In a globally identifiable Bayesian system identification problem, the uncertainty of model parameters can be quantified by their posterior covariance matrix calculated for a particular data set. When the data is modeled to be distributed as the likelihood function (i.e., no modeling error), a statistical law analogous to the law of large numbers results, where the posterior covariance matrix is asymptotic to a deterministic quantity that depends on the information content of data rather than its particular (stochastic) details. This was referred as the uncertainty law in a recent study of the achievable precision of modal parameters in operational modal analysis (OMA). Deriving the uncertainty law involves asymptotics techniques and leveraging on the mathematical structure of the likelihood function, which was found to be tedious. As a sequel to the development, this work shows that for long data and up to a Gaussian approximation of the posterior distribution, the uncertainty law is asymptotic to the inverse of the Fisher information matrix, which coincides with the tightest Cramer-Rao bound in classical statistics. A parametric study is presented to illustrate the theoretical results in the context of OMA. As a direct application with practical relevance, the relationship provides a systematic means for deriving the uncertainty laws in OMA. Applied and interpreted properly, the posterior covariance matrix (for given data), uncertainty law, and Cramer-Rao bound can provide a powerful means for quantifying and managing the uncertainties in structural health monitoring.

Brownjohn, James Mark William
Au, Siu-Kui
Zhu, Yichen
Sun, Zhen
Li, Binbin
Bassitt, James
Hudson, Emma
Sun, Hongbin

Vibration testing of long span bridges is becoming a commissioning requirement, yet such exercises represent the extreme of experimental capability, with challenges for instrumentation (due to frequency range, resolution and km-order separation of sensor) and system identification (because of the extreme low frequencies). The challenge with instrumentation for modal analysis is managing synchronous data acquisition from sensors distributed widely apart inside and outside the structure. The ideal solution is precisely synchronised autonomous recorders that do not need cables, GPS or wireless communication. The challenge with system identification is to maximise the reliability of modal parameters through experimental design and subsequently to identify the parameters in terms of mean values and standard errors. The challenge is particularly severe for modes with low frequency and damping typical of long span bridges. One solution is to apply 'third generation' operational modal analysis procedures using Bayesian approaches in both the planning and analysis stages. The paper presents an exercise on the Jiangyin Yangtze River Bridge, a suspension bridge with a 1385 m main span. The exercise comprised planning of a test campaign to optimise the reliability of operational modal analysis, the deployment of a set of independent data acquisition units synchronised using precision oven controlled crystal oscillators and the subsequent identification of a set of modal parameters in terms of mean and variance errors. Although the bridge has had structural health monitoring technology installed since it was completed, this was the first full modal survey, aimed at identifying important features of the modal behaviour rather than providing fine resolution of mode shapes through the whole structure. Therefore, measurements were made in only the (south) tower, while torsional behaviour was identified by a single measurement using a pair of recorders across the carriageway. The modal survey revealed a first lateral symmetric mode with natural frequency 0.0536 Hz with standard error +/- 3.6% and damping ratio 4.4% with standard error +/- 88%. First vertical mode is antisymmetric with frequency 0.11 Hz +/- 1.2% and damping ratio 4.9% +/- 41%. A significant and novel element of the exercise was planning of the measurement setups and their necessary duration linked to prior estimation of the precision of the frequency and damping estimates. The second novelty is the use of the multi-sensor precision synchronised acquisition without external time reference on a structure of this scale. The challenges of ambient vibration testing and modal identification in a complex environment are addressed leveraging on advances in practical implementation and scientific understanding of the problem. (C) 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license

Abstract Modern engineering systems are becoming increasingly complex. Assessing their risk by simulation is intimately related to the efficient generation of rare failure events. Subset Simulation is an advanced Monte Carlo method for risk assessment and it has been applied in different disciplines. Pivotal to its success is the efficient generation of conditional failure samples, which is generally non-trivial. Conventionally an independent-component Markov Chain Monte Carlo (MCMC) algorithm is used, which is applicable to high dimensional problems (i.e., a large number of random variables) without suffering from ‘curse of dimension’. Experience suggests that the algorithm may perform even better for high dimensional problems. Motivated by this, for any given problem we construct an equivalent problem where each random variable is represented by an arbitrary (hence possibly infinite) number of ‘hidden’ variables. We study analytically the limiting behavior of the algorithm as the number of hidden variables increases indefinitely. This leads to a new algorithm that is more generic and offers greater flexibility and control. It coincides with an algorithm recently suggested by independent researchers, where a joint Gaussian distribution is imposed between the current sample and the candidate. The present work provides theoretical reasoning and insights into the algorithm.

