Apex-shifted Radon transform for baseline-subtraction-free (BSF) damage scattered wave extraction

Abstract

In ultrasonic structural health monitoring (SHM) and nondestructive evaluation (NDE), the scattered waves caused by damage sites and defects are the key to damage diagnosis. However, structural components and boundaries also interact with traveling waves, creating events that can bury damage scattered waves. The baseline subtraction method, which directly subtracts the waveform of a damage signal from that of a pristine baseline signal, is a common processing technique to separate these damage scattered waves. However, baseline subtraction is less effective when a component is measured in different environmental/loading conditions from when its baseline was recorded. For instance, baseline subtraction can be ineffective in aerospace structural parts because such parts expect significant and routine changes in ambient temperature, pressure, and humidity. To overcome the limitations of baseline subtraction, this paper proposes to develop an spex-shifted Radon transform (ASRT)-based damage scattered event extraction technique without baseline subtraction. Our proposed ASRT method converts the original time-space $(t - x)$ domain signals to a time delay–curvature–apex offset $(τ - q - y)$ domain, which targets specific geometries of scattered/reflected waves in the $t - x$ domain and compresses them into a single point-like region in the new domain. In this domain, identifying and isolating targeted events becomes significantly easier. To benchmark the performance of this baseline-subtraction-free (BSF) method, ASRT is applied for signals acquired from (1) spectral finite element-based simulations to scan a water-immersed steel specimen (2) 3-D wave simulations in a bent aluminum plate, and (3) a full matrix capture for experimentally scanning a high-density polyethylene (HDPE) with multiple holes. The goal in each case is to target and separate events of interest from the original signals using the proposed algorithm. The results suggest the ASRT method is effective as a damage scattered wave extraction tool for ultrasonic SHM and NDE.

Keywords

Radon transform apex-shifted Radon transform guided waves ultrasonic testing damage scattered baseline-free complex structures

Introduction

Identifying the existence and propagation of cracks and other material defects is one of the central goals of structural health monitoring (SHM) and nondestructive evaluation (NDE). Ultrasonic wave-based methods are one of the most commonly used techniques in SHM and NDE.^1–7 In ultrasound imaging, actuators are used to excite ultrasonic waves in a structure, and sensors are used to record the signals. When the propagating wave reaches a material defect/damage site, the interaction between the two alters the wave propagation, resulting in damage scattered events. Structural damage can then be identified through the detection of these damage scattered events.

Coping with the limitations of guided wave-based damage detection is a popular subject of recent SHM research.^8,9 One major limitation is the effect that structural and material complexities have on wave propagation. Structural complexity refers to reinforcements such as stringers, spars, and ribs in aerospace applications that cause additional wave scattering which buries damage scattered waves. Material complexity on the other hand refers to effects such as the anisotropy in composite laminates that causes directional distortions to wave propagation patterns. The result of both of these complexities is that damage scattered events are often buried in the time-space $(t - x)$ representation of signals, making damage detection more difficult.

A common strategy to recover buried events is the baseline-subtraction method.^5,10 This method uses baseline signals measured from the undamaged specimen, which is saved to compare with later evaluations. To detect damage, the data from subsequent evaluations is then subtracted by the baseline, and the difference of the two signals is assumed to be the damage scattered events. However, there are challenges to baseline subtraction. Firstly, baseline measurements are also not always available. Also, while simple baseline subtraction is suitable for evaluating components in a stable environment, it is less so for those that experience changes in their ambient conditions. For instance, aircraft components experience temperature, pressure, and humidity fluctuation during flight that causes deviation from the baseline, making baseline subtraction unreliable without compensation strategies.^11,12

To overcome the limitations of baseline subtraction methods, several alternative SHM methods have been developed. One strategy involves compensating for changes in the ambient conditions of the test specimen using prior knowledge of how the waveforms respond to temperature variations.¹³ However, this method requires knowledge about how the ambient conditions have changed, and how each condition (e.g., temperature, humidity) affects the wave propagation. Another method uses pairs of dual PZT transducers to detect Lamb wave mode conversion, and a reference-free damage classification scheme to differentiate actual damage from false-positives.¹⁴ This method, however, relies on the use of this arrangement of dual PZTs. The instantaneous baseline method^15,16 can also be used to overcome the limitations of typical baseline subtraction. This technique entails the comparison of measurements taken along similar paths. However, this proves challenging where the structural components present on the two paths are different. Advanced imaging techniques such as reverse time migration^17,18 and full waveform inversion are also available options.^19,20 These methods accurately image damage sites based on solver-based inversion but the valid imaging results require accurate modeling, which might be nontrivial to achieve for complex structures.

