Statistical distributions of daily rainfall regime in Europe for the period 1951–2000

Abstract

Amount and time distributions, X and Y, of daily rain amounts in Europe along the second half of 20th century have been studied from 267 rain gauge records. Different geographical features, such as latitude, vicinity to Mediterranean Sea or the Atlantic Ocean or altitude above sea level, cause the averages of daily rain and annual number of rainy days to vary within a wide range. The largest daily percentiles of amount and time distributions are reached at latitudes south of 50°N and in southwestern Norway. The amount of distribution, X, is well-modelled by the exponential function, with parameters derived from probability graphs. Time distributions, Y, are well-fitted by Pearson type III (Gamma) and Weibull models, their parameters being estimated by L-moments. Normalised rainfall curves (NRC) have been modelled by the analytical function $X = Y \cdot \exp \left\{ { - b{{\left( {1 - Y} \right)}^c}} \right\}$, with b and c parameters depicting spatial variability. Alternatively, the beta distribution also describes quite well the empirical NRCs, with parameters estimated by statistical moments. The coordinates of the average daily amount (X _r , Y _r ) and the values of X^* and Y^*, which are defined as the fraction of rain amount for a half of rainy days and the fraction of number of rainy days accounting for a half of total rain amount, respectively, depict very similar spatial distribution throughout Europe. In fact, X _r and X^* keep a linear relationship, as well as Y _r and Y^*, the four coordinates depending on the coefficient of variation of daily rain amounts. A similar linear relationship is found for the pair (X^*, Y^*). Finally, the Average Linkage algorithm applied to the coordinates X _r , Y _r , X^* and Y^* characterising every one of the 267 NRCs permits to group the rain gauges into several spatial clusters, each of them related to a different normalised daily pluviometric regime.

Appendix

Theoretical distributions applied to the different random variables studied (Benjamin and Cornell 1970; Hosking and Wallis 1997).

• (a) Exponential distribution

$$ F\left( {x;{{\lambda }}} \right) = 1 - \exp \left[ { - {{\lambda }}\left( {x - {{\varepsilon }}} \right)} \right]\quad \quad \quad x \geqslant {{\varepsilon }}\,;\;{{\lambda }} > 0 $$

(A.1)

• (b) Pearson type III (Gamma) distribution

$$ {{F}}\,\left( {x;{{\alpha }},{{\beta }},{{\xi }}} \right) = {{G}}\,\left( {{{\alpha }},\frac{{x - {{\xi }}}}{{{\beta }}}} \right)/\Gamma \left( {{\alpha }} \right)\quad \quad x\, \geqslant {{\xi }};\;{{\alpha }},{{\beta }}\, > \,0 $$

(A.2)

where

$$ G\left( {\alpha, \;x} \right) = \int_0^x {\,{t^{\alpha - 1}}\,{e^{ - t}}\,dt} $$

(A.3)

And Γ(·) denotes the gamma function

• (c) Weibull distribution

$$ F\left( {x;k,u,\varepsilon } \right) = 1 - \exp \left[ { - {{\left( {\frac{{x - \varepsilon }}{{u - \varepsilon }}} \right)}^k}} \right]\quad \quad \quad x \geqslant \varepsilon; \;u,k > 0 $$

(A.4)

• (d) Beta type I distribution

$$ F\left( {x;\alpha, \beta } \right) = \frac{{\Gamma \left( {\alpha + \beta } \right)}}{{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}\int_0^x {{x^{\alpha - 1}}{{\left( {1 - x} \right)}^{\beta - 1}}dx\quad \quad \quad 0 < x < 1;\quad \alpha, \;\beta > 0} \quad $$

(A.5)

where Γ(·) again denotes the gamma function