Regional analysis of annual maximum rainfall using TL-moments method

An approach based on regional frequency analysis using L moments and LH moments are revisited in this study. Subsequently, an alternative regional frequency analysis using the partial L moments (PL moments) method is employed, and a new relationship for homogeneity analysis is developed. The results were then compared with those obtained using the method of L moments and LH moments of order two. The Selangor catchment, consisting of 37 sites and located on the west coast of Peninsular Malaysia, is chosen as a case study. PL moments for the generalized extreme value (GEV), generalized logistic (GLO), and generalized Pareto distributions were derived and used to develop the regional frequency analysis procedure. PL moment ratio diagram and Z test were employed in determining the best-fit distribution. Comparison between the three approaches showed that GLO and GEV distributions were identified as the suitable distributions for representing the statistical properties of extreme rainfall in Selangor. Monte Carlo simulation used for performance evaluation shows that the method of PL moments would outperform L and LH moments methods for estimation of large return period events.

In the first part of this study, five homogeneous regions in view of topographical and geographically homogeneous aspects except Jeju and Ulreung islands in Korea were accomplished by K-means clustering method. A total of 57 rain gauges were used for the regional frequency analysis with minimum rainfall series for the consecutive durations. Generalized Extreme Value distribution was confirmed as an optimal one among applied distributions. Drought rainfalls following the return periods were estimated by at-site and regional frequency analysis using L-moments method. It was confirmed that the design drought rainfalls estimated by the regional frequency analysis were shown to be more appropriate than those by the at-site frequency analysis. In the second part of this study, LH-moment ratio diagram and the Kolmogorov-Smirnov test on the Gumbel (GUM), Generalized Extreme Value (GEV), Generalized Logistic (GLO) and Generalized Pareto (GPA) distributions were accomplished to get optimal probability distribution. Design drought rainfalls were estimated by both at-site and regional frequency analysis using LH-moments and GEV distribution, which was confirmed as an optimal one among applied distributions. Design rainfalls were estimated by at-site and regional frequency analysis using LH-moments, the observed and simulated data resulted from Monte Carlotechniques. Design drought rainfalls derived by regional frequency analysis using L1, L2, L3 and L4-moments (LH-moments) method have shown higher reliability than those of at-site frequency analysis in view of RRMSE (Relative Root-Mean-Square Error), RBIAS (Relative Bias) and RR (Relative Reduction) for the estimated design drought rainfalls. Relative efficiency were calculated for the judgment of relative merits and demerits for the design drought rainfalls derived by regional frequency analysis using L-moments and L1, L2, L3 and L4-moments applied in the first report and second report of this study, respectively. Consequently, design drought rainfalls derived by regional frequency analysis using L-moments were shown as more reliable than those using LH-moments. Finally, design drought rainfalls for the classified five homogeneous regions following the various consecutive durations were derived by regional frequency analysis using L-moments, which was confirmed as a more reliable method through this study. Maps for the design drought rainfalls for the classified five homogeneous regions following the various consecutive durations were accomplished by the method of inverse distance weight and Arc-View, which is one of GIS techniques.

