Probability Of Extreme Events Research Articles

The influence of microstructure on the probability of early failure in aluminum-based interconnects

For electromigration in short aluminum interconnects terminated by tungsten vias, the well known “short-line” effect applies. In a similar manner, for longer lines, early failure is determined by a critical value Lcrit for the length of polygranular clusters. Any cluster shorter than Lcrit is “immortal” on the time scale of early failure where the figure of merit is not the standard t50 value (the time to 50% failures), but rather the total probability of early failure, Pcf. Pcf is a complex function of current density, linewidth, line length, and material properties (the median grain size d50 and grain size shape factor σd). It is calculated here using a model based around the theory of runs, which has proved itself to be a useful tool for assessing the probability of extreme events. Our analysis shows that Pcf is strongly dependent on σd, and a change in σd from 0.27 to 0.5 can cause an order of magnitude increase in Pcf under typical test conditions. This has implications for the web-based two-dimensional grain-growth simulator MIT/EmSim, which generates grain patterns with σd=0.27, while typical as-patterned structures are better represented by a σd in the range 0.4 – 0.6. The simulator will consequently overestimate interconnect reliability due to this particular electromigration failure mode.

Changes in climate variability and seasonal rainfall extremes: a case study from San Fernando (Spain), 1821-2000

¶Small changes in the mean and standard deviation values can produce relatively large changes in the probability of extreme events. The seasonal precipitation record in San Fernando (SW Spain) for 1821–2000 is used to investigate how much the relative frequency of dry and wet seasons changes with changes in mean value and standard deviation. The percentiles P10, P25, P75 and P90 of the reference period 1961–1990 are used to define dry and wet seasons. The probability of extreme seasons as function of mean and standard deviation is analysed. The main conclusion is a non-linear relationship between changes in mean and standard deviation values and extreme seasons probability. With these threshold values, the main influence corresponds to changes in mean value. Results are discussed bearing in mind projections of General Circulation Models on future climate in southern Iberian Peninsula.

FRACTALITY IN A MACROECONOMIC MODEL: NONLINEAR OSCILLATION AROUND A LONG-TERM EQUILIBRIUM

Recent studies have established that macroeconomic time series exhibit fractal properties. Empirical tests here demonstrate that interest rates, exchange rates, output and prices all show evidence of a non-integer fractal dimension. Several classes of volatility models widely used in econometrics can give rise to fractality. In the paradigm proposed here, fractality results from multiplicative relationships between residual noise terms in simultaneous equation systems. The emergence of fractality in a large-scale econometric model is analyzed. The model uses well-established structural equations, so that all variables converge toward their equilibrium paths in the long run. The forecasted paths are then embedded in noise, and the model is re-simulated at a higher frequency. The simultaneity of the model equations causes the embedding noise to take on fractal properties. Multi-scaling demonstrates that the model simulations reproduce the fractal properties of the real-world time series reasonably well. Finally, it is possible to forecast at short horizons using an algorithm that exploits two aspects of fractality, scaling symmetries and intermittency. Ratios of rates of change capture proximate symmetries. A logit regression is used to predict the conditional probability of extreme events.

The characterization of landslide size distributions

Landslide size distributions generally exhibit power‐law scaling over a limited scale range. The range is set by the mapping resolution, by the number of observed events, and by the slope failure process itself. This property of self‐similarity is an important insight into the physics of hillslope failure. Typically, however, a large proportion of the landslide data does not fit a simple power law. These data are always ignored in order to characterize the scaling. We show that landslide data sets from New Zealand and Taiwan exhibit two scaling regimes, separated by a crossover scale that is purely an artefact of mapping resolution. Below this scale the landslide data are undersampled. We propose a general model for the size distribution of observed landslides which can account for the whole population of mapped slope failures. The model quantifies the undersampling of smaller landslides and provides an improved estimation of the power‐law scaling of larger landslides. Estimates of this scaling suggest that the area disturbed by landsliding, and perhaps the landslide sediment yield, are essentially dependent on the frequency of smaller landslides. Higher resolution landslide maps will be required in order to quantify these fluxes. Our results also indicate that the probability of extreme landslide events is less than previous studies would predict.

