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Return period maps of dry spells for Catalonia (northeastern Spain) based on the Weibull distribution / Périodes de retour des périodes sèches en Catalogne (nord-est de l'Espagne) à partir de la distribution de Weibull X. LANA , M. D. MARTÍNEZ , A. BURGUEÑO & C. SERRA To cite this article: X. LANA , M. D. MARTÍNEZ , A. BURGUEÑO & C. SERRA (2008) Return period maps of dry spells for Catalonia (northeastern Spain) based on the Weibull distribution / Périodes de retour des périodes sèches en Catalogne (nord-est de l'Espagne) à partir de la distribution de Weibull, Hydrological Sciences Journal, 53:1, 48-64, DOI: 10.1623/hysj.53.1.48 To link to this article: http://dx.doi.org/10.1623/hysj.53.1.48

Published online: 18 Jan 2010.

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48

Hydrological Sciences–Journal–des Sciences Hydrologiques, 53(1) February 2008

Return period maps of dry spells for Catalonia (northeastern Spain) based on the Weibull distribution X. LANA1, M. D. MARTÍNEZ2, A. BURGUEÑO3 & C. SERRA1 1 Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Av. Diagonal 647,

E-08028 Barcelona, Spain [email protected] 2 Departament de Física Aplicada, Universitat Politècnica de Catalunya, Av. Diagonal 649, E-08028 Barcelona, Spain 3 Departament de Meteorologia i Astronomia, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain

Abstract A detailed description of the spatial distribution of dry spell lengths in Catalonia (northeastern Spain) for the years 1950–2000 is obtained from a statistical analysis. The database is derived from the daily records of 75 raingauges, and a dry spell is defined as a set of consecutive days with daily rainfall amounts below thresholds of 0.1, 1.0, 5.0 and 10.0 mm/d. The Weibull model fits the distribution of dry spell lengths well, whatever the raingauge and threshold. The Weibull parameters estimated by L-moments and the L-skewness–kurtosis diagrams allow quantification of the goodness of fit between the model and the empirical distribution. Dry spell lengths for return periods of 2, 5, 10, 25 and 50 years indicate the areas where drought phenomena might be more severe, as well as how often they might occur. A regional homogeneity method is used to test whether a single set of Weibull parameters could describe the series of dry spells for the whole region. Key words dry spells; spatial patterns; L-moments; Weibull distribution; return periods; regional homogeneity; Catalonia (northeastern Spain)

Périodes de retour des périodes sèches en Catalogne (nord-est de l’Espagne) à partir de la distribution de Weibull Résumé Une analyse statistique des longitudes des périodes sèches en Catalogne (nord-est de l’Espagne) a permis la description détaillée de leur distribution spatiale. Les résultats ont été obtenus à partir des données journalières de 75 pluviomètres pour la période 1950–2000. La période sèche est définie comme la succession des jours consécutifs avec quantités de pluie au dessous de 0.1, 1.0, 5.0 et 10.0 mm/jour. La répartition observée des longueurs des périodes sèches est bien modelée par la distribution de Weibull pour l’ensemble des pluviomètres et pour les différents seuils. Les paramètres de cette distribution ont été estimés à partir des moments L et, avec l’aide des diagrammes dissymétrie–curtosie-L, ils ont permis la quantification du degré de bonté de l’ajustement des répartitions observées à la distribution de Weibull. La grande variabilité spatiale du régime des pluies journalières en Catalogne est montrée à travers des longueurs des périodes sèches avec périodes de retour de 2, 5, 10, 25 et 50 années. Les résultats signalent où la sécheresse peut arriver à être plus rigoureuse et avec quelle fréquence celle-ci devient. Une méthode pour étudier l’homogénéité régionale est appliquée afin de savoir si un ensemble unique de paramètres de Weibull peut décrire tous les périodes sèches observées dans la région. Mots clefs periode sèche; distribution spatiale; moments L; distribution de Weibull; periodes de retour; homogeneite régionale; La Catalogne (nord-est de l’Espagne)

INTRODUCTION The drought phenomenon is a very important climatic hazard for many areas of the Earth, such as the Mediterranean region, because it affects human activities, conditions water resource policies and, as a whole, constrains the economic development of a territory (AMS Statement, 2004). A meteorological drought, a long rainfall shortage period, can be defined as successive dry spells separated by wet spells, which are neither long enough nor of outstanding daily amounts to balance previous rain deficits. Besides the analysis of persistent synoptic situations leading to drought episodes, it is also important to obtain statistical models in order to quantify probabilities concerning dry spell lengths and their repetition. Additionally, if public administrations are interested in a coherent water policy, a spatial description of the most outstanding patterns of dry spells is necessary, especially if the region analysed is characterised by strong spatiotemporal climatic irregularity. There are many studies focused on the drought phenomenon in terms of dry spells deduced from daily amounts, often using statistical models based on Markov chains (Douguédroit, 1980; Berger & Goosens, 1983; Conesa & Martín-Vide, 1993; Gómez Navarro, 1996; Lana & Burgueño, Open for discussion until 1 August 2008

