Abstract

Conventional ultrasonic approaches for corrosion surveillance such as large-scale C-scans and continuous localised monitoring face inherent limitations in either temporal resolution or spatial coverage. Recent advances in sensor technology and low-cost autonomous robotics enables a hybrid approach that combine their strengths. However, critical questions remain about optimising such approaches, including determining the optimal inspection intervals, spatial coverage, and number of sensors. This study presents a multi-stage framework that addresses these challenges through integrated modelling: degradation simulation for spatiotemporal evolution of component surfaces, physics-based surrogate models for ultrasonic thickness measurements, data subsampling to replicate inspection and monitoring procedures, and reliability assessments against surveillance targets. Through Monte Carlo simulations on synthesised data calibrated to field measurements, we demonstrate the potential benefits of the hybrid strategy. Results show that strategic sensor placement and periodic repositioning can enhance reliability while reducing coverage requirements and extending inspection intervals. Importantly, this framework provides quantitative guidance for corrosion surveillance planning and can be adapted to different degradation phenomena across various asset types.

1. Introduction

Ultrasonic non-destructive evaluation (NDE) techniques serve as vital tools for corrosion surveillance and assessment in industrial assets. The reliability of these methods is critical for ensuring the safety and operational availability of industrial infrastructure, pipelines, vehicles, and other critical systems. Depending on the type of corrosion anticipated, NDE techniques and procedures can generally be categorised into two main approaches: time-based regular inspection and continuous monitoring [1].

Specifically in this study, inspection refers to periodic and independent surveys of a structure which provides accurate information about an asset’s condition (e.g., remaining thickness) at the time of the survey, but offer limited insight into changes between successive inspections. An example of the inspection technique is ultrasonic thickness measurement. These can be used for corrosion mapping of larger surfaces, either by manually acquiring data from a drawn on grid or using more advanced automated C-scanning techniques [2–5]. However, due to challenges in gaining access and the potential disruption to asset operations, inspections are typically conducted months or even years apart. As a result, the inspection approach is often constrained by its limited temporal resolution [6,7].

Alternatively, to address the high access costs associated with traditional inspection methods, the monitoring approach to corrosion surveillance involves installing sensors that provide continuous, automated measurements to track corrosion progression in real time. Among ultrasonic monitoring systems, ultrasonic guided wave systems [8–11] offer large detection range and extensive spatial coverage [12]. However, guided wave methods may fall short in sensitivity, particularly on complex structures, for small defects, and over long distances [13]. In contrast, bulk wave based point thickness measurements [14–16] offer excellent repeatability and sensitivity at the measurement location, allowing accurate estimate of corrosion trend and rate. However, their effective coverage is very limited. Additionally, to ensure long-term stability, these sensors are typically installed in a ‘‘permanent’’ manner — via welding or mechanical bonding — making relocation after installation practically impossible.

It is evident that a trade-off between sensitivity and spatial coverage inherently exists. However, recent progress in autonomous robotic systems and sensor technology has introduced a new dimension that helps address this compromise. Specifically, the availability of autonomous robotics provides new means for sensor delivery, significantly reducing the access costs to traditionally hard-to-reach areas that hinder frequent inspections [17–22]. At the same time, this also effectively increases the spatial coverage of point thickness measurement sensors. Additionally, advancements in sensing technology, such as inexpensive multi-channel electronics, have made these data acquisition systems more affordable and scalable, enabling their deployment in large quantities [23,24]. The emergence of compact and low-power electromagnetic acoustic sensors (EMATs) eliminate the need for couplant between the sensor and the structure, simplifying installation, removal, and reinstallation [25–28]. The integration of sensors with robotic sensor delivery platforms allow an inspection robot to perform ultrasonic surface scans (e.g., C-scans) more frequently while simultaneously managing a fleet of monitoring sensors. This capability provides data with enhanced spatial coverage and temporal resolution, effectively blurring the boundary between inspection and monitoring.

Fig. 1. Schematic workflow of the multi-stage reliability assessment framework.

The presented study introduces a simulation approach for evaluating inspection and monitoring strategies, with a focus on the potential benefits of adopting hybrid approaches enabled by resident inspection robot concepts. A multi-stage evaluation framework, as shown in Fig. 1, is introduced to systematically assess and compare the performance of various NDE procedures. The framework comprises four key stages: (i) modelling the degradation process of interest, (ii) simulating NDE data acquisition techniques and the associated errors, (iii) subsampling the data based on prescribed inspection and monitoring procedures, and (iv) conducting a quantitative assessment of the selected approaches in terms of system reliability. For proof of concept, normal-incident ultrasonic thickness measurements are employed as the primary NDE technique, serving as the foundational method for both ultrasonic C-scans and point thickness monitoring systems. This study highlights how lifting the temporal and spatial restrictions inherent in traditional inspection and monitoring approaches can enhance the reliability of NDE systems for corrosion assessment.

This paper is organised as follows: Section 2 presents the method-ology to model corrosion progression, establishing the ground truth surface morphology for subsequent analysis. A physics-based surrogate model is introduced in Section 3 that generates realistic ultrasonic thickness maps, including errors based on surface morphology and sensor characteristics. Section 4 simulates three data acquisition strate-gies — inspection scanning, monitoring with permanently installed sensors, and hybrid scanning and monitoring. The efficacy of these approaches are compared through quantitative metrics at varying spatio-temporal resolutions. Section 5 discusses practical applications, current limitations, and future research directions.

2. Modelling of corrosion evolution

The evolution of thickness loss and morphological changes is divided into two distinct processes: general corrosion and pitting corrosion. These two processes are modelled independently, and their effects are superimposed to calculate the total thickness loss. The remaining thickness of a sample is determined by subtracting the accumulated thickness loss from its initial nominal thickness.

