Fitting probability distribution to sample data

In practice, the main focus of the flood frequency analysis is to select an appropriate probability distribution to describe the relationship between the magnitude and frequency of floods. The selected distribution is fitted to a sample of observed flood data in order to enable extrapolation of the relationship between the magnitude and frequency of floods beyond the range of observed magnitudes and frequencies. However, it should be kept in mind that such an extrapolation is always a guess: we will never know "true" frequencies of extreme floods.

Calculation of X(P) or P(X)

Depending on the tasks of a particular flood-related project, the selected probability distribution for floods can be used in two ways. One task may be to calculate a design flood associated with some level of risk (e.g. probability of exceedence of the design flood), while another task may be to make a risk assessment for a specific flood magnitude (e.g. to find the probability of exceeding certain flow rate).

For the task of estimating a design flood, the selected distribution of floods is used to calculate quantiles (values of the random variable X) for a given probability. In the case of the risk assessment, the selected distribution of floods is used to calculate probabilities for given values of flood variables. Essentially, the procedure of the flood frequency analysis for the two tasks is the same, but the calculations are performed in two opposite directions.

Uncertainty in flood frequency analysis

Once the results of applying the selected distribution are obtained, the next step in the flood frequency analysis can be to analyze uncertainty inherent in these results. The reasons for uncertainties in flood frequency analysis are two facts: first, we deal with relatively small samples of observed floods, and second, we have to make assumptions about probability distribution governing these floods. Therefore, the uncertainties in flood frequency analysis are inevitable, but it is useful to have an idea on their amounts. The usual way to express the uncertainty of the results of flood frequency analysis is to construct confidence intervals of these results.