---
title: "Double mass curve, station X and 25 stations, 1941-1976"
subject: "Engineering Hydrology"
chapter: 2
key: "semester_5/engineering_hydrology:dmc-x-25stn-36yr"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/dmc-x-25stn-36yr
site: BCT Engineering
---

# Double mass curve, station X and 25 stations, 1941-1976

Engineering Hydrology · Chapter 2: Precipitation

- Asked in: 2074 Asoj, 6+1+4+3 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/dmc-x-25stn-36yr
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-2

## Question

The annual rainfall at station X and the average of annual rainfall at 25 surrounding base stations in cm are given below for the period of 36 years starting from 1941.

i) Check whether the data of station X is consistent.
ii) In which year a change in regime is indicated?
iii) Compute the mean annual rainfall for station X at its present site for the given 36 year period, first without adjustment and secondly with the data adjusted for the change in regime.
iv) Compute the adjusted annual rainfall at station X for the affected period.

| Year | Rainfall at X | Average rainfall of base stations |
|---|---|---|
| 1941 | 163 | 135 |
| 1942 | 119 | 111 |
| 1943 | 121 | 124 |
| 1944 | 129 | 111 |
| 1945 | 126 | 123 |
| 1946 | 120 | 90 |
| 1947 | 153 | 138 |
| 1948 | 172 | 119 |
| 1949 | 127 | 108 |
| 1950 | 108 | 107 |
| 1951 | 126 | 111 |
| 1952 | 190 | 142 |
| 1953 | 112 | 112 |
| 1954 | 97 [?] | 99 |
| 1955 | 86 | 93 |
| 1956 | 111 | 131 |
| 1957 | 68 | 92 |
| 1958 | 88 | 142 |
| 1959 | 112 | 123 |
| 1960 | 95 | 142 |
| 1961 | 106 | 92 |
| 1962 | 81 | 91 |
| 1963 | 116 | 131 |
| 1964 | 112 | 104 |
| 1965 | 80 | 97 |
| 1966 | 88 | 111 |
| 1967 | 85 | 114 |
| 1968 | 90 | 92 |
| 1969 | 120 | 146 |
| 1970 | 72 | 93 |
| 1971 | 113 | 138 |
| 1972 | 82 | 112 |
| 1973 | 116 | 117 |
| 1974 | 122 | 152 |
| 1975 | 73 | 90 |
| 1976 | 74 | 104 |

## Short answer

A double mass curve reveals inconsistency with a regime change after 1952. Unadjusted mean is 109.8 cm; adjusted to the present regime it is 97.2 cm. The 1941–1952 data are corrected by a factor of 0.726.

## Answer

The double mass curve plots cumulative X against cumulative base-station average. A single straight line means consistent data.

#### (i) Cumulative values and consistency
| Year | X | Base avg | Cum. base | Cum. X |
|---|---|---|---|---|
| 1941 | 163 | 135 | 135 | 163 |
| 1942 | 119 | 111 | 246 | 282 |
| 1943 | 121 | 124 | 370 | 403 |
| 1944 | 129 | 111 | 481 | 532 |
| 1945 | 126 | 123 | 604 | 658 |
| 1946 | 120 | 90 | 694 | 778 |
| 1947 | 153 | 138 | 832 | 931 |
| 1948 | 172 | 119 | 951 | 1103 |
| 1949 | 127 | 108 | 1059 | 1230 |
| 1950 | 108 | 107 | 1166 | 1338 |
| 1951 | 126 | 111 | 1277 | 1464 |
| 1952 | 190 | 142 | 1419 | 1654 |
| 1953 | 112 | 112 | 1531 | 1766 |
| 1954 | 97 | 99 | 1630 | 1863 |
| 1955 | 86 | 93 | 1723 | 1949 |
| 1956 | 111 | 131 | 1854 | 2060 |
| 1957 | 68 | 92 | 1946 | 2128 |
| 1958 | 88 | 142 | 2088 | 2216 |
| 1959 | 112 | 123 | 2211 | 2328 |
| 1960 | 95 | 142 | 2353 | 2423 |
| 1961 | 106 | 92 | 2445 | 2529 |
| 1962 | 81 | 91 | 2536 | 2610 |
| 1963 | 116 | 131 | 2667 | 2726 |
| 1964 | 112 | 104 | 2771 | 2838 |
| 1965 | 80 | 97 | 2868 | 2918 |
| 1966 | 88 | 111 | 2979 | 3006 |
| 1967 | 85 | 114 | 3093 | 3091 |
| 1968 | 90 | 92 | 3185 | 3181 |
| 1969 | 120 | 146 | 3331 | 3301 |
| 1970 | 72 | 93 | 3424 | 3373 |
| 1971 | 113 | 138 | 3562 | 3486 |
| 1972 | 82 | 112 | 3674 | 3568 |
| 1973 | 116 | 117 | 3791 | 3684 |
| 1974 | 122 | 152 | 3943 | 3806 |
| 1975 | 73 | 90 | 4033 | 3879 |
| 1976 | 74 | 104 | 4137 | 3953 |

