ASM Metals HandBook Vol. 17 - Nondestructive Evaluation and Quality Control

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Fig. 1 Example crack detection probabilities from 60 eddy current inspections of each crack

The POD(a) function is defined as the proportion of all cracks of size a that will be detected in a particular application of

an NDE system. Assume that each crack of size a in the potential population of cracks has its own distinct crack detection

probability, p, and that the probability density function of the detection probabilities is given by f

(p). Figure 2 shows a

schematic representation of this density. The conditional probability of a randomly selected crack from the population

having detection probability of p and being detected at the inspection is given by p f

(p) dp. The unconditional probability

of a randomly selected crack from the population being detected is the sum of the conditional probabilities over the range

of p, that is:

(Eq 1)

Therefore, POD(a) is the average of the detection probabilities for cracks of size a.

Fig. 2

Schematic of distribution of detection probabilities for cracks of fixed

length

Equation 1 implies that the POD(a) function is the curve through the averages of the individual density functions of the

detection probabilities. This curve is the regression equation and provides the basis for testing assumptions about the

applicability of various POD(a) models. In Ref 4, seven different functional forms were tested for applicability to

available POD data, and it was concluded that the log-logistics (log odds) function best modeled the data and provided an

acceptable model for the data sets of the study. Note that the log odds model is commonly used in the analysis of binary

(hit/miss) data because of its analytical tractability and its close agreement with the cumulative log normal distribution

(Ref 8).

Two mathematically equivalent forms of the log odds model have subsequently been used. The earliest form is given by:

(Eq 2)

This parametrization can also be expressed as:

(Eq 3)

In the Eq 3 form, the log of the odds of the probability of detection (the left-hand side of Eq 3) is expressed as a linear

function of ln (a) and is the source of the name of the log odds model. Note that given the results of a large number of

independent inspections of a large number of cracks, the parameters of the model can be fit with a regression analysis. As

an example, Fig. 3 shows Eq 3 fit to the data of Fig. 1. This regression approach will not be discussed further, because the

maximum likelihood estimates (see the section "Analysis of Hit/Miss Data" in this article) can be applied to much smaller

samples of inspection results and can give equivalent answers for large sample sizes.

Fig. 3 Example linear relation between log odds of crack detection and log crack size

Although the parametrizations of Eq 2 and 3 are sensible in terms of estimation through regression analyses, and are

not easily interpretable in physical terms. A mathematically equivalent form of the log odds POD(a) model is given by

(Ref 8):

(Eq 4)

In this form, μ= ln a

0.5

, where a

0.5

is the flaw size that is detected 50% of the time, that is, the median detectable crack

size. The steepness of the POD(a) function is inversely proportional to ; that is, the smaller the value of σ, the steeper

the POD(a) function. The parameters of Eq 2 and 4 are related by:

(Eq 5)

(Eq 6)

The log odds POD(a) function is practically equivalent to a cumulative log normal distribution with the same parameters,

μ and σ of Eq 4. Figure 4 compares the log odds and cumulative log normal distribution functions for μ= 0 and σ= 1.

Equation 4 is the form of the log odds model that will be used in the section "Analysis of Hit/Miss Data" in this article.

Fig. 4 Comparison of log odds and cumulative log normal models

POD(a) From Signal Response Data. The NDE flaw indications are based on interpreting the response to a

stimulus. In eddy current or ultrasonic systems, the response might be a peak voltage referenced to a calibration. In

fluorescent penetrant inspections, the response would be a combination of brightness and size of the indication. Assume

the response can be quantified and recorded in terms of a parameter, , that is correlated with flaw size. Then

summarizes the information for determining if a positive flaw indication will be given. Only if exceeds a defined

decision threshold,

dec

, will a positive indication be given.

As an example of the concept, Table 2 summarizes the results of highly automated eddy current inspections of 28 cracks

in flat plate specimens. The three data sets resulted from the use of three probes, with all other factors held constant. The

values in Table 2 are the depth of each crack and the peak voltage in counts recorded by the system. Figure 5 shows a plot

of the versus a data for probe A. No signal was recorded for 2 of the cracks, because their values were below the

recording signal threshold,

. These points are indicated by a down arrow at

, indicating that the response was at an

indeterminable value below the recording signal threshold. Similarly, for 5 of the cracks exceeded the saturation limit,

sat

, of the recording system. These points are indicated by an up arrow at

sat

, indicating that the response was at an

indeterminable value above the recording saturation limit. In Fig. 5, the decision threshold is set at 250 counts. Only those

cracks whose value is above 250 would have been flagged (detected).

