Regional rainfall and flood frequency analysis for Sicily...

Hydrological Sciences -Journal- des Sciences Hydrologiques,40,l, February 1995 19

Regional rainfall and flood frequency analysis for Sicily using the two component extreme value distribution

M. CANNAROZZO Istituto di Idraulica, Université di Palermo, Viale de lie Scienze, 90128 Palermo, Italy

F. D'ASARO Dipartimento Tecnico-Economico, Università délia Basilicata, Via N, Sauro, 85100 Potenza, Italy

V. FERRO Istituto di Genio Rurale, Università di Reggio Calabria, Piazza San Francesco 4, 89061 Gallina di Reggio Calabria, Italy

Abstract This paper first presents a duration-dependent hypothesis regarding the parameters of the two component extreme value (TCEV) distribution and proposes a simple model to obtain the rainfall depth-duration relationship both at ungauged sites and at short record gauged sites in the Sicilian region. Then, by using the annual maximum peak flood and the annual maximum mean daily discharge recorded in Sicily, a TCEV hierarchical regional procedure is developed. An empirical estimation criterion which links the mean annual flood with the watershed area, and a modified rational formula in which a mean runoff coefficient is introduced, are proposed. Finally, two relationships are established for estimating the mean runoff coefficient, making use of the permeable watershed area and the wooded area.

Analyse régionales des pluies et des crues en Sicile grâce à la loi des valeurs extrêmes à deux composantes Résumé Ayant admis l'hypothèse d'une dépendance des paramètres de la loi des valeurs extrêmes à deux composantes à la durée considérée, les auteurs proposent un modèle simple permettant d'établir une relation hauteur-durée aussi bien dans les zones dépourvues de stations pluvio-métriques que dans celles qui en sont pourvues mais qui ne disposent que de brèves séries d'observations. Ensuite, utilisant les chroniques des maximums des débits de pointe annuels et des maximums des débits moyens journaliers, ils appliquent la loi des valeurs extrêmes à deux composantes à l'échelle régionale. Un critère empirique d'estimation reliant la crue moyenne annuelle et l'aire du bassin est alors proposé, ainsi qu'une formule rationnelle, modifiée par l'introduction d'un coefficient de débit moyen. Les auteurs présentent enfin deux relations prenant en compte la surface perméable et de la couverture forestière du bassin afin d'estimer le coefficient moyen de débit.

Open for discussion until I August 1995

20 M. Cannarozzo et al.

INTRODUCTION

The aim of hydrological frequency analysis is to interpret a past sample of the hydrological variable in terms of future probabilities of occurrence. In other words, by using a sample of the annual maximum values of a given hydrological variable x, having a sample size N, the aim is to estimate xT, the value of the variable for a given return period T (statistical inference). This process needs a choice of the theoretical distribution function to be used and the estimation of its parameters.

In the past, the procedure followed was to check and compare the suitability of several candidate distributions. The choice is based on an a posteriori examination of the goodness of fit of the theoretical cumulative distribution function (CDF) to the empirical one in the range of the observations. This procedure has a high uncertainty if the size of the at-site sample is short and it can lead to inconsistent results, i.e. different theoretical CDFs can be fitted to sample series of neighbouring gauges (recording rain gauges or stream-gauges) (Versace et al., 1989). Experience shows that when the tail of the distribution is of interest, as for extreme rainfall and flood analysis, quantités estimated with different theoretical models can differ substantially, even if each theoretical CDF fits the data well in the central area (for T < 10 years) (Landwehr et al., 1978, 1980). The quantile estimate xT obtained by using a single sample has always an high uncertainty when the return period T is much higher than the sample size N.

In order to reduce the uncertainty of the estimated parameters of the chosen theoretical distribution, i.e. to improve at-site estimates based on limited data and to estimate xT for ungauged sites, a regional estimation technique is needed (Wiltshire, 1986; Cunnane, 1988). Regionalization is a tool to extend the length of the historical samples and to reduce time sampling errors. Regionalization introduces errors due to space disturbance (Matalas & Gilroy, 1968) and inter-station correlation (Stedinger, 1983) leads to estimates of the hydrological variable which are less accurate then they would be if the samples were independent.

The time sampling variability increases with the order m of the moments, while the ratio between the space disturbance and the time sampling variability decreases with m (Fiorentino et al., 1987; Rossi & Villani, 1992). Stedinger (1983) showed that the influence of the inter-station correlation decreases with m.

Regionalization has to be applied for the statistical analysis of extreme hydrological events because of the large influence that the higher moments (skewness) exert on the shape of the tail of the distribution (Wallis et al., 1974).

In this paper, after a brief review of the two component extreme value (TCEV) distribution (Rossi et al., 1984) and its hierarchical regionalization procedure (Versace et al., 1989), a regional flood analysis for the Sicilian region is presented. This analysis also uses the results of the regionalization

Regional rainfall and flood frequency analysis for Sicily 21

technique applied to the annual maximum rainfall with fixed duration (1, 3, 6, 12 and 24 h). A duration dependent hypothesis for the TCEV parameters is shown and a simple model is proposed for obtaining the rainfall depth-duration (RDD) relationship both at ungauged sites and at gauged sites with short records. The regional flood analysis also provides three sub-regional growth curves and two different criteria for estimating the index flood by using regression or physically based models.

