R Miorelli
email: roberto.miorelli@cea.fr

F Le Bourdais

X Artusi

Assessing Performance of Flaw Characterization Methods through Uncertainty Propagation

In this work, we assess the inversion performance in terms of crack characterization and localization based on synthetic signals associated to ultrasonic and eddy current physics. More precisely, two different standard iterative inversion algorithms are used to minimize the discrepancy between measurements (i.e., the tested data) and simulations. Furthermore, in order to speed up the computational time and get rid of the computational burden often associated to iterative inversion algorithms, we replace the standard forward solver by a suitable metamodel fit on a database built offline. In a second step, we assess the inversion performance by adding uncertainties on a subset of the database parameters and then, through the metamodel, we propagate these uncertainties within the inversion procedure. The fast propagation of uncertainties enables efficiently evaluating the impact due to the lack of knowledge on some parameters employed to describe the inspection scenarios, which is a situation commonly encountered in the industrial NDE context.

INTRODUCTION

In the field of Nondestructive Testing & Evaluation (NdT&E), parametric forward solvers [START_REF]CIVA -CIVA -Non destructive testing at CEA List[END_REF] are employed to solve a wide variety of problems such as probe design, inspection method design, analysis of inspection results, etc. The main advantage in employing parametric solvers consists in the ease of use and in the high computational efficiency (since semi-analytical solutions or accurate approximate models are often available) and a user-friendly description of the NdT scenario. Recently, the generation of databases and their exploitation through associated metamodels (or surrogate models) has been successfully demonstrated for eddy current testing inspection problems [START_REF] Bilicz | Solution of Inverse Problems in Nondestructive Testing by a Kriging-Based Surrogate Model[END_REF][START_REF] Bilicz | Sparse Grid Surrogate Models for Electromagnetic Problems With Many Parameters[END_REF][START_REF] Miorelli | Database generation and exploitation for efficient and intensive simulation studies[END_REF]. The database technology has shown considerable potential by allowing fast sensitivity analyses, modelassisted probability of detection and inversion studies [START_REF] Miorelli | Database generation and exploitation for efficient and intensive simulation studies[END_REF]. However, in spite of the results obtained in the aforementioned works, many practical problems remain to be addressed, most of them concerning inversion applications. Very recently, metamodels have been proposed as tools for fast propagation of uncertainties in the inversion scheme in order to provide simulated results closer to real inspection scenarios [START_REF] Aldrin | Model-based Inverse Methods for Bolt-holt Eddy Current (BHEC) Inspection[END_REF][START_REF] Shell | Demonstration of Model-Based Inversion of Electromagnetic Signals for Crack Characterization[END_REF][START_REF] Aldrin | Model-based probe state estimation and crack inverse methods addressing eddy current probe variability[END_REF][START_REF] Storn | Differential Evolution -A Simple and Efficient Heuristic for global Optimization over Continuous Spaces[END_REF]. In the aforementioned works, crack characterization was based on numerical simulation of Eddy Current Testing (ECT) signals while inversion was performed on experimental data. Moreover, the propagation of uncertainties in the inversion process, the so-called Uncertainty Quantification (UQ), associated to fluctuations on nominal inspections parameters (i. e., the ones that will not be inverted) has been also performed. These sources of uncertainty are normally associated to some discrepancies between nominal and real-inspection set-up parameters e.g., probe (lift-off, skew, tilt, etc.) or specimen parameters (parametric medium description such as conductivity, wave velocity, etc.).

In this paper, we describe how a database and the associated metamodel can be employed to propagate the uncertainties within the inversion process in the case of ECT and Ultrasonic Testing (UT) signals. We have also quantified the impact of the uncertainties on the obtained results. The inversion strategy consisted in minimizing the distances between measurements and simulation, the so-called misfit function. This misfit function is minimized iteratively by using a state-of-the-art deterministic (quadratic programming) and a stochastic (differential evolution) algorithm. The results obtained through these algorithms have been compared from the perspective of inversion performance in terms of accuracy and CPU time efficiency with and without UQ. Moreover, an analysis on the illposedness of the studied inversion problems is carried out for both physics applications.