He et al.²¹ proposed the use of a linear Radon transform (LRT)-based extraction algorithm as a BSF damage detection approach. Promising initial results were obtained using a LRT, which converts the signals to a 2-dimensional intercept-slowness $(τ - q)$ domain where linear events are compressed into a point-like region. In this domain, it is easy to identify and isolate events of linear shape, which can then be returned to the $t - x$ domain for evaluation. However, the LRT is insufficient for compressing curved events, which appear distorted and “smeared” in the linear Radon domain.²¹ Since damage scattered waves in NDE and SHM often have curved shapes, the LRT alone is not ideal for general use.

In geophysics, where curved events from seismic data often need to be separated, higher order Radon transform (RT)-based wave extraction algorithms are employed, for instance, the parabolic RT (PRT) method for separating multiple seismic signals and reflections.^22–25 However, conventional PRTs can only compress parabolas aligned along a single, fixed axis. It is common to refer to this axis as the event’s “apex” location, which refers to the location of the curved event’s local extrema, such as the vertex of a parabola or hyperbola. A PRT focusing at a fixed apex location would be insufficient for NDE applications, where curved events may appear anywhere in the $t - x$ domain of the signals. In order to address this challenge, a more general form of RT called the apex-shifted Radon transform (ASRT) is proposed in this paper. The ASRT produces a three-dimensional Radon domain where the additional third dimension represents the varying transform apex locations.^26–30 In this transform, events are compressed into the Radon domain at a location corresponding to their time coordinate, curvature, and apex location, resulting in a time delay–curvature–apex offset $(τ - q - y)$ domain.

Inspired by the features of ASRT, this paper proposes to develop an ASRT-based method to separate buried damage scattered events from direct arrivals or scattered signals produced by complex geometries. This allows us to compress and extract curved events at any location in the $t - x$ domain, corresponding a wide range of sensor array distributions in SHM and NDE. This paper is organized as the follows: First, the mathematical theory is discussed. This is followed by a step-by-step description of the algorithm, along with a description of two proposed additional processing steps: the “intercept-shifted LRT mute” and the “crop and threshold” techniques. Next, the algorithm’s performance is demonstrated on an ideal synthetic case designed to demonstrate the working principle. Then, the algorithm is applied to two different numerical simulations (i.e., a water medium with a steel specimen and a simulated bent aluminum plate) and an experimental phased array measurement from a high-density polyethylene specimen. After the results are presented, a discussion section is included to address some of the effects we observe and possible avenues to mitigate them. Finally, the results and promising benefits of the algorithm are summarized in the conclusion.

Theory of the ASRT for ultrasonic waves

In array-based ultrasonic imaging, signals are generated using an actuator or a source, which sends ultrasonic waves through a test specimen or structure, and an array of sensors record the waveforms. The time-domain signals are often collected and stored in the form of a matrix, with dimensions $n_{t} \times n_{x}$ , which are the number of samples and channel numbers, respectively. Knowing the sampling interval at which the signals are recorded and the physical spacing of the sensor array allows for the $t - x$ domain of the signals to be represented as a 2-dimensional field. In this data representation, events often form a recognizable shape. For instance, parabolic events such as the direct arrivals of a ring array^31,32 or the scattered waves from point-like objects (damage/defects/rivets) are very common in SHM and NDE. Events of parabolic shape have an “apex,” which refers to the local extreme of the shape. Such an “apex” is also commonly referred to as the “vertex” on parabolas and hyperbolas.

The discrete forward ASRT maps data from the original domain to the Radon domain where the $t - x$ signals are represented by d, and the transformed $τ - q - y$ domain signals are represented by m. The conceptual forward ASRT is performed by a summation of values along a curve in the $t - x$ domain. The shape of the curve is specified by the type of RT being performed. For instance, the PRT sums along a parabolic curve. With this process, events that follow the specified curve shape of the RT are compressed into a point-like region in the 3-D Radon domain at a location corresponding to its apex’s time coordinate, the event’s curvature, and the apex spatial coordinate. The more closely an event matches the specified curve shape, the better compressed it is in the Radon domain. The 3-D apex-shifted Radon domain is shown in Figure 1. The point-like event in Figure 1 represents a transformed event of the selected shape from the $t - x$ domain, which has been compressed into a small region according to its time delay, curvature, and offset.

