Modeling Non-Ergodic Path Effects Using Conditional Generative Model for Fourier Amplitude Spectra

TL;DR

The paper tackles non-ergodic path effects in ground-motion modeling and the computational limits of GP-based approaches for large-scale, multi-frequency predictions. It introduces CGM-FAS, a Conditional Variational Autoencoder that learns the distribution of within-site residuals conditioned on earthquake and station coordinates, enabling simultaneous, interfrequency-aware estimation of spatially varying path terms. Results show CGM-FAS produces path predictions with comparable mean and variability to GP-based methods while delivering smoother spatial patterns, learnable interfrequency correlations, and rapid predictions (e.g., maps across many sites and frequencies in seconds) with modest memory. While CGM-FAS demonstrates strong practical potential and reduced computational burden, it currently lacks quantified epistemic uncertainty in path terms, and future work will address uncertainty characterization and extension to jointly model source, site, and path effects across frequencies.

Abstract

Recent developments in non-ergodic ground-motion models (GMMs) explicitly model systematic spatial variations in source, site, and path effects, reducing standard deviation to 30-40% of ergodic models and enabling more accurate site-specific seismic hazard analysis. Current non-ergodic GMMs rely on Gaussian Process (GP) methods with prescribed correlation functions and thus have computational limitations for large-scale predictions. This study proposes a deep-learning approach called Conditional Generative Modeling for Fourier Amplitude Spectra (CGM-FAS) as an alternative to GP-based methods for modeling non-ergodic path effects in Fourier Amplitude Spectra (FAS). CGM-FAS uses a Conditional Variational Autoencoder architecture to learn spatial patterns and interfrequency correlation directly from data by using geographical coordinates of earthquakes and stations as conditional variables. Using San Francisco Bay Area earthquake data, we compare CGM-FAS against a recent GP-based GMM for the region and demonstrate consistent predictions of non-ergodic path effects. Additionally, CGM-FAS offers advantages compared to GP-based approaches in learning spatial patterns without prescribed correlation functions, capturing interfrequency correlations, and enabling rapid predictions, generating maps for 10,000 sites across 1,000 frequencies within 10 seconds using a few GB of memory. CGM-FAS hyperparameters can be tuned to ensure generated path effects exhibit variability consistent with the GP-based empirical GMM. This work demonstrates a promising direction for efficient non-ergodic ground-motion prediction across multiple frequencies and large spatial domains.

Figures (9)

• Figure 1: Station locations (a) and source locations (b) from the selected dataset from Lacour2025c for the East-West component. Red triangles and blue stars show the selected station and source locations, respectively. The black box defines the spatial domain shown in later generated maps, which contains all selected stations and their associated recordings. The thin black lines indicate mapped faults in the USGS quartenary database. The earthquakes range between magnitude 1 and 4 and were recorded by broadband seismometers. More information on the dataset is available in Lacour2025b.
• Figure 2: Illustration of the architecture of CGM-FAS. The encoder $p_{\boldsymbol{\mathbf{\theta}_{\mathrm{encoder}}}}$ uses convolutional layers to extract features from observed within-site residuals ${\mathbf{\mathbf{\boldsymbol{\delta WS}}_{obs}}}$ conditioned on event and site coordinates ($\mathbf{te}$, $\mathbf{ts}$) and map them to a low-dimensional latent space . A Multi-Layer Perceptron (MLP) layer outputs output the parameters of a latent distribution: mean $\boldsymbol{\mu}$ and standard deviation $\boldsymbol{\sigma}$. A latent variable $\mathbf{z} \sim \mathcal{N}( \boldsymbol{0}, \mathbf{{I})}$ is sampled from a standard normal distribution and passed through the decoder $p_{\boldsymbol{\mathbf{\theta}_{\mathrm{decoder}}}}$ to generate the reconstructed observation $\hat{\mathbf{\mathbf{\boldsymbol{\delta WS}}_{obs}}}$ conditioned on the same event and site coordinates. Here, $\boldsymbol{\theta_{\mathrm{encoder}}}$ and $\boldsymbol{\theta_{\mathrm{decoder}}}$ represent the hyperparameters of the encoder and decoder neural networks, respectively.
• Figure 3: Comparison of within-path residuals from LANN25 (red) and CGM-FAS (blue). (a) Mean residuals versus frequency, (b) standard deviation $\phi_{SP,NE}$ versus frequency, (c) histograms at 10 Hz with mean (solid lines) and standard deviation (dashed lines). Note that $\phi_{SP,NE}$ of the GP in (b) is computed from the within-path residuals at the available stations rather then the value estimated using Equation \ref{['eq:FAS_correlation_path']}.
• Figure 4: Comparison of path effects prediction between LANN25 (a) and CGM-FAS (b) at 10 Hz. (ec Histogram of within-path residuals from LANN25 (red) and CGM-FAS (blue).
• Figure 5: Spatial correlation lengths of predicted path effects. (a) Semivariogram of the CGM-FAS prediction (black) and fitted squared-exponential correlation function (blue) for the Berkeley event at 10 Hz as a function of site separation distance. The fitted correlation length is equal to 18 km. (b) Frequency dependence of correlation lengths for all events. The red line shows LANN25 from Lacour2025b. The blue lines show correlation lengths from CGM-FAS predictions for each available earthquake in the dataset. The solid line represents the mean correlation length, while the dashed lines show the standard deviation.