Frequency Domain Method for Resolution of Two Overlapping Ultrasonic Echoes

Transcription

1 Frequency Domain Method for Resolution of Two Overlapping Ultrasonic Echoes by Chi-Hang Kwan A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Mechanical and Industrial Engineering University of Toronto Copyright by Chi-Hang Kwan 2017

2 ii Frequency Domain Method for Resolution of Two Overlapping Ultrasonic Echoes Abstract Chi-Hang Kwan Doctor of Philosophy Department of Mechanical and Industrial Engineering University of Toronto 2017 The ability to identify and resolve overlapping echoes is crucial to the enhancement of axial scan resolution in ultrasonic testing. Overlapping echoes are frequently encountered in the inspection of shallow and/or short cracks in Time-of-Flight Diffraction and normal incidence reflection inspection of near surface flaws. Dictionary-based parametric representation (DBPR) has been proposed as a powerful framework to separate overlapping echoes of different shapes. However, the large solution space in DBPR renders the optimization process difficult. We propose a new echo separation method named Trigonometric Echo Identification (TEI) that exploits the consistent frequency domain amplitude and phase relationships of two overlapping ultrasonic echoes to reduce the number of optimization parameters. In TEI, frequency amplitude profiles are entered as inputs and the corresponding set of frequency phase profiles are reconstructed as outputs. The optimality of the output phase profiles is then used as a metric to determine the accuracy of the trial amplitude inputs. By reconstructing the phase information instead of explicitly specifying the phase profiles, we can reduce the number of unknowns in the problem of identifying two overlapping ultrasonic echoes. Compared to DBPR, TEI can describe more complex ultrasonic echoes using the same number of optimization ii

3 iii parameters. In addition, since the phase profiles are reconstructed using the acquired data, TEI would perform more reliably in the presence of noise. Simulation tests were conducted to assess the relative performance of TEI and DBPR. Echo parameters including center frequency, phase shift and relative amplitudes were systematically varied to yield different test configurations. The standard deviation of timing errors obtained from TEI were 50% lower compared to DBPR. The difference in algorithm performance is especially evident in low SNR signals and signals containing echoes of complex shapes. The TEI algorithm was also verified on experimental ultrasound testing data containing overlapping echoes. The echo arrival times extracted using TEI agree with the values obtained using geometric calculations. iii

4 iv Acknowledgments Firstly, I would like to express my gratitude to my supervisor, Prof. Anthony Sinclair, for his guidance and support throughout the course of my thesis project. Thank you for placing your confidence in me to pursue my own research directions. Secondly, I am grateful to the Natural Sciences and Engineering Research Council of Canada (NSERC), Ontario Graduate Scholarship (OGS) and Olympus NDT Canada for sponsoring my research. I am fortunate to have the opportunity to work on various interesting industrial research projects at Olympus NDT and use their laboratory facilities to conduct my experimental work. I would also like to express my gratitude to my colleagues at Ultrasonic Nondestructive Evaluation Laboratory (UNDEL) and Olympus NDT for their collaboration and the valuable discussions we ve had together. Many of the ideas pursued in this research project stemmed directly or indirectly from our many long conversations. Finally, I would like to thank my friends and family for their encouragement, patience and love during this long and at times arduous journey. This thesis would not have been possible without your support. iv

5 v Table of Contents Acknowledgments... iv Table of Contents...v List of Tables... viii List of Figures... ix List of Symbols... xiv List of Appendices... xvi Chapter 1 Introduction Introduction and Motivation Thesis Objectives Thesis Overview...4 Chapter 2 Background and Literature Review Ultrasonic Inspection System Pulser-Receiver Piezoelectric Transducers Ultrasonic Testing Data Representation Resolution Limits in Ultrasonic Testing Modeling of Ultrasonic Echoes One-Dimensional Piezoelectric Transducer Models Complete Transfer Function Modeling of Ultrasonic Echoes Single Reference Deconvolution Basic Assumptions Direct Deconvolution Schemes Iterative Deconvolution Schemes Technique Limitations...26 v

6 vi 2.4 Dictionary-based Parametric Representation Mathematical Formulation Sparsity-Promoting Algorithms...28 Chapter 3 Basis and Assumptions of TEI Algorithm Frequency Domain Assumptions Amplitude Profile Assumption Phase Profile Assumption Justification of Amplitude Profile Assumption Applicability Limits of Echo Assumptions...36 Chapter 4 Trigonometric Echo Identification Algorithm Algorithm Overview Trigonometric Phase Profile Reconstruction Components of TEI Algorithm Echo Optimality Metrics Determination of the Correct Set of Phase Profiles Phase Slope Inequality Constraint Implementation as Constrained Optimization Problem Constrained Optimization Formulation Implementation Details Summary of Novelty and Advantages of the TEI Algorithm...50 Chapter 5 Results and Discussions Simulation Tests and Comparison Benchmark Synthetic Echoes with Symmetric Envelope Echo Parameter Tests Signal to Noise Ratio Tests Results Summary and Discussion...66 vi

7 vii 5.3 Synthetic Echoes with Asymmetric Envelope Echo Parameter Tests Signal to Noise Ratio Tests Results Summary and Discussion Experimental Verification TOFD Test on Notched Sample Phased Array Test on Side-Drilled Hole Sample...90 Chapter 6 Conclusions Thesis Summary Research Findings Future Work...97 References Appendix 1: Two-way Impulse Response of Van-Dyke Model Appendix 2: KLM Model of Broadband Transducer Appendix 3: Source Code of KLM model vii

8 viii List of Tables Table 3.1: Properties for broadband KLM simulation Table 4.1: Phase reconstruction chart Table 5.1: Baseline parameters for symmetric echoes Table 5.2: Performance table (phase difference vs time separation for symmetric echoes) Table 5.3: Performance table (frequency difference vs time separation for symmetric echoes).. 58 Table 5.4: Performance table (amplitude ratio vs time separation for symmetric echoes) Table 5.5: Baseline parameters for asymmetric echoes Table 5.6: Performance table (phase difference vs time separation for asymmetric echoes) Table 5.7: Performance table (center frequency difference vs time separation for asymmetric echoes) Table 5.8: Performance table (amplitude ratio vs time separation for asymmetric echoes) viii

9 ix List of Figures Figure 1.1: Acoustic travel paths for two adjacent defects... 1 Figure 1.2: Configuration of a TOFD scan... 2 Figure 2.1: Schematic diagram of ultrasonic inspection system... 6 Figure 2.2: Two models of Pulser-Receiver... 7 Figure 2.3: Voltage pulse of analog pulser... 7 Figure 2.4: Voltage pulse of digital pulser... 8 Figure 2.5: Schematic diagram of a single element piezoelectric transducer (courtesy of [11])... 9 Figure 2.6: Steering of phased array transducers Figure 2.7: Focusing of phased array transducers Figure 2.8: A-scan representation from TOFD data Figure 2.9: TOFD B-scan containing 4 flaws Figure 2.10: C-scan of back surface of a coin (from [16]) Figure 2.11: S-scan of three side-drilled holes (from [17]) Figure 2.12: Lateral resolution in ultrasound imaging Figure 2.13: Van Dyke approximate transducer model Figure 2.14: Frequency amplitude response predicted by Van Dyke model Figure 2.15: Schematic diagram of KLM model Figure 2.16: Transmission matrix model of transducer (Operated as transmitter) Figure 2.17: Thevenin's equivalent circuit ix

10 x Figure 2.18: Two-way impulse and frequency response for two points in a pressure field Figure 2.19: Single reference convolution Figure 3.1: Asymmetric Q-Gaussian distribution Figure 3.2: First harmonic impulse response of KLM model Figure 3.3: KLM model of broadband transducer Figure 3.4: Pitch-catch backwall echo acquisition configuration Figure 3.5: Experimental pitch-catch backwall echo Figure 3.6: Fourier transform of experimental pitch-catch backwall echo Figure 3.7: Echo distortion due to wavefield diffraction Figure 4.1: Flowchart of TEI algorithm Figure 4.2: Vector representation of overlapping echoes Figure 4.3: Alternative vector addition configuration Figure 4.4: 50% taper Tukey window Figure 5.1: Baseline configuration for symmetric echoes Figure 5.2: Percentage timing error (phase difference vs time separation for symmetric echoes)56 Figure 5.3: Percentage reconstruction error (phase difference vs time separation for symmetric echoes) Figure 5.4: Percentage timing error (frequency difference vs time separation for symmetric echoes) Figure 5.5: Percentage reconstruction error (frequency difference vs time separation for symmetric echoes) x

11 xi Figure 5.6: Percentage timing error (amplitude ratio vs time separation for symmetric echoes). 59 Figure 5.7: Percentage reconstruction error (amplitude ratio vs time separation for symmetric echoes) Figure 5.8: Overlapped echoes at SNR = 40 db (symmetric echoes) Figure 5.9: Percentage timing error (40 db for symmetric echoes) Figure 5.10: Overlapped echoes at SNR = 25 db (symmetric echoes) Figure 5.11: Percentage timing error (25 db for symmetric echoes) Figure 5.12: Overlapped echoes at SNR = 15 db (symmetric echoes) Figure 5.13: Percentage timing error (15 db for symmetric echoes) Figure 5.14: Overlapped echoes at SNR = 10 db (symmetric echoes) Figure 5.15: Percentage timing error (10 db for symmetric echoes) Figure 5.16: Performance vs SNR (symmetric echoes) Figure 5.17: Overlapped signal with time separation of 0.2 µs Figure 5.18: Quadratic modulation frequency Figure 5.19: Baseline configuration for asymmetric echoes Figure 5.20: Phase slope difference of two asymmetric echoes (nominal time separation at 0.2 µs) Figure 5.21: Percentage timing error (phase difference vs time separation for asymmetric echoes) Figure 5.22: Percentage reconstruction error (phase difference vs time separation for asymmetric echoes) xi

12 xii Figure 5.23: Percentage timing error (center frequency difference vs time separation for asymmetric echoes) Figure 5.24: Percentage reconstruction error (center frequency difference vs time separation for asymmetric echoes) Figure 5.25: Percentage timing error (amplitude ratio vs time separation for asymmetric echoes) Figure 5.26: Percentage reconstruction error (amplitude ratio vs time separation for asymmetric echoes) Figure 5.27: Overlapped echoes at SNR = 40 db (asymmetric echoes) Figure 5.28: Percentage timing error (40 db for asymmetric echoes) Figure 5.29: Overlapped echoes at SNR = 25 db (asymmetric echoes) Figure 5.30: Percentage timing error (25 db for asymmetric echoes) Figure 5.31: Overlapped echoes at SNR = 15 db (asymmetric echoes) Figure 5.32: Percentage timing error (15 db for asymmetric echoes) Figure 5.33: Overlapped echoes at SNR = 10 db (asymmetric echoes) Figure 5.34: Percentage timing error (10 db for asymmetric echoes) Figure 5.35: Performance vs SNR (asymmetric echoes) Figure 5.36: Test sample containing vertical notches Figure 5.37: TOFD configuration for notch sample Figure 5.38: B-scan of notch sample TOFD scan Figure 5.39: Overlapping echoes in TOFD scan of notch sample Figure 5.40: Notch sample time series data analyzed by TEI and DBPR xii

13 xiii Figure 5.41: Reconstructed echoes for notch sample (TEI using phase linearity condition) Figure 5.42: Reconstructed echoes for notch sample (DBPR) Figure 5.43: Frequency phase profiles of TEI reconstructed echoes (notch sample) Figure 5.44: Lateral wave reference echo for TEI Figure 5.45: Reconstructed echoes for notch sample (TEI using cross-correlation condition) Figure 5.46: Test sample for pitch-catch matrix probe scan Figure 5.47: Phased array pitch-catch testing of SDH sample Figure 5.48: Indirect path for SDH Figure 5.49: Overlapping echoes for SDH pitch-catch test Figure 5.50: SDH sample time series data analyzed by TEI and DBPR Figure 5.51: Reconstructed echoes for SDH sample (TEI) Figure 5.52: Reconstructed echoes for SDH sample (DBPR) xiii

14 xiv List of Symbols Symbol Definition A Amplitude scaling parameter in time domain echo model C Capacitance (F) D Diameter of transducer (m) F Force output from transducer (N) G(ω) Frequency domain Wiener filter H x (ω) Transfer function of system x (variable) I Electrical current (A) L Electrical Inductance (H) M A (ω) Frequency amplitude profile of constituent echo A M B (ω) Frequency amplitude profile of constituent echo B M T (ω) Frequency amplitude profile of total signal N Number of data points N(ω) Frequency noise P(ω) Frequency pressure response (N/m 2 ) R Electrical resistance (Ω) REF(ω) Frequency domain reference signal S Amplitude scaling parameter for frequency domain Q-Gaussian model SIG(ω) Fourier transform of total signal SNR(ω) Frequency domain signal-to-noise ratio T x Transfer matrix of layer x in KLM model (variable) V x (ω) Frequency domain voltage response of system x (V) Z Electrical or acoustic impedance (Ω or Rayl) a Exponential time decay parameter (1/s) b Width parameter for frequency domain Q-Gaussian model (s 2 ) c Speed of sound (m/s) e(t) Reconstruction error echo(t) Echo waveform in the time domain env(t) Echo amplitude envelope in the time domain f Frequency (Hz) h x (t) Impulse response for system x (variable) k Wavenumber (1/m) l Thickness (m) m Order for series in mathematics (positive integer) n(t) Time domain noise p x (t) Time domain pressure response of system x (N/m 2 ) p Vector of optimization parameters used in Dictionary-based Parametric Represenation (DBPR) model q Tail-heaviness parameter for frequency domain Q-Gaussian model r Radial direction (m) ref(t) Time domain reference signal sig(t) Total ultrasound time series signal t Time (s) xiv

15 xv Δt u v(t) w j x z α(ω) β(ω) γ(ω) θ A (ω) θ B (ω) θ B (ω) κ(ω) λ μ ξ spread ρ σ τ φ χ ψ ω Arrival time difference between two ultrasound echoes (s) Frequency power in attenuation for materials Time domain voltage signal (V) Weights used in auto-regressive extrapolation model Vector of optimization parameters used in Trigonometric Echo Identification (TEI) model Axial distance (m) Interior angle opposite of M A (ω) in phase reconstruction triangle (rad) Interior angle opposite of M B (ω) in phase reconstruction triangle (rad) Interior angle opposite of M T (ω) in phase reconstruction triangle (rad) Frequency phase profile of constituent echo A (rad) Frequency phase profile of constituent echo B (rad) Frequency phase profile of total signal (rad) Transformer ratio in KLM model (V/N) Wavelength (m) Penalty parameter in Augmented Lagrangian (ALAG) constrained optimization algorithm Beam spread angle of ultrasound transducer (rad) Envelope asymmetry parameter in time domain echo model Sparsity control parameter in Basis Pursuit (also known as L1-norm deconvolution) Time shift parameter in time domain echo model (s) Constant phase shift parameter in time domain echo model (rad) Lagrange multiplier in Augmented Lagrangian (ALAG) constrained optimization algorithm Quadratic modulation frequency variation parameter in time domain echo model (1/s 3 ) Angular frequency (rad/s) xv

16 xvi List of Appendices Appendix 1: Two-way Impulse Response of Van-Dyke Model Appendix 2: KLM Model of Broadband Transducer Appendix 3: Source Code of KLM model xvi

17 1 Chapter 1 Introduction 1.1 Introduction and Motivation Ultrasonic testing is a Non-Destructive Testing (NDT) method to characterize the internal structure of a test sample using high frequency sound waves. The typical frequencies employed in ultrasonic inspection systems range from 200 khz up to 100 MHz [1]. An advantage of ultrasonic testing is that sound waves can propagate in a multitude of solids and liquids. Consequently, ultrasonic testing can be performed on test samples made of metals, plastics, ceramics, polymers, composite materials and biomedical materials [2]. In ultrasonic testing, a voltage waveform originating from an ultrasonic wave scattered by a discontinuity in the test sample is called an echo. The shapes and time durations of ultrasonic echoes are determined by the design of the transducer, the characteristics of the electronics of the inspection system and the characteristics of the defect present in the test sample [1]. If two defects in the test sample are located adjacent to each other, as shown in Figure 1.1, the difference in acoustic travel times for the two acquired echoes might be shorter than the time duration of the individual echoes. In such situations, the two echoes will overlap in the time domain and it would be difficult to accurately determine the arrival times of each echo. Although Figure 1.1 shows that separate transducers are used for transmitting and receiving the acoustic waves, there are many NDT applications where a single transducer is used for both roles, in what is known as a pulse-echo configuration. Figure 1.1: Acoustic travel paths for two adjacent defects Overlapping ultrasonic echoes are frequently encountered in applications where the examined features have characteristic dimensions comparable to the wavelength in the material. NDT examples of such applications include characterization of shallow and/or short cracks in Time-

18 2 of-flight Diffraction (TOFD) studies [3], testing of adhesive bonds between thin structures [4] and normal incidence inspection of subsurface corrosion [5]. In these applications, multiple overlapping echoes with similar frequency content may be picked up by the receiving transducer. As will be explained in Section 2.1, overlapping echoes is the limiting factor for the axial resolution in ultrasonic cross-section imaging techniques. In addition, the presence of overlapping echoes can also directly affect the accuracy of time-of-flight based ultrasonic testing measurements. As an illustrative example, consider the TOFD scan for a weld sample containing a vertical crack shown in Figure 1.2. In Figure 1.2, the cone in the center represents the weld area and each coloured line represents the ray path of an ultrasonic wave travelling from transmitter to receiver, and is assigned a name indicative of the path followed by the wave. Figure 1.2: Configuration of a TOFD scan For the test configuration shown in Figure 1.2, we expect to obtain four return echoes corresponding to the lateral wave, the top tip diffracted echo, the bottom tip diffracted echo and the back-wall reflection echo. If the speed of sound in the sample is known, then simple trigonometry will yield the vertical position and size of the cracks from the arrival times of each echo at the receiving transducer. However, if the vertical extent of the crack is small and/or if the crack is located close to the top or bottom surface of the sample, overlapping echoes would be acquired and it would not be possible to obtain accurate estimate of the arrival time of each echo, nor the location and size of the crack.

