Issue 50

A. Kostina et alii, Frattura ed Integrità Strutturale, 50 (2019) 667-683; DOI: 10.3221/IGF-ESIS.50.57 676 where e Q is the absorbed energy,  e c  is the effusivity, t is the time. To find the temperature response of the surface in case of the presence of subsurface defects, the following relation can be applied [ 13 ]:                  2 (t) 1 2 e d r Q L T R Exp t e t   (10) where r R is the reflection coefficient, L is the depth of the subsurface defect,  / ( c)    is the thermal diffusivity. Therefore, the temperature contrast can be evaluated as:        2 2 (t) r e R Q L C Exp t e t   (11) Hence, evolution of the temperature contrast can be written in the form:          2 2 (t) 2 (t) 2 C L t C t   (12) Using (12) and finite-difference form of the time derivative we can present Eqn. (7) in the form:                     2 1 2 2 1 2 k k k t L t C C w t   (13) where  t is the step size. Thus, our system is described by Eqns. (13) and (8) which are the base for the Kalman filtration technique. According to the Kalman algorithm [19], the initial values of the optimal state 0 a C and the error covariance 0 a P should be provided:  0 0 a C C (14)  0 0 a P P (15) where 0 C is the initial state, 0 P is the initial error covariance. The next step is the prediction where values f k C and f k P are calculated:   1 f a k k k C A C (16)    1 f k k k k P A P A Q (17) where  2, k n ,  1 a k C is the optimal estimation of k C on the k -1 time step, Q is the process noise covariance parameter. Correction step of the Kalman filter includes calculation of the Kalman gain k K :   2 f k k f k P H K P H R (18)