Multi-site modelling and reconstruction of past extreme skew surges along the French Atlantic coast

This study proposes a novel multi-site modelling framework combining a multivariate generalized Pareto distribution and an extreme regression approach to reconstruct historical extreme skew surge time series along the French Atlantic coast by leveraging long-term data from key stations to predict events at locations with limited records.

The Gist

Imagine the French Atlantic coast as a long, winding bathtub. Sometimes, the water doesn't just rise with the normal tide; it gets pushed up violently by storms, creating a "skew surge." These surges are like sudden, dangerous waves that can flood the shore.

The problem scientists faced is that some parts of this "bathtub" have been watching the water for over 150 years (like the city of Brest), while other spots only started watching recently (like Port Tudy, which only started in 1999). If you want to know how dangerous the water was at Port Tudy in 1870, you have a gap in your data. You can't just guess; you need to reconstruct the history.

This paper is about building a statistical time machine to fill in those missing years. Here is how they did it, using simple analogies:

1. The Core Idea: The "Neighborhood Watch"

The researchers realized that storms don't hit just one city; they hit the whole region. If a massive storm hits Brest (a city with a long history of records), it almost certainly hit Port Tudy (a city with a short history) at the same time, even if Port Tudy didn't have a gauge to record it yet.

They used the long, reliable records from Brest and Saint-Nazaire as "teachers" to guess what happened at the "students" (Port Tudy, Concarneau, and Le Crouesty) during the years before the students had their own measuring tools.

2. The Two Methods: The "Ruler" vs. The "Crystal Ball"

To make these guesses, the team built two different types of mathematical engines. Think of them as two different ways to predict the future based on the past.

Method A: The "Crystal Ball" (ROXANE - Machine Learning)

• How it works: This method uses a computer algorithm (specifically, a type of machine learning) to look at the shape of the storm data. Imagine you are looking at a storm from a distance. You don't care exactly how high the water is in meters; you care about the angle or direction of the storm's energy.
• The Trick: The computer learns the relationship between the "angle" of the storm at Brest and the "angle" of the storm at Port Tudy. Once it learns this pattern, it can look at a storm at Brest from 1870, figure out the angle, and instantly guess the angle at Port Tudy.
• Best for: It is excellent at predicting the absolute worst events (the biggest, most dangerous surges). It gives you a single, very sharp number for what the water level likely was.

Method B: The "Crystal Ball with a Safety Net" (MGPRED - Parametric Model)

• How it works: This method uses strict mathematical rules (statistics) to build a complete map of how the water behaves. Instead of just guessing one number, it builds a "cloud" of possibilities.
• The Trick: It says, "Based on the storm at Brest, the water at Port Tudy could be anywhere between 2 meters and 4 meters." It doesn't just give you a guess; it gives you a confidence interval (a safety net).
• Best for: It is better at understanding the whole picture, including smaller surges, and it tells you how confident it is in its guess. It's like saying, "I think it rained 2 inches, but it could have been anywhere from 1.5 to 2.5 inches."

3. The "Threshold" Problem: When is a Wave a "Surge"?

A major challenge was deciding what counts as an "extreme" event. Is a 1-meter wave extreme? What about 1.5 meters?

• The Innovation: The authors invented a new, automatic way to draw the line. They used a special mathematical curve (called an EGP distribution) to find the exact point where the data starts behaving like a "wild" storm rather than a normal day. It's like a smart sensor that automatically decides, "Okay, anything above this specific height is a storm we need to study."

4. The Results: Filling in the Gaps

The team tested their methods on data they already had (the years where Port Tudy did have a gauge) to see if they could correctly "predict" the past.

• The Verdict: Both methods worked well.
  • The Machine Learning (ROXANE) method was slightly better at predicting the very largest surges (the ones that cause the most damage).
  • The Statistical (MGPRED) method was better at predicting the smaller surges and gave them a range of uncertainty, which is crucial for risk management.
• The Time Travel: They successfully used these models to reconstruct the history of Port Tudy all the way back to 1846. They found that the biggest storm they predicted happened on New Year's Eve, 1876/1877. This matched historical records of a massive storm that caused flooding in Brittany, proving their "time machine" was accurate.

Summary

In short, this paper teaches us how to use the long history of one city to "fill in the blanks" for its neighbors. By using two different mathematical tools—one focused on the sharpest peaks and the other on the full range of possibilities—they created a reliable history of extreme water levels. This helps coastal managers understand how often dangerous floods might happen, even in places where we don't have old records.

Technical Summary

Technical Summary: Multi-site modelling and reconstruction of past extreme skew surges along the French Atlantic coast

Problem Statement
The paper addresses the critical need to model and reconstruct historical extreme sea level events, specifically skew surges, along the French Atlantic coast. Skew surges, defined as the difference between observed maximum sea levels and astronomically predicted tides, are primary drivers of coastal flooding. A significant challenge arises at tide gauge stations with short observational records (e.g., Port Tudy, Concarneau, Le Crouesty), which lack the historical depth required to accurately estimate long-range return periods. The objective is to reconstruct missing extreme skew surge time series at these short-record stations by leveraging spatial statistical dependence with nearby stations possessing extensive historical data (e.g., Brest and Saint-Nazaire, with records spanning over 150 years).