Characteristics of large snowfall events in the montane western United States as examined using snowpack telemetry (SNOTEL) data

Daily snow water equivalent records from the snowpack telemetry archive are used to assess spatiotemporal characteristics of large snowfall events over the montane western United States. The largest mean annual (leading) events are found in the Pacific Northwest and Sierra Nevada. The mean leading event lasting up to 72 hours typically accounts for 10–23% of the water equivalent of annual snowfall, with the largest contribution in the Arizona/New Mexico sector. For most of the West, snowfall events in the top quartile of station distributions are most common during midwinter, but those for the Rocky Mountain states and Utah are more common during late winter or spring. Colorado also shows a secondary peak in large events during November. Large midwinter snowfall events in the marine sectors, Idaho, and Arizona/New Mexico are spatially coherent in that when observed at one station, they tend to occur at surrounding stations. Large events are less spatially coherent for drier inland regions. When annual snowfall is anomalously positive, there tends to be an increase in the number of snow days as well as a shift in the distributions toward the larger event sizes. Opposite relationships are observed for negative annual snowfall anomalies. These findings are in accord with recent studies using lower elevation data, demonstrating that the probability of extreme precipitation events is altered during El Nino or La Nina conditions.

Modeling the hydroclimatic disturbance of soil landscapes in the southern Canadian plains: the problems of scale and place.

The sensitivity of soil landscapes to climatic variability and hydroclimatic events can be expressed as a landscape change safety factor, the ratio of potential disturbance to resistance to change. The use of a geographic information system (GIS) enables the spatially-explicit modeling of landscape sensitivity, but also raises the risk of violating the characteristic scales of disturbance and resistance, because the GIS technically simplifies the extrapolation of models, and associated concepts, to landscapes and scales not represented by the digital data base. Embedding landscape sensitivity into hierarchy theory, the formal analysis of the hierarchical structure of complex systems, provides a conceptual framework for the transfer of models and variables among landscape scales. In the subhumid southern Canadian plains, major hydroclimatic events (strong winds, intense rain, rapid snow melt) cause much of the physical disturbance of soil landscapes and terrestrial ecosystems. Prolonged dry or wet weather influences the resistance of soil and vegetation to these events. The potential disturbance of soil landscapes therefore can be derived from the probabilities of extreme events and seasonal conditions, as recorded in instrumental and proxy climate records. This time series analysis can be linked to the modeling of landscape sensitivity by establishing the probabilities of hydroclimatic events and climatic conditions which may exceed or lower the resistance of individual soil landscapes.

Simulation of Indian summer monsoon rainfall and its intraseasonal variability in the NCAR climate system model

The broad climatological features associated with the Asian monsoon circulation, including its mean state and intraseasonal and interannual variability over the Indian subcontinent as simulated in the National Center for Atmospheric Research (NCAR) global coupled climate system model (CSM) in its control reference experiment, are presented in this paper. The CSM reproduces the seasonal cycle as well as basic observed patterns of key climatic parameters reasonably well in spite of some limitations in simulation of the monsoon rainfall. However, while the seasonality in rainfall over the region is simulated well, the simulated area-averaged monsoon rainfall is underestimated to only about 60% of the observed rainfall. The centers of maxima in simulated monsoon rainfall are slightly displaced southward as compared to the climatological patterns. The cross-equatorial flow in simulated surface wind patterns during summer is also stronger than observed with an easterly bias. The transient experiment with a 1% per year compound increase in CO2 with CSM suggests an annual mean area-averaged surface warming of about 1.73 °C over the region at the time of CO2 doubling. This warming is more pronounced in winter than during the monsoon season. A net increase in area-averaged monsoon rainfall of about 1.4 mm day–1, largely due to increased moisture convergence and associated convective activity over the land, is obtained. The enhanced intraseasonal variability in the monsoon rainfall in a warmer atmosphere is confined to the early part of the monsoon season which suggests the possibility of the date of onset of summer monsoon over India becoming more variable in future. The enhanced interannual and intraseasonal variability in the summer monsoon activity over India could also contribute to more intense rainfall spells over the land regions of the Indian subcontinent, thus increasing the probability of extreme rainfall events in a warmer atmosphere.

