<figure><img src="image_1.png" /></figure>

# Structural Health Monitoring with Guided Ultrasonic Waves for Assets like Pressure Vessels and Pipes for Oil and Gas Applications Regarding Corrosion Under Insulation

Robert NEUBECK<fnref n="1" />, Tobias GAUL<fnref n="1" />, Bianca WEIHNACHT<fnref n="1" />, Lars SCHUBERT<fnref n="1" />, Arne-Ulrik BINDINGSBØ<fnref n="2" />,<fnref n="3" />

## Abstract

Pressure vessels and pipelines are the structural backbone of the petrochemical industry. However, due to typical operating conditions, they are often exposed to high corrosion and abrasion. This leads to costly inspection, maintenance and repair cycles, especially if the plant components are located in poorly accessible or specially protected areas, such as offshore, under insulation or in an ATEX zone. In such a scenario Structural Health Monitoring (SHM) by a permanently installed sensor system has the potential to greatly reduce costs and increase the inspection and maintenance efficiency. In this contribution, the development of a guided ultrasonic wave SHM system for pressure vessels in the oil and gas industry is presented. The system was developed as part of a research project between Equinor ASA and Fraunhofer IKTS to detect typical damage patterns in the millimetre to centimetre range under the insulation of operating pressure vessels by adapted high-resolution imaging routines. Furthermore, the algorithms were adapted to monitor critical regions around typical geometrical features as welded parts or nozzles. The system was validated through several laboratory and field tests, and its robustness with respect to the sensor network density has been investigated.

**Keywords:** Guided Ultrasonic Waves, Structural Health Monitoring, Pressure Vessels, Structural Features, Corrosion Monitoring

## 1. Introduction

Over the last two decades, guided ultrasonic waves (GUW) have established themselves as an alternative to conventional non-destructive testing methods for pipelines in the oil and gas industry [1]. Their main advantages are their damage detection capabilities and long propagation distances which result in the ability to investigate regions between or around sensors rather than underneath as for conventional ultrasonic testing. As a result, the technique is suitable for Structural Health Monitoring (SHM), since larger areas can be covered by a limited number of permanently installed sensors, especially if the network is applied before damages occur (baseline approach) [2]. Pipelines and pipe-like structures were addressed by the scientific community over recent years [3], whereas vessels are becoming the focus of more recent research.

In [4] the propagation of GUW in vessels is investigated by finite element simulations including structural features as in-/outlets (nozzles). [5] implemented a damage localization by elliptical imaging for vessel head and body separately. The structure was divided into two separate simplistic geometries wherefore Time-Of-Flight maps were calculated analytically by using a constant group velocity. [6] also used the elliptical imaging approach but optimized the sensor positioning around vessel head and nozzle. More recently, [7] considered several sensor arrangements and a damage localization by a reconstruction algorithm for probabilistic inspection of damage (RAPID) on an anisotropic vessel without any additional structural features. In latter time resolved signals are reduced to parameters before imaging and stacking, without requiring a velocity model. This method therefore implies data reduction and simplification which is easier to implement for complex geometries. On the other hand, sensitivity and interpretability are relaying on the chosen parameter (damage index). Nowadays

<footnote n="1">¹ Fraunhofer Institute for Ceramic Technologies and Systems (IKTS), Dresden, Germany; Phone: +49 351 88815-663; robert.neubeck@ikts.fraunhofer.de</footnote>

<footnote n="2">² Equinor ASA, Sandsli, Norway; abin@equinor.com</footnote>

<footnote n="3">³ Department of Marine Technology, Norwegian University of Science and Technology (NTNU), Trondheim, Norway</footnote>
machine and deep learning methods are either used supplementary or alternatively for damage localization and identification tasks [8]. For example, [9, 10] provided damage localization algorithms for vessels by convolutional neural networks, whereas [11] used a neural network to create a digital twin of a pipeline by combining GUW imaging, finite element simulations and conventional non-destructive testing data, integrated in augmented reality environment. [12] used deep learning to generate an imaging operator for an aluminum plate specimen, providing accurate thickness reduction maps. For such approaches a sufficient amount of data is required whereas data augmentation [13] can be applied or additional numerical data as for the probability of detection estimation shown on [14].

