Intensity-duration-frequency curve

2.1   What does precipitation frequency mean?

Rainfall frequency

The probability distribution specifying the exceedance probability of different rainfall depths for a given duration (such as 1 hour, or a 24-hour day).

The expected number of times, during a specified time period, that a given precipitation depth will be exceeded.

Rainfall Frequency

Rainfall Frequency means the probability of a rainfall event of defined characteristics occurring in any given year.

2.2   Return period

The average time between events, $T$. Also known as a recurrence interval.

The reciprocal of its corresponding exceedance probability $P=\frac{1}{T}$.

For example,

• A 2-year storm has an average time of 2 years between their occurrences statistically.
• It has an exceedance probability of 50% ($\frac{1}{2}$).

2.3   Exceedance probability

Try it yourself!

The probability of a variable being greater than or equal to a certain value.

What does a 100-year storm mean?

1. Once every 100 years?
2. Statistically once every 100 years? Yes
3. 100% exceedance probability?
4. 1% exceedance probability? Yes

3.1   See Figure 4.7

• Statistically, a rainfall event with a uniform intensity of 4 in/hr and a duration of 20 minutes has a return period of 5 years.
• Or a 5-yr rainfall event lasting 20 minutes would have a uniform intensity of 4 in/hr.
• For the 20-yr, 90-min design storm, the intensity and depth would be 2 in/hr and 3 inches, respectively.
• For the 20-yr, 30-min design storm, they would be 4.6 in/hr and 2.3 inches, respectively.
• Both design storms have the same frequency.
• What is the exceedance probability of these design storms?

3.2   Development of IDF curves

The annual exceedance probability (AEP) or $P(x)$ is the complement (a sum-to-1 relationship) of the cumulative distribution function (CDF) $F(x)$. That is, $F(x)=1-P(x)$.

3.3   100-yr 1-hr storm AEP vs. CDF

For example, a 100-yr 1-hr storm has a 1% AEP ($P=\frac{1}{100}=0.01$) and a 99% CDF ($F=0.99$).

What does it mean?

There is only a 1% chance of any 1-hr storm events exceeding the intensity of this 100-yr 1-hr storm.

99% recorded 1-hr storm events would have a lower intensity than this 100-yr 1-hr storm.

3.4   Estimating the AEP and return period

For the AEP ($P$), we commonly use \begin{equation} P=\frac{m}{n+1} \label{eq:p} \end{equation} where $m$ is a rank number in decreasing order of magnitude and $n$ is the number of observations.

The return period can now be estimated using the Weibull formula \begin{equation} T=\frac{1}{P}=\frac{n+1}{m} \label{eq:weibull} \end{equation}

3.5   Partial-duration series

• A complete series consists of all observed data.
• An annual series consists of annual extremes.
• A partial series consists of any subset of the complete series.

Any non-greatest event in one year can exceed the greatest event in another year. Such events are neglected in the annual series.

These extreme events are analyzed in a partial-duration series without regard for a return period.

3.6   Example 4.2

Perform a frequency analysis of the 30-min Baltimore rainfall data in Table 4.3 as an annual and a partial-duration series and plot the results.

3.7   Problem 4.17

For a 60-year record of precipitation intensities and durations, a 30-min intensity of 2.50 in/hr was equaled or exceeded a total of 85 times. All but 5 of the 60 years experienced one or more 30-min intensities equaling or exceeding the 2.50-in/hr value. Use the Weibull formula to determine the return period of this intensity using (a) a partial series and (b) an annual series.

3.8   General form of IDF relationships

The following general form is used to express IDF relationships: \begin{equation} i=\frac{a}{(t+b)^c} \label{eq:general_idf} \end{equation} where $a$, $b$, and $c$ are characteristic constants of both the rainfall region and frequency.

3.9   Example 4.3

Fit the following rainfall data to determine the 10-year intensity-duration-frequency curve. Use $c=1$, and determine $a$, $b$, and the correlation coefficient of the IDF curve. You may use Excel.

Use the correlation coefficient equation: \begin{equation} r_{xy}=\frac{\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^n(x_i-\bar{x})^2}\sqrt{\sum_{i=1}^n(y_i-\bar{y})^2}} \end{equation}

3.10   Example 4.4

Table 4.5 catalogs data for numerous storms having a range of intensities and durations. The entire record spans 40 years. By interpolating the values in the table, estimate the time versus intensity values for the 5-year storm.

See this figure again.

3.11   Example 4.5

You are given average annual values of precipitation (inches) for the period from 1889 to 1997 for Tallahasse, Florida (see table-4.6.csv). Arrange the data in order of magnitude from highest to lowest and calculate the exceedance probability for each annual value. Use Eq. 4.3 and determine the values as percentages. Plot the exceedance probabilities versus average annual precipitation.