A Dimension-Reduced Multivariate Spatial Model for Extreme Events: Balancing Flexibility and Scalability

<p>Alaba Boluwade*</p> <p>School of Climate Change & Adaptation, University of Prince Edward Island, Charlottetown, Canada; aboluwade@upei.ca; abolu2013@gmail.com</p> <p>*Correspondence: aboluwade@upei.ca</p> <p><strong>Abstract </strong></p> <p>Hydrological risk assessment, such as flood protection, requires estimates of variables (e.g., precipitation) measured from several weather stations. The spatial modeling of average rainfall estimates differs from extreme precipitation analysis. This is because extremes are focused on the tail of the probability distribution and the assumption of Gaussianity may not be suitable. Extreme Value Theory (EVT) application to univariate weather variables measured at weather stations has been well documented; however, extreme precipitation at closer stations tend to show trends and dependencies (similar values). It is, therefore, crucial to quantify the dependency structure and trend surface of weather stations in space. The max-stable process has been well documented to model spatial extremes. The objective of this study is to quantify the spatial dependency and trend of an annual maxima precipitation (annual highest daily precipitation, from 1970-2020) across selected weather stations in the Northern Great Plains (i.e., Nelson Churchill River Basin (NCRB)) of North America. The annual maxima data were extracted from the Global Historical Climatology Network Daily (GHCNd) and Environment and Climate Change Canada (ECCC). NCRB covers four states and four provinces in the United States and Canada. A heterogenous rainfall pattern characterizes NCRB. This is due to enormous quantities of orographic rainfall in the west and the convective precipitation in the Prairies (which is dominated by short-duration, sporadic, extreme rain), causing millions of dollars in damages. This study uses max-stable processes to examine spatial extremes of annual maxima precipitation.</p> <p>The results show that topography, time, and geographical coordinates were important covariates in reproducing the stochastic extreme precipitation field using the spatial generalized extreme value (SPEV). Takeuchi’s information criterion (TIC) shows that the SPEV model with all the covariates above superseded the one without the covariates.   The inclusion of time as a covariate further confirms the impacts of climate change on extreme precipitation in the NCRB. The fitted Extremal-t max-stable model captured the spatial dependency and equally predicted the 50-year return period levels. Furthermore, ten realizations (equal probable) were simulated from the max-stable model. The study is relevant in quantifying the spatial trend and dependency of extreme precipitation in the Northern Great Plains. The result will help as a decision-support system in climate adaptation strategies in the United States and Canada.</p> <p><em> </em></p> <p><em>Keywords: extreme events; Max-Stable processes; flood protection; maxima annual rainfall; flash flood protection; Canada, United States</em></p>

Radley Horton,1,a Daniel Bader,1,a Yochanan Kushnir,2 Christopher Little,3 Reginald Blake,4 and Cynthia Rosenzweig5 1Columbia University Center for Climate Systems Research, New York, NY. 2Ocean and Climate Physics Department, Lamont-Doherty Earth Observatory, Columbia University, Palisades, NY. 3Atmospheric and Environmental Research, Lexington, MA. 4Physics Department, New York City College of Technology, CUNY, Brooklyn, NY. 5Climate Impacts Group, NASA Goddard Institute for Space Studies; Center for Climate Systems Research, Columbia University Earth Institute, New York, NY

It is well recognized that the generalized extreme value (GEV) distribution is widely used for any extreme events. This notion is based on the study of discrete choice behavior; however, there is a limit for predicting the distribution at ungauged sites. Hence, there have been studies on spatial dependence within extreme events in continuous space using recorded observations. We model the annual maximum daily rainfall data consisting of 25 locations for the period from 1982 to 2013. The spatial GEV model that is established under observations is assumed to be mutually independent because there is no spatial dependency between the stations. Furthermore, we divide the region into two regions for a better model fit and identify the best model for each region. We show that the regional spatial GEV model reflects the spatial pattern well compared with the spatial GEV model over the entire region as the local GEV distribution. The advantage of spatial extreme modeling is that more robust return levels and some indices of extreme rainfall can be obtained for observed stations as well as for locations without observed data. Thus, the model helps to determine the effects and assessment of vulnerability due to heavy rainfall in northeast Thailand.