This work focuses on isotropic pressure vessels by using enhanced imaging routines for damage localization which can integrate structural features by approximation of a physical model. The method relies on a baseline approach and a known velocity model. Because the imaging kernel is implemented in frequency domain, the algorithm accounts for the typical dispersive behavior of guided waves and does not rely on non-dispersive frequency bands or modes. This article is based on a research project between Equinor ASA and Fraunhofer IKTS from 2017 till 2022 which provided in parallel a newly developed multiphase level monitoring system as result [15].

## 2. Data processing

A GUW-SHM system is assumed as a mathematical abstract distribution of transducers, which can receive and/or excite guided waves in a structure. After excitation at transducer $x_{TX}$ each sensor is measuring a wavefield $u(x, t)$ with respect of the time t and space x at the specific sensor location $x_{RX}$. Generally, the wavefield consist of an undamaged state (baseline) $u_B$ and a scattered wavefield $u_S$, which can be obtained by:

$$u_S(x, t) = u(x, t) - u_B(x, t) \qquad (1)$$

if the linear wave propagation is valid. In SHM this situation is often given, whereby the remaining issue is the additional influence of environmental or process parameters, which is neglected at first for the sake of simplicity. Since the scattered wavefield is only known at the sensor location $x_{RX}$ the concept of backpropagation and stacking is applied:

$$u_S(x, t) = \sum_{RX} \mathcal{F}^{-1}\{u_S(x_{RX}, \omega)G^*(x_{RX}, x, \omega)\} \qquad (2)$$

by using the complex conjugated Greens function $G^*$ and the inverse Fourier transform $\mathcal{F}^{-1}$ to approximate the scattered wavefield in the whole observation domain for a single measurement [16, 17]. In geophysics this concept is known as wavefield extrapolation, the underlaying concept of every migration operator. It is often treated as boundary value problem or given in more complex formula in time domain instead. The 4D information is usually reduced by collapsing the time or frequency axis in form of an imaging condition. For example, often cross correlation with forward propagated source wavefield is used to image reflectivity [18]. In this work we apply autocorrelation, which can be interpreted as calculating the signal energy for the whole observation domain:

$$I(x) = \sum_{TX} \sum_{t} (u_S(x, t))^2 \qquad (3)$$

Stacking of the extrapolated data for each excitation $TX$ is done implicitly after applying the imaging condition. In [16] it is additionally proposed to use temporal deviations by additional lags in the correlation, whereby [17] suggested a geometric mean value instead.

However, Greens function is usually unknown for more complex geometries like real structures and the approximation by numerical methods is required. In [19] we proposed a combination
of Fast-Marching-Method and a transport equation solver to calculate an approximation of the
Greens function for triangulated geometries. In this work we used this method in the frequency
domain to consider the frequency-dependent phase velocity of the investigated A0-Lamb mode.
Therefore, amplitude and Time-Of-Flight maps were calculated for the first arrivals of every
transducer location and relevant frequency. For each geometry and sensor network this was a
one-time effort. After storing, those maps were used efficiently for wavefield extrapolation
later. As the approximation applies for A0 first arrivals, mode conversion, multiples and
reflections on boundaries or welds are neglected. But at least multiples and reflections can be
added to the propagator, as for example shown by [20] in the context of tomographic imaging.
To further reduce artefacts, Fresnel-volume filtering was applied within the propagator [21, 22],
so that only the first three Fresnel volumes contribute to the wavefield extrapolation. Due to the
pitch-catch arrangement of the sensor network, Fresnel volumes were used in a first step along
direct propagation paths. In section 3.1 and 3.2 we show the improving capabilities of focused
Fresnel volumes. Therefore, the image centroid with 80 % threshold from a first imaging
iteration with direct Fresnel volumes was used as an estimated damage location, afterwards the
focused Fresnel volumes were constructed by mirroring on an assumed point reflector location
(image centroid) comparable to [21, 22]. Please note that all following imaging results are
normalized for better comparability.

