The effect of corrosion induced surface morphology changes on ultrasonically monitored corrosion rates

Corrosion rates obtained by very frequent (daily) measurements with permanently installed ultrasonic sensors have been shown to be highly inaccurate when changes in surface morphology lead to ultrasonic signal distortion. In this paper the accuracy of ultrasonically estimated corrosion rates (mean wall thickness loss) by means of standard signal processing methods (peak to peak—P2P, first arrival—FA, cross correlation—XC) was investigated and a novel thickness extraction algorithm (adaptive cross-correlation—AXC) is presented. All of the algorithms were tested on simulated ultrasonic data that was obtained by modelling the surface geometry evolution coupled with a fast ultrasonic signal simulator based on the distributed point source method. The performance of each algorithm could then be determined by comparing the actual known mean thickness losses of the simulated surfaces to the values that each algorithm returned. The results showed that AXC is the best of the investigated processing algorithms. For spatially random thickness loss 90% of AXC estimated thickness trends were within −10 to +25% of the actual mean loss rate (e.g. 0.75–1.1 mm year−1 would be measured for a 1 mm year−1 actual mean loss rate). The other algorithms (P2P, FA, XC) exhibited error distributions that were 5–10 times larger. All algorithms performed worse in scenarios where wall loss was not distributed randomly in space (spatially correlated thickness loss occured) and where the overall rms of the surface was either growing or declining. However, on these surfaces AXC also outperformed the other algorithms and showed almost an order of magnitude improvement compared to them.


Introduction
Corrosion is a major issue that is limiting the life of infrastructure all around the world. It is estimated that corrosion costs developed countries 2%-4% of GDP [1] and the annual cost of corrosion to the US oil and gas industry alone has been estimated to be about US$ 8B [2]. Traditionally, large industrial plants use manual or semi-automatic inspection techniques to check that their pipework is fit for service. In the recent past rapid development of ultrasonic online monitoring sensors and wireless communcation has made it possible to permanently install autonomous sensors on pipe work [3]. These sensors regularly (hourly or daily) take measurements and automatically store them in a database. Besides the practical advantages of not needing to re-access and prepare a particular plant location for re-inspection, one of the main advantages of permanently installed measurements is that they have an exceptional repeatability (precision). This is due to the removal of any coupling and probe positioning errors in the measurement process. Sub-micrometre precision has been reported for some laboratory measurements [4][5][6], whereas the repeatibilty of manual ultrasonic measurements is in the range of millimetres [7]. Furthermore, very frequent measurements are possible which makes it easy to extract wall loss rates with a very good response time. The determination of a wall loss rate, or corrosion rate, with a short response time is very important when one is interested in mitigating the effects of corrosion. Accurate information can be used to fine tune production processes in many ways, e.g. by feedstock variation, change of operating conditions or the addition of corrosion inhibitor chemistries [8,9].
However, ultrasonically monitored wall thickness data acquired on real plants has shown that sudden, unrealistically rapid changes in wall thickness can be reported by online monitoring systems. An example of this is shown in the solid blue line of figure 1, where a component that loses wall thickness at a steady rate of about 0.3 mm year −1 suddenly shows an apparent thickness increase followed by a large drop in thickness. After 2 months of large deviation the monitored wall thickness then re-joins the overall steady trend of 0.3 mm year −1 .
There is a clear physics based explanation for the effect that is observed in figure 1. The wall thickness estimate that is reported is extracted from the temporal separation of the ultrasonic wavepackets that bounce up and down in the material whose thickness is being measured. Usually a simple algorithm such as the timing between consecutive peaks (peak to peak-P2P) is used to evaluate the travel time and hence the wall thickness (see figure 2). However, if the inner wall surface of the instrumented pipe changes shape, such as during some corrosion/errosion processes, then the ultrasonic wave packets can become distorted [10,11]. This can lead to substantial changes/errors in the estimated wall thickness.
