## Uncertainty in estimating floods by frequency analysis

Statistical analysis of any hydrologic variable, including floods, is based on series of observed data. These data samples are of rather limited size from the statistical point of view. Parameters of probability distributions that are estimated from sample data on floods at a particular site may not be "true" parameters of distribution of flood population at that site (so called parent population). Therefore, the question is always raised how much uncertainty there is in flood estimates that result from statistical analysis.

A commonly used measure of uncertainty of a quantile estimate xp (flood with probability of exceedance equal to p) is its standard error SE[xp], which is equal to the square root of variance of quantile estimate, var[xp].

Confidence intervals for quantile estimates xp are constructed using the standard error SE[xp]. In general, under assumption of large sample size, the quantile estimate xp is normally distributed. Lower and upper limit of the 100(1 – a)% confidence interval for xp are calculated from:

• upper limit:

• lower limit:

where z1–a/2 = –za/2 is standard normal variable for levels corresponding to confidence interval 1 – a.

Normal distribution.  Standard error of quantile estimate is:

where zp is standard normal variable corresponding to probability of nonexceedance p, sx is standard deviation of normal variable X, i.e. of sample data, and n is sample size.

Log-normal distribution.  For a log-normally distributed variable X, variable Y = ln X (or Y = log X) is normally distributed. From results for normal distribution, standard error of a normal quantile estimate yp is:

where sy is standard deviation of logarithms of sample data. Confidence limits for yp are then:

From the above, approximate confidence limits for the original variable X = eY are:

Alternatively, if base 10 logarithms were used, then the approximate confidence limits for the original variable X = 10Y are:

Pearson type 3 distributionApproximate standard error for the Pearson 3 quantile is can be calculated using the following rather cumbersome expression:

where sx is standard deviation of X, n is sample size, and d is calculated as square root from:

where Kp is frequency factor the Pearson 3 distribution, which depends only on coefficient of skewness Cs, and K'p is its derivative. Frequency factor can be read from statistical tables, while the following analytical approximation is often used:

where zp is standard normal variable corresponding to the nonexceedance probability p. Derivative of Kp can be found from the above expression and it is equal to:

Log-Pearson type 3 distribution.  Similar to log-normal distribution, the easiest way to construct confidence interval for log-Pearson 3 distribution is to find confidence limits for the transformed variable Y = ln X (or Y = log X) and to use standard error for Pearson 3 distribution with standard deviation and coefficient of skewness of logarithms (sy and Csy). An inverse transformation is then needed to obtain confidence limits for the original variable X.

Gumbel (EV1) distributionFor the Gumbel distribution with parameters estimated by the method moments, standard error of the p-quantile is:

where Kp is frequency factor for the Gumbel distribution, sx is standard deviation of X, and n is sample size. Gumbel frequency factor depends only on nonexceedance probability since it can be expressed using Gumbel standard variable yp: