Non-stationary Extreme Value Models Research Articles

Projection of snowfall extremes in the French Alps as a function of elevation and global warming level

Abstract. Following the projected increase in extreme precipitation, an increase in extreme snowfall may be expected in cold regions, e.g., for high latitudes or at high elevations. By contrast, in low- to medium-elevation areas, the probability of experiencing rainfall instead of snowfall is generally projected to increase due to warming conditions. Yet, in mountainous areas, despite the likely existence of these contrasted trends according to elevation, changes in extreme snowfall with warming remain poorly quantified. This paper assesses projected changes in heavy and extreme snowfall, i.e., in mean annual maxima and 100-year return levels, in the French Alps as a function of elevation and global warming level. We apply a recent methodology, based on the analysis of annual maxima with non-stationary extreme value models, to an ensemble of 20 adjusted general circulation model–regional climate model (GCM–RCM) pairs from the EURO-CORDEX experiment under the Representative Concentration Pathway 8.5 (RCP8.5) scenario. For each of the 23 massifs of the French Alps, maxima in the hydrological sense (1 August to 31 July) are provided from 1951 to 2100 and every 300 m of elevations between 900 and 3600 m. Results rely on relative or absolute changes computed with respect to current climate conditions (corresponding here to +1 ∘C global warming level) at the massif scale and averaged over all massifs. Overall, daily mean annual maxima of snowfall are projected to decrease below 3000 m and increase above 3600 m, while 100-year return levels are projected to decrease below 2400 m and increase above 3300 m. At elevations in between, values are on average projected to increase until +3 ∘C of global warming and then decrease. At +4 ∘C, average relative changes in mean annual maxima and 100-year return levels, respectively, vary from −26 % and −15 % at 900 m to +3 % and +8 % at 3600 m. Finally, for each global warming level between +1.5 and +4 ∘C, we compute the elevation threshold that separates contrasted trends, i.e., where the average relative change equals zero. This elevation threshold is shown to be lower for higher return periods, and it is projected to rise from 3000 m at +1.5 ∘C to 3350 m at +4 ∘C for mean annual maxima and from 2600 to 3000 m for 100-year return levels. These results have implications for the management of risks related to extreme snowfall.

Trend and Return Level Analysis of Extreme Rainfalls in Senegal

In recent years, Senegal has been confronted with increasingly frequent and damaging extreme events. In the context of climate change, we conducted this study to characterize the trends of rainfall extremes in Senegal. In this work, we used daily rainfall data from 27 stations in Senegal from the period 1951 to 2005 (55 years). To study their linear trends, non-stationary extreme value models with time as a covariate are fitted to evaluate them. Our results indicate a decreasing trend of extreme rainfalls at most of the stations except for 5 stations. However, the decreasing trends are only significant for two stations (Thiès and Kidira), however, this can only be taken as information that climate change may have already impacted extreme rainfalls. For the 20-year and 30-year return periods, the results show that they have undergone changes, in fact for almost all stations, the trends in return periods are decreasing.

Elevation-dependent trends in extreme snowfall in the French Alps from 1959 to 2019

Abstract. Climate change projections indicate that extreme snowfall is expected to increase in cold areas, i.e., at high latitudes and/or high elevation, and to decrease in warmer areas, i.e., at mid-latitudes and low elevation. However, the magnitude of these contrasting patterns of change and their precise relations to elevation at the scale of a given mountain range remain poorly known. This study analyzes annual maxima of daily snowfall based on the SAFRAN reanalysis spanning the time period 1959–2019 and provided within 23 massifs in the French Alps every 300 m of elevation. We estimate temporal trends in 100-year return levels with non-stationary extreme value models that depend on both elevation and time. Specifically, for each massif and four elevation ranges (below 1000, 1000–2000, 2000–3000, and above 3000 m), temporal trends are estimated with the best extreme value models selected on the basis of the Akaike information criterion. Our results show that a majority of trends are decreasing below 2000 m and increasing above 2000 m. Quantitatively, we find an increase in 100-year return levels between 1959 and 2019 equal to +23 % (+32kgm-2) on average at 3500 m and a decrease of −10 % (-7kgm-2) on average at 500 m. However, for the four elevation ranges, we find both decreasing and increasing trends depending on location. In particular, we observe a spatially contrasting pattern, exemplified at 2500 m: 100-year return levels have decreased in the north of the French Alps while they have increased in the south, which may result from interactions between the overall warming trend and circulation patterns. This study has implications for natural hazard management in mountain regions.