Table 2 Example of a summary data sheet of versus a data

The example is based on eddy current inspections of flat plates.

Crack depth

Peak voltage in counts

Crack identification

in. Probe A

Probe B

Probe C

11 0.33

0.013

1052 884

1282

30 1.40

0.055

4095 4095

3831

42 0.38

0.015

1480 1182

1699

2 0.25

0.010

723 624

840

21 0.74

0.029

4095 4095

2249

13 0.48

0.019

2621 2401

1101

19 0.30

0.012

377 809

350

26 0.23

0.009

223 205

277

15 0.56

0.022

1654 3319

1289

29 1.65

0.065

4095 4095

2648

33 0.08

0.003

(a)

25 0.25

0.010

669 565

824

32 0.18

0.007

374 379

407

34 0.03

0.001

(a)

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

28 0.23

0.009

638 454

782

27 0.15

0.006

533 385

631

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

(a)

Peak voltage below the recording level threshold

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 g

( ) 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 g

( ), that is:

= (a) +

(Eq 8)

where (a) is the mean of g

( ) and is a random error term accounting for the differences between and (a). The

distributional properties of δ determine the probability density g

( ) 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 ( ) =

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 β

, β

, 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 β

, β

and are also included in the section "Signal Response Analysis."

References cited in this section

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

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

A.P. Berens and P.W. Hovey, Statistical Methods for Estimating Crack Detection Probabili

ties, 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,

p 79-94

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

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

1984

D.R. Cox, The Analysis of Binary Data, Methuen and Co., 1970

NDE Reliability Data Analysis

Alan P. Berens, University of Dayton Research Institute

Design of NDE Reliability Experiments

An NDE reliability experiment comprises a test matrix of inspections on a set of specimens with known flaw locations

and sizes. The specimens are inspected under conditions that simulate as closely as practical the actual application

conditions. The experimental design determines the test matrix, and there are four major analysis concerns to be

addressed in the experimental design. These are:

• The method of controlling the factors to be evaluated in the experiment

• The method of accounting for the uncontrolled factors in the experiment

• The number of flawed and unflawed inspection sites

• The sizes of the flaws in the specimens

These topics are addressed in the following sections.

Controlled and Uncontrolled Factors

The primary objective of NDE reliability experiments has been to demonstrate efficacy for a particular application by

estimating the POD(a) function and its lower 95% confidence bound. (Although NDE reliability experiments can also be

conducted to optimize a system, analyses to meet this objective are beyond the scope of this article.) To demonstrate

capability, it is assumed that the protocol for conducting the inspections is well defined for the application, that the

inspection process is under control (hit/miss decisions are stable over time), and that all other factors introducing

variability into the inspection decision will be representative of the application. The representativeness of these other

factors can be ensured either by controlling the factors during the inspection or randomly sampling the factors to be used

in the experiment. The methods of accounting for these factors are important aspects of the statistical design of the

experiment and significantly influence the statistical properties of the estimates of the POD(a) function parameters. Of

particular note in this regard is that k inspections on n flaws is not equivalent to inspections on n · k different flaws, even

if the inspections are totally independent.

The most important of the factors introducing variation are:

• Differences in physical properties of cracks of nominally identical sizes

•

The basic repeatability of the magnitude of the NDE signal response when a specific crack is

independently inspected by a single inspector using the same equipment

•

The summation of all the human factors associated with the particular inspectors in the population of

interest

• Differences introduced by changes in inspection hardware

These factors must be addressed explicitly or implicitly in every NDE reliability experiment.

In general, the specimens used in NDE reliability experiments are very expensive to obtain and characterize in terms of

the sizes of the flaws in the specimens. Therefore, each experiment is based on one set of specimens containing flawed

and unflawed inspection sites. Because the results are significantly influenced by the specimens, it must be assumed that

the flaws are representative of those that will be present in the structural application. If other factors are to be included in

the experiment, they will be based on repeated inspections of the same flaws. From a statistical viewpoint, this restriction

on the experimental design limits the sample size to the number of flaws in the specimen set. Because different cracks of

the same size can have significantly different crack detection probabilities, multiple inspections of the same crack provide

information about the detection probability of only that crack.