Bayesian Estimation for Crack Monitoring of a Resonant Pipe using Acoustic Emission Method
Mohd Fairuz Shamsudin, Cristinel Mares, Yoann Lage*, Graham Edwards*, Tat-Hean Gan*
Brunel University,
School of Engineering and Design, Uxbridge, Middlesex UB8 3PH, UK
*TWI Ltd,
Granta Park, Great Abington, Cambridge CB21 6AL, UK email: mohd.fairuz.shamsudin@gmail.com
ABSTRACT
Vibration induced fatigue is a well-known problem in the oil and gas piping and pipeline systems. However the use of vibration data to detect damage is not an easy task without a priori knowledge about the undamaged condition. In this paper, acoustic emission monitoring of a resonance fatigue test of a welded carbon steel pipe has been carried out to monitor the weld condition from healthy until failure. The information provided by acoustic emission monitoring is useful in evaluating the condition of the pipe during the test and the occurrence of cracking before failure. However, acoustic emission signals are also susceptible to extraneous noises.
Therefore in this work, the acoustic emission signals from different combination of sensors were recursively correlated, which provides parameter for Bayesian estimation to discriminate the true signals and avoid false detection and localization errors. The method provides indicator for damage detection, which shows a strong relationship of the acoustic emission energy and the estimated coefficients. High correlation of signals was found to be associated with cracking, which has an effect of the increase of acoustic emission energy. Likewise low correlation of signals was found to be associated with random signals or noises. The method will be useful in monitoring piping or pipelines condition to reduce the risk of vibration induced fatigue failure.
KEYWORDS: Bayesian filter, acoustic emission, pipe fatigue.
1. INTRODUCTION
Vibration induced fatigue is a well-known problem for the oil and gas piping and pipeline systems [1]. Various vibration-based monitoring techniques have been implemented in the field of condition and structural health monitoring. However, the major challenge in monitoring technology is to detect failure with high confidence. In principle, vibration-based techniques measure the change of physical or dynamic characteristics to evaluate structural condition; particularly crucial to detect the onset of damage as well as prediction of the remaining life.
However, the use of vibration data to locate damage is complex since piping or pipeline vibration is random. The vibration mechanisms can be originated from mechanical equipment, fluid flows and other external factors.
Damage due to fatigue cracks are difficult to be identified in reality due to negligible change of structural stiffness, thus, no change of natural frequencies could be observed. Therefore, relying solely on the information from the dynamic response of a system may not be able to provide useful information about damage. In this paper the acoustic emission technique is used for damage detection and localization of damage in a welded pipe subjected dynamic loading by resonance fatigue testing. Acoustic emission technique is susceptible to extraneous noises and thus, a Bayesian filtering method is used. Bayesian filtering by recursive method is able to provide qualitative information about the state of the pipe and filter unwanted signals from noisy background.
2. SIGNAL PROCESSING METHOD
There are several studies [2–5] published summarizing different techniques and approaches regarding vibration- based damage detection in structural health monitoring. Generally, selection of features can be categorized as parametric-based or signal-based [6]. The signal-based method employs signal processing techniques which acquire parameters directly in the time domain, frequency domain or time-frequency domain. These include Auto-Regressive Model (AR) model and Auto-Regressive Moving Average (ARMA) [7–10] model for linear or non-linear problems. The parametric-based method uses explicit physical-dynamic parameters from measurements [11–13] or from model updating by numerical methods to evaluate structural modal parameters
[14–18]. There are many uncertainties associated with the measurement output, such as noise, variability in the measuring device and material properties. Thus, in this paper Bayesian estimation on acoustic emission signals is used to reduce the uncertainties rising from the data acquired during the extreme condition of a resonance fatigue test.
