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Fusing Sparse Observations and Dense Simulations for Spatial Extreme Value Analysis: Application to U.S. Coastal Sea Levels

This paper introduces a two-stage frequentist framework implemented in the R package `evfuse` that fuses sparse observational data with dense physics-based simulations via a linear model of coregionalization to significantly improve the accuracy of spatial extreme value analysis, demonstrated by a 35% reduction in return level estimation error for U.S. coastal sea levels.

Original authors: Brian N. White, Brian Blanton, Rick Luettich, Richard L. Smith

Published 2026-03-04

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CC BY 4.0

Original authors: Brian N. White, Brian Blanton, Rick Luettich, Richard L. Smith

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 trying to draw a map of the highest water levels a storm could ever reach along the entire U.S. coastline. This is crucial for building sea walls, planning evacuations, and protecting homes.

The problem is that we have two very different types of information, and neither is perfect on its own:

The "Real" Observations (The Sparse Dots): We have actual tide gauges (like weather stations for the ocean) at 29 locations. They are real, reliable, and measured by humans. But they are very far apart, and they haven't been measuring for very long compared to how rare a "once-in-a-century" storm is. It's like trying to guess the temperature of an entire country by only checking thermometers in three cities.
The "Simulated" Data (The Dense Cloud): We have a supercomputer model (ADCIRC) that simulates ocean physics. It can spit out water levels for 100 different spots along the coast, filling in all the gaps between the real gauges. It's like having a high-resolution satellite photo of the whole country. But, it's just a computer guess. It might be slightly "off" because the physics in the code aren't 100% perfect.

The Big Idea: Fusing the Two

This paper introduces a clever statistical method called "Data Fusion" to combine these two sources. Think of it like a detective solving a mystery.

The Detective's Logic: The detective knows the real witnesses (tide gauges) are trustworthy but few in number. The detective also has a very detailed, high-tech surveillance video (the simulation) that covers everywhere, but the video might have a slight color filter making things look a bit different.
The Solution: Instead of ignoring the video or ignoring the witnesses, the detective uses the video to fill in the gaps between the witnesses, but uses the witnesses to "correct" the video's color filter.

How the Method Works (The Two-Stage Process)

The authors use a two-step process to mix these ingredients:

Stage 1: The Individual Check-Up
First, they look at each location separately.

For the Real Gauges, they calculate the "worst-case scenario" statistics. Because the records are short, these calculations are a bit shaky (like guessing a coin's bias after only flipping it 40 times).
For the Simulations, they do the same math. These are also shaky, but in a different way.

Stage 2: The Group Hug (The "Coregionalization")
This is the magic part. They realize that even though the numbers might be slightly different between the real gauges and the computer, the patterns are almost identical.

If the real gauge in Florida says "storms get worse as you go south," the computer simulation says the exact same thing.
If the real gauge in Maine says "storms are milder," the computer agrees.

The authors build a mathematical bridge (a "Linear Model of Coregionalization") that says: "We know the computer and the real gauge are looking at the same underlying reality, just with different lenses."

They use the 100 computer points to "smooth out" the shaky estimates from the 29 real gauges. It's like taking a blurry photo (the real gauges) and using a high-definition reference image (the simulation) to sharpen the blurry parts without losing the truth of the original photo.

The Results: Why It Matters

When they tested this method, the results were impressive:

Sharper Maps: The uncertainty (the "fog" around the prediction) dropped by 35%. Imagine trying to aim a dart; this method made the target 35% clearer.
Better Guesses in Empty Spots: In areas where there were no real gauges (like parts of the Gulf Coast), the computer simulation helped pull the estimates in the right direction. Without the fusion, the model would have been too low; with the fusion, it corrected itself to match reality.
The "Shape" Factor: The most important part of predicting extreme storms is the "tail" of the data (how crazy the worst storms get). The computer and the real gauges agreed very strongly on this "shape," allowing the method to borrow strength from the computer to fix the shaky real-world data.

The Analogy: The Orchestra

Think of the Real Gauges as a small, talented string section (violins). They play the true melody, but there are only a few of them, so you can't hear the full sound of the orchestra.

Think of the Computer Simulation as a massive, loud brass section (trumpets and tubas). They play the same melody, but they are slightly out of tune and too loud.

If you only listen to the strings, the music is quiet and you miss the big picture. If you only listen to the brass, it's loud but inaccurate.

This paper's method is the conductor who listens to both. It realizes the brass section is playing the same song as the strings, just slightly off-key. The conductor uses the loud brass to fill in the gaps where the strings are missing, but uses the strings to tune the brass. The result? A full, loud, and accurate symphony of storm predictions.

The Bottom Line

This research gives us a new tool to predict dangerous coastal flooding. By combining the "truth" of a few real sensors with the "coverage" of a powerful computer model, we can create much safer, more accurate maps for protecting our coastal cities. It turns a sparse, shaky map into a dense, reliable guide for the future.

1. Problem Statement

Estimating spatial extremes (e.g., 100-year return levels for sea levels) is challenging due to the sparsity of observational networks.

Observational Data: NOAA tidal gauges provide reliable long-term records but are spatially sparse (29 stations in the study) and often too short relative to the return periods of interest (43 years of data vs. 100-year return levels). This leads to high uncertainty in tail parameter estimates, particularly the shape parameter ( $\xi$ ).
Simulation Data: Physics-based hydrodynamic models (e.g., ADCIRC) can generate dense spatial outputs (100+ sites) but represent model physics rather than direct observations. They may contain systematic biases or lack long-term trends present in real-world data.
The Gap: Existing methods struggle to fuse these heterogeneous sources effectively. Traditional spatial extreme value models (like max-stable processes) are computationally intractable for moderate station counts, and standard interpolation fails to leverage the structural information from simulations to improve tail inference.

