adaptive dispersion compensation for guided wave imaging
ADAPTIVE DISPERSION COMPENSATION FOR GUIDED WAVE IMAGING James S. Hall and Jennifer E. Michaels School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250
ABSTRACT. Ultrasonic guided waves offer the promise of fast and reliable methods for interrogating large, plate-like structures. Distributed arrays of permanently attached, inexpensive piezoelectric transducers have thus been proposed as a cost-effective means to excite and measure ultrasonic guided waves for structural health monitoring (SHM) applications. Guided wave data recorded from a distributed array of transducers are often analyzed and interpreted through the use of guided wave imaging algorithms, such as conventional delay-and-sum imaging or the more recently applied minimum variance imaging. Both imaging algorithms perform reasonably well using signal envelopes, but can exhibit significant performance improvements when phase information is used. However, the use of phase information inherently requires knowledge of the dispersion relations, which are often not known to a sufficient degree of accuracy for high quality imaging since they are very sensitive to environmental conditions such as temperature, pressure, and loading. This work seeks to perform improved imaging with phase information by leveraging adaptive dispersion estimates obtained from in situ measurements. Experimentally obtained data from a distributed array is used to validate the proposed approach. Keywords: Ultrasonic, Lamb Waves, Nondestructive Evaluation, Structural Health Monitoring, Distributed Arrays, Sparse Arrays, Array Geometry, Dispersion PACS: 43.35.Zc, 43.40.Le, 43.60.Fg, 43.60.Lq, 43.60.Mn, 43.60.Uv, 81.70.Cv.
INTRODUCTION Ultrasonic guided waves represent a promising tool for structural health monitoring (SHM) applications. Guided waves are sensitive to both surface and subsurface features, and maintain this sensitivity over long propagation distances, making them particularly attractive for damage detection in large, plate-like structures such as aircraft skins, storage tank walls, ship hulls, etc. By permanently attaching multiple, inexpensive piezoelectric transducers throughout the structure, a large aperture distributed (or sparse) array can be created that presents a cost-effective method to perform both damage detection and localization throughout the structure. Guided wave imaging algorithms convert the data from a sensor array, such as the distributed arrays considered here, into a graphical format, which can then be analyzed for both damage detection and localization. Several guided wave imaging algorithms have been developed for this purpose [1-3]; however, this paper will focus almost exclusively on minimum variance imaging [4]. Minimum variance imaging is closely related to the more conventional delay-and-sum imaging, in that the algorithm can be described as delay-andReview of Progress in Quantitative Nondestructive Evaluation AIP Conf. Proc. 1430, 623-630 (2012); doi: 10.1063/1.4716285 © 2012 American Institute of Physics 978-0-7354-1013-8/$30.00
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sum imaging with carefully chosen weighting coefficients. Alternatively, minimum variance imaging degenerates to conventional imaging when the regularization factor is taken to have an infinite value. Although both imaging algorithms have been shown to be capable of detecting and locating damage in isotropic aluminum plates [4], quasi-isotropic carbon-fiber reinforced plastics [5], and even complex structures [6], the experimental examples have been restricted to using the envelope of narrowband signals with a center frequency chosen in a minimally dispersive frequency range. Such operating limitations are largely related to an inability to accurately account for the phase of the recorded data, which is influenced by several factors: transducer transfer functions (encompassing guided wave setup as well as excitation and measurement transduction), dispersion (propagation effects), and scattering. All three of these factors change in response to the operating environment, such as ambient temperature, pressure, and loading, further complicating attempts to account for these operational parameters. By using the signal envelope for imaging, phase information is discarded and these factors become manageable. Even with the use of signal envelopes, operation with narrowband signals in a minimally dispersive frequency regime is still necessary to minimize wave packet spreading [7]. Simulated, non-dispersive data were used in [4] to demonstrate that significant imaging improvement is possible when phase information is incorporated into the imaging algorithm. For experimental data, however, this requires that the imaging algorithm account for all factors that affect the phase of the signal, which in turn requires an accurate estimate of these operational parameters. Since the system parameters vary with environmental conditions, it is important to estimate them at the time of test. Recently, the model-based parameter estimation (MBPE) algorithm was introduced [8], enabling a distributed array to adaptively characterize the propagation environment. Parameters such as dispersion relations, transducer transfer functions, propagation loss, and relative sensor distances can be estimated using in situ measurements from the array at the time of test. The measured data can then be compensated by deconvolving the signals with the transducer transfer function estimates and performing dispersion compensation with the frequency-wavenumber (ω-k) mapping algorithm [9]. This paper demonstrates with experimental data the significant imaging performance that can be achieved by incorporating phase information into the imaging algorithm and the necessity for adaptive parameter compensation. First, conventional delay-and-sum and minimum variance imaging results are shown for the case when signal envelopes are used to discard phase information. Minimum variance imaging is then performed with phase information to motivate the need for parameter compensation. Adaptive estimates of dispersion relations and transducer transfer functions are obtained with the MBPE algorithm, and the steps required to compensate for these parameters are introduced. Finally, imaging results are presented with phase information using the conditioned, experimental signals followed by a brief summary and conclusion. GUIDED WAVE IMAGING WITHOUT PARAMETER COMPENSATION Imaging artifacts are an inherent part of guided wave imaging. These artifacts are due to a wide range of factors, including imperfect baseline subtraction, mode conversion, and reflections from geometric features such as boundaries. Minimum variance imaging is a derivative of conventional delay-and-sum imaging in which the weighting coefficients are chosen to (1) maximize the pixel value when damage is present and (2) minimize the pixel value when damage is absent. In the interest of space, technical details are not presented here; instead readers are referred to [4,10] for additional information.
