Modeling and projecting the occurrence of bivariate extreme heat events using a non-homogeneous common Poisson shock process

A joint model is proposed for analyzing and predicting the occurrence of extreme heat events in two temperature series, these being daily maximum and minimum temperatures. Extreme heat events are defined using a threshold approach and the suggested model, a non-homogeneous common Poisson shock process, accounts for the mutual dependence between the extreme events in the two series. This model is used to study the time evolution of the occurrence of extreme events and its relationship with temperature predictors. A wide range of tools for validating the model is provided, including influence analysis. The main application of this model is to obtain medium-term local projections of the occurrence of extreme heat events in a climate change scenario. Future temperature trajectories from general circulation models, conveniently downscaled, are used as predictors of the model. These trajectories show a generalized increase in temperatures, which may lead to extrapolation errors when the model is used to obtain projections. Various solutions for dealing with this problem are suggested. The results of the fitted model for the temperature series in Barcelona in 1951–2005 and future projections of extreme heat events for the period 2031–2060 are discussed, using three global circulation model trajectories under the SRES A1B scenario.     

The Random Distribution of the Loading and Unloading Response Ratio Under the Assumptions of Poisson Models

This paper discussed the random distribution of the loading and unloading response ratio(LURR) of different definitions(Y_1～Y_5)using the assumptions that the earthquakes occurfollowing the Poisson process and their magnitudes obey the Gutenberg-Richter law.Theresults show that Y_1～Y_5 are quite stable or concentrated when the expected number of eventsin the calculation time window is relatively large(>40);but when this occurrence ratebecomes very small,Y_2～Y_5 become quite variable or unstable.That is to say,a high value ofthe LURR can be produced not only from seismicity before a large earthquake,but also from arandom sequence of earthquakes that obeys a Poisson process when the expected number ofevents in the window is too small.To check the influence of randomness in the catalogue tothe LURR,the random distribution of the LURR under Poisson models has been calculated bysimulation.90%,95% and 99% confidence ranges of Y_1 and Y_3 are given in this paper,which is helpful to quantify the random influe     

Observed advantage for negative binomial over Poisson distribution in partial duration series

The goodness of fit of the negative binomial and the Poisson distributions to partial duration series of runoff events is tested. The data have been recorded by eight hydrometric stations located on ephemeral rivers in Isreal. For each station, a number of threshold discharges are considered, by that series of nested subsamples are formed. Owing to size limitations, a Chi-square test is conducted on samples associated with low to moderate threshold discharges. Positive results, at a 5% significance level, are obtained in 30 out of the 53 tests of the Poisson distribution, and in 22 out of the 28 tests of the negative binomial distribution. The fit of the Poisson distribution to samples of conventionally recommended sizes (of 2 to 3 events per year) is found positive for five rivers and negative for the three other rivers The fit of the negative binomial distribution to these samples is found positive for six rivers, inconclusive for one river and short of data for the eighth river. Mixed results are obtained as the threshold level is raised. Therefore, no direct extrapolation is possible to samples associated with high thresholds.An indirect extrapolation is drawn through a comparison of the actual properties of the samples with those expected under a perfect fit of the distribution functions. Ranges of such properties are defined with respect to the properties of the tested samples and to the test results. The actual properties of nine of the eleven samples associated with high thresholds (i.e. mean number of events <-0.1year ^–1) are found within these ranges. This provides a hint for a probable good fit of either distribution, and particularly the negative binomial, to the occurrence frequency of high events.     

