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Analysing Extreme Rainfall via a Geometric Framework

Motivated by the EVA 2025 Data Challenge, this paper proposes a non-stationary geometric framework for extreme-value analysis to predict the spatial extent and temporal duration of extreme rainfall in the eastern United States by leveraging large climate model ensembles and accounting for topographical and seasonal effects.

Original authors: Ryan Campbell, Kristina Grolmusova, Lydia Kakampakou, Jeongjin Lee

Published 2026-03-20

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CC BY 4.0

Original authors: Ryan Campbell, Kristina Grolmusova, Lydia Kakampakou, Jeongjin Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a city planner trying to build a floodwall. You need to know: How big could the worst possible flood get? How wide would it spread? And how long would it last?

The problem is, we've never seen a flood that big before. If you only look at the history books, you'll never predict a "once-in-a-million-year" storm. You have to guess what lies beyond the edge of our known data.

This paper is about a team of statisticians who built a new, smarter "crystal ball" to make that guess for extreme rainfall in the Eastern United States. Here is how they did it, explained without the heavy math jargon.

1. The Problem: The "Unseen" Storm

The team was given data from a super-computer climate model. Think of this model as a giant video game that simulates weather. It ran 50 different "what-if" scenarios (called ensemble runs).

However, these simulations don't perfectly match real history. They are like rough sketches of the weather. The challenge was to use these sketches to predict the worst-case scenario for three specific things:

The "Super-Flood": How often will rain be so heavy that every single one of 25 different towns gets flooded at once?
The "Widespread Deluge": How often will rain be heavy enough to flood at least 6 towns simultaneously?
The "Long Haul": How often will at least 3 towns get flooded for two days in a row?

2. The Old Way vs. The New Way

The Old Way (The Rigid Ruler):
Traditional statistics often assume the world is "neat and tidy." They assume that if it rains hard in New York, it's equally likely to rain hard in Boston, and that the weather patterns don't change from summer to winter.

The Flaw: Nature is messy. Rainfall depends on mountains, seasons, and wind. A "neat" model breaks when the weather gets weird.

The New Way (The Geometric Framework):
The team used a method called Geometric Extremes. Imagine the rainfall data not as a list of numbers, but as a cloud of points floating in 3D space.

The Shape of the Cloud: Most rain falls in the middle of the cloud (normal days). The "extreme" rain is at the very edge of the cloud.
The Magic Trick: Instead of trying to guess the weather, they studied the shape of the cloud's edge. If you know the shape of the edge of a balloon, you can predict how big the balloon will get if you blow more air into it, even if you've never blown it that big before.

3. The Three Big Hurdles (and How They Cleared Them)

To make their "shape predictor" work, they had to fix three major problems:

A. The "Seasonal Shift" (Time)

Rain in July looks different from rain in January. The "average" rain changes throughout the year.

The Fix: They acted like a photo editor. Before analyzing the shape, they "normalized" the data. They stripped away the seasonal trends (like removing the winter coat and summer shirt) so that the underlying "shape" of the storm remained consistent, regardless of the date.

B. The "Bumpy Map" (Space)

The Eastern US isn't flat. Mountains and valleys change how rain behaves. A storm might hit a mountain and dump water on one side but not the other.

The Fix: They used Spatial Deformation. Imagine the map of the US is made of stretchy rubber. They physically stretched and warped the rubber map until the "bumps" (mountains) smoothed out. On this new, warped map, the weather patterns looked "stationary" (consistent), making it much easier to calculate the shape of the extreme edge.

C. The "Time Travel" (Duration)

The challenge asked about storms lasting two days. Standard models often look at one day at a time, forgetting that a storm doesn't just vanish overnight.

The Fix: They used a Block Sampling technique. Instead of picking random days, they grabbed "blocks" of 4 days at a time. This kept the "memory" of the storm intact, allowing them to simulate how long a heavy rain event might actually last.