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(a) (b) FIGURE 1. (Color online) Conventional delay-and-sum image using the envelope of experimental timedomain data [11]. (a) Pixel values are shown on a 20 dB color scale normalized so that maximum pixel value in vicinity of 5 mm diameter through-hole (white ‘+’ symbol) is 0 dB. Sensors are indicated with white ‘o’ symbols. (b) Normalized pixel values are shown on linear scale arranged as function of distance from known damage location with exponential fit shown in black.
(a) (b) FIGURE 2. (Color online) Minimum variance image using the envelope of experimental time-domain data [11].
To provide a quantitative comparison between imaging results, the performance metric introduced in [4] will be used. This performance metric is obtained by computing an exponential least-squares fit to the pixel values expressed as a function of distance from the known damage location. The metric provides a single value with which to compare images, and simultaneously incorporates the quantity of imaging artifacts as well as their magnitude and proximity to the known damage location. Figures 1 and 2 illustrate experimental imaging results for both conventional delayand-sum imaging and minimum variance imaging, respectively, when the envelopes of the differenced signals are used to image a structure. The dimensions of the 6061 aluminum interrogation structure are 914 × 914 × 3.175 mm. Six piezoelectric transducers have been attached to the structure in an arbitrary pattern, as indicated by white ‘o’ markers in the left-hand images. The excitation signal was a 7-cycle Hanning-windowed sinusoid centered at 300 kHz, which produces an S0 dominant Lamb wave in this material. A 5 mm through-hole was drilled in the top-left corner of the plate, which is designated by a white ‘+’ marker. Figures 1(a) and 2(a) are color-coded on a 20 dB scale normalized such that the maximum pixel value located within a 15 mm radius of the known damage location has a pixel value of 0 dB. Figures 1(b) and 2(b) depict the pixel values on a linear scale expressed as a function of distance from the known damage location along with the performance metrics for each image, which are 3.31 and 7.86, respectively.
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(a) (b) FIGURE 3. (Color online) Minimum variance image using experimental time-domain data (with phase information) [11].
Figures 1 and 2 highlight the dramatic imaging performance improvements that minimum variance imaging offers over conventional delay-and-sum imaging. Minimum variance imaging has significantly decreased the magnitude of the imaging artifacts throughout the structure. Even so, it is desirable to further reduce the imaging artifacts since they directly affect a system’s ability to accurately detect and locate defects. In [4] significant imaging improvements were demonstrated by using the phase information in the imaging algorithm. When the same approach is used with experimental data, however, the resulting images are of limited utility, as shown in Figure 3. In Figure 3, a large number of imaging artifacts are present that far exceed the pixel value at the known damage location. This result is primarily due to the fact that the imaging algorithm is not accounting for all of the factors affecting the phase of the signal, specifically dispersion, and, to a lesser degree, variations between transducers. As mentioned in the introduction, however, an accurate estimate of these parameters must be obtained in order to compensate for them. PARAMETER ESTIMATION AND COMPENSATION Model-based parameter estimation (MBPE) is an algorithm that is capable of adaptively estimating dispersion relations, transducer transfer functions, relative transducer distances, and propagation loss using direct arrivals recorded from the distributed array at the time of test. By estimating these parameters at the time of test and minimizing the amount of a priori information used in the algorithm, the estimated parameters are likely to be more accurate than either measurements made at some other point in time or potentially erroneous assumptions. Additionally, by using the MBPE algorithm, no additional equipment or measurements are required to obtain the parameter estimates. As with minimum variance imaging, technical details are omitted here in the interest of space. Readers are referred to [8] for the algorithmic details of MBPE. The MBPE algorithm was performed using two distinct sets of assumptions. First, parameter estimates were computed under the assumption that all transducers are identical. Then, the algorithm was repeated by allowing the transducers to have unique transducer transfer functions. When all transducers are assumed identical, the combined transmitterreceiver transfer function obtained from the MBPE algorithm is depicted as a thick red line in Figure 4(a). This can be contrasted with the 15 individual transfer functions (shown as thin blue lines) obtained when the transducers are allowed to vary. Although the transfer functions are very similar, there are clear variations in both magnitude and phase.