For many years now, forward solvers have been employed instead of expensive measurements campaigns to obtain Probability of Detection (POD) curves, Sensitivity Analysis (SA) indexes, to perform probe optimization and inversion in terms of crack characteristics, etc. In recent years, a new trend has emerged concerning the most computationally demanding tasks. This trend consists of fitting a metamodel (also known in the literature as a surrogate model) using a database of NdT signals obtained by well-verified and validated forward solvers. Here, the metamodel is used as an almost real-time replacement of the forward solver for the most time consuming tasks. More details on this topic can be found in [START_REF] Bilicz | Solution of Inverse Problems in Nondestructive Testing by a Kriging-Based Surrogate Model[END_REF][START_REF] Bilicz | Sparse Grid Surrogate Models for Electromagnetic Problems With Many Parameters[END_REF][START_REF] Miorelli | Database generation and exploitation for efficient and intensive simulation studies[END_REF]. In the following, we show how metamodels can be employed in order to propagate uncertainties during the inversion process in two inspection cases: crack characterization and crack localization.

Metamodel formulation in a nutshell

A database 𝐷 is defined as a collection of input/output (IO) couples, also known as a training set. In particular, 𝐷 = [(𝑥̅ 1 , 𝑦 ̅ 1 ), (𝑥̅ 2 , 𝑦 ̅ 2 ), … , (𝑥̅ 𝑁 , 𝑦 ̅ 𝑁 )] where the i-th input is a real valued vector 𝑥̅ 𝑖 and the corresponding i-th output is the vector 𝑦 ̅ 𝑖 . The outputs 𝑦 ̅ 𝑖 normally consists in a functional output which can be real or complex-valued regardless of the considered NdT method and the associated inspection technique. In this work, the database is built based on functional complex valued signals in the case of ECT and functional real signals in the case of UT problems. Based on the database training set, the metamodel (𝑀) is fit by employing suitable kernel-based methods (e.g., kriging, radial basis function (RBF)) [START_REF] Bilicz | Solution of Inverse Problems in Nondestructive Testing by a Kriging-Based Surrogate Model[END_REF][START_REF] Miorelli | Database generation and exploitation for efficient and intensive simulation studies[END_REF] or sparse grid methods [START_REF] Bilicz | Sparse Grid Surrogate Models for Electromagnetic Problems With Many Parameters[END_REF]. Thus predictions 𝑦 ̅ ̃ are obtained by evaluating the metamodel on a unseen test set such that 𝑦 ̅ ̃= 𝑀(𝑥̅ ).

In the uncertainty propagation framework, a subset of the database parameters is modelled as the presence of uncertainties on the measurements. The rest of the parameters are used to model the problem itself and are associated to the quantities directly investigated. Therefore, given a set of parameters, the metamodel predictions (i.e., the output signals) will vary in a certain range of values accordingly to the statistical laws associated to the uncertain parameters. More formally, let the parameter space be split into two sets 𝑥̅ 𝑖 = [𝑥̅ 1 , … , 𝑥̅ 𝑃 ] ∪ [𝑥̅ 𝑃+1 , … , 𝑥̅ 𝑈 ] = 𝑥̅ 𝑃 ∪ 𝑥̅ 𝑈 such that the 𝑃 inputs are employed to describe the variation of parameters and the other 𝑈 parameters are employed to model the system uncertainties. Fast propagation of uncertainties through the use of matamodel is conducted, for a given sample observation, as 𝑦 ̅ ̃= 𝑀(𝑥̅ 𝑃 |𝑥̅ 𝑈 ) where the number of generated outputs depend on the number of samples collected on the Probability Density Functions (PDFs) associated to the uncertain parameters.

