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Home/Biophysics, Fluids & Geoscience/Flood Frequency (Gumbel)

Flood Frequency (Gumbel)

Annual maxima ~ Gumbel EV1: return period T, exceedance probability, design flood x_T, and method-of-moments fit to a synthetic sample.

Gumbel flood frequency

120 m³/s
35 m³/s
100 yr
40 yr

Annual maxima ~ Gumbel (EV1). Exceedance probability = 1/T. Design flood x_T = μ − β ln(−ln(1−1/T)). Gray dots use Weibull plotting positions on the sample.

Shortcuts

  • •Tune μ, β and return period T; reseed to draw a new annual-maxima sample.

Measured values

Exceedance P(Q>x_T)0.0100
Non-exceedance F0.9900
Design flood x_T281.0m³/s
MOM-fit x_T311.0m³/s
Sample mean141.4m³/s
Sample σ54.1m³/s

About this model

Flood frequency analysis models annual peak discharges with an extreme-value distribution. The Gumbel (EV1) law has CDF F(x)=exp(−exp(−(x−μ)/β)). A T-year flood has exceedance probability 1/T and quantile x_T=μ−β ln(−ln(1−1/T)). The simulator shows the PDF with a synthetic annual-maxima sample, the Q–T frequency curve, and a method-of-moments fit for comparison.

Who it's for: Hydrology, water resources, civil/environmental engineering, and statistics of extremes courses.

Key terms

  • Gumbel distribution
  • Return period
  • Exceedance probability
  • Annual maxima
  • Design flood
  • Method of moments

How it works

Flood frequency analysis with the Gumbel (Extreme Value Type I) distribution for annual maxima. Set location and scale, pick a return period, and compare the true quantile with a method-of-moments fit to a synthetic sample.

Key equations

F(x) = exp(−exp(−(x−μ)/β))
T = 1/(1−F) · x_T = μ − β ln(−ln(1−1/T))
MOM: β̂ = σ√6/π , μ̂ = x̄ − γ β̂

Frequently asked questions

What is a 100-year flood?
It is the discharge with annual exceedance probability 1/100 — not a flood that occurs exactly once per century. In any given year the chance is still 1%.
Why Gumbel?
It is the classical Type-I limit for maxima of many parent distributions and remains a standard first model in flood frequency analysis before GEV/LP3.