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(a) (b) FIGURE 4. (Color online) MBPE results under the assumption that all transducer are identical (thick red line) and that each transducer is independent of the others (thin blue lines) [11]. (a) Estimated transducer transfer functions. (b) Phase velocity estimates.
Figure 4(b) compares the phase velocity estimates under each of the assumptions along with the nominal dispersion curves computed by solving the Rayleigh-Lamb equations. Interestingly, regardless of assumptions regarding the transducers, the phase velocity estimates are in agreement and deviate from the nominal value by ~1%. To underscore the significance of this discrepancy, the 1% difference in phase velocity shown here will produce an 180o phase difference in a signal propagating just over 900 mm. Once estimated, the effects of each parameter can be removed from the measured data. The transducer transfer functions shown in Figure 4(a) can be removed from each differenced signal through deconvolution. For this paper, deconvolution is performed with frequency-domain division and additional band-pass filtering to address numerical instabilities that result from division by small numbers. Dispersion compensation is performed using a frequency-wavenumber (ω-k) mapping algorithm [9] that converts dispersive time-domain signals to the non-dispersive distance-domain. The (ω-k) mapping algorithm is an attractive dispersion compensation method because of its computational efficiency, particularly for this application. The algorithm requires a Fourier transform, a resampling operation, and an inverse Fourier transform for each transducer pair. This is a dramatic improvement over more conventional frequencydomain shifting operations that require a separate inverse Fourier transform for each transducer pair at each pixel location. GUIDED WAVE IMAGING WITH PARAMETER COMPENSATION Elliptical guided wave imaging algorithms, such as minimum variance imaging, are easily adapted to use distance-domain signals. With time-domain elliptical imaging, each pixel value is computed using a shifted version of the time-domain signals. The time shifts are a function of the total propagation time required for a signal to propagate from transmitter to pixel location to receiver (total propagation distance divided by group velocity). With distance-domain signals, however, the shift is simply the total propagation distance itself. This approach is identical to that used in [12] for dispersion compensation of data from a compact guided wave array. Figure 5 demonstrates minimum variance imaging with phase information when distance-domain signals are used. The distance domain signals used to generate Figure 5 were obtained from the (ω-k) mapping algorithm using nominal dispersion estimates (gray line in Figure 4(b)). The defect location is clearly identifiable and imaging artifacts have
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been significantly reduced in comparison to Figure 3, which used dispersive time-domain signals. It should be noted that although the overall artifact level in Figure 5(b) appears larger than that of Figure 2(b), the performance metric for Figure 5 is actually higher than that of Figure 2. This a reflection of the choice of performance metric and is due to the fact that Figure 5 provides significantly better resolution at the defect location and the nonartifact pixels of Figure 5 (with magnitude less than –20 dB) are generally much lower than those of Figure 2. Figure 6 represents minimum variance imaging when dispersion compensation is performed with the adaptive dispersion estimates obtained from the MBPE algorithm. In addition to a high-resolution estimate of the defect location, the overall artifact level from Figure 5 has been significantly reduced and the performance metric increased from 8.76 to 10.73. It should be pointed out that the overall absolute artifact level of Figure 6 is actually very similar to that of Figure 5; however, the improved accuracy of the dispersion estimate has increased the pixel value at the defect location, which is responsible for the imaging improvements.
(a) (b) FIGURE 5. (Color online) Minimum variance image using dispersion-compensated distance-domain data (with phase information) [11]. Dispersion compensation was performed using nominal dispersion relations.
(a) (b) FIGURE 6. (Color online) Minimum variance image using dispersion-compensated distance-domain data (with phase information) [11]. Dispersion compensation was performed using an adaptive estimate of the dispersion relations obtained with the MBPE algorithm.
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(a) (b) FIGURE 7. (Color online) Minimum variance image using dispersion-compensated distance-domain data (with phase information) after deconvolution with transducer transfer functions [11]. Dispersion compensation and deconvolution were performed using adaptive estimates of the dispersion relations and transducer transfer functions obtained with the MBPE algorithm.