BRIEF REVIEW OF THE TCEV DISTRIBUTION AND ITS HIERARCHICAL REGIONALIZATION TECHNIQUE

The choice of a theoretical CDF should be made taking into account certain criteria: (a) information about the physical phenomena (rainfall, flood) can be used

to establish the structure of the model (theoretical basis); (b) the theoretical CDF has to be able to reproduce the statistical charac

teristics (skewness, coefficient of variation, mean) of the observed samples (descriptive ability);

(c) the model parameters have to have physical meaning in order to make the regionalization easy (physical basis); and

(d) the T-year variable estimate, xT, must be efficient (minimum mean square error) and robust (resistant to departures from the hypothesis) (predictive ability). According to these criteria the Italian National Research Group for the

Prevention of Hydro-Geological Disasters proposed the use of the TCEV distribution to analyse annual rainfall series (ARF) and annual flood series (AFS) measured in Italy. The TCEV distribution takes into account that one or more annual maximum values of the observed hydrological variable x stand much higher than the bulk of the remaining data. Two components are distinguishable: the basic component, which takes into account the usual values, and the outlying component which takes into account the extreme values. This choice is justified if maximum rainfalls and floods are due to storms with different meteorological characteristics (Rossi & Villani, 1992).

The CDF of the TCEV distribution Fix) is the following:

-Xjexp -x

-X2exp -x

A in which X1; X2 are the shape parameters and 6U 62 are the scale parameters, respectively, of the basic (component 1) and outlying (component 2) components.

Since X,- (i = 1, 2) is the mean number of events (maximum rainfall, flood) which belong to each component i, while 0;- represents the at-site central value of the hydrological variable, X! > X2 and 0l < 02. In other words the


basic component is characterized by a high number of events and by values of the hydrological variable less than those corresponding to the outlying component.

Introducing the standardized variable y = (x/6^ - In X1; equation (1) assumes the following form:

F(y) = exp -exp(-v) -A* exp -y e*

(2)

in which 9* = 62/dl and A* = \2/\xl/e*.

The A* and 9* parameters therefore synthetically represent, at the regional scale, the relationship between the basic and the outlying component.

In order to develop the regionalization procedure, the dimensionless variable x', equal to the ratio between x and the mean value [x of the TCEV distribution, is introduced. The CDF of the x' variable (growth curve) is the following:

F(x') = expl-XjCexpar^'-A*

in which:

a = fi/ei = 0.5772+lnX!-£

n i ; e * exp a 9*

(-iy'A*-T(//0*)

} (3)

(4) f-

According to equation (4) a is dependent on A*, 9* and X, (Beran et al., 1986) and only these three parameters are necessary to define F(x').

The regionalization procedure is hierarchical and broken down into three sequential levels.

At the first level the parameters A* and 9* are assumed constant over the whole region examined and according to equation (2), the standardized variable, y, is identically distributed over the region. The spatial homogeneity hypothesis for the TCEV parameters A* and 9* also implies the hypothesis of statistical constancy for the skewness coefficient G (Beran et al., 1986).

At the second level the region is divided in smaller areas (homogeneous sub-regions) in which the mean annual number of events coming from the basic component, Xj, is constant. At this level of the regional analysis the standardized variable x' is identically distributed in a sub-region, i.e. the distribution function F(x') is completely defined in each sub-region.

At the third level regression models or other methods are used to achieve estimates of the mean value, ji, at ungauged sites or at gauged ones with short records.

For each regionalization level the assumed hypothesis (for example A* = constant and 9* = constant) has to be verified. In other words, the ability to reproduce the statistical characteristics of the observed series in the region (for example the skewness) has to be verified. The statistical behaviour of the regional model is verified by comparing the theoretical regional distribution with the observed regional distribution of the statistic considered (skewness,


coefficient of variation) (Cunnane, 1989). According to Matalas & Gilroy (1968) this is equivalent to verifying that the spatial disturbance errors are negligible compared to the sampling errors (Rossi & Villani, 1992).

In order to verify the spatial homogeneity hypothesis of the regional (A* and 9*) or sub-regional (Xj) parameters, the hierarchical regional procedure has to be performed. In other words, the homogeneity verifications need the parameter estimates and can only be carried out at the end of each level of the regionalization procedure by Monte Carlo techniques.

REGIONAL RAINFALL ANALYSIS

Rainfall data

The rainfall data used in this research are the annual maximum rainfalls with 1, 3, 6, 12 and 24 h duration published by the Italian Hydrographie Service. The rainfall data used were recorded at 172 recording raingauges (one rain-gauge per 150 km2) (Fig. 1) in the period 1928-1981. The sample size N varies from 10 to 45 years and the mean sample size equals 23 years. The statistics of the annual maximum daily rainfall (Ferrari, 1985) are also used.

Fig. 1 Sicilian recording raingauges.

First level

At the first level of the hierarchical regionalization procedure, Sicily is considered to be a homogeneous region with regard to the skewness coefficient.


