# A model assisted probability of detection for guided wave SHM systems: comparative study on recent statistical developments

Bengisu Yilmaz, Arnaud Recoquillay, Bastien Chapuis, Pierre Calmon

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Bengisu Yilmaz, Arnaud Recoquillay, Bastien Chapuis, Pierre Calmon. A model assisted probability of detection for guided wave SHM systems: comparative study on recent statistical developments. e-Journal of Nondestructive Testing, 2024, 29 (7), pp.1435-4934. 10.58286/29839. cea-05508084

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# A model assisted probability of detection for guided wave SHM systems: comparative study on recent statistical developments

Bengisu YILMAZ¹, Sanjay SHARMA¹, Arnaud RECOQUILLAY¹, Bastien CHAPUIS¹, Pierre CALMON¹

¹Université Paris Saclay, CEA-List, F-91120 Palaiseau, France, bengisu.yilmaz@cea.fr

**Abstract.** Despite the potential benefits, industrial implementation of guided wave monitoring systems remains limited today. One of the reasons behind this is the lack of consolidated reliability assessment techniques. Recent works clearly illustrated that the classical statistical methodology adopted from nondestructive evaluation to estimate POD is not directly applicable to structural health monitoring (SHM): the hypothesis of independence does not hold for the data obtained from permanently attached sensors and growing defects. The experimental determination of POD for SHM systems requires extensive and costly studies, in addition to the numerous representative structures. Involving numerical studies can complement experimental trials, inform POD evaluation and reduce this cost drastically. We propose in this work a comparative study to demonstrate the application of Model Assisted Probability Of Detection approach (MAPOD) for elastic guided wave monitoring systems. We apply classical and recently developed statistical methods proposed in the literature, using simulated data obtained thanks to the CIVA SHM module. The MAPOD estimation is validated in comparison to experimental studies.

**Keywords:** model assisted probability of detection, guided waves, numerical models, statistical models, length at detection, random effects model

## Introduction

The performance validation of structural health monitoring (SHM) systems is critical for their acceptance in the industry [1], [2]. Statistical methodologies behind the most commonly used reliability estimation techniques used in the non-destructive testing (NDT) field, are not directly applicable to SHM studies [3]. To develop and adapt reliability assessment methods
for SHM, we should address the essential differences between NDT and SHM. Some of these
differences can be summarized, for example, as the following:

* While NDT uses one time measurements, SHM relies on periodic or almost continuous monitoring campaigns over a long period. In SHM, the prior state is known and a growing defect can be monitored. Whereas in NDT, the prior knowledge is not necessarily used for the upcoming inspections.

* Since the structures and the monitoring systems are integrated together, the data collected over time are dependent, unlike in NDT where the data are collected separately in each periodic inspection.

* The variations that affect the results in NDT and SHM are inherently different. In SHM, the environmental and operational conditions (EOCs), sensor implementation, sensor aging and degradation are usually the main factors inducing changes in the acquired data [3]. On the contrary, human factors such as inspector's experience plays an important role in NDT inspection results.

* The damage detection using SHM usually requires a pristine state of measurement, where the damage index is estimated in relation to a baseline – which is more complicated than NDT inspections based on signal response amplitude or time of flight.

Several researchers proposed modified evaluation solutions specific to the performance of SHM systems [4]. Probability of Detection (POD) estimation with modified statistical methods, such as length at detection (LaD), linear mixed model (LMM) and random effects model (REM) are becoming popular in the field.

POD is a robust tool that enables inspectors to estimate the reliability of the inspection process [5]. The SHM system performance analysis should incorporate the EOCs in order to fully capture the potential validity of the system [6], [7]. Although it is possible to capture EOCs during monitoring, the results might take a long time and can be experimentally costly. The model assisted probability of detection (MAPOD) methods can help to overcome these challenges by the ability to simulate variations that are difficult and costly to demonstrate in real life [8].

