Rotation-Invariant Ground-Motion as Directional Selection Operators: A Closed-Form Framework for RotD Response Spectra

Horizontal earthquake ground motion is inherently two-dimensional, yet most engineering applications rely on scalar intensity measures. Rotation-invariant response spectra such as RotD50, MaxRotD50, and RotD100 are widely used to remove dependence on sensor orientation. They are often treated as direction-free scalars, which they are not. In this study, directional pseudo-spectral acceleration is treated instead as a stochastic field on the circle. Rotation-invariant measures are interpreted as directional selection operators acting on that field. Building on empirical evidence that the squared directional response is strongly dominated by its first admissible angular harmonic, we derive closed-form approximations for RotD50, MaxRotD50, and RotD100. This removes the need for explicit directional sampling while making clear what directional information is retained by each operator. Validation against a large ground-motion dataset shows that the closed-form predictions are essentially unbiased across periods and preserve record-to-record variability. A key finding is that max-type rotation-invariant measures depend on an effective anisotropy that includes a stochastic contribution arising from directional peak variability, even when geometric anisotropy is weak. As a result, ratios such as RotD100/RotD50 can be elevated in motions that are nearly isotropic in the root mean square sense. The results identify anisotropy of pseudo-spectral acceleration (PSA), evaluated along the principal directions of root mean square response, as a latent directional state variable. It controls the behaviour of rotation-invariant measures and their ratios across records and periods. Rather than removing directionality, rotation-invariant operators transform it. Recognizing this distinction provides a more transparent and physically interpretable basis for the use of rotation-invariant intensity measures in ground-motion modelling and seismic design.