Data Breach CAT Bonds: Modeling and Pricing

Data breaches cause millions of dollars in financial losses each year. The insurance industry has been exploring the ways to transfer such extreme risk. In this work, we investigate data breach catastrophe (CAT) bonds via developing a multiperiod pricing model. It is found that the nonstationary extreme value model can capture the statistical pattern of the monthly maximum of data breach size very well and, in particular, a positive time trend is discovered. For the financial risks, data-driven time series approaches are proposed to model the complex patterns exhibited by the financial data, which are different from those in the literature. Simulation studies are performed to determine the bond prices and cash flows. Our results show that the data breach CAT bond can be an attractive financial product and an effective instrument for transferring the extreme data breach risk.

Non-stationary extreme value analysis of ground snow loads in the French Alps: a comparison with building standards

Abstract. In a context of climate change, trends in extreme snow loads need to be determined to minimize the risk of structure collapse. We study trends in 50-year return levels of ground snow load (GSL) using non-stationary extreme value models. These trends are assessed at a mountain massif scale from GSL data, provided for the French Alps from 1959 to 2019 by a meteorological reanalysis and a snowpack model. Our results indicate a temporal decrease in 50-year return levels from 900 to 4200 m, significant in the northwest of the French Alps up to 2100 m. We detect the most important decrease at 900 m with an average of −30 % for return levels between 1960 and 2010. Despite these decreases, in 2019 return levels still exceed return levels designed for French building standards under a stationary assumption. At worst (i.e. at 1800 m), return levels exceed standards by 15 % on average, and half of the massifs exceed standards. We believe that these exceedances are due to questionable assumptions concerning the computation of standards. For example, these were devised with GSL, estimated from snow depth maxima and constant snow density set to 150 kg m−3, which underestimate typical GSL values for the snowpack.

Assessment of return value estimates from stationary and non-stationary extreme value models

This article compares the accuracy of return value estimates from stationary and non-stationary extreme value models when the data exhibits covariate dependence. The non-stationary covariate representation used is a penalised piecewise-constant (PPC) model, in which the data are partitioned into bins defined by covariates and the extreme value distribution is assumed to be homogeneous within each bin. A generalised Pareto model is assumed, where the scale parameter can vary between bins but is penalised for the variance across bins, and the shape parameter is assumed constant over all covariate bins. The number and sizes of covariate bins must be defined by the user based on physical considerations. Numerical simulations are conducted to compare the performance of stationary and non-stationary models for various case studies, in terms of quality of estimation of the T-year return value over the full covariate domain. It is shown that a non-stationary model can give improved estimates of return values, provided that model assumptions are consistent with the data. When the data exhibits non-stationarity in the generalised Pareto tail shape, the use of non-stationary model assuming a constant shape parameter can produce biases in return values. In such cases, a stationary model can give a more accurate estimate of return value over the full covariate domain as only the most extreme observations (regardless of covariate) are used to estimate tail shape. In other cases, the assumption of a stationary model will ignore key features of the data and be less reliable than a non-stationary model. For example, if a relatively benign covariate interval exhibits a long (or heavy) tail, extreme values from this interval may influence the T-year return value for very large T. However the sample of peaks over threshold, with high threshold, used to estimate a stationary model in this case may not include sufficient observations from this interval to estimate the return value adequately.

On the Smoothing of the Generalized Extreme Value Distribution Parameters Using Penalized Maximum Likelihood: A Case Study on UVB Radiation Maxima in the Mexico City Metropolitan Area