RESEARCH: Assessing Uncertainty in Estimates of Nitrogen Loading to Estuaries for Research, Planning, and Risk Assessment.

/ There can be considerable uncertainty associated with calculations of nutrient loading to estuaries from their watersheds, arising from uncertainty in the variables used in the calculation. Analysis of uncertainty is particularly important in the context of planning and management, where such information can be useful in helping make decisions about development in the coastal zone and in risk assessment, where probability of worse-case extremes may be relevant. This fact has been largely ignored when loading calculations have been made, presumably because both uncertainty estimates for the input variables and a standard method were lacking. Parametric (propagation for normal error estimates) and nonparametric methods (bootstrap and enumeration of combinations) to assess the uncertainty in calculated rates of nitrogen loading were compared, based on the propagation of uncertainty observed in the variables used in the calculation. In addition, since such calculations are often based on literature surveys rather than random replicate measurements for the site in question, error propagation was also compared using the uncertainty of the sampled population (e.g., standard deviation) as well as the uncertainty of the mean (e.g., standard error of the mean). Calculations for the predicted nitrogen loading to a shallow estuary (Waquoit Bay, MA) were used as an example. The previously estimated mean loading from the watershed (5,400 ha) to Waquoit Bay (600 ha) was 23,000 kg N yr(-1). The mode of a nonparametric estimate of the probability distribution differed dramatically, equaling only 70% of this mean. Repeated observations were available for only 8 of the 16 variables used in our calculation. We estimated uncertainty in model predictions by treating these as sample replicates. Parametric and nonparametric estimates of the standard error of the mean loading rate were 12-14%. However, since the available data include site-to-site variability, as is often the case, standard error may be an inappropriate measure of confidence. The standard deviations were around 38% of the loading rate. Further, 95% confidence intervals differed between the nonparametric and parametric methods, with those of the nonparametric method arranged asymmetrically around the predicted loading rate. The disparity in magnitude and symmetry of calculated confidence limits argue for careful consideration of the nature of the uncertainty of variables used in chained calculations. This analysis also suggests that a nonparametric method of calculating loading rates using most frequently observed values for variables used in loading calculations may be more appropriate than using mean values. These findings reinforce the importance of including assessment of uncertainty when evaluating nutrient loading rates in research and planning. Risk assessment, which may need to consider relative probability of extreme events in worst-case scenarios, will be in serious error using normal estimates, or even the nonparametric bootstrap. A method such as our enumeration of combinations produces a more reliable distribution of risk.

Assessment of the probability of extreme weather events and their potential effects in large conurbations

The likelihood of occurrence of extreme high-temperature run events is estimated for different values of the event intensity and persistence from very long synthetic time series of daily maximum temperatures generated by Monte Carlo simulations using a first-order autoregressive or Markov model. A theoretical analysis reveals a higher relative sensitivity of the simulated extreme event probabilities to changes in the variability of climate than to changes in its mean state. Moreover, this sensitivity relatively increases at a nonlinear rate the more extreme the event. The developed probabilistic model is applied in order to derive local scenarios of extreme high-temperature run events for a large conurbation like the city of Berlin assuming both arbitrary hypothetical and physically based new climate states described by changes in the model parameters (e.g. the mean, the standard deviation and the first-order autocorrelation of the daily maximum temperature time series). As a consequence of a 1.7°C increase in the mean as well as a 19% increase in the temperature variability in July as predicted by the climate model ECHAM_1/LSG assuming an unrestricted future increase in the global atmospheric concentration of climate relevant greenhouse gases according to the IPCC Scenario A (“Business as usual”) the intensity as well as the persistence of extreme high-temperature run events will rise considerably up to the end of the next century. In particular, intense hot spells characterized by at least five consecutive daily maximum temperatures equaling or exceeding 33°C are expected to occur every eight years under the new climate conditions compared to a current repetition time of about 47 years. The potential environmental effects might be a significant increase in the heat-stress-related morbidity and mortality rate, an aggravation of the summer smog situation and a destabilization of the urban ecosystems.