Additional improvement in imaging performance can be achieved by compensating for the estimated transducer transfer functions through deconvolution. This compensation step accounts for variation between transducers, which can be seen in the slight magnitude and phase discrepancies of the blue lines in Figure 4. The minimum variance image generated with both adaptive dispersion and transducer transfer function estimates is shown in Figure 7. Figure 7 represents only an incremental improvement over Figure 6, increasing the performance metric by 0.75, but does hint at the potential impact of transducer variations. A close inspection of Figures 6 and 7 reveals that the peak location of the defect in the image has shifted towards the known damage location. This small improvement is reasonable, since the transducer transfer functions in Figure 4 are so similar. For systems with larger variability between transducers, the improvement achieved by compensating for the transfer functions can be expected to increase proportionately. SUMMARY AND CONCLUSIONS This paper has explored the use of phase information in guided wave imaging applications and the need for adaptive parameter compensation. Minimum variance imaging was first compared to conventional delay-and-sum imaging using the envelope of the measured, differenced signals. Next, phase information was incorporated into the imaging algorithm, resulting in degraded imaging performance. MPBE was then employed to adaptively estimate the dispersion relations and transducer transfer functions using in situ measurements from the distributed array at the time of test. Finally, minimum variance imaging results were presented using parameter-compensated signals with both nominal and adaptive parameter estimates obtained with the MBPE algorithm. The tradeoff when minimum variance imaging is used over conventional delay-andsum imaging, and the subsequent use of phase information in the imaging algorithm, is a greater reliance on the accuracy of the assumed propagation model. Although errors in propagation velocity (group or phase) result in image degradation for both imaging algorithms, the effects of small errors are more pronounced with minimum variance imaging. In this context, the MBPE algorithm represents an enabling technology for robust minimum variance imaging and imaging with phase information, since it is currently the
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only method available to obtain parameter estimates, such as dispersion relations, from a distributed sparse array using in situ measurements obtained at the time of test. This combination of MBPE and minimum variance imaging represents a significant step forward towards robust, reliable damage detection and localization with distributed arrays using ultrasonic guided waves. Future work remains in incorporating additional adaptive parameter estimates from the MBPE algorithm, such as propagation distances and propagation loss, into the imaging algorithm and evaluating performance in more complex structures and materials. ACKNOWLEDGEMENTS The authors would like to thank NASA’s Graduate Student Research Program (GSRP) for their support through Grant No. NNX08AY93H, and the Air Force Office of Scientific Research (AFOSR) for their support through Grant No. FA9550-08-1-0241. REFERENCES 1. J. E. Michaels, "Detection, localization and characterization of damage in plates with an in situ array of spatially distributed sensors," Smart Mater. Struct., 17 (3) p 035035 (2008). 2. A. J. Croxford, P. D. Wilcox, and B. W. Drinkwater, "Guided wave SHM with a distributed sensor network," Proc. SPIE, 6935 (69350E) pp. 1-9 (2008). 3. C. H. Wang, J. T. Rose, and F.-K. Chang, "A synthetic time-reversal imaging method for structural health monitoring," Smart Mater. Struct., 13 (2) pp. 415-423 (2004). 4. J. S. Hall and J. E. Michaels, "Minimum variance ultrasonic imaging applied to an in situ sparse guided wave array," IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 57 (10) pp. 2311-2323 (2010). 5. J. S. Hall, P. McKeon, L. Satyanarayan, J. E. Michaels, N. F. Declercq, and Y. H. Berthelot, "Minimum variance guided wave imaging in a quasi-isotropic composite plate," Smart Mater. Struct., 20 (2) p 025013 (2011). 6. T. Clarke, P. Cawley, P. D. Wilcox, and A. J. Croxford, "Evaluation of the damage detection capability of a sparse-array guided-wave SHM system applied to a complex structure under varying thermal conditions," IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 56 (12) pp. 2666-2678 (2009). 7. P. Wilcox, M. Lowe, and P. Cawley, "The effect of dispersion on long-range inspection using ultrasonic guided waves," NDT & E Int., 34 (1) pp. 1-9 (2001). 8. J. S. Hall and J. E. Michaels, "Model-based parameter estimation for characterizing wave propagation in a homogeneous medium," Inverse Prob., 27 (3) p 035002 (2011). 9. P. D. Wilcox, "A rapid signal processing technique to remove the effect of dispersion from guided wave signals," IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 50 (4) pp. 419-427 (2003). 10. J. S. Hall and J. E. Michaels, "Computational efficiency of ultrasonic guided wave imaging algorithms," IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 58 (1) pp. 244-248 (2011). 11. J. S. Hall, Adaptive Dispersion Compensation and Ultrasonic Imaging for Structural Health Monitoring. Ph.D. Dissertation, Georgia Institute of Technology, 2011. 12. P. D. Wilcox, "Omni-directional guided wave transducer arrays for the rapid inspection of large areas of plate structures," IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 50 (6) pp. 699-709 (2003).
