Water 2010, 2, ; doi: /w OPEN ACCESS

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1 Water 200, 2, ; doi:0.3390/w OPEN CCESS water ISSN rticle Best Fit and Selection of Theoretical Flood Frequency Distributions Based on Different Runoff Generation Mechanisms Vito Iacobellis, *, Mauro Fiorentino 2, ndrea Gioia and Salvatore Manfreda 2 2 Diartimento di Ingegneria delle cque e di Chimica, Politecnico di Bari, Camus Universitario, Via E. Orabona 4, 7025 Bari, Italy; a.gioia@oliba.it Diartimento di Ingegneria e Fisica dell mbiente, Università degli Studi della Basilicata, Via N. Sauro 85, 8500 Potenza, Italy; s: fiorentino@unibas.it (M.F.); salvatore.manfreda@unibas.it (S.M.) * uthor to whom corresondence should be addressed; v.iacobellis@oliba.it; Tel.: ; Fax: Received: 6 March 200; in revised form: 3 May 200 / cceted: 9 May 200 / Published: 28 May 200 bstract: Theoretically derived distributions allow the detection of dominant runoff generation mechanisms as ey signatures of hydrologic similarity. We used two theoretically derived distributions of flood ea annual maxima: the first is the IF distribution, which exloits the variable source area concet, couled with a runoff threshold having scaling roerties; the second is the Two Comonent-IF (TCIF) distribution, which generalizes the IF distribution, and is based on two different threshold mechanisms, associated with ordinary and extraordinary events, resectively. By focusing on the alication of both models to two river basins, of sub-humid and semi-arid climate in Southern Italy, we resent an ad hoc rocedure for the estimation of arameters and we discuss the use of aroriate techniques for model selection, in the case of nested distributions. Keywords: runoff thresholds; frequency distributions; selection criteria; lielihood; model comarison

2 Water 200, List of Model Parameters, Units (Parameters without Units are Dimensionless), and Short Descrition (m 2 ): basin area; h): lag-time of basin area ; : routing factor; : scale arameter of Gamma distribution; E[i, ] (mm/h): average rainfall intensity referred to the entire basin area ; : scale arameter of the relationshi between average rainfall intensity E[i a, ] and source area a; q o (m 3 /s): base flow; : mean annual number of indeendent rainfall events; : shae arameter of the Weibull distribution of the rainfall intensity; f (mm/h): average hydrologic loss referred to the entire basin area ; : scale arameter of the relationshi between average hydrologic loss (f a ) and source area a; r: ratio of the mean contributing area E[a] to the total basin area ; q : mean annual number of indeendent flood events; f,l (mm/h): lower runoff threshold referred to the entire basin area ; f, (mm/h): higher runoff threshold referred to the entire basin area ; L : scale arameter of the relationshi between average hydrologic loss (f a,l ) and source area a; : scale arameter of the relationshi between average hydrologic loss (f a, ) and source area a; r L : ratio of the L-tye mean contributing area E[a L ] to the total basin area ; r : ratio of the -tye mean contributing area E[a ] to the total basin area ; L : mean annual number of indeendent flood events for L-tye; : mean annual number of indeendent flood events for -tye.. Introduction The identification of dominant rocesses in flood generation reresents the main route for building models able to reroduce real rocesses and reduce the uncertainty of flood rediction with articular reference to ungauged basins. In this context, the detection of the dynamics resonsible for runoff generation and the suitability of a given concetual scheme may rovide interesting insights into basin classification and regionalization. With this aim, in the recent ast, much effort has been sent by hydrologists in order to maximize the exloitation of different inds of information useful for understanding the hydrological regimes. Basically, in the framewor of flood frequency analysis, the uncertainty of rediction is strongly affected by the scarcity of historical data usually due to oor quality or quantity in ea discharge time series. One of the most oular strategies for coing with data scarcity is rovided by regional analysis whose urose is to transfer hydrological information from gauged to ungauged watersheds by identifying hydrologically homogeneous regions and allowing for imroved redictions in ungauged basins []. In the last few years many studies have been devoted to the analysis of satial variability in soil roerties and land use. They investigate relationshis between basin hysical roerties, model

