Gumbel Extreme Value Distribution Research Articles

Quantifying regional rainfall dynamics in southern India: Unravelling monsoon characteristics and intense precipitation using satellite and observed data records

Rainfall patterns are unpredictable, which has implications for planning irrigation systems, constructing farm ponds and drainage systems, and conserving water and soil. The research findings have been compiled over 36 years, utilizing gridded daily precipitation data from the IMD and PWD (1984–2019). This study investigates the frequency of consecutive rainy days, the yearly rainfall pattern and the precipitation threshold value that triggers the flood. The Mann-Kendall approach is used to analyze trends in the statistical application XLSTAT. According to the trend analysis, rainfall and wet days during the Northeastern Monsoon (NEM) throughout the year show significant and nonsignificant increasing trends at most stations. Statistical or empirical approaches can be used to establish the threshold value of precipitation that causes a flood by examining historical flood episodes and the precipitation that occurred. The Gumbel extreme value distribution method was used to determine the recurrence intervals of days 1, 2, 3, and 4 that are consecutive and equal to or above 5 days. The study's findings will show that rainfall patterns can be used to forecast floods and estimate their return and recurrence intervals. As a result, implementing this research with these precipitation threshold values could help with flood warning systems and the infrastructure required for emergency responses to flooding disasters.

Effects of damage initiation points of depth-damage function on flood risk assessment

The flood depth in a structure is a key factor in flood loss models, influencing the estimation of building and contents losses, as well as overall flood risk. Recent studies have emphasized the importance of determining the damage initiation point (DIP) of depth-damage functions, where the flood damage is assumed to initiate with respect to the first-floor height of the building. Here we investigate the effects of DIP selection on the flood risk assessment of buildings located in Special Flood Hazard Areas. We characterize flood using the Gumbel extreme value distribution’s location (μ) and scale (α) parameters. Results reveal that average annual flood loss (AAL) values do not depend on μ, but instead follow an exponential decay pattern with α when damage initiates below the first-floor height of a building (i.e., negative DIP). A linear increasing pattern of the AAL with α is achieved by changing the DIP to the first-floor height (i.e., DIP = 0). The study also demonstrates that negative DIPs have larger associated AAL, thus contributing substantially to the overall AAL, compared to positive DIPs. The study underscores the significance of proper DIP selection in flood risk assessment.

Flood frequency analysis of Asa River, Ilorin, Nigeria

Flooding poses a risk to human lives, livelihoods, infrastructure, and the environment. Therefore, understanding the flood frequency patterns of the Asa River is crucial in developing effective flood management strategies. This study explores the statistical technique to understand the nature and magnitude of high discharge of floods in the Asa River, aiming at relating the magnitude of floods and their frequency of occurrence through probability distribution. The Gumbel Extreme Value Distribution method was used to analyze the 30 years (1991-2020) flood discharge data of River Asa which was collected from Lower Niger River Basin and Development Authority, Ilorin, and flood discharge for the return periods of 5 years, 10 years, 20 years, 30 years, 50 years, 100 years, and 150 years, was predicted. From the observations of Gumbel’s distribution, the R2 value acquired from the trend line equation was 0.9227, indicating that Gumbel's extreme value distribution is suitable for estimating predicted river flood flow. The expected flood discharge for the return periods of the upcoming 5 years, 10 years, 15 years, 20 years, 25 years, 50 years, 100 years, and 150 years, was estimated to be 43.19 m3/s, 51.12 m3/s, 55.60 m3/s, 58.73 m3/s, 63.11 m3/s, 68.58 m3/s, 75.96 m3/s, and 80.27m3/s respectively. The results obtained in this research would adequately serve as a planning guide for the decision makers on the development and flood risk management along Asa River, to achieve sustainable development and management, with a view of protecting lives and properties downstream of the watershed.

Extremes in incomplete samples from moving averages of random variables from the domain of attraction of the Gumbel distribution

We are interested in further extremal properties of the moving average process defined from an i.i.d. noise sequence with common distribution function belonging to the maximum domain of attraction of the Gumbel extreme value distribution and satisfying additionally two specific conditions. In particular, we consider the limiting behavior of a sequence of random vectors consisted of maxima in complete and incomplete samples from this class of moving averages. The point process approach is used to derive the corresponding results.

