Fast Bayesian ambient modal identification in the frequency domain, Part II: Posterior uncertainty

This paper investigates the determination of the posterior covariance matrix of modal parameters within the framework of a Bayesian FFT approach for modal identification using ambient vibration data. The posterior covariance matrix is approximated by the inverse of the Hessian of the negative log-likelihood function (NLLF) with respect to the modal parameters. To suppress the growth of computational effort with the number of measured dofs, a condensed form of the NLLF is derived that only involves matrix computation of dimension equal to the number of modes. Issues associated with the singularity of the Hessian due to mode shape scaling are discussed and a strategy is presented to properly evaluate its inverse. The theory described in Parts I and II of this work is applied to modal identification using synthetic and field data with a moderate to large number of measured dofs. It is demonstrated that using the proposed method Bayesian modal identification can be performed in a matter of seconds in typical cases, which is otherwise prohibitive based on the original formulation.