Return level estimation
Annual maxima or peaks over a threshold are fitted with generalised extreme value or generalised Pareto distributions, and return levels are reported with confidence intervals and record length.
GCRI · Climate and Hazards
Extreme value statistics for hazards
Extreme value statistics estimate how large and how frequent rare events can be from limited records. In this field they give return levels for rainfall, floods, heat, wind and surge, and increasingly allow those levels to change with a warming climate.
01Contents
Extreme value statistics for hazards in five parts.
02Application
What extreme value theory and tail risk does here.
Extreme value theory and tail risk study the behaviour of rare, high-impact events. They provide methods to estimate the magnitude and frequency of extremes beyond the range of observed data. In climate and hazards and disaster and crisis, extreme value methods set design levels and capital buffers for rare events. Tail dependence analysis reveals when extremes in different systems tend to occur together. The canonical description is on the STEM Commons.
Non-stationary extremes
Distribution parameters are allowed to vary with a covariate such as global mean temperature, so that return levels can be expressed for present and future climates.
Regional frequency analysis
Records from similar sites are pooled to estimate extremes where single records are short, with homogeneity of the region tested.
03Geography
Climate signals worldwide.
Daytime surface heat, rain and flood water across the world, from NASA GIBS. Each layer carries its own date.
The interactive map needs JavaScript; the layer list and sources are below.
Layers 3
Climate signals, for this page
About this layer
- Evidence
- Observed (persistent cloud still leaves gaps)
- Source
- VIIRS NOAA-20 Level 3, 8-day (NASA LANCE)
- Provider
- NASA LANCE
- Licence
- NASA open data
- Vintage
- 8-day; Satellite 8-day composite; persistent cloud still leaves gaps
- Newest at build
- 2026-09-14
- Resolution
- about 1 km grid
- Uncertainty
- Satellite skin temperature of the ground over the 8 days ending on the date shown, a heat exposure indicator; not air temperature, heat index or a heat warning.
- Reached through
- gibs.earthdata.nasa.gov, only after you allow the live map
About this layer
- Evidence
- Estimated
- Source
- GPM IMERG (NASA GES DISC)
- Provider
- NASA GES DISC
- Licence
- NASA open data
- Vintage
- daily; Satellite estimate, near real time
- Newest at build
- 2026-10-12
- Resolution
- about 2 km grid
- Uncertainty
- Satellite estimate, near real time. Satellite or model raster as the provider publishes it. Clouds, swath gaps and retrieval error leave gaps and noise; values are indicators, not measurements at a place.
- Reached through
- gibs.earthdata.nasa.gov, only after you allow the live map
About this layer
- Evidence
- Observed (clouds leave gaps)
- Source
- VIIRS Global Flood Product (NASA LANCE)
- Provider
- NASA LANCE
- Licence
- NASA open data
- Vintage
- daily; Satellite detection; clouds leave gaps
- Newest at build
- 2026-10-14
- Resolution
- about 250 m grid
- Uncertainty
- Satellite detection; clouds leave gaps. Satellite or model raster as the provider publishes it. Clouds, swath gaps and retrieval error leave gaps and noise; values are indicators, not measurements at a place.
- Reached through
- gibs.earthdata.nasa.gov, only after you allow the live map
About this layer
- Evidence
- Observed
- Source
- GHRSST MUR25 (NASA JPL PO.DAAC)
- Provider
- NASA JPL PO.DAAC
- Licence
- NASA open data
- Vintage
- daily; Satellite analysis
- Newest at build
- 2026-10-12
- Resolution
- about 2 km grid
- Uncertainty
- Satellite analysis of how far sea temperature departs from its usual value for the date shown; a marine heat indicator, not a coral bleaching alert.
- Reached through
- gibs.earthdata.nasa.gov, only after you allow the live map
About this layer
- Evidence
- Observed
- Source
- VIIRS NOAA-20 corrected reflectance (NASA LANCE)
- Provider
- NASA LANCE
- Licence
- NASA open data
- Vintage
- daily; Satellite image
- Newest at build
- 2026-10-14
- Resolution
- about 250 m grid
- Uncertainty
- Satellite image. Satellite or model raster as the provider publishes it. Clouds, swath gaps and retrieval error leave gaps and noise; values are indicators, not measurements at a place.
- Reached through
- gibs.earthdata.nasa.gov, only after you allow the live map
About this layer
- Evidence
- Modelled (latest year GIBS publishes)
- Source
- MODIS Terra and Aqua land cover (MCD12Q1) (NASA LP DAAC)
- Provider
- NASA LP DAAC
- Licence
- NASA open data
- Vintage
- yearly; Satellite classification; latest year GIBS publishes
- Newest at build
- 2024-01-01
- Resolution
- about 500 m grid
- Uncertainty
- Satellite land cover classes for the year shown, including croplands; a classification with known errors, not a farm register.