19 3 For the reasons listed above, a method to separate overlapping ultrasonic echoes would be highly beneficial for accurate location and sizing of small defects. High-frequency high-bandwidth ultrasound transducers have been designed to reduce the time duration of the ultrasonic echoes in order to minimize the problem of overlapping echoes[6]. However, such transducers have weak output and limited penetration depth since acoustic attenuation increases with wave frequency [2]. Consequently, hardware solutions to mitigate overlapping echoes are limited for many NDT applications. Due to the limitation of hardware solutions, a software solution to separate overlapping echoes is proposed to enhance the axial resolution in ultrasonic imaging and provide an improved estimate of the size and location of any defects present. In this thesis, we will present a novel postprocessing algorithm designed to separate two overlapping echoes that are present in ultrasonic testing time series data. 1.2 Thesis Objectives The major objectives of this research project are introduced in the section. Development of Trigonometric Echo Identification (TEI) Algorithm The main objective of this thesis is to develop a novel algorithm for separation of two overlapping ultrasonic echoes. The name of the proposed algorithm is called Trigonometric Echo Identification (TEI). The proposed algorithm is designed to identify and separate two ultrasonic echoes that overlap partially in time and also possess similar frequency content. (If two ultrasonic echoes have distinctly different spectral content, conventional time-frequency transform methods such as the continuous wavelet transform [7] can be used to separate the two echoes.) Since the shapes of ultrasonic echoes are highly dependent on the configurations of the inspection system, it is not feasible to develop a generic algorithm that can successfully process echoes acquired from all possible test configurations. Consequently, the proposed algorithm is targeted to separate echoes acquired from bulk (longitudinal and/or shear) wave inspection of metallic samples using high bandwidth piezoelectric transducers. The proposed algorithm

20 4 should also be sufficiently flexible to handle two overlapping echoes that have differently shaped amplitude envelopes and different phase shifts. It should be stressed that goal of this research is not to develop a new ultrasound imaging technique. The proposed echo separation algorithm is a signal processing tool that can be incorporated in existing ultrasound testing methods to improve the resolution in defect size and location estimates. Evaluation of Algorithm Performance After the development of the echo separation algorithm, its performance is to be evaluated using simulation and experimental tests. The performance of the proposed algorithm will also be compared to that of an existing state-of-the-art echo separation algorithm. Simulation tests are valuable because we have exact information of the properties of the individual echoes. Simulation tests also allow us to vary the shapes of the individual echoes and the signal-to-noise ratio (SNR) level of the input signals to obtain statistical metrics of algorithm performance. Experimental tests will also be conducted on the proposed algorithm to verify that the assumptions made during the algorithm development process are actually applicable for real world NDT applications. Results obtained from simulation and experimental tests will allow us to determine the advantages and limitations of the proposed algorithm. 1.3 Thesis Overview In this section, we present an outline of the material that will be presented in the remaining chapters of the thesis. In Chapter 2, we present a literature review of the relevant background topics. The chapter begins with a description of ultrasonic inspection systems and the different representations of ultrasonic testing data. The importance of separation of overlapping echoes for axial resolution enhancement is also discussed. The chapter then introduces linear models that can be used to predict the voltage-to-voltage frequency response of ultrasonic inspection systems. Next, two main categories of conventional echo separation algorithms are reviewed: Single Reference Deconvolution and Dictionary-based Parametric Representation (DBPR).

21 5 In Chapter 3, we introduce the assumptions used in our new TEI algorithm. Since TEI is a frequency-domain method, the algorithm assumptions are expressed in terms of the frequency amplitude and phase profiles. The justifications for these assumptions are then described based on both empirical data and theoretical models. Chapter 4 is dedicated to the detailed presentation of the TEI algorithm. The chapter begins with a high-level overview of the complete algorithm. The chapter then introduces a trigonometrybased method to recover the phase profiles from the amplitude profiles of two overlapping echoes. This phase profile reconstruction method is an integral part of the TEI algorithm and contributes to its unique properties. Next, details of the different components of the TEI algorithm are described. Using the described components and assumptions presented in Chapter 3, TEI is then formulated as a constrained-optimization problem. This problem formulation allows TEI to be solved using existing optimization methods. The chapter concludes with a summary of the novel ideas and advantages of the TEI algorithm. In Chapter 5, we present results from simulation and experimental evaluations of the TEI algorithm. The echo separation performance of TEI is compared to that of DBPR, which we select as the benchmark method. For the simulation tests, the percentage timing and reconstruction errors are used as performance metrics. Signal parameters including phase shift, frequency difference, amplitude ratio and noise level are varied to obtain different test configurations. For the experimental tests, we evaluate the applicability of the TEI algorithm for processing of ultrasound testing data acquired from two NDT applications. The echo separation performance of TEI is assessed by comparing its extracted arrival time difference between the two echoes with the arrival time difference estimated using geometric calculations. Thesis conclusions are presented in Chapter 6, which begins with a review of the major tasks completed in the research project. This review is followed by a summary of the most important research findings. The thesis concludes with a list of suggestions for future research directions.

22 6 Chapter 2 Background and Literature Review 2.1 Ultrasonic Inspection System A schematic drawing of a typical ultrasonic inspection system is shown in Figure 2.1. In this schematic drawing, the computer controls the pulser which sends a high voltage pulse to the transmitting transducer. The voltage pulse is transformed into a mechanical vibration in the transmitting transducer and leads to a propagating ultrasonic wave being sent into the sample. If a flaw or discontinuity is present in the sample, a portion of the propagating ultrasonic wave would be reflected or scattered, and a portion of these deflected waves could then be captured by the receiving transducer. The receiver then amplifies the output voltage signal, and sends the analog waveform signal to the oscilloscope. Finally, the oscilloscope converts the analog signal into digital data and sends the data to the computer for further processing and storage. Although in Figure 2.1 separate transducers are used for the transmission and reception paths, in many NDT applications a single transducer can be used in pulse-echo mode to both transmit and capture the reflected ultrasonic wave. Figure 2.1: Schematic diagram of ultrasonic inspection system Pulser-Receiver The Pulser-receiver is an electronic device used for both the creation of a voltage drive pulse for the transmitting transducer and the reception and amplification of the voltage signal from the receiving transducer. Since the drive voltage is usually many orders of magnitude stronger than the received signal (hundreds of volts compared to millivolts), protection circuits must be in

23 7 place to prevent voltage cross-talk between the two compartments [8]. Two different models of pulser-receivers are shown in Figure 2.2. Figure 2.2: Two models of Pulser-Receiver The top pulser-receiver shown in Figure 2.2 is a newer model with digital drive pulse control while the bottom model uses analog control circuits. In pulsers that use analog control circuits, the voltage drive pulse is created from a sudden release of electrical energy stored in a capacitor. Consequently, the voltage waveform would follow an exponential decay as shown in Figure 2.3. Decay Time Figure 2.3: Voltage pulse of analog pulser From Figure 2.3, we see that the voltage pulse has a characteristic decay time which is controlled by the amount of electrical energy stored in the capacitor and the amount of damping in the circuit. This decay time has a significant impact on the transducer pressure waveform output. Using a linear model, the pressure waveform transmitted from the transmitting transducer can be modeled as the convolution of input voltage pulse with the transducer voltage-to-pressure impulse response [1]:

24 8 p transducer (t) = v pulser (t) h transducer (t) (2.1) In pulsers with analog control circuits, there are typically energy and damping settings which can changed independently to modify the shape of the output echo. However, in practice it is often difficult to use these settings to obtain an output voltage pulse that matches with the bandwidth of the transducer. In contrast, for newer pulsers with digital pulse control, the output voltage signal is a square pulse as shown in Figure 2.4. In addition, both the pulse amplitude and width can be specified. Consequently, digital ultrasound pulsers offer a more powerful method to fine-tune the output pressure waveform of a transducer. For this reason, digital pulsers can be used to drive different transducers across a wide range of design center frequencies. Pulse width Figure 2.4: Voltage pulse of digital pulser Piezoelectric Transducers Despite recent developments in electromagnetic[9] and capacitive [10] ultrasound transducers, piezoelectric transducers based on the direct and inverse piezoelectric effects remain the most commonly used type of transducers used for ultrasonic testing. There exist two main types of piezoelectric transducers: single element transducers and phased array transducers. Single Element Transducers Single element transducers are the simplest ultrasonic transducers; they consist of only one active piezoelectric element used to transmit and/or receive pressure waves. A schematic diagram of the components of a single element transducer is shown in Figure 2.5.

25 9 Figure 2.5: Schematic diagram of a single element piezoelectric transducer (courtesy of [11]) In Figure 2.5, we see that there is an electrical connector that sends and receives electrical signal to and from the piezoelectric element. The piezoelectric element is usually in the shape of a disc; it is in contact with a backing element on one face and a matching layer on the other. The purpose of the backing element is to attenuate excessive ringing from the piezoelectric element to increase the frequency bandwidth of the transducer. The purpose of the matching layer is to maximize the wave energy transfer from the piezoelectric element to the test sample. Usually a quarter-wavelength matching layer design is employed [12]. The center frequency of the transducer is controlled by the thickness of the piezoelectric element. The thickness of the element is typically selected to be 1/2 of the wavelength at the design center frequency. The beam spread of the transmitted pressure wave can be related to the diameter of the piezoelectric element using the following formula [13]: sin(ξ spread ) = 1.22c Df (2.2) In Eq. (2.2), ξ spread is the beam divergence angle from transducer centerline to point where signal is at half strength, c is the speed of sound in the propagation medium, D is the transducer active diameter and fis the pressure wave frequency. From Eq. (2.2), we see that transducer beam spread can be reduced by increasing the active element diameter and/or the transducer center frequency.

26 10 Phased Array Transducers Phase array transducers are constructed from arranging multiple active piezoelectric elements in a geometrical array. Even though rectangular matrix [14] and annular [15] array transducers have been tested, linear arrays where the active elements are arranged along a single direction remain the most popular phased array transducer design. A great advantage of phased array transducers is that the steering angle and the focal depth of the output ultrasonic wave can be changed by controlling the relative firing time delays of the individual elements. Figure 2.6 and Figure 2.7 demonstrate the time delay patterns used to achieve beam steering and focussing. Steering and focussing can also be performed simultaneously by combining the two time delay patterns. Figure 2.6: Steering of phased array transducers Figure 2.7: Focusing of phased array transducers

27 11 Compared to single element transducers, phased array transducers offer much more flexibility. Different areas of the test sample can be scanned without physically moving the transducer by electronically changing the steering angles. In addition, the effective aperture size of the transducer can be changed by altering the number of firing elements. Despite these advantages, phased array transducers have yet to replace single element transducers in many NDT applications due to their increased equipment cost, larger physical size and increased complexity in data processing Ultrasonic Testing Data Representation In this section, we will introduce different representation methods used to display the data collected from ultrasound testing. The most basic data representation method used in ultrasound testing is the A-scan, which is simply a 1D plot of the receiving transducer s output voltage signal as a function of time. Figure 2.8 shows an example of an A-scan using the data collected from a TOFD experiment featuring a test piece with a vertical crack. Lateral wave Back wall Top tip Bottom tip Figure 2.8: A-scan representation from TOFD data Looking at Figure 2.8, we see that there are four return echoes which correspond to the lateral wave, top tip diffracted echo, bottom tip diffracted echo and the specular backwall reflection echo. The presence of these echoes corresponds well with the expected signal from a TOFD scan of a vertical crack shown in Figure 1.2. In Figure 2.8, we also see that the lateral wave echo overlaps with the diffracted echo from the top tip of the crack. If the two echoes interfere with each other, then it becomes impossible to visually determine the exact temporal location of the two echoes such that the TOFD technique cannot yield a good estimate of the crack size. In such

28 Scan Time [us] 12 situations, echo identification algorithms can be employed to separate the two echoes and determine the time difference between them. Another form of ultrasonic testing data representation is the B-scan. B-scan representations are created by stacking A-scans line-by-line adjacent to each other in order to create a rudimentary 2D image. Figure 2.9 is an example of a B-scan obtained from translation of a pair of TOFD probes along the direction of the weld. In Figure 2.9, the horizontal axis represents the probe translation direction and the vertical axis is the time axis of the stacked A-scans. Consequently, the A-scans obtained along the probe translation direction are stacked column by column in Figure 2.9. From Figure 2.9, we see that the lateral wave and back wall echoes are continuous along the scan direction. This is expected as the weld sample has continuous top and bottom surfaces. There are also four distinct echoes in Figure 2.9; these echoes correspond to localized flaws along the length of the weld. From this example, we see that B-scans can be used to locate both the lateral and axial (along the sound propagation path) locations of a flaw. 0 B-scan of TOFD scan Lateral wave Flaw echoes Back wall Scan Distance [mm] Figure 2.9: TOFD B-scan containing 4 flaws Another form of ultrasonic testing data representation is the C-scan. C-scans are 2D maps of a test sample, where the color of each pixel represents the arrival time of the echo or the strength of the reflected signal. C scans are obtained by mechanically translating a single element

29 13 transducer over the scan region. Each time the transducer is moved to a new (x, y) co-ordinate, an A-scan is performed and the return echo time of flight or return echo amplitude is recorded. Figure 2.10 shows a C-scan from recording the echo amplitude reflected from the back surface of the coin. Since the front surface topology also affects the strength of the transmitted signal, features on both the top and bottom of the coin can be seen. Figure 2.10: C-scan of back surface of a coin (from [16]) The final ultrasonic data representation method that we introduce in this section is the S-scan. Similar to the B-scan, S-scan produces a 2D slice image that shows both the lateral and axial locations of any discontinuities. However, S-scans are obtained by electronically changing the beam steering angle of a phased array transducer instead of mechanically moving the probe. Figure 2.11 is an example of a S-scan performed for a test sample containing three side-drilled holes. Compared to the B-scan, S-scans are more convenient to acquire since it does not require physical repositioning of the transducer. Figure 2.11: S-scan of three side-drilled holes (from [17])

30 Resolution Limits in Ultrasonic Testing As seen in Section 2.1.4, B-scan and S-scan are the two most commonly used representations to obtain 2D slice images of the test sample. In ultrasound images, resolution is defined as the minimum spatial separation of two flaws that can be clearly identified as two distinct features. For both the B-scan and the S-scan, the resolution in the lateral direction is limited by the width of the acoustic beam that is used to illuminate the flaw. This concept is shown in Figure In Figure 2.12, due to spreading of the beam, a point defect would be detected over a finite lateral displacement. This displacement constitutes the lateral imaging resolution of the scan configuration. Figure 2.12: Lateral resolution in ultrasound imaging The lateral resolution of an ultrasonic scan can be improved by focusing of the probe. A recent development for the improvement of lateral scan resolution is the Total Focusing Method (TFM) [18]. TFM uses post-processing to focus at every point within a desired scan region by summing delayed unfocussed A-scans acquired from a phased array transducer. However, the size of the focal zone in TFM is still limited by physical laws. The theoretical minimum size of the focal zone of a transducer is determined by the Abbe diffraction limit [19]: diffraction limit = 2(1.22 λ z f) D (2.3) In Eq. (2.3), λ is the wavelength in the medium, z f is the focal depth and D is the diameter of the aperture of the transducer. In actual applications, the focal zone of a phased array transducer is usually much larger than the theoretical limit expressed in Eq. (2.3) due to time-delay quantization errors and non-uniform performance of the active piezo elements [20].

