed in Table 1, while Fig. 1 shows the watershed DEM with a drainage-network and soil-use map (CORINE 2000). Temperature, rainfall and discharge time series are available for the selected area. Temperature data from 1983–2003 were registered three times per day (07:00, 14:00 and 21:00 h) at the Wattener Lizum gauging station in the watershed at 1970 m a.s.l. Rainfall data observed at the same station are available for 1983–2008 with a 15-min resolution. Discharge data at the outlet are accessible for 1977–2008 at a 15-min resolution. Because the elevation of 48% of the watershed area is above 2000 m, snow is expected to be continuously present on the ground during the year. The soil surface is free of snow only in the summer. Therefore,

Table 1 Main characteristics of the case study watershed (Wattenbach, Austria). Watershed properties DEM source Cell size (m) Precision Outlet coordinate N (lat.) Outlet coordinate E (long.) Area (km2 ) Min elevation (m a.s.l.) Max elevation (m a.s.l.) Mean elevation (m a.s.l.) Max slope (◦ ) Mean slope (◦ ) Main channel length (km) Maximum divide-outlet distance (km)

ASTER GDEM 30 integer 47◦ 17& 48&& 11◦ 35& 29&& 71.58 551 2824 1910 67 25 20.16 20.81

the years characterized by missing data in the summer season were discarded, resulting in 17 years of rainfall–runoff data from which 50 rainfall–runoff events were selected. Direct runoff was obtained by applying a recursive filter on total runoff (Lyne and Hollick 1979, Nathan and McMahon 1990). The direct runoff peaks of all selected events are greater than 2 m3 /s and are observed between the end of May and the middle of September. Given the total volume of the net runoff, the net rainfall was estimated using the SCS-CN method (USDA-SCS 1986, Chow et al. 1988). The initial abstraction ratio (λ) was estimated by considering the concurrent synchronized records of both rainfall and discharge (Mishra and Singh 2003, 2004, Woodward et al. 2003). For each event, the CN value was estimated by balancing the total net rainfall volume and the total net discharge volume. Furthermore, for each event, the Tc was estimated as the time interval from the end of rainfall excess to the end of the net hydrograph (McCuen 2009). The properties of the 50 rainfall–runoff events are listed in the Appendix. Despite the fact that only summer season data were considered in the analysis, an underestimation of the selected flood-event properties is expected. For example, during summer storms, sudden temperature decreases can generate snow that melts after a few hours. This behaviour is confirmed in the recorded data set by examining the temperature during these events. Temperature data are available only for 35 of the selected 50 events. It is verified that during 21 of the 35 selected floods, the temperature decreased below 6◦ C at 2000 m and, consequently, snow is expected during the event for over 48% of the watershed area (see Appendix). This particular behaviour affects the event properties, providing an

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Fig. 1 Case-study watershed DEM with (a) drainage network and (b) soil-use map.

Fig. 2 Snow-effect analysis. Secondary (upper) x-axis: net rainfall at a temperature higher or lower than 6◦ C in black or grey, respectively. Primary x-axis: observed net discharge (solid black line); WFIUH discharge simulated with the higher-temperature “black” rainfall (dashed black line); and WFIUH discharge simulated with the entire “black + grey” rainfall (grey line).

underestimation of the flood peak and volume with respect to the values that would be obtained without the occurrence of snow. A simple example that can be useful to roughly quantify the effect of snow is described in Fig. 2, on the upper x-axis, where rainfall is reported in black (grey) when occurring at

