Statistical Hydrology

Introduction

Hydrologic events (floods, droughts, storms) are stochastic (random) in nature. Statistical Hydrology uses probability theory to analyze historical data and predict the likelihood of future events.

Return Period ( $T$ )

The Return Period (or Recurrence Interval) is the average time interval between events equal to or exceeding a certain magnitude.

T = \frac{1}{P}

Where $P$ is the exceedance probability (probability that an event of magnitude $\ge x$ will occur in any given year).

100-year flood: $T = 100$ , so $P = 0.01$ (1% chance each year).

Risk ( $R$ )

The probability that an event with return period $T$ will occur at least once in $n$ years.

R = 1 - (1 - P)^n = 1 - (1 - \frac{1}{T})^n

Reliability

\text{Reliability} = 1 - R = (1 - P)^n

Step-by-Step Solution

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Frequency Analysis

Used to relate the magnitude of extreme events to their frequency of occurrence.

General Equation:

x_T = \bar{x} + K \cdot \sigma

Where:

$x_T$ = Value of variate with return period $T$ .
$\bar{x}$ = Mean of the data series.
$\sigma$ = Standard deviation.
$K$ = Frequency factor (depends on distribution and $T$ ).

Gumbel's Extreme Value Distribution

Commonly used for flood frequency analysis.

K = \frac{y_T - \bar{y}_n}{S_n}

y_T = -\ln [-\ln (1 - \frac{1}{T})]

Where $\bar{y}_n$ and $S_n$ are functions of sample size $N$ .

Log-Pearson Type III Distribution

Standard method for flood frequency analysis in the US. Uses the logarithm of discharge values.

\log x_T = \overline{\log x} + K_z \cdot \sigma_{\log x}

Where $K_z$ depends on the skewness coefficient ( $C_s$ ).

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