# Embedded passive guided wave tomography. Application to corrosion monitoring in multilayered pipes

Arnaud Recoquillay, Antoine Gallet, Tom Druet, Julien Albertini, Théo

Duval, Guillaume Laffont, Nicolas Roussel, Bastien Chapuis

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## To cite this version:

Arnaud Recoquillay, Antoine Gallet, Tom Druet, Julien Albertini, Théo Duval, et al. Embedded passive guided wave tomography. Application to corrosion monitoring in multilayered pipes. EWSHM 2024 - 11th European Workshop on Structural Health Monitoring, Jun 2024, Potsdam, Germany. 10.58286/29855. cea-04892586

HAL Id: cea-04892586

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Submitted on 16 Jan 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Embedded passive guided wave tomography.
## Application to corrosion monitoring in multilayered pipes

Arnaud RECOQUILLAY¹, Antoine GALLET¹, Tom DRUET¹, Julien ALBERTINI¹, Théo DUVAL¹, Guillaume LAFFONT¹, Nicolas ROUSSEL¹, Bastien CHAPUIS¹

¹Université Paris-Saclay, CEA, List, F-91120, Palaiseau, France

**Abstract.** We present here guided wave tomography methods applied to the imaging of defects in multi-layered structures using ambient noise data acquired with Fiber Bragg Gratings on optical fibers. After presenting the principle of the methods and the instrumentation, imaging results on numerical data are shown to assess the possibility to image in multi-layered structures. Experimental results of passive methods acquired using FBGs on a single optical fiber are then presented.

**Keywords:** guided waves, tomography, fiber Bragg gratings, passive methods, multi-layered structure

## Introduction

Ultrasonic guided waves have been studied for several years for structural health monitoring due to their high sensitivity to changes in many industrial structures, such as pipe-like or plate-like structures, as well as their large propagation distance, enabling the use of a limited number of sensors to monitor large areas. Among techniques relying on guided waves, tomography algorithms enable a precise quantitative imaging of thickness reductions like corrosion or erosion defects [1]. They can also be used without a reference state [2], increasing their robustness to varying environmental and operational conditions [3] and hence limiting the risk of false alarms. These techniques have been, up until now, used for homogeneous materials like aluminium panels or steel pipes. Some industrial structures on the other hand may be multi-layered, as is the case of concrete/steel/concrete pipes occurring in nuclear power plants, among other applications, the middle layer of which are subject to corrosion. Previous studies have shown that ultrasonic guided waves can be emitted and measured in these structures [4], opening the way for their monitoring if the methods can be adapted to their specificities. A first aim of this paper is then to adapt tomography methods to these multi-layered structures.
However, such techniques rely on a rather large number of sensors compared to other SHM methods, inducing a bigger added burden to the structure. A possibility to circumvent this issue is to use Fiber Bragg Gratings (FBG) inscribed on optical fibers, which have a limited intrusivity due to their small dimensions and the fact that, thanks to their multiplexing capability, several sensors can be inside a single optical fiber. Furthermore, they can cope with many harsh environments such as extreme temperatures or pressures, explosive atmosphere or radiative environments. FBG can measure ultrasonic waves using the so-called edge filtering technique [8]. As of today, there is however no mature solution to emit ultrasonic waves using optical fibers. It is then necessary to either use complementary actuators, or monitoring techniques not needing emission. The solution explored here is the use of passive methods, which rely on the acquisition of the ambient noise inherent to the structure to recover signals equivalent to the ones acquired with a sensor acting as an actuator. By doing so, an imaging system using solely FBGs is feasible and the ambient noise, which is usually an hindrance, becomes an asset. Furthermore, optical fibers with proper coating can easily be embedded into cementitious materials to acquire the data as close as possible to the metal layer. Note that the feasibility of guided wave tomography based on passive data has already been demonstrated [7]. Moreover, the use of FBG to recover a signal from passive measurement has also been studied but with one FBG per fiber [8]. Thus, a second part of this paper is then dedicated to a first passive acquisition using two FBGs on a single optical fiber.