**3. Experimental validation**

**3.1 Laboratory validation**

The artificial defects were introduced in a 1.5x2.5 m² steel plate with a thickness of 16 mm.
Whereby the guided waves were excited by piezoelectric transducer from the company PI-
Ceramics, of the type PTYY-0605, with a diameter of 10 mm and a thickness of 3 mm. As
sensor the radial transducer of the type DuraAct P-876.SP1, with a diameter of 10 mm and a
thickness of 0.2 mm was used. Those transducers were chosen according to their linear power
spectra in the desired bandwidth and high energy output, verified by experimental
measurements on undamaged plates.

<figure><img src="image_2.png" /><figcaption>Figure 1. Experimental setup for validation in lab on 1.5x2.5 m² steel plates; distances are given in millimetre.</figcaption></figure>
The transducers were glued on the beforehand cleaned plate surface. During the hardening of the couplant, a reproducible and constant contact pressure was ensured by magnets. The sensor network consisted of 5 actuators (transmitters) on a line arrangement with an equidistant spacing of 25 cm and two sensor lines of 7 sensors (receivers) with an equidistant spacing of 20 cm as depicted in Figure 1. Note that for the data processing and imaging in this article only the first sensor line was used.

As waveform a 5-cylced cosine and Hanning windowed impulse (RC5) with centroid frequencies of 50, 70 or 100 kHz was excited. Therefore, each dataset contains all transmitter-receiver combinations (5x7) for a certain excitation frequency, which were measured 64 times, bandpass filtered and averaged subsequently.

<table><caption>Table 1. Experimental validated damage types and dimensions.</caption><thead><tr><th>Damage</th><th>Depth [mm]</th><th>Length [mm]</th><th>Width [mm]</th><th>Diameter [mm]</th><th>Rotation [°]</th><th>Realisation</th></tr></thead><tbody><tr><td rowspan="3">Corrosion damage</td><td>1, 2 and 3</td><td>30</td><td>30</td><td></td><td></td><td rowspan="3">Surface removal of a rectangular area using an angle grinder</td></tr><tr><td>1, 2 and 3</td><td>70</td><td>70</td><td></td><td></td></tr><tr><td>1, 2 and 3</td><td>110</td><td>110</td><td></td><td></td></tr><tr><td rowspan="3">Pitting in shell wall</td><td>2, 5 and 8</td><td></td><td></td><td>4.0</td><td></td><td rowspan="3">Drill to specific dimensions</td></tr><tr><td>2, 5 and 8</td><td></td><td></td><td>8.5</td><td></td></tr><tr><td>2, 5 and 8</td><td></td><td></td><td>12.0</td><td></td></tr><tr><td rowspan="2">Group of pitting in weld seam</td><td>2 and 5</td><td></td><td></td><td>6</td><td></td><td rowspan="2">Multiple drillings to specific dimensions in a single v-weld</td></tr><tr><td>2 and 5</td><td></td><td></td><td>12</td><td></td></tr><tr><td>Hole through</td><td>16</td><td></td><td></td><td>2,4,6,8 and 10</td><td></td><td>Drill through complete plate</td></tr><tr><td>Crack</td><td>3 and 4</td><td>30</td><td>0.8</td><td></td><td>0, 45 and 90</td><td>Cut with angle grinder</td></tr></tbody></table>

In total, eleven steel plate specimen were used to generate data of 5 different damage types with various variations in dimension and orientation, as given in Table 1. The damages were introduced artificially by either drilling or using an angle grinder. Once an artificial damage had been applied, its dimensions were checked with a tape measure. However, especially the damages introduced manually by an angle grinder with an 115 mm disc have some uncertainties. The maximum depth is specified for the crack damage, whereby the crack depth decreases evenly at the beginning and end due to the curvature of the 115 mm disc. For all plate specimen at least one undamaged baseline dataset was measured before the damages were introduced. The measurements took place in a laboratory environment with minimal environmental changes. Therefore, no compensations were necessary in this respect.