temperatures higher (lower) than 6◦ C. On the lower x-axis, the observed direct runoff is shown. Two WFIUH-1par simulations have been developed, and the resulting floods are also shown in Fig. 2. The first simulation is obtained assuming that precipitation occurring at temperatures lower than 6◦ C is in the form of snow and, thus, does not contribute to the runoff and the estimation of SCS-CN parameters. The second simulation is obtained assuming that all rainfall is liquid and applying the WFIUH model with the same SCS-CN parameters estimated in the first simulation. The two simulations give the same results in the time interval from 0 to 20 h (Fig. 2, grey lines and black dashed lines are superimposed). As expected, assuming the absence of snow, a new, higher peak and greater volume are obtained. Because the shown snow effect determines an approximation on the observations that is difficult to quantify, we prefer to evaluate the proposed model comparing the results with a corrected runoff scenario and not with the observed data. To create the corrected scenario, we proceed as follows. For each rainfall–runoff-event data point without the presence of snow, the model parameters λ, CN and Tc are calibrated. The obtained mean values (λ = 0.1, CN = 45, Tc = 5) are assumed to be optimal for the analysed watershed. With this optimal parameter set (assuming

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Fig. 3 Comparison between the observed and corrected hydrograph peaks (a), volumes (b) and durations (c) in a Gumbel probability plot. The secondary x-axis represents the probability of non-exceedence.

Ts = 24 h, see Section 5), the proposed model is applied on the 17 years of 15-min observed rainfall data. The corrected runoff time series obtained will clearly include not only the snow effect but also the inherent approximation of the linear theory; however, it seems reasonable for the proposed application and analysis. Figure 3 compares the peak, volume and duration related to the extreme flood events resulting from the corrected and observed runoff series. The two samples are sorted considering the peak as a driving variable and by selecting up to three hydrographs per year with the largest peaks greater than 2 m3 /s. The peak and volume differences are high, but credible considering the watershed dimension and slope values (Table 1). It is of note that the event durations appear similar in the two samples, mainly because the snow effect is reduced for this hydrograph parameter. 4 CASE STUDY APPLICATION In this section, we describe the application of the COSMO4SUB framework, providing a practical example of SDH estimation with an assigned return period. The case study is developed using the optimal set of λ, CN and Tc parameter values (λ = 0.1, CN = 45, Tc = 5), as described in the previous section, while, for Ts , a value of 24 h is assumed. The procedure is applied, simulating 500 years of direct runoff, and compared to the corrected streamflow series. The first step is the calibration of the rainfall simulator and the generation of 500 years of 15-min synthetic rainfall time series. For the data on hand, the daily model is specified using a two-parameter

Weibull distribution for Ψ , and a survival Clayton copula for C, with parameters being estimated on a monthly basis; daily data are disaggregated by the CUM model with α = 1.5 and C 1 = 0.4. Figure 4 compares eight physical attributes computed on the observed and simulated series. The simulated 15-min positive rainfall shows an upper tail heavier than the observed one, due to the unbounded singularities resulting from α values >1 (e.g. Lovejoy and Schertzer 1995). The rainfall models reproduce the event rainfall depth, the wet- and dry-spell length, and the mean and standard deviation of the annual maxima—across time scales from 15 min (scale ratio = 1) to 1920 min = 1.33 days (scale ratio = 128)—quite well. The wet-spell length tends to be overestimated, whereas the auto-correlation function (ACF) and the probability of no rain are slightly underestimated. Given the synthetic rainfall time series, after having applied the SCS-CN method and the WFIUH-1par model, a direct runoff signal is estimated, and the annual maximum hydrographs are selected. These hydrographs are sorted using the flood peak as a driving variable. The hydrograph is identified as a continuous sequence starting with a non-zero runoff and ending with a null runoff value. For each hydrograph, the peak, volume and duration are quantified. Figure 5 shows a Monte Carlo analysis comparing the 17 corrected maximum annual hydrograph properties and the simulated runoff series. From the 500 synthetic years, 1000 groups of 17 years are sampled, and the maximum annual hydrograph peaks and related volume and duration values are selected for each group. The white line in Fig. 5 represents

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Fig. 4 Comparison of physical attributes computed on the observed and simulated rainfall time series: (a) rainfall intensity at 15-min resolution; (b) event amount; (c) wet spell length; (d) dry spell length; (e) probability of no rain; (f) autocorrelation function (ACF); (g) mean of annual maxima; and (h) standard deviation of annual maxima.