Mesoscopic pore characteristics analysis of aged bridge concrete based on X-ray computed tomography

The characteristics of mesoscopic pores in concrete bridges have significant influence on structural mechanical properties. In this study, 18 core specimens, taken from three decommissioned concrete bridges serving for about 25 years, were investigated using X-ray computed tomography (X-ray CT). After segmentation and 3D reconstruction with CT scanning images, the spatial characteristics for the porosity, volume, and shape of the concrete mesoscopic pores were analysed. The mesoscopic porosity decreases with the increase of concrete strength. The logarithm of the pore volume obeyed the Gumbel extreme value distribution, whose location and scale parameters had obvious bilinear degradation with depth. Ellipsoidal approximation was employed to precisely characterise the individual pore structure, and the Gaussian copula function could describe the joint distribution among the pore shape parameters. It was found that correlation coefficients of ellipsoidal shape parameters remained stable with depth. The proposed model quantitatively describes the mesoscopic pores in decommissioned concrete bridges and can shed certain light on concrete pore simulation.

Frequency Analysis of Flood Flow in Markanda Basin of Ghaggar River System in North Western India

Abstract This study aims to estimate the probabilities of occurrence and return periods of peak flood discharges over the Markanda basin of Ghaggar river system in north western India. For this purpose, two most frequently employed probability distribution models, namely Gumbel Extreme Value (GEV) and Log-Pearson Type III (LP-III) have been used to estimate the future flood discharges by means of annual extreme flood series (Qmax) data from 1990 to 2013, available at eight gauge and discharge sites. Two goodness-of-fit tests, i.e., Kolmogorov-Smirnov and Anderson-Darling have been applied to the fitted probability distributions to identify the best-fit model. The Qmax for several return periods, for example 2, 5, 10, 25, 50, 100, and 200 years have been estimated and compared. The return period for the highest Qmax recorded at Jhansa (2670 m3/s) gauge and discharge site has been computed as 9.5 and 8.2 years using the GEV and LP-III distribution model, respectively. The analyses have shown that GEV distribution model has produced the overestimated results in comparison to LP-III distribution model. Also, flood index has been computed for flood regionalization. The flood index values have been found variable at different gauge and discharge sites with an increase in return periods. Finally, the finding of this study will be valuable for water resource engineers in designing the hydraulic structures for the management of recurrent floods.

A data-driven spatial approach to characterize the flood hazard.

Model output of localized flood grids are useful in characterizing flood hazards for properties located in the Special Flood Hazard Area (SFHA-areas expected to experience a 1% or greater annual chance of flooding). However, due to the unavailability of higher return-period [i.e., recurrence interval, or the reciprocal of the annual exceedance probability (AEP)] flood grids, the flood risk of properties located outside the SFHA cannot be quantified. Here, we present a method to estimate flood hazards that are located both inside and outside the SFHA using existing AEP surfaces. Flood hazards are characterized by the Gumbel extreme value distribution to project extreme flood event elevations for which an entire area is assumed to be submerged. Spatial interpolation techniques impute flood elevation values and are used to estimate flood hazards for areas outside the SFHA. The proposed method has the potential to improve the assessment of flood risk for properties located both inside and outside the SFHA and therefore to improve the decision-making process regarding flood insurance purchases, mitigation strategies, and long-term planning for enhanced resilience to one of the world's most ubiquitous natural hazards.

Climate change simulation and trend analysis of extreme precipitation and floods in the mesoscale Rur catchment in western Germany until 2099 using Statistical Downscaling Model (SDSM) and the Soil & Water Assessment Tool (SWAT model).

