log-Pearson Type 3 Distribution Research Articles

Low-Flow Identification in Flood Frequency Analysis: A Case Study for Eastern Australia

Design flood estimation is an essential step in many water engineering design tasks such as the planning and design of infrastructure to reduce flood damage. Flood frequency analysis (FFA) is widely used in estimating design floods when the at-site flood data length is adequate. One of the problems in FFA with an annual maxima (AM) modeling approach is deciding how to handle smaller discharge values (outliers) in the selected AM flood series at a given station. The objective of this paper is to explore how the practice of censoring (which involves adjusting for smaller discharge values in FFA) affects flood quantile estimates in FFA. In this regard, two commonly used probability distributions, log-Pearson type 3 (LP3) and generalized extreme value distribution (GEV), are used. The multiple Grubbs and Beck (MGB) test is used to identify low-flow outliers in the selected AM flood series at 582 Australian stream gauging stations. It is found that censoring is required for 71% of the selected stations in using the MGB test with the LP3 distribution. The differences in flood quantile estimates between LP3 (with MGB test and censoring) and GEV distribution (without censoring) increase as the return period reduces. A modest correlation is found (for South Australian catchments) between censoring and the selected catchment characteristics (correlation coefficient: 0.43), with statistically significant associations for the mean annual rainfall and catchment shape factor. The findings of this study will be useful to practicing hydrologists in Australia and other countries to estimate design floods using AM flood data by FFA. Moreover, it may assist in updating Australian Rainfall and Runoff (national guide).

Beyond Simple Trend Tests: Detecting Significant Changes in Design‐Flood Quantiles

AbstractChanges in annual maximum flood (AMF), which are usually detected using simple trend tests (e.g., Mann‐Kendall test (MKT)), are expected to change design‐flood estimates. We propose an alternate framework to detect significant changes in design‐flood between two periods and evaluate it for synthetically generated AMF from the Log‐Pearson Type‐3 (LP3) distribution due to changes in moments associated with flood distribution. Synthetic experiments show MKT does not consider changes in all three moments of the LP3 distribution and incorrectly detects changes in design‐flood. We applied the framework on 31 river basins spread across the United States. Statistically significant changes in design‐flood quantiles were observed even without a significant trend in AMF and basins with statistically significant trend did not necessarily exhibit statistically significant changes in design‐flood. We recommend application of the framework for evaluating changes in design‐flood estimates considering changes in all the moments as opposed to simple trend tests.

Flood Frequency Analysis of Betwa River, Madhya Pradesh India

Flood is common phenomena worldwide since time immemorial. Recently the change in climatic parameters has drastically affected the pattern and magnitude of flood. India being one of the tropical country face flood and drought situations every year, therefore it needs accurate assessment and forecast of flood for proper management of natural resources. An attempt has been made through the present study which consists frequency analysis on maximum daily discharge data in Betwa river at Basoda, Mohana and Shahijina gauging stations in Madhya Pradesh state using Gumbel’s Extreme value distribution and Log Pearson Type-3 distribution for 20 years period (1993-2012). The result shows that Log Pearson Type-3 distribution is better suited for Betwa basin. The results can be used by civil engineers for deciding the dimensions of hydraulic structures such as spillways, dams, bridges etc. Floods are forecasted for the different return periods for Betwa river.

Forecasting flood risk in the Indus River system using hydrological parameters and its damage assessment

Hydrological parameters are among the widely used parameters in assessing flood risk. On the other hand, anticipated flood damages, in case of flooding, are estimated with the help of expected losses in areas nearer to the watercourse. The major source of almost every-year flooding in Pakistan is the Indus River system that comprises the major rivers of Pakistan. We first use observed data to construct simulated data models based on various probability distributions namely normal, lognormal, Weibull, largest extreme value, gamma-3, and log-Pearson type-3 distributions and thereby compute probable maximum flood. Secondly, we perform log-Pearson type-3 analysis with and without historic adjustment on the observed data series of 17 years to forecast floods with return periods T of 2, 5, 10, 25, 50, 100, and 200 years. We also categorize the river structures based on the risk of flooding. Lastly, we estimate risk of flood damages in terms of expected losses based on observed data. The present study reveals that the log-Pearson type-3 distribution is relatively better for estimating probable maximum flood. We use exceedence probability to assess the risk of flooding in the various structures of the said rivers. The analysis shows that flood damages in Pakistan may be reduced by increasing the design capacity of the structures and also by giving awareness to people about the flood-generating factors.

