The space industry has developed many composite materials that have high durability in proportion to their weights. Many of these materials have a likelihood for flaws that is higher than in traditional metals. There are also coverings (such as paint) that develop flaws that may adversely affect the performance of the system in which they are used. Therefore there is a need to monitor the soundness of composite structures. To meet this monitoring need, many nondestructive evaluation (NDE) systems have been developed. An NDE system is designed to detect material flaws and make flaw measurements without destroying the inspected item. Also, the detection operation is expected to be performed in a rapid manner in a field or production environment. Some of the most recent video-based NDE methodologies are shearography, holography, thermography, and video image correlation

Temple, Enoch C.

English

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1994
N95-19010
NASA/ASEE SUMMER FACULTY FELLOWSHIP PROGRAM
MARSHALL SPACE FLIGHT CENTER
THE UNIVERSITY OF ALABAMA
THE PROBABILITY OF FLAW DETECTION AND THE PROBABILITY OF FALSE
CALLS IN NONDESTRUCTIVE EVALUATION EQUIPMENT
Prepared By:
Academic Rank:
Institution and Department:
Enoch C. Temple, Ph.D.
Associate Professor
Alabama A&M University
Mathematics Department
NASA/MSFC:
Laboratory:
Division:
Branch:
Materials and Processes
Engineering Physics
Nondestructive Evaluation
MSFC Colleague: Sam Russell, Ph.D.
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1. INTRODUCTION
The space industry has developed many composite materials that have been designed to
have high durability in proportion to their weights. Many of these materials have a likelihood for
flaws that is higher than in traditional metals. There are also material coverings (such as paint)
that develop flaws that may adversely affect the performance of the system in which they are
used. Therefore, there is a need to monitor the soundness of composite structures. To meet this
monitoring need, many nondestructive evaluation (NDE) systems have been developed. An NDE
system is designed to detect material flaws and make flaw measurements without destroying the
inspected item. Also, the detection operation is expected to be performed in a rapid manner in a
field or production environment.
•Within the last few years, several video-based optical NDE methodologies have been intro-
duced. Some of the most recent of these methodologies are shearography, holography, thermog-
raphy, and video image correlation. A detailed description of these may be found in Chu et al.
(1985), Hung (1982), and Russell and Sutton (1989).
The NDE Branch of the Materials and Processes Laboratory at Marshall Space Flight
Center has contracted to purchase video optical NDE equipment. Therefore, that Branch is now
interested in qualifying the performance capability of this equipment. Qualification standards are
found in MIL-STD (1989, draft No. 2) which gives the probability of detection (POD) curve as
the primary qualification measurement. This curve summarizes the POD of a particular equip-
ment type for a wide range of flaw sizes.
Section 2 will discuss what techniques are now available to estimate the POD curve from
sample data. Section 3 will develop needed extensions for shortcomings in POD analysis. Sec-
tion 4 shows how these extensions may be used to solve problems in the NDE Branch.
2. METHODS FOR COMPUTING POD
Figure 1 shows an example of a POD curve along with a lower 95-percent confidence
boundary. The horizontal axis represents flaw size a and the vertical axis expresses probability as
a percentage. As expected, the POD increases with flaw size. The darker curve is labeled POD
and is sometimes called the 50-percent confidence boundary or the estimated mean POD. The
lower lighter curve is called (in this figure), the lower 95-percent confidence boundary.
100 ._- _ _ _ 7 _ _ _ T .__ _ 7 _,_ _ _ _:_ _. _ _,_ T ...t_,_ _ ........
90 : L , i , i ; : t . i !lii__:i i . , . . , . , , . , . i J t i * i , f
50
" i i " : " ' ', ' , . : " ' : .............. i --Connd ....