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ABSTRACT. Ultrasonic guided waves offer the promise of fast and reliable methods for interrogating large, plate-like structures. Distributed arrays of permanently attached, inexpensive piezoelectric transducers have thus been proposed as a cost-effective means to excite and measure ultrasonic guided waves for structural health monitoring (SHM) applications. Guided wave data recorded from a distributed array of transducers are often analyzed and interpreted through the use of guided wave imaging algorithms, such as conventional delay-and-sum imaging or the more recently applied minimum variance imaging. Both imaging algorithms perform reasonably well using signal envelopes, but can exhibit significant performance improvements when phase information is used. However, the use of phase information inherently requires knowledge of the dispersion relations, which are often not known to a sufficient degree of accuracy for high quality imaging since they are very sensitive to environmental conditions such as temperature, pressure, and loading. This work seeks to perform improved imaging with phase information by leveraging adaptive dispersion estimates obtained from in situ measurements. Experimentally obtained data from a distributed array is used to validate the proposed approach. Keywords: Ultrasonic, Lamb Waves, Nondestructive Evaluation, Structural Health Monitoring, Distributed Arrays, Sparse Arrays, Array Geometry, Dispersion PACS: 43.35.Zc, 43.40.Le, 43.60.Fg, 43.60.Lq, 43.60.Mn, 43.60.Uv, 81.70.Cv.
INTRODUCTION Ultrasonic guided waves represent a promising tool for structural health monitoring (SHM) applications. Guided waves are sensitive to both surface and subsurface features, and maintain this sensitivity over long propagation distances, making them particularly attractive for damage detection in large, plate-like structures such as aircraft skins, storage tank walls, ship hulls, etc. By permanently attaching multiple, inexpensive piezoelectric transducers throughout the structure, a large aperture distributed (or sparse) array can be created that presents a cost-effective method to perform both damage detection and localization throughout the structure. Guided wave imaging algorithms convert the data from a sensor array, such as the distributed arrays considered here, into a graphical format, which can then be analyzed for both damage detection and localization. Several guided wave imaging algorithms have been developed for this purpose [1-3]; however, this paper will focus almost exclusively on minimum variance imaging [4]. Minimum variance imaging is closely related to the more conventional delay-and-sum imaging, in that the algorithm can be described as delay-andReview of Progress in Quantitative Nondestructive Evaluation AIP Conf. Proc. 1430, 623-630 (2012); doi: 10.1063/1.4716285 © 2012 American Institute of Physics 978-0-7354-1013-8/$30.00
623 Downloaded 07 Jun 2012 to 130.207.50.37. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
sum imaging with carefully chosen weighting coefficients. Alternatively, minimum variance imaging degenerates to conventional imaging when the regularization factor is taken to have an infinite value. Although both imaging algorithms have been shown to be capable of detecting and locating damage in isotropic aluminum plates [4], quasi-isotropic carbon-fiber reinforced plastics [5], and even complex structures [6], the experimental examples have been restricted to using the envelope of narrowband signals with a center frequency chosen in a minimally dispersive frequency range. Such operating limitations are largely related to an inability to accurately account for the phase of the recorded data, which is influenced by several factors: transducer transfer functions (encompassing guided wave setup as well as excitation and measurement transduction), dispersion (propagation effects), and scattering. All three of these factors change in response to the operating environment, such as ambient temperature, pressure, and loading, further complicating attempts to account for these operational parameters. By using the signal envelope for imaging, phase information is discarded and these factors become manageable. Even with the use of signal envelopes, operation with narrowband signals in a minimally dispersive frequency regime is still necessary to minimize wave packet spreading [7]. Simulated, non-dispersive data were used in [4] to demonstrate that significant imaging improvement is possible when phase information is incorporated into the imaging algorithm. For experimental data, however, this requires that the imaging algorithm account for all factors that affect the phase of the signal, which in turn requires an accurate estimate of these operational parameters. Since the system parameters vary with environmental conditions, it is important to estimate them at the time of test. Recently, the model-based parameter estimation (MBPE) algorithm was introduced [8], enabling a distributed array to adaptively characterize the propagation environment. Parameters such as dispersion relations, transducer transfer functions, propagation loss, and relative sensor distances can be estimated using in situ measurements from the array at the time of test. The measured data can then be compensated by deconvolving the signals with the transducer transfer function estimates and performing dispersion compensation with the frequency-wavenumber (ω-k) mapping algorithm [9]. This paper demonstrates with experimental data the significant imaging performance that can be achieved by incorporating phase information into the imaging algorithm and the necessity for adaptive parameter compensation. First, conventional delay-and-sum and minimum variance imaging results are shown for the case when signal envelopes are used to discard phase information. Minimum variance imaging is then performed with phase information to motivate the need for parameter compensation. Adaptive estimates of dispersion relations and transducer transfer functions are obtained with the MBPE algorithm, and the steps required to compensate for these parameters are introduced. Finally, imaging results are presented with phase information using the conditioned, experimental signals followed by a brief summary and conclusion. GUIDED WAVE IMAGING WITHOUT PARAMETER COMPENSATION Imaging artifacts are an inherent part of guided wave imaging. These artifacts are due to a wide range of factors, including imperfect baseline subtraction, mode conversion, and reflections from geometric features such as boundaries. Minimum variance imaging is a derivative of conventional delay-and-sum imaging in which the weighting coefficients are chosen to (1) maximize the pixel value when damage is present and (2) minimize the pixel value when damage is absent. In the interest of space, technical details are not presented here; instead readers are referred to [4,10] for additional information.
