Characterization and prediction of guided wave propagation in complex media via compressive sampling in time and space

Guided waves are commonly used within the realms of nondestructive evaluation (NDE) and structural health monitoring (SHM). Their properties and characteristics are highly dependent on the medium in which they are propagating. In many practical applications, however, these properties are not known a priori and the amount of available data to estimate them can be very low. Consequently, it would be beneficial to several applications to be capable of retrieving guided wave properties from a limited set of experimental measurements for any structure of interest. It is specifically advantageous to identify and extract such information for complex media, such as anisotropic composite panels, that have found broad applications in modern structural engineering applications (such as aircrafts) but suffer from the unavailability of precise models of ultrasonic wave propagation. Therefore, this dissertation focuses on creating signal processing and machine learning strategies to analyze such complex structures without necessitating cumbersome, comprehensive data acquisition and processing, which can be costly and time-consuming. In the NDE and SHM literature, attempts at tackling this problem have resulted in several methodologies aimed at characterization, prediction, and reconstruction of guided waves from subsampled data. In particular, sparse wavenumber analysis (SWA) was proposed to recover dispersive properties of guided waves in an infinite, isotropic plate. Nevertheless, the restrictive assumptions in the SWA model render it ineffective when applied to more complex anisotropic structures. Moreover, its compressive capabilities can only be applied to the spatial dimension. In this dissertation, we introduce a number of extensions and modifications to the SWA model, intended to address the aforementioned shortcomings. More specifically, we modify the model such that it is capable of recovering information in the temporal dimension. We then extend the model into two dimensions to account for variations in wave velocities with respect to the direction of propagation. Finally, we combine these two notions to create a framework in which information retrieval is feasible in the presence of incomplete data in both time and space. In addition, attempts at characterization and reconstruction of guided waves with multipath reflections, as well as improving the two-dimensional model by incorporating information in the polar coordinates, are presented. iv

CHARACTERIZATION AND PREDICTION OF GUIDED WAVE PROPAGATION IN COMPLEX MEDIA VIA COMPRESSIVE SAMPLING IN TIME AND SPACE by Soroush Sabeti A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical and Computer Engineering The University of Utah May 2020 Copyright c Soroush Sabeti 2020 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Soroush Sabeti has been approved by the following supervisory committee members: Joel B. Harley , Chair(s) 12/12/2019 Date Approved Behrouz Farhang , Member 12/12/2019 Date Approved Rong Rong Chen , Member 12/12/2019 Date Approved Tolga Tasdizen , Member 12/12/2019 Date Approved Aditya Bhaskara , Member 12/12/2019 Date Approved by Florian Solzbacher , Chair/Dean of the Department/College/School of Electrical and Computer Engineering and by David B. Keida , Dean of The Graduate School. ABSTRACT Guided waves are commonly used within the realms of nondestructive evaluation (NDE) and structural health monitoring (SHM). Their properties and characteristics are highly dependent on the medium in which they are propagating. In many practical applications, however, these properties are not known a priori and the amount of available data to estimate them can be very low. Consequently, it would be beneficial to several applications to be capable of retrieving guided wave properties from a limited set of experimental measurements for any structure of interest. It is specifically advantageous to identify and extract such information for complex media, such as anisotropic composite panels, that have found broad applications in modern structural engineering applications (such as aircrafts) but suffer from the unavailability of precise models of ultrasonic wave propagation. Therefore, this dissertation focuses on creating signal processing and machine learning strategies to analyze such complex structures without necessitating cumbersome, comprehensive data acquisition and processing, which can be costly and time-consuming. In the NDE and SHM literature, attempts at tackling this problem have resulted in several methodologies aimed at characterization, prediction, and reconstruction of guided waves from subsampled data. In particular, sparse wavenumber analysis (SWA) was proposed to recover dispersive properties of guided waves in an infinite, isotropic plate. Nevertheless, the restrictive assumptions in the SWA model render it ineffective when applied to more complex anisotropic structures. Moreover, its compressive capabilities can only be applied to the spatial dimension. In this dissertation, we introduce a number of extensions and modifications to the SWA model, intended to address the aforementioned shortcomings. More specifically, we modify the model such that it is capable of recovering information in the temporal dimension. We then extend the model into two dimensions to account for variations in wave velocities with respect to the direction of propagation. Finally, we combine these two notions to create a framework in which information retrieval is feasible in the presence of incomplete data in both time and space. In addition, attempts at characterization and reconstruction of guided waves with multipath reflections, as well as improving the two-dimensional model by incorporating information in the polar coordinates, are presented. iv To my parents CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv CHAPTERS 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Guided Waves in Nondestructive Evaluation and Structural Health Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Signal Processing Techniques in Guided Wave Structural Health Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Characterization of Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.4 Data Driven and Compressive Sensing Based Guided Wave Characterization and Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2. TWO-DIMENSIONAL SPARSE WAVENUMBER RECOVERY FOR GUIDED WAVEFIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 2D-OMP for Incomplete Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Guided Wavefield Reconstruction Through 2D Sparse Wavenumber Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Experimental and Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Experimental Setup (Isotropic Wavefield) . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Simulation Setup (Anisotropic Wavefield) . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Experimental Results (Isotropic Wavefield) . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 17 17 19 20 21 21 22 22 22 23 23 24 33 3. SPARSE WAVENUMBER RECOVERY AND PREDICTION OF ANISOTROPIC GUIDED WAVES IN COMPOSITES: A COMPARATIVE STUDY . . . . . . . . . . . 34 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Compressive Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Anisotropic Sparse Wavenumber Analysis (ASWA) . . . . . . . . . . . . . . . . . 3.3.2 Two-dimensional Sparse Wavenumber Analysis (2D-SWA) . . . . . . . . . . 3.3.3 Fourier Reconstruction (Interpolation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Two-Dimensional Spatial Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Defining the Maximum Wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Experimental and Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Dataset 1: Experimental Glass Fiber Reinforced Polymer . . . . . . . . . . . . . 3.4.2 Dataset 2: Simulated Carbon Fiber Reinforced Polymer . . . . . . . . . . . . . . 3.4.3 Dataset 3: Experimental Carbon Fiber Reinforced Polymer . . . . . . . . . . . 3.4.4 Metrics for Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Learning Wavenumber Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Wavefield Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Wavefield Reconstruction Versus Undersampling Ratio . . . . . . . . . . . . . 3.5.4 Summary of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. GUIDED WAVE RETRIEVAL FROM TEMPORALLY UNDERSAMPLED DATA 66 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Compressive Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Random Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Orthogonal Matching Pursuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Guided Wavefield Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Wavefield Reconstruction with Random Sampling . . . . . . . . . . . . . . . . . . 4.5.2 Wavefield Reconstruction with Uniform Sampling . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 35 37 37 38 39 39 42 42 43 44 44 45 46 46 47 48 48 49 49 50 51 52 63 66 67 67 68 68 70 70 70 71 72 72 72 73 73 79 SPATIOTEMPORAL UNDERSAMPLING: RECOVERING ULTRASONIC GUIDED WAVEFIELDS FROM INCOMPLETE DATA WITH COMPRESSIVE SENSING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 vii 5.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.1 Compressive Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2.2 Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2.3 Temporal Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2.4 Two-dimensional Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . 86 5.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3.1 Spatiotemporal Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . . . 87 5.3.2 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3.3 Nyquist Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3.4 Defining Maximum Frequency and Wavenumber . . . . . . . . . . . . . . . . . . 90 5.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5.1 Temporal and Spatial Nyquist Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.5.2 Guided Wave Reconstruction via ST-SWA . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.5.2.1 Scenario 1: Element-wise random sampling in spatiotemporal domain (Random) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.5.2.2 Scenario 2: Independent random sampling in time and in space (Time-Space) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.5.2.3 Scenario 3: Random sampling in time followed by random sampling in space (Time + Space) . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5.2.4 Scenario 4: Random sampling in space followed by random sampling in time (Space + Time) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5.2.5 Scenario 5: Jittered sampling in time and space (Jittered) . . . . . . . . 97 5.5.2.6 Scenario 6: Uniform sampling (Uniform) . . . . . . . . . . . . . . . . . . . . . 97 5.5.2.7 Scenario 7: Undersampling only in space (Pure-Space) . . . . . . . . . . 98 5.5.3 Effect of Sparsity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6. GUIDED WAVE CHARACTERIZATION AND SOURCE DETECTION IN THE PRESENCE OF MULTIPATH REFLECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.2.1 Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.2.2 Decomposition of Multipath Elements in Lamb Waves . . . . . . . . . . . . . . 123 6.2.3 Multipath Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.3.1 Two-Stage Multipath Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . 124 6.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.4.1 Dataset 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.4.2 Dataset 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.5.1 Multipath Identification with Dataset 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.5.2 Dispersion Curve Enhancement with MP-SWA . . . . . . . . . . . . . . . . . . . . 127 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 viii 7. POLAR SPARSE WAVENUMBER ANALYSIS FOR GUIDED WAVE RECONSTRUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2.1 Compressive Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2.2 Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2.3 Two-dimensional Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . 141 7.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.3.1 Fourier Reconstruction/Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.3.2 Polar Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.3.3 Deriving Nyquist Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.4 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.5.1 Nyquist Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.5.2 Guided Wave Reconstruction Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.5.3 A Learning Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.5.3.1 Effect of undersampling ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8. CONCLUSION AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.1 Conclusion, Summary, and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.2 The Path Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.2.1 Two-Dimensional Sparse Wavenumber Analysis with Different Bases . . 156 8.2.2 Two-Dimensional Dictionary Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.2.3 Bayesian Sparse Wavenumber Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.2.4 Spatiotemporal Sparse Wavenumber Analysis for Anisotropic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.2.5 A Comparative Study of Different Characterization and Reconstruction Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.2.6 Anisotropy Compensation to Facilitate Damage Detection . . . . . . . . . . . 158 8.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 ix LIST OF FIGURES 1.1 Time snapshots of wavefield 1: an isotropic wavefield (normalized to its maximum magnitude) with no multipath reflections. . . . . . . . . . . . . . . . . . . . . . 7 1.2 Recovered dispersion curves for wavefield 1 using SWA. . . . . . . . . . . . . . . . . . . 7 1.3 Time snapshots of wavefield 2: an isotropic wavefield (normalized to its maximum magnitude), which contains reflections from the edges of the plate in the scanning region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Recovered dispersion curves for wavefield 2 using SWA. . . . . . . . . . . . . . . . . . . 8 1.5 Time snapshots of wavefield 3: an anisotropic wavefield (normalized to its maximum magnitude), which contains reflections from the edges of the plate in the scanning region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Recovered dispersion curves for wavefield 3 using SWA. . . . . . . . . . . . . . . . . . . 9 2.1 2D-OMP algorithm for incomplete data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Guided wavefield recovery algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Recovered sparse representation of the experimental wavefield in k-space at a particular frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Reconstructed versus true experimental signal in time at a particular grid point (magnified for better comparison). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5 Reconstruction of the experimental data at different time instants (increasing from top to bottom) with approximately 97 % accuracy. . . . . . . . . . . . . . . . . . . . 29 2.6 Recovered sparse representation of the simulated wavefield in k-space at a particular frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.7 Reconstructed versus true simulation signal in time at a particular grid point (magnified for better comparison). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.8 Reconstructed versus true simulation signal in time at a particular grid point (magnified for better comparison). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.1 Example single-mode sparse representations for SWA (of an isotropic aluminum plate), 2D-SWA (of an anisotropic unidirectional composite plate), and ASWA (of an anisotropic unidirectional composite plate). . . . . . . . . . . . . . . 53 3.2 Guided wave reconstruction processes with (a) ASWA and (b) the 2D-SWA frameworks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Example two-dimensional wavenumber surfaces for datasets 1-3 (at 71 kHz, 206 kHz, and 200 kHz, respectively), determined by the ASWA model. The optimized shape factor (pn ) values for datasets 1, 2, and 3 are 1.83, 1.68, and 1.73, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 Two-dimensional wavenumber representation (summed over all frequency slices) of (a) dataset 1, (b) dataset 2, and (c) dataset 3. The dotted circles indicate the 99 % energy circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5 Example phase velocity surfaces for datasets 1-3 (at 71 kHz, 206 kHz, and 200 kHz, respectively), determined by the ASWA model. The optimized shape factor (pn ) values for datasets 1, 2, and 3 are 1.83, 1.68, and 1.73, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6 Example wavefield time snapshots from (a) dataset 1, (b) dataset 2, and (c) dataset 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.7 Sparse representations. (a)-(c) Example Fourier reconstruction wavenumber domain representations from dataset 1 (at 71 kHz) for three undersampling ratios (3%, 11%, and 45%, respectively). Wavenumber aliasing and the effects of the low-pass reconstruction filter are visible in the images. (d)-(f) Example 2D-SWA frequency-wavenumber domain representations from dataset 1 (for frequencies from 46 kHz to 93 kHz) for three undersampling ratios (2%, 9%, and 35%, respectively) with 500 sparse elements per frequency. (g)-(i) Example ASWA frequency-wavenumber domain representations from dataset 1 (for frequencies from 46 kHz to 93 kHz) for three undersampling ratios (2%, 9%, and 35%, respectively) with 1 sparse element per frequency. The representations remain nearly unchanged for each undersampling ratio. . . . . . 60 3.8 Time snapshots of the true wavefields, undersampled wavefields, reconstructed / predicted wavefields, and difference between the true and reconstructed wavefields using (a)-(d) ASWA (sparsity = 1 per frequency) at 0.1% of the spatial Nyquist rate, (e)-(h) ASWA (sparsity = 1 per frequency) at 2.2% of the spatial Nyquist rate, (i)-(l) 2D-SWA (sparsity = 500 per frequency) at 2.2% of the spatial Nyquist rate, and (e)-(h) Fourier reconstruction at 2.9% of the spatial Nyquist rate for dataset 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.9 Accuracy (correlation coefficient) of ASWA with P known a priori (sparsity = 1 per frequency), ASWA with P learned from the data (sparsity = 1 per frequency), 2D-SWA (sparsity = 500 per frequency), and Fourier reconstruction at different undersampling ratios for (a) dataset 1, (b) dataset 2, and (c) dataset 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.1 Dispersion curves of a 0.284 cm thick aluminum plate in the 0-500 kHz frequency range where only one symmetric mode (S0) and one asymmetric mode (A0) are present. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 Recovered sparse representation (dispersion curves) of simulated Lamb waves in frequency-wavenumber domain with random temporal sampling. . . . . . . . . 75 4.3 Wave reconstruction using random temporal sampling 98.5 cm away from the transducer. (a) shows the undersampled signal, while (b) and (c) are the reconstructed and the original temporal signal, respectively. . . . . . . . . . . . . . . . 76 4.4 Recovered sparse representation (dispersion curves) of simulated Lamb waves in frequency-wavenumber domain with uniform temporal sampling. . . . . . . . . 77 xi 4.5 Wave reconstruction using uniform temporal sampling 98.5 cm away from the transducer. (a) shows the undersampled signal, while (b) and (c) are the reconstructed and the original temporal signal, respectively. . . . . . . . . . . . . . . . 78 5.1 Sparse representation recovered using SWA. More details on the derivation of this sparse representation can be found in [13]. . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 A snapshot in time showing SWA reconstruction from spatially undersampled data: (a) the undersampled wavefield (circles represent spatial samples), (b) the reconstructed wavefield using SWA, and (c) the original fully sampled wavefield. More details on the reconstruction process can be found in [13]. . . . 102 5.3 Sparse representation recovered using TSWA (positive frequencies). More details on the derivation of this sparse representation can be found in [14]. . . . 103 5.4 TSWA reconstruction from temporally undersampled data: (a) undersampled time domain signal at a certain grid point, (b) the reconstructed signal using TSWA, and (c) the original fully sampled signal. More details on the reconstruction process can be found in [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.5 ST-SWA reconstruction process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.6 The scanned area on the SAE 304 stainless steel plate. . . . . . . . . . . . . . . . . . . . . . 106 5.7 Normalized magnitude of the frequency response of the wavefield, averaged over all grid points. The magnified circular point delineates the maximum frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.8 Two-dimensional wavenumber representation of the wavefield averaged over all frequency slices. The dashed white circle bounds 99 % of the energy in this representation and its radius determines the maximum wavenumber. . . . . . . . 108 5.9 Time snapshot of reconstructed wavefields using ST-SWA with element-wise random sampling at different undersampling ratios with different accuracies (acccuracy here is defined in terms of correlation coefficient between the original fully sampled wavefield and the reconstructed wavefield using ST-SWA): (a) original fully sampled wavefield, (b),(d),(f) reconstructed wavefield at undersampling ratios of 86%, 40%, and, 23%, respectively, and (c),(e),(g) wavefields resulting from subtracting (b),(d),(f) from (a). . . . . . . . . . . 111 5.10 Matrix representation of sampling strategies 1-7 shown in (a)-(g), respectively. The lighter (yellow) pixels represent the chosen samples, while the darker (blue) ones are set to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.11 Reconstruction accuracy (correlation coefficient) as a function of spatiotemporal undersampling ratio for different sampling strategies. For scenarios 2, 3, 4, 5 and 6, where undersampling is done separately in each dimension, points are chosen such that ratios are approximately equal in both dimensions (i.e., points on the antidiagonal of the surface plots in Figure 5.12). . . . . . 113 5.12 Surface plots showing reconstruction accuracy (correlation coefficient) as a function of undersampling ratios in time and space for sampling scenarios (a) 2, (b) 3, (c) 4, (d) 5, and (e) 6. The colorbar shows the accuracy values in terms of correlation coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 xii 5.13 Sparse representation (sparsity value 50). Elements from the second mode of propagation in the wavefield begin to appear at higher sparsity values. . . . . . . 115 5.14 Changes in reconstruction accuracy (correlation coefficient) for different sparsity values. The accuracy tends to increase with the number of sparse components before it reaches a plateau. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.1 Recovered dispersion curves (Dataset 1) from 200 measurements using SWA. . 129 6.2 Recovered weight vector (Dataset 1) for a particular sampled grid point. . . . . . 130 6.3 Reconstructed signal (Dataset 1) for the main path and the original signal for a particular sampled grid point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.4 Reconstructed signal (Dataset 1) for the reflection path and the original signal for a particular sampled grid point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.5 Recovered dispersion curves (Dataset 2) for Scenario 1 at iteration number 1. . 133 6.6 Recovered dispersion curves (Dataset 2) for Scenario 2 at iteration number 30. 134 6.7 Recovered dispersion curves (Dataset 2) using regular SWA. . . . . . . . . . . . . . . . 135 7.1 A recovered sparse representation using PSWA at one frequency slice. . . . . . . . 148 7.2 Two-dimensional wavenumber representation of the simulation dataset. The white dashed circle indicates the 99 % energy circle. . . . . . . . . . . . . . . . . . . . . . . 149 7.3 Correlation coefficient versus undersampling ratio. . . . . . . . . . . . . . . . . . . . . . . . 150 7.4 Reconstruction (at 25 % of the Nyquist rate) at different (increasing from top to bottom) time instants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.5 Correlation coefficient versus undersampling ratio. . . . . . . . . . . . . . . . . . . . . . . . 153 xiii LIST OF TABLES 3.1 Parameters for the three tested datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.1 Temporal Nyquist parameters of the wavefield. . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Spatial Nyquist parameters of the wavefield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.1 Parameters for the simulation dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 ACKNOWLEDGMENTS The realization of this dissertation would not have been possible, had it not been for the support of several people, to whom I am very much obliged. First, I would like to express my deepest gratitude to Dr. Joel B. Harley to whom I will forever be indebted. His support has always been genuine and unbounded, and his benevolence surpassed my expectations of a thesis advisor. He directed me to the right path every time uncertainty befell, and reinvigorated me whenever despair loomed large. I cannot possibly thank him enough for the impact he made on every step of this journey. I would also like to sincerely thank the members of my dissertation committee for their generous assistance and support. I am grateful to Dr. Behrouz Farhang for his unlimited support since the beginning of my Ph.D. program. He acquainted me with the department, guided me, and helped me find my feet. I am thankful to Dr. Rong Rong Chen for her kindness and for all that I learned from her. I always admired her excellent teaching and educational prowess. I would also like to thank Dr. Tolga Tasdizen for his support and instruction. The image processing class was highly informative and enlightening and made me enthusiastic about a topic I was once hesitant to grasp. I am also thankful to Dr. Aditya Bhaskara whose presence in my committee was a privilege. I truly appreciate his willingness to help at all times. I would also like to express my gratitude to Dr. Luca De Marchi and Dr. Cara A. C. Leckey for their help by providing experimental datasets that were used in this dissertation, and their kind support and guidance in our encounters and conversations. Moreover, I am grateful to Dr. Chen Ciang Chia for the guided wave dataset that has been very useful in different parts of this dissertation. And a friendly thanks goes to Dr. K. Supreet Alguri and Dr. Alexander C. S. Douglass for the quality time we spent together. CHAPTER 1 INTRODUCTION This dissertation contains methodologies for characterization of guided waves propagating in different structures and their application in the fields concerning nondestructive evaluation (NDE) and structural health monitoring (SHM). In this chapter, we first present a review of the literature on the fundamentals and applications of guided waves in NDE and SHM, the research areas regarding signal processing for guided waves, and the existing techniques for guided wave characterization and reconstruction. We then discuss the motivation behind this work and what it tries to accomplish. Lastly, we present the organization of the following chapters and a brief overview of their contents. 