2.1. General corrosion

General corrosion, under a specific environmental condition, typically results in a relatively uniform thickness loss over time and the affected surfaces can be modelled using Gaussian or exponential distributions. Building on this understanding, a modelling approach similar to that proposed in literature has been employed to represent general corrosion as follows:

S_{t+1}(x, y) = S_t(x, y) + max[P_t(x, y, r_p, cl_p) + Δh, 0]

Fig. 2. An illustrative example of the progression of uniform corrosion.

In Eq. (1), 𝑆𝑡(𝑥, 𝑦) and 𝑆𝑡+1(𝑥, 𝑦) denote the thickness loss over a surface at two consecutive time steps 𝑡 and 𝑡 + 1, respectively. The for-mulation implies that at each time step, the thickness loss is governed by a constant term, 𝛥ℎ, and a spatially varying component described by a Gaussian perturbation, 𝑃𝑡. The perturbation term is characterised by its root-mean-square (RMS) height, 𝑟𝑝, and the correlation length, 𝑐𝑙𝑝. The 𝑚𝑎𝑥 operation ensures that positive thickness loss is enforced, consistent with the nature of the corrosion phenomena. An example of this process is illustrated in Fig. 2.

2.2. Evolution of pitting corrosion

The modelling of pitting corrosion is divided into two aspects: temporal evolution and spatial distribution. To describe the temporal evolution of instantaneous pitting corrosion rate, d(t), a probabilistic-deterministic model described by Wang et al. is adopted. A brief summary of the model is provided below.

The model describing the temporal evolution of the defect growth rate is divided into three distinct phases, separated by two key time instances: the pit initiation time, 𝑡0, and the steady state time, 𝑡𝑠𝑠. The model is summarised in Eq. (2).

For 𝑡 ∈ [0, 𝑡0), the defect growth rate, 𝑑(𝑡), is zero as no pit appears on the sample surface before the initiation time, 𝑡0. The initiation time can follow either a normal distribution or a Wielbull distribution in a Monte Carlo simulation.

For 𝑡 ∈ (𝑡𝑠𝑠, ±∞), the defect growth rate follows a deterministic evolution path, 𝛬(𝑡), as described by the power-law function shown in Eq. (3). Where 𝛼 is a positive growth constant controlling the rate of change of growth rate. 𝜏 is a positive constant governing the temporal scaling of defect growth rate. 𝑑0 represents the growth rate at pit initiation which follows a probabilistic distribution.

For t in [t0, t_ss], the defect growth rate exhibits stochastic behaviour and is modelled as a geometric Brownian bridge process (also known as the Wiener process). The stochastic term W'(t) introduces probabilistic deviations from the deterministic path while ensuring smooth convergence to the deterministic path at 𝑡 = 𝑡0 and 𝑡 = 𝑡𝑠𝑠. Equation (4) gives the expression of the stochastic term:

where 𝑊 (𝑡) is a standard Wiener process, defined over 𝑡0 ≤ 𝑡 ≤ 𝑡𝑠𝑠, with variance 𝜎2 . The Wiener process is expressed as:

Once the temporal evolution of pit growth rate, 𝑑(𝑡), is defined, the depth of a corrosion pit at an arbitrary time, 𝐷(𝑡), can be calculated by the integrating the growth rate over time:

Fig. 3. Temporal evolution of (a) corrosion rate and (b) pit depth, comparing deterministic predictions (red dashed line) against stochastic realisations (30 blue solid lines). The following parameters are used to generate given illustration: 𝑡0 = 10, 𝑡𝑠𝑠 = 100, 𝜎𝑤 = 0.05, 𝑑(𝑡0 ) = 0.01, 𝛼 = 1.8, 𝜏 = 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3 illustrates the defect growth evolution process with an example. For visual clarity, all of the pits begin to grow at 𝑡0 = 10, with an initial rate of 𝑑(𝑡0) = 0.01. The red dashed lines indicate the deterministic growth trajectory predicted by the model, characterised by pit growth parameters 𝛼 = 1.8 and 𝜏 = 4. The solid blue lines represent 30 stochastic realisations driven by a Wiener process which starts at 𝑡0 = 10 and ends at 𝑡𝑠𝑠 = 100.

Fig. 4. An illustration of pit geometry for AR = 0.5, Depth = 1/R. (a) Top view of the surface. (b) Cross-sectional view along the red dotted line shown in (a) at 𝑦 = 50. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2.3. Pit geometry and spatial distribution

The spatial distribution of corrosion pits and their exact geometries are known to impact pit detectability and ultrasonic thickness measurements. In the model, the location of pit seeds can follow either a uniform or normal distribution along the x- and y-axes, with the distributions set independently for each axis.

As pit depth increases over time, thickness loss extends beyond the pit centre and affects the surrounding area. The geometry of thickness loss is modelled by convolving the pit depth with a predefined spatial filter. The filter size is linked to the pit depth and controlled by the pit aspect ratio (AR), defined as the ratio of pit diameter to pit depth. The pit's extent and geometry are determined by the filter kernel.

Fig. 4 gives an example of the pit geometry where thickness loss is inversely proportional to the distance from the pit centre following ‘‘Inverse radial (1/R)’’ relation and with an AR of 0.5.

2.4. Comparing the simulation output to real world field measurements

To validate the simulation methodology, we compare the empirical cumulative distribution function (ECDF) of field measurements from a pipeline affected by pitting corrosion with that of the simulation outputs. The ECDF represents the cumulative probability distribution of the remaining wall thickness below a given threshold, and is commonly used as a metric for identifying corrosion patterns and assessing the associated risks.