Figure 2 shows, for each duration t (1, 3, 6, 12 and 24 h), the empirical cumulative distribution function (CDF) for the skewness of the rainfall samples. The Figure shows that each CDF is clearly distinguished from the others and that the skewness values generally increase with increasing duration t. This empirical result supports the duration-dependent (DD) hypothesis for the TCEV parameters A* and 9*. The maximum likelihood (ML) estimates of the parameters A* and 6*, listed in Table 1, confirm the DD hypothesis which other authors (Fiorentino et al., 1984) discarded because few recording raingauges were installed in their region. For Sicily, the DD hypothesis does not allow one to assume, as an estimate of the A* and 6* parameters, the values obtained from the regional analysis of the mean daily rainfalls. Figure 3(a) shows that the relationship 0*(f) between the regional parameter G* and duration t is represented by a straight line:

9 * = 2.1+0.0284? (5>

while the relationship A*(t) has the following mathematical shape (Fig. 3(b)):

A* = 0.467 f0301 (6)

0 1 2 3 4 G 5

Fig. 2 Comparisons between the empirical distribution of the skewness of hourly (N = 35) and daily (N = 58) samples.

" [-"

o eq. (6!

y*^ -i A

• ,

^^^"e 3.(5)

^^^£L

0 5 10 15 ?0 tSW 25 1 10 { ( M 100

Fig. 3 Relationships between (a) ©*; (b) A* and duration t.


Table 1 Regional parameters A and 6 for each duration, t

f(h)

A*

e*

3

0.6100

2.1765

6

0.8040

2.2310

12

1.1728

2.5264

24

1.0770

2.7555

Table 1 does not list the ML estimates of A* and O* for t = 1 h because for this duration the iterative estimation procedure proposed by Fiorentino et al. (1987) does not converge. The ML estimation procedure failed to converge because either the historical sequences have a different sample size or they are characterized from a small number of outliers. Since for the Sicilian recording raingauges the sample size does not change with t, the failure of the ML estimation procedure to converge is probably due to the small number of outliers. The estimated regional parameters, A* and 9*, for the mean daily rainfalls (Ferrari, 1985) are:

A* = 2.6319 (7)

Q* = 0.4551 ( 8 )

A comparison between equations (7) and (8) and the 24 h values listed in Table 1 shows a remarkable difference. In Sicily daily rainfall and 24 h rainfall are generally due to the same rainfall storms and therefore the empirical skew-ness CDF of the 24 h rainfalls would be coincident with the empirical skewness CDF of the mean daily rainfall. Figure 2 shows that this last distribution is clearly distinguished from the empirical skewness CDF of the rainfalls of fixed duration. This result can be dependent on the sample size of the fixed duration rainfall (N = 35 years) which is shorter than the daily rainfall (N = 58 years) sample size. In order to verify this hypothesis, 1000 samples of daily rainfall

0 1 2 3 4 G 5

Fig. 4 Comparisons between empirical skewness CDF of t hour rainfall and skewness CDF of daily rainfall generated by a TCEV distribution.


were generated by a Monte Carlo technique, with a sample size of 35 years distributed according to a TCEV distribution with regional parameter estimates given by equations (7) and (8). Figure 4 shows that the skewness CDF of the generated samples coincides with the empirical skewness CDF of the 24 h rainfall historical sequences. Therefore the difference in the parameter estimate between the daily rainfall and the 24 h rainfall can be attributed to the different sample sizes (58-35 years). The difference between the skewness CDF of the daily rainfall and the skewness CDF of the t h rainfall, with t = 1, 3, 6 and 12 h, confirms the DD hypothesis.

In order to take into account the hydrological information of the daily rainfall samples with a greater sample size, an indirect estimate procedure is proposed based on the following two hypotheses: (a) the A* and 6* estimates of the 24 h rainfall are respectively equal to A*

and Q*; and (b) the variability of A* and 6* with duration is given from equations (5)

and (6). According to these hypotheses the following relationships can be

deduced:

6* = 1.95+0.0284f (9)

A* = 0.175r0-301 (10)

Monte Carlo experiments (Fig. 5) show that the TCEV distribution, having regional parameter values chosen according to equations (9) and (10), reproduces the empirical CDF of the skewness coefficient. Figure 5(a) shows the comparison between the skewness CDF of historical and Monte Carlo generated samples for the duration t = 1 h. This comparison verifies the goodness of the proposed indirect estimation procedure because equations (9) and (10) are deduced without the A* and 9* estimates for 1 h duration. Figure 6 shows comparisons between the empirical CDF of the standardized variable y and the theoretical one (equation (2)) for which the regional parameters are estimated by equations (7) and (8) or by equations (9) and (10). Figures 6(c) and 6(d) confirm that the agreement between the empirical and the theoretical F(y) is better using the DD hypothesis.

Fig. 5 Comparisons, for t = 1 h and 6 h, between the skewness CDF of historical sequences and of samples generated by a TCEV distribution.


1 ' F ( y )

0.9

| t z ) h f

< 8

$

4>

/

(a)

F(y) |t=1h

(0 y 16 " y

F(y) I

| U 6 h r |

/

À

'

?P\

(b)

F ( y )

03

I N 6 h r |

/is /""

, r - ~ - '

(d)

^ * b 3 '0 12 1* Hi 18 v ^ 2 -1 6 6 10 12 1* y 16

Fig. 6 Comparisons between the empirical CDF of y and the theoretical one with regional parameters estimated by equations (7) and (8), (a) and (b); or by equations (9) and (10), (c) and (d).

