188
7 Integration of Functionals, PCM and Stochastic IntegralEquations
0
5e-05
0.0001
0.00015
0.0002
0.00025
0.0003
0.00035
0.0004
-4
-3
-2
-1
0
1
2
3
4
Scan position (mm)
Variance in Z12 computed with 10 sample functions
R
X
Fig. 7.16 Variance in Z 12 (ξ 1 , ξ 2 ) computed using the ten sample functions of Fig. 7.15
0
5e-05
0.0001
0.00015
0.0002
0.00025
0.0003
-4
-3
-2
-1
0
1
2
3
4
R
Scan Position (mm)
Variance in Rij computed from ten sample functions
12
13
15
23
25
35
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
-4
-3
-2
-1
0
1
2
3
4
X
Scan Position (mm)
Variance in Xij computed from ten sample functions
12
13
15
23
25
35
Fig. 7.17 Variance in Z ij (ξ i , ξ j ) computed using ten sample functions. The legend denotes the
index-pair, ij
The problem that we are going to address in this section is that of determining
metrics for estimating the ‘detectability’ of a flaw located at, say, 1.625 mm in
Fig. 7.18, compared with one located at the center of the scan. This is a problem of
‘signal detection,’ as described, for example, in [48]. Following [48], we introduce
the notion of ‘hypothesis testing,’ in which we designate by H 0 the ‘null hypothesis’,
that we have detected only random noise in our measurement, and the alternate
hypothesis that we have detected a flaw by H 1 . There is a probability associated
with a true response for each hypothesis, and that is what we will determine next.