The relative role of hillslope and river network routing in the hydrologic response to spatially variable rainfall fields

Accepted Manuscript The relative role of hillslope and river network routing in the hydrologic response to spatially variable rainfall fields Davide Zoccatelli, Marco Borga, Giovanni Battista Chirico, E.I. Nikolopoulos PII: DOI: Reference:

S0022-1694(15)00573-9 http://dx.doi.org/10.1016/j.jhydrol.2015.08.014 HYDROL 20633

To appear in:

Journal of Hydrology

Please cite this article as: Zoccatelli, D., Borga, M., Chirico, G.B., Nikolopoulos, E.I., The relative role of hillslope and river network routing in the hydrologic response to spatially variable rainfall fields, Journal of Hydrology (2015), doi: http://dx.doi.org/10.1016/j.jhydrol.2015.08.014

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

The relative role of hillslope and river network routing in the hydrologic response to spatially variable rainfall fields

Davide Zoccatelli1*, Marco Borga1, Giovanni Battista Chirico2, E.I. Nikolopoulos1 1

Dipartimento Territorio e Sistemi Agro-Forestali, University of Padova, via dell’Università, 16 Legnaro, IT-35020 2

Dipartimento di Ingegneria Agraria, Università di Napoli Federico II, Naples, Italy

*corresponding author e-mail: [email protected]

Submitted to Journal of Hydrology August 2014 Revised: July 2015

1

Abstract This paper introduces a new methodology and a rainfall spatial organization index to examine the relative role of hillslope and channel residence times in the analysis of the significance of spatial rainfall representation in catchment flood response modelling. The relationship between the flood response, the hillslope and channel residence time representation and the spatial organisation of the rainfall fields is obtained by extending the ‘spatial moments of catchment rainfall’ statistics (Zoccatelli et al., 2011) to the hillslope system. The flood prediction error generated by assuming spatially uniform rainfall is related to the spatial organisation of the rainfall fields by means of the scaled spatial moment of order one for the channel network and the hillslope system. The methodology provides a basis for a more general consideration of the relationship between the flood response dependence to spatial rainfall organisation and catchment size. The methodology is illustrated based on data from five extreme flash floods occurred in various European regions in the period 2002-2007. Discharge data are available either from streamgauges or from post-flood surveys for 27 catchments, ranging in size between 36 and 982 km2. High space-time resolution radar rainfall fields are also available for the analyses. These data are used to implement a distributed hydrological model simulating the runoff generation by infiltration excess and explicitly representing the surface flow paths across both the hillslopes and the river network. The hydrological model is alternatively forced with spatially-distributed and spatially-uniform rainfall input, to analyse the factors controlling the sensitivity of the model output to the spatial rainfall data. Our results show that the spatial variability of the rainfall can influence the flash-flood hydrographs for catchments as small as 50 km2 , and that the dependence of flood hydrograph shape to spatial rainfall variability cannot be treated as scale dependent relative to the size of the catchment. The rainfall index can be exploited as similarity index for classifying catchments and flood events according to the hillslopes / channel residence times and to provide guidance on the space and time resolution of the rainfall monitoring system required to predict the flood response.

2

Keywords: Flash flood; rainfall spatial statistics; rainfall organisation; basin morphology; hillslope

Introduction Runoff propagation processes along hillslopes and flow dynamics along the river network combine to shape the hydrologic response of a basin (D'Odorico and Rigon, 2003; Giannoni et al., 2003; Naden, 1992; Robinson et al., 1995; Saco and Kumar, 2004; Snell and Sivapalan, 1994; Viglione et al., 2010; Yen and Lee, 1997). The relative contribution of hillslope processes and network geomorphology to the catchment’s hydrologic response has been investigated by several researchers (Saco and Kumar, 2004; Viglione et al., 2010). A general result is that the relative role and mutual interactions of hillslope and channel network transport change substantially with catchment size (Beven and Wood, 1993; Di Lazzaro, 2009; Kirkby, 1976; Robinson et al., 1995; Saco and Kumar, 2002). In small catchments, the time delay to rainfall forcing tends to be dominated by the routing of surface and subsurface flow on the hillslopes, whereas in large catchments the flow routing through the river network controls the hydrograph shape (Beven and Wood, 1993; Botter and Rinaldo, 2003). Less attention has been devoted to examine to what extent the relative role of hillslope and river network processes affects the sensitivity of the flood response to rainfall spatial variability. Some hypotheses have been put forward to identify the mechanisms through which rainfall spatial variability may affect catchment response, with an emphasis on hydrologic partitioning processes (Anquetin et al., 2010; Brath and Montanari, 2003; Gabellani et al., 2007; Shah et al., 1996; Winchell et al., 1998). Several works have focused on the relation between the spatial rainfall organization and the heterogeneities embedded in the basin geomorphic structure, mostly by examining the rainfall variability relative to a distance metric imposed by the drainage network

3

(Lobligeois et al., 2014; Smith et al., 2005; Smith et al., 2002; Zhang et al., 2001). Nicótina et al. (2008) focused on the effects of transport processes along the hillslopes and the channel network as a key element to clarify the extent of the possible influence of rainfall spatial variability on the hydrologic response. They used a geomorphological model of the runoff response and analyzed the distribution of travel times, showing that the sensitivity of the hydrologic response to rainfall distribution increases with decreasing hillslope residence time. By considering relatively large storm events, they found that rainfall spatial variability does not significantly influence the flood response for basin areas up to about 3500 km2. These results confirm earlier results by a number of studies (Andreassian et al., 2001; Beven and Hornberger, 1982; Naden, 1992; Obled et al., 1994; Smith et al., 2004a; Smith et al., 2004b), which were carried out in relatively small to medium-size catchments where the catchment response is expected to be dominated by the hillslope response. However, these results are at odds with other empirical findings which show that the rainfall spatial variability can play a major role in flood modelling even at the scale of a small basin (Faurès et al., 1995; Michaelides and Wainwright, 2002; Michaud and Sorooshian, 1994; Schuurmans and Bierkens, 2007), hence implying that the sensitivity to spatial rainfall variability cannot be treated as scale dependent relative to the size of the catchment. There is therefore a need for a more general approach, which is able to characterize the relative role of the hillslope and river network routing in the hydrologic response to spatially variable rainfall fields. A number of papers have focused on the interaction between rainfall spatial organisation and flow distance, i.e. the distance along the runoff flow path from a given point to the outlet, to predict the impact of spatial and temporal variability of precipitation on the flood hydrograph (Emmanuel et al., 2012a; Emmanuel et al., 2012b; Smith et al., 2002; Viglione et al., 2010; Woods and Sivapalan, 1999; Zhang et al., 2001). Based on these works, Zoccatelli et al. (2011) proposed a series of statistics, termed ‘spatial moments of catchment rainfall’, which are able to isolate the effect of rainfall spatial variability on mean and variance of runoff time. These authors reported large impacts of rainfall spatial variability on hydrologic response for catchments as small as 50 km2. In

