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

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bound. In fact, the least squares estimates from the regression analysis also happen to be the maximum likelihood

estimates. The analysis presented in this section is applicable to the more general case in which some of the values are

censored at the recording threshold or the saturation limit. This more general analysis will give answers identical to those

of the regression analysis if all values are available for all of the flaws (Ref 15).

In the response signal analysis, it is assumed that the values for a flaw of size a have a normal distribution, with mean

and standard deviation given by:

ln (a)

(Eq 39)

(Eq 40)

where does not depend on the crack size. To simplify the notation, let Y

= ln (

) and X

= ln (a

). The random

variable:

(Eq 41)

has a standard normal distribution. Let (z) represent the density function of the standard normal distribution:

(Eq 42)

and (z) represent the cumulative normal distribution:

(Eq 43)

The likelihood function is partitioned into three regions:

• Region R, for which values were recorded

• Region T, for which only a maximum value is known (the

values fall below the recording signal

threshold and cannot be recorded)

• Region S, for which only a minimum value is known (the

values fall above the saturation limit and

cannot be recorded)

These regions are identified by the open circles, the down arrows, and the up arrows, respectively, in Fig. 5. The

likelihood function for the entire sample is the product of the likelihood functions for the three regions:

, ) =

(Eq 44)

But (suppressing the dependency of L on

, and ):

(Eq 45)

(Eq 46)

(Eq 47)

because 1/ (Z

) dz is the probability of observing

for the ith flaw in R,

) is the probability of obtaining an ln

value below the recording threshold for the ith flaw in S, and 1 -

sat

) is the probability of obtaining an value

above the saturation limit for the ith flaw in T. The log of the likelihood function is:

(Eq 48)

where r is the number of cracks in R, that is, the number of cracks for which values were recorded.

Parameter Estimation in Analysis. The maximum likelihood estimates are given by the solutions to:

(Eq 49)

(Eq 50)

(Eq 51)

where:

(Eq 52)

(Eq 53)

Standard numerical methods, such as the Newton-Rhapson iterative procedure (Ref 14), can be used to find the solutions

to Eq 49, 50, and 51. Excellent choices for the initial estimates of iterative procedures are the intercept, slope, and

standard deviation of residuals obtained from a standard regression analysis of only those values for which a valid

response was recorded.

Confidence Bound Calculation in Analysis. Because the POD(a) parameters are calculated from the estimates of

the versus a data, the calculation of the lower confidence bound is a five-step process:

• The information matrix for the estimates of

, and is obtained using Eq 21

• The variance-covariance matrix of

, and is obtained by inverting the information matrix (

20)

• The variance-covariance matrix of the estimates of and are calculated based on a first-

order Taylor

series expansion of the equations relating

, and to and (Eq 12 and 13)

• The information matrix for and is obtained by inverting the variance-covariance matrix to obtain

• The calculated values are substituted into Eq 23, 24, and 25 to obtain the lower confidence bound

The elements of the information matrix for = (

, ) are given by (dropping the subscripts):

(Eq 54)

(Eq 55)

(Eq 56)

(Eq 57)

(Eq 58)

(Eq 59)

(Z) = V(Z) [V(Z) - Z]

(Eq 60)

(Z) = -W(Z) [W(Z) + Z]

(Eq 61)

Let V (

, ) represent the variance-covariance matrix of the maximum likelihood estimates of the ln ( ) versus ln

(a) analysis. The value V (

, ) is obtained from the inverse of the information matrix. Let the elements of V (

, ) be defined by:

(Eq 62)

Using a Taylor series expansion about the true values of and to linearize the relations expressed by Eq 12 and 13, the

variance-covariance matrix of and is given by (Ref 16):

(Eq 63)

and the transformation matrix T is defined by:

(Eq 64)

Multiplying the matrices yields the variances and covariance of and as:

(Eq 65)

(Eq 66)

(Eq 67)

Inverting this variance-covariance yields the values of k

, k

, and k

required in Eq 25 to calculate the lower confidence

bound on the POD(a) function.

Multiple Inspections Per Flaw. Repeat values for the same flaw can be analyzed to estimate the magnitude of the

total variability being introduced by factors other than the flaws in the experiment. In essence, the random term, , of Eq

9, can be partitioned into components that can be estimated if the experiment is properly designed. The relative magnitude

of the components of variance indicates potential areas for improving the system. However, the methods for using the

values to generate POD(a) functions and confidence bounds from complex experiments with censored values are still

under development.

If there are no censored values, the following analysis provides valid estimates of both the POD(a) function and its

confidence bound. If there are censored values, the POD(a) parameter estimates are valid, but the confidence bound is

approximate. Much of the data recorded in the format is from automated systems. In these systems, the variability in

values has been dominated by that associated with different flaws. This variability is correctly analyzed in the following

analysis. In applications to date, the approximation to account for the secondary sources of variability has been judged to

be negligible.