31 15 In contrast to the lateral resolution which is limited by the size of the focal zone, the axial resolution of both B-scans and S-scans is primarily limited by the ability to separate two defect features in the time domain A-scan signal. The time-duration of an ultrasonic echo is determined by the bandwidth of the transducer and cannot be reduced through beam focusing [2]. Efforts have been made to design high-frequency high-bandwidth transducers to reduce the time duration of the output echo in order to improve the axial resolution [6]. However, such transducers have weak amplitude output and have limited penetration depth since acoustic attenuation increases with wave frequency [2]. Consequently, hardware solutions to improve the axial resolution of ultrasound images are limited for many NDT applications. For these applications, a post-processing algorithm to separate overlapping echoes is the most viable method to enhance the axial resolution in ultrasonic testing and provide an improved estimate of the height and depth of any defects present. 2.2 Modeling of Ultrasonic Echoes In this section, we will review various physical models designed to analyze the shapes of ultrasonic echoes. Some of these models will be used in Chapter 3 and 4 for the development of the Trigonometric Echo Identification (TEI) algorithm One-Dimensional Piezoelectric Transducer Models As mentioned in Section 2.1.2, piezoelectric transducers are the most commonly used type of ultrasonic transducers in industrial NDT applications. For this reason, we will tailor the TEI algorithm for separation of echoes acquired from piezoelectric transducers. The voltage-to-voltage two-way impulse response of piezoelectric transducers is often modelled using one-dimensional equivalent circuit models [1], [2], [6]. The one-dimensional approximation is valid if the thickness of piezoelectric element is much smaller than its lateral dimensions. For typical piezoelectric transducers used in NDT applications, the thickness of the piezoelectric element is of the order of 0.5 mm while the lateral dimensions are of the order of 10 mm. Consequently, the one-dimensional assumption can be applied. For lightly loaded piezoelectric transducers, the Van Dyke approximate model can be used [2], [6]. In the Van Dyke equivalent circuit model, a transformer is used to transform the electrical voltage into mechanical force in the acoustic path. In the acoustic path, the transducer is

32 16 modelled by an RLC circuit. The capacitance C is inversely proportional to the stiffness of the piezoelectric material; the inductance L is proportional to the vibration mass and the resistance R is proportional to damping of the transducer. A schematic diagram of the Van Dyke approximate model is shown below: Figure 2.13: Van Dyke approximate transducer model As shown in Appendix 1, when an impulse voltage is applied to the transducer, the face velocity (analogous to electrical current) of the transducer takes on the general form of an exponentially enveloped sinusoid: velocity(t) = const e at cos(ω d t + φ) (2.4) Here φ is a constant phase shift. The exponential decay rate a and the damped frequency ω d are dependent on the RLC parameters: a = R 2L ; ω o = 1 LC ; ω d = ω o 2 a 2 (2.5) As derived in Appendix 1, the frequency domain amplitude profile of the two-way voltage-tovoltage transfer function predicted by the Van Dyke model can be expressed as: V out(ω) V in (ω) = const a 2 (ω ω d) 2 (2.6) Looking at Eq. (2.6), we see that the predicted amplitude profile of the two-way transducer transfer function is a symmetric distribution with its peak located at ω d, the damped oscillation frequency. The bandwidth of the distribution is determined by the decay rate a. The larger the

33 17 value of a, the wider the frequency bandwidth of the amplitude profile. Figure 2.14 shows examples of this amplitude profile with const =1, ω d = 3 [rad/s] and a = 0.5 [1/s] and 1.0 [1/s]. Figure 2.14: Frequency amplitude response predicted by Van Dyke model For transducers that are coupled to acoustic media which have acoustic impedance values comparable to the piezoelectric element, the lightly-loaded assumption of the Van Dyke model is no longer valid. For these transducers, the exact KLM one-dimensional model can be used [2], [6]. A schematic drawing of the KLM model is shown in Figure Figure 2.15: Schematic diagram of KLM model In Figure 2.15, C o and C are the input capacitances and κ(f) is the ratio of the electromechanical transformer that converts electrical voltage and current into mechanical forces and velocities. The definitions of these parameters can be found in [2] and [6]. In addition, F 1 and F 2

34 18 are respectively the forces present at the front and back faces of the transducer. The mechanical ports (the front and back faces of the transducer) are connected to the center transformer through a pair of quarter-wave transmission lines. The lengths of these transmission lines are determined by the thickness of the piezoelectric element. Although the KLM model shown in Figure 2.15 does not include matching layers, the KLM model can be extended using the method of transmission matrices [21]. In this method, all components in the KLM model are replaced by a 2 2 transmission matrix. The definitions of these transmission matrices are summarized in Eq. (2.7). Discrete series impedance: [ 1 Z series 0 1 ] 1 0 Discrete parallel impedance: [ 1/Z parallel 1 ] cos(kl) Transmission line: [ sin(kl) Z a jz a sin(kl) ] cos(kl) (2.7) In the definitions shown above, Z a, k and l are respectively the impedance, angular wavenumber and length of the transmission line. Note that for all three types of transmission matrices, the determinant of the matrix is equal to one. Consequently, these matrices have reciprocal properties and can be used in both the transmission and reception paths. Figure 2.16 shows a schematic representation of how the method of transmission matrices can be used to model the transmission path of a transducer with the addition of electrical and acoustic matching.

35 19 Figure 2.16: Transmission matrix model of transducer (Operated as transmitter) The overall transmission matrix can be found by multiplying the transmission matrices shown in Figure The voltage-to-force frequency transfer function of the transmission is equal to the inverse of the (1, 1) element of the overall transmission matrix: [T tr ] = [ 1 Z s 0 1 ] [Telt ][T C o][t C ][T xf ][T P ][T T ][T M 1 0 ] [ 1/Z T 1 ] (2.8) H tr (ω) tr = 1/T 11 Similar transmission matrix multiplication procedures can be conducted to find the reception force-to-voltage frequency response of the piezoelectric transducer. Finally, multiplying the transmission and reception transfer functions would yield the two-way voltage-to-voltage frequency response of the transducer Complete Transfer Function Modeling of Ultrasonic Echoes In the previous section, we examined in detail two different models that can be used to predict the frequency response of a piezoelectric ultrasound transducer. Although modeling the transducer frequency response is crucial to predicting the expected echo shape, other factors such as wave diffraction and defect scattering can also greatly influence the echo shapes in NDT ultrasonic testing.

36 20 According to [1] and [22], the frequency response of each echo can be expressed as a cascade multiplication of frequency transfer functions: H echo (ω) = H elec (ω)h diff (ω)h att (ω)h disp (ω)h sc (ω) (2.9) Here H elec (ω) is the total electrical transfer function including the piezoelectric transducers and the pulser/receiver system; H diff (ω) is the transducer diffraction transfer function; H att (ω) is the attenuation transfer function, H disp (ω) is the dispersion transfer function and H sc (ω) is the defect scattering transfer function. In this section, we will provide an overview of how these factors can be modelled. Electrical System Transfer Function The models introduced in the previous section can be used to predict the frequency response of the piezoelectric transducers. However, the frequency response of the pulser/receiver system is usually measured experimentally. Although the pulser/receiver circuits contain nonlinear elements, they can be approximated by a Thevenin equivalent circuit shown in Figure 2.17 [23]. Figure 2.17: Thevenin's equivalent circuit The Thevenin equivalent voltage source V th (ω) and equivalent resistance R th (ω) can be experimentally determined using two simple measurements [23]. However, it should be noted that both V th (ω) and R th (ω) can vary with energy and gain settings of the pulser/receiver system. Once V th (ω) and R th (ω) are determined, the Thevenin s equivalent circuit can be incorporated in the KLM model introduced in the previous section to obtain the complete transfer function of the electrical system. Transducer Wavefield Diffraction Transfer function

37 21 The transducer wavefield diffraction transfer function describes the pressure field radiated into an acoustic medium from an ultrasound transducer. For an idealized circular piston transducer, exact analytical expressions of the pressure field have been solved in the time domain using the impulse response method [24]. According to [24], the resultant pressure field is axially symmetric and therefore only dependent on the axial distance z (measured from the plane of transducer) and the radial distance r (measured from the central-axis of the transducer). In Figure 2.18, we plot the normalized two-way impulse response and its Fourier transform for two points in the pressure field using the expressions developed in [24]. For these calculations, the radius of the circular transducer is set at 4 mm while the axial distance z is set at 60 mm. The radial distance away from the central axis of the transducer are set at r = 0 and r = 15 mm. These observation points are located in the far-field for frequencies below 21 MHz. Figure 2.18: Two-way impulse and frequency response for two points in a pressure field Looking at Figure 2.18, we see that as we move laterally away from the central-axis, the impulse response becomes broader. It is also time delayed because the point is located further away from the transducer. From the frequency plot, we see that the r = 15 mm response has a much smaller passband compared to r = 0. These results are consistent with the well-known acoustic property that the beam spread of a transducer is inversely proportional to its center frequency. As we move away from the central-axis of the transducer, the transducer diffraction transfer function suppresses the spectral content of the higher frequencies.

38 22 Attenuation Transfer Function Over the typical frequency range used in ultrasonic testing (~from 1 MHz to 20 MHz), the attenuation coefficient of most materials follows an approximate power law relationship with frequency [1]. attenuation(f) = const 1 + const 2 f u (2.10) According to [25], the frequency power u varies from 1.8 to 2.2 for different grades of lowcarbon steel. Since attenuation is frequency dependent, the Fourier transform of ultrasonic echoes are in general asymmetric with respect to the center frequency. Dispersion Transfer Function The effects of dispersion can be safely neglected for bulk wave ultrasonic testing measurements. Dispersion effects can arise either by material property of the acoustic medium or by the mode of wave propagation. In contrast to plate waves such as lamb wave or the shear-horizontal wave, bulk shear or longitudinal wave propagation is not inherently dispersive [26]. Consequently, any dispersion effects present must be attributed to the material property of the acoustic medium. Acoustic attenuation and dispersion of a medium can be related by the Kramers-Kronigs equations [2]. From the Kramers-Kronigs equations, it can be shown that materials which follow a quadratic attenuation curve are not dispersive. Since the attenuation power of steel varies from 1.8 to 2.2 in the frequency range from 1 to 20 MHz, we can conclude that its dispersion effects are negligible. This theoretical conclusion is also corroborated by experimental results [1]. Scattering Transfer Function The scattering coefficients for simple defect geometries such as cylindrical and spherical voids, point reflectors, cracks and flat surfaces have been investigated in [1] and [27] using ray methods. In general, the scattering coefficient of a defect is both frequency-dependent and angledependent. For example, for Rayleigh scattering of small particles, the amplitude of the scattering coefficient is proportional to the fourth power of frequency. However, there also exists defects which have frequency-independent scattering responses. Examples of these defects include diffraction from sharp crack tips and specular reflection from

39 23 flat surfaces [28], [27]. For such cases, the scattering transfer function would be a constant and therefore would not introduce any shape distortion to the ultrasonic echoes. 2.3 Single Reference Deconvolution Before we begin development of the Trigonometric Echo Identification (TEI) algorithm, it is necessary to first examine the existing techniques that have been investigated for identification of overlapping ultrasonic echoes. Among the different techniques, single reference deconvolution is one of the most commonly investigated methods [29],[30]. In this section, we will describe the assumptions of this technique and its limits of applicability Basic Assumptions In single reference deconvolution, it is assumed that each return echo can be modelled by a timeshifted and amplitude-scaled version of a reference echo [29]. Mathematically, this assumption can be expressed as: sig(t) = echo i (t) + n(t) = ref(t) h(t) + n(t) (2.11) In Eq. (2.11), ref(t) is the reference echo, h(t) is the scattering impulse response of the defects present in the test sample, n(t) is the noise present in the system and is the convolution operator in the time domain. A schematic representation of a convolution operation without the addition of noise is shown in Figure Figure 2.19: Single reference convolution From Figure 2.19, it is clear that if h(t) is recovered, we would obtain information regarding the location and scattering strength of each defect present. Using the convolution-multiplication

40 24 duality property of the Fourier transform [31], the equivalent frequency domain expression of Eq. (2.11) can be written as: SIG(ω) = REF(ω)H(ω) + N(ω) (2.12) Using Eq. (2.12), we see that the scattering response H(ω) can be estimated using a simple frequency domain division operation: H est (ω) = SIG(ω) REF(ω) (2.13) Note that H est (ω) is different from the true scattering response H(ω) because it neglects the effect of the noise term in the signal. Eq. (2.13) is the fundamental single reference deconvolution equation and in the next sub-section we will introduce various modifications that have been investigated to improve the performance of this technique Direct Deconvolution Schemes Direct deconvolution schemes are modifications made to the spectral division equation expressed in Eq. (2.13) to improve its performance. One of the earliest modifications introduced is the Wiener deconvolution [32]. The Wiener deconvolution is designed to minimize the impact of deconvolved noise at frequencies with low SNR. In Wiener deconvolution, the scattering response is estimated using the following formula: H est (ω) = G(ω)SIG(ω) (2.14) Here G(ω) is the Wiener filter and is defined as: G(ω) = 1 REF(ω) [ REF(ω) 2 ] REF(ω) SNR(ω) (2.15) Looking at Eq. (2.15), we see that when the SNR(ω) is low, the denominator in the square bracket would have a high value and therefore G(ω) would reduce the contribution from these frequencies. Conversely, when SNR(ω) approaches infinity, G(ω) would approach 1/REF(ω) and we would revert to the basic spectral division equation of Eq. (2.13).

41 25 For Wiener deconvolution to work effectively, we need to have an accurate estimate of the noise spectral density N(ω) or equivalently the signal-to-noise ratio SNR(ω). Although SNR(ω) is in theory frequency-dependent, in practice it is often replaced by a constant SNR value [33] since the frequency dependence of the noise level is difficult to estimate. Another direct deconvolution scheme investigated by researchers is Auto-Regressive Spectral Extrapolation [34], [35]. From Eq. (2.13), we can see that the functional bandwidth of H est (ω) is limited by the frequency bandwidth of REF(ω). At frequencies where REF(ω) is small, H est (ω) cannot be accurately determined even with the adoption of the Wiener deconvolution scheme. Auto-Regressive Spectral Extrapolation is designed to address this problem. In Auto-Regressive Spectral Extrapolation, it is assumed that H est (ω) can be modeled by a sum of complex sinusoids [35]. If this assumption is valid, the spectral content of H est (ω) at frequencies where the SNR is low can be extrapolated from a weighted sum of the spectral content of H est (ω) at frequencies where the SNR is deemed to be sufficiently large. Mathematically, this can be expressed as: m H est (ω) = w j H est (ω i j ) j=1 (2.16) In Eq. (2.16), w j are the weights of each frequency point and m is the order of the autoregressive process. Both w j and m are parameters that need to be optimized. A successful application of the Auto-Regressive Spectral Extrapolation method can extend the useful bandwidth of H est (ω) and therefore sharpen the time-domain scattering response h est (t). A sharpened time-domain scattering response can lead to a more accurate estimate of the locations of each defect Iterative Deconvolution Schemes Iterative deconvolution schemes are not based on the spectral division operation of Eq. (2.13). Instead, an initial guess of h est (t) is made and subsequently improved upon as an optimization problem. A major advantage of iterative deconvolution schemes is that the solution can be optimized to better satisfy the preconceived assumptions of h est (t). However, iterative deconvolution schemes are more computation-intensive compared to direct methods.

42 26 One of the most commonly investigated iterative deconvolution schemes is L1-norm deconvolution. This scheme is designed to recover a h est (t) that consists of sparse spike train [36]. A sparse spike train scattering response is ideal because it provides accurate timing information for all identified defects. L1-norm deconvolution is mathematically formulated to minimize the following expression: min h est (t) [ sig(t) ref(t) h est(t) 2 t + σ h est (t) ] t (2.17) From this equation, we see that there is a sum of two terms that needs to be minimized. The first term is the L2-norm of the deconvolution error. By minimizing this term, we can obtain a convolved response ref(t) h est (t) that best approximates the observed signal sig(t). The second term of Eq. (2.17) is the scaled L1-norm of the scattering response. By minimizing the second term we can obtain a h est (t) that is sparse (contains a small number of non-zeros values). Since Eq. (2.17) has two conflicting minimization criteria, it is not possible to obtain a solution that is optimal for both terms for real signals that contain some noise. By adjusting the value of the σ in Eq. (2.17), we can vary the relative importance of the two terms. Details for choosing the value of σ can be found in [37] Technique Limitations Despite the many improvements made to the single reference deconvolution technique, its application is still limited by the fundamental assumption that all return echoes can be modelled by a scaled and time-delayed copy of a reference echo. As explained in Section 2.2.2, differences in defect location and scattering properties can yield ultrasonic echoes with different center frequencies, envelopes and phase shift. Consequently, single reference deconvolution often performs poorly in configurations where the ultrasonic echoes are of significantly different shapes. In the next section, we will introduce a parametric model approach that addresses this major limitation of single reference deconvolution.

