Question

Which of the following relations exists in terms of their better performance if rainfall data points are small in number (\(<25\)) in frequency analysis of rainfall?

• A. California > Weibull > Foster
• B. California > Foster > Weibull
• C. Foster > California > Weibull
• D. Weibull > Foster > California

Hint:

For small sample sizes (\(N < 25\)), the California method (\(P = m/N\)) provides a better fit for empirical return periods of extreme events compared to the Weibull method, which tends to underestimate the probability of the highest-ranked events.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Plotting positions are used in hydrology to assign empirical probabilities to rainfall or flood events of different magnitudes from a historical record.
Different empirical formulas (like California, Weibull, and Hazen) are used for frequency analysis.
When the sample size of rainfall data points is small (\(N < 25\)), the bias and performance of these methods vary significantly.

Step 2: Detailed Explanation:
Let us examine the common plotting position formulas:
- California Method:
\[ P = \frac{m}{N} \]
Where \(m\) is the rank of the event (1 for the largest) and \(N\) is the total number of data points.
- Weibull Method:
\[ P = \frac{m}{N+1} \]
- Foster Method (or others):
Various other forms exist, such as Hazen's formula:
\[ P = \frac{m - 0.5}{N} \]
For small data sets (\(N < 25\)), the California method is historically shown to perform better in terms of assigning empirical return periods that closely align with observed regional probabilities, despite its theoretical limitation at \(m = N\) (where \(P = 1.0\)).
Comparative studies on small sample sizes show that the empirical return periods calculated by the California method yield more realistic predictions for extreme events than the Weibull and Foster methods.
Therefore, the performance hierarchy for small rainfall datasets is:
\[ \text{California} > \text{Weibull} > \text{Foster} \]
Thus, Option (A) is the correct relation.

Step 3: Final Answer:
The correct option is (A).