Fast Bayesian ambient modal identification in the frequency domain, Part I: Posterior most probable value

A Bayesian theory for modal identification using the fast Fourier transform (FFT) of ambient vibration data has been formulated previously. It provides a rigorous means for obtaining modal properties as well as their uncertainties by operating in the frequency domain, which allows a natural partitioning of information according to frequencies. Since there is a one-to-one correspondence between the time-domain data and its FFT, the method can make full use of the relevant information contained in the data. In the context of Bayesian inference, the identification results are in terms of a posterior distribution given the data, which can be characterized by the most probable value and covariance matrix. Determining these quantities, however, requires solving a numerical optimization problem whose dimension grows with the number of measured degrees of freedom; and whose objective function involves repeated inversion of ill-conditioned matrices. These have so far made the approach impractical for applications. For well-separated modes, an efficient algorithm has been developed recently. As a sequel to the development, this work considers the general case of multiple, possibly close modes. This paper focuses on the most probable values and develops an efficient iterative procedure for their determination. Asymptotic behavior of the modal identification problem is also investigated for high signal-to-noise ratios. The companion paper focuses on the posterior covariance matrix and applies the proposed method to simulated and field data.