43 Dictionary-based Parametric Representation To separate overlapping echoes with different shapes, researchers have developed the Dictionary-based Parametric Representation (DBPR) approach. We will begin a review of this technique with an overview of its mathematical formulation Mathematical Formulation In DBPR, it is assumed the acquired signal can be represented as a sum of echoes. In addition, each echo is modelled by a parametric mathematical expression whose parameter values can be adjusted [38], [39]. Mathematically, this can be expressed as: sig(t) = echo(x i, t) + e(t) i (2.18) In Eq. (2.18), e(t) is the reconstruction residual error. The notation echo(x i, t) indicates that while each echo is expressed in the time-domain, its shape is controlled by the parameter vector x i. The values of the parameters in each x i are optimized by minimizing the amplitude of the reconstruction error e(t): min e(t) x 2 = min [ sig(t) echo(x i x i, t) ] i t i 2 (2.19) For DBPR to be effective, it is necessary to use parametric mathematical models that accurately describe the shapes of actual ultrasound echoes. The most commonly used parametric model for ultrasound signals is the Gabor dictionary [38], [39], [40]. In a Gabor dictionary, each echo is modelled as a Gaussian enveloped oscillation: echo(x i, t) = A exp[ a 2 (t τ) 2 ] cos[2πf c (t τ) + φ] (2.20) Looking at Eq. (2.20), we see that each parameter vector in the Gabor dictionary contains 5 different variables x i = [A, a, τ, f c, φ]. These five parameters respectively control the amplitude, width, time shift, oscillation frequency and the constant phase shift of the echo. The Gabor dictionary is chosen because it is empirically found to be an adequate model of the backscattered echo from a flat surface reflector in pulse-echo ultrasonic testing [39].

44 28 Despite the popularity of the Gabor dictionary, it often does not perform adequately in situations where it is necessary to obtain an accurate time difference measurement between two overlapping echoes [41]. This is because an accurate reconstruction of the echo envelope is critical for timing measurements. The Gabor dictionary which uses Gaussian-enveloped oscillations is often inadequate for this task. To address this problem, researchers have developed more complicated parametric models to describe ultrasonic echoes. For example, the asymmetric Gaussian chirplet model has been proposed [42]: echo(x i, t) = A env (t τ)cos[2πf c (t τ) + ψ(t τ) 2 + φ] (2.21) env(t) = exp[ a 2 (1 ρ tanh(mt))t 2 ] Note that in Eq. (2.21), m is a fixed positive integer that determines the rate of transition in the hyperbolic tangent function. Neglecting the predetermined parameter m, each parameter vector now contains 7 parameters x i = [A, a, τ, f c, φ, ρ, ψ ]. The additional two parameters, ρ and ψ respectively control the asymmetry of the echo envelope and the frequency chirp factor. Since the value of tanh(mt) varies from -1 to +1, the envelope function can also be expressed as: env(t) = exp[ a 2 (1 + ρ)t 2 ], t < ε env(t) = exp[ a 2 (1 ρ)t 2 ], t > ε (2.22) Here ε is the transition period of the hyperbolic tangent function and is determined by the choice of m. The asymmetric Gaussian chirplet model allows one to model ultrasonic echoes that are both asymmetric and have a non-constant modulation frequency. Despite the enhanced modelling flexibility, the asymmetric Gaussian chirplet model has not been widely adopted for DBPR because the increased number of parameters makes it more difficult to obtain a stable solution for the optimization problem shown in Eq. (2.19) Sparsity-Promoting Algorithms Looking at Eq. (2.19), we see that the reconstruction error e(t) can be progressively reduced by increasing the number of parametric echoes used to represent the signal sig(t). Although such an approach is useful for ultrasonic data compression applications [38], it is not appropriate for NDT applications because it can lead to the detection of extraneous echoes. False positive echoes in NDT testing incur time and monetary costs as a more thorough scan or a destructive test would need to be conducted to assess the condition of the test specimen. Consequently, for NDT

45 29 applications it is crucial to obtain a sparse solution which suppresses the occurrence of extraneous echoes. One of the methods to obtain a sparse solution is the L1-norm minimization approach introduced in Section When L1-norm minimization is used to solve for the parameters in DBPR, the technique is also known as Basis Pursuit [37], [43]. The minimization objective of Basis Pursuit can be expressed as: min x i [ sig(t) echo(x i, t) t i 2 + σ A i ] i (2.23) In Eq. (2.23), A i is the absolute value of the amplitude parameter of each echo identified. By minimizing the sum of the amplitude parameters, one can reduce the number of echoes that are used to represent the signal. Once again, the relative importance of the two terms in the minimization objective can be adjusted by changing the value of σ. Another sparse-solution promoting algorithm is the Matching Pursuit technique [40], [44], [45]. At the first iteration of the matching pursuit algorithm, the echo(x 1, t) that best matches the obtained signal is found by maximizing the inner product between the two: max x i [ echo(x 1, t) sig(t) echo(x 1, t) 2 ] t (2.24) The division in Eq. (2.25) is needed to normalize the energy of the echo. Notice that we use the subscript 1 for the parameter vector because it is the first echo identified. After this echo is found, it is subtracted from the signal and a residual signal e 1 (t) remains. e 1 (t) = sig(t) echo(x 1, t) (2.25) The process is then repeated by finding the next echo that has the largest inner product with the residual echo. The Matching pursuit algorithm ends when the L2-norm the residual e i (t) is below a threshold value. Compared to Basis Pursuit, Matching Pursuit is less computationally expensive and is guaranteed to converge since it is always possible to find an echo that reduces the residual signal. However, the algorithm is also short-sighted (only one echo is identified at each iteration) and therefore its performance is often inferior than that of Basis Pursuit [37].

46 30 Chapter 3 Basis and Assumptions of TEI Algorithm 3.1 Frequency Domain Assumptions During the development of an echo separation algorithm, it is necessary to introduce assumptions of the expected properties of an ultrasonic echo. These assumptions enable us to correctly decompose the ultrasound signal into its constituent echoes. In this section, we will first introduce the frequency domain assumptions used in the TEI algorithm. The justifications and the limits of applicability of these assumptions will be detailed in Sections and Amplitude Profile Assumption As will be shown in chapter 4, TEI is a frequency-domain algorithm where we iteratively improve our estimates of the frequency amplitude profiles of the two constituent echoes. In order to formulate the algorithm as an optimization problem, the amplitude profiles must be first described as a parametric mathematical model. After analyzing both the theoretical and experimental frequency response of piezoelectric transducers, it was decided to model the amplitude response of each echo as an asymmetric Q- Gaussian distribution. Compared to the standard Gaussian distribution, the Q-Gaussian has an extra degree of freedom which allows it to vary the decay rate at the tails of a distribution [46]. This extra degree of freedom is important for modeling of the frequency amplitude profile of an ultrasonic echo. Amplitude Profile Assumption: The frequency amplitude profile of an ultrasonic echo can be adequately modelled by an asymmetric Q-Gaussian distribution expressed as: for ω < ω c : M(ω) = S[1 (1 q 1 )b 1 (ω ω c ) 2 ] 1 1 q1 for ω > ω c : M(ω) = S[1 (1 q 2 )b 2 (ω ω c ) 2 ] 1 1 q2 (3.1) In Eq. (3.1), S is the amplitude scaling parameter; ω c is the center frequency; b 1 and b 2 are the width scaling parameters and q 1 and q 2 are the tail-heaviness control parameters. When the value of q is less than 1, the distribution is less tail-heavy than the normal distribution and viceversa. The normal distribution is recovered when q approaches 1.

47 31 By introducing independent width and tail-heaviness parameters below and above the center frequency ω c, the mathematical expression in Eq. (3.1) can be used to model asymmetric frequency-domain amplitude responses. With this definition, the frequency amplitude information of each echo can be fully defined using six parameters p = (S, ω c, b 1, b 2, q 1, q 2 ). Figure 3.1 shows an example of a Q-Gaussian distribution with S = 1, ω c = 4 [rad/s], b 1 = 3 [s 2 ], b 2 = 2.5 [s 2 ], q 1 = 1, q 2 = 2. Figure 3.1: Asymmetric Q-Gaussian distribution Phase Profile Assumption In the frequency domain, the TEI algorithm assumes that one echo has an earlier arrival time for all frequencies in which both echoes have significant spectral content. If the assumption is satisfied, then one can unambiguously identify which echo arrives at an earlier time. This assumption is important because for a non-dispersive medium the arrival time difference of two echoes is directly proportional to the difference in their acoustic path travel distance. This time difference can therefore be used to identify the exact location and size of a material flaw. Using the well-known Fourier transform property that a positive time delay corresponds to an increase of negative phase slope in the frequency domain [31], the assumption of sequential spectral arrival time can be expressed as a phase slope inequality condition.

48 32 Phase Profile Assumption: Considering the frequency domain phase profiles of two echoes that overlap in the time domain, the earlier arriving ultrasonic echo has a less negative phase slope for all frequencies in which both echoes have significant spectral content: if echo A arrives first: if echo B arrives first: θ A (ω) ω θ A (ω) ω > θ B(ω) ω < θ B(ω) ω + t min t min (3.2) In Eq. (3.2), θ A (ω) and θ B (ω) are the frequency domain phase profiles of echo A and B and t min is the minimum allowable time separation between the two echoes. t min is added in Eq. (3.2) because from a practical standpoint it is extremely difficult to separate two echoes which have a time separation that can be infinitely small. In the simulation tests of the TEI algorithm that will be presented in Chapter 5, the value of t min is set to be 1/2 of the inverse of the estimated center frequency of the two echoes and represents half of a period of the characteristic modulation frequency of the echoes. In other words, we assume that the smallest defect feature size that we can detect is half of the wavelength of the center frequency of the transducer. In actual applications, the phase slope inequality assumption presented above will be satisfied if there exists sufficient time separation between the two echoes. In Chapter 5, we will explore how the performance of the TEI algorithm is affected when the time separation between the two echoes approaches the minimum time-separation t min Justification of Amplitude Profile Assumption Theoretical Justifications In Section 2.2.1, we introduced two one-dimensional piezoelectric models. For a lightly loaded transducer, the Van Dyke approximate model can be used to obtain an analytical expression of the two-way voltage-to-voltage transducer transfer function. The transfer function expression was shown in Eq. (2.6) and will be repeated here: V out(ω) V in (ω) = const a 2 (ω ω d) 2 (3.3)

49 33 Comparing Eq. (3.3) with the asymmetric Q-Gaussian distribution shown in Eq. (3.2), it can be easily seen that the Van Dyke transfer function can be exactly modelled by a symmetric Q- Gaussian distribution with b 1 = b 2 = 1 and q a 2 1 = q 2 = 2. Therefore, we can conclude that the two-way transfer function of a lightly-loaded transducer can be exactly described by our amplitude profile model. For highly-damped broadband transducers typically used in arrival time sensitive NDT applications, the Van Dyke approximate model is no longer adequate and the KLM model can be used. As detailed in Section 2.2.1, the two-way transfer function of the KLM model can be calculated through a cascade multiplication of transmission matrices. Since many of these transmission matrices contain complex frequency-dependent terms, a convenient analytical transfer function expression cannot be obtained except for the simplest configurations. However, one can still evaluate the validity of the asymmetric Q-Gaussian amplitude profile model using a demonstrative model of a broadband piezoelectric transducer. For this demonstrative model, we used the extended KLM model described in Section to model the frequency response of a broadband piezoelectric transducer. The geometric and material properties of the model are shown in Table 3.1. The values of the parameters in Table 3.1 were selected based on piezoelectric transducer design guidelines [12]. The center frequency of the transducer is designed to be at 2 MHz. By substituting these properties into the KLM model, we can obtain its two-way transfer function. The fundamental harmonic impulse response of the KLM model is shown in Figure 3.2 and its Fourier transform is shown in Figure 3.3. Only the fundamental harmonic is displayed because in typical NDT applications, the higher harmonics of the transducer are suppressed by the bandwidth of the excitation pulse and frequency-dependent attenuation effects in the propagation path. Details of implementation of this model can be found in Appendix 2 and 3.

50 34 Property Value Impedance of Piezoelectric Material PZT-5H: Z = 34.6 MRayl Material Coupling Factor 0.49 Piezo Clamped Permittivity F/m Piezo Density 7500 kg/m 3 Piezo Active Area 1 cm 2 Piezo Thickness 1 mm Backing Material Assumption Perfectly Matched Layer Matching Layer Acoustic Impedance Z = 40 MRayl Matching Layer Density 9000 kg/m 3 Matching Layer Thickness 550 µm Impedance of Propagation Medium Steel: Z = 46 MRayl Electrical Impedance of Pulser 50 Ω Electrical Impedance of Receiver 50 Ω Table 3.1: Properties for broadband KLM simulation Figure 3.2: First harmonic impulse response of KLM model Figure 3.3: KLM model of broadband transducer

51 35 Looking at Figure 3.3, we see that the amplitude profile of a simple one-dimensional transducer as simulated by the KLM model can be accurately approximated by an asymmetric Q-Gaussian distribution. We can also see that the phase profile predicted by the KLM model is approximately linear in the bandwidth of the first harmonic response. Experimental Verification In addition to the theoretical justifications introduced above, it is also worthwhile to verify the validity of our frequency domain assumptions with experimental data. For this purpose, we examine the backwall obtained from a TOFD experiment. In this experiment, we used a pair of 5 MHz, 3 mm diameter piezoelectric transducers attached to 60 Rexolite 1 wedges. The small active area diameter is chosen to provide divergent beams that cover a wide scan area. The test setup schematic is shown in Figure 3.4. Following TOFD protocols [47], the transducers are spaced horizontally such that the intersection of the central propagation axes of the transducers occurs at the bottom 1/3 of the sample thickness as shown in Figure 3.4. The shaded region in Figure 3.4 indicates the -6dB beam spread at 5 MHz. Figure 3.4: Pitch-catch backwall echo acquisition configuration The acquired backwall echo and its Fourier transform are shown in Figure 3.5 and Figure 3.6. From Figure 3.6, we see that the amplitude peak of the backwall echo is located at approximately 3.8 MHz. This is lower than the expected center frequency of 5 MHz and is likely caused by 1 Rexolite is a trademark plastic made by C-Lec Plastics Inc. It is a material often used for acoustic lenses due to its low acoustic attenuation coefficient and stable chemical properties.

52 36 frequency downshifting due to off-axis diffraction (see Section for details). Despite the influence of diffraction effects, we see that the amplitude profile can still be accurately modelled by an asymmetric Q-Gaussian distribution. This experimental result is consistent with the frequency domain assumptions employed by the TEI algorithm. In Figure 3.6, we also see that the phase profile is again approximately linear within the -6dB bandwidth of the signal. Figure 3.5: Experimental pitch-catch backwall echo Figure 3.6: Fourier transform of experimental pitch-catch backwall echo Applicability Limits of Echo Assumptions The assumption justifications provided in the previous sections are mainly based on the transfer function of the piezoelectric transducer. However, as explained in Section 2.2.2, the complete transfer function model of an ultrasonic echo includes other contributing factors such as

53 37 attenuation, dispersion, transducer wavefield diffraction and flaw scattering. In this section, we will discuss the applicability limits of our echo assumptions with respect to these contributing factors. Attenuation As explained in Section 2.2.2, ultrasound attenuation in steel follows a near quadratic frequency dependence in the Rayleigh scattering regime. Consequently, higher frequencies would be attenuated at a greater rate and the resultant frequency amplitude profile would be asymmetric. This type of smooth amplitude profile asymmetry can be adequately modelled by an asymmetric Q-Gaussian amplitude model having independent width and tail-heaviness parameters below and above the center frequency. Since most engineering materials including metals, polymers and plastics demonstrate a smooth power law acoustic attenuation frequency dependence [2], acoustic attenuation should not be a limiting factor in the application of the TEI algorithm. Dispersion TEI is designed as an algorithm to enhance the axial resolution in ultrasonic time-of-flight based size estimates of defects and assumes that the speed of sound in the test sample is constant. If the speed of sound were not constant, there would be distortion of the waveform group delay and it would be difficult to relate the arrival time of the echo to the physical location of a flaw. In Section 2.2.2, we showed that dispersion effects in bulk wave ultrasound testing of low-carbon steel specimens are negligible. Consequently, TEI is applicable for longitudinal and shear wave inspection of steel test pieces. However, the constant speed of sound assumption of TEI will not be applicable for inspection of highly dispersive materials or for guided wave applications where the wave propagation mode is inherently dispersive [26]. Transducer Wavefield Diffraction As shown in Figure 2.18 of Section 2.2.2, the frequency response of transducer wavefield diffraction can act as a low-pass filter if the point of observation is displaced laterally from the central axis of the transducer. If the bandwidth of the transducer is only marginally higher than the pass band of the diffraction low-pass filter, transducer wavefield diffraction will cause a downshift of the central frequency of the transducer as observed in the pitch-catch backwall echo

54 38 of Section For such cases the frequency assumptions of the TEI algorithm can still be valid as demonstrated in the experimental backwall echo demonstrated in Section However, if the center frequency of the transducer is much higher than the pass band of the diffraction filter, the wavefield diffraction response would cause significant distortions to the phase and amplitude profiles of the ultrasound echo. To demonstrate this idea, in Figure 3.7 we multiply a simulated transducer response which has a center frequency of 4 MHz with the waveform diffraction frequency response shown in in Figure 2.18 (calculated using a = 4 mm, z = 60 mm, and r = 15 mm). Figure 3.7: Echo distortion due to wavefield diffraction In Figure 3.7, we see that wavefield diffraction has created a double peak in the resultant echo amplitude profile. In addition, the resultant echo phase profile is also heavily distorted. Consequently, in such cases the asymmetric Q-Gaussian amplitude model of TEI would not be able to accurately portray the echo amplitude profile. Distortion due to wavefield diffraction can be reduced by using a small aperture transducer which has a wide beam spread pattern. Flaw Scattering The geometry of the scattering defect can have a significant impact on the shape of the return echo. For example, for a cylindrical or spherical void with a radius comparable to the wavelength

55 39 of the incoming ultrasonic wave, the scattered wave response would have a creeping wave component that has travelled around the round defect [1]. Such scatterers would cause a substantial distortion in both the amplitude and phase profiles of the return echo. Consequently, the frequency assumptions of the TEI algorithm are only applicable for the ultrasonic testing of scattering defects that have an approximately frequency-independent scattering response within the bandwidth of the transducer. Examples of frequency independent scattering defects include sharp cracks and plane reflectors [28], [27]. NDT applications that are expected to contain these defects include among others Time of Flight Diffraction [3] and normal-incidence testing of adhesive bonds between thin structures [4].