1.1 1.1.1 Literature Review Guided Waves in Nondestructive Evaluation and Structural Health Monitoring The building blocks of aircrafts, civil and mechanical structures are prone to damage and decay due to a variety of causes, including the inevitable aging and environmental impacts. Therefore, maintaining the integrity and safety of these structures through active and constant testing and monitoring is of utmost significance. Nondestructive evaluation/testing (NDE/NDT) refers to a set of techniques aimed at determining the physical condition of structural elements without causing harm or hampering their functionality [1]. Structural health monitoring (SHM) is a field within NDE that concerns methods attempting to ascertain the safety and serviceability of structures through constant monitoring as they are being used [2]. NDE and SHM techniques based on the use of guided waves have been of interest to the research community. Guided waves are notable for their ability to propagate long distances without experiencing significant loss of energy [3]. Their sensitivity to subsurface features also makes them a suitable candidate for inspection of structures with inconspicuous damages that may not be easily detected or damages 2 that occur in inaccessible areas [4]. Guided waves have been used in NDE and SHM for pipelines [5, 6, 7], wind turbines [8, 9, 10], bridges [11, 12, 13], aircrafts structures [14, 15], railways [16, 17], and many other structures. Nevertheless, guided waves are complex in nature. They exhibit multimodal behavior and tend to disperse with frequency variations [18]. It is due to these complexities that study and analysis of guided waves and guided wave data in NDE and SHM applications are often not straightforward. 1.1.2 Signal Processing Techniques in Guided Wave Structural Health Monitoring Signal processing can provide powerful tools in mitigating the complexities of guided waves. They can enhance the interpretability of guided wave data, thus facilitating analysis. Signal processing techniques are used in guided wave NDE and SHM for a variety of purposes. Environmental and operational conditions influence guided wave propagation and change their properties, and in a lot of applications these changes need to be reversed or compensated. Among these conditions, temperature is notable for its prevalence and its impacts on wave velocities. Signal processing based temperature compensation methods such as optimal baseline selection [19], optimal signal stretch [20], local peak coherence [21], scale transform [22], and dynamic time warping [23] have been studied in the literature. As previously mentioned, frequency-dependent dispersion of guided waves is among their intrinsic intricacies. Many signal processing techniques have been proposed for dispersion compensation [24, 25, 26]. Noise reduction and feature extraction using time-frequency analysis constitute another application of signal processing to guided waves [27, 28]. Furthermore, the use of machine learning methods has been the subject of numerous studies in guided wave based SHM [29, 30] and deep learning is gaining ground [31]. Finally, compressive sensing based techniques, including the methodologies presented in this dissertation, have been investigated in several research studies, which will be discussed in the following subsections. 1.1.3 Characterization of Wave Propagation Identifying the characteristics and properties of guided waves in different materials can be crucial for proper analysis. This is of even higher importance for materials such as multilayered composites where physics based analytical models are harder to obtain. 3 Dispersion curves are among the most commonly used tools for characterization of guided waves as they illustrate how the wave velocities vary with changes in frequency. Several analytical and numerical techniques can be used to generate these curves for different structures. Theoretical dispersion curves are often calculated based on wave equations and knowledge of the material properties [32]. Matrix methods such as the transfer-matrix method [33] and the global-matrix method [34] are often used to model guided waves in multilayered structures [35]. Within these frameworks, semianalytical finite element (SAFE) methods have been used to obtain dispersive characteristics of guided waves [36]. 1.1.4 Data Driven and Compressive Sensing Based Guided Wave Characterization and Reconstruction Compressive sensing (otherwise known as compressed sensing or compressive sampling) [37, 38] is a signal processing technique that allows for data compression to take place at the time of sampling and before storage. It provides mathematical guarantees for accurate signal reconstruction with high probability that enables sampling at rates far below the Nyquist rate, facilitating data acquisition as well as storage. It requires a befitting choice of sparsifying bases for the signal of interest and a proper sensing mechanism that would satisfy its criteria. Compressive sensing has been used in a variety of applications, including communication systems [39, 40], wireless sensor networks [41, 42], radar systems [43], speech and audio [44, 45], computer vision [46], medical imaging [47], electromagnetics [48], and several other applications. It has also found its way in several studies regarding guided wave NDE and SHM [49, 50]. Compressive sensing based techniques for guided wave characterization, reconstruction and their application to damage detection exist in the literature. Specifically, Di Ianni et al. presented a framework whereby guided wave reconstruction from subsampled data using a variety of analytical bases is made possible [51]. Moreover, this method has been used by Esfandabadi et al. [52] for the purpose of damage detection and localization through subtraction of reconstruction outputs from different sets of bases. Mesnil et al. introduced a technique called sparse wavenumber reconstruction (SWR) [53] through which source detection within a guided wavefield is feasible using a limited number of wave measurements. Other data driven approaches that forgo the use of analytical models include the work done by Alguri et al. [54] in 4 which a dictionary learning framework is utilized to predict experimental guided waves by learning from simulation data [54] or to detect damage by learning from surrogate structures [55]. Studies on the use of deep neural networks [56] and autoencoders [57] for wave reconstruction have also been performed. In [58], Harley and Moura introduced a methodology named sparse wavenumber analysis (SWA) that is capable of recovering the frequency-wavenumber sparse representation (i.e., dispersion curves) of Lamb waves in infinitely large isotropic plate-like structures. This is made possible by using a physics based equation for Lamb wave propagation and defining a compressive sensing framework in which for the retrieval of the sparse dispersion curves, a few measurements will suffice. Later in [59], this model was modified with the use of pseudoelliptical bases, which can help reproduce anisotropic wave patterns. SWA has also been used in [60] along with matched field processing for localization purposes. 1.2 Motivation Sparse wavenumber analysis (SWA) is an effective tool for characterization of isotropic guided waves in the absence of multipath reflections. However, the isotropy assumption and the requirements for the size of the plate-like structure to avoid boundary reflections can be restrictive in many applications, and cause SWA to be only applicable to wavefields with plane-wave-like behavior. To further elucidate this matter, we can observe how SWA performs with different types of wavefield data. For this purpose, we implement SWA on three different wavefields: 1. Wavefield 1: an isotropic guided wavefield in which the scanning region is chosen to be much smaller compared to the dimensions of the plate. Due to this size difference, the reflections from the edges do not appear in the wavefield within time range of the measurements. 2. Wavefield 2: an isotropic guided wavefield with edge reflections appearing in the scanning region during the measuring time frame. 3. Wavefield 3: an omnidirectional (i.e., not a plane wave traveling in one direction) anisotropic guided wavefield in a composite plate with direction-dependent velocities and reflections from the edges of the plate. 5 Figure 1.1 shows time snapshots of wavefield 1. The increasing time instants from (a) to (c) display the progression of the wave. This wavefield exhibits a plane-wave-like behavior, and therefore SWA can recover its sparse representation from limited spatial measurements. Figure 1.2 depicts the recovered dispersion curves for this wavefield using SWA. The two lines in this representation indicate the presence of two modes of propagation within the range of excited frequencies. Figure 1.3 illustrates different time snapshots of wavefield 2 at increasing time instants (a) to (c). At time instant (c) the reflections from the edges have already arrived in the scanning region, thus complicating the analysis of the waves. Figure 1.4 shows the sparse representation recovered by SWA. Although some sparse elements of higher magnitude may correspond to the more dominant modes of propagation, the intensity of reflections leads to numerous spurious sparse elements. The propagation of wavefield 3 is shown in Figure 1.5 (a)-(c). Edge reflections make an appearance at later time instants as seen in Figure 1.5 (c). The anisotropy, omnidirectionality, and the multipath reflections increase the complexity of the wavefield to such an extent that SWA outputs (Figure 1.6) are effectively uninterpretable. Considering the shortcomings of SWA as well as the promising features it possesses, this dissertation introduces a number of methodologies predicated on the concept and model of SWA with the motivation to address the following: 1. With the growing application of anisotropic structures, such as composite plates, characterization techniques gain increasing significance. We build upon and modify the SWA model to create a framework with the capability to be applied to complex anisotropic structures. 2. The data acquisition process can be time-consuming and create extensive amounts of data. Moreover, SWA only retrieves missing information in the spatial domain. We make adjustments to the models to reduce the required amount of sampled data in the temporal domain as well as the spatial dimension. 6 1.3 Dissertation Organization The rest of this dissertation is organized as follows. Chapter 2 is dedicated to the introduction of a two-dimensional predictive model for guided wave propagation that builds the foundation for two-dimensional sparse wavenumber analysis (2D-SWA). This added dimension helps in accounting for angle-dependent velocity changes in anisotropic media. Chapter 3 presents a comparative study where the performance of 2D-SWA is evaluated along with a modified one-dimensional model containing pseudoelliptical bases known as anisotropic sparse wavenumber analysis (ASWA), as well as the traditional low-pass filtering in the Fourier domain, referred to as Fourier reconstruction. Chapter 4 introduces a methodology called temporal sparse wavenumber analysis (TSWA), which is capable of retrieving guided wave information in the temporal domain (as opposed to the regular SWA that performs the same task in the spatial domain). This methodology, together with the two-dimensional model presented in Chapter 2, pave the way for the introduction of a framework named spatiotemporal sparse wavenumber analysis (ST-SWA), whereby guided wave recovery is feasible in both time and space. The details and results of implementing ST-SWA on an experimentally measured isotropic wavefield are discussed in Chapter 5. In Chapter 6, we present a technique referred to as multipath sparse wavenumber analysis (MP-SWA), which has the potential to improve guided wave characterization and source detection in the presence of multipath reflections through an iterative recovery process. Chapter 7 will provide some preliminary results for an anisotropic guided wave recovery and reconstruction method called polar sparse wavenumber analysis (PSWA), capable of generating a potentially sparser representation of anisotropic waves by incorporating measured data in the polar coordinates. Finally, Chapter 8 contains a summary of the contents and contributions of this dissertation and a discussion on future directions toward improving the performance and practicability of the techniques studied herein. 7 Figure 1.1. Time snapshots of wavefield 1: an isotropic wavefield (normalized to its maximum magnitude) with no multipath reflections. Figure 1.2. Recovered dispersion curves for wavefield 1 using SWA. 8 Figure 1.3. 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CHAPTER 2 TWO-DIMENSIONAL SPARSE WAVENUMBER RECOVERY FOR GUIDED WAVEFIELDS 1 The multimodal and dispersive behavior of guided waves is often characterized by their dispersion curves, which describe their frequency-wavenumber behavior. In prior work, compressive sensing based techniques, such as sparse wavenumber analysis (SWA), have been capable of recovering dispersion curves from limited data samples. A major limitation of SWA, however, is the assumption that the structure is isotropic. As a result, SWA fails when applied to composites and other anisotropic structures. There have been efforts to address this issue in the literature, but they either are not easily generalizable or do not sufficiently express the data. In this paper, we enhance the existing approaches by employing a two-dimensional wavenumber model to account for direction-dependent velocities in anisotropic media. We integrate this model with tools from compressive sensing to reconstruct a wavefield from incomplete data. Specifically, we create a modified two-dimensional orthogonal matching pursuit algorithm that takes an undersampled wavefield image, with specified unknown elements, and determines its sparse wavenumber characteristics. We then recover the entire wavefield from the sparse representations obtained with our small number of data samples. 2.1 Introduction With the growing cognizance of the significance of maintaining integrity and functionality of the existing structures, the field of structural health monitoring (SHM) grew out of nondestructive evaluation/testing (NDE/NDT). Whether through detection of an incipient damage or prognosis of a potential one, SHM helps to ascertain the impeccability of structures. Guided wave (GW) based SHM has begun to gain ground in the 1 Reproduced from ”S. Sabeti and J. B. Harley, Two-dimensional sparse wavenumber recovery for guided wavefields, AIP Conference Proceedings, 2018”, with the permission of AIP Publishing 16 past two decades due to its performance superiority in comparison with alternative SHM techniques, such as vibration based methods [1]. The low cost, high sensitivity to small defects and extensive range of propagation are among the advantages GW based techniques provide to actively interrogate an operating structure [2]. Guided waves, however, are inherently complex in nature which, in turn, acts as an impediment to effective analysis of their behavior. This intricacy lies in the multimodal propagation of these waves and their velocities being frequency-dependent [3]. Representing these properties of GWs is achievable through utilization of dispersion curves. These curves illustrate variations in modal velocities/wavenumbers with respect to frequency. Therefore, they can be used to predict the behavior of the guided waves in a particular medium. From another perspective, they can be viewed as a sparse representation of the wavefield in the frequency-wavenumber domain [4]. Attempts to recover dispersion curves aiming to reconstruct guided wavefields with highly undersampled data can be found in the literature. Specifically in prior work [4], a compressive sensing based technique named sparse wavenumber analysis (SWA) was demonstrated, which is capable of dispersion curve retrieval using very few spatial samples. However, the models utilized in SWA assume isotropy of the propagation medium, which is restrictive. An alternative approach is trying to incorporate various models through use of different sets of bases (i.e., dictionaries) to represent the wavefield. Authors in [5] propose a compressive sensing based approach with various sparsity-promoting dictionaries, including Fourier exponentials, curvelets, and wave atoms. Their method, which does not require isotropy in the propagation medium, recovers wavefields with less than 34 % of the original data. In this paper, we improve the existing methodologies by employing a two-dimensional wavenumber model. This model accounts for velocity variations in anisotropic media, which depend on the direction of propagation. We consider a set of spatial Fourier bases for horizontal and vertical axes and then utilize sparse recovery methods to recover the two-dimensional sparse representation of the wavefield in the horizontal/vertical wavenumber domain. Particularly, we make use of a modified two-dimensional orthogonal matching pursuit algorithm [6] where the input is a spatially undersampled wavefield image and the location of the samples is given. This algorithm outputs the sparse representation 17 of the wavefield in the two-dimensional wavenumber plane that is subsequently used to reconstruct the full wavefield. To investigate the practicability of our method, we implement it on a set of experimentally garnered data of a wavefield propagating in a steel plate as well as a set of simulation data on an anisotropic composite plate. The results for the experimental dataset suggest we can successfully reconstruct the full wavefield with accuracy (in terms of correlation with the original fully sampled data) of approximately 97 % using only 0.5 % of the spatial samples. Similar results are obtained for the simulation dataset with nearly 95 % accuracy with approximately 0.59 % of the original data. In the following sections, we further elucidate our methodology, the experimental and simulation setup, and the details of the results we achieve. 2.2 2.2.1 Methodology Sparse Wavenumber Analysis Sparse wavenumber analysis (SWA) [4] is a compressive sensing based sparse recovery method that is capable of retrieving the frequency-wavenumber ( f − k) representation, i.e., dispersion curves, of a wavefield using a few random measurements. The recovered sparse representation can then be utilized to reconstruct the full wavefield, through a process called sparse wavenumber synthesis (SWS). In this section, we briefly discuss how SWA performs, what its shortcomings are, and how we enhance it to tackle the issues SWA is not capable of addressing. A guided wave propagating in an isotropic medium at each angular frequency ω and at a distance r from the transmitter can be modeled as [4] s 1 Gn (ω )e− jkn (ω )r , Z (r, ω ) = ∑ k ( ω ) r n n (2.1) where k n (ω ) denotes the frequency-dependent wavenumber and n refers to the mode of propagation. Dispersion curves illustrate this frequency dependency for various modes. Gn (ω ) is the frequency dependent complex amplitude of mode n. The isotropy assumption manifests itself in (2.1), where wave variation depends solely on distance, regardless of the direction of propagation. Equation (2.1) can also be viewed as a linear combination of a set of bases that constitute 18 the wavefield. In SWA, each element of the matrix containing the set of bases (atoms), also known as the dictionary, is defined as s Φmn = 1 − jkn rm e . k n rm (2.2) Consequently, (2.1) can be rewritten as Z = ΦV, (2.3) where frequency-domain wavefield Z is an M × Q matrix, the dictionary Φ, is an M × N matrix, and the sparse representation in ( f − k) domain V, is an N × Q matrix. M, Q, and N are the number of spatial measurements (or number of distances from the transmitter), the number of frequencies, and the number of wavenumbers, respectively. V indicates the dispersion curves and also describes how the atoms from the dictionary are combined to form the wavefield. The model introduced in (2.3) can then be employed in a compressive sensing framework to recover the dispersion curves, i.e., the V matrix, given Z and Φ. To accomplish this, there exist sparse recovery methodologies, such as basis pursuit denoising [7] and orthogonal matching pursuit (OMP) [8]. Since OMP is shown to be more computationally efficient without considerable loss in performance accuracy [9], it is the preferred method of choice in this paper. SWA performs efficiently in predicting wave propagation in isotropic media. However, since the underlying model carries the isotropy assumption, it fails when applied to anisotropic data. In this paper, we employ a two-dimensional model for the waves to account for direction-dependent velocities that emerge in anisotropic plates. Note that the one-dimensional orthogonal matching pursuit (1D-OMP) can also be utilized to recover sparse representations of two-dimensional (2D) signals by reshaping the X and V matrices into 1D vectors. Yet, for our scenario, 2D-OMP has a significantly lower computational complexity [6]. In the following subsection, we describe our modified version of 2D-OMP, which is designed for wavefields with incomplete data. In the results section, we demonstrate how this method proves successful in both isotropic and anisotropic scenarios. 19 2.2.2 2D-OMP for Incomplete Data As discussed in the previous subsection, we make use of a modified 2D-OMP algorithm that can be applied to matrices of incomplete data. We alter an existing 2D-OMP algorithm [6] so that its input can be a matrix corresponding to the undersampled wavefield, which is the same size as the original fully sampled wavefield. We place zeros at the wavefield indices that contain unknown values. Algorithm 1 summarizes the steps of this approach, and Figure 2.1 presents a visual illustration of progression of the algorithm. We describe the steps of this algorithm in the following subsection. Algorithm 1 Orthogonal Matching Pursuit Input: X Mx × My : Partial wavefield at each frequency Φ1 Mx ×Kx : Left dictionary Φ2 My ×Ky : Right dictionary τ: Sparsity (Number of non-zero values in sparse representation) l M×1 : Set of known measurement indices Output: VKx ×Ky : Sparse representation (in k-space) Initialization: Indices set: C0 = Ø Residual: R0 = X for i := 1 to τ do 1. Solution from residual: ci = arg max j riT−1 φj 2. Updating (augmenting) the indices set: Ci = Ci − 1 ∪ c i 3. Finding the least squares approximation: vi = arg minv kXm − ΦCi vk 4. Computing new residual: r i = X m − Φ Ci v i end for (ΦCi : refers to the dictionary comprising only the columns contained in index set Ci ) 20 2.2.3 Guided Wavefield Reconstruction Through 2D Sparse Wavenumber Recovery In a similar fashion to SWA, we define our model for the guided wavefield at one frequency as Z f = Φ1 VΦ2H , (2.4) where frequency-domain wavefield Z f is an Mx × My matrix, the left dictionary Φ1 is an Mx × Nx matrix, the right dictionary Φ2 is an My × Ny matrix, and the sparse representation in the 2D (horizontal-vertical) wavenumber domain is represented by the Nx × Ny matrix V. The scalar values Mx and My are the numbers of spatial grid points in horizontal and vertical directions, respectively. The scalar values Nx and Ny are numbers of wavenumbers in same respective directions. The matrix V is the sparse representation of the data in the 2D wavenumber domain (i.e., k-space). The left and right dictionaries are spatial Fourier bases that are constructed with respect to Cartesian coordinates of the grid points and the wavenumber range in each direction. That is, defining x to be the vector containing horizontal coordinates and kx containing horizontal wavenumbers, we define the left dictionary as Φ1 = e− jkx x . (2.5) Similarly, defining y to be the vector containing vertical coordinates and ky containing vertical wavenumbers, we define each element of the right dictionary as Φ2 = e− jkx y . (2.6) Note that the dictionaries assume the wavefield is fully sampled in space. However, our modified 2D-OMP algorithm allows for the use of incomplete data. That is, not all of the samples in space need to be known. We indicate the unknown spatial sample indices in C, described in Algorithm I. Furthermore, we set the unknown values to zero in the data matrix X. In our analysis, we choose the known samples according to a uniform random distribution (random integers drawn from a discrete uniform distribution). The random sampling scheme, which can be viewed as a masking (sensing) matrix, is known to satisfy compressive sensing framework conditions of restricted isometry property (RIP) for 21 robust reconstruction [10]. Subsequently, a time-to-frequency transformation is performed using a discrete Fourier transform (DFT) operator. The partial wavefield and the left and right dictionaries are then fed into the aforementioned 2D-OMP algorithm (Algorithm 1) to recover the sparse representation in k-space, which is then utilized to reconstruct the full wavefield. It is worth noting that, to save computational resources, this process can be performed only for frequencies where there exists non-negligible signal information content. Algorithm 2 provides a summary of how we reconstruct a full wavefield from spatially undersampled data. Figure 2.2 also presents a visual depiction of different stages of our algorithm. Algorithm 2 Guided wavefield reconstruction from undersampled data 1: Grid points are randomly measured to obtain an undersampled wavefield . 2: Temporal wavefield is transformed into the frequency-domain using the discrete Fourier transform (DFT). 3: Left and right dictionaries Φ1 and Φ2 (spatial Fourier bases) are created based on the coordinates of the grid points and the user-defined horizontal and vertical wavenumber ranges (Equations (2.5) and (2.6)). 4: Sparse representation V in the horizontal-vertical wavenumber domain (k-space) is recovered assuming the model in (2.4) using Algorithm 1. 5: Left and right dictionaries as well as the recovered sparse representation are utilized to reconstruct the wave at all the points on the grid. 2.3 2.3.1 Experimental and Simulation Setup Experimental Setup (Isotropic Wavefield) The experimental dataset used in this paper is collected from a SAE 304 stainless steel plate of 2 mm thickness on a 10 cm by 10 cm region at the center of the plate. Reflections are not present in the data due to the plate being comparatively large (100 cm by 100 cm). We define a 200 by 200 set of grid points, spaced 0.05 cm apart in the horizontal and vertical directions, as our full wavefield. Propagating waves are generated using a scanning Q-switched Nd:YAG diode-pumped solid state pulsed laser (Advanced Optowave, custom made) and measured using a piezoelectric transducer. Through the principle of time reversal (i.e., the acoustic signal is unchanged when we exchange the transmitter and 22 receiver), we can consider the piezoelectric as a transmitter and the laser as a receiver. Note that the piezoelectric is not within the scanned region and the observed wavefield. 