Fig. 5. Representative cumulative distribution functions (ECDFs) of field measurements exhibiting different pitting corrosion morphologies, compared against matching simulation data. Blue circles: the ECDF of field measurements reported in the literature. Black solid line: the averaged ECDF from 100 simulated realisations of the fitted corrosion models. (a)m Wall thickness distribution on an oil line affected by pitting corrosion. Field measurements extracted and reproduced from Figure 13 in [29]. (b) Wall thickness distribution in a corroded area of a condenser vessel. Field measurements extracted and reproduced from Figure 17 in [29]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5 presents two representative ECDF curves of field measurements, each exhibiting distinct pitting corrosion morphologies. In sub-plot (a), the ECDF of wall thickness measurements from an oil line affected by pitting corrosion is shown, with the field data extracted from Figure 13 in [29]. The curve features a sharp transition to a near-exponential distribution around a remaining thickness of 26.5 mm, followed by a pronounced drop-off in the tail region at approximately 12.5 mm. In subplot (b), the ECDF of wall thickness measurements from a corroded area of a condenser vessel is presented, with the field data extracted from Figure 17 in [29]. In the second case, the distribution corresponding to remaining thickness less than 15.8 mm can be reasonably approximated by either a normal or Weibull distribution.

A total of 100 simulation realisations are generated and fitted to each of the two corrosion morphologies using the parameters listed in Table 1. Each simulation case evolves over 51 temporal steps, with the model parameters tuned such that the simulated ECDF at 𝑡 = 45 (averaged across the 100 realisations) closely matches the corresponding field data. Fig. 6 illustrates the spatio-temporal evolution of corrosion progression in a representative realisation.

For the case shown in Fig. 5 subplot (a), each realisation contains 30 corrosion pits uniformly distributed across both the 𝑥 and 𝑦 directions. For the case shown in subplot (b), each realisation contains 150 corrosion pits, uniformly distributed in the 𝑥-direction and following a Gaussian distribution in the 𝑦-direction. The averaged ECDF curve of the simulation cases (represented by the black solid line) is overlaid on the field measurement data (represented by the blue markers) in Fig. 5, demonstrating that the proposed corrosion model can successfully reproduce the observed corrosion patterns.

Beyond matching the averaged field trends, individual realisations exhibit variability arising from Gaussian perturbations in general corrosion and stochastic pit growth governed by the Wiener process. This variability reflects the inherent uncertainty and diversity in corrosion development.

As such, the degradation model supports generation of any observed surface morphologies in statistically significant quantities, enabling reliability assessment studies in two key aspects: (1) potential variations in surface morphology beyond limited field measurements, and (2) temporal evolution paths both preceding and following the snapshot available from field data.

3. Simulation of ultrasonic thickness measurements

In the second stage of the evaluation process, normal-incidence, pulse-echo ultrasonic thickness measurements (i.e. corrosion mapping) are simulated using the surface profile generated by the corrosion model. The selection of this NDE method was driven by its practical compatibility with robotic platforms, as demonstrated by several early prototypes reported worldwide. [19,22]. This simulation step enables the analysis of how measurement errors influence the effectiveness of inspection and monitoring strategies, as explored in subsequent sections (see Fig. 7). As with all measurement techniques, ultrasonic thickness measurements are subject to errors and uncertainties. The HOIS Joint Industry Projects conducted evaluations of multiple corrosion mapping and wall thickness measurement systems, with results published in technical reports [35,36].

For the thickness error of corrosion mapping over plates (Table 7 in Ref. [36]), the tested systems showed a mean error of 0.143 mm (i.e. overestimation of residual thickness) and a standard deviation of 0.153 mm. For wall thickness loss measurements over small pitting flaws (Table 6 in Ref. [36]), the tested configurations demonstrated a mean error of −0.062mm (i.e. underestimation of thickness loss, which corresponds to overestimation of residual thickness) and a standard deviation of 0.168 mm. The worst-performing system showed a mean error of 0.91 mm with a standard deviation of 0.24 mm. These studies revealed substantial performance variations among systems, with measurement accuracy and precision influenced by the surface condition, measurement hardware (e.g. equipment types) as well as data processing routine (e.g. signal processing algorithms and parameter settings).

To physically represent measurement uncertainties under vary-ing conditions, we employ the surrogate model proposed by Burch et al. [37], which simulates normal-incidence ultrasonic thickness measurements based on the acoustic beam-spread effect. The model takes into account of correlations between surface roughness and ultra-sonic sensor characteristics, including sensor dimensions and operating frequencies. The following assumptions are assumed in building the beam-spread surrogate model:

1. The corroding surface (i.e., the sample back wall) is assumed to be in the far-field regime of the ultrasonic system.
2. The contribution of residual wall thickness to the ultrasonic mea-surement is assumed to be proportional to distribution of the acoustic pressure.

Fig. 6. Representative spatio-temporal evolution of a sample surface undergoing mixed general and pitting corrosion.

Fig. 7. Schematic of the surface scanning process: comparison between the true surface profile and ultrasonic measurements.

The output of the beam-spread model, ℎ 𝐵𝑆 , consists of the summa- tion of expected model output, ℎ 𝑐𝑜𝑛𝑣, and measurement uncertainties due to scattering from rough surfaces, 𝛥ℎ 𝑟𝑚𝑠. The expected ultrasonic measurements ℎ 𝑐𝑜𝑛𝑣 are spatially weighted averages of the underlying surface, where the weighting kernel, 𝐻 𝑝𝑟𝑜𝑏𝑒, is determined by probe frequency and dimensions using an acoustic beam spread model. The transformation is implemented through convolution operations, ∗ , as formulated in Eq. ( 8 ).

The radius, 𝑅 𝐻, of the kernel 𝐻 𝑝𝑟𝑜𝑏𝑒 can be calculated based on the sample thickness and the beam divergence angle, 𝜃, as illustrated in Fig. 8 . The beam divergence angle is defined as the angle at which the acoustic intensity drops to a specified threshold relative to the central axis intensity, assuming the sample thickness, ℎ, is greater than the Fresnel distance of the chosen ultrasonic probe and excitation frequency. Equation ( 10 ) [ 38 ] is used to calculate the beam divergence angle, where 𝑉 denotes the ultrasonic velocity in the inspected material, and 𝐷 and 𝑓 denote the diameter and operating frequency of ultrasonic probe, respectively. The coefficient 𝐾 is obtained by solving the Bessel function of the first kind. For a beam spread threshold set to one-half ( − 6 dB) of the central intensity, 𝐾 takes a value of 0.514.