56 40 Chapter 4 Trigonometric Echo Identification Algorithm 4.1 Algorithm Overview TEI is an iterative algorithm where the amplitude control parameters are repeatedly updated in order to obtain optimal phase response. The TEI algorithm begins with an initial estimate of the amplitude control parameters of the two echoes. These amplitude control parameters are then substituted into Eq. (3.1) to create the initial trial amplitude profiles. Next, the corresponding phase profiles are reconstructed according to the phase reconstruction procedure that will be introduced in Section 4.2. After the phase profiles are calculated, they are evaluated for violation of the phase assumption that was introduced in Chapter 3. The recovered echoes are also evaluated according to additional optimality metrics that will be introduced in this Section If the phase profiles are optimal according to these criteria, the algorithm is considered converged. If the phase profiles are suboptimal, the amplitude control parameters of the two echoes are updated, and so begins the second iteration of the algorithm. A flowchart of the TEI algorithm is shown in Figure 4.1. Figure 4.1: Flowchart of TEI algorithm

57 Trigonometric Phase Profile Reconstruction At each frequency, the spectral information of a signal can be represented by a complex vector with a length equal to its amplitude and an orientation equal to its phase value. Using this concept, the Fourier transforms of two overlapping echoes (ECHO A (f), ECHO B (f)) and the resultant total signal SIG(f) can be graphically represented by a vector addition diagram as shown in Figure 4.2. Note that in this section we do not explicitly show the frequency dependence of the amplitude and phase values for convenience in representation. Figure 4.2: Vector representation of overlapping echoes In Figure 4.2, M A and M B are the amplitudes of the overlapping echoes while M T is the amplitude of the total signal. Similarly, θ A and θ B are the phases of the overlapping echoes while θ T is the phase of the total superimposed signal. Note that we adopt the convention that a positive phase change is an angular displacement in the counter-clockwise direction. The values M T and θ T are obtained by the Fourier transform of the recorded time-domain ultrasonic data and therefore are known for all frequencies within the bandwidth of the total signal. It should be stressed that Figure 4.2 only shows the magnitude and phase information of the two echoes and the total signal at one particular frequency. As one sweeps through the frequency range, the amplitude (length) and phase (orientation) values of each component would vary. From Figure 4.2, we can see that if the values of M A and M B are known (or estimated), the interior angles (α, β, γ) of the vector addition triangle can be solved using the cosine law. Once

58 42 the interior angles are solved, the phase angles of the two echoes can be easily calculated. For example, in the configuration shown in Figure 4.2, θ A = θ T + β and θ B = θ T α. However, the vector addition orientation shown in Figure 4.2 is not unique. For a given set of amplitude values (M A, M B, M T ), there exists two possible phase configurations. Figure 4.3 shows an equally valid vector addition configuration with the same set of component amplitude values (M A, M B, M T ). In this phase configuration, it can be seen that θ A = θ T β and θ B = θ T + α. Figure 4.3: Alternative vector addition configuration In order to select the correct vector addition configuration, one also needs to know the relative rotation of the echo phasor vectors (ECHO A (ω), ECHO B (ω)) and whether the interior angle between the two vectors is increasing or decreasing. These two attributes along with the vector amplitude values (M A, M B, M T ) are sufficient to define a unique vector addition configuration. To determine the change in the interior angle, one can use trigonometry to calculate the value of γ for the frequency range of interest and then calculate its derivative γ. In contrast, the relative rotation of the echo phasor vectors can be determined using the sequence of arrival time of the two echoes. From the phase assumption expressed in Eq. (3.2), we see that the phase slope of the earlier arriving echo is less negative than the phase slope of the second echo. Since the phase slope is by definition the rate of change of the phase angle, in the complex plane it can be represented by the rate of rotation of the complex phasor vector in the counter-clockwise direction. Consequently, ω

59 43 the earlier arriving echo would have a phasor vector that rotates counter-clockwise relative to the phasor vector of the second echo. Summarizing the concepts described above, we can develop the phase profile reconstruction chart shown in Table 4.1. γ ω > 0 γ ω < 0 Table 4.1: Phase reconstruction chart Echo A arrives earlier θ A = θ T β θ B = θ T + α θ A = θ T + β θ B = θ T α Echo B arrives earlier θ A = θ T + β θ B = θ T α θ A = θ T β θ B = θ T + α From Table 4.1, we see that the two separate sets of reconstructed phase profiles are obtained depending on which echo arrives first. Since it is not possible to know in advance the sequence of echo arrival, we need to examine the two sets of reconstructed phase profiles to select the correct set of phase profiles. The phase profile selection procedure will be described in Section Components of TEI Algorithm Echo Optimality Metrics As seen from the flowchart shown in Figure 4.1, one needs to assess both the optimality of the reconstructed ultrasonic echoes and their violation of the phase slope inequality in order to determine the state of convergence at each iteration of the TEI algorithm. Assessment of echo optimality is needed because the phase slope inequality condition imposes only a restriction on the relative arrival times of the two echoes. In order to obtain echoes with the desired shapes, it is necessary to introduce additional echo optimality conditions. The exact form of the echo optimality metric employed is chosen depending on the prior knowledge available for the return ultrasonic echoes. If the approximate echo shape for one or both of the return echoes is known, a cross-correlation based method can be used to assess the similarity between the recovered and reference echoes. Mathematically this can be formulated as:

60 44 CC(ref 1 (t), echo 1 (t)) CC(ref 2 (t), echo 2 (t)) optimality = max [ ] max [ ] t echo 1 (t) 2 t ref 1 (t) 2 t echo 2 (t) 2 t ref 2 (t) 2 (4.1) In Eq. (4.1), the subscripts 1 and 2 stand for the time order of echo arrival and CC(ref(t), echo(t)) is the cross-correlation operation between the reference echo and the recovered echo. The denominators in Eq. (4.1) are required to normalize the energy of the crosscorrelation. We take the maximum value of each of the normalized cross-correlation functions, which corresponds to the time-shift between the reference and the recovered echo. A negative sign is required because optimization problems are typically formulated as minimization problems. Although Eq. (4.1) is shown to use two references, the cross-correlation optimality metric can also be applied if only one reference is known (either the first or the second arriving echo). If reference waveforms estimates are not available, more general echo optimality metrics can be used. From Figure 3.3 and Figure 3.6, we see that both the simulation and experimental Fourier transforms of broadband piezoelectric transducers exhibit near linear phase responses within the transducer bandwidth. This is not a mere coincidence but in fact a conscious design goal of piezoelectric transducer designers to obtain a near linear phase response to reduce echo shape distortions [6]. For this reason, the linearity of the echo phase response could be used as an echo optimality metric. The linearity of the phase response can be evaluated using the following statistical measurement of the standard deviation of the phase slope: nonlinearity(θ(ω)) = θ (ω) + t M(ω)dω M(ω)dω (4.2) In Eq. (4.2), M(ω) is the amplitude profile of the echo and t is negative of the spectrallyaveraged phase slope and can also be interpreted as the spectrally-averaged echo arrival time: t = θ (ω)m(ω)dω M(ω)dω (4.3) Consequently, an alternative optimality metric can be formulated as the sum of the phase nonlinearity of the two phase profiles: optimality = nonlinearity(θ A (ω)) + nonlinearity(θ B (ω)) (4.4)

61 45 The echo optimality metrics differ from the amplitude and phase assumptions listed in chapter 3 in that they are not conditions that are strictly enforced. The first priority of the TEI algorithm is to satisfy the amplitude and phase assumptions listed in chapter 3. Once these conditions are satisfied, TEI would attempt to minimize the selected optimality metric in order to recover echoes with the desired shapes Determination of the Correct Set of Phase Profiles As shown in Table 4.1 of Section 4.2, there exists two possible sets of reconstructed phase profiles depending on which of the two echoes arrives first. Since the order of echo arrival cannot be determined a priori, we must examine the two sets of reconstructed phase profiles after each iteration to determine which set should be selected. The TEI algorithm selects the set of phase profiles which has the smallest violation of the phase slope inequality assumption that is used for its reconstruction. For example, if the first set of phase profiles are reconstructed using the assumption that echo A arrives first, then from Eq. (3.2) we see that the phase profile of echo A should have a less negative phase slope compared to the phase profile of echo B. If the reconstructed phase profiles show that echo A has a more negative phase slope than echo B, then the assumption used for the reconstruction of the phase profiles is violated. For any one set of phase profiles, the violation of its reconstruction phase slope inequality assumption can be calculated using the following metric: VIOL = {window(ω) max [0, ( θ 2(ω) ω ω θ 1(ω) ω )] } 0 (4.5) In Eq. (4.5), θ 1 (ω) and θ 2 (ω) are the unwrapped phase profile of first and second echoes. The max[ ] function is used to avoid penalizing frequency points that satisfy the phase slope assumption. The presence of a window function is needed to limit the applicability of the constraint to frequencies for which the two echoes overlap. We also take the L0 norm of the max[ ] function because the number of violation points is a more stable measurement of the phase slope violation. (If we instead took the L2 norm of the max[ ] function in Eq. (4.5), the stability of the metric would be greatly affected by the unwrapping errors that occur near the 0 and 2π phase crossover points.)

62 46 By selecting the set of phase profiles with the smallest violation, we can determine which arrival time assumption is actually correct (i.e. which echo arrives earlier) Phase Slope Inequality Constraint To ensure that the phase slope inequality assumption of Eq. (3.2) is satisfied, we need to enforce a constraint such that the violation of Eq. (3.2) must be less than a small tolerance value: VIOL[θ 1 (ω), θ 2 (ω), t min ] < tolerance (4.6) Where the violation of Eq. (3.2) can be calculated as: VIOL = {window(ω) max [0, ( θ 2(ω) ω ω θ 1(ω) ω ) + t min] } 0 (4.7) Note that the violation metric of Eq. (4.7) is a modified version of Eq. (4.5) with the addition of the minimum allowable time separation t min. The reason why t min is included in Eq. (4.7) but not in Eq. (4.5) is because these two phase slope violation metrics serve different purposes. Equation (4.5) is used to select the correct set of phase profiles and in the process also determine which of the two echoes arrive first. After we establish the time order of echo arrival, Eq. (4.7) can then be used to verify whether the reconstructed phase responses actually satisfy the TEI phase profile assumptions. 4.4 Implementation as Constrained Optimization Problem Constrained Optimization Formulation Having introduced the phase slope inequality violation constraint of Eq. (4.6), we are finally in the position to formulate the TEI algorithm as a constrained-optimization problem to satisfy all frequency-domain assumptions while minimizing the echo optimality metric. A flexible method to solve a constrained optimization problem is the Augmented Lagrangian (ALAG) method. ALAG transforms a constrained problem into a series of unconstrained optimization problems through the use of additional penalty terms that are proportional to the violation of any constraints [48].

63 47 With the adoption of ALAG, the TEI algorithm is divided into inner and outer loops. In the inner loop, we solve an unconstrained optimization problem with a cost function that contains both the echo optimality metric and the violation of the phase slope inequality: Cost = optimality χ2 4μ + μ [max (0, VIOL + χ 2μ )] 2 (4.8) In Eq. (4.8), VIOL is the phase slope inequality violation metric expressed in Eq. (4.7) and χ and μ are respectively the Lagrange multiplier and penalty parameters. The values of χ and μ do not change within the inner loop. Once the inner loop is converged, the ALAG algorithm would check for the value of VIOL. If the value of VIOL is less than the tolerance, the outer loop and hence the entire TEI algorithm is considered converged. Otherwise, the values of χ and μ will be adjusted in the outer loop and we will go back inside the inner loop to solve a new unconstrained optimization problem with an adjusted cost function. Since any inner loop solution that has a VIOL value less than the tolerance is considered the final solution, it is important that we initialize the values of χ and μ to small values. This can ensure we do not place too large an initial penalty on VIOL and obtain a suboptimal solution that prematurely ends the ALAG algorithm. By using small initial values, we can allow the ALAG algorithm to update the values of χ and μ and obtain a more optimal solution that minimizes the optimality metric while ensuring VIOL is less than the tolerance. The following pseudo-code shows the constrained-optimization implementation of the TEI algorithm using ALAG. Again note that the inner while-loop is the unconstrained optimization problem and the outer while-loop updates the χ and μ parameters.

64 48 # Begin Procedure 1 Initialize p A, p B, χ, μ 2 While (VIOL > tolerance) 3 While (not converged) 4 Generate M A (ω), M B (ω) by substituting p A, p B into Eq. (3.1) 5 Reconstruct [θ A (ω), θ B (ω)] (1) and [θ A (ω), θ B (ω)] (2) 6 Select [θ A (ω), θ B (ω)] (correct) using VIOL[θ A (ω), θ B (ω)] of Eq. (4.5) 7 Calculate optimality using Eq. (4.1) or Eq. (4.4) 8 Calculate VIOL[θ A (ω), θ B (ω), t min ] using Eq. (4.7) 9 Calculate Cost[optimality, VIOL, χ, μ ] using Eq. (4.8) 10 Check convergence 11 Update p A, p B 12 End While 13 Update χ, μ using VIOL[θ A (ω), θ B (ω), t min ] in ALAG 14 End While # End Procedure Implementation Details The final TEI algorithm is implemented in the MATLAB programming environment. Since MATLAB is optimized for matrix and vector operations, code vectorization is employed extensively in the implementation in order to improve the performance of the TEI algorithm. In the time domain, each simulated test signal is composed of 2048 data points with a time step of 10 ns. With these time sampling settings, the corresponding Fourier transform would have a frequency resolution of 48.8 khz and a maximum frequency of 51 MHz. As will be shown in Chapter 5, the simulated echoes have center frequencies near 3 MHz and -6dB percentage bandwidths of approximately 50%. Consequently, the chosen sampling settings ensure that we can utilize approximately 100 frequency domain data points to represent the amplitude and phase profiles. For experimental signals, the sampling time is determined by the data acquisition rate of the hardware. Since the TEI algorithm can only handle two overlapping echoes, experimental signals must be time-windowed to remove additional echoes. In the current implementation, a Tukey window with 50% taper width (shown in Figure 4.4) is chosen for windowing the two overlapping echoes. A Tukey window is used because it can suppress transition side-lobes without affecting the amplitude at the center of the signal. After windowing the overlapping echoes, the signal is zero-padded so that it contains 2048 data points. Zero-padding is used to ensure that there are sufficient data points in the frequency domain to perform the TEI algorithm.

65 49 Figure 4.4: 50% taper Tukey window For transforming the time-domain data into the frequency domain, we use the built-in Fast Fourier Transform (FFT) function of MATLAB. The FFT algorithm in MATLAB is internally based on the FFTW library [49]. The FFTW library automatically chooses the Fourier transform method which is expected to provide the best performance depending on the processing hardware and the length of the time series N. However, all Fourier methods employed by the FFTW library have computational complexity of O(N log N). To solve the unconstrained optimization problem in the inner-loop of the TEI algorithm, we use the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) optimization method [50]. CMA-ES is chosen as the optimization method because it does not depend on the gradient of the cost function. Since the cost function shown in Eq. (4.8) consists of both the echo optimality metric and the phase slope inequality violation, the gradient may not be continuous in the solution space. Another advantage of CMA-ES is that it is a population-based approach, meaning at each iteration the cost function is evaluated at multiple points. For this reason, CMA-ES is less sensitive to the initial guess of the optimization parameters. However, the use of CMA-ES must be conducted carefully. Since CMA-ES is a probabilitybased algorithm, the exact search locations at each iteration are randomly distributed based on a calculated multi-variate probability density distribution. Although this randomness allows the CMA-ES algorithm to search through a multi-dimensional space efficiently [51], it can also lead to different solutions for the same input parameters. In chapter 5 we will discuss the methodology we employed to improve the robustness of our solutions.

