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Spatially-Heterogeneous Causal Bayesian Networks for Seismic Multi-Hazard Estimation: A Variational Approach with Gaussian Processes and Normalizing Flows

Xuechun Li, Shan Gao, Runyu Gao, Susu Xu

TL;DR

Spatial-VCBN addresses the challenge of post-earthquake multi-hazard estimation by modeling spatially varying causal relationships using a Gaussian Process prior over latent spatial effects, combined with invertible normalizing flows to capture non-Gaussian distributions of causal parameters. The coefficients controlling hazards-to-damage relationships, $v^{l}$, are generated as $v^{l} = f_{K_{v}} \circ \cdots \circ f_1(z_{v}^{l})$ with $oldsymbol{z}_{v} \sim \mathcal{GP}(m_v(\mathbf{GF}), k_v(\mathbf{GF},\mathbf{GF}'))$, enabling location-dependent, non-linear causal behavior. Inference uses a sparse GP with inducing points and a stochastic variational framework, achieving near $0.94$ seconds per $km^2$ on GPU, and delivering up to $35.2\%$ AUC improvements over priors and $5.5\%$ over state-of-the-art baselines across Haiti, Puerto Rico, and Turkey-Syria earthquakes. The method decouples co-located hazards, improves robustness to noisy remote-sensing signals, and offers practically actionable, spatially-resolved hazard assessments for rapid disaster response and risk reduction.

Abstract

Post-earthquake hazard and impact estimation are critical for effective disaster response, yet current approaches face significant limitations. Traditional models employ fixed parameters regardless of geographical context, misrepresenting how seismic effects vary across diverse landscapes, while remote sensing technologies struggle to distinguish between co-located hazards. We address these challenges with a spatially-aware causal Bayesian network that decouples co-located hazards by modeling their causal relationships with location-specific parameters. Our framework integrates sensing observations, latent variables, and spatial heterogeneity through a novel combination of Gaussian Processes with normalizing flows, enabling us to capture how same earthquake produces different effects across varied geological and topographical features. Evaluations across three earthquakes demonstrate Spatial-VCBN achieves Area Under the Curve (AUC) improvements of up to 35.2% over existing methods. These results highlight the critical importance of modeling spatial heterogeneity in causal mechanisms for accurate disaster assessment, with direct implications for improving emergency response resource allocation.

Spatially-Heterogeneous Causal Bayesian Networks for Seismic Multi-Hazard Estimation: A Variational Approach with Gaussian Processes and Normalizing Flows

TL;DR

Spatial-VCBN addresses the challenge of post-earthquake multi-hazard estimation by modeling spatially varying causal relationships using a Gaussian Process prior over latent spatial effects, combined with invertible normalizing flows to capture non-Gaussian distributions of causal parameters. The coefficients controlling hazards-to-damage relationships, , are generated as with , enabling location-dependent, non-linear causal behavior. Inference uses a sparse GP with inducing points and a stochastic variational framework, achieving near seconds per on GPU, and delivering up to AUC improvements over priors and over state-of-the-art baselines across Haiti, Puerto Rico, and Turkey-Syria earthquakes. The method decouples co-located hazards, improves robustness to noisy remote-sensing signals, and offers practically actionable, spatially-resolved hazard assessments for rapid disaster response and risk reduction.

Abstract

Post-earthquake hazard and impact estimation are critical for effective disaster response, yet current approaches face significant limitations. Traditional models employ fixed parameters regardless of geographical context, misrepresenting how seismic effects vary across diverse landscapes, while remote sensing technologies struggle to distinguish between co-located hazards. We address these challenges with a spatially-aware causal Bayesian network that decouples co-located hazards by modeling their causal relationships with location-specific parameters. Our framework integrates sensing observations, latent variables, and spatial heterogeneity through a novel combination of Gaussian Processes with normalizing flows, enabling us to capture how same earthquake produces different effects across varied geological and topographical features. Evaluations across three earthquakes demonstrate Spatial-VCBN achieves Area Under the Curve (AUC) improvements of up to 35.2% over existing methods. These results highlight the critical importance of modeling spatial heterogeneity in causal mechanisms for accurate disaster assessment, with direct implications for improving emergency response resource allocation.

Paper Structure

This paper contains 25 sections, 27 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Prior Geospatial Models
  4. InSAR Data and Damage Proxy Maps
  5. Geophysical Features
  6. Causal Modeling for Disaster Impacts
  7. Modeling Spatially Heterogeneous Causal Effects Using Normalizing Flows with Gaussian Process
  8. Stochastic Variational Inference with Spatial-variant Causal Parameters
  9. Training and Stochastic Optimization with Sparse Gaussian Processes
  10. Results
  11. Data Description
  12. Evaluation Metrics and Benchmarks
  13. Spatially-Varying Causal Influence Mapping between Multi-Hazards and Structural Damage
  14. Robust Spatially-Varying Causal Inference Between Hazards and Remote Sensing Observations
  15. Cross-Event Validation and Baseline Comparison
  16. ...and 10 more sections

Figures (9)

  • Figure 1: After the 2023 Turkey-Syria earthquake sequence, the USGS produced example ground failure models for landslide and liquefaction Fig3. The probability of ground failure models is what the legend colors represent.
  • Figure 2: Damage proxy maps (DPM) generated by the AIRA team after (a) Haiti, (b) Puerto Rico, (c) Turkey-Syria earthquakes. DPMs show surface change detection with brighter areas indicating greater surface deformation. The areas covered by DPMs are also our study regions.
  • Figure 3: Overview of our causal Bayesian inference framework for seismic multi-hazard and impacts estimation. $l$ in the figure refers to the $l^{\text{th}}$ location in a target area. Blue circles refer to latent hazard/impact variables. Green cirles refer to the spatial-varying causal coefficients. Orange rectangles refer to the observations or known information.
  • Figure 4: Spatial distribution of hazards, causal parameters, and damage following the 2020 Puerto Rico earthquake. (a) Google satellite imagery of the study area with extent of $-66.945 \degree W$, $17.956 \degree N$ to $-66.876 \degree W$, $17.998 \degree N$. (b) DPM derived from satellite imagery. (c) Spatial distribution of $\gamma_{LS}$ values, showing the causal parameter strength from landslides (LS) to building damage (BD). (d) Spatial distribution of $\gamma_{LF}$ values, showing the causal parameter strength from liquefaction (LF) to building damage (BD). (e) Posterior probability of landslide occurrence. (f) Posterior probability of building damage occurrence. (g) Posterior probability of liquefaction occurrence.
  • Figure 5: Spatial distribution of causal parameters and DPM signals in a high-deformation area. (a) Damage Proxy Map (DPM) showing surface deformation with brighter areas indicating higher change detection values in the study area with extent of $-66.946 \degree W$, $17.958 \degree N$ to $-66.890 \degree W$, $18.009 \degree N$. (b) Spatial distribution of $\lambda_{LF}$, quantifying the causal relationship strength from liquefaction to DPM signals. (c) Spatial distribution of $\lambda_{LS}$, quantifying the causal relationship strength from landslides to DPM signals. (d) Spatial distribution of $\lambda_{BD}$, quantifying the causal relationship strength from building damage to DPM signals.
  • ...and 4 more figures