Univariate Frequency Analysis Research Articles

Evaluation of trends and multivariate frequency analysis of droughts in three meteorological subdivisions of western India

ABSTRACTThis article presents evaluation of trends and multivariate frequency analysis of droughts in three meteorological subdivisions of western India, namely, western Rajasthan, Saurashtra and Kutch and Marathwada regions. These regions are frequently affected by droughts and there is an urgent need for effective planning and management of droughts. Meteorological drought is modelled using Standardized Precipitation Index (SPI) at a time scale of 6 months over 110 years during 1896–2005. Trends in SPI time series are investigated by using nonparametric Mann–Kendall trend test for different time windows: entire study period of 1896–2005 and then splitting into three time windows of 1896–1931, 1932–1966 and 1967–2005. For total study period, the long‐term trend in SPI time series is found to be in an upward direction for the three regions. However, statistically significant downward trend is observed during 1932–1966 for western Rajasthan region, and Saurashtra and Kutch region in the month of June, indicating increase in number of drought occurrences during this period. Further, drought is a multivariate natural calamity characterizing severity, duration and peak; hence, probabilistic assessment of drought characteristics is investigated using copula method. The joint distribution of drought properties is modelled using three fully nested forms of Archimedean copulas: Clayton, Gumbel–Hougaard and Frank and one elliptical class of Student's t copula. On performing various statistical tests as well as upper tail dependence test it is found that Student's t copula better represents trivariate drought properties when compared with other copula families. The joint distribution obtained from the copula is utilized for computation of conditional probabilities and joint return periods. The importance of trivariate frequency analysis of drought is elucidated over univariate and bivariate frequency analysis. Overall, the results of the study could provide valuable insight towards regional drought risk management under changing climate.

Risk and uncertainty analysis for dam overtopping – Case study: The Doroudzan Dam, Iran

There is a growing tendency to assess the safety levels of existing dams by using mathematical and statistical methods. In this study, the application of risk and uncertainty analysis to dam overtopping is presented for Doroudzan Reservoir located at the south part of Iran. The main objective of the overtopping analysis of dams is estimating the height of water in the reservoir under various inflows and comparing the computed results with the dam crest elevation. Hence, the main steps of this study include univariate flood frequency analysis of annual maximum inflows to estimate the peak flows in various return periods, generate inflow hydrographs based on the estimated peak flows, and route the hydrographs through the reservoir to compute the maximum height of the water in reservoir. In this study, the spillway discharge coefficient, quantile of peak flows, and initial water surface level are subject to uncertainty, and the Monte-Carlo simulation (MCS) and Latin hypercube sampling (LHS) are applied to perform the uncertainty analysis. In addition to inflows, the effect of different wind speeds on the probability of overtopping has been considered. The results demonstrated that both increasing water level and wind speed have significant impact on the risk of overflowing.

Copula 모형을 이용한 이변량 강우빈도해석

The estimation of the rainfall quantile is of great importance in designing hydrologic structures. Conventionally, the rainfall quantile is estimated by univariate frequency analysis with an appropriate probability distribution. There is a limitation in which duration of rainfall is restrictive. To overcome this limitation, bivariate frequency analysis by using 3 copula models is performed in this study. Annual maximum rainfall events in 5 stations are used for frequency analysis and rainfall depth and duration are used as random variables. Gumbel (GUM), generalized logistic (GLO) distributions are applied for rainfall depth and generalized extreme value (GEV), GUM, GLO distributions are applied for rainfall duration. Copula models used in this study are Frank, Joe, and Gumbel-Hougaard models. Maximum pseudo-likelihood estimation method is used to estimate the parameter of copula, and the method of probability weighted moments is used to estimate the parameters of marginal distributions. Rainfall quantile from this procedure is compared with various marginal distributions and copula models. As a result, in change of marginal distribution, distribution of duration does not significantly affect on rainfall quantile. There are slight differences depending on the distribution of rainfall depth. In the case which the marginal distribution of rainfall depth is GUM, there is more significantly increasing along the return period than GLO. Comparing with rainfall quantiles from each copula model, Joe and Gumbel-Hougaard models show similar trend while Frank model shows rapidly increasing trend with increment of return period.

