Gumbel Probability Distribution Research Articles

Hydrological Modelling with HEC-HMS in Krueng Peudada Sub-Watershed Bireuen Regency

The flood that occurred in Krueng Peudada was influenced by several factors, such as watershed conditions. Heavy rains caused Krueng Peudada to overflow so that a number of villages adjacent to the river were submerged. Based on these problems, to minimize the occurrence of flooding in Krueng Peudada building, namely by carrying out occurrence of flooding in Krueng Peudada to be used to obtain peak discharge as a reference in building planning in Krueng Peudada using HEC-HMS (Hydrologic Engineering Center’s Hydrologic Modelling System). The benefit of the research is that the resulting model can be useful as a guide in calculating flood discharge considerations. The scope of this research includes the research location at Krueng Peudada using annual maximum daily rainfall data, calculating the return period of rainfall using of Gumbel probability distribution, Normal Log, Normal, and Pearson III Log as the basis for calculating the discharge and continuing the calculation of the hours with Manonobe and ABM. The results of calculations of the Nakayasu Synthetic Unit Hydrograph Method and running from HEC-HMS found that the flow that occurred in the Krueng Peudada two sub-watershed was 2903m3/second and 2723,0 m3/second, it can be concluded that in determining watershed management the flood discharge obtainded has error as much as 6,2%. The difference in the results obtained is due to the number of sub-watersheds in the watershed based on different flows and the value of the curve number for each area of the sub-watershed.

Analysis of Annual Rainfall Distribution and Planned Rain Intensity at 11 (Eleven) Rain Post Stations in Serang District

This research aimed to find the best rainfall probability model and planned rain intensity using the Mononobe formula for 11 rain stations in Serang Regency. Annual rainfall data is processed and evaluated as secondary data using the rain data consistency test method, the Gumbel probability distribution, the Normal probability distribution, the Log Normal probability distribution, and the Pearson Type III Log probability distribution. The consistency test results reveal that the rainfall data is consistent. The Gumbel probability distribution is used to examine the design rain intensity based on the analysis. This study is intended to guide estimating the design rainfall intensity to develop the design flood discharge and run-off calculations. Suggestions for future research include improving the accuracy of rain data processing and making longer-term data available to provide a more accurate frequency analysis

Influence of welded pores on fatigue behavior of TC17 titanium alloy welded joints subjected to gigacycle regime at room and high temperatures

Welded structures in aero-engines commonly operate in high-temperature environments, making them susceptible to reduced fatigue life and premature failure due to welding defects within the structure. Thus, the gigacycle fatigue behaviors of titanium alloy welded joints at both room temperature (RT) and 400 °C were investigated, aiming to uncover the mechanism behind the formation of fine granular area (FGA) surrounding welded pores. The research findings demonstrate that the S-N curves of TC17 titanium alloy electron beam welded joint undergo a transition from a single linear decline at RT to a bilinear decline at 400 °C. However, the fatigue failure mode remains unaffected by temperatures, and crack initiation is attributed to welded pores. By utilizing the Chapetti model curve to modify the Kitagawa–Takahashi (K–T) diagram, the lower threshold stress amplitude is introduced, enabling the determination of a safe size for welded pores at 400 °C, which is calculated to be 11.3 µm. Additionally, the Gumbel probability distribution function is employed to assess the maximum size of welded pores. Finally, based on dislocation interactions, the formation mechanism of the FGA consisting of discontinuous nanograins with high-density dislocations is elucidated.