## 3.2 Field test

After successful lab tests, a field test was performed on a decommissioned insulated pressure vessel in a non-ATEX zone at the Equinor laboratories at Kårstø (K-Lab), Norway. The test specimen was made of S235 steel, had a wall thickness of 16 mm, 9954 mm length and an outer diameter of 2268 mm. The vessel had several nozzles as well as other structural features and was almost completely insulated. The sensor network was instrumented on the inside of the vessel. The instrumented section of 1.5x1.0 m² had no significant geometrical features within the relevant area. The piezoelectric transducers/sensors were glued and positioned in the same manner as in the lab (section 3.1) after polishing the surface by an angle grinder. The network consisted of a line of 12 transmitting PTYY-0605 elements with an equidistant spacing of
125 mm and a parallel line of 25 receiving DuraAct P-876.SP1 with an equidistant spacing of 62.5 mm, 1 m apart from each other. Additionally, two temperature sensors were attached close to the sending and receiving array, respectively. The instrumented section and the measured surface temperature during the trial are shown in Figure 2.

After successful instrumentation, several undamaged states (baseline) were measured before starting to introduce artificial damages. The baseline data do not cover the whole temperature range of the field test. A single pitting was introduced by drilling and a 45° to the sensor network rotated crack was cut by an angle grinder. As in section 3.1 the dimension in Table 2 maximum values, acquired by subsequent control measurements.

To reduce the influence of temperature dependency a combined approach of optimal baseline selection and baseline stretch was applied. An artificial baseline was generated for the temperature of the damaged state by a signal interpolation in the frequency domain based on third order L1-regression of each frequency component individually. For the last two pitting and all crack damages an extrapolation was required due to the missing temperature coverage of the baseline.

<figure><img src="image_3.png" /><figcaption>Figure 2. Experimental setup of the field test with instrumented 1.5x1.0 m² section (left), close ups of the two damage types (middle) and measured surface temperature with the measurement time of each dataset (right).</figcaption></figure>

<table><caption>**Table 2. Dimensions and positions of the artificially introduced damages in the field test.**</caption><thead><tr><th>Damage</th><th>Designation</th><th>Depth [mm]</th><th>Length [mm]</th><th>Width [mm]</th><th>Diameter [mm]</th><th>Centre position (x,y) [m]</th><th>Realisation</th></tr></thead><tbody><tr><td rowspan="4">Pitting</td><td>Pitting 1</td><td>3.0</td><td></td><td></td><td>5.0</td><td rowspan="4">(1.00, 0.40)</td><td rowspan="4">drill to specific dimensions</td></tr><tr><td>Pitting 2</td><td>4.0</td><td></td><td></td><td>5.0</td></tr><tr><td>Pitting 3</td><td>5.0</td><td></td><td></td><td>5.5</td></tr><tr><td>Pitting 4</td><td>8.0</td><td></td><td></td><td>7.0</td></tr><tr><td rowspan="2">45° Crack</td><td>Crack 1</td><td>4.0</td><td>60.0</td><td>1.0</td><td></td><td rowspan="2">(0.52, 0.65)</td><td rowspan="2">cut with angle grinder</td></tr><tr><td>Crack 2</td><td>7.0</td><td>80.0</td><td>2.0</td><td></td></tr></tbody></table>

### 3.3 Investigation of nozzles in lab

To investigate the influence of structural features, two additional plate specimens with a nozzle were manufactured (nozzle 1 and nozzle 2). Similar as the lab test in section 3.1, the steel plates had a dimension of 1.5x2.5 m² and a thickness of 16 mm. In the middle section of the plate an artificial nozzle was attached by a curricula cut-out and subsequent welding of a pipe section with an outer diameter of 219 mm, a wall thickness of 10.31 mm and length of 250 mm. As in previous lab tests, a line of 12 transmitting PTYY-0605 elements with an equidistant spacing of 125 mm and a line of 25 receiving DuraAct P-876.SP1 with an equidistant spacing of 60 mm were attached. This implies an increased transducer density by factor of 2 and sensor density of
~~2.1. The exact arrangement of investigation can be obtained from Figure 3~~.

<figure><img src="image_4.png" /><figcaption>Figure 3. Experimental setup for investigation of structural features in the lab on 1.5x2.5 m² steel plates; distances are given in millimetre</figcaption></figure>

The measurement setup regarding the hardware, the excited waveform and the pre-processing was the same as in section 3.1. In each plate a crack damage was introduced and increased in its dimensions. Afterwards a new baseline was measured, and a pitting damage was introduced and increased subsequently. By integrating the previous damages to the baseline, two plates were sufficient for a series of crack and pitting damages at two different locations each, under the assumption of linear wave propagation and an insignificant influence of the prior damage on the wavefield propagation. The exact damage combinations and positions can be obtained from Table 3.

