Risk Analysis of Structures in Presence of Stochastic Fields of Deterioration: Flowchart for Coupling Inspection Results and Structural Reliability

Abstract Inspection by non-destructive testing techniques of existing structures is not perfect and it has become a common practice to model their reliability in terms of receiver operating characteristic (ROC) curves. This paper suggests a method of building ROC curves in case of random fields of defects on a structure by using polynomial chaos decomposition. Knowledge of spatial distribution of defects allows a reliability analysis to be performed. When selecting stochastic finite element analysis to solve this problem, the format is the same as the one chosen for modelling inspections results. The paper shows how to link these quantities (ie. reliability and inspection results) in a risk analysis by using polynomial chaos decomposition as a common language.


INTRODUCTION
Reassessment of existing structures generates a need for up-dated materials properties. Commonly, onsite inspections are needed and in some cases visual inspections are not suffi cient for an accurate sizing or detection. For example, non-destructive testing (NDT) tools are required for the inspection of coastal and marine structures where marine growth acts as a mask. In these fi elds, the cost of inspection can be prohibitive and an accurate description of the onsite performance of NDT tools must be provided. Inspection of existing structures by a NDT tool is not perfect and it has become a common practice to model its reliability in terms of probability of detection (PoD), probability of false alarms (PFA) and receiver operating characteristic (ROC) curves. These quantities are generally the main inputs needed by owners of structures in view to achieve Inspection, maintenance and repair plans (IMR) (Sheils et al, 2008). The assessment of PoD and PFA comes either from inter-calibration of NDT tools or from the modelling of the probability density functions of the noise caused by the NDT imperfection and the signal. In this last case, if the noise and the signal depend on the location on the structure then PoD and PFA are spatially dependent. The decomposition on polynomial chaos (PC) is then used to represent the underlying stochastic fi elds. It is then natural to perform reliability analysis in the presence of a stochastic fi eld of defects by using the same format. It is shown that PC decomposition offers in this case a common language for the two inputs of the RBI: the modelling of inspection results and the computing of the structural reliability. This paper presents fi rstly how to defi ne PoD and PFA when damage and detection are stochastic fi elds, ie. spatially dependent (section 2). Section 3 deals with PC decomposition of the marginal distributions of the stochastic fi elds obtained at given locations along the structure. It is shown how to identify the coeffi cients of the PC decomposition from data. However, it has been shown (Rouhan & Schoefs, 2003) that PoD and PFA cannot be directly linked with consequences when an IMR policy has been defi ned. From the decision theory and a Bayesian description of inspection results, it is shown in section 4 how to introduce new events for risk-based inspection (RBI). Expressions of their likelihood as functions of PoD, PFA and γ, the probability of defect presence are presented. Results are then generalised to use ROC curves for RBI. By knowing the spatial stochastic fi eld for the signal and the noise, ROC curves are deduced in each point of the fi eld. This section is illustrated with the ultrasonic measurement of the spatially distributed corrosion of a sheet pile in a marine environment.
Finally, section 5 gives the last output needed for risk assessment based on reliability analysis: the computing of the probability of failure. From loading modelling and by knowing the stochastic fi eld of defects, a stochastic computation can be performed in view to estimate the spatial distribution of probability of failure. This section ends with a complete application of risk analysis of a corroded sheet-pile in steel.

Hazards and uncertainties when assessing defect sizes on structures
Let us consider the assessment of a defect d through a NDT tool. Inspection is generally not perfect. We consider the case where harsh conditions affect the measurement of the d. Then, depending on the environment and the location of the inspected area, the measurement of the size is affected by hazards. We model this quantity as a stochastic fi eld d (X, t, θ), where X denotes the spatial coordinates, t the time and θ the hazard. When assessing this defect, the chain of measurement (accessibility of the inspected area, calibration of NDT tool, experience and tiredness of the operator) introduces uncertainties, called noise in the signal processing theory (Rouhan & Schoefs, 2003). The result of the inspection, ie. the measured defect size, is a stochastic fi eld denoted ˆ( , , ) d X t . In section 2.2, d(X, t, θ) and ˆ( , , ) d X t are written as d and d for simplicity.