Methodology
The study employs a Peaks-Over-Threshold (POT) framework within a multivariate extreme value theory (EVT) context. Unlike block-maxima approaches, the authors define an extreme event as occurring whenever at least one input station records a value exceeding a high threshold, acknowledging that a single large surge at a neighboring station can cause flooding at the target site regardless of local conditions.

The methodology is divided into three main components:

1. Marginal Modelling and Threshold Selection:

   • The authors model marginal distributions using the Extended Generalized Pareto (EGP) distribution, which allows for modeling the entire positive part of the distribution while asymptotically converging to a Generalized Pareto (GP) distribution in the tail.
   • A novel, automated method is proposed for selecting the threshold $t$. Instead of relying on visual stability plots, the threshold is defined as the lowest value above which the fitted EGP density becomes strictly convex. This leverages the property that GP densities are strictly convex for shape parameters $\xi > -1/2$.

2. Multivariate Prediction Approaches:
   Two complementary approaches are developed and compared to predict the target variable $Y$ (skew surge at the short-record station) given covariates $X$ (skew surges at long-record stations):

   • Nonparametric Machine Learning (ROXANE): This approach utilizes the ROXANE algorithm (Clémençon et al., 2025). It transforms data to unit Pareto margins and focuses on predicting the "angular" component ($\Theta_Y$) of the target based on the angular component of the covariates ($\Theta_X$), exploiting the asymptotic independence between radial and angular components in the tail. The algorithm uses off-the-shelf regression models (tested with Ordinary Least Squares and Random Forest) trained on these angular variables.
   • Parametric Modelling (MGPRED): This approach transforms data to unit Exponential margins and fits a Multivariate Generalized Pareto (MGP) distribution to the shifted exceedances. It employs a censored likelihood approach to fit parametric models (specifically, independent Gumbel components with a dependence parameter) to the extreme data. Prediction is performed via Monte Carlo sampling from the conditional distribution, allowing for the generation of prediction intervals.

Key Contributions

• Novel Threshold Selection: The paper introduces an automated, convexity-based criterion for selecting univariate thresholds in the EGP framework, reducing reliance on subjective visual inspection.
• Comparative Framework: It provides a rare direct comparison between a model-agnostic machine learning approach (ROXANE) and a fully parametric multivariate EVT approach (MGPRED) for the specific task of reconstructing missing extreme components.
• Theoretical Connection: The authors elucidate the theoretical link between the two methods, demonstrating that despite their methodological differences (angular regression vs. conditional density modeling), they rely on essentially equivalent statistical assumptions regarding the distributional tail.
• Reconstruction of Historical Data: The methods are successfully applied to reconstruct the extreme skew surge time series at Port Tudy prior to 1966, utilizing data from Brest and Saint-Nazaire.

Results

• Marginal Fit: The EGP distribution provided a robust fit for the marginal data, and the proposed threshold selection method effectively identified the region where the GP approximation holds (verified by comparing EGP and GP fits above the threshold).
• Predictive Performance:
  • MGPRED demonstrated superior overall performance in terms of Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) across the entire test set. It provided better modeling of smaller extreme values and offered the advantage of generating prediction intervals (0.95 coverage probability was 0.89).
  • ROXANE (specifically with OLS) performed slightly better in the most extreme regions (the upper tail), yielding more accurate point estimates for the largest skew surges.
  • Both methods struggled with the lower tail of the extreme distribution (smaller extreme values), a limitation attributed to the sensitivity of the back-transformation to errors in the shape parameter $\kappa$ and the use of censored likelihoods that prioritize the most extreme events.
• Historical Reconstruction: The reconstructed time series for Port Tudy (pre-1966) successfully captured major historical events, such as the significant skew surge recorded on the night of December 31, 1876, to January 1, 1877, which corresponded to major storms in Brest and Saint-Nazaire.

Significance and Claims
The paper claims that both the nonparametric and parametric approaches yield valid and practically significant results for reconstructing extreme sea levels.

• Practical Utility: The MGPRED procedure is recommended for scenarios requiring overall performance and uncertainty quantification (prediction intervals), while the ROXANE procedure is preferred when the primary goal is the most accurate point prediction of the absolute largest events.
• Methodological Advancement: The work promotes the use of multivariate extreme value theory in geoscience, specifically highlighting the utility of the EGP distribution for marginal modeling and the ROXANE algorithm for extreme regression.
• Limitations and Future Work: The authors modestly note that the models are less precise for older years, potentially due to unmodeled temporal trends (e.g., global warming) that may differ across stations. They also acknowledge that the current models do not incorporate meteorological covariates (like wind), which could improve the modeling of smaller extreme events. The construction of statistically sound prediction intervals for extreme covariates is identified as a topic for future research.

The study concludes that these methods are broadly applicable to other spatial extreme problems and can be extended to include additional covariates or applied to missing data imputation for other variables like wind or pressure.