2.1. Bayesian Estimation
The Bayes’s theorem is attributed Reverend Thomas Bayes (1702-1761). The premise of using Bayesian methods is based upon the ‘degree-of-belief’ interpretation of probability such that the underlying probability distribution of random parameters can be assumed known from historical data, or experience. The ‘degree-of- belief’ is updated in the process upon ‘seeing’ evidence, which result in the updated distribution of the parameters. Statistical inference can be made from the updated distribution either by point or interval estimates [19], [20], such as the mean values of the posteriors, or the parameter values that maximize the posteriors. The method requires a mathematical model to describe the estimation process, which consists of prior probability distribution (i.e. P(θ) ) or initial hypothesis of random parameters, which cannot be measured directly. The prior probability distribution is estimated to represent the distribution of the sought-after parameters before considering any observations (i.e. the input data). The quality of the process depends on the likelihood function i.e. P(D|θ) . In this case, the likelihood function finds the most probable data set for different values of parameter θ . Therefore, the likelihood function is a conditional probability, which is based upon a parameterised distribution. Bayes’s theorem can be written as follows;
P(θt|Dt)= P(Dt|θt)p(θt)
−∞∫
∞
P(Dt|θt)p(θt)d θt (1)
where;
θt - The predicted variables Dt - The observed data
Acoustic emission signals are susceptible to extraneous noises and such signals may consist of burst-like signals, which are normally associated with cracks, or continuous signals that correspond to plastic deformation, boundary sliding and dislocation movement in crystal lattices [21], [22], or noise resulting from continuous rubbing of materials in contact during dynamic motion. Taking advantage of these facts, the signals can be discriminated using Bayesian methods such that the generated events from a set of sensors can be further filtered so that only those having high probability of crack related events are retained. This method is useful in discriminating signals by using the underlying distribution of the data. In this paper, the Bayesian method is applied to make an inference from time series signals which are represented by the maximum cross correlation coefficient (i.e. normalized cross correlation) of signals given all possible sensor combination. The normalization is done by the autocorrelation of the signals with the most power at zero lag which is given as follows;
ϕ´i , j= maxϕi , j(τ)
√ϕi ,i(0)ϕj , j(0) (2)
where;
ϕi , j(τ) is the cross correlation function of lag τ
ϕ´i , j is the cross correlation coefficient of signals i and j
ϕi ,i(0) is the autocorrelation of signals for sensor i at lag zero
The significance of using cross correlation for acoustic emission signals is that typical burst-like signals can be distinguished from other form of signals which may be detected at the same instance by other sensors. The motivation of the proposed technique is to reduce falsely produced acoustic emission events.
2.1.1. MCMC and the Metropolis-Hastings Algorithm
The integral in the denominator given in Eq. 1 i.e. “evidence” is solved using Markov Chain Monte Carlo (MCMC) theory and the Metropolis-Hastings algorithm [23], [24]. The MCMC method allows for the solution
of integrals over high dimensional probability distributions to infer the values of the chosen parameters. In many applications the integrals, like the one in equation (2) must be carried out over a high-dimensional space and may not be tractable. Monte-Carlo sampling is used so that the integration of continuous random variable , θ is replaced by summation of large number of discrete random variables θ(i)i=0,1,2,.. , N which is independent and identically distributed (i. i .d) . The approximate solution will be computed most accurately by concentrating the samples θ(i) in the regions of high probability. A Markov chain is a stochastic process where the algorithm concentrates at high probability sampling regions to give high accuracy results. The stochastic process of Markov Chain Monte Carlo (MCMC) [24], is referred to as a guided random walk which combines multiple samples sequentially. In the estimation of posterior distribution by many samples of independent random variables θ(i) , a chain of distribution of the random variables is constructed. The sequence of distribution tends to equilibrium, independent of the initial estimate. The convergent to posterior distributions i.e. p(θ|D) is termed as the stationary distribution of the Markov chain such that
f(θ(τ+1)|θ(τ), … , θ(0))=f(θ(τ+1)∨θ(t)) given all preceding θ . This means that the parameter distribution
θ(t+1) at step t+1 only depends on θ(t) and f(θ(τ+1)∨θ(t)) is independent of t .