Definition of the misfit function and the studied iterative inversion strategy

In this paper, we deal with crack characterization and/or localization based on an iterative process in which the discrepancy between measurements 𝑦 ̅ 𝑚𝑒𝑎𝑠𝑢𝑟 and simulated data is to be minimized. The misfit function we will use is

𝑚(𝑥) = ||𝑦 ̅ 𝑚𝑒𝑎𝑠𝑢𝑟 -𝑦 ̅|| 2
2 , which can be minimized by employing standard deterministic or stochastic algorithms. In this work, we have chosen to employ the MATLAB® routine fmincon with the Sequential Quadratic Programing (SQP) option activated as a deterministic minimizer. Differential Evolution (DE) has been employed as global stochastic minimizer [START_REF] Ahmed | Real Time Groove Characterization Combining Partial Least Squares and SVR Strategies: Application to Eddy Current Testing[END_REF]. In order to speed-up the inversion process, the direct solver employed within the iterative minimization loop has been replaced by a sufficiently precise metamodel. Therefore, an approximated estimation of the misfit function is obtained as

    2 2 ~x x x x U P measur U P M y m   (1)
When the UQ is considered the above expression turns into 𝑚 ̅ ̃(𝑥̅ ) = ‖𝑦 ̅ 𝑚𝑒𝑎𝑠𝑢𝑟 -𝑀(𝑥̅ 𝑃 |𝑥̅ 𝑈 )‖ 2 2 , where the 𝑥̅ 𝑃 are obtained from the optimization procedure. The vector 𝑥̅ 𝑈 is not involved in the minimization stage since it belongs to the uncertain parameters. The whole minimization loop is sketched in FIGURE 1.

NUMERICAL ASSESSMENT

In this section, we deal with the numerical results obtained for the ECT and UT test cases. In order to approach more realistic noisy data sets, Additive White Gaussian Noise (AWGN) has been added to metamodel interpolations using the following definition for the Signal to Noise Ratio (SNR)

   K k k k dB A SNR 1 2 2 log 10  , ( 2
)
where 𝐴 𝑘 represents the amplitude of the signal for the k-th measured feature, e.g., the variation of coil impedance or the amplitude of sound pressure for ECT and UT physics, respectively. 𝜎 𝑘 represents the associated noise amplitude.

To quantitatively evaluate the inversion performance, we employ two metrics commonly used in the literature to assess the prediction performance. The first is the Normalized Mean Square Error (NMSE) defined as

𝑁𝑀𝑆𝐸 = 1 𝑁 ⁄ • ∑ (𝑥 𝑛 -𝑥 𝑛 ̃) (𝑥̅ • 𝑥̅ ̃) ⁄ 𝑁 𝑛=1 , ( 3
)
where 𝑥 𝑛 and 𝑥 𝑛 ̃ correspond to the n-th "true" value and prediction, respectively. The mean value is computed on the 𝑁 test samples. Moreover, we define the correlation coefficient

𝑅 2 = 1 -∑ (𝑥 𝑛 -𝑥 𝑛 ̃)2 (𝑥 𝑛 -𝑥̅ ) 2 𝑁 𝑛=1 . ( 4
)