# 1. Ultrasonic measurements using FBGs

Fiber Bragg Gratings are local periodic perturbations of the refractive index of the core of an optical fiber. This specific perturbation leads to a reflection peak around a typical optical wavelength named the Bragg wavelength, which depends on the period of the grating and the refractive indices. Solicitations like strain or temperature lead to a shift in this peak. In the same way, ultrasonic waves, which are high frequency strain waves, induce an oscillation of this peak. Most measuring techniques rely on the detection of the peak value, which cannot be done today at a high enough sampling frequency, preventing its use for ultrasonic waves. Instead, the so-called edge filtering technique is usually used [9] (see Figure 1): a constant power narrowband light source is emitted on either side of the Bragg peak, leading to only a fraction of the optical power being reflected. If an ultrasonic wave impacts the FBG, the peak oscillates, inducing an oscillation in the reflected power as the relative position of the light source to the Bragg peak changes. A photodiode with bandwidth higher than the frequencies of interest, can then convert this optical power into an electrical signal.
<figure><img src="image_3.png" /><figcaption>Figure 1: Principle of edge filtering</figcaption></figure>

As each FBG is characterized by a localized peak in the wavelength domain, the position of the peak depending mostly on the period of the FBG, several FBGs can be put in a single optical fiber by changing their step. Each FBG can then be interrogated by taking a light source at the right wavelength.

The full optoelectronic setup is then the following: for each FBG, a narrowband light source is set on the side of the peak. All the light sources are connected to a multiplexer feeding a circulator. The circulator is then connected to the optical fiber with the FBGs, the reflected light going back into the circulator and through its second output to a wavelength demultiplexer to separate the optical power corresponding to each FBG. The separated optical powers are then fed to different photodiodes.

Note however that the edge filtering as described is sensitive to environmental and operational conditions (EOC) like strain and temperature: changes in these parameters will induce a low frequency shift of the Bragg reflection peak, leading to a change in the relative position of the light source and the side of the peak. First, this change will induce a change in sensitivity: as the local slope is not the same, the variation in reflected power induced by the wave also changes. For higher EOC variations, the sensitivity may be lost as the support of the light source and the peak become disjoint. To circumvent this issue, a tracking method was used: the mean value of the reflected optical power is indeed an indication of the relative position of the light source and the peak. By tracking this value, using a wavelength-tunable light source, the relative position can be kept constant. To do so, a discrete mode laser diode with a Peltier module was used, and the tracking is then performed on the DC value measured by the photodiode.

It is well known that these sensors' sensitivity depends on the angle between the fiber direction and the polarization of the incident wave [9], which may have an effect on data processing. As an illustration, simulations were performed on a simplified concrete / steel / concrete pipe of outer diameter 720 mm, each layer being respectively, from outside to inside, 36 mm, 2 mm and 22 mm thick (see Figure 2). The concrete is considered homogeneous, with material parameters $ρ = 2440$ kg.m⁻³, $v_L = 4950$ m.s⁻¹ and $v_T = 3150$ m.s⁻¹. The parameters of the steel layer are $ρ = 7800$ kg.m⁻³, $v_L = 5710$ m.s⁻¹ and $v_T = 3220$ m.s⁻¹. A ring of 32 PZT actuators is positioned at axis position 650 mm and a ring of 32 receivers, either PZT or FBG, is at axis position 1350 mm. The transducers of both rings are equally spaced along the circumference. PZT transducers have a 10 mm diameter and FBG are 10
mm long. Simulations were performed with the SHM module of CIVA [5] using a pin force model for PZT transducers. The FBGs were simulated using a one-way coupling strategy. Strain values at their positions were extracted in the ultrasonic simulation to compute the FBG response using a transfer matrix approach [6]. The optical wavelength on the left of the peak, corresponding to half of the maximum amplitude, is chosen for edge filtering. By using this one-way coupling, FBG response can be computed with a limited impact on computation performances. The emission signal is a 5 cycle Hann burst of center frequency 100 kHz.