4. The Result: Predicting the Impossible

Once they had their "warped, normalized, 4-day block" data, they applied their geometric model.

They looked at the "edge" of the data cloud.
They extended that edge outward to see what would happen if the rain got 10x or 100x heavier than anything recorded.
They ran millions of simulations to count how often these "Super-Floods" would happen.

The Outcome:
Their model successfully predicted the frequency of these extreme events. Interestingly, they found that their first attempt (which didn't account for all the "bumps" and "seasons") was too pessimistic (it thought floods would happen way too often). After applying their geometric fixes, the predictions became much more realistic and aligned with the broader data.

The Big Picture Analogy

Imagine you are trying to predict how high a tree will grow if you give it unlimited water.

Old Method: You measure the tree every day and draw a straight line. If the tree hits a rock (a mountain) or the sun moves (seasons), your line is wrong.
This Paper's Method: You study the shape of the tree's branches. You realize that even if the tree grows in a weird direction because of the wind, the pattern of how the branches spread out follows a specific geometric rule. By understanding that rule, you can accurately predict how tall the tree will get, even if it grows in a direction you've never seen before.

In short: They turned a messy, unpredictable weather problem into a clean geometric shape problem, allowing them to safely "extrapolate" (guess) the future of extreme rainfall with much higher confidence.

1. Problem Statement

The study addresses the challenge of predicting extreme rainfall events in the eastern United States, specifically focusing on the spatial extent and temporal duration of these events. The data source is the CESM2 Large Ensemble Community Project (LENS2), a climate model simulation covering a $5 \times 5$ spatial grid centered near Asheville, NC.

The core challenge involves three specific quantities of interest (CTQs) that require extrapolation beyond observed historical levels:

CTQ1: The expected number of times all 25 grid locations simultaneously exceed a high threshold (1.7 Leadbetters).
CTQ2: The expected number of times at least 6 of the 25 locations exceed a high threshold (5.7 Leadbetters).
CTQ3: The expected number of times at least 3 locations exceed a threshold (5 Leadbetters) for a duration of at least two consecutive days.

The data exhibits complex non-stationarity (both spatial and temporal) and extremal dependence structures that vary across the domain, making classical stationary models insufficient.

2. Methodology

The authors employ a geometric framework for extreme-value analysis, which offers flexibility in capturing complex joint tail behaviors and enables extrapolation in any direction of the multivariate tail. The methodology proceeds in four main stages:

A. Handling Non-Stationarity

Temporal Non-Stationarity: The authors adapt the approach of Eastoe and Tawn (2009). They model the marginal distribution of precipitation $Y_t$ $Y_{t}$ using a location-scale model: $Y_t = \mu(x_t) + \sigma(x_t)Z_t$ $Y_{t} = μ (x_{t}) + σ (x_{t}) Z_{t}$ .
- $\mu(x_t)$ and $\sigma(x_t)$ are modeled using Generalized Additive Models (GAMs) with linear/cyclic trends and autoregressive (AR) terms to account for daily cycles, long-term trends, and lagged precipitation.
- The standardized residuals $Z_t$ are then modeled using a non-stationary Generalized Pareto Distribution (GPD) for exceedances above a high threshold.
Spatial Non-Stationarity: To address anisotropy and spatial non-stationarity (where dependence structures vary by location), the authors use a spatial deformation approach (Richards and Wadsworth, 2021).
- A thin-plate spline maps the original geographical coordinates (G-plane) to a latent "D-plane" where stationarity is assumed.
- The deformation parameters are estimated by minimizing the discrepancy between theoretical and empirical pairwise extremal dependence coefficients.

B. Marginal Transformation

After preprocessing, the data is transformed to have standard exponential margins. This is achieved via a semi-parametric transformation: empirical distribution functions below the threshold and the fitted non-stationary GPD above the threshold, followed by an inverse probability integral transform.

C. The Geometric Framework

The core of the model relies on the convergence of scaled sample clouds to a limit set $G$ .