Extreme hydrological events, palaeo-information and climate change

Extreme flood events have been and continue to be one of the most important natural hazards responsible for deaths and economic losses. Extreme floods result in direct destructive effects during the time of the event, and they also may be followed by a related chain of indirect calamities such as famines and epidemics that produce additional damages and suffering. Extreme hydrological events that have occurred in the historical past may also occur in the future. Knowledge about magnitudes and recurrence frequencies of past extreme hydrological events in most regions are too short to adequately evaluate potential magnitudes and recurrence frequencies of extreme hydrological events. Stationary climate in which the mean and variance do not change over time is a basic underlying assumption of standard methodological procedures for estimating recurrence probabilities of extreme hydrological events. Palaeo-archives contained in river and lake sediments, fossil plant and animal matter, ice layers, and other natural archives show that the assumption of stationary climate is not valid when the time scale is extended beyond centuries and millennia. Records of past extreme floods that occurred long before the period of instrumentation can be reconstructed from the distribution of slackwater flood deposits or from derivation of water depths competent to transport the largest rocks found in flood deposited sediment. Palaeoflood records reconstructed from the Upper Mississippi and Lower Colorado River systems in the United States confirm nonstationary behaviour of the mean and variance in hydrological time series. These stratigraphic records have shown that even very modest climatic changes have resulted in very important changes in the magnitudes and recurrence frequencies of extreme floods. A close relationship was found between the palaeo-flood record of extreme floods in the Upper Mississippi River system and a palaeo-record of stable isotopes of oxygen and carbon preserved in speleothem calcite from a local cave. The relationship suggests that isotopic records elsewhere might be calibrated to provide insight about how future potential climate changes might impact extreme flood magnitudes and recurrence frequencies there. Atmospheric global circulation models (GCMs) mainly simulate average climatic conditions and are presently inadequate sources of information about how future climate changes might be represented at the extreme event scale. Palaeo-flood archives, however, provide basic information about how magnitudes and recurrence frequencies of extreme hydrological events responded to past climate changes and they also provide a reference base against which GCM simulations can be calibrated regionally and be better interpreted to decipher hydrological information at the extreme event scale.

USE OF A STOCHASTIC WEATHER GENERATOR IN THE DEVELOPMENT OF CLIMATE CHANGE SCENARIOS

Climate change scenarios with a high spatial and temporal resolution are required in the evaluation of the effects of climate change on agricultural potential and agricultural risk. Such scenarios should reproduce changes in mean weather characteristics as well as incorporate the changes in climate variability indicated by the global climate model (GCM) used. Recent work on the sensitivity of crop models and climatic extremes has clearly demonstrated that changes in variability can have more profound effects on crop yield and on the probability of extreme weather events than simple changes in the mean values. The construction of climate change scenarios based on spatial regression downscaling and on the use of a local stochastic weather generator is described. Regression downscaling translated the coarse resolution GCM grid-box predictions of climate change to site-specific values. These values were then used to perturb the parameters of the stochastic weather generator in order to simulate site-specific daily weather data. This approach permits the incorporation of changes in the mean and variability of climate in a consistent and computationally inexpensive way. The stochastic weather generator used in this study, LARS-WG, has been validated across Europe and has been shown to perform well in the simulation of different weather statistics, including those climatic extremes relevant to agriculture. The importance of downscaling and the incorporation of climate variability are demonstrated at two European sites where climate change scenarios were constructed using the UK Met. Office high resolution GCM equilibrium and transient experiments.

Parametric time series analysis of geoelectrical signals: an application to earthquake forecasting in Southern Italy

An autoregressive model was selected to describe geoelectrical time series. An objective technique was subsequently applied to analyze and discriminate values above (below) an a priorifixed threshold possibly related to seismic events. A complete check of the model and the main guidelines to estimate the occurrence probability of extreme events are reported. A first application of the proposed technique is discussed through the analysis of the experimental data recorded by an automatic station located in Tito, a small town on the Apennine chain in Southern Italy. This region was hit by the November 1980 Irpinia-Basilicata earthquake and it is one of most active areas of the Mediterranean region. After a preliminary filtering procedure to reduce the influence of external parameters (i.e. the meteo-climatic effects), it was demonstrated that the geoelectrical residual time series are well described by means of a second order autoregressive model. Our findings outline a statistical methodology to evaluate the efficiency of electrical seismic precursors.