the 50th percentile, while the dark- and light-grey areas denote the 50% and 95% Monte Carlo confidence intervals, respectively. Except for a single volume value, the results confirm an acceptable performance of the model in reproducing annual extreme hydrographs. As a proof of concept, the flood hydrograph information was synthesized using SDHs derived through the procedure described in Section 2.4. Figure 6(a) shows the dimensionless cumulative SDHs corresponding to the 500 simulated Qp annual maxima (point (a) in Section 2.4) and the three GSTSP hydrographs with the expected parameters

(expected shapes, point (d) in Section 2.4) corresponding to the peak flow values (Qp ) with 50-, 100and 200-year return periods. As the Monte Carlo approach allows us to generate series with an arbitrary size, these Qp values are directly derived based on the empirical distribution of the simulated annual maxima. Figure 6(b) shows the three final hydrographs with 50-, 100- and 200-year Qp values. Because the 500 simulated hydrographs exhibit highly variable shapes, without a predominant asymmetric behaviour, and the average shape corresponding to Qp with a fixed return period is selected, the final SDHs are rather symmetrical.

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Fig. 5 Comparison between the corrected and simulated maximum annual hydrographs using Monte Carlo analysis in a Gumbel probability plot. The secondary abscissa represents the non-exceedence probability; black dots are the corrected values; the white line is the 50th percentile; and the dark grey and light grey areas represent the 50% and 95% Monte Carlo confidence intervals, respectively.

Fig. 6 (a) Dimensionless cumulative SDHs and the three GSTSP hydrographs corresponding to peak flow values of Qp with 50-, 100- and 200-year return periods. (b) Three SDHs with 50-, 100- and 200-year Qp values.

5 PARAMETER SENSITIVITY ANALYSIS AND REMARKS In the case study described in Section 4, the values of λ, CN and Tc can be considered optimal as they are estimated on the observed hydrograph. However, in fully ungauged basins, the parameter estimates must rely on geomorphological and land-use information, resulting in considerable uncertainty. Therefore, a sensitivity analysis was performed to verify the output variability related to the parameter values, and to analyse their role. With this aim, the model is applied assuming 10 000 parameter combinations varying in a wide interval of realistic values. For each variable, 10 values are fixed: λ from 0.0 to 0.45 with a step size of 0.05, CN from 30 to 75 with a step size of 5, Tc

from 3.5 to 8 h with a step size of 0.5 h, and Ts from 18 to 36 h with a step size of 2 h. Practically, these combinations provide all of the possible hydrographs that the procedure is able to generate for the selected watershed. For example, within the CN range, the soil use varies from the hypothesis of a brush-covered surface to gravel streets, while the different Tc values, which affect the maximum WFIUH-1par abscissa, determine a large variability of scenarios (see Fig. 7). For each parameter combination, a 500-year runoff time series is simulated, and the maximum annual events are selected using the flood peak as a driving variable. In the following, the results are compared considering peak, volume and duration values related to the 0.998 peak quantile. Similar behaviours

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Fig. 7 Watershed WFIUHs estimated with Tc values ranging from 3.5 to 8 h and a smoothing window of 15-min.

are also evident for other quantiles, not shown here for brevity. In Fig. 8 the global variability of each parameter is compared to that of the other parameters. For each fixed parameter triplet, 10 combinations of the fourth parameter are available. Estimating the coefficient of variation (CV: the ratio between the standard deviation and the mean) on the 0.998 quantile of the peak discharge, volume and duration obtained for these 10 combinations, it is possible to quantify the variability related to the fourth parameter. In Fig. 8, the grey line represents the 1000 CV values estimated for the 0.998 quantile of the peak discharges, volume and duration obtained from the 10 Ts values and fixing Tc , CN and λ. The plot is drawn with the data organized in increasing order. Rotating the parameters, the