o__ "7 _ .... _ ............ ,...... _ .... _ ....... _ -:- _ --_ - _-_ _ -:-- .-1 •
0
a axis
Figure 1
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Whenqualifying anNDE equipmentsystem,two majortasksmustbeperformed.Theyare:
(a) usesampledata (perhapsfrom a demonstrationtest) to estimatethe POD curve, and (b)
determineaconfidenceboundaryon theestimatedPODcurve.MIL-STD specifiesa 95-percent
confidenceboundary.
ResearchershavealwaysclassifiedPOD(a)asa randomvariablethat dependson a. This
classification means that if the probability distribution of POD(a) s known, then mean POD and
confidence boundaries are easily computed. Direct approaches used to identify the probability
distribution of POD(a) have met with varying degrees of success. At least three different direct
approaches are discussed by Berens and Hovey (1984). They also described alternative
approaches that avoid the difficulties normally encountered with direct POD(a) distribution
identification procedures. One is the pass/fail method and the other is labeled as the a-hat
method.
Pass/Fail Method
The first method used to estimate the POD curve is described in Packman etal. (1976) and
is based on the binomial distribution. To describe this method, let us assume that there are Ni
identically fabricated flaws for each fixed flaw size ai. An NDE system examines the Ni flaws
ni
and detects ni of them. Binomial theory tells us that Pi = _// is an estimate of POD for flaw size
ai. Packman uses a slightly different definition for Pi and uses a collection of Pi's, i = 1,2 .... k, to
estimate a POD curve and its confidence boundary. A NASA supported study (Yee et al. (1976))
improves Packman's method by using a smoothing procedure. The improved method appears in
the USAF POD software system as subroutine FF. The NDE Branch at MSFC uses this software
system (see Berens et al. (1988)).
A-hat Method
An alternative to the direct characterization of the distribution of POD(a) is to find a proba-
bility distribution for the output signal of the NDE equipment. In fact, if we let fi denote equip-
ment response to the examination of a flaw of size a, experimentation by Berens et al. (1984) has
shown that 8 and a are related by
lna= O_o+oqlna+e , (2.1)
where e denotes a random error term.
Berens also showed that In _ has an approximate normal probability distribution. Hence,
to summarize the probability behavior of equation (2.1), we say that e ~ N(O,cr2), In fi ~
N(/.t(a),cr2) where/.t(a) = OCo+_ 1 In a and N(,) denotes the normal distribution. The value of
POD(a) is computed by
POD(a) = P(ln 8 > a th) , (2.2)
where ath is called a threshold value and
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a th = I.t(a)+ Z p or. (2.3)
Here, Zp is the Pth percentile of the standard normal distribution.
From sample to sample, sample estimates fi(a) and & of parameters in equation (2.3)
change in a random manner. The random behavior of fi(a) and _ makes POD(a) a random vari-
able. (It appears that ath changes from sample to sample. Actually, the probability distribution
changes relative to ath where ath is fixed.) Hence, POD(a) as defined in equation (2.2) has a
probability distribution which, if known, may be used to determine a confidence boundary for
POD(a). However, Cheng and Iles (1983) showed that if/z(a) and a are estimated and their
confidence region determined, a corresponding confidence boundary for POD(a) may be
determined without actually knowing the probability distribution of POD(a). Their approach
used Lagrange multipliers to find maximum and minimum values of ath where _, al, and o" are
restricted to some confidence region. The maximum and minimum values of ath are used to find
corresponding confidence boundaries for POD(a).
3. SHORTCOMING OF POD
The probability of false calls (POF) is the probability that an NDE system will indicate the
presence of a flaw when a flaw does not exist. The POD(a) curve does not assess this situation.
In fact, MIL-STD (1989) left POF evaluation as an open question. The next section uses the rela-
tionship in equation (2.1) to recommend a computation procedure for POD and POF.