Estimation of possible climate change impacts on water availability, extreme flow events and soil moisture in the Goulburn and Ovens Basins, Victoria

A rainfall-runoff model (IHACRES) is applied on a daily timestep to a large area of the state of Victoria, Australia. Successful calibrations of this dynamic lumped parameter model were performed for 5 rivers contributing streamflow to the Ovens Basin, and for 9 rivers of the Goulburn Basin. This is the first application of the model on such a scale, involving two basins where the total drainage area of the catchments modelled is about 6,500 km2. The models were tested by simulation over the entire common period of observation for the 14 catchments under consideration. The results show that the models closely simulate the observed streamflow. The effect of historical climate variability on streamflow was investigated. The models were used for estimation of the potential impact of climatic change on water availability for irrigation for different climate scenarios developed in the Division of Atmospheric Research, CSIRO. This allows conditional estimates to be made of water supply in these basins for the periods 2030 and 2070 under current vegetation conditions. Projecting the future hydrologic regime in this region is extremely important, in particular for supporting irrigation management of the Basin. The problem of estimating the impact of climate change on the probability of extreme events of the hydrological regime was analysed. Flood frequency was found to increase for the scenarios providing the maximum amount of water; to 50% at 2030 and 100% at 2070. The probability of flood events for the ‘dry’ scenarios rapidly decreases for these dates. Drought frequency, as defined by a soil wetness index, increased 35% for the ‘dry’ scenario at 2030 and 80% for this scenario at 2070.

Coping with uncertainty in forest wildlife planning

Forest planning rarely incorporates uncertainty about the incidence of disturbance and how ecosystems respond to disturbance. This omission may bias perceptions about the impact of disturbance. Human impacts, such as timber harvesting, are often deterministic, with disturbance occurring at prescribed intervals, at a single spatial scale, and at a single intensity. Such deterministic processes are likely to have very different impacts from natural stochastic processes that operate at random over a range of spatial and temporal scales. It is insufficient to describe disturbance in terms of mean values such as frequency, because variation around the mean is important for determining the impact of disturbance. Uncertainty is ubiquitous in natural systems and the importance of including uncertainty in forest management is discussed. Methods are described for integrating uncertainty of various types into forest planning, while accounting for timber and wildlife values. The methods involve placing bounds on predicted impacts of disturbance and determining probabilities of extreme events.

ESTIMATING PROBABLE MAXIMUM FLOOD PROBABILITIES1

ABSTRACT: Methods of computing probabilities of extreme events that affect the design of major engineering structures have been developed for most failure causes, but not for design floods such as the probable maximum flood (PMF). Probabilities for PMF estimates would be useful for economic studies and risk assessments. Reasons for the reluctance of some hydrologists to assign a probability to a PMF are discussed, and alternative methods of assigning a probability are reviewed. Currently, the extrapolation of a frequency curve appears to be the most practical alternative. Using 46 stations in the Mid‐Atlantic region, the log‐gamma, log‐normal, and log‐Gumbel distributions were used to estimate PMF probabilities. A 600,000‐year return period appears to be a reasonable probability to use for PMFs in the Mid‐Atlantic region. The coefficient of skew accounts for much of the variation in computed probabilities.

Extreme High-Temperature Events: Changes in their probabilities with Changes in Mean Temperature

Abstract Most climate impact studies rely on changes in means of meteorological variables, such as temperature, to estimate potential climate impacts, including effects on agricultural production. However, extreme meteorological events, say, a short period of abnormally high temperatures, can have a significant harmful effect on crop growth and final yield. The characteristics of daily temperature time series, specifically mean, variance and autocorrelation, are analyzed to determine possible ranges of probabilities of certain extreme temperature events [e.g., runs of consecutive daily maximum temperatures of at least 95°F (35°C)] with changes in mean temperature of the time series. The extreme temperature events considered are motivated primarily by agricultural concerns, particularly, the effects of high temperatures on corn yields in the U.S. Corn Belt. However, runs of high temperatures can also affect, for example, energy demand or morbidity and mortality of animals and humans. The relationships betw...