<figure><img src="image_4.png" /><figcaption>Figure 2: Scheme of the simulated multi-layered pipe. Blue circles represent PZT transducers.</figcaption></figure>

Obtained data are plotted in Figure 3 for the emitter of index 5. Note that signals are normalized in each sub-figure with respect to the maximum of all signals. In both cases, we see direct and helical wave packets reaching the receivers, the first wave packet being for the receiver directly in front of the emitter (index 5, shortest distance). For PZT receivers, the amplitude of the wave packets is only slowly decreasing with propagation distance while, for FBG receivers, large amplitude differences can be seen: they correspond to the directive sensitivity, leading to an amplitude decrease as the angle increases. It is especially visible for helical packets, which correspond to higher angles.

<figure><img src="image_5.png" /><figcaption>Figure 3: Simulated data with PZT receivers (left) and FBG receivers (right).</figcaption></figure>
## 2. Guided wave tomography for multi-layered structures

Guided wave tomography methods are based on a single mode propagation assumption, enabling to interpret data using the corresponding dispersion curves to reconstruct a thickness map and to consider an acoustic propagation model, that is the Helmholtz equation in the frequency domain. The iterative method briefly presented here is available in more details in [2] and is based on [1]. It relies on the Lippman-Schwinger equation and a Born approximation to express the scattered field $\varphi^s$ for a source at $x_i$ and a receiver at $x_j$:

$$ \varphi^s(x_j, x_i) = \int G_0(x_j, x)O(x)\varphi_0(x, x_i)dx, $$

where $G_0$ is the Green's function of the Helmholtz equation in the current domain $\Omega$, $\varphi_0$ is the unperturbed field and $O(x) = k^2(x) - k_0^2$ is the object function, where $k$ is the wave number and $k_0$ is the wave number for the initial thickness. Inverting this equation then enables to reconstruct the object function:

$$ \mathbf{O}(x) = \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \frac{\varphi^s(x_j; x_i)}{G_0(x_j; x)\varphi_0(x; x_i)} W(\theta_i, \theta_j) d\theta_i d\theta_j, \quad (1) $$

where $W$ is a weighting function coming from a variable change and $\theta_i$ and $\theta_j$ are the angles associated to the emitter and the receiver, respectively. In a single material domain, it is well known that wave number variations can be directly linked to the thickness through the dispersion curves. As this expression is based on a Born approximation, some discrepancy can remain between the reconstructed domain and the true domain. To circumvent this issue, the object is updated iteratively, replacing in (1) by $O$ by $\delta O^n$, $G_0$ by $G_{n-1}$ and $\varphi_0$ by $\varphi_{n-1}$, which are respectively the Green's function and the solution corresponding to $O_{n-1} = \sum_{l=0}^{n-1} \delta O_{n-1}$.

As mentioned, the thickness reconstruction is obtained through the dispersion curves, linking the wave number to the thickness. Even though this link is direct for a homogenous material, the same relations can be obtained for a multi-layered material by computing the dispersion curves with respect to the thickness of one of the layers, the defect being expected to affect this layer.

Note that an autocalibration method, described in [2], removes the need for a reference state for this method, making it more robust to environmental and operational condition changes.

This is illustrated on the same configurations as in section 2, where a thickness loss of diameter 140 mm and maximum thickness reduction 5 mm in the external concrete layer was added. Note that the defect was added on the exterior of the pipe for simplicity of the simulation for this proof of concept, but the study aims at imaging a defect on the steel layer. The thickness loss has a tukey shape along the diameter, the flat bottom being as long as the varying sides. This defect is centered at the axis position 1050 mm and circumferential angle 240° with respect to the position of the first sensor. Tomography results considering only transmission data, that is for a ring of actuators and a ring of receivers as described previously, are plotted in Figure 4 for PZT and FBG receivers, respectively. Note that these results are plotted for the unrolled pipe, the red dots corresponding to the sensors. In this representation, the defect is expected at position (-0.38; 0.1) m, (0; 0) corresponding to the angular position of the first sensor along the circumference and the middle of the two transducer rings. We can see that, in the PZT case, the defect is well located and the thickness loss is well estimated, even though the defect is overestimated along the axis, which is well known due to a lack of sensors in this direction. In the FBG case, the defect is once again well located, but the extension and the thickness loss are overestimated. This is most probably
due to the directivity of the sensitivity of these sensors, leading to an amplitude variation depending on the relative position of the sensors, as already seen in Figure 3. Future works will focus on compensating this effect to improve the imaging results.