Operational modal analysis (OMA) has gained popularity for identifying the modal properties of a structure for its high economy and feasibility. Conventionally, time synchronisation among data channels is required to determine mode shape. OMA can be conducted more flexibly if synchronisation is not required. The power spectral density (PSD) matrix of data and its spectral properties are often used for analysing potential modes. Conventionally known properties assume synchronous data and do not carry over to asynchronous data. This paper investigates the spectral properties of asynchronous OMA data. A stationary process with imperfect coherence is proposed that is conducive to OMA while capturing the key asynchronous characteristics. The theoretical properties of PSD matrix are derived and validated using synthetic and experimental data. Although conventional methods do not allow mode shape to be determined from asynchronous data, the present work reveals the possibility by noting that the data are measured under the same excitation and hence share a common PSD in the modal force. On this basis, a simple method is proposed for determining the mode shape. For perfectly incoherent data channels, it is not possible to determine the relative sense of their mode shape values, which is a fundamental limitation of such data. In implementation, the sense can be determined from intuition or estimated from the residual coherence between channels. Experimental application reveals practical issues in OMA with asynchronous data. This work aspires to provide the pathway for more flexible implementation of OMA, for example, using asynchronous data from multiple smart phones.

Ambient vibration tests have attracted increasing attention over the last few decades because they can be performed economically with the structure under working condition without artificial loading. Ambient modal identification techniques do not require knowledge of the loading but they assume that it is statistically random. A Bayesian approach provides a fundamental means for extracting the information in the data to yield information about the modal parameters consistent with modeling assumptions. Issues do exist in the implementation and interpretation of results. This paper presents an overview of a Bayesian frequency-domain approach for ambient modal identification. Issues of theoretical, computational and practical nature are discussed, drawing experience from field applications. [All rights reserved Elsevier].

This paper presents a study on the Bayesian spectral density method for operational modal analysis. The method makes Bayesian inference of the modal properties by using the sample power spectral density (PSD) matrix averaged over independent sets of ambient data. In the typical case with a single set of data, it is divided into non-overlapping segments and they are assumed to be independent. This study is motivated by a recent paper that reveals a mathematical equivalence of the method with the Bayesian FFT method. The latter does not require averaging concepts or the independent segment assumption. This study shows that the equivalence does not hold in reality because the theoretical long data asymptotic distribution of the PSD matrix may not be valid. A single time history can be considered long for the Bayesian FFT method but not necessarily for the Bayesian PSD method, depending on the number of segments. (C) 2015 Elsevier Ltd. All rights reserved.

Slope failure mechanisms (e.g., why and where slope failure occurs) are usually unknown prior to slope stability analysis. Several possible failure scenarios (e.g., slope sliding along different slip surfaces) can be assumed, leading to a number of scenario failure events of slope stability. How to account rationally for various scenario failure events in slope stability reliability analysis and how to identify key failure events that have significant contributions to slope failure are critical questions in slope engineering. In this study, these questions are resolved by developing an efficient computer-based simulation method for slope system reliability analysis. The proposed approach decomposes a slope system failure event into a series of scenario failure events representing possible failure scenarios and calculates their occurrence probabilities by a single run of an advanced Monte Carlo simulation (MCS) method, called generalized Subset Simulation (GSS). Using GSS results, representative failure events (RFEs) that are considered relatively independent are Identified from scenario failure events using probabilistic network evaluation technique. Their relative contributions are assessed quantitatively, based on which key failure events are determined. The proposed approach is illustrated using a soil slope example and a rock slope example. It is shown that the proposed approach provides proper estimates of occurrence probabilities of slope system failure event and scenario failure events by a single GSS run, which avoids repeatedly performing simulations for each failure event. Compared with direct MCS, the proposed approach significantly improves computational efficiency, particularly for failure events with small failure probabilities. Key failure events of slope stability are determined among scenario failure events in a cost-effective manner. Such information is valuable in making slope design decisions and remedial measures. (C) 2017 Elsevier Inc. All rights reserved.