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(a) (b) FIGURE 1. (Color online) Conventional delay-and-sum image using the envelope of experimental timedomain data [11]. (a) Pixel values are shown on a 20 dB color scale normalized so that maximum pixel value in vicinity of 5 mm diameter through-hole (white ‘+’ symbol) is 0 dB. Sensors are indicated with white ‘o’ symbols. (b) Normalized pixel values are shown on linear scale arranged as function of distance from known damage location with exponential fit shown in black.
(a) (b) FIGURE 2. (Color online) Minimum variance image using the envelope of experimental time-domain data [11].
To provide a quantitative comparison between imaging results, the performance metric introduced in [4] will be used. This performance metric is obtained by computing an exponential least-squares fit to the pixel values expressed as a function of distance from the known damage location. The metric provides a single value with which to compare images, and simultaneously incorporates the quantity of imaging artifacts as well as their magnitude and proximity to the known damage location. Figures 1 and 2 illustrate experimental imaging results for both conventional delayand-sum imaging and minimum variance imaging, respectively, when the envelopes of the differenced signals are used to image a structure. The dimensions of the 6061 aluminum interrogation structure are 914 × 914 × 3.175 mm. Six piezoelectric transducers have been attached to the structure in an arbitrary pattern, as indicated by white ‘o’ markers in the left-hand images. The excitation signal was a 7-cycle Hanning-windowed sinusoid centered at 300 kHz, which produces an S0 dominant Lamb wave in this material. A 5 mm through-hole was drilled in the top-left corner of the plate, which is designated by a white ‘+’ marker. Figures 1(a) and 2(a) are color-coded on a 20 dB scale normalized such that the maximum pixel value located within a 15 mm radius of the known damage location has a pixel value of 0 dB. Figures 1(b) and 2(b) depict the pixel values on a linear scale expressed as a function of distance from the known damage location along with the performance metrics for each image, which are 3.31 and 7.86, respectively.
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(a) (b) FIGURE 3. (Color online) Minimum variance image using experimental time-domain data (with phase information) [11].
Figures 1 and 2 highlight the dramatic imaging performance improvements that minimum variance imaging offers over conventional delay-and-sum imaging. Minimum variance imaging has significantly decreased the magnitude of the imaging artifacts throughout the structure. Even so, it is desirable to further reduce the imaging artifacts since they directly affect a system’s ability to accurately detect and locate defects. In [4] significant imaging improvements were demonstrated by using the phase information in the imaging algorithm. When the same approach is used with experimental data, however, the resulting images are of limited utility, as shown in Figure 3. In Figure 3, a large number of imaging artifacts are present that far exceed the pixel value at the known damage location. This result is primarily due to the fact that the imaging algorithm is not accounting for all of the factors affecting the phase of the signal, specifically dispersion, and, to a lesser degree, variations between transducers. As mentioned in the introduction, however, an accurate estimate of these parameters must be obtained in order to compensate for them. PARAMETER ESTIMATION AND COMPENSATION Model-based parameter estimation (MBPE) is an algorithm that is capable of adaptively estimating dispersion relations, transducer transfer functions, relative transducer distances, and propagation loss using direct arrivals recorded from the distributed array at the time of test. By estimating these parameters at the time of test and minimizing the amount of a priori information used in the algorithm, the estimated parameters are likely to be more accurate than either measurements made at some other point in time or potentially erroneous assumptions. Additionally, by using the MBPE algorithm, no additional equipment or measurements are required to obtain the parameter estimates. As with minimum variance imaging, technical details are omitted here in the interest of space. Readers are referred to [8] for the algorithmic details of MBPE. The MBPE algorithm was performed using two distinct sets of assumptions. First, parameter estimates were computed under the assumption that all transducers are identical. Then, the algorithm was repeated by allowing the transducers to have unique transducer transfer functions. When all transducers are assumed identical, the combined transmitterreceiver transfer function obtained from the MBPE algorithm is depicted as a thick red line in Figure 4(a). This can be contrasted with the 15 individual transfer functions (shown as thin blue lines) obtained when the transducers are allowed to vary. Although the transfer functions are very similar, there are clear variations in both magnitude and phase.