Due to climate change and global warming, speed and intensity of the hydrological cycle will accelerate. In order to carry out regional risk assessment, integrated water resources management and flood protection, far reaching predictions and future scenarios of climate change effects on extreme precipitation and flooding are of particular relevance. In this study, trends in frequencies of extreme precipitation and floods until 2099 are analysed for the German Rur catchment, which is half located in highlands and half in lowlands and therefore has a high topographical and climatological contrast. To predict future trends, coupled modeling is performed based on NCEP reanalysis data and a General Circulation Model (GCM). Assuming HadCM3 future emission scenarios A2a and B2a, an empirical Statistical Downscaling Model (SDSM) is developed and daily precipitation amounts are projected until 2099 by a stochastic weather generator. The generated precipitation data are used as an input for the ecohydrological Soil & Water Assessment Tool (SWAT model) to simulate daily water discharge until 2099. Statistical trend analyses are implemented based on three annual extreme precipitation indices (EPIs) and the magnitudes of ten flood return periods derived with GEV and Gumbel extreme value distributions for 109 30-year moving periods using regression analyses and Mann-Kendall tendency tests to check for significant trends in the frequencies until 2099. As a result, it could be demonstrated for all EPIs that the frequency of extreme precipitation in the upper Rur catchment will significantly increase by +33% to +51% until 2099 compared to the base period 1961-1990, whereas mostly non-significant negative trends of extreme precipitation can be projected in the lowlands. For runoff, it was found that the magnitudes of the ten flood return periods will significantly increase by +31% for B2a to +36% for A2a until 2099 compared to the base period.

Improved building-specific flood risk assessment and implications of depth-damage function selection

Average annual loss (AAL) is traditionally used as the basis of assessing flood risk and evaluating risk mitigation measures. This research presents an improved implementation to estimate building-specific AAL, with the flood hazard of a building represented by the Gumbel extreme value distribution. AAL is then calculated by integrating the area under the overall loss-exceedance probability curve using trapezoidal Riemann sums. This implementation is compared with existing AAL estimations from flood risk assessment. A sensitivity analysis is conducted to examine the variability in AAL results based on depth-damage function (DDF) choice. To demonstrate the methodology, a one-story single-family residence is selected to assess the financial benefits of freeboard (i.e., increasing lowest floor elevations). Results show that 1 ft. of freeboard results in annual flood risk reduction of over $1,000, while 4 ft of freeboard results in annual flood risk reduction of nearly $2,000. The sensitivity result suggests that the DDF selection is critical, as a large proportion of flood loss is counted below the top of the first floor. The findings of this paper will enhance DDF selection, improve flood loss estimates, encourage homeowners and communities to invest in flood mitigation, and provide government decision-makers with improved information when considering building code changes.

Flood Frequency Analysis of Araniar Medium Irrigation Project in Chittoor District by using Gumbel's Distribution

Flood frequency analysis is one of the most important statistical technique to understand the nature and magnitude of high discharge of floods in the river. The objective of flood frequency analysis was to relate the magnitude of floods and their frequency of occurrence through probability distribution. Flood frequency analysis is essential to reduce the impact of flood damage by predicting the floods by adopting the suitable flood prediction model. The Araniar reservoir was constructed across the Araniar river in the Chittoor district of Andhra Pradesh. The flood frequency analysis was conducted on the Araniar reservoir using Gumbel’s extreme value distribution method in the year 2019-2020. Daily maximum inflow data from the Department of Water Resources, Andhra Pradesh, were collected for the period 1990-2019 and used for flood frequency analysis of Araniar Reservoir using Gumbel's extreme value distribution method. From the observations of Gumbel’s distribution, the R2 value acquired from the trend line equation was 0.9803, indicating that Gumbel's extreme value distribution was suitable for estimating predicted reservoir flood flow. The estimated flood flow for upcoming years of 2 years, 10 years, 50 years, 100 years,150 years, 200 years, 300 years and 400 years were found to be 38.29 Mm3/year, 66.96 Mm3/year, 92.08 Mm3/year, 102.71 Mm3/year, 108.90 Mm3/year, 113.29 Mm3/year, 119.47 Mm3/year and 123.85 Mm3/year respectively. The mean instantaneous flow in the reservoir was 40.88149 Mm3/year which was nearly equal to a return period of about 2 years. The estimated flood flows of the Araniar reservoir was useful for constructing important dam hydraulic structures, advising agricultural patterns in the command area, and protecting lives and property downstream of the catchment region.