Assembling (or 'gluing') mode shapes identified from multiple setups is a problem frequently encountered in full-scale modal tests that cover a large number of locations with a limited number of sensors. Mode shapes identified in individual setups can have different sense and scaling. Depending on the number of reference degrees-of-freedom (dofs) and the quality of identified mode shapes, implementation issues can arise when determining the optimal mode shape that compromise among different setups. This paper presents a theory with an automated procedure for determining the optimal mode shape that fits the mode shapes identified from multiple setups in a least square sense. The measure-of-fit function is defined as the squared difference between the theoretical and identified mode shapes suitably oriented and scaled to the same norm. Due to the nonlinear nature of the objective function, the optimal mode shape cannot be determined analytically as in conventional least square problems. A fast iterative procedure is proposed, making use of partially optimal solutions that can be derived analytically. The proposed method can be implemented in an automated manner without the need to select the reference dof or setup for scaling purpose. It is applied to assembling mode shapes identified from ambient vibration tests of two full-scale structures. (C) 2010 Elsevier Ltd. All rights reserved.

Pollutants that are chemically inert flow with the carrier fluid passively while diffuse at the same time. In this study, the stochastic diffusion behavior of the passive pollutant in a progressive or standing wave field is examined with analytical means. Our focus is on the nonlinear interactions between the stochastic diffusion and the deterministic wave motions, and we limit the scope to cases whereby a small parameter, e, exists between the advective and diffusive displacements, which then allows a perturbation analysis to be performed. With a sinusoidal progressive wave, the results show that the deterministic wave motion can either increase or decrease the embedded stochastic diffusion depending on the wave characteristics. Longer wave lengths and shorter wave periods tend to promote diffusion significantly, while shorter wave lengths and longer wave periods act in the opposite manner but with a much smaller effect. An analysis of the standing wave motion, represented by a combination of left and right moving progressive waves, shows that the effects due to two opposing waves to the stochastic diffusion can be superimposed. (C) 2010 Elsevier Ltd. All rights reserved.

This paper presents observations on the identified modal properties of two tall buildings using ambient vibration data collected during strong wind events. A recently developed fast Bayesian frequency domain method is used for modal identification based on the measured ambient data. The approach views modal identification as an inference problem where probability is used as a measure for the relative plausibility of outcomes given a model of the structure and measured data. Focusing on the first three modes, the modal properties of the buildings are identified on non-overlapping time windows during the strong wind events, spanning periods of normal to high wind speeds. Investigation of the identified natural frequencies and damping ratios versus the modal root-mean-square value indicates a significant trend that is statistically repeatable across events. (C) 2011 Elsevier Ltd. All rights reserved.

This paper develops a more general reliability-based design approach for drilled shafts that formulates the design process as an expanded reliability problem in which Monte Carlo simulations (MCS) are used in the design. Basic design parameters, such as the shaft diameter (B) and depth (D), are formulated as discrete uniform random variables. Then the design process becomes one in which failure probabilities are developed for various combinations of B and D [i.e., conditional probability p(Failure vertical bar B, D)] and are compared with a target probability of failure p(T). Equations are derived for this expanded reliability-based design (RBD(E)) approach, and criteria are established for the minimum number of MCS samples to ensure a desired level of accuracy. Its usefulness is illustrated using a drilled shaft design example. This RBD(E) approach has the following advantages: (1) it gives results that agree well with current RBD designs, but it improves the resolutions of the designs; (2) it offers design engineers insight into how the expected design performance level changes as B and D change; (3) it gives design engineers the ability to adjust p(T), without additional calculation effort, to accommodate specific needs of a particular project; and (4) it is transparent and "visible" to design engineers who are given the flexibility to include uncertainties deemed appropriate. Finally, the effects of uncertainties in the at-rest horizontal soil stress coefficient (K(0)) and allowable displacement (y(a)) are illustrated using this approach.

This paper describes a process for seismic risk assessment and identification of critical links of water supply systems subjected to earthquakes. Probabilistic performance of water supply systems is reflected by the system serviceability index (SSI), damage consequence index (DCI), and upgrade benefit index (UBI). The process is illustrated using a hypothetical water supply system subjected to a seismic damage scenario, where direct Monte Carlo simulation is used for estimating the performance indices. It is shown that probabilistic characteristics of SSI can be attributed to system characteristics (e.g., demand distribution pattern) of the water supply system. UBI is shown to be the primary index in seismic mitigation, and critical links are pipes with relatively large UBI values. It is recognized that the values of UBI corresponding to different upgrade scenarios of pipes can be estimated using conditional samples from a single run of direct Monte Carlo simulation instead of repeated runs. The concept of efficient frontier is employed to identify the system critical links. It is found that, a group of links that have the largest UBI individually do not necessarily have the largest group UBI, nor are they necessarily the group of critical links.