Further evidence for this explanation of the thickness deviations in figure 1 can be found when inspecting the raw ultrasonic waveforms (A-scans) associated with thickness measurements at different stages during the thickness deviation. Figure 3 shows the A-scans that were recorded at the times corresponding to the locations marked by the letters A, B, C and D in figure 1. The A-scans clearly show a strong distortion of the wave packet that is reflected from the inner pipe surface (second wave packet) indicating strong shape changes (corrosion activity) at that location. This is clearly an ultrasonic phenomenon which is a result of the interaction of the ultrasonic wave with the non-uniform surface morphology of the component that is being monitored. There is literature that has described the effect of rough surface scattering on ultrasonic signals and thickness measurements for both shear [10,11] and longitudinal waves [12][13][14][15].
The data of figure 1 obviously casts doubts on the accuracy of ultrasonic corrosion rate estimates that are based on P2P timing algorithms. This paper addresses this issue by suggesting a more advanced signal processing algorithm. The algorithm is called adaptive cross-correlation (AXC). Its working principle is described in detail and it is compared to standard travel time estimation techniques (P2P, first arrival -FA, cross correlation-XC). The performance of all algorithms was tested on simulated ultrasonic signals reflecting from sequences of evolving, rough surfaces. Figure 2(b) shows a typical waveform that is measured. It also illustrates that the thickness is extracted from the arrival time year (1 measurement every 12 h). A distinct deviation in monitored wall thickness is clearly visible 3 months after monitoring commenced. Thickness estimates shown were calculated using a peak to peak (P2P) timing algorithm. (a) Illustration of transmitter and receiver configuration of the particular ultrasonic monitoring sensor (Permasense WT100) [3] that was investigated and the signal paths that the ultrasonic shear horizontal wave takes in the material (not to scale, usually  D T ). (b) A typical ultrasonic signal (A-scan) that is received by a permanently installed sensor. Several wavepacket arrivals are clearly visible. The wavepackets correspond to signals that have travelled from the transmitter to the receiver via different paths as indicated in (a). There is a long delay before the first signal arrives, this is due to an intentionally added delay caused by the length of the particular (waveguide) transducers that were used in order to thermally isolate the transduction assembly. The time difference between the arrival of the different wavepackets is used to calculate the wall thickness. difference between different wavepackets. Once the arrival times of individual wave packets have been established and the wave path is known this is a straightforward excercise. Equation (1) describes how to calculate the wall thickness based on the arrival times for the example geometry that is shown in figure 2(a):

Thickness calculation and arrival time extraction
where c is the velocity of the ultrasonic wave, dt is the time difference between t 1 and t 2 the arrival time of the first and second wave packet respectively and D is the separation between the transmitting and receiving transducer. In this paper D=1.7 mm and c=3250 ms −1 unless otherwise stated. Estimation of the arrival time of the first and second wavepackets from the ultrasonic waveform is therefore the key problem. There are a number of signal processing methods commonly used for this purpose. In this study the P2P, XC and FA algorithms have been implemented and are briefly described here.
P2P methods ignore the phase information of a signal and rely on computing an envelope function for the measured waveform. One way to achieve this is via the Hilbert transform. The Hilbert transform applies a 90°phase shift to all frequency components of the signal. This allows the envelope to be calculated: is the Hilbert transform of f(t) and E(t) is the computed signal envelope. Once the envelope is computed for a waveform, its maximum peaks are assumed to be the arrival times of the various wavepackets. This is illustrated in figure 4(a). FA is another common technique to estimate arrival times. It also relies on calculating an envelope for the waveform; FA then finds the highest peak for each wavepacket. Following this, a threshold is applied to each peak as a function of the amplitude of that peak (e.g.: −6 dB). The first crossing of the given threshold with the envelope signal is then taken as the arrival time for the given wavepacket. This is shown in figure 4 XC is also a popular arrival time estimation method. The XC process for real valued functions is defined by equation (3): where h(t) is the XC of function f(t) with a kernel function g (t). The peaks of the resulting correlation function are then determined and taken as arrival times. This is because at those particular time offset values the received signal is most similar to the transmitted signal. An example signal, its XC function and the extracted arrival times are shown in figure 5. The above commonly used arrival time extraction techniques are not expected to perform well on ultrasonic signals that are reflected from surfaces with rough and evolving surface morphology. This is most easily explained by looking at the ultrasonic signal that is reflected from a rough surface  figure 3(c) has become distorted by the rough surface reflection so that none of the techniques based on the arrival time of the maximum amplitude (P2P), the FA or the XC with an idealised toneburst will result in a reasonable estimate for the arrival time of the backwall echo wave packet. Since the standard signal processing methods are not expected to perform adequately for gradually changing rough backwall surfaces, a new method-AXC-is proposed.
