Toward characterizing the effects of environmental and operational

conditions on diffuse-field ultrasonic guided-waves in pipes

Matineh Eybpoosh

1, Mario Berges

2, Hae Young Noh

3

Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

ABSTRACT

One of the main challenges in real-world application of guided-waves based nondestructive evaluation (NDE) of

pipelines is their sensitivity to changes in environmental and operational conditions (EOC) that these structures are

subject to. In spite of many favorable characteristics of guided-waves for NDE of pipes, their multi-modal,

dispersive, and multi-path characteristics result in complex signals whose interpretation is a difficult task.

Studies that have considered the effects of EOC variations either fail to reflect realistic EOC scenarios (e.g., limited

to particular effects of specific EOCs, like time shifting effects of temperature in plates) or lack the necessary

understanding of the effects of EOC variations on different aspects of the developed damage detection approaches.

Such gaps limit the extensibility of these approaches to pipeline applications outside of controlled environments.

This paper motivates the idea of analytically incorporating the effects of temperature and flow rate variations into

damage diagnosis of pipes, through a number of case studies. A review of the existing literature on guided-wave

based testing is also provided. For damage detection, a linear supervised classification method, namely linear

discriminant analysis (LDA), is applied to experimental guided-wave data recorded from a hot water piping system

under regular operation. Principal components, obtained through principal component analysis (PCA), and Fourier

transforms of the signals are two sets of damage-sensitive features (DSF) that are examined for LDA-based

classification. The effects of temperature and flow rate difference among testing and training datasets on (A)

detection performance and (B) goodness of fit of the method to the data are investigated.

Keywords: Pipeline Monitoring, Guided-waves, Nondestructive Evaluation, Structural Health Monitoring, Effects of

Temperature and Flow Rate, Damage Detection, Linear Discriminant Analysis.

1. RESEARCH BACKGROUND

Pipelines are crucial infrastructure components in several applications. Reliable estimate of structural status of pipes

throughout their life time is important to ensure delivery of expected services, and decrease the costs and

environmental/human-safety risks that can be caused by not/late detected damages. According to Department of

Transportation’s report in 20101, pipelines transmit about 71% of oil and refined petroleum products, and 100% of

the natural gas consumed in United States. If not detected on time, damages in them can lead to large costs, safety

risks, and/or environmental effects. For example, according to a study conducted in 20132, the ratio of the leakage to

the average hourly natural gas production can reach 14% in Uintah County, Utah. Reliable, timely and continuous

monitoring of pipelines has been the subject matter of research for decades. The field of nondestructive evaluation

(NDE) of pipelines has been actively benefiting from the increased availability and cost-effectiveness of sensing

technologies, as well as advances in data analysis techniques. However, according to Department of Transportation,

Pipeline and Hazardous Materials Safety Administration3, 1,337 release incidents in transmission pipelines were

reported during 2010 to 2012, while less than 10% of these reported leakages were identified by data acquisition and

testing systems.

1 PhD student, matineh@cmu.edu 2 Assistant professor, marioberges@cmu.edu 3 Assistant professor, noh@cmu.edu

Guided-waves have been widely considered for NDE of pipes during the past decades in a number of applications

ranging from water-supply to gas/oil transmission and power generation 4–6. Guided-waves can travel long distances

through the pipe (up to tens of meters), providing full coverage of the pipe surface, with minimum attenuation7–10.

Guided-waves are sensitive to different types of damage, such as cracks, corrosion or through-thickness notches, at

different sizes. Moreover, guided-wave based systems are cost/time efficient since the large portion of the pipe can

be tested using a small number of sensors (in some methods, only one sensor is used)4,9,10.

However, practical application of guided-waves for NDE of pipelines is still limited8,10. Guided-waves propagate in

multi-modal, multi-path and dispersive fashion making the interpretation of the resulting complex signals

challenging. Adding to these complexities, variations in environmental and operational conditions (EOCs) under

which these pipes operate degrade damage detection performance in real-world applications11,12. Effects of EOCs

range from change in the velocity of wave modes to energy leakage or mode conversion12–14.

