Flood Risk of a Homestead Mound

What is modelled

The highest flood in each year is modelled by a Gumbel distribution, the usual law for the largest of many values. A random record of annual maxima is drawn from it, and the model refits the law to that record, because a real planner only has the record and not the true law. The chance that a mound is overtopped in a year gives its return period, and the chance over many years gives the lifetime risk. A taller mound costs more earth, so there is a height that makes build cost plus expected loss smallest.

Equations

• Distribution F(x) = exp[−exp(−(x − μ)/β)], return level xT = μ − β ln(−ln(1 − 1/T)).
• Moments fit β = s √6 / π, μ = mean − 0.5772 β.
• Lifetime risk 1 − (1 − p)N with p = 1 − F(H).
• Mound volume as a conical frustum, cost 0.4 V + L (1 − F(H)N).

Checks run in the page

• The quantile and the distribution function against each other.
• The moment fit recovering the true parameters from a very long record.
• The likelihood fit satisfying both score equations and beating the moment fit.
• The lifetime risk against a simulation of many lifetimes.

Honest simplifications

Floods in different years are independent with a fixed climate, so there is no trend, no river regulation and no ice jams; the mound is a stable cone with no erosion and the loss is a single number. The cost weights (0.4 person-days per cubic metre, a 12 m radius platform) are illustrative, not historical values, and a short record gives wide uncertainty in the long return levels, as the dotted band shows.

• Press Short record and watch the fitted line wander away from the true one.
• Press Very variable river and see how the mound has to grow for the same risk.
• Change the random record number to draw another history.