2.2
Modelling capability of NDT tools in a probabilistic context The most common concept that characterises inspection tool performance is the PoD. Let a d be the minimal defect size, under which it is assumed that no detection is done. Parameter a d is called the detection threshold in the following. Thus, the PoD is defi ned as: The detection threshold a d is a deterministic parameter or a random variable. This defi nition implies that PoD is a monotonic increasing function.
Detection theory allows for the introduction of another useful quantity, the PFA. From the signal theory, d is called the complete signal or "signal + noise" with a probability density function (pdf) f SN . The real defect size d is unknown and it is deduced from the knowledge of the noise η by the relationship d d . The pdf of noise is denoted f N . Noise depends on environmental conditions, human interference and the nature of what is being measured. Let's assume that noise and signal amplitudes are independent random variables, then  (Rouhan & Schoefs, 2003).
PoD and PFA have the following expressions (2) and (3): where δ denotes the increment. Figure 1 illustrates the corresponding pdf in the case where variables are normally distributed and the computation of PFA and PoD for a given detection threshold.
For a given detection threshold, the couple (PoD, PFA) allows the performance of a NDT to be defi ned; this is the ROC. This couple can be considered as the coordinates of a point. Let us consider that a d is unknown and takes values in the range [-∞; +∞], this point belongs to a curve called ROC curve. It is a parametric curve defi ned by equations (2) and (3) with parameter a d .
The ROC curve, which is plotted in figure 2, is computed using the pdf presented on figure 1. From a theoretical point of view, this is a convex curve corresponding to a monotonically increasing function, always lying above the diagonal line, where the fi rst derivative is closely linked to the sensitivity of the receiver (Arques, 1982;Fücsök et al, 2000). The diagonal line running from the lower left to the upper right (curve "PoD=PFA") is the line of no "performance", since in that case the inspection PoD equals PFA whatever the detection threshold (Rouhan & Schoefs, 2003).
inspections is reasonable and allows a rich data base of information to be accessed.
We follow the second approach in this paper. Note that the PFA is named PFI (probability of false indication) too. The defi nition of discrete values for PFI is generally expressed as a percentage of false indications on the inspected length (Silk, 1996;Rudlin et al, 2006). Finally, as the number of samples is limited, authors provide confi dence bounds: 90% PoD for example (Barnouin et al, 1993;Rudlin et al, 2006).
In some cases, the performance of NDT tools depends on the location of the point to be inspected on the structure. For example, when locations of defects are on welded joints in the sea, the corresponding PoD and PFA should be changed according to the access, the luminosity and the wave shaking for instance. When defects are continuous fi elds on the structure, the PoD and PFA should be indexed by the coordinates X of the inspected point M. We consider here that the defect is induced by a deterioration mechanism indexed by time t and can be modelled by a stochastic fi eld: d(X, t, θ).

Defi nitions of PoD and PFA for stochastic deterioration model
As described in the previous section, after inspection with a NDT tool, the size of the defect d(X, t, θ) becomes ˆ( , , ) d X t . Then the noise η(X, t, θ) is defi ned from the knowledge of these two stochastic fi elds by equation (4).
For each location X and at each time t, the PoD and PFA are, respectively, computed from equations (2) and (3). Thus the PoD and PFA, as well as the ROC curves, are functions indexed by X and t. We denote PoD(X, t) and PFA(X, t) as functions of time and space. The fi eld d at a given time t being assessed from inspection, the computation of PoD(X, t) and PFA(X, t) requires the knowledge of one of the other stochastic fi elds in equation (4): d(X, t, θ) or η(X, t, θ). Two situations can be considered: (i) The noise is known because it is constant whatever the location of the NDT tool on the structure or because it is constant on given areas on the structure. It is generally time invariant and zero mean. (ii) The real size is known because it has been measured before on-site inspections as in the ICON project (Barnouin et al, 1993) or because an assumption is made.
In both situations, the defi nition of continuous spatial functions needs the complete characterisation of the stochastic fi elds by their joint distribution. Practically, no models of joint distribution are available for onsite inspection and almost all NDT tools give data on Looking for the best detection performance, the PoD should always take larger values than the PFA (low noise sensitivity). We have then PoD ≥ PFA. When analysing ROC curves, one must keep in mind that the PFA depends on the noise and detection threshold only. It does not depend on defect unless the noise depends on defect size. That is the case for instance if the operator adjusts the device to detect smaller defects when the current adjustment does not give any signal. The PoD is a function of the detection threshold, the defect size and the noise. Thus, for a given detection threshold, the PFA is a constant, but the PoD is an increasing function of the defect size. The ROC curve is a fundamental characteristic of the NDT tool performance for a given defect size. A perfect tool is represented by a ROC curve reduced to a single point whose coordinates are (PoD, PFA) = (1, 0). The distance between this "best performance point" and the ROC curve is a measure of the NDT ability (Schoefs & Clement, 2004); we call it NDT-PI (NDT Performance Index). Theoretical ROC curves are presented in Rouhan & Schoefs (2003), where each one is obtained for several signal/ noise ratios of a NDT tool.
When assessing a ROC curve, two approaches can be considered: • the statistical approach, which aims to provide discrete values for PFA and PoD and requires the knowledge of the real size; this is performed within specifi c inter-calibration projects such as the ICON project in the fi eld of offshore platform analysis (Barnouin et al, 1993), when on-site measurements on real structures are costly or when on-site conditions are affected by a lot of factors as for the detection of corrosion pitting (Pakrashi et al, 2008) • the probabilistic approach, which requires a modelling of the (signal + noise) and noise pdf; it is generally preferred when the cost of on-site specifi c locations; thus, only marginal distributions can be assessed. Moreover, the owner generally does not base its inspection planning on RBI methods; then the inspection campaign aims to give a global overview of the state of the structure. Thus the distance between measurements is generally larger than the distance of correlation and additional assumptions on the structure of correlation for the stochastic fi elds are needed.
Finally, note that the knowledge of ageing laws for d allows predicting evolution of ROC curves with time. This can be implemented in a RBI method for optimising the period between inspection, as well as the location of inspections. For example, for corrosion processes, several models are available (Melchers, 2003;Paik et al, 2003a;Paik et al, 2003b;Guedes Soares & Garbatov, 1999). In the fi eld of uniform corrosion, spatially dependent ROC curves are available (Schoefs et al, 2007a).

