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

Abstract

ABSTRACT Directional variability of horizontal earthquake response spectra is commonly characterized using rotation‐based measures such as RotD50 and RotD100, yet the stochastic mechanisms underlying this variability remain incompletely understood. This study develops a covariance‐based framework for directional oscillator response under isotropic excitation. A geometric anisotropy measure, , is defined from the normalized eigenvalue contrast of the two‐component response covariance matrix. Under independent isotropic Gaussian sampling, is the complement of the classical bivariate sphericity statistic and follows an exact Beta distribution. The trace–anisotropy formulation used here recovers this classical result directly in coordinates relevant to directional oscillator response. For temporally correlated and amplitude‐modulated oscillator response, the independent‐sample law is used as a stochastic baseline through an effective‐sample‐size closure relating anisotropy to oscillator bandwidth, energetic duration, and period. Monte Carlo simulations verify this approximation under controlled conditions, while analysis of a large set of real ground motion records shows systematic period‐ and duration‐dependent covariance anisotropy consistent with finite‐sample covariance effects. The results further demonstrate that RotD100/RotD50 is not a direct measure of anisotropy: covariance anisotropy develops before peak extraction, upon which directional peak‐selection effects subsequently act. The framework therefore provides a stochastic covariance baseline against which additional physical directionality in recorded motions may be interpreted.