176 C. Rainieri et al. • Level 3: identification of the type of damage; • Level 4: quantification of damage severity; • Level 5: prediction of the remaining service life of the structure (prognosis). Modal based damage detection starts by recognizing that the modal parameters depend on the physical parameters (mass, stiffness and damping). Assuming that damage yields a change in the physical properties of the structure, this is reflected by a change in the modal properties. Thus, it is theoretically possible to identify damage from the analysis of the variations of the modal parameters. A number of damage sensitive features have been, therefore, defined in terms of modal parameters. Damage sensitive features can be defined in terms of natural frequencies and mode shapes. Natural frequency variations provide the easiest way to detect the presence of damage, because they can be accurately estimated even in the presence of a few sensors. However, the information they provide is limited to Level 1 damage detection. Thus, other features have been defined in terms of mode shapes and mode shape curvatures, because mode shapes can provide information also for damage location. However, they are typically estimated with lower accuracy with respect to natural frequencies. One of the main drawbacks of modal based damage detection is related to the sensitivity of natural frequency estimates to environmental and operational conditions that can cause changes of the same order of magnitude of those induced by damage. As a consequence, the estimates have to be depurated from the effects of environmental factors in order to effectively detect damage. Using a large number of experimental data, models relating modal properties and environmental and operational factors can be set. However, the selection of the factors to be measured is typically not straightforward. As an alternative, statistical tools can be used to correct the estimates without the need to measure those factors. Once the damage sensitive features have been filtered by removing the environmental effects, a number of tools can be applied for feature discrimination. They can be broadly classified as supervised and unsupervised learning approaches [13]. The former are applied when data are available for both the undamaged and damaged structure, while the latter are applied when reference data are available only for the structure in healthy state. In the present paper, after a review of the available approaches to quantify the influence of environmental and operational factors, the opportunity of applying robust blind source separation techniques in this field is assessed. 16.2 Removal of the Influence of Environmental Factors from Natural Frequency Estimates Environmental effects (temperature, humidity, wind, : : : ) and operational factors influence the natural frequency estimates. Such an influence has to be quantified and removed in order to ensure the ability of the continuous monitoring system to detect slight changes induced by damage. Temperature has a major influence on natural frequency estimates, as demonstrated in several studies [14, 15]. In order to remove its effect on natural frequencies, an attractive approach consists in the definition of models able to represent the physical phenomenon behind the frequency changes. In particular, it is possible to rely on black-box models, whose parameters are tuned by collecting a large number of observations, to establish a relation between the natural frequencies and a set of environmental and operational factors. These have to be measured as well. However, the selection of these factors is not straightforward. Whenever the factors influencing the estimates cannot be clearly identified or they cannot be measured, approaches based on statistical tools can be profitably applied to correct the natural frequency estimates without the need of measuring the environmental and operational factors. If measurements of the parameters influencing the estimates are available, approaches based on regression analysis can be applied. For instance, the Multiple Linear Regression can be used to identify the relation between a single dependent variable and several independent variables, the so-called predictors. The established model allows the prediction of future values of the dependent variable when only the predictors are known. In the context of SHM, after the quantification of the influence of environmental factors based on a given set of observations, the predicted natural frequencies are compared with the values directly estimated from the collected acceleration time series in order to remove the environmental influence on the estimates. The following equation characterizes the regression model: fygDŒZ fˇgCf"g (16.1) where the vector fyg collects the n observations of the dependent variable, [Z] is the n-by-p matrix collecting the corresponding n values of p selected independent variables, and f“g is the vector of the unknown coefficients relating the dependent variable to each predictor; finally, f"g is the vector that takes into account the effect of measurement errors and other variables not explicitly considered in the model. The error vector is assumed to have zero mean and constant variance:
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