Fig. 8. Illustration of the ultrasonic beam spread, defining the beam divergence angle, 𝜃 and the acoustic kernel radius, 𝑅𝐻.

The kernel weights are assigned to follow a Gaussian distribution with a mean of zero and a standard deviation determined using Equation (11), which results in a distribution where the value at 𝑟 = 𝑅 is half of the value at 𝑟 = 0. The kernel weights are then normalised by their summation to ensure proper scaling.

Equation ( 12 ) describes the modelling of measurement uncertain- ties, 𝛥ℎ 𝑟𝑚𝑠 , arising from roughness-induced wave scattering. In this study, the model only accounts for effects caused by surface roughness, quantified using root-mean-square (RMS) height, while excluding the influence of localised pitting defects. 𝛥ℎ 𝑟𝑚𝑠 is modelled as a normal distribution scaled by a coefficient, 𝑓 ( 𝑆 ( 𝑥, 𝑦 ) | 𝐻 𝑝𝑟𝑜𝑏𝑒 ) , which is a function of the surface roughness within the sonified area. The exact value of the coefficient is determined from the finite element (FE) simulations described below.

30 surface realisations were generated using the approach outlined by Gajdacsi [ 32 ] for RMS heights ranging from 0 to 0.3 mm in intervals of 0.1 mm. The surface profiles were transformed into node coordinates of a three-dimensional finite element (FE) model. General-purpose linear brick elements with reduced integration (C3D8R) were used to construct the simulation domain, which has nominal dimensions of 30 mm × 30 mm × 10 mm ( 𝑥 × 𝑦 × 𝑧 ).

An ultrasonic probe is positioned on the flat surface at z = 0, while the surface profile representing the back wall is offset by 10 mm. The coordinates of the intermediate nodes were determined using the mesh- squashing technique described by Zimmermann [ 39 ]. The mechanical properties of typical steel materials were assigned to the simulation do- main and element size was set to 1/20 of the wavelength of ultrasonic waves propagating in the material. For compressional waves with a central frequency of 5 MHz, the element size is approximately 0.06 mm. Absorbing regions based on a stiffness reduction method were added to the xz and yz surfaces surrounding the domain [ 40 ].

A 3-cycle Hann-windowed sinusoidal toneburst signal is applied as a force load to nodes within the footprint of the selected probe size at z = 0. The time-domain response (i.e. A-scan) of ultrasonic waves reflected from the rough surfaces are simulated using the FE solver Pogo [ 41 ] and the Hilbert envelope of the signal is calculated. Ultrasonic thickness measurements are calculated based on the A-scan signals using an envelope peak detection (EPD) algorithm described in Ref. [ 42 ]. The standard deviation of ultrasonic thickness measurements for each RMS values are computed.

In Fig. 9 , the standard deviations of ultrasonic thickness measure- ments by a virtual ultrasonic probe with a diameter of 6 mm and a central frequency of 5 MHz are plotted against the ratio of surface RMS height to wavelength of the transmitted ultrasonic wave. The FE values, denoted by blue circles, are found to be in good agreement with experimental measurements, denoted by black crosses, from previous studies by Benstock et al. [ 42 ].

To obtain the appropriate scaling coefficients, the moving standard deviation of the surface profile is calculated, and the scaling coefficients are estimated through interpolation based on the anchoring points derived from the FE studies.

The presented wave scattering model, and the derived coefficient, inherit the limitations of the FE simulations on which it is based — specifically, that only surface roughness profiles with a root mean square (RMS) height less than one-third of the wavelength have been tested. The FE studies should be repeated if a different transducer dimension or central frequency is employed.

Fig. 10 presents a representative example which compares the 2D image of a corroded surface (i.e. the ground truth (GT)) and the ultra- sonic C-scan of the underlying surface predicted by the beam spread (BS) surrogate model (circular sensor diameter D = 6 mm, longitudinal wave speed = 5900 m/s, central frequency f = 5 MHz).

4. Simulation of inspection and monitoring procedures and performance evaluation

Having established methods to simulate ultrasonic measurement errors and uncertainties, this section describes a data subsampling methodology that replicates typical inspection and monitoring protocols. The subsampled datasets enable quantitative evaluation of different corrosion assessment strategies: surface scanning during scheduled inspections, monitoring with permanently installed sensors, and mixed scanning and monitoring approaches. The data subsampling process can be performed on either the ultrasonic thickness maps described in Section 3, or directly on the underlying ground truth data described in Section 2. Alternatively, a comparison can be conducted to assess the impact of measurement uncertainties. The data presented in Sections 4.2 to 4.4 are generated based on the ground truth data to illustrate different data subsampling scenarios. A comparison is provided in Section 4.5 to demonstrate the effect of measurement errors and uncertainties on strategy effectiveness.

Fig. 9. Standard deviation of ultrasonic thickness measurements as a function of surface RMS normalised by the wavelength. Values from the FE studies are denoted by blue circles and values from experimental studies are denoted by black crosses. Experimental data extracted from Figure 19 in [ 42 ] with permission. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.

Fig. 10. Comparison of surface profile between the ground truth and beam- spread model (circular sensor diameter D = 6 mm, longitudinal wave speed = 5900 m/s, central frequency f = 5MHz)

4.1. Evaluation objective and metrics

The primary objective of inspection and monitoring is to ensure structural integrity by tracking the most critical degradation features. In this study, the primary focus is on tracking the minimum thickness of a surface when it falls below a predefined trigger threshold. Alternative objectives, such as identifying defects exceeding size thresholds, could be similarly implemented.