3 Water 200, 2 24 arameters and hydrological resonse (assuming that catchments with the same hysical characteristics have similar hydrological resonse) with the aim of finding basin descritors reresentative of hydrological signatures (e.g., [2-8]). Others articularly focus on the issue of rediction in ungauged basins [9-3]. Notwithstanding the availability of numerous studies in this field, today no clear guidance is available regarding which model or model structure is aroriate for any articular catchment or management question. Similarly, no clear guidance is available regarding which dominant rocesses and mechanisms are oerating in a given catchment tye [4]. romising oortunity for hydrologists arises from the introduction of hysical concets in the construction of the flood frequency curve by means of derived distributions (e.g., [5-2]). These models, with a simle and hysically consistent structure, may rovide a valuable comromise between the comlexity of real rocesses and the need for model consistency. In this context, several authors (e.g., [22-24]) have exlored the effects due to the coexistence of different runoff rocesses in flood generation. For instance, Sivaalan et al. [22] assumed that floods may be roduced by both infiltration excess and saturation excess in the same basin. nother case of coexistence of different runoff rocesses is given by llamano et al. [24] who roosed a distribution where runoff includes both rainfall and snowmelt contributions. The derived distribution aroach rovides the oortunity to bridge the ga between urely statistical aroaches and hysically based (more or less concetual) simulation models. The first frequently involve the use of distributions that are characterized by many arameters (e.g., [25-29]) and most of them totally lac hysical interretation. On the other hand, advanced nowledge of real rocesses has driven the construction of several hydrological models used to derive the flood frequency curve based on Monte-Carlo simulations (e.g., [30-32]). These models, in order to achieve reliable redictions, usually require more or less comlex rocedures for calibration of some ey model arameters. In fact, in most cases, direct evaluation of arameters through field observations is not feasible because the scale of measurement is usually much smaller than the effective scale at which the model arameter is alied (e.g., [33,34]). novel theoretically derived robability distribution of floods was introduced [20], based on the assumtion that two distinct runoff mechanisms are resonsible for ordinary and extraordinary flood events. This distribution, called TCIF, is based on the theoretical framewor of the IF model (from Iacobellis and Fiorentino [7], where a single runoff mechanism is adoted. The TCIF model generalizes the IF model, for cases which arise when the second comonent does not exist for return times of technical interest. Comaring models of different comlexity, the simler model is nested within the more comlex model if it is a secial case or restricted version of the other one (e.g., [35,36]). Then, the IF and TCIF models, which are briefly described in Section 2 (with more details in the aendix) are nested distributions. The TCIF distribution was tested in several river basins of Southern Italy characterized by high sewness of the annual maximum flood series (MFS), roviding good erformances. Following such results, Gioia et al. [20] stated that non-linearity in hydrological rocesses may be due to the coexistence of different threshold-driven mechanisms of runoff generation. In this aer we investigate basins characterized by less sewed MFS and dry climatic conditions. We revise the rocedure for the estimation of arameters of the IF and TCIF distributions with resect to the reviously mentioned alications introducing a different algorithm

4 Water 200, for the estimation of the set of arameters which rovides the maximum lielihood. More imortantly, thans to the hysical meaning associated with the two nested theoretical distributions, we imlemented a faster rocedure for the estimation of the TCIF arameters which exloits constraints rovided by results obtained for the IF arameter values. Section 3 reorts the erformances of the TCIF and IF distributions in two river basins of Southern Italy, roviding interesting insights into the behavior of different runoff thresholds tyical of semi-arid and sub-humid climatic conditions. The issue of model selection for nested distributions is also addressed for the case studies, in Section 4. lthough this aer does not aim at exhaustively treating the general roblem of model selection, the results obtained are of great interest in the statistical hydrology of extreme events which maes a large use of nested distributions, including, for examle, the Generalized Extreme Value (GEV), which generalizes both the Gumbel and Frechet distributions (e.g., [35,37]). 2. Derived Flood Frequency Distributions 2.. IF Model The IF model (from Iacobellis and Fiorentino [7]) is based on the concet of artial contributing (or source) area and the ea of direct streamflow Q is considered the roduct of two random variables strongly correlated, the source area contributing to runoff ea a and the runoff ea er unit of a, u a, which is where ua ( ia, fa ), () i a, τ is the sace-time average areal rainfall intensity concerning the contributing area a in the lag-time a and, f a is the corresonding sace-time average hydrologic loss in the area a and in the interval of time a equal to the lag-time of a. The hydrologic loss includes, in general, evaoration of water from the land and vegetative leaf surface, intercetion of rainfall by vegetation, deression storage on the land surface and infiltration of water into the soil matrix. While considering extreme rainfall-runoff events, here it is mainly referred to infiltration. The exceedance robability function of the ea of direct streamflow Q, G Q (q), is found as the integral of the oint robability density function (PDF) of a and u a The IF model assumes that both average rainfall intensity (E[i a,t ]) and average hydrologic loss (f a ) have imortant scaling roerties: E [ ia ] E[ i, ] a /, (2) f a f a / ' (3) where E[i, ] and f are the average rainfall intensity and the average hydrologic loss referred to the entire basin area. It is useful to remar that introducing the average rainfall intensity (E[i a,t ]) and the average hydrologic loss (f a ), we aly the E[] oerator (exected value) only to rainfall intensity because it is considered a random variable whose entire distribution is exloited in the model. On the other hand, f a is a quantity that deterministically scales with area and time.