Rainfall intensity model with spatialization of intensity-duration-frequency curve parameters - A case study for the state of Maranhão, Brazil

The characterization of intense rainfall in engineering projects is fundamental, mainly regarding the estimate of design flows necessary for designing hydraulic works. Intense rainfall events are commonly measured by Equations and curves that relate their intensity, duration, and frequency. Such relations, known as IDF, enable the hydrological characterization of a given region. This article presents a methodological design and results from both determination and spatialization of IDF curve parameters for the state of Maranhão. Historical series of maximum daily rainfalls obtained from National Water and Sanitation Agency (ANA) were used in 126 rainfall gauge stations and the Gumbel probability distribution estimated the maximum daily rainfall for 5, 10, 15, 25, 50, and 100 return periods. The Isozonal Method obtained the IDF correlations of intense rainfall events for 0,1. 1, and 24 h durations, and their performance were conducted by Nash-Sutcliffe R2 coefficient and Root Mean Square Relative Error (RMSE). “K, a, b, and c” parameters of intense rainfall equations were determined by optimization and convergence processes and their spatialization was carried out by interpolation by Inverse Distance Weighted (IDW), which enabled to determine the values of each parameter in regions without physical measurements of rainfall. Similarly, rainfall intensity was spatialized for the entire state. According to the results, the rainfall distribution in the state of Maranhão shows a variation in the indexes of precipitation, with the highest values found in areas located in central-southern, southwestern, and southeastern regions.

FREQUENCY ANALYSIS OF DELI RIVER FLOOD DISTRIBUTION PLAN USING THE GUMBEL PROBABILITY DISTRIBUTION METHOD

The Deli river basin is one of the rivers in North Sumatra Province which is in Medan City with an area of ​​394.88 km2 and a length of 166.01 km. Medan Maimun District which has an area of ​​2.98 km2 and a population density of 16,520 people/km2. Medan Mimun sub-district is one of the flood-prone areas which is drained by the Deli River. The cause of the flooding that occurred in Medan Maimun District was high rainfall so that the river flow rate increased and drainage was poor. So that the problem of flooding in the Deli river watershed, Medan Maimun sub-district, can hamper community activities, these floods can also harm and endanger the community. The flood discharge plan for each repeat period where the variable used is the maximum daily rainfall for 10 years From 2013-2022 which is sourced from the BMKG Deli Serdang. In this study using frequency analysis and then proceed with the Gumbel Probability Distribution method, Normal Distribution Log Person Type III. The result of this study with hydrological data and distribution test, the suitable method for analyzing the planned flood discharge in the Deli River is Log Person Type III that it can be flood discharge for a 2 year retention period is 12.7684672 m3/sec, 5 year return period is 12.56275855 m3/sec, 10 year return period is 15.91400029 m3/sec, 25 year return period is 16.87871678 m3/sec, 50 year return period is 17.43836748 m3/sec and a 100 year return period is 17.89619614 m3/sec.
 
 Keywords: Rainfall, Deli River, Planned Flood Discharge

Bias-Corrected IDF Curves From Satellite-Based Rainfall for HoaBinh Province, Vietnam

Satellite-based rainfall is extremely valuable data to quantify the probability of occurrence of rainfall events but is still uncertain to a certain extent. Therefore, this study firstly evaluates the performance of the satellite-based rainfall, PERSIANN-CCS, with rain gauge rainfall. The power transformation method is then applied to correct the satellite-based rainfall. Importantly, the Intensity-Duration-Frequency (IDF) curves are then constructed with the return periods of 5, 10, 25, 50, 100 and 200 years using the Gumbel probability distribution. The results show an efficiency of power transformation on satellite-based rainfall for both rainfall amount and events. It is especially noticed that it is well matched between bias-corrected satellite-based and rain gauge IDF curves duration 12-, 24-, 48- and 72-hour as a particular. For the duration of less than 12-hour, satellite-based IDF curves without bias correction significantly fit the rain-gauge IDF curves within the considered periods.