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black line shows the same analysis with variable Tc and, consequently, fixing Ts , CN and λ; in the analysis represented by the dashed black line, λ is variable, while Tc , Ts and CN are fixed; and, finally, the dashed grey line represents CN being variable with Tc , Ts and λ being fixed. Figure 8 allows us to conclude that Ts is less influential than the other parameters. Except for a few unrealistic combinations, the CV values related to Ts are always considerably lower than those estimated for the other parameters. This behaviour reflects the role of Ts in the model. As explained in Section 2.1, Ts is introduced to define the dry interval time to reset the cumulative gross and net rainfall in the SCS-CN module. At the end of the Ts interval, the beginning of a new rainfall event is assumed, so the initial infiltration and antecedent moisture condition (AMC) in the SCS-CN module are quantified. The longer this interval is, the longer the rainfall-event duration and the lower the infiltration toward the end of the event will be. Therefore, the first expected effect would be the presence of higher peaks. In contrast, the longer this interval is, the lower the AMC will be at the beginning of the selected event (because a longer dry sequence is present before the beginning of the rainfall event). Therefore, CN could decrease for that event and, consequently, the infiltration increases, reducing the peak. The described effects are opposite in sign and, as confirmed by Fig. 8, they balance each other, providing nearly stable results. Based on these first results of the sensitivity analysis, only the combinations corresponding to Tc , CN and λ, fixing Ts = 24 h, are further analysed.

Fig. 8 Coefficients of variation (CV) estimated for the 0.998 quantile of: (a) peak discharge, Q; (b) volume, V ; and (c) duration, D, obtained from 1000 combinations of three assigned variables: Tc , CN and λ (grey lines); Ts , CN and λ (black lines); Tc , Ts and CN (dashed black lines); and Tc , Ts and λ (dashed grey lines).

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Fig. 9 Estimated values related to 0.998 peak quantile for the 10 000 combinations of Tc , CN and λ: (a) hydrograph peak (Q0.998 |Ts = 24 h; m3 s-1 ); (b) volume (V 0.998 |Ts = 24 h; hm3 ); and (c) duration (D0.998 |Ts = 24 h; h). Each sub-panel refers to a constant Tc value shown in the strips. Black and white contoured pixels with internal numbers denote the results obtained with optimal and empirical parameter combinations, respectively.

Figure 9 indicates the 0.998 quantile of peak, volume and duration of the hydrographs obtained by varying the three parameters. For each sub-panel, a Tc value is assigned. Figure 9 leads to the following general comments:

• Using a wide range of parameter combinations, the obtained hydrograph properties are rather variable, and, consequently, the model is able to provide all of the possible physical values related to the studied watershed (0.4 m3 /s < Q0.998 < 545.9 m3 /s,

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•

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•

•

•

6.5 h < D0.998 < 169.2 h, 7.6 × 103 m3 < V 0.998 < 6.6 × 106 m3 ). This result is apparently obvious, but it confirms that the model is quite flexible, varying only three/four parameters. With respect to discharges (Fig. 9(a)), when the Tc values are increased, the peak decreases, as expected according to Fig. 7. However, for the range of λ > 0.2 and CN < 50, this variation is reduced, and consequently, the infiltration parameters are dominant over the other parameters for these combinations. With respect to volumes (Fig. 9(b)), as expected, there is no variability related to Tc , and when λ increases, the volume decreases. However, for CN > 65, it seems that the variation in the volume is minor, suggesting that CN is prominent for values greater than 65. With respect to the duration (Fig. 9(c)), as shown in Fig. 7, when the Tc values increase, the duration increases. This variation is relevant in the range λ < 0.2 and CN > 50. This behaviour is probably due to reduced infiltration, which leads to longer rainfall events and longer hydrographs for high Tc values. These effects could merge events that are independent when applying the procedure with other parameter set. Consequently, the range of λ > 0.2 and CN < 50 also appears dominant over the other parameters in this case. In general, combinations of diagonal values of CN-λ appear to be stable. This finding is probably due to the compensation of these two parameters in the SCS-CN rainfall-event volume estimation.

