A two-component generalized extreme value distribution for precipitation frequency analysis

Previous research shows that for limited sample sizes applying the simplified metastatistical extreme value (sMEV) distribution instead of the more commonly used general extreme value (GEV) distribution can significantly reduce the associated uncertainty in rainfall return levels. Recent literature has also highlighted the possibility to analyze the effects of climate change using the non-stationary version of the sMEV distribution. Thus, the objective of this study is to test the performance of the sMEV and GEV for hourly precipitation using a convection-permitting regional climate model. The global climate model MIROC5 is employed to drive the regional climate model COSMO over the greater Germany area for the past, near future, and distant future. It is set up at a high temporal and spatial resolution allowing it to explicitly resolve deep convection, which is important when assessing extreme hourly precipitation. No comparable time series from a convection-permitting model has previously been analyzed using the sMEV distribution. The results show that the sMEV performs much better than the GEV in terms of the uncertainty for almost all return periods regardless of the observational years available. In addition, there is a north-south gradient in the return level difference, the uncertainty difference and the adequacy of the left-censoring threshold chosen for the sMEV. Investigating non-stationary versions of the sMEV and GEV shows that the non-stationary sMEV is more suitable to describing the change in return levels. However, both implemented versions of the non-stationary distributions are limited by the complexity of the temperature dependency. Therefore, we recommend a careful application for the prediction of return levels under higher temperatures. 

Contrary to common belief, Fisher‐Tippett's extreme value (EV) theory does not typically apply to annual rainfall maxima. Similarly, Pickands' extreme excess (EE) theory does not typically apply to rainfall excesses above thresholds on the order of the annual maximum. This is true not just for long‐averaging durationsdbut also for shortdand in the high‐resolution limit asd→ 0. We reach these conclusions by applying large‐deviation theory to multiplicative rainfall models with scale‐invariant structure. We derive several asymptotic results. One is that, asd→ 0, the annual maximum rainfall intensity ind,Iyr,d, has generalized extreme value (GEV) distribution with a shape parameterkthat is significantly higher than that predicted by EV theory and is always in the EV2 range. The value ofkdepends not on the upper tail of the marginal distribution but on regions closer to the body. Under the same conditions, the excesses above levels close to the annual maximum have generalized Pareto distribution with parameterkthat is always higher than that predicted by Pickands' EE theory. For finited, the distribution ofIyr,dis not GEV, but in accordance with empirical evidence is well approximated by a GEV distribution with shape parameterkthat increases asddecreases. We propose a way to estimatekunder preasymptotic conditions from the scaling properties of rainfall and suggest a near‐universalk(d) relationship. The new estimator promises to be more accurate and robust than conventional estimators. These developments represent a significant conceptual change in the way rainfall extremes are viewed and evaluated.

Abstract. Anticipating risks related to climate extremes often relies on the quantification of large return levels (values exceeded with small probability) from climate projection ensembles. Current approaches based on multi-model ensembles (MMEs) usually estimate return levels separately for each climate simulation of the MME. In contrast, using MME obtained with different combinations of general circulation model (GCM) and regional climate model (RCM), our approach estimates return levels together from the past observations and all GCM–RCM pairs, considering both historical and future periods. The proposed methodology seeks to provide estimates of projected return levels accounting for the variability of individual GCM–RCM trajectories, with a robust quantification of uncertainties. To this aim, we introduce a flexible non-stationary generalized extreme value (GEV) distribution that includes (i) piecewise linear functions to model the changes in the three GEV parameters and (ii) adjustment coefficients for the location and scale parameters to adjust the GEV distributions of the GCM–RCM pairs with respect to the GEV distribution of the past observations. Our application focuses on snow load at 1500 m elevation for the 23 massifs of the French Alps. Annual maxima are available for 20 adjusted GCM–RCM pairs from the EURO-CORDEX experiment under the scenario Representative Concentration Pathway (RCP) 8.5. Our results show with a model-as-truth experiment that at least two linear pieces should be considered for the piecewise linear functions. We also show, with a split-sample experiment, that eight massifs should consider adjustment coefficients. These two experiments help us select the GEV parameterizations for each massif. Finally, using these selected parameterizations, we find that the 50-year return level of snow load is projected to decrease in all massifs by −2.9 kN m−2 (−50 %) on average between 1986–2005 and 2080–2099 at 1500 m elevation and RCP8.5. This paper extends the recent idea to constrain climate projection ensembles using past observations to climate extremes.