 To quantitatively evaluate different inspection and monitoring strategies, we introduce the Unreliability Function (URF) as formulated in Eq. ( 13 ). For a given surface at time t, the URF returns 0 when the error of minimum thickness estimates falls within the measurement tolerance, and 1 if the error exceeds the tolerance. Averaging URF values across multiple realisations yield an ensemble URF, which serves as a probabilistic metric of strategy performance, ranging from 0 (all estimated minimum thickness measurements are within tolerance) to 1 (all measurements exceed the tolerance). In all subsequent analyses, a fixed trigger threshold of 25 mm and measurement tolerance of 0.3 mm are applied consistently.

where ℎ mm is the minimum measured thickness, ℎ gm is the true global minimum thickness, 𝜖 ℎ is the acceptable measurement tolerance, and ℎ threshold is the trigger threshold for thickness tracking.

4.2. Simulation of surface scanning during scheduled inspections

To simulate a time-based inspection scanning strategy, the spatiotemporal dataset is subsampled in both the temporal and spatial domains. The simulation accounts for two key factors: the inspection interval and the spatial coverage ratio.

Fig. 11. (a) Distribution of minimum remaining thickness across 100 realisations visualised using a boxplot. The bottom and top edges of the box represent the 25th and 75th percentiles of remaining thickness, while the whiskers extend to the most extreme non-outlier data points. The most critical cases for inspection intervals of 1 unit and 10 units are connected by lines with black circles and blue triangles, respectively. For an inspection interval of 10, measurements acquired at scheduled inspections are shown with filled triangles, while projected thickness estimates between inspections (calculated via linear extrapolation) are shown with empty triangles. (b) Comparison of URF curves for different inspection intervals (3-, 5-, and 10-unit periods). Normalised area under the curve (NAUC) = 0.006 (interval=3), 0.068 (interval=5), 0.35 (interval=10). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

4.2.1. URF versus inspection scanning intervals

In Fig. 11 (a), the distribution of minimum remaining thickness across all 100 realisations are visualised using boxplots plotted in black. The bottom and top edges of the box indicates the 25th and 75th percentiles of remaining thickness, while the whiskers extend to the most extreme non-outlier data points. As shown in the graph, the distribution of minimum thickness gradually widens over time, indicating increasingly diverse surface morphology driven by stochastic pitting corrosion.

The minimum values across the 100 realisations are connected using a line with black circular markers, highlighting the most critical case which can only be captured if complete spatial coverage and infinitely small temporal resolution (i.e.inspection interval = 1) can be achieved with inspection scanning. However, practical implementation often imposes operational constraints, leading to scheduled inspections at fixed intervals. The trace shown in blue represents one such scenario with a 10 unit inspection interval (i.e. subsampled in time for every 10 simulation steps). The filled blue markers represent minimum thickness at the scheduled inspections (t = 10, 20, 30, . . . ) whilst the empty blue triangles represent inter-inspection thickness estimates projected using linear extrapolation of the two preceding measured thickness minima.

The linear extrapolation between inspection points introduces discrepancies between the black and blue traces, representing measurement errors caused by the limited temporal resolution in scheduled surface scanning during inspections. Once this discrepancy exceeds the selected measurement tolerance, it results in non-zero Unreliability Function (URF) values, as shown in Fig. 11 (b). The characteristic sawtooth pattern of the URF curve demonstrates the periodic resetting of estimates during inspections, followed by progressive error accumulation until the next measurement.

To quantitatively evaluate the effectiveness of scheduled inspection scanning, we compute the area under the URF curve (AUC), where lower values indicate better performance. For an inspection interval of 10 unit, we obtain an AUC of 17.5. The AUC is normalised by the maximum possible area (1 × simulation duration = 50), yielding a normalised area under the curve (NAUC) of 17.5/50 = 0.35. The NAUC represents the overall probability that measurements remain within tolerance throughout the duration and across all realisations, with lower values indicating higher reliability. For shorter inspection intervals of 5- and 3-unit periods, the NAUC decreases to 0.068 and 0.006, respectively. These results demonstrate that reduced inspection intervals significantly improve system reliability in this test case, illustrating the fundamental trade-off between inspection frequency and reliability.

Fig. 12. Effect of coverage ratio (CR) on inspection reliability (inspection interval fixed at 3-unit period). (a-c) Measurement patterns at CR=10%, 30%, and 50%. (d) Variation of URF curves at different CR values. Normalised area under the URF curve (NAUC) = 0.006 at CR = 1; NAUC = 0.044 at CR = 0.5; NAUC = 0.106 at CR = 0.3, and NAUC = 0.409 at CR = 0.1.

4.2.2. URF versus spatial coverage ratio

This subsection examines the relationship between spatial coverage and inspection scanning reliability. Spatial coverage is parametrised using the Coverage Ratio (CR), defined as the fraction of sampled grid points to total grid points (CR = 𝑁 𝑠𝑎𝑚𝑝𝑙𝑒𝑑 ∕ 𝑁 𝑡𝑜𝑡𝑎𝑙 ). Fig. 12 (a-c) illustrate representative inspection scanning patterns at CR = 10%, 30%, and 50%. The initial scanning position are randomised to at each instance of inspection. These patterns reflect practical inspection constraints such as finite ultrasonic scanning resolution (pitch between measurement points) and robotic platform mobility limitation (path planning constraints).

Fig. 12 (d) examines the coverage-reliability relationship using a fixed 3-unit inspection interval. The results reveal several key trends: Full coverage (CR = 1) achieves best performance with an NAUC of 0.006, while reduced coverage leads to progressively worse reliability: CR = 0.5 yields an NAUC of 0.044, and CR = 0.3 produces an NAUC of 0.106. Most notably, for this particular data set, severe performance degradation is observed at CR = 0.1, where the NAUC jumps to 0.409. These results demonstrate a non-linear relationship between coverage ratio and inspection reliability.