Many damage detection studies to date have either ignored or controlled the effects of EOCs4,6,15. Although, there

are a growing number of research studies investigating the effects of different EOCs on guided-waves, the results of

literature review conducted by the authors16 suggest that important gaps still need to be addressed to move toward

robust guided-wave based damage detection of pipelines. First, in the majority of these works, the effects of EOCs

are investigated in plate-like structures. However, although governing wave equations, solution methods, and

boundary conditions in pipes are similar to those in plates, the propagation and dispersion behavior in cylindrical

media shows significant difference compared to plates. For example, pipe and plate dispersion curves are very

different in low frequency ranges17. Here, it is worth noting that the majority of guided-wave based NDE

applications are restricted to lower frequencies8,14. Moreover, in the case of cylindrical wave-guide, three different

types of modes need to be studied, as opposed to two in plates, namely longitudinal (i.e. L(0,m) modes), torsional

(i.e. T(0,m) modes), and flexural (i.e. F(n,m) modes), where n is the circumferential order (n=1,2,3,…), and m is the

mode order (m=1,2,3,..). Each of these modes has different propagation, dispersion characteristics, and sensitivity to

different types of EOCs, which adds to the complexity of the wave behavior in pipes compared to plates.

Among all EOCs, the effects of temperature variation in plates are most widely studied12,18. This is because

temperature is the most commonly encountered varying EOC in different structures. Moreover, guided-wave

systems can be very vulnerable to temperature variations, as several components of the system can be affected by

those changes. The majority of these studies are focused on damage detection through baseline subtraction, for

which the time-shift effects due to temperature difference between baseline and monitored signal need to be

compensated9,18–20. However, time-shift is not the only effect of temperature variation on guided-waves. Changing

the dispersion properties of wave modes, temperature variations can cause changes in the shape of the propagated

signals. Moreover, temperature is only one of the EOCs that structures such as pipelines are subject to. For example,

as the authors’ preliminary work suggests16, variations in the flow rate inside the pipe can also lead to effects such as

time-shift and/or amplitude change of the propagated waves.

Studies addressing the effects of EOCs in pipes are mostly focused on investigating the effects of a second medium

coupling with the pipe, such as fluid inside the pipe13,21. However, unrealistic assumptions of static and

unpressurized nature of the fluid carried by the pipe still remain unaddressed. Recent studies have shown that inner

pressure22,23 and flow rate16 can mask the effects of damage in pipes.

Finally, the majority of the research on guided-wave testing of pipes is based on excitation of individual modes,

through either narrow-band excitation in non-dispersive regions, and/or specifically designed transducer setups4,6,14.

However, in real-world applications obtaining such pure signals is difficult, if not impossible. Multiple reflections

and refractions of wave modes from the wave-guide’s boundary, welding, junctions and/or damages, and effects of

EOC variations prevent the collected signal to maintain the purity of the excited signal9.

Although the summarized research gaps remain to be addressed, a growing number of studies have been benefitting

from recent advances in data analysis techniques (e.g. machine learning and/or signal processing methods) to

overcome the EOC challenges5,24,25. The main motivation for this trend is to develop robust damage detection

methods, using damage-sensitive features (DSFs), without needing to comprehend, in detail, the complex

characteristics of the EOC effects. As the authors’ literature review in the field of guided-wave based NDE reveals,

generally, there is a disconnect between the studies investigating the effects of EOCs and the ones developing

damage detection methods. The missing physical and analytical intuition about the way EOCs affect different

aspects of a damage diagnosis approach, limits its extensibility to different EOC scenarios.

This paper summarizes the results of case studies motivating damage detection approaches that analytically

incorporate the effects of temperature and flow rate variations in pipe operations. A linear supervised classification

method, namely linear discriminant analysis (LDA), is used for damage detection using guided-wave data recorded

from a hot water supply piping system under operation. Principal components, obtained through principal

component analysis (PCA), and Fourier transforms of the signals are two sets of damage-sensitive features (DSF)

that are examined for LDA-based classification. The effects of temperature and flow rate difference among testing

and training datasets on (A) detection performance and (B) goodness of fit of the method to the data are

investigated.

