Flood Frequency Analysis (FFA)
Overview
Flood Frequency Analysis (FFA) uses a probability distribution fitted to extreme streamflow observations (e.g., annual maxima) to estimate the recurrence likelihood of floods. To perform FFA, we require a probability model and corresponding parameter estimates based on the data.
FFA relates flood peak magnitudes \(Q\) to their expected frequency of occurrence, expressed as a return period. For example, a flood with a 10-year return period—commonly referred to as a 10-year flood—has a 1-in-10 chance of being equalled or exceeded in any given year. This corresponds to an annual exceedance probability of \(p_e = 0.1\). Since the FFA Framework uses annual maxima data, this equates to the 90th percentile (i.e., the \(0.90\) quantile) of the fitted probability distribution.
Here is a summary of the return periods, exceedance probabilities, and associated quantiles used by default in the FFA framework:
| Return Period (\(T\)) | Exceedance Probability (\(p_e\)) | Quantile ( \(F(q)\) ) |
|---|---|---|
| 2 Years | 0.50 | 0.50 |
| 5 Years | 0.20 | 0.80 |
| 10 Years | 0.10 | 0.90 |
| 20 Years | 0.05 | 0.95 |
| 50 Years | 0.02 | 0.98 |
| 100 Years | 0.01 | 0.99 |
Let \(F(q)\) be the cumulative distribution function (CDF) of the fitted model. This function maps flood magnitudes to exceedance probabilities: \(p_e = 1 - F(q)\). To estimate flood magnitudes for a given exceedance probability, we use the inverse of the CDF, better known as the quantile function: \(\hat{q} = F^{-1}(p_e)\).
Example Plot
FFA results are typically visualized with return period on the \(x\)-axis and flood magnitude on the \(y\)-axis. These plots can be interpreted in two directions:
-
Estimate flood magnitude for a given return period. For example, A 50-year flood is estimated to be about \(85\ \text{m}^3/\text{s}\).
-
Estimate return period for a given flood magnitude. For example, A streamflow of \(50\ \text{m}^3/\text{s}\) is expected to occur roughly every 4 years.
Note: For an explanation of the confidence bounds in this plot, see Uncertainty Quantification.
Handling Nonstationarity
A probability model is considered nonstationary if its statistical properties (e.g., location or scale) change over time. In such cases, the quantile function becomes time-dependent: \(F^{-1}(p_e, t)\).
As a result, return levels and exceedance probabilities vary with time, and a static return period curve is no longer valid.
To address this, the FFA framework computes effective return periods, which yield flood estimates for a specific year based on the time-varying distribution.
Example Plot
The plot below illustrates effective return levels for the year 2017. Remember, a 100-year effective return level does not imply that such a flood is expected to occur once in the next 100 years. It means that in the year 2017, the probability of exceeding that flood magnitude is 1 in 100.
