⚡ electrical engineering

The Geometry of Earthquake Shaking: A Probabilistic Model for Response-Spectrum Directionality

This study demonstrates that the directional variability of horizontal earthquake response spectra is primarily a finite-sample stochastic effect governed by the duration-to-period ratio, rather than intrinsic physical anisotropy, and presents a parsimonious probabilistic model that accurately reproduces observed anisotropy patterns to improve ground-motion modeling and seismic hazard analysis.

Original authors: Rajesh Rupakhety, Victor Moises Hernández-Aguirre

Published 2026-10-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rajesh Rupakhety, Victor Moises Hernández-Aguirre

Original paper licensed under CC BY 4.0 (https://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

When the ground shakes during an earthquake, the motion is not a simple back-and-forth push. It is a complex, two-dimensional wave that moves in every horizontal direction at once. For a building standing on that ground, the damage it suffers depends heavily on which way the building is facing relative to the direction of the shaking. A structure might be perfectly safe if the strongest part of the wave hits its side, but vulnerable if that same wave strikes its front. Engineers have long known that this directional variability is real, but they have struggled to predict it. Traditionally, they have treated the shaking as a single number or a simple average, effectively ignoring the fact that the ground moves differently depending on the angle you look at it. This simplification works for rough estimates, but it fails to capture the true geometry of the danger, leaving a gap in how we assess risk for cities and critical infrastructure.

A new study by researchers at the University of Iceland seeks to fill this gap by treating earthquake shaking not as a mystery of complex physics, but as a problem of geometry and statistics. The researchers analyzed thousands of recordings of actual ground motion to understand why the shaking appears stronger in some directions than others. They discovered that much of what looks like a special, physical preference for a certain direction is actually an illusion created by the shortness of the earthquake itself. Just as a coin flipped only a few times might land on heads by chance, a short earthquake record might appear to shake mostly in one direction simply because there were not enough independent waves to smooth out the randomness. The team found that the key to predicting this behavior lies in a simple ratio: the length of the shaking compared to the time it takes for a building to sway back and forth.

The researchers examined 4,690 recordings of strong ground motion from a European database, covering a wide range of earthquake sizes and distances. Their goal was to separate the true, physical reasons for directional shaking from the statistical noise that arises when we try to measure a complex process with limited data. They focused on a specific measure of how "stretched" the shaking is in one direction versus another. In their analysis, they identified three distinct behaviors that occur depending on the duration of the shaking. When the shaking is very short, the measurement is unreliable and tends to look highly directional by pure chance. When the shaking is very long, the measurement becomes stable and the directionality settles into a predictable pattern. In the middle range, the directionality grows steadily as the shaking lasts longer.

By combining these observations, the team built a probabilistic model that acts as a baseline for what we should expect from random shaking. This model relies on a single, calibrated parameter to describe the maximum amount of independent information available in a ground motion record. When they tested this model against the real data, it successfully reproduced the observed patterns of directional variability across the entire range of building swaying times, from very fast vibrations to slow, rolling movements. The model suggests that for most earthquakes, the apparent directionality is not a fixed property of the earthquake source or the path the waves traveled, but rather a consequence of how much time the shaking lasts.

The study also looked for exceptions to this rule, searching for cases where the physical nature of the earthquake forces a specific direction regardless of the duration. They found that near-fault earthquakes, where the ground ruptures directly toward a recording station, do produce genuine, physical directionality. However, even in these cases, the researchers showed that a large portion of the observed effect is still due to the short duration of the pulse-like motion, which limits the amount of data available to measure the shaking. Once the researchers accounted for the duration, a smaller, genuine directional signal remained, confirming that rupture directivity does matter, but that its influence is often masked by the statistical noise of short records.

The team also investigated whether the size of the earthquake or the distance from the source played a direct role in creating directionality. They found that these factors do not influence the shaking direction directly. Instead, their influence is entirely indirect: larger earthquakes and those further away tend to last longer, and it is this increased duration that reduces the random noise and reveals the underlying geometry. When the researchers removed the effect of duration from their analysis, the apparent connection between earthquake size, distance, and directionality vanished. This finding suggests that engineers do not need to invent complex new rules for every type of earthquake; they only need to understand how long the shaking lasts to predict its directional behavior.

Finally, the researchers tested their model by simulating thousands of artificial directional shaking patterns. They combined their new model for the geometric shape of the shaking with an existing model for the random peaks of the motion. The simulations produced results that closely matched the real-world data, capturing the way shaking becomes more directional as the period of the building increases. The only significant difference appeared at the extreme edges of the data, where the simulations occasionally predicted shaking that was physically impossible, exceeding the theoretical maximum limit. The researchers noted that this is a minor flaw in how they combined the two models, not a failure of the core idea. The study concludes that the chaotic, directional nature of earthquake shaking can be understood as a bounded, random geometric phenomenon, primarily governed by the duration of the event. This perspective offers a clearer, more physically grounded way to model ground motion, moving away from treating directionality as a mysterious extra variable and toward understanding it as a natural consequence of how long the earth shakes.

Technical Summary: The Geometry of Earthquake Shaking: A Probabilistic Model for Response-Spectrum Directionality

Problem Statement
Horizontal ground motion is inherently a two-component vector process, meaning the intensity experienced by a structure depends on its orientation relative to the ground-motion field. While rotation-invariant measures like RotD50 (median) and RotD100 (maximum) are standard in seismic hazard analysis, they reduce the directional field to scalar summaries, discarding the underlying geometric structure. Existing empirical models for directionality (e.g., the ratio RotD100/RotD50) often treat directional variability as an additional aleatory term or a descriptive function of period, without identifying the physical state variables governing the evolution of the directional field. Furthermore, it remains unclear how much of the observed anisotropy is due to persistent physical effects (e.g., rupture directivity, site effects) versus statistical artifacts arising from finite-sample covariance estimation.