The organization of this paper is as follows. Section 2 gives a brief overview of the experimental testbed and the

characteristics of the data used for the case studies. Section 3 briefly introduces the LDA-based damage detection

approach used in this paper. Section 4.1 summarizes the results of case #1 with similar temperature and flow rate

variation ranges of test and train datasets. Section 4.2 describes case #2 through which the effects of different

temperature and flow rate ranges on detection performance and on goodness of fit of the model are examined.

Concluding remarks and future work are given in section 5.

2. EXPERIMENTAL TESTBED

Ultrasonic guided-waves are recorded from a campus building hot water piping system through pitch-catch method.

The piping system operates continuously, 24 hours a day, 7 days a week, with varying flow temperature, flow rate,

and flow pressure, among other possible EOCs. A broadband excitation with sinc signals with frequency band of

100 KHz to 300 KHz is used. These characteristics of the data collected from this testbed enable examining the

research gaps discussed in previous section.

The pipe segment under study is a schedule 40 steel pipe with 10 (in.) inner diameter and 0.365 (in.) wall thickness,

covered by fiberglass insulation. Hot water is periodically pumped to the system leading to flow rate variations in

the range of 200 to 450 gpm. Water temperature also fluctuates between 100 to 140 . For each pitch-catch record,

flow rate and temperature values are measured and recorded using the in-situ sensors. The resolution of temperature

sensors is 1 .

Damage is simulated by coupling a mass scatterer to the pipe’s wall surface. Over 100,000 records have been

collected from both damaged and undamaged pipes in this testbed, over 1,000 hours of operation. More detailed

information about the testbed and data collection process can be found in Liu et al.11

3. LINEAR DISCRIMINANT ANALYSIS

Similar to PCA, LDA seeks for linear combinations of n features that best represent a set of multi-dimensional

observations ( ). However, in contrast to PCA, the objective of LDA is to preserve class

discriminatory information as much as possible, by finding the linear subspace that provides best separation among

K classes (i.e., a discriminant function). Generally, for a K-class problem, K-1 discriminant functions exist. Damage

detection is a two-class problem (i.e., K=2), as the class label (Y) of an observation can be either damaged (i.e., Y=1)

or undamaged (i.e., Y=0).

Assuming that the n-dimensional observations follow a multi-variate normal distribution, LDA uses class means ( )

as the measure of separation, and finds discriminant function (w) that maximizes the distance between projected

means, that is . For example, for a two-class problem, the objective function is as given in equation 1.

(1)

This method, however, doesn’t take into account the within-class variance ( ). Fisher26 suggest maximizing the

between-class variance ( ) while minimizing the variance within each class ( ). That is, finding the linear

combinations of n features (w) that represents the largest difference among the class means relative to .

Within-class variance is measured by the covariance of each class, weighted by the prior probability of each class

happening (i.e., ). Between-class variance is measured by the sum of the variance of each class

with respect to the mean of the whole data. Equations 2 and 3 summarize these measures, in which is the data

matrix of n-dimensional observations in the ith class, is the mean matrix of the observations in the ith class, and

is the mean matrix of all observations in the dataset.

∑

(2)

∑ (3)

Therefore, Fisher criterion for identifying discriminant function (w) is to maximize the following function:

(4)

Solving

leads to identification of Fisher’s discriminant function w, as given in equation 5 for a two-class

problem:

(5)

Each row of the matrix of coefficients described in equation 6 contains the coefficients for the

marginal linear function closest to each class, parallel to the Fisher’s discriminant function.

[

] (6)

Projecting test observations ( ) on each of the marginal functions given in matrix C can be used to score

the class probabilities. The higher the score, the more it is likely that an observation belongs to a particular class.

For the two-class problem at hand, the score matrix and the matrix of class probabilities are

calculated as described in equations 7 and 8. For more detailed discussions on LDA, please refer to references such

as Alpaydin27.

[

]

[

] (7)

[

]

(8)

4. CASE STUDIES

The dataset used in this paper contains a total of 1132 pitch-catch measurements, including 840 measurements from

undamaged pipes, and 292 measurements from damaged pipes. Temperature (T) variation for this dataset is between

103 and 109 , and flow rate (F) varies between 272 gpm and 435 gpm. Figure 1 shows the fluctuation of these

parameters for all observations, as well as the histogram of the number of observations at each T and F.

