--- title: "Flood of return period T using log-normal distribution" subject: "Engineering Hydrology" chapter: 6 key: "semester_5/engineering_hydrology:log-normal-flood-procedure" source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/log-normal-flood-procedure site: BCT Engineering --- # Flood of return period T using log-normal distribution Engineering Hydrology · Chapter 6: Flood Hydrology - Asked in: 2082 Bhadra, 6 marks - Video: [Flood Estimation using Microsoft Excel: Logarithm Methods](https://www.youtube.com/watch?v=y8n_NobkwHg) — Revit for Civil Engineers - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/log-normal-flood-procedure - Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-6 ## Question If the annual flood series data for a catchment is available for N consecutive years, explain a procedure to determine a flood discharge with a return period of T (where T>N), by using log-normal distribution. ## Short answer Take logs of annual maxima, compute mean ȳ and standard deviation S_y. For return period T with probability P=1/T, find the standard normal variate z_T. Then log flood y_T = ȳ + z_T S_y, and x_T = 10^(y_T). ## Answer **Log-normal distribution.** The logarithm of the annual flood is assumed to be normally distributed. The flood of a given return period is then found using the standard normal variate in place of the frequency factor. #### Procedure (N years of annual maxima, T > N) 1. Arrange the annual flood series $x_1,\dots,x_N$ and take logarithms, $y_i=\log_{10}x_i$ (or ln). 2. Compute the mean and standard deviation of the log series: $$ \bar y=\frac1N\sum y_i,\qquad S_y=\sqrt{\frac{\sum(y_i-\bar y)^2}{N-1}} $$ 3. Compute the exceedance probability $P=1/T$ and the non-exceedance probability $F=1-1/T$. 4. Find the standard normal variate $z_T$ corresponding to $F$ from the standard normal table (the frequency factor of the log-normal distribution, $K_T=z_T$). A good approximation is $z=w-\dfrac{2.515517+0.802853w+0.010328w^2}{1+1.432788w+0.189269w^2+0.001308w^3}$ with $w=\sqrt{\ln(1/P^2)}$ for $0 N), so confidence limits should be considered; the log-normal distribution is a special case of log-Pearson III with zero skew. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the log-normal method assume? A. The logarithm of the annual flood is normally distributed B. The flood itself is normally distributed C. Floods follow Gumbel D. Skewness of logs is large Answer: A. Log-normal means log x is normal, so normal-distribution tools apply to the logs. (Key point: Assumption) 2. What is the frequency factor K_T in the log-normal method? A. The standard normal variate z_T for F = 1 - 1/T B. Cs-based factor from Pearson tables C. y_T of Gumbel D. 1/T Answer: A. K_T = z_T, e.g. 2.054 for T = 50 and 2.326 for T = 100. (Key point: Frequency factor) 3. After y_T = y-bar + z_T S_y, what is the last step? A. Divide by N B. x_T = log y_T C. x_T = y_T D. x_T = 10^(y_T) (or e^(y_T) for natural logs) Answer: D. The result is in log units and must be converted back to a flood. (Key point: Back-conversion) 4. How is log-normal related to log-Pearson III? A. It is the infinite-skew case B. They are unrelated C. It uses raw data skew D. It is the zero-skew special case Answer: D. With Cs of the logs equal to zero, K_T equals z_T. 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