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1998a; Martín-Vide & Gómez, 1999; Anagnostopoulou et al., 2003), Poisson models accounting for the repetition of long dry episodes (Lana & Burgueño, 1998b), the negative binomial method (Douguédroit, 1987, 1991; Galloy et al., 1982), or the combination of geometric and Poisson distributions (Dobi-Wantuch et al., 2000). Studies on extreme dry spell lengths have also been undertaken. Annual extremes series (AES) are usually modelled by the generalized extreme value (GEV) distribution and the partial duration series (PDS) often adapted to a generalised Pareto (GPA) distribution (Perzyna, 1994; Vicente & Beguería, 2003; Lana et al., 2006). An introduction to the statistical modelling of extreme values can be found in Coles (2001), where it is pointed out that the Gumbel, Fréchet and Weibull distribution families, deriving from the GEV model, are appropriate for the analysis of blocks of maxima. Similarly, the GPA distribution, also consistent with the GEV, is a good option for the study of extreme events, defined as those exceeding a threshold level. Besides the annual extremes analysis, drought phenomena can be analysed at the monthly scale through the standardised precipitation index (SPI) (McKee et al., 1993; LloydHughes & Saunders, 2002), also applied to the Iberian Peninsula (Lana et al., 2001; VicenteSerrano, 2006a,b). A different point of view is adopted in the present study. A daily rainfall database of Catalonia (northeastern Spain) is analysed to obtain different sets of dry spell lengths for several daily rain amount thresholds. A dry spell is defined as a set of consecutive days with daily amounts below a given threshold level. The analysis of the drought phenomenon is not restricted to dry periods with amounts below 0.1 mm/d (the resolution of the pluviometers). It must be taken into account that, for practical purposes, it would be more useful to evaluate the possibility that a rainfall shortage due to a long and severe drought period could be balanced by a considerable wet spell (amounts of 10 mm/d or even higher) instead of only requiring a threshold of 0.1 mm/d. This is the principal reason why the distinction between different threshold levels is relevant. As a main result, the dry spell series are modelled by the Weibull distribution and dry spell lengths are derived for different return periods from 2 to 50 years for the whole spatial domain, making use of the L-moments parameter estimation. In this way, places where rainfall shortage could be persistent, possibly causing severe drought, are detected. In fact, the present analysis is intended to complement previous studies of dry spells in Catalonia, which have been performed with a smaller number of raingauges and using different statistical tools, such as the distribution of extreme dry spells (Lana et al., 2006) and the estimation of time trends of dry spells (Serra et al., 2006). There are several objectives in the present study that differ from those in Lana et al. (2006). All available dry spell lengths are considered, not only block of maxima (AES) or events exceeding certain lengths (PDS). To distinguish the new approach from the extreme sample methods, the present strategy is referred to herein as all dry spells (ADS). It must be emphasized that including all spell length time series could introduce dependency among the events and, consequently, some bias in the statistical moments. Return period maps from 2 to 50 years obtained for threshold levels of 0.1, 1.0 and 5.0 mm/d are quite similar to those in Lana et al (2006), but these maps now refer to any spell length, not only to extreme events. In addition, a new threshold level of 10.0 mm/d can be investigated, as the number of empirical dry spells for this level is large enough for the present methodology, but not for sampling strategies based on the AES or PDS, which were discarded due to high uncertainties (AES) or a very low number of retained samples (PDS). For water management proposals, it should be more advisable to consider return period maps derived from PDS or AES strategies, when available. However, for large daily thresholds, the shortness of data series does not usually permit sampling strategies based on extremes. Then, it is recommended to consider the return period maps derived from the Weibull distribution, especially for high return periods, as a good enough alternative. Finally, discrepancies between dry spell lengths linked to several return periods, estimated by moments and L-moments, are also investigated. The spell length series fit the Weibull distribution well, whatever the daily amount threshold and gauge emplacement. Although the Weibull distribution is usually considered for extreme analyses, the L-skewness–kurtosis diagrams suggest that it should also be the right option at present, and numerous examples in climatology and other scientific fields support this possibility. Copyright © 2008 IAHS Press

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Besides the use of the Weibull distribution in engineering applications, such as random failures, fatigue and reliability of materials, it is also applied in several fields of climatology, such as wind regimes (Justus et al., 1978; Tackle & Brown, 1978; Tuller & Brett, 1985; Bauer, 1996), thunderstorms (Schütte et al., 1987) and rainfall statistics (Swift & Schreuder, 1981; Wilks, 1989; Groisman et al., 1999; Stephenson et al., 1999; Burgueño et al., 2004). In particular, the Weibull distribution has also been applied to model the time intervals between successive events in the same storm cluster, while the lognormal distribution is applied to fit the elapsed time between successive storm clusters (Cowpertwait, 2001). It is well known that the rainfall regime of Catalonia is governed by the synoptic circulation, its complex orography and its vicinity to the Mediterranean Sea (Serra et al., 1998; Sotillo et al., 2003; Lana et al., 2004). These facts, together with the time irregularity of the Mediterranean climate (Barry & Chorley, 2003), generate a set of complex patterns of the rainfall regime that makes an analysis of the drought phenomena relatively difficult, especially if attention is focused on dry spell lengths derived from daily data. All these facts, together with soil moisture, streamflow and groundwater conditions would allow a complete description of the drought risk in Catalonia to be drawn according to the complete methodology proposed in Tallaksen & van Lanen (2004). The contents of the paper are structured as follows. The Database Section introduces the raingauge network and establishes the daily rain amount thresholds considered. The Weibull cumulative distribution is described in the Methodology Section, together with parameter estimation by moments and L-moments and the return period formulation. The applications are given in the next section, paying attention to the spatial variability of the Weibull distribution parameters and to the comparison between the return periods obtained by moments and L-moments. Additionally, a test of regional homogeneity evaluates whether a single set of Weibull parameters could describe the dry spell distribution for the whole network of raingauges. The Discussion deals with previous analyses for the Iberian Peninsula and the Conclusions outline the main results.

