On the Origin of Directional Variability in Earthquake Response Spectra: A Stochastic Covariance Framework

Rajesh Rupakhety, Victor Moises Hernández Aguirre 

Directional variability of horizontal earthquake response spectra is commonly described using rotation-based measures such as RotD50 and RotD100, yet its physical and statistical origin remains unclear. This study shows that directional anisotropy arises fundamentally from finite-sample fluctuations of the covariance matrix of filtered ground-motion response. Even under perfectly isotropic excitation, covariance estimation over finite-duration records produces random polarization governed by Wishart statistics, establishing an intrinsic stochastic baseline for anisotropy. A geometric anisotropy measure, κ, is introduced based on the eigenvalue contrast of the response covariance matrix. It is shown that κ² follows an exact Beta distribution determined by the effective number of statistically independent realizations. Extension to oscillator response reveals that this effective number depends on oscillator bandwidth and energetic duration, leading to a closed-form scaling that explains the observed increase of anisotropy with period. A record-conditioned isotropic surrogate framework is developed to isolate this stochastic baseline in real ground motions while preserving spectral content and temporal envelope. Simulations and surrogate ensembles confirm that the Wishart-based model accurately captures the variability of directional anisotropy.The results show that rotation-based measures such as RotD100 and RotD50 do not directly quantify anisotropy, but combine covariance-driven geometric effects with stochastic peak selection. The proposed formulation provides a stochastic foundation for interpreting and modelling directional ground-motion fields in seismic hazard analysis.