---
title: "Gumbel floods from 25 and 250-year values"
subject: "Engineering Hydrology"
chapter: 6
key: "semester_5/engineering_hydrology:gumbel-25-250-to-50-500"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/gumbel-25-250-to-50-500
site: BCT Engineering
---

# Gumbel floods from 25 and 250-year values

Engineering Hydrology · Chapter 6: Flood Hydrology

- Asked in: 2072 Kartik, 7 marks
- Video: [Gumbel's method in Hydrology for flood estimation | Flood Routing | GATE 2022 |](https://www.youtube.com/watch?v=gtUiLfn_E8U) — TechnoCivil
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/gumbel-25-250-to-50-500
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-6

## Question

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.

## 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.

## Answer

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.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Two known floods suffice because x_T is:
   A. Linear in T
   B. Independent of y_T
   C. Linear in the reduced variate y_T
   D. Quadratic in T
   Answer: C. x_T = a + b y_T. (Key point: Linearity)

2. The slope is:
   A. (550-90)/(250-25)
   B. 550/90
   C. (550+90)/2
   D. (550-90)/(5.5195-3.1985) = 198.2
   Answer: D. Difference in x over difference in y. (Key point: Slope)

3. The 50-year flood is about:
   A. 229 m3/s
   B. 90 m3/s
   C. 550 m3/s
   D. 688 m3/s
   Answer: A. 90 + 198.2 x (3.9019 - 3.1985). (Key point: x50)

4. The 1000-year flood is about:
   A. 688 m3/s
   B. 825 m3/s
   C. 550 m3/s
   D. 1000 m3/s
   Answer: B. 90 + 198.2 x (6.9073 - 3.1985). (Key point: x1000)

## Related in Engineering Hydrology

- [Gumbel 500-year flood, 95% limits, 30 years](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-30yr-100-50-1200-1060-2) — Engineering Hydrology, chapter 6
- [Gumbel 500-year flood from two risk-based floods](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-risk-63-6-4200-5800) — Engineering Hydrology, chapter 6
- [Gumbel analysis from 100 and 50-year floods](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-30yr-100-50-1200-1060) — Engineering Hydrology, chapter 6
- [Gumbel 200-year flood from 50 and 100-year floods](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-35yr-200yr-flood) — Engineering Hydrology, chapter 6
- [Gumbel 500-year flood and 80% limits, Kamalamai River](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-kamalamai-80cl) — Engineering Hydrology, chapter 6
- [Gumbel flood at given risk, 50-year project life](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-risk-63-58-4200) — Engineering Hydrology, chapter 6

## Related in other subjects

- [Cross drainage: Q 50, HFD 550](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Acd-q50-bed30-fsd21-hfd550) — Irrigation and Drainage Engineering, chapter 8, asked 2×