Observed advantage for negative binomial over Poisson distribution in partial duration series

The goodness of fit of the negative binomial and the Poisson distributions to partial duration series of runoff events is tested. The data have been recorded by eight hydrometric stations located on ephemeral rivers in Isreal. For each station, a number of threshold discharges are considered, by that series of nested subsamples are formed. Owing to size limitations, a Chi-square test is conducted on samples associated with low to moderate threshold discharges. Positive results, at a 5% significance level, are obtained in 30 out of the 53 tests of the Poisson distribution, and in 22 out of the 28 tests of the negative binomial distribution. The fit of the Poisson distribution to samples of conventionally recommended sizes (of 2 to 3 events per year) is found positive for five rivers and negative for the three other rivers The fit of the negative binomial distribution to these samples is found positive for six rivers, inconclusive for one river and short of data for the eighth river. Mixed results are obtained as the threshold level is raised. Therefore, no direct extrapolation is possible to samples associated with high thresholds.An indirect extrapolation is drawn through a comparison of the actual properties of the samples with those expected under a perfect fit of the distribution functions. Ranges of such properties are defined with respect to the properties of the tested samples and to the test results. The actual properties of nine of the eleven samples associated with high thresholds (i.e. mean number of events <-0.1year ^–1) are found within these ranges. This provides a hint for a probable good fit of either distribution, and particularly the negative binomial, to the occurrence frequency of high events.     

Statistical analysis of some volcanologic data regarded as series of point events

Summary The case histories of some active volcanoes in various parts of the world are analyzed from the standpoint of their being observations of point events in a time continuum. The eruptive histories of the three Japanese volcanoes included show trend in the rate of occurrence of outbreaks. The possible existence of trend in rate of occurrence of events was found for certain Lower Cretaceous bentonites of Wyoming. The data investigated for Etna derive from a period of persistent activity and here also trend in the rate of occurrence of ejections could be identified. The remaining volcanoes studied do not display significant trend in the rate of occurence of outbreaks over the time interval available. Various statistical tests indicate, that although some of the non-trend volcanoes may be fairly closely approximated as regards rate of occurrence of eruptions by the plausible Poisson model, none agree in all respects with the requirements of this process. The patterns of activity of volcanoes found to differ greatly from the Poisson model are complicated kinds of point processes, but owing to the shortness of the series available and their rather unsatisfactory accuracy, it is not possible to be explicit as to their precise nature. In order to elucidate some aspects of the analysis, a simulated series of outbreaks with exponentially distributed intervals between events was produced. The general scheme of analysis adopted has been firstly to test for trend; if trend in the rate of occurrence of events does not occur, the series have been tested for dependence. If there is no dependence between events, tests for agreement with a Poisson model have been carried out, with a negative conclusion leading to a test for agreement with some kind of renewal process. In order to provide a comparison with another type of natural phenomenon of a random nature, the earthquakes occurring in Fennoscandia over the period 1891 to 1950 were analyzed by the same methods. Perhaps surprisingly, the 322 shocks registered during this time (shocks3.0 on the Gutenberg-Richter scale) show an indication of trend with a tendency for a decrease in the rate of occurrence of shocks. The eruption pattern of Mauna Loa is thought to be approximately a simple Poisson process. The patterns for Semeru, Bromo and Peak of Ternate seem to be reasonably consistent with a renewal process model, but appear to differ from a Poisson process. The Indonesian volcanoes have several features in common, among these a high coefficient of variation for the times between eruptions. It is tentatively suggested that this may be of some genetic significance. It is possible, that the Indonesian volcanoes erupt in accordance with a pattern approximating to some kind of stationary point process.     

Model of earthquake occurrence in the European area

Summary The time distribution of earthquake occurrence in the European area is investigated by statistical laws. The original data of shallow-focus earthquakes are taken from the European catalogue 1901–1967. Evidence is given that the process with the negative binomial entries as a model describing the occurrence of shallow-focus earthquakes is better than the Poisson process. Further, the influence of magnitude classes and magnitude threshold value on the time distribution of earthquake occurrence is examined.Communication presented at the XIII General Assembly of the European Seismological Commission in Brasov in 1972.     

