186
7 Integration of Functionals, PCM and Stochastic IntegralEquations
z 11 = 0.25[Z(−0.5, −0.5) + Z(0.5, −0.5) + Z(−0.5, 0.5) + Z(0.5, 0.5)]
−[Z(0.0, −0.5) + Z(−0.5, 0.0) + Z(0.5, 0.0) + Z(0.0, 0.5)]
+4.00Z(0.0, 0.0)
z 12 = −0.25[Z(−0.5, −0.5) + Z(0.5, −0.5)]
+[Z(0.0, −0.5) + Z(−0.5, 0.0) + Z(0.5, 0.0)]
−4.00Z(0.0, 0.0) − 1.25[Z(−0.5, 0.5) + Z(0.5, 0.5)] + 5.00Z(0.0, 0.5)
z 20 = 1.25[Z(−0.5, −0.5) + Z(0.5, 0.5)] − 5.00[Z(0.0, −0.5) + Z(0.5, 0.0)]
+6.25Z(0.5, −0.5)
−[Z(−0.5, 0.0) + Z(0.0, 0.5)] + 4.00Z(0.0, 0.0) + 0.25Z(−0.5, 0.5)
z 21 = −0.25[Z(−0.5, −0.5) + Z(−0.5, 0.5)] + [Z(0.0, −0.5) + Z(−0.5, 0.0)
+Z(0.0, 0.5)]
−1.25[Z(0.5, −0.5) + Z(0.5, 0.5)] − 4.00Z(0.0, 0.0) + 5.00Z(0.5, 0.0)
z 22 = 0.25Z(−0.5, −0.5) − [Z(0.0, −0.5) + Z(−0.5, 0.0)] + 1.25[Z(0.5, −0.5)
+Z(−0.5, 0.5)]
+4.00Z(0.0, 0.0) − 5.00[Z(0.5, 0.0) + Z(0.0, 0.5)] + 6.25Z(0.5, 0.5) (7.56)
We compute the nine nodal responses for each of the six two-dimensional
functions listed above (7.51) using VIC-3D®, and in Fig. 7.15 we show ten
sample functions of R and X for Z 12 (ξ 1 , ξ 2 ) computed following this procedure.
Figure 7.16 shows the variance computed from these ten sample functions.
In a similar manner, we compute the variances of Z 13 (ξ 1 , ξ 3 ), Z 15 (ξ 1 , ξ 5 ),
Z 23 (ξ 2 , ξ 3 ), Z 25 (ξ 2 , ξ 5 ), and Z 35 (ξ 3 , ξ 5 ) using ten sample functions of each, and
plot the results in Fig. 7.17. It is interesting to note that only those pairs of random
variables that involve ξ 2 contribute a non-null response, and this indicates why this
algorithm is referred to as ‘analysis of variance.’ It clarifies which variables, whether
singly or jointly, contribute significantly to the variance of the process.
The total variance associated with the one- and two-dimensional functions is
obtained by adding the results of Figs. 7.13 and 7.17, and is shown in Fig. 7.18.
7.10 Probability of Detection and the Chebychev Inequality
If we assume, as is typical, that a flaw is ‘detected’ if its response exceeds the
uncertainty in the background, as in Fig. 7.19, then it is clear from Fig. 7.18 that
a flaw whose peak signal is away from the center will be obscured by the random
clutter to a greater extent than a flaw whose peak signal is centered on the random
surface. We assume that the variance of a random process defines its uncertainty or
‘noise level.’
7 Integration of Functionals, PCM and Stochastic IntegralEquations
z 11 = 0.25[Z(−0.5, −0.5) + Z(0.5, −0.5) + Z(−0.5, 0.5) + Z(0.5, 0.5)]
−[Z(0.0, −0.5) + Z(−0.5, 0.0) + Z(0.5, 0.0) + Z(0.0, 0.5)]
+4.00Z(0.0, 0.0)
z 12 = −0.25[Z(−0.5, −0.5) + Z(0.5, −0.5)]
+[Z(0.0, −0.5) + Z(−0.5, 0.0) + Z(0.5, 0.0)]
−4.00Z(0.0, 0.0) − 1.25[Z(−0.5, 0.5) + Z(0.5, 0.5)] + 5.00Z(0.0, 0.5)
z 20 = 1.25[Z(−0.5, −0.5) + Z(0.5, 0.5)] − 5.00[Z(0.0, −0.5) + Z(0.5, 0.0)]
+6.25Z(0.5, −0.5)
−[Z(−0.5, 0.0) + Z(0.0, 0.5)] + 4.00Z(0.0, 0.0) + 0.25Z(−0.5, 0.5)
z 21 = −0.25[Z(−0.5, −0.5) + Z(−0.5, 0.5)] + [Z(0.0, −0.5) + Z(−0.5, 0.0)
+Z(0.0, 0.5)]
−1.25[Z(0.5, −0.5) + Z(0.5, 0.5)] − 4.00Z(0.0, 0.0) + 5.00Z(0.5, 0.0)
z 22 = 0.25Z(−0.5, −0.5) − [Z(0.0, −0.5) + Z(−0.5, 0.0)] + 1.25[Z(0.5, −0.5)
+Z(−0.5, 0.5)]
+4.00Z(0.0, 0.0) − 5.00[Z(0.5, 0.0) + Z(0.0, 0.5)] + 6.25Z(0.5, 0.5) (7.56)
We compute the nine nodal responses for each of the six two-dimensional
functions listed above (7.51) using VIC-3D®, and in Fig. 7.15 we show ten
sample functions of R and X for Z 12 (ξ 1 , ξ 2 ) computed following this procedure.
Figure 7.16 shows the variance computed from these ten sample functions.
In a similar manner, we compute the variances of Z 13 (ξ 1 , ξ 3 ), Z 15 (ξ 1 , ξ 5 ),
Z 23 (ξ 2 , ξ 3 ), Z 25 (ξ 2 , ξ 5 ), and Z 35 (ξ 3 , ξ 5 ) using ten sample functions of each, and
plot the results in Fig. 7.17. It is interesting to note that only those pairs of random
variables that involve ξ 2 contribute a non-null response, and this indicates why this
algorithm is referred to as ‘analysis of variance.’ It clarifies which variables, whether
singly or jointly, contribute significantly to the variance of the process.
The total variance associated with the one- and two-dimensional functions is
obtained by adding the results of Figs. 7.13 and 7.17, and is shown in Fig. 7.18.
7.10 Probability of Detection and the Chebychev Inequality
If we assume, as is typical, that a flaw is ‘detected’ if its response exceeds the
uncertainty in the background, as in Fig. 7.19, then it is clear from Fig. 7.18 that
a flaw whose peak signal is away from the center will be obscured by the random
clutter to a greater extent than a flaw whose peak signal is centered on the random
surface. We assume that the variance of a random process defines its uncertainty or
‘noise level.’