The αδ method for reliability modelling and improvement of NDT-tools for Risk Based Inspection: application to corroded structures

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The αδ method for reliability modelling and

improvement of NDT-tools for Risk Based Inspection:

application to corroded structures

Franck Schoefs, Jérôme Boéro

To cite this version:

Franck Schoefs, Jérôme Boéro. The αδ method for reliability modelling and improvement of

NDT-tools for Risk Based Inspection: application to corroded structures. 11th International Conference on

Applications of Statistics and Probability in Civil Engineering (ICASP’11), 2011, Zurich, Switzerland.

�hal-01009013�

The

αδ method for reliability modeling and improvement

of NDT-tools for Risk Based Inspection (RBI): Application

to corroded structures

F. Schoefs

LUNAM Université, Université de Nantes-Ecole Centrale Nantes, GeM,

Institute for Research in Civil and Mechanical Engineering, CNRS UMR 6183, Nantes, France

J. Boéro

Oxand S.A, Avon, France

First, on the basis of several previous works, this paper reviews the theoretical aspects coming from detection theory and probabilistic modeling of inspections results. The objective is to pro-vide inputs in the computation of mathematical expectation of RBI cost models. It is shown how these models highlight the role of the probabil-ity of defect presence. Expert judgment or the knowledge of ageing laws allows quantifying this probability (Rouhan & Schoefs, 2003). The paper introduces the polar coordinates of NDT-BPP for characterizing ROC curves, which allows us to perform parametric studies and improve NDT techniques with various assumptions on costs. The application concerns corroded steel sheet-piles in harbours.

2 PROBABILISTIC MODELING OF INSPECTION BASED ON DETECTION THEORY

The most common concept which characterizes inspection tool performance is the probability of detection. Let ad be the minimal defect size, under which it is assumed that no detection is done. Parameter ad is called detection threshold. Thus, the Probability of Detection (PoD) is defined as Eq. 1:

ˆ ( a )d

PoD= (PP (1)

where ˆd is the measured defect size.

Let’s assume that noise and the signal amplitude are independent random variables, then PoD and PFA have the following expressions:

ˆ ˆ ( ) d signal a PoD f_s d +∞ ∂ ( ) signal f_s )

∫∫

Replacement of engineering structures results in high economic and environmental costs, thus increasing the interest in maintaining these structures with efficient management plans. Therefore, the challenge for the owners consists in guaranteeing the operation and safety of age-ing structures, while ensurage-ing reasonable costs and availability conditions. Harbor structures meet all these stakes.

Reassessment of existing structures generates a need for updated material properties. In a lot of cases, on-site inspections are necessary and in some cases visual inspections are not sufficient. Non Destructive Testing (NDT) tools are required for the inspection of coastal and marine structures where marine growth acts as a mask or where immersion gives poor conditions for inspection (visibility, …). In these fields, the cost of inspec-tion can be prohibitive and an accurate descripinspec-tion of the on-site performance of NDT tools must be provided. Inspection of existing structures by a NDT tool is not perfect and it has become a com-mon practice to model their reliability in terms of probability of detection (PoD), probability of false alarms (PFA) and Receiver Operating Char-acteristic (ROC) curves (Rouhan & Schoefs 2003, Straub & Faber 2003, Pakrashi et al., 2008). These quantities are generally the main inputs needed by owners of structures who are looking to achieve Inspection, Maintenance and Repair plans (IMR) through Risk Based Inspections methods (RBI). The assessment of PoD and PFA is even deduced from inter-calibration of NDT tools or from the modeling of the noise and the signal. Both due to their great economic interest and the cost (direct and indirect) of inspection, authors have selected steel harbor structures for the application (not detailed in this paper).

PFAFF fff_noise ad ∂ f_noise ff +∞

∫∫

where fsignal and fnoise are respectively the probability

density functions of ‘signal + noise’ (or measured defect) and ‘noise’.

Thus, PoD is a function of the detection thresh-old, the defect size and the noise, while PFA depends on the detection threshold and noise only. Noise is due to the decision-chain “physical measurement-decision on defect measurement transfer of information”, the harsh environment of inspection and the complexity of testing pro-cedure (link diver-inspector for underwater inspec-tions for instance).