In order to provide a complete evaluation of the model performance, two additional numeric results are included in Fig. 9. The black-contour pixels report the model results obtained by using the optimal parameter set estimated on the observed data, whereas the white-contour pixels display the results

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corresponding to the worst ungauged condition, when the model parameters are empirically quantified. The values of the latter empirical parameter set (λ, CN, Tc ) are estimated as follows: λ is assumed to be equal to 0.2 as request by the USDA-NRCS (2004, page 10-3) and detailed by Woodward et al. (2010) in the case of an absence of additional local calibrations, CN is deduced from CN tables, and Tc is computed through the empirical formula suggested by NRCS (1997). The differences are relevant, particularly for the peak flow (−40% for Qp , −22% for V , −18% for D), but the results appear to be acceptable considering the similar application results of the continuous models mentioned in the Introduction and considering that the proposed procedure is developed using empirical parameter calibration, while the other approaches are calibrated using regionalization methods. This practical application leads to further discussion. Without any discharge measurements or other observations useful for the calibration of SCS-CN method, the procedure needs and offers only two parameters: CN and Tc (λ being fixed as 0.2 and Ts fixed as 24 h). In this case, the results corresponding to 100 combinations of Tc and CN (shown in Fig. 10) highlight that the model flexibility decreases and the variability consequently increases. For instance, for Tc = 4 h, in the three-parameter model, 100 peak values are obtainable, ranging from 0.8 m3 /s to 482.1 m3 /s, while in the two-parameter model, only 10 peak values are possible, ranging from 52.8 m3 /s to 457.1 m3 /s. This result clearly suggests that the infiltration module should be improved for future development of the framework. 6 CONCLUSION In this paper, a continuous procedure for estimating a design hydrograph with an assigned return time for small and ungauged watersheds,

Fig. 10 Estimated values related to the 0.998 peak quantile for the 100 combinations of Tc , CN and λ = 0.2: (a) hydrograph peak (Q0.998 |Ts = 24 h, λ = 0.2; m3 s-1 ); (b) volume (V 0.998 |Ts = 24 h, λ = 0.2; hm3 ); and (c) duration (D0.998 |Ts = 24 h, λ = 0.2; h).

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named COSMO4SUB, is described and analysed. The proposed approach consists of three modules: (a) a rainfall simulation and disaggregation model, (b) a geomorphological rainfall–runoff model, and (c) a flood-frequency analysis to define the synthetic design hydrograph on the simulated direct runoff time series. The aim of the framework is to improve design hydrograph estimation in critical ungauged areas where the total absence of discharge observations and regional studies leads to the adoption of empirical procedures. The proposed approach is useful both to define the flood peak and to obtain coherent information on the design hydrograph volume, which is crucial in many hydrological analyses. The input data requested by the proposed model are limited to daily and sub-daily raingauge observations, watershed DEM, and soil-use maps. The model requires only four parameters that should be guesstimated when the streamflow observations are not available: λ and CN, related to the SCS-CN infiltration scheme, Ts , used to separate rainfall events, and Tc , the concentration time. A small gauged watershed (71.6 km2 ) was selected to describe the proposed procedure. Using a parameter set estimated based on rainfall runoff observations, a complete application of the procedure is described. Considering the empirical nature of the approach, a detailed sensitivity analysis is provided by applying the procedure using 10 000 combinations of parameter values. The obtained results suggest the following conclusions. The effect of Ts is negligible, and therefore, it can be assumed a priori, reducing the number of parameters. CN and λ are prominent compared to the other parameters, especially for λ > 0.2 and CN < 50. Finally, the differences among the obtained values using the optimal and empirical parameter set are acceptable considering the context of the application of the proposed procedure. In practical applications, when local λ values are not available, the procedure requires only two empirical parameters: CN and Tc , as the original version of SCS-CN implies λ = 0.2. With only two parameters, the proposed procedure is already able to provide useful results, but it is clear that further improvements concerning the infiltration scheme are necessary. These improvements should be related to either the empirical nature of the SCS-CN approach, which is not appropriate for continuous modelling, or to the use of a single parameter, which increases the output variability. These improvements are the subject of ongoing research.

