Issue 50

A. Kostina et alii, Frattura ed Integrità Strutturale, 50 (2019) 667-683; DOI: 10.3221/IGF-ESIS.50.57

where 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 ]: e Q is the absorbed energy,  e

   

   

  

  

2

Q

L

e



 1 2



T

R Exp

(t)

(10)

d

r

t 

e t 

 / ( c)  

 is the thermal diffusivity.

r R is the reflection coefficient, L is the depth of the subsurface defect,

where

Therefore, the temperature contrast can be evaluated as:

2

 

 

2 (t) r e R Q

L



 

C

Exp

(11)

t 

e t 

Hence, evolution of the temperature contrast can be written in the form:

  2 2 C L t t   (t) 2 2

 

 C

 

(12)

(t)

 

Using (12) and finite-difference form of the time derivative we can present Eqn. (7) in the form:

   

   

  

  

 L t  2



t

2

(13)

  1

 C w  1 k

C

k

k

2



2

t

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:

 a C C

(14)

0

0

 0 a P P 0

(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

f

C and

k P are calculated:

f

a

 C A C

(16)



k

k k

1

f

 k k k P A P A Q   1 k

(17)

where  2, k n ,

a

C on the k -1 time step, Q is the process noise covariance parameter.

C

is the optimal estimation of k

 1

k

Correction step of the Kalman filter includes calculation of the Kalman gain k K :

f

P H K

k



(18)

k

 2 P H R f

k

676