AXC was conceived specifically for the purpose of accurately estimating the mean wall thickness loss rate of gradually changing backwall surfaces that lead to distortion of the backwall echo signal. The method is based on the standard XC algorithm, however it uses an alternative reference signal for the XC process. This is because the transmitted toneburst that is used in the standard XC technique is not a good model for distorted backwall echo signals. Corrosion is assumed to take place on the backwall only, this is a reasonable assumption when measuring internal corrosion in pipes with a sensor on the outside of the pipe. AXC relies on the XC function to determine arrival times, however it accounts for distortion of the backwall signal by updating the signal that is used in the XC process with the received ultrasonic waveform. The updated signal is the distorted backwall reflection that was previously recorded from the rough internal surface. Careful windowing of the backwall signal is required and then only small distortions of the signal are expected to occur between consecutive measurements and therefore the overall temporal shift of the wavepacket is extracted more reliably. Consequently, AXC is expected to provide more accurate mean wall thickness loss rate measurements. In the absensce of a previous measurement, i.e. for the first measurement a standard XC measurement is required.
The signal processing protocol of AXC to determine the arrival times of a sequence of waveforms (w n 1 .. ) can be formally summarised as follows: ) is the XC of signals a and b, ¬ 1 denotes the extraction of the time of the highest peak in the first wavepacket of a signal, ¬ 2 denotes the extraction of the time of the highest peak in the second wavepacket of a signal, w n is the nth waveform, w t t : ) denotes windowing a waveform between times t a and t b , and S n BW is the windowed backwall wavepacket for the nth measurement. The superscripts SW and BW refer to surface wave and backwall respectively. The plots at the bottom of figure 5 graphically illustrate this (AXC) process and contrast it to standard cross corelation with a constant kernel function which is shown in the top plots of figure 5. In this implementation of AXC only the backwall signal is windowed and updated in order to obtain an improved estimate of the arrival time of the first wavepacket. This is because corrosion is assumed to only take place on the backwall surface (i.e. the internal surface of a pipe), the surface to which the transducer is attached is not corroding, transducer coupling is assumed to be constant and therefore surface wave signal distortions are not expected and do not need to be compensated for.

Surface morphology and a model to describe the evolution of surface morphology
Corrosion is a very complex phenomenon and can produce very different surface morphologies. It can be spatially uniform such as in etching, or spatially non-uniform, which is commonly described as pitting. Figure 6 illustrates the difference between spatially uniform and non-uniform corrosion processes by depicting several 2D wall thickness profiles throughout a corrosion process from start to end. During spatially uniform corrosion all spatial locations along the Figure 5. Illustration of the cross correlation (XC) algorithm (top) and adaptive cross correlation algorithm (bottom). In both cases two consecutively received raw ultrasonic signals are shown as well as the kernel functions that they are cross correlated with. In the standard XC algorithm the kernel function is not updated, whereas in AXC the backwall signal of the previous measurement is used as kernel function in order to account for potential signal distortions due to surface morphology changes. horizontal axis have the same probability of getting thinner, whereas in spatially non-uniform corrosion there are some sites where material is preferentially lost. In practice, even if some non-uniformity is present, most plants quote an allowable corrosion rate for engineering components in service. This inherently assumes averaging of the material lost over a larger area. Because reporting of a corrosion rate is standard in industry and because it inherently averages over an area it makes sense to report mean wall thickness loss as an output parameter of an ultrasonic measurement system. Therefore the mean thickness described by the surface is the quantity that we are trying to measure in order to compute a wall thickness/corrosion rate in this paper. It is noted that for some spatially non-uniform corrosion processes (very sharp pits) it would be more appropriate to track the minium thickness within a profile and its rate of change, however this not comparable to corrosion rates that are generally quoted and which are an area average.