4

the development of the ‘spatial moments of catchment rainfall’, Zoccatelli et al. (2011) disregarded the differentiation between hillslopes and channel network contribution to the total runoff travel time. However, for most river basins the contribution of hillslopes to the total residence time is relevant to the proper representation of the basin response (D'Odorico and Rigon, 2003; Nicótina et al., 2008; Rinaldo et al., 1995). The correct description of hillslope contribution is even more important in the small to medium catchments (less than 1000 km2), which are more frequently impacted by flash floods (Yakir and Morin, 2011). In this work, we aim to identify how large the hillslope residence time has to be relative to the channel residence time, to smooth out the effects of a certain degree of spatial rainfall organisation - quantified by the spatial first moment of catchment rainfall - on the flood hydrograph. This objective is achieved by extending the concept of ‘spatial moments of catchment rainfall’ by incorporating both hillslope and channel contributions to the travel time in the moment formulations. This formulation leads to an index of spatial rainfall organisation which is used to gain insight into the relative role of the hillslope and channel residence time. This approach provides a basis for a more general consideration of the relationship between the flood response sensitivity to spatial rainfall organisation and catchment size. The role of the hillslope residence time in damping the rainfall spatial variability is illustrated by analyzing five extreme flash floods occurred in various European regions in the period 2002–2007. The size of the study catchments ranges between 36 to 982 km2. These events are characterized by high intensity and large space-time variability of rainfall, which have a strong effect on the flood response even at small spatial basin size. Extended spatial moments are evaluated by exploiting high resolution, carefully controlled, radar rainfall fields. Results obtained by means of the extended spatial moments are compared with those provided by using a spatially distributed hydrological model of the flood response. The model is alternatively forced with spatially-distributed and spatially-uniform rainfall input, to analyse the factors controlling the sensitivity of the model output

5

to the spatial rainfall data and to assess the relevance of the results obtained by using the index of spatial rainfall organisation.

6

1. Spatial moments of catchment rainfall: extension to the hillslope processes

‘Spatial moments of catchment rainfall’ (Zoccatelli et al., 2011, referred to as Z2011 hereinafter) provide a description of the spatial rainfall organisation at a certain time t as a function of the rainfall field r(x,y,t) value at any position (x,y) within a catchment, and of the flow distance d(x,y) to the catchment outlet measured along the flow path. Similar statistics have been introduced in previous works by Smith et al. (2002) and Smith et al. (2005) to describe the rainfall spatial variability from the perspective of a distance metric imposed by the drainage network. In Z2011 the spatial moments of catchment rainfall have been defined under the assumption of flood routing with constant flow velocity , in both space and time. In this section, the spatial moments are extended to include the hillslope processes. The runoff transport is described by using two different time invariant values of velocity, vh and vc, characterizing the hillslope and the channel system, respectively. This assumption has been used in a number of flood modelling works (Marchi et al., 2010; Nicótina et al., 2008; Zanon et al., 2010 , among the others). Evaluation of simulation results reported in these works supports the assumption that models of the hydrologic response employing basin-constant channel celerity explain observed travel time distributions, at least for high flows conditions as observed in (Pilgrim, 1976). The invariant hillslope celerity assumption is more conceptual in nature (Botter and Rinaldo, 2003). In fact, great variability in hillslope transport properties is expected, particularly when it is driven by local topographic gradients as subsurface runoff through partially saturated areas and in the presence of preferential flow paths (Beven and Wood, 1983; Dunne, 1978). The term dh(x,y) identifies here the distance from any point in the basin to the channel network following the steepest descent path, while dc(x,y) identifies the length of the subsequent drainage path through the streams down to the watershed outlet. Then, the following definitions are provided for the spatial moments of catchment rainfall of order n for the channel (pn,c(t)) and hillslope systems (pn,h(t)) at time t, respectively:

7

(1)

where A indicates the catchment area. It is easy to verify that the zero-th order spatial moment along hillslope and channel flow paths are both equal to the catchment average rainfall at time t, p0(t). Analogously to Z2011, the terms Pn,c and Pn,h are employed to indicate the corresponding means of pn,c and pn,h over the storm duration Ts:

(2)

where R(x,y) is the cumulated precipitation over the time interval Ts at point x,y. In a similar way, the moments of the flow distance along the channel and hillslope flowpaths are given by:

(3)

The first order moments

and

are the catchment average distance of hillslopes from the

channel network and channels from the catchment outlet, respectively. Dimensionless first order spatial moments of catchment rainfall over the hillslope and river system can be obtained by taking the ratio of the spatial moments of catchment rainfall to the moments of the flow distance, as shown below: 8

(4)

(5)

The scaled moment of order one δ1,c describes the rainfall weighted distance along the river network, with respect to the average value of the flow distance along the river network. A spatially uniform rainfall or a rainfall concentrated towards the position of the river network centroid will result in a value of δ1,c close to one. Values of δ1,c less than one indicate that rainfall is distributed near the basin outlet, whereas values greater than one indicate that rainfall is distributed towards the headwaters. Values of

less than one describe cases with stronger rainfall near the channel

network, while values greater than one indicate that rainfall is concentrated near hillslopes crests. Analogously, the weighted scaled moments Δn,h and Δn,c are defined with a formalism similar to δn,h and δn,c, but for a mean rainfall intensity over the finite temporal interval equal to Ts:

(6)

It can be shown that the overall spatial moment of first order Δ1, introduced in Z2011 and expressed as a function of the whole flow distance d(x,y)=dh(x,y)+dc(x,y), is a weighted function of the channel and hillslope moments as follows:

(7)

9

2. Relationship between the extended spatial moments of catchment rainfall and the flood hydrograph shape

In this section we relate the extended spatial moments to the mean time of catchment runoff, i.e. the center of mass of the flood hydrograph, under the assumption that the runoff coefficient is uniform in space and time (see Z2011 for the implications of this assumption). According to this assumption, the space and time variability of the runoff is the same as that of the rainfall. The catchment runoff time (Tq) measures the time from the runoff generation until a drop of water exits the catchment. This time includes the runoff transport in the hillslope system and in the channel system: (8) where Tc and Th are the holding times for channel travel and hillslope travel, respectively.