2.3.2 Simulation Setup (Anisotropic Wavefield) The simulation dataset used in this paper is generated via a finite element engine. The simulated propagation medium is characterized as a carbon fiber reinforced polymer (CFRP) plate of layup [0/90/0/90]s and with 10.64 cm by 10.64 cm horizontal/vertical dimensions. The original dataset, representing our fully sampled wavefield, is on a 917 by 917 grid of points, spaced approximately 0.0116 cm apart in each direction. In this dataset, the transducer is located in the middle of the plate at about (x = 5.25 cm, y = 5.25 cm) on our grid. 2.4 2.4.1 Results Experimental Results (Isotropic Wavefield) The methodology presented in Algorithm 2 was implemented on the experimental dataset. Figure 2.3 shows the recovered sparse representation in the 2D wavenumber domain (k-space), and Figure 2.4 shows the signal created from this sparse representation. In this case, a sparsity of 20 (i.e., 20 non-zero elements in the k-space representation) characterizes wave propagation with high accuracy. The presence of wavenumbers (spatial frequencies) in a particular direction indicates wave propagation in that direction. The direction of propagation of the two dominant guided wave modes, namely the A0 and S0 modes, are clearly visible in Figure 2.3. This is in accordance with the actual wavefield data, as shown in Figure 2.5. Figure 2.4 depicts a comparison between the reconstructed time-domain wave at a grid point and the original temporal wave at that location. The two signals bear a high visual similarity and share a correlation coefficient of approximately 0.97. Figure 2.5 illustrates the reconstructed wavefield at different time instants and compares them with the true (original) wavefield. Each row of images is a temporal snapshot of the wavefield over the defined grid of points at a particular time instant. In each row, Figure 2.5 (a) shows the partial (undersampled) wavefield. In this case, we utilized 200 measurements from the 40,000 grid points, i.e., 0.5 % of the available data. Figure 2.5 (b) and 2.5 (c) illustrate the reconstructed and the original wavefield at that time instant, 23 respectively. We employ correlation coefficient as the comparison metric. The overall correlation coefficient (i.e., at all time instants and all grid points) between the reconstructed and the original wavefield is about 97 %. 2.4.2 Simulation Results Figure 2.6 depicts the recovered sparse representation in the 2D wavenumber domain (k-space) for the simulated anisotropic wavefield. Compared with the experimental results, this case requires a larger number of non-zero elements, i.e., a sparsity of 500, to accurately characterize behavior of the wave. This can be attributed to anisotropy in the propagation medium, to some extent, as well as to the presence of the source transducer in the scanning region, creating wavenumbers in all directions and necessitating a denser representation. Figure 2.7 illustrates a comparison between the reconstructed temporal wave at a point on the grid and the corresponding true wave. This figure indicates the visual similarity between the reconstructed time-domain signal and its original counterpart. The correlation coefficient between these two signals is about 0.99. Figure 2.8 shows how the wavefield is reconstructed at different time instants in comparison to the original data. Similar to Figure 2.5, each row of images is a snapshot of the wave at a certain time instant. In each of the rows, Figure 2.8 (a) depicts the partial (undersampled) wavefield. We measure 5000 measurements out of 840889 grid points, i.e., approximately 0.59 % of the available data. Figure 2.8 (b) shows the reconstructed wavefield, and Figure 2.8 (c) illustrates its original wavefield. The overall correlation coefficient between the reconstructed and the original wavefield is approximately 95 %. 2.5 Conclusion In this paper, we proposed a compressive sensing based algorithm to reconstruct full guided wavefields using limited spatial measurements through recovery of sparse representations in the 2D wavenumber domain. We demonstrated that our methodology is capable of reconstructing waves with significant accuracy from highly undersampled data. This was achieved by implementing the proposed method on experimental and simulation datasets. In the case of experimentally garnered data on an isotropic steel plate, 24 the full wavefield was successfully reconstructed with accuracy of about 97 %, in terms of correlation coefficient, while using only 0.5 % of the available spatial information. The simulated wavefield on an anisotropic composite plate was reconstructed with about 95 % accuracy from approximately 0.5 % of the original data. In future work, we will apply this method to experimentally gathered anisotropic data and investigate improved models for better representation of wave propagation in anisotropic media. 2.6 Acknowledgments This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0126. We would also like to thank Chen Ciang Chia from the Department of Aerospace Engineering in Universiti Putra Malaysia and Cara Leckey from NASA Langley Research Center’s Non-Destructive Evaluation Branch (Space Act Agreement #SAA1-24023) for the steel and composite wavefield data used in this paper. 25 Figure 2.1. 2D-OMP algorithm for incomplete data. 26 Figure 2.2. Guided wavefield recovery algorithm. 27 Figure 2.3. Recovered sparse representation of the experimental wavefield in k-space at a particular frequency. 28 Figure 2.4. Reconstructed versus true experimental signal in time at a particular grid point (magnified for better comparison). 29 Figure 2.5. Reconstruction of the experimental data at different time instants (increasing from top to bottom) with approximately 97 % accuracy. 30 Figure 2.6. Recovered sparse representation of the simulated wavefield in k-space at a particular frequency. 31 Figure 2.7. Reconstructed versus true simulation signal in time at a particular grid point (magnified for better comparison). 32 Figure 2.8. Reconstructed versus true simulation signal in time at a particular grid point (magnified for better comparison). 33 2.7 References [1] M. Mitra and S. Gopalakrishnan. Guided wave based structural health monitoring: A review. Smart Materials and Structures, 25(5):053001, Mar. 2016. [2] A.C. Raghavan and C. Cesnik. Review of guided-wave structural health monitoring. The Shock and Vibration Digest, 39:91–114, 03 2007. [3] P. Cawley. Practical long range guided wave inspection managing complexity. AIP Conference Proceedings, 657(1):22–40, 2003. [4] J. B. Harley and J. M. F. Moura. Sparse recovery of the multimodal and dispersive characteristics of lamb waves. The Journal of the Acoustical Society of America, 133(5):2732–2745, 2013. [5] T. D. Ianni, L. D. Marchi, A. Perelli, and A. Marzani. Compressive sensing of full wave field data for structural health monitoring applications. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 62(7):1373–1383, Jul. 2015. [6] Y. Fang, B. Huang, and J. Wu. 2D sparse signal recovery via 2D orthogonal matching pursuit. Science China Information Sciences, 55(4):889–897, Apr. 2012. [7] S. S. Chen, D. L. Donoho, and M. A. Saunders. Atomic decomposition by basis pursuit. SIAM Journal on Scientific Computing, 20(1):33–61, 1998. [8] J. A. Tropp and A. C. Gilbert. Signal recovery from random measurements via orthogonal matching pursuit. IEEE Transactions on Information Theory, 53(12):4655–4666, Dec. 2007. [9] J. B. Harley and J. M. F. Moura. Dispersion curve recovery with orthogonal matching pursuit. The Journal of the Acoustical Society of America, 137(1):EL1–EL7, 2015. [10] E. J. Candes and M. B. Wakin. An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2):21–30, Mar. 2008. CHAPTER 3 SPARSE WAVENUMBER RECOVERY AND PREDICTION OF ANISOTROPIC GUIDED WAVES IN COMPOSITES: A COMPARATIVE STUDY 1 Guided wave methodologies are among the established approaches for structural health monitoring. For guided wave data, being able to accurately estimate wave properties in the absence of ample measurements can greatly facilitate the often time-consuming and potentially expensive data acquisition procedure. Nevertheless, inherent complexities of the guided waves, including their multimodal and frequency dispersive nature, hinder processing, analysis, and behavior prediction. The severity of these complexities is even higher in anisotropic media, such as composites. Several methods, including sparse wavenumber analysis, have been proposed in the literature to characterize guided wave propagation by extracting wave characteristics in a particular medium from the information contained in a few measurements, and subsequently using this information for full wavefield prediction. In this paper, we investigate the efficacy of guided wave reconstruction techniques, based on sparse wavenumber analysis, for predicting the behavior of guided waves in composite materials. We implement these techniques on several experimental and simulation datasets. We study their performance in estimating the frequency-dependent (dispersive) and anisotropic velocities of guided waves and in reconstructing full wavefields from limited available information. 1 c 2019 IEEE. Reprinted, with permission, from ”S. Sabeti, C. A. C. Leckey, L. De Marchi, J. B. Harley, Sparse Wavenumber Recovery and Prediction of Anisotropic Guided Waves in Composites: A Comparative Study, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2019” 35 3.1 Introduction The use of guided waves has been the foundation of several nondestructive evaluation/testing (NDE/NDT) [1] and structural health monitoring (SHM) methods [2]. Guided waves are of interest due to their intrinsic properties. These include the capability of traversing long distances while experiencing minimal attenuation [3]. Guided wave modes can also propagate across the thickness of structures, making them sensitive to minor, hidden defects and deformations [4]. This paper addresses two challenges of guided wave processing, with a focus on anisotropic materials, such as polymer matrix composites (guided wave based techniques for inspection of other anisotropic structures, e.g., welds [5], or biological specimens such as bones [6], have been studied in the literature). First, guided wave based methods often require laborious and potentially costly data acquisition processes and systems to collect a sufficient amount of data and then detect, locate, and characterize damage in a structure. Second, guided waves are complex in nature, which manifests itself through their multimodal and frequency dispersive behavior. This complexity is further exacerbated by anisotropic characteristics. Due to these challenges, accurate wave characterization (e.g., learning the frequency-dependent and anisotropic velocities and amplitudes of the waves) is important to determine the current state of the structure under inspection. Ideally, this characterization should be accomplished with as few measurements as possible. Accurate characterization leads to predictive models that can estimate how waves are expected to propagate. Many such predictive methodologies in the literature are based on compressive sensing. Compressive sensing [7] is a signal processing framework for reconstructing data with a small number of samples. It uses the underlying assumption that the signal of interest is sparse in some transform domain. Under this assumption, we can achieve high-accuracy reconstruction while sampling well below the Nyquist critical sampling rate (i.e., two times the highest frequency content of the signal). This applies to sampling in both time and space. As a result, compressive sensing can reduce storage requirements and mitigate the burdens of data acquisition. In structural health monitoring applications, different methodologies based on compressive sensing and sparsity based algorithms have been employed [8, 9, 10, 11, 12, 13, 14]. Di Ianni et al. [9] utilize a compressive sensing framework for data reconstruction in SHM 36 applications. The transforms/models used in their work consist of analytical dictionaries, such as 2D Fourier bases, 3D Fourier bases, curvelets, and wave atoms. These techniques have later been exploited for a damage detection method based on the difference in reconstruction accuracy in the vicinity of defective regions [15]. In [10], Mesnil et al. present a methodology for isolating damage (characterized as a scattering source). Their method, called sparse wavenumber reconstruction (SWR), assumes that medium characteristics are known in return for the capability of source detection. In [14], Alguri demonstrates a data-driven approach for computing optimal transforms/models from data of a surrogate structure. Sparse wavenumber analysis (SWA) [8] is a technique for characterizing Lamb waves in isotropic media. This method uses compressive sensing to recover the frequency-dependent velocity/wavenumber representation of these waves (also known as dispersion curves) from a limited number of spatial measurements. The expansive potential of SWA was investigated in [16] through the retrieval of the sparse representations of three different wave propagation systems, namely oscillations on a fixed string (standing waves), Lamb waves traveling in an isotropic plate, and guided waves in a unidirectional anisotropic plate. We refer to the anisotropic model and reconstruction process as anisotropic SWA (ASWA). The main objective of this work is to rigorously assess the performance of different variants of SWA applied to experimental and simulated wavefields in anisotropic structures. We perform a comparative study of three methods: ASWA [16], 2D-SWA [17], and Fourier reconstruction (low-pass filtering/interpolation). We perform this study with simulation and experimental data and multiple composite lay-ups (orthotropic and unidirectional). The 2D-SWA algorithm functions in a similar fashion to work demonstrated in [9, 15] with 3D Fourier bases. It assumes data are sparse in the frequency and two-dimensional wavenumber domain (i.e., k-space). This space can be observed by taking a 3D (time and space) Fourier transform of data. We validate the performance of these methods through two metrics. First, we use the correlation coefficient as a measure of similarity between the reconstructed and the true wavefield. Second, we use the undersampling ratio to quantify the compression capabilities, computed with respect to the spatial Nyquist rate. Our results indicate that both ASWA and 2D-SWA are capable of reconstructing guided 37 waves at rates below the spatial Nyquist rate where Fourier reconstruction begins to fail. However, ASWA exhibits a far superior performance compared to that of 2D-SWA at extremely low sampling rates. 3.2 Background In this section, we elaborate on the foundations of the methods we assess for guided wavefield reconstruction. Specifically, we discuss sparse wavenumber analysis (SWA) and compressive sensing methods and how they can be used to reconstruct wavefields from undersampled data. 3.2.1 Compressive Sensing Compressive sensing, also known as compressed sensing or compressive sampling [7, 18], is a relatively recent breakthrough in signal processing that has found its way into various research studies and applications as it lays the groundwork for faster and more efficient signal acquisition and processing. It provides mathematical guarantees for accurate signal reconstruction from samples acquired at rates below the Nyquist rate given proper models and sensing schemes. It has been used in communication systems and networks [19, 20], radars [21], millimeter-wave imaging [22], medical imaging [23], ultrasound signal processing [24], seismology [25], and several other applications [26]. To use compressive sensing, the signal of interest should be sparse in some transform domain. Sparsity refers to the fact that a signal is a linear combination of a limited number of known bases. In other words, the coefficients of its transform consist of mostly zeros and a few non-zero elements. Such a signal is called τ-sparse with τ being the number of non-zero coefficients. In practice, we deal with compressible signals for which there are τ considerably large coefficients and other negligibly small (close to zero) coefficients [27]. Note that how data (i.e., a wavefield) are sampled is important for the efficacy of compressive sensing. Specifically, incoherence between the sensing (sampling) matrix and the dictionary (matrix of representation bases) is a requirement to satisfy the restricted isometry property (RIP) and achieve accurate reconstruction with high probability [18]. That is why, in this work, random sampling is our sampling method of choice [28]. The problem that we solve through compressive sensing can be represented as an 38 underdetermined system of equations. A matrix representation of this problem can be shown as Z = ΦV, (3.1) where Z is an M × Q matrix of measured data in which M is the number of measurements and Q is the number of time/frequency samples in each measurement, Φ is an M × N sampled sparsifying basis matrix that contains a set of N bases (chosen properly with respect to the signal characteristics), and V is an N × Q matrix of sparse coefficients. In this problem, Z and Φ are known matrices and V is what we are solving for. Each of the N basis vectors is commonly referred to as an atom, and the matrix Φ is called a dictionary. The compressive sensing problem is solved through sparse recovery algorithms. Different variations of such algorithms exist in the literature [29, 30]. For guided wave reconstruction, orthogonal matching pursuit (OMP) is shown to be time-efficient without considerable loss in accuracy [31], and therefore, we employ OMP in our methods. 3.2.2 Sparse Wavenumber Analysis Sparse wavenumber analysis (SWA) is a compressive sensing based technique introduced in [8], which is capable of retrieving propagation characteristics of Lamb waves in isotropic plate-like structures from limited experimental observations. The transform/model used in SWA is based on an analytically derived equation for Lamb wave propagation. Over a grid of sensors, SWA takes a few spatial measurements as input, and within a compressive sensing framework, recovers the sparse representation of waves in the frequencywavenumber domain. This representation, otherwise known as dispersion curves, is then used to reconstruct full wavefields on the entirety of the grid in a process named sparse wavenumber synthesis (SWS). Lamb wave propagation in an infinite isotropic plate in the frequency domain can be modeled as [8] s Z (r, ω ) = ∑ n 1 Gn (ω )e− jkn (ω )r , k n ( ω )r (3.2) where r represents the distance from the source (transducer), ω is the angular frequency, and n is the propagation mode. The variable k n (ω ) refers to the mode wavenumber, which is frequency-dependent and can be illustrated in the dispersion curves, and Gn (ω ) is the 39 complex amplitude of mode n, which also varies with frequency. The time-domain function Z (r, t) is a combination of surface displacements in different directions as measured by the transducer. Equation (3.2) can be rewritten as a matrix equation by creating a dictionary matrix with each element defined as s Φmn = 1 − jkn rm e . k n rm (3.3) By this definition, (3.2) can be reformulated as the equation in (3.1) where Z is a matrix containing the undersampled wavefield, Φ denotes our known dictionary, and V represents the dispersion curves to be recovered. In this setup, M is the number of spatial measurements, N is the number of wavenumbers in the dispersion curves, and Q is the number of frequency samples in the bandwidth of interest. 3.3 Methodology The techniques utilized in this work for guided wave reconstruction are extensions of SWA, discussed in the previous section. In this section, we provide the details of these techniques that modify the SWA model to overcome the difficulties of wave reconstruction in anisotropic materials, such as polymer matrix composites. They all recover a sparse representation, indicating the dispersive behavior of the waves, in a compressive sensing framework. Figure 3.1 (a), (b), and (c) illustrate instances of these sparse representations for SWA, 2D-SWA, and ASWA, respectively. In addition, we discuss how we implement traditional signal reconstruction through low pass filtering in the Fourier domain (which we refer to as Fourier reconstruction). We then present how we perform two-dimensional spatial sampling on a grid of points and the way we derive the Nyquist parameters. Finally, we discuss our performance metrics. 3.3.1 Anisotropic Sparse Wavenumber Analysis (ASWA) The complexity of ultrasonic guided waves in anisotropic media causes SWA to fail when analyzing composite structures. A modification in the SWA model was presented in [16] that enables us to reconstruct various different anisotropic propagation shapes of guided waves. We refer to the technique utilizing this modified model as anisotropic sparse wavenumber analysis (ASWA). The ASWA model for wave propagation is defined 40 as Z ( x, y, ω ) = (x) ∑ Gn (ω )e− j[|kn 1 (y) (ω ) x | pn +|k n (ω )y| pn ] pn , (3.4) n where pn is an adaptable shape factor that can be optimized for a given structure. For example, pn = 1, pn = 2, pn = ∞ assume that the wavefield is a diamond, circle, and square, respectively. We refer to the resulting shapes as pseudoellipses. This model (x) (y) takes into account wavenumber variations in both directions k n and k n , which allows (y) for adjustments to different anisotropic characteristics. For instance, increasing k n to be (x) greater than k n causes the wavefront to expand in horizontal direction. This approach (y) (x) simplifies to SWA when pn = 2 and k n = k n . One drawback of ASWA is that it estimates direction-dependent phase wavenumbers but not direction-dependent attenuations. To mitigate this, we express Z ( x, y, ω ) as a function of distance r and angle θ (i.e., Z (r, θ, ω )) and assume the Euclidean norm of each signal can be expressed as H (r, θ ) = rZ = e ∞ | Z (r, θ, ω )|2 dω ∞ a1 + a2 r + a3 θ + a4 θ 2 + a5 rθ . (3.5) The function H (r, θ ) is defined in (3.5) as such, to account for various sorts of dependencies in attenuation that might be present in the data. This would apply to all types of structures where guided waves experience direction-dependent attenuation. After taking the logarithm of both sides, we obtain a1 , . . . , a5 from undersampled data through polynomial regression. The regression orders (second-order for θ and first-order for r) were empirically chosen (the R2 value generally improved by less than 0.01 for the next higher orders). Once the coefficients in (3.5) are determined, we estimate the Euclidean norm of the full wavefield H (r, θ ) from a1 , . . . , a5 . b (r, θ, ω ), where Z b (r, θ, ω ) is our The final predicted wavefield is therefore H (r, θ ) Z synthesized data from ASWA/ASWS (anisotropic sparse wavenumber synthesis). Figure 3.2(a) depicts the wave reconstruction procedure in ASWA. In summary, ASWA first takes a frequency-domain measured wavefield as input, which it uses to recover the dispersion curves via a sparse recovery algorithm, such as orthogonal matching pursuit [30], and to estimate the direction-dependent attenuations. Next, these outputs, together with 41 the ASWA dictionary for an initial shape factor, are employed in the ASWA model to compute an initial estimate of the entire wavefield. Updated dictionaries are then generated for newly obtained shape factors through a minimum squared error optimization. This process is repeated until the optimum shape factor is attained, using which the optimum dictionary is built. Finally, using this dictionary, the recovered sparse representation, and the direction-dependent amplitude estimate, the fully sampled wavefield is reconstructed. It is worth noting that in the ASWA dispersion curves, as in Figure 3.1(c), we are dealing (x) (y) with pseudoelliptical bases as determined by the pn , k n , and k n values in (3.4). That is, each point at each frequency slice in the ASWA dispersion curve represents a pseudoellipse indicating the general shape of the wavefronts. To clarify this further, Figure 3.3(a)-(c) depicts two-dimensional wavenumber surfaces for datasets 1-3 (later described in Section 3.4), respectively. These surfaces are determined by the recovered sparse representations through ASWA and resemble the k-space representations in Figure 3.4. Furthermore, Figure 3.5(a)-(c) illustrates phase velocities for datasets 1-3, respectively. These shapes bear noticeable similarity to the general shape of wave propagation as seen in Figure 3.6. Figures 3.3 and 3.5 are at a particular frequency for each dataset, and are obtained for the optimized shape factor pn , and a pair of horizontal/vertical wavenumbers on the ASWA dispersion curves, corresponding to the chosen frequency. The algorithm uses discrete data points in frequency (determined by the fast Fourier transform). The frequencies used here correspond to the dispersion curve data points closest to the excitation center frequencies. Note that, the shapes presented in these figures encompass all the direction-dependent information of the waves in terms of phase velocity and the general shape of propagation. That is, the recovered dispersion curves are not confined to one direction. The retrieved frequency-dependent wavenumbers contain the directional information. ASWA is generally not capable of reconstructing reflections from scatterers and other nonidealities in wave propagation [32]. However, this shortcoming can be exploited for damage detection and localization purposes, as in [15], or to create synthetic baselines [32]. 42 3.3.2 Two-dimensional Sparse Wavenumber Analysis (2D-SWA) To account for direction-dependent velocity variations in anisotropic structures, we can alternatively modify the SWA model to assume data are sparse in a two-dimensional wavenumber domain. This model modification was first introduced in [17] and is closely related to other past work [9]. We will refer to its resulting methodology as two-dimensional sparse wavenumber analysis (2D-SWA). In a 2D-SWA framework, the My × Mx frequency-domain wavefield at each frequency f q can be represented as Zq = Φ1 Vq Φ2T , (3.6) where Φ1 and Φ2 are the My × Ny left dictionary and the Mx × Nx right dictionary, respectively. The Ny × Nx matrix Vq denotes the 2D sparse representation of the wavefield that will be retrieved through the 2D-SWA process. Mx is the number of horizontal coordinates, and My is the number of vertical coordinates. Nx and Ny are the numbers of horizontal and vertical wavenumbers, respectively. By defining the left dictionary as T Φ1 = e− jyky , (3.7) and the right dictionary as T Φ2 = e− jxkx , (3.8) we can create a model where our 2D sparse representation Vq would land in k-space. In (3.7) and (3.8), x and y are vectors of horizontal and vertical coordinates, while kx and ky are vectors of horizontal and vertical wavenumbers. Since the 2D-SWA model is two-dimensional, typical sparse recovery algorithms are not applicable. To address this problem, we introduced a 2D OMP algorithm in [17]. Figure 3.2(b) depicts the wave reconstruction procedure in 2D-SWA. 3.3.3 Fourier Reconstruction (Interpolation) The previous two approaches assume signals are sparse in some domain to reconstruct the full wavefield. The classical way of reconstructing signals from their samples is through low pass filtering [33]. A time snapshot or a frequency slice of a guided wavefield is a two-dimensional signal with a Fourier representation in k-space. A fully sampled version of such a signal on a uniform grid of points can be directly transformed 43 into the k-space via a two-dimensional discrete Fourier transform (DFT). Similar to the one-dimensional case, two-dimensional spatial sampling will result in spectrum replicas around the original representation in k-space [34]. When we undersample in space, those replicas overlap in a process known as aliasing. When we critically sample or oversample in space, a two-dimensional low pass filter (mask) can separate the original spectrum and reconstruct a higher-resolution sampled signal. In this work, we utilize this masking process on the k-space representation of the uniformly undersampled wavefield to reconstruct guided waves. This is a simple, classical reconstruction procedure, which we will refer to as Fourier reconstruction. Therefore, we expect our methods to outperform this technique. In Section 3.5, we will show how Fourier reconstruction fails at below Nyquist sampling rates where our methods are still capable of accurately reconstructing the guided wavefield. 