Doubly stochastic Poisson pulse model for fine-scale rainfall

Stochastic rainfall models are widely used in hydrological studies because they provide a framework not only for deriving information about the characteristics of rainfall but also for generating precipitation inputs to simulation models whenever data are not available. A stochastic point process model based on a class of doubly stochastic Poisson processes is proposed to analyse fine-scale point rainfall time series. In this model, rain cells arrive according to a doubly stochastic Poisson process whose arrival rate is determined by a finite-state Markov chain. Each rain cell has a random lifetime. During the lifetime of each rain cell, instantaneous random depths of rainfall bursts (pulses) occur according to a Poisson process. The covariance structure of the point process of pulse occurrences is studied. Moment properties of the time series of accumulated rainfall in discrete time intervals are derived to model 5-min rainfall data, over a period of 69 years, from Germany. Second-moment as well as third-moment properties of the rainfall are considered. The results show that the proposed model is capable of reproducing rainfall properties well at various sub-hourly resolutions. Incorporation of third-order moment properties in estimation showed a clear improvement in fitting. A good fit to the extremes is found at larger resolutions, both at 12-h and 24-h levels, despite underestimation at 5-min aggregation. The proportion of dry intervals is studied by comparing the proportion of time intervals, from the observed and simulated data, with rainfall depth below small thresholds. A good agreement was found at 5-min aggregation and for larger aggregation levels a closer fit was obtained when the threshold was increased. A simulation study is presented to assess the performance of the estimation method.     

The first and the second e of the extreme value distribution,EV1

In the peak over threshold model resulting in the Extreme-value distribution, type I, (EV1) the firste of the distribution function is based on the Poisson number of exceedances, and the seconde arises from the Exponentially distributed magnitudes.This paper, on the one hand, generalises the Poisson model to the (positive and negative) Binomial distribution, and, on the other hand, the Exponential distribution is generalised to the Generalised Pareto distribution. Lack of fit with respect to the Poisson and Exponential distribution is measured by statistics derived from those which would be locally most powerful if the estimates of the location and scale parameter were equal to the true parameter values. Ways of combining both statistics are discussed.     

On the annual maximum distribution in dependent partial duration series

As a basis for development of the annual maximum distribution the so-called partial duration series with Poissonian occurrence times and exponentially distributed peak exceedance values has been selected. The model is generalized by allowing for a Markov dependence between succeeding peak values. Correlation values from p=0 to p=1 can be accounted for by introducing the Marshall-Olkin bivariate exponential distribution, which is presented in detail. The developed distribution function for the annual maximum is throughly analysed and a variety of distribution forms depending on the value of the correlation coefficient and the intensity in the Poisson process is hereby recognized. To a certain extent this might be considered as parallel to the scattering of hydrological regions with different generating mechanisms for the annual maxima.     

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. The author wishes to thank the two anonymous reviewers for their constructive comments that improving the quality of this work.     

Non‐ergodicity and PEER's framework formula

A framework formula for performance‐based earthquake engineering, advocated and used by researchers at the Pacific Earthquake Engineering Research (PEER) Center, is closely examined. The formula was originally intended for computing the mean annual rate of a performance measure exceeding a specified threshold. However, it has also been used for computing the probability that a performance measure will exceed a specified threshold during a given period of time. It is shown that the use of the formula to compute such probabilities could lead to errors when non‐ergodic variables (aleatory or epistemic) are present. Assuming a Poisson model for the occurrence of earthquakes in time, an exact expression is derived for the probability distribution of the maximum of a performance measure over a given period of time, properly accounting for non‐ergodic uncertainties. This result is used to assess the approximation involved in the PEER formula for computing probabilities. It is found that the PEER approximation of the probability has a negligible error for probabilities less than about 0.01. For larger probabilities, the error depends on the magnitude of non‐ergodic uncertainties and the duration of time considered and can be as much as 20% for probabilities around 0.05 and 30% for probabilities around 0.10. The error is always on the conservative side. Copyright © 2005 John Wiley & Sons, Ltd.     