Fatigue Assessment of Wire and Arc Additively Manufactured Ti-6Al-4V

Wire and arc additively manufactured (WAAM) parts and structures often present internal defects, such as gas pores, and cause irregularities in the manufacturing process. In order to describe and assess the effect of internal defects in fatigue design, this research study investigates the fatigue strength of wire arc additive manufactured structures covering the influence of imperfections, particularly gas pores. Single pass WAAM structures are manufactured using titanium alloy Ti-6Al-4V and round fatigue, tensile specimen are extracted. Tensile tests and uniaxial fatigue tests with a load stress ratio of R = 0.1 were carried out, whereby fatigue test results are used for further assessments. An extensive fractographic and metallographic fracture surface analysis is utilized to characterize and measure crack-initiating defects. As surface pores as well as bulk pores are detected, a stress intensity equivalent ∆Keqv transformation approach is presented in this study. Thereby, the defect size of the surface pore is transformed to an increased defect size, which is equivalent to a bulk pore. Subsequently, the fatigue strength assessment method by Tiryakioğlu, commonly used for casting processes, is applied. For this method, a cumulative Gumbel extreme value distribution is utilized to statistically describe the defect size. The fitted distribution with modified data reveals a better agreement with the experimental data than unmodified. Additionally, the validation of the model shows that the usage of the ∆K modified data demonstrates better results, with a slight underestimation of up to about −7%, compared to unmodified data, with an overestimation of up to about 14%, comparing the number of load cycles until failure. Hence, the presented approach applying a stress intensity equivalent transformation of surface to bulk pores facilitates a sound fatigue strength assessment of WAAM Ti-6Al-4V structures.

Interpolation of Precipitation Extremes on a Large Domain Toward IDF Curve Construction at Unmonitored Locations

An intensity–duration–frequency (IDF) curve describes the relationship between rainfall intensity and duration for a given return period and location. Such curves are obtained through frequency analysis of rainfall data and commonly used in infrastructure design, flood protection, water management, and urban drainage systems. However, they are typically available only in sparse locations. Data for other sites must be interpolated as the need arises. This paper describes how extreme precipitation of several durations can be interpolated to compute IDF curves on a large, sparse domain. In the absence of local data, a reconstruction of the historical meteorology is used as a covariate for interpolating extreme precipitation characteristics. This covariate is included in a hierarchical Bayesian spatial model for extreme precipitations. This model is especially well suited for a covariate gridded structure, thereby enabling fast and precise computations. As an illustration, the methodology is used to construct IDF curves over Eastern Canada. An extensive cross-validation study shows that at locations where data are available, the proposed method generally improves on the current practice of Environment and Climate Change Canada which relies on a moment-based fit of the Gumbel extreme-value distribution.

Urban flash flood and extreme rainfall events trend analysis in Bamako, Mali

One of the consequences of climate change is the increasing rainfall intensity and frequency leading to flooding. Lately, some major cities in West Africa have been experiencing high frequency of flooding due to extreme rainfall. This study investigated the temporal trends in flash flood and extreme rainfall events in Bamako, Mali for the period 1982–2019. Rainfall observation data from both Mali Meteorological Agency and Climate Hazards Group Infrared Precipitation, and flood ground information’ collected from the Mali Civil protection service were used for this study. Five rainfall extreme indices established by Expert Team for Climate Change Detection Monitoring and Indices (ETCCDI) were analyzed to characterize extreme rainfall intensity and frequency indices. The Gumbel extreme value distribution was used to estimate the return period of flood and extreme rainfall for the period of 5 to 100 years based on the annual maximum daily rainfall. All the five extreme rainfall indices displayed an upward trends except the consecutive wet days (CWD) that showed a decreasing trend. Both the intensity and frequency of extreme rainfall were found to have increased over the study period. The analysis of the flood reports showed an increasing trend in Bamako from 1982 to 2019. The returned period revealed that about 58% of flood events in Bamako are caused by normal rainfall, with 33.3% being caused by extreme rainfall. It was also determined that most flood events were recorded in the years that had higher number of extremely wet days. The findings from this study have demonstrated that floods in Mali are prevalent and adaptation and mitigation strategies are needed.