Extreme precipitation events are a major natural hazard and cause significant socio‐economic damages. Precipitation events are spatially extended and, thus, can cause large water accumulations, which can lead to flooding events. In order to help design flood protection infrastructure, a detailed investigation of the temporal and spatial dependencies of extreme precipitation is essential. Here, we use a statistical spatial extremes framework to systematically study the historical and projected spatial–temporal characteristics of extreme precipitation in Germany. For this purpose, we use data from 10 high‐resolution global climate model‐regional climate model (GCM‐RCM) combinations from the EURO‐CORDEX initiative and derive a statistical spatial extremes precipitation model. Our results show that there are large spreads in reproducing the temporal–spatial characters of extreme precipitation. Few climate simulations can well present the temporal clustering of observed extreme precipitation in both summer and winter. In reproducing the spatial dependencies of the observations, most GCM‐RCM combinations behave well in summer, while in winter most RCMs produce too many spatially localized extreme precipitation events. The derived statistical model, which accounts for both the spatial and temporal variability, performs well in representing the spatial dependency and intensity characteristics in summer. Furthermore, global warming will have a significant impact on the temporal and spatial dependencies of extreme precipitation in Germany. There will be more temporal‐dependent and homogeneous extreme precipitation in summer; and more temporal‐independent and localized extreme precipitation in winter. The intensity quantified by the 25‐year return level of the 10 GCM‐RCM combinations is increasing; with relative changes ranging from 5.33% to 53.24% in summer and from −15.38% to 32.33% in winter under RCP8.5. A future projection by our statistical spatial extremes model using projected temperature from GCM‐RCM combinations as a covariate shows that the 25‐year return level will increase by 3.02% under RCP2.6 and 4.16% under RCP8.5 in winter.

Daily maximum/minimum temperature and precipitation data from 35 weather stations in Xinjiang during 1961-2010 were examined using kriging spatial analysis, linear tendency estimation, and correlation analysis. Temporal trends and spatial distribution patterns of extreme temperature and precipitation in this area were then analyzed using 12 extreme temperature and 7 extreme precipitation indices. The following results were obtained. 1) Over the past 50 years, extreme cold indices, excepting the monthly maximum temperature minimum value and monthly extreme minimum temperature, showed slight decreasing trends. These indices include the maximum number of consecutive frost days, icy days, cold-nighttime days, and cold-daytime days. 2) Extreme warm events generally showed significant increasing trends (P < 0.01), including the indices of summertime days, warm-nighttime days, warm-daytime days, monthly extreme maximum temperature, and monthly minimum temperature maximum value. 3) The spatial distributions of threshold values of extreme warm and cold events showed notable regional differences. A reducing trend of extreme cold events and an increase in extreme warm events has occurred mainly in northern Xinjiang. 4) For the past 50 years, six extreme precipitation indices, aside from consecutive dry days, showed significant increasing trends in Xinjiang (P < 0.05) and notable differences in spatial distribution. The increase in extreme precipitation events was more rapid at northern than at southern sites. Extreme precipitation intensity was greater in mountainous areas, and precipitation frequency increased in the plain region. 5) Factor analysis revealed good correlations among extreme temperature indices, excepting extreme temperature days.

ABSTRACTTo model extreme spatial events, a general approach is to use the generalized extreme value (GEV) distribution with spatially varying parameters such as spatial GEV models and latent variable models. In the literature, this approach is mostly used to capture spatial dependence for only one type of event. This limits the applications to air pollutants data as different pollutants may chemically interact with each other. A recent advancement in spatial extremes modelling for multiple variables is the multivariate max-stable processes. Similarly to univariate max-stable processes, the multivariate version also assumes standard distributions such as unit-Fréchet as margins. Additional modelling is required for applications such as spatial prediction. In this paper, we extend the marginal methods such as spatial GEV models and latent variable models into a multivariate setting based on copulas so that it is capable of handling both the spatial dependence and the dependence among multiple pollutants. We apply our proposed model to analyse weekly maxima of nitrogen dioxide, sulphur dioxide, respirable suspended particles, fine suspended particles, and ozone collected in Pearl River Delta in China.