5 Water 200, The variable contributing area distribution has arameters and which resectively control osition and scale, while the following relationshi holds: where r E[ a]. r/ (4) Thus, under the hyothesis that the annual maximum floods arise from a comound Poisson rocess, Iacobellis and Fiorentino [7] derived the cumulative distribution function (CDF) of the annual maximum flood ea Q by means of the relationshi: CDFQ q ex G ' q q Q, (5) where G Q is the exceedance robability function of ea flow Q, q the mean annual number of indeendent flood events, which is related to the mean annual number of indeendent rainfall events ( ), the average rainfall intensity (E[i a,t ]) and the average hydrologic loss (f ) referred to the entire basin area : q ex f / E[ i, ] (6) The IF robability density function exressed as the derivative of the CDF is: q o q qo / ξal fa fa da E[ ia, ]/Γ / - q PDF Q q CDFQ q Λq g a f a, ex 0 ξa E[ i, ]/Γ / ξa (7) a 2.2. Two Comonent IF Model (TCIF) Gioia et al. [20] generalized the IF theoretical robability distribution introducing a two-comonent derived distribution called Two Comonent IF distribution (TCIF). They identified two different resonse tyes, lined to different runoff thresholds, starting from the consideration that different mechanisms may arise, in any basin, with different frequency and magnitude (e.g., [22]). The two different threshold-driven rocesses are defined as: - L-tye (frequent) resonse, occurring when a lower threshold f a,l is exceeded, and resonsible of ordinary floods liely roduced by a relatively small ortion of the basin a L : u a,l = ( i f a,l ) with L fa, L f, L al / (8) - -tye (rare) resonse, occurring when a higher threshold f a, is exceeded, and roviding extraordinary floods mostly characterized by larger contributing areas a : u a, = ( i f a, ) with fa, f, a / (9) The flood-ea contributing areas a L and a are assumed, in analogy with the IF model, as Gamma distributed, with = 4, and different mean values. Therefore, two dimensionless arameters are introduced: r L = E[a L ]/ and r = E[a ]/ with r r L. (0)

6 Water 200, ssuming that L-tye and -tye events are indeendent and that both rates of occurrence are Poisson distributed, the overall rocess of exceedances is also a Poisson rocess and the CDF of the annual maximum floods is q ex G' q G q CDFQ L Q, L ' Q,, () where G Q,L and G Q, are the exceedance robability functions of ea flow corresonding resectively to L-tye events and -tye events; L and are resectively the mean annual number of indeendent flood events for L-tye and for -tye rocesses and are related to the runoff thresholds by means of the following relationshis:. f, L q L ex and E[ i, ] f, ex, (2) E[ i, ] PDF The TCIF cumulative distribution function and its robability density function are: Q CDF Q q ex Λ L ga L 0 ex ex Λ q q o / ξa L f a,l f a,l E[ i ]/Γ / 0 g a ex al, q qo q CDFQ q ΛL g a L ξa E[ i ]/Γ / ξa Λ 0 L al, q qo g a ξa E[ i ]/Γ / ξa 0 a, L f a, da q q o / ξa f a, f a, da E[ i ]/Γ / f - a,l - ex a, ex L q qo / ξal fa,l f a,l E[ i ]/Γ / al, da q qo / ξa fa, fa, da E[ ia, ]/Γ / L (3) (4) 3. Case Studies and lication In this section, we reort results of the alication of the IF and TCIF models to two gauged catchments in Southern Italy: the Caraelle river at Caraelle, in Puglia, and the Bradano river at Ponte Colonna, in Basilicata. Puglia and Basilicata are regions in Southern Italy, reresented in Figure with the studied basins, their stream networ and a 90 m 90 m digital elevation model (D.E.M.) grid. The main features of the two basins are reorted in Table, where is basin area, Cv, Cs and N are, resectively mean, coefficient of variation, coefficient of sewness and samle size of the observed MFS, I is the Thornthwaite climatic index [38,39], which comares annual reciitation P and annual otential evaotransiration E, I = (P E )/E. The climatic index distinguishes, in general, between dry (I < 0) and humid (I > 0) basins. In articular, Caraelle at Caraelle is classified as semi-arid ( 0.4 I < 0.2) and Bradano at Ponte Colonna as dry subhumid ( 0.2 I < 0). They were selected with the aim of finding the most aroriate model structure for each river basin and, consequently, of detecting the resence of different runoff thresholds affecting the rocesses resonsible for runoff generation.