2.1 Effect of spatial rainfall variability on the temporal pattern of the flood hydrograph Using the mass conservation property (see V2010) we can write the mean of Tq, i.e. the first temporal moment of the flood hydrograph, as follows:

(9)

The first term E(Tc) represents the average time to route the effective rainfall from the geographical centroid of the rainfall spatial pattern to the catchment outlet through the channel system. The second term E(Th) is the average hillslope residence time. By using the spatial moments, the term E(Tc) may be expressed as follows: (10)

10

where

is the rainfall that fell at point (x,y) and at time . Details concerning the derivation

of Eq. (13) are reported in Zoccatelli et al. (2011). The term E(Th) is written in a corresponding way, by considering the properties of the hillslope system and the interaction between rainfall spatial organization and the hillslope system. Therefore, Eq. (9) may be written as follows:

(11)

By using the definition of mean residence time in the river network hillslope system

and in the

, Eq (14) can be written as:

(12)

A key dimensionless indexdescribing the dependence of the first temporal moment of the flood hydrograph to the rainfall spatial organisation is the following:

(13)

where τtot=τc + τh . The index

represents the ratio between E(Th)+E(Tc) and the mean runoff

propagation time through the hillslope and the channel system. Index

is positive definite, and it is equal to one either in case of spatially uniform precipitation or

in case of spatially variable rainfall concentrated in areas with travel time equal to the basin average. When the hillslope transport processes are negligible with respect to the total runoff travel time, values of

less than one indicate that rainfall is concentrated towards the outlet, and values

larger than one indicate that rainfall is concentrated towards the headwater portion of the basin.

11

The effect of spatial rainfall distribution on the hydrograph shape can be isolated by comparing the hydrograph generated by spatially distributed rainfall (termed ‘reference hydrograph’ hereinafter) with that generated by spatially uniform rainfall. The time error between the two hydrographs is quantified by a statistic, termed “normalised time difference” dTn. The normalised time difference dTn is computed as the ratio of the time difference between the two hydrograph centroids to the mean response time of the catchment, as explained in (Zoccatelli et al., 2011). Combining Eq. (11) and Eq (13) with Eq. (18) of Zoccatelli et al. (2011) we can write:

(14)

where E(Tq Dist) and E(Tq Unif) are the centroids of the reference hydrograph and of the hydrograph generated by uniform rainfall, respectively. A positive (negative) value of dTn implies a positive (negative) shift in time of the reference hydrograph with respect to the one produced by using uniform precipitation. Eq. (14) shows that the normalised time error is related in a simple way to the spatial organisation of the rainfall fields by means of the scaled spatial moment of order one for the channel and the hillslope system. Accordingly with Eq. (14), we expect that a rainfall spatial distribution characterized by

less than one causes a reduction of lag time between rainfall and

runoff in comparison with the case of a uniform rainfall distribution. This means that when rainfall is concentrated towards the outlet, the hydrograph is anticipated relative to the case of spatially uniform rainfall. The opposite is true for rainfall concentrated towards the periphery of the catchment, with the hydrograph delayed relative to the case of a spatially uniform rainfall.

12

2.2 Derivation of the simplified index of rainfall spatial organization In this section, we develop a simplified index of rainfall spatial organization which is explicitly written based on the overall spatial moment of first order Δ1 and the mean channel and hillslope residence times. This offers an opportunity to quantify the damping effect of the hillslope system on the sensitivity of the flood response to the spatial rainfall variability. The simplified index is developed based on the assumption that rainfall fields are not correlated with hillslope flow distance. This is a rather common situation, with mean hillslope distances ranging between 100m and 300m (Marchi et al., 2010), i.e. much shorter than the mean hourly rainfall correlation length for flood producing storms, commonly quantified between 4 km and 10 km (Berne et al., 2004; Ciach and Krajewski, 2006). Under this hypothesis, we have:

(15)

Then: (16) Under this assumption, Eq. (13) can be written in the following way:

(17)

Eq. (17) may be written as follows by using Eq. (7):

(18)

Eq. (18) allows to relate the

index of Z2011 with

, which has a similar conceptual meaning but

includes the effects of hillslopes. Eq. (18) offers the advantage of predicting the sensitivity of the

13

flood response to the spatial rainfall variability by varying the geomorphologic and flow propagation parameters of a catchment. This can be used to identify those conditions which may dampen the effect of rainfall spatial organization on the flood response. Eq. (18) may be further simplified by considering the generally negligible amount of the term g1,h/vc, which is ranging around a few minutes for typical values of mean hillslope length (100m300m) and channel velocity (1m s -1- 3m s-1). This permits obtaining the following relationship:

(19)

Examination of Eq. (19) shows that the index of spatial rainfall organization

is function of two

terms: the first term depends on the spatial organization of the rainfall over the catchment (summarized in the term Δ1) weighted by the ratio between the channel network residence time and the mean catchment response time, whereas the second term exclusively depends on the ratio between the hillslope residence time and the mean catchment response time. An equivalent form of Eq. (19) is derived by introducing a velocity parameter parameter

and a morphological

, which are related with the residence times τc and τh by the following relation:

(20)

Hence, Eq. (19) can be written as follows:

(21)

Eq. (21) has a similar meaning to Eq. (18), but it separates the specific effects of the velocity and morphological parameters. Eq. (21) shows that the difference between Δ1 and

i.e. the hillslope 14

impact on the sensitivity of the flood response to the spatial rainfall variability, tends to vanish when the average distance of the hillslope from the channel network is negligible with respect to the average distance of the channel network from the outlet, as it generally occurs for relatively large basins. On the other hand, when the value of the parameter V* vanishes, i.e. when the hillslope velocity is negligible with respect to the channel velocity,

tends to 1, indicating a negligible

effect of rainfall spatial variability on the first temporal moment of the flood hydrograph. As it will be shown more extensively in Section 4.4, the value of

is always closer to 1 with respect to Δ1 ,

i.e., smaller (larger) than Δ1 when Δ1 is larger (smaller) than 1, and the difference between Δ1 and quantifies the damping effect of the hillslope system on the sensitivity of the flood response to the spatial rainfall variability.

3. Analysis of the index of rainfall spatial organization for extreme flash floods We analyzed the dependence of the flood hydrograph shape to the rainfall spatial variability and to the hillslope and river system residence times for five flood events in Europe for which observational and modeling analyses are available. The application of the framework to real flash floods characterized by very high intensity and variability of rainfall, will be a test on extreme but still real cases where rainfall variability is thought to have played a crucial role in the development of the flood.