- Reached through
- gibs.earthdata.nasa.gov, only after you allow the live map
Keyboard, with the map focused
Hold Ctrl or ⌘ and scroll to zoom
Latest available
NASA GIBS / EOSDIS · ASTER GDEM (METI, NASA) · Blue Marble (NASA) · © OpenStreetMap contributors (ODbL) · Natural Earth
Loading the latest NASA imagery.
Drag to move. Ctrl or ⌘ and scroll, pinch or the buttons to zoom; hover for the values under the pointer; reset returns to the region.
Climate signals, framed on the world.Satellite and model products as NASA publishes them, each with its own date. They are indicators, not assessments of any place.
Data: NASA GIBS · ASTER GDEM · Blue Marble · © OpenStreetMap contributors · Natural Earth · NASA EONET · NASA JPL OPERA DIST-ALERT · NASA AIRSSources and licences
- We acknowledge the use of imagery provided by services from NASA’s Global Imagery Browse Services (GIBS), part of NASA’s Earth Science Data and Information System (ESDIS). Colours redrawn from the published colour map; values averaged into hexagonal cells where shown (modified). NASA open data
- We acknowledge the use of imagery provided by services from NASA’s Global Imagery Browse Services (GIBS), part of NASA’s Earth Science Data and Information System (ESDIS). Colours redrawn from the published colour map; values averaged into hexagonal cells where shown (modified). NASA open data
- VIIRS near-real-time products (flood, land surface temperature, snow cover, corrected reflectance), NASA LANCE; imagery via NASA GIBS. Colours redrawn from the published colour map; values averaged into hexagonal cells where shown (modified). NASA open data
- GPM IMERG precipitation, NASA Precipitation Processing System; distributed by the NASA Goddard Earth Sciences Data and Information Services Center (GES DISC); imagery via NASA GIBS. Colours redrawn from the published colour map; values averaged into hexagonal cells where shown (modified). NASA open data
- GHRSST Level 4 MUR analyses, NASA Jet Propulsion Laboratory; distributed by the NASA Physical Oceanography DAAC (PO.DAAC); imagery via NASA GIBS. Colours redrawn from the published colour map; values averaged into hexagonal cells where shown (modified). NASA open data
- MODIS land products (vegetation indices, gross primary production, land cover), NASA LP DAAC and LANCE; imagery via NASA GIBS. Colours redrawn from the published colour map; values averaged into hexagonal cells where shown (modified). NASA open data
- ASTER GDEM is a product of METI and NASA (ASTER Global Digital Elevation Model V003, NASA LP DAAC); imagery via NASA GIBS. Colours redrawn from the published colour map; values averaged into hexagonal cells where shown (modified). Cite as: NASA/METI/AIST/Japan Spacesystems and U.S./Japan ASTER Science Team (2019). ASTER Global Digital Elevation Model V003. doi:10.5067/ASTER/ASTGTM.003. NASA open data
- Made with Natural Earth. Free vector and raster map data @ naturalearthdata.com. Public domain
- © OpenStreetMap contributors (ODbL), via NASA GIBS (coastlines, boundaries, roads and place names). Colours redrawn from the published colour map; values averaged into hexagonal cells where shown (modified). ODbL
- Blue Marble, NASA Earth Observatory; imagery via NASA GIBS. Colours redrawn from the published colour map; values averaged into hexagonal cells where shown (modified). NASA open data
- Context on the globe and flat map: coastlines, boundaries, roads and place names from NASA GIBS reference layers (OpenStreetMap data, ODbL, © OpenStreetMap contributors); Blue Marble imagery as an optional base; 3D terrain and the elevation read-out from the ASTER GDEM elevation colour index (5 m to 200 m classes, exaggerated for legibility). Boundaries are drawn as published and imply no position on any territory.
- Live events: NASA Earth Observatory Natural Event Tracker (EONET); events are indicative and keep their original source links.
- Live: NASA EONET Events are indicative, not authority warnings.
- Observed (EO): NASA JPL OPERA DIST-ALERT (contains modified Copernicus Sentinel data 2026). We acknowledge the use of imagery provided by services from NASA’s Global Imagery Browse Services (GIBS), part of NASA’s Earth Science Data and Information System (ESDIS).
- Observed (EO): NASA AIRS. We acknowledge the use of imagery provided by services from NASA’s Global Imagery Browse Services (GIBS), part of NASA’s Earth Science Data and Information System (ESDIS).
- NASA GIBS: We acknowledge the use of imagery provided by services from NASA’s Global Imagery Browse Services (GIBS), part of NASA’s Earth Science Data and Information System (ESDIS).
