39 0.18 0.007 409 387 586
12 0.28 0.011 895 690 677
38 0.20 0.008 374 301 549
20 0.79 0.031 4095 4095 1778
27 0.15 0.006 533 385 631 1 0.08 0.003 150 136 135 1 0.08 0.003 150 136 135 35 0.28 0.011 749 660 989 40 0.20 0.008 433 378 591 31 0.36 0.014 879 888 1402 3 0.23 0.009 286 211 352 7 0.23 0.009 298 163 215 16 0.41 0.016 1171 1110 1628 37 2.54 0.100 4095 4095 4095
Fig. 5 Example inspection signal response as a function of crack depth
The POD(a) function can be obtained from the relation between and a. If ga( ) represents the probability density of the values for fixed crack size a, then:
(Eq 7)
This calculation is illustrated in Fig. 6, in which the shaded area under the density functions represents the probability of detection.
Fig. 6 Schematic of POD(a) calculation from versus a relation
In general, the correlating function between and a defines the mean of ga( ), that is:
= (a) + (Eq 8)
where (a) is the mean of ga( ) and is a random error term accounting for the differences between and (a). The distributional properties of δ determine the probability density ga( ) about μ (a), as will be shown.
In the data analyzed to date, a linear relation between ln ( ) and ln (a) with normally distributed deviations has proved satisfactory (for example, Fig. 5). This model is expressed by:
ln ( ) = 0 + 1 ln (a) + (Eq 9)
where δ is normally distributed with zero mean and constant standard deviation, . Data have been observed that flatten at the large crack sizes. However, because the decision threshold was far below the non-linear range, restricting the range of cracks to smaller sizes permitted the application of Eq 9. The normality of has proved to be an acceptable assumption.
Assuming that the versus a relation is modeled by Eq 9 and that is normally distributed with zero mean and standard deviation of , the POD(a) function is calculated as:
(Eq 10)
where is the standard normal distribution function. Using the symmetry properties of , Eq 10 can be reduced to:
(Eq 11)
Equation 11 is a cumulative log normal distribution function with mean and standard deviation of log crack length given by:
(Eq 12)
(Eq 13)
In the section "Signal Response Analysis" in this article, maximum likelihood methods for estimating β0, β1, and σ from versus a data will be presented. Note that the values below the recording threshold and above the saturation limit must be properly accounted for in these analyses. Note also that data from multiple inspections of the same cracks require analysis methods that are dependent on the design of the reliability experiment. Methods for placing lower confidence bounds on the estimated POD(a) function using the sampling distributions of the maximum likelihood estimates of β0, β1, and are also included in the section "Signal Response Analysis."
References cited in this section
3. W.H. Lewis, W.H. Sproat, B.D. Dodd, and J.M. Hamilton, "Reliability of Nondestructive Inspections--Final Report," SA-ALC/MME 76-6-38-1, San Antonio Air Logistics Center, Kelly Air Force Base, Dec 1978 Report," SA-ALC/MME 76-6-38-1, San Antonio Air Logistics Center, Kelly Air Force Base, Dec 1978 4. A.P. Berens and P.W. Hovey, "Evaluation of NDE Reliability Characterization," AFWAL-TR-81-4160, Vol
1, Air Force Wright-Aeronautical Laboratories, Wright-Patterson Air Force Base, Dec 1981
5. A.P. Berens and P.W. Hovey, Statistical Methods for Estimating Crack Detection Probabilities, in
Probabilistic Fracture Mechanics and Fatigue Methods: Applications for Structural Design and Maintenance, STP 798, J.M. Bloom and J.C. Ekvall, Ed., American Society for Testing and Materials, 1983, Maintenance, STP 798, J.M. Bloom and J.C. Ekvall, Ed., American Society for Testing and Materials, 1983, p 79-94
6. D.E. Allison et al., "Cost/Risk Analysis for Disk Retirement--Volume I," AFWAL-TR-83-4089, Air Force Wright-Aeronautical Laboratories, Wright-Patterson Air Force Base, Feb 1984 Wright-Aeronautical Laboratories, Wright-Patterson Air Force Base, Feb 1984
7. A.P. Berens and P.W. Hovey, "Flaw Detection Reliability Criteria, Volume I--Methods and Results," AFWAL-TR-84-4022, Air Force Wright-Aeronautical Laboratories, Wright-Patterson Air Force Base, April AFWAL-TR-84-4022, Air Force Wright-Aeronautical Laboratories, Wright-Patterson Air Force Base, April 1984