Accurate estimation of extreme (i.e., 1-day maximum) rainfall is essential for water resources management, flood forecasting, agricultural planning, and climate change impact studies. This can be achieved through fitting the extreme value family of probability distributions (EVD) that consists of Extreme Value Type-1 (EV1), Extreme Value Type-2 (EV2), Generalized Extreme Value (GEV), and Generalized Pareto (GPA) to the series of observed annual 1-day maximum rainfall (AMR), whereas the parameters are determined by the Method of L-Moments (LMO). This paper presents a study on the comparison of LMO estimators of EVD for the determination of design rainfall depth at Akkalkuwa, Kamrej, Navapur, Sakri, Shahada, and Taloda rain gauge sites. For this purpose, the AMR series was generated from the daily rainfall data observed at the sites during the period 1960 to 2022 and used. The adequacy of fitting LMO of EVD to the AMR series was examined through Goodness-of-Fit (GoF) tests, viz., Chi-Square and Kolmogorov-Smirnov (KS), while the selection of the best-fit distribution was made through model performance analysis with various indicators, viz., correlation coefficient (CC), Nash- Sutcliffe model efficiency (NSE), root mean squared error (RMSE), and cross-correlation matrix analysis (CCMA). The Chi- Square test results uniformly supported the use of EV1 and GEV for modelling the AMR data of six sites, whereas KS test results supported all four distributions of EVD for all six sites. The results indicated that the CC values obtained from four distributions vary between 0.960 and 0.994. The study showed that the NSE computed by EV1, GEV, and GPA varies from 91.7% to 98.7%. The outcomes of CCMA showed that there is a perfect correlation between the estimated rainfall by EV1 and GEV, and also nearer to 1.000. On the basis of evaluation of the results with quantitative (viz., CC, NSE, and RMSE) and qualitative assessments, it was found that GEV is the best choice for rainfall estimation for Akkalkuwa, Kamrej, Navapur, Sakri, Shahada, and Taloda. The estimated extreme rainfall by GEV distribution could be considered as a design rainfall depth while planning water resources management projects and their related activities in the respective sites.

Rainfall extreme value estimates in Canada have historically been based on fitting the Gumbel distribution to annual maxima at individual sites by the method of moments (MOM). Studies have, however, shown that regional frequency analyses (RFA) may perform better than at-site methods. Also, the Fréchet rather than Gumbel form of the generalized extreme value (GEV) distribution may better describe the distribution of annual extremes. In this study, at-site Gumbel MOM and GEV extreme value analyses based on L-moment, maximum likelihood (MLE), and generalized maximum likelihood (GML) estimators are compared against RFA with L-moment methods at stations in southern British Columbia, Canada via cross-validation and Monte Carlo simulations. While GEV shape parameter estimates are predominately negative, qualitatively showing weak evidence for the Fréchet form of the GEV distribution, field significant differences from the Gumbel distribution in the region are not found. Regional frequency analysis leads to substantial reductions in error relative to at-site methods, especially for the GEV distribution and small samples. While Gumbel estimators exhibit lower variance than GEV estimators, they are also more biased, underestimating 100 year return levels. Of the at-site GEV estimators, GML tended to perform better than the L-moment estimator, in some cases nearing performance of RFA. Maximum likelihood performed worst, especially for small samples sizes.

The quantiles of annual maximum wind speed (AMWS) can be estimated for different meteorological stations of interest by using at‐site frequency analysis and extreme value theory. These estimates are of immense importance for the codification of wind speed. However, the historical data of wind speed at the number of meteorological stations are sometimes unavailable and often insufficient due to the shorter length, especially in developing countries like Pakistan. The scarcity of the data increases the uncertainty of the quantiles estimates regarding policy implications. To cope with the problem, an approach of Regional Frequency Analysis (RFA) is opted here. In this study, RFA of AMWS using linear‐moments (L‐moments) is carried out by considering wind speed data of nine meteorological stations of province Punjab, Pakistan. No station is found to be discordant. A single homogenous region is constituted from these nine stations using a subjective approach based on their geographical locations. Heterogeneity measures justify that these nine stations of Punjab form a single homogeneous region. Regional quantiles estimates are found through the most appropriate probability distribution among generalized normal (GNO), generalized logistic (GLO), Pearson Type 3 (P3), generalized Pareto (GPA), Weibull (WEI), log Pearson Type 3 (LP3) and generalized extreme value (GEV) distributions. Z‐statistic and L‐moment ratio diagram suggest that GLO and GNO distributions are better choices than others. Robustness of both distributions is evaluated through relative bias (RB) and relative root mean square error (RRMSE). Findings indicate that overall, GLO distribution is better than GNO. Further, we also find at‐site quantiles from dimensionless quantities (regional quantiles) using the sample mean and median as scaling factors. Quantiles' estimates calculated from this study can be used in codified structural designs for policy implications.