<figure><img src="image_5.png" /><figcaption>Figure 4. Lab test on influence of nozzles on guided waves localization capabilities: measurement setup (right), close-up of nozzle (left, top) and close-up of frist crack damage in weld (left, bottom).</figcaption></figure>
<table><caption>Table 3. Dimension and positions of the investigated artificial damages in the two steel plates with a nozzle.</caption><thead><tr><th>Damage</th><th>Plate</th><th>Depth [mm]</th><th>Length [mm]</th><th>Width [mm]</th><th>Diameter [mm]</th><th>Orientation to nozzle [°]</th><th>Realisation</th></tr></thead><tbody><tr><td rowspan="12">Crack</td><td>Nozzle 1</td><td>1.8</td><td>30.0</td><td>0.8</td><td></td><td>270</td><td rowspan="12">cut with angle grinder</td></tr><tr><td>Nozzle 1</td><td>2.5</td><td>35.0</td><td>0.8</td><td></td><td>270</td></tr><tr><td>Nozzle 1</td><td>3.3</td><td>40.0</td><td>0.8</td><td></td><td>270</td></tr><tr><td>Nozzle 1</td><td>4.2</td><td>45.0</td><td>0.8</td><td></td><td>270</td></tr><tr><td>Nozzle 1</td><td>7.7</td><td>60.0</td><td>0.8</td><td></td><td>270</td></tr><tr><td>Nozzle 2</td><td>2.2</td><td>33.0</td><td>0.8</td><td></td><td>180</td></tr><tr><td>Nozzle 2</td><td>3.0</td><td>38.5</td><td>0.8</td><td></td><td>180</td></tr><tr><td>Nozzle 2</td><td>4.2</td><td>45.0</td><td>0.8</td><td></td><td>180</td></tr><tr><td>Nozzle 2</td><td>6.3</td><td>54.5</td><td>0.8</td><td></td><td>180</td></tr><tr><td>Nozzle 2</td><td>7.5</td><td>59.5</td><td>0.8</td><td></td><td>180</td></tr><tr><td rowspan="12">Pitting</td><td>Nozzle 1</td><td>2.0</td><td></td><td></td><td>4.0</td><td>90</td><td rowspan="12">drill to specific dimensions</td></tr><tr><td>Nozzle 1</td><td>2.0</td><td></td><td></td><td>6.0</td><td>90</td></tr><tr><td>Nozzle 1</td><td>2.0</td><td></td><td></td><td>8.0</td><td>90</td></tr><tr><td>Nozzle 1</td><td>5.0</td><td></td><td></td><td>8.0</td><td>90</td></tr><tr><td>Nozzle 1</td><td>8.0</td><td></td><td></td><td>8.0</td><td>90</td></tr><tr><td>Nozzle 2</td><td>1.3</td><td></td><td></td><td>2.0</td><td>0</td></tr><tr><td>Nozzle 2</td><td>3.5</td><td></td><td></td><td>4.0</td><td>0</td></tr><tr><td>Nozzle 2</td><td>4.5</td><td></td><td></td><td>4.0</td><td>0</td></tr><tr><td>Nozzle 2</td><td>4.8</td><td></td><td></td><td>6.0</td><td>0</td></tr><tr><td>Nozzle 2</td><td>6.4</td><td></td><td></td><td>6.0</td><td>0</td></tr><tr><td>Nozzle 2</td><td>6.6</td><td></td><td></td><td>8.0</td><td>0</td></tr><tr><td>Nozzle 2</td><td>7.9</td><td></td><td></td><td>8.0</td><td>0</td></tr><tr><td>Nozzle 2</td><td>8.5</td><td></td><td></td><td>10.0</td><td>0</td></tr></tbody></table>

# 4. Results

## 4.1 Laboratory validation

<figure><img src="image_6.png" /><figcaption>Figure 5. Lab test results with direct Fresnel volume and 100 kHz for a selection of three single pitting damages.</figcaption></figure>

From all investigated damages in the lab, the single pitting with a diameter of 4 mm was the smallest. Each depth increment from 2 mm till 8 mm produced a high contrast and an accurately localization by means of the image centroid, as seen for example for 100 kHz in Figure 5. For 100, 70 and 50 kHz the expected wavelengths of the investigated A0 mode in a 16 mm steel plate are 25, 34 mm and 44 mm, respectively. The remaining point spreading function correlated with that dimension.
The rectangular corrosion produced comparable results in Figure 6, although the increasing the depth improved the imaging significantly.