MODELLING RANDOM FIELDS OF DEFECTS FROM NDT MEASUREMENTS
Only the spatial dependence is addressed here. We suggest a representation with PC decomposition for modelling marginal distributions or random fi elds. It allows for the systematic identifi cation of random variables or random fi elds. We choose the estimate of maximum likelihood for the identifi cation of PC decomposition (Desceliers et al, 2007). This method has already been applied for the identification of random variables from structural monitoring (Schoefs et al, 2007b). The question is to identify the coeffi cients d i , ˆi d and η i of the one-dimensional PC decomposition for every random variable. For generality, let us denote x(θ) as the random variable and x i as the PC coeffi cients. Equation (5) gives the general form of a one-dimensional PC decomposition.
where p is the order of the PC decomposition; ξ(θ) is the Gaussian germ, ie. a standardised normal variable; and h i is the Hermite polynomial of degree i. By using the maximum likelihood method, coeffi cients x i are solutions of the optimisation problem: where is the vector of components x i ( = [x 0 , …, x p ]) with dimension (p+1), and L is the likelihood function: where x(θ j ) is the j th realisation (here measurement) of x; and p x (.; ) is the probability distribution of x, parameterised by .
The likelihood function (7) takes very fair values close to the numerical precision. Then the problem (6) is replaced by equation (8): The algorithm for solving the optimisation problem (8) is detailed in Schoefs et al (submitted) and only the main steps are described here. Basically, the question is to build a solution by stating conditions (9): where μ x and σ x are, respectively, the statistical average and standard deviation of variable x(θ). They are computed from the N measurements x(θ j ). In equation (9), the fi rst condition reduces the number of unknown PC coeffi cients to p and the second one facilitates the search for other coeffi cients on an hyper-sphere with radius σ x . Moreover, by denoting x i * the quantity x i /σ x , the second condition of (9) becomes equation (10) .. x p * on a hyper-sphere with radius 1. This last condition is interesting for the optimisation process. Let us now consider a PC of order 3. To parametrise the hyper-sphere, we can introduce two angular parameters φ 1 and φ 2 : A two-step optimisation fl owchart is used solving equation (8) by knowing equation (11): • a first localisation of the minimum is found through Monte Carlo simulations (size 100 and 1000, respectively, for PC of order 2 and 3) • starting from this point, the Nelder-Mead Simplex Method is used (Lagarias et al, 1998).
This process allows us to avoid a pseudo-convergence around minimum values. Figure 3(a) presents the fi tting of distributions in the case of uniform corrosion of piles in harbours for the two variables: measured loss of thickness d and noise η with PC for several orders p. Figure 3