Topography based flood susceptibility prediction by using the GIS platform and historical flood data of lower Ajay river basin

Flood inundation mapping of lower Ajay River basin was implemented in the Geographic Information System (GIS) using the historical flood data and statistical frequency analysis by following the Gumbel probability distribution method. The historical data proofs the magnitudes of the flood intensity by various recurrence intervals on the basis of the projected flood height and present topographic character inundation mapping done in the GIS platform. Most importantly, due to the lack of hydrological past gauge height data are used in different gauge stations for projecting flood inundation mapping. Open access SRTM DEM data provides the topographic characteristics of the basin region and by the combined data of flood magnitudes and topographic data, flood inundation-mapping may help in the identification of the hazards prone area. The study area is regularly flooded and due to high population density huge damage takes place. So for the better management planning and risk estimation, hazards mapping is essential. Flood hazard mapping is done on the basis of 2, 10 and 25-year return period. The result depicts that 44% area in the lower part of the basin is flood prone and almost more than 474000 population and almost 59000 child population will be affected during the flood. The model is validated followed by overall accuracy assessment and KAPPA statistics methods and both results above 80% proof the model suitability.

Method for Prediction of Extreme Wave Loads Based on Ship Operability Analysis Using Hindcast Wave Database

The method for the prediction of extreme vertical wave bending moments on a passenger ship based on the hindcast database along the shipping route is presented. Operability analysis is performed to identify sea states when the ship is not able to normally operate and which are likely to be avoided. Closed-form expressions are used for the calculation of transfer functions of ship motions and loads. Multiple operability criteria are used and compared to the corresponding limiting values. The most probable extreme wave bending moments for the short-term sea states at discrete locations along the shipping route are calculated, and annual maximum extreme values are determined. Gumbel probability distribution is then fitted to the annual extreme values, and wave bending moments corresponding to a return period of 20 years are determined for discrete locations. The system reliability approach is used to calculate combined extreme vertical wave bending moment along the shipping route. The method is employed on the example of a passenger ship sailing across the Adriatic Sea (Split, Croatia, to Ancona, Italy). The contribution of the study is the method for the extreme values of wave loads using the hindcast wave database and accounting for ship operational restrictions.

Evaluation of generalized extreme value and Gumbel distributions for estimating maximum daily rainfall

Extreme rain events can cause social and economic impacts in various sectors. Knowing the risk of occurrences of extreme events is fundamental for the establishment of mitigation measures and for risk management. The analysis of frequencies of historical series of observed rain through theoretical probability distributions is the most commonly used method. The generalized extreme value (GEV) and Gumbel probability distributions stand out among those applied to estimate the maximum daily rainfall. The indication of the best distribution depends on characteristics of the data series used to adjust parameters and criteria used for selection. This study compares GEV and Gumbel distributions and analyzes different criteria used to select the best distribution. We used 224 series of annual maximums of rainfall stations in Santa Catarina (Brazil), with sizes between 12 and 90 years and asymmetry coefficient ranging from -0.277 to 3.917. We used the Anderson–Darling, Kolmogorov-Smirnov (KS), and Filliben adhesion tests. For an indication of the best distribution, we used the standard error of estimate, Akaike’s criterion, and the ranking with adhesion tests. KS test proved to be less rigorous and only rejected 0.25% of distributions tested, while Anderson–Darling and Filliben tests rejected 9.06% and 8.8% of distributions, respectively. GEV distribution proved to be the most indicated for most stations. High agreement (73.7%) was only found in the indication of the best distribution between Filliben tests and the standard error of estimate.

Probabilistic Analysis as a Method for Ground Freezing Depth Estimation

Accurate frost depth prediction is an important aspect in different engineering designs such as for pavements, buildings, bridge foundations, and utility lines. This paper presents a probabilistic method of assessment of the depth of soil freezing. Annual (winter) maxima of the position of the zero centigrade temperature measured in the soil were approximated by Gumbel probability distribution. Its parameters were estimated using maximum likelihood method. The results received on the basis of data from 36 meteorological stations in Poland and 50 years of observations, as characteristic values with 50-year return period, reflect the influence of the climatic conditions on the freezing depth. On the other hand, the soil structure and its conditions also play an important role in freezing. Nowadays they may be taken into account using correction coefficients. It is concluded that this method is more precise than a method using the air freezing index because through the use of direct measurements it takes into account additional factors affecting the actual depth of freezing. The obtained results are not the same as those given in the older Polish Standard which was based on the simplified and limited data. The results confirm the impact of climate change on ground freezing depth.

