Question

The normal annual rainfall of the stations A, B, C and D in a basin are 75, 80, 96 and 72 cm, respectively. In the year 2020, the station A was inoperative and th stations B, C and D recorded annual precipitations of 70, 72 and 63 cm, respectively. Estimate the rainfall at station A in that year.

• A. 50 cm
• B. 55 cm
• C. 60 cm
• D. 62.5 cm

Hint:

- If the difference in normal precipitation is \(< 10%\), use the simple Arithmetic Average Method.
- If the difference is \(> 10%\), always use the weighted Normal Ratio Method.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When a rain gauge station fails during a storm or over a year, hydrology engineers estimate the missing data using records from surrounding stations.
We select the estimation method based on how much the normal annual rainfall varies between the stations.
Key Formula or Approach:
If the normal annual precipitation at any of the index stations differs from the missing station's normal precipitation by more than \(10%\), we must use the Normal Ratio Method: \[ P_x = \frac{N_x}{m} \sum_{i=1}^{m} \frac{P_i}{N_i} \] where:
- \(P_x\) is the estimated missing precipitation.
- \(N_x\) is the normal annual precipitation of the missing station.
- \(m\) is the number of surrounding index stations.
- \(P_i\) and \(N_i\) are the recorded and normal annual precipitations at the index stations.

Step 2: Detailed Explanation:
Let's list the given parameters:
- Missing Station \(A\): \(N_A = 75 \text{ cm}\)
- Surrounding Stations \(B\), \(C\), and \(D\) (\(m = 3\)):
- Station \(B\): \(N_B = 80 \text{ cm}\), \(P_B = 70 \text{ cm}\)
- Station \(C\): \(N_C = 96 \text{ cm}\), \(P_C = 72 \text{ cm}\)
- Station \(D\): \(N_D = 72 \text{ cm}\), \(P_D = 63 \text{ cm}\)
Since \(N_C\) (\(96 \text{ cm}\)) differs from \(N_A\) (\(75 \text{ cm}\)) by more than \(10%\) (\(|96 - 75|/75 = 28%\)), we must use the Normal Ratio Method: \[ P_A = \frac{N_A}{3} \left[ \frac{P_B}{N_B} + \frac{P_C}{N_C} + \frac{P_D}{N_D} \right] \] Substitute the values into the equation: \[ P_A = \frac{75}{3} \left[ \frac{70}{80} + \frac{72}{96} + \frac{63}{72} \right] \] Simplify the fractions inside the bracket: \[ P_A = 25 \left[ 0.875 + 0.75 + 0.875 \right] \] \[ P_A = 25 \left[ 2.50 \right] = 62.5 \text{ cm} \]

Step 3: Final Answer:
The estimated rainfall at station A for the year 2020 is 62.5 cm.