Recent studies have found that the El Niño–Southern Oscillation (ENSO) has statistically significant influences on extreme precipitation. A limitation of most existing work is that a separate generalized extreme value (GEV) distribution is fitted for each individual site. Such models cannot address important questions that involve events jointly defined across multiple sites; for instance, what is the probability that the 50 year return levels of three sites in the vicinity of a city occur in the same season? With the latest statistical methodology for spatial extremes, we fit max‐stable process models to winter maximum daily precipitation of 192 sites in California over 55 years. A composite likelihood approach is used since the full likelihood is unavailable either analytically or numerically. In addition to latitude, longitude, and elevation, the Southern Oscillation Index (SOI) is incorporated into the parameters of the marginal GEV models. We find that, in a spatial context, the ENSO has a significant influence on the extreme precipitation in California by shifting the location parameter of the GEV distributions, with higher values of the SOI corresponding to lower maximum winter daily precipitation. The joint spatial model is used to assess risks concerning joint extremal events at network of sites with spatial dependence properly accounted for. The probability of extremal events occurring at multiple sites in the same season is found to be much higher than what would be expected under the independence assumption.

Uncertainty in return level estimates for rare events, like the intensity of large rainfall events, makes it difficult to develop strategies to mitigate related hazards, like flooding. Latent spatial extremes models reduce the uncertainty by exploiting spatial dependence in statistical characteristics of extreme events to borrow strength across locations. However, these estimates can have poor properties due to model misspecification: Many latent spatial extremes models do not account for extremal dependence, which is spatial dependence in the extreme events themselves. We improve estimates from latent spatial extremes models that make conditional independence assumptions by proposing a weighted likelihood that uses the extremal coefficient to incorporate information about extremal dependence during estimation. This approach differs from, and is simpler than, directly modeling the spatial extremal dependence; for example, by fitting a max-stable process, which is challenging to fit to real, large datasets. We adopt a hierarchical Bayesian framework for inference, use simulation to show the weighted model provides improved estimates of high quantiles, and apply our model to improve return level estimates for Colorado rainfall events with 1% annual exceedance probability. Supplementary materials accompanying this paper appear online.

Spatial Modelling of Extreme Rainfall in Northeast Thailand

Synoptic‐scale patterns associated with daily temperature and precipitation extremes in Alaska are identified and evaluated for daily variability in order to understand consistency in forcing mechanisms associated with extreme events as well as the tendency for each pattern to produce an extreme event. Daily station data at five locations for the 29‐year period from 1982 to 2010 are used. The widely recognized ClimDex indices are used to identify extreme high temperature, low temperature, and single‐day precipitation events. Pressure patterns during extreme events are evaluated seasonally for summer (JJA) and winter (DJF) at mean sea level pressure, 700 and 500‐hPa geopotential heights. Temperature extremes are largely characterized by synoptic‐scale patterns associated with substantial warm and cold air advection, and precipitation extremes are associated with moisture advection from the North Pacific. Extreme warm and cold temperature events are generally associated with more consistent daily synoptic‐scale patterns than precipitation. Considerable dissimilarities among daily synoptic‐scale patterns associated with extreme events based on location, type of extreme event and season highlight the need for careful consideration in the practice of applying synoptic‐scale forcing patterns to forecast or downscale extreme weather events in Alaska.