4.3. Simulation of monitoring with permanently-installed sensors

This section evaluates the performance of fixed-sensor monitoring systems. Following an initial inspection phase, sensors are permanently deployed at identified hotspots corresponding to local thickness minima, enabling continuous thickness monitoring with high temporal resolution. The methodology assumes that continuous monitoring of critical defects can effectively replace the need for periodic inspections.

Fig. 13. Graphical illustration of the automated corrosion sensor placement workflow showing: (a) the original corroded surface profile. (b) Corrosion pits (highlighted in red asterisks) extracted from the surface profile. (c) Grouping of pit clusters using a K-mean clustering algorithm. 𝑁 = 3 = available monitoring sensors. The three clusters are denoted by blue circles, cyan diamonds and black square markers, respectively. (d) Final sensor positions (red dot markers) at the local minima within each cluster. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.

4.3.1. Automated pit identification and sensor placement

To perform statistical analysis via the Monte Carlo simulation, an automated sensor placement algorithm is needed to process thou- sands of corrosion surfaces realisations while meeting the following objectives: (1) autonomous identification of corrosion features without manual labelling, (2) accommodation of diverse pit geometries and dimensions, and (3) intelligent sensor distribution among multiple pit clusters that prevents local over-concentration while maintaining sufficient spatial coverage.

Fig. 13 illustrates an automated decision-making workflow for sen- sor deployment. This workflow optimises sensor locations based on both the spatial distribution of corrosion features and the number of available sensors. The process follows the steps outlined below.

1. Pit Identification: Localised corrosion defects are isolated through surface coordinate analysis, evaluating both in-plane and out-of-plane distances with appropriate dimensional scaling to accurately distinguish pits from general corrosion.
2. Defect Clustering: Identified pits are grouped into N spatial clusters, where N corresponds to the number of sensors available for deployment, ensuring strategic sensor distribution across the interrogated area.
3. Sensor Placement: Within each cluster, the algorithm identifies the location of the minimum thickness (the deepest pit) and installs monitoring sensors at these locations.

4.3.2. URF vs sensor installation time

Fig. 14 illustrates the progression of both the global corrosion minimum (denoted by a red star) and the optimal sensor positions (denoted by black circles) at each time step. As the surface evolves due to corrosion effects, the minimum remaining thickness location shifts, necessitating continuous adaptation of the optimal sensor positions.

Fig. 14. Spatio-temporal evolution of global corrosion minima (red star) versus the optimal sensor locations (black circles) at installation time t = 10, 20, 30, and 40. Full spatial coverage inspection for pit identification is assumed at the sensor installation time. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 15. Variation of URF versus sensor installation time (t = 10, 20, 30, and 40). The corresponding area under the URF curve (NAUC) = 0.254, 0.085, 0.170, and 0.251, respectively.

However, most existing ultrasonic monitoring systems are permanently installed at fixed locations on a structure, this presents a complex optimisation challenge for monitoring stochastic corrosion evolutions. Premature transition from inspections to monitoring may miss critical developing pits, whereas excessive delay wastes resources by requiring unnecessary inspections.

The impact of transition time on the performance of a permanently installed monitoring system is evaluated in Fig. 15, which compares four different transition strategies through their corresponding URF curves and NAUC values. The earliest sensor deployment (t = 10) yields a NAUC value of 0.254 because the monitoring sensors are installed too early to detect pits that initiate subsequently. Intermediate transition times (t = 20 and t = 30) show significantly improved reliability, with NAUC values reduced to 0.086 and 0.170, respectively. Notably, the latest deployment (t = 40) yields a NAUC of 0.251 which is worse than the intermediate cases. This reveals a non-monotonic relationship between transition timing and monitoring effectiveness, emphasising the critical role of installation timing in maintenance planning.

Fig. 16. Variation of URF versus number of monitoring sensors (K = 1, 3, 5, 10). The normalised area under the URF curve (NAUC) = 0.161, 0.085, 0.076 and 0.076, respectively.

4.3.3. URF versus number of sensors

Monitoring reliability is also evaluated for different numbers of sensors being deployed (K = 1, 3, 5, and 10) at t = 20, yielding NAUC values of 0.161, 0.085, 0.076 and 0.076, respectively. The 47% improvement in reliability from 1 to 5 sensors demonstrates significant gains, while the identical NAUC values for 5 and 10 sensors indicate a performance plateau. This suggests strongly diminishing returns when the sensor-to-pit ratio exceeds a certain threshold — approximately 16.7% (5 sensors/30 pits) in this case. Although the exact threshold ratio may vary across datasets depending on pit growth characteristics, the demonstrated non-linear relationship provides crucial insight for designing cost-effective monitoring systems that balance reliability and resource allocation (see Fig. 16 ).

4.3.4. URF vs coverage ratio at sensor installation

Monitoring effectiveness is found to be strongly influenced by the spatial coverage ratio (CR) during initial inspection, as limited coverage may fail to identify critical defects and result in suboptimal sensor placement. Fig. 17 demonstrates the impact of CR on the URF curve and NAUC. Similar to the trend observed in Fig. 12, reducing CR shifts the URF curve of monitoring strategies upwards, with changes that are disproportionate to the corresponding reductions in CR. Specifically, reducing CR from 1.0 to 0.5 and 0.3 causes modest NAUC increases from 0.085 to 0.114 ( + 34.1%) and 0.137 ( + 61.2%), respectively. A further reduction to CR=0.1 leads to a drastic increase in NAUC to 0.369 ( + 334.1%), highlighting the non-linear relationship between spatial coverage and monitoring reliability

4.4. Simulation of mixed inspection and monitoring

This section evaluates a mixed corrosion assessment strategy that integrates scheduled inspection scanning with movable monitoring sensors, addressing the limitation of traditional fixed monitoring systems. The methodology employs an iterative process where initial partial inspections identify critical hotspots for sensor deployment, followed by periodic re-inspections that update sensor locations to track evolving damage patterns.