There is limited information in the literature about the actual surface geometry changes during corrosion processes, a rare example being presented by Strutt et al [16]. There are also some models and computer algorithms that can be used to simulate the progression of the surface morphology during corrosion, however these require the input of many parameters (which are not readily available) to describe the particular type of corrosion [17][18][19]. Therefore in this paper the surface statistics of a real retired pipe sample were measured using a surface profilometer (Talysurf, Taylor Hobson, UK). The pipe was retired from an environment where it was exposed to high temperature sulfidation corrosion, a generally uniform corrosion mechanism. It was found that the statistics of the surface morphology were close to those of a Gaussian rough surface with rms in the range from 0.1 to 0.4 mm and correlation lengths in the range from 1 to 10 mm.
Based on this information a simple geometrical model of the evolution of surface morphology during the corrosion processes could be constructed so that at least once in the evolution of the surface geometry a surface with similar surface characteristics as measured on the retired pipe sample can be observed. In this model a sequence of 50 backwalls is constructed. The profile of each backwall within the sequence is different and is constructed by addition of random Gaussian profiles with correlation length C L and surface roughness R rms . In order to simulate the effect of different surface roughness values two independent R rms values, r i and r p are defined. r i defines the R rms value of the initial surface, where as r p defines the R rms value of all subsequent surfaces that are added or that perturb the initial surface (hence subscript p). All Gaussian profiles that are generated have zero mean and they are offset by T n , n=1-50 so that a predetermined rate of wall thickness change exists for each backwall sequence. = T 10 mm, 9 The mean thickness loss of the whole profile per step in the backwall sequence therefore is r p . This can potentially lead to small temporary (between consecutive backwall steps) local thickness increases. One can argue for or against this being physical. A line of thought suggests that corrosion always requires a loss of wall thickness. However, on a microscopic level there are phenomena such as the formation of passivation films and oxide scales that can result in small thickness gains. The proposed model, allows both of these; small local thickness increases and an overall mean wall thickness loss are modelled. We do not claim that it is an accurate model of any particular corrosion process but it does roughly describe the variations in surface geometry that are to be expected. Finally, it is important to describe the parameter s. This parameter was introduced because the addition of two Gaussian surfaces of the same R rms value will result in a third Gaussian surface with different R rms value. If the resulting surface is scaled by s, which remains constant for each backwall sequence, then the R rms value of all surfaces can be kept constant (for s≈1) or varied by a controlled amount, e.g. so that the initial backwall surface has an R rms value of r i =100 μm and the R rms steadily increases throughout the simulation to R rms =300 μm for B 50 . The control of s enables the simulation of spatially random perturbations to the surface (s≈1) so that the R rms value remains roughly constant throughout the backwall sequence and all spatial locations are equally likely to be attacked. If s is larger or smaller than one, the perturbation will not be spatially random and thinner parts will preferentially thin and thicker parts will preferentially stay thick (this is essentially what happens in pitting). Based on the above, the algorithm that describes the surface evolution can be summarised by the following equations: ) is an array of Gaussian distributed points with zero mean, R rms , correlation length C L and x is the horizontal coordinate or index (if discretised) and B m is the mth backwall, = m 1 .. 50.