DATABASE The daily pluviometric data (years 1950–2000) have been compiled from 75 raingauges in Catalonia (northeastern Spain), belonging to the Instituto Nacional de Meteorología (Spanish Ministry of Environment). The homogeneity of every raingauge record and a certain lack of data for some gauges have been extensively analysed, quantified and discussed by Lana et al. (2004) and Burgueño et al. (2005), who have used the same database to investigate different patterns of the daily rainfall regime. Figure 1 depicts the main orographic features of Catalonia and Fig. 2 shows the location of the 75 raingauges, providing a visual estimation of the density of the pluviometric network. In order to obtain a population of dry spell lengths allowing an accurate analysis of the rainfall shortage, the first step is to define the threshold levels of daily rainfall used to generate the different series of dry spells. Since water management is often adapted to the normal hydroclimatological conditions in a region, the thresholds should be defined as a deviation from normal conditions, e.g. by using percentiles, for instance, the 25th, 50th, 75th, 90th and 95th percentiles of cumulative rainy days. However, when there is empirical evidence of a great spatial variability of the daily rainfall, it is much more advisable to define a set of common threshold values for all gauges. Thus, a common percentile for all raingauges leading to very different daily amounts is avoided. The threshold values were chosen by establishing a gradation of daily amounts from 0.1 to 10.0 mm/d, which is an outstanding rain amount for the area analysed, with intermediate values of 1.0 and 5.0 mm/d. Higher threshold levels do not generate long enough series of empirical dry spells for a representative number of gauges and were not considered because statistical models would be subjected to notable uncertainties.

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Return period maps of dry spells for Catalonia (northeastern Spain)

FRANCE

Western Pyrenees

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Fig. 1 Main orographic features of Catalonia and its location in the Iberian Peninsula.

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METHODOLOGY The Weibull distribution In 1939, the Swedish engineer Waloddi Weibull introduced a distribution for life-testing of materials (Weibull, 1939, 1951), which is known as the Weibull distribution. Its cumulative expression (Benjamin & Cornell, 1970) can be written as: Copyright © 2008 IAHS Press

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⎧⎪ ⎛ x − ε ⎞ k ⎫⎪ F ( x; k , u , ε ) = P( x ≤ X ) = 1 − exp⎨− ⎜ ⎟ ⎬ ; x ≥ ε ; u, k > 0 ⎪⎩ ⎝ u − ε ⎠ ⎪⎭

(1)

and its parameters are location ε, scale (u – ε) and shape k. The two first statistical moments of the Weibull distribution are given by: x = ε + (u − ε ) Γ (1 + 1 / k )

[

σ x2 = (u − ε ) 2 Γ (1 + 2 / k ) − Γ 2 (1 + 1 / k )

(2a)

]

(2b)

Because of the exponential structure of the Weibull distribution, the scale and shape parameters can be derived from Weibull probability paper by means of a least squares regression of log{–log(1 – F)} with respect to log(x – ε). The location parameter ε represents the minimum possible value for the variable x, which in our dry spell analysis is 1 d. Alternatively, the parameter estimation can be done by the method of moments. Then, the system of equations (2a) and (2b) has to be solved to obtain parameters u and k, again assuming ε equal to 1 d. The maximum likelihood estimation of the parameters is not appropriate here, given that a number of observations are often equal to ε, and thus log(xi – ε) is not defined, so that a high number of dry spell lengths of 1 d must be discarded. Least square regressions also have the same problem. Here, the parameters of the Weibull distribution are estimated by L-moments (Hosking & Wallis, 1997; Jawitz, 2004; Mahdi & Ashkar, 2004), due to the robustness of this procedure in comparison to the other two proposed. In terms of the L-moments formulation, the probability weighted moments of order r for the Weibull distribution can be expressed as: Γ(1 + 1/k ) (r + 1)1+1/k

αr = ε/( r + 1) + (u − ε )

(3)

The first four L-moments are then given by:

λ1 = α0 = ε + (u − ε ) Γ (1 + 1/k )

(4a)

{

λ2 = α0 − 2α1 = (u − ε ) Γ (1 + 1/k ) 1 − 2 −1 /k

{

}

(4b)

λ3 = α0 − 6α1 + 6α2 = (u − ε ) Γ (1 + 1/k ) 1 − 3 ⋅ 2 −1/k + 2 ⋅ 3−1 /k λ4 = α0 − 12α1 + 30α2 − 20α3 =

{

= (u − ε ) Γ (1 + 1/k ) 1 − 6 ⋅ 2 −1 /k + 10 ⋅ 3−1 /k − 5 ⋅ 4 −1/k

}

}

(4c) (4d)

and the derived L-skewness, τ3, and L-kurtosis, τ4, are given by:

τ 3 = λ3 /λ2 =

1 − 3 ⋅ 2 −1/k + 2 ⋅ 3−1 /k 1 − 2 −1/k

(5a)

1 − 6 ⋅ 2 −1 /k + 10 ⋅ 3−1 /k − 5 ⋅ 4 −1/k (5b) 1 − 2 −1/k Then, it is straightforward to derive that the shape parameter k can be numerically estimated by: τ 4 = λ4 /λ2 =

k=

− ln(2) ln{1 − l 2 / (l 1 − ε )}

(6a)

and scale u – ε by: u−ε =

l1 − ε Γ (1 + 1/k )

(6b)

where l 1 and l 2 are estimations of λ1 and λ2 computed from the sample values arranged in ascending order {xi, i = 1, …, n}, according to the expressions (Hosking & Wallis, 1997): Copyright © 2008 IAHS Press

Return period maps of dry spells for Catalonia (northeastern Spain) n

b0 = n −1 ∑ x j

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(7a)

j =1

( j − 1) xj j = 2 ( n − 1) n

b1 = n −1 ∑

(7b)

l 1 = b0

(7c)

l 2 = 2b1 − b0

(7d)

After revision of equations (5a) and (5b), it is observed that the L-skewness and L-kurtosis depend solely on the shape parameter k. This fact permits an easy and accurate validation of the closeness of the empirical distribution of dry spells to the Weibull model. This validation is based on the Euclidean distance D of empirical L-moment coordinates, t3 and t4, and L-moment coordinates computed according to equations (5a) and (5b) after a previous estimation of parameter k from equation (6a) and the first two empirical L-moments l 1 and l 2 . In other words:

{

D = (t3 − τ 3 (k )) 2 + (t 4 -τ 4 (k )) 2

}

1/ 2

(8)

is assumed as a measure of goodness of fit (Lana et al., 2006). Return periods

The return period Tr can be defined as:

Tr = 1 / β{1 − F ( xr )}

(9)

where β is the average number of observations per unit time, expressed here in years (or otherwise the inverse of the mean interarrival time). For the Weibull distribution it can be expressed as: Tr = exp[( xr − ε ) /(u − ε )] k / β

(10)

where xr is the corresponding dry length. This can be solved for xr (expressed in days) to give: xr = ε + (u − ε )[ln( βTr )]

1/ k

(11)

STATISTICAL DISTRIBUTION OF DRY SPELLS AND SPATIAL PATTERNS The Weibull distribution

The Weibull model, with a location parameter ε of 1 d, satisfies very well the empirical distributions of dry spell lengths whatever the threshold level and raingauge considered. Figure 3 depicts the fits for different raingauges and threshold levels of 0.1, 1.0, 5.0 and 10.0 mm/d. The selected raingauges are located in areas of Catalonia with different rainfall regime. The very small discrepancy between the empirical distributions and the Weibull model is noticeable. Even though the Kolmogorov-Smirnov test is specifically for parameters values known a priori, but not estimated from the sample data, and the results of the test are purely indicative, the empirical distribution is only outside the 95% confidence bands given by the Kolmogorov-Smirnov test (Benjamin & Cornell, 1970) for ten raingauges and for very short lengths (less than 3–4 d) and threshold levels of 0.1 and 1.0 mm/d. It is very likely that the reason for this behaviour is the discrete resolution of 1 d for the spell length and that the 95% confidence bands are much more restrictive for 0.1 and 1.0 mm/d than for 5.0 and 10.0 mm/d. It should be remembered that Kolmogorov-Smirnov 95% confidence bands tend to 1.36 / n for a high enough n, with n the sample size of dry spells for every raingauge, and the number of dry spells diminishes with the increasing threshold level. Copyright © 2008 IAHS Press

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0.999

CUMULATIVE DISTRIBUTION

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RAIN GAUGE 16 (0.1 mm) u = 4.35 k=0.708

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Fig. 3 Weibull model and dry spell distribution for raingauges 16 (0.1 mm/d), 21 (1.0 mm/d), 5 (5.0 mm/d) and 25 (10 mm/d) together with the Kolmogorov-Smirnov 95% confidence bands.

A global overview of the fit between empirical data and the Weibull distribution is given by the Euclidean distance D of equation (8). Figure 4 shows the empirical L-skewness–kurtosis diagrams for the four thresholds and several theoretical distributions; i.e. the generalised logistic (GLO), generalised Pareto (GPA), Pearson type 3 (PE3), exponential (E) and Weibull L-skewness–kurtosis curves are plotted. The GLO and E distributions obviously must be discarded for all the threshold levels. The GPA model could only be acceptable in a few cases. Finally, the Weibull and PE3 distributions depict a quite similar evolution and it is also worth mentioning the departure of five dry spell series from the proposed models for 10.0 mm/d. The close vicinity of empirical Lmoments to theoretical L-skewness–kurtosis is well highlighted and, after a detailed revision of distances given by equation (8), it can be reasonably assumed that the Weibull distribution fits the empirical data slightly better than PE3. Average (standard deviations) L-moment distances for the Weibull distribution are 0.050 (0.017), 0.046 (0.016), 0.039 (0.016), and 0.035 (0.020) for 0.1, 1.0, 5.0 and 10.0 mm/d, respectively, and maximum distances of 0.090 are seldom detected. Copyright © 2008 IAHS Press

Return period maps of dry spells for Catalonia (northeastern Spain)

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Fig. 4 L-skewness L-kurtosis diagrams for empirical data and Weibull distribution for thresholds of 0.1, 1.0, 5.0 and 10.0 mm/d. Solid line represents the Weibull distribution. Dashed lines indicate GLO, GPA, E and PE3 distributions.