ECT flaw characterization in steam generator tube

In this section, we present the synthetic inversion results obtained on a problem encountered in the nuclear sector. Referring to FIGURE 2 (a), a steam generator tube is inspected by a +Point-like probe. The tube has an inner radius of 8.255 mm and a thickness of 1.05 mm with a conductivity of 1.39 MS/m. Embedded within the tube and surface breaking to the coil side,is a rectangular flaw having a given length, depth and width. A database of 1795 samples has been created for the four probe parameters and the three crack parameters by using the adaptive sparse grid method [START_REF] Bilicz | Sparse Grid Surrogate Models for Electromagnetic Problems With Many Parameters[END_REF]. The number of complex valued measured features considered, i.e. the number of coil positions, was 7321, which corresponds to a signal map made of 121x61 points collected around the crack zone, as shown in FIGURE 2 (b). The probe parameters, which are supposed to be known at the inspection stage, have been employed in order to propagate uncertainties through the inversion process. The considered parameters were probe lift-off which varies between 0.2 mm and 1.2 mm, probe pitch which varies between -10° and 10° andprobe yaw which may take values between 0° and 45°. In order to simulate some winding inhomogeneity due to the manufacturing process, the wire thickness varies between 0.1 mm and 0.35 mm. The crack parameters have been selected such that their variation range spans from 3.0 mm to 10.0 mm concerning the crack length (Parameter #1), from 0.25 mm to 1.05 mm for the crack height (Parameter #2) and between 0.02 mm and 0.2 mm for the crack gap (Parameter #3). In FIGURE 3, we show the inversion results obtained through SQP and DE algorithms on a synthetic test set made of 250 samples with nominal probe parameters fixed to the mean value of the variation range (i.e., no propagation of uncertainties is considered). On these samples, a SNR equal to 10 dB has been applied according to equation 2. We notice that the prediction accuracy of DE is largely superior compared to that of the SQP optimization algorithm for all parameters. This behavior is very likely due to the fact that stochastic algorithm is able to escape the basin of attraction of the global minima thanks to its global search over the entire solution space. Conversely, the SQP algorithm tends to be stuck in local optima when the guessed solution is within one of the multiple local minima basins, therefore prediction results are scattered all over the plots as shown on the top of FIGURE 3 (a). In TABLE 1, we have expressed the obtained crack characterization results in terms of correlation coefficient and normalized mean error. To propagate the uncertainties associated to the inspection, a test set with a fixed set of the crack parameters was generated, with twenty different values sampled from the PDF laws associated to the probe parameters. To be precise, for the probe lift-off a normal PDF defined as N(µ,  2 ) = [0.7 mm, 0.25] was used. For probe pitch and yaw N(µ,  2 ) = [0.0°, 0.25] and N(µ,  2 ) = [22.5°, 0.25] distributions were chosen, respectively. A uniform PDF law U(a, b) = [0.10625 mm, 0.2875 mm] was used for the winding thickness. In FIGURE 3 (b) we show, in terms of error bars, the distribution of these twenty results obtained by uncertainty propagation using DE algorithm solutions on a noisy (SNR = 20 dB) test set. We can notice that the prediction accuracy concerning the crack width is quite low due to the uncertainty in the measurement (see FIGURE 3 (a)). This illustrates the fact that uncertainties on probe parameters can worsen the overall prediction accuracy.

UT crack characterization and localization of nearby weld bevel

In this section, we present the results obtained for crack characterization and localization in the UT test case. The studied scenario consists of a single-element probe working at the center frequency of 2 MHz inspecting a 2D extruded CAD geometry representative of two homogeneous stainless steel specimens connected together by a weld having the same material characteristics as the base metal. The inspection angle is 45°, using shear waves. We used a B-scan along a direction perpendicular to the weld (see, FIGURE 4 (a)) as input data. The total number of real-valued measured features was equal to 65559 which corresponds to 41 probe positions times 1599 time samples (see, FIGURE 4 (b)). A database having five parameters and 3500 samples were generated and a metamodel based on radial basis functions has been fit on it. The crack height (Parameter #1) varied between 2 mm and 6 mm, crack length (Parameter #2) was set to vary between 8 mm and 26 mm, the crack position (laying parallel to the bevel) within the specimen (Parameter #3) varied from 11 mm to 25 mm. Furthermore, the probe yaw ranges from 0° to 15° and the shear wave velocity in the medium varies between 3180 m/s to 3280 m/s. These last two parameters, are supposed to be known when conducting this inspection, even though probes and real specimen characteristics slightly vary from one inspection case to another. Due to this reason we propagate these as uncertainties in the inversion process. In FIGURE 5 (a) we show the prediction results obtained with SQP and DE algorithms on a noisy test set (SNR = 2 dB) having 250 samples, the uncertain parameter having been set to the mean values of the respective variation ranges. We observe that performance of DE is much better than the SQPsince fewer outliers appear in the FIGURE 5 (a) bottom plot. This is very likely due to the fact that DE is globally exploring the solution domain (i.e., the misfit function space) which leads to the discovery of the basin of attraction of the global minima. SQP on the other side remains blocked in local minima and thus wrong solutions were found. This behaviour is summed up quantitatively in TABLE 2 in terms of R 2 coefficient and NMSE. The obtained results change drastically when uncertainties are propagated in the inversion process. In order to propagate uncertainties, we generated a test set by fixing the crack parameters and we randomly sampled twenty values from the PDF of the laws associated to the remaining two database parameters. A normal PDF having N(µ,  2 ) = [7.5°, 0.25] was used for the probe yaw whereas a uniform law U(a, b) = [3219 m/s, 3241 m/s] has been chosen for the wave velocity in the medium. As one can remark in FIGURE 5 (b), considering uncertainties at the inversion stage leads to very inaccurate estimations of crack height and length, and no meaningful values can be deduced for these parameters. On the other hand, the inversion results associated to the third parameter (the crack localization in the depth along the bevel) are accurate.