Radial-Angular Decomposition: The data vector $Z$ is transformed into a radial component $R = \sum Z(s_i)$ and an angular component $W = Z/R$ .
Gauge Function: The limit set is characterized by a gauge function $g_Z(w)$ . The authors utilize a generalized Gaussian gauge function:
$g_Z(w; \gamma) = \left[ (w^{1/\gamma})^\top \Sigma^{-1} w^{1/\gamma} \right]^{\gamma/2}$
where $\Sigma$ is a correlation matrix with powered exponential components (controlled by range $\phi$ and shape $\kappa$ ).
Truncated Gamma Model: Conditional on the angular component $W=w$ and exceeding a high threshold $r_\tau(w)$ , the radial component $R$ is modeled using a Truncated Gamma (TG) distribution. This allows for flexible modeling of the tail decay.

D. Extrapolation and Sampling

To estimate probabilities for events far beyond observed data:

The authors simulate extreme points by drawing angular components from the empirical distribution (re-weighted for higher thresholds) and radial components from the fitted Truncated Gamma distribution.
Temporal Duration (CTQ3): To estimate the duration of events, the authors employ a block sampling procedure. They sample temporal blocks of angular components (block size = 4 days) based on spatio-temporal tail dependence analysis, ensuring the temporal persistence of extreme events is captured.

3. Key Contributions

Application of Geometric Extremes: This work demonstrates the application of the recently developed geometric framework (Wadsworth and Campbell, 2024; Kakampakou and Wadsworth, 2025) to a high-dimensional ( $d=25$ ) spatial environmental dataset, a setting where such methods are rarely tested.
Integrated Non-Stationary Modeling: The paper successfully integrates temporal non-stationarity (via GAMs/GPD) and spatial non-stationarity (via thin-plate spline deformation) into a unified geometric extreme value framework.
Temporal Duration Estimation: A novel sampling procedure is proposed to retain temporal information, allowing for the estimation of the duration of extreme rainfall events (CTQ3), which is critical for hydrological risk assessment.
Methodological Refinement: The authors explicitly address discrepancies in their initial competition submission by refining the selection of the extrapolation parameter $k$ , adjusting the temporal block size, and improving the handling of non-stationarity, leading to more robust estimates.

4. Results

Model Fit: Diagnostic plots (PP and QQ plots) indicate a satisfactory fit for the Truncated Gamma model across the four climate model runs. The model-based tail dependence coefficients align well with empirical estimates in the deformed D-plane.
Target Quantities (CTQs):
- CTQ1 & CTQ2: The model provides reasonable estimates for the frequency of widespread extreme rainfall. For runs 2, 3, and 4, the estimates closely match empirical values derived from 50 model runs. Run 1 showed slight overestimation, attributed to specific tail behavior in that realization.
- CTQ3: Estimates for the duration of events showed slightly more bias compared to empirical values, suggesting that while the block sampling approach captures the general trend, a fully spatio-temporal extension of the gauge function could further improve accuracy.
Comparison to Submission: The refined methodology in this paper corrected an overestimation in CTQ1 found in the initial competition submission, demonstrating the importance of rigorous non-stationarity preprocessing and careful selection of extrapolation parameters.

5. Significance

Risk Mitigation: The ability to accurately predict the spatial extent and temporal duration of extreme rainfall is vital for designing defense systems and mitigating disaster risks (e.g., flooding similar to Hurricane Helene).
Methodological Advancement: The study validates the geometric framework as a powerful tool for high-dimensional spatial extremes, particularly in scenarios involving asymptotic independence (AI), which is common in environmental data.
Future Directions: The paper highlights the potential for extending this framework to fully spatio-temporal models using separable or non-separable correlation matrices and more flexible mixtures of gauge functions to better capture complex physical mechanisms driving extreme weather.

In conclusion, the paper presents a robust, flexible statistical framework that successfully bridges the gap between complex climate model data and the practical need for extrapolating extreme weather risks in both space and time.