This paper shows that the POD computed from experimental data using several statistical methods, can be estimated using simulated data representative of an SHM system. Different statistical methods are illustrated, namely signal response model for a classic NDT not taking into account the data dependency, the length at detection and the random effects model. Although intrinsically the simulation datasets are independent, a dependency is introduced by adapting the data sampling strategy.

# 1. Problem definition and experimental dataset

To mimic a crack growing from a rivet hole that can be subject to fatigue in a simplified test case, we consider a crack detection scenario of a notch initiated on a hole in the center of an aluminum plate, as is represented in Fig.1. The experiments were performed on 12 aluminum plates with 600 mm x 600 mm edge dimensions and 3 mm thickness. Four piezoelectric sensors were placed around the defect, their nominal positions being described in Table 1. Five-cycle tone burst actuation signal at 100 kHz were generated using Keysight Technologies wave generator, have been amplified to 40 V with power amplifier and filtered with a first order band-pass filter between 10 kHz and 1 MHz using a preamplifier (Stanford Research Systems). All sensors acted as emitter and receptor (round robin setting). The digitized measurement signals were recorded with 2 MHz sampling frequency over the period of 21 days monitoring the growing crack example around a hole [9]. Before the crack initiation procedure, pristine states were monitored with the hole at the center for each 12 plates. The cracks in each plate were manually sawed each day, increasing their length by
approximately 1.5 mm every time. Crack lengths between 0 mm and 30 mm were hence obtained. The ambient temperature was recorded using thermocouple and showed a variance between 20°C and 23°C. Data corresponding to different crack lengths of a given plate are hence dependent in this setup as the other parameters of the plate, especially the sensors, are the same.

<figure><img src="image_3.png" /><figcaption>**Fig. 1** The schematic for the aluminum plate with through hole and growing crack as well as piezoelectric sensors.</figcaption></figure>

## 2. Numerical simulation with CIVA SHM

The simulation have been performed with the SHM module of the CIVA software. The SHM module of CIVA is based on a parametric description of the domain and high order spectral finite elements in space to achieve fast computations with a low memory consumption, enabling parametric studies for simple configurations, as is the case here; or more complex ones, arbitrary sensor models can be used [10], [11]. In this study, the sensor definition is determined as radial excitation using pin-force model where an in-plane mechanical load is distributed equally around the outer diameter of the sensor [12]. As the finite element description of the problem needs a surface and not only a line on which to input the load, the sensor is in the end represented by an annulus of fixed width 0.6 mm. The general simulation parameters related to material (aluminum), sensors and defect definition are given at Table 1.

<table><caption>**Table 1.** Simulation parameters (nominal)</caption><thead><tr><th>Name</th><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td rowspan="2">Aluminum</td><td>Acoustic wave velocity</td><td>Longitudinal: 6132 m/s<br>Transverse: 3130 m/s</td></tr><tr><td>Density</td><td>2.66 g/cm³</td></tr><tr><td rowspan="4">Sensor</td><td>Location (center)</td><td>X: 220 mm, Y: 310 mm<br>X: 370 mm, Y: 310 mm<br>X: 340 mm, Y: 240 mm<br>X: 265 mm, Y: 370 mm</td></tr><tr><td>Active radius</td><td>4.4 mm</td></tr><tr><td>Hole location (center)</td><td>X: 300 mm<br>Y: 300 mm</td></tr><tr><td>Crack length</td><td>[0 – 30] cm</td></tr></tbody></table>
In this work, we created a numerical database of 1500 simulations. Computation time for each simulation was between 193 and 1770 seconds, increasing by the size of the crack length. We used a cluster made of 38 nodes, 8 of them being pairs of E5-2620 v4 2.1 GHz sharing 32 GB RAM and 30 of them pairs of R410 X5660 2.8 GHz sharing 24 GB RAM. These simulations take into account multiple EOC variations. Table 2 indicates the details of these variations. The variation in the sensor position is representing the instrumentation uncertainty linked to the manual positioning of the SHM system. The uncertainty of the quality of sensor bonding is represented with the changes in the active radius [13]. Last but not least, temperature change is indicating possible changes in the environmental condition and reflecting the observed experimental variations.