66 Summary of Novelty and Advantages of the TEI Algorithm To our knowledge, TEI is the only echo separation algorithm that calculates the phase information using trial amplitude profiles and the frequency transform of the acquired signal. Due to this important distinction, TEI possesses the following advantages: Compared to DBPR using the same number of optimization parameters, TEI would have smaller echo reconstruction errors because the phase information is calculated from the frequency transform of the acquired signal Compared to DBPR using the same number of optimization parameters, TEI can describe more complex ultrasonic waveforms because the variable phase profiles offer extra degrees of freedom Since phase information is adapted to the acquired data, TEI performs more reliably when the echo waveforms are not perfectly described by the mathematical form of the model (i.e. the parametric model used to describe the frequency domain amplitude profiles) However, when compared to DBPR, the new TEI algorithm also possesses two disadvantages: Since TEI is formulated as a constrained optimization problem, for the same number of optimization parameters it would require more iterations to converge compared to DBPR, which for the special case of two overlapping echoes is formulated as an unconstrained optimization problem. However, the difference in convergence time can be reduced by having suitable initial values of the Lagrange multiplier and penalty parameters χ and μ. Since TEI relies on phase reconstruction using trigonometric relationships, the algorithm would not work properly if the spectral content of the two echoes are vastly different from each other. However, for such situations a time-frequency transform such as the continuous wavelet transform [7] can be easily applied to separate the two echoes.

67 51 Chapter 5 Results and Discussions 5.1 Simulation Tests and Comparison Benchmark Once the TEI algorithm was implemented in MATLAB, simulation tests were conducted to assess the performance of this novel technique. Since simulation signals were created with known parameters, it was possible to determine precisely both the timing and reconstruction errors of the algorithm. To benchmark the performance of the TEI algorithm against existing techniques, the DBPR technique was also implemented in MATLAB. In particular, we implemented a DBPR model that is a modified version of Eq. (2.21). echo(x, t) = A env (t τ)cos[2πf c (t τ) + φ] (5.1) env(t) = exp[ a 2 (1 ρ tanh(mt))t 2 ] This particular model was chosen because it uses six parameters, x = [A, a, τ, f c, φ, ρ ], to describe each echo. Using the same number of parameters for both TEI and DBPR allows a more direct comparison of the two techniques. Looking at Eq. (5.1), we see that this particular DBPR model can describe ultrasonic echoes with asymmetric envelopes but cannot accurately model echoes with non-constant modulation frequency. It should be noted that DBPR is a time-domain based algorithm while TEI is a frequency-domain based algorithm. However, the relative performance difference between the two algorithms can still be assessed with the use of appropriate performance metrics. In this study, percentage echo timing and reconstruction errors were selected as the critical performance metrics. To obtain the arrival time of the extracted echoes, we used the spectrally-averaged arrival time as defined in Eq. (4.3). Using the spectrally-averaged arrive time, we calculated the percentage echo timing error as: %Error(timing) = ( t 2 t 1 calc. t 2 t 1 actual ) t 2 t 1 actual 100 (5.2) In Eq. (5.2), t 1 and t 2 are respectively the spectrally-averaged arrival times of the first and second echoes. To calculate the percentage reconstruction error, we used the following definition:

68 52 %Error(reconst. ) = [sig(t) (echo 1 (t) + echo 2 (t)) calc. ] 2 t sig(t) 2 t 100 (5.3) The denominator of Eq. (5.3) is needed to normalize the reconstruction error by the energy of the total signal. In the implementation of DBPR, we use the Eq. (2.19) as the optimality metric in the resultant unconstrained optimization problem. To ensure that the TEI and DBPR algorithms were compared objectively, we used the same CMA-ES optimization solver for solving the optimization problems in both algorithms. In addition, we selected the same population size (500) and the same maximum number of iterations (350) for both techniques. In CMA-ES, population size refers to the number of search locations at every iteration. Since TEI is solved using the ALAG approach, the maximum number of iterations refers to the number of iterations inside the inner loop. As mentioned in Chapter 4, CMA-ES is a probability-based algorithm and therefore the converged solution can vary even if the same starting parameters are used. To improve the quality of the converged solutions, we ran the CMA-ES solver 6 times for each test configuration and chose the solution with the lowest value for the cost function of Eq. (4.8). The reason why we selected the best solution is that we are interested in the finding the global minimum in the parameter solution space. If we instead take the average solution, we would be averaging a number of local minima which may not provide us with physically meaningful results. At each restart of the CMA-ES solver, we created a vector of random numbers within the search space to use as the initial values for the optimization parameters. Random numbers were used because CMA-ES is a population-based probabilistic optimization method that is not sensitive to the initial guess for the parameters. For TEI the optimization parameters were the frequency domain Q-Gaussian distribution amplitude parameters for the two echoes p = [(S, ω c, b 1, b 2, q 1, q 2 ) A, (S, ω c, b 1, b 2, q 1, q 2 ) B ]; for DBPR the optimization parameters were the time domain echo parameters x = [(A, a, τ, f c, φ, ρ ) A, (A, a, τ, f c, φ, ρ ) B ]. Since we used random vectors as the initial parameter values, the converged solutions for each run can be considered statistically-independent observations of a random variable. In other words, the converged solution from one run is independent of the results from another run. For this reason, each

69 53 converged solution would have a 50% chance of being better than the mean of the solutions. By repeating the same solver 6 times, the probability of obtaining a converged solution that is better than the mean solution can be calculated using the binomial distribution: probabity(sol. > mean) = = 98.4% (5.4) The probability of obtaining a better solution of course increases with the number of restarts. However, setting the number of restarts to 6 offers a good compromise between the computational time and the quality of the solution. 5.2 Synthetic Echoes with Symmetric Envelope For the first set of simulation tests, we used synthetic echoes that are of the form: echo(x, t) = A exp[ a 2 (t τ) 2 ] cos[2πf c (t τ) + φ] (5.5) Consequently, each echo for this set of simulation experiment was a constant frequency oscillation multiplied by a Gaussian envelope. Comparing Eq. (5.5) with Eq. (5.1), we see that the Gaussian modulated echoes used are simply a subset of the DBPR model. For this reason, in theory DBPR should be able to identify the two echoes perfectly without the presence of noise. In addition, the Gaussian modulated echoes of Eq. (5.5) should also perfectly satisfy the frequency domain assumptions of the TEI algorithm. The Fourier transform of Eq. (5.5) can easily be calculated to be [31]: ECHO(x, f) = π a A exp [ π2 (f f c ) 2 a 2 ] exp[ j (2π(f f c )τ + φ)] (5.6) Since the amplitude profile of Eq. (5.6) is another Gaussian distribution, it can be fully described by the Q-Gaussian distribution amplitude model. In addition, the phase response of Eq. (5.6) is linear, consequently the phase nonlinearity measurement of Eq. (4.4) can be used as the TEI optimality metric Echo Parameter Tests The phase shift, center frequency difference and amplitude ratio between the two echoes and the overall SNR of the signal can all affect the performance of both the TEI and DBPR algorithms.

70 54 For this reason, we varied the shape parameters of the individual echoes in order to obtain statistically meaningful comparisons of algorithm performance. Since there are many echo shape parameters that control the overall composition of the overlapping echoes, it is not practical to examine all possible combinations of shape parameters. Instead, for each test we only varied two parameters simultaneously and the other shape parameters were fixed at their baseline values. The baseline echo parameter values for the symmetric envelope synthetic echoes are summarized in Table 5.1. A multiplication factor of 10 6 is present in the envelope width parameter a because the time shift is in the order of µs. Parameter Echo A Echo B Center frequency f c 3.0 MHz 2.76 MHz Amplitude scaling A Phase shift φ 0 1.0π Time shift τ 3.0 µs 3.6 µs Envelope width a [1/s] [1/s] Table 5.1: Baseline parameters for symmetric echoes The overlapping echoes created from the parameter values listed in Table 5.1 are shown in Figure 5.1. The linear phase echo optimality metric was used for TEI in this section. In addition, the value of t min in the phase slope inequality condition of Eq. (3.2) was set to be 0.17 µs because it is approximately half of the time period at 3 MHz. Lastly, for all the tests in this section, the SNR level was set to be 100 db to simulate noise-free test cases.

71 55 Figure 5.1: Baseline configuration for symmetric echoes Phase Difference vs Time Separation In this test, we varied the phase difference (φ A φ B ) and the time separation (τ A - τ B ) between the two echoes. The values of (φ A φ B ) were varied from 0 to 1.8π in increments of 0.2 π; the values of (τ A - τ B ) were varied from 0.2 µs to 1.0 µs in increments of 0.08 µs. Consequently, a total of = 110 different signal configurations were examined. When the time separation between the two echoes is only 0.2 µs, we are approaching the t min value that we was set for the phase inequality condition. Graphical representations of the results are shown in Figure 5.2 and Figure 5.3 where we summarize the results of all 110 test configurations into color-coded image plots. Each square represents a different configuration with its x-coordinate being the time separation and its y- coordinate being the phase difference. The color of each square represents the best solution obtained after 6 runs of the CMA-ES solver. In addition, we also calculated the means and standard deviations of the 110 data points shown in Figure 5.2 and Figure 5.3 and summarized the results in Table 5.2. Table 5.2 provides a statistical summary of the results obtained for this test and allows for quantifiable comparisons of the two echo separation algorithms.

72 56 Figure 5.2: Percentage timing error (phase difference vs time separation for symmetric echoes) Figure 5.3: Percentage reconstruction error (phase difference vs time separation for symmetric echoes) TEI DBPR Mean (Timing Error) % % Standard Deviation (Timing Error) % % Mean (Reconstruction Error) 0.67 % 1.05 % Standard Deviation (Reconstruction Error) 0.87 % 2.28 % Table 5.2: Performance table (phase difference vs time separation for symmetric echoes) Looking at Figure 5.2, we see that no apparent trend existed between the phase difference and the percentage timing error. However, it can be seen that TEI performed more reliably compared to DBPR. In this noise-free test, the performance of DBPR was bimodal, either there was perfect reconstruction (zero timing error) or there was a large percentage timing error.

73 57 From Figure 5.3, we see that for TEI the reconstruction error actually decreased with decreasing time separation between the two echoes. In fact, this result can also be observed for the subsequent test cases that will be presented in this section. The cause of this seemingly counterintuitive observation will be explained in Section The reliability and performance advantage of TEI is clearly shown in Table 5.2. The standard deviation of timing errors was approximately 40% lower than DBPR while the standard deviation for reconstruction errors was 60% lower. For this test, the mean of both timing and reconstruction errors were also lower for TEI. Lower timing errors indicates a more accurate estimate of the location and size of a defect. Frequency Difference vs Time Separation In this test, we varied the frequency difference (f c,a f c,b ) and the time separation (τ A - τ B ) between the two echoes. The values of (f c,a f c,b ) were varied from -0.6 MHz to +0.6 MHz in increments of 0.12 MHz; the values of (τ A - τ B ) were varied from 0.2 µs to 1.0 µs in increments of 0.08 µs. Consequently, a total of = 121 different signal configurations were examined. Graphical representations of the results are shown in Figure 5.4 and Figure 5.5. In addition, a table of the means and standard deviations of the timing and reconstruction errors is shown in Table 5.3. Figure 5.4: Percentage timing error (frequency difference vs time separation for symmetric echoes)

74 58 Figure 5.5: Percentage reconstruction error (frequency difference vs time separation for symmetric echoes) TEI DBPR Mean (Timing Error) % % Standard Deviation (Timing Error) % % Mean (Reconstruction Error) 1.28 % 1.11 % Standard Deviation (Reconstruction Error) 2.60 % 2.94 % Table 5.3: Performance table (frequency difference vs time separation for symmetric echoes) Looking at Figure 5.4 and Figure 5.5, we see that DBPR performed marginally better when there was a large difference between the center frequencies of the two echoes. In contrast, such a trend was not observed for TEI. This result may be explained by the fact that the trigonometric phase reconstruction algorithm of TEI can only be applied at frequencies where both echoes contain significant spectral content. If two echoes have a large difference in center frequency, the spectral overlap between the echoes would be limited and TEI would be forced to use fewer frequency data points in the optimization algorithm. In contrast, a large center frequency difference would lead to rapid cycling of constructive/destructive oscillation interference in the time domain signal. This rapid cycling allows DBPR to identify the time region of echo overlap and therefore DBPR can separate the two echoes more effectively for such configurations. In Figure 5.4 and Figure 5.5 we also observe occasional outlier points where the timing and reconstruction errors are much larger than the adjacent points. These outlier points are present due to the probabilistic CMA-ES solver used to solve the optimization problems. The probability

75 59 of the occurrence of outliers can be reduced by repeating the solver many times at the same test configuration but can never be eliminated. From Table 5.3, it is again evident that the performance TEI was more consistent within the range of time separation tested. The standard deviation of the timing errors was approximately 50% lower for TEI compared to DBPR. The standard deviation for the reconstruction error was also 12% lower for TEI compared to DBPR. Amplitude Ratio vs Time Separation In this test, we varied the amplitude ratio (A B /A A ) and the time separation (τ A - τ B ) between the two echoes. The values of (A B /A A ) were varied from 0.4 to 1.0 in increments of 0.06; the values of (τ A - τ B ) were varied from 0.2 µs to 1.0 µs in increments of 0.08 µs. Consequently, a total of = 121 different signal configurations were examined. Graphical representations of the results are shown in Figure 5.6 and Figure 5.7. In addition, a table of the means and standard deviations of the timing and reconstruction errors is shown in Table 5.4. Figure 5.6: Percentage timing error (amplitude ratio vs time separation for symmetric echoes)

76 60 Figure 5.7: Percentage reconstruction error (amplitude ratio vs time separation for symmetric echoes) TEI DBPR Mean (Timing Error) % % Standard Deviation (Timing Error) % % Mean (Reconstruction Error) 0.48 % 0.65 % Standard Deviation (Reconstruction Error) 0.66 % 2.24 % Table 5.4: Performance table (amplitude ratio vs time separation for symmetric echoes) Looking at the results presented above, we see that the performances of both DBPR and TEI were both less affected by changing the amplitude ratio compared to varying the frequency difference and phase shift. The mean timing errors for this test was roughly 11% for both techniques. This error percentage was lower than both the phase shift variation test (13% for TEI; 16% for DBPR) and the center frequency variation test (14% for TEI; 18% for DBPR). This result may be explained by the lack of change in echo oscillation interference when only the amplitude ratio of the two echoes is varied. When the phase and frequency difference of the two echoes are varied, the oscillation patterns formed by the two echoes are shifted and therefore the amount of constructive and destructive interference is affected. Even though the mean timing errors were similar for the TEI and DBPR methods, again it is evident that TEI performed more consistently. The standard deviation of the timing errors for TEI was 50% lower compared to DBPR while the standard deviation for the reconstruction errors was 70% lower.

