Variability of the statistical characteristics of extreme rainfall values in the French Alps
For many development projects, it is important to have some idea of the magnitude of extreme precipitation events that may occur for different probability levels and for time steps of less than 24 hours. Unfortunately, most existing rain gauge networks measure precipitation on only a daily basis. In the French Alps, 65 rain gauge stations provide precipitation data over short time steps (1 to 24 hours). This very diverse network, managed jointly by the French electrical utility (Electricité de France), the national weather office (Météorologie Nationale) and the regional water resources service (SRAE), provides a valuable basis for investigating possible relationships between the characteristics of extreme precipitation for 24-hour periods and those for shorter time periods. The results of such a study, although of course valid only for the investigated area, should provide an indication of whether or not it is possible to calculate the characteristics of rainfall over short time steps from much denser 24-hour rain gauge networks. A statistical analysis was carried out to estimate extreme rainfall values for return periods of 2, 5, 10, 20, 50 and 100 years and for time steps of 1, 2, 3, 6, 12 and 24 hours. Each station is therefore associated with 36 precipitation values as a function of return period and duration. A variable referred to as the gradex (gradient of the exponential) is defined, reflecting the change in precipitation values as a function of the return period for each time step and each station. The definition of this variable is based on the fact that Gumbel's law is used to represent the frequency distribution of extreme rainfalls over time intervals extending from 1 hour up to several days, which is equivalent to assuming an exponentially decreasing frequency distribution for extreme rainfalls for a given time step and a given location. When plotted on Gumbel paper, the right-hand part of this distribution has a slope equal to the parameter "a" of Gumbel's law: F(x)=exp{-exp{-(x-x>indice>0/a}} where F(x) is the probability of occurrence of a value less than x. The parameter "a" is the gradex, and has the same dimensions as x. It can be determined with the method of moments : a(t)=0.78xσx where σxis the standard deviation of the sample. This definition is equivalent to taking the slope of the line passing through the points corresponding to T=20 and 100 years on a Gumbel plot. For each of the stations, we can evaluate six gradex values, i.e. one for each time step. In this way, for each of the 65 stations and for each time step, we obtain the gradex values and estimated precipitation values for return periods from 2 to 100 years. Several types of curves were tested in order to determine possible relationships among the gradex values for different time steps, including linear, power law, exponential and logarithmic relationships. For the French Alps, the best fit was obtained with a linear relationship and we calculated the corresponding correlation coefficients. We found that the gradex values were well correlated for adjacent time steps, but not for those that were very different. In particular, it would appear to be impossible to deduce gradex values for very small time steps (1 to 6 hours) from the 24-hour gradex. The 24-hour gradex accounts for only 17% of the variance of the 1-hour gradex, while it accounts for 92% of the variance of the 12-hour gradex. Using a linear relationship, the only gradex values that can be estimated with any degree of accuracy from the 24-hour value are those corresponding to time steps greater than 6 hours. To check these results, we carried out a similar study after dividing the test area into four regions. The extreme precipitation values for these regions presented similar characteristics (same order of magnitude of precipitation and gradex values). For each region, we looked for significant linear relationships between the gradex values for the different time steps. The conclusions were the same as when we considered the entire area, i.e. the relationship between the gradex values of short time steps and the 24-hour values is very poor. We have shown that no simple relationship exists to deduce values for short time steps from those measured for long time steps. The problem we posed at the outset therefore appears to have no straight-forward solution. A network of rain gauges measuring daily precipitation values cannot be used to determine the statistical characteristics of the precipitation for much shorter time steps, i.e. less than 6 hours. The only solution would be to use devices capable of measuring the precipitation over short time intervals, for instance recording rain gauges or automatic stations linked to data acquisition systems. Unfortunately such devices have not been in use for a long time and provide records for periods rarely exceeding ten years. In conclusion, this study reveals the limits for the extrapolation of extreme daily rainfall characteristics to shorter time steps.