Rainfall Estimation Calculation
The problem requires estimating the rainfall at station C for a specific year when its rain gauge failed. We are provided with the normal annual rainfall data for five stations (A, B, C, D, E) and the actual rainfall recorded at stations A, B, D, and E during the year C's gauge malfunctioned.
Normal Rainfall Data
| Station | Normal Annual Rainfall (cm) |
| A | 112.7 |
| B | 120.4 |
| C | 118.3 |
| D | 125.2 |
| E | 110.6 |
Actual Rainfall Data (Year of Failure at C)
| Station | Actual Annual Rainfall (cm) |
| A | 114.9 |
| B | 118.3 |
| D | 122.6 |
| E | 114.5 |
Methodology for Rainfall Estimation
Since the actual rainfall at the surrounding stations (A, B, D, E) deviates from their respective normal values, the Normal Ratio Method is the appropriate technique for estimating the rainfall at station C. This method adjusts the estimate based on the ratio of actual to normal rainfall at surrounding stations.
Normal Ratio Method Formula
The formula used for estimation is:
$ P_C = \frac{N_C}{k} \times \left( \frac{P_A}{N_A} + \frac{P_B}{N_B} + \frac{P_D}{N_D} + \frac{P_E}{N_E} \right) $
Where:
- $P_C$ represents the estimated rainfall at station C in the year of failure.
- $N_C$ is the normal annual rainfall at station C (118.3 cm).
- $k$ is the number of surrounding stations used for estimation (here, k=4).
- $P_A, P_B, P_D, P_E$ are the actual recorded rainfalls at stations A, B, D, E in the year of failure.
- $N_A, N_B, N_D, N_E$ are the normal annual rainfalls at stations A, B, D, E.
Calculation Steps
- Input Data:
- Normal Rainfall ($N_C$) at Station C: 118.3 cm
- Actual Rainfall ($P$) and Normal Rainfall ($N$) for surrounding stations:
- Station A: $P_A = 114.9$ cm, $N_A = 112.7$ cm
- Station B: $P_B = 118.3$ cm, $N_B = 120.4$ cm
- Station D: $P_D = 122.6$ cm, $N_D = 125.2$ cm
- Station E: $P_E = 114.5$ cm, $N_E = 110.6$ cm
- Calculate Rainfall Ratios:
- Ratio A: $\frac{P_A}{N_A} = \frac{114.9}{112.7} \approx 1.0195$
- Ratio B: $\frac{P_B}{N_B} = \frac{118.3}{120.4} \approx 0.9826$
- Ratio D: $\frac{P_D}{N_D} = \frac{122.6}{125.2} \approx 0.9792$
- Ratio E: $\frac{P_E}{N_E} = \frac{114.5}{110.6} \approx 1.0352$
- Sum the Ratios:
Total Ratio Sum $= 1.0195 + 0.9826 + 0.9792 + 1.0352 = 4.0165$
- Compute Estimated Rainfall ($P_C$):
Using the formula: $ P_C = \frac{N_C}{4} \times (\text{Sum of Ratios}) $
$ P_C = \frac{118.3}{4} \times 4.0165 $
$ P_C = 29.575 \times 4.0165 \approx 118.77 \text{ cm} $
Estimated Rainfall Value
The calculated estimated rainfall at station C is approximately 118.77 cm. This value falls within the range specified (118.5 cm to 119 cm).
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