2. Methodology

The authors propose a two-stage frequentist framework that fuses sparse observations and dense simulations using a Linear Model of Coregionalization (LMC).

Stage 1: Marginal GEV Fitting

Approach: Generalized Extreme Value (GEV) distributions are fitted independently at each site using Maximum Likelihood Estimation (MLE).
Nonstationarity: For NOAA gauge sites, a nonstationary location parameter ( $\mu(t) = \mu_0 + \mu_1(t - t_{ref})$ ) is used to accommodate observed sea-level rise trends. ADCIRC sites, which lack long-term trends in the simulation forcing, retain stationary fits.
Output: A vector of parameter estimates $\hat{\theta}$ (location, log-scale, shape) for each site.

Stage 2: Multivariate Spatial Modeling (LMC)

Latent Process: The GEV parameters from both sources are modeled as a single high-dimensional (6-dimensional) latent spatial process $\theta(s)$ . This vector includes parameters for both the "NOAA" source and the "ADCIRC" source at every location, even though only one is observed at any given site.
Coregionalization: The process is defined as $\theta(s) = \beta + A \delta(s)$ $θ (s) = β + A δ (s)$ , where:
- $\beta$ is the spatial mean.
- $A$ is a lower-triangular matrix of loadings.
- $\delta(s)$ consists of six independent Gaussian processes with exponential covariance and distinct range parameters ( $\rho_i$ ).
Information Transfer: Cross-source correlations (e.g., correlation between NOAA location and ADCIRC location) are estimated from spatially interspersed networks without requiring co-located sites. This is possible because the LMC is identifiable under heterotopic sampling if range parameters are distinct and networks are sufficiently intermixed.
Partial Observations: A selection matrix ( $S$ ) extracts observed components from the full latent vector, allowing the model to handle sites with only NOAA data, only ADCIRC data, or different subsets of parameters.
Likelihood & Optimization: The model uses a Gaussian likelihood for the Stage 1 estimates (treating them as noisy observations of the latent process). An analytic gradient derived from the LMC structure is used to optimize the 33-parameter log-likelihood, reducing computation time from ~90 minutes to <3 minutes.
Uncertainty Quantification: Stage 1 estimation errors are estimated via a nonparametric block bootstrap and incorporated as a fixed noise term ( $W$ ) in the Stage 2 covariance.

3. Key Contributions

Generalized LMC for Data Fusion: Extends the LMC framework from single-source spatial smoothing to a multi-source setting, enabling the fusion of heterogeneous data (observations vs. simulations) without co-located sites.
Computational Efficiency: Derives an analytic gradient for the high-dimensional likelihood, making multi-start optimization and bootstrap inference computationally feasible for spatial extremes.
Quantification of Fusion Benefits: Provides rigorous cross-validation evidence demonstrating that fusing simulations with sparse observations significantly reduces uncertainty in return level estimates, even when the simulation data has systematic biases.
Software Implementation: The method is implemented in the open-source R package evfuse.

4. Results

The framework was applied to U.S. coastal sea levels (1979–2021), fusing 29 NOAA gauges with 100 ADCIRC simulation sites.

Cross-Source Correlations:
- Location ( $\mu$ ): Near-perfect correlation ( $r = 0.995$ ). Both sources capture the same spatial pattern of extreme sea levels.
- Shape ( $\xi$ ): Strong correlation ( $r = 0.837$ ), crucial for tail inference.
- Scale ( $\log \sigma$ ): Moderate correlation ( $r = 0.443$ ).
Performance Metrics:
- Leave-One-Out Cross-Validation (LOO-CV): The joint model reduced the Root Mean Square Error (RMSE) of 100-year return levels by 35% compared to a NOAA-only model.
- Geographic Block CV: Under spatial extrapolation (holding out entire coastal regions), the joint model still achieved an 18% RMSE reduction.
- Uncertainty Reduction: Confidence bands narrowed significantly along the Atlantic coast (where ADCIRC density is high). Along the Gulf Coast, fusion primarily corrected point estimates (shifting them upward) where gauge coverage was sparse.
Saturation Analysis: The benefit of fusion saturates with approximately 30 ADCIRC sites (roughly one site per minimum correlation range), suggesting the 100-site network used was more than sufficient for the large-scale patterns analyzed.

5. Significance and Implications

Risk Assessment: The method provides more precise and spatially continuous maps of extreme sea levels, which are critical for coastal engineering, flood risk management, and climate adaptation planning.
Handling Sparse Data: It offers a robust solution for regions with sparse observational networks by leveraging dense, albeit potentially biased, simulation outputs.
Bias Correction vs. Information Transfer: The study demonstrates that even if a simulation model has a systematic bias (e.g., ADCIRC underestimates levels), the spatial pattern of the simulation can still be used to correct and refine estimates from sparse observations, provided the cross-source correlations are strong.
Generalizability: The framework is applicable to other environmental domains where sparse ground-truth data coexists with dense model outputs (e.g., precipitation extremes with radar data, air quality with satellite retrievals).

Conclusion: The paper successfully demonstrates that a two-stage frequentist approach using LMC can effectively fuse sparse observations and dense simulations. By exploiting cross-source correlations, the method significantly improves the precision of spatial extreme value estimates, offering a computationally tractable and statistically rigorous alternative to fully Bayesian hierarchical models.