In conclusion, assuming Sicily is a homogeneous region and accepting the DD hypothesis of the regional parameters (equations (9) and (10)), the CDF

of the y variable is the following:

Fiy) = exp - exp( -v ) -0 .175r J U i exp -y 1.95+0.0284r

(11)

Second level

At the second level the region is divided into smaller areas, named homogeneous sub-regions, in which the \ parameter is assumed constant. In order to establish the pluviométrie sub-regions the coefficient of variation of the basic component, CV\, can be used. In fact the relationship between CV\ and Xj is the following:

CV1 = 0.577 logXt+0.251

(12)

The CV1 statistic is used because it is less variable (referred to its mean value) than Xj. For grouping recording raingauges, for each duration t, a cluster analysis technique was used. In particular each site was characterized by its geographical coordinates (x1, x2) and the CV1 value; for each duration the rain-gauges having the minimum distance in the three-dimensional space (xh x2, CVl) were grouped. For each duration the same three sub-regions, named as


A, B and C in Fig. 7, were obtained. The homogeneous hypothesis of a given sub-region was accepted if the coefficient of variation CDF of the historical samples was coincident with the CV\ CDF of Monte Carlo generated samples. For each duration t, generating samples by a Monte Carlo technique, a Kx

parameter was used equal to the mean value of the \ parameters of all the recording raingauges located in each sub-region (Table 2).

Fig. 7 Sicilian pluviométrie sub-regions.

Table 2 A, values for each pluviométrie sub-region and for each duration, t

Sub-region

A

B

C

f(h)

1

14.87

12.92

12.15

3

19.05

14.22

12.93

6

21.37

16.52

14.24

12

26.27

20.72

15.34

24

32.50

21.76

16.25

The Aj parameters are dependent on the duration t according to the following relationships (Fig. 8):

Sub-region A:

Sub-region B:

Sub-region C:

A, = 14.55 t024i9

Ax = 12.40?01802

Aj = 11.96 f ' 0.0960

(13a)

(13b)

(13c)


35

30

25

20

15

10

• A

-O B

A C

/C ft-

^

-tr——

o

•

0 5 10 15

Fig. 8 Relationships between A! and duration t.

20 25

t(h)

Since for each sub-region A! is dependent only on the duration t (see equations (13)), the x' variable is identically distributed according to equation (3). The a parameter, defined by equation (4), is also dependent on duration t according to these rough relationships:

Sub-region A:

Sub-region B:

a = 3.5208 f0-1034

a = 3.3536 f00945

a = 3.3081100765

(14a)

(14b)

(14c) Sub-region C:

Since the distribution function F(x') is implicit as regards x' for each sub-region and T > 10 years the following rough relationships are proposed:

Sub-region A:

xtT = 0.5391 -0.001635? + (0 .00022m 2 +0.00 im + 0.9966)logr (15a)

Sub-region B:

vr,r 0.5135 -0.002264r + (0.0001980f2+0.00329f+ 1.0508)logr (15b)

Sub-region C:

xtT = 0.5015-0.003516f + (0.0003720r2+0.00102z' + 1.1014)logr (15c)

in which x'tT is the dimensionless rainfall of duration t and return period T. The rainfall depth-duration relationship is obtained by multiplying x\ T

with the mean, nR, of the TCEV distribution:

*t,T ~ '^•tJ'V'Ky' (16)

in which xuT is the rainfall depth of fixed duration t and return period T and fiR(f) is the relationship between the mean and duration.


Fig. 9 Sicilian iso-a map.

Fig. 10 Sicilian iso-« map.

Third level

This last step of the hierarchical procedure aims to determine a regional criterion to estimate fxR for ungauged sites or for sites with short records. For the Sicilian recording raingauges the theoretical ixR values were calculated by equation (4) in which 8X parameter values are ML estimates related to known


regional (A*, 8*) and sub-regional (A{) parameters. Since the theoretical ixR

values are equal to the empirical one, mc, the third regionalization level was developed via mc values. For each recording raingauge, mc depends on duration t according to the relationship:

m„ at" (17)

in which a and n are constants. In Figs 9 and 10 the iso-a and iso-n maps (maps in which the contour lines with constant a or n values are plotted) are represented. The two maps allow estimation of the two parameters at ungauged sites and also at gauged ones with short records.

REGIONAL FLOOD ANALYSIS

Flood data

The flood data used in this research are the annual maximum peak flood (AMPF) recorded at 27 streamgauges in the period 1936-1982 (Fig. 11). The sample size N is variable from 10 to 46 years and the mean sample size is 22 years.

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The annual maximum mean daily discharge (AMDD) data, published by the Italian Hydrographie Service and measured at 33 streamgauges (Fig. 11), were also used. The AMDD sequences have sample sizes varying from 10 to 53 years and the mean sample size is 22 years.