COMPUTATIONAL STUDY OF EVOLUTION AND FATIGUE DISPERSITY OF MICROSTRUCTURES BY ADDITIVE MANUFACTURING

In order to predict the correlation among the processing parameters, microstructures, and mechanical properties for additive manufacturing, a computational framework integrating discrete element method, phase-field simulation, crystal plasticity finite element method, and extreme value statistics is proposed. The framework is applied to reveal the influence of laser scanning velocity on the microstructure evolution, yield stress, fatigue indicator parameter (FIP) distribution, and fatigue dispersity, in order to show its capability in simulating additive manufacturing process and the resultant mechanical properties. Firstly, discrete element method simulations are carried out to spread the powder bed layer by layer with the consideration of powder size distribution. The spreading is performed on the curved surface of the previously solidified layer. Secondly, the heat-melt-microstructure coupled non-isothermal phase-field simulations are performed to obtain the temporal and spatial evolution of melt, pore, grain boundary, grain distribution/orientation, etc., as well as the final polycrystal microstructure. Thirdly, crystal plasticity finite element method is utilized to attain the macroscopic mechanical response and stress/strain distribution of the additively manufactured polycrystal microstructure (AMPM) and FIP which is a surrogate measure for the driving force to form fatigue cracks. Fourthly, extreme value statistics are carried out to analyze the extreme value distribution of FIPs and the fatigue dispersity of the AMPM. Comprehensive simulations are put into practice for the selective laser melting based additive manufacturing of a typical metallic material 316L stainless steel. The simulation results indicate that the macroscopic yield stress of the AMPM is anisotropic and decreases with the increasing laser scanning velocity. The extreme value of FIPs from the AMPM with random distribution of grain orientations correlates well with the Gumbel extreme value distribution. The increase of laser scanning velocity could decrease the fatigue dispersity of the AMPM, but increase the FIP extremum and the associated driving force for fatigue crack initiation and thus notably decrease the fatigue life.

Statistical analysis of extreme value of meteorological elements observed for the last 31 years (1989–2019) at Narora site

Understanding of extreme weather conditions at the site of interest is essentially required in the design of engineering structures so that the structures can withstand weather stresses. This paper presents an analysis of extreme values of meteorological elements observed at Narora site for the last three decades (1989–2019). The elements considered are extreme air temperature (°C), minimum relative humidity (%), extreme wind gust (km/h), maximum rainfall (mm) in a day and a month, and annual rainfall. The extreme value analysis reveals that the maximum air temperature, maximum wind gust at 30 m, maximum monthly rainfall, and maximum annual rainfall obey Fisher-Tippett Type-1/Gumbel extreme value distribution, whereas minimum air temperature, minimum relative humidity (%), annual daily maximum rainfall (mm), and annual minimum rainfall (mm) obey Fisher-Tippett Type-2/Frechet extreme value distribution function. Distribution function parameters, i.e., location, scale, and shape parameter for each variable, have been determined. Extreme values corresponding to return periods of 50, 100, and 1000 years are worked out using best fit linear regression curve as a compliance of the Atomic Energy Regulatory Board Safety Guide Recommendations. The derived extreme values are particularly useful to designer for arriving at suitable design basis values of different elements to ensure the safety of the reactors and other civil structures in Narora region, with respect to stresses due to weather conditions. Extreme values corresponding to return periods of 50 and 100 years at Narora are compared with corresponding values at other three nuclear reactor sites in India, namely, Tarapur, Kalpakkam, and Trombay. In addition, the time series pattern analysis of rainfall for 31 years at the Narora site closely following the 2-year moving average rainfall data pattern. These results can be used for water harvesting, irrigation, and floods management plans in the future.

Application of HEC-RAS and HEC-GeoRAS model for Flood Inundation Mapping, the case of Awash Bello Flood Plain, Upper Awash River Basin, Oromiya Regional State, Ethiopia