Non-stationary frequency analysis of heavy rainfall events in southern France

Abstract

Heavy rainfall events often occur in southern French Mediterranean regions during the autumn, leading to catastrophic flood events. A non-stationary peaks-over-threshold (POT) model with climatic covariates for these heavy rainfall events is developed herein. A regional sample of events exceeding the threshold of 100 mm/d is built using daily precipitation data recorded at 44 stations over the period 1958–2008. The POT model combines a Poisson distribution for the occurrence and a generalized Pareto distribution for the magnitude of the heavy rainfall events. The selected covariates are the seasonal occurrence of southern circulation patterns for the Poisson distribution parameter, and monthly air temperature for the generalized Pareto distribution scale parameter. According to the deviance test, the non-stationary model provides a better fit to the data than a classical stationary model. Such a model incorporating climatic covariates instead of time allows one to re-evaluate the risk of extreme precipitation on a monthly and seasonal basis, and can also be used with climate model outputs to produce future scenarios. Existing scenarios of the future changes projected for the covariates included in the model are tested to evaluate the possible future changes on extreme precipitation quantiles in the study area.

Editor Z.W. Kundzewicz; Associate editor K. Hamed

Citation Tramblay, Y., Neppel, L., Carreau, J., and Najib, K., 2013. Non-stationary frequency analysis of heavy rainfall events in southern France. Hydrological Sciences Journal, 58 (2), 280–294.     

A statistical method linking geological and historical eruption time series for volcanic hazard estimations: Applications to active polygenetic volcanoes

The probabilistic analysis of volcanic eruption time series is an essential step for the assessment of volcanic hazard and risk. Such series describe complex processes involving different types of eruptions over different time scales. A statistical method linking geological and historical eruption time series is proposed for calculating the probabilities of future eruptions. The first step of the analysis is to characterize the eruptions by their magnitudes. As is the case in most natural phenomena, lower magnitude events are more frequent, and the behavior of the eruption series may be biased by such events. On the other hand, eruptive series are commonly studied using conventional statistics and treated as homogeneous Poisson processes. However, time-dependent series, or sequences including rare or extreme events, represented by very few data of large eruptions require special methods of analysis, such as the extreme-value theory applied to non-homogeneous Poisson processes. Here we propose a general methodology for analyzing such processes attempting to obtain better estimates of the volcanic hazard. This is done in three steps: Firstly, the historical eruptive series is complemented with the available geological eruption data. The linking of these series is done assuming an inverse relationship between the eruption magnitudes and the occurrence rate of each magnitude class. Secondly, we perform a Weibull analysis of the distribution of repose time between successive eruptions. Thirdly, the linked eruption series are analyzed as a non-homogeneous Poisson process with a generalized Pareto distribution as intensity function. As an application, the method is tested on the eruption series of five active polygenetic Mexican volcanoes: Colima, Citlaltépetl, Nevado de Toluca, Popocatépetl and El Chichón, to obtain hazard estimates.     

Statistical properties of aftershock rate decay: Implications for the assessment of continuing activity

Aftershock rates seem to follow a power law decay, but the assessment of the aftershock frequency immediately after an earthquake, as well as during the evolution of a seismic excitation remains a demand for the imminent seismic hazard. The purpose of this work is to study the temporal distribution of triggered earthquakes in short time scales following a strong event, and thus a multiple seismic sequence was chosen for this purpose. Statistical models are applied to the 1981 Corinth Gulf sequence, comprising three strong (M = 6.7, M = 6.5, and M = 6.3) events between 24 February and 4 March. The non-homogeneous Poisson process outperforms the simple Poisson process in order to model the aftershock sequence, whereas the Weibull process is more appropriate to capture the features of the short-term behavior, but not the most proper for describing the seismicity in long term. The aftershock data defines a smooth curve of the declining rate and a long-tail theoretical model is more appropriate to fit the data than a rapidly declining exponential function, as supported by the quantitative results derived from the survival function. An autoregressive model is also applied to the seismic sequence, shedding more light on the stationarity of the time series.     