First level

At the first level, Sicily was considered homogeneous with regard to skewness. At first the ability of TCEV, LN2 and EVl distributions to reproduce the observed distribution function of the skewness coefficient of AMPF samples was verified. The descriptive ability of the LN2 distribution was verified because it is a simple model which has often been applied in Italy (Lazzari, 1967; Cicioni et al, 191 A) and in Sicily (Modica et al, 1988, 1990). The EVl distribution was also considered because it is a physically based probabilistic model and it is probably the most widely applied distribution for the analysis of extreme events. Figures 12 and 13 compare the observed CDF of the skewness of non-overlapping sequences having a fixed size (N = 10, 20 years) with a theoretical CDF. The theoretical skewness CDF was obtained by Monte Carlo experiments: 1000 sequences of fixed size (10, 20 years) were generated,

Fig. 12 Comparisons between the observed CDF of non-overlapping sequences with N = 10 and a theoretical CDF.

Fig. 13 Comparisons between the observed CDF of non-overlapping sequences with N = 20 and a theoretical CDF.


distributed according to a theoretical law (LN2, EV1, TCEV). In particular Figs 12(a) and 13(a) compare the empirical skewness CDF of the logarithms of the AMPF values with the skewness CDF of generated sequences of the same variable. Figures 12 and 13 show that the TCEV distribution is the best law among the candidate distributions in being able to reproduce the observed CDF of skewness. Figure 14 compares the skewness CDF of 27 samples, having the same sample size, for which the AMPF and AMDD sequences were available. Figure 14 shows that the two CDFs agree and supports the hypothesis of using the Sicilian AMDD sequences, as these have a larger sample size than the AMPF sequences, for developing a regional flood analysis. The ML estimates of the parameters A* and 6* are the following:

A* = 0.8320 (18)

9* = 3.3110 (19)

Figure 15 shows the ability of the TCEV distribution, with ML estimated parameters (equations (18) and (19)), to describe the empirical CDF of the standardized variable y.

P(G)

02

0

O

$

*

À t

•

o r>

• AMDD

0 AMPF

Fig. 14 Comparison of the skewness CDF for the AMPF and AMDD sequences.

Ply)

0.92 -

0.9

O ' Jo^

Jieoretieal

° empirical

| 1

—

6 8 10 ' 2 14 16 13 20 22 24

Fig. 15 Comparison between the empirical CDF of y and the theoretical one with regional parameters estimated by equations (18) and (19).


Second level

In dividing Sicily into homogeneous hydrometric sub-regions, the following criteria were used: (a) grouping basins by a purely geographical criterion; (b) grouping basins by a morphological criterion (Wiltshire, 1985, 1986); (c) grouping basins by cluster analysis; and (d) grouping basins by using the hydrological information obtained by a

regional rainfall analysis (Versace et al., 1989). In particular, the geographical criterion was applied either considering

Sicily as a homogeneous region in which the \ parameter was assumed constant, or assuming three homogeneous sub-regions coincident with the three sides of the island.

The cluster analysis was carried out characterizing the outlet of each basin by its geographical coordinates and the CV1 values.

The criterion (d) is based on the hypothesis that the pluviométrie sub-regions coincide with the hydrometric ones, i.e. for each sub-region the mean annual number of rainfall and flood events coming from the basic component is constant. This statistical hypothesis is also convincing from a physical point of view because for the Sicilian region the flood events are due to rainfall events. In each sub-region the rainfall Xj parameter, \lR, was greater than the flood Xj parameter, X1F. The difference between \lR and X1F increased with increasing aridness of the region considered. Gabriele et al. (1990) showed that if the rainfalls are distributed according to a Poisson law with a mean annual number of events, \lR, and broadly supposing that the basin permeability characteristics are uniform (i.e. the permeability is not spatially dependent) then the floods are also distributed according to a Poisson law with a X1F parameter which decreases with increasing aridness of the region.

Each criterion determined a sub-division which was examined by verifying the ability of the TCEV distribution to reproduce, in each hydro-metric sub-region, the empirical CDF of the CV\ coefficient. For Sicily the criterion (d) was the only one which verified the descriptive ability of the TCEV distribution.

Table 3 TCEV regional and sub-regional parameters for the AMPF and AMDD sequences

AMPF sequences AMDD sequences

_ _ _ _ o.8320

B* 2.7998 3.3110

Sub-region X, a X, a

A 6.6402 4.3636 9.6289 4.7415

B 3.7551 3.7936 6.7115 4.3805

C 3.1100 3.6051 4.1923 3.9099


Table 3 lists the A*, 9*, Aj and a parameters for the AMPF and AMDD sequences for each sub-region. Figure 16(a) and (b) plots the comparison between the empirical and theoretical growth curves for sub-region A. For the theoretical growth curve the AMPF parameters (i.e. the parameters estimated by the AMPF sequences) and the AMDD parameters were used. The empirical growth curve was always plotted using the AMPF discharge sequences. The comparison shows that the theoretical growth curve with parameters estimated by the AMDD sequences has the best agreement with the empirical growth curve. Since the distribution function F(x') is implicit for x', for each sub-region and T > 5 years the following rough relationships are proposed:

Sub-region A: xQT = 0.3232 +1.6171 logT

Sub-region B: xQT = 0.2670 + 1.7503logT

(20a)

(20b)

Sub-region C: xQT = 0.1785+ 1.961 l logr (20c)

in which XQT is the dimensionless flood with a return period of T years. Since for the Sicilian streamgauges the theoretical mean values, y.Q, are equal to the empirical one, mQ, QT, the peak discharge having a return period T, can be evaluated from the relationship:

Qi XQJMQ XQJUQ

The third level of the hierarchical procedure aims to determine a regional criterion to estimate the mean, mQ, used as an index flood for ungauged sites or for sites with short records.