2.2. Selection of the Prior and Likelihood
The likelihood is a function of the parameter that evaluates the data set to maximize the posterior. Each updating variable is assumed to vary in Gaussian law. The likelihood in this case;
ϕ´ N(μ ,´τ);´τ= 1
σ2 (3)
where;
μ is the mean of the distrubution
´τ is the precision or inverse of variance σ2
WinBUGS (i.e. Bayesian Inference Using Gibbs Sampling [25]) environment is used to generate sampling from full conditional distribution of the posterior by Markov chain [26]. In this work “non-informative” prior is used without considering similar previous measurement data. The prior is assumed to be normally distributed given the fact that non-correlated signals are random. The prior for mean and precision are set in WinBUGS [27] such that;
μ0 N(0.0,1.0e−06);τ´0 Γ(0.001,0.001) (4)
3. EXPERIMENTAL SETUP
A 15 inch diameter pipe with a 50mm wall thickness and made from carbon steel was installed and tested in a resonance fatigue test machine [28]. The pipe was 7.2 m long and contained a girth weld at mid-length. During the test, the pipe was filled with water. This has the effect of increasing the overall mass and so reducing the pipe length as compared to the length theoretically required for the same resonant frequency in an empty pipe. The pipe length was chosen based on the mass of the pipe including internal medium and both end attachments to achieve pipe resonance in the first bending mode at around 30 Hz . The resonance fatigue test rig comprises a drive unit at one end as shown in Figure 1-a. A ‘balance’ unit is present at the other end for symmetry. The drive unit contains an eccentric mass which provides a rotating eccentric force to the system when it is driven by a motor. The driving frequency was set to run the test at a particular strain range (as measured by strain gauges located at mid-length of the pipe). The pipe mid-section, where the welded joint is located, is subjected to the highest cyclic strain and hence highest cyclic stress resulting around the pipe circumference during the fatigue test. The stress varies from maximum and minimum with stress ratio R = - 1. This is superimposed on the axial stress generated by pressurising the water. The water pressure is chosen to produce an axial stress that is half of the applied stress range so that the resulting R ratio is greater than zero (R > 0). The pipe is supported at the nodes of the first vibration mode as illustrated in Figure 1-b where the displacement is almost negligible.
(a) (b)
Figure 1: The test set up. (a) Cyclic excitation is driven by a drive unit containing an eccentric mass at the drive end. There is a similar mass at the other (Balance) end for symmetry. (b) Two supporting systems are positioned at the nodes where the
pipe displacement is almost negligible.
3.1. Instrumentation setup
Four broad band acoustic emission sensors model VS900-RIC with built in pre-amplifier propriety of Vallen Systeme were used with AMSY-6 multi-channels system. The system was set with 5 MHz signal sampling rate and band-pass filter between 100 kHz and 850 kHz . Acoustic emission signals are highly contaminated by extraneous noise such as fretting at the pipe supports. Other sources of noise may be generated due to the mechanisms of fatigue cracking such as crack closure or crack face rubbing. Therefore, appropriate sensor positioning is crucial to avoid recording unwanted signals. In order to remove the effect of noise generated from the pipe supports itself and mechanical noise from both pipe ends, a guard sensor (i.e. the noises will hit the guard sensor before hitting the data sensors) was placed at 6 o’clock position at either side of the nodes close to the pipe supports. The elastic deformation resulting from repetitive bending creates additional noise by friction between sensors and the pipe surface. To overcome these problem two sensors were located at each node where local displacement is almost negligible, as illustrated in Figure 2. The asymmetric sensor arrangement was such that two rings rotated by 90° against each other to avoid two possible locations exist if all sensors are on the same plane.
Figure 2: The location setup of acoustic emission sensors (numbered 1-4) at 12, 6, 9 and 3 o’clock positions respectively. A guard sensor (G) was mounted at 6 o’clock position close to each node and an accelerometer was mounted at 12 o’clock
position in close proximity to the weld joint.