<figure><img src="image_6.png" /><figcaption>Figure 4: Tomography results on a multi-layered structure with PZT receivers (left) and FBG receivers (right). Color scale is the thickness in mm.</figcaption></figure>

## 3. Passive guided wave measurement using FBGs on a single optical fiber

### 3.1 Passive techniques

Passive techniques rely on the measure of a diffuse field [10], that is a space-time stationary random field with constant spectral density, with no correlation for two different measurement points. Under these assumptions, it can be shown that the cross-correlation of the time signals acquired at two different points converge to the impulse response of the system. In more details, for two measurement points $x$ and $y$, let $V_x(t)$, respectively $V_y(t)$, be the acquired signal over the time interval $(0, T)$. The cross-correlation is defined as

$$C(x, y, t; T) = \int_{0}^{T} V_{x}(\tau)V_{y}(t + \tau)d\tau.$$

It can then be shown [10][11] that

$$\lim_{T \to +\infty} \frac{dC}{dT}(x, y, t; T) \propto H_{xy}(t) - H_{yx}(-t),$$

where $H_{xy}$, respectively $H_{yx}$, is the impulse response of the system for the sensor in $x$ acting as actuator and the one in $y$ acting as receiver, respectively for the sensor in $y$ acting as actuator and the one in $x$ acting as receiver. Note that the method is written here under the assumption of a constant spectral density, which is not true in practice. However, the method remains valid as long as the ambient noise has a significant energy level at the frequencies of interest. Among these techniques, the passive inverse filter [12] has shown promising features: it can be seen as a regularization of the cross-correlation result, whitening the noise source in space and time. It can be expressed as a post-process of the cross-correlation: let $C$ be the matrix containing the cross-correlation result for all sensor couples at a given frequency. The passive inverse filter consists in computing the singular value decomposition of this matrix, cancel small singular values (corresponding to noise) and replace others by 1. The threshold below which values are cancelled can be determined either using a threshold on the value or their decrease speed. Also, as demonstrated in [13], cross-correlation results can be normalized by the auto-correlation of the data of each sensor of the pair, reducing the effect of the noise source over the reconstructed transfer function. This is called hereafter the intra noise portion normalization. Note that this method is still affected by the acquisition noise, and an improvement would be to normalize by the auto-correlation of the sensors but for different noise portions [13]. These methods have already been coupled to guided wave tomography using piezoelectric transducers [7] and with FBGs on different optical fibers using classic Delay And Sum imaging [8].
## 3.2 Experimental validation using two FBGs on a single optical fiber

As described in section 1, multiple FBGs with different Bragg wavelengths can be inscribed and interrogated in the same optical fiber. Here, two 10 mm-long FBGs were inscribed in a single fiber, 50 cm apart, respectively at 1549.8 and 1569.5 nm. Two tunable discrete mode laser diodes with matching wavelengths were used. The chosen wavelengths enable the use of a standard coarse wavelength division multiplexing MUX/DEMUX for injecting the light in the circulator and separate each grating reflexion. Furthermore, the 20 nm spacing between the two peaks prevents any spectral overlap of the two signals.