Assessment of wind speed and spatial mapping across the Indian subcontinent

Site-specific assessment of wind speed is essential in analysing the susceptibility of any slender structure. With the increasing number of occurrences of extreme wind events in the Indian subcontinent, a quantitative measure of wind hazard based on recent developments has become the need of the hour for India. The Indian standard (IS 875: Part 3) is a national code for reference to the basic wind speed in India, essentially for infrastructural planning and design of mid-rise to tall structures, and the specified basic wind speeds in the code are computed based on a 50-year return period by using the Gumbel probability distribution. However, recently reported cyclones were extreme wind events that already exceeded the specified value in the code. The Gumbel distribution (type I) is applied in the present study by using the method of moments on the long-term wind data (yearly maxima) considering the past few decades (40 years) available for forecasting the wind speed at different return periods. The Anderson–Darling test results are also accommodated to establish the accuracy of the tail behaviour of the statistical distribution used, where the p-value is found to be less than 0.05. Further, in the present study, a site-specific basic wind speed map is also proposed considering latest event data. Indeed, based on the present study, a 100-year return period for basic wind speed is recommended instead of 50 years to ensure enhanced structural safety and serviceability during the design life.

A New Left Truncated Gumbel Distribution: Properties and Estimation

Gumbel probability distribution is an important probability distribution that has wide applications in various fields such as engineering, physics, hydrology, geology, etc. The study aims to generate a new truncated distribution by truncate the probability density function of the first type Gumbel distribution from the left side to obtain the Truncated Gumbel distribution from the left side within the period (0, ∞). As we work in this paper on the derivation of the new truncated distribution formula and its characteristics. As well as discussing different methods of estimating its parameters, represented by the maximum likelihood method and Percentiles Estimation Method, then making a simulation of the comparison between methods of estimation.

Flood frequency analysis of Casimcea river

Flood frequency analysis is the most important statistical technique in understanding the nature and magnitude of high discharge in a river. This technique is used for establishing a relationship between flood magnitude and frequency of occurrence (return period). This study performs the flood frequency analysis of Casimcea river. The Casimcea catchment is located in the Dobrogea-Littoral basin of south-eastern Romania, and covers parts of two counties, Tulcea and Constanta. The main river of Casimcea catchment flows into the Tasaul lake and has two main tributaries (Ramnic and Cartal). In this study, Log-Normal, Log-Pearson Type III and Gumbel probability distributions have been applied on maximum discharge time series data of four hydrological stations, Casimcea and Cheia on Casimcea River and Pantelimonu on its main tributaries Ramnic and Cartal. For this study have been used the annual maximum series data spreading from 29 to 52 years. The frequency analysis will be performed with Hydrognomon. open-source software for the analysis of hydrological data. Hydrognomon is free software licensed under the General Public License (GPLv3).

Development of Intensity Duration Frequency (IDF) Curves for Rainfall Prediction within the Middle Niger River Basin