The performance enhancement of the hybrid approach stems from two mechanisms. First, deploying monitoring sensors alongside inspec- tion scanning creates a data augmentation effect, as these sensors are strategically positioned in high-hazard zones identified during prior inspections. This targeted placement yields a greater-than-proportional increase in the effective coverage ratio for subsequent scanning. Sec- ond, sensor relocation preserves these multiplicative effects over time by adapting to shifting damage patterns during each inspection cycle.

Fig. 17. Variation of URF versus inspection coverage ratio at sensor installa- tion (CR = 1, 0.5, 0.3, and 0.1). The normalised area under the URF curve (NAUC) = 0.085, 0.114, 0.137, and 0.369, respectively.

This adaptive capability is unavailable to permanent monitoring systems, where fixed sensors inevitably become misaligned with evolving corrosion patterns. Fig. 18 illustrates the data subsampling workflow underpinning this hybrid strategy, providing direct comparison with conventional inspection-only and monitoring-only approaches.

4.4.1. URF: Comparison of inspection, monitoring, and hybrid approaches 

Fig. 19 quantifies the effectiveness of the three corrosion assessment approaches. The inspection scanning-only strategy (CR = 0.5, interval = 10) yields a NAUC = 0.386, while the permanently-installed monitoring system (PIMS) (3 sensors deployed at t = 20) achieves NAUC = 0.114. The hybrid approach demonstrates far superior performance (NAUC = 0.030), representing an order of magnitude improvement relative to inspection-scanning strategy. This improvement is achieved by combin- ing scheduled scanning with only three additional movable monitoring sensors, underscoring the substantial performance gains offered by the hybrid method.

Fig. 20 presents a second comparison, where the hybrid strategy (CR = 0.5, interval = 8, 3 sensors, NAUC = 0.017) outperforms both inspection scanning (CR = 1, interval = 5, NAUC = 0.068) and PIMS (5 sensors at t = 20, NAUC = 0.048) approaches. Notably, this superior performance is achieved with two operational advantages: (1) 60% longer inspection intervals and 50% reduced spatial coverage com- pared with the inspection scanning, (2) 40% fewer sensors than the monitoring-only baseline.

4.5. URF: impact of measurement noise and uncertainties

Fig. 21 compares URF curves calculated using ultrasonic beam- spread (BS) measurements versus the ground truth. Compared to the ground-truth-based results shown in Fig. 19 , all strategies exhibit significant performance deterioration due to measurement errors. Specifi- cally, the NAUC values increase from 0.068 to 0.214 for the scheduled inspection scanning, from 0.048 to 0.200 for the PIMS approach, and from 0.017 to 0.112 for the hybrid scanning and monitoring strategy.

The performance degradation observed in the BS model stems from its convolution-based smearing effect, which causes thickness overestimation at pit centres and underestimation at pit edges. Notably, the URF curve based on BS measurements lacks the sawtooth features present in the ground-truth URF curves between 𝑡 = 15 and t=30. This phenomenon also stems from the BS model’s spatial averaging effect, which enables the detection of pits even when sensors are not directly above them but are positioned nearby.

5. Discussion

5.1. Practical applications

The proposed simulation framework offers several principal bene- fits for corrosion management: (1) an integrated workflow covering the entire maintenance planning cycle — from damage progression modelling to data acquisition vis non-destructive testing (NDT), and to final reliability assessment; (2) quantitative evaluation of inspection and monitoring strategies using standardised metrics (URF/NAUC); (3) optimisation of sensor deployment parameters (quantity, position- ing, installation timing); and (4) sensitivity analysis of measurement uncertainties to inform future sensor development.

As demonstrated in the study, since the corrosion model can be calibrated to match observed surface profiles, this simulation framework offers particular value when experimental data is limited or costly to obtain. It allows: (1) simulate long-term damage progression and predicting future corrosion assessment capabilities; and (2) analyse stochastic variations in corrosion morphology while preserving consistent growth trajectories.

To facilitate practical application and wider adoption, the framework can be deployed as dedicated software that enables the full simulation workflow and rapid scenario analysis — particularly valuable when expertise in one or more stages of the evaluation process is limited. A demonstrative implementation is available as the Mon Ami graphical user interface [43].

Case study results indicate that the monitoring-only approach with permanently installed sensors cannot fully replace time-based inspection scanning. On the other hand, by lifting the constraint of post- installation sensor immobility, the hybrid scanning and monitoring strategy demonstrates clear advantages over both scanning-only and monitoring-only methods. The practical feasibility of physical sensor repositioning is central to the realisation and real-world impact of the hybrid approach, highlighting the need for further development of adaptable sensors and deployment platforms. Future work should also consider the cost implications associated with sensor relocation and redeployment.

The framework also enables analysis of measurement uncertainty impacts, as demonstrated by evaluations of URF and NAUC metrics using both ground-truth data and simulated ultrasonic measurement outputs. Furthermore, through techniques such as value-of-information (VoI) analysis [44], the URF and NAUC metrics can be linked to economic costs, enabling the study of their influence on final decision- making. However, a detailed cost–benefit analysis for specific implementations is beyond the scope of this study.

5.2. Limitations and future outlook

5.2.1. Modelling of material degradation

This work presents a proof-of-concept demonstration of a multi- stage evaluation framework, with several avenues for future refinement. The employed corrosion model is empirical and designed to describe the evolution of surface morphology based on field observations. We intentionally decouple the resulting surface changes from the underlying corrosion mechanisms, recognising that ultrasonic wave reflection and scattering are primarily governed by surface geometry. As such, provided the surface profile is clearly defined, the model can be refined or replaced with alternative formulations. This flexibility allows the framework to be applied across a wide range of materials, corrosion mechanisms, and inspection scenarios.