<table><caption>**Table 2.** Variation parameters used in simulation</caption><thead><tr><td>Name</td><td>Distribution</td><td>Extreme values</td><td>Notes</td></tr></thead><tbody><tr><td>Sensor position</td><td>N(μ: X, σ:2mm)</td><td>[X-5 mm, X+5mm]</td><td>X being the planned centre position</td></tr><tr><td>Sensor radius</td><td>N(μ: 4.4 mm, σ:0.1mm)</td><td>[4 mm, 4.8 mm]</td><td>Modeling of sensor coupling uncertainty</td></tr><tr><td>Temperature</td><td>N(μ: 20°C, σ:5°C)</td><td>[10°C, 30°C]</td><td>Effect on acoustic wave velocity [14].</td></tr><tr><td>Crack length</td><td>Uniform random</td><td>[3 mm, 30 mm]</td><td>30 crack lengths for a growing defect.</td></tr><tr><td>Crack orientation</td><td>N(μ: 0°, σ: 5°)</td><td>[-15°, +15°]</td><td>The change in crack position to the centerline of the hole</td></tr></tbody></table>

To mimic the data dependency of the experimental case, the sampling to generate the data was performed in two steps: first, 50 samples of sensor positions and radii were taken, which correspond to our different simulated plates. For each of these plates, a single value for crack orientation is assigned, temperature and crack length were then sampled 30 times. 50 separate pristine states with the samples configurations were modelled with a hole at the center of the plate and without any crack propagation.

## 3. Statistical methods for POD Estimation

### 3.1 POD using $\hat{a}$ versus a method

One of the most commonly used POD analysis in NDE field is $\hat{a}$ vs a analysis, that is based on a signal response analysis where $\hat{a}$ refers to measurement output (e.g. maximum amplitude, time of flight) and a is the damage size (e.g. crack length). $\hat{a}$ is a monotonous function of the measurement output and a is a monotonous function of the damage size. In practice, $\hat{a}$ is either directly the measurement output or its log, and the same for a regarding the damage size. This method is based on the regular regression statistical model that there is a population of cracks determining the POD and assumes that the observed data is uncorrelated and holds a uniform variance, the error follows a normal distribution, and there is a linear relationship between $\hat{a}$ and a [15]. POD of the regression model is estimated by using a threshold value ($a_{th}$). The linear regression model parameters ($\beta_0, \beta_1, \varepsilon$) are usually estimated using maximum likelihood estimation methods. Please note that $\Phi_{Norm}()$ is the normal cumulative distribution function.

$$ \hat{a} = \beta_0 + \beta_1 a + \varepsilon, \quad \varepsilon \sim N(0, \tau); \qquad (\text{Eq.1}) $$
$$ POD(a) = 1 - \Phi_{Norm} \left( \frac{a_{th} - (\beta_0 + \beta_1 a)}{\tau} \right). \qquad (\text{Eq.2}) $$

## 3.2 POD using length at detection method

Length at detection (LaD) analysis has been proposed in order to overcome the issue related to the dependency of the data measured by permanently mounted sensors. It uses the first crack length that is detected using the SHM systems [3], that is the first crack length for which the system output is above a given threshold. Under the assumption that the observed cracks follow a normal distribution, the POD can be computed as follows:

$$ POD(length) = \Phi_{Norm} \left( \frac{a - \bar{x}}{s} \right), \qquad (\text{Eq.3}) $$

where $\bar{x}$ is the mean and $s$ is the standard deviation of the observed crack lengths ($x_0, x_a, ... x_n$) for the $n$ observed cracks.