Following the catastrophic floods that struck Slovenia in 2023, public awareness of the impacts of climate change has significantly increased. As a result, there is growing demand for information on future climate extremes—particularly changes in extreme precipitation, and to a lesser extent, extreme temperatures. The majority of these requests come from engineers who need to design climate-resilient infrastructure. More recently, municipalities—either individually or as regional consortia—have also begun seeking assessments of their resilience to future climate change.In 2016, the Slovenian Environment Agency (ARSO) launched the project Assessment of Climate Change by the End of the 21st Century, analysing regional climate model outputs from the Euro-CORDEX project. The analysis included three Representative Concentration Pathways (RCPs): RCP2.6, RCP4.5, and RCP8.5. For each scenario, we evaluated an ensemble of models to calculate projected changes and their statistical robustness for various climate variables, focusing on future 30-year periods relative to the 1981–2010 reference period. For temperature and precipitation, we also assessed trends in extremes using the Generalized Extreme Value (GEV) distribution.Trends in annual maximum values were used to estimate changes in extreme one-day precipitation and temperature (both maximum and minimum), based on the GEV location parameter. However, we found that one-day precipitation trends tend to underestimate the changes in shorter-duration events (e.g., 5–30 minutes). To better estimate changes in sub-daily extreme rainfall, we now derive trends from mean temperature changes and apply the Clausius–Clapeyron relation. These temperature trends are calculated from annual averages using simple linear regression, rather than from extremes or GEV analysis.When users request an analysis, they typically provide location coordinates, a future target year, and a preferred RCP scenario. Most analyses are focused on projections to 2050 under RCP4.5, though some users request longer-term assessments under RCP8.5. For temperature, the most common request is the projected change in the 50-year return level of maximum temperature. For precipitation, users frequently request 100-year return levels for one-day and sub-daily extreme rainfall (e.g., 5, 10, 15, 20, 30 minutes). Present-day conditions are derived from meteorological station data interpolated to a ~1 km grid. We then apply model climate trends to these current extremes to estimate future values.Approximately 90 % of requests involve extreme precipitation projections, followed by 7 % for municipal or regional climate impact assessments, and the remaining 3 % for extreme temperature projections.

Knowledge about long‐term changes in climate extremes is vital to better understand multidecadal climate variability and long‐term changes and to place today's extreme events in a historical context. While global changes in temperature and precipitation extremes since the midtwentieth century are well studied, knowledge about century‐scale changes is limited. This paper analyses a range of largely independent observations‐based data sets covering 1901–2010 for long‐term changes and interannual variability in daily scale temperature and precipitation extremes. We compare across data sets for consistency to ascertain our confidence in century‐scale changes in extremes. We find consistent warming trends in temperature extremes globally and in most land areas over the past century. For precipitation extremes we find global tendencies toward more intense rainfall throughout much of the twentieth century; however, local changes are spatially more variable. While global time series of the different data sets agree well after about 1950, they often show different changes during the first half of the twentieth century. In regions with good observational coverage, gridded observations and reanalyses agree well throughout the entire past century. Simulations with an atmospheric model suggest that ocean temperatures and sea ice may explain up to about 50% of interannual variability in the global average of temperature extremes, and about 15% in the global average of moderate precipitation extremes, but local correlations are mostly significant only in low latitudes.