Figure 1. Temperature and flow rate variation for all observations in the dataset

4.1 Case #1: Similar ranges of EOC variation in train and test datasets

In this case, temperature and flow rate fluctuate in both test and train datasets; however, the range of these variations

are similar, although not identical. A 10-fold cross validation (CV) is used. Using the original signals for training

the LDA algorithm results in poor damage detection performance. Two damage-sensitive features (DSF) are

examined, namely principal components extracted through PCA, and Fourier transforms of the signals.

Damage detection is based on the class probabilities obtained for test observations. It is found that, for both

approaches, the probability of being undamaged P(Y=0) drops as damage occurs. The average amount of this drop

for all test observations at all 10 folds is 4 times larger when using principal components (PCs) as DSFs compared to

Fourier transforms. In continuous NDE practices, this drop in P(Y=0) can be used as a damage detection criterion.

That is, the instant of damage occurrence in pipe is when P(Y=0) drops beyond a certain value. The larger drops

observed in PCA-based approach suggest that, in continuous monitoring, using PCA for DSF extraction would be

preferred.

However, when NDE is applied at discrete times throughout the pipe’s life-time, such drops in P(Y=0) are not

applicable. In such cases, damage detection based on these class probabilities is only possible through their

comparison (i.e., ). The damage detection algorithm is as follows: for the observations in the

test dataset ( ), if , ̂ , otherwise ̂ . This is equivalent to checking

whether it is more likely for the observation to be damaged or undamaged.

Figure 2 summarizes the prediction accuracies for all 10 folds of CV using this algorithm. As illustrated in this

figure, using the first 500 PCs as DSFs leads to an average accuracy of 98%, whereas using magnitudes of frequency

spectrum results in an average accuracy of 83%.

Figure 2. Detection accuracies for 10-fold CV for test and train datasets with similar temperature and flow

rate variation ranges

4.2 Case #2: Investigating the effects of temperature on LDA-based damage detection

The approach followed in case #1 for validation of the detection algorithm is common among supervised damage

detection methods developed to date to overcome the EOC challenges(e.g 22,25.). That is, the test dataset is selected

from the same pool as is the train dataset. Therefore, test and train datasets possess similar ranges of EOC variations,

although they may not be identical. The way that the variations of EOCs can affect the performance of these

approaches is not studied.

In this case study, the sensitivity of the developed LDA-based damage detection approach to the range of

temperature variations in test and train datasets are investigated.

4.2.1 Effects of temperature on detection performance

Train dataset is selected uniformly from the observations falling in the lower half of the temperature histogram given

in Figure 1. That is, . Several test datasets ( ) with different temperatures are generated,

so that the temperatures of m observations in one dataset vary within 1 (i.e., for every test dataset,

).

The difference between each test observation’s temperature and the mean temperature of train dataset (i.e.,

) varies as . It is notable that, for the purpose of studying the effects of

temperature, the range of flow rate variations are allowed to be similar among test and train datasets, but not

restricted to any particular value.

Although using PCs for LDA-based damage detection result in higher overall detection accuracies than using

Fourier transforms, as illustrated in Figure 3, these accuracies tend to decrease, almost linearly, as increases,

with an approximate rate of per . Detection accuracies for the second approach are less sensitive to the

temperature difference between test and train datasets, with a decrease rate of 5.1% per .

As explained before, detection algorithm is based on the comparison of class probabilities. Investigating the reasons

for the decrease in detection accuracies reported in Figure 3, it is observed that class probabilities of undamaged

cases are independent from temperature difference between test and train datasets ( ). However, for damaged

observations, a positive correlation between P(Y=0) and is observed. That is, as the temperature difference

increases, it is more likely for damaged cases to be misclassified as undamaged, hence, the false negative alarm rate

increases. This highlights the masking effects of temperature difference that prevents the effects of damage to be

extracted by PCA, and detected by LDA algorithm.

Figure 4 depicts the variation of P(Y=0) for damaged test cases over the range of temperature difference ( )

between train and test datasets. Assuming a linear relation among the two, the increase rate of P(Y=0) is about 0.07

per 1 increase in . For 90% of the observations with , the calculated P(Y=0) is above 50%,

meaning that they would be misclassified by the LDA-based damage detection algorithm explained in case #1.