Figure 4:
Comparison between ROC curves coming from polynomial chaos with order 3 identifi cation (left) and experimental data (right) for several inspections in depth.
where p ξ is the probability density function of ξ (standard normal pdf). Practically, integrals ℑ i (a d ) in equations (12) and (13) are computed through Monte-Carlo simulations using 10 6 samples. These quantities are independent of the study: they can be preprocessed once for all and used for each application.
The stochastic fi eld of corrosion for sheet piles in harbours is spatially indexed by the vertical abscissa only (Schoefs et al, 2007a). ROC curves at six depths between -1 m and +2 m along a pile are plotted in fi gure 4 for a PC chaos of order 3. Abscissa (+1 m) and (+2 m) represent points located in the tidal area where others refer to locations in the immersion area.
Note that the identifi cation of a variance-covariance matrix is generally more complex due to the distance of inspected points that is much larger than the correlation length as mentioned previously in section 2.3. In this case we suggest using independent germs ξ i (θ) for each inspected point and to interpolate PC coeffi cients between two inspected points (14).
where φ k (X) are linear functions for interpolation from knowledge of coefficients at coordinate X j . Note that this representation needs at least measurements where the average trajectory of the process takes its minimum and maximum value.
In the case of corrosion of coastal structures, shape of profiles and position of extreme values are generally known and inspections are carried out at these positions (see fi gure 5). In view to compare this method with another one based on the fi tting with a classical pdf, we used the NDT-PI: it is the distance between the best performance point in (PFA, PoD) space of coordinates (0, 1) and the ROC curve. Table 1 gives a comparison between the NDT-PI obtained with predefi ned pdf (table 1(a)) (Normal, Generalised Extreme Value (GEV), Student pdf) and PC identifi cations (table 1(b)) at a given position along the vertical axis (X = +2 m). The value of NDT-PI deduced from the original curve (experimental) being 0.054, only PC chaos of order 3 allows us to reach this value (0.052) when both lower order PC decomposition and predefi ned pdf leads to values of around 0.075. The tables give the detection threshold a d at the best performance point. Values of coeffi cients are given at each inspected level on fi gure 6 for PC decomposition of order 3. We notice that the normal pdf and order 1 PC decomposition are the same, leading to the same result. & Schoefs (2003) further developed this methodology by focusing on the probability that a defect exists after an inspection has been carried out. The PoD is the probability that an existing defect is detected, whereas the decision is based on the event that a defect exists, given the results of an inspection. We consider the events:
From Bayesian point of view, PoD can then be defi ned as the conditional probability of the event "(D(δ) = 1|δ = 1)". The conditional probability of the event of interest "(δ = 1|D(δ) = 1)" can be evaluated using Bayes Theorem. It is subsequently introduced into cost functions, which are used to investigate the effect of cost overrun due to inaccurate inspection results. Four events E i (i =1:4) are then defi ned. For example, the expression of the probability of event E 3 "(δ = 1|D(δ) = 0)" is given in equation (15). This event is of fi rst importance in risk analysis because it can lead to a failure.
They can be associated to costs and subsequently introduced in a complete RBI planning (Sheils et al, 2008). The occurrence of these events also depends on another parameter, γ, the probability of presence of a defect. The probability density function of γ is related to the natural size of existing defects and their spatial variation (which comes from expert judgment or an historical data base).
An alternative approach is to introduce a parameter known as the probability of indication (PoI): where PFI is the probability of false indication (Straub & Faber, 2003). The parameter PFI must be defi ned relative to a specifi c size of the inspected area (Straub & Faber, 2003).
From equations (12) and (13), P(E i ) and PoI are expressed as functions of spatial dependant quantities d and . Figure 7 gives the evolution of P(E 2 ) and P(E 3 ) for three assumptions of expert judgement and the result after selection of a standard expert judgement. This factor strongly affects the shape of these curves.