>(*')

J.Jb

0.9 i

/<

li r

/f f

^ ^

cal

£> empirical

(a) <*f

/

/

(b)

Fig. 16 Comparisons between empirical and theoretical growth curves (a) with AMPF parameters and (b) with AMDD parameters.

Estimating the index flood

At the third level of the regionalization procedure, the variability of the mean annual flood (MAF), mQ, was analysed. In order to determine a regional relationship for estimating the MAF, two different models can be used:


(a) a regression model; and (b) a conceptual model.

The regression model linked the MAF with measurable basin characteristics (Canuti & Moisello, 1980; Cao et al, 1988; Lazzari, 1967). The model structure was generally identified by an empirical approach based on a multiple regression analysis technique. This estimation criterion for the MAF was purely empirical and it led to relationships which are applicable for the region examined and for the data range used. For Sicily the best empirical relationship was the following (Fig. 17):

mQ = 3.0937S0'7351 (21)

in which 5 (km2) is the basin area. Equation (21) is characterized by a correlation coefficient of 0.9047 and a mean square error of 17795.

A conceptual model allows use of the a priori knowledge of the phenomena which affect the physical process. A conceptual model is also a more general relationship which can be used for regions different from the examined one. According to the rational formula (Kuichling, 1889; Ben-Zvi, 1989):

Q = ' K r 5 (22) T 3.6rc

in which 4> is the runoff coefficient, tc is the basin time of concentration, and xtjT is the rainfall depth corresponding to a duration tc and a return period of T years. From equation (22) 4> can be deduced:

= 3.6f cg r ^ 3.6tcXgTtiQ 2^,

The ratio XQ Tlx't T is a frequency factor, K{T,t), which depends only on both rainfall and flood growth curves. For the three Sicilian sub-regions K{T,t) could be calculated by equations (15) and (20). The frequency factor can be

[mS/s]

10 100 1000 ,. , 10000 S [Km2]

Fig. 17 Relationship between MAF and basin area for the Sicilian streamgauges.


also expressed by the relationship K = Km (27100)", in which Km is the K value corresponding to a return period of 100 years and a is a coefficient which assumes a value depending on the specific region (for Sicily a = 0.025).

Defining \p by:

yj, = 3-6t^Q (24) SfiR

equation (23) becomes:

4> = \pK(T,t) <25>

Therefore, according to equation (25), the runoff coefficient is the product of two factors. The first one, named mean runoff coefficient, is a scale factor which depends on the mean values of the hydrological variables. The second factor is a frequency factor which amplifies \p and depends on the duration and the return period of the rainfall event. The ip coefficient takes into account, by the ratio /xg//^, the characteristics (soil permeability, initial moisture) which affect the rainfall-runoff process.

Introducing equation (25) into equation (22) the following modified rational formula is obtained:

3.6,,

If \p is known, equation (24) may be used for estimating the MAF. In order to establish a criterion for evaluating ip, one introduces equation (17), in which fiR = mc, into equation (24):

^ = 3.6*>6 = 3 . 6 ^ ' " ( 2 ? )

Sat" Sa

In Table 4, for 12 Sicilian streamgauges, the basin area, the historical MAF and the a and n coefficients are listed.

For evaluating \[/ via equation (27), a tc estimation criterion is also necessary. The following relationship was used:

t = (3js (28>

This was obtained (see Appendix) by GIUH theory (Gupta & Waymire, 1983; Agnese & D'Asaro, 1990). For an "average basin", i.e. a basin having morphological features which occur frequently, (3 equals 0.35. Replacing equation (28) in equation (27) gives the following relationship:

j , = 3 - 6 ^ (29)

aS0.5(Un)


Table 4 Characteristic data of some Sicilian streamgauges

Streamgauge

Pollina ad Aquileia

Eleuterio a Risalaimi

Eleuterio a Lupo

Eleuterio a Rossella

V. Dell'Acqua a Serena

Nocella a Zucco

Fastaia a La Chinea

Delia a Pozzillo

Imera Mer. a Petralia

Girgia a Case Celso

Dittaino a Bozzetta

Crisa a Case Carella

N

10

16

33

14

20

21

10

20

11

20

18

22

S

(km2)

52

53

10

11

22

57

33

139

28

25

79

47

(m3 s4)

29

41

11

14

24

63

31

79

37

139

263

108

a

25.6

24.9

24.9

24.9

21.6

21.2

21.3

23.3

23.6

31.1

30.6

31.1

n

0.3451

0.3216

0.2819

0.2819

0.3511

0.3120

0.2548

0.2703

0.3674

0.2990

0.4012

0.2990

i>

0.145

0.212

0.160

0.211

0.251

0.367

0.268

0.246

0.301

0.953

0.770

0.490

In Table 4, \p values, calculated by equation (29), are also listed. For each basin the percentage of permeable basin area, S , was estimated using geological information. Using îp and Sp values, the following empirical relationship (Fig. 18(a)) was deduced:

\f> = 9.25/Sp (30)

with a standard estimate error equal to 0.19. Another expression was deduced by analysing the influence of the crop

cover on the \p estimate. For each basin the percentage of wooded area was determined (Indelicato, 1988; Assessorato Agricoltura e Foreste, 1980). By multiple regression analysis the following relationship was obtained (Fig. 18(b)):

12 13

* = i^f (31) with a standard estimate error equal to 0.13.