COMMENTS AND REMARKS ON THE OBTAINED INVERSION RESULTS

In this section, we analyze the ill-posedness of the inversion problem considering the misfit function landscape in three-dimension space associated to the all possible crack configurations. At the end of this section a brief analysis on the performance in terms of CPU time is also given.

On the multimodal behavior of the misfit function

For the sake of brevity, only the UT test case considered in this paper is analyzed in this paragraph. However, similar conclusions also hold for the ECT problem. In FIGURE 6, we sketched the misfit function in the threedimension crack parameter space. This cube of data is obtained from equation ( 1) by evaluating the metamodel on a fine grid. That is, the figure represents an approximated image of all possible values among which, the solution (i.e., the minimum) appears. The purpose of the employed inversions algorithm is to try to reach this minimum performing a "clever" search in the misfit space. In FIGURE 6 (a), the darker iso-surfaces represent the lower misfit function values. These values occupy different locations in the parameter space (see FIGURE 6 (b)). This behavior clearly indicates that the considered misfit function is multimodal. Therefore, iterative solutions of a deterministic inversion algorithm like the SQP can be trapped in the basin of attraction of these minima and remain stuck inside. If the solution stays in a local minimum, then wrong results are found. With global search algorithms like DE, the whole misfit solution space is stochastically sampled, which leads to a more exhaustive and robust minimization process which allows discovering the true solution in almost all cases.

Computational time efficiency of the employed metamodel-based approach

In this work metamodels have been employed to replace standard forward solver to vastly increase inversion performance in terms of CPU time efficiency. In particular, a metamodel based on adaptive sparse grid algorithm [START_REF] Bilicz | Sparse Grid Surrogate Models for Electromagnetic Problems With Many Parameters[END_REF] has been employed in for ECT test case, whereas a kernel based method based on RBF [START_REF] Miorelli | Database generation and exploitation for efficient and intensive simulation studies[END_REF] was used for the UT case. The inversion performance depends mainly on both the metamodel employed and the number of iterations needed to accomplish the inversion procedure. It worth to be underlined that metamodel computational times almost independent of the number of evaluated test samples. In fact, parallel evaluation of multiple samples can be performed easily with both demonstrated metamodels. This characteristic is very appealing when global optimization algorithms are used in the optimization stage since the population can be initialized with a very low computational cost. On the other hand, the metamodel computational performance are influenced by the number of measured features. In fact, the volume of data to be handled at the interpolation stage can be bit cumbersome for the entire CPU processing pipeline (RAM memory storage, internal transfer of data and computing results on the CPU).

In the addressed problems, 250 inversion results for the ECT problem required about 3 minutes with SQP algorithms and about 5 minutes with DE. The number of evaluations required for the SQP algorithm was about 200 for each sample and about 7000 for the stochastic counterpart. It is worth underlining that a call to the ECT forward solver takes ~9 minutes. This fact by itself largely justify the use of metamodels in parametric inversion. Concerning the UT test case, the inversion task has required about 45 seconds and 2.5 minutes for SQP and DE, respectively. The number of evaluations required for the SQP and DE were 170 and 7000, respectively. Also in this case, the use of metamodel allows a big leap in the inversion computational time performance since the forward solution is calculated in about one minute.