4. GENERAL SOLUTION FOR POD_ POF
Let y = In a where a is the output of NDE equipment when a flaw exists. Let X 1 "" In a
where a is actual flaw size. Assume
Y = f(X1, £Zl)+el , (4.1)
where y is normal, X1 is a vector that contains xl as a first component, and E 1 - N(0,cr). In a
similar manner, let w denote the logarithm of the output of NDE equipment when flaw does not
exist. Assume w = g(Xl,ae2)+e 2 where w has normal distribution and e 2 ~ N(O,a2). In func-
tions f and g, parameters _1, a2, and cr are to be estimated from sample data. POD and POF are
based on equation (2.2). That is
POD(a) = P(y > ath) , (4.2)
and
POF(a) = P(w > ath) • (4.3)
After parameters are estimated, confidence bounds for POD(a) and POF(a) may be determined
by procedures outlined in section 2.
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Problem
Qualification standards require that the camera used in video optical equipment be qualified
at a fixed distance between the camera lens and the object being examined. However, the NDE
Branch of the Materials and Processes Laboratory is interested in the POD performance of this
equipment at the tested distance as well as at untested distances. The solution to this problem
requires the use of equation (4.2) which has the ability to predict POD performance of NDE
equipment.
5. RECOMMENDATION
A solution to the problem posed above may be obtained by using equations (4.1) and (4.2).
In this discussion, original variables of equation (2.1) will be used instead of the general vector
variables of equation (4.1). That is, using equation (2.1) along with d which represents the dis-
tance of the camera lens from the examined object, a may be related to a and d by
In a = OCo+O_1 In a+ ot2 d + E 1 = f l (a,d)+ el , (5.1)
or
In a = O_o+C 1 d In a+e 2 = f2(a,d)+e 2 (5.2)
Data inspection will provide a more realistic relationship. Furthermore, analysis of experimental
data will determine if d in equations (5.1) or (5.2) is a significant predictor of In 8. If d is judged
to be a significant variable, then POD(a) and its confidence bounds may be determined for any
fixed distance d. Confidence boundaries are determined as discussed in section 2. However, if
the functions fl or f2 are nonlinear in their parameters, the confidence boundary computation
procedure will depend on the nonlinear function. Actual data for this problem will be analyzed at
a later date.
.
.
.
*
REFERENCES
Berens, A. P., and Hovey, P. W. (1984), "Flaw Detection Reliability Criteria," vol. 1,
Method and Results, AFWAL-TR-84-4022, Air Force Aeronautical Labs.
Berens, A. P., Hovey, P. W., Donahue, R. M., and Craport, W. N. (1988), "User's Manual
for Probability of Detection Software System," UDR-TR-88-12, University of Dayton
Research Institute.
Cheng, R. C. H., and Iles, T. C. (1983), "Confidence Bands for Cumulative Distribution
Functions of Continuous Random Variables," Technometrics, pp. 77-86.
Chu, T. C., Ranson, W. F., Sutton, M. A., and Peters, W. H. (1985), "Applications of
Digital Image Correlation Techniques to Experimental Mechanics," Experimental
Mechanics, pp. 232-244.
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o°
.
8.
.
Hovey, P. W., and Berens, A. P. (1988), "Statistical Evaluation of NDE Reliability in the
Aerospace Industry," 1988 Review of Progress in Quantitative NDE, Williamsburg, VA.
Hung, Y. Y. (1982), "A New Optical Method for Strain Measurement and Nondestructive
Testing," Optical Engineering, pp. 391-395.
MIL-STD (1989, Draft No. 2).
Packman, P. F. et al. (1976), "Reliability of Flaw Detection by Nondestructive Inspection,"
ASM Mental Handbook, vol. 11, 8th edition, pp. 214-224.
Russell, S. S., and Sutton, M. A. (1989), "Strain Field Analysis Acquired Through Correla-
tion of X-Ray Radiographs of Fiber Reinforced Composite Laminates," Experimental
Mechanics.
10. Yee, B. G. et al. (1976), "Assessment of NDE Reliability Data," NASA CR-134991.
XHII-5


The probability of flaw detection and the probability of false calls in nondestructive evaluation equipment

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