- Air quality forecast, 5 days, PM2.5 (CAMS via Open-Meteo): Contains modified Copernicus Atmosphere Monitoring Service information 2026.
- Past year, daily temperature (ERA5 via Open-Meteo): Contains modified Copernicus Climate Change Service information 2026.
- Past year, daily temperature and sunshine (NASA POWER): These data were obtained from the NASA Langley Research Center (LaRC) POWER Project funded through the NASA Earth Science/Applied Science Program..
- Weather data by Open-Meteo.com (https://open-meteo.com/), licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
- Optional layers (22 sources in the layer list: live, forecast, observed and context) print their provider credit on the map and here when switched on; forecasts are model output and derived indices are estimates.
Layers and dates
- Land surface temperature, day, 8-day composite: VIIRS NOAA-20 Level 3, 8-day, NASA LANCE · 14 Sep 2026
- Precipitation rate, daily mean: GPM IMERG, NASA GES DISC · 12 Oct 2026
- Flood water, 3-day composite: VIIRS Global Flood Product, NASA LANCE · 14 Oct 2026
- Sea surface temperature anomaly: GHRSST MUR25, NASA JPL PO.DAAC · 12 Oct 2026
- True colour: VIIRS NOAA-20 corrected reflectance, NASA LANCE · 13 Oct 2026
- Land cover type (IGBP), yearly: MODIS Terra and Aqua land cover (MCD12Q1), NASA LP DAAC · 2024
04Data
What the method consumes.
Datasets and records used by extreme value theory and tail risk on this platform, with the role each plays and the label its values carry. Sources are listed in the source register.
| Data | Role | Label |
|---|---|---|
| Station and gauge annual maxima | Fitting inputs | OBSERVED |
| Reanalysis extremes | Spatial coverage | MODELLED |
| Model large ensembles | Rare-event sampling | MODELLED |
| Historical and palaeo flood records | Extending records | ESTIMATED |
05Monitor
Tail fit.
The screen a team would use to follow this method: its measures, states and decisions. Names are examples and values are illustrative.
Tail fit
Climate and hazards · Extreme value statistics · Run R-4223
Shape parameter uncertainGCRI · Climate and Hazards
Return level curve
Method card
- Method
- Extreme value theory and tail risk
- Approach
- peaks over threshold
- Threshold
- 93th percentile
- Label
- MODELLED
Parameters
generalised Pareto fit
| Shape ξ | 0.18 | −0.07 | 0.59 | heavy tail if above zero |
| Scale σ | 2.6 | 0.9 | 6.1 | units of the variable |
| Exceedances | 144 | declustered |
06Validation
How results are checked.
Fits are checked with diagnostic plots, goodness-of-fit tests and out-of-sample comparison. Sensitivity to threshold and record length is reported.
| Check | How it is done |
|---|---|
| Goodness of fit in the tail | Fits are checked with quantile plots focused on the largest values |
| Confidence intervals | Intervals are reported with every return level, using profile likelihood or bootstrap |
| Stationarity tests | Trends are tested before assuming a stationary distribution |
Extrapolation far beyond the data is highly uncertain. Changing climates and markets violate stationarity assumptions.
07Twin and cascade simulator
Model extreme value statistics for hazards as a system. Run the cascade.
Assembled from the Nexus Standards area profile Climate risk standards: the hazards it names, the systems and sectors that carry the loss, the controls that dampen it and the data that observe it. Choose the components, pick an initiating hazard and step through the cascade. The Map view places the run on an example geography with open data. Every coupling weight is a stated assumption you can change. A teaching model, not a forecast, an assessment or a certification. To model a place of your own choosing, open the full System Builder.