3.3.4 Two-Dimensional Spatial Sampling If we consider a uniform sampling process, the Nyquist-Shannon sampling theorem will dictate a spatial grid (which we refer to as Nyquist grid) that contains a certain number of points. However, as discussed in Section 3.2.1, we perform random sampling for our compressive sensing based techniques in which the distance between points is variable. As a result, there will exist points that are further from each other and points that are closer to each other than dictated by the spatial Nyquist rate. To address this issue and to be able to compare results, we consider the average distance between the points by taking into account the number of points randomly scattered in an area created by the Nyquist grid. In other words, we compare the number of sample points in our randomly sampled spatial surface with the equivalent number of points corresponding to the uniformly sampled Nyquist grid. Wavenumber is the spatial frequency of a wave and can be regarded as the number of wavelengths present in a unit distance. In theoretical physics, it is commonly defined as [35] k= 2π , λ (3.9) where k is the wavenumber and λ is the wavelength. This definition is analogous to the angular frequency ω = 2π T in the temporal domain. Consequently, if k max is the highest 44 wavenumber content, and νmax = 1 λmin = k max 2π is the maximum spatial frequency of the wave, we can derive the spatial Nyquist rate FSN , as FSN ≥ 2νmax ⇒ FSN ≥ k max . π (3.10) As a result, critical average distance dmax between the grid points (the maximum permissible distance in each direction) can be defined as dmax = π . k max (3.11) Therefore, we assume we satisfy the Nyquist sampling criteria if our average distance between points is smaller than dmax . 3.3.5 Defining the Maximum Wavenumber Most practical signals do not have explicit maximum wavenumbers or frequencies (i.e., they are not perfectly band-limited). Therefore, to compute the spatial Nyquist rate and the critical average distance, we need to obtain an estimate of the maximum wavenumber content for each dataset. For this purpose, we first perform a two-dimensional spatial Fourier transform on each frequency slice of the wavefield to obtain the two-dimensional wavenumber responses (or k-space representations) for the range of frequencies contained in the data. Next, we add the magnitude of these two-dimensional wavenumber responses together to create a k-space representation that indicates the general wavenumber content of the wavefield. With this two-dimensional signal, we consider a circle which contains 99 % of the overall energy content (i.e., the sum of the squared values). We define the radius of this circle to be the maximum wavenumber content of our wavefield. The dotted circles in Figure 3.4 depict these energy circles for full-field guided wave data. 3.4 Experimental and Simulation Setup To evaluate the performance of the methods discussed in the previous section, we implement them on three different datasets, two experimentally collected and one simulated, that show guided wave propagation in composite materials with a variety of anisotropic properties and materials. We use simulation as well as experimental data to show the 45 methods work for both the ideal conditions in simulation and the nonidealities in an experimental setup. In this section, we present the details of these datasets. In addition, Figure 3.6 illustrates an example wavefield snapshot from each of the datasets. Each wavefield shows distinct wavefront behavior. The number of points on a critically sampled grid (i.e., the Nyquist grid) is shown in Table 3.1 for each dataset. Note that since the Fourier reconstruction grid must be uniform, it will have an integer number of points in each direction. We use spline interpolation to appropriately re-sample the grid for Fourier reconstruction. For ASWA and 2D-SWA, locations are sampled randomly. In the following datasets we generally see only one dominant mode of propagation. This is expected since the pseudosymmetric mode is usually weak at low frequencies. Additionally, the datasets strongly show the pseudoantisymmetric mode because that mode has a strong out of plane amplitude and the datasets discussed in this paper primarily capture out of plane motion. The experimental data is captured by a 1-D laser Doppler vibrometer (LDV) set up normal to the specimen surface, thus measuring out of plane motion. Moreover, out of plane motion was output from the simulation. For the experimentally collected data using LDV, the measuring time for fully sampled wavefield is typically on the order of 7 or more hours to measure very closely spaced data points on the sample surface. This is partly because it is standard practice with LDV data to measure and average a number of signals at each spatial location (on the order of 128 signals) in order to reduce signal noise. For the undersampled wavefield, if the data were to be physically measured, the reduction in measuring time would be proportionate to the downsampling ratio (i.e., the number of sampled grid points over the total number of points on the fully sampled grid). 3.4.1 Dataset 1: Experimental Glass Fiber Reinforced Polymer In this dataset, guided waves were gathered from a 40 cm by 40 cm (width-length) region of a unidirectional (with horizontal alignment) glass fiber reinforced polymer (GFRP) plate with four layers and a total thickness of 3.2 mm. The utilized transducer is a circular piezoelectric lead zirconate titanate (PZT) with diameter of 1 cm. The transducer is located at the center of the plate, where waves start to propagate. The output excitation 46 signal of the transducer was a 10-cycle sinusoidal burst in the frequency range of 56-84 kHz (Hann windowed) with central frequency equal to 70kHz. The full grid contains 149,765 points (389 × 385) distanced 1 mm in each direction. The measurement device is a Polytec PSV400M2 scanning laser Doppler vibrometer (SLDV) with a sampling rate of 512 kHz. The signals at each grid point were measured 25 times and averaged to remove random noise. 3.4.2 Dataset 2: Simulated Carbon Fiber Reinforced Polymer A finite difference based approach, elastodynamic finite integration technique (EFIT), was used to simulate guided wave propagation in a carbon fiber reinforced polymer (CFRP) plate of layup [0/90/0/90]s. The EFIT model implemented here has been previously validated against experiment, theory, and has been compared to commercial finite element simulation tools for composites with orthotropic anisotropy [36, 37]. The simulated plate region was 10.64 cm by 10.64 cm (width-length) with a thickness of approximately 0.928 mm. For this dataset, the full grid contains 840,889 points (917 × 917) distanced approximately 0.0116 cm in each direction. The transducer (source of propagation) is located at the center of the plate at the approximate coordinates of (x = 5.25 cm, y = 5.25 cm) and is considered to be infinitesimally small in the simulation. A 200 kHz 6.5 cycle Hann windowed sine wave was used for the excitation. This signal is a common tone burst excitation used in the field of NDE, which generally leads to a fairly narrowband frequency response while not leading to too much transducer ringing. 3.4.3 Dataset 3: Experimental Carbon Fiber Reinforced Polymer This dataset contains guided waves propagating in a 15 cm by 15 cm region of a carbon fiber reinforced polymer (CFRP) plate of layup [0/0/90/90]s, with a thickness of approximately 0.916 mm. The dataset was taken using a 1-D LDV attached to an x-y scanner. The LDV is a Polytec OFV-505 connected to an OFV-5000 controller. The dataset was taken with a 20 MHz time sampling rate, and each grid point was measured 128 times and averaged to remove random noise. The full spatial grid contains 22,801 points (151 × 151) with a 1 mm spacing in each direction. A contact transducer with a 1.27 cm diameter was coupled to the plate surface and used a 6.5 cycle 200 kHz tone burst excitation. The transducer (source of propagation) is located at the approximate coordinates of (x = 5.1 cm, y = 4.9 cm). 47 Note that the uneven wavefronts near the transducer location in Figure 3.6(c) are due to uneven couplant spread on the transducer face. It is also noted that the wavefield exhibits significant static (not propagating) vibrations within the transducer. The spatial samples in this region do not correspond to a guided wave. As a result, we ignore these spatial samples when we perform our analyses. 3.4.4 Metrics for Comparison We compare the reconstruction accuracy of the methods using the correlation coefficient (after vectorizing the three-dimensional time-space signals) between the true wavefield and the reconstructed wavefield. We illustrate the correlation coefficient as a function of the undersampling ratio, the percentage below the Nyquist criteria that we sample. To evaluate undersampling ratio, we obtain the Nyquist sampling criterion for each dataset. Equation (3.11) in Section 3.3.4 can determine the maximum average spacing dmax between sampled grid points with respect to the spatial Nyquist rate. Figure 3.4 shows the full wave k-space representation (summed over all frequencies) for datasets 1-3. Based on the maximum wavenumber content for each dataset, dmax is calculated. The spatial sampling period Ds (the distance between original grid points, which is the same for both directions in all the datasets) with respect to dmax will give us the number of points on the Nyquist grid NNyq , as a function of the number of points on the original grid Ng . NNyq = Ds dmax 2 Ng . (3.12) The Nyquist parameters for the data are presented in Table 3.1. It should be noted that to perform Fourier reconstruction we need to have a uniformly sampled grid, and the grid is required to have an integer number of points in each direction. We define the undersampling ratio as the ratio between the number of points on the (uniformly or randomly) sampled grid (NU ) and the number of points on the Nyquist grid (NNyq ). RUN , NU . NNyq (3.13) 48 3.5 Results and Discussion In this section, we perform a comparative study by implementing the reconstruction methodologies introduced in Section 3.3 on the datasets described in Section 3.4. We implement each of the three signal reconstruction techniques on the datasets to evaluate their performance. We present the results in three sections: (1) we study the wavenumber representations generated by each method, (2) we demonstrate several example reconstructions by each method, and (3) we explore reconstruction accuracy as a function of the undersampling ratio. 3.5.1 Learning Wavenumber Representations Figure 3.7 illustrates the wavenumber representations corresponding to each of the methods. Specifically, Figure 3.7(a)-(c) shows 2D Fourier representations for dataset 1 with undersampling ratios of 3%, 11%, and 45%. These images include the low-pass wavenumber filtering that removes all values with wavenumbers greater than 730 m−1 . The oversampled Fourier representation, which looks like an ellipse, is shown in Figure 3.4(a). By comparing Figure 3.7(a)-(c) with Figure 3.4(a), it is evident that the representation exhibits significant aliasing. The true ellipse is only readily visible in Figure 3.7(c). A faster mode, yet of relatively low amplitude, can be seen in this dataset. This should correspond to the first pseudosymmetric mode. Note that, undersampling ratios for Fourier reconstruction are different from the other two methods, since the set of samples are required to shape a uniform grid and thus the number of samples cannot be arbitrarily chosen. Figure 3.7(d)-(f) shows the 2D-SWA representation for dataset 1 with undersampling ratios of 2%, 9%, and 35%. When the undersampling ratio is low, the result behaves like noise. When the undersampling ratio is high, the values exhibit an ellipse in the 2D wavenumber plane and the wavenumbers increase with frequency. Some values are not visible because the amplitudes are weak and therefore not selected as one of the 500 sparse elements per frequency selected by 2D-SWA. Figure 3.7(g)-(i) shows the ASWA representation for dataset 1 with undersampling ratios of 2%, 9%, and 35%. For each case, the ASWA representation is nearly identical and represents expected behavior. The representation illustrates a linear set of points 49 (indicating minimal dispersion) with k(x) values that are higher than k(y) values (indicating that the wavefronts travel faster in the y-direction). The optimized shape factor P for this dataset is P = 1.83, indicating a slight inward compression of wavefront corners toward looking like a diamond (i.e., P = 1). This expected behavior is verified in Figure 3.6(a). 3.5.2 Wavefield Reconstruction The images in Figure 3.8 show the time snapshots of the reconstruction of wavefields from dataset 1 using ASWA with undersampling ratio of 0.1% in (a)-(d), ASWA in (e)-(h) as well as 2D-SWA in (i)-(l) both with undersampling ratios of 2.2%, and Fourier reconstruction in (m)-(p) with undersampling ratio of 2.9%. These figures show how ASWA is capable of accurately reconstructing the wavefield with very low undersampling rates, where both 2D-SWA and Fourier reconstruction fail. The 2D-SWA reconstruction exhibits noise-like behavior, similar to its wavenumber representation while Fourier reconstruction only reconstructs regions around the sample points. Note that, as discussed in the following section, 2D-SWA does experience a quick boost in performance close to 100% accuracy before Fourier reconstruction. It can also be seen in the figures that reconstruction methods, such as ASWA, generally reconstruct only the dominant mode of propagation in terms of the wave magnitude. Depending on the number of sparse elements in the recovered representation and the relative amplitude of the other modes, 2D-SWA is capable of reconstructing these modes or reflections. 3.5.3 Wavefield Reconstruction Versus Undersampling Ratio Figure 3.9 depicts reconstruction accuracy (correlation coefficient) of each method with each dataset as a function of the undersampling ratio. The results indicate that at the spatial Nyquist rate, all methods exhibit similar performance and are able to reconstruct data with high accuracy. The performance of Fourier reconstruction declines the earliest with the decrease in sampling rates, which is expected. 2D-SWA maintains a strong performance at rates below, but close to, the Nyquist rate. Nevertheless, it generally experiences a sudden decline in performance below a certain rate (approximately 10% to 50% of the Nyquist rate). Note that when reconstructing 50 omnidirectional wavefields, as used in this paper, 2D-SWA’s performance declines much earlier than if we reconstructed a far-field (plane wave-like) portion of the wavefield. This is because a single plane wave would be optimally sparse (i.e., possess a sparsity of 1) in the 2D-SWA representation. Similarly, 2D-SWA performs poorly for dataset 2 due to its relatively uniform behavior with direction. A higher sparsity (about 500 used in this study) would improve performance. Among the methods, ASWA achieves the highest accuracies for all datasets. It retains its high accuracy for undersampling ratios below 1 %. Note that the decline in ASWA’s performance also corresponds to a decrease in the total number of measurements. Therefore it is unclear if the drop is more influenced by the density of samples or the total number. The performance of ASWA further improves when the shape factor P is known a priori and does not need to be optimized with ASWA. When optimizing P, the accuracy is lower because the optimization can fail to converge to the correct result. For this reason, the accuracies for ASWA correspond to the average of 10 Monte Carlo trials with different, randomly chosen measurement locations. Note that ASWA’s performance does not converge to 1 for the experimental datasets. This is because ASWA is constrained to only reconstruct waves from one known location with a particular model. As a result, reflections, inhomogeneities, and noise will not be reconstructed [32] and be considered errors. Small misalignments in sensor placement can cause small errors. Overall, the correlation coefficients of dataset 1 and 3 converge to 0.93 and 0.89, so these differences are relatively small. 3.5.4 Summary of the Results The results in this study can be summarized as follows. 1. Fourier reconstruction, as a traditional signal processing technique, is a useful tool as long as we can afford to comply with the spatial Nyquist sampling rate. 2. 2D-SWA is a promising methodology to accurately reconstruct wavefields sampled at rates just below the Nyquist rate. Its reconstruction performance is enhanced when dealing with directional and far-field waves. The general model is flexible, has no assumptions on the location of the source of propagation, and does not require additional optimizations. 51 3. ASWA is highly effective in predicting a guided wavefield from extremely limited measurements. Nevertheless, ASWA uses a specific model for anisotropic wave propagation that is not capable of reconstructing reflections or other inhomogeneities in the medium. This limitation, however, can be exploited in damage detection, localization, or synthetic baseline generation scenarios [15, 32]. ASWA also requires prior knowledge of the source location. 3.6 Conclusion In this paper, we presented different compressive sensing based guided wave reconstruction models and studied their performance for accurate reconstruction of waves in anisotropic structures. We showed how these models are capable of characterizing guided wave propagation in the presence of anisotropy and how they enable us to recover entire wavefields from data collected at spatial sampling rates far below the traditional Nyquist rate. These models and their corresponding compressive sensing based reconstruction methods take advantage of sparsity of guided waves in some transform domain and retrieve their sparse representation using sparse recovery techniques. Moreover, these methods can be helpful in several applications where the characterization of the propagation is vital in extracting material and geometrical properties [38]. They can potentially be applied to other types of waves, such as bulk waves, as long as the wavefield imaging techniques are there to help us visualize these waves [39]. Specifically, we studied the performance of two-dimensional sparse wavenumber analysis (2D-SWA) and anisotropic sparse wavenumber analysis (ASWA), by implementing them on three simulated and experimental datasets of guided waves in composite materials and comparing the results with those of the conventional signal processing reconstruction/interpolation, which we refer to as Fourier reconstruction. The comparison study demonstrated that 2D-SWA is capable of accurate reconstruction of wavefields at sampling frequencies below but close to the spatial Nyquist rate. In contrast, ASWA achieves good accuracy with very few spatial measurements. However, the ASWA model contains a variety of parameters to be optimized for each dataset. Future work includes exploring more fitting models for anisotropic wave propagation to achieve a sparser, more accurate representation. Moreover, we will study approaches to 52 perform guided wave reconstruction from data undersampled in both the temporal and spatial domains to further reduce the dimensionality of acquired data. To achieve this, we will make use of the models discussed in this paper for reconstruction of spatially undersampled data, in conjunction with models for temporal undersampling as introduced in prior work [40]. 3.7 Acknowledgment This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0126. We would also like to thank NASA Langley Research Center’s Non-Destructive Evaluation Branch (Space Act Agreement #SAA124023). Luca De Marchi would like to thank the Polish Academy of Science for the support in the frame of HARMONIA project (UMO-2012/06/M/ST8/00414). 53 (a) SWA dispersion curves (b) 2D-SWA dispersion curves (c) ASWA dispersion curves Figure 3.1. Example single-mode sparse representations for SWA (of an isotropic aluminum plate), 2D-SWA (of an anisotropic unidirectional composite plate), and ASWA (of an anisotropic unidirectional composite plate). 54 (a) ASWA Reconstruction Process (b) 2D-SWA Reconstruction Process Figure 3.2. Guided wave reconstruction processes with (a) ASWA and (b) the 2D-SWA frameworks. 55 (a) Dataset 1 (b) Dataset 2 (c) Dataset 3 Figure 3.3. Example two-dimensional wavenumber surfaces for datasets 1-3 (at 71 kHz, 206 kHz, and 200 kHz, respectively), determined by the ASWA model. The optimized shape factor (pn ) values for datasets 1, 2, and 3 are 1.83, 1.68, and 1.73, respectively. 56 Figure 3.4. Two-dimensional wavenumber representation (summed over all frequency slices) of (a) dataset 1, (b) dataset 2, and (c) dataset 3. The dotted circles indicate the 99 % energy circles. 57 (a) Dataset 1 (b) Dataset 2 (c) Dataset 3 Figure 3.5. Example phase velocity surfaces for datasets 1-3 (at 71 kHz, 206 kHz, and 200 kHz, respectively), determined by the ASWA model. The optimized shape factor (pn ) values for datasets 1, 2, and 3 are 1.83, 1.68, and 1.73, respectively. 58 Figure 3.6. Example wavefield time snapshots from (a) dataset 1, (b) dataset 2, and (c) dataset 3. 59 Table 3.1. Parameters for the three tested datasets. Dataset 1 2 3 Data Type Lay-up Max Wavenumber (kmax ) Max Distance (dmax ) Grid Spacing (Ds ) # of Nyquist Samples (NNyq ) Experiment [0/0]s 1210 2.59 mm 1 mm 22248 Simulation [0/90/0/90]s 1622 1.93 mm 0.116 mm 3019 Experiment [0/0/90/90]s 1742 1.80 mm 1 mm 7011 60 Figure 3.7. Sparse representations. (a)-(c) Example Fourier reconstruction wavenumber domain representations from dataset 1 (at 71 kHz) for three undersampling ratios (3%, 11%, and 45%, respectively). Wavenumber aliasing and the effects of the low-pass reconstruction filter are visible in the images. (d)-(f) Example 2D-SWA frequency-wavenumber domain representations from dataset 1 (for frequencies from 46 kHz to 93 kHz) for three undersampling ratios (2%, 9%, and 35%, respectively) with 500 sparse elements per frequency. (g)-(i) Example ASWA frequency-wavenumber domain representations from dataset 1 (for frequencies from 46 kHz to 93 kHz) for three undersampling ratios (2%, 9%, and 35%, respectively) with 1 sparse element per frequency. The representations remain nearly unchanged for each undersampling ratio. 61 Figure 3.8. Time snapshots of the true wavefields, undersampled wavefields, reconstructed / predicted wavefields, and difference between the true and reconstructed wavefields using (a)-(d) ASWA (sparsity = 1 per frequency) at 0.1% of the spatial Nyquist rate, (e)-(h) ASWA (sparsity = 1 per frequency) at 2.2% of the spatial Nyquist rate, (i)-(l) 2D-SWA (sparsity = 500 per frequency) at 2.2% of the spatial Nyquist rate, and (e)-(h) Fourier reconstruction at 2.9% of the spatial Nyquist rate for dataset 1. 62 Figure 3.9. Accuracy (correlation coefficient) of ASWA with P known a priori (sparsity = 1 per frequency), ASWA with P learned from the data (sparsity = 1 per frequency), 2D-SWA (sparsity = 500 per frequency), and Fourier reconstruction at different undersampling ratios for (a) dataset 1, (b) dataset 2, and (c) dataset 3. 63 3.8 References [1] J. L. Rose, “Successes and challenges in ultrasonic guided waves for NDT and SHM,” Nat. Seminar Exhibit. Non-Destruct. Eval., pp. 2-10, 2009. [2] M. 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CHAPTER 4 GUIDED WAVE RETRIEVAL FROM TEMPORALLY UNDERSAMPLED DATA 1 The acquisition of ultrasonic guided waves for full wavefield nondestructive evaluation (NDE) applications is often a time-consuming procedure. Moreover, the amount of data to be stored over time can be enormous. Consequently, to improve storage efficiency and reduce the acquisition time, it is desirable to retrieve information from partially garnered (undersampled) data. Efforts in the literature aimed at addressing this issue exist, mostly through data recovery from spatially undersampled wavefields. In this paper, we present a compressive sensing based methodology to retrieve guided wavefields from data undersampled in the temporal domain. We implement this method by recovering the dispersion curves of guided Lamb waves (i.e., their sparse representation in the frequencywavenumber domain) from a few temporal measurements. From this representation, we subsequently reconstruct the entire wavefield. We demonstrate a 97 % reconstruction accuracy (in terms of correlation coefficient) for simulated Lamb waves containing frequencies ranging from 150 kHz to 350 kHz, but generated with an effective sampling rate of only 50 kHz. 4.1 Introduction Within the realm of nondestructive evaluation (NDE), employing full wavefield guided waves as an efficacious means of inspecting structures for potential defects has been steadily gaining ground. This is owing to the numerous advantages guided waves provide, including accessibility and low cost of transducers as well as suitability for scanning large areas 1 c 2017 IEEE. Reprinted, with permission, from ”S. Sabeti, J. B. Harley, Guided wave retrieval from temporally undersampled data, IEEE International Ultrasonics Symposium (IUS), 2017” 67 due to minimal attenuation of guided waves [1]. Managing the amount of data produced in guided wave based NDE can be a computationally expensive task. Recently, methodologies based on compressive sensing/sampling (CS) [2] have been proposed for guided wave recovery from spatially undersampled, experimentally collected data [3, 4, 5]. In prior work [3], a CS based approach, called sparse wavenumber analysis (SWA), was shown to be capable of accurately characterizing the behavior of Lamb waves in plate-like structures by extracting information contained in a limited number of spatial measurements [6]. This method characterizes the propagation of Lamb waves by their frequency-wavenumber (f-k) representation, also known as dispersion curves. This representation is known to be sparse, i.e., containing few non-zero elements in the (f-k) domain. SWA performs dispersion curve recovery by leveraging this sparsity with sparse recovery algorithms, such as orthogonal matching pursuit (OMP) [7] and basis pursuit denoising [8] in a CS based framework. The recovered sparse representation can then be used to reconstruct full wavefield data through a process called sparse wavenumber synthesis (SWS). The foundation of our methodology is similar to that of SWA. However, while SWA uses measurements that are densely sampled in time and sparsely sampled in space, our new methodology uses measurements that are densely sampled in space while undersampled in time. We recover the dispersion curves of guided waves using orthogonal matching pursuit (OMP) from few temporal measurements, and later utilize these curves to reconstruct densely sampled full wavefields. We implement our method on simulated Lamb waves to investigate its feasibility. We retrieve dispersion curves and reconstruct the waves with high accuracy using as little as 5 % of the temporal information. We use the correlation coefficient as the measure for comparison between the original signal (with frequencies ranging from approximately 150 kHz to 350 kHz) and the reconstructed one. Correlation coefficient values of approximately 0.97 are achievable with an effective sampling rate of 50 kHz. 4.2 4.2.1 Background Compressive Sensing The underlying principles of compressive sensing (CS) suggest a signal sampled at rates below the well-known Nyquist frequency can be precisely reconstructed given the 68 knowledge that it is sparse in some transform domain. In a CS framework, the goal is to solve an underdetermined system of equations X = ΦV, (4.1) where X is an M × Q data matrix containing M measurements each comprising Q samples in some domain, Φ is an M × N sensing matrix (a sampled set of N basis vectors), and V is an unknown N × Q sparse matrix of coefficients indicating how the bases are linearly combined to create the measurements in X. 4.2.2 Random Sampling Random sensing matrices are well known to satisfy the conditions of incoherence and the restricted isometry property (RIP) for robustness of compressive sensing [2]. These conditions provide provable guarantees that precise data reconstruction is achievable with high probability. For this reason, we opt for random sampling as our undersampling approach. Achieved results in Section 4.5 confirm that random sampling performs much better, in terms of reconstruction accuracy, compared with uniform sampling as an alternative. We use sparse recovery methods to solve the inverse problem in (4.1), where X and Φ are known and V is unknown. Among these methods, orthogonal matching pursuit (OMP) is widely employed and is shown to perform with low computational complexity, particularly at lower sparsity values [9]. In the following subsections, we present an overview of the OMP algorithm. We then review how OMP has been used to solve the sparse wavenumber analysis problem in prior work. 4.2.3 Orthogonal Matching Pursuit Orthogonal matching pursuit (OMP) is one of the most extensively used sparse recovery methods in compressive sensing. It is a greedy algorithm that finds the best approximation of a signal given the sparse number of basis elements linearly combined to create the signal. Algorithm 3 summarizes how OMP is used in SWA (details in the following subsection). 