To model an additional source of variability, Eq 9 is rewritten as:

= ln (

) =

ln (a

) + c

+ r

i = l, . . ., n

j = l, . . ., k

(Eq 68)

where

represents the jth observation on the ith flaw, c

is the random deviation of flaw i from the fit, and r

is the

random deviation about c

introduced by replicate j of the source of the repeated observations. For this model of ln ( )

versus ln (a):

= c

+ r

(Eq 69)

Because it is reasonable to assume that the variability introduced by flaws is independent of that introduced by other

factors:

= +

(Eq 70)

where can be estimated as the pooled-within variance of

values for each flaw using those flaws for which values

were recorded (Ref 17):

(Eq 71)

where n* is the number of flaws with uncensored values and k

is the number of uncensored values for flaw i.

Between-crack variability, , cannot be estimated directly, but can be estimated indirectly from a censored regression

analysis. First, the mean log response for each flaw is calculated. This mean response may be a simple average if all

values are available, but a mean based on an analysis of censored data (as previously discussed) will be required for the

flaws for which values were censored at the decision threshold or saturation limit. The analysis in the section

"Parameter Estimation in Analysis" in this article is then used on the model:

+ c

(Eq 72)

to obtain estimates

, and * . However:

(Eq 73)

Solving Eq 73 for and substituting into Eq 70 yields:

(Eq 74)

The value ( * )

is obtained from the censored regression analysis, and is obtained directly from the ln ( ) values.

Equation 74 provides the estimate of to be used in the POD(a) function.

To date, the lower confidence bound has been placed on the POD(a) function using the information matrix derived from

the censored regression through the average of the ln ( ) values for each crack. This procedure does not account for the

added uncertainty resulting from the estimate of . When is significantly greater than , the error introduced by

neglecting this variation is judged to be negligible. (Most of the applications of the analysis have been on highly

automated eddy current systems. In these applications, the variability of ln ( ) values within cracks has been significantly

less than that between cracks.) The method of constructing exact confidence bounds is under development. This process

can be extended to account for more than one component of the variability of ln ( ) values.

Examples of Analysis. The data in Table 2 resulted from three eddy current inspections of 28 fatigue cracks in flat

plates simulating the web/bore of an aircraft engine disk. The three inspections are from the use of three different probes,

with all other factors being held fixed. Table 5 presents a summary of the parameters of the POD(a) function for each data

set in Table 2 and the combination of the three data sets in a single analysis. The ln ( ) versus ln (a) data from probe A

are presented in Fig. 5. This data set had two inspection results below the recording threshold and five results above the

saturation limit. The POD(a) function for probe A and its lower 95% confidence bound are shown in Fig. 10. The

combined results from the three probes yield the composite POD(a) function and its lower 95% confidence bound, which

are also shown in Fig. 10. If the probes were in some sense selected at random from the population of all probes

applicable to this equipment and if other factors were assumed to be representative of the application, this composite

POD(a) function would be representative of the inspection system. Figure 11 shows the individual POD(a) functions for

all three probes and their composite.

Table 5 Example of POD(a) parameters determined from the data in Table 2

The decision threshold is 250 counts.

Probe Parameter

A B C

Composite

-1.89 -1.80 -2.12

-1.96

0.24 0.23 0.40

0.33

, mm (in.)

(a)

0.15 (0.006) 0.17 (0.007)

0.12 (0.005)

0.14 (0.0055)

, mm (in.)

(b)

0.21 (0.0083)

0.22 (0.009)

0.20 (0.008)

0.21 (0.0083)

(a)

= exp( ) = estimate of crack size at 50% POD.

(b)

= exp( + 1.282 ) = estimate of crack size at 90% POD.

(c)

90/95

= upper 95% confidence bound on the estimate of a

Fig. 10 POD(a) function and lower 95% confidence bound from signal response analysis of the data in Table 2

with one inspection per crack (with probe A) and with three inspections per crack (from the composite result of

probes A, B, and C)

Fig. 11 POD(a) functions from signal response analysis of the data in Table 2

Comments on Analysis. There are several advantages of the analysis over the hit/miss analysis in estimating the

POD(a) function. These accrue primarily because of the added information contained in the values.

It is not as critical in the experiment to have flaws with sizes in the range of increase of the POD(a) function. Because the

analysis does not depend on whether or not flaws were detected, the decision threshold can be arbitrarily set after the

experiment. Even if the recording threshold is close to the decision threshold, the method permits extrapolation of the

results from larger flaw sizes to the smaller sizes of interest. The extrapolation is reasonable over the flaw size range for

which the ln ( ) versus ln (a) relation is linear, and deviations from the fit are normally distributed with constant

variance.

A principal advantage of the analysis is that statistical tests of the underlying assumptions are readily available (Ref 17).