Fig. 18. Data subsampling workflow comparing three corrosion surveillance strategies: (1) periodic inspection-only, (2) fixed-sensor monitoring following initial inspection, and (3) hybrid approach combining periodic inspections with relocatable monitoring sensors.

Fig. 19. Comparison of URF curves for three scanning and monitoring approaches. Inspection scanning-only (CR = 0.5, interval = 10, NAUC = 0.386), permanently installed monitoring (3 sensors, t = 20, CR = 0.5, NAUC = 0.114) and hybrid inspection and monitoring approaches (CR = 0.5, interval = 10, 3 sensors, sensor deployed at t = 10, NAUC = 0.030).

5.2.2. Evaluation objective and metrics

This study introduces the unreliability function (URF) as a metric to evaluate the effectiveness of various inspection and monitoring strategies. URF shares conceptual alignment with established reliability measures such as false negatives (Type II errors) and the broader probability of detection (POD) framework. However, conventional POD analysis is typically defined for discrete, idealised flaws — such as flat- bottom holes — with clearly specified geometry and binary detection outcomes (i.e., hit or miss). In contrast, natural corrosion defects are often irregular and temporally evolving, making them difficult to discretise or label consistently for POD-style evaluation. For instance, a 30% wall loss caused by widespread general corrosion is significantly more detectable than the same depth loss resulting from an isolated narrow pit — despite both sharing identical scalar flaw metrics. Such cases illustrate the limitations of applying scalar-based POD evaluations to complex damage morphologies.

Fig. 20. Comparison of URF curves for three inspection and monitoring approaches. Inspection scanning-only (CR = 1, interval = 5, NAUC = 0.068), permanently installed monitoring (5 sensors deployed at t = 20, NAUC = 0.048) and hybrid scanning and monitoring (CR = 0.5, interval = 8, 3 sensors deployed at t = 10, NAUC = 0.017).

URF addresses these challenges by directly assessing the accuracy of minimum wall thickness estimates, thereby capturing both detection and quantification in a single metric. This makes it particularly well suited to the hybrid inspection and monitoring strategies proposed in this work, where defects may evolve gradually over time. As such, URF supports continuous structural integrity assessment and enables data-informed operational optimisation.

5.2.3. Selection and modelling of NDE techniques

Although the evaluation framework is inherently generic and sup- ports NDE models of varying fidelity, the current implementation is designed to enable statistically meaningful assessment of NDE strategies across diverse corrosion morphologies and data acquisition protocols. To support Monte Carlo simulations over a large number of scenarios, a lightweight and computationally efficient NDE simulation model is essential. While finite element (FE) simulations provide a rigorous and versatile modelling approach, their high computational cost particularly in the three-dimensional cases - renders them imprac- tical for simulations that span both spatial and temporal domains. For this reason, the beam-spread and wave scattering surrogate mod- els are employed to simulate normal-incidence ultrasonic thickness measurements. Although each of these models has been individually corroborated by experimental results in previous studies [ 37 , 42 ], a key direction for future work will be the systematic benchmarking of the surrogate model against FE simulations and experimental data across diverse surface conditions and inspection configurations.

Fig. 21. URF curves based on the beams spread (BS) ultrasonic measurements. Inspection scanning-only (CR = 1, interval = 5, NAUC = 0.214), permanently installed monitoring (5 sensors deployed at t = 20, NAUC = 0.200) and hybrid scanning and monitoring (CR = 0.5, interval = 8, 3 sensors deployed at t = 10, NAUC = 0.112).

Alternative NDE techniques could be incorporated into the framework to broaden its applicability, provided suitable surrogate models are available. For instance, Zhang et al. [ 45 ] demonstrated a robot- enabled guided wave inspection approach, where defect detection was associated with variations in echo amplitude, modelled using an analytical wave scattering formulation [ 46 ]. To extend the framework to pitch-catch or angled-beam configurations, the surrogate model proposed by Paialunga et al. [ 47 ] offers a promising direction. Their approach combines finite element modelling with machine learning to simulate time-of-flight diffraction from rough defects. With suitable adaptation, this model could be extended to 3D and integrated into the current framework to support additional NDE modalities.

It should also be noted that NDE techniques such as most guided wave methods are primarily suited for defect detection and screening. In such cases, evaluation metrics such as probability of detection and receiver operating characteristic analysis are more appropriate. Future developments could integrate these detection-oriented methods alongside point-based measurements within the framework, enabling the simulation of a two-stage NDE process involving initial screening followed by targeted quantitative assessment.

6. Conclusions

This paper presents a multistage framework for evaluating corrosion inspection and monitoring strategies, integrating four critical components: degradation modelling, simulation of NDE data acquisition, design of subsampling protocols, and system reliability analysis.

The use of movable monitoring sensors overcomes the temporal and spatial limitations of traditional inspection protocols. By strategically placing and periodically repositioning sensors in high-hazard zones, the hybrid approach demonstrates significant reliability improvement while requiring fewer sensors, reduced spatial coverage, and longer inspection intervals.

A physics-based surrogate model has been implemented to characterise the distribution of measurement errors and uncertainties, revealing their substantial effect on system reliability and underscoring the importance of accounting for them. While the current models employ simplified representations, the framework’s modular architecture permits future integration of more sophisticated corrosion models and ultrasonic measurement simulations.

This framework lays a foundation for interdisciplinary collaboration across materials science, NDE development, and statistical analysis. Its refinement and adoption can advance inspection technologies, support method validation and certification, and improve infrastructure reliability and safety.

Credit Authorship contribution statement

 Yifeng Zhang: Writing – original draft, Software, Methodology, Formal analysis, Data curation, Conceptualization. Frederic Cegla: Writing – review & editing, Supervision, Methodology, Funding acquisition, Formal analysis, Conceptualization.

Declaration of competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement:

The authors are grateful to the UK Research Centre in Non-Destructive Evaluation (RCNDE) for providing the funding for this research as part of their core research programme.

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