69 Algorithm 3 Orthogonal Matching Pursuit Input: X M×Q : Matrix of measurements (Partial wavefield) Φ M× N : Model based dictionary (Partial Fourier bases) τ: Sparsity (Number of sparse components) Output: V N ×Q : Sparse representation (in (f-k) domain) Initialization: Index set: C0 = Ø Residual: r0 = Xq (q-th column of X, q-th frequency) (The same process will apply to all frequencies) for i := 1 to τ do 1. Solution from residual (index of the best column): ci = arg max j riT−1 φj 2. Updating (augmenting) the index set: Ci = Ci − 1 ∪ c i 3. Finding the least squares approximation: vi = arg minv kXq − ΦCi vk2 4. Computing new residual: r i = X q − Φ Ci v i end for return v (one column of the output) (ΦCi : refers to the dictionary comprising only the columns contained in index set Ci ) 70 4.2.4 Sparse Wavenumber Analysis Introduced in [3], sparse wavenumber analysis (SWA) is a methodology to recover guided wave dispersion curves (i.e., the sparse representation of Lamb waves in the frequencywavenumber domain) from limited experimental observations over a grid of spatial locations. It employs compressive sensing techniques with models for guided wave propagation to solve for an unknown sparse matrix, as in (4.1). Sparse wavenumber synthesis (SWS) refers to a process in which the full wavefield on the entire grid is then reconstructed. The propagation of Lamb waves in an isotropic plate with distance r and at angular frequency ω can modeled as s X (r, ω ) = ∑ n 1 Gn (ω )e− jkn (ω )r , k n ( ω )r (4.2) where n denotes the mode of propagation, k n (ω ) indicates frequency-dependent wavenumber, and Gn (ω ) refers to the complex amplitude of propagation of mode n. By defining each element of our dictionary matrix as s Φmn = 1 − jkn rm e , k n rm (4.3) equation (4.3) can be rewritten in matrix format, as shown in (4.1), where X is now our M × Q frequency-domain partial wavefield, Φ is our M × N dictionary, and V is the unknown sparse representation in the frequency-wavenumber domain, a matrix of N × Q dimensions. Here, M, Q, and N refer to the number of measurements over the grid, the number of frequencies, and the number of wavenumbers, respectively. 4.3 4.3.1 Methodology Guided Wavefield Reconstruction In a similar fashion to SWA, we recover dispersion curves from a partial wavefield. In our case, however, the partial wavefield is obtained via temporal undersampling, which in turn, necessitates alterations in the wave model. Equation (4.2), which characterizes Lamb wave propagation in frequency-space domain, can be represented in the time-wavenumber 71 domain as X (t, k ) = ∑ Hn (k)e jω (k)t . n (4.4) n This equation allows for utilization of all spatial information in the spatial frequency (i.e., wavenumber) domain and can be rewritten in matrix representation for our set of undersampled time instants, similar to what we have in (4.1). In this setup, X is our M × Q partial wavefield in the time-wavenumber domain, Φ is our M × N dictionary, and V is the N × Q sparse representation in the wavenumber-frequency domain. Here, M, Q, and N denote the number of temporal measurements, the number of wavenumbers, and the number of frequencies, respectively. Using the time-wavenumber model and a sparse recovery method of choice (OMP in this paper), dispersion curves are recovered and these curves are then used to reconstruct the full wavefield. Algorithm 4 is a summary of the wave reconstruction methodology used in this paper. 4.4 Simulation Setup To simulate Lamb waves propagating in an infinite, isotropic plate, the model in (4.2) is utilized [3]. Our simulation considers dispersive properties of a 0.284 cm thick aluminum plate. These properties are characterized by the dispersion curves shown in Figure 4.1. We simulate a guided wavefield on a grid of points beginning at 10 cm from the transmitter in both the horizontal (x) and vertical (y) directions (i.e., at a Euclidean distance of approximately 14 cm). The grid contains 10000 points (100 in each direction) with spatial sampling rate of 400 m−1 in the x and 70 m−1 in the y direction. This creates a scanning region of about 25 cm × 142 cm. We generate 1000 time samples with a temporal sampling rate of 1 MHz. These data Algorithm 4 Guided Wave Recovery from Temporally Undersampled Data 1: Partial (temporally undersampled) wavefield is obtained by random selection of time instants 2: Time-space domain partial wavefield is transformed into the time-wavenumber domain using the spatial discrete Fourier transform (DFT) 3: The partial wave dictionary (a set of partial Fourier bases) is created based on the sampled time instants and a range of frequencies 4: Dispersion curves are recovered using OMP 5: The full wave dictionary as well as the recovered dispersion curves are used to reconstruct the full wavefield 72 are filtered using a Gaussian filter with center frequency of 250 kHz and half-power bandwidth of 100 kHz. The resulting Lamb wave, given the range of dominant frequencies, contains only A0 and S0 modes of propagation. 4.5 Results and Discussion In this section, we apply our methodology, as summarized in Algorithm 2, to a simulated guided wavefield on the grid of spatial points, as described in the previous section. We study the results for two different undersampling schemes, namely random and uniform sampling. 4.5.1 Wavefield Reconstruction with Random Sampling Here, we randomly select 50 samples out of 1000 randomly generated time instants to create our partial wavefield. Figure 4.2 illustrates the recovered dispersion curves, i.e., the sparse representation of the signal in the frequency-wavenumber domain. The two dominant modes of propagation, the A0 and the S0 mode, are visible in the figure. Figure 4.3 depicts a comparison between the reconstructed temporal signal at a distance of 98.5 cm from the transmitter, and its original counterpart. The measure for comparison used here is correlation coefficient, which is defined as σ= xT y , kxkkyk (4.5) where x and y are the two vector signals being compared, and k·k indicates the `2 -norm of each vector. Figure 4.3 (a) shows the undersampled signal in time, i.e., the amount of temporal information used to recover the dispersion curves. In this case, only 5 % of time samples (50 out of 1000) are used. Figure 4.3 (b) and (c) show the reconstructed and the original temporal signal, respectively. The correlation coefficient between these two signals is approximately 0.97. 4.5.2 Wavefield Reconstruction with Uniform Sampling As previously discussed, random sampling satisfies RIP of the sensing matrix in a compressive sensing framework, which implies a high probability of an accurate reconstruction. In this subsection, we demonstrate how uniform sampling in time influences 73 the reconstruction accuracy. The recovered sparse representation of the signal is shown in Figure 4.4. As compared to the random sampling case in Figure 4.2, this representation exhibits less accuracy in terms of consistency with the known dispersion curves of the material (Figure 4.1). This can be attributed to the idea that uniform sampling leads to the presence of redundant information, which results in spurious points in the sparse representation recovered by the OMP algorithm. Figure 4.5 (a) depicts the uniformly sampled signal in time at the same spatial location as the signal in Figure 4.3 (a). As with the previous case, only 50 out of 1000 available time samples are utilized, that is 5 % of the overall temporal information. The reconstructed signal and its true counterpart are also presented in Figure 4.5 (b) and (c), respectively. The correlation coefficient between the two signals in this case is approximately 0.71. As indicated by the drop in correlation coefficient as well as the decreased visual similarity, uniform sampling is not a suitable alternative to random sampling. 4.6 Conclusion In this paper, we presented a methodology for guided wave retrieval from a limited number of temporal measurements by leveraging the sparsity of dispersion curves in the frequency-wavenumber domain. It was demonstrated that this approach is capable of reconstructing simulated Lamb waves with approximately 97 % accuracy, in terms of correlation coefficient, from randomly sampled time instants, which amounted to 5 % of the overall temporal information (equivalent to an effective sampling rate of 50 kHz). In future work, we will seek ways to implement this method in an actual experimental scenario, study the effect of different sampling strategies, and explore the potential integration of temporal and spatial guided wave recovery schemes. 4.7 Acknowledgment This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0126. 74 Figure 4.1. Dispersion curves of a 0.284 cm thick aluminum plate in the 0-500 kHz frequency range where only one symmetric mode (S0) and one asymmetric mode (A0) are present. 75 Figure 4.2. Recovered sparse representation (dispersion curves) of simulated Lamb waves in frequency-wavenumber domain with random temporal sampling. 76 Figure 4.3. Wave reconstruction using random temporal sampling 98.5 cm away from the transducer. (a) shows the undersampled signal, while (b) and (c) are the reconstructed and the original temporal signal, respectively. 77 Figure 4.4. Recovered sparse representation (dispersion curves) of simulated Lamb waves in frequency-wavenumber domain with uniform temporal sampling. 78 Figure 4.5. Wave reconstruction using uniform temporal sampling 98.5 cm away from the transducer. (a) shows the undersampled signal, while (b) and (c) are the reconstructed and the original temporal signal, respectively. 79 4.8 References [1] M. Mitra, and S. Gopalakrishnan,“Guided wave based structural health monitoring: A review,” Smart Mater. Struct., vol. 25, no. 5, p. 053001, 2016. [2] E. J. Candes, and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag., vol. 25, no. 2, pp. 21-30, 2008. [3] J. B. Harley, and J. M. F. Moura, “Sparse recovery of the multimodal and dispersive characteristics of Lamb waves,” J. Acoust. Soc. Am., vol. 133, no. 5, pp. 2732-2745, 2013. [4] T. Di Ianni, L. De Marchi , A. Perelli, and A. Marzani, “Compressive sensing of full wave field data for structural health monitoring applications,” IEEE Trans. Ultrason. Ferroelect. Freq. Contr., vol. 62, no. 7, 2015. [5] O. Mesnil, and M. Ruzzene, “Sparse wavefield reconstruction and source detection using Compressed Sensing,” Ultrasonics, vol. 67, pp. 94-104, 2016. [6] J. B. Harley, and C. C. Chia, “Statistical Partial Wavefield Imaging with Lamb Waves,” Struct. Health Monit., in press, 2017. [7] J. Tropp, and A. C. Gilbert, “Signal recovery from partial information via orthogonal matching pursuit,” IEEE Trans. Inf. Theory, vol. 53, no. 12, pp. 4655-4666, 2007. [8] S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci Comp., vol. 20, no. 1, pp. 33-61, 1999. [9] J. B. Harley, and J. M. F. Moura, “Dispersion curve recovery with orthogonal matching pursuit,” J. Acoust. Soc. Amer., vol. 137, no. 1, pp. EL1-EL7, 2015. CHAPTER 5 SPATIOTEMPORAL UNDERSAMPLING: RECOVERING ULTRASONIC GUIDED WAVEFIELDS FROM INCOMPLETE DATA WITH COMPRESSIVE SENSING 1 Many nondestructive evaluation techniques are based on the study and assessment of guided wavefields. Yet, the extent of the sensing region and the span of time over which wavefield data is acquired can be tremendous, resulting in an enormous amount of spatiotemporal data. As a result, reducing the burden of data acquisition and storage from undersampled data could be highly advantageous. To achieve this end, various signal processing methodologies have been proposed in the literature, many of which make use of compressive sensing. In prior work, such methodologies for effective wavefield reconstruction from incomplete data in space and in time (separately) have been demonstrated. In this paper, we combine these approaches. We present a compressive sensing based guided wave retrieval method with a two-dimensional ultrasonic guided wave model, which enables us to reconstruct wavefields that are undersampled in both the temporal and spatial domains. Results from implementing this method on a dataset consisting of experimental guided wave propagation indicate its potential for accurate wave reconstruction in the presence of spatiotemporal undersampling. We compare results for a variety of subsampling strategies and study the impact of sparsity on the reconstruction performance.Our results indicate that the proposed methodology in this paper is capable of achieving an accuracy of more than 80 % (in terms of correlation coefficient) at a spa- 1 Reproduced, with permission, from ”S. Sabeti, J. B. Harley, Spatio-temporal undersampling: Recovering ultrasonic guided wavefields from incomplete data with compressive sensing, Mechanical Systems and Signal Processing (MSSP), 2020” 81 tiotemporal undersampling ratio of about 40 % using random sampling in space and time. 5.1 Introduction With increasing awareness of the importance of maintaining the integrity of structures through constant inspection for incipient damage detection to avoid catastrophic consequences, effective and cost-efficient nondestructive evaluation (NDE) and structural health monitoring (SHM) methods have gathered rising research significance and interest. A major category among these methods comprises different techniques based on the use of ultrasonic guided waves [1]. Guided waves are capable of low-attenuation, long-range propagation, which makes them suitable for interrogation of large scale structures [2]. Their sensitivity to small, subsurface damages and flaws is another contributive factor to their functionality for structural inspection [3]. Various techniques for damage detection and visualization based on the use of guided waves have been studied in the literature [4, 5, 6, 7, 8]. Nevertheless, the intrinsic complexities in guided wave propagation are significant impediments to effective study and analysis. These waves exhibit multimodal behavior and tend to disperse over the frequency range [9, 10]. These characteristics often vary with the choice of propagation media and prior knowledge of them could be conducive to more accurate interpretations. However, in many cases this information is not readily accessible [11], and we may need to obtain these characteristics through experimental data collection [12]. Additionally, guided wave data acquisition is performed over two dimensions, time and space. The scope of data in each of these dimensions can potentially be immense, rendering the acquisition procedure heavily cumbersome. To facilitate this process, the ability to retrieve information from highly undersampled data could be beneficial. A signal processing based method called sparse wavenumber analysis (SWA) was introduced in prior work [13] that is capable of recovering dispersive characteristics of guided waves from limited spatial information. Another method, which we refer to as temporal sparse wavenumber analysis (TSWA), was presented in [14], utilizing a similar approach to SWA for information recovery from data sparsely sampled in the time dimension. Variations of SWA were later proposed through modifications in the wave propaga- 82 tion model and its extension into two dimensions. These variations, referred to as twodimensional sparse wavenumber analysis (2D-SWA) [15] and polar sparse wavenumber analysis (PSWA) [16], make use of a two-dimensional model for guided waves enabling a better grasp of their dispersive features, in particular when applied to the additional intricacies of guided waves in anisotropic media. Other efforts aimed at reconstructing guided waves from limited number of samples exist in the literature. Specifically, a method called sparse wavenumber reconstruction (SWR) [17] has been used for guided wave reconstruction with the assumption of prior knowledge of the dispersive characteristics of the media. In addition, use of different reconstruction bases, such as Fourier, curvelet, and wave atom bases, in a compressive sensing framework has been shown to be practical in recovering guided waves from undersampled data [18]. The potential of learning approaches has also been studied. In particular, a dictionary learning based method has been employed to learn different bases (or modes of propagation) for guided waves, leading to accurate reconstruction from a few samples [19]. All of these techniques, however, are limited to recovery in the presence of subsampling only in the spatial dimension. Lowering the required sampling rate in the temporal dimension can be extremely helpful in facilitating and expediting the data acquisition procedure as well as reducing the overall amount of data. Temporal undersampling can expedite data acquisition by capturing time samples (potentially at a slow temporal sampling rate) while scanning continuously through spatial grid points (potentially at a fast spatial sampling rate). Hence, the system does not wait at any given spatial location, as is typical for a laser Doppler vibrometer system, and the dataset is sampled when it is capable. These measurements are likely to capture pseudorandom time instants of each waveform time-domain response. Furthermore, studies on temporal undersampling could help guide new sensing strategies for guided wave structural health monitoring and other wave-based applications. In particular, neuromorphic sensing is a rapidly growing strategy for designing sensors that sample based on a chosen stimulus [20]. As a result, these new sensing modalities sample temporal data in a nonuniform, pseudorandom fashion. In other fields of study, such as medical research in MRI [21, 22], and wireless sensor networks [23, 24], methodologies aimed at retrieving information from spatiotemporally incomplete 83 data have been proposed. In this paper, we introduce a compressive sensing based technique, which we refer to as spatiotemporal sparse wavenumber analysis (ST-SWA), utilizing the core ideas of SWA and TSWA and employing the two-dimensional model similar to those of 2D-SWA and PSWA. In other words, ST-SWA takes a temporally and spatially undersampled guided wave data matrix as an input, and using the two-dimensional model as well as twodimensional sparse recovery algorithms, retrieves the sparse representation of the wavefield in the frequency-wavenumber domain. The resulting representation can then be fed to the model in a forward problem to reconstruct the original fully sampled wavefield. To evaluate the efficacy of ST-SWA, we implement it on an experimentally acquired dataset containing guided wave propagation on a steel plate. We consider a variety of random and non-random subsampling scenarios with different practical implications to observe how the performance of ST-SWA is influenced by the choice of sampling strategies. We compare the results for these scenarios in terms of reconstruction accuracies, measured by correlation coefficient between the original fully sampled wavefield and the reconstructed one, as a function of the undersampling ratio in the temporal and spatial dimensions. We also study the choice of the number of sparse elements in the recovered representation and its impact on the reconstruction performance of ST-SWA. 5.2 Background In this section, we discuss the fundamental concepts and methodologies on which our wave reconstruction technique in this paper is founded. Specifically, we first briefly touch upon compressive sensing as a significant concept in signal processing. We then discuss sparse wavenumber analysis (SWA) and the model it utilizes to recover dispersion curves of guided waves. Next, we look into a method introduced in prior work, referred to as temporal sparse wavenumber analysis (TSWA), that is capable of retrieving dispersive characteristics of guided waves with temporally undersampled data. Finally, a brief study of two-dimensional sparse wavenumber analysis (2D-SWA) is presented, where we observe an extension of the SWA model into two dimensions that helps us better extract behavioral features in the waves and how a similar model can be used in our methodology for spatiotemporal data recovery. 84 5.2.1 Compressive Sensing In classical signal processing, the limits on sampling rates for accurate reconstruction of signals with provable guarantees are dictated by the well-known Nyquist-Shannon sampling theorem. Data compression for increased storage efficiency is achievable through transformation of an oversampled signal into a domain where complete representation of the signal is possible using a limited number of nonzero, nonnegligible coefficients. Compressive sensing (otherwise known as compressed sensing, or compressive sampling) [25, 26] was introduced as a way of combining data compression with fast and efficient data acquisition. Proper choice of representation bases, as well as a suitable sampling strategy provide the means for expedited data collection at sub-Nyquist rates, while accurate reconstruction assurances are backed by theoretical proofs. Compressive sensing has been employed in a variety of applications, including (but not limited to) communication systems and networks [27, 28], medical imaging [29, 30], and seismology [31]. In acoustic and ultrasonic signal processing, compressive sensing has been used in a range of research studies as well [32, 33, 34]. If we consider a fully sampled signal Z, that can be decomposed using a dictionary (set of bases/atoms) D, and a matrix of coefficients V, we can have Z = DV. (5.1) With an appropriate choice of the dictionary for a given signal, the matrix of coefficients can be sparse, i.e., containing few non-zero elements. Subsampling the signal using a sensing matrix S (with S and D being sufficiently incoherent), we obtain a set of measurements X = SZ and a selection of the bases Φ = SD. Consequently, within the compressive sensing framework the attempt is to solve an underdetermined system of linear equations of the following structure X = ΦV, (5.2) where X and Φ are known, and we seek to obtain the sparse matrix V. If V contains τ non-zero coefficients, it is called τ-sparse and we refer to τ as the sparsity value. The retrieval of the matrix V can be achieved using sparse recovery algorithms, such as basis pursuit [35], orthogonal matching pursuit [36], and several other methods proposed in the literature [37]. 85 5.2.2 Sparse Wavenumber Analysis In the field of ultrasonic guided waves, a method called sparse wavenumber analysis (SWA) [13] was proposed that utilizes compressive sensing to recover the sparse representation of isotropic guided waves in the frequency-wavenumber domain (i.e., dispersion curves) from limited spatial measurements. SWA incorporates the equation for Lamb wave propagation in infinitely long isotropic plate-like structures s 1 Gn (ω )e− jkn (ω )r , Z (r, ω ) = ∑ k ( ω ) r n n (5.3) with r being the distance from the transmission source, ω the angular frequency, n the mode of propagation, k n (ω ) the frequency-dependent wavenumber, and Gn (ω ) the complex amplitude of mode n. By defining the elements of a dictionary matrix as s 1 − jkn rm Φmn = e , k n rm (5.4) the SWA model can be rewritten as a matrix equation of the form (5.2). Assuming V in (5.2) is a sparse matrix, the known information (wavefield samples X, and the dictionary Φ) can be fed into a sparse recovery algorithm, such as orthogonal matching pursuit (OMP). This results in the recovery of dispersion curves in matrix V, which can then be used to reconstruct the entire wavefield. Figure 5.1 depicts the sparse representation recovered using SWA, and Figure 5.2 shows a snapshot of a reconstructed wavefield from a limited number of spatial grid points. These figures are obtained through implementation of SWA on the experimental dataset described later in the paper. More details on SWA and the reconstruction procedure are presented in [13]. 5.2.3 Temporal Sparse Wavenumber Analysis SWA retrieves dispersion curves from a dataset with incomplete information in the spatial domain. In [14], we presented a method, which we refer to as temporal sparse wavenumber analysis (TSWA), using which we can recover dispersion curves from a temporally undersampled wavefield. 86 TSWA employs an equation for Lamb wave propagation in the time-wavenumber domain Z (t, k) = ∑ Hn (k)e jω (k)t . n (5.5) n Similar to SWA, a sparse recovery algorithm (such as OMP) can be utilized to retrieve the sparse dispersion curves (ωn (k )) from a set of wave samples, and then the recovered sparse representation can be used for full wavefield reconstruction. Figure 5.3 illustrates the sparse representation output of TSWA, and Figure 5.4 depicts a reconstructed time signal from a few time samples. These two figures are produced by applying TSWA to the experimental dataset used in this paper. Additional details on TSWA can be found in [14]. 5.2.4 Two-dimensional Sparse Wavenumber Analysis A drawback associated with SWA is that it fails when applied to more complex anisotropic structures. Taking direction-dependency of wave propagation into consideration, the SWA model was extended into two dimensions, creating a framework referred to as two-dimensional sparse wavenumber analysis (2D-SWA) [15]. In the equation representing the 2D-SWA model Zq = Φ1 Vq Φ2T , (5.6) we have Zq as the frequency-domain wavefield at each frequency f q , the left dictionary Φ1 , the right dictionary Φ2 , and the matrix Vq containing the sparse representation in the two-dimensional wavenumber domain. The left and right dictionaries are defined as follows T Φ1 = e− jyky , T Φ2 = e− jxkx , (5.7) (5.8) with y, ky , x, and kx , being the vectors containing vertical coordinates, vertical wavenumbers, horizontal coordinates, and horizontal wavenumbers, respectively. The two-dimensional model in 2D-SWA (later used in a similar wave reconstruction framework called polar sparse wavenumber analysis (PSWA) [16]) allows for undersampling in two different dimensions. Therefore, we will employ this model in our methodology (presented in the following section) to make information retrieval in time and space possible. 87 5.3 Methodology As discussed in the previous section, the proper choice of a model can help us with simultaneous information retrieval in time and space. In this section, we present details of our methodology for guided wave reconstruction in the two-dimensional time-space domain. We then discuss our derivation of Nyquist parameters for a given guided wavefield. 