Plotting cum. X against cum. base average shows a straight line up to 1952 and a flatter straight line afterwards.

- Slope up to 1952: $M_a = \dfrac{1654}{1419} = 1.1656$
- Slope after 1952: $M_c = \dfrac{2299}{2718} = 0.8458$

The slope changes, so the **data of station X are not consistent**.

#### (ii) Change in regime
The break occurs at the 1952 point, so the regime changed after 1952 (the changed conditions apply from 1953). (The 1954 value of 97 is taken as printed.)

#### (iii) Mean annual rainfall of X (36 years)
- Without adjustment: $\bar P = \dfrac{3953}{36} = 109.8$ cm
- With adjustment to the present site (recent regime), the 1941-1952 data are corrected by $M_c/M_a = 0.8458/1.1656 = 0.7257$. Corrected total for 1941-52 $= 1200.3$ cm; total for 1953-76 $= 2299$ cm.

$$
\bar P_{adj} = \frac{1200.3 + 2299}{36} = 97.2\ \text{cm}
$$

#### (iv) Adjusted annual rainfall for the affected period (1941-1952)
$$
P_{cx} = P_x \times 0.7257
$$

| Year | Observed (cm) | Adjusted (cm) |
|---|---|---|
| 1941 | 163 | 118.3 |
| 1942 | 119 | 86.4 |
| 1943 | 121 | 87.8 |
| 1944 | 129 | 93.6 |
| 1945 | 126 | 91.4 |
| 1946 | 120 | 87.1 |
| 1947 | 153 | 111.0 |
| 1948 | 172 | 124.8 |
| 1949 | 127 | 92.2 |
| 1950 | 108 | 78.4 |
| 1951 | 126 | 91.4 |
| 1952 | 190 | 137.9 |

**Answer: Inconsistent; change in regime after 1952; mean = 109.8 cm (unadjusted) and 97.2 cm (adjusted); adjusted values for 1941-52 are listed above (factor 0.726).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What do the slopes 1.1656 (to 1952) and 0.8458 (after) show?
   A. X is inconsistent with a regime change after 1952
   B. X is consistent
   C. The base stations are wrong
   D. The gauge was added in 1941
   Answer: A. A change in slope means the X record is not consistent; the break is at 1952. (Key point: Break in slope)

2. What factor corrects the 1941-52 data to the recent regime?
   A. Ma/Mc = 1.378
   B. Mc/Ma = 0.7257
   C. Ma x Mc = 0.986
   D. Mc - Ma = -0.32
   Answer: B. P_cx = P_x x Mc/Ma = 0.8458/1.1656 = 0.7257. (Key point: Correction factor)

3. How is the adjusted 36-year mean calculated?
   A. 3953/36 = 109.8 cm
   B. (1654 + 2299)/36
   C. (1200.3 + 2299)/36 = 97.2 cm
   D. Average of both slopes x 36
   Answer: C. Corrected 1941-52 total plus unchanged 1953-76 total, divided by 36. (Key point: Adjusted mean)

4. Why is the adjusted mean (97.2) lower than the unadjusted (109.8)?
   A. Later data were scaled down
   B. The record length was reduced
   C. Base stations were removed
   D. Earlier data were relatively too high and are scaled down by 0.726
   Answer: D. The earlier regime overestimates relative to the present site, so those values are reduced. (Key point: Unadjusted vs adjusted)

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