CONCLUSION AND PERSPECTIVES

This paper studied the inversion performance of standard algorithms on two NdT synthetic problems which are a matter of industrial interest for real-world applications. That is, crack characterization in steam generation tubes inspected by a +Point-like probe and crack characterization and localization of cracks nearby weld bevel through UT where addressed. We have tested the performance of two standard iterative inversion algorithms dealing with a deterministic gradient free (SQP) and stochastic approach (DE). Moreover, the propagation of uncertainties linked to perturbation of nominal parameters associated to probes and medium has also been accounted for. We have shown that inspection uncertainties heavily influence the inversion performance of certain parameters as well as the overall inversion results. This observation has two-fold consequences. The first and obvious one is that, within the inversion loop, uncertainties should be taken into account in order to have more confidence in the obtained results. This leads to a better understanding on how inversion results vary with respect to the variation of other parameters "separately" or "globally" (this kind of analysis could be seen as an equivalent "empirical" sensitivity analysis for inversion). The second remark concerns the inversion procedure itself. We have found that the inverse problem can be ill-posed. This means that the misfit function might have a multimodal behavior. The ill-posedness implies poor inversion performance for deterministic algorithms which may get trapped in local minima. In order to address this problem a suitable feature extraction stage (e.g., partial least squares, principal component analysis, kernel-methods, etc.) could be added before performing inversion. Since feature extraction enables to work with less features, this also means that a lighter volume of data is employed at the inversion stage. Working with features can thus decrease the inversion time and increase the robustness to measurements noise corruption [START_REF] Ahmed | Real Time Groove Characterization Combining Partial Least Squares and SVR Strategies: Application to Eddy Current Testing[END_REF]. In our next works we will focus on increasing the performance of iterative inversion procedures by adding a suitable feature extraction block in the implemented schema. Furthermore, a second research axis will consist in assessing the performance of inversion by applying noniterative methods borrowed from the statistical learning theory as recently proposed in the framework of ECT [START_REF] Ahmed | Real Time Groove Characterization Combining Partial Least Squares and SVR Strategies: Application to Eddy Current Testing[END_REF]. Furthermore, comparisons with available ECT and UT experimental data are also envisaged.

FIGURE 1 .

 1 FIGURE 1. Sketch of the iterative minimization loop based on the use of a metamodel.

FIGURE 2 .

 2 FIGURE 2. In (a) the steam generator tube is inspected by a +Point-like probe. In (b), from left to right, examples of the measured features in terms of amplitude of the coil impedance variation are shown for SNRs equal to noiseless, 20 dB and 10 dB, respectively.

FIGURE 3 .

 3 FIGURE 3. In (a), inversion results associated to the crack parameters are shown (top) for SQP algorithm and (below) for the DE algorithm. In (b), the obtained results in terms of crack characterization when uncertainties associated to the probe parameters are propagated, using the DE algorithm solutions. For all plots a normalized scale from zero to one has been employed.

FIGURE 4 .

 4 FIGURE 4. In (a) the weld bevel inspected by a single-element UT probe. In (b), from left to right, an example of the measured features in terms of amplitude of pressure field is shown for SNR equal to noiseless and 2 dB, respectively.

FIGURE 5 .

 5 FIGURE 5. In (a), inversion results associated to the crack parameter are shown on the top for SQP algorithm and below for DE one. In (b), the obtained results in terms of crack characterization when uncertainties associated to the probe parameters are shown by employing DE algorithm.

FIGURE 6 .

 6 FIGURE 6. In (a), the misfit function space is shown for a test set laying in a three-dimensional parameter space (shown with normalized axes). The true solution in the misfit space is identified by a white circle "o" in the three dimensional data space as well as on the cutting planes passing through the solution points. In (b), a detail view of the lowest misfit values is shown.

TABLE 1 .

 1 Quantitative comparisons between SQP and DE results for crack characterization shown in FIGURE3(a), are expressed in terms of correlation coefficient and normalized mean square error.

	Crack Parameter	MATLAB® fmincon SQP R 2 | NMSE	DE R 2 | NMSE
	Length	0.82 | 0.02	~1.0 | ~0.0
	Depth	0.88 | 0.04	~1.0 | ~0.0
	Width	0.50 | 0.75	~0.99 | 0.0008

TABLE 2 .

 2 Quantitative comparisons between SQP and DE results for crack characterization shown in FIGURE5(a), are expressed in terms of correlation coefficient and normalized mean square error.

	Crack Parameter	MATLAB® fmincon SQP R 2 | NMSE	DE R 2 | NMSE
	Height	0.91 | 0.15	0.99 | 0.014
	Length	0.82 | 0.17	0.99 | 0.03
	z-Position	0.99 | 0.01	~1.0 | 0.0001

ACKNOWLEDGMENTS

This work was done in the framework of the international collaboration CIVAMONT 2012 and the French project ANR ByPASS.