Issue 50

A. Kostina et alii, Frattura ed Integrità Strutturale, 50 (2019) 667-683; DOI: 10.3221/IGF-ESIS.50.57 677 where R is the measurement noise covariance. Updated values of the optimal estimation a k C and error covariance k P are calculated as:      a f f k k k k k C C K z HC (19)     1 f k k k P K H P (20) In order to improve values of the temperature contrast obtained by Kalman filter the Rauch-Tung-Steilbel smoother algorithm [20] was applied as a postprocessing treatment of the obtained data. The algorithm can be summarized as:  ˆ / f k k k k K P A P (21)        1 1 ˆ ˆ ˆ a f k k k k k C C K C C (22)       1 ˆ ˆ ˆ ˆ f k k k k k k P P K P P K (23) where   1,1 k n , ˆ k K is the smoothed gain, ˆ k C is the smoothed value of the temperature contrast, ˆ k P is the error covariance. Let’s illustrate the efficiency of the proposed filtration algorithm. In the previous section, we have obtained reference (noise-free) signals. However, experimental values of temperature contrast are distorted by surface noise induced by non- uniform heating, variation of emissivity, reflections from the environment and so on. In order to simulate experimentally obtained signals we add Gaussian white noise to the reference signal. In this case, Eqn. (8) has the form:   k k k z C v (24) where k z represents noisy signal, k C is the reference signal obtained by numerical simulation, k v denotes white Gaussian noise. It is easy to notice that  1 H . Reliability of non-destructive techniques can be characterized by the signal to noise ratio. The higher this value is the smaller defects can be detected. Therefore, the main goal of the developed technique is to provide a relatively high value of the signal to noise ratio which can help to detect the presence of the defect. Signal to noise ratio ( SNR ) is defined as [21]:  signal noise P SNR P (21) where    2 1 n signal k k P X is the power of the signal,       2 ' 1 n noise k k k P X X is the power of the noise, k X is the k -th point of the discrete reference signal without noise, ' k X is the k -th point of the discrete signal obtained after the processing, n is the number of points in the discrete signal. Fig. 11 illustrates application of the proposed filtration technique to the signal with random additive Gaussian noise. Evolution of the temperature contrast obtained for the defect of 8 mm located at the depth of 0.6 mm (a green curve in Fig. 7 (a)) was used as a reference signal. Values of R =5 and Q =0.001 allows us to achieve a high SNR value which is equal to 929 and reconstruct the shape of the reference signal (Fig. 11 (a)). Fig. 11 (b) displays one more application of the proposed technique to the filtration of the same reference signal corrupted with another noise. Results of the filtration also demonstrate a high value of SNR which is equal to 1150. Therefore, the proposed technique is insensitive to the initial noise and can provide accurate reconstruction of the signal for large sizes of defects.