Fundamental Statistical Properties and Characterization of Transmissibility Estimates

There is a need for sound theoretical methods to assess uncertainty in transmissibility functions estimated from real-world data, particularly given inherent randomness introduced by measurement errors, estimation variability, and environmental conditions. This chapter presents a unified statistical framework to quantify uncertainties in raw transmissibility estimates modeled as complex ratio variables. The approach is grounded in rigorous mathematics, introducing new theorems on multivariate circularly symmetric complex Gaussian ratio distributions derived through probabilistic transformations of random vectors. Building on this, the framework offers explicit, computationally efficient probabilistic models for local transmissibility estimates at any frequency, applicable to both Gaussian and non-Gaussian stationary processes. It provides analytical marginal probability density functions for the real and imaginary parts, as well as the magnitude and phase, of a univariate local transmissibility estimate. The framework’s effectiveness is validated using real-world case studies, including a high-rise tower and a multi-story building, demonstrating statistical model accuracy. By advancing the quantification of uncertainty in transmissibility estimates, this chapter lays the groundwork for uncertainty-aware algorithms introduced in subsequent chapters. The results bridge theoretical rigor with practical reliability, delivering more insightful tools for structural identification and health monitoring that account for multiple uncertainty sources in engineering contexts.