독립 호우사상에 대한 이변량 강우빈도해석 및 강우-유출해석

본 연구에서는 독립 호우사상을 이용하여 이변량 빈도해석 및 유출해석을 수행하고, 이를 기존 단변량 빈도해석 결과와 비교 평가하였다. 본 연구는 규모가 다른 중랑천, 청계천, 우이천유역 등 세 유역에 대해 수행되었다. 유출모형으로 Clark 모형을 이용하였고, 유효우량은 SCS 방법을 적용하여 계산하였다. 강우의 시간분포 모형으로 교호블록 방법 및 Huff 방법을 적용하여 그 결과가 비교될 수 있도록 하였다. 그 결과를 정리하면 다음과 같다. (1) 연최대치 독립 호우사상계열의 이변량 빈도해석 결과, 지속기간이 짧은 경우에는 단변량 빈도해석 결과와의 차이가 매우 크나 지속기간이 길어짊에 따라서 그 차이가 현저히 줄어드는 것으로 나타났다. 아울러 지속기간이 짧은 경우, 단변량 빈도해석 결과가 이변량 빈도해석 결과보다 더욱 크게 나타났으나 특정 지속기간 이상부터 그 결과가 역전되는 것으로 나타났다. (2) 교호블록 시간분포 방법을 적용하는 경우가 Huff 방법을 적용한 경우보다 더욱 큰 첨두유출량을 발생시키는 것으로 나타났다. 아울러 교호블록 방법을 적용하는 경우에는 강우 지속기간의 증가에 따라서 첨두유출량이 점차 증가하는 것으로 나타났으나, 강우 지속기간이 대략 24시간 정도 되었을 때 그 값이 거의 수렴하는 것으로 나타났다. (3) 중랑천 유역에 대해 Huff 방법을 적용하여 유출해석을 수행한 결과에서는 이변량 설계강우를 적용한 경우가 단변량 설계강우를 적용한 경우보다 더욱 큰 홍수량을 발생시키는 것으로 나타났다. 반면에 청계천 및 우이천 유역의 경우에는 이변량 설계강우를 적용한 경우보다 단변량 설계강우를 적용한 경우의 홍수량이 다소 큰 것으로 나타났다. 그러나 교호블록 방법을 적용한 경우에서는 모든 유역에 대해 이변량 설계강우를 적용한 경우가 단변량 설계강우를 적용한 경우보다 큰 홍수량을 발생시키는 것으로 나타났다. In this study, the bivariate frequency analysis of the independent annual rainfall event series was done to be used for the runoff analysis, whose results were also compared with those from the conventional univariate frequency analysis. This study was applied to three differently-sized basins such as the Joongryang Stream, Chunggye Stream, and Ooyi Stream. The Clark model was used as the runoff model, and the SCS method was applied for the calculation of the effective rainfall. The alternating block method and the Huff method were considered to be compared for the temporal distribution of rainfall event. Summarizing the results are as follows. (1) The difference between the univariate and bivariate frequency analysis results were large when the rainfall duration was short, but significantly decreased as the rainfall duration increased. The univariate frequency analysis results were bigger when the rainfall duration was short, but smaller in opposite case. (2) The peak flow derived by applying the alternating block method was bigger than that by the Huff method. Also, the peak flow when applying the alternating block method increased as the rainfall duration increased, but converged smoothly around the rainfall duration of 24 hours. (3) For the Joongryang Stream, when applying the Huff method, the peak flow derived for the bivariate frequency analysis was bigger than that for the univariate case, but for the other two basins, the results were opposite. When applying the alternating block method, the results were consistent for all three basins that the peak flow derived by applying the bivariate frequency analysis was bigger than those by the univariate frequency analysis.

Evaluation of dam overtopping risk based on univariate and bivariate flood frequency analyses

There is a growing tendency to assess safety of dams by mathematical and statistical methods in hydrosystem engineering. This research presents the application of risk and uncertainty analysis to dam overtopping based on univariate and bivariate flood frequency analyses by applying Gumbel logistic distribution. The bivariate frequency analyses produced six inflow hydrographs with a joint return period of 100 years. Afterward, the overtopping risk of the Doroudzan Dam was evaluated for all six inflow hydrographs by considering quantile of flood peak discharge, initial depth of water in the reservoir, and discharge coefficient of spillway as uncertain variables and using two uncertainty analysis methods; Monte Carlo simulation and Latin hypercube sampling. Finally, the results of both univariate and bivariate frequency analyses were compared to show the significance of bivariate analysis on dam overtopping.