In most of the experiments that have been analyzed by this method, the assumptions could not be rejected. However, for

very large flaws, a tendency for the ln ( ) versus ln (a) relation to bend down has been observed on occasion. In this

region, the values are very large, and all of the flaws are easily detected. One method of linearizing the ln ( ) response

is to restrict the analysis to a range of smaller flaw sizes by censoring values at a lower saturation limit. This is

equivalent to deleting data points and may reduce the sample size to an unacceptable number. In this case, the flaw sizes

in the experiment were not appropriate for the application. It should be noted that ignoring this type of nonlinearity tends

to produce a nonconservative POD(a) function, because the effect is to produce a smaller value of the median detectable

crack size, .

When the data do not fit the model, it is possible to obtain results that have no relation to reality. This can occur if ln ( )

is not an increasing linear function of ln (a) or if there are outliers in the data set that have a significant influence on the

analysis. The effects of these anomalies are sometimes manifested in unreasonable values of or a or by a lower

confidence bound that eventually decreases for large crack sizes. As a minimum, it is recommended that a plot of ln ( )

versus ln (a) be obtained for all experiments. This will permit at least a subjective judgment concerning the assumptions

of the analysis.

If the data do not fit the model, it is necessary to ensure that the process is in control and that the experiment was properly

designed and executed. It may also be possible to use other relations between and a as the basis of analysis, but these

have not yet been explored.

References cited in this section

14.

A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, 1965

15.

M. Glaser, Regression Analysis With Dependent Variable Censored, Biometrics, Vol 21, June 1965, p 300-

307

16.

S.R. Searle, Linear Models, John Wiley & Sons, 1971

17.

W.J. Dixon and F.J. Massey, Jr., Introduction to Statistical Analysis, McGraw-Hill, 1957

NDE Reliability Data Analysis

Alan P. Berens, University of Dayton Research Institute

References

B.G.W. Yee, F.H. Chang, J.C. Couchman, G.H. Lemon, and P.F. Packman, "Assessment of NDE

Reliability Data," NASA CR-134991, National Aeronautics and Space Administration, Oct 1976

W.D. Rummel, Recommended Practice for Demonstration of Nondestructive Evaluation (NDE)

Reliability on Aircraft Production Parts, Mater. Eng., Vol 40, Aug 1982, p 922-932

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

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

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

Probabilistic Fracture Mechanics a

nd 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

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

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 1984

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

A.P. Berens and P.W. Hovey, The Sample Size and Flaw Size Effects in NDI Reliability Experiments, in

Review of Progress in Quantitative Nondestructive Evaluation 4B,

D.O. Thompson and D.E. Chimenti,

Ed., Plenum Press, 1985

10. H. Cramer, Mathematical Methods of Statistics, Princeton University Press, 1946

11. J.F. Lawless, Statistical Models and Methods for Lifetime Data, John Wiley & Sons, 1982

12.

R.C.H. Cheng and T.C. Iles, Confidence Bands for Cumulative Distribution Functions of Continuous

Random Variables, Technometrics, Vol 25 (No. 1), Feb 1983, p 77-86

13. R.C.H. Cheng and T.C. Iles, One Sided Confidence Bands for Cumulative Distr

ibution Functions,

Technometrics, Vol 32 (No. 2), May 1988, p 155-159

14. A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, 1965

15. M. Glaser, Regression Analysis With Dependent Variable Censored, Biometrics,

Vol 21, June 1965, p

300-307

16. S.R. Searle, Linear Models, John Wiley & Sons, 1971

17. W.J. Dixon and F.J. Massey, Jr., Introduction to Statistical Analysis, McGraw-Hill, 1957

Models for Predicting NDE Reliability

J.N. Gray, T.A. Gray, N. Nakagawa, and R.B. Thompson, Center for NDE, Iowa State University

Introduction

FRACTURE CONTROL PHILOSOPHIES, as discussed in previous articles in this Section, depend on a damage-tolerant

design. The essential feature of such an approach is the incorporation of redundant load paths so that, even if local failures

do occur, the structure will be safe for a period of time and can either be removed from service or repaired. Parts

containing readily detectable damage can be used if that damage will remain in stable condition until the next inspection

opportunity.

The implementation of such an approach requires the knowledge and integration of stress, flaw size, and failure

mechanisms. As shown in Fig. 1 for the case in which fatigue can be modeled by linear-elastic fracture mechanics, cyclic

stress excursions, Δσ, and flaw size, a, are required as inputs to fracture mechanics to predict a stress intensity range, ΔK.

If the stress intensity exceeds a critical value known as the plane-strain fracture toughness, K

, catastrophic failure is

imminent. Otherwise, crack growth laws of the form da/dN = A(ΔK)

(where N is the number of stress excursions and A

and m are constants) can be used to estimate the safe life available before catastrophic failure.

Fig. 1 Methodology of lifetime prediction for metal parts undergoing cyclic fatigue. Source: Ref 1

Implementation of this damage-tolerant design approach rests on three methodologies: stress analysis, nondestructive

evaluation (NDE), and failure modeling. Incorporation of these methodologies in the design of damage-tolerant