10 Stochastic Wavenumber Estimation: Damage Detection Through Simulated Guided Lamb Waves 115 40 60 80 100 120 30.5 31 31.5 32 32.5 33 33.5 Wavenumber (m-1) Intensity (dB) 0 10 20 30 40 50 60 Intensity (dB) 0 20 40 60 80 100 120 200 180 160 140 120 100 80 60 40 20 0 Wavenumber (m-1) Frequency (kHz) x True wavenumber Spurious peak Fig. 10.9 Experimental variability in on site dataset (left) frequency-wavenumber map and (right) wavenumber-intensity at 80 kHz Fig. 10.10 Hypothetical posterior for a noisy likelihood with (a) uniform prior and (b) informative prior applied to the thickness parameter potential corrosion. The noise manifests itself in the data almost always as spurious peaks at high wavenumbers. Therefore, if there are multiple peaks, those at lower wavenumbers are expected to be representative of the true damage and thus, should be weighted more by the prior. This knowledge is incorporated by implementing a prior distribution to the thickness parameter of the LWP model, effectively providing more weight to low wavenumber peaks and less weight to high wavenumber peaks (which are likely to be related to noise), resulting in a more realistic posterior distribution. One example of the effect of defining such prior knowledge is shown in Fig. 10.10. In this example, the likelihood function, as determined by the MCMC is shown at the top of the figure. The likelihood is shown to have two peaks, representative of the case where multiple peaks occur in the wavenumber-intensity measurements. Two prior distributions of the thickness are shown: one in which a uniform prior is assumed (Fig. 10.10a), representing prior knowledge only of the upper and lower bound on the parameter but no expected behavior in between, and another where expert knowledge regarding the low wavenumber peaks being associated with noise is incorporated (Fig. 10.10b). It is emphasized that for damage to be detected with the prior distribution of Fig. 10.10b, the majority of experimental evidence must be related to severe damage in the structure in order for the posterior to infer severe damage. Thus, in the case where experimental evidence indicates little to no damage with one peak and spurious peaks relate to severe damage,
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