Simulation of the reflection of ultrasonic signals from rough surfaces, simulation procedure and parameters
Equations (13)-(11) describe the actual surface geometry evolution. For all simulations the wall surface changes were described by 50 discrete surfaces. In order to compare the performance of different thickness estimation algorithms an ultrasonic signal corresponding to the reflection from each surface is required. Therefore it was also necessary to simulate a sequence of realistic ultrasonic signals that are reflected from each one of those 50 rough surfaces. This then needed to be repeated many times because many surface evolution sequences need to be simulated so that a distribution of wall thickness trends can be determined. Therefore a fast simulation tool was required. The distributed point source method (DPSM) [20] was chosen as it is particularly fast and has been shown to simulate realistic ultrasonic signals reflecting of rough surfaces [10,11]. For the particular transducer geometry and wave mode (shear horizontal SH wave) that was simulated it was also shown that the statistics of 2D simulations can be related to those of the full 3D case [21]. In this paper 2D simulations are carried out, but they are not adjusted to match the expected statistics of 3D simulations. This is because the 2D assumptions can be treated as the worst case scenario, since averaging over the orthogonal direction (as in the 3D case) in general reduces the distortion of the ultrasonic signal due to scattering from the rough surface. In addition, 2D simulations only take of the order of minutes per signal rather than several hours for 3D simulations. This makes it possible to simulate thousands of ultrasonic 2D signals that are representative of the worst case real life signals over a timeframe of weeks rather than years. Figure 7 shows an illustration of the DPSM model geometry that was used for the simulation. 100 active point sources were used to model the transmitter transducer. These point sources were distributed with a separation of 10 μm and were offset from the transducer/sample interface by 5 μm. The backwall surface was represented by 1200 passive point sources with a separation of 50 μm offset from the surface by 25 μm altogether spanning the width of the 60 mm backwall surface. The receiver transducer was simulated by a single receiver point at the centre of the coupled transducer. The implementation of DPSM used in this paper is identical to that of [11], which has been validated against Finite element simulation results and experiments and the interested reader is referred to this publication for exact details on the DPSM model implementation.
To generate a backwall sequence using equations (13)-(11), C r r , , L i p and s need to be defined. C L was chosen to be l 1 mm 0.6 for all simulations. Initial simulations showed that this causes the largest changes in the ultrasonic signal and therefore we expect that it will lead to conservative results and conclusions. All of the remaining parameters for the simulations are shown in table 1. It is highlighted that the remaining parameters are broken down into two separate simulation sets. The first set of simulations are intended to create backwall surfaces with constant R rms (no change in rms throughout the sequence of 50 backwall surfaces, s was calculated according to equation (11), so that the R rms would not change throughout a backwall sequence). Three Here the scaling coefficient was chosen so that it would result in an R rms increase from 100 to 300 μm (where r i =100 μm) and an R rms decrease from 300 to 100 μm (where r i =300 μm). The numerical values for s to achieve the intended amount of R rms change are a function of both r i and r p and they are summarised in table 1.
For each parameter set, 200 backwall sequences were simulated, with 50 backwall samples each. Ultrasonic signals were simulated for all of the backwalls, which were then evaluated with each of the discussed signal processing methods. This resulted in 50 thicknesses per backwall sequence. Backwall sequences are therefore linked to a sequence of thickness estimates as produced by the signal processing techniques. For each backwall sequence and its corresponding thicknesses a thickness trend could be extracted using a linear least squares line fit. These trend lines were denoted m 1 . The performance of signal processing methods were then compared based on their thickness trend error distributions. In order to represent this visually for a large number of parameter sets, trend error distributions are shown as boxplots, where the boxes represent the data between the 25th and 75th percentile, whereas the whiskers represent data between the 5th and 95th percentiles. A visual representation of the meaning of the box plots is shown in figure 8.

Backwall evolution without rms change
The results of the mean wall thickness loss trend error distribution plots for AXC, XC, P2P and FA methods under constant R rms conditions are shown in figure 9. Overall, the effects of both the initial surface R rms (r i ) and the size of the R rms that the surface is perturbed with (r p ) are as expected, increasing initial R rms and perturbation increases the error bars of any signal processing method. This aligns with the conclusions of previous reports suggesting that in general ultrasonic thickness measurements are sensitive to changes of backwall morphology [10,11].
