A Bayesian Approach to Determine the Minimum Detectable Damage

This paper proposes an approach to the evaluation of the minimum detectable damage, which takes advantage of the Bayes Theorem and of Bayesian Hypothesis Testing. Assuming that some model outputs depending on random parameters are observed, a special application of the Kalman Filter to stationary inverse problems is applied, also called Linear Bayesian Filter, which allows to obtain an analytic formulation of the posterior distribution. A method called HDI+ROPE is used, which is based on a decision rule considering a range of plausible values indicated by the highest density interval of the posterior distribution, and its relation to a region of practical equivalence around the null value. The analytic formula for the minimum detectable damage derives from the limit condition for which it is possible to establish with certainty the presence of damage. In order to validate the formula, an application is developed to a simple linear abstract problem and to a single degree of freedom system, in which the results obtained analytically are compared with those obtained by simulation. This approach could represent a significant step forward in the design of non-destructive tests for existing infrastructures since it allows to put in relationship structural reliability with the reliability of the measurement system, allowing also, in the particular case of Structural Health Monitoring, to consider static and dynamic measurements.