Annual maximum daily rainfall trends in the Midwest, southeast and southern Brazil in the last 71 years

Probable maximum precipitation (PMP) is widely used as an input in rainfall‐runoff modelling to estimate probable maximum flood (PMF), which is required for the design of spillways of large dams. In relation to dam safety studies at 32 dam sites, an analysis was done of extreme rainfalls in Sri Lanka, which is influenced by the Asian monsoon system and cyclonic effects. Accordingly, the Hershfield's statistical method, which is one of the recommended procedures of the World Meteorological Organization (WMO), was used for estimating PMP. It is also a technique that can be readily used in design office practice. However, it is of significant value and scientific interest to explore what the outcome would have been if the physically based hydro‐meteorological procedure had been adopted. For this purpose, long series of annual maximum daily rainfall data from seven stations were analysed. The research shows that the maximum moisture and corresponding wind run or the use of maximum moisture alone are two scenarios, which yield results compatible with the statistical PMP. Although globally there is a tendency to exclude wind maximization, this does not automatically apply to a country like Sri Lanka, which experiences cyclonic rainfall and special attention should be paid to precipitation efficiency. These results are further confirmed by a frequency analysis based on the generalized extreme value (GEV) distribution. An approximate value of PMP could be in the order of 2.0–3.0 times the maximum observed daily rainfall. However, this aspect needs further investigation when observed maxima are near PMP storms. Copyright © 2010 Royal Meteorological Society

Information on extreme rainfall events for a given duration and a given frequency of occurrence is commonly required for the design of various urban hydraulic structures. More specifically, this information is extracted from the rainfall “intensity–duration–frequency” (IDF) relations. In Canada, these IDF relations are provided by Environment Canada for approximately 650 stations across Canada with at least 10-year record by fitting the Gumbel distribution to the annual maximum (AM) rainfall data for nine different rainfall durations (from 5 min to one day) using the method of moments. However, it has been widely known that the Generalized Extreme Value (GEV) distribution using the L-moment estimation method can provide more accurate extreme rainfall estimates. In addition, rather than fitting the GEV distribution to AM rainfall series for each duration independently as commonly used in the traditional method, the scale-invariance GEV model has been shown to provide more robust estimates of extreme rainfalls since it can take into account the relationship between the statistical properties of the AM rainfall processes over different durations. Therefore, this study presents the development of more accurate and more robust site-specific IDF relations as well as regional extreme rainfall maps for different locations across Canada based on all available historical AM rainfall data and using the improved scale-invariance GEV model. It is expected that these newly developed IDF relations and regional extreme rainfall maps could produce more cost-effective design of urban infrastructure in Canada.

Estimation of precipitation amounts associated with different return periods is an important task for the planning and design of many types of infrastructures. In this study, regional frequency analysis based on L-moments is proposed to estimate the annual maximum daily precipitation quantiles in the Taihu basin, China. The Generalized Extreme Value (GEV) distribution is used to describe the frequency distributions of extreme rainfall events. At-site frequency analysis results based on L-moments are compared with those obtained from regional analysis. The 95% confidence intervals of estimated precipitation quantiles are calculated using Monte Carlo simulations (MCS). Uncertainty assessment results indicate that regional analysis is more robust and more accurate than at-site analysis. Furthermore, when conducting regional frequency analysis, the estimation of precipitation quantile confidence intervals can be simplified by assuming normality for the MCS results. The variation of the precipitation quantiles’ sample statistics for different return periods is expressed as a function of the return period. The proposed methods are useful for the Taihu Basin and are recommended for other regions.

This study presents a probabilistic model for daily extreme rainfall. The Annual Maximum Series (AMS) data of daily rainfall in Makurdi was fitted to Generalized Extreme Value (GEV) distribution using Maximum Likelihood Estimation (MLE) and Bayesian Markov Chain Monte Carlo (Bayesian MCMC) simulations. MLE is a reliable principle to derive an efficient estimator for a model as sample size approaches infinity. Results in this study show that despite the asymptotic requirement of the MLE, its performance can be improved when adopting Bayesian MCMC. The comparison between the performance of MLE and Bayesian MCMC methods using Percent Bias (PBIAS), Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) proved Bayesian MCMC is the better method to estimate the distribution parameters of extreme daily rainfall amount in Makurdi. Based on the 36-year record of rainfall (1979-2014) in Makurdi, return levels for the next 10, 100, 500, 1000 and 10000 years were derived. Key words: Extreme daily rainfall, generalized extreme value distribution, parameter estimation, t-year return level, Makurdi.