Rainfall Intensity-Duration-Frequency (IDF) relationship remains one of the mostly used tools in hydrology and water resources engineering, especially for planning, design and operations of water resource projects. IDF relationship can provide adequate information about the intensity of rainfall at different duration for various return periods. The focus of this research was to develop IDF curves for the prediction of rainfall intensity within the middle Niger River Basin (Lokoja and Ilorin) using annual maximum daily rainfall data. Forty (40) year’s annual maximum rainfall data ranging from 1974 to 2013 was employed for the study. To ascertain the data quality, selected preliminary analysis technique including; descriptive statistics, test of homogeneity and outlier detection test were employed. To compute the three hours rainfall intensity, the ratio of rainfall amount and duration was used while the popular Gumbel probability distribution model was employed to calculate the rainfall frequency factor. To assess the best fit model that can be employed to predict rainfall intensity for various return periods at ungauged locations, four empirical IDF equations, namely; Talbot, Bernard, Kimijima and Sherman equations were employed. The model with the least calculated sum of minimized root mean square error (RMSE) was adopted as the best fit empirical model. Results obtained revealed that the Talbot model was the best fit model for Ilorin and Lokoja with calculated sum of minimized error of 1.32170E-07 and 8.953636E-08. This model was thereafter employed to predict the rainfall intensity for different durations at 2, 5, 10, 25, 50 and 100yrs return periods respectively.

Hydro-climatology Characterization of Degraded Lwamunda Forest Catchment Based on Probability Distributions

Hydroclimatology assessment is conventionally based on area data for identification of change patterns and trends. In this paper, monthly averages, maximum seasonal and maximum annual hydro- climatology data series from Lwamunda forest catchment area in central Uganda have been analyzed in order to determine the appropriate probability distribution models for the underlying climatology (i.e. rainfall, soil moisture content, evapotranspiration and temperature). A total of 7 probability distributions were considered and three goodnessof- fit tests were used to evaluate the best-fit probability distribution model for each hydro-climatology data series. They were Lilliefors (D), Anderson-Darling (AD), and Cramer-Von Mises (W2). A ranking metric based on the test statistic from the three GoF tests was used to select the most appropriate probability distribution model capable of reproducing the statistics of the hydroclimatological data series. The best fit probability distribution was selected based on the minimum sum of the three test statistic. Results showed that different best fit probability distribution models were identified for the different data series depending on location and on temporal scales which corroborate with those reported in literature. With the exception of soil moisture content for annual and seasonal maximum series who have the same best fit model. The same applied to evapotranspiration seasonal maximum and near surface temperature seasonal maximum as well as monthly near surface temperatures have the same best fit model. The soil moisture content data series was best fit by the Weibull probability distribution, rainfall series was best fit by Chi square and Gamma probability distributions. The evapotranspiration data series was best fit by Logistic and Extreme value maximum (Gumbel) probability distributions. Finally for near surface temperature it was best fitted by Logistic and Gumbel probability distributions. The contribution of this study lies in the use of hydroclimatological data series including soil moisture content from the area that had forest cover change to analyzeits impact on water resources patterns. The contribution is important for agricultural planning and forest managers’ simulation of forest degradation impacts.

Heavy rainfall maps in Brazil to 5 year return period

This study created thematic heavy rainfall maps for Brazil, with durations of 5-, 30-, 60- and 120 minutes and 5 years of return period (T). The intensity-duration-frequency (IDF) relationships used were compiled from studies found in the literature (798 locations) and derived for 4411 rainfall gauges available in the Hidroweb information system, totaling 5209 rainfall data collected. To derive IDF relationships, Gumbel's probability distributions were used, with parameters estimated by the method of moments. Distribution adequacy was verified by the Kolmogorov-Smirnov test. Rainfall-intensity values obtained by IDF relationships were spatialized in Geographic Information Systems, allowing elaboration of the thematic maps. Thematic maps enable obtain rainfall intensities for places without rain gauge data and/or precarious time-series data. Therefore, these maps are a great tool for the design of hydraulic structures related to urban and rural micro-drainage.