This paper concerns the use and implementation of penalized maximum likelihood procedures to fitting smoothing functions of the generalized extreme value distribution parameters to analyze spatial extreme values of ultraviolet B (UVB) radiation across the Mexico City metropolitan area in the period 2000–2018. The model was fitted using a flexible semi-parametric approach and the parameters were estimated by the penalized maximum likelihood (PML) method. In order to investigate the performance of the model as well as the estimation method in the analysis of complex nonlinear trends for UVB radiation maxima, a simulation study was conducted. The results of the simulation study showed that penalized maximum likelihood yields better regularization to the model than the maximum likelihood estimates. We estimated return levels of extreme UVB radiation events through a nonstationary extreme value model using measurements of ozone (O3), nitrogen oxides (NOx), particles of 10 μm or less in diameter (PM10), carbon monoxide (CO), relative humidity (RH) and sulfur dioxide (SO2). The deviance statistics indicated that the nonstationary generalized extreme value (GEV) model adjusted was statistically better compared to the stationary model. The estimated smoothing functions of the location parameter of the GEV distribution on the spatial plane for different periods of time reveal the existence of well-defined trends in the maxima. In the temporal plane, a presence of temporal cyclic components oscillating over a weak linear component with a negative slope is noticed, while in the spatial plane, a weak nonlinear local trend is present on a plane with a positive slope towards the west, covering the entire study area. An explicit spatial estimate of the 25-year return period revealed that the more extreme risk levels are located in the western region of the study area.

On the relationship of daily rainfall extremes and local mean temperature

Extreme rainfall is the most common cause of flooding and likelihood of such events is found to increase in recent studies. Large scale natural variability, human induced global warming and local atmospheric warming are important drivers of extreme rainfall. Understanding the association of these drivers with extreme rainfall and finding out the most significant physical driver will provide a foundation for risk assessment. This study presents a methodology to investigate the association of extreme rainfall events with physical drivers and to model their dependence structure. The methodology is demonstrated with application to Mahanadi River basin, India. Non-stationary extreme value models with multivariate ENSO index, global average temperature anomalies and local mean temperature anomalies as covariates are fitted at non-stationary rainfall grids of the basin. Statistical modelling confirms localized temperature to be the main driver influencing rainfall extremes. Clausius-Clapeyron (CC) scaling curves relating daily rainfall extremes and local mean temperature are developed at all the grids to obtain insights into warming induced changes in rainfall extremes. Significant deviations from the expected CC scaling behaviour are observed. Daily rainfall extremes show peaks at low (25 °C) to medium (30 °C) range of temperature, but exhibit a decrease at high temperatures for most of the grids. No change in this pattern of relationship is observed when the extreme rainfall is lagged by 1–9 days with temperature. This time lag is selected based on the concept of residence time of water in the atmosphere. The Copula theory is applied to capture the dependence of annual maximum daily rainfall and temperature extremes. Conditional probabilities of extreme rainfall are estimated by conditioning on local mean temperature values of strongest extreme rainfall events. Comparison of conditional probabilities suggests that stronger rainfall events are occurring at 30 °C. This study emphasizes the necessity of understanding the dependence structure of extreme rainfall and local mean temperature.

Future evolution of extreme precipitation in the Mediterranean

Mediterranean basins can be impacted by severe floods caused by extreme rainfall, and there is a growing awareness about the possible increase in these heavy rainfall events due to climate change. In this study, the climate change impacts on extreme daily precipitation in 102 catchments covering the whole Mediterranean basin are investigated using nonstationary extreme value model applied to annual maximum precipitation in an ensemble of high-resolution regional climate model (RCM) simulations from the Euro-CORDEX experiment. Results indicate contrasted trends, with significant increasing trends in Northern catchments and conversely decreasing trends in Southern catchments. For most cases, the time of signal emergence for these trends is before the year 2000. The same spatial pattern is obtained under the two climate scenarios considered (RCP4.5 and RCP8.5) and in most RCM simulations, suggesting a robust climate change signal. The strongest multi-model agreement concerns the positive trends, which can exceed + 20% by the end of the twenty-first century in some simulations, impacting South France, North Italy, and the Balkans. For these areas, society-relevant strong impacts of such Mediterranean extreme precipitation changes could be expected in particular concerning flood-related damages.

Non-stationary statistical modeling of extreme wind speed series with exposure correction

Extreme wind speed analysis has been carried out conventionally by assuming the extreme series data is stationary. However, time-varying trends of the extreme wind speed series could be detected at many surface meteorological stations in China. Two main reasons, exposure change and climate change, were provided to explain the temporal trends of daily maximum wind speed and annual maximum wind speed series data, recorded at Hangzhou (China) meteorological station. After making a correction on wind speed series for time varying exposure, it is necessary to perform non-stationary statistical modeling on the corrected extreme wind speed data series in addition to the classical extreme value analysis. The generalized extreme value (GEV) distribution with time-dependent location and scale parameters was selected as a non-stationary model to describe the corrected extreme wind speed series. The obtained non-stationary extreme value models were then used to estimate the non-stationary extreme wind speed quantiles with various mean recurrence intervals (MRIs) considering changing climate, and compared to the corresponding stationary ones with various MRIs for the Hangzhou area in China. The results indicate that the non-stationary property or dependence of extreme wind speed data should be carefully evaluated and reflected in the determination of design wind speeds.