Spatial patterns of the Weibull parameters

Figure 5(a) represents the spatial distribution of average dry spell lengths, in days, for the four threshold levels. Figure 5(b) depicts the spatial behaviour of parameter u (days) of the Weibull distribution, estimated by L-moments. It is noticeable that both figures show a very similar spatial distribution. They are characterised by a gradient from north to south for 0.1 and 1.0 mm/d. A nucleus of maximum values, which drifts from the south Mediterranean coast to the Central Basin as threshold level increases, appears for all threshold levels. The spatial distribution of parameter k is not shown for reason of its lack of organization. Its values range within a narrow band (0.7–0.9) for the four threshold levels, the highest values being reached in the Central Basin for the level of 10.0 mm/d, as happens with parameter u. Given that the range of k is relatively close to 1.0, whatever the threshold level, the spatial distribution of parameter u is a reasonable approximation of the average dry spell lengths, according to equation (2a). Copyright © 2008 IAHS Press

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(a)

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Average dry spell length (days)

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Fig. 5 Spatial distribution of (a) the average dry spell length and (b) parameter u of the Weibull distribution estimated by L-moments for threshold levels of 0.1, 1.0, 5.0 and 10.0 mm/d.

Return period maps of dry spells for Catalonia (northeastern Spain)

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Spatial patterns of the return periods

Figures 6 and 7 depict the spatial patterns of the dry spell lengths for all threshold levels and return periods of 2, 5, 10, 25 and 50 years, and considering parameters u and k estimated by L-moments. Maps of return period levels are obtained by kriging interpolation. Standard deviations of interpolated values range from 2 to 6 d for most of the region. A first common pattern is a north-tosouth gradient with minimum dry spell lengths in the Pyrenees domain and maximum lengths in the south or southwest of Catalonia, whatever the threshold. A second outstanding feature is the clockwise rotation of this gradient as threshold level increases, whatever the return period, the longest dry spells thus shifting from the southern coast towards western Catalonia. In the Pyrenees, the shortest lengths shift from the western to eastern Pyrenees as the threshold level increases. Again, this pattern is observed for all return periods.

Fig. 6 Return period maps of dry spell lengths for 2, 5, 10, 25 and 50 years and a threshold level of 0.1 mm/d, according to a Weibull distribution with parameters estimated by L-moments. Copyright © 2008 IAHS Press

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There is not a simple spatial pattern for the dry spell lengths in the littoral and pre-littoral domains. On the one hand, besides the latitudinal gradient just mentioned, there are some singularities, such as a relative maximum length detected at the northeast corner of Catalonia for rainfall thresholds of 5.0 and 10.0 mm/d. On the other hand, the effects of mountain ranges near the Mediterranean coast (pre-littoral, littoral and transversal chains) do not impose a uniform dry spell length for the littoral fringe and neighbouring domains, whatever the return period analysed. Additionally, it was studied how increasing the number of raingauges from 39 (Lana et al, 2006) to 75 influences the uncertainty on return period levels. The standard deviation of the interpolated values always ranges from 2 to 6 d throughout the domain, but, as expected, areas assigned to standard deviations between 4 and 6 d are slightly wider for 39 gauges than for 75.

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Fig. 7 Return period maps of dry spell lengths for 2, 5, 10, 25 and 50 years and a threshold level of 10.0 mm/d, according to a Weibull distribution with parameters estimated by L-moments. Copyright © 2008 IAHS Press

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Return period maps of dry spells for Catalonia (northeastern Spain)

Comparison of dry spell lengths derived from moments and L-moments

A comparison of parameters u and k derived by moments (equations (2a) and (2b)) and L-moments (equations (6a) and (6b)) manifests that, as expected, the results are similar. Nevertheless, a comparison between spell lengths derived from the two estimation methods is advisable in order to investigate the range of differences and their possible spatial patterns. Table 1 shows, for each threshold and return period, the ranges of differences between the spell lengths derived by L-moments and those by moments. For a better understanding of their relevance, the ranges of the spell lengths estimated by L-moments are also included in the table. As a general overview, there is an increase of the discrepancies with the threshold level and the return period, with a maximum difference of 19 days for 10.0 mm/d and 50 years. For 0.1 and 1.0 mm/d the maximum differences are observed in the west of the country, with values ranging from 3 d, for a return period of 2 years, to 12 d for 50 years. The differences are negative only in a few small areas and their values range from 1 to 4 d. For 5.0 and 10.0 mm/d, the largest discrepancies appear in some coastal zones and in the centre of the territory, with maximum values ranging from 4–5 d for a return period of 2 years, to 13–19 d for 50 years. As for 0.1 and 1.0 mm/d, negative differences are detected in a few small areas. Table 1 Ranges of differences, Δxr, of dry spell lengths (days) deduced from L-moments minus those deduced from moments, and dry spell lengths (days), xr, derived from L-moments.

