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Application of Bayes Statistics for Reliability Assessment of NDT Systems
Abstract
The assessment of the Probability of Detection (POD) is used to evaluate the reliability of the non-destructive testing (NDT) system. The POD is required in industries, where a missed flaw might cause grave consequences. If only the artificial defects are evaluated, the POD could lead to wrong conclusion or even be invalid. The POD based on real flaws is needed. A small amount of real flaws can lead to a not statistically significant result or even to incorrect results. This work presents an approach to obtain to a significant result for the POD of the current dataset, despite the small amount of real defects. Two steps are necessary to assess a NDT system based on real flaws. First we evaluated the correlation between the NDT signal and the real size of the flaw. Second we use a statistical approach based on the Bayesian statistics to assess a POD in spite of the small amount of data. The approach allows including information of the POD evaluation of artificial defects in the assessment of the POD of real flaws.
The evaluation of non-destructive testing (NDT) methods in terms of reliability is an increasing demand in various industries and applications. The probability of detection (POD) is the most frequently used method for this task. In the testing of holes with low-frequency eddy current
testing, the frequently-used one-parametric POD approach cannot be used because the requirements, mainly the linearity, cannot be met. Therefore, the use of a multi-parametric non-linear regression approach to calculate the POD is proposed. Instead of numerical simulations, commonly used
in multi-parametric approaches, an analytical model is used. The goal of this work is to evaluate the reliability of the eddy current system for the testing of electron-beam welded copper canisters by calculating a POD with the help of different artificial hole-like defects. In this example,
the multi-parametric non-linear regression approach is shown to be successful, enabling the combination of depth and diameter in the POD calculation.
In an earlier article, a method of calculating two-sided confidence bands for cumulative distribution functions was suggested. In this article, the construction of one-sided confidence bands is described. The case of the genera1 location-scale parameter mode1 is discussed, and formulas for the normal and extreme-value models are given as illustrations. A simple numerical example is also included.
The Bayesian approach to uncertainty evaluation is a classical example of the fusion of information from different sources. Basically, it is founded on both the knowledge about the measurement process and the influencing quantities and parameters. The knowledge about the measurement process is primarily represented by the so-called model equation, which forms the basic relationship for the fusion of all involved quantities. The knowledge about the influencing quantities and parameters is expressed by their degree of belief, i.e. appropriate probability density functions that usually are obtained by utilizing the principle of maximum information entropy and the Bayes theorem. Practically, the Bayesian approach to uncertainty evaluation is put into effect by employing numerical integration techniques, preferably Monte-Carlo methods. Compared to the ISO-GUM procedure, the Bayesian approach does not have any restrictions with respect to nonlinearities and calculation of confidence intervals.
During the last decade, the field of assessing the reliability of nondestructive testing (NDT) has blossomed. One area that has received attention is the relationship between the reliability assessment of a test technique undertaken under laboratory conditions and one undertaken in the field. For that reason, attempts were developed to base test reliability measures on field test data only. These attempts were not totally successful. The present study deals with the determination of the reliability of test methods based on scarce field test data but employs a novel strategy that uses bayesian techniques in order to obtain estimates of the probability of detection of discontinuities. In bayesian methods, prior information is updated in light of new data. The bayesian framework developed works explicitly in the domain of the parameters of the log/logistic model for the probability of detection. This paper develops the concepts and provides the theoretical grounding for the procedures proposed to estimate probability of detection curves based on field test data. It also goes beyond the theory and reports on the results of a determination of the probability of detection curve as a function of discontinuity size based on specific field test data. Scenario analysis is used to explore the effect of different assumptions of the prior information on the probability of detection. The conclusions of this study indicate that particular care has to be taken in order to obtain defensible prior information on the reliability of the test technique.
International Journal of Pressure Vessels and Piping Vol.87 Nr. 2 - 3, 111 - 116 The European methodology for qualification of non-destructive testing is a well-established approach adopted by nuclear utilities in many European countries. According to this methodology, qualification is based on a combination of technical justification and practical trials. The methodology is qualitative in nature, and it does not give explicit guidance on how the evidence from the technical justification and results from trials should be weighted. A Bayesian model for the quantification process was presented in a previous paper, proposing a way to combine the “soft” evidence contained in a technical justification with the “hard” evidence obtained from practical trials. This paper describes the results of a pilot study in which such a Bayesian model was applied to two realistic Qualification Dossiers by experienced NDT qualification specialists. At the end of the study, recommendations were made and a set of guidelines was developed for the application of the Bayesian model.
In recent years, Bayesian model updating techniques based on measured data have been applied to system identification of structures and to structural health monitoring. A fully probabilistic Bayesian model updating approach provides a robust and rigorous framework for these applications due to its ability to characterize modeling uncertainties associated with the underlying structural system and to its exclusive foundation on the probability axioms. The plausibility of each structural model within a set of possible models, given the measured data, is quantified by the joint posterior probability density function of the model parameters. This Bayesian approach requires the evaluation of multidimensional integrals, and this usually cannot be done analytically. Recently, some Markov chain Monte Carlo simulation methods have been developed to solve the Bayesian model updating problem. However, in general, the efficiency of these proposed approaches is adversely affected by the dimension of the model parameter space. In this paper, the Hybrid Monte Carlo method is investigated (also known as Hamiltonian Markov chain method), and we show how it can be used to solve higher-dimensional Bayesian model updating problems. Practical issues for the feasibility of the Hybrid Monte Carlo method to such problems are addressed, and improvements are proposed to make it more effective and efficient for solving such model updating problems. New formulae for Markov chain convergence assessment are derived. The effectiveness of the proposed approach for Bayesian model updating of structural dynamic models with many uncertain parameters is illustrated with a simulated data example involving a ten-story building that has 31 model parameters to be updated.