Coherent Source Subsampling: A Data-Driven Strategy for Restoring Causal-Acausal Symmetry in Ambient Seismic Wavefield Correlations

TL;DR

Ambient-noise cross-correlations often suffer from biased Green's-function estimates due to nonuniform and time-varying source distributions, breaking causal–acausal symmetry. The authors introduce Coherent Source Subsampling (CSS), a data-driven conditioning framework that uses a discrete latent state $\lambda$ learned by a variational autoencoder to identify stationary-zone sources and compute conditional averages over coherent windows. By averaging only windows with high probability of belonging to the stationary state ($\lambda=1$) and branch-specific coherence, CSS substantially restores symmetry between causal and acausal components and yields more stable Rayleigh-wave dispersion curves, aligning with earthquake-based references for several station pairs. The approach reduces data requirements (typically 10–50% of windows) and is scalable to dense networks and DAS, offering a physically interpretable link between source statistics and noise-based Green's-function retrieval with broad potential for robust ambient-noise tomography in challenging environments.

Abstract

Ambient noise tomography relies on the assumption that the seismic wavefield is equipartitioned, meaning that energy is uniformly distributed among all directions. However, in practice, ambient noise sources are highly non-uniform in both spatial and temporal dimensions, resulting in biased estimation of the Green's function between stations. We introduce a data-driven method, Coherent Source Subsampling (CSS), which selects and averages only those cross-correlation time windows that are associated with the excitation of sources in the stationary-zone. By confining the ensemble average to these coherent subsets, CSS effectively mitigates the influence of anisotropic or intermittent sources and restores causal-acausal symmetry in the retrieved Green's functions. Applications to regional-scale ambient noise datasets demonstrate that CSS boosts inter-station coherence and enhances the reliability of surface-wave dispersion measurements, providing a physically interpretable bridge between source statistics and noise correlation theory.

Figures (4)

• Figure 1: Coherent source subsampling (CSS) concept. a) Ambient noise sources on the boundary, distributed non-uniformly, excite receivers at $\mathbf{x}_i$ and $\mathbf{x}_j$. The dashed line marks the source state $\lambda$=$1$ associated with the stationary‑phase zone. For this state, red (acausal) and cyan (causal) stars denote subsampled sources that are considered for averaging. Diamonds and a dotted line represent the second state ($\lambda$=$2$). CSS excludes the remaining blue-dot sources while averaging as they bias the estimation of interstation response. b) Latent‑variable autoencoder model for CSS. For each branch $b$ and pair $(i,j)$, the encoder maps input cross-correlations to three latent-variable components (orange boxes). The source-state probabilities act as data‑driven selectors that identify stationary‑phase source subsets. An example with three cross-correlation windows and two source states is depicted, although their quantities can be chosen arbitrarily.
• Figure 2: Cross-correlation gathers for the eastern United States. (a) Before subsampling: The linearly averaged cross-correlations, $\widehat{\mathbb{E}}\left[ \prescript{b}{}{\mathbf{w}}_{ij} \right]$, show large amplitudes close to zero lag and pronounced causal-acausal asymmetry, indicating a directionally biased ambient noise field. (b) After subsampling: The conditional averages, $\widehat{\mathbb{E}}\left[ \prescript{b}{}{\mathbf{w}}_{ij} \mid \lambda=1 \right]$, obtained after subsampling, display enhanced waveform coherence and substantially improved causal–acausal symmetry. The percentages of time-window allocation (in red) indicate the fraction of windows selected for the source state $\lambda\text{ = 1}$. The correlation values (in blue) measure the degree of symmetry between the acausal and causal components for each station pair. The black lines mark the time lags corresponding to velocities of $2$ and $5\,$km/s.
• Figure 3: Temporal and dispersion analysis for the station pairs (GOGA, NCB) and (BLA, PKME). (a–b, i–j) Acausal and causal time-window selections obtained from CSS for source states $\lambda\text{ = \{1, 2\} }$. (c–e, k–m) Before subsampling: standard linear averaging produces dispersion images and cross-correlations with low coherence and pronounced causal–acausal asymmetry. (f-h, n-p) After subsampling: source state $\lambda\text{ = 1}$ results in dispersion images and cross-correlations with substantially improved causal–acausal symmetry.
• Figure 4: Group‐velocity dispersion curves for nine station pairs in the eastern United States. Solid lines represent the causal and acausal dispersion curves obtained for source state $\lambda\text{ = 1}$, while dotted lines indicate the corresponding dispersion curves based on linear averaging. Earthquake‐based dispersion curves are shown for comparison for station pairs (GOGA, NCB) and (ACSO, CNNC). For all station pairs, the dispersion curves retrieved using source state $\lambda\text{=1}$ show markedly reduced scatter between $10\,$ and $50\,$s, whereas the linearly averaged curves display substantial variability and systematically underestimate velocities. This closer agreement demonstrates the effectiveness of CSS for recovering symmetric dispersion information.