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(a) (b) FIGURE 4. (Color online) MBPE results under the assumption that all transducer are identical (thick red line) and that each transducer is independent of the others (thin blue lines) [11]. (a) Estimated transducer transfer functions. (b) Phase velocity estimates.
Figure 4(b) compares the phase velocity estimates under each of the assumptions along with the nominal dispersion curves computed by solving the Rayleigh-Lamb equations. Interestingly, regardless of assumptions regarding the transducers, the phase velocity estimates are in agreement and deviate from the nominal value by ~1%. To underscore the significance of this discrepancy, the 1% difference in phase velocity shown here will produce an 180o phase difference in a signal propagating just over 900 mm. Once estimated, the effects of each parameter can be removed from the measured data. The transducer transfer functions shown in Figure 4(a) can be removed from each differenced signal through deconvolution. For this paper, deconvolution is performed with frequency-domain division and additional band-pass filtering to address numerical instabilities that result from division by small numbers. Dispersion compensation is performed using a frequency-wavenumber (ω-k) mapping algorithm [9] that converts dispersive time-domain signals to the non-dispersive distance-domain. The (ω-k) mapping algorithm is an attractive dispersion compensation method because of its computational efficiency, particularly for this application. The algorithm requires a Fourier transform, a resampling operation, and an inverse Fourier transform for each transducer pair. This is a dramatic improvement over more conventional frequencydomain shifting operations that require a separate inverse Fourier transform for each transducer pair at each pixel location. GUIDED WAVE IMAGING WITH PARAMETER COMPENSATION Elliptical guided wave imaging algorithms, such as minimum variance imaging, are easily adapted to use distance-domain signals. With time-domain elliptical imaging, each pixel value is computed using a shifted version of the time-domain signals. The time shifts are a function of the total propagation time required for a signal to propagate from transmitter to pixel location to receiver (total propagation distance divided by group velocity). With distance-domain signals, however, the shift is simply the total propagation distance itself. This approach is identical to that used in [12] for dispersion compensation of data from a compact guided wave array. Figure 5 demonstrates minimum variance imaging with phase information when distance-domain signals are used. The distance domain signals used to generate Figure 5 were obtained from the (ω-k) mapping algorithm using nominal dispersion estimates (gray line in Figure 4(b)). The defect location is clearly identifiable and imaging artifacts have
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been significantly reduced in comparison to Figure 3, which used dispersive time-domain signals. It should be noted that although the overall artifact level in Figure 5(b) appears larger than that of Figure 2(b), the performance metric for Figure 5 is actually higher than that of Figure 2. This a reflection of the choice of performance metric and is due to the fact that Figure 5 provides significantly better resolution at the defect location and the nonartifact pixels of Figure 5 (with magnitude less than –20 dB) are generally much lower than those of Figure 2. Figure 6 represents minimum variance imaging when dispersion compensation is performed with the adaptive dispersion estimates obtained from the MBPE algorithm. In addition to a high-resolution estimate of the defect location, the overall artifact level from Figure 5 has been significantly reduced and the performance metric increased from 8.76 to 10.73. It should be pointed out that the overall absolute artifact level of Figure 6 is actually very similar to that of Figure 5; however, the improved accuracy of the dispersion estimate has increased the pixel value at the defect location, which is responsible for the imaging improvements.
(a) (b) FIGURE 5. (Color online) Minimum variance image using dispersion-compensated distance-domain data (with phase information) [11]. Dispersion compensation was performed using nominal dispersion relations.
(a) (b) FIGURE 6. (Color online) Minimum variance image using dispersion-compensated distance-domain data (with phase information) [11]. Dispersion compensation was performed using an adaptive estimate of the dispersion relations obtained with the MBPE algorithm.
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(a) (b) FIGURE 7. (Color online) Minimum variance image using dispersion-compensated distance-domain data (with phase information) after deconvolution with transducer transfer functions [11]. Dispersion compensation and deconvolution were performed using adaptive estimates of the dispersion relations and transducer transfer functions obtained with the MBPE algorithm.