Single station and regional analysis of daily rainfall extremes

A peaks over threshold (POT) method of analysing daily rainfall values is developed using a Poisson process of occurrences and a generalised Pareto distribution (GPD) for the exceedances. The parameters of the GPD are estimated by the method of probability weighted moments (PWM) and a method of combining the individual estimates to define a regional curve is proposed.     

地震活动性正常与异常的群体概率特征及其在地震预测中的应用以汶川地震为例

地震事件一般具有离散性, 即它们的发生和转移具有一定的时间和空间间隔。 当孕震系统处于平衡(或正常)状态时, 系统状态变量X(这里以地震活动性为例)的时间序列概率分布表现为单峰型(如泊松分布); 其时间间隔(或等候时间)Δt分布可用负指数分布来描述。 当系统处于非平衡(或异常)状态时, X的概率分布表现为双峰型或多峰型, 其时间间隔Δt(或空间间隔ΔS)可采用幂函数关系来描述。 可以计算X的1~4阶矩统计参数识别X的概率分布性质, 进一步判断系统所处的状态。 根据这个思路, 本文尝试针对地震活动性的群体概率特征, 探索研究地震活动系统时间和空间结构的演变以及正常和异常状态的辨别, 并将这些方法应用于2008年汶川8.0级地震及其余震趋势和强余震预测实践。     

A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations

We extend the particle-tracking method to simulate general multi-rate mass transfer (MRMT) equations. Previous methods for single-rate equations used two-state Markov chains and found that the time a particle spends in the mobile state between waiting time epochs is random and exponentially distributed. Using Bochner’s subordination technique for Markov processes, we find that the random mobile times are still exponential for the stochastic process that corresponds to the MRMT equations. The random times in the immobile phase have a distribution that is directly related to the memory function of the MRMT equation. This connection allows us to interpret the MRMT memory function as the rate at which particles of a certain age, measured by residence time in the immobile zone, exit to become mobile once again. Because the exact distributions of mobile and immobile times are known from the MRMT equations, they can be simulated very simply and efficiently using random walks.     

Direct relationships of discharges to the mean and standard deviation of the intervals between their exceedences

Abstract

This article paves a way for assessing flood risk by the use of two-parameter distributions, for the intervals between threshold exceedences rather than by the traditional exponential distribution. In a case study, the apparent properties of intervals between exceedences of runoff events differ from those anticipated for exponentially distributed series. A procedure is proposed to relate two statistical parameters of the intervals to threshold discharges. It considers partial duration series (PDS) with thresholds equal to all high enough observed discharges. To avoid unnecessary assumptions on the behaviour of those parameters and effects of dependence between parameters for different PDS, a non-parametric trend-free pre-whitened scheme is applied. It leads to power-law relationships between a discharge and the mean and standard deviation of the intervals between its exceedences. Predicted mean inter-exceedence intervals, for the highest observed discharges at the stations, are closer to the observational periods than those predicted by GEV distributions fitted to AMS, and by GP distributions to fitted PDS. In the present case, the latter predictions are longer than the observational periods whereas some of the predicted mean inter-exceedences are shorter than the corresponding observational periods and some others are longer.

Citation Ben-Zvi, A. & Azmon, B. (2010) Direct relationships of discharges to the mean and standard deviation of the intervals between their exceedences. Hydrol. Sci J. 55(4), 565–577.     

On the probability of exceeding a given river flow threshold

ABSTRACT

To obtain estimates of the probability that a river flow will exceed a given threshold at time t + 1, given the flow value at time t, two stochastic models are considered: a filtered Poisson process and a diffusion process with jumps. Estimates derived from linear regression are also considered. The model parameters are assumed to depend on the flow value. An application to the Delaware River is presented.

Editor D. Koutsoyiannis; Associate editor S. Grimaldi