7 Water 200, Figure. Basins of Southern Italy selected as case studies. Parameter Estimation and Results Table. River Basin Characteristics. n. (m 2 ) I (m 3 /s) Cv Cs N Caraelle at Caraelle Bradano at Ponte Colonna The IF distribution has twelve arameters: baseflow (q o ), four arameters deendent on basin geomorhology (,, ), four rainfall arameters (E[i, ],,, ), and three arameters (, q, r), which are strictly related to runoff generation mechanisms. It is worth mentioning that all arameters, with the excetion of r and q, are not calibrated on the available MFS. We erformed for them a a riori evaluation by using rainfall statistics and other information. Once all the other arameters are nown, only two of them (namely r and q ) are estimated by means of the maximum lielihood function evaluated on MFS. The TCIF distribution includes the following fifteen arameters: nine of them are already in the IF model (q o,,, E[i, ],,, ), six more arameters ( L,,, L, r L, r ) are strictly related to runoff generation mechanisms. Even in this case, four of them (, L, r L, r,), are obtained from at-site estimation based on the maximum lielihood function evaluated on MFS, while all remaining arameters are a riori evaluated from information other than MFS. We first estimate all arameters of the IF and TCIF distributions which deend on a riori information other than MFS. mong these, all arameters deendent on reciitation which is

8 Water 200, analyzed by means of standard regional methods alied to the observed series of annual maxima of rainfall records. Then, the remaining two unnown arameters of the IF model ( q, r) are calibrated using the observed MFS. Finally, the remaining four unnown arameters of the TCIF model are calibrated by exloiting the IF arameter estimates as initial guess. Most of the model arameters were estimated in revious studies [7,20,39]. In articular, in Fiorentino and Iacobellis [39], the IF model was alied to several basins in Puglia and Basilicata, including Bradano and Caraelle. Nevertheless we briefly include here the rocedures adoted, results are in Table 2. We first describe the evaluation of arameters common to IF and TCIF models. The base flow q o was estimated as the average monthly flow measured at-site in January and February. There are four arameters deendent on rainfall (E[i, ],,, ) and they were estimated by means of regional frequency analysis of rainfall annual maximum series (MS) based on the flood index rocedure with hierarchical estimation of arameters [39]: was deendent on the unique regional coefficient of sewness of rainfall MS; was deendent on the regional estimates of the coefficient of variation (different for Basilicata and Puglia). The exected value of the sace-time average rainfall intensity E[i, ] was evaluated exloiting the intensity-duration-frequency (IDF) curve of the exected annual maximum rainfall intensity, obtaining the average of the base rocess from the annual maxima of a Poisson Process and the US Weather Bureau areal reduction factor [20]. The analysis of the regional scaling of E[i, ], rovided the regional estimates of the exonent (different for Basilicata and Puglia) Parameters deendent on basin geomorhology are,,. Basin area and lag-time were available in regional studies of basins in Puglia and Basilicata [39], = 4 and = 0.7 were assigned as described in the aendix (see also [7] for further details). The loss threshold scaling factors, L, deserve articular attention. For the IF model, was equal to 0.5, assuming that in dry basins the revalent mechanism is of the storage tye [39]. On the other hand, the lower and the higher runoff thresholds of the TCIF model are characterized by Equations (8) and (9), resectively, and articularly by the exonents L and. In articular, Gioia et al. [20] assumed L = 0 and = 0.5, roviding a constant infiltration rate for the lower threshold f a,l, and a storage behavior for the higher threshold f a,.. With this aer being devoted to the analysis of basins in a dry climate, we assumed for both thresholds a caacitive behavior ( L = = 0.5) assuming that a dry state characterizes the antecedent soil moisture conditions of both mechanisms. Such an assumtion was confirmed by [40] where the analysis was extended to several other basins of Southern Italy. Table 2. Estimated arameter values of the IF and TCIF models. Site q o (m 3 /s) E[i, ] (mm/h) (h) q Caraelle at Caraelle Bradano at Ponte Colonna Site f (mm/h) r L L f,l (mm/h) f, (mm/h) r L r Caraelle at Caraelle Bradano at Ponte Colonna