독립호우사상의 확률론적 해석 : 2. 호우사상의 재현기간

본 연구에서는 이변량 극치분포를 이용하여 연최대치 호우사상을 평가하였다. 이를 위해 특정 재현기간을 가지는 호우사상의 강우량을 비동시결합 재현기간, 동시결합 재현기간 그리고 구간조건부 결합재현기간의 세 가지를 이용하여 산정하였다. 이때, 결합재현기간별 호우사상의 값의 크기가 서로 다르게 산정되는 이유를 이변량 분포의 확률특성을 보여주는 사분면을 이용하여 설명하였다. 호우지속기간 24시간인 경우에 동시결합재현기간을 이용하여 산정한 확률강우량은 전통적인 방법으로 얻어진 강우지속기간 24시간의 확률강우량과 유사하게 나타났다. 이러한 결과는 전통적인 강우빈도해석의 제약사항을 극복하는데 도움이 될 것으로 보여진다. 이변량 빈도해석으로 얻어진 확률호우사상은 저류시설물의 계획시 통계적으로 보다 유용하면서도 간단한 설계 호우사상을 제공할 수 있을 것으로 보여진다. In this study, annual maximum storm events are evaluated by applying the bivariate extremal distribution. Rainfall quantiles of probabilistic storm event are calculated using OR case joint return period, AND case joint return period and interval conditional joint return period. The difference between each of three joint return periods was explained by the quadrant which shows probability calculation concept in the bivariate frequency analysis. Rainfall quantiles under AND case joint return periods are similar to rainfall depths in the univariate frequency analysis. The probabilistic storm events overcome the primary limitation of conventional univariate frequency analysis. The application of these storm event analysis provides a simple, statistically efficient means of characterizing frequency of extreme storm event.

Multivariate quantiles in hydrological frequency analysis

Several hydrological phenomena are described by two or more correlated characteristics. These dependent characteristics should be considered jointly to be more representative of the multivariate nature of the phenomenon. Consequently, probabilities of occurrence cannot be estimated on the basis of univariate frequency analysis (FA). The quantile, representing the value of the variable(s) corresponding to a given risk, is one of the most important notions in FA. The estimation of multivariate quantiles has not been specifically treated in the hydrological FA literature. In the present paper, we present a new and general framework for local FA based on a multivariate quantile version. The multivariate quantile offers several combinations of the variable values that lead to the same risk. A simulation study is carried out to evaluate the performance of the proposed estimation procedure and a case study is conducted. Results show that the bivariate estimation procedure has an analogous behaviour to the univariate one with respect to the risk and the sample size. However, the dependence structure between variables is ignored in the univariate case. The univariate estimates are obtained as special combinations by the multivariate procedure and with equivalent accuracy

Copula-based depth-duration-frequency analysis of typhoons in Taiwan

Typhoons are an inevitable and frequently occurring natural hazard in Taiwan which cause severe economic damage and loss of life. The common practice for flood-mitigation planning and design uses univariate frequency analysis. However, separate univariate analysis cannot reveal the significant relationship among correlated variables. This study therefore employs copulas to construct the joint distribution of rainfall depth and duration for typhoon data. Using copulas to construct a multivariate distribution means that the effects of marginal variables can be separated from that of dependent variables. We derive the depth-duration-frequency (DDF) formula based on using copulas to represent the joint distribution of rainfall depth and duration. Typhoon data recorded at the Kaohsiung Weather Station located in southern Taiwan are used as an example to illustrate the proposed methodology. The marginal distributions for rainfall depth and duration are fitted as the three-parameter gamma and Gumbel distributions, respectively. The Plackett copula is selected to construct the DDF curves. The DDF allows rainfall depth for a specific rainfall duration and return period to be estimated. This DDF formula improves the understanding of complex hydrologic processes and enhances the design safety criterion of hydraulic structures.

Index flood–based multivariate regional frequency analysis

Because of their multivariate nature, several hydrological phenomena can be described by more than one correlated characteristic. These characteristics are generally not independent and should be jointly considered. Consequently, univariate regional frequency analysis (FA) cannot provide complete assessment of true probabilities of occurrence. The objective of the present paper is to propose a procedure for regional flood FA in a multivariate framework. In the present paper, the focus is on the estimation step of regional FA. The proposed procedure represents a multivariate version of the index flood model and is based on copulas and a multivariate quantile version with a focus on the bivariate case. The model offers increased flexibility to designers by leading to several scenarios associated with the same risk. The univariate quantiles represent special cases corresponding to the extreme scenarios. A simulation study is carried out to evaluate the performance of the model in a bivariate framework. Simulation results show that bivariate FA provides the univariate quantiles with equivalent accuracy. Similarity is observed between results of the bivariate model and those of the univariate one in terms of the behavior of the corresponding performance criteria. The procedure performs better when the regional homogeneity is high. Furthermore, the impacts of small variations in the record length at gauged sites and the region size on the performance of the proposed procedure are not significant.