The vibration measurement was evaluated by the input and output signals. The input i.e. excitation frequency from the motor was closely monitored and compared against the output vibration response measured by an industrial accelerometer mounted near the weld joint. The measurement output was fed into the Vallen system as parametric input with sampling rate of 500 Hz .
3.2. Calibration measurement
A standard calibration measurement by Pencil Lead Breakage (PLB) [29], [30], which transmits the out of plane and in-plane waves from fractured lead, were conducted at the girth weld to examine whether acoustic emission
Nodes
Balance End
Drive End
signals could be detected by the sensors with the proposed sensor arrangement. The average group velocity, Vg was determined by pulsing between acoustic emission sensors (using one as a transmitter and the others as receivers). The calculated velocity was 3000m/s at 70dB threshold setting. It has been noted that using a single wave velocity can be erroneous due to the dispersion effect and mode conversion resulting from reflection from boundaries. The crack orientation could also influence the propagating waves[31]. In this work, the calculated velocity was assumed similar to that when the crack appears given that a high threshold setting was used to only capture the out-of-plane displacement i.e. dominant wave mode as crack propagates through thickness of the pipe. PLBs were done at 3, 6, 9 and 12 o’clock positions. Using the localization method provided by the Vallen Systeme, the source locations were almost accurately located as shown in Figure 3.
Figure 3: The Located events from PLB sources. Sensor 2 is labelled twice which marks a complete pipe revolution.
4. RESULTS AND DISCUSSION
The resonance test lasted in this experiment from start-up to failure for six days (corresponding to 14,106,360 applied cycles). The frequency of pipe vibration was consistent with the frequency of the motor throughout the test. The driving motor was immediately stopped by a trip system as soon as the internal pressure in the pipe dropped rapidly resulting from loss of containment due to through-wall cracking. The failure location was identified at 6 o’clock position where a 4mm crack length was present on the pipe outer wall. The crack had initiated on the inner surface and propagated through the pipe thickness. The acoustic emission has recorded significant events in the last three days of the test, where cracks were believed to be actively propagating. A large number of data points were generated from the beginning of the test and hence only the data set from the last three days were post-processed to demonstrate the practicality of using the proposed method. Figure 4-a shows the acoustic emission energy, where some energy peaks were recorded. However, the source locations as shown in Figure 4-b are distributed across the entire mid-section of the pipe circumference and span about 100cm in each direction along the pipe length from the weld joint. This shows the localization of the potential weak locations is uncertain due large false events being recorded.
(a) (b)
Figure 4: (a) The acoustic emission energy recorded for the last three days of the fatigue test (b) Located events on the pipe.Sensor 2 is labelled twice which marks a complete pipe revolution.
6 O’clock 3 O’clock 12 O’clock 9 O’clock 6 O’clock
6 O’clock 3 O’clock 12 O’clock 9 O’clock 6 O’clock
It was identified that the acoustic emission signals associate with the located sources was not only from the burst-like signals that is normally observed from crack-related activities but also some random signals which is continuous and have long signal duration. Examples of these signals are illustrated in Figure 5 and Figure 6. The burst-like signals have higher correlation compared with the random signals. Therefore, Bayesian filtering based on this parameter will be used to identify less correlated signals recursively.
(a) (b)
Figure 5: (a) Typical random signals recorded at sensor 1 – 3. (b) Low correlations of the random signals calculated for different signals combination.
(a) (b)
Figure 6: (a) Typical burst like signals recorded at sensor 1 – 3. (b) High correlations of the burst-like signals calculated for different signals combination.
Twenty equally spaced time intervals were set for the Bayesian estimation to predict the most probable ϕ´ . A number of signals from the located events as shown in Figure 4-b were cross correlated for each time intervals in a MATLAB environment. The data sets were then loaded into WinBUGS for parameter estimation. It was noted that the first few iterations were discarded during the MCMC process. These correspond to the “burn-in” stage that is a non-stationary period of iteration in the early stage of calculation, as illustrated in Figure 7.