The FBGs were coupled to a square 3 mm thick aluminium panel of side 1.2 m using shear gel. Ambient noise was generated using an air flow moving randomly above the panel. This noise was recorded during 30 s into 0.5 s portions at a sampling frequency of 500 kHz and was then processed using the Passive Inverse Filter and the intra noise portion normalization described above. The signals are then reconstructed by convoluting the result with a classic source signal for guided wave experiments, that is a 5 cycles Hann burst of varying center frequency. The results are presented in Figure 5. Note that the signals are plotted for positive and negative times, corresponding to switched roles between emitter and receiver. We can see that, after a first packet at $t = 0$ µs, corresponding to electronic noise, the wave packet corresponding to the A0 mode is well seen, up to a center frequency of 90 kHz. Note however that the noise level seems to be increasing at higher frequencies, characterized by residual wave packets before the A0 mode. This is expected, as natural ambient noise energy decreases with the frequency. Other wave packets are visible at later time, probably linked to edge reflections as they show coherent variations in their time of flight with respect to the evolution of the A0 wave speed with frequency.

<figure><img src="image_7.png" /><figcaption>Figure 5: Reconstructed signals for ambient noise between two FBGs on the same optical fiber for different center frequencies. Black line: theoretical time of flight of the A0 mode.</figcaption></figure>

# 4. Conclusion and perspectives

We have shown in this paper results related to the use of passive guided wave tomography using Fiber Bragg Gratings on optical fibers. New results regarding the generalization of the guided wave tomography to multi-layered structures and passive measurements using FBGs on the same optical fiber were shown. Based on these preliminary results, future works will handle an experimental tomography using solely FBGs on a test
sample with a defect on the steel layer and the compensation of the directive sensitivity of
the FBGs to improve the imaging results.

Acknowledgements

The research presented in this paper is part of the FO-US project, supported by the French
National Research Agency (ANR) under grant number ANR-21-CE04-0007.

References

[1] Huthwaite, P., & Simonetti, F. (2013). High-resolution guided wave tomography. Wave Motion, 50(5), 979-993.

[2] Druet, T. et al. (2019). Autocalibration method for guided wave tomography with undersampled data. *Wave Motion*, **89**, 265-283.

[3] Druet, T. et al. (2023). Passive guided wave tomography for corrosion monitoring under varying environmental conditions. NDE in Nuclear 2023, June, Hall in Sheffield, United Kingdom. e-Journal of Nondestructive Testing. https://doi.org/10.58286/28283.

[4] Raveendran T. et al. (2022). Inspection of Concrete/Steel/Concrete pipes by ultrasonic methods. In: NDT-CE 2022-The International Symposium on Nondestructive Testing in Civil Engineering.

[5] Mesnil O. et al. (2021). Experimental validation of transient spectral finite element simulation tools dedicated to guided wave-based structural health monitoring. Journal of Nondestructive Evaluation, Diagnostics and Prognostics of Engineering Systems, vol. 4, no 4, p. 041003.

[6] Fiborek P. et al. (2022). Spectral element modeling of ultrasonic guided wave propagation in optical fibers. Ultrasonics, vol. 124, p. 106746.

[7] Druet, T. et al. (2019). Passive guided wave tomography for structural health monitoring. The Journal of the Acoustical Society of America, 146(4), 2395-2403.

[8] Recoquillay A. et al. (2020). Guided wave imaging of composite plates using passive acquisitions by fiber Bragg gratings. The Journal of the Acoustical Society of America, 147(5), 3565-3574.

[9] Betz D. et al. (2007). Structural damage location with fiber Bragg grating rosettes and Lamb waves. Structural health monitoring, vol. 6, no 4, p. 299-308.

[10] Lobkis, O. I., & Weaver, R. L. (2001). On the emergence of the Green's function in the correlations of a diffuse field. The Journal of the Acoustical Society of America, 110(6), 3011-3017.

[11] Garnier J. et Papanicolaou G. (2016). Passive imaging with ambient noise. Cambridge University Press.

[12] Gallot T. et al. (2012). A passive inverse filter for Green's function retrieval. The Journal of the Acoustical Society of America, vol. 131, no 1, p. EL21-EL27.

[13] Lanza Di Scalea F. et al. (2018). Robust passive reconstruction of dynamic transfer function in dual-output systems. The Journal of the Acoustical Society of America, vol. 143, no 2, p. 1019-1028.