## 3.3 POD using random effects generalization of $\hat{a}$ versus $a$

Random effects model (REM) can be seen as the generalized $\hat{a}$ vs $a$ signal response analysis. As opposed to LaD, REM considers all data points. We assume multiple linear relationships between $\hat{a}$ and $a$, and the regression parameters differ from one SHM system (mock-up) to another and follow a normal distribution. The idea is to consider that the response $\hat{a}_i$ of each SHM system can be written as a linear regression with respect to the defect parameter $a_i$. These assumptions lead to an expression of the POD similar to the $\hat{a}$ vs $a$ POD, where the $Z$ function is not a function of the parameters of the regression themselves but is a function of the parameters of the distributions of these parameters:

$$ \hat{a}_{ij} = \beta_{0i} + \beta_{1i} (a_{ij} - \bar{a}) + \varepsilon_{ij}, \qquad (\text{Eq.4}) \\ i = 1, \dots, n, \quad j = 1, \dots, m_i, \quad \varepsilon_{ij} \sim N(0, \tau); $$

$$ POD(a) = 1 - \Phi_{Norm}(Z), \\ Z = \frac{a_{th} - (\mu_{\beta_0} + \mu_{\beta_1}(a - \bar{a}))}{(\sigma_{\beta_0}^2 + (a - \bar{a})^2\sigma_{\beta_1}^2 + 2(a - \bar{a})\sigma_{\beta_0}\sigma_{\beta_1}\rho + \tau^2)^{1/2}}, \qquad (\text{Eq.5}) $$

where $n$ is the number of systems and $m_i$ is the number of specific reading for system $i$; $\beta_{0i}$ and $\beta_{1i}$ are the intercepts and slopes respectively; $\mu_{\beta_0}$ and $\mu_{\beta_1}$ are their mean values; $\sigma_{\beta_0}$, $\sigma_{\beta_1}$, are their standard deviation; $\rho$ is the correlation between them. REM method is described in detail by Meeker et al [3] as a possible variation of linear mixed model proposed by Kabban et al [16].

# 4. Case study

## 4.1 Damage index and path selection

The used damage index (DI) is the Euclidean distance of the damaged state to the pristine state with a time window on the A0 mode, normalized by the norm of the pristine signal for the sensor pair 1-2. Pristine state (P) is considered as a plate with a hole without any crack propagation. For the experimental dataset, each sample (#12) had their own baseline
measurement. In the case of simulation, the same the sensor locations and radii are used. Damaged state (D) follows 20 different crack lengths for experimental case, and 30 different size for simulations. The DI is then defined as:

$$DI = \frac{||P - D||_2}{||P||_2}.$$

Based on the estimated damage indicators shown in Fig.2, the dataset interval is limited to 4 mm to 20 mm of the crack length to comply with the linearity and homoscedasticity rules. Based on the graphical representation of Fig. 2, the log-linear scale was found to be the most linear, hence the POD calculations were done in log-linear scale. Note that a good fit is obtained between data from the experiments, in red, and from simulation, in black.

<figure><img src="image_4.png" /><figcaption>Fig. 2. a vs. $\hat{a}$ graphs with DI estimations through the experimental and numerical database (a) linear-linear scale, (b) log-linear scale, (c) linear-log scale, (d) log-log scale.</figcaption></figure>

## 4.2 POD estimation with $\hat{a}$ versus a model

The probability of detection curve and its 95% confidence interval were first estimated using log-linear estimations based on Delta method as described in MH-HDBK 1823 [15] and ENIQ Report No.41 [17], following a classic NDT methodology. Two separate POD curves are shown in Fig. 3 and Fig. 4, for experimental and numerical database, respectively. It should be noted that this representation is given for comparison purposes, since SHM systems do not generate uncorrelated observations. It is also seen that homoscedasticity principle that should be complied with to use this method is not fully satisfied as the datasets are not distributed around the regression line equally.
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** The probability of detection curve and its 95% Wald confidence bound using the Delta method calculated using experimental results from 12 samples resulting in $a_{90} = 11.23$ mm and $a_{90/95} = 11.44$ mm.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Model based probability of detection curve and its 95% confidence interval calculated using (>1000) CIVA SHM simulation results resulting in $a_{90} = 11.35$ mm and $a_{90/95} = 11.44$ mm.</figcaption></figure>

Note that the obtained shape and values of the curves are close for experimental and simulated data. The slight change in the $a_{90}$ value may come from a wider variation range in the simulation, for example in sensor variability.