Figure 7: Example of MCMC samples of the estimated parameters
The estimated parameters for each time intervals are shown in Table 1. The minimum and maximum coefficient ϕ´ is 0.258 and 0.432 respectively. A number of burst emissions are recorded during these intervals followed by continuous emissions. It is understood that continuous emission is associated with various mechanisms such as plastic deformation, crack initiation, deformation twinning, or phase transformation [22], [32], [33], thus, are not quantified in this work. The continuous emissions give less correlated signals which can be represented by the small values of ϕ´ . Figure 8 illustrates the relationship of ϕ´ and acoustic emission energy. It can be seen from this figure that the energy and ϕ´ shows some similarity at some peaks. After the 3rd interval, the energy increased almost linearly with time, but at approximately constant ϕ´ . During the 9th interval the energy recorded another peak with the highest ϕ´ and sustained it for a few hours before it started to decrease.
However, the energy did not show significant indicators a few hours before failure, apart from a few smaller peaks despite higher ϕ´ .
Figure 8: The relationship of unfiltered acoustic emission energy and ϕ´
Table 1- The estimated parameters ϕ´ and standard deviation σ . Higher color intensity indicates higher values.
Interval Density ϕ´ σ
1 0.392 0.01024
2 0.349 0.00686
3 0.332 0.00538
4 0.274 0.00843
5 0.271 0.00735
6 0.266 0.00774
7 0.258 0.00649
8 0.264 0.00815
9 0.432 0.00875
10 0.376 0.00867
11 0.382 0.00884
12 0.392 0.0063
13 0.392 0.00564
14 0.351 0.01094
15 0.266 0.01176
16 0.286 0.01282
17 0.375 0.01656
18 0.376 0.0117
19 0.414 0.00857
20 0.351 0.00912
The weld joint is mid-length between sensors 1, 2 and 3, 4. Therefore maximum time of arrival ∆ tmax can be limited in the acquisition system to only capture the events generated near the weld joint area. ∆ tmax limits the arrival time of subsequent sensors upon detection of signals by the trigger sensor. The trigger sensor is the first sensor that receives signal above threshold. Limiting the arrival time prevents the reflected signals from features away from the weld joint or random signals being recorded. Figure 9 shows the located events after limiting the time to 50 μs . There are a number of located sources recorded around the weld joint apart from the location of failure at 6 o’clock position, as a result of the applied cyclic stress around pipe circumference.
The located sources (marked in green) as shown in Figure 9 were repeatedly recorded before the crack finally emerged on the pipe.
Figure 9: Located acoustic emission sources filtered by ∆ tmax . Sensor 2 is labelled twice which marks a complete pipe revolution. The marked areas are the located events immediately before failure.
The energy associated with these events were plotted with ϕ´ as shown in Figure 10. The relationship between both parameters is more pronounce with a clear indication of the energy peaks for different stages of damage.
Likewise the lower correlations of signals mostly come from the events outside of weld areas. This also means the increase of energy at some intervals may also be related to other events apart from cracking.
Figure 10: The relationship of filtered acoustic emission energy and ϕ´
Due to the dynamic motion of the pipe there are also events associated with other mechanisms which may be generated by fretting at the pipe supports. Such signals have larger in amplitudes and longer durations. It has been identified that sensor 2, which is located very close to the pipe support recorded a high number of hits. An example of such signals is illustrated in Figure 11. The generated signals are larger in amplitudes and have larger energy mostly, above 1 M V2s .
6 O’clock 3 O’clock 12 O’clock 9 O’clock 6 O’clock
Figure 11: An example of signal recorded by sensor 2 which was generated by pipe fretting.