0.1 mm/d 1 mm/d 5 mm/d 10 mm/d

2 years Δxr –1/+3 –1/+3 –1/+4 –1/+5

xr 22/58 22/58 32/70 42/90

5 years Δxr –2/+5 –2/+5 –1/+6 –2/+8

xr 28/75 28/75 42/95 54/125

10 years Δxr –3/+7 –2/+6 –2/+8 –2/+11

xr 33/90 33/90 49/114 63/152

25 years Δxr –4/+10 –3/+9 –2/+10 –3/+15

xr 40/110 41/110 59/140 76/189

50 years Δxr –4/+12 –3/+10 –2/+13 –4/+19

xr 46/125 47/125 68/160 86/218

Comparisons of return period spells derived from ADS, AES and PDS strategies

Bearing in mind that the dry spell regime in Catalonia has been recently analysed from the point of view of AES and PDS extreme sample strategies (Lana et al., 2006), a comparison with return period spells derived from ADS would be convenient. According to Vicente-Serrano & Beguería (2003), a root-mean-square error, RMSE, is used to evaluate the best return period spell among different sampling methods. For return periods of 2, 5, 10 and 25 years, according to equations (9)–(11), differences between empirical return period spells, xer, and those computed from equation (12), xr, are used to quantify the corresponding RMSE: RMSE (Tr ) =

1 NG ∑ {xe − xr }2 NG i =1 r

(12)

with NG the number of common gauges, 39, considered in Lana et al. (2006) and here. For a direct comparison between sampling strategies, the remaining gauges up to 75 are not considered. It must also be remembered that the extreme sample strategies do not permit us to analyse dry spells for threshold levels of 10.0 mm/d. Results are given in Table 2. It is observed that, for 0.1, 1.0 and 5.0 mm/d and all return periods, the RMSE is systematically lower for ADS than for AES and PDS. Test of regional homogeneity

Besides the analyses of the spatial distribution of average dry spell lengths, parameters u and k and spell lengths associated with different return periods, the regional homogeneity of the whole set of dry spells, distinguishing among threshold levels, can be investigated in a similar way to that proposed by Hosking & Wallis (1997) or Tallaksen & van Lanen (2004). Regional L-moments are Copyright © 2008 IAHS Press

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Table 2 Root-mean-square error, in days, between empirical and theoretical return period spells obtained from extreme sample strategies (AES and PDS) and all dry spells (ADS), for return periods, Tr, of 2, 5, 10 and 25 years. Tr 2 years

AES 4.968 7.084 19.100 4.765 6.879 19.639 6.119 8.894 18.137 9.184 10.799 24.724 -

0.1 mm 1.0 mm 5.0 mm 10.0 mm 0.1 mm 1.0 mm 5.0 mm 10.0 mm 0.1 mm 1.0 mm 5.0 mm 10.0 mm 0.1 mm 1.0 mm 5.0 mm 10.0 mm

5 years

10 years

25 years

PDS 4.317 6.544 19.624 5.924 7.961 22.827 6.801 10.061 21.647 9.186 11.774 27.951 -

ADS 2.550 4.314 17.109 39.430 3.726 4.924 18.555 36.275 4.857 6.681 15.841 33.498 8.487 8.961 21.373 41.255

introduced as the weighted average of the L-moments estimated for the different raingauges, the weighting factor being the ratio of the number nj of dry spells detected in a raingauge with respect to the total number of dry spells of the region, NT. According to Hosking & Wallis (1997), it is interesting to compute the weighted standard deviation, V, of the L-CVs, τ2, as: 1/ 2

⎧⎪ m ⎫⎪ V = ⎨∑ n j {τ 2, j − τ 2R }2 / N T ⎬ ⎪⎩ j =1 ⎪⎭

(13)

with τ 2R the regional L-CV and τ2,j the L-CV for raingauge j. The hypothesis of regional homogeneity can be tested by assuming that, at a regional scale, the whole set of spell lengths for the m raingauges fits a statistical distribution with a single set of parameters. Given that the Weibull distribution seems to be the right option for every raingauge, it is assumed that this distribution would also properly describe the dry spells at a regional scale. Regional parameters u and k are estimated from equations (6a) and (6b), considering now regional l 1R and l R2 as weighted averages of estimations l 1 and l 2 of the m raingauges. After that, sets of dry spells following a Weibull distribution are randomly generated, keeping invariant the original number of dry spells for every raingauge. Simulated L-CVs are different of those obtained empirically for every raingauge, but τ 2R is preserved for all simulations. Then, a set of weighted standard deviations, Vsim, are computed according to equation (13). Finally, after a large enough number of simulations, the ratio of times for which the empirical V exceeds Vsim can be considered as a measure of the level of confidence for regional heterogeneity. The proper number of simulations is chosen by observing the evolution of the average and standard deviation of Vsim towards an asymptotic value. It is assumed that if V exceeds Vsim for at least 95% of simulations, it is very likely that the distribution of dry spell lengths should be heterogeneous. Conversely, a low percentage of simulations for which V exceeds Vsim suggests a high probability of a regional homogeneity of the dry spells. A measure of heterogeneity, H, can also be obtained from (Hosking & Wallis, 1997): H=

