--- title: "Preparing Gumbel probability paper" subject: "Engineering Hydrology" chapter: 6 key: "semester_5/engineering_hydrology:gumbel-probability-paper" source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/gumbel-probability-paper site: BCT Engineering --- # Preparing Gumbel probability paper Engineering Hydrology · Chapter 6: Flood Hydrology - Asked in: 2067 Mangsir (old course), 6 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/gumbel-probability-paper - Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-6 ## Question Prepare a Gumbel probability paper from an ordinary graph paper provided to you. ## Short answer Use ordinary arithmetic axes: vertical for discharge, horizontal uniformly scaled for reduced variate y_T = –ln[ln(T/(T–1))]. Mark T values at their y_T positions (e.g., T=5 at 1.50) to create a non-uniform T scale; Gumbel data plot as a straight line. ## Answer **Gumbel probability paper** is a plotting sheet on which data that follow Gumbel's distribution plot as a straight line. It can be prepared from ordinary arithmetic graph paper because Gumbel's equation $x_T=\bar x+K\sigma$ is linear in the reduced variate $y_T$. #### Steps 1. Choose the **vertical axis** as an ordinary arithmetic scale for the variable (flood discharge or rainfall depth), covering the range of data and predictions. 2. For selected return periods compute the reduced variate $$ y_T=-\ln\left[\ln\frac{T}{T-1}\right] $$ | T (years) | 2 | 5 | 10 | 20 | 50 | 100 | 200 | 500 | 1000 | |---|---|---|---|---|---|---|---|---|---| | $y_T$ | 0.367 | 1.500 | 2.250 | 2.970 | 3.902 | 4.600 | 5.296 | 6.214 | 6.907 | 3. Draw the **horizontal axis** with a uniform scale for $y_T$ (for example 1 cm = 1 unit of $y_T$ from −1 to 7). 4. At the positions of the computed $y_T$ values mark the return periods T (and the probability of exceedance $P=1/T$) as labels. This gives the non-uniform return period scale along the top or bottom of the sheet. 5. For plotting data: arrange in descending order, assign $T=(N+1)/m$, and plot the flood against T (that is, against its $y_T$ position). 6. Draw the best-fit straight line and extend it to read the flood for any desired T. ```text Q | o | o | o | o | o +--+----+----+----+----+---- y_T 0.37 1.50 2.25 3.90 4.60 6.21 T=2 T=5 T=10 T=50 T=100 T=500 ``` Two points are enough to fix the straight line, since for Gumbel's distribution the line is $Q = \bar x + \sigma\,(y_T-\bar y_n)/S_n$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why can Gumbel paper be made from ordinary graph paper? A. All data are normally distributed B. Discharge is linear in T C. x_T = mean + K sigma is linear in the reduced variate y_T D. The horizontal axis is logarithmic in x Answer: C. Linearity in y_T means a uniform y_T axis makes Gumbel data plot straight. (Key point: Why it works) 2. How is the horizontal axis drawn? A. Uniform scale in T B. Uniform in P only C. Uniform scale in y_T, with T labels at the computed positions D. Uniform scale in log T Answer: C. Return periods are marked at their y_T positions, making the T labels non-uniform. (Key point: Horizontal axis) 3. Which pair of y_T values is correct? A. T = 2: 2.0; T = 100: 100 B. T = 10: 1.0; T = 100: 2.0 C. T = 2: 0.367; T = 100: 4.600 D. T = 2: 0.693; T = 100: 4.605 negative Answer: C. y_T = -ln(ln(T/(T-1))) gives 0.367 for T = 2 and 4.600 for T = 100. (Key point: Reduced variate values) 4. How is observed data plotted? A. Plot in chronological order against year B. Plot ascending with T = m/N C. Arrange in descending order, T = (N+1)/m, plot flood against T, draw best-fit line D. Plot against m only Answer: C. Plotting positions give T, which fixes the y_T position, then a best-fit line is extended. 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