Application of bivariate frequency analysis to the derivation of rainfall–frequency curves

Bivariate distributions have been recently employed in hydrologic frequency analysis to analyze the joint probabilistic characteristics of multivariate storm events. This study aims to derive practical solutions of application for the bivariate distribution to estimate design rainfalls corresponding to the desired return periods. Using the Gumbel mixed model, this study constructed rainfall–frequency curves at sample stations in Korea which provide joint relationships between amount, duration, and frequency of storm events. Based on comparisons and analyses of the rainfall–frequency curves derived from univariate and bivariate storm frequency analyses, this study found that conditional frequency analysis provides more appropriate estimates of design rainfalls as it more accurately represents the natural relationship between storm properties than the conventional univariate storm frequency analysis.

A multivariate model of sea storms using copulas

Sea storms are customarily characterized in terms of significant wave height, peak period, and storm direction. Much less investigated is the storm duration, which is also very important when dealing with coastal dynamics and durability of large structures at sea. Moreover, these random variables are often treated using univariate frequency analysis, even if they are generally non-independent. A thorough statistical description requires the joint use of their marginal laws to obtain a multivariate distribution. In this paper we focus the attention on the multidimensional frequency analysis of sea storm significant wave height ( H), storm duration ( D), storm direction ( A), and storm interarrival time ( I) (i.e. the calm period separating two successive storms). In particular, we explain how copulas can be used for this purpose, for they constitute a novel and efficient mathematical tool for the analysis of multivariate events. An accurate model of these four random variables would undoubtedly be useful in aiding decision making processes with regard to marine operations, and this work could provide a guide to the oceanographic community not already familiar with copulas. Several important issues are considered, and the corresponding practical problems are solved using new copula-based techniques. (1) The construction of a bivariate model for the pair ( H, D): in turn, this yields the statistics of the sea storm magnitude M. (2) The calculation of the return period of multivariate events: this gives the possibility to calculate the probability of occurrence of super-critical events, and yields an estimation of the minimum energetic content of sea storms having assigned (multivariate) return period. (3) The construction of a trivariate model for the triple ( H, D, A): this provides useful indications about the relation between sea storm magnitude and direction. (4) The extension to storm interarrival duration I: this yields a trivariate model for the triple ( D, I, A), that casts new light on the relation between sea storm timing and direction. (5) The construction of a global model for the vector ( H, D, I, A): the overall structure is that of a reward alternating renewal process, whose dynamics develops along a random direction. In turn, this gives the possibility to simulate a sequence of sea storm events, accounting for all the variables of interest and their mutual relations. A case study is presented, based on real data collected by the Italian Sea Wave Measurement network. These statistical analyses are very important when dealing with coastal dynamics, marine structures reliability, or the planning of operations at sea. A proper evaluation of the main sea state parameters is crucial for a correct structural design, for reliable prognosis of the evolution of coastal environments, and for proper decision making in marine operations, due to the nonlinearities and the multidimensional structure of most of the design or failure function formulations.