Acknowledgements The authors gratefully acknowledge Dr Magdalena Rogger, Dr Alberto Viglione and Prof. Günter Blöschl from the Vienna University of Technology for providing the rainfall, runoff and temperature case-study data and related information. The authors also thank Francesco Laio (Politecnico di Torino, Italy), Andreas Efstratiadis (National Technical University of Athens, Greece) and one anonymous reviewer for their valuable comments. REFERENCES Alfieri, L., Laio, F., and Claps, P., 2008. A simulation experiment for optimal design hyetograph selection. Hydrological Processes, 22, 813–820. ASTER GDEM, 2009. Available from: http://www.gdem.aster.ersdac. or.jp/ [Accessed 22 June 2012]. Blazkova, S. and Beven, K., 2009. A limits of acceptability approach to model evaluation and uncertainty estimation in flood frequency estimation by continuous simulation: Skalka catchment, Czech Republic. Water Resources Research, 45, W00B16. Boughton, W. and Droop, O., 2003. Continuous simulation for design flood estimation–a review. Environmental Modelling and Software, 18 (49), 309–318. Calver, A., Stewart, E., and Goodsell, G., 2009. Comparative analysis of statistical and catchment modelling approaches to river flood frequency estimation. Journal of Flood Risk Management, 2, 24–31. Castiglioni, S., et al., 2010. Calibration of rainfall–runoff models in ungauged basins: a regional maximum likelihood approach. Advances in Water Resources, 33 (10), 1235–1242. Chow, V.T., Maidment, D.R., and Mays, L.W., eds., 1988. Applied hydrology. New York: McGraw-Hill Civil Engineering Series. CORINE (Coordination of Information on Environment) Database, 2000. A key database for European integrated environmental assessment. Programme of the European Commission, European Environmental Agency (EEA). Dooge, J.C.I., 1973. Linear theory of hydrologic systems. Technical Bulletin 1468. United States Department of Agriculture. EU (European Union), 2007. Directive 2007/60/EC of the European Parliament and of the Council of 23 October 2007 on the assessment and management of flood risk. Official Journal of the European Union, L288, 27–34. FEMA, 2009. Guidelines and Specifications for Flood Hazard Mapping Partners, Appendix C: Guidance for Riverine Flooding Analyses and Mapping. Available from: http://www. fema.gov/library/viewRecord.do?id=2206 [Accessed 22 June 2012]. Faulkner, D. and Wass, P., 2005. Flood estimation by continuous simulation in the Don catchment, South Yorkshire, UK. Journal of the Chartered Institution of Water and Environmental Management, 19 (2), 78–84. Grimaldi, S., et al., 2005. A 3-copula function application for design hyetograph analysis. IAHS-AISH Publication, 293, 203–211. Grimaldi, S., et al., 2007. A physically based method for removing pits in digital elevation models. Advances in Water Resources, 30, 2151–2158. Grimaldi, S., et al., 2010. Flow Time estimation with variable hillslope velocity in ungauged basins. Advances in Water Resources, 33 (10), 1216–1223. Grimaldi, S., Petroselli, A., and Nardi, F., 2012. A parsimonious geomorphological unit hydrograph for rainfall-runoff modelling in