The ROC (Receiver Operating Characteristic) curve links the Probability of Detection and the Probability of False Alarm. For a given detec-tion threshold, the couple (PFA, PoD) allows defining NDT performance. This couple can be considered as coordinates of a point in R2 _with

axes representing PFA and PoD. Let us consider that ad takes values in the range [−∞; + ∞], these points belong to a curve called Receiver Operat-ing Characteristic (ROC) which is a parametric curve with parameter ad and defined by equations

(2) and (3).

3 THE α−δ METHOD

A simple geometric characterization of ROC curves is the distance δ between the curve and the Best Performance Point (BPP) of coordinates (PFA = 0, PoD = 1). By definition, the greater the distance is, the worse the performance. The cor-responding point on the ROC curve is called the performance point of the NDT tool (NDT-BPP). However as the configuration of ROC curves for the same distance are varied due to the ‘noise’ and ‘signal + noise’ pdf, we extant this measure of per-formance by using the polar coordinates of the NDT-BPP. The αδ-method lies on this characte-rization. The NDT-BPP polar coordinates are then defined by:

• the radius δNDT is the performance index

(NDT-PI) (distance between the best perform-ance point and the ROC curve) (Schoefs & Clément 2004, Schoefs et al., 2007);

• the angle αNDT between axis (PFA = 0) and the

line (BPP, NDT-BPP).

The first objective of the αδ-method was to perform parametric studies lying on these two parameters to analyze the effect of the shape of ROC curves on the decision process relating to

inspection, maintenance and repair. To achieve this goal, the influence of the performance of ROC curves, represented by δNDT and αNDT, was

appreciated through the costs and (Schoefs et al., 2010). It was shown that these parameters were sufficient to describe the effect of the ROC curve on the costs.

4 α−δ METHOD

Figure 1 presents mappings of extra cost of no detection according to the position of NDT-BPP in polar (δNDT, αNDT) coordinates, for γ = 0.9 and

following cost assumptions: Failure 1, Repair 0.1, Inspection 0.01. For this large probability of defect presence (γ = 0.9), extra costs of no detection for increase when αNDT decreases and δNDT increases.

Note that when considering extra costs of detec-tion, they are maximum in the zone defined by values of δNDT higher than 0.3 and αNDT higher

than 50°. Outside this area, extra cost of detection increases mainly with δNDT.

Maximum extra costs of detection are located in zone with high values of δNDT and αNDT. It is

also interesting to note that for values of δNDT

lower than 0.1, the influence of αNDT is negligible.

In addition, extra costs of no detection for γ = 0.1 are maximum for values of δNDT superior than

0.1 and for values of αNDT minor than 30°. When

αNDT is higher than 30°, extra costs of no

detec-tion increase mainly with δNDT. Let us consider

NDT-BPP A in the following (α = 30°, δ = 0.5) for γ = 0.9.

Then if the cost of inspection is negligible in comparison to cost of repair several strategies of improvement (reduction of risk) can be compared in terms of “displacement” of the original position A of the NDT-BPP on the mapping (see positions of A’, A”, and A”’ on Figure 1).

Figure 1. Mapping of extra cost of no detection

_E

nd

( )

C

in polar plane for γ = 0.9.

REFERENCES

Pakrashi, V., Schoefs, F., Memet, J.B. & O’Connor, A. (2010). “ROC dependent event isolation method for image processing based assessment of corroded har-bour structures”, NSIE, vol. 6(3), pp. 365–378. Rouhan, A. & Schoefs, F. (2003). “Probabilistic

model-ling of inspections results for offshore structures”, Structural safety, vol. 25, pp. 379–399, 20 pages, Elsevier Ltd.

Schoefs, F., Clément, A., Boéro, J. & Capra, B. (2010). “The αδ method for modeling expert Judgment and combination of NDT tools in RBI context: applica-tion to Marine Structures, NSIE, accepted January 26th 2010, in Press.

Straub, D. & Faber, M.H. (2003). “Modelling dependency in inspection performance”, Proc. Application of Sta-tistics and Probability in Civil Engineering, ICASP 2003 – San Franncisco, Der Kiureghian, Madanat and Pestana eds., Millpress, Rotterdam, ISBN 90 5966 004 8. pp. 1123–1130.