Anticipating risks related to climate extremes often relies on the quantification of large return levels (values exceeded with small probability) from climate projection ensembles. Current approaches based on multi-model ensembles (MMEs) usually estimate return levels separately for each chain of the MME. By contrast, using MME obtained with different combinations of general circulation model (GCM) and regional climate model (RCM), our approach estimates return levels together from the past observations and all GCM-RCM pairs, considering both historical and future periods. The proposed methodology seeks to provide estimates of projected return levels accounting for the variability of individual GCM-RCM trajectories, with a robust quantification of uncertainties. To this aim, we introduce a flexible non-stationary generalized extreme value (GEV) distribution that includes i) piecewise linear functions to model the changes in the three GEV parameters ii) adjustment coefficients for the location and scale parameters to adjust the GEV distributions of the GCM-RCM pairs with respect to the GEV distribution of the past observations. Our application focuses on snow load at 1500 m elevation for the 23 massifs of the French Alps, which is of major interest for the structural design of roofs. Annual maxima are available for 20 adjusted GCM-RCM pairs from the EURO-CORDEX experiment, under the scenario RCP8.5. Our results show with a model-as-truth experiment that at least two linear pieces should be considered for the piecewise linear functions. We also show, with a split-sample experiment, that eight massifs should consider adjustment coefficients. These two experiments help us select the GEV parameterizations for each massif. Finally, using these selected parameterizations, we find that the 50-year return level of snow load is projected to decrease in all massifs, by −2.9 kN m−2 (−50 %) on average between 1986–2005 and 2080–2099 at 1500 m elevation and RCP8.5. This paper extends to climate extremes the recent idea to constrain climate projection ensembles using past observations.

Abstract. Occurrence probabilities of extreme sea levels required in coastal planning, e.g. for calculating design floods, have been traditionally estimated individually at each tide-gauge location. However, these estimates include uncertainties, as sea-level observations typically have only a small number of extreme cases such as annual maxima. Moreover, exact information on sea-level extremes between the tide-gauge locations and incorporation of dependencies between the adjacent stations is often lacking in the analysis. In this study, we use Bayesian hierarchical modelling to estimate return levels of annual maxima of short-term sea-level variations related to storm surges in the Finnish coastal region. We use the generalised extreme value (GEV) distribution as the basis and compare three hierarchical model structures of different complexity against tide-gauge-specific fits. The hierarchical model structures allow us to share information on annual maximum sea levels between the neighbouring stations and also provide a natural way to estimate uncertainties in the theoretical estimates. The results show that compared to the tide-gauge-specific fits, the hierarchical models, which pool information across the tide gauges, provide narrower uncertainty ranges for both the posterior parameter estimates and the corresponding return levels in most locations. The estimated shape parameter of the GEV model is systematically negative for the hierarchical models, which indicates a Weibull type of behaviour for the extremes along the Finnish coast. The negative shape parameter also allows us to calculate the theoretical upper limit for the annual maximum sea levels on the Finnish coast. Depending on the tide gauge and hierarchical model considered, the median value of the theoretical upper limit was 47–73 cm higher than the highest observed sea level.

Rainfall intensity–duration–frequency relationships derived from large partial duration series

Extreme value modeling for extreme rainfall is one of the most important processes in the field of hydrology. For the improvement of extreme value modeling and its physical meaning, large-scale climate modes have been widely used as covariates of distribution parameters, as they can physically account for climate variability. This study proposes a novel procedure for extreme value modeling of rainfall based on the significant relationship between the long-term trend of the annual maximum (AM) daily rainfall and large-scale climate indices. This procedure is characterized by two main steps: (a) identifying significant seasonal climate indices (SCIs), which impact the long-term trend of AM daily rainfall using statistical approaches, such as ensemble empirical mode decomposition, and (b) selecting an appropriate generalized extreme value (GEV) distribution among the stationary GEV and nonstationary GEV (NS-GEV) using time and SCIs as covariates by comparing their model fit and uncertainty. Our findings showed significant relationships between the long-term trend of AM daily rainfall over South Korea and SCIs (i.e., the Atlantic Meridional Mode, Atlantic Multidecadal Oscillation in the fall season, and North Atlantic Oscillation in the summer season). In addition, we proposed a model selection procedure considering both the Akaike information criterion and residual bootstrap method to select an appropriate GEV distribution among a total of 59 GEV candidates. As a result, the NS-GEV with SCI covariates generally showed the best performance over South Korea. We expect that this study can contribute to estimating more reliable extreme rainfall quantiles using climate covariates.

Rainfall extreme value estimates in Canada have historically been based on fitting the Gumbel distribution to annual maxima at individual sites by the method of moments (MOM). Studies have, however, shown that regional frequency analyses (RFA) may perform better than at-site methods. Also, the Fréchet rather than Gumbel form of the generalized extreme value (GEV) distribution may better describe the distribution of annual extremes. In this study, at-site Gumbel MOM and GEV extreme value analyses based on L-moment, maximum likelihood (MLE), and generalized maximum likelihood (GML) estimators are compared against RFA with L-moment methods at stations in southern British Columbia, Canada via cross-validation and Monte Carlo simulations. While GEV shape parameter estimates are predominately negative, qualitatively showing weak evidence for the Fréchet form of the GEV distribution, field significant differences from the Gumbel distribution in the region are not found. Regional frequency analysis leads to substantial reductions in error relative to at-site methods, especially for the GEV distribution and small samples. While Gumbel estimators exhibit lower variance than GEV estimators, they are also more biased, underestimating 100 year return levels. Of the at-site GEV estimators, GML tended to perform better than the L-moment estimator, in some cases nearing performance of RFA. Maximum likelihood performed worst, especially for small samples sizes.

