Flood estimation by Log Pearson III method

Log Pearson Type III distribution

In this distribution, the logarithms of the annual flood peaks are assumed to follow the Pearson Type III (gamma-type) distribution, which has three parameters: the mean, standard deviation and coefficient of skew of the log series. It is recommended in the USA (US Water Resources Council) for flood frequency. When the skew of the logs is zero, it reduces to the log-normal distribution.

For a variate $z = \log_{10} x$, the magnitude at return period T is

$$z_T = \bar z + K_T\, S_z \qquad\Rightarrow\qquad x_T = 10^{\,z_T}$$

where $\bar z$ = mean of the log values, $S_z$ = standard deviation of the logs, and $K_T$ = frequency factor, which depends on T and on the skew coefficient $C_s$ of the logs (given in tables).

Procedure (N years of data, estimate for T > N)

1. Arrange the annual maximum flood series $x_1,\dots,x_N$ and convert each to $z_i = \log_{10}x_i$.
2. Compute the mean:

$$\bar z = \frac{1}{N}\sum z_i$$

3. Compute the standard deviation:

$$S_z = \sqrt{\frac{\sum (z_i-\bar z)^2}{N-1}}$$

4. Compute the coefficient of skew:

$$C_s = \frac{N\sum (z_i-\bar z)^3}{(N-1)(N-2)\,S_z^3}$$

5. For the required return period T and the computed $C_s$, read the frequency factor $K_T$ from the table of Pearson Type III (for $C_s = 0$, $K_T$ is the standard normal variate; Wilson–Hilferty approximation for other skews). Interpolate if necessary.
6. Compute $z_T = \bar z + K_T S_z$.
7. Flood of return period T: $x_T = 10^{z_T}$ m³/s.
8. (Optional) Plot the points with Weibull plotting positions $T=(N+1)/m$ on log-probability paper, draw the fitted line $x_T$ against T and extrapolate to the required T. Confidence limits can also be computed.

Because T > N, the result is an extrapolation of the fitted distribution; its reliability decreases when T is much larger than N (rule of thumb T ≤ 2N).