Some aspects of fluctuating vertical wind shears

Fluctuating vertical shears of wind speed have been measured using anemometers on an array of towers. The statistical distributions of these shears are compared with formulas proposed by Fichtl and good agreement is found. A comparison of Fichtl's formula for σ Δu , the standard deviation of the fluctuating shears, with a more empirical one proposed by Ramsdell shows that the latter is consistent with the former under the proper conditions. The probability of occurrence of extreme shears in speed is discussed. Fluctuating shears two or more times larger than the mean values are shown to occur frequently, and their likelihood increases as the mean measuring height increases if Δz is held fixed. These results are applicable for slightly unstable to neutral conditions over level, homogeneous terrain, and in the surface boundary layers. The normal distribution gives a useful lower bound on shear values. Probabilities of extreme shears as function of Δz z are given. σ Δu is also shown to be affected by τ, the averaging time used for the shear measurements. Increasing τ from 0.2 to 4 sec can decrease σ Δu by 25 per cent or more. The effects of τ on the probability of extreme events is shown.

Some properties of variance reduction techniques where hydrological extremes are estimated by Monte Carlo Simulation

An estimate of a water resource system's performance, when it is derived by simulation using synthetic streamflow sequences, is subject to at least three errors: first, model errors, arising from the approximation to the ‘true’ streamflow mechanism which the model represents; second, sampling errors in the model parameters θ when they are calculated from the historic records; and third, errors introduced by the Monte Carlo calculation from which is derived. This paper presents some observations on the effects of the first two types of error on the estimate but concentrates on the application of variance reduction techniques to the derivation of Monte Carlo estimates for given θ. These techniques are, first, the use of control variates and, second, the use of antithetic variates, and their application is illustrated by using some hypothetical examples of the calculation of probabilities of extreme hydrological events and of the calculation of reliability measures for a much oversimplified storage system. Considerable reduction in the variance of resulted from the application of the control variate method; the reduction in Var resulting from the use of antithetic variates was much less but still probably worth the small additional programing effort required. It is concluded that the promise of the control variate method suggests that it should be applied to assist in the efficient simulation of more realistic water resource systems than the trivial ones considered in this paper.

A Note on the Average Probability of Extreme Events

In a recent paper, Benson [1967] has shown that the limiting value of the average probability of all floods above the k‐year flood is equal to 1/2k; when the California, Weibull, or Hazen formulas are used to determine plotting positions. The purpose of this article is to show that the limiting value of the average probability is 1/2k; for any plotting‐position‐probability formula of the form p = (m + a) /(n + b) where m is the rank with the highest value taken as 1, n is the number of ranked events, and a and b can be selected arbitrarily, except that b cannot be a negative integer.Table 1, adapted from Chow [1964], gives some important plotting‐position formulas.

Misconceptions in hydrology and their consequences

Several misconceptions have been introduced into hydrology that have had adverse consequences on the science and on water‐resources development. Persistent use of the concept of maximum precipitation and maximum probable precipitation, for which there is no physical proof, has discouraged research into the structure and probability of extreme events, and may have led to an unwarranted sense of security concerning flood‐control works. Attempts at long range forecasts of water supply based entirely on meteorological processes have misdirected research and raised false expectations. A misconception that hydrologic processes are composed of a limited number of hidden periodicities has retarded the use of modern stochastic analyses of time series. Cloud‐seeding to increase numbers of raindrops rather than their coalescence may have had a negative effect on precipitation in arid regions. Considering ground water, instead of the aquifer, as a resource has stressed the technology of withdrawal at the expense of study of methods of recharge and has led to much unplanned overdraft of ground water. Use of descriptive instead of numerical variables has delayed the application of modern statistical methods to planning the exploration of the ground‐water environment. As a result of the neglect of the stochastic properties of porous mediums, fluid mechanics theory has not departed far from the original works of Darcy. Hydrology has developed slowly because it has been considered an appendage of hydraulic engineering rather than a natural science.