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Nardi, F., et al., 2008. Hydrogeomorphic properties of simulated drainage patterns using DEMs: the flat area issue. Hydrological Science Journal, 53 (6), 1176–1193. Nathan, R.J. and McMahon, T.A., 1990. Evaluation of automated techniques for baseflow and recession analysis. Water Resources Research, 26 (7), 1465–1473. NRCS, 1997. Ponds-planning, design, construction. Agriculture Handbook N. 590. Washington, DC: U.S. Natural Resources Conservation Service. Orlandini, S., et al., 2003. Path-based methods for the determination of non-dispersive drainage directions in grid-based digital elevation models. Water Resources Research, 39 (6), 1144. PAI, 2006. Piano Stralcio di Assetto Idrogeologico. Autorità di Bacino del Fiume Tevere. Available from: http://www.abtevere.it/ attivita/piani_approvati/PAI_NEW/pai.htm [Accessed 22 June 2012]. Pramanik, N., Panda, R.K., and Sen, D., 2010. Development of design flood hydrographs using probability density functions. Hydrological Processes, 24 (4), 415–428. Rahman, A., et al., 2002. Monte Carlo simulation of flood frequency curves from rainfall. Journal of Hydrology, 256, 196–210. Rodríguez-Iturbe, I. and Rinaldo, A., 1997. Fractal river networks: chance and self-organization. Cambride: Cambridge University Press. Santini, M., et al., 2009. Pre-Processing algorithms and landslide modelling on remotely sensed DEMs. Geomorphology, 113 (1–2), 110–125. Schertzer, D. and Lovejoy, S., 1987. Physical modeling and analysis of rain and clouds by anisotropic scaling of multiplicative processes. Journal of Geophysical Research, 92, 9693–9714. Schertzer, D. and Lovejoy, S., 1997. Universal Multifractals do Exist!: Comments on “A statistical analysis of mesoscale rainfall as a random cascade”. Journal of Applied Meteorology, 36, 1296–1303. Serinaldi, F., 2008. Analysis of inter-gauge dependence by Kendall’s τ, upper tail dependence coefficient, and 2-copulas with application to rainfall fields. Stochastic Environmental Research and Risk Assessment, 22 (6), 671–688. Serinaldi, F., 2009a. A multisite daily rainfall generator driven by bivariate copula-based mixed distributions. Journal of Geophysical Research, 114, D10103. Serinaldi, F., 2009b. Copula-based mixed models for bivariate rainfall data: an empirical study in regression perspective. Stochastic Environmental Research and Risk Assessment, 23 (5), 677–693. Serinaldi, F. and Grimaldi, S., 2011. Synthetic design hydrographs based on distribution functions with finite support. Journal of Hydrologic Engineering, 16 (5), 434. Sherman, L.K., 1932. Streamflow from rainfall by the unit-graph method. Engineering News-Record, 108, 501–505. Shu, C. and Ouarda, T.B.M.J., 2007. Flood frequency analysis at ungauged sites using artificial neural networks in canonical correlation analysis physiographic space. Water Resources Research, 43, W07438, 1–12. Soczyñska, U., et al., 1997. Prediction of design storms and floods. IAHS-AISH Publication, 246, 297–303. Tarboton, D.G. and Ames, D.P., 2001. Advances in the mapping of flow networks from digital elevation data. In: World water and environmental resources congress. Orlando, FL: ASCE. Tarboton, D.G., Bras, R.L., and Rodriguez-Iturbe, I., 1991. On the extraction of channel networks from digital elevation data. Hydrological Processes, 5 (1), 81–100. Tessier, Y., Lovejoy, S., and Schertzer, D., 1993. Universal multifractals: theory and observations for rain and clouds. Journal of Applied Meteorology, 32, 223–250. Tramblay, Y., et al., in press. Assessment of initial soil moisture conditions for event-based rainfall–runoff modelling. Journal of Hydrology, DOI: 10.1016/j.jhydrol.2010.04.006.

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Wilson, J., Lovejoy, S., and Schertzer, D., 1991. Physically based cloud modelling by scaling multiplicative cascade processes. in: Scaling, fractals and non-linear variability in geophysics, Dordrecht, The Netherlands: Kluwer, 185–208. Woodward, D.E., et al., 2003. Runoff curve number method: Examination of the initial abstraction ratio. World Water and Environmental Resources Congress, 691–700. Woodward, D.E., et al., 2010. Discussion of “Modifications to SCSCN Method for Long-Term Hydrologic Simulation” by K. Geetha, S.K. Mishra, T.I. Eldho, A.K. Rastogi, and R.P. Pandey. Journal of Irrigation and Drainage Engineering, 136 (6), 444–446. Yue, S., et al., 2002. Approach for describing statistical properties of flood hydrograph. Journal of Hydrologic Engineering, 7 (2), 147–153.