Projecting future nonstationary extreme streamflow for the Fraser River, Canada

We describe an efficient and flexible statistical modeling framework for projecting nonstationary streamflow extremes for the Fraser River basin in Canada, which is dominated by nival flow regime. The framework is based on an extreme value analysis technique that allows for nonstationarity in annual extreme streamflow by relating it to antecedent winter and spring precipitation and temperature. We used a representative suite of existing Variable Infiltration Capacity hydrologic model simulations driven by Coupled Model Intercomparison Project Phase 3 (CMIP3) climate simulations to train and evaluate a nonlinear and nonstationary extreme value model of annual extreme streamflow. The model was subsequently used to project changes under CMIP5-based climate change scenarios. Using this combination of process-based and statistical modeling, we project that the moderate (e.g., 2–20-year return period) extreme streamflow events will decrease in intensity. In contrast, projections of high intensity events (e.g., 100–200-year return period), which reflect complex interactions between temperature and precipitation changes, are inconclusive. The results provide a basis for developing a general understanding of the future streamflow extremes changes in nival basins and through careful consideration and adoption of appropriate covariates, the methodology could be employed for basins spanning a range of hydro-climatological regimes.

Non-stationary extreme value models to account for trends and shifts in the extreme wave climate due to climate change

The extreme values of wave climate data are of great interest in a number of different ocean engineering applications, including the design and operation of ships and offshore structures, marine energy generation, aquaculture and coastal installations. Typically, the return values of certain met-ocean parameters such as significant wave height are of particular importance. There exist many methods for estimating such return values, including the initial distribution approach, the block maxima approach and the peaks-over threshold approach. In a climate change perspective, projections of such return values to a future climate are of great importance for risk management and adaptation purposes. However, many approaches to extreme value modelling assume stationary conditions and it is not straightforward how to include non-stationarity of the extremes due to for example climate change. In this paper, various non-stationary GEV-models for significant wave height are developed that account for trends and shifts in the extreme wave climate due to climate change. These models are fitted to block maxima in a particular set of wave data obtained for a historical control period and two future projections for a future period corresponding to different emission scenarios. These models are used to investigate whether there are trends in the data within each period that influence the extreme value analysis and need to be taken into account. Moreover, it will be investigated whether there are significant inter-period shifts or trends in the extreme wave climate from the historical period to the future periods. The results from this study suggest that the intra-period trends are not statistically significant and that it might be reasonable to ignore these in extreme value analyses within each period. However, when it comes to comparing the different data sets, i.e. the historical period and the future projections, statistical significant inter-period changes are detected. Hence, the accumulated effect of a climatic trend may not be negligible over longer time periods. Interestingly enough, such statistically significant shifts are not detected if stationary extreme value models are fitted to each period separately. Therefore, the non-stationary extreme value models with inter-period shifts in the parameters are proposed as an alternative for extreme value modelling in a climate change perspective, in situations where historical data and future projections are available.

Estimation of Storm Peak and Intrastorm Directional–Seasonal Design Conditions in the North Sea

Specification of realistic environmental design conditions for marine structures is of fundamental importance to their reliability over time. Design conditions for extreme waves and storm severities are typically estimated by extreme value analysis of time series of measured or hindcast significant wave height, HS. This analysis is complicated by two effects. First, HS exhibits temporal dependence. Second, the characteristics of HSsp are nonstationary with respect to multiple covariates, particularly wave direction, and season. We develop directional–seasonal design values for storm peak significant wave height (HSsp) by estimation of, and simulation under a nonstationary extreme value model for HSsp. Design values for significant wave height (HS) are estimated by simulating storm trajectories of HS consistent with the simulated storm peak events. Design distributions for individual maximum wave height (Hmax) are estimated by marginalization using the known conditional distribution for Hmax given HS. Particular attention is paid to the assessment of model bias and quantification of model parameter and design value uncertainty using bootstrap resampling. We also outline existing work on extension to estimation of maximum crest elevation and total extreme water level.