9 Water 200, For the remaining arameters of the IF model, we derived from Equation (6) the relationshi f Λ E[i,τ ] log (5) Λ q roviding f as a function of q, once the a riori estimates of,, E[i, ] are available, and we carried out an at-site evaluation rocedure of arameters q and r based on minimizing a negative log lielihood function evaluated on MFS. The rocedure was erformed by exloring the domain of feasible arameter values on a regular grid. In articular, the grid-dataset values was reared with r ranging from 0.0 to, with ste 0.0, and q from 0. to, with ste 0.. For each test basin, the best arameters dataset was chosen as the one minimizing a negative log lielihood function of the observed samle of annual maximum floods. nalogously, for the remainder arameters of the TCIF model, we used Equations (2) in order to obtain the exression of f,l and f,, using the a riori estimates of,, E[i, ]: f,l E[i,τ ] Λ log ΛL Λ and f, Λ E[i,τ ] log Λ (6) then, arameters L,, r L, r were calibrated adoting a maximum lielihood rocedure. In this case, in order to avoid the cumbersome exloration of the entire arameters domain on a regular grid, we used as initial guess values r = r L = r, = 0 and L = q. These values corresond to the hyothesis that TCIF distribution collases into IF distribution. Starting from the initial guess values, the maximum lielihood was found exloring the four-dimensional sace of arameters r L, r,, L by following the direction of maximum sloe of the negative log-lielihood function. In articular, consistently with definitions of the L-tye and -tye events we assumed that r L may only decrease from r to 0, while r ranges from r to, both with ste 0.0; also L ranges from q to and varies, accordingly (i.e., always resecting the condition L + ) from 0 to. The selected arameter-datasets are reorted in Table 2, where it is worth noting that the lower threshold f a,l of the TCIF model corresonds to the single threshold of the IF model. Slight differences are found with resect to the estimates of r and f reorted in Fiorentino and Iacobellis [39], which used regional estimates of q for evaluating f, and only r was calibrated by equating an aroximate exression of the IF first order moment to the observed mean annual flood. In Figure 2 we dislay the TCIF-CDF, the IF-CDF and the Weibull lotting ositions of the MFS of test basins in a Gumbel robability lot. From the visual comarison, one may note the difference between the two distributions which is slight for the semi-arid Caraelle basin while it is more ronounced in the case of the dry-subhumid Bradano basin, the latter being characterized by a more sewed distribution (e.g., [4,42]). In order to obectively assess the right model for each test basin, we investigated, as reorted in next section, the use of statistical techniques for the best model selection.

10 Gumbel reduced variate Gumbel reduced variate Water 200, Figure 2. Comarison between TCIF and IF CDFs and the Weibull lotting ositions of the annual maximum flood series: (a) Caraelle at Caraelle and (b) Bradano at Ponte Colonna. 6 5 Caraelle at Caraelle TCIF Plotting Position IF 4 3 (a) q (m 3 /s) Bradano at Ponte Colonna 5 TCIF Plotting Position IF (b) q (m 3 /s) 4. Model Selection Procedure The selection of the best statistical model, for a given samle series, is often based on inference tests deending, for examle, on the significance level chosen. In statistical hydrology they are often used in the articular case of nested distributions (e.g., the Gumbel, Frechet and GEV distributions). With the increase of comuter caabilities many methods have been roosed and develoed for model selection on cross-validation techniques, also in fields other than hydrology (e.g., [43,44]). In this wor, five different methods for model selection criteria are used with the aim of finding the most aroriate model structure between the IF and TCIF models. In articular, we first considered the log-lielihood criterion (LLC), which does not use any enalty term, the aie information criterion (IC) roosed by aie [45], who introduced the rincile of

