Directional Peak Factors of Strong-Motion Response Spectra: A Stochastic Field Representation on the Circle

Directional variability in horizontal earthquake ground motions is often addressed using orientation-independent intensity measures obtained by rotating the two horizontal components and summarizing the resulting response spectra. In contrast, the stochastic structure of directional peak factors, which connect second-order response measures to extreme response levels, has received limited attention. This paper presents a systematic empirical characterization of the directional peak-factor field associated with horizontal elastic response spectra. Using a large, uniformly processed strong-motion dataset, directional peak factors are analyzed as a function of oscillator period and orientation. Working in logarithmic space, we decompose the directional peak-factor field into a period-dependent mean and a zero-mean directional fluctuation field that is π-periodic in orientation. Angular dependence is quantified using correlation diagnostics and harmonic decomposition. The results show that directional peak-factor variability exhibits smooth angular structure, with strong correlation across nearby orientations and a consistent zero crossing of angular autocorrelation at 45°. Harmonic analysis demonstrates dominance of the lowest admissible angular mode across all periods considered. The dominant angular harmonic exhibits a consistently nonzero amplitude, weak period dependence, and a phase that is effectively uniform across records. The associated harmonic coefficients are approximately Gaussian and isotropic in the cosine–sine coefficient space, allowing the stochastic structure of directional peak-factor variability to be summarized by a single, smoothly varying scale parameter. This characterization clarifies the dependence structure underlying rotation-based intensity measures and how uncertainty should be interpreted when peak response is summarized across orientation.