5. CONCLUSIONS
In this work Bayesian estimation of the cross correlation coefficients were undertaken for a resonance fatigue test of a welded pipe. The premise of using Bayesian was to estimate the similarity of signals received by 4 different sensors attached to the pipe. A large number of data points were generated during the test. However the acquired results did not produce good indications of damage evolution apart from some energy peaks were recorded, thus these energy peaks were quantified using the proposed method. It was found that high correlation signals are most likely associated with crack since some of the located sources were identified to be the actual crack location emerged on the pipe. The energy level identified from crack signals are much lower compared to the overall energy generated from fretting at the pipe supports although the displacement is almost negligible at the pipe nodes. The proposed method is able to distinguish high correlated signals despite low energy signals which can be potentially used for monitoring of crack growth for resonance fatigue test in particular.
6. ACKNOWLEDGEMENT
This publication was made possible by the sponsorship and support of Fatigue Integrated Management and Condition & Structural Monitoring sections at TWI, Brunel University London, and Majlis Amanah Rakyat (MARA). The work was enabled through, and undertaken at, the National Structural Integrity Research Centre (NSIRC).
7. REFERENCES
[1] Health & Safety Executive, “Offshore Hydrocarbon Releases Statistics And Analysis,” Health & Safety Executive, 2003.
[2] E. P. Carden and P. Fanning, “Vibration based condition monitoring: a review,” Structural health monitoring, vol. 3, no. 4, pp. 355–377, 2004.
[3] S. W. Doebling, C. R. Farrar, M. B. Prime, and D. W. Shevits, “A summary review of vibration-based damage identification methods,” Shock and vibration digest, vol. 30, no. 2, pp. 91–105, 1998.
[4] W. Fan and P. Qiao, “Vibration-based damage identification methods: a review and comparative study,”
Structural Health Monitoring, vol. 10, no. 1, pp. 83–111, 2011.
[5] H. Sohn, C. R. Farrar, F. M. Hemez, D. D. Shunk, D. W. Stinemates, B. R. Nadler, and J. J. Czarnecki, A review of structural health monitoring literature: 1996-2001. Los Alamos National Laboratory Los Alamos, NM, 2004.
[6] L. Qiao, A. Esmaeily, and H. G. Melhem, “Signal pattern recognition for damage diagnosis in structures,”
Computer-Aided Civil and Infrastructure Engineering, vol. 27, no. 9, pp. 699–710, 2012.
[7] E. P. Carden and J. M. Brownjohn, “ARMA modelled time-series classification for structural health monitoring of civil infrastructure,” Mechanical systems and signal processing, vol. 22, no. 2, pp. 295–
314, 2008.
[8] H. Sohn, J. A. Czarnecki, and C. R. Farrar, “Structural health monitoring using statistical process control,” Journal of Structural Engineering, vol. 126, no. 11, pp. 1356–1363, 2000.
[9] H. Sohn and C. R. Farrar, “Damage diagnosis using time series analysis of vibration signals,” Smart materials and structures, vol. 10, no. 3, p. 446, 2001.
[10] H. Sohn, K. Worden, and C. R. Farrar, “Statistical damage classification under changing environmental and operational conditions,” Journal of Intelligent Material Systems and Structures, vol. 13, no. 9, pp.
561–574, 2002.
[11] C. Huang, S. Hung, C. Lin, and W. Su, “A Wavelet-Based Approach to Identifying Structural Modal Parameters from Seismic Response and Free Vibration Data,” Computer-Aided Civil and Infrastructure Engineering, vol. 20, no. 6, pp. 408–423, 2005.
[12] J.-T. Kim, Y.-S. Ryu, H.-M. Cho, and N. Stubbs, “Damage identification in beam-type structures:
frequency-based method vs mode-shape-based method,” Engineering structures, vol. 25, no. 1, pp. 57–
67, 2003.
[13] J. B. Kosmatka and J. M. Ricles, “Damage detection in structures by modal vibration characterization,”
Journal of Structural Engineering, vol. 125, no. 12, pp. 1384–1392, 1999.