11 Water 200, maximum entroy for model selection, and the Bayesian information criterion (BIC) roosed by Schwarz [46]. Both IC and BIC adot a enalty term accounting for the number of model arameters. The fourth method is the log-lielihood ratio test (LLR). Finally, the generalization criterion roosed by Busemeyer and Wang [36]. Mosier [43] was the first to resent a clear definition of the cross-validation criterion. Others have shown that the cross-validation criterion is asymtotically equivalent to the IC [47,48]. Nevertheless, the generalization criterion is based on a riori redictions made before observing the data. Thus it obectively assesses the model caability to redict states different from those observed and used for model calibration. The LLC for the th oerational model is simly evaluated as: n LLC 2ln[L (, x)] (7) where L (, x) g ( x, ) is the lielihood function, evaluated at the oint =, corresonding i i to the maximum lielihood estimator of the arameter vector [49]. The IC for the th oerational model may be comuted as: n IC 2ln[ L (, x)] 2 (8) where L (, x) g ( x, ) is the lielihood function, evaluated at the oint =, corresonding i i to the maximum lielihood estimator of the arameter vector [49], and is the number of estimated arameters of the th oerational model. By analyzing Equation (8), one can see that the first term on the right-hand side tends to decrease as more arameters are added to the aroximating model, while the second term tends to increase. Note that the enalty term tends to select simler models under the rincile of arsimony. Sugiura [50] derived a second-order variant of IC, called ICc, valid in the case of a small samle size n, with resect to the number of estimated arameters (n/ < 40): n ICc 2ln[ L (, x)] 2 (9) n BIC for the th oerational model is evaluated as follows: BIC ) 2ln[ L (, x)] ln( n (20) In ractical alication one selects the model with the minimum value of the discreancy measure LLC, IC or BIC. Table 3 shows the results of these three selection criteria alied to IF and TCIF. The two criteria (ICc and BIC) accounting for model arsimony suggests reection of the hyothesis that the FMS of the two river basins investigated are extracted from the TCIF model which has more arameters, while the negative log-lielihood selection criteria chooses, always, the TCIF model.

12 Water 200, Table 3. lication of model selection techniques. Site LLC ICc BIC LLR IF TCIF IF TCIF IF TCIF ( IF, TCIF ) Caraelle at Caraelle Bradano at Ponte Colonna The log-lielihood ratio test, which is secifically suited for comarisons among nested models, introduces the log-lielihood ratio statistic for two different models i and : LLR i, 2ln[L ( i, x) / L (, x)] = LLC i LLC i whose robability distribution can be aroximated by a chi-square ( ) distribution with ( i ) degrees of freedom. In this test, if LLR(i,), where model i is nested within model, exceeds a cutoff ( ), which deends on the test significance level, then the null hyothesis that imlies no significant 2 model differences ( 0 ), is reected. The results in Table 3 show that even the log-lielihood ratio test, which accounts for model arsimony by mean of the chi-square degrees of freedom, always select the IF distribution as being the LLR values well below the cutoff value = 5.99 obtained for significance level = 0.05 and degrees of freedom = 2. These results ose a serious question which, for the case of theoretically derived distributions, also has imlications on the individuation of the main rocesses that affect runoff generation. Is it suitable, in this case, to recourse to such selection criteria accounting for model arsimony? n obective answer was found by introducing the generalization criterion which is based on a riori redictions made by mean of a slit-samle rocedure. More recisely, the rocedure is structured as follows: () For each river basin, the samle of MFS is divided into two sub-samles statistically indeendent of sizes N and N 2 (where the total number N = N + N 2 ). (2) During the first calibration stage, best fitting arameter estimates IF and TCIF are obtained from the sub-samle N, resectively by selecting arameters of the IF and the TCIF models that minimize the discreancy (D) evaluated as negative log-lielihood function ( 2ln[ L(, N )] and 2ln[ L(, N )]) alied DIF,N IF 2 DTCIF,N TCIF to the observed sub-samle N. (3) During the second validation stage the reviously estimated arameters, IF and TCIF, are exloited to comare the two models in terms of their redictive erformance calculating the negative log-lielihood function (D IF,N2 and D TCIF,N2 ) with resect to the second indeendent samle N 2. (4) The difference between the redictive erformance of the two models ( = D TCIF,N2 D IF,N2 ) is calculated. (5) Stes 4 are reeated 00 times by randomly selecting different sub-samles to roduce mean and standard deviation ( ) and )) of the factor. If ) > 0 the IF model is selected and, otherwise, TCIF. The alication of this criterion to test basins selected the TCIF model as a better model for both as reorted in Table 4 where ) and ) are shown. These results suggest that when cross-validation techniques cannot be alied due to small samle size, the log-lielihood criterion, without any enalty factor accounting for model arsimony, should be referred when dealing with such nested distributions.

