--- title: "Double mass curve, station X and 18 stations" subject: "Engineering Hydrology" chapter: 2 key: "semester_5/engineering_hydrology:dmc-x-18" source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/dmc-x-18 site: BCT Engineering --- # Double mass curve, station X and 18 stations Engineering Hydrology · Chapter 2: Precipitation - Asked in: 2079 Baisakh, 8 marks - Video: [Double Mass Curve Analysis Using Excel | Hydrology](https://www.youtube.com/watch?v=GlRb3JydBtg) — ENG-School - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/dmc-x-18 - Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-2 ## Question The annual rainfall at station X and the average annual rainfall at 18 surrounding stations are given below. Check the consistency of the record at station X and determine the year in which a change in regime has occurred. Determine the average annual rainfall for the period 1952-1970 for the changed regime. | Year | Station X (cm) | 18-stations' average (cm) | |---|---|---| | 1952 | 30.5 | 22.8 | | 1953 | 38.9 | 35 | | 1954 | 43.7 | 30.2 | | 1955 | 32.2 | 27.4 | | 1956 | 27.4 | 25.2 | | 1957 | 32 | 28.2 | | 1958 | 49.3 | 36.1 | | 1959 | 28.4 | 18.4 | | 1960 | 24.6 | 25.1 | | 1961 | 21.8 | 23.6 | | 1962 | 28.2 | 33.3 | | 1963 | 17.3 | 23.4 | | 1964 | 22.3 | 36 | | 1965 | 28.4 | 31.2 | | 1966 | 24.1 | 23.1 | | 1967 | 26.9 | 23.4 | | 1968 | 20.6 | 23.1 | | 1969 | 29.5 | 33.2 | | 1970 | 28.4 | 26.4 | ## Short answer A double mass curve shows a slope change after 1959, so the record is inconsistent with a regime change in 1960. The adjusted mean annual rainfall for 1952–1970 under the changed regime is 24.9 cm. ## Answer A double mass curve plots the cumulative rainfall of station X against the cumulative average rainfall of the 18 base stations. A change in slope shows an inconsistency. #### Step 1: Cumulative values | Year | X (cm) | Base avg (cm) | Cum. base | Cum. X | |---|---|---|---|---| | 1952 | 30.5 | 22.8 | 22.8 | 30.5 | | 1953 | 38.9 | 35 | 57.8 | 69.4 | | 1954 | 43.7 | 30.2 | 88.0 | 113.1 | | 1955 | 32.2 | 27.4 | 115.4 | 145.3 | | 1956 | 27.4 | 25.2 | 140.6 | 172.7 | | 1957 | 32 | 28.2 | 168.8 | 204.7 | | 1958 | 49.3 | 36.1 | 204.9 | 254.0 | | 1959 | 28.4 | 18.4 | 223.3 | 282.4 | | 1960 | 24.6 | 25.1 | 248.4 | 307.0 | | 1961 | 21.8 | 23.6 | 272.0 | 328.8 | | 1962 | 28.2 | 33.3 | 305.3 | 357.0 | | 1963 | 17.3 | 23.4 | 328.7 | 374.3 | | 1964 | 22.3 | 36 | 364.7 | 396.6 | | 1965 | 28.4 | 31.2 | 395.9 | 425.0 | | 1966 | 24.1 | 23.1 | 419.0 | 449.1 | | 1967 | 26.9 | 23.4 | 442.4 | 476.0 | | 1968 | 20.6 | 23.1 | 465.5 | 496.6 | | 1969 | 29.5 | 33.2 | 498.7 | 526.1 | | 1970 | 28.4 | 26.4 | 525.1 | 554.5 | #### Step 2: Plot and slopes Plotting cum. X (y-axis) against cum. base average (x-axis), the points follow one straight line up to 1959 and another, flatter line afterwards. - Slope before the break (to 1959): $M_a = \dfrac{282.4}{223.3} = 1.265$ - Slope after the break (1960-1970): $M_c = \dfrac{272.1}{301.8} = 0.902$ The slopes differ, so the record of station X is **not consistent**. The **change in regime occurred in 1960** (the break appears at the 1959-60 point). #### Step 3: Adjust to the changed (present) regime $$ P_{cx} = P_x \times \frac{M_c}{M_a} = P_x \times \frac{0.902}{1.265} = 0.713\,P_x $$ Adjusted total for 1952-1959 $= 0.7129 \times 282.4 = 201.3$ cm; the total of 1960-1970 is unchanged $= 272.1$ cm. $$ \bar P_{1952-70} = \frac{201.3 + 272.1}{19} = 24.92\ \text{cm} $$ **Answer: Record is inconsistent; change in regime in 1960; mean annual rainfall for 1952-1970 under the changed regime = 24.9 cm** (compared with the unadjusted mean of 29.2 cm). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the double mass curve used here, what is plotted against what? A. Annual rainfall of X against the year B. Cumulative rainfall of X against cumulative average of the 18 base stations C. Cumulative base average against the year D. Rainfall of X against rainfall of one nearby station Answer: B. A double mass curve plots cumulative X (y) against cumulative base-station average (x); a break in slope shows inconsistency. (Key point: Double mass curve axes) 2. What shows that the record at station X is inconsistent? A. The points form one straight line B. Rainfall at X is always higher than the base average C. The slope changes from 1.265 to 0.902 D. The cumulative values increase every year Answer: C. Two different slopes mean the relationship between X and the base stations changed, so the record is not consistent. (Key point: Detecting inconsistency) 3. To bring the 1952-59 data to the changed (recent) regime, by what factor is P_x multiplied? A. M_a / M_c = 1.40 B. M_c x M_a = 1.14 C. M_c - M_a = -0.363 D. M_c / M_a = 0.713 Answer: D. Corrected P_cx = P_x x M_c/M_a = 0.902/1.265 = 0.713 times P_x. (Key point: Correction factor) 4. How is the mean for 1952-70 under the changed regime obtained? A. Adjusted 1952-59 total (201.3) plus unchanged 1960-70 total (272.1), divided by 19 B. Unadjusted total 554.5 divided by 19 C. Average of the two slopes times 19 D. Adjust the 1960-70 data and keep 1952-59 unchanged Answer: A. Only the earlier period is corrected; (201.3 + 272.1)/19 = 24.92 cm. 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