<figure><img src="image_7.png" /><figcaption>Figure 6. Lab test results with direct Fresnel volume approach and 100 kHz for all sizes of the 1 mm and 3 mm deep rectangular corrosion damages.</figcaption></figure>

The crack damage in Figure 7 is given for the smallest crack size, for all three orientations, for the direct and the focused Fresnel volume approach. All cracks were localized within an error in the centimetre range. The focusing improved the accuracy, reduced the point spreading and lowered the background significantly. From all three damages, the crack with 0° orientation was expected to have the lowest interaction with the guided waves. Nevertheless, the three crack orientations are comparable.

<figure><img src="image_8.png" /><figcaption>Figure 7. Lab test results: Comparison of direct and focused Fresnel volume approach, for three orientations of a 30x0.8x3 mm³ crack and 100 kHz excitation frequency.</figcaption></figure>
## 4.2 Field test

In the field test a higher sensor density was used compared to the previous lab test, but the measurements were influenced by temperature changes and the increased surface roughness of the vessel. Additionally, sensor failures occurred as seen by the grey marked and not used transducers/sensors in Figure 8 and Figure 9.

Despite the additional challenges, the imaging localized all damages for all excitation frequencies, as seen in Figure 8. Compared to the lab, the imaging contained not a single but multiple indications close to the real pitting location (black dot). As a result, the image centroid was in between those indications (red dot). Focused Fresnel volumes improved that behaviour and were increasing the accuracy especially for 50 kHz.

<figure><img src="image_9.png" /><figcaption>Figure 8. Exemplary results for pitting 1(φ=3 mm & depth=3 mm) and an excitation frequency of 50 kHz (left) and 100 kHz (right), respectively.</figcaption></figure>

Compared to the pitting the crack damages produced even better results, similar to the quality achieved in the lab. The re-alignment of the Fresnel volumes did not improve the accuracy in the millimetre range but reduced background noise (Figure 9).

<figure><img src="image_10.png" /><figcaption>Figure 9. Exemplary results for crack 1(60x1x4 mm³) and an excitation frequency of 50 kHz (left) and 100 kHz (right), respectively.</figcaption></figure>

## 4.3 Investigation of nozzles in lab

The proposed method localized the investigated damages on the weld in general even for sensor failures during the measurement. In case of the pitting damage, the accuracy was higher as shown in Figure 10. The accuracy increased with the damage size whereas the background noise reduced, as expected.
<figure><img src="image_11.png" /><figcaption>Figure 10: Various 0° pitting damages, for 100 kHz and direct Fresnel volume approach, measured on a nozzle plate 2.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 11: Various 270° crack damages, for 100 kHz and direct Fresnel volume approach, measured on a nozzle plate 1.</figcaption></figure>

The localization of crack damages was less stable for the plates with structural features than for previous investigations on plates without any additional features. Increasing the damage size did not improve the result evidently in any case. In certain cases, imaging artefacts appeared, or the image centroid was shifted along the weld, as shown exemplary in Figure 11. This behaviour is most likely caused by two interacting effects. Firstly, the nozzle act as an acoustical lense and secondly, mode conversion as well as multiple reflections are not considered by the approximation. Therefore, damage positions along the weld differ in illumination strength for the given arrangement, which is not compensated perfectly by the asymptotic amplitude factor in the used Greens operator.