ABSTRACTExtreme precipitation and temperature have large socioeconomic and human health impacts. This study aims to analyse the projected changes of extreme precipitation and temperature indices at 1.5°C and 2°C of warming over the Mississippi River Basin (MRB) under Shared Socio‐economic pathways (SSP) 2‐4.5 and SSP5‐8.5. We used a technique named bias correction constructed analogues with quantiles mapping reordering (BCCAQ) to downscale daily precipitation, minimum and maximum temperature from a set of 12 Coupled Models Intercomparison Project phase 6 (CMIP6) over MRB. The changes in extreme precipitation and temperature indices such as very heavy rainfall (R95p), warm days (TX90p), and warm spell duration (WSDI) are sensitive to warming targets and emission scenarios. Results indicate that both warming targets are expected to exacerbate R95p whilst intensifying extreme precipitation and temperature as a whole except for cumulative wet days (CWD) (many parts of MRB are experiencing reduced CWD at both warming targets and scenarios). However, the rainfall intensity (SDII) is more reduced under SSP5‐8.5 compared to SSP2‐4.5 with an additional 0.5°C highlighting the sensitivity of SDII to the emission scenario. An additional 0.5°C (from 1.5°C to 2°C) climate warming is expected to: (1) increase TX90p and WSDI by 50% under SSP2‐4.5 and nearly 100% under SSP5‐8.5 over much of the MRB subregions, (2) reduce extreme precipitation in the centre of the MRB. Uncertainty superimposes on the magnitude of changes with more than 75% contribution from internal climate variability to total variance, nearly 20% from climate models, and marginal contribution from climate scenarios. The predominance of natural climate variability underscores a decreased predictability in future extreme precipitation and extreme temperature due to anthropogenic forcings, particularly at the regional scale. So, a deep understanding of what drives climate and its variability on a local and regional scale is critical for future generations of climate models and climate projections assessment. However, climate warming will pose serious challenges to water availability over the MRB, with consequences for agriculture, crop yields, and ecosystems.

Quantification of precipitation extremes is important for flood planning purposes, and a common measure of extreme events is the T year return level. Extreme precipitation depths in Belgium are analyzed for accumulation durations ranging from 10 min to 30 days. Spatial generalized extreme value (GEV) models are presented by considering multisite data and relating GEV parameters to geographical/climatological covariates through a common regression relationship. Methods of combining data from several sites are in common use, and in such cases, there is likely to be nonnegligible intersite dependence. However, parameter estimation in GEV models is generally done with the maximum likelihood estimation method (MLE) that assumes independence. Estimates of uncertainty are adjusted for spatial dependence using methodologies proposed earlier. Consistency of GEV distributions for various durations is obtained by fitting a smooth function to the preliminary estimations of the shape parameter. Model quality has been assessed by various statistical tests and indicates the relevance of our approach. In addition, a methodology is applied to account for the fact that measurements have been made in fixed intervals (usually 09:00 UTC–09:00 UTC). The distribution of the annual sliding 24 h maxima was specified through extremal indices of a more than 110 year time series of 24 h aggregated 10 min rainfall and daily rainfall. Finally, the selected models are used for producing maps of precipitation return levels.

Under global warming, the rising frequency and intensity of extreme climate events pose challenges to disaster prevention and sustainable development. Based on daily meteorological observations from 1970 to 2020 in Jilin Province, this study analyzes the spatiotemporal evolution and driving mechanisms of extreme temperature and precipitation events. Linear trend analysis and the Mann–Kendall test were employed to examine temporal trends and abrupt change years in extreme temperature and precipitation indices. Wavelet analysis was used to identify dominant periodicities and multi-scale variability. Empirical Orthogonal Function Analysis (EOF) revealed the spatial distribution characteristics of variability in extreme precipitation and temperature across Jilin Province, identifying high-incidence zones for extreme temperature and precipitation events. Additionally, Pearson correlation analysis was to investigate the correlation patterns between extreme climate indices in Jilin Province and geographical environmental factors alongside atmospheric circulation indicators. Results show that: (1) Warm-related temperature indices display significant upward trends, while cold-related indices generally decline, with abrupt changes mainly occurring in the 1980s–1990s and dominant periodicities of 3–5 years. Precipitation indices, though variable, show general increases with 3–4year cycles. (2) Spatially, most indices follow an east–high to west–low gradient. Temperature indices exhibit spatial coherence, while precipitation indices vary, especially between the northwest and central-southern regions. (3) The Arctic Oscillation (AO) exhibits a significant negative correlation with the extreme cold index, with correlation coefficients ranging from −0.31 to −0.46. It shows a positive correlation with the extreme warm index, with correlation coefficients between 0.16 and 0.18, confirming its regulatory role in cold air activity over Northeast China, particularly elevation and latitude, influence the spatial distribution of precipitation. These findings enhance understanding of extreme climate behaviors in Northeast China and inform regional risk management strategies.