3.1 Data The data sources used for this study are represented by rainfall and discharge data from five extreme storms and ensuing floods which have been observed in Europe in the period between 2002 and 2007. The case studies are the following: Sesia at Quinto (North-western Italy, 982 km2) occurred on 04/06/2002, Sora at Vester (Slovenia, 212 km2), occurred on 18/09/2007, Feernic at Simonesti (Romania, 168 km2), occurred on 23/08/2005, Clit at Arbore (Romania, 36 km2), occurred on 30/06/2006 and Grinties at Grinties (Romania, 51 km2), occurred on 04/08/2007 (Fig. 15

1). The main features of the storms and the sizes of the examined sub-catchments, together with the references to the works describing the individual events, are reported in Table 1. These storms were selected because they represent a large range of catchment areas (from 36 to 982 km2) and storm durations (from 5h30’ to 21 hours) and their rainfall fields are characterized by significant spacetime variability. The data concerning the events were derived from the flash flood data archive developed in the frame of the EU Project HYDRATE (www.hydrate.tesaf.unipd.it) (Borga et al., 2010). The hydro-meteorological data includes high-resolution rainfall patterns, flow type processes (either liquid flow or debris flow or hyperconcentrated flows) and hydrographs or peak discharges. Climatic information and data concerning morphology, land use and geology are also included in the database. These data enable the identification and analysis of the hydrometeorological causative processes and the individual reconstruction of the events by using hydrologic and hydraulic modelling. A total of 27 sub-catchments have been examined for modelling applications (see Table 1 for details). For the five events, both original raw radar reflectivity values and raingauge data were made available for rainfall estimation. The quantitative precipitation estimation problem is particularly crucial and difficult in the context of flash-floods since the causative rain events may develop at very short space and time scales (Bouilloud et al., 2010; Krajewski and Smith, 2002). The methodology implemented here for radar rainfall estimation is based on the application of correction procedures exploiting the understanding of radar observation physics (Heistermann and Kneis, 2011). The methodology is based on (Bouilloud et al., 2010): (1) detailed collection of data and metadata about the radar systems and the raingauge networks (including raingauge data from amateurs and from bucket analysis), (2) analysis of the detection domain and of the ground/artificial clutters for the considered case (Kirstetter et al., 2010; Pellarin et al., 2002; Seo et al., 2010), (3) implementation of corrections for range-dependent errors (e.g. screening, attenuation, vertical profiles of reflectivity) and (4) optimisation of the rainfall estimation procedure by means of radar-

16

raingauge comparisons at the event duration scale. The methodology was applied consistently in the same form over the five storm events.

3.2 The distributed hydrological model A spatially distributed hydrological model was calibrated for each of the five floods. The hydrological model is described in Zanon et al. (2010) and only an outline is provided here. The discharge Q(t) is computed by the model at any location along the river network as follows:





Q(t )   q x, y, t   ( x, y ) dA

(22)

A

where A indicates the area draining to the specified outlet location, (x,y) is the routing time from the location (x,y) to the outlet of the basin specified in Eq. (22) and q(x,y,t) is the runoff rate at time t and location (x,y). The runoff rate is computed from the rainfall rate r(x,y,t) using the Green-Ampt infiltration model with moisture redistribution (Ogden and Saghafian, 1997). The routing time is computed by using two invariant hillslope (vh) and channel (vc) velocity as reported in the Sections above. The distinction between hillslope and channel systems is based on the use of a channelization support area (As) which is kept constant at the scale of the basins. Table 2 lists the flow propagation model parameters used for simulating catchment storm responses. The values of the morphological parameters g1c and g1h derived for the various basins according to the As values used in the flood simulations, are reported in Fig. 2a and Fig 2b, respectively. A power equation, with exponent equal to 0.52, fits the relationship between the values of g1,c and drainage area. This equation is a version of the classical Hack’s law, Hack (1957). The exponent in this equation is almost in the range 0.53-0.59 reported by Rigon et al. (1996) for the Hack’s law in basins ranging from 50 to 2000 km2. Explanations for the exponent being larger than 0.5 (implying positive allometry) emphasise the role of basin elongation as well as the fractal characteristic of

17

river networks (Rigon et al., 1996). The values of the parameter g1,h are in the range of 100-300 m, exhibiting almost no correlation with the drainage area, as expected.

3.3 Influence of space-time rainfall distribution on flood response To isolate the role of rainfall distribution for each catchment considered in the analysis, we compared the reference hydrograph with the hydrograph generated by considering spatially uniform rainfall. The comparison is carried out by using in both cases a space-time constant runoff coefficient, evaluated based on the ratio between the observed runoff and rainfall volume. The advantage of this type of analysis is to separate the role of rainfall spatial variability in the runoff generation process from that played in the runoff propagation process. An example of this analysis is provided in Fig. 3a for the flood of 04/06/2002 on the 501 km2-wide catchment of Sesia at Busonengo in the Sesia river system (Italy). The runoff model parameters are reported in Table 3, together with the values of the spatial moments and of the index of rainfall spatial organization. The simulations carried out by applying a space-time constant runoff coefficient, for spatially uniform and spatially distributed rainfall, are reported in Fig. 3a. The comparison between the two hydrographs allows identifying a considerable reduction of lag time between rainfall and runoff obtained by using spatially uniform rainfall with respect to that obtained by using spatially distributed rainfall. The reduction of lag time reflects the specific features of the rainfall fields, which exhibit a concentration over the catchment headwaters, due to the considerable orographic effect on convective precipitation (Sangati et al., 2009). The rainfall concentration is quantified by the overall spatial moment of order one (Δ1), with a value of 1.38. The rainfall spatial organization index Θ1 is equal to 1.27, a value smaller than Δ1, due to the damping effect of the hillslope system. The time error dTn is equal to 0.27, as indicated by Eq. (14). This is not surprising, since the rainfall-runoff model applied in this way reproduces exactly the assumptions used in the development of the spatial moments.

18

Fig. 3b displays the hydrographs generated by the distributed hydrological model with the variable space-time infiltration by using both spatially distributed and spatially uniform rainfalls. The general behaviour reported in Fig. 3b is similar to that shown in Fig. 3a, even though a more pronounced reduction of lag time (i.e. a larger time error) can be identified. The time error dTn is equal to 0.65, which is more than twice the value obtained by using a space and time constant runoff coefficient. Of course, the spatial moments are unchanged with respect to the previous case. This observation suggests that a magnification of the time error occurs when the infiltration processes are considered in the modelling phase. These effects are more broadly discussed later in this Section. The dTn values obtained for the 27 cases by using a rainfall-runoff model with space-time constant runoff coefficient are reported in Fig. 4 together with the corresponding Θ1 values. As expected, the relationship between dTn and Θ1 accurately follows the relationship provided by Eq. (14), since in this case the model application exactly reproduces the assumptions used in the development of the relationship. The values of dTn range between -0.05 and 0.28, with many more positive values than negative values. This shows that in general, the effect of disregarding the spatial rainfall variability translates into a reduction of lag time between rainfall and runoff with respect to that obtained by considering the rainfall spatial variability in the flood modelling phase. This is due to concentration of rainfall towards the headwater basins, which reflects the orographic enhancement of convection characterising some of the cases analysed here. Also, the figures show that positive values of dTn characterise medium-size basins larger than 150 km2, whereas smaller basins are characterised by very low or negative values of dTn. Fig. 5 reports the results for the 27 sub-catchments obtained by removing the assumption of spacetime constant runoff coefficient (i.e., by using the whole modelling chain, including the infiltration module). The organisation of the dTn values follows the same pattern already reported in Fig. 4, with the larger basins characterised by larger positive values of dTn. Of course, some scatter may be noted in the comparison, even though the predictive power of the relationship is very high, with an