In addition, it is apparent from figure 9 that on every plot the width of trend error distributions for AXC is narrowest. This is most noticeable on the right column of results on figure 9, where r i =300 μm. Here the trend error distribution width of all standard methods (XC, P2P and FA) span between ±100%, while the trend error distribution width of AXC is close to an order of magnitude narrower, spanning between +25% and −10% with a mean of +7.5%. This means that AXC has a slight bias to overestimate the thickness (or underestimate thickness loss rate), but this is negligible compared to the error of other methods.
During the study it also became clear that AXC has limitations. AXC is based on XC, and so its failings can be similar to the erratic behaviour of XC. XC is sensitive to backwall roughness as shown by figure 9. The breakdown in accuracy is caused by the distortion of the backwall echo wavepackets when the backwall surface is rough. When the backwall surface is rough and the signal is distorted, the synthesised toneburst used by XC does not correlate well with the received signal. Since XC relies on determining the biggest peak in the signal, in these cases a peak that is not representative of the mean wall thickness may be the biggest. Consequently, the wrong peak is often found for the purposes of the thickness measurement. This failure mode of XC is referred to as peak jumping.
AXC avoids this problem by using the backwall echo wavepacket from the previous measurement in the XC process, as it is much more likely to correlate well with the received signal. However, when the backwall surface changes significantly between measurements (which could occur in practice if ultrasonic signals are not acquired sufficiently frequently), excessive signal distortion may occur. In this case the current signal will not correlate well with the previous backwall echo sample and AXC will be affected by peak jumping. For this reason AXC is expected to perform  similarly to XC when applied to uncorrelated realisations of backwall surface geometries as evaluated by Jarvis et al [11].
Although peak jumping may introduce large errors, it is simple to detect, since the error it causes is an integer multiple of ∼λ/2. It is also easily avoided by frequent measurements, as in a short time the backwall geometry is unlikely to change excessively. In addition, when measurements are carried out frequently, the thickness is not likely to change significantly and therefore the large error caused by peak jumping is even more straightforward to detect. Permanently installed monitoring is therefore well suited for AXC as it allows for frequent data acquisition.
The results of figure 9 only show trends where AXC peak jumping does not occur. The number of trends out of the 200 simulated sequences that match this criterion is shown above each of the plots on the figure. It is apparent from the figure that although the distribution of trend errors is not affected significantly by increasing perturbation, the number of peak jumps is affected. This observation is in agreement with the concept that excessive change in backwall geometry causes peak jumps. This finding therefore confirms that frequent measurements are recommended when using AXC in order to ensure reliable and accurate thickness loss trends. In contrast to this when standard XC algorithms are used increasing the measurement frequency does not improve matters because the signal shape of the kernel function remains constant and does not adapt to surface morphology introduced changes.

Backwall evolution with rms change
The mean wall thickness trend error distribution plots with R rms scaling applied are shown in figure 10. It should be noted that the axes of the plots in figure 10 are 5 times larger than those of figure 9. This larger range was chosen as the trend error distributions are substantially larger when the R rms is continously increasing or decreasing. In order to better understand the reason for this, the behaviour of R rms scaling in the backwall sequence generator model is considered.
The R rms scaling was defined in the model as a factor that scaled the backwall geometry at every step. This was initially used to keep the surface rms constant, however if it is used to continously increase or decrease the rms then it effectively introduces spatially correlated thickness changes. This means that with each step in the backwall sequence thinner parts of the component will become thinner and thicker parts will stay thicker (or the other way round) relative to the mean thickness of the component. It is important to point out that this correlated perturbation caused by R rms scaling also leads to distortion in the ultrasonic signal. This distortion is in addition to that introduced by random perturbation. However, in the backwall sequence generator model, mean wall thickness loss is linked to random perturbation alone. Because of this, when random perturbation is small, the mean wall loss will still be small even if the correlated perturbation is large. The error introduced by large correlated perturbation will however be large relative to the small mean wall loss. This can be observed on the top row of figure 10, where the random perturbation r p term is small but trend error distributions are large.