V − μV σV

(14)

with μV and σV the mean and standard deviations of the simulations. For an acceptable Copyright © 2008 IAHS Press

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Return period maps of dry spells for Catalonia (northeastern Spain)

Table 3 Regional parameters of the Weibull distribution of dry spells and H values. Bold type indicates regional homogeneity. ε (d) 1.0 1.0 1.0 1.0

0.1 mm/d 1.0 mm/d 5.0 mm/d 10.0 mm/d

u (d) 5.68 6.49 10.50 16.30

k 0.741 0.759 0.813 0.857

H 13.0 8.9 –0.4 0.6

homogeneous region, H is less than 1.0. Values of H ranging from 1.0 to 2.0 suggest possible heterogeneity and H greater than or equal to 2.0 implies clear heterogeneity. After the simulations for the four daily threshold levels, dry spells derived for 0.1 and 1.0 mm/d show regional heterogeneity at a confidence level of 99% and H equal to 13.0 and 8.9, respectively. Conversely, regional homogeneity is much more likely for 10.0 mm/d and, especially, for 5.0 mm/d. Whereas only 35% of simulated Vsim are exceeded by empirical V for 5.0 mm/d, 70% of simulations are surpassed for 10.0 mm/d, in agreement with H equal to −0.4 and 0.6, respectively. Thus, a quite different spatial behaviour is observed, depending on the daily threshold level. Table 3 gives regional Weibull parameters and values of H for the four thresholds.

DISCUSION OF THE RESULTS

The spatial patterns concerning the dry spell lengths analysed can be compared with a simpler description of the pluviometric regime given by the average annual amounts (Fig. 8). Firstly, a gradient from south to north is observed for the average annual amount, which is opposite to gradients detected for the parameter u of the Weibull distribution and for dry spells with different return periods. Secondly, the location of the maximum of the parameter u for 10.0 mm/d in the Central Basin coincides with a minimum of average annual rainfall amounts. Major discrepancies in the location of the highest values of u and patterns of the annual average amount are observed for levels of 0.1, 1.0 and 5.0 mm/d. As a general feature, as expected, places where dry spells can reach extreme lengths are also characterised by low annual amounts. Consequently, it is very likely that a long dry period in southern and western Catalonia would not be immediately balanced

Average annual rainfall (mm/year)

300

500

700

900

A A i f Fig. 8 Average annual rainfall in Catalonia.

1100

(

1300

/

) Copyright © 2008 IAHS Press

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by episodes of high rain, thus possibly leading to drought episodes. In contrast, drought episodes can be more easily balanced in places such as the Pyrenees and pre-Pyrenees ranges and northeastern Catalonia, where annual amounts above 800 mm are expected and the shortest values of u are obtained. One of the most relevant features depicted by dry spells on the different return period maps is the clockwise rotation of the dry spell length gradient when threshold level increases. This feature is then clearly related to daily rainfall intensity. From the synoptic point of view, eastern advections would be the main cause of this pattern. Under this synoptic situation, the highest rain amounts are usually recorded in the littoral and pre-littoral areas and eastern Pyrenees, while the Central Basin is sheltered by the littoral and pre-littoral ranges. Additionally, the effects of eastern advections on the western Pyrenees are strongly mitigated by the remoteness to the Mediterranean coast. Dry spell length patterns for the whole Iberian Peninsula (years 1951–1990), including Catalonia, have been studied by Martín-Vide & Gómez (1999) who considered Markov chains instead of the Weibull distribution. Comparisons can be made for threshold levels of 0.1, 1.0 and 10.0 mm/d, even though some discrepancies should be expected due to the very different density of raingauges. Whereas just four of the 36 raingauges covering the Iberian Peninsula were located in Catalonia, 75 emplacements have been used here. The main dry spell patterns for Catalonia found here are in agreement with the overall features deduced by Martín-Vide & Gómez (1999) for the Iberian Peninsula. They obtained for Catalonia average dry spells from 5 to 7 d (0.1 mm/d level), which are very similar to values derived here for parameter u (Fig. 5(a),(b)). A similar situation is observed for the threshold of 1.0 mm/d. The parameter u ranges from 6 to 10 days throughout Catalonia, in agreement with Martín-Vide & Gómez (1999), excepting for the western Pyrenees (3–5 d) and a very small region in the southern corner (9–13 d). For 10.0 mm/d, the average lengths obtained range from 10 to above 30 d, which is an interval almost coincident with that of u deduced here. In short, both studies led to similar average spell lengths for Catalonia, but a more detailed spatial description is now obtained. Maximum lengths derived for the Iberian Peninsula can be compared with dry spell lengths for the 50-year return period, which constitute a good approximation to maximum lengths in Catalonia because the recording period is also 50 years. Martín-Vide & Gómez (1999) obtained dry lengths ranging from 50 to 80 d in Catalonia for 0.1 mm/d, which approximately correspond to values deduced for the northern half of Catalonia for a return period of 50 years, whereas longer dry spells, from 85 to 125 d, are obtained for the southern half of Catalonia. For 1.0 mm/d, lengths vary from 45 to 125 d for a return period of 50 years, while lengths of 65–105 d were obtained by Martín-Vide & Gómez (1999). Finally, dry spell lengths range from 80 (western Pyrenees) to 220 (southwestern corner of Catalonia) days for 10.0 mm/d. This range is close to that of 115–235 d derived by Martín-Vide & Gómez (1999).