Climate change and non-stationary flood risk for the upper Truckee River basin

Abstract. Future flood frequency for the upper Truckee River basin (UTRB) is assessed using non-stationary extreme value models and design-life risk methodology. Historical floods are simulated at two UTRB gauge locations, Farad and Reno, using the Variable Infiltration Capacity (VIC) model and non-stationary Generalized Extreme Value (GEV) models. The non-stationary GEV models are fit to the cool season (November–April) monthly maximum flows using historical monthly precipitation totals and average temperature. Future cool season flood distributions are subsequently calculated using downscaled projections of precipitation and temperature from the Coupled Model Intercomparison Project Phase 5 (CMIP-5) archive. The resulting exceedance probabilities are combined to calculate the probability of a flood of a given magnitude occurring over a specific time period (referred to as flood risk) using recent developments in design-life risk methodologies. This paper provides the first end-to-end analysis using non-stationary GEV methods coupled with contemporary downscaled climate projections to demonstrate the evolution of a flood risk profile over typical design life periods of existing infrastructure that are vulnerable to flooding (e.g., dams, levees, bridges and sewers). Results show that flood risk increases significantly over the analysis period (from 1950 through 2099). This highlights the potential to underestimate flood risk using traditional methodologies that do not account for time-varying risk. Although model parameters for the non-stationary method are sensitive to small changes in input parameters, analysis shows that the changes in risk over time are robust. Overall, flood risk at both locations (Farad and Reno) is projected to increase 10–20% between the historical period 1950 to 1999 and the future period 2000 to 2050 and 30–50% between the same historical period and a future period of 2050 to 2099.

Extreme wave climate changes in Central-South America

The spatial and temporal variability of extreme wave climate along the Central- South American continent is analyzed. The study evaluates changes in the intensity of extreme significant wave height (SWH) throughout the year over the 1980-2008 period, using a calibrated long-term wave reanalysis database forced with NCEP/NCAR reanalysis. A non-stationary extreme value model, based on monthly maxima with a new approach for long-term trends, has been applied. Results show a common positive trend in the Pacific basin throughout all seasons and a significant decreasing trend pattern in the area of Surinam and north of Brazil, on the Atlantic border (up to �1.5 cm/year in March-April-May). A higher increase of the extreme wave heights is found in the austral summer (December- January-February) at Tierra de Fuego and the Falkland Islands, reaching 6.5 cm/year (which means 1.82 m for the 28 years studied). Furthermore, the complete reanalysis period (1948- 2008) is analyzed in order to compare results with the assimilation data period (1980-2008), resulting in some discrepancies, especially in the Atlantic basin.

Dynamical versus statistical downscaling methods for ocean wave heights

AbstractIn this study, dynamical and statistical downscaling methods for estimating seasonal statistics of significant wave heights (SWH) were intercompared, with the downscaling results being evaluated against the ERA40 wave data in terms of climatological characteristics and interannual variability. It was also shown that biases in climate‐model‐simulated climate and variability of the atmospheric circulation (or predictors in general) can result in large biases in the estimated climate and variability of SWH (or the predictand in general), and that such biases can be effectively diminished by using standardized predictor quantities in statistical downscaling models. In dynamical downscaling, however, model variability biases remain to be dealt with, whereas the effects of model climate biases can be reduced to some extent by replacing the climate‐model‐simulated wind climate with the observed one. Therefore, the dynamical approach was found to be not as good as the statistical methods in terms of reproducing the observed climate and interannual variability of the predictand, although it bears substantial similarity to the statistical methods in terms of projected possible future changes. Also, it was shown that the observed interannual variability of seasonal statistics (including extremes) can be better reproduced by using 12‐hourly, rather than seasonal, data in statistical downscaling. This stresses the importance of availability of higher‐resolution data from climate model outputs. Nevertheless, a non‐stationary extreme value model with covariates was found to be the best in reproducing the observed climate of extremes. All the statistical downscaling methods and the intercomparison results are applicable to other climate variables (not limited to ocean wave heights). Copyright © 2009 Crown in the right of Canada. Published by John Wiley & Sons, Ltd.