ABSTRACT
The paper makes a research into the uncertainty in the flaw determination and constructs the statistical relationship between the measured flaw sizes by NDT and the true flaw sizes. According to the statistical relationship, the distribution of true flaw sizes is obtained at a given measured flaw size. Furthermore, a quantitative method for determining the true flaw size is also successfully developed by making full use of the measured flaw sizes.Keywords: Inspection, flaw size, distribution, quantitative method, reliability
1. INTRODUCTION
When the integrity of the engineering structure with flaws is assessed, the flaw size in the structures must firstly be inspected accurately. But owing to the effect of random and fuzzy factors, nondestructive inspections, such as ultrasonic testing, carried out in an industrial environment to reveal hidden flaws in structures,' are imperfect'. There are obvious errors between the measured and true flaw sizes, and the measured sizes of the identical flaw are different because of different inspectors. The reliability of ultrasonic inspection depends on several factors. The most important among these are, e.g. the type, size and orientation of the defect, crack morphology, location of the crack, type of material inspected, geometrical restrictions(inaccessibility) in scanning, human factors, inspection procedure and inspection equipment, and environmental conditions in inspection work.
The paper makes a research into the uncertainty in the flaw determination and constructs the mathematical relation between the measured flaw sizes by NDT and the true flaw sizes. The distribution of true flaw sizes is obtained at a given measured flaw size. Furthermore, A quantitative method for determining the true flaw size is also successfully developed by making full use of the measured flaw sizes from different inspectors.
2. THE PROBABILITY MODEL FOR FLAW SIZING
In analyses of a large number of nondestructive size data, a linear relation between the logarithm of the measured flaw size and the logarithm of the true flaw size with normally distributed deviations has proved satisfactory[1]. Furthermore, the use of lognormal distribution is mathematically convenient. and the parameters of the corresponding regression model can also be interpreted physically.
In the lognormal model , the true flaw size, a, is related to the measured flaw size, a^{'}, in the following way[24]:
 (1)

Where e is a normally distributed random error term with zero mean and variance s^{2}, and are the regression parameters.
According to this model, the expected value of the logarithm of the true flaw size is
 (2)

Which means that the median of the true flaw size is equal to exp( l_{o} + l_{1} ln a'). The variance of the logarithm of the true flaw size is assumed to be independent of the flaw size:
 (3)

The model definition given in eqns(1)(3) corresponds to the fact that the true flaw size is a lognormally distributed random variable.
The parameters and s^{2} can be determined by estimating the parameters of the corresponding linear regression model from data consisting of known flaw sizes and corresponding measurement results. Before performing a linear regression, the data must be reduced to a set of n pairs, ( a_{i}', a_{i}), where a_{i}' is the measured flaw size for the ith flaw and a_{i}' is the true flaw size for the ith flaw. Given the n pairs of ( a_{i}', a_{i}) data points to be fit by the regression analysis, the estimates and for l_{o} and l_{1} can be obtained respectively. The formulas for and are
 (4)

 (5)

Where and are given by
 (6)

 (7)

Then the regression equation can be written as
 (8)

The distributions of the estimates and are[2]:
 (9)

 (10)

The covariance of the and are
 (11)

Therefore the mean and variance of are:
 (12)

 (13)

Then
 (14)

Where
 (15)

The estimate ^{2} for the variance s^{2} can be written as
 (16)

Assuming n = n2 then
 (17)

3. THE TRUE FLAW SIZE DISTRIBUTION
4. THE EXPRESSION METHOD OF THE FLAW SIZE
5. EXAMPLE
The ultrasonic inspection results of the stainless steel component is given in Table 1. The true length of the crack can be obtained by anatomizing specimen[5].
No.i
 1
 2
 3
 4
 5
 6
 7
 8
 9
 10
 11
 12

True length a_{i}
 7.6
 2.9
 9.0
 1.6
 9.2
 2.1
 6.6
 1.9
 1.8
 1.8
 1.3
 2.9

Inspection length a_{i}
 9.0
 1.5
 10.0
 1.0
 8.0
 1.5
 6.0
 1.5
 1.5
 2.0
 1.0
 1.0

Table 1: The ultrasonic inspection results of the stainless steel component(mm) 
According to the linear regression analysis method, the regression equation of crack true length a_{i} and inspection length a_{i}' can be written as
 (33)

and the standard deviation estimator for s is 0.2728。
If the identical crack with the same weld is inspected ultrasonically m times independently, the crack length are respectively, according to the method mentioned above, the distribution of the true crack length can be expressed as
 (34)

Then the crack length a_{R} with reliability R is
 (35)

 (36)

Likewise, the oneside confidence upper limit of the crack length a_{RU} with confidence g and reliability R can be obtained by
 (37)

 (38)

6. CONCLUSION
In this paper, the uncertainty in the flaw determination was studied and the model for flaw sizing on the basis of statistical relationship between the measured and true flaw size was constructed. The model discussed is based on the lognormal model. Using the model, it is possible to take into account the prior information of the flaw size and combine it with the measurement results.
Through the model, the distribution of true flaw sizes is obtained at a given measured flaw size Furthermore, A quantitative method for determining the true flaw size is also successfully developed by making full use of the measured flaw sizes from different inspectors. According to the method, the flaw size with confidence g and reliability R can be obtained easily. One example is given to illustrate the effectiveness of the model and the method.
REFERENCES
 Berens,A. P., NDE reliability data analysis. In Metals Handbook, 9th edn, Vol.17, ASM Int., 1989,pp.689701.
 Ding Keqin, Study on fuzzy R6 failure assessment method for the pressure vessel in service. Postdoctoral research report, Beijing University of Aeronautics & Astronautics, 1999.10. (in Chinese)
 Fu Huiming,Ding Keqin and Liu Dengdi, The fuzzy expression method of crack size. Journal of Mechanical Strength, 2000, 22(1) (in Chinese)
 Fu Huiming Liu dengdi, Fuzzy theory of nondestructive testing and its application. 1999, 14(1), 225231.(in Chinese)
 Kaisa Simoda, Urho Pulkkinen, Models for nondestructive inspection data. Reliability Engineering & System Safety. 1998,60: 112.