Regional flood frequency analysis in eastern Australia: Bayesian GLS regression-based methods within fixed region and ROI framework – Quantile Regression vs. Parameter Regression Technique

Summary In this article, an approach using Bayesian Generalised Least Squares (BGLS) regression in a region-of-influence (ROI) framework is proposed for regional flood frequency analysis (RFFA) for ungauged catchments. Using the data from 399 catchments in eastern Australia, the BGLS-ROI is constructed to regionalise the flood quantiles (Quantile Regression Technique (QRT)) and the first three moments of the log-Pearson type 3 (LP3) distribution (Parameter Regression Technique (PRT)). This scheme firstly develops a fixed region model to select the best set of predictor variables for use in the subsequent regression analyses using an approach that minimises the model error variance while also satisfying a number of statistical selection criteria. The identified optimal regression equation is then used in the ROI experiment where the ROI is chosen for a site in question as the region that minimises the predictive uncertainty. To evaluate the overall performances of the quantiles estimated by the QRT and PRT, a one-at-a-time cross-validation procedure is applied. Results of the proposed method indicate that both the QRT and PRT in a BGLS-ROI framework lead to more accurate and reliable estimates of flood quantiles and moments of the LP3 distribution when compared to a fixed region approach. Also the BGLS-ROI can deal reasonably well with the heterogeneity in Australian catchments as evidenced by the regression diagnostics. Based on the evaluation statistics it was found that both BGLS-QRT and PRT-ROI perform similarly well, which suggests that the PRT is a viable alternative to QRT in RFFA. The RFFA methods developed in this paper is based on the database available in eastern Australia. It is expected that availability of a more comprehensive database (in terms of both quality and quantity) will further improve the predictive performance of both the fixed and ROI based RFFA methods presented in this study, which however needs to be investigated in future when such a database is available.

Regional flood frequency analysis using Bayesian generalized least squares: a comparison between quantile and parameter regression techniques

AbstractRegression‐based regional flood frequency analysis (RFFA) methods are widely adopted in hydrology. This paper compares two regression‐based RFFA methods using a Bayesian generalized least squares (GLS) modelling framework; the two are quantile regression technique (QRT) and parameter regression technique (PRT). In this study, the QRT focuses on the development of prediction equations for a flood quantile in the range of 2 to 100 years average recurrence intervals (ARI), while the PRT develops prediction equations for the first three moments of the log Pearson Type 3 (LP3) distribution, which are the mean, standard deviation and skew of the logarithms of the annual maximum flows; these regional parameters are then used to fit the LP3 distribution to estimate the desired flood quantiles at a given site. It has been shown that using a method similar to stepwise regression and by employing a number of statistics such as the model error variance, average variance of prediction, Bayesian information criterion and Akaike information criterion, the best set of explanatory variables in the GLS regression can be identified. In this study, a range of statistics and diagnostic plots have been adopted to evaluate the regression models. The method has been applied to 53 catchments in Tasmania, Australia. It has been found that catchment area and design rainfall intensity are the most important explanatory variables in predicting flood quantiles using the QRT. For the PRT, a total of four explanatory variables were adopted for predicting the mean, standard deviation and skew. The developed regression models satisfy the underlying model assumptions quite well; of importance, no outlier sites are detected in the plots of the regression diagnostics of the adopted regression equations. Based on ‘one‐at‐a‐time cross validation’ and a number of evaluation statistics, it has been found that for Tasmania the QRT provides more accurate flood quantile estimates for the higher ARIs while the PRT provides relatively better estimates for the smaller ARIs. The RFFA techniques presented here can easily be adapted to other Australian states and countries to derive more accurate regional flood predictions. Copyright © 2011 John Wiley & Sons, Ltd.

Log-Pearson Type 3 Distribution and Its Application in Flood Frequency Analysis. I: Distribution Characteristics

Since the adoption of the log-Pearson Type 3 (LP3) distribution by U.S. federal agencies, it has been widely used in hydrology, but its properties are not well understood. This paper explores the characteristics of the LP3 distribution in both real space and log space, and their relationship. Comparisons with U.S. flood data summaries reveal that the LP3 distribution provides a reasonable model of the distribution of annual flood series from unregulated watersheds for log space skews ∣ γx ∣⩽1.414 (though ∣ γx ∣⩽1 is more realistic), and for γx =0 with standard deviations in the range 0.1 to 1.0 with base-e natural logarithms (0.04 to 0.43 with base-10 common logarithms). L-moment ratio relationships for the LP3 distribution are also developed so they can be compared to summary statistics for a region, and to several other distributions frequently recommended for modeling hydrometeorological extremes.