Analysis of Average Annual Rainfall and Average Maximum Annual Temperature for a Period of 30 years to Establish Trends in Kieni, Central Kenya

The estimation of precipitation quantileshas always been an area of great importance to meteorologists, hydrologists, planners and managers of hydrotechnical infrastructures. In many cases, it is necessary to estimate the values relating to extreme events for the sites where there is little or no measurement, as well as their return periods. A statistical approach is the most used in such cases. It aims to find the probability distribution that best fitsthe maximum daily rainfall values. In our study, 231 rainfall stations were used to regionalize and find the best distribution for modeling the maximum daily rainfall in Northern Algeria. The L-moments method was used to perform a regionalization based on discordance criteria and homogeneity test. It gave rise to twelve homogeneous regions in terms of L-Coefficient of variation(L-CV), L-Skewness (L-CS)and L-Kurtosis (L-CK). This same technique allowed us to select the regional probability distribution for each group using the Z statistic. The generalized extreme values distribution (GEV) was selected to model the maximum daily rainfall of 10 groups located in the north of the steppe region and the generalized logistic distribution (GLO) for groups representing the steppes of Central and Western Algeria. The study of uncertainty by the bias and RMSE showed that the regional approach is acceptable. We have also developed maximum daily rainfall maps for 2, 5, 10, 20, 50 and 100 years return periods. We relied on a network of 255 rainfall stations. The spatial variability of quantiles was evaluated by semi-variograms. All rainfallfrequencymodelshave a spatial dependence with an exponential model adjusted to the experimental semi-variograms. The parameters of the fitted semi-variogram for different return periods are similar,throughout, while the nugget is more important for high return periods. Maximum daily rainfall increases from South to North and from West to East, andis more significant in the coastal areas of eastern Algeria where it exceeds 170 mm for a return period of 100 years. However,it does not exceed 50 mm in the highlands of the west.

Climatic drivers of floods have been widely used to improve nonstationary flood frequency analysis (FFA). However, the forecast ability of nonstationary FFA with out-of-sample prediction has not been comprehensively evaluated. We use 379 flood records from Brazil to assess the ability of process-informed nonstationary models for out-of-sample FFA using the generalized extreme value (GEV) distribution. Five drivers of floods are used as covariates: annual temperature, El Nino Southern Oscillation, annual rainfall, annual maximum rainfall, and annual maximum soil moisture content. Our results reveal that a nonstationary model is preferable when there is a significant correlation between flood and climate covariates in both the training period and full record. The rainfall-based covariates lead to better out-of-sample nonstationary FFA models. These findings highlight that using climate information as covariates in nonstationary FFA is a promising approach for estimating future floods and, hence, better infrastructure design, risk assessment and disaster preparedness.

Local and regional flood frequency analysis based on hierarchical Bayesian model in Dongting Lake Basin, China

The LQ-moment proposed by Mudholkar, et al. (1998) is used for regional flood frequency analysis of the North-Bank region of the river Brahmaputra, India. Five probability distributions are used for the LQmoment: generalized extreme value (GEV), generalized logistic (GLO) and generalized Pareto (GPA), lognormal (LN3) and Pearson Type III (PE3). The same regional frequency analysis procedure proposed by Hosking (1990) for the L-moment is used for the LQ-moment. Based on the LQ-moment ratio diagram and |Zidist| -statistic criteria, the PE3 distribution is identified as the robust distribution for the study area. For estimation of floods of various return periods for both gauged and ungauged catchments of the study area, regional flood frequency relationships are developed using the LQ-moment based PE3 distribution.