Hazards
- Sea-level rise
- Coastal flood (set off downstream)
- Storm surge (set off downstream)
- High-tide flooding (set off downstream)
- Coastal erosion (set off downstream)
- Saltwater intrusion (set off downstream)
- Estuary flood (set off downstream)
- Coastal habitat squeeze (set off downstream)
- Wetland loss (set off downstream)
- Mangrove loss (set off downstream)
- Groundwater flood (set off downstream)
- Climate-driven migration and relocation (set off downstream)
- Mass displacement (set off downstream)
Systems and couplings
Controls
NS-0Constitution and principlesNS-1The UNI and the artefact catalogue (grammar v0.2: domain, sector, region and environment segments)NS-3Risk record UNIs (grammar, legacy crosswalk, allocation authority)NS-3bDataset and product metadata profile (STAC, DCAT, PROV-O)NS-9Contextual UNIs (deployments, instances, programme instances)NS-11Verification and monitoring (evidence, attestations, zero-trust conformance observation)NS-14Conformance and crosswalk to external standardsNRS-H65Hazard family: Planetary boundary transgressions (NHS-1 planet.boundary)NRS-H62Hazard family: Compound and multi-hazard events (NHS-1 sys.compound)NRS-H71Hazard family: Deep-time and cosmic catastrophes (NHS-1 planet.deep-time)NRS-H79Hazard family: Mental health crises (NHS-1 health.mental)NRS-H75Hazard family: Climate-sensitive illness (NHS-1 health.climate-sensitive)NRS-H7Hazard family: Large-scale circulation anomalies (NHS-1 atmo.climate-mode)
Observation
- ERA5 reanalysis (Copernicus Climate Data Store) (listed for reference)
- CMIP6 climate projections (Copernicus Climate Data Store) (listed for reference)
- NGFS Scenario Explorer (listed for reference)
- INFORM Risk Index (listed for reference)
- Number of deaths, missing persons and directly affected persons attributed to disasters per 100,000 population
- Total greenhouse gas emissions per year
- Direct economic loss attributed to disasters in relation to global GDP
Example geography
- Pacific Island states: 150 hexagonal cells
- Night lights tonight, outages (VIIRS Day/Night Band), NASA GIBS
- Surface water through cloud, 30 m (OPERA DSWx-S1 radar), NASA GIBS
- Rain rate, last half hour (IMERG), NASA GIBS
- Vegetation disturbance alerts, 30 m (OPERA DIST-ALERT), NASA GIBS
08Connections
Topics that rely on this method.
09Methods and evidence
How extreme value statistics for hazards is modelled, measured and tested.
The model, its assumptions, its data and its uncertainty, in the form a scientist or engineer would check. Structures follow published methods. The values drawn are illustrative, not findings.
Uncertainty bands
How far the outlook for extreme value statistics for hazards could spread, shown as ranges rather than a single line.
- Central estimate
- median of the ensemble
- Ranges
- likely and very likely, in IPCC calibrated language
- Ensemble
- members, scenarios and initial conditions stated
- Read with
- the sensitivity of the result to each input
FAIR guiding principles · GO FAIR and FORCE11W3C PROV · World Wide Web ConsortiumGuidance on consistent treatment of uncertainties · IPCCCEOS Analysis Ready Data · Committee on Earth Observation Satellites
10Stewardship
Who maintains the method.
Reinsurance and Catastrophe stewards extreme value theory and tail risk for the Consortium and reviews how it is applied across platforms.
11Process
How the process runs.
Process
Extreme value statistics for hazards: Catastrophe model pipeline
Catastrophe models usually have hazard, vulnerability, exposure and financial modules, and produce loss tables, average annual loss, exceedance curves, PML and TVaR.
01 Hazard
- 01Event catalogueDataset · Model developerThe hazard module holds a large catalogue of simulated events that represents a wide range of plausible scenarios.
- 03Hazard moduleModel · Model developerFor each event the hazard module gives the local hazard intensity that the vulnerability module reads.
02 Exposure and damage
- 02ExposureDataset · National meteorological serviceThe exposure module houses the portfolio data, such as location-specific information.
- 04Vulnerability moduleModel · Model developerThe vulnerability module quantifies the expected damage to the exposures from an event.
03 Financial
- 05Financial moduleModel · National meteorological serviceThe financial module calculates losses from all event scenarios. Ground-up, gross and net losses differ as shown below.
- 06Event loss tableDataset · National meteorological serviceAn event loss table lists each event with its annual occurrence probability and its loss for the portfolio.
- 07Year loss tableDataset · National meteorological serviceSome models also sample simulated years from event losses. This outline does not describe their format.
04 Metrics
- 08Exceedance probability curvesModel · National meteorological serviceThe occurrence curve gives the chance that the largest loss in a year exceeds an amount. The aggregate curve does so for the sum of losses.
- 09AAL, PML and TVaRModel · National meteorological serviceAverage annual loss sums event probability times loss. PML is one point on the occurrence curve.
05 Use of results
- 10Use of resultsDecision · National meteorological serviceThe tail metrics are used when considering a portfolio’s exposure and the level of catastrophe reinsurance an insurer requires.
Outline of an external standard or method, from its public sources. It describes what the framework says, not a record of any run. It does not show that Nexus or any partner performs, certifies or approves a step. A gate is a step of the method, never a Nexus approval.
12Connected
Where this page leads across the Consortium.
13Questions
Questions and answers.
What are the limits of extreme value theory and tail risk in this field?
Extrapolation far beyond the data is highly uncertain. Changing climates and markets violate stationarity assumptions.
How are results checked?
Fits are checked with diagnostic plots, goodness-of-fit tests and out-of-sample comparison. Sensitivity to threshold and record length is reported.
Are the values on this page real?
The data names, roles and checks are real descriptions of practice. The canvas shows a design of the method’s use, not a run; no result is reported here.
Where else is this method applied?
The canonical method page lists every platform that applies it.
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