Development of plotting position for the general extreme value distribution

On the value of hourly precipitation extremes in regional climate model simulations

The study examines trends in characteristics of convective and stratiform precipitation in the Czech Republic over 1982–2010. The spatially averaged trends in convective precipitation are rising for indices of mean precipitation, and the increases are significant in all seasons except for winter. For extremes, the trends are spatially much more variable and insignificant, but increases tend to prevail as well. The trends in convective precipitation are larger in the western part of the country where Atlantic influences are stronger. For characteristics of stratiform precipitation, the trends are usually smaller compared to those of convective precipitation, but increases prevail too. They are significant in autumn, especially for extremes, and larger in the eastern part of the country where Mediterranean cyclones play more important role. The trends in convective precipitation tend to be more pronounced at lowland than higher-elevated stations while an opposite pattern prevails for stratiform precipitation. The results suggest that in spring and summer, when convective precipitation represents an important fraction of the total amounts in central Europe (around 30% and 50%, respectively), the observed increases in total precipitation are mainly due to increases in convective precipitation. In autumn, increases in both convective and stratiform precipitation are important, and the trends are weakest and least pronounced in winter.

The generalized extreme value (GEV) distribution is suitable for modeling the extreme events, such as the maximum value of an asymptotic distribution. This paper presents results of an experimental investigation where the distribution of peak-to-average power ratio (PAPR) in downlink MC-CDMA systems is modeled by the GEV distribution. Two orthogonal sets of sequences, Walsh-Hadamard and Golay complementary sequences, are used in spreading processes in the system. Then the parameters of the GEV distribution are estimated for the PAPR distribution. Through intensive numerical results, it is shown that the GEV distribution is an accurate model of the PAPR distribution of MC-CDMA systems. Also, the statistically estimated GEV distribution parameters for the PAPR reveal that when the number of subcarriers increases, the PAPR distributions converge to the Gumbel distribution.

Trends of annual maximum 1-day precipitation in three major Hawaiian Islands are investigated using a nonparametric Mann-Kendall method and Sen's test (MKS). The records are from 24 stations on Oahu, Maui, and Hawaii, and the period of analysis ranges from 1960 to 2009. To complement the MKS method, a non-stationary three- parameter generalized extreme value (GEV) distribution is also used to detect trends in precipitation extremes. Both methods demonstrate that negative trends prevail for Oahu and Maui but positive trends dominate the Island of Hawaii. The influence of the location and the scale parameter in the GEV model on different return levels (2-year, 20-year, and 100-year) are explicitly described. The return-level threshold values are found to change with time considerably. As a result, a rare storm with daily precipitation of 300 mm (20-year return period) in 1960 has become a rather common storm event (3-5-year return period) in 2009 on the Island of Hawaii. The opposite trend behavior in extreme events is observed on Oahu and Maui, where rainfall extremes have become less frequent in the last five decades. A positive relationship is found between the precipitation extremes and Southern Oscillation Index (SOI), implying greater extreme events during La Ni˜ na years and the opposite for El Niyears.

Climate change has brought about unprecedented new weather patterns, one of which is changes in extreme rainfall. In Kenya, heavy rains and severe flash floods have left people dead and displaced hundreds from their settlements. In order to build a resilient society and achieve sustainable development, it is paramount that adequate inference about extreme rainfall be made. To this end, this research modelled and predicted extreme rainfall events in Kenya using Extreme Value Theory for rainfall data from 1901-2016. Maximum Likelihood Estimation was used to estimate the model parameters and block maxima approach was used to fit the Generalized Extreme Value Distribution (GEVD) while the Peak Over Threshold method was used to fit the Generalized Pareto Distribution (GPD). The Gumbel distribution was found to be the optimal model from the GEVD while the Exponential distribution gave the optimal model over the threshold value. Furthermore, prediction for the return periods of 10, 20, 50 and 100 years were made using the return level estimates and their corresponding confidence intervals were presented. It was found that increase in return periods leads to a corresponding increase in return levels. However, the GPD gave higher return levels for 10 and 20 years compared to GEVD. While, for higher return periods 50 and 100 years, the GEVD gave higher return levels compared to the GPD. Model diagnostics using probability, density, quantile and return level plots indicated that the models provided were a good fit for the data.

ABSTRACTThe principle of return periods or frequencies of natural hazards is adopted in many countries as the basis of eligibility for the compensation of associated losses. For adequate risk management and eligibility in Belgium, hazard maps for drought events with a 20‐year return period are needed. The maximum precipitation deficit in the summer half‐year (1 April–30 September) was taken as a good indicator for atmospheric drought severity during a particular year. Precipitation deficit is calculated as a cumulative balance between precipitation and evapotranspiration for a free open water surface, short grass, deciduous forest and coniferous forest. Precipitation deficit maxima were modelled with the generalized extreme value (GEV) distribution. Mapping precipitation deficit return levels is based on a GEV distribution where the parameters vary smoothly in space as a function of altitude and distance to the sea. The final products are return level maps of extreme precipitation deficit, relevant for insurance companies providing cover for the forestry and agriculture sector.