---
title: "Flood estimation by Log Pearson III method"
subject: "Engineering Hydrology"
chapter: 6
key: "semester_5/engineering_hydrology:log-pearson-iii-procedure"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/log-pearson-iii-procedure
site: BCT Engineering
---

# Flood estimation by Log Pearson III method

Engineering Hydrology · Chapter 6: Flood Hydrology

- Asked in: 2071 Chaitra, 8 marks; 2070 Chaitra, 5 marks; 2068 Chaitra, 5 marks (asked 3 times)
- Video: [Extreme Value Analysis - Log Pearson Type III](https://www.youtube.com/watch?v=fNspbC5i08Q) — Kenneth Lamb
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/log-pearson-iii-procedure
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-6

## Question

Explain the Log Pearson Type III distribution and the procedure to estimate a flood of return period T (T > N) from N years of annual flood data.

## Short answer

Take base-10 logs of annual peak flows, compute their mean, standard deviation, and skew coefficient. For return period T, obtain frequency factor KT from Pearson III table using the skew, then compute log flood zT = mean + KT × standard deviation; the flood magnitude is 10^zT.

## Answer

#### 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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In the Log Pearson Type III method, the Pearson Type III distribution is assumed for:
   A. The logarithms of the annual flood peaks
   B. The flood peaks themselves
   C. The square roots of flood peaks
   D. The return periods
   Answer: A. The log series z = log10 x is assumed Pearson III. (Key point: What is fitted)

2. The flood of return period T is found from:
   A. x_T = mean(x) + K_T S_x with no log
   B. z_T = mean(z) + K_T S_z, then x_T = 10^(z_T)
   C. z_T = mean(z) x K_T
   D. x_T = ln(z_T)
   Answer: B. Compute z_T with the frequency factor then take the antilog. (Key point: Main equation)

3. The frequency factor K_T depends on:
   A. Only the sample size N
   B. Only the mean of the logs
   C. The return period T and the skew coefficient Cs of the logs
   D. The area of the catchment
   Answer: C. K_T is tabulated against T and Cs. (Key point: K_T dependence)

4. When the skew of the logs is zero, the distribution reduces to:
   A. Gumbel
   B. Exponential
   C. Uniform
   D. Log-normal, with K_T the standard normal variate
   Answer: D. With Cs = 0 the Pearson III becomes the normal distribution of the logs. (Key point: Zero skew)

## Related in Engineering Hydrology

- [Log-Pearson III 50-year flood, 1990-2006](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Alog-pearson-50yr-1990-2006) — Engineering Hydrology, chapter 6
- [Flood of return period T using log-normal distribution](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Alog-normal-flood-procedure) — Engineering Hydrology, chapter 6
- [Prove mean annual flood has 2.33-year return period](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-mean-flood-2-33yr) — Engineering Hydrology, chapter 6
- [Return period of 350 m3/s design flood by Gumbel](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-return-period-350-25yr) — Engineering Hydrology, chapter 6, asked 2×
- [Gumbel flood for 500-year return period](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-100-50-500) — Engineering Hydrology, chapter 6
- [Gumbel return period and confidence limit](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-return-period-300) — Engineering Hydrology, chapter 6, asked 2×

## Related in other subjects

- [Demand of rural municipality, base period 3 years](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Ademand-bk79) — Water Supply Engineering, chapter 3