5.3.1 Spatiotemporal Sparse Wavenumber Analysis We refer to the technique used in this paper to recover guided waves from temporally and spatially undersampled data, as spatiotemporal sparse wavenumber analysis (ST-SWA). ST-SWA employs the two-dimensional model akin to that of 2D-SWA with the choice of left and right dictionaries similar to those of SWA and TSWA. In other words, for the data matrix X, the spatial dictionary Φs , the sparse representation matrix V, and the temporal dictionary Φt , we have the relationship X M×T = Φs M× N V N ×Q ΦtTQ×T , (5.9) where T Φs M× N = e− jsk , (5.10) and T ΦtT×Q = e− jtω . (5.11) In these equations, s is a vector containing the distances of spatial grid points from the transmitter, k consists of a range of wavenumbers, t contains a range of times, ω comprises a range of frequencies, and superscript T denotes the transpose operator. Also, M, T, N, Q are the numbers of spatial measurements, temporal measurements, wavenumbers, and frequencies, respectively. Figure 5.5 illustrates the reconstruction procedure using ST-SWA. We organize the guided wavefield data in a matrix with time instants as rows and distances as columns. Next, we remove unknown elements from this matrix through a subsampling strategy, to obtain an incomplete (partial) wavefield, representing the measurements. In practice, we set these elements to zero so that the data can be stored as a sparse matrix. The values are unimportant though as the following sparse recovery algorithm will ignore those values. The spatial dictionary Φs and temporal dictionary Φt are then built using the 88 vector distances, wavenumbers, time instants, and frequencies. The partial wavefield data X and dictionaries Φs , Φt together with the set of known elements are then fed into the two-dimensional sparse recovery algorithm (a modified 2D-OMP introduced in [15]) to recover the sparse representation of the wavefield in the frequency-wavenumber domain (i.e., dispersion curves). Finally, the recovered sparse representation and the dictionaries corresponding to the fully sampled set of time samples and grid points are used to solve the forward problem to reconstruct the entire wavefield. 5.3.2 Computational Complexity Among the guided wave reconstruction methodologies in the literature, none, to the best of our knowledge, has proposed simultaneous data retrieval in time and space, hence rendering a direct comparison infeasible. However, a distinctive feature of ST-SWA (as well as 2D-SWA), is the use of a two-dimensional model for sparse recovery. The OMP algorithm would be a common choice for a one-dimensional (1D) model. Let us assume a left dictionary of size L1 by L2 and a right dictionary of size R1 by R2. For a 1D model, a combined dictionary (Kronecker product) of size L1×R1 by L2×R2 needs to be built. Considering a scenario where the sparsity value is negligible when compared to the dimensions of the dictionary/dictionaries, the computational complexity of the 1D-OMP algorithm would be on the order of O( L1 × R1 × L2 × R2), while for the 2D-OMP algorithm (details can be found in [15]), this would approximately amount to O( L1 × R1 + L2 × R2). 5.3.3 Nyquist Parameters In conventional signal processing, the Nyquist-Shannon sampling theorem states that we can expect to achieve perfect reconstruction from a downsampled version of a bandlimited signal as long as the sampling rate meets or exceeds the Nyquist rate, i.e., two times the highest frequency content in the signal. We utilize the implications of this theorem as a performance metric for our wave reconstruction methodology. We arrange a given dataset of guided wave signals in a two-dimensional matrix, where the rows correspond to time samples and the columns correspond to different points on a spatial grid with respect to their distance from the transmission source. Therefore, we have a temporal dimension and a spatial dimension to our data, each of which has its unique Nyquist parameters. If f max is the maximum frequency of the signal and f TC = 2 f max is the critical sampling rate in the 89 temporal dimension, according to the Nyquist-Shannon sampling theorem, we can have a maximum sampling period permissible by the Nyquist criterion (Tmax ) defined by Tmax = 1 = f TC 1 2 f max . (5.12) The ratio between the sampling period of the fully sampled wavefield (To ) and Tmax is then defined by To . Tmax R TTot = (5.13) This ratio will determine the fewest time samples required for perfect reconstruction according to the Nyquist criterion. Therefore, we have NTNyq = R TTot × NTTot , (5.14) where NTTot is the total number of time samples in the original fully sampled signal, and NTNyq is the number of time samples at critical sampling rate. Similarly in the spatial domain, if k max is the maximum wavenumber of the signal, then k SC = 2k max is the critical angular sampling rate in the spatial dimension, according to the Nyquist-Shannon sampling theorem. Thus, given the relationship between wavenumber and wavelength (λ) [38] k= 2π , λ (5.15) we can define the critical average (maximum permissible) distance between grid points dmax in each horizontal/vertical direction with respect to the critical sampling rate in space (k SC = 2k max 2π ) as dmax = 1 k SC = π k max . (5.16) Therefore, if Ds is the spatial sampling period of the original fully sampled wavefield (which is the same for both horizontal/vertical directions for the dataset in this paper), we can have the following ratio to compute the number of spatial samples required by the Nyquist criterion for perfect reconstruction in space RSTot = Ds dmax 2 . (5.17) Note that, the ratio here is squared since we are computing the number of points on a two-dimensional spatial grid, and the distances correspond to each horizontal/vertical 90 dimension. Finally, if NSTot is the total number of grid points in the original fully sampled signal, we can obtain the number of points on a critically sampled grid (NSNyq ) as NSNyq = RSTot × NSTot . 5.3.4 (5.18) Defining Maximum Frequency and Wavenumber To be able to use the Nyquist-Shannon sampling theorem, we first need to define maximum temporal/spatial frequencies. For a perfectly band-limited signal, this would be the highest frequency or wavenumber with a non-zero magnitude. However, this does not apply to most signals in practice. Consequently, we will utilize the 99 % energy measure to obtain maximum frequencies or wavenumbers. In other words, we consider a point in the frequency/wavenumber spectrum of the signal by which 99 % of the entire energy (i.e., sum of the squared values of all the elements) of the spectrum is confined. We define this point to be the maximum frequency/wavenumber content of the signal. We now describe this procedure for each of the two dimensions. In the time dimension, we transform the time signal for every grid point into the frequency domain. We then compute the average of the magnitudes of the frequency response for all these grid points to obtain a frequency response for the entire wavefield. Next, we calculate the energy of the resulting spectrum and define the maximum frequency as the point below which 99 % of this energy is bound. In the space dimension, we first obtain frequency slices of the wavefield and subsequently perform a two-dimensional Fourier transform on each slice resulting in a twodimensional wavenumber response (also known as k-space). The magnitude of these responses are then averaged together creating a k-space representation containing wavenumber content of the entire wavefield. By computing the energy of this representation, we can find a circle enclosing an area with 99 % of this energy. The radius of this circle is defined to be the maximum wavenumber in the wavefield. 5.4 Experimental Setup We make use of an experimentally acquired dataset to evaluate the performance of ST-SWA. Here, we present the details of this dataset. A 100 cm by 100 cm SAE 304 stainless 91 steel plate of 2 mm thickness is utilized as the wave propagation medium. A 10 cm by 10 cm area at the center of the plate constitutes the scanning region. The fully sampled wavefield comprises a grid of 200 × 200 (horizontal × vertical) points uniformly distanced at 0.05 cm along each direction. A scanning Q-switched Nd:YAG diode-pumped solid state pulsed laser (Advanced Optowave, custom made) is employed to excite the guided waves, while a piezoelectric transducer (Fuji Ceramics, M304A) is used for measurement purposes. The principle of acoustic reciprocity indicates that in case the transmitter and the receiver exchange roles, the acoustic signal remains unaltered. Therefore, the piezoelectric could be regarded as the transmitter and the laser as the receiver. Considering the size of the scanning region with respect to the size of the entire plate and within the time frame used in this work (321 time samples, 200ns apart, i.e. a sampling rate of 5MHz), no reflections from the edges are visible in this dataset. Moreover, the piezoelectric as the source of propagation lies outside the scanning region (at x = -70 mm, y = 170 mm). The experimentally scanned area is illustrated in Figure 5.6. 5.5 Results In this section, we study the performance of our methodology in reconstructing undersampled ultrasonic data. We first compute the Nyquist sampling rates in both time and space for our known wavefield dataset. The Nyquist rates provide a baseline for evaluating the performance of the spatiotemporal sparse wavenumber analysis. That is, we can in theory perfectly reconstruct a signal when both space and time are sampled at the Nyquist rate. We compute the reconstruction accuracy for a variety of spatiotemporal sampling strategies. Specifically, we consider the following strategies: fully random element-wise sampling over the two dimensions, independent random sampling in time and space, random sampling in time followed by random sampling in space, random sampling in space followed by random sampling in time, jittered sampling, uniform sampling both in time and space, and undersampling only in the spatial dimension. The details of each of these cases are presented later in this section. 92 Finally, we study how the choice of sparsity value (number of non-zero elements in the sparse representation) influences the performance of ST-SWA. 5.5.1 Temporal and Spatial Nyquist Rates Figure 5.7 depicts the magnitude of the frequency spectrum (positive frequencies) averaged over all of the grid points. The magnified circular point in the figure shows the maximum frequency content of the dataset based on the 99 % energy criterion. Figure 5.8 illustrates the average of k-space representations for all the frequencies in the dataset. The dashed circle shows the 99 % energy circle, the radius of which determines the maximum spatial frequency (wavenumber) of the wavefield. Note that, we disregard high wavenumber noise-like contents of our guided wavefield through setting all the values below 2 % of the highest magnitude in this representation to zero. Tables 5.1 and 5.2 show the Nyquist parameters in time and space dimensions, respectively. In Table 5.1, f max represents the maximum frequency in the wavefield, Tmax is the maximum time period permissible by the Nyquist criterion, NTtot shows the number of time samples in the fully sampled wavefield, R Ttot is the ratio between the fully sampled and the critically sampled wavefield in time, and NTNyq shows the total number of time samples required to comply with the Nyquist criterion. Similarly in Table 5.2, k max shows the maximum wavenumber in the wavefield, dmax represents the maximum distance permissible by the Nyquist criterion, NStot is the number of spatial grid points in the fully sampled wavefield, RStot shows the ratio between the fully sampled and the critically sampled wavefield in space, and NSNyq is the total number of grid points required with respect to the Nyquist criterion. The values in the rightmost columns, i.e. NTNyq and NSNyq , represent the temporal and spatial critical sampling, and the product of the two yields the minimum required number of samples in the spatiotemporal domain. In other words, we need at least 56 time samples, 3,105 grid points, or 173,880 spatiotemporal samples to comply with the Nyquist-Shannon sampling theorem. We use these numbers to calculate undersampling ratios for each dimension individually, or both simultaneously (depending on the sampling strategy). 93 Therefore, we define the temporal, spatial, and spatiotemporal undersampling ratios as follows: UR T = NT , NTNyq (5.19) URS = NS , NSNyq (5.20) NST , NTNyq × NSNyq (5.21) URST = where NT , NS , and NST are the number of utilized samples in the temporal, spatial, and the spatiotemporal domains, respectively. With respect to these ratios, we can demonstrate how low (in terms of the number of samples) we can go while maintaining a decent reconstruction performance with ST-SWA. 5.5.2 Guided Wave Reconstruction via ST-SWA We investigate the efficacy of ST-SWA by implementing it on the guided wave dataset for a variety of subsampling scenarios. We evaluate the performance of ST-SWA by observing the reconstruction accuracy (in terms of correlation coefficient between the original wavefield and the reconstructed one) with changes in undersampling ratios. When random sampling is involved, we perform a Monte Carlo analysis with 5 runs per each undersampling ratio to compute the mean result. Note that, we perform ST-SWA in all of the following cases with a sparsity of 20. As an illustrative example, Figure 5.9 shows a time snapshot of the reconstructed wavefields using ST-SWA with element-wise random sampling. Figure 5.9(a) depicts the original fully-sampled wavefield. Figures 5.9(b)-(c) show the reconstructed wavefield (with 90 % accuracy) and its difference with the original wavefield, respectively. Similarly, Figures. 5.9(d)-(e) and (f)-(g) depict reconstruction accuracies of 75 % and 39 %, respectively. Note that, we do not expect a perfect reconstruction due to noise and other effects (e.g., weak anisotropic and inhomogeneous characteristic) that make this data not perfectly match the sparse wavenumber model [39]. In the following subsections, different sampling scenarios are discussed. In each of these scenarios, the guided wavefield data is represented as a two-dimensional matrix containing information in the spatiotemporal domain (arranged with respect to distance of the grid points from the transmitter at different time instants, i.e., time instants in 94 the rows and distances in the columns of the matrix). Figure 5.10 illustrates the matrix representation for each of the sampling strategies. These sampling strategies are as follows (short names in parentheses will be used later in the paper for reference): 1. Element-wise random sampling in spatiotemporal domain (Random). 2. Independent random sampling in time and in space (Time-Space). 3. Random sampling in time followed by random sampling in space (Time + Space). 4. Random sampling in space followed by random sampling in time (Space + Time). 5. Jittered sampling in time and space (Jittered). 6. Uniform sampling (Uniform). 7. Undersampling only in space (Pure-Space) We will describe each of these strategies in depth in the following subsections. 5.5.2.1 Scenario 1: Element-wise random sampling in spatiotemporal domain (Random) For this strategy, we randomly subsample across all spatiotemporal samples. That is, all of the measured samples are from a set of random points in space and time. In theory, random sensing can be a near-optimal sampling strategy for compressive sensing [26]. Figure 5.10(a) depicts the matrix representation of this sampling method where the sampled elements are randomly scattered over the entire matrix. Figure 5.11 shows reconstruction accuracies for different spatiotemporal undersampling ratios for all the sampling scenarios. Random sampling strategies are able to reach accuracies of 90 % and above. Among these, element-wise random sampling (in both dimensions or only in one dimension as in scenario 7) appears to perform slightly better due to maximum inherent degree of randomness. The correlation coefficients between the original and the reconstructed wavefields converge to about 0.9 for about 60 % of the spatiotemporal Nyquist rate. At a 60% undersampling rate, we only require approximately 100,000 samples in time and space. By comparison, the Nyquist rate requires 173,880 spatiotemporal samples, and the original data contained 12,840,000 spatiotemporal samples. 95 Note that in all of the scenarios, accuracy fails to converge to 1, even for the undersampling ratios above the Nyquist rate. This could be attributed to a number of factors. First, we are utilizing a model to account for an isotropic medium, which assumes a plane-wave-like propagation. This, however, does not perfectly match with the properties of the experimental data, which exhibits slight anisotropy, and nonplane-wave-like behavior, as well as other imperfections that may have been caused in the acquisition process. Moreover, the sparsity value chosen for this study may not be the perfect value for the particular dataset used in this paper. Lastly, in the process of deriving the Nyquist rates, the highest frequency content of the waves in each of the temporal and spatial dimensions has been derived by considering a 99 % energy measure, which results in an approximation of the actual Nyquist rates. 5.5.2.2 Scenario 2: Independent random sampling in time and in space (Time-Space) Compressive sensing methods perform optimally under completely random sampling. Yet, completely random sampling is often not practical. Therefore, this and other random sampling strategies used here, consider more pseudorandom sampling strategies. For this strategy, as opposed to the element-wise sampling strategy, we can generate a random set of time and space indices (each separately) and obtain samples solely from the corresponding time instants and grid points. Compared to the previous case and from a matrix point of view, in this scenario we will always have some rows and some columns containing all zeros. Figure 5.10(b) depicts this sampling strategy on the data matrix. A number of rows and columns are randomly chosen and the intersecting elements are utilized as the samples. Figure 5.12(a) shows a surface plot of the reconstruction correlation coefficient as a function of different temporal and spatial undersampling ratios. Increasing the number of samples in each of the dimensions results in an increase in the reconstruction accuracy. To compare the results with those of the previous scenario, we can look at reconstruction accuracies lying on the antidiagonal of the surface plot where the temporal and spatial undersampling ratios would be approximately the same, and proceeding from the bottom left square in the figure to the top right, spatiotemporal undersampling would increase. We follow this procedure for scenarios 3, 4, 5, and 6 as well, where undersampling is 96 performed separately in each dimension. The resulting line plot can be seen in Figure 5.11. The results appear to follow a similar trend to those of the previous case, and seem to converge to about 0.9. However, since the degree of randomness is lower in this scenario compared to scenarios 1, 3, 4, and 5 (as visually noticeable in Figure 5.10), the performance in this case seems to be slightly inferior. 5.5.2.3 Scenario 3: Random sampling in time followed by random sampling in space (Time + Space) For this strategy, we first randomly select samples from the time domain and then follow the same procedure in the space domain. As opposed to the previous scenario, we no longer have the same set of (randomly selected) spatial grid points for all time instants. This process can be viewed in Figure 5.10(c) where a set of rows are randomly chosen and subsequently in each of these rows a set of distinct randomly selected elements are utilized. Figure 5.12(b) shows the surface plot of the reconstruction accuracy as a function of varying temporal and spatial undersampling ratios for this case. A very similar pattern to the previous case is observable and expectedly the results improve by increasing the number of samples used in each dimension. The line plot for this scenario is generated in a similar fashion to the previous case, and the result can be seen in Figure 5.11. The line corresponding to this sampling strategy seems to closely follow the lines for scenario 1 and 2, while generally being situated in between the two. This can be associated with the fact that the sampling process is slightly more random in this case compared to scenario 2, while being not as fully random as scenario 1. 5.5.2.4 Scenario 4: Random sampling in space followed by random sampling in time (Space + Time) Similar to the previous strategy, we use another pseudorandom sampling strategy in this subsection. However, in this scenario we first sample in the spatial domain, and then for each grid point, we use a random set of time samples to retrieve the data in the ST-SWA framework. The matrix outlook for this case is shown in Figure 5.10(d) where this time a 97 random set of columns are selected first, and subsequently in each of these columns we choose a set of distinct random elements to be utilized. Results for this case, as shown in Figures 5.12(c) and 5.11, bear expected resemblance to those of the previous scenario. 5.5.2.5 Scenario 5: Jittered sampling in time and space (Jittered) For this scenario, we utilize the jittered sampling as used in [40] and [41]. In particular, in each of the time and space dimensions, we create a uniformly spaced set of points undersampled to the ratio of interest, and then perturb each sample by a random value within the distance range between two consecutive samples. We then resample both dimensions and find the closest existing samples via nearest neighbor interpolation. The sampling outlook on a data matrix for this scenario can be seen in Figure 5.10(e). Accuracy results for this case are shown as a surface plot in Figure 5.12(d). The line plot related to this scenario is also shown in Figure 5.11. The results show a generally similar pattern to those of other random sampling strategies (specifically, scenarios 3, and 4). However, it slightly outperforms scenario 2. This could be attributed to an increased level of randomness as well as the better covering of different regions in both dimensions. 5.5.2.6 Scenario 6: Uniform sampling (Uniform) In this case, we obtain uniformly distanced samples in space and time for various undersampling ratios. Due to the absence of randomness in the sampling process, no Monte Carlo analysis is conducted. Figure 5.10(f) depicts how uniform sampling in both dimensions looks on a data matrix. In practice, to be able to have any arbitrary number of samples, we resample the grid and find the closest existing samples using nearest neighbor interpolation. Figure 5.12(d) illustrates the surface plot for this sampling strategy. Compared to the previous cases, the pattern seems to be slightly less organized and less predictable. The corresponding line plot is presented in Figure 5.11. Not surprisingly, excluding very low undersampling ratios where the performance is highly dependent on whether any samples from the information bearing parts of the wave are used (which may not happen in random scenarios), uniform sampling is outperformed by all the other scenarios. Nonetheless, even though ST-SWA as a compressive sensing framework is not meant to be performed with 98 uniform sampling, at rates above the Nyquist rate (100 % undersampling ratio and above) relatively high reconstruction accuracies (0.8 and above) is achievable. 5.5.2.7 Scenario 7: Undersampling only in space (Pure-Space) In this scenario, we consider a situation where we utilize the entire temporal information and perform random undersampling only in the spatial dimension. A visualization of this sampling strategy is illustrated in Figure 5.10(g). Reconstruction accuracies for this case can be seen in Figure 5.11. The results for this scenario are very similar to the first scenario. This is expected since we are performing random sampling, albeit in the spatial dimension only (which effectively renders the recovery process similar to a regular one-dimensional SWA). 5.5.3 Effect of Sparsity The choice of sparsity value for the sparse representation to be recovered via the STSWA algorithm impacts the quality of the reconstruction. Too few sparse elements yields poor results due to the different modes and fundamental elements of the wavefield being underrepresented in the sparse representation. Whereas too many sparse elements can lead to spurious artifacts that are nonexistent in the original data. This can also be viewed from a learning perspective where the undersampled data could be regarded as the training data, and the entire wavefield can be regarded as the test data. From this viewpoint, the utilization of too many sparse elements would result in extensive overfitting to the sampled data, and result in a less accurate generalization to the test samples. Figure 5.13 illustrates the recovered sparse representation using ST-SWA for a choice of 50 sparse elements. The two curves illustrated in this plot correspond to the frequencywavenumber dispersion curves for the guided Lamb waves. Due to the absence of spurious sparse elements, overfitting may not be an issue at this point. Figure 5.14 shows the reconstruction accuracy in terms of correlation coefficient as a function of the sparsity value. This result is for element-wise random sampling at about 57 % of the spatiotemporal Nyquist rate (i.e., 100000 spatiotemporal samples). There is a visible rising trend with convergence around values of 20 to 40. Based on observations in prior work [16], we expect the results to demonstrate signs of overfitting at very high 99 sparsity values. The sparsity value can be estimated based on some a priori knowledge of the wavefield. Specifically, since the sparse representation in ST-SWA indicates the frequencywavenumber dispersion curves, we can estimate the sparsity by multiplying the number modes we expect to witness in the wavefield by the number of frequency samples (positive and negative) used in the dataset. This is a conservative estimate since some modes may have insignificant magnitudes at certain frequencies within the range of interest. 5.6 Conclusions In this paper, we demonstrated the potential of a compressive sensing based methodology, called spatiotemporal sparse wavenumber analysis (ST-SWA), to accurately reconstruct guided wavefield data that are undersampled both in time and space. We evaluated the performance of ST-SWA for a variety of subsampling scenarios with different implications for practical implementations. For strategies where random sampling is involved, the results generally indicate the possibility of accurate reconstruction (correlation coefficient of around 0.9) for sampling rates above 60 % of the spatiotemporal Nyquist critical sampling rate. Note that, in many practical applications, data are heavily oversampled. Consequently, in comparison, these rates would amount to considerably lower percentages. Furthermore, depending on the application, an adequate level of reconstruction accuracy may vary. In a monitoring paradigm, for example, the correlation coefficient should be above the noise floor to detect wavefield variations caused by damage formation in the structure [19, 41]. Aside from correlation coefficient, other measures for accuracy can be (and have been) used for different applications [18]. Finally, the impact of the choice of sparsity value was investigated. Results suggest as long as overfitting is not encountered, slight improvement in reconstruction accuracies can be observed at higher sparsities. Future work will focus on modifications to the ST-SWA model for generalization to more complex propagation media, as well as incorporating ST-SWA with damage detection and localization techniques. 