19

explained variance of 0.83. However, the range of dTn values is much larger than that reported in Fig. 4, with values spanning between -0.31 and 0.72. Correspondingly, the regression line dTn =2.74 Θ1-2.82 is much steeper in this figure than in Fig. 4. As reported in the comment to Fig. 3b, this effect is due to the nonlinearity incorporated into the runoff generation modelling: proportionally more runoff is generated with increasing rainfall accumulation. This leads to a magnification of the dTn statistics with respect to those obtained under the assumption of a spacetime constant runoff coefficient. Correspondingly, this shows that the dTn statistics obtained by means of Eq. (14), i.e. by using the rainfall field as a proxy for the runoff space and time distribution, provides a lower bound for the time error distribution. Research is underway to include runoff generation space-time variability in the model.

3.4 Assessing the hillslope influence on the sensitivity of flood response to the spatial variability of the rainfall As previously stated, the rainfall organization index of Eq. (19) chiefly relies on the assumption that rainfall fields do not exhibit a significant correlation between the rainfall spatial distribution and the hillslope flow distance, which leads to consider the value ∆1,h equal to one. This assumption is assessed over the sample of cases considered in this study. Fig. 6a and Fig. 6b report the frequency distribution of the values of ∆1,c and of ∆1,h, showing that the values of ∆1,h are distributed very close to 1. On the contrary, the values of ∆1,c are distributed over a wider range of values comprised between 0.95 and 1.4. Fig. 7 shows that there is no relationship between the values of ∆1,c and those of ∆1,h, as expected. The accuracy of the relationship based on Eq. (19) has been tested in a series of numerical experiments in which the flood simulations over the 27 sub-catchments have been repeated by modifying only the hillslope system characteristics and using a space-time constant runoff coefficient (equal to one). The runoff propagation parameters include: the channel velocity vc and the support area As, which have been kept equal to 3 ms-1 and to 0.02 km2, respectively and three 20

hillslope velocity (vh) scenarios, corresponding to 0.5 ms-1, 0.1 ms-1 and 0.01 ms-1. This allows to examine three scenarios where the hillslope residence time is progressively more important with respect to the catchment runoff response time. The relationship between dTn and Θ*1 is reported in Fig. 8 for the three scenarios. Two features are noteworthy in this figure. First of all, the figure clearly shows that the simplified index of rainfall spatial organization Θ*1 is able to effectively reproduce the distribution of dTn for the three cases. The quality of the description slightly decreases with increasing hillslope residence time. However, the correlation between dTn and Θ*1 is very high in all cases, with the lowest correlation value equal to 0.94 for the scenario characterized by a hillslope velocity equal to 0.01 ms-1. This means that Eq. (19) provides a robust description of the relative effects of the hillslope and river network routing on the time error over a wide range of scenarios. Moreover, it is interesting to analyze the distribution of dTn in the two extreme scenarios: dTn ranges between -0.05 and 0.38 for the highest hillslope velocity, whereas it ranges between -0.025 and 0.25 for the lowest hillslope velocity. This means that the time error is generally reduced by 63% when decreasing the hillslope velocity from 0.5 ms-1 to 0.01 ms-1, implying that the higher the hillslope routing time, the more accurately the hydrograph temporal distribution is represented by spatially uniform rainfall conditions. The effect of the hillslope residence time on the runoff time error dTn , i.e. on the significance of rainfall spatial organisation in flood response modelling, is captured by the ratio Θ1*/∆1 based on Eq. (21). The relationship between the ratio Θ1*/∆1 and the parameters V* and G*, as derived from Eq. (20), is exemplified in the colored grid of Fig. 9 drawn for ∆1 equal to 0.5. The red (blue) color means that the ratio Θ1*/∆1 is closer to (farther from) one, implying that the hillslope systems have a negligible (considerable) effect on the first temporal moment of the flood hydrograph. The symbols on the grid are representative of the 27 sub-catchments, based on the respective values of parameters V* and G*. The colored grid shows that, for a given value of V*, the effect of the hillslope system increases by increasing the value of G*. On the other hand, for a given value of G*, the effect of the hillslope system increases by decreasing the value of V*. One very small basin

21

(5 km2) belonging to the Feernic river system, exhibits the highest impact of the hillslope system on the basin sensitivity to the spatial rainfall distribution, as expected. However, the size of the basin is not always a good indicator of the hillslope impact on the flood response sensitivity. Indeed, the three lowest Θ1*/∆1 ratios (i.e., closest to one) are verified for three basins (Clit, Feernic and Sesia) of widely different sizes, ranging from less than 50 km2 to more than 500 km2. This provides a clear indication that we need to take both geomorphological and flow propagation parameters into account, to assess the potential damping effect of the hillslope system on the rainfall spatial variability. The interrelated influence of the various factors is a likely reason for the failure of empirical approaches based only on basin size to explain the impact of rainfall spatial variability on the flood response, as revealed in the literature review section. In the same vein, Eq. (21) can be used to quantify the catchment similarity as far as the effect of hillslope system on the flood response sensitivity is of concern. The basin symbols mapped in the grid of Fig. 9 show for instance that the sensitivity of the largest basin of the Sesia (982 km2) river system is similar to that of the largest basin of the Grinties (51 km2) river system, in spite of the rather great difference in terms of size. Fig. 10 shows the relationship between the ratio Θ1*/∆1 and the parameters V* and G* for ∆1 equal to 1.5. As expected, the pattern is exactly the same as the one reported in Fig. 9, the only difference being the range of the ratio Θ1*/∆1 from 1.0 to 0.7.

4. Conclusions In this work we presented an analytical framework, which permits to identify how large the hillslope residence time has to be relative to the channel residence time, to smooth out the effects of a certain degree of spatial rainfall organisation - quantified by the spatial first moment of catchment rainfall - on the flood hydrograph. This objective was achieved by extending the concept of ‘spatial moments of catchment rainfall’ by incorporating both hillslope and channel contributions to the travel time distribution in the moment formulations. 22

The framework summarises the dependence of the first temporal moment of the flood hydrograph to the spatial variability of the rainfall by means of a simple dimensionless index

, based on the

first extended spatial moment of catchment rainfall and the parameters characterizing the residence time and the morphology of the hillslope system and the channel network. The difference in catchment response time (

between flood hydrographs obtained by using spatially variable and

uniform rainfall patterns is linearly related to

.