The latest empirical results (equations (30) or (31) and equation (29), with jS = 0.35, led to evaluation of the MAF by the following relationship:

aS°-5iU,f>

fi0 = —\P (32) u 3.6(0.351_")

Equation (32) with ip estimated according to equation (30) was characterized by a mean square error equal to 1037. Finally, equation (32) with \p estimated by equation (31) was characterized by the lowest mean square error, equal to 289.


0 tO 20 30 40 50 60 70 0 20 40 60 80 100 120 Sp(%) Sp • Sb |%]

Fig. 18 Relationships between $ and (a) the percentage of permeable basin area; (b) Sp + Sb.

CONCLUSIONS

For estimating xT, the value of a hydrological variable, x, of fixed return period, T, the choice of a theoretical distribution function and estimates of its parameters are necessary. In order to improve the at-site estimates based on limited data and to estimate xT for ungauged sites, a regional estimation technique can be used. This paper presents, for the Sicilian region, a regional rainfall and flood analysis which used the TCEV distribution and a hierarchical regionalization technique.

The rainfall analysis, carried out with rainfall data recorded by Sicilian raingauges, showed a duration dependence of the TCEV parameters and allowed a determination of the rainfall depth-duration relationship at ungauged sites and also at gauged sites with short records.

The flood data used in this research were the annual maximum peak flood (AMPF) and the annual maximum mean daily discharge (AMDD) samples. The AMDD sequences were used because they have sample sizes greater than those of the AMPF series. At the first level, the flood analysis showed that the TCEV distribution has a descriptive ability better than the other candidate distributions (LN2, EV1). The first and second levels of the hierarchical procedure for estimating the TCEV flood parameters, were carried out by with AMPF and AMDD sequences. The analysis showed that the theoretical growth curve with A*, 0*, Xj parameters estimated with AMDD sequences had the best agreement with the AMPF empirical growth curve.

The third level was carried out in order to determine a regional criterion to estimate the mean annual flood (MAF) used as an index flood. An empirical estimation criterion which links MAF with basin area was proposed. A conceptual model based on the rational formula was also proposed. This last criterion (equation (32)) allows the MAF to be estimated via the basin area, the n coefficient of RDD relationship and a mean runoff coefficient. For estimating the latter, two relationships were proposed in which the permeable basin area and the wooded area were introduced.


REFERENCES

Assessorato Agricoltura e Foreste Delia Regione Siciliana (1980) Piano di difesa dei boschi dagli incendi e di ricostituzione forestale. Carta dei boschi.

Agnese, C. & D'Asaro, F. (1987) Comparing die hydrologie response deduced by different analysis of network géomorphologie structures. IAHS Workshop no. 1, 19th IUGG General Assembly, (Vancouver, Canada).

Agnese, C. & D'Asaro, F. (1990) On derivation of the IUH from the geomorphology of link structured channel networks. Excerpta 5.

Ben-Zvi, A. (1989) Toward a new rational method. J. Hydraul. Engng ASCE 115(9). Beran, M., Hosking, J. R. M. & Arnell, N. W. (1986) Comment on "TCEV distribution for flood

frequency analysis". Wat. Resour. Res. 22(2). Canuti, P. & Moisello, U. (1980) Indagine régionale sulle portate di massima piena in Liguria e Toscana

(in Italian). Geologia applicata e Idrogeologia 15. Cao, C , Sechi, G. M. & Becciu, G. (1988) Analisi régionale per la valutazione probabilistica delle piene

(in Italian). Proc. XXI Convegno di Idraulica e Costruzioni Idraulkhe Appendice, L'Aquila. Cicioni, G. B., Giuliano, G. & Spaziani, F. M. (1974) Indagine sulla distribuzione di probabilité delle

portate di piena dei corsi d'acqua italiani (in Italian). Proc. XIV Convegno di Idraulica e Costruzioni Idraulkhe, Napoli.

Cunnane, C. (1988) Methods and merits of regional flood frequency analysis. J. Hydrol. 100. Cunnane, C. (1989) Statistical Distributions for Flood Frequency Analysis. WMO no. 718, Operational

Hydrology Report no. 33, Geneva. Ferrari, E. (1985) Modelli idrologici per lo studio delle massime altezze di pioggia giornaliere.

Un'applicazione al caso della Sicilia (in Italian). Unpublished manuscript, Università délia Calabria, Cosenza, Italy.

Fiorentino, M., Gabriele, S., Rossi, F. & Versace, P. (1987) Hierarchical approach for regional flood frequency analysis. In: Regional Flood Frequency Analysis ed. by V. P. Singh. Reidel, Dordrecht, The Netherlands.

