Circularly-symmetric complex normal ratio distribution for scalar transmissibility functions. Part II: Probabilistic model and validation

• Probabilistic models are established for raw scalar transmissibility functions.
• Explicit formulas are derived analytically for multivariate and univariate cases.
• The statistical structures of the probabilistic models are concise and compact.
• The accuracy is verified using numerical study as well as field test data.
• The models pave the way for developing new statistical methods in structural health monitoring.

In Part I of this study, some new theorems, corollaries and lemmas on circularly-symmetric complex normal ratio distribution have been mathematically proved. This part II paper is dedicated to providing a rigorous treatment of statistical properties of raw scalar transmissibility functions at an arbitrary frequency line. On the basis of statistics of raw FFT coefficients and circularly-symmetric complex normal ratio distribution, explicit closed-form probabilistic models are established for both multivariate and univariate scalar transmissibility functions. Also, remarks on the independence of transmissibility functions at different frequency lines and the shape of the probability density function (PDF) of univariate case are presented. The statistical structures of probabilistic models are concise, compact and easy-implemented with a low computational effort. They hold for general stationary vector processes, either Gaussian stochastic processes or non-Gaussian stochastic processes. The accuracy of proposed models is verified using numerical example as well as field test data of a high-rise building and a long-span cable-stayed bridge. This study yields new insights into the qualitative analysis of the uncertainty of scalar transmissibility functions, which paves the way for developing new statistical methodologies for modal analysis, model updating or damage detection using responses only without input information.