The dimensionless index has been developed under the hypothesis of uniform runoff coefficient, to avoid the complexities related to the prediction of its spatial and temporal variability. Under this assumption, results obtained with the proposed dimensionless index likely apply to rainfall events characterized by large rain rates and accumulations, so that the runoff is produced all across the catchment and infiltration excess runoff is dominating the hydrograph response. Further simplification of the dimensionless index

has been achieved to permit the direct use of

the non-extended spatial moment of catchment rainfall in the index formulation. This simplification is important because it provides a direct way to analyse the relative impact of hillslope and channel residence time, for a given scenario of rainfall spatial organisation, on the flood hydrograph shape. The capability of

to predict the effect of the spatial variability on catchment response time has

been successfully verified by simulating five extreme historical flash floods in 27 catchments with a distributed hydrological model, also accounting for spatially variable runoff coefficients. In large catchments the effect of the rainfall variability on flood response tends to be dominated by the spatial organization of the rainfall pattern with respect to the catchment flow path pattern. However, the effect of the rainfall gradients within the catchment can be damped when the spatial extent of the hillslopes increases or the celerity of the hillslope runoff decreases with respect to those analogous quantities of the river network. While this confirms earlier results, we have also shown that the damping effect of the hillslope system can be similar in catchments characterized by very different size. This implies that the dependence of flood hydrograph shape to spatial rainfall variability cannot be treated as scale dependent relative to the size of the catchment. The

23

formulation of the index

provides therefore a generic approach, which is able to characterize the

relative role of the hillslope and river residence times in the hydrologic response to spatially variable rainfall fields, for rainfall scenarios characterized by the spatial moments of catchment rainfall. The index

can be exploited as similarity index for classifying catchments and flood

events according to the relative role of the hillslopes and the channel network and to provide guidance on the space and time resolution of the rainfall monitoring system required to predict the flood response.

Acknowledgements. The work presented in this paper has been carried out as part of the European Union FP6 Project HYDRATE (Project no. 037024) under the thematic priority, Sustainable Development, Global Change and Ecosystems and by the Research Project GEO-RISKS (University of Padova, STPD08RWBY).

24

References Andreassian, V., Perrin, C., Michel, C., Usart-Sanchez, I., Lavabre, J., 2001. Impact of imperfect rainfall knowledge on the efficiency and the parameters of watershed models. Journal of Hydrology, 250(1-4): 206-223. Anquetin, S. et al., 2010. Sensitivity of the hydrological response to the variability of rainfall fields and soils for the Gard 2002 flash-flood event. Journal of Hydrology, 394(1-2): 134-147. DOI:10.1016/j.jhydrol.2010.07.002 Berne, A., Delrieu, G., Creutin, J.D., Obled, C., 2004. Temporal and spatial resolution of rainfall measurements required for urban hydrology. Journal of Hydrology, 299(3-4): 166-179. DOI:10.1016/j.jhydrol.2004.08.002 Beven, K., Wood, E.F., 1983. Catchment geomorphology and the dynamics of runoff contributing areas. Journal of Hydrology, 65(1): 139-158. Beven, K., Wood, E.F., 1993. Flow routing and the hydrological response of channel networks, Channel network hydrology. John Wiley, Chichester, pp. 99-128. Beven, K.J., Hornberger, G.M., 1982. Assessing the effect of spatial pattern of precipitation in modeling stream flow hydrographs. JAWRA Journal of the American Water Resources Association, 18(5): 823-829. Borga, M., Anagnostou, E.N., Bloschl, G., Creutin, J.D., 2010. Flash floods Observations and analysis of hydro-meteorological controls. Journal of Hydrology, 394(1-2): 1-3. DOI:10.1016/j.jhydrol.2010.07.048 Botter, G., Rinaldo, A., 2003. Scale effect on geomorphologic and kinematic dispersion. Water Resources Research, 39(10): 1286. DOI:10.1029/2003wr002154 Bouilloud, L., Delrieu, G., Boudevillain, B., Kirstetter, P.E., 2010. Radar rainfall estimation in the context of post-event analysis of flash-flood events. Journal of Hydrology, 394(1): 17-27. Brath, A., Montanari, A., 2003. Sensitivity of the peak flows to the spatial variability of the soil infiltration capacity for different climatic scenarios. Phys Chem Earth, 28(6-7): 247-254. DOI:10.1016/S1474-7065(03)00034-2 Ciach, G.J., Krajewski, W.F., 2006. Analysis and modeling of spatial correlation structure in smallscale rainfall in Central Oklahoma. Advances in Water Resources, 29(10): 1450-1463. DOI:10.1016/j.advwatres.2005.11.003 D'Odorico, P., Rigon, R., 2003. Hillslope and channel contributions to the hydrologic response. Water Resources Research, 39(5): 1113. DOI:10.1029/2002wr001708 Di Lazzaro, M., 2009. Regional analysis of storm hydrographs in the rescaled width function framework. Journal of Hydrology, 373(3): 352-365. Dunne, T., 1978. Field studies of hillslope flow processes. Hillslope hydrology, 227: 293. Emmanuel, I., Andrieu, H., Leblois, E., Flahaut, B., 2012a. Temporal and spatial variability of rainfall at the urban hydrological scale. Journal of Hydrology, 430: 162-172. DOI:10.1016/j.jhydrol.2012.02.013 Emmanuel, I., Andrieu, H., Leblois, E., Janey, N., 2012b. Influence of rainfall spatial variability on hydrological modelling: study by simulations. Iahs-Aish P, 351: 514-519. Faurès, J.M., Goodrich, D., Woolhiser, D.A., Sorooshian, S., 1995. Impact of small-scale spatial rainfall variability on runoff modeling. Journal of Hydrology, 173(1): 309-326. Gabellani, S., Boni, G., Ferraris, L., von Hardenberg, J., Provenzale, A., 2007. Propagation of uncertainty from rainfall to runoff: A case study with a stochastic rainfall generator. Advances in Water Resources, 30(10): 2061-2071. DOI:10.1016/j.advwatres.2006.11.015 Giannoni, F., Smith, J.A., Zhang, Y., Roth, G., 2003. Hydrologic modeling of extreme floods using radar rainfall estimates. Advances in Water Resources, 26(2): 195-203. Hack, J., T., 1957. Studies of longitudinal profiles in Virginia and Maryland. U.S. Geol. Surv. Prof. Pap., 1: 294-B.