Figure 5 plots the distribution of false negative alarms for damaged observations with respect to . To obtain this

plot, P(Y=0) values at each in Figure 3 are assumed to be normally distributed, with linearly increasing means

(i.e., red dashed line in Figure 3). The cumulative density for the normal distribution at each for is the expected false negative alarms at that . As can be seen in Figure 4, false negative alarm rates tend to

increase as increases.

Figure 3. Effects of temperature difference among test and train data ( ) on detection accuracy of LDA

Figure 4. Variation of undamaged class probabilities, calculated for damaged test cases, with respect to

temperature difference among test and train data

One possible reason for relatively scattered distribution of P(Y=0) versus given in Figure 4 is the variation of

other EOCs such as flow rate. Although for this case study the flow rate ranges are not restricted to be different

between test and train datasets, differences still can exist, and can contribute to the degradation of the detection

performance. Here, it is worth noting that 0.3 correlation coefficient between flow rate and temperature variation of

test observations suggest that these parameters are not significantly related, so their effects can be studied

independently. Figure 6a plots variation of P(Y=0) for damaged test observations with respect to both temperature

and flow rate variations (i.e., and ). In this figure, the brightness of the plot reflects the size of P(Y=0). That

is, damaged observations with brighter color are associated with larger undamaged class probability. It is observed

that larger values of P(Y=0) happens when and gpm.

Interestingly, while distribution of P(Y=0), shown in Figure 6b, is almost independent of flow rate variations for

, for , there is a positive, relatively linear, relationship between P(Y=0) and ,

with a rate of 0.002 increase in undamaged class probability for every 1 gpm deviation of flow rates. This slight

dependency is a potential reason for relatively scattered distribution of P(Y=0) vs. , shown in Figure 4.

Figure 5. False negative rates for test datasets with different temperature ranges than train dataset

Figure 6. Variation of undamaged class probabilities, calculated for damaged test observations, with respect

to flow rate and temperature difference among test and train data

4.2.2 Effects of temperature on the goodness of the fit of the method to the data

The objective of this section is to test, for every test dataset, whether the identified discriminant function leads to a

significant relationship among the features and the class labels. In other words, the goal is to test whether the LDA-

based approach explained in case #1 is a good method for damage detection at different ranges of temperature.

In discriminant analysis, the key metric for evaluating the fitness of the calculated discriminant function to the data

is the significant test for Wilk’s Lambda (Λ)28. Wilk’s Lambda statistic tests the null hypothesis that all features

have the same mean for different classes. If is true, class means will not be a good basis for classification, as is

used in LDA29. In other words, Λ measures the ratio of the variance in the discriminant scores that is NOT due to

differences between the classes.

In order to calculate Λ for each test dataset, the first step is to calculate the maximum eigenvalue (λ) of the ratio of

between-class variance ( ) to the within-class variance ( ). Wilk’s Lambda (Λ) is then defined as28:

(9)

It is notable that the statistic Wilk’s Lambda is not necessarily an indicator of the classification accuracy of the

discriminant function, but is a measure of goodness of fit of the model to the data to explain class membership28.

It is observed that the Λ statistics are statistically insignificant for test datasets with temperature differences ( )

larger than ~2 . That is, for these observations, p-values of Wilk’s Lambda’s Chi-square is less than 0.001.

However, no particular relationship between the magnitude of Λ and for observations with is

observed. This suggests that the discriminant function obtained through LDA do not represent a significant

relationship among features and class labels when , hence, classification using this function may not

perform better than chance.

The magnitudes of discriminant coefficients reported in C matrix, given in equation 6, define the contribution of the

corresponding features in classification of the observations. That is, the feature with the largest coefficient is the first

principal feature for classification. For the train dataset used in this case study, the 11th and 74th principal

components are found as the two most effective features for classification, with standardized discriminant

coefficients of and respectively.