Additional improvement in imaging performance can be achieved by compensating for the estimated transducer transfer functions through deconvolution. This compensation step accounts for variation between transducers, which can be seen in the slight magnitude and phase discrepancies of the blue lines in Figure 4. The minimum variance image generated with both adaptive dispersion and transducer transfer function estimates is shown in Figure 7. Figure 7 represents only an incremental improvement over Figure 6, increasing the performance metric by 0.75, but does hint at the potential impact of transducer variations. A close inspection of Figures 6 and 7 reveals that the peak location of the defect in the image has shifted towards the known damage location. This small improvement is reasonable, since the transducer transfer functions in Figure 4 are so similar. For systems with larger variability between transducers, the improvement achieved by compensating for the transfer functions can be expected to increase proportionately. SUMMARY AND CONCLUSIONS This paper has explored the use of phase information in guided wave imaging applications and the need for adaptive parameter compensation. Minimum variance imaging was first compared to conventional delay-and-sum imaging using the envelope of the measured, differenced signals. Next, phase information was incorporated into the imaging algorithm, resulting in degraded imaging performance. MPBE was then employed to adaptively estimate the dispersion relations and transducer transfer functions using in situ measurements from the distributed array at the time of test. Finally, minimum variance imaging results were presented using parameter-compensated signals with both nominal and adaptive parameter estimates obtained with the MBPE algorithm. The tradeoff when minimum variance imaging is used over conventional delay-andsum imaging, and the subsequent use of phase information in the imaging algorithm, is a greater reliance on the accuracy of the assumed propagation model. Although errors in propagation velocity (group or phase) result in image degradation for both imaging algorithms, the effects of small errors are more pronounced with minimum variance imaging. In this context, the MBPE algorithm represents an enabling technology for robust minimum variance imaging and imaging with phase information, since it is currently the
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only method available to obtain parameter estimates, such as dispersion relations, from a distributed sparse array using in situ measurements obtained at the time of test. This combination of MBPE and minimum variance imaging represents a significant step forward towards robust, reliable damage detection and localization with distributed arrays using ultrasonic guided waves. Future work remains in incorporating additional adaptive parameter estimates from the MBPE algorithm, such as propagation distances and propagation loss, into the imaging algorithm and evaluating performance in more complex structures and materials. ACKNOWLEDGEMENTS The authors would like to thank NASA’s Graduate Student Research Program (GSRP) for their support through Grant No. NNX08AY93H, and the Air Force Office of Scientific Research (AFOSR) for their support through Grant No. FA9550-08-1-0241. REFERENCES 1. J. E. Michaels, "Detection, localization and characterization of damage in plates with an in situ array of spatially distributed sensors," Smart Mater. Struct., 17 (3) p 035035 (2008). 2. A. J. Croxford, P. D. Wilcox, and B. W. Drinkwater, "Guided wave SHM with a distributed sensor network," Proc. SPIE, 6935 (69350E) pp. 1-9 (2008). 3. C. H. Wang, J. T. Rose, and F.-K. Chang, "A synthetic time-reversal imaging method for structural health monitoring," Smart Mater. Struct., 13 (2) pp. 415-423 (2004). 4. J. S. Hall and J. E. Michaels, "Minimum variance ultrasonic imaging applied to an in situ sparse guided wave array," IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 57 (10) pp. 2311-2323 (2010). 5. J. S. Hall, P. McKeon, L. Satyanarayan, J. E. Michaels, N. F. Declercq, and Y. H. Berthelot, "Minimum variance guided wave imaging in a quasi-isotropic composite plate," Smart Mater. Struct., 20 (2) p 025013 (2011). 6. T. Clarke, P. Cawley, P. D. Wilcox, and A. J. Croxford, "Evaluation of the damage detection capability of a sparse-array guided-wave SHM system applied to a complex structure under varying thermal conditions," IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 56 (12) pp. 2666-2678 (2009). 7. P. Wilcox, M. Lowe, and P. Cawley, "The effect of dispersion on long-range inspection using ultrasonic guided waves," NDT & E Int., 34 (1) pp. 1-9 (2001). 8. J. S. Hall and J. E. Michaels, "Model-based parameter estimation for characterizing wave propagation in a homogeneous medium," Inverse Prob., 27 (3) p 035002 (2011). 9. P. D. Wilcox, "A rapid signal processing technique to remove the effect of dispersion from guided wave signals," IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 50 (4) pp. 419-427 (2003). 10. J. S. Hall and J. E. Michaels, "Computational efficiency of ultrasonic guided wave imaging algorithms," IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 58 (1) pp. 244-248 (2011). 11. J. S. Hall, Adaptive Dispersion Compensation and Ultrasonic Imaging for Structural Health Monitoring. Ph.D. Dissertation, Georgia Institute of Technology, 2011. 12. P. D. Wilcox, "Omni-directional guided wave transducer arrays for the rapid inspection of large areas of plate structures," IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 50 (6) pp. 699-709 (2003).
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