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APPENDIX

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Table A1 Main properties of the 50 selected observed events. λ and CN: SCS-CN parameters; V : volume net runoff; Qp : net peak discharge; T c : concentration time; D: net hydrograph duration; T min : minimum temperature during gross rainfall event; ESN: estimated snow percentage during gross rainfall event. Event number

λ

CN

V (mm)

Qp (m3 /s)

Tc (h)

D (h)

T min (◦ C)

ESN (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.00 0.01 0.05 0.01 0.02 0.01 0.20 0.15 0.02 0.01 0.20 0.07 0.02 0.00 0.00 0.00 0.00 0.02 0.01 0.01 0.40 0.10 0.05 0.00 0.00 0.01 0.20 0.20 0.10 0.01 0.05 0.02 0.05 0.20 0.30 0.01 0.01 0.03 0.00 0.15 0.05 0.00 0.01 0.03 0.01 0.01 0.20 0.10 0.30 0.03

7.2 23.8 47.6 7.0 18.9 15.3 75.3 65.3 24.0 18.1 56.5 45.6 27.4 11.6 32.8 11.7 22.2 28.7 16.9 80.7 92.6 54.1 57.1 18.2 8.3 16.2 66.7 71.7 63.6 21.1 52.8 40.1 33.7 76.9 92.7 45.3 24.3 40.2 25.2 63.8 72.7 26.6 22.5 17.9 86.9 25.7 74.7 57.5 81.6 25.6

1.7 0.7 0.5 2.1 9.5 2.3 0.6 0.7 2.7 2.9 1.5 1.1 0.9 0.9 1.2 0.5 0.7 0.7 1.2 0.8 0.6 1.1 0.7 1.3 1.4 0.7 1.2 0.6 0.9 1.3 0.5 1.1 3.0 0.7 0.4 0.8 2.3 3.4 1.0 0.9 0.6 1.5 5.5 6.0 0.8 1.0 0.3 1.1 1.2 1.3

2.5 3.1 2.1 6.2 20.6 6.1 4.2 4.4 7.2 7.1 6.7 5.0 2.9 2.8 3.0 3.6 4.9 5.9 6.0 4.8 3.9 4.7 4.9 5.0 5.5 6.3 4.4 3.9 4.3 5.0 3.5 5.3 7.3 4.3 5.2 4.0 4.3 6.1 2.6 4.2 7.0 5.6 11.2 7.2 5.0 4.3 3.7 5.3 7.5 4.3

5.3 6.5 5.0 3.5 4.5 5.5 3.5 7.5 10.8 6.8 4.3 5.8 6.5 9.3 6.3 5.0 4.0 2.0 4.3 4.3 8.0 3.5 3.3 5.8 5.0 6.3 3.3 6.0 5.3 4.3 4.0 4.3 3.5 3.8 3.8 5.3 6.0 7.0 6.5 4.0 2.8 6.5 5.3 4.5 4.5 4.3 7.0 3.8 4.5 8.5

40.5 12.0 8.5 20.3 34.8 35.3 19.0 9.8 27.0 23.5 8.0 14.0 12.0 19.8 16.0 11.0 15.0 7.3 16.5 10.5 9.0 11.0 9.5 23.8 21.3 14.0 13.8 26.0 8.3 18.5 6.0 14.8 22.3 8.3 4.3 12.5 29.8 44.0 36.0 8.3 4.8 12.0 33.0 36.5 9.3 16.3 6.5 9.8 9.3 21.0

−2.0 7.0 4.6 −0.5 −1.0 3.0 11.0 0.4 4.5 4.0 9.0 5.0 1.6 −0.5 3.0 10.5 4.2 8.2 1.8 10.0 7.0 4.8 5.0 2.8 3.8 6.2 1.0 9.0 0.0 5.2 2.0 1.0 1.4 5.6 10.0 −0.5 2.9 – 4.7 – – – – – – – – – – –

95.4 0.0 0.0 81.4 41.3 24.2 0.0 66.4 60.9 0.0 0.0 5.5 50.7 58.4 95.4 0.0 0.0 0.0 61.6 0.0 0.0 5.9 0.0 41.2 80.3 0.0 25.3 0.0 86.4 25.4 100.0 0.0 0.0 0.0 0.0 100.0 10.9 – 0.4 – – – – – – – – – – –