**5. Conclusions and outlook**

The work provides a damage localization method in the context of an GUW SHM system tested on lab and field data for pressure vessels in the oil and gas industry. The algorithm performed a wavefield extrapolation based on the velocity model of a single GUW mode considering the dispersive phase velocity. In the implemented data processing, a baseline was used to isolate the damage signature from the measurement. It was shown that typical structural features can be included in the model, which leads to the possibility of monitoring larger or more complex real-world structures. Together with the selected imaging condition, it has proven a stable and accurate localisation of damages in the sub-wavelength range for most of the cases investigated. However, the implemented extrapolator could be used in combination with other imaging conditions, e.g. cross-correlation to account for different damage types or contrasts. An additional pulse-echo (reflection) or complete full-matrix-capture arrangement is expected to
enhance results even further and most likely to illuminate structural features more evenly. But
in this case only consecutive focused Fresnel volume filtering after a non-filtered imaging is
possible instead of initial direct Fresnel volume filtering. Since the wave field extrapolator is
based on an acoustic approximation that does not consider multiple modes or mode conversions,
structures exhibiting such behaviour are more likely to produce additional noise and image
artefacts, even if multiple reflections for a single mode are included. An extrapolator modelling
the complete elastic wavefield is the next logical step to include such effects but will require
additional signal processing or sensors providing the complete three-dimensional elastic
displacement field as for example given the case of nowadays three-dimensional Doppler laser
vibrometers. As an alternative additional post-processing could decrease imaging artefacts and
with sufficient data machine learning or even deep learning approaches are a valid option, also
when it comes to automated interpretation of the imaging results.

The sensitivity of the method relies on the baseline approach and compensation of
environmental or process conditions. However, the compensation does not yet take into account
the changing behaviour in the interaction between the wavefield extrapolator and the
environment, for example due to a change in wave velocity caused by temperature changes.
Therefore, multiple approximations of Greens function for the expected temperature
distributions must be calculated beforehand.

Acknowledgements

The authors would like to express Equinor ASA their gratitude for funding the research as well
as the collaboration over several years by sharing industry experience and access to the K-Lab
facilities. The authors like to thank especially Dr. Jan-Magnus Østvik, Dr. Magnus Ottøy and
Dr. Jan Richard Sagli for their immense support in this project

References

[1] A. Ghavamian, F. Mustapha, B. T. H. T. Baharudin and N. Yidris, “Detection, Localisation and Assessment of Defects in Pipes Using Guided Wave Techniques: A Review,” *Sensors*, vol. 18, no. 12, 2018.

[2] M. Mitra and S. Gopalakrishnan, “Guided wave based structural health monitoring: A review,” *Smart Materials and Structures*, vol. 25, no. 5, p. 053001, 2016.

[3] R. Guan, Y. Lu, W. Duan and X. Wang, “Guided waves for damage identification in pipeline structures: A review,” *Structural Control and Health Monitoring*, vol. 24, no. 11, p. e2007, 2017.

[4] F. Li, Z. Liu, X. Sun, H. Li and G. Meng, “Propagation of guided waves in pressure vessel,” *Wave Motion*, vol. 52, p. 216–228, 2015.

[5] B. Yang, Y. Xiang, F.-Z. Xuan, C. Hu, B. Xiao, S. Zhou and C. Luo, “Damage localization in hydrogen storage vessel by guided waves based on a real-time monitoring system,” *International Journal of Hydrogen Energy*, vol. 44, no. 40, p. 22740–22751, 2019.

[6] C. Hu, B. Yang, B. Xiao, F.-Z. Xuan and Y. Xiang, “Damage localization in pressure vessel using guided wave-based techniques: Optimizing the sensor array configuration to mitigate nozzle effects,” *Applied Acoustics*, vol. 185, p. 108393, 2022.

[7] J. Heimann, S. Mustapha, B. Yilmaz and J. Prager, “Guided Waves in Composite Overwrapped Pressure Vessels and Considerations for Sensor Placement Toward Structural Health Monitoring—An Experimental Study,” *Journal of Nondestructive Evaluation, Diagnostics and Prognostics of Engineering Systems*, vol. 8, no. 3, 2025.

[8] Z. Yang, H. Yang, T. Tian, D. Deng, M. Hu, J. Ma, D. Gao, J. Zhang, S. Ma, L. Yang, H. Xu and Z. Wu, “A review on guided-ultrasonic-wave-based structural health monitoring:
From fundamental theory to machine learning techniques,” *Ultrasonics*, vol. 133, p. 107014, 2023.

