Understanding Flood Frequency and Return Period

Flood frequency analysis is a common hydrological tool used to estimate the probability of future flood events of a certain magnitude. A key concept in this analysis is the 'return period', which helps engineers and planners design structures like bridges, culverts, and dams to withstand potential floods.

What is an Annual Flood Series?

An annual flood series is created by taking the maximum flood peak discharge that occurred in each year over a significant period (e.g., 30, 50, or more years) at a specific location on a river or stream. This series of maximum annual flood events is then used for statistical analysis.

Return Period Explained

The return period ($T$) of a hydrological event (like a flood of a certain magnitude) is the average time interval between events of that magnitude or larger. It's often expressed in years. A flood with a return period of 100 years means that, on average, a flood of that size or larger is expected to occur once every 100 years. This is a statistical average, not a guarantee; such a flood could occur in consecutive years or not for several hundred years.

The probability of an event with return period $T$ occurring in any given year is $P = \frac{1}{T}$. Conversely, the return period is the reciprocal of the probability, $T = \frac{1}{P}$.

Calculating Return Period using Plotting Positions

When analyzing an annual flood series, we often rank the flood magnitudes and assign them a 'plotting position'. The plotting position is used to estimate the empirical probability of exceedance for each ranked event, which then helps calculate the return period.

Different formulas exist for estimating plotting positions. A commonly used one is the Weibull formula. For an annual flood series arranged in decreasing order of magnitude (from largest to smallest), the rank 'm' represents the position in this sorted list. The largest flood has rank m=1, the second largest m=2, and so on.

The Weibull formula for estimating the probability of exceedance ($P$) for an event with rank 'm' in a series of 'N' total entries, when sorted in decreasing order, is given by:

$\qquad P = \frac{m}{N + 1}$

This formula estimates the probability that a flood of a certain magnitude (corresponding to rank 'm') will be equaled or exceeded in any given year.

Since the return period ($T$) is the reciprocal of the probability of exceedance ($P$), we can find the return period using the formula:

$\qquad T = \frac{1}{P}$

Substituting the Weibull probability formula into the return period formula:

$\qquad T = \frac{1}{\frac{m}{N + 1}} = \frac{N + 1}{m}$

This formula gives the estimated return period for a flood event ranked 'm' in an annual flood series of 'N' events, when the series is arranged in decreasing order of magnitude.

Let's check the given options against this derived formula:

1. m/N: This is not the standard formula for return period. It relates more to probability in some contexts but not return period directly like this.
2. m/(N + 1): This is the probability of exceedance (P), not the return period (T).
3. (N + 1)/m: This matches our derived formula for return period ($T$).
4. N/(m + 1): This is another plotting position formula (Gringorten formula) but it estimates probability. The corresponding return period would be $(m+1)/N$, which is not given and also not the formula derived from the standard Weibull plotting position for return period.

Therefore, the correct formula for the return period of an annual flood event ranked 'm' in a series of 'N' entries, arranged in decreasing order, is $\frac{N+1}{m}$.

Here is a summary of the terms:

Summary of Flood Frequency Analysis

• Collect annual maximum flood peak data for N years.
• Arrange the data in decreasing order of magnitude.
• Assign ranks (m=1, 2, ..., N) to the ordered data.
• Calculate the return period for each flood magnitude using $T = \frac{N+1}{m}$.
• Plot flood magnitude vs. return period or probability on appropriate graph paper (e.g., log-probability paper) to develop a flood frequency curve.

Revision Table: Key Flood Frequency Concepts

Additional Information on Flood Frequency Analysis

• Other Plotting Positions: Besides Weibull, other formulas exist like California (m/N), Hazen (m-0.5)/N, and Gringorten (m-0.44)/(N+0.12). Weibull's formula is commonly used due to its theoretical basis related to order statistics.
• Risk and Reliability: The probability of an event occurring at least once in 'n' years (risk) can be calculated. Reliability is 1 - Risk. For example, the risk of a 100-year flood occurring in 30 years is $1 - (1 - \frac{1}{100})^{30}$.
• Limitations: Flood frequency analysis assumes that the hydrological system is stationary (i.e., the statistical properties don't change over time). Climate change, land use changes, and upstream reservoir construction can violate this assumption. Extrapolating far beyond the period of record (N) can be uncertain.
• Frequency Distribution: Statistical distributions like Gumbel (Extreme Value Type I), Log-Pearson Type III, and Log-Normal distributions are often fitted to the annual flood series data to provide more robust estimates, especially for rare events, compared to just using plotting positions directly for extrapolation.