Flood is the devastating natural events as it causes massive destructive of life, economy and infrastructure. The main objective of this study was flood inundation mapping using HEC-RAS model: the case of Awash Bello flood plain, Upper Awash River basin, Ethiopia. River geometry, annual peak flood, and boundary conditions are the most important input parameters for RAS preprocessing. The river cross-sections along the flood plain were extracted using HEC-Geo RAS. Annual peak flood frequency analysis for different recurrence intervals (i.e., 2, 5, 10, 25, 50, and 100 years) was computed using the calibrated and validated HEC-HMS model based on a 25 years (1990–2015) hydro-meteorological data collected from Ministry of Water Resources Irrigation and Electricity and National Meteorological Agency, respectively. The model result depicted the flood extreme value for the respective recurrence interval was 526, 610, 828.8, 1072.8, 1263.6, and 1461.3 m3/s, respectively. For more reliability, the HEC-HMS model outputs were compared with the flood extreme value obtained from Log Pearson type-III, General Pareto and Gumbel extreme value distribution function and it was found the HEC-HMS model result for every recurrence interval was higher than the extreme value distribution function outputs. After verifying the acceptability of the HEC-HMS model result, the verified peak flood was inserted into HEC-RAS model and flood inundation mapping for different recurrence intervals were executed. The HEC-RAS model outcome indicated that the flood inundation mapping area for 2, 5, 10, 25, 50, and 100-year recurrence intervals, respectively, was 71.475, 76.630, 89.150, 100.290, 105.160, and 109.462 km2. Finally, it was realized that the whole Awash Bello flood plain is under the influence of flood inundation due to the intensive rainfall event.

Estimation of the return periods of maxima rainfall and floods at the Pra River Catchment, Ghana, West Africa using the Gumbel extreme value theory

The Pra river catchment in Ghana is adversely affected by perennial flooding from high-intensity rainfall events. To aid in flood management at the catchment, the Gumbel extreme value distribution has been used to estimate the return periods of maxima rainfall, flood, and consecutive dry and wet days (CDD and CWD) for a period of 5 to 100 years. The results revealed an expected increase in maxima rainfall, CDD and CWD. Maxima rainfall favours the south of the catchment while the CDD decreases northward. Furthermore, an increase in the magnitude of CWD observed at the centre of the catchment had a maximum of approximately 30 days for the 100 year return period, while lower flood volumes had a higher recurrence of 50% to 100% for 1 to 2 year return periods. The inclusion of a projected increase in anthropogenic activities and climate factors at the catchment will slightly affect the magnitude of these variables for the various return periods. Nonetheless, the findings in this study will be of essential input to policy implementation of the Integrated Water Resource Management Plan for river catchments in Ghana, West Africa.

The Three Extreme Value Distributions: An Introductory Review

The statistical distribution of the largest value drawn from a sample of a given size has only three possible shapes: it is either a Weibull, a Fréchet or a Gumbel extreme value distributions. I describe in this short review how to relate the statistical distribution followed by the numbers in the sample to the associate extreme value distribution followed by the largest value within the sample. Nothing I present here is new. However, from experience, I have found that a simple, short and compact guide on this matter written for the physics community is missing.

New insights from probabilistic modelling of corrosion in structural reliability analysis

Infrastructure intended to contain or exclude fluids or gasses, or reinforced with glass or other fibres may be subject to failure caused by pitting corrosion of the containment system or of the reinforcement. To assess the reliability of the infrastructure then requires probabilistic models for the deterioration processes involved. Typically the critical deterioration is through localized pitting and in particular the most extreme pit depth as this governs containment capability and also fibre strength. The ‘arch-typical’ distribution for representing maximum pit depth is the Gumbel extreme value distribution, for which complying data plot linearly on a so-called Gumbel plot. However, there is increasing evidence that large, homogeneous datasets show significant deviations from linearity. This is demonstrated for steel plates continuously immersed in seawater, for the interior corrosion of water injection pipelines and for crude oil production pipelines. Further examples are given for the maximum pit depth on stainless steel rollers in the papermaking industry and for localized corrosion of glass fibre reinforcement in concrete structures. In each case the non-linear trends obtained permit a re-interpretation of, and new insights for, the underlying physico-chemical-material mechanisms. The results are important for accurate representation of deterioration processes and for best-practice structural reliability analyses.

Testing for the change of the mean-reverting parameter of an autoregressive model with stationary Gaussian noise

The likelihood ratio test for a change in the mean-reverting parameter of a first order autoregressive model with stationary Gaussian noise is considered. The test statistic converges in distribution to the Gumbel extreme value distribution under the null hypothesis of no change-point for a large class of covariance structures including long-memory processes as the fractional Gaussian noise.