[9] C. Hu, B. Yang, J. Yan, Y. Xiang, S. Zhou and F.-Z. Xuan, “Damage Localization in Pressure Vessel by Guided Waves Based on Convolution Neural Network Approach,” *Journal of Pressure Vessel Technology*, vol. 142, no. 6, 2020.

[10] J. Zhang, C. Hu, J. Yan, Y. Hu, Y. Gao and F. Xuan, “Guided Wave Damage Location of Pressure Vessel Based on Optimized Explainable Convolutional Neural Network for Multivariate Time Series Classification Neural Network,” *Journal of Pressure Vessel Technology*, vol. 145, no. 4, 2023.

[11] T. Wang, K. Feng, J. Ling, M. Liao, C. Yang, R. Neubeck and Z. Liu, “Pipeline condition monitoring towards digital twin system: A case study,” *Journal of Manufacturing Systems*, vol. 73, p. 256–274, 2024.

[12] X. Wang, M. Lin, J. Li, J. Tong, X. Huang, L. Liang, Z. Fan and Y. Liu, “Ultrasonic guided wave imaging with deep learning: Applications in corrosion mapping,” *Mechanical Systems and Signal Processing*, vol. 169, p. 108761, 2022.

[13] C. Polle, S. Bosse and A. S. Herrmann, “Damage Location Determination with Data Augmentation of Guided Ultrasonic Wave Features and Explainable Neural Network Approach for Integrated Sensor Systems,” *Computers*, vol. 13, no. 2, p. 32, 2024.

[14] K. Tschoke, I. Mueller, V. Memmolo, M. Moix-Bonet, J. Moll, Y. Lugovtsova, M. Golub, R. S. Venkat and L. Schubert, “Feasibility of Model-Assisted Probability of Detection Principles for Structural Health Monitoring Systems Based on Guided Waves for Fiber-Reinforced Composites,” *IEEE transactions on ultrasonics, ferroelectrics, and frequency control*, vol. 68, no. 10, p. 3156–3173, 2021.

[15] R. Neubeck, M. Stephan, T. Gaul, B. Weihnacht, L. Schubert, A. U. Bindingsbø and J.-M. Østvik, “Estimation of the Filling Distribution and Height Levels Inside an Insulated Pressure Vessel by Guided Elastic Wave Attenuation Tomography,” *Sensors (Basel, Switzerland)*, vol. 21, no. 1, 2020.

[16] P. Sava and I. Vasconcelos, “Extended imaging conditions for wave-equation migration,” *Geophysical Prospecting*, vol. 59, no. 1, p. 35–55, 2011.

[17] N. Nakata and G. C. Beroza, “Reverse time migration for microseismic sources using the geometric mean as an imaging condition,” *Geophysics*, vol. 81, no. 2, pp. KS51-KS60, 2016.

[18] R. Neubeck, C. Kexel and J. Moll, “Matrix techniques for Lamb-wave damage imaging in metal plates,” *Smart Materials and Structures*, vol. 29, no. 11, p. 117003, 2020.

[19] R. Neubeck, S. Buske and L. Schubert, *Berechnung von Laufzeit- und Amplitudenverteilung durch simultane Lösung der Eikonal- und Transportgleichung mittels einer adaptierten Fast-Marching-Methode*, Leipzig: DGZfP Jahrestagung, 2018.

[20] N. Rawlinson and M. Sambridge, “The Fast Marching Method: An Effective Tool for Tomographic Imaging and Tracking Multiple Phases in Complex Layered Media,” *Exploration Geophysics*, vol. 36, no. 4, p. 341–350, 2005.

[21] S. Buske, S. Gutjahr and C. Sick, “Fresnel volume migration of single-component seismic data,” *Geophysics*, vol. 74, no. 6, pp. WCA47-WCA55, 2009.

[22] M. M. Malatesta, R. Neubeck, J. Moll, K. Tschoke and L. Marchi, “Double-Stage DMAS With Fresnel Zone Filtering in Guided Waves Damage Imaging,” *IEEE transactions on ultrasonics, ferroelectrics, and frequency control*, vol. 69, no. 5, p. 1751–1762, 2022.