Fiorentino, M., Gabriele, S. & Versace, P. (1984) Stima della pioggia di progetto per il proportionamento delle fognature pluviali. Un modello régionale (in Italian). Atti del Seminario Deflussi Ursani, Cosenza, Italy.

Gabriele, S., Ferrari, E., Versace, P., Rossi, F. & Villani, P. (1990) Pluviométrie information in regional flood studies. Proc. XV EGS General Assembly (Copenhagen, Denmark).

Gupta, V. K. & Waymire, E. (1983) On the formulation of an analytical approach to hydrologie response and similarity at the basin scale. J. Hydrol. 65.

Indelicato, S. (1988) Verifica modelli di valutazione del rischio idraulico-geologico ed efficacia degli intervenu' (in Italian). Rapport» '88 dell'U.O. 3.10 del GNDCI, Roma, Italy.

Kuichling, E. (1889) The relation between the rainfall and the discharge of sewers in populous districts. Trans. ASCE 20.

Landwehr, J. M., Matalas, N. C. & Wallis, J. R. (1978) Some comparison of flood statistics in real and log space. Wat. Resour. Res. 14(5).

Landwehr, J. M , Matalas, N. C. & Wallis, J. R. (1980) Quantile estimation with more or less flood like distribution. Wat. Resour. Res. 16(3).

Lazzari, E. (1967) Studio probabilistico delle piene con particolare riferimento ai corsi d'acqua della Sardegna (in Italian). L'Energia Elettrica 4.

Matalas, N. C. & Gilroy, E. J. (1968) Some comments on regionalization in hydrologie studies. Wat. Resour. Res. 4(6).

Melton, M. A. (1957) An analysis of the relation among elements of climate, surface properties and geomorphology. Off. Nav. Res. (USA) Geogr. Branch. Project 389-042, Tech. Rep. 11.

Modica, C , Reitano, B. & Rossi, G. (1988) Hydrological homogeneity criteria for regional analysis of flood flows. Proc. Int. Conference on Fluvial Hydraulics, Budapest.

Modica, C , Reitano, B. & Rossi, G. (1990) Individuazione di gruppi omogenei di bacini per l'analisi régionale delle portate di piena (in Italian). Proc. XXII Convegno di Idraulica e Costruzioni Idraulkhe, Cosenza.

Rossi, F., Fiorentino, M. & Versace, P. (1984) Two-component extreme value distribution for flood frequency analysis. Wat. Res. Res. 20(7).

Rossi, F. & Villani P. (1992) Regional methods for flood estimation. Pre-Proc. of the NATO ASI on Coping with Floods, Erice.

Soil Conservation Service (1972) Hydrology. In: National Engineering Handbook, Section 4. US Department of Agriculture, Washington, USA.

Stedinger, J. R. (1983) Estimating a regional flood frequency distribution. Wat. Resour. Res. 19(5). Versace, P., Ferrari, E., Gabriele, S. & Rossi, F. (1989) Valutazione delle piene in Calabria, Geodata 30. Wallis, J. R., Matalas, N. C. & Slack, J. R. (1974) Just a moment! Wat. Resour. Res. 10(2). Wiltshire, S. E. (1986) Regional flood frequency analysis. J. Hydrol. Sci. 31(3).

Received 27 September 1993; accepted 7 July 1994


APPENDIX

A relationship for evaluating the concentration time for a basin with average morphological features can be deduced applying the Geomorphological Instantaneous Unit Hydrograph (GIUH) theory to a channel network ordered according to Shreve's scheme (Gupta & Waymire, 1983; Agnese & D'Asaro, 1985, 1987).

For a channel network with /x links at the first level and a network with a common morphological shape, Agnese & D'Asaro (1990) deduced the following expression for the time lag, tL:

= ÇL\i)hl ( 1 A ) L v

in which v is the space-time average velocity, and / is the mean links length. Since 2ti is the total number of links of a channel network (approxi

mating 2[i — 1 = 2/u.), the topological variable \K is linked to the basin area S and to the total channel length Sv according to the relationships:

S = 2 n lb. (2A)

Sv = 2fil (3A)

in which the ratio be = SISy is the width of the link-associated drainage area. Deducing 2/x from equation (2A) and replacing it in equation (1A) gives:

h

si (4A)

In order to evaluate the mean value of the ratio l/be, i.e. the most frequent natural value, one can use equations (2A) and (3A) and the definitions of drainage density, Dd, and drainage frequency, F, giving:

1 - Hi Hi tL - ®k (5A) be 2fi S S F

in which Dd = Sv/S and F is the ratio between the total channel links number (2/x) and the basin area. Using Horton's order scheme (total channel network number approximately equal to 1.2/x), according to Melton (1957) the ratio FIDd

2 is a constant equal to 0.694. If we use Shreve's order scheme, the Melton constant is roughly equal to 1.

According to the Soil Conservation Service (1972), the ratio between the lag time and the time of concentration, tLltc, is equal to 0.6, so from equations (4A) and (5A) one obtains:

1


t - IL - fi - 0Mfi (6A) c 0.6 3.6(0.6v) v

in which S is measured in km2 and tc in h. Finally, considering velocity values variable between 1 and 2 m s"1, /3 values equal to 0.35 may be used in equation (6A) to represent mean kinematic conditions.

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