Section 4: Soil and Water Conservation Engineering
Hydrology
Subtopic: Analysis of precipitation data
Question Statement:
In Gumbel's Extreme Value Type-I distribution for flood frequency analysis, the reduced variate $y_T$ associated with an annual recurrence interval (return period) of $T$ years is defined as:
Probability of non-exceedance in any single year is $P = 1 - \frac{1}{T}$.
The cumulative distribution function of Gumbel's Type-I extreme value distribution is $P = e^{-e^{-y_T}}$.
Taking natural logarithms twice:
$\ln(P) = -e^{-y_T} \implies -\ln(P) = e^{-y_T}$
$y_T = -\ln\left( -\ln(P) \right) = -\ln\left[ -\ln\left( 1 - \frac{1}{T} \right) \right]$
Option A is correct.