### 4.3 POD Estimation with Length at Detection model

Length at detection as described in [3] was then applied to both datasets.

Experimental dataset for each sample (#12) resulted in a single length at detection value as shown in different fitted lines in Fig.5. On the other hand, numerical dataset with fixed variations (sensor position, sensor radius, crack angle) are considered acting as the same structure. While the number of data points for LaD estimation is limited by the cost of increasing the number of samples in the case of the experimental dataset, this can be easily increased using simulation to allow a convergence in POD estimation. It is seen that the probability density of the LaD values follows a normal distribution, however the bell-shape is better represented in numerical dataset as the number of inputs is significantly higher (Fig 5b, Fig6b).
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** LaD using 12 sample experimental dataset each having 20 crack lengths (a) POD curve with 95% confidence bound $a_{90} = 10.53$ mm, $a_{90/95} = 11.05$ mm, (b) corresponding fitted length-at-detection model, each color corresponds to a plate sample.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** LaD using 50 variant simulation settings each having 30 crack lengths (a) POD curve with 95% confidence bound $a_{90} = 11.32$ mm and $a_{90/95} = 11.72$ mm, and (b) corresponding LaD model, each color corresponds to a simulated sample.</figcaption></figure>

Once again, the obtained curves are close. However, it should be noted that the $a_{90}$ and $a_{90/95}$ values are significantly higher in the simulation in this case. Based on the probability distribution graphs shown on the LaD model, the 12 experimental LaD values might not be enough to converge to a complete POD representation. A thorough study of convergence might be necessary.

## 4.4 POD estimation with random effects model

Finally, random effects model was applied to both experimental and numerical datasets. The POD values are estimated by building a Bayesian model with the PyMC python package [18]. Two level hierarchical random intercept and random slope linear regression model is built with suggested priors. The posterior marginal distributions were sampled using a NUTS sampler using Hamiltonian Monte Carlo method with four chains, resulting in more than $2 \times 10^6$ draws. As described in [3], while the 95% confidence bound is obtained putting 0.05th quantiles of model parameters to the Eq(5), the mean values of model parameters are used
for PoD estimation. The variance of error term ($\tau^2$) for the POD curve is estimated based on the fitted regression lines.

<figure><img src="image_9.png" /><figcaption>Fig. 7. REM using 12 sample experimental dataset each having 20 crack lengths (a) POD curve with 95% confidence bound $a_{90} = 11.55$ mm and $a_{90/95} = 12.77$ mm, (b) corresponding fitted random-effects model, each color corresponds to a plate sample.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. REM using 50 simulated samples each having 30 crack lengths (a) POD curve with 95% confidence bound $a_{90} = 11.97$ mm and $a_{90/95} = 12.97$ mm, (b) corresponding fitted random-effects model, each color corresponds to a simulated sample.</figcaption></figure>

As in previous cases, computation on the simulation dataset results in slightly higher values than the ones on the experimental dataset. Furthermore, the characteristic values of this model are higher than the ones obtained with the LaD model and the classic NDT approach.

## 4. Summary

This work presented a case study on several statistical techniques for POD estimation of SHM systems using numerical and experimental datasets. Specifically, the growing notch from a hole in the aluminum plate has been monitored with an example of guided wave structural health monitoring system. The numerical representation of the setting has been simulated using CIVA SHM. POD estimations showed that the numerical database can be used to estimate the reliability of the SHM system considering the possible influences that can be numerically simulated, as the computed damage indices showed a good fit with the
experimental ones and a great quantity of data could be generated in a reasonable time. It is
also seen that, in this case study, *a*<sub>90/95</sub> values estimated with REM were more conservative
than classic *â* vs *a* and LaD analysis. A more thorough convergence study based on
simulation is needed to confirm this observation in the future.

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