A real life example of a similar phenomenon is pitting. With pitting-type degradation mechanisms the backwall of the sample loses wall thickness in a spatially non-uniform fashion as individual pits grow (see figure 6(a). The continuous growth of a pit is a type of correlated perturbation, which may occur without significant mean wall thickness loss. Over time substantial changes in backwall geometry may occur, without much mean wall thickness loss, but still introducing large amounts of distortion in the ultrasonic signal.
The results of figure 10 show quantitatively that the trend error distributions for all standard methods (XC, P2P, FA) extend beyond the ±100% mark for almost all simulated scenarios. The worst case scenario is the top row of the figure, where the correlated perturbation is most significant. AXC still performs better than any other signal processing method in all scenarios, however its performance is not as accurate as when uncorrelated backwall changes occur. The widths of normalized trend error distribution of AXC are as high as ±100%, where error is quantified as the width of trend error distributions between the 5th and 95th percentiles. In comparison, the width of trend error distributions for all other methods (XC, P2P and FA) are of the order of ±500%. It is worth noting however, that when random perturbation is applied in higher proportion compared to correlated perturbation (bottom two rows of figure 10), the error of all four methods (AXC, XC, P2P and FA) decreases significantly.
Another interesting feature of the displayed plots is that under increasing R rms conditions (left column of plots on figure 10) XC, P2P and FA methods tend to overestimate the thickness. Under decreasing R rms conditions however (right column of plots on figure 10) the same methods consistently underestimate thickness. This is a consequence of the interaction of the scattered wavefield from the backwall and the coherent backwall echo wavepacket: with increasing R rms the relative amplitude of the scattered wavefield increaseseffectively delaying energy within the received wavepacket. With decreasing R rms the opposite effect is observed, as expected. It is important to note that the model that introduces scaling of the rms of the backwall has severe limitations: (1) correlated perturbation in our model is simulated as scaling the backwall shape vertically. Consequently, no horizontal changes are introduced. A real pit would however be expected to grow both in the vertical and horizontal dimensions. Because of this, it is expected that the vertical scaling only may not be realistic to simulate pits. (2) For surface evolutions that show severely correlated backwall changes (i.e. isolated pits) the determination of a mean wall thickness does not make sense and is expected to always lead to large errors. This is because in the limit a zero width full depth pit (pin hole) does not affect the mean wall thickness but it is a critical defect. The presented results therefore only give an insight into the effect that different (non-random) backwall change scenarios might have on ultrasonic measurements.
It should also be noted that this study was carried out for a particular transducer geometry and wave mode (SH waves) that are used in practice for thickness monitoring. Results would be slightly different for other transducer geometries and other wave modes that are employed, but they are likely to show the same trends as presented here. The ultrasonic scattering phenomena and interactions with the rough backwall remain similar for other transducer geometries. For example the study by Benstock and Cegla [12] has shown that variation of thickness measurements with round transducers and compressional waves is of a similar order to that of waveguide transducers (see figure 2) [10]. Simply the size of the surface over which the wave field interacts with the surfaces will be different. Furthermore, it is expected that the relative performance differences between various signal processing methods are similar.

AXC results on field data
In addition to the wall thickness data that was processed by a P2P algorithm figure 11 also shows the same data processed by the AXC algorithm. It is clearly visible that AXC produces a trend that is not influenced by the signal distortion due to backwall surface morphology changes and gives a more representative wall thickness trend/corrosion rate. Overall this results in less variability in the extracted corrosion rate and thus in improvement in the response time and confidence with which significant changes in corrosion behaviour can be picked up.
It is worth noting that this study explicitly focused on the accuracy of rate measurements, we did not analyse the overall accuracy of the wall thickness measurement (see e.g. Jarvis et al [11]). AXC enables better tracking of the arrival time of distorted wavepackets. This does not necessarily mean that The green boxes represent the results for adaptive cross-correlation (AXC), the red boxes are for cross-correlation (XC), the blue boxes are for peak-to-peak (P2P) and the black boxes are for first arrival (FA) methods. Axes on all plots are identical for comparability. The numbers shown above each plot are the numbers of trends that have been evaluted and excludes trends that include peak jumps. the overall wall thickness measurement has become more accurate (there might be a constant systematic error/offset). However, as the simulations in this paper show clearly, the rate of change (corrosion rate) prediction is markedly improved.

