--- title: "Gumbel 500-year flood and 80% limits, Kamalamai River" subject: "Engineering Hydrology" chapter: 6 key: "semester_5/engineering_hydrology:gumbel-kamalamai-80cl" source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/gumbel-kamalamai-80cl site: BCT Engineering --- # Gumbel 500-year flood and 80% limits, Kamalamai River Engineering Hydrology · Chapter 6: Flood Hydrology - Asked in: 2081 Bhadra, 8 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/gumbel-kamalamai-80cl - Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-6 ## Question 30 years flood data of Kamalamai River has been used for frequency analysis and after fitting the Gumbel distribution the 100-year and 50-year floods are predicted equal to 1200 m$^3$/s and 1060 m$^3$/s respectively. Calculate the magnitude of flood for 500 years return period if reduced mean and reduced standard deviation in Gumbel's distribution for available data series are 0.5362 and 1.1124 respectively. Also calculate 80% confidence limits of 500-year flood if $f(c)$ for 80% is 1.28. ## Short answer From 100-yr and 50-yr: σ=223.1 m³/s, mean=385.1 m³/s. 500-yr flood: K=5.104, discharge=1524 m³/s. 80% limits: 1524±1.28×245.3 = 1210 to 1838 m³/s. ## Answer #### Statistics of the series $\bar y_n=0.5362$, $S_n=1.1124$, $y_T=-\ln[\ln(T/(T-1))]$, $K=(y_T-\bar y_n)/S_n$, $x_T=\bar x+K\sigma$. | T (yr) | $y_T$ | $K$ | $x_T$ (m³/s) | |---|---|---|---| | 100 | 4.600 | 3.653 | 1200 | | 50 | 3.902 | 3.026 | 1060 | Subtract the two equations: $$ \sigma=\frac{1200-1060}{3.653-3.026}=223.1 \text{ m}^3/\text{s} $$ $$ \bar x=1200-3.653\times223.1=385.1 \text{ m}^3/\text{s} $$ **Answer: mean = 385.1 m³/s, standard deviation = 223.1 m³/s.** #### 500-year flood $$ y_{500}=-\ln[\ln(500/499)]=6.214,\qquad K=\frac{6.214-0.5362}{1.1124}=5.104 $$ $$ x_{500}=385.1+5.104\times223.1=1524 \text{ m}^3/\text{s} $$ **Answer: $x_{500}$ ≈ 1524 m³/s.** #### 80% confidence limits of the 500-year flood $$ b=\sqrt{1+1.3K+1.1K^2}=\sqrt{1+1.3(5.104)+1.1(5.104)^2}=6.024 $$ $$ S_e=\frac{b\sigma}{\sqrt N}=\frac{6.024\times223.1}{\sqrt{30}}=245.3 \text{ m}^3/\text{s} $$ $$ x_T'=x_T\pm f(c)S_e=1524\pm1.28\times245.3=1524\pm314 $$ **Answer: 80% confidence limits are 1210 m³/s and 1838 m³/s.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What are the mean and standard deviation of the series? A. 1200 and 1060 B. 3.653 and 3.026 C. 385.1 and 223.1 m^3/s D. 223.1 and 385.1 m^3/s Answer: C. sigma = 140/(3.653 - 3.026) = 223.1, mean = 1200 - 3.653 x 223.1 = 385.1. (Key point: Finding sigma and mean) 2. x500 is A. 1200 + 500 B. 385.1 + 5.104 x 223.1 = 1524 m^3/s C. 385.1 x 5.104 D. 223.1 x 5.104 Answer: B. Use x_T = mean + K sigma with K = 5.104. (Key point: 500-year flood) 3. How is S_e computed? A. b/sqrt(N) B. sigma/N C. b sigma x sqrt(N) D. b sigma/sqrt(N) = 245.3 Answer: D. b = 6.024, sigma = 223.1, N = 30. (Key point: Standard error) 4. Why is the interval narrower than a 95% one? A. N differs B. S_e is smaller C. f(c) = 1.28 is smaller than 1.96 D. The mean is different Answer: C. Limits are x_T ± f(c) S_e = 1524 ± 314. (Key point: 80% limits) ## 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 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 analysis, 80-year and 1.4-year floods](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-80yr-620000-1-4yr) — 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 30-year flood, Lothar River 1989-2009](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-lothar-1989-2009) — Engineering Hydrology, chapter 6