Heavy rainfall equations for the “Manuel Alves da Natividade” river watershed, Tocantins, Brazil

Knowledge of the equation that relates intensity, duration and frequency of heavy rainfall is of great technical interest in hydraulic projects. The aim of this study was to obtain the intensity-duration-frequency (IDF) relation­ship of heavy rainfall in the Manuel Alves da Natividade river watershed, located in the southeastern region of the state of Tocantins, Brazil. We used the rainfall disaggregation model, with adjustment of the heavy rainfall equation by nonlinear multiple regression, in addition to historical series of six pluviometric stations, obtained from the Hydrological Information System of the National Water Agency (Hidroweb-ANA). Gumbel probability distribution was adequate to model the frequency distribution of annual maximum daily rainfall series, according to the Kolmogorov-Smirnov test at 20% significance level. The heavy rainfall equations presented good statistical quality, with coefficients of determination above 0.92, which characterize them as useful tools in the context of water resource management in the Manuel Alves da Natividade river watershed.

Extreme Statistics of Storm Surges in the Baltic Sea

Statistical analysis of the extreme values of the Baltic Sea level has been performed for a series of observations for 15–125 years at 13 tide gauge stations. It is shown that the empirical relation between value of extreme sea level rises or ebbs (caused by storm events) and its return period in the Baltic Sea can be well approximated by the Gumbel probability distribution. The maximum values of extreme floods/ebbs of the 100-year recurrence were observed in the Gulf of Finland and the Gulf of Riga. The two longest data series, observed in Stockholm and Vyborg over 125 years, have shown a significant deviation from the Gumbel distribution for the rarest events. Statistical analysis of the hourly sea level data series reveals some asymmetry in the variability of the Baltic Sea level. The probability of rises proved higher than that of ebbs. As for the magnitude of the 100-year recurrence surge, it considerably exceeded the magnitude of ebbs almost everywhere. This asymmetry effect can be attributed to the influence of low atmospheric pressure during storms. A statistical study of extreme values has also been applied to sea level series for Narva over the period of 1994–2000, which were simulated by the ROMS numerical model. Comparisons of the “simulated” and “observed” extreme sea level distributions show that the model reproduces quite satisfactorily extreme floods of “moderate” magnitude; however, it underestimates sea level changes for the most powerful storm surges.

Daily rainfall disaggregation for Tocantins State, Brazil

In order to design effective Brazilian hydraulic structures, it is necessary to obtain data relating to short-duration intense rainfall from historical series of daily rainfall. This recurring need can be fulfilled by rainfall disaggregation methodology. The objective of this study was to determine the intense rainfall disaggregation constants for the State of Tocantins and to compare these constants with those obtained for other regions of Brazil. For the modeling of the frequency of intense rainfall of different durations of less than 24 hours, the Gumbel probability distribution (GPD) was employed using rainfall series from 10 locations in Tocantins state. The results showed that the GPD was adequate by the Kolmogorov-Smirnov and Chi-square tests. The disaggregation constants presented low variability values for different return periods (from 10 to 100 years); the values for Tocantins state are: h12h/h24h=0.93, h6h/h24h=0.86, h4h/h24h=0.82, h3h/h24h=0.78, h2h/h24h=0.72, h1h/h24h=0.61, h50min/h1h=0.92, h40min/h1h=0.83, h30min/h1h=0.68, h20min/h30min=0.76 e h10min/h30min=0.46. The comparison of the results with those from studies developed for other Brazilian regions showed variations of up to -62.30%, allowing us to conclude that the use of local constants is important in the process of rainfall disaggregation.

Extreme value statistics for pitting corrosion of old underground cast iron pipes

Many major city water supply distribution networks consist of buried cast iron pipes. In many cases the pipes are internally cement-lined and the predominant corrosion is by external pitting. This may cause leakage and eventual structural failure. It is conventional to use the Gumbel extreme value distribution to represent the statistics of maximum pits depth and to use it to estimate the probability of pipe wall perforation. Herein data obtained for maximum pit depths for large-sized (1–2m long) samples of 10 pipes exhumed from different, apparently randomly selected, locations after 34–129 years of service are examined for consistency with the Gumbel probability distribution. This was the case for the deepest pits, but the data for less deep pits show a consistent pattern of departure from the Gumbel distribution. Some extreme pit depth data, inconsistent with the rest are interpreted as possibly caused by material imperfections.