77 Signal to Noise Ratio Tests In order to test the performance of each echo separation method in the presence of noise, we repeated the amplitude ratio test at four different SNRs. The SNR was varied by adding white Gaussian noise to the signal containing the overlapping echoes. For these simulation tests, the percentage reconstruction error was mainly dominated by the noise variance and therefore these metrics did not provide useful information regarding the relative performances of the TEI and DBPR algorithms. For this reason, the color-coded image plots for the reconstruction errors are not presented in this section. SNR = 40 db At 40 db, the average SNR amplitude ratio is equal to 100 and therefore the additive noise is hardly visible from a visual inspection of the time domain signal containing the overlapped echoes. A plot of a representative test configuration with (A B /A A ) set at 0.7 is shown in Figure 5.8. The percentage timing error comparison between TEI and DBPR is shown in Figure 5.9. Figure 5.8: Overlapped echoes at SNR = 40 db (symmetric echoes)

78 62 Figure 5.9: Percentage timing error (40 db for symmetric echoes) Comparing Figure 5.9 with the noise-free case shown in Figure 5.6, we see that the addition of a negligible level of noise was sufficient to influence the performance of DBPR. The timing error of DBPR has noticeably increased at small time separation values compared to the noise-free test. In contrast, the timing error of TEI was not noticeably influenced. SNR = 25 db At 25 db, the average SNR amplitude ratio is equal to 17.78; at this noise level the additive noise can be visually detected in the simulated signal. A plot of a representative test configuration with (A B /A A ) set at 0.7 is shown in Figure The percentage timing error comparison between TEI and DBPR is shown in Figure Figure 5.10: Overlapped echoes at SNR = 25 db (symmetric echoes)

79 63 Figure 5.11: Percentage timing error (25 db for symmetric echoes) Comparing Figure 5.11 with Figure 5.9, we see that the timing errors of DBPR at small time separations have significantly increased while the timing errors of TEI were largely unaffected by the decrease in SNR. With the introduction of noise, the performance of DBPR became less bimodal but showed a gradual degradation in time difference estimation accuracy with decreasing time separation between the two echoes. SNR = 15 db At 15 db, the average SNR amplitude ratio is equal to 5.62; at this noise level the additive noise significantly affects the oscillation waveform in the simulated signal. A plot of a representative test configuration with (A B /A A ) set at 0.7 is shown in Figure The percentage timing error comparison between TEI and DBPR is shown in Figure Figure 5.12: Overlapped echoes at SNR = 15 db (symmetric echoes)

80 64 Figure 5.13: Percentage timing error (15 db for symmetric echoes) It is clear from inspection of Figure 5.13 that TEI had lower timing errors compared to DBPR at small time separations. This result is drastically different from the noise-free case shown in Figure 5.6 where DBPR was capable of obtaining near zero timing errors at small time separations. This observation indicates that the TEI algorithm is more robust in the presence of noise. SNR = 10 db At 10 db, the average SNR amplitude ratio is equal to 3.16; at this noise level the additive noise causes severe distortion of the oscillation waveform in the simulated signal. A plot of a representative test configuration with (A B /A A ) set at 0.7 is shown in Figure The percentage timing error comparison between TEI and DBPR is shown in Figure Figure 5.14: Overlapped echoes at SNR = 10 db (symmetric echoes)

81 65 Figure 5.15: Percentage timing error (10 db for symmetric echoes) From Figure 5.15, we see that the timing errors for both TEI and DBPR have increased compared to results obtained at higher SNR levels. However, the degradation in the performance of DBPR was much more significant compared to TEI. This again shows that TEI is more robust to the addition of noise. SNR Tests Summary To summarize the results in this section, we plot the mean and standard deviation of the percentage timing error as a function of SNR in Figure The 100 db data points are plotted using the results from the amplitude test in Section From Figure 5.16, it is evident that the performance of TEI was almost independent of SNR within the noise range that was examined. The mean timing error remained constant at approximately 12% and the timing error standard deviation was close to 15%. In contrast for DBPR, we see a sharp increase in both the mean and standard deviation of the percentage timing errors when the SNR was decreased below 40 db.

82 66 Figure 5.16: Performance vs SNR (symmetric echoes) Results Summary and Discussion One of the most significant trends observed from the results presented in sections and is that the TEI algorithm performed much more consistently than to DBPR. The standard deviations of timing errors of TEI was approximately 40% lower than those of DBPR. This trend can be explained by understanding the difference between the working principles of the two algorithms. From Eq. (2.19), we see that DBPR obtains the optimal echo parameters by minimizing the reconstruction residual error between the total signal and the sum of parametric echoes. Consequently, if the optimization solver fails to find the correct value for one or more of the parameters, the other echo parameters would need to adjust in incorrect ways to reduce the residual error. Since it is highly possible to obtain a local minimum of the residual error with incorrect time shift parameters, we see many outlier points in the output of DBPR where the timing errors are very large. In contrast, for the simulation tests in sections and 5.2.2, TEI adapts the amplitude profiles in order to obtain phase profiles that are as linear as possible. If the optimization solver fails to find the correct value for one or more of the amplitude parameters, the other amplitude parameters would need to adjust in incorrect ways to maximize the linearity of the reconstructed phase profiles. However, it is in general difficult to reconstruct near-linear phase profiles which have phase slope values that are drastically different from the correct ones. For this reason, even if the converged amplitude profiles are not strictly correct, the reconstructed phase profiles often

83 67 have phase slopes values that are close to the correct ones. Since the arrival time of an echo is determined by its phase slope, TEI is therefore less likely to produce outliers in the estimate of echo arrival time difference. Another major trend that we can observe from Section is that the performance of TEI was less influenced by decreasing SNR compared to DBPR. For DBPR there was a sharp increase in both the mean and standard deviation of the timing errors when the SNR was lowered below 40 db. In comparison, the performance metrics of TEI remained approximately constant when SNR was decreased. This trend can also be explained by understanding the optimization goals of each algorithm. DBPR aims to reduce the L2 norm of the difference between the total signal and the sum of parametric echoes. Any noise present in the total signal would be directly entered into the residual metric calculation as shown Eq. (5.7): [ sig(t) + n(t) echo(x i, t) ] t i 2 (5.7) In Eq. (5.7), the echo parameters need to be adjusted to compensate for the noise and minimize the residual. Consequently, additive noise has a direct impact on the ability of DBPR to recover the correct parametric echoes. In contrast, TEI aims to reduce the nonlinearity in the phase profiles θ A (ω) and θ B (ω). Using the phase reconstruction table shown in Table 4.1, we can decompose the noisy phase profiles θ A,noisy (ω) and θ B,noisy (ω) into the following components: θ A,noisy (ω) = θ A (ω) + noise[β(ω)] + noise[θ T (ω)] (5.8) θ B,noisy (ω) = θ B (ω) + noise[α(ω)] + noise[θ T (ω)] Where θ T (ω) is the phase profile of the total signal and α(ω) and β(ω) are the interior angles calculated from the vector addition triangle shown in Figure 4.2. In Eq. (5.8), noise[θ T (ω)] represents the phase noise present in the total signal. In contrast, noise[α(ω)] and noise[β(ω)] stem from the amplitude noise noise[m T (ω)] because the amplitude of the total signal M T (ω) is used in the calculation of α(ω) and β(ω).

84 68 When one measures the nonlinearity of Eq. (5.8), noise[θ T (ω)] adds a baseline level of nonlinearity to the phase profiles. However, since the phase noise is common to all reconstructed phase profiles, its influence on the selection of the most linear phase profile is minimal. In addition, the noise[α(ω)] and noise[β(ω)] are not directly proportional to noise[m T (ω)] because the trial amplitudes M A (ω) and M B (ω) are also used in the calculation of α(ω) and β(ω). For example, using the vector addition triangle shown in Figure 4.2, the interior angle α(ω) can be calculated using the cosine law as follows: α(ω) = cos 1 [ M A(ω) 2 + M B (ω) 2 M T (ω) 2 ] 2M A (ω)m B (ω) (5.9) Looking at Eq. (5.9), we see the influence of noise[m T (ω)] is reduced by the trial amplitude profiles of the two echoes. For the reasons explained above, we can understand why TEI is less sensitive to the decreasing SNR compared to DBPR. Yet another trend that we can observe from the results presented in Section is that for TEI the timing error increased while the reconstruction error decreased with decreasing time separation. This apparently contradictory trend can be explained by examining two representative echoes that are spaced 0.2 µs apart (the other echo parameters follow the baseline parameter values shown in Table 5.1). Figure 5.17: Overlapped signal with time separation of 0.2 µs

85 69 Looking at Figure 5.17, we see that the overlapped signal at a small time separation strongly resembles a single echo. The frequency amplitude profile closely follows a Gaussian distribution and the phase profile is close to being linear. Since the overlapped signal already satisfies the assumptions of TEI, it is relatively easy to decompose the signal into two arbitrary echoes that also satisfy the TEI assumptions. This is the reason why the reconstruction error is small for small echo time separations. However, the decomposed echoes may not actually be the correct ones as multiple valid solutions exist. This is the reason why the timing error increases with decreasing echo time separation. From this explanation, we can deduce that the time difference estimate performance of TEI decreases when the signal containing the overlapped echoes resembles a single echo. This result is consistent with our intuitive understanding of the echo separation problem. 5.3 Synthetic Echoes with Asymmetric Envelope For the second set of analyses of simulated signals, we employed echoes with asymmetric envelopes and non-constant modulation frequencies. Each asymmetric echo used in this study is mathematically described by the following expression: echo(x, t) = A env (t τ)cos[2πf c (t τ) + 2πψ(t τ) 3 + φ] (5.10) env(t) = exp[ a 2 (1 ρ tanh(mt))t 2 ] The asymmetric echoes described above present a challenging case for both the TEI and DBPR models. Comparing Eq. (5.10) with Eq. (5.1), we see that Eq. (5.10) has an extra ψ parameter multiplied by a cubic time delay. Since the instantaneous modulation frequency is proportional to the time derivative of the argument of the cosine function, the modulation frequency is not constant but rather a quadratic function of time. Consequently, in this test the constituent echoes are no longer perfectly described by the DBPR mathematical model. These asymmetric echoes also present multiple challenges for TEI. Firstly, using the multiplication-convolution duality property, the Fourier transform of Eq. (5.10) is the frequency domain convolution of the Fourier transform of envelope function A env(t τ) with the Fourier transform of the nonlinear oscillation. In general, an analytical expression cannot be obtained for the Fourier transform of an oscillation with a non-linear frequency and numerical

86 70 methods (such as the FFT) are used to estimate it. Consequently, it is clear that the frequency amplitude profile of the constituent echoes will not be perfectly described by the Q-Gaussian distribution model shown in Eq. (3.1). Secondly, since the time envelope of Eq. (5.10) is asymmetric, its phase profile in the frequency domain will be non-linear [31]. Consequently, the linear phase assumption of Eq. (4.4) used for the echo optimality metric will not be strictly valid. The purpose of this set of simulation tests is to compare the performance of the echo identification algorithms in situations where the mathematical models employed do not perfectly describe the actual echoes. It is important to analyze the performance of the algorithms in such situations because real life ultrasonic echoes cannot be perfectly described by simple parametric mathematical models Echo Parameter Tests The asymmetric echo simulation tests closely followed the procedure outlined in Section In each test, we only varied two parameters while the other parameters were held constant at their baseline values. The baseline values for the echo parameters are shown in Table 5.5: Parameter Echo A Echo B Center frequency f c 3.0 MHz 2.76 MHz Amplitude scaling A Phase shift φ 0 1.0π Time shift τ 3.0 µs 3.6 µs Envelope width a [1/s] [1/s] Envelope asymmetry ρ Frequency nonlinearity ψ [1/s 2 ] f C,A [1/s 2 ] f C,B Table 5.5: Baseline parameters for asymmetric echoes Looking at Table 5.5, we see that the nonlinear modulation frequency factor was set at the value [1/s 2 ] f C. Since the instantaneous frequency is defined as 1/2π multiplied by the time derivative of the argument of the cosine function, the modulation frequency of each echo described in Eq. (5.10) would have the form: f(t) = f c [1/s 2 ] f c (t τ) 2 = f c ( [1/s 2 ] (t τ) 2 ) (5.11)

87 71 In Eq. (5.11), the modulation frequency is a quadratic function of time. The instantaneous frequency is highest at t = τ and is reduced when we move away from the center of the echo. A factor of is needed in the definition of ψ because the time shift τ is in the order of µs and the center frequency is in the order of MHz. A demonstrative frequency profile for f c = 3.0 MHz and τ = 3.0 us is shown below in Figure The overlapping echoes created from the parameter values listed in Table 5.5 are shown in Figure Figure 5.18: Quadratic modulation frequency Figure 5.19: Baseline configuration for asymmetric echoes From Figure 5.19, we see that the effects of variation in modulation frequency is moderate since the durations of the echoes were short. The linear phase echo optimality metric was again used for TEI even though this assumption was not perfectly satisfied by the asymmetric echoes. In addition, the value of t min in the phase slope inequality condition of Eq. (3.2) was set to be

88 µs because it is approximately half of the time period at 3 MHz. Once again, for all the tests in this section, the SNR level was set to be 100 db to simulate noise-free test cases. Phase Difference vs Time Separation In this test, we varied the phase difference (φ A φ B ) and the time separation (τ A - τ B ) between the two echoes. The values of (φ A φ B ) were varied from 0 to 1.8π in increments of 0.2 π; the values of (τ A - τ B ) were varied from 0.2 µs to 1.0 µs in increments of 0.08 µs. Consequently, a total of = 110 different signal configurations were examined. For all asymmetric echo tests in this section, we set the t min value for Eq. (3.2) in the TEI algorithm as 0.17 µs, which is approximately half the time period at 3 MHz. However, since the phase profiles of the non-asymmetric echoes are non-linear, the value of the phase slope difference, d(θ A θ B ), may be lower than 0.17 µs at some frequencies. As an illustrative example, dω when we set the nominal time shift difference between the two echoes at 0.2 µs and retain all other echo parameters at their baseline values listed in Table 5.5, we can obtain the phase slope difference profile shown in Figure Figure 5.20: Phase slope difference of two asymmetric echoes (nominal time separation at 0.2 µs) From Figure 5.20, we see that there are frequencies at which the value of d(θ A θ B ) is below the value of t min set at 0.17 µs. Consequently, the phase slope assumptions of Eq. (3.2) are no longer satisfied at all frequencies where both echoes have significant spectral content. In this section, we will examine how this violation of the phase slope assumption would affect the timing and reconstruction errors of the TEI algorithm. In addition, since the phase slope dω

89 73 difference is non-constant, it is important that we use the spectrally-averaged arrival time defined in Eq. (4.3) to calculate the arrival-time difference between the two echoes. If we instead use the nominal time shift difference as the true arrival time difference, it would lead to inaccuracies in the estimation of the timing errors for the two algorithms. Graphical representations of the results of varying phase difference against time separation are shown in Figure 5.21 and Figure In addition, a table of the means and standard deviations of the timing and reconstruction errors is shown in Table 5.6. Figure 5.21: Percentage timing error (phase difference vs time separation for asymmetric echoes) Figure 5.22: Percentage reconstruction error (phase difference vs time separation for asymmetric echoes)

90 74 TEI DBPR Mean (Timing Error) % % Standard Deviation (Timing Error) % % Mean (Reconstruction Error) 0.53 % 4.63 % Standard Deviation (Reconstruction Error) 0.27 % 2.29 % Table 5.6: Performance table (phase difference vs time separation for asymmetric echoes) Looking at Figure 5.21 and Figure 5.22, we see that TEI had much smaller timing and reconstruction errors compared to DBPR. This observation is confirmed in Table 5.6 where TEI outperformed DBPR in every statistical performance metric. For this test, there was not an apparent trend between the percentage timing error and the phase difference between the echoes. However, it is apparent that percentage timing error increased with decreasing time separation between the two echoes. This result is expected because a small timing error can produce a large percentage timing error at small time separations. In addition, as shown in Figure 5.20, the phase slope inequality assumption used by TEI in this test is not strictly satisfied when the time separation is only 0.2 µs. Consequently, by enforcing the phase slope assumption we can introduce errors in the echo reconstruction process. Comparing Table 5.6 with the symmetric echo results summarized in Table 5.2, we see that the performance of DBPR has deteriorated significantly. The mean and standard deviation of the timing errors have increased from 18% and 29% to 31% and 37% respectively. In comparison, the timing errors of TEI have actually decreased. The mean and standard deviation of the timing errors were approximately 12% and 14%; these values compare well with the previous values of 14% and 18% observed for the separation of symmetric echoes. This observation suggests that TEI is more robust than DBPR in situations where the actual echo shapes are not perfectly described by the mathematical forms of the chosen model. Frequency Difference vs Time Separation In this test, we varied the center frequency difference (f c,a f c,b ) and the time separation (τ A - τ B ) between the two echoes. The values of (f c,a f c,b ) were varied from -0.6 MHz to +0.6 MHz in increments of 0.12 MHz; the values of (τ A - τ B ) were varied from 0.2 µs to 1.0 µs in increments of 0.08 µs. Consequently, a total of = 121 different signal configurations

91 75 were examined. Graphical representations of the results are shown in Figure 5.23 and Figure In addition, a table of the means and standard deviations of the timing and reconstruction errors is shown in Table 5.7. Figure 5.23: Percentage timing error (center frequency difference vs time separation for asymmetric echoes) Figure 5.24: Percentage reconstruction error (center frequency difference vs time separation for asymmetric echoes) TEI DBPR Mean (Timing Error) % % Standard Deviation (Timing Error) % % Mean (Reconstruction Error) 0.95 % 4.57 % Standard Deviation (Reconstruction Error) 1.11 % 2.40 % Table 5.7: Performance table (center frequency difference vs time separation for asymmetric echoes)

92 76 Looking at Figure 5.23 and Figure 5.24, we see that DBPR performed better when there was a large difference between the center frequencies of the two echoes. In contrast, such a trend was not observed for TEI. The explanation for this result had already been presented in Section In this test, it is again evident that TEI had smaller timing and reconstruction errors than the DBPR method. Comparing Table 5.7 with the symmetric echo results summarized in Table 5.3, we see that the performance of DBPR has deteriorated significantly. The mean of the timing errors has increased from 16% to 26%. In comparison, the mean timing error of TEI has only increased marginally from 13% to 15%. Changing the center frequency was a challenging test for both TEI and DBPR. As shown in Eq. (5.11), the time variation in modulation frequency is designed to be proportional to the center frequency. Consequently, the rate of modulation frequency variation is not constant for two overlapping echoes when they possess different center frequencies. Amplitude Ratio vs Time Separation In this test, we varied the amplitude ratio (A B /A A ) and the time separation (τ A - τ B ) between the two echoes. The values of (A B /A A ) were varied from 0.4 to 1.0 in increments of 0.06; the values of (τ A - τ B ) were varied from 0.2 µs to 1.0 µs in increments of 0.08 µs. Consequently, a total of = 121 different signal configurations were examined. Graphical representations of the results are shown in Figure 5.25 and Figure In addition, a table of the means and standard deviations of the timing and reconstruction errors is shown in Table 5.8. Figure 5.25: Percentage timing error (amplitude ratio vs time separation for asymmetric echoes)

93 77 Figure 5.26: Percentage reconstruction error (amplitude ratio vs time separation for asymmetric echoes) TEI DBPR Mean (Timing Error) % % Standard Deviation (Timing Error) % % Mean (Reconstruction Error) 0.62 % 4.58 % Standard Deviation (Reconstruction Error) 0.37 % 2.60 % Table 5.8: Performance table (amplitude ratio vs time separation for asymmetric echoes) From the results presented above, we see that the percentage timing error of DBPR was smaller for the amplitude ratio test compared to changing frequency difference and phase shift. The mean percentage timing error of this test for DBPR was 21%, which was smaller than 26% for the center frequency variation test and 31% for the phase difference test. This trend was also observed for the symmetric echo tests presented in Section and may be explained by the lack of change in echo oscillation interference when only the amplitude ratio of the two echoes is varied. In comparison, the mean percentage timing error for TEI was relatively constant for all three tests. This suggests that the performance of TEI is more robust to variation in echo shape. Despite the fact that DBPR was less affected by change in amplitude ratio, its mean and standard deviation of the percentage timing errors (21% and 28%) were still much larger compared to TEI (14% and 14%).