Floodflow frequency model selection in Australia

Uniform flood frequency guidelines in Australia and the United States recommend the use of the log Pearson type 3 (LP3) distribution in flood frequency investigations. Many investigators have suggested alternate models such as the Generalized Extreme Value (GEV) distribution as an improvement over the LP3 distribution. Using floodflow data at 61 sites across Australia, we explore the suitability of various flood frequency models using L-moment diagrams. We also repeat the experiment performed in the original US Water Resource Council report (Bulletin 17B) which led to the LP3 mandate in the United States. Our evaluations reveal that among the models tested, the GEV and Wakeby distributions provide the best approximation to floodflow data in the regions of Australia that are dominated by rainfall during the winter months, such as southwest Western Australia and Tasmania. For the remainder of the continent, the Generalized Pareto (GPA) and Wakeby distributions provide the best approximation to floodflow data. The two- and three-parameter log-normal models and the LP3 distribution performed satisfactorily, yet not as well as either the GEV or GPA distributions. Other models such as the Gumbel, log-normal, normal, Pearson, exponential, and uniform distributions are shown to perform poorly. Recent research indicates that regional index-flood type procedures should be more accurate and more robust than the type of at-site procedures evaluated here. Nevertheless, this study reveals that index-flood procedures should not be restricted to a single distribution such as the GEV distribution because other distributions such as the GPA distribution perform significantly better in the most densely populated regions of Australia.

A comparative evaluation of the estimators of the log Pearson type (LP) 3 distribution

The log Pearson type 3 (LP3) distribution, recommended by the U.S. Water Resources Council (USWRC) in 1967, and subsequently updated in 1975, 1977 and 1981 as the base method of flood frequency analysis in the United States, has been widely used in many parts of the world. However, the estimation procedure for the LP3 distribution recommended by the USWRC has been shown by many investigators to perform rather poorly. In this study several estimation methods for the LP3 distribution, some quite recent, are investigated. The performance of the various methods is then compared via Monte Carlo simulation with the objective of identifying the most robust estimator among them. It is found that besides the USWRC method, the methods based on maximum likelihood and entropy perform poorly. The method based on the moments in real space, and the relatively recent so-called method of mixed moments perform markedly superior to other methods in terms of both resistance and efficiency of estimation or robustness. Therefore, the USWRC guidelines are in need of revision.

Generalized Method of Moments Applied to LP3 Distribution

The log-Pearson type 3 (LP3) distribution is recommended in the United States by the Water Resources Council (WRC) as the parent distribution to maximum annual flood series. The parameter estimation technique proposed for this distribution consists in applying a logarithmic transformation to the observed flood sample and then applying the method of moments to these logarithmic values using moments of order 1, 2, and 3 (mean, variance, and coefficient of skew) in log space. Other methods have been proposed which use only moments in real space such as the mean, variance, geometric mean, harmonic mean, etc. of the observed sample, or combination of moments in real and log space (method of “mixed moments”). There are many other versions of the method of moments which have yet to be explored and evaluated. To help motivate this kind of exploration, we derive a general formula for the variance of the \IT\N-year event \IX\dT\N obtained by a method called the Generalized Method of Moments (GMM) which combines any three moments of the LP3 distribution. We also present a special case of the GMM which we all the Sundry Averages Method (SAM) which uses the harmonic, geometric, and arithmetic means of the distribution.

Variance of the T-year event in the log Pearson type-3 distribution — Reply

Variance of the T-year event in the log Pearson type-3 distribution — Reply

Variance of the T-year event in the log Pearson type-3 distribution — Comment

Abstract The computation of the variance of the T -yr. event evaluated by means of the log Pearson type- 3 distribution is described. This computation involves the variances and covariances of the parameter estimates. Explicit formulae for these variances and covariances are analytically derived in this study for the case where the parameters are estimated by the methods of direct and mixed moments.

Variance of the T-year event in the log Pearson type-3 distribution

Variance of the T-year event in the log Pearson type-3 distribution

Applications of the log Pearson type-3 distribution in hydrology

Although the log Pearson (LP3) type-3 distribution was recommended specifically for flood-frequency analysis, its applicability to annual maximum rainfall, annual streamflow and annual rainfall was also considered in this study. It was found that the LP3 distribution was applicable in most cases; however, for annual flood and maximum rainfall intensity, the existence of an upper bound to the distribution, in some cases, may cause some concerns, while this fact may indicate the suitability of the LP3 distribution for other variables such as annual streamflow and annual rainfall.

Log Pearson type-3 distribution: Parameter estimation

Four additional methods for estimating the parameters of the log Pearson type-3 distribution are provided and appropriate computational schemes are suggested for them as well as for the method of maximum likelihood and method of (direct and indirect) moments. Simulation results indicate that the method based on the first two moments of the original data and the variance of the log-transformed values provides the best estimates, which are slightly better than those obtained from the method of maximum likelihood.