RELIABILITY ANALYSIS BASED ON STOCHASTIC FINITE ELEMENT IN CASE OF RANDOM FIELDS OF DEFECTS
The stochastic (or probabilistic) finite element method refers to finite elements methods that account for uncertainties in the geometry or material properties of a structure, as well as the applied loads. Since the 1980s, quite a lot of developments have been made. They are mainly concerned with uncertainties of materials properties or hazards in loading. The question of the random geometry due to defects during building or the effect of ageing laws (corrosion) is still a challenge. Methods such as X-SFEM are effi cient in this context 2008a;2008b;Clément et al, 2008). In this paper, the computation is based on the so-called non-intrusive stochastic fi nite element analysis (Berveiller, 2005). We consider in this section that the loss of thickness d (X, t, θ) can be introduced in reliability through "macro" variables, ie. quantities such as diameter, section or inertia are explicit function of d. Then no specifi c method that deals with random geometries such as X-SFEM is needed.
Firstly, a quantity of interest or a limit state is selected.
Here, we are interested in displacement of each point of the structure displacements u(X). The limit state is then expressed: G(X) = u(X)u c , where u c is the acceptable critical displacement. The structural computation is made by using deterministic software. The question is then to fi nd an approximation ( , ) u X of the PC decomposition u(X, ξ). By knowing the decomposition of basic input variables such as d(X, t, θ), on a PC decomposition with germ ξ, ( , ) u X is expressed in equation (17) and can be assessed by computing u α through the quadrature formula (18).
where α is a multi-index, P is a measure of probability associated with ξ, and H α is an Hermite polynomial associated to this multi-index. θ k are the elementary events where the deterministic computation is needed (generally called point of integration or Gauss points in the case of normal measure of probability) and ω k is the corresponding weights associated with the measure of probability P.
A post-processing allows the computation of the probability of failure P f to be carried out with very low computational costs. This is of primary interest when carrying out risk analysis where a good assessment of P f is needed.
For illustration, we can consider an application where an explicit form of the solution can be obtained by the static balance of the structure assuming an elastic behaviour of the material (see fi gure 5).
Considering here a limit state relying on yield momentum, the probability of failure with depth is plotted on fi gure 8, with the loss of matter on fi gure 6 and the confi guration of loading on fi gure 5.

Figure 7:
Evolution of P(E 2 ) and P(E 3 ) with depth (top) for several expert judgements γ (in bold line 0.9, in fi ne line 0.5 and in dotted line 0.1); standard expert judgement (middle); and corresponding evolution of P(E 2 ) and P(E 3 ) with depth with this standard judgement (bottom).

CONCLUSION: FLOWCHART FOR COUPLING RESULTS OF INSPECTION AND STRUCTURAL RELIABILITY
Sections 4 and 5 provide the main steps for a fl owchart that couples the results of inspection and structural reliability with a formulation based on PC decomposition. From inspection results and noise modelling, it is based on the PC decomposition of basic variables describing the risk based modelling of inspection results ˆ( , , ) d X t and η(X, t, θ) on the one hand, and a PC decomposition of geometrical basic variables for the reliability study d(X, t, θ) on the other hand. The corresponding terms of risk analysis P(E 3 ) and P f can be assessed. The fl owchart in fi gure 9 resumes the steps of methodology. When considering only event E3 (there is a defect knowing we do not detect it), which leads to failure and the probability of failure, fi gure 8 presents the result of the integrated risk analysis: the evolution of risk with depth. The PC decomposition is shown to be an interesting common language between reliability analysis and RBI when stochastic fi elds of degradation occur. Note that quite a lot of uncertainties and hazards are introduced: (i) The natural hazard of the corrosion process: d(X, t, θ) (ii) The uncertainty in the inspection results from on-site measurements: η(X, t, θ) (iii) The uncertainty of the previous knowledge on the corrosion process: γ (X, t).

FRANCK SCHOEFS
Dr Franck Schoefs is an assistant professor at the Institute for Research in Civil and Mechanical Engineering (GeM) within the Nantes Atlantic University in France since 1998. He graduated in 1992 from Ecole Normale Superieure de Cachan with the best student in civil engineering award. The received his PhD in 1996, which was on "Response Surface of Wave Loading for Reliability of Marine Structures". Franck's interests include stochastic fi nite element of random geometries (X-SFEM); matrix response surfaces for the modelling of wave and wind extreme loading on framed structures (storms); probabilistic deterioration modelling of steel and concrete in coastal areas; probabilistic modelling of inspection results; risk analysis of deteriorated structures; and probabilistic modelling of loading from structural monitoring. Since 2006, he is President of the French association for RIsk MAnagement in Civil Engineering (@RIMAGIN) and Chair of the Scientifi c Committee of the French scientifi c network for RIsk Anagement in Civil EnGINeering). He received the 2008 prize of "the scientifi c collaboration" during the MEDACHS 08 conference and was President of the Organizing Committee and member of the Scientifi c Committee of the 5 th French Workshop on Reliability of Materials and Structures in Nantes, France.