Return period curves for extreme 5-min rainfall amounts at the Barcelona urban network

Abstract

Heavy rainfall episodes are relatively common in the conurbation of Barcelona and neighbouring cities (NE Spain), usually due to storms generated by convective phenomena in summer and eastern and south-eastern advections in autumn. Prevention of local flood episodes and right design of urban drainage have to take into account the rainfall intensity spread instead of a simple evaluation of daily rainfall amounts. The database comes from 5-min rain amounts recorded by tipping buckets in the Barcelona urban network along the years 1994–2009. From these data, extreme 5-min rain amounts are selected applying the peaks-over-threshold method for thresholds derived from both 95% percentile and the mean excess plot. The return period curves are derived from their statistical distribution for every gauge, describing with detail expected extreme 5-min rain amounts across the urban network. These curves are compared with those derived from annual extreme time series. In this way, areas in Barcelona submitted to different levels of flood risk from the point of view of rainfall intensity are detected. Additionally, global time trends on extreme 5-min rain amounts are quantified for the whole network and found as not statistically significant.

Appendix. Kendall-τ test of independence

If a set of {z_i}, i = 1,…,n, peaks are selected from a time series and their associated ranks, {R_i} are determined, the Kendall-τ test can be applied to verify the independence of the series, usually with a probability exceeding 95%. After obtaining the set of rank pairs {(R₁, R₂), (R₂, R₃), ….., (R_n−1, R_n)}, the number of discordances n_d or, in other words, the number of pairs (R_i, R_i+1) and (R_j, R_j+1) accomplishing either R_i < R_j and R_i+1 > R_j+1 or R_i > R_j and R_i+1 < R_j+1, with i = 1,…,n−1, j = 1,…,n−1 and i ≠ j leads to the empiric Kendall-τ statistic

$$ {\tau}_{\mathrm{emp}}=1-\frac{4{n}_d}{\left(n-1\right)\left(n-2\right)} $$

(9)

The null hypothesis of independent {z_i} peaks approaches τ to a normal random variable, provided that the number of samples n exceeds 10. The expected value of τ will be

$$ <\tau >=-\frac{2}{3\left(n-1\right)} $$

(10)

and its variance

$$ {\sigma}^2\left(\tau \right)=\frac{20{n}^3-74{n}^2+54n+148}{45{\left(n-1\right)}^2{\left(n-2\right)}^2} $$

(11)

If τ_emp, given by Eq. (9), does not exceed

$$ {\tau}_{0.95}=<\tau >+1.65\sigma \left(\tau \right) $$

(12)

the one-sided 95% test permits accepting the null hypothesis of {z_i} independence with 95% confidence.