2072 Kartik · 7 marks

Gumbel floods from 25 and 250-year values

Short answer

Flood magnitudes are found by linear interpolation in Gumbel reduced variate y_T. The line through (25yr,90 m³/s) and (250yr,550 m³/s) gives x_50≈229 m³/s, x_500≈688 m³/s, x_1000≈825 m³/s.

The flood discharge for 25 and 250 years from fitted Gumbel distribution are 90 and 550 m $^3$ /sec respectively. Estimate the flood magnitudes for 50, 500 and 1000 years by Gumbel analytically.

Gumbel's equation $x_T=\bar x+K\sigma$ is linear in the reduced variate $y_T$ , so two known floods fix the line and the other floods follow by proportion (no need for the sample size).

y_T=-\ln\left[\ln\frac{T}{T-1}\right]

Slope of the line:

b=\frac{x_{250}-x_{25}}{y_{250}-y_{25}}=\frac{550-90}{5.5195-3.1985}=198.20 \text{ m}^3/\text{s}

x_T=90+198.20\,(y_T-3.1985)

T (yr)	y_T	x_T (m³/s)
25	3.1985	90
50	3.9019	229.4
250	5.5195	550
500	6.2136	687.6
1000	6.9073	825.1

Answer: $x_{50}$ ≈ 229 m³/s, $x_{500}$ ≈ 688 m³/s, $x_{1000}$ ≈ 825 m³/s.

✓ Answer checked by Claude Sonnet 5.5

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