Methodology
The study utilizes a dataset of 4,690 horizontal-component strong-motion records from the European Strong Motion (ESM) database, covering 417 events and 1,304 stations. The authors develop a probabilistic framework based on the decomposition of the directional response spectrum into two components: a geometric RMS ellipse (governed by covariance anisotropy) and a stochastic peak-factor field.

Geometric Anisotropy Variable ( $\kappa_{rms}$ ): The authors define $\kappa_{rms}$ as the normalized contrast between the principal eigenvalues of the response covariance matrix. This variable isolates geometric elongation independent of response amplitude and peak formation.
Effective Sample Size ( $N_{eff}$ ): The core of the methodology posits that the distribution of $\kappa_{rms}$ is governed by the effective number of statistically independent response cycles ( $N_{eff}$ ). Under an isotropic Gaussian null model, $\kappa_{rms}^2$ follows a Beta distribution determined by $N_{eff}$ .
Composite Model for $N_{eff}$ : The study derives a composite model for $N_{eff}$ $N_{e f f}$ as a function of the dimensionless ratio $\psi = \ln(D_R^*/T)$ $ψ = ln (D_{R}^{*} / T)$ , where $D_R^*$ $D_{R}^{*}$ is the energetic duration of the response and $T$ $T$ is the oscillator period. The model incorporates three physical regimes:
- A narrowband scaling regime where $N_{eff} \propto \psi$ (intermediate periods).
- A short-period saturation imposed by the finite bandwidth of the ground motion.
- A long-period floor ( $N_{min} = 3$ ) representing the minimum degrees of freedom for a 2D covariance estimate.
Calibration and Validation: The model is calibrated using maximum likelihood estimation on the ESM dataset. The authors validate the model using probability integral transform (PIT) diagnostics to generate a "duration-conditioned" normal score ( $z_\psi$ ), which isolates anisotropy deviations from the stochastic baseline.
Residual Analysis: The study examines the $z_\psi$ residuals to identify physical effects (pulse-like motions, magnitude, distance, and site amplification) that persist after accounting for the duration-controlled stochastic baseline.
Simulation: The anisotropy model is combined with an existing stochastic model for directional peak factors to simulate full directional response spectra and compare them against observed data.

Key Contributions

Probabilistic Framework: The paper establishes that a substantial fraction of observed directional variability can be explained as a finite-sample effect driven by the ratio of energetic duration to oscillator period.
Composite $N_{eff}$ Model: It introduces a parsimonious, single-parameter model for the effective sample size that successfully reproduces the observed distributions of $\kappa_{rms}$ across the full period range (0.01s to 10s).
Decomposition of Variability: By conditioning on the stochastic baseline, the study separates "apparent" anisotropy (caused by short durations and few cycles) from "genuine" physical anisotropy.
Generative Simulation: The authors demonstrate a method to simulate directional response spectra using only two calibrated population parameters ( $N_\infty$ and the peak-factor scale $\sigma$ ), reproducing the principal features of observed directional variability.

Results

Stochastic Baseline: The composite model accurately reproduces the empirical distributions of $\kappa_{rms}$ , including the progressive rightward shift and broadening of the distribution as oscillator period increases. The PIT diagnostics confirm the model provides a satisfactory marginal description of the data ( $\mu \approx -0.09, \sigma \approx 1.03$ ).
Source and Path Effects: The apparent dependence of anisotropy on earthquake magnitude ( $M_w$ ) and source-to-site distance ( $R_{JB}$ ) is found to be overwhelmingly mediated through energetic duration. Once conditioned on $\psi$ , the explanatory power of magnitude and distance collapses (median $R^2$ drops from ~0.038 to ~0.001), suggesting these factors influence anisotropy primarily by controlling the number of independent response cycles.
Pulse-like Motions: Near-fault pulse-like motions exhibit elevated anisotropy. While a significant portion of this is explained by the reduced energetic duration of pulses (a stochastic effect), a smaller, period-matched residual persists, indicating a genuine polarization component associated with rupture directivity.
Site Effects: Station-level variability accounts for 25–30% of the residual variance at short periods. Stations with higher short-period HVSR amplification show systematically elevated anisotropy relative to the stochastic baseline, suggesting a link between site resonance and directional variability.
Simulation Performance: The combined simulation framework reproduces the population-level distributions of PSA/RotD50 and RotD100/RotD50 well. However, at long periods, the model slightly overestimates the upper tail (violating the physical $\sqrt{2}$ ceiling) because it treats the covariance anisotropy and peak-factor fields as independent.

Significance and Claims
The paper argues that directional variability should be reinterpreted as a "bounded stochastic geometric phenomenon" governed primarily by the duration-to-period ratio. This perspective offers a physically grounded alternative to treating directionality solely through conversion factors or additional aleatory variability terms in Ground Motion Models (GMMs).

The authors claim that compact, low-dimensional stochastic representations of horizontal ground-motion fields are achievable. By identifying the stochastic baseline, the study provides a framework for record selection and probabilistic seismic hazard analysis that distinguishes between statistical artifacts of finite sampling and persistent physical directional structures. The work suggests that future improvements should focus on predictive models for energetic response duration and joint modeling of covariance anisotropy and peak-factor structure to better capture extreme directional responses.

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