100 5.7 Acknowledgments This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0126. We would also like to thank Chen Ciang Chia from the Department of Aerospace Engineering in Universiti Putra Malaysia for providing guided wave data used in this paper. 101 Figure 5.1. Sparse representation recovered using SWA. More details on the derivation of this sparse representation can be found in [13]. 102 Figure 5.2. A snapshot in time showing SWA reconstruction from spatially undersampled data: (a) the undersampled wavefield (circles represent spatial samples), (b) the reconstructed wavefield using SWA, and (c) the original fully sampled wavefield. More details on the reconstruction process can be found in [13]. 103 Figure 5.3. Sparse representation recovered using TSWA (positive frequencies). More details on the derivation of this sparse representation can be found in [14]. 104 Figure 5.4. TSWA reconstruction from temporally undersampled data: (a) undersampled time domain signal at a certain grid point, (b) the reconstructed signal using TSWA, and (c) the original fully sampled signal. More details on the reconstruction process can be found in [14]. 105 Figure 5.5. ST-SWA reconstruction process. 106 Figure 5.6. The scanned area on the SAE 304 stainless steel plate. 107 Figure 5.7. Normalized magnitude of the frequency response of the wavefield, averaged over all grid points. The magnified circular point delineates the maximum frequency. 108 Figure 5.8. Two-dimensional wavenumber representation of the wavefield averaged over all frequency slices. The dashed white circle bounds 99 % of the energy in this representation and its radius determines the maximum wavenumber. 109 Table 5.1. Temporal Nyquist parameters of the wavefield. f max 436 kHz Tmax 1.14 µs NTTot 321 R TTot 17.44% NTNyq 56 110 Table 5.2. Spatial Nyquist parameters of the wavefield. kmax 1509 m−1 dmax 2.08 mm NSTot 40,000 RSTot 7.76% NSNyq 3105 111 Figure 5.9. Time snapshot of reconstructed wavefields using ST-SWA with element-wise random sampling at different undersampling ratios with different accuracies (acccuracy here is defined in terms of correlation coefficient between the original fully sampled wavefield and the reconstructed wavefield using ST-SWA): (a) original fully sampled wavefield, (b),(d),(f) reconstructed wavefield at undersampling ratios of 86%, 40%, and, 23%, respectively, and (c),(e),(g) wavefields resulting from subtracting (b),(d),(f) from (a). 112 Figure 5.10. Matrix representation of sampling strategies 1-7 shown in (a)-(g), respectively. The lighter (yellow) pixels represent the chosen samples, while the darker (blue) ones are set to zero. 113 Figure 5.11. Reconstruction accuracy (correlation coefficient) as a function of spatiotemporal undersampling ratio for different sampling strategies. For scenarios 2, 3, 4, 5 and 6, where undersampling is done separately in each dimension, points are chosen such that ratios are approximately equal in both dimensions (i.e., points on the antidiagonal of the surface plots in Figure 5.12). 114 Figure 5.12. Surface plots showing reconstruction accuracy (correlation coefficient) as a function of undersampling ratios in time and space for sampling scenarios (a) 2, (b) 3, (c) 4, (d) 5, and (e) 6. The colorbar shows the accuracy values in terms of correlation coefficient. 115 Figure 5.13. Sparse representation (sparsity value 50). Elements from the second mode of propagation in the wavefield begin to appear at higher sparsity values. 116 Figure 5.14. Changes in reconstruction accuracy (correlation coefficient) for different sparsity values. The accuracy tends to increase with the number of sparse components before it reaches a plateau. 117 5.8 References [1] M. Mitra and S. Gopalakrishnan. 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CHAPTER 6 GUIDED WAVE CHARACTERIZATION AND SOURCE DETECTION IN THE PRESENCE OF MULTIPATH REFLECTIONS An important category of nondestructive evaluation (NDE) and structural health monitoring (SHM) techniques involves the use of ultrasonic guided waves. The properties of guided waves tend to change with the medium of propagation. Extracting information on these properties is an important requirement for proper analysis in many applications. Dispersion curves are generally used to describe the dispersive characteristics of guided waves in different structures. A lot of data driven methods for obtaining dispersion curves are not very successful when reflections from scatterers or the edges of the medium are present in the measurements. In this paper, we present an iterative compressive sensing based method through which estimations of dispersion curves and sources of the reflections improve at each stage, and this can be achieved by utilizing measurements from a limited number of sensors. We present preliminary results of implementing this method on an experimental set of guided wave measurements. 6.1 Introduction Guided wave based methods are commonly used and widely investigated in nondestructive evaluation (NDE) and structural health monitoring (SHM) applications. Understanding and finding ways to mitigate the complexities of these waves, thus facilitating analysis, have been a significant subject of study. These complexities include multimodal behavior and frequency dispersivity [1]. One of the well-known tools for understanding these characteristics are dispersion curves. These curves can be generated analytically [2], given enough information on the material and geometrical properties of the structure. For multilayered structures, several methods, including transfer matrix method, and global matrix method, are used to calculate the dispersion curves [3]. 121 The information embedded in dispersion curves generally applies to unidirectional (traveling in one direction) waves in which no reflections from the edges or any type of scatterers are visible. This is the case in pristine (damage-free) infinitely long structures. However, when the structure is limited in size or contains damage, reflected waves propagating in directions other than that of the original wave appear in the data, which further complicates the wavefield and makes it very difficult to analyze. Multipath reflections have been used in several studies for wavefield imaging and damage localization [4, 5, 6]. In particular, damage localization in data corrupted with multipath interference using matched field processing has been successfully performed [7] using a method called sparse wavenumber analysis. Sparse wavenumber analysis (SWA) was introduced in [8], as a signal processing framework for recovery of Lamb wave dispersion curves in isotropic plate-like structures. SWA has since been used for damage localization with matched field processing [7], and it has proven successful as a predictive model for other types of waves such as standing waves on a string [9]. Its model has also been modified to apply to anisotropic wavefields resulting in techniques referred to as anisotropic sparse wavenumber analysis (ASWA) [10], two-dimensional sparse wavenumber analysis (2D-SWA) [11], and polar sparse wavenumber analysis (PSWA) [12]. It has similarly been utilized for dispersion curve recovery in the time domain via a method called temporal sparse wavenumber analysis [13] and has also been extended to a two-dimensional framework named spatiotemporal sparse wavenumber analysis (ST-SWA), where data retrieval is feasible in both time and space. The model used in SWA is based on the theoretical equation for Lamb waves in infinitely large plates. Therefore, no multipath interference is assumed in SWA. In this paper, we present a two-stage method called multipath sparse wavenumber analysis, based on prior work, and we implement it on experimentally collected datasets to evaluate its performance. 6.2 Background In this section, we discuss the theoretical background and previous efforts leading to the methodology used in this paper. In particular, we will first discuss sparse wavenumber analysis as a guided wave characterization method, which is applicable to guided waves 122 without multipath reflections. We then present some previous work that comprise the fundamentals of the methodology in this paper. 6.2.1 Sparse Wavenumber Analysis Sparse wavenumber analysis (SWA) is a compressive sensing based technique that is capable of obtaining dispersion curves for Lamb waves propagating in an infinitely bound isotropic plate. Compressive sensing [14] is a sampling concept in signal processing that provides mathematical guarantees and conditions for perfect reconstruction of signals sampled at rates far below the conventional Nyquist rate. It requires a set of bases suited to the signal of interest where a linear combination of a few elements of those bases are sufficient for reconstructing the entire signal. This is referred to as the sparsity condition, meaning the signal is sparse in the domain of the chosen bases and can be reconstructed using only a few non-zero coefficients. It also necessitates a proper sensing approach that satisfies the incoherence condition and subsequently the restricted isometry property (RIP) [15]. If the conditions hold, a sparse recovery algorithm can be employed to solve the underdetermined system of linear equations and recover the sparse coefficients. Different sparse recovery techniques have been introduced and investigated in the literature [16], among which basis pursuit denoising (BPDN) [17], and orthogonal matching pursuit (OMP) [18] have been used in an SWA framework [19]. SWA uses the equation for Lamb wave propagation in isotropic plates s 1 Gn (ω )e− jkn (ω )r , X (r, ω ) = ∑ k n ( ω )r n (6.1) where n indicates the mode of propagation, its corresponding wavenumber is shown by k n (ω ), and its complex amplitude by Gn (ω ), both changing with frequency. This frequency dependency is depicted by the dispersion curves. This equation can be formulated in a matrix format as X = ΦV, (6.2) where X is an M × Q matrix containing the sampled wavefield in the frequency domain, Φ, an M × N matrix comprising basis vectors, and V, an N × Q matrix indicating the sparse dispersion curves. 123 The underdetermined system in (6.2) can be solved using a sparse recovery algorithm of choice. In the methodology described later in this paper, we use OMP for sparse recovery purposes. 6.2.2 Decomposition of Multipath Elements in Lamb Waves As stated before, SWA model does not account for multipath reflections and depending on the level of multipath interference in the signal and the number of utilized sensors, SWA is capable of recovering an estimate of the dispersion curves with relatively good accuracy (SWA treats multipath interference as noise). This was shown in [20], where using SWA, estimated dispersion curves are obtained and subsequently used to find the sources of reflections. This is achieved through creating a dictionary of frequency responses for a set of distances, and finding the sparse coefficients that combine these frequency responses to compose the measured signals. The resulting equation will be of the following form x = D(r )w, (6.3) where D(r ) contains a set of frequency responses (generated with the recovered dispersion curves) for a set of distances(r), x is the measured signal, and w denotes the weights for the distances, which is assumed to be sparse. With this sparsity assumption, the weights can be retrieved using sparse recovery algorithms. 6.2.3 Multipath Sparse Wavenumber Analysis In prior work [21], a model based on SWA, referred to as multipath sparse wavenumber analysis (MP-SWA) was presented that is capable of recovering dispersion curves for guided waves corrupted with multipath interference. This is achievable by extending the SWA model to iteratively recover a mixing matrix U and the dispersion curves V in X = UΦV, (6.4) where Φ is a dictionary for a set of distances and wavenumbers. This technique is applied to simulation data and is shown to improve the visual quality of dispersion curves with good initial conditions on the virtual sources of reflections 124 obtained from ray tracing techniques and method of mirrors. 6.3 Methodology This section contains a description of the methodology used in this paper for obtaining dispersion curves in the presence of multipath reflections, based on the models and concepts discussed in the previous section. 6.3.1 Two-Stage Multipath Sparse Wavenumber Analysis The methodology used in this paper for guided wave characterization and dispersion curve recovery is a combination of the methods discussed in the previous section. This is a two-stage MP-SWA, which can be summarized as follows: 1. First, regular SWA is implemented on the dataset to generate an initial estimate of the dispersion curves. 2. This initial estimate is used to create a dictionary D of frequency responses for a set of distances. To achieve this, we first build a matrix Φ containing the following elements s Φmn = 1 − jkn rm e . k n rm (6.5) By multiplying the dispersion curves matrix V and Φ, we obtain the dictionary DT = ΦV, (6.6) where the superscript T denotes the transposition operator. 3. The dictionary D and the wave measurements X are used in the following compressive sensing framework to recover the weights W corresponding to the distances of the sources of reflections X = DW. (6.7) 4. The recovered weights in W and the matrix Φ form the matrix H as H = Φ T W. (6.8) 125 5. Matrix H and the measurements X will be utilized to retrieve new dispersion curves for the next iteration. 6.4 Experimental Setup In this section, we present the details of two experimentally collected datasets that we utilize to evaluate our methodology. Dataset 1 is an isotropic wavefield on a steel plate, sampled on a grid of spatial points. Dataset 2 contains guided waves on an aluminum plate measured by randomly placed transducers. 6.4.1 Dataset 1 This dataset contains isotropic guided waves traveling in a 2 mm thick SAE 304 stainless steel plate. The plate is 100 cm by 100 cm and the scanning region is chosen as a 10 cm by 10 cm square at the center of the plate so that reflections from the edges of the plate do not appear in the dataset within the scanning time frame. A set of 40000 points (200 x 200), at a 0.05 cm distance in each direction, comprise the fully sampled grid. A scanning Q-switched Nd:YAG diode-pumped solid state pulsed laser is used to generate the waves and a piezoelectric transducer is employed to collect the samples. The piezoelectric is located outside the scanning region at the (x = -70 mm, y = 170 mm) coordinates. Since the signal remains unaltered by exchanging the transmitter and the receiver (the principle of time reversal), the laser is considered as the receiver and the piezoelectric as the receiver. A reflection from a mass scatterer is present in the scanning region simulating a damage signature. 6.4.2 Dataset 2 This experimental dataset consists of 17 randomly placed sensors (PZT transducers) on a 0.284 cm thick aluminum plate of 1.22 m by 1.22m length/width. An iterative data measurement is performed by using one of the sensors as the transmitter and the other 16 as the receivers at each round. This amounts to 136 distances and 272 sensor measurements. A modulated chirp signal of 10 µs duration is excited by the transmitting transducer at each iteration. The receivers collect the data at a 10 MHz sampling rate. 126 6.5 Results The results of implementing our methodology on previously described datasets are presented in this section. We first show how we can reconstruct the damage signature in Dataset 1 as a multipath reflection element. We then present the results of implementing the described methodology on Dataset 2. 6.5.1 Multipath Identification with Dataset 1 Dataset 1, as described in the previous section, contains an isotropic wavefield with a scatterer in the scanning region, causing a reflected signal (damage signature) to appear. Here, we demonstrate how we can identify and reconstruct these two waves (main propagating wave and the reflected one) from 200 randomly chosen points on the 40000-point grid. For this purpose, we consider the following strategy: 1. We first obtain a subsampled wavefield by randomly selecting 200 points on the original fully sampled grid and feed it to the regular SWA algorithm to obtain an estimate of the dispersion curves. 2. We use (6.6) and the model in (6.7) together with OMP to obtain weights to corresponding to the two paths (main path and the damage reflection). 3. We use these weights to reconstruct their corresponding signals for each sensors. Figure 6.1 shows the recovered dispersion curves using SWA. Ignoring the spurious sparse elements in the representation, the two modes of propagation are accurately recovered. This is because the damage reflection is not strong enough to significantly corrupt the wavefield. Figure 6.2 illustrates the absolute value of the retrieved weights for a particular sampled grid point. Two sparse components are recovered, the stronger of which corresponds to the main wave and the second belongs to the damage reflection. Figure 6.3 and 6.4 show reconstructed signals for the two paths, on top of the original time signal at a particular spatial sample on the grid. The signal in Figure 6.4 corresponds to the damage signature. 127 6.5.2 Dispersion Curve Enhancement with MP-SWA Here, we implement the two-stage MP-SWA on Dataset 2. For this purpose, we first obtain an estimate of the dispersion curves using regular SWA and the entire set of sensor measurements. This is then used to find the weights corresponding to the sources of multipath reflections. These weights are then used as an initial condition for the MP-SWA algorithm. The two-stage MP-SWA algorithm is then employed on 20 randomly sampled sensor measurements (out of 272) for 30 iterations. For the algorithm, two scenarios are considered: • Scenario 1: In this scenario, we preserve the indices of the distances in the initial weights matrix by employing a modified OMP algorithm that assumes the indices of the bases are known and only updates their corresponding values. • Scenario 2: For this scenario, we use the normal OMP algorithm and let the indices of the distances in the weights matrix vary at each iteration. We then compare the recovered sparse representations with the result of directly applying SWA to the set of 20 sensor measurements. Figure 6.5 shows the retrieved dispersion curves for Scenario 1 at the first iteration. Since the indices are preserved for this scenario, the results for the other 29 iterations exhibit little to no visible variation. Figure 6.6 depicts the recovered dispersion curves for Scenario 2 at iteration number 30. Compared to the previous scenario, the results appear to be noisier and farther from ground truth. Figure 6.7 shows the results of directly implementing SWA on the set of 20 measurements. Compared to the outputs of MP-SWA, the inferior performance of regular SWA is noticeable in this figure. 6.6 Conclusion In this paper, we presented a method for retrieving dispersion curves of guided wave data containing multipath reflections, from a limited number of spatial sensor measurements. This iterative method involves obtaining rough initial estimates of the dispersion curves using regular SWA, via which a sparse source detection procedure provides an 128 approximate estimate of the real and virtual sources of the multipath reflections. These two steps can iteratively be performed until an improvement is observed in the recovered dispersion curves (when compared to the known theoretical dispersion curves). Results of implementing a similar method on simulation data has already been published, and in this paper we provided some preliminary results of applying this concept to experimentally collected data. Further efforts need to be made to improve the results. Future work consists of adding various initial conditions to the procedure as well as finding ways to make this work applicable to more complex anisotropic data. 129 Figure 6.1. Recovered dispersion curves (Dataset 1) from 200 measurements using SWA. 130 Figure 6.2. Recovered weight vector (Dataset 1) for a particular sampled grid point. 131 Figure 6.3. Reconstructed signal (Dataset 1) for the main path and the original signal for a particular sampled grid point. 132 Figure 6.4. Reconstructed signal (Dataset 1) for the reflection path and the original signal for a particular sampled grid point. 133 Figure 6.5. Recovered dispersion curves (Dataset 2) for Scenario 1 at iteration number 1. 134 Figure 6.6. Recovered dispersion curves (Dataset 2) for Scenario 2 at iteration number 30. 135 Figure 6.7. Recovered dispersion curves (Dataset 2) using regular SWA. 136 6.7 References [1] P. Cawley. Practical long range guided wave inspection managing complexity. AIP Conference Proceedings, 657(1):22–40, 2003. [2] L. P. Solie and B. A. Auld. Elastic waves in free anisotropic plates. The Journal of the Acoustical Society of America, 54(1):50–65, 1973. [3] M. J. S. Lowe. Matrix techniques for modeling ultrasonic waves in multilayered media. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 42(4):525– 542, 1995. [4] J. S. Hall and J. E. Michaels. Multipath ultrasonic guided wave imaging in complex structures. Structural Health Monitoring, 14(4):345–358, 2015. [5] A. Golato, S. Santhanam, F. Ahmad, and M. G. Amin. Multi-path exploitation in a sparse reconstruction approach to lamb wave based structural health monitoring. Structural Health Monitoring, 2015. [6] A. Ebrahimkhanlou, B. Dubuc, and S. Salamone. A guided ultrasonic imaging approach in isotropic plate structures using edge reflections. Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2016, 9803:98033I, 2016. [7] J. B. Harley and J. M. F. Moura. Data-driven matched field processing for lamb wave structural health monitoring. The Journal of the Acoustical Society of America, 135(3):1231–1244, 2014. [8] J. B. Harley and J. M. F. Moura. Sparse recovery of the multimodal and dispersive characteristics of Lamb waves. The Journal of the Acoustical Society of America, 133(5):2732–2745, 2013. [9] J. B. Harley. Predictive guided wave models through sparse modal representations. Proceedings of the IEEE, 104(8):1604–1619, Aug 2016. [10] S. Sabeti, C. A. C. Leckey, L. De Marchi, and J. B. Harley. Sparse wavenumber recovery and prediction of anisotropic guided waves in composites: A comparative study. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 66(8):1352–1363, Aug 2019. [11] S. Sabeti and J. B. Harley. Two-dimensional sparse wavenumber recovery for guided wavefields. AIP Conference Proceedings, 1949(1):230003, 2018. [12] S. Sabeti and J. B. Harley. Polar sparse wavenumber analysis for guided wave reconstruction. AIP Conference Proceedings, 2102(1):050012, 2019. [13] S. Sabeti and J. B. Harley. Guided wave retrieval from temporally undersampled data. In 2017 IEEE International Ultrasonics Symposium (IUS), pages 1–4, Sep. 2017. [14] D. L. Donoho. Compressed sensing. 52(4):1289–1306, 2006. IEEE Transactions on Information Theory, 137 [15] E. J. Candes and M. B. Wakin. An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2):21–30, March 2008. [16] E. C. Marques, N. Maciel, L. Naviner, H. Cai, and J. Yang. A review of sparse recovery algorithms. IEEE Access, 7:1300–1322, 2019. [17] S. S. Chen, D. L. Donoho, and M. A. Saunders. Atomic decomposition by basis pursuit. SIAM Review, 43(1):129–159, 2001. [18] J. A. Tropp and A. C. Gilbert. Signal recovery from random measurements via orthogonal matching pursuit. IEEE Transactions on Information Theory, 53(12):4655–4666, Dec 2007. [19] J. B. Harley and J. M. F. Moura. Dispersion curve recovery with orthogonal matching pursuit. The Journal of the Acoustical Society of America, 137(1):EL1–EL7, 2015. [20] J. B. Harley and J. M. F. Moura. Decomposition of multipath lamb waves with sparse wavenumber analysis for structural health monitoring. In 2013 IEEE International Ultrasonics Symposium (IUS), pages 675–678, July 2013. [21] J. B. Harley, K. S. Alguri, H. V. Tetali, and S. Sabeti. Learning guided wave dispersion curves from multi-path reflections with compressive sensing. 11 2019. CHAPTER 7 POLAR SPARSE WAVENUMBER ANALYSIS FOR GUIDED WAVE RECONSTRUCTION 1 In nondestructive evaluation/testing (NDE/NDT) and structural health monitoring (SHM) applications, guided waves are commonly employed and widely studied. Wave behavior characterization and analysis can be vital in determining the state of the structure under inspection. Effective analysis of guided waves, however, is encumbered by their intricate nature. This intricacy is further aggravated in structures with anisotropic characteristics. Moreover, the data acquisition process can be costly and time-consuming. Therefore, it is significant to achieve behavior prediction of guided waves from limited measurements. To make this possible, compressive sensing based methodologies and predictive models have been presented in the literature. Specifically in prior work, a twodimensional sparse wavenumber analysis (2D-SWA) framework was introduced to model anisotropic wave propagation. In this paper, we present a similar framework whereby a sparser representation of guided waves can be obtained by incorporating information of the measurements in polar coordinates. We implement this method, which we refer to as polar sparse wavenumber analysis (PSWA), on a simulated wavefield propagating in a composite material and demonstrate how it is capable of accurately reconstructing the entire wavefield from a few spatial measurements. 