Conclusion
From field data it is known that surface morphology changes can introduce substantial errors into ultrasonically measured corrosion rates (thickness trends). In this paper a new signal processing method, AXC to overcome these problems was presented. The effect of continuously changing surface morphology on the accuracy of ultrasonically monitored corrosion rates was investigated. This was achieved by means of a backwall sequence generator model that simulates gradual perturbation of backwall surface geometries. This model was then used to generate Gaussian rough backwall sequences characterised by a range of parameters, including constant and changing surface R rms values and a varying size of perturbation between each surface in the sequence. Instances of both spatially random and spatially correlated perturbation were generated to simulate phenomena such as spatially statistically uniform corrosion and spatially non-uniform corrosion. Ultrasonic signals were simulated for all generated backwall geometries. These were then analysed using 3 standard signal processing methods: XC, P2P and FA and also with the newly developed AXC technique. Corrosion rates (wall thickness loss trends) were computed using all methods and the accuracy of estimated mean wall thickness loss trends was compared to the real simulated value. Figure 10. Distribution of normalised trend error e 1 .. 200 for each backwall generator parameter set shown for each signal processing method with R rms scaling. The green boxes represent the results for adaptive cross-correlation (AXC), the red boxes are for cross-correlation (XC), the blue boxes are for peak-to-peak (P2P) and the black boxes are for first arrival (FA) methods. Axes on all plots are identical for comparability within the figure, however they are 5 times larger compared to figure 9. The numbers shown above each plot are the numbers of trends that have been evaluted and excludes trends that include peak jumps. Figure 11. Ultrasonically monitored wall thickness over the period of 1 year (1 measurement every 12 hrs). A distinct deviation in monitored wall thickness is clearly visible 3 months after monitoring commenced. Thickness estimates shown by the solid blue line were calculated using a peak to peak (P2P) timing algorithm, thickness estimates shown by the dashed black line were calculated using the newly developed adpative cross correlation algorithm (AXC) that is described in this paper.
It was found that the accuracy of trend predictions varies significantly with signal processing methods. When the backwall geometry was perturbed at random spatial locations, the trend errors of the XC, P2P and FA methods were as high as ±100%, where error is quantified as the width of trend error distributions between the 5th and 95th percentiles. For the same ultrasonic signals the worst trend error of AXC was 7.5%±18%, close to an order of magnitude less than other methods. A slight underestimation of the AXC estimated wall thickness loss rate was also observed, but this was small compared to the error of other methods and the width of the distribution. Based on this data it is expected that use of AXC on spatially randomly distributed corrosion with corrosion rate of 1 mm year −1 would result in estimates of corrosion rate of 0.75-1.1 mm year −1 whereas the estimates of other algorithms would record rates between 0 and 2 mm year −1 for the same ultrasonic information.
When a spatially correlated perturbation (i.e. continously growing or shrinking R rms ) was added to a spatially random perturbation, the error of all signal processing methods increased compared to the case with a random perturbation only. AXC still performed best under these conditions. In the worst case scenario, where the spatially correlated perturbation was much larger than the spatially random perturbation AXCʼs 5th to 95th percentile trend error width was ±100% compared to about ±500% of other methods. However these reduced to ±20% and ±70% respectively when the random perturbation was much larger than the spatially correlated perturbation. Therefore for corrosion mechanisms that result in correlated backwall changes (pitting-type) larger errors to the estimated mean wall loss trend are to be expected. This is because mean wall loss is not a good measure of spatially correlated (pitting-type) corrosion.
The improved capability to extract corrosion rates was verified on measurement data that was acquired by an industrial sensor in the field. It was shown that AXC greatly reduces the susceptibilty of the sensor to surface morphology induced changes in ultrsonic reflected signal and therefore enhances the corrosion rate measurement capabilities of ultrasonic monitoring systems.