25

Heistermann, M., Kneis, D., 2011. Benchmarking quantitative precipitation estimation by conceptual rainfall-runoff modeling. Water Resources Research, 47(6): W06514. DOI:10.1029/2010wr009153 Kirkby, M., 1976. Tests of the random network model, and its application to basin hydrology. Earth Surface Processes, 1(3): 197-212. Kirstetter, P.-E., Delrieu, G., Boudevillain, B., Obled, C., 2010. Toward an error model for radar quantitative precipitation estimation in the Cévennes–Vivarais region, France. Journal of Hydrology, 394(1): 28-41. Krajewski, W.F., Smith, J.A., 2002. Radar hydrology: rainfall estimation. Advances in Water Resources, 25(8-12): 1387-1394. DOI:10.1016/S0309-1708(02)00062-3 Lobligeois, F., Andréassian, V., Perrin, C., Tabary, P., Loumagne, C., 2014. When does higher spatial resolution rainfall information improve streamflow simulation? An evaluation using 3620 flood events. Hydrology and Earth System Sciences, 18(2): 575-594. Marchi, L., Borga, M., Preciso, E., Gaume, E., 2010. Characterisation of selected extreme flash floods in Europe and implications for flood risk management. Journal of Hydrology, 394(12): 118-133. DOI:DOI 10.1016/j.jhydrol.2010.07.017 Michaelides, K., Wainwright, J., 2002. Modelling the effects of hillslope–channel coupling on catchment hydrological response. Earth Surface Processes and Landforms, 27(13): 14411457. Michaud, J.D., Sorooshian, S., 1994. Effect of Rainfall-Sampling Errors on Simulations of Desert Flash Floods. Water Resources Research, 30(10): 2765-2775. Naden, P.S., 1992. Spatial Variability in Flood Estimation for Large Catchments - the Exploitation of Channel Network Structure. Hydrolog Sci J, 37(1): 53-71. Nicótina, L., Alessi Celegon, E., Rinaldo, A., Marani, M., 2008. On the impact of rainfall patterns on the hydrologic response. Water Resources Research, 44(12): W12401. Obled, C., Wendling, J., Beven, K., 1994. The Sensitivity of Hydrological Models to Spatial Rainfall Patterns - an Evaluation Using Observed Data. Journal of Hydrology, 159(1-4): 305-333. Ogden, F.L., Saghafian, B., 1997. Green and Ampt infiltration with redistribution. J Irrig Drain EAsce, 123(5): 386-393. DOI:Doi 10.1061/(Asce)0733-9437(1997)123:5(386) Pellarin, T. et al., 2002. Hydrologic visibility of weather radar systems operating in mountainous regions: Case study for the Ardèche catchment (France). Journal of Hydrometeorology, 3(5): 539-555. Pilgrim, D.H., 1976. Travel times and nonlinearity of flood runoff from tracer measurements on a small watershed. Water Resources Research, 12(3): 487-496. Rigon, R. et al., 1996. On Hack's law. Water Resources Research, 32(11): 3367-3374. Rinaldo, A., Vogel, G.K., Rigon, R., Rodriguez-Iturbe, I., 1995. Can one gauge the shape of a basin? Water Resources Research, 31(4): 1119-1127. Robinson, J.S., Sivapalan, M., Snell, J.D., 1995. On the relative roles of hillslope processes, channel routing, and network geomorphology in the hydrologic response of natural catchments. Water Resources Research, 31(12): 3089-3101. Saco, P.M., Kumar, P., 2002. Kinematic dispersion in stream networks - 1. Coupling hydraulic and network geometry. Water Resources Research, 38(11): 1244. DOI:10.1029/2001wr000695 Saco, P.M., Kumar, P., 2004. Kinematic dispersion effects of hillslope velocities. Water Resources Research, 40(1): W01301. DOI:10.1029/2003wr002024 Sangati, M., Borga, M., Rabuffetti, D., Bechini, R., 2009. Influence of rainfall and soil properties spatial aggregation on extreme flash flood response modelling: an evaluation based on the Sesia river basin, North Western Italy. Advances in Water Resources, 32(7): 1090-1106. Schuurmans, J.M., Bierkens, M.F.P., 2007. Effect of spatial distribution of daily rainfall on interior catchment response of a distributed hydrological model. Hydrology and Earth System Sciences, 11(2): 677-693.

26

Seo, D.J., Seed, A., Delrieu, G., 2010. Radar and multisensor rainfall estimation for hydrologic applications. Rainfall: State of the science: 79-104. Shah, S., O'Connell, P., Hosking, J., 1996. Modelling the effects of spatial variability in rainfall on catchment response. 1. Formulation and calibration of a stochastic rainfall field model. Journal of Hydrology, 175(1): 67-88. Smith, J.A. et al., 2005. Field studies of the storm event hydrologic response in an urbanizing watershed. Water Resources Research, 41(10): W10413. DOI:10.1029/2004wr003712 Smith, J.A. et al., 2002. The regional hydrology of extreme floods in an urbanizing drainage basin. Journal of Hydrometeorology, 3(3): 267-282. Smith, M.B. et al., 2004a. Runoff response to spatial variability in precipitation: an analysis of observed data. Journal of Hydrology, 298(1-4): 267-286. DOI:DOI 10.1016/j.jhydrol.2004.03.039 Smith, M.B. et al., 2004b. The distributed model intercomparison project (DMIP): motivation and experiment design. Journal of Hydrology, 298(1-4): 4-26. DOI:DOI 10.1016/j.jhydrol.2004.03.040 Snell, J.D., Sivapalan, M., 1994. On Geomorphological Dispersion in Natural Catchments and the Geomorphological Unit-Hydrograph. Water Resources Research, 30(7): 2311-2323. Viglione, A., Chirico, G.B., Woods, R., Blöschl, G., 2010. Generalised synthesis of space–time variability in flood response: An analytical framework. Journal of Hydrology, 394(1): 198212. Winchell, M., Gupta, H.V., Sorooshian, S., 1998. On the simulation of infiltration- and saturationexcess runoff using radar-based rainfall estimates: Effects of algorithm uncertainty and pixel aggregation. Water Resources Research, 34(10): 2655-2670. DOI:Doi 10.1029/98wr02009 Woods, R., Sivapalan, M., 1999. A synthesis of space-time variability in storm response: Rainfall, runoff generation, and routing. Water Resources Research, 35(8): 2469-2485. Yakir, H., Morin, E., 2011. Hydrologic response of a semi-arid watershed to spatial and temporal characteristics of convective rain cells. Hydrology and Earth System Sciences, 15(1): 393404. DOI:DOI 10.5194/hess-15-393-2011 Yen, B., Lee, K.T., 1997. Unit Hydrograph Derivation for Ungauged Watersheds by Stream-Order Laws. Journal of Hydrologic Engineering, 2(1): 1-9. DOI:Doi 10.1061/(Asce)10840699(1997)2:1(1) Zanon, F. et al., 2010. Hydrological analysis of a flash flood across a climatic and geologic gradient The September 18, 2007 event in Western Slovenia. Journal of Hydrology, 394(1-2): 182197. DOI:DOI 10.1016/j.jhydrol.2010.08.020 Zhang, Y., Smith, J.A., Baeck, M.L., 2001. The hydrology and hydrometeorology of extreme floods in the Great Plains of Eastern Nebraska. Advances in Water Resources, 24(9-10): 10371049. Zoccatelli, D., Borga, M., Viglione, A., Chirico, G.B., Bloschl, G., 2011. Spatial moments of catchment rainfall: rainfall spatial organisation, basin morphology, and flood response. Hydrology and Earth System Sciences, 15(12): 3767-3783. DOI:DOI 10.5194/hess-153767-2011 Zoccatelli, D., Borga, M., Zanon, F., Antonescu, B., Stancalie, G., 2010. Which rainfall spatial information for flash flood response modelling? A numerical investigation based on data from the Carpathian range, Romania. Journal of Hydrology, 394(1-2): 148-161.

