Understanding Return Period in Hydrology

Key Concepts

• Return Period: Refers to the average interval of time between events like rainfall or floods exceeding a certain magnitude.
• Used in hydrology for:
  • Rainfall: Must specify the duration (e.g., 2-hour or 24-hour rainfall).
  • Flood: Related to the magnitude of discharge (e.g., 5000 m³/s).

Calculating Return Period for Floods

1. Data Collection: Gather maximum discharge data from a stream gauging site over a period (e.g., 30 years).
2. Determining Frequency: Count how many times discharge exceeds a specified value (e.g., 5000 m³/s).
   • Example: Discharge exceeds 5000 m³/s five times in 30 years.
3. Time Intervals: Calculate the years between each exceedance. These intervals may vary.
   • E.g., intervals of 5, 4, 4, 2 years.
4. Calculate Average: Average time interval = Total period / Number of intervals.
   • E.g., 30 years / 4 intervals = 7.5 years (average return period).
5. Interpretation: The average does not guarantee the event occurs every 7.5 years; it's a probabilistic average.

Probability and Return Period

• Probability of Occurrence: Probability that discharge exceeds a value in a given year = 1/T.
• Independence: Assumes yearly event independence, suitable for binomial distribution modeling.

Probability Modeling

• Binomial Distribution: Used to find the probability of an event occurring R times in N years.
  • Formula: P(N, R) = nCr * p^R * q^(N-R)
  • p = 1/T, q = 1-p

Definitions

• Risk: Probability of event occurring at least once in N years.
  • Formula: R = 1 - q^n
• Reliability: Probability of event not occurring in N years.
  • Formula: Reliability = 1 - Risk = q^n

Example Problem

• Given: 280 mm rainfall in one day with a 50-year return period.
• Tasks:
  1. Once in 20 Years: Use binomial distribution with N=20, R=1.
     • Probability: ~0.272
  2. Twice in 15 Years: Use binomial distribution with N=15, R=2.
     • Probability: ~0.0323
  3. At Least Once in 20 Years: Calculate risk.
     • Probability: ~0.332

Additional Note

• Rainfall events are also tied to specific durations, affecting return period calculations.
• Different durations (e.g., one day vs. 12-hour events) result in different return periods.