13 Water 200, 2 25 Table 4. lication of generalization criterion for model selection. Caraelle at Caraelle Bradano at Ponte Colonna N N 2 ) ) N N 2 ) ) Conclusions n imrovement in flood rediction is exected by selecting the most aroriate model scheme for reresenting real rocesses and, consequently, for detecting dominant mechanisms resonsible for non-linearity in flood distributions. comarison between the IF and TCIF models was made using data from two river basins in semi-arid and dry-subhumid climate. The TCIF model, which generalizes the IF model, introduces two different threshold mechanisms as resonsible for ordinary and extraordinary events, in analogy with the theory of the TCEV distribution [26]: The first one is characterized by frequent occurrences and lower average of exceedances (L-tye), the second one includes rare events and higher average of exceedances (-tye). Results of this wor show that two different mechanisms of runoff generation may be observed in dry-subhumid and semi-arid climates. In fact, while it is already recognized in the international hydrologic literature that non linear effects in flood frequency distribution may deend on the alternation of infiltration and saturation excess (e.g., [22]), it is less common to observe those different runoff mechanisms in dry climate. We have shown that a high-frequency behavior may be rovided by a storage threshold affecting smaller areas of the basin while the low-frequency comonent may arise when a higher storage threshold is exceeded in large areas of a basin. Such imortant results are also comatible with the occurrence of a saturation excess rocess at the lower comonent being sometimes modeled as a storage rocess too (e.g., [22,29]). We introduced a novel and faster rocedure for the estimation of arameters of the TCIF distribution. This is based on the individuation of the maximum-lielihood arameter dataset and is lined to the estimated arameters of the IF distribution, which are used as initial guess while the exloration of the lielihood function is erformed resecting the hysical constrains r L r, r r and L + on the domain of arameter values. Such a rocedure is of general interest and may be alied at any climate in any basin were floods are tyically rainfall-driven. Finally, interesting results were obtained in the framewor of model selection, in the case of nested distributions IF and TCIF. The comarison was made through the use of selection criteria able to account for the more aroriate model structure. We observed that the selection criteria based on the log-lielihood function, without enalty term, tends to refer the TCIF model even in the Caraelle basin that does not dislay a high non-linearity. On the other hand, criteria accounting for a enalty factor related to the number of arameters, such as the IC, the BIC and the chi-square test, systematically select the IF distribution. This haens also for the dry-subhumid Bradano basin, notwithstanding the TCIF distribution rovides a clear better fit to the right tail of the observed distribution (see Figure 2). Then, we referred to the generalized criterion based on a slit-samle rocedure that obectively tests the a riori redictive caability of the model. In both test basins the generalized criterion selected the TCIF distribution, thus roviding significant suort for its structural