이차원 가뭄빈도해석을 통한 서울지역의 가뭄 평가

가뭄에 대한 대책을 수립하기 위해서는 가뭄의 심도 및 지속기간 등 가뭄특성을 산정하여 가뭄을 정량화하는 것이 필요하다. 본 연구에서는 표준강수지수(SPI)를 도입하여 가뭄특성을 정량적으로 산정하였다. 가뭄은 일정기간 지속되는 특성을 지니고 있기 때문에 일정 관측기간동안 제한된 가뭄사상을 가지게 된다. 본 연구에서는 조선시대에 기록된 측우기 자료(1770년 <TEX>${\sim}$</TEX> 1907년)를 이용하여 관측자료를 확장하였고, 이를 이용하여 보다 많은 가뭄특성자료를 획득하여 가뭄빈도해석을 실시하였다. 측우기 자료와 관측자료를 이용하여 산정된 SPI로부터 절단수준법 개념을 이용하여 SPI의 -1이하를 가뭄으로 정의하고, 가뭄 심도 및 가뭄 지속기간을 구하였다. 가뭄의 심도와 지속기간은 가뭄을 특성화하는데 있어서 중요한 인자로서 최근 이 두 특성을 함께 고려하는 빈도해석법들이 제안되고 있다. 기존의 단일변수 빈도해석을 가뭄에 적용하였을 경우, 가뭄의 심도와 지속기간을 이용하여 구한 각각의 재현특성이 상이하게 나타나는 경우가 있는데, 이는 하나의 가뭄사상의 재현특성을 표현하는데 적절하지가 않다. 본 연구에서는 이변수 감마분포를 이용하여 가뭄심도와 지속기간의 결합확률밀도함수를 추정하고, 이를 통하여 가뭄의 이변수 재현 기간을 산정하여 분석하였다. Drought characteristics need to be preceded before establishing a drought mitigation plan. In this study, using a Standardized Precipitation Index (SPI), a hydrologic drought was defined as an event during which the SPIs are continuously below a certain truncation level. Then, a methodology of drought frequency analysis was performed to quantitatively characterize droughts considering drought duration and severity simultaneously. The theory of runs was used to model drought recurrence and to determine drought properties like duration and severity. Short historical records usually do not allow reliable bivariate analyses. However, more than hundred years of precipitation data (1770 <TEX>${\sim}$</TEX> 1907) collected in Chosun Kingdom Age using an old Korean rain gage called Chukwooki can provide valuable information about past events. It is shown that a bivariate gamma distribution well represented the joint probabilistic properties of Korean drought duration and severity. The overall results of this study show that the proposed bivariate drought frequency analysis overcomes the drawbacks of the conventional univariate frequency analysis by providing a consistent representation of the drought recurrent property.

Asymmetric copula in multivariate flood frequency analysis

The univariate flood frequency analysis is widely used in hydrological studies. Often only flood peak or flood volume is statistically analyzed. For a more complete analysis the three main characteristics of a flood event i.e. peak, volume and duration are required. To fully understand these variables and their relationships, a multivariate statistical approach is necessary. The main aim of this paper is to define the trivariate probability density and cumulative distribution functions. When the joint distribution is known, it is possible to define the bivariate distribution of volume and duration conditioned on the peak discharge. Consequently volume–duration pairs, statistically linked to peak values, become available. The authors build trivariate joint distribution of flood event variables using the fully nested or asymmetric Archimedean copula functions. They describe properties of this copula class and perform extensive simulations to highlight differences with the well-known symmetric Archimedean copulas. They apply asymmetric distributions to observed flood data and compare the results those obtained using distributions built with symmetric copula and the standard Gumbel Logistic model.

Return period of bivariate distributed extreme hydrological events

Extreme hydrological events are inevitable and stochastic in nature. Characterized by multiple properties, the multivariate distribution is a better approach to represent this complex phenomenon than the univariate frequency analysis. However, it requires considerably more data and more sophisticated mathematical analysis. Therefore, a bivariate distribution is the most common method for modeling these extreme events. The return periods for a bivariate distribution can be defined using either separate single random variables or two joint random variables. In the latter case, the return periods can be defined using one random variable equaling or exceeding a certain magnitude and/or another random variable equaling or exceeding another magnitude or the conditional return periods of one random variable given another random variable equaling or exceeding a certain magnitude. In this study, the bivariate extreme value distribution with the Gumbel marginal distributions is used to model extreme flood events characterized by flood volume and flood peak. The proposed methodology is applied to the recorded daily streamflow from Ichu of the Pachang River located in Southern Taiwan. The results show a good agreement between the theoretical models and observed flood data.

On the analysis of sector-related and gender-related stresses at the workplace — An analysis of the AET data bank

A database of stress analyses collected by an ergonomic job analysis procedure (AET) was evaluated. The database was collected between 1979 and 1993 in 350 organizations by 150 analysts trained in ergonomics and includes 3893 job analyses covering 216 stress factors. The methods used in the evaluation included univariate profile and frequency analyses. The data was subjected to an ex-post quota check by economic sector and gender. The stress factors working posture, heavy dynamic and active light work, static work, information reception and processing, physical and chemical working environment were investigated by economic sector and gender. Analysis shows that heavy dynamic work predominates in the building industry and in agriculture and active light work in manufacturing industry, whilst work in the private household involves high proportions of both heavy dynamic and active light work. Information reception via proprioceptors is of particular importance in the building industry and public corporations (e.g. municipal workers). The analysts gave high ratings to informatory and mental stress factors in jobs in the services sector and in public corporations. In contrast, physical and chemical environmental stress factors predominate in mining, manufacturing industry and the building industry. Significant gender-specific differences were found in the stresses arising from physical work, sensomotor work and the physical and chemical environmental factors. Investigations scheduled for the future are described.