94 Signal to Noise Ratio Tests In order to test the echo separation performances of TEI and DBPR for asymmetric echoes in the presence of noise, we followed the test procedure in Section and repeated the amplitude ratio test at four different levels of SNR. The SNR level was again varied by adding white Gaussian noise to the signal containing the overlapping echoes. Once again, the reconstruction error for these tests were dominated by the noise variance and did not provide useful information regarding the relative performances of the TEI and DBPR algorithms. For this reason, the colorcoded image plots for the reconstruction errors are not be presented in this section. SNR = 40 db At 40 db, the average SNR amplitude ratio is equal to 100 and therefore the additive noise is hardly visible from a visual inspection of the time domain signal containing the overlapped echoes. A plot of a representative test configuration with (A B /A A ) set at 0.7 is shown in Figure The percentage timing error comparison between TEI and DBPR is shown in Figure Figure 5.27: Overlapped echoes at SNR = 40 db (asymmetric echoes)

95 79 Figure 5.28: Percentage timing error (40 db for asymmetric echoes) Comparing Figure 5.28 with the noise-free case shown in Figure 5.25, there was not any significant difference in performance for both methods. This indicates that at this SNR level noise is not an important factor in the echo identification performance of both methods. SNR = 25 db At 25 db, the average SNR amplitude ratio is equal to 17.78; at this noise level the additive noise can be visually detected in the simulated signal. A plot of a representative test configuration with (A B /A A ) set at 0.7 is shown in Figure The percentage timing error comparison between TEI and DBPR is shown in Figure Figure 5.29: Overlapped echoes at SNR = 25 db (asymmetric echoes)

96 80 Figure 5.30: Percentage timing error (25 db for asymmetric echoes) Comparing Figure 5.30 with Figure 5.28, we see that there was a small increase in timing error for both the TEI and DBPR methods. For TEI, there was a clear increase of percentage timing error with decreasing time separation. For DBPR, the overall trend was more random although larger percentage timing errors occurred more frequently at smaller time separations. SNR = 15 db At 15 db, the average SNR amplitude ratio is equal to 5.62; at this noise level the additive noise significantly affects the oscillation waveform in the simulated signal. A plot of a representative test configuration with (A B /A A ) set at 0.7 is shown in Figure The percentage timing error comparison between TEI and DBPR is shown in Figure Figure 5.31: Overlapped echoes at SNR = 15 db (asymmetric echoes)

97 81 Figure 5.32: Percentage timing error (15 db for asymmetric echoes) Comparing Figure 5.32 with Figure 5.30, we see that there was an increased in percentage timing error for both TEI and DBPR. The maximum timing errors for TEI and DBPR have risen to 115% and 140% at echo time separation of 0.2 µs. Once again, for TEI there was a gradual transition to larger timing errors with decreasing time separation whereas the DBPR timing errors were more randomly distributed. SNR = 10 db At 10 db, the average SNR amplitude ratio is equal to 3.16; at this noise level the additive noise causes severe distortion of the oscillation waveform in the simulated signal. A plot of a representative test configuration with (A B /A A ) set at 0.7 is shown in Figure The percentage timing error comparison between TEI and DBPR is shown in Figure Figure 5.33: Overlapped echoes at SNR = 10 db (asymmetric echoes)

98 82 Figure 5.34: Percentage timing error (10 db for asymmetric echoes) Comparing Figure 5.34 with Figure 5.32, we see that the timing errors have increased again with decreasing SNR level. The maximum timing error for TEI and DBPR were respectively 140% and 240%. Both maximum timing errors occurred at an echo time separation of 0.2 µs. The large percentage error at 0.2 µs is expected for TEI since the phase slope difference assumption is not strictly satisfied as shown in Figure SNR Tests Summary To summarize the results in this section, we plot the mean and standard deviation of the percentage timing error as a function of SNR in Figure The data points corresponding to a SNR of 100 db are plotted using the noise-free results summarized in Table 5.8. From Figure 5.35, we can see that the timing error means and standard deviations for both TEI and DBPR increased significantly when the SNR was decreased below 40 db. However, the time difference estimation performance of TEI was still superior to DBPR for all noise levels tested. It is interesting to note that the performance of TEI at a SNR level of 40 db was marginally better than the results obtained from 100 db. This is likely caused by small random fluctuations in the solutions obtained from the CMA-ES solver.

99 83 Figure 5.35: Performance vs SNR (asymmetric echoes) Results Summary and Discussion One of the significant trends observed from Section is that TEI outperformed DBPR significantly for the separation of two asymmetric echoes with non-constant modulation frequencies. The means of the timing errors (13-15% compared to 21-31%) and the reconstruction errors (~1% compared to ~4%) were both significantly lower for TEI. This result can be explained by understanding the parametric modeling aspects of the two algorithms. From Eq. (5.1), we see that this implementation of DBPR uses 6 parameters to describe the shape of each echo. In contrast, from Eq. (3.1) we see that TEI uses all 6 parameters to describe the frequency domain amplitude profile of each echo, and then uses trigonometry to solve for the phase profiles. Consequently, echoes described by the TEI algorithm can have more complex amplitude profile shapes compared to DBPR. In addition, since the phase profiles of the TEI echoes are reconstructed using the total signal amplitude M T (ω) and the total signal phase θ T (ω), the phase profiles can adapt to the acquired data and are not governed by fixed mathematical expressions. Due to these two unique aspects of the algorithm, TEI can describe more complex ultrasonic echoes using the same number of modeling parameters compared to DBPR and adapt the shapes of the echoes to fit the acquired data. This is the primary reason why TEI outperforms DBPR in this set of simulation experiments. Another important trend we can observe from Section is that the performance of TEI was no longer independent of SNR level for the separation of asymmetric echoes. This is different from our observation in Section where the performance of TEI was not affected by change

100 84 in SNR level for the separation of symmetric echoes. This result can be explained using the following argument. For an asymmetric echo with a non-constant modulation frequency, its frequency phase profile is in general non-linear even without the presence of noise. Consequently, the phase linearity metric employed in this section is only an approximate measurement for the optimality of the reconstructed echoes. When noise is added to the signal, there will be fluctuations added to the reconstructed phase profiles as explained in Section Since it is not possible to differentiate between the fluctuations introduced by noise and the inherent non-linearity in the reconstructed phase profiles, it is likely for TEI to converge to a suboptimal solution when the SNR is decreased. This is the primary reason why the timing estimation performance of TEI deteriorated with decreasing SNR. 5.4 Experimental Verification Having compared the echo separation performance of TEI and DBPR for various simulated echoes, in this section we evaluate the performance of the two algorithms for the separation of ultrasonic echoes in signals obtained from experiments. These experimental results will verify whether the assumptions of the TEI algorithm are applicable for actual NDT applications TOFD Test on Notched Sample Test Configuration For the first experimental test, we seek to separate two overlapping echoes obtained from TOFD inspection of a sample containing a vertical notch. A photograph of the notched sample is shown in Figure The test sample is made of low carbon steel and has a thickness of 0.5. There are four vertical notches cut into the sample which are 0.3, 0.2, 0.1 and 0.05 deep. The TOFD measurement was conducted on the 0.3 deep notch because the small distance between the notch tip and the top surface leads to the creation of overlapping echoes in the A-scan data. The adjacent notches are spaced sufficiently far apart that they did not interfere with the TOFD inspection.

101 85 Figure 5.36: Test sample containing vertical notches A schematic diagram of the TOFD scan configuration is shown in Figure From Figure 5.37, we see that the beam entry points of the two probes were spaced 30 mm apart such that the intersection of the central propagation axes of the transducers occurred at the bottom 1/3 of the sample thickness. This configuration was chosen according to standard TOFD measurement protocol [47]. The design center frequency of the probes used was 5 MHz. Figure 5.37: TOFD configuration for notch sample During the TOFD acquisition, we translated the two probes parallel to the direction of the notch to obtain the B-scan image shown in Figure In Figure 5.38, the x-axis represents the position of the inspection system in the scan direction and the y-axis represents time in the individual A-scans. From this figure, we can see that there was an overlap between the lateral wave and the notch tip diffraction echoes in the central portion of the scan along the x-axis. An A-scan extracted from the scan location of 35 mm showing these overlapping echoes is shown in Figure 5.39.

102 86 Back wall Notch tip Lateral wave Extracted A-scan Figure 5.38: B-scan of notch sample TOFD scan Lateral wave Notch tip diffraction Figure 5.39: Overlapping echoes in TOFD scan of notch sample From the configuration of the inspection system, we know that the probe separation is 30 mm, the notch tip distance from the surface is 5.08 mm and the speed of sound in the steel sample is 5890 m/s. Using these parameters, the theoretical time difference between the two echoes can be estimated based on simple trigonometry: t geometry = 30mm 2 (15mm)2 + (5.08mm) m/s = us (5.12) Results using Phase Linearity as Optimality Metric Since TEI is designed to separate two overlapping echoes, the first processing step was to crop the A-Scan signal so that only two overlapping echoes remained in the time series data. For this reason, we cropped the A-Scan data as shown in Figure 5.39 from 1.7 µs to 3.1 µs. In addition, we also multiplied the cropped signal by a Tukey window in order to reduce transition effects

103 87 when we transform the time-series data into the frequency domain. The cropped signal and the Tukey window are shown in Figure Figure 5.40: Notch sample time series data analyzed by TEI and DBPR The cropped and windowed time series data was then passed into the TEI and DBPR algorithms to separate the two echoes. For DBPR, we again used the parametric model shown in Eq. (5.1). Therefore, each echo was described using six different shape parameters. For TEI, we first used the phase linearity measurement of Eq. (4.4) as the echo optimality metric. In addition for TEI, we set the value of t min to be 0.14 µs in the phase slope inequality constraint of Eq. (4.7). This value was chosen because it was approximately half of the time difference between the apparent peaks of the two echoes. Although currently the value of t min is chosen heuristically, in the future the value of t min should be calculated automatically using the estimated bandwidth and center frequency of the overlapped signal. The echo separation results for TEI and DBPR are shown respectively in Figure 5.41 and Figure In addition, the frequency phase profiles of the TEI reconstructed echoes are shown in Figure 5.43.

104 88 Figure 5.41: Reconstructed echoes for notch sample (TEI using phase linearity condition) Figure 5.42: Reconstructed echoes for notch sample (DBPR) Figure 5.43: Frequency phase profiles of TEI reconstructed echoes (notch sample)

105 89 Looking at the results of Figure 5.41 to Figure 5.43, we can see that both algorithms were able to reconstruct echoes with a distinct order of arrival times. Using the reconstruction error formula defined in Eq. (5.3), the percentage reconstruction error for TEI was 1.5% which is lower than the percentage reconstruction error of DBPR at 7.5%. This indicates that the echoes reconstructed from TEI in fact better described the input signal. From Figure 5.43, we see that the TEI reconstructed phase profiles were near linear. This suggests that the phase linearity assumption is appropriate for the piezoelectric transducers used in this experiment. Using the spectrally averaged arrival time formula shown in Eq. (4.3), the estimated time differences between the two separated echoes were 0.30 µs for TEI and 0.29 µs for DBPR. Both time difference estimates were in line with the value of µs obtained in Eq. (5.12). Results using Cross-Correlation as Optimality Metric From the B-Scan shown in Figure 5.38, we see that the A-scans near the edges of the sample had lateral wave echoes that did not overlap with the notch tip diffracted echo. Consequently, we were able to use the clean lateral wave echo as a reference for the cross-correlation optimality metric shown in Eq. (4.1). However, since we only had a reference for the lateral wave which is the first echo, we had to modify Eq. (4.1) so that it only maximized the cross-correlation of that single reference with the first echo: optimality = max [ CC(ref 1 (t), echo 1 (t)) t echo 1 (t) 2 t ref 1 (t) 2 ] (5.13) By maximizing the cross-correlation between the first echo and the reference echo, we are in essence performing an adaptive background subtraction to remove the lateral wave echo. The lateral wave reference echo used for Eq. (5.13) is shown in Figure We also multiplied the reference echo with a Tukey window to reduce the edge effects in the FFT. The reconstructed echoes determined by the TEI algorithm using the optimality constraint of Eq. (5.13) are shown in Figure 5.45.

106 90 Figure 5.44: Lateral wave reference echo for TEI Figure 5.45: Reconstructed echoes for notch sample (TEI using cross-correlation condition) Despite the use of different optimality metrics, the reconstructed echoes shown in Figure 5.45 were similar to the ones obtained in Figure The percentage reconstruction error of this test was 2.1%, indicating the sum of the two reconstructed echoes accurately described the acquired signal. The spectrally averaged time difference between the two echoes was 0.29 µs; once again this extracted time difference value agreed with the theoretical value of µs obtained in Eq. (5.12) based on trigonometry Phased Array Test on Side-Drilled Hole Sample Test Configuration For the second experimental test, we seek to separate two overlapping echoes obtained from a pitch-catch phased array scan. The engineering diagram for the test sample used is shown in Figure 5.46.

107 91 Figure 5.46: Test sample for pitch-catch matrix probe scan From Figure 5.46, we see that the test sample has two machined side-drilled holes (SDH). The top SDH is spaced 5.0 mm from the top surface while the bottom SDH is spaced 2.5 mm from the bottom surface. The top SDH yields an echo that overlaps with the lateral wave echo; the bottom SDH yields an echo that overlaps with the backwall echo. For this test, we used a pair of phased array probes to transmit and receive a propagating wave that covered the bottom SDH. A schematic drawing of the phased array testing configuration is shown in Figure Figure 5.47: Phased array pitch-catch testing of SDH sample From Figure 5.47, we see that the first 12 elements of each phased array transducer were used for transmission and reception. The element timing delays were adjusted such that the refracted angle of each beam inside the sample was 63 with respect to the surface normal. Note that the ultrasound wave propagated into the sample was a mode-converted shear wave because the angle of incidence at the wedge/sample interface was greater than the critical angle for longitudinal

108 92 waves. The active aperture for each phased array probe was 12 mm 10 mm and the design center frequency of each active element was 5 MHz. Using a ray tracing program, the round trip travel time difference from the center of the aperture to the SDH and to the back wall was calculated to be 0.71 µs. Echo Separation Results Using the test configuration shown in Figure 5.47, we obtained the A-scan data shown in Figure From Figure 5.49, we see that there were three overlapping echoes. The third overlapping echo was created by the propagating wave travelling in a W path reflecting from the bottom surface twice as shown in Figure Since the TEI algorithm can only separate two overlapping echoes, we had to crop the time series so that the input signal to the echo separation algorithms contained only the direct SDH and backwall echoes. The cropped time series is shown in Figure Once again, we had to multiply the cropped signal with a Tukey window to minimize transition edge effects when performing the FFT. Figure 5.48: Indirect path for SDH SDH Backwall SDH ( W path) Figure 5.49: Overlapping echoes for SDH pitch-catch test