CONCLUSIONS

The analysis, developed from a database of daily records for a relatively dense network of raingauges, has enabled us to obtain the main statistical and spatial patterns of dry spell lengths for Catalonia. To improve the analysis of dry spells, four different threshold levels of daily rainfall (0.1, 1.0, 5.0 and 10.0 mm/d) have been considered. Besides the results directly concerning the dry spell regime in Catalonia, two methodological aspects should be mentioned. First, it is confirmed that, according to a previous study (Lana et al., 2006), the Euclidean distances in terms of L-skewness–kurtosis can be assumed as an indicator of the goodness of fit between empirical distributions and statistical models, especially when the number of available data is small and the Kolmogorov-Smirnov 95% confidence bands are not very restrictive. Second, confirming previous remarks by Hosking & Wallis (1997), discrepancies between return periods deduced from moments and L-moments are not negligible for some raingauges and the use of the most robust estimation method (L-moments) is recommended. Copyright © 2008 IAHS Press

Return period maps of dry spells for Catalonia (northeastern Spain)

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The most remarkable results, from an applied point of view, are: (a) Instead of extreme spell lengths (AES and PDS), the whole set of dry spells (ADS) for every raingauge is now fitted to a single statistical distribution. The inclusion of all spell lengths allows maps of average dry spell length to be obtained for the different threshold levels, thus providing additional information about areas subjected to severe drought episodes. Moreover, as the number of available spells has increased notably with respect to that used for AES and PDS, a new threshold of 10 mm/d can be analysed. (b) The parameter k and, in particular, the parameter u of the Weibull distribution change with the gauge emplacement and threshold levels. Additionally, the spatial distributions of the parameter u and of the average dry spell lengths are quite similar given that k is often close to 1.0. (c) Return period maps for threshold levels of 0.1, 1.0 and 5.0 mm/d depict similar spatial patterns to those deduced for extreme events (Lana et al., 2006). Thus, it would be expected that the new return period maps for 10 mm/d would be quite similar to those deduced from AES and PDS, if they were available. Additionally, RMSE values deduced for AES, PDS and ADS strategies are quite similar, but the lowest values of RMSE are always observed for the ADS sampling strategy proposed herein. Consequently, the use of ADS and the Weibull distribution should not be discarded in favour of the classical analysis based on extreme sampling strategies. (d) A nucleus of high values of dry spell lengths is detected on the southern Mediterranean coast for threshold levels of 0.1 and 1.0 mm/d and all return periods. This nucleus drifts towards the Central Basin for threshold levels of 5.0 and 10.0 mm/d. A similar behaviour is observed for the parameter u of the Weibull distribution. (e) The nucleus of minimum dry spell lengths in the western Pyrenees for thresholds of 0.1 and 1.0 mm/d drifts towards the eastern Pyrenees for levels of 5.0 and 10.0 mm/d, whatever the return period. A quite similar evolution is observed for the parameter u. (f) A comparison of the main geographical features of the dry spell patterns with average annual amounts confirms, particularly for the highest threshold levels, that the areas with the most difficult recovery to normal rainfall amounts after a drought period are usually characterised by low annual amounts. Conversely, areas with the fastest recovery of rainfall amounts are associated with high annual amounts. (g) Finally, although the average dry spell lengths, the parameter u of the Weibull distribution and the return period maps depict clear spatial variability, the proposed test of regional homogeneity suggests signs of spatial homogeneity for thresholds of 5.0 and 10.0 mm/d. Conversely, the dry spell distributions seem to be spatially heterogeneous for thresholds of 0.1 and 1.0 mm/d. As a main conclusion, it is emphasized that the statistical tools applied and the results derived here would allow better handling of the problems concerning drought and water resources management in a domain of the Iberian Peninsula, where water supply and crops could be affected by severe drought periods. Acknowledgements We are indebted to the Instituto Nacional de Meteorología (Spanish Ministry of the Environment) for providing the pluviometric data series. REFERENCES AMS Statement (2004) AMS Statement on meteorological drought. Bull. Am. Met. Soc. (May), 771–773. Anagnostopoulou, Chr., Maheras, P., Karacostas, T. & Vafiadis, M. (2003) Spatial and temporal analysis of dry spells in Greece. Theor. Appl. Climatol. 74, 77–91. Barry, R. G. & Chorley, R. J. (2003) Atmosphere, Weather and Climate (eighth edn). Rouledge, London, UK. Bauer, E. (1996) Characteristic frequency distribution of remotely sensed in situ and modelled wind speeds. Int. J. Climatol. 16, 1087–1102. Benjamin, J. R. & Cornell, C. A. (1970) Probability, Statistics and Decision for Civil Engineers. McGraw-Hill, New York, USA. Copyright © 2008 IAHS Press

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Received 22 February 2006; accepted 29 August 2007 Copyright © 2008 IAHS Press