14 Water 200, validity and further reinforcing its concetual reresentation of the hydrologic rocesses dominant at basin scale. Such results confirm what was already stated by Busemeyer and Wang [36]: both the aie (IC) information criterion and the Bayesian information criterion (BIC), as well as other methods including a enalty factor, should not be considered aroriate for selecting between nested distributions. They also observed that it is well nown that the chi-square criterion tends to ic the oversimlified model, with small samle sizes that suffer from a lac of statistical ower, and it tends to ic the overly comlex model in large samle sizes that enoy extremely high statistical ower. Only the generalized criterion, which is based on cross-validation, erforms a riori redictions and rovides an obective assessment of the model redictive ability. Obviously, the generalized criterion is alicable only to MFS with minimum length of 30 years, with sub-samles of 5 years used for calibration and validation. Thus, the use of a log-lielihood criterion, without any enalty factor accounting for model arsimony, is recommended when dealing with nested distributions and small samle size. Further investigation on techniques for model selection with diagnostic ability (i.e., able to evaluate model structure validity) is the obect of ongoing research and future develoments by the authors. cnowledgements The uthors are grateful to two anonymous reviewers for their aroriate and useful comments. The wor was realized with suort of PRIN Cubist CoFin2007 of the MIUR (Italian Ministry of Instruction, University and Research). References and Notes. Sivaalan, M.; Taeuchi, K.; Frans, S.W.; Guta, V.K.; Karambiri,.; Lashmi, V.; Liang, X.; McDonnell, J.J.; Mendiondo, E.M.; O Connell, P.E.; Oi, T.; Pomeroy, J.W.; Schertzer, D.; Uhlenbroo, S.; Zehe, E. IS Decade on Predictions in Ungauged Basins (PUB), : Shaing an exciting future for the hydrological sciences. ydrol. Sci. J. 2003, 48, bdulla F..; Lettenmaier, D.P. Develoment of regional arameter estimation equations for a macro scale hydrologic model. J. ydrol. 997, 97, Seibert, J. Regionalisation of arameters of a concetual rainfall-runoff model. gric. For. Meteorol. 999, 98-99, Fernandez, W.; Vogel R.M.; Sanarasubramanian,. Regional calibration of a watershed model. ydrol. Sci. J. 2000, 45, undecha, Y.; Bardossy,. Modeling of the effect of land use changes on the runoff generation of a river basin through arameter regionalization of a watershed model. J. ydrol. 2004, 292, Merz, B.; Blöschl, G. Regionalisation of watershed model arameters. J. ydrol. 2004, 287, Wagener, T.; Wheater,.S. Parameter estimation and regionalization for continuous rainfall-runoff models including uncertainty. J. ydrol. 2006, 320, euvelmans, G.; Muys, B.; Feyen, J. Regionalisation of the arameters of a hydrological model: Comarison of linear regression models with artificial neural nets. J. ydrol. 2006, 39,

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17 Water 200, Stone, M. On asymtotic equivalence of choice of model by cross-validation and aie s criterion. J. Roy. Stat. Soc. Ser. B 977, 39, Browne, M.; Cudec W. Single Samle cross-validation indices for covariance structures. Multivariate Behav. Res. 989, 24, Linhart,.; Zucchini, W. Model Selection; John Wiley: oboen, NJ, US, Sugiura, N. Further analysis of the data by aie s information criterion and the finite corrections. Commun. Stat. Theory Methods 978, 7, endix Main features of the IF model are summarized in the following oints: both random variables a and u a are controlled by: (i) rainfall intensity, duration and areal extension; (ii) runoff concentration; (iii) hydrological losses. The runoff ea er unit area, u a, is linearly deendent on the areal net rainfall intensity in a time interval equal to a with a constant routing factor Then, the robability distribution of u a, can be derived from the robability distribution of rainfall intensity i a,t conditional on a duration equal to a, lag-time of a. The areal rainfall intensity i a,t is assumed Weibull distributed with two arameters a, and. The mean areal rainfall intensity is: i Ei / E a, a, a, / (2) The routing factor is a ey model arameter which in reality aeared very stable. In fact,, it was found to vary in a narrow range (0.6, 0.8) with an average value close to 0.7 which has been used in all the alications of the IF and TCIF models made since they were introduced. The lag-time a scales with a according to a ower law with exonent 0.5. The variable contributing area a follows a mixed distribution with a continuous art which is a two arameter gamma distribution, valid for 0 < a < and a discrete robability P g a a a ex a P (22) P rob a /, a a ex da (23) The gamma function arises as the distribution of the sum of stochastic (indeendent) variables exonentially distributed with equal mean value Thus, being any flood ea due to the suerosition of flows coming from sub-basins whose exected number is equal to the number N of sub-basins of orton order immediately smaller than that of the whole basin, we identified to E[N. N tends to be invariant at any scale and assumes values ranging between 3 and 5 [50] with exected value close to 4 [5].

18 Water 200, The annual maximum floods arise from a comound Poisson rocess and the following relationshis hold for the flood ea q, the ea of direct streamflow Q, and the exceedance robability function of the ea of direct streamflow G Q (q): q = Q + q o, (24) q qo / a f a f a GQ ' q g a ex da, (25) 0 E[ ia, ] with base flow q o 200 by the authors; licensee MDPI, Basel, Switzerland. This article is an Oen ccess article distributed under the terms and conditions of the Creative Commons ttribution license (htt://creativecommons.org/licenses/by/3.0/).