27

List of figures: Figure 1: Study catchments and their location in Europe. .............................................................. 29 Figure 2: Relationship between drainage area and a) g1,c and b) g1,h. ............................................. 30 Figure 3: Flood hydrographs generated with spatially uniform and distributed rainfall inputs for the basin of Sesia river at Busonengo (501 km2) (insert). a) Simulations obtained by using a space and time constant runoff coefficient; b) simulations obtained by using the complete distributed rainfall-runoff model. ....................................................................................... 31 Figure 4: Relationship between and for hydrological simulations with constant runoff coefficient. The continuous line is the line dT n= Θ1-1.0, given by Eq. (14)......................... 32 Figure 5: Relationship between and for hydrological simulations obtained by using the complete hydrological model. The continuous line is the regression line dTn=2.71 Θ1-2.80, which is characterized by r2=0.83. ..................................................................................... 33 Figure 6: Frequency distribution of the values of a) ∆ 1,c and (b) ∆1,h . .......................................... 34 Figure 7: Relationship between the values of ∆ 1,c and ∆1,h . ......................................................... 35 Figure 8: Relationship between the timing error and with decreasing hillslope residence time. a) vh=0.01 ms-1; b) vh=0. 1 ms-1 ; c) vh=0.5 ms-1. ....................................................... 36 Figure 9: Relationship between the ratio Θ1*/∆1 and the parameters V* and G* for ∆1 =0.5 and a) for the study catchments, b) for specific classes of catchment area. .................................... 37 Figure 10: Relationship between the ratio Θ1*/∆1 and the parameters V* and G* for ∆1 =1.5 and a)for the study catchments, b) for specific classes of catchment area. ................................. 38

List of tables: Table 1: Main features of the flood events (from Marchi et al., 2010, modified). ........................... 39 Table 2: Flood propagation parameters used in the flood simulations ............................................ 40 Table 3: Geomorphological parameters, flow propagation parameters, spatial moments and rainfall organization index for the case of Sesia river at Busonengo (501 km2) ............................... 41

28

Figure 1: Study catchments and their location in Europe.

29

a)

b) Figure 2: Relationship between drainage area and a) g1,c and b) g1,h.

30

a)

b)

Figure 3: Flood hydrographs generated with spatially uniform and distributed rainfall inputs for the basin of Sesia river at Busonengo (501 km2) (insert). a) Simulations obtained by using a space and time constant runoff coefficient; b) simulations obtained by using the complete distributed rainfall-runoff model.

31

Figure 4: Relationship between

and

for hydrological simulations with constant runoff

coefficient. The continuous line is the line dTn= Θ1-1.0, given by Eq. (14).

32

Figure 5: Relationship between

and

for hydrological simulations obtained by using the

complete hydrological model. The continuous line is the regression line dTn=2.71 Θ1-2.80, which is characterized by r2=0.83.

33

a)

b)

Figure 6: Frequency distribution of the values of a) ∆1,c and (b) ∆1,h .

34

Figure 7: Relationship between the values of ∆1,c and ∆1,h .

35

a)

b)

c)

Figure 8: Relationship between the timing error

and

with decreasing hillslope residence

time. a) vh=0.01 ms-1; b) vh=0. 1 ms-1 ; c) vh=0.5 ms-1.

36

a)

b)

Figure 9: Relationship between the ratio Θ1*/∆1 and the parameters V* and G* for ∆1 =0.5 and a) for the study catchments, b) for specific classes of catchment area.

37

a)

b)

Figure 10: Relationship between the ratio Θ1*/∆1 and the parameters V* and G* for ∆1 =1.5 and a)for the study catchments, b) for specific classes of catchment area.

38

Tables Table 1: Main features of the flood events (from Marchi et al., 2010, modified).

Basin river system

Date

Sesia at Quinto (Italy) Selška Sora at Vester (Slovenia) Feernic at Simonesti (Romania) Clit at Arbore (Romania) Grinties at Grinties (Romania)

5/6/2002

1

No. of studied watersheds

Range in watershed area [km2]

9

Range of unit peak discharge

NashSutcliffe

References

75 - 982

Storm duration[h] Total rainfall[mm] 22 - 126

1.33-4.79

0.87

Sangati et al. (2009)

18/9/2007 4

31.9 - 212

16.5 – 157

1.66-3.77

0.86

Zanon et al. (2010)

23/8/2005 9

36 - 168

5.5 - 76

2.2

0.78

Zoccatelli et al. (2010)

30/6/2006 2

12 - 36

4 - 81

4.7

-1

4/8/2007

11 - 52

4 - 67

1.8

-1

Zoccatelli et al. (2010) Zoccatelli et al. (2010)

3

Only post-event estimates of flood peak discharge and timing are available for model calibration

39

Table 2: Flood propagation parameters used in the flood simulations

Sesia Clit Grinties Feernic Sora

As [m2] 20000-50000 6400 2000 6000 10000-40000

vc [m s-1] 2.5-3.5 3 3 2.5 4-1.22

vh [m s-1] 0.05-0.1 0.1 0.1 0.05 0.2-0.05

40

Table 3: Geomorphological parameters, flow propagation parameters, spatial moments and rainfall organization index for the case of Sesia river at Busonengo (501 km2)

g1,c , g 1,h

29.8 km; 0.255 km

vc, vh , As

3.5m s-1 ; 0.1m s-1; 20000 m2 1.38, 1.06, 1.38 1.26

41

Highlights The damping hillslope effect on flood sensitivity to spatial rain distribution is quantified Flood response sensitivity to spatial rainfall variability cannot be treated as scale dependent An index quantifies similarity in flood modelling sensitivity to rainfall spatial organisation

42