Frequency Analysis

Flood Frequency Analysis (FFA) is the act of using a fitted probability distribution to make predictions about the frequency of extreme streamflow events (i.e. floods). To do FFA, we require a probability model with suitably chosen parameters based on the data. This section will assume that these requirements are met.

Typically, we describe the severity of floods in terms of their return period. Suppose we have a flood, which I will refer to as \(A\). If we expect to see a flood at least as severe as \(A\) every ten years, then we say that \(A\) is a ten-year flood. Since our framework uses annual maximum streamflow data, a ten-year flood corresponds to an exceedance probability of \(0.1\). Note that an exceedance probability of \(0.1\) corresponds to the \(1 - 0.1 = 0.90\) quantile of our probability distribution. Here is a table of the return periods, exceedance probabilities, and quantiles used in the FFA framework:

Return Period	Exceedance Probability	Quantile
\(2\) 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	\(0.01\)	\(0.99\)

Suppose our fitted probability distribution has cumulative distribution function \(F(x)\). The function \(F(x)\) maps annual maximum streamflow values to quantiles. However, we want to determine the streamflow from the quantiles, so we use the inverse of the cumulative distribution \(F^{-1}(x)\) instead. The function \(F^{-1}(x)\) is also known as the Quantile Function.

Note: The FFA framework uses the quantile functions implemented in the lmom package.

We typically present the results of a flood frequency analysis as a graph with time on the \(x\)-axis and annual maximum streamflow on the \(y\)-axis. There are two ways to read a graph like this:

1. Estimate the severity of a flood for a given time period. From the graph below, we could determine that every 50 years, we can expect a streamflow event of approximately \(4500\text{m}^3/\text{s}\).
2. Estimate the frequency of a flood for a given severity. From the graph below, we could determine that a streamflow event of \(3000\text{m}^3/\text{s}\) or higher will occur every 6-7 years.

Note: For information about how the confidence bounds in this figure are computed, see here.

Handling Non-Stationarity

We say that a distribution is Non-Stationary if its mean or variance (or both) are changing over time. Under non-stationarity, the quantile function of our fitted probability distribution is also a function of time \(F^{-1}(x, t)\).

An Idea for Estimating Return Periods

Suppose we want to estimate the severity of a flood with return period \(t_{0}\) at time \(t^{*}\).

1. Generate a sample \(x_{1}, \dots, x_{t_{0}}\) where \(x_{i} \sim \text{Unif} [0, 1]\).
2. Compute simulated streamflow events \(y_{1}, \dots, y_{t_{0}}\) where \(y_{i} = F^{-1}(x_{i}, t^{*} + i)\).
3. Let \(z = \max (y_{1}, \dots, y_{t_{0}})\) be the most severe streamflow event in the sample.
4. Repeat steps 1-3 \(n\) times to generate a sequence \((z_{1}, \dots, z_{n})\).
5. The mean \((z_{1}, \dots, z_{n}) / n\) is an accurate estimate of the severity.