--- title: "Double mass curve, station X and 15 stations, 2013-2022" subject: "Engineering Hydrology" chapter: 2 key: "semester_5/engineering_hydrology:dmc-x-15stn-2013-2022" source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/dmc-x-15stn-2013-2022 site: BCT Engineering --- # Double mass curve, station X and 15 stations, 2013-2022 Engineering Hydrology · Chapter 2: Precipitation - Asked in: 2080 Bhadra, 8 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/dmc-x-15stn-2013-2022 - Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-2 ## Question Annual Precipitation of station X and average of the 15 surrounding stations are listed below. Examine the consistency of the station X. In which year, the change in regime observed? Make adjustments in the rainfall data wherever applicable. | Year | 2022 | 2021 | 2020 | 2019 | 2018 | 2017 | 2016 | 2015 | 2014 | 2013 | |---|---|---|---|---|---|---|---|---|---|---| | Rainfall at station X (mm) | 173 | 290 | 225 | 246 | 260 | 493 | 320 | 305 | 340 | 437 | | Average rainfall at surrounding stations (mm) | 234 | 333 | 236 | 251 | 284 | 361 | 282 | 240 | 265 | 302 | ## Short answer The record is inconsistent with a break after 2017. Earlier data (2013–2017) are multiplied by 0.683 to adjust to the present regime: 298.4, 232.2, 208.3, 218.5 and 336.6 mm. ## Answer The data are first arranged in chronological order (2013 to 2022) and the cumulative values are found. | Year | X (mm) | 15-stn avg (mm) | Cum. avg | Cum. X | X / avg | |---|---|---|---|---|---| | 2013 | 437 | 302 | 302 | 437 | 1.45 | | 2014 | 340 | 265 | 567 | 777 | 1.28 | | 2015 | 305 | 240 | 807 | 1082 | 1.27 | | 2016 | 320 | 282 | 1089 | 1402 | 1.13 | | 2017 | 493 | 361 | 1450 | 1895 | 1.37 | | 2018 | 260 | 284 | 1734 | 2155 | 0.92 | | 2019 | 246 | 251 | 1985 | 2401 | 0.98 | | 2020 | 225 | 236 | 2221 | 2626 | 0.95 | | 2021 | 290 | 333 | 2554 | 2916 | 0.87 | | 2022 | 173 | 234 | 2788 | 3089 | 0.74 | #### Consistency check Plot cumulative X (y) against cumulative average of the 15 stations (x). The ratio X/avg is about 1.1 to 1.45 up to 2017, and below 1 from 2018. So the points follow one straight line up to the 2017 point and a flatter line after it. - Slope up to 2017: $M_a = \dfrac{1895}{1450} = 1.307$ - Slope after 2017: $M_c = \dfrac{3089 - 1895}{2788 - 1450} = \dfrac{1194}{1338} = 0.892$ The slopes differ, so the record of station X is **not consistent**. The **change in regime occurred after 2017 (from 2018)**. #### Adjustment The earlier data (2013-2017) are adjusted to the present regime (slope $M_c$): $$ P_{cx} = P_x\times\frac{M_c}{M_a} = P_x\times\frac{0.892}{1.307} = 0.683\,P_x $$ | Year | Observed (mm) | Adjusted (mm) | |---|---|---| | 2013 | 437 | 298.4 | | 2014 | 340 | 232.2 | | 2015 | 305 | 208.3 | | 2016 | 320 | 218.5 | | 2017 | 493 | 336.6 | Data for 2018-2022 are unchanged. **Answer: Station X is inconsistent; change in regime from 2018 (break after 2017); the 2013-2017 values are multiplied by 0.683 to give 298.4, 232.2, 208.3, 218.5 and 336.6 mm.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the consistency of station X checked? A. Plot annual X against year and look for a trend B. Plot cumulative X against cumulative average of the surrounding stations and look for a change in slope C. Compare the mean of X with the mean of one neighbour D. Plot X against its own standard deviation Answer: B. A double mass curve of cumulative X versus cumulative neighbour average is a straight line if X is consistent; a break in slope shows a change in regime. (Key point: Double mass curve) 2. In which year did the regime of station X change? A. From 2013 B. From 2015 C. From 2017 D. From 2020 Answer: C. The points follow one line up to 2016 and a flatter line from 2017, so the change is from 2017. (Key point: Year of change) 3. What is the slope Ma of the line up to 2016? A. 1402/1089 = 1.287 B. 1687/1699 = 0.993 C. 3089/2788 = 1.108 D. 437/302 = 1.447 Answer: A. Ma = cumulative X / cumulative average at 2016 = 1402/1089 = 1.287. (Key point: Slope before change) 4. How are the 2013-2016 data adjusted? A. Multiplied by Ma/Mc = 1.30 B. Multiplied by Mc/Ma = 0.771 to match the present regime C. Reduced by 0.771 mm D. Left unchanged and 2017-2022 data are changed Answer: B. Old data are brought to the present regime by P_cx = P_x x Mc/Ma = 0.771 P_x, e.g. 437 gives 337.0 mm. 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