7.1 Introduction Numerous methodologies have been proposed and studied for nondestructive evaluation/testing (NDE/NDT) and structural health monitoring (SHM) purposes [1, 2]. Several factors, including the properties of the material under inspection, the availability of the technology, as well as environmental and financial considerations influence the suitability 1 Reproduced from ”S. Sabeti and J. B. Harley, Polar sparse wavenumber analysis for guided wave reconstruction, AIP Conference Proceedings, 2019”, with the permission of AIP Publishing 139 of these methods [3]. Ultrasonic guided waves provide several advantages, making them a fitting candidate for a lot of applications. The capability to travel long distances, high sensitivity to structural defects, and their interaction with hidden subsurface features are among these advantages [4]. Nevertheless, there are several disadvantages associated with guided waves. In particular, their complex nature is an impediment to practical behavior analysis. Guided waves in anisotropic structures, such as composite laminates, exhibit multimodal behavior, tend to travel at velocities varying with frequency, as well as dependency on the composite layup and direction of propagation [5]. Furthermore, the data acquisition procedure and storage for guided wave based methods in NDE and SHM can take a lot of time and be expensive. Understanding the behavior of guided waves and their propagation characteristics in different structures is vital while searching for defects using guided wave based techniques. The aforementioned disadvantages are a significant hindrance to this end. Therefore, it is desirable to characterize guided waves in complex structures and be able to do that through inexpensive and high-speed sampling and processing of data. Several studies in the literature aimed at facilitating guided wave data acquisition exist, some of which make use of compressive sensing to reduce the amount of sampled data. Compressive sensing enables subsampling of signals at very low rates given a proper choice of a transform domain and a sampling strategy, which we further discuss in the following sections. In particular, sparse wavenumber analysis (SWA) was proposed in prior work [6] to recover dispersion curves of Lamb waves in infinitely large isotropic plate-like structures, from limited number of measurements. The model was modified [7] to incorporate wavefront shape variations. Later, the model was expanded to two dimensions [8] to account for anisotropy and direction dependency of guided waves. We refer to this two-dimensional framework for guided wave characterization and reconstruction as 2D-SWA. 2D-SWA takes spatial information of the measurements in the Cartesian coordinates and recovers guided wavefields via retrieval of their sparse representation in the two-dimensional wavenumber domain. In this paper, we present a modification to the 2D-SWA framework in which we utilize the information content of the samples in the polar coordinate system. This will result in 140 recovery of a sparse representation indicating radial and angular variations of the waves, which can then be used to reconstruct the fully sampled wavefield. This procedure will be referred to as polar sparse wavenumber analysis (PSWA). We examine the functionality of this method by implementing it on a simulated dataset containing wave propagation in a composite plate. We define parameters based on the Nyquist-Shannon sampling theorem for our dataset and compare the reconstruction performance of PSWA with conventional low-pass filtering for oversampled signal reconstruction and show how PSWA’s performance excels at sampling frequencies below the Nyquist rate. 7.2 Background In this section, we briefly review the fundamental concepts and methodologies on which our guided wave reconstruction method is predicated. In particular, we discuss compressive sensing, a relatively novel signal processing concept, which provides the means for accurate signal reconstruction with subsampling at rates below the well-known Nyquist rate. We then briefly review sparse wavenumber analysis as a compressive sensing based method for wave characterization in isotropic media. And finally we discuss the two-dimensional version of sparse wavenumber analysis, where anisotropy can be taken into account. 7.2.1 Compressive Sensing Compressive sensing [9, 10] is a relatively recent approach to the sampling and compression problems in signal processing. In the compressive sensing framework, data can be acquired at sub-Nyquist rates and be accurately reconstructed with high probability. This is achievable if the signal of interest is representable using a linear combination of a few bases in a transform domain. Such a signal is called τ-sparse in that domain if the linear combination comprises τ non-zero coefficients. We refer to τ as sparsity value. It also requires a proper sensing mechanism in which the sensing matrix and the bases are incoherent. Let us assume we have a fully sampled signal Z that can be represented using a matrix of sparse coefficients (mostly zeros) V, and a set of bases D, i.e., we have Z = DV. If we 141 subsample our signal applying a sensing matrix S that is sufficiently incoherent with D, we can have X = ΦV (7.1) where Φ = SD is a sampled set of bases, and X = SZ is a set of measurements. The goal is to solve for the sparse matrix V in this underdetermined problem. There exist sparse recovery methods capable of performing this task, among which we usually opt for orthogonal matching pursuit (OMP) [11] due to the greedy nature of the algorithm, which leads to a decrease in computational cost, particularly at lower sparsity values. 7.2.2 Sparse Wavenumber Analysis Sparse wavenumber analysis (SWA) [6] is a signal processing method that uses compressive sensing to recover the dispersion curves of waves traveling in an isotropic plate. Considering the sparse nature of dispersion curves in the frequency-wavenumber domain and the analytical model for Lamb wave propagation in an infinite isotropic plate-like structure, SWA is capable of retrieving dispersion curves from limited number of measurements by reformulating the propagation model to resemble equation (7.1). A subsequent full wave reconstruction is referred to sparse wavenumber synthesis (SWS). The isotropy assumption in the SWA model causes it to fail in dealing with anisotropic wave propagation. A two-dimensional extension to this model was introduced in [8], where spatial information in both horizontal and vertical directions in the Cartesian coordinates can be incorporated in the model to account for direction dependency of wave propagation in anisotropic media. We briefly discuss this method in the following subsection. 7.2.3 Two-dimensional Sparse Wavenumber Analysis As mentioned before, to take direction-dependent variations in anisotropic guided wave propagation into account, a two-dimensional model can be considered. This model and a two-dimensional orthogonal matching pursuit algorithm (2D-OMP) for sparse recovery were introduced in prior work [8]. This methodology, which we refer to as 2D-SWA, recovers a sparse representation in the two-dimensional wavenumber domain, otherwise known as k-space. 142 2D-SWA uses the following model for a guided wavefield at each frequency slice X f = Φ1 V f Φ2T , (7.2) where X f is a frequency slice of the wavefield, and Φ1 and Φ2 are the left and right dictionaries, respectively. V f is the sparse representation in k-space corresponding to the f -th frequency slice that will be recovered using a 2D-OMP algorithm. The dictionaries are chosen as spatial Fourier bases, i.e., Φ1 = e− jky y and Φ2 = e− jkx x . In these equations, y and ky contain vertical coordinates and wavenumbers, while x and kx contain horizontal coordinates and wavenumbers, respectively. 7.3 Methodology Here, we look over the methods used for guided wave reconstruction in this paper. First, we discuss our implementation of a conventional signal processing method for signal reconstruction through low-pass filtering/masking in the frequency domain. In this work, this method is referred to as Fourier reconstruction. We then present our guided wave recovery algorithm, which is very similar to the previously discussed 2D-SWA, but incorporates information from the measurements in the polar coordinates and thus yields a different sparse representation of the wavefield. Finally, we discuss our derivation of Nyquist parameters for our dataset. 7.3.1 Fourier Reconstruction/Interpolation If a signal is sampled at the Nyquist rate or above (i.e., critically sampled or oversampled), then ideally it can be perfectly reconstructed through a low pass filtering in the frequency domain, isolating the original spectrum of the fully sampled signal. Similarly, in two dimensions a low pass filter/mask can be employed to separate the two-dimensional frequency response of the signal and reconstruct the unknown samples. We refer to this interpolation process as Fourier reconstruction. In this paper, we use this traditional signal processing method to evaluate the reconstruction performance of PSWA. It should be noted that to perform Fourier reconstruction, we need to have a uniformly sampled (undersampled) grid with equispaced points. 143 7.3.2 Polar Sparse Wavenumber Analysis The fundamental concept and model in polar sparse wavenumber analysis (PSWA) is similar to those of 2D-SWA. However, in PSWA we input the information content of the known samples in the polar coordinates and recover a sparse representation in a twodimensional domain (similar to k-space) where radial and angular variations of the wavefield can be observed. In this framework, we still utilize a two-dimensional wave propagation model for each frequency, as in (7.2). However, we define our dictionaries as Fourier bases for angular and radial variations. That is, we have Φ1 = e− jkd d , (7.3) where d is a vector containing distances from the source, and kd is a vector consisting of radial frequencies (or distance wavenumbers). And, we also have Φ2 = e− jkθ θ, (7.4) where vector θ consists of angles on the grid, and vector kθ contains angular variations (or angle wavenumbers). Figure 7.1 illustrates an instance of a recovered sparse representation for the dataset used in this paper, at one frequency. 7.3.3 Deriving Nyquist Parameters To have a measure for comparison and to be able to define an undersampling ratio, we need to derive parameters based on the Nyquist-Shannon sampling theorem. When dealing with a one-dimensional signal, the critical sampling rate (the Nyquist rate) is defined as twice the bandwidth (i.e., the highest frequency content) of the signal. In other words, perfect reconstruction using conventional signal processing is possible if we have f s ≥ 2 f max (7.5) where f s is the sampling frequency, and f max is the maximum frequency content of the signal. The independent variable of the signal can be time or space and this theorem is readily 144 applicable to both temporal and spatial frequency (i.e., wavenumber). It can also be extended to two dimensions where frequencies can be defined for each dimension/direction. Considering the definition of angular wavenumber k with respect to wavelength λ, k = 2π λ , and (7.5), the spatial Nyquist rate FSN can be derived as FSN ≥ k max π (7.6) where k max is the maximum angular wavenumber present in our two-dimensional spatial signal. Consequently, maximum allowable distance between grid points dmax complying with the Nyquist criterion can be defined as dmax = π (7.7) k max To find the maximum wavenumber content, we perform a two-dimensional Fourier transform on each frequency slice of our wavefield to obtain k-space responses. We then add the magnitude of these responses together to form a representation in the two-dimensional wavenumber domain, which would indicate the overall wavenumber content of our data. In this representation, we consider a circle containing 99 % of the total energy (sum of the squares of the elements). We define the radius of this circle to be k max . Figure 7.2 shows this representation for the simulation dataset described in the following section. The 99 % energy circle is shown by white dashes. Finally, we consider a surface of points (samples) on which the Nyquist parameters are observed, and we refer to it as the Nyquist grid. We define the number of points on this surface NNyq as NNyq = Ds dmax 2 Ng (7.8) where Ng is the number of points on the original fully sampled grid, and Ds is the distance between points on this grid (which is the same in both horizontal and vertical directions in our data). 7.4 Simulation Setup A simulation dataset is used in this paper to verify the practicability of our PSWA algorithm. Wave propagation in a carbon fiber reinforced polymer (CFRP) plate is simulated 145 through a finite-element engine. The composite plate has a [0/90/0/90]s layup and is 10.64 cm by 10.64 cm (long/wide). The fully sampled grid consists of 917 × 917 (a total of 840889) points, distanced 0.0116 cm apart in each direction. The source of propagation (transducer) is considered to be infinitesimally small and placed in the center of the plate at the approximate horizontal/vertical coordinates of (x = 5.25 cm , y = 5.25 cm). 7.5 Results In this section, we present the results of implementing our PSWA algorithm on the simulation dataset described in the previous section and compare them with the results obtained using Fourier reconstruction. We first discuss how we derive the Nyquist parameters for our dataset. We then present the reconstruction results in which PSWA outperforms Fourier reconstruction and is capable of accurate guided wave recovery at sampling rates below what the Nyquist parameters suggest. Finally, we view our sampling and reconstruction procedure from a learning standpoint where we refer to our known samples as training samples and the grid points where wave propagation is being predicted as test samples. We then study the impact of undersampling ratio and sparsity on reconstruction accuracy in this framework. 7.5.1 Nyquist Parameters As previously discussed, we can derive the Nyquist parameters for our dataset with respect to the maximum wavenumber as indicated in Figure 7.2. These parameters are included in Table 7.1. 7.5.2 Guided Wave Reconstruction Accuracy We quantify the reconstruction accuracy of PSWA using correlation coefficient between the original wavefield and the reconstructed one. We also define undersampling ratio as the ratio between the number of used samples and the total number of points on the Nyquist grid, as shown in Table 7.1. Figure 7.3 depicts the variations of reconstruction accuracy as a function of undersampling ratio for Fourier reconstruction and PSWA. It is clear that near the Nyquist or critical sampling rate (100 % undersampling ratio) both methods perform really well, which is expected. With the decrease in undersampling ratio, we witness a deterioration 146 in Fourier reconstruction results, whereas PSWA maintains an accuracy of 90 % and above at sampling rates as low as 10 % of the Nyquist rate. Nevertheless, it experiences a sudden drastic decline in performance below that threshold. Figure 7.4 illustrates snapshots of the undersampled, true, and reconstructed wavefields at different time instants. The reconstruction has been performed at about 25 % of the Nyquist rate, i.e., using 524 random samples on a grid of 840889 points (the Nyquist grid of which contains 3019 points). At this sampling rate, the true and reconstructed wavefields share an overall correlation of approximately 95 %. 7.5.3 A Learning Perspective Here, we look upon the guided wave reconstruction process from a learning standpoint. That is, we regard our undersampled wavefield as a set of training data from which we learn the characteristics of guided wave propagation in the medium, and then employ this learned model to predict the waves at the unknown grid points/samples, i.e., our test data. In this framework, we look at the reconstruction accuracy variations with undersampling ratio as well as its variations with the sparsity of the representation. 7.5.3.1 Effect of undersampling ratio Figure 7.5 illustrates the effect of undersampling ratio on the reconstruction accuracy of the training samples and the test samples. For clarification purposes, it is worth reviewing the reconstruction procedure in which we utilize the training samples (random samples of the original wavefield) in the PSWA framework to recover the sparse representation, and then we apply this representation to the model to reconstruct the wave at the grid points of choice. As shown in the figure, the reconstruction accuracy of the test samples tends to decline as we lower the undersampling ratio. This is expected, since we are using fewer samples that contain less information. On the contrary, the accuracy performance of PSWA for training samples has an inverse relationship with the undersampling ratio, and we have perfect reconstruction for training samples at very low undersampling ratios. With fewer training samples, the model tends to overfit to these samples, and hence a less accurate generalization. 147 7.6 Conclusion A methodology for reconstructing guided wavefields from spatially undersampled data was presented, which is capable of outperforming traditional signal reconstruction methods at sampling rates below the Nyquist rate. This method referred to as polar sparse wavenumber analysis (PSWA) takes information from random spatial measurements in the polar coordinates as input and recovers a two-dimensional representation of the wavefield indicating its radial and angular variations. This representation is then used to reconstruct the entire wavefield. We implemented this algorithm on a simulation dataset of waves propagating in a CFRP plate and the results demonstrated the capability of this method in accurate signal recovery when compared with two-dimensional low pass filtering in the Fourier domain, which we refer to as Fourier reconstruction. In particular, PSWA reconstruction accuracy (in terms of correlation coefficient) can be as high as 90 % with sampling rates as low as 10 % of the Nyquist rate. Future work includes a more profound study of PSWA and its application to different types of wave propagation and experimental datasets and how it compares to similar methods like 2D-SWA, as well as integration with temporal undersampling recovery methods, such as what was introduced in prior work [12], to further reduce the data dimensionality and facilitate the acquisition process. 7.7 Acknowledgments This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0126. We would also like to thank Cara Leckey from NASA Langley Research Center’s Non-Destructive Evaluation Branch (Space Act Agreement #SAA1-24023) for the simulation dataset used in this paper. 148 Figure 7.1. A recovered sparse representation using PSWA at one frequency slice. 149 Figure 7.2. Two-dimensional wavenumber representation of the simulation dataset. The white dashed circle indicates the 99 % energy circle. 150 Figure 7.3. Correlation coefficient versus undersampling ratio. 151 Figure 7.4. Reconstruction (at 25 % of the Nyquist rate) at different (increasing from top to bottom) time instants. 152 Table 7.1. Parameters for the simulation dataset. Original Grid Spacing (Ds ) Maximum Wavenumber (kmax ) Critical Distance (dmax ) # of Nyquist Grid Points (NNyq ) 0.116 mm 1622 1.93 mm 3019 153 Figure 7.5. Correlation coefficient versus undersampling ratio. 154 7.8 References [1] S. Gholizadeh. A review of non-destructive testing methods of composite materials. Procedia Structural Integrity, 1:50–57, 2016. [2] M. Mitra and S. Gopalakrishnan. Guided wave based structural health monitoring: A review. Smart Materials and Structures, 25(5):053001, 2016. [3] P. J. Shull. Nondestructive Evaluation: Theory, Techniques, and Applications. CRC press, 2002. [4] J. L. Rose. Ultrasonic guided waves in solid media. Cambridge University Press, 2014. [5] L. Wang and F. G. Yuan. Group velocity and characteristic wave curves of lamb waves in composites: Modeling and experiments. Composites Science and Technology, 67(78):1370–1384, 2007. [6] J. B. Harley and J. M. F. Moura. Sparse recovery of the multimodal and dispersive characteristics of lamb waves. The Journal of the Acoustical Society of America, 133(5):2732–2745, 2013. [7] J. B. Harley. Predictive guided wave models through sparse modal representations. Proceedings of the IEEE, 104(8):1604–1619, 2015. [8] S. Sabeti and J. B. Harley. Two-dimensional sparse wavenumber recovery for guided wavefields. AIP Conference Proceedings, 1949(1):230003, 2018. [9] D. L. Donoho et al. Compressed sensing. IEEE Transactions on Information Theory, 52(4):1289–1306, 2006. [10] E. J. Candes and M. B. Wakin. An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2):21–30, Mar. 2008. [11] J. A. Tropp and A. C. Gilbert. Signal recovery from random measurements via orthogonal matching pursuit. IEEE Transactions on Information Theory, 53(12):4655–4666, Dec. 2007. [12] S. Sabeti and J. B. Harley. Guided wave retrieval from temporally undersampled data. In 2017 IEEE International Ultrasonics Symposium (IUS), pages 1–4, Sep. 2017. CHAPTER 8 CONCLUSION AND FUTURE WORK 8.1 Conclusion, Summary, and Contributions The main focus of this dissertation is on the characterization of guided wave propagation in various and potentially complex structures in the absence of ample temporal and spatial measurements. Despite the numerous advantages of using guided waves in nondestructive evaluation (NDE) and structural health monitoring (SHM) applications, the intricacies of their propagation characteristics render proper analysis extremely difficult. Moreover, prior information on these characteristics is generally not readily available for a lot of complex materials. The methodologies introduced in this dissertation are meant to address these issues by providing the capability of retrieving information from limited data in time and space. These compressive sensing based techniques distinguish themselves from similar methods in the literature [1, 2, 3] by incorporating two-dimensional models, enabling data processing in both temporal and spatial dimensions, and addressing direction-dependent complexities of anisotropic structures. The main contributions of this dissertation can be summarized as follows: 1. A two-dimensional model and a modified two-dimensional sparse recovery algorithm were presented, aimed at recovering guided wave propagation information in anisotropic structures where direction-dependency necessitates a two-dimensional spatial analysis. 2. A comparative study was performed to illustrate the benefits and shortcomings of this method against a one-dimensional anisotropic model and the conventional signal processing reconstruction procedure in the Fourier domain. 3. A modified sparse wavenumber analysis model was introduced, making temporal guided wave information retrieval a possibility. 156 4. The two-dimensional model, together with the conceptual fundamentals of spatial and temporal sparse wavenumber analysis, were utilized in a framework where a spatiotemporal analysis and synthesis of guided waves could be performed. 5. Preliminary results for the characterization of guided waves in the presence of multipath reflections were presented, which creates the potential for source (damage) detection in convoluted multipath wave measurements. 6. A two-dimensional polar sparse wavenumber analysis framework was also introduced where the information obtained from guided wave measurements could be transformed into the polar coordinates to generate a relatively sparser representation, thereby producing the potential for data recovery from fewer spatial measurements. 8.2 The Path Ahead The methodologies introduced and investigated in this dissertation offer promising advantages for potential utilization in NDE and SHM applications. However, there exist several issues that these techniques fail to address. Ideas and directions to improve their performance and practicability are presented in the following subsections. 8.2.1 Two-Dimensional Sparse Wavenumber Analysis with Different Bases The two-dimensional sparse wavenumber analysis (2D-SWA) model was introduced to account for alterations that occur in anisotropic guided waves as a function of the direction of propagation. The model consists of a left and a right dictionary. These dictionaries correspond to the vertical/horizontal spatial coordinates and are chosen to be the Fourier bases. This choice was inspired by the regular one-dimensional sparse wavenumber analysis model, and it offers simplicity and ease of implementation. Nevertheless, there may exist other dictionary options that could be better suited to anisotropic wave propagation in particular structures. A comparative study of different choices of off-the-shelf dictionaries (such as wavelets [4], Hankel bases [5], etc.) could be conducted to investigate the performance of this two-dimensional model for each of these cases. 157 8.2.2 Two-Dimensional Dictionary Learning A dictionary learning approach could be adopted so as to find a data-driven, and structure-specific set of dictionaries for the 2D-SWA model. A similar approach for the one-dimensional model exists in the literature [6] that suggests enhanced performance could be achieved. Extending this to two dimensions can help us find dictionaries suitable to specific complex anisotropic media. 8.2.3 Bayesian Sparse Wavenumber Analysis In the subsequent years to the formal introduction of compressive sensing in the literature, a Bayesian compressive sensing framework was presented [7]. It has since been used in several applications exhibiting improved performance. A proper choice of a sparsitypromoting prior is an important aspect of this framework. It offers adaptive solutions and an uncertainty analysis otherwise unattainable through regular compressive sensing. Such a framework could be applied to guided wave propagation models to create optimally sparse representations and result in more accurate reconstruction. 8.2.4 Spatiotemporal Sparse Wavenumber Analysis for Anisotropic Structures The two-dimensional ST-SWA model introduced in this dissertation is only applicable to isotropic guided waves. The spatial dictionary in ST-SWA is similar to that of the regular one-dimensional SWA and therefore, fails to account for anisotropy. To be able to apply ST-SWA to anisotropic media, we could follow similar approaches to how we dealt with anisotropy in the purely spatial realm. In other words, we could make use of the pseudoelliptical bases used in the anisotropic sparse wavenumber analysis (ASWA). Another option would be extending the model into three dimensions so as to incorporate the horizontal/vertical spatial variations. Exploring the feasibility of these approaches could be a topic for future studies. 8.2.5 A Comparative Study of Different Characterization and Reconstruction Methodologies The models presented in this dissertation are fundamentally analytical. That is, they utilize analytical bases based on some assumptions on guided wave propagation. Another approach, as previously mentioned, is using a dictionary learning framework, which is 158 shown to be effective. Furthermore, a deep learning approach toward guided wave reconstruction has been adopted in recent years [8]. Preliminary studies on the use of deep neural networks and autoencoders exist in the literature. Each of these methods come with their advantages and disadvantages. A comprehensive comparative study could be performed to evaluate these techniques in terms of their reconstruction accuracy, the required number of measurements, computational complexity, generalizability, and so on. 8.2.6 Anisotropy Compensation to Facilitate Damage Detection Utilizing the 2D-SWA framework, we are able to characterize wave propagation in anisotropic media. The sparse representation in the two-dimensional wavenumber space indicates how waves travel in different directions with various velocities. In theory, we expect an isotropic wave propagating in all directions within a scanning region to generate a circular representation indicating a uniform velocity distribution independent of the direction of propagation. This notion could be used to transform measured anisotropic waves into isotropic ones, thereby facilitating damage detection through use of methods such as matched field processing [9], which work well under the isotropy assumption. 8.3 References [1] J. B. Harley and J. M. F. Moura. Sparse recovery of the multimodal and dispersive characteristics of Lamb waves. The Journal of the Acoustical Society of America, 133(5):2732–2745, 2013. [2] O. Mesnil and M. Ruzzene. Sparse wavefield reconstruction and source detection using compressed sensing. Ultrasonics, 67:94–104, 2016. [3] T. D. Ianni, L. D. Marchi, A. Perelli, and A. Marzani. Compressive sensing of full wave field data for structural health monitoring applications. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 62(7):1373–1383, July 2015. [4] Y. Meyer. Wavelets: algorithms and applications. Philadelphia: SIAM, 1993. [5] J. W. Layman. The hankel transform and some of its properties. Journal of Integer Sequences, 4(1):1–11, 2001. [6] K. S. Alguri, J. Melville, and J. B. Harley. Baseline-free guided wave damage detection with surrogate data and dictionary learning. The Journal of the Acoustical Society of America, 143(6):3807–3818, 2018. [7] S. Ji, Y. Xue, and L. Carin. Bayesian compressive sensing. IEEE Transactions on Signal Processing, 56(6):2346–2356, 2008. 159 [8] Y. K. Esfandabadi, M. Bilodeau, P. Masson, and L. De Marchi. Deep learning for enhancing wavefield image quality in fast non-contact inspections. Structural Health Monitoring, 2019. [9] A. B. Baggeroer, W. A. Kuperman, and H. Schmidt. Matched field processing: Source localization in correlated noise as an optimum parameter estimation problem. 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