Issue 50

A. Kostina et alii, Frattura ed Integrità Strutturale, 50 (2019) 667-683; DOI: 10.3221/IGF-ESIS.50.57 675 Figure 10 : Effect of the shape of heating pulse on temperature contrast (1 – Dirac pulse without smoothing, 2 – Dirac pulse with smoothed descending branch, 3 – smoothed Dirac pulse). The conducted research has shown that pulsed thermography is sensitive to the size of the defect, its depth and the heating time. The smaller the defect is and the deeper it lies the higher power of heating is required to identify it. On the other hand, the noise that affects the signal will increase too [15]. Thus, it is necessary to develop simple and effective approaches for its reduction. The results obtained in this section will be used as a “pure” (reference) signals for the investigation of the efficiency of the filtration technique proposed in the next section. D EVELOPMENT OF APPROACH FOR NOISE REDUCTION Kalman-based filtration technique he developed methodology is a continuation of the work [16] on Kalman-based filtration technique for signals obtained by thermal non-destructive testing. The main idea of Kalman filter is to estimate the current state of the dynamic system using information about its previous state and the value of its current measurement. The algorithm can be divided into two steps. In the first step (prediction) filter extrapolates state variables and their uncertainties. In the second step (correction) the result is refined. The algorithm can be used for the in-situ estimation of the object’s state using only current measurements, data on its previous state and its uncertainty. The detailed description of Kalman filter can be found, for example, in [17]. In our case, the state variable is the temperature contrast C . The value of the temperature contrast k C on the k th time step can be estimated according to the following equation [18]:       1 1 k k k C A t C w (7) where   A t is the state transition function,  1 k C is the state of the temperature contrast on the previous step k -1, k w is the process noise. In general, A is the m x m matrix. A measurement k z of the k C is presented in the form:   k k k z HC v (8) where H is a variable that relates the state to the measurement k z , k v is the measurement noise. In general, H is the l x m matrix which can change its value with each time step. Effective application of Kalman algorithm requires formulation of the model for the considered process. The model is based on two analytical solutions which are widely applied to the problems of pulsed thermal non-destructive testing. In case of Dirac pulse heating of the plate without subsurface defects, the temperature on the front surface can be evaluated according to equation [2]:  (t) e s Q T e t  (9) T