Figure 7. Variation of separation power of discriminant function as among test and train datasets changes

Beside Wilk’s Lambda statistic, another way to examine the goodness of fit of the discriminant function to the test

datasets at different temperatures is to investigate how the separation power of these principal features varies as

changes. Figure 7a, shows the distribution of damaged and undamaged observations for the two principal

components, denoted PC1 and PC2, with respect to . The brightness of the plotted points reflects the size of the

associated with that observation. The line in Figure 7b is the discriminant function in the PC1-PC2 subspace that

maximizes between-class scatter while minimizing the within-class variance. 86% of the damaged observations

whose projection on the discriminant function merge with undamaged observations; therefore, they cannot be

classified by this model, are cases with . Among all observations with , 13% of them fall in

this region. These cases are marked with the dashed box in Figure 6.

5. DISCUSSION AND CONCLUDING REMARKS

Guided-wave based testing of pipelines operating under varying EOCs is growingly benefiting from recent advances

in data analysis techniques in machine learning and signal processing. However, the disconnect between different

aspects of these approaches and the effects of EOCs limits the applicability of the developed approaches in pipelines

operating under different conditions. This paper summarizes the results of case studies investigating the effects of

temperature and flow rate on detection performance and goodness of fit of the model to a supervised damage

detection method based on LDA.

It is observed that using principal components as DSFs for LDA results in an average detection accuracy of 98%,

when train and test datasets have similar ranges of temperature and flow rate fluctuations. Although this approach is

robust to fluctuations in temperature and flow rate values, further investigations show that its performance can be

affected by the range of these fluctuations.

Investigating the detection performance, it is observed that detection accuracy decreases, almost linearly, as the

distance among the temperature of test observations and the mean temperature of the observations in train dataset

increases. The source for the decrease in detection accuracy is found to be misclassification of damaged cases, that

is, increase in false negative rate. This finding highlights the masking effects of temperature. In other words, as the

distance among the test temperature and the mean temperature of training observations increase, the effects of

damage are likely to be surpassed by the effects of temperature. It is notable that one of the factors affecting accurate

identification of the beyond which the effects of temperature become dominant is sensor resolution. In the

studies reported in this paper, temperature sensors’ resolution is 1 . Therefore, the threshold of obtained

in the studies reported in this paper can actually be any value in the range of . Moreover, size

and/or type of damage can also affect the temperature difference threshold. For the datasets used in this paper, a

grease coupled mass is used to simulate damage, however, same experiments need to be done with different sizes of

longitudinal, circumferential, and through-thickness damages, as different wave modes are sensitive to different

types of damage. Also, effects of other EOCs can contribute to the masking of the effects of damage. For example,

as briefly investigated in this paper, exceeding a certain level, flow rate difference can also contribute to the false

negative alarms. It is notable that the dataset used in this study is recorded from a piping system under operation,

with no control over variation of EOCs. Therefore, the reported results may not purely reflect the effects of

particular EOC. However, the findings reported in this paper motivate investigations on controlled datasets through

which the effects of every EOC can be studied in isolation, and analytically incorporated into damage detection

approach. Comparing these results with those of such controlled studies would also provide insight on the

superimposed effects of EOCs, hence, enhancing extensibility of the results to different EOC scenarios.

In addition to detection accuracy, in order to ensure the extensibility of a damage detection approach, it is important

to test whether the method is still a good fit to the data at hand, and whether it classifies the data better than chance.

The goodness of fit of the discriminant function is tested using the Wilk’s Lambda statistic. For the cases used in

this paper, Wilk’s Lambda is found to be statistically insignificant for test datasets with temperature differences

above ~2 . Examining the source of this misfit, separation power of the discriminant function using the first two

principal components is investigated. It is shown that for larger values of temperature difference, the principal

components identified using LDA lose their power in representing damaged cases, and the discriminant function can

no longer separate two classes.

The findings reported in this paper motivate controlled experiments in order to formally investigate the effects of

different EOCs typical in pipeline applications, and to analytically incorporate such effects into damage detection.

Such studies will address the existing gap in linking the effects of EOCs to the guided-wave based damage detection

approaches developed for NDE of pipes.

6. ACKNOWLEDGEMENT

The authors would like to acknowledge Julia and Michael Ellegood research fellowship for partially

supporting this study. The authors would also like to thank Dr. Irving Oppenheim and his group at

Carnegie Mellon University, and the PhD student Chang Liu, for the data used in this paper.

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