[14] C.-P. Fritzen, D. Jennewein, and T. Kiefer, “Damage detection based on model updating methods,”
Mechanical systems and signal processing, vol. 12, no. 1, pp. 163–186, 1998.
[15] C. Mares, B. Dratz, J. Mottershead, and M. Friswell, “Model updating using Bayesian estimation,” in International Conference on Noise and Vibration Engineering, ISMA2006, Katholieke Universiteit Leuven, 2006, pp. 18–20.
[16] C. Mares, J. Mottershead, and M. Friswell, “Stochastic model updating: part 1—theory and simulated example,” Mechanical Systems and Signal Processing, vol. 20, no. 7, pp. 1674–1695, 2006.
[17] J. Mottershead and M. Friswell, “Model updating in structural dynamics: a survey,” Journal of sound and vibration, vol. 167, no. 2, pp. 347–375, 1993.
[18] J. E. Mottershead, M. Link, and M. I. Friswell, “The sensitivity method in finite element model updating:
a tutorial,” Mechanical systems and signal processing, vol. 25, no. 7, pp. 2275–2296, 2011.
[19] J. L. Beck and S.-K. Au, “Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation,” Journal of Engineering Mechanics, vol. 128, no. 4, pp. 380–391, 2002.
[20] J. L. Beck and L. S. Katafygiotis, “Updating models and their uncertainties. I: Bayesian statistical framework,” Journal of Engineering Mechanics, vol. 124, no. 4, pp. 455–461, 1998.
[21] K. ONO, H. CHO, and M. Takuma, “THE ORIGIN OF CONTINUOUS EMISSIONS,” Journal of Acoustic Emission, vol. 23, p. 206, 2005.
[22] A. Rao, “Acoustic emission and signal analysis,” Defence Science Journal, vol. 40, no. 1, p. 55, 1990.
[23] W. K. Hastings, “Monte Carlo sampling methods using Markov chains and their applications,”
Biometrika, vol. 57, no. 1, pp. 97–109, 1970.
[24] N. Metropolis and S. Ulam, “The monte carlo method,” Journal of the American statistical association, vol. 44, pp. 335–341, 1949.
[25] S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” Pattern Analysis and Machine Intelligence, IEEE Transactions on, no. 6, pp. 721–741, 1984.
[26] E. I. George and R. E. McCulloch, “Variable selection via Gibbs sampling,” Journal of the American Statistical Association, vol. 88, no. 423, pp. 881–889, 1993.
[27] D. J. Lunn, A. Thomas, N. Best, and D. Spiegelhalter, “WinBUGS-a Bayesian modelling framework:
concepts, structure, and extensibility,” Statistics and computing, vol. 10, no. 4, pp. 325–337, 2000.
[28] Y.-H. Zhang and S. Maddox, “Development of fatigue testing of full-scale girth welded pipes under variable amplitude loading,” in ASME 2011 30th International Conference on Ocean, Offshore and Arctic Engineering, 2011, pp. 481–486.
[29] ASTM, “Standard for Leak Detection and Location using Surface-Mounted Acoustic Emission Sensors, E1211-02,” ASTM International, Book of Standards Vol. 03.03.
[30] ASTM, “Standard Practice for Continuous Monitoring of Acoustic Emission from Metal Pressure Boundaries, E1139-02,” ASTM International, Book of Standards Vol. 03.03.
[31] M. R. Gorman and W. H. Prosser, “AE source orientation by plate wave analysis,” J. Acoust. Soc. Am., vol. 90, 1991.
[32] C. Heiple and S. Carpenter, “Acoustic emission produced by deformation of metals and alloys-A review.,” Journal of Acoustic Emission, vol. 6, pp. 177–204, 1987.
[33] M. Huang, L. Jiang, P. K. Liaw, C. R. Brooks, R. Seeley, and D. L. Klarstrom, “Using acoustic emission in fatigue and fracture materials research,” JOM 50.11, vol. 1–14, 1998.