Equation (1) is the ASRT of the $t - x$ signals, d, which results in the model $τ - q - y$ domain m.

\begin{matrix} m (τ, q_{k}, y_{i}) = \sum_{j = 0}^{n_{x} - 1} d (t = τ + q_{k} (y_{i} - y_{0})^{n}, x_{j}), \\ k = 0, . . ., n_{q} - 1 \\ i = 0, . . ., n_{y} - 1, \end{matrix}

(1)

where the parameter $τ$ is the time delay, and the parameter q is the Radon coefficient that defines the event curvature. For an LRT, parameter q is defined as $time / distance$ , and in a PRT, it is $time / distanc e^{2}$ . Parameter y is the spatial coordinate for the event apex locations used in the transform, where each $y_{i}$ is a specific apex location, and $y_{0}$ is a reference y-coordinate, which is a spatial location used to define how far each apex is offset from the left edge of the $t - x$ domain. The allowance of y parameter selection is the key difference between a conventional RT and the proposed ASRT. Because the ASRT does not assume a fixed apex location, events of the selected shape located anywhere in the $t - x$ can be compressed using ASRT into a point-like region in $τ - q - y$ corresponding to their apex location, as long as that apex location is contained within the range of y. Generally, $y_{0}$ corresponds to the leftmost apex location, such that a single-apex ASRT about $y_{0}$ is equivalent to a conventional RT. To analyze the Radon domain distribution, we must choose the amount of apex locations in our y according to the specific task. For instance, if multiple events with many different apexes must be separated, y will be selected to span the entire domain, sampling many $y_{i}$ . If only one event with a known apex location is being targeted, then a single $y_{i}$ is then chosen at the apex location. The parameters $n_{x}$ , $n_{q}$ , and $n_{y}$ are the number of discrete increments of the x, q, and y axes.

The inverse RT, which returns the Radon domain representation of the data to its original domain, is performed according to

d (t, x_{j}) = \sum_{k = 0}^{n_{q} - 1} \sum_{i = 0}^{n_{y} - 1} m (τ, q_{k}, y_{i}), j = 0, . . ., n_{x} - 1,

(2)

where the value of d at a time t and spatial coordinate $x_{j}$ .

By taking its temporal Fourier transform, Equation (2) can be represented in the frequency domain as Equation (3).

D (ω, x_{j}) = \sum_{k = 0}^{n_{q} - 1} \sum_{i = 0}^{n_{y} - 1} M (ω, q_{k}, y_{i}) e^{- i ω q_{k} {(x_{j} - y_{i})}^{n}} .

(3)

Equation (3) can also be represented in a matrix form for each frequency $ω$ , which helps us to develop the algorithm for digital data processing.

D = LM .

(4)

Here, $L$ is the apex-shifted PRT operator. This operator is a horizontal concatenation of smaller $L_{i}$ matrices of dimension $n_{x} \times n_{q}$ . Each submatrix $L_{i}$ corresponds to apex location $y_{i}$ . Since there are a $n_{y}$ number of $y_{i}$ locations, this results in a full $L$ operator of dimension $n_{x} \times (n_{q} \times n_{y})$ . The structure of $L$ is shown in Equation (5)

[\begin{matrix} L_{1} & L_{2} & \dots & L_{n_{y}} \end{matrix}] .

(5)

and $L_{i}$ , a submatrix of $L$ , is defined according to Equation (6).

L_{i} = [\begin{matrix} e^{- i ω q_{0} {(x_{0} - y_{i})}^{n}} & e^{- i ω q_{1} {(x_{0} - y_{i})}^{n}} & \dots & e^{- i ω q_{n_{q} - 1} {(x_{0} - y_{i})}^{n}} \\ e^{- i ω q_{0} {(x_{1} - y_{i})}^{n}} & e^{- i ω q_{1} {(x_{1} - y_{i})}^{n}} & \dots & e^{- i ω q_{n_{q} - 1} {(x_{1} - y_{i})}^{n}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ e^{- i ω q_{0} {(x_{n_{x} - 1} - y_{i})}^{n}} & e^{- i ω q_{1} {(x_{n_{x} - 1} - y_{i})}^{n}} & \dots & e^{- i ω q_{n_{q} - 1} {(x_{n_{x} - 1} - y_{i})}^{n}} \end{matrix}] .

(6)

Because the ASRT is discretized for this application, both the range and spacing of apex locations (y-coordinates) must be selected to encompass all target events. Apex locations used in the transform are represented by each $L_{i}$ submatrix, which individually correspond to the $y_{i}$ locations that are chosen. These locations can be selected arbitrarily and do not have to be spaced evenly. Some applications may benefit from an increased resolution in certain apex ranges, such as trying to separate scattered events from two very close sources.

Using this matrix representation, Equation (1) can be rewritten as

M = L^{H} D,

(7)

where H represents a complex conjugate transpose. However, $L$ and $L^{H}$ do not form an inverse pair, and so

D \neq L L^{H} D .

(8)

Therefore, we treat this as an inverse problem, where M must be approximated using a least-squares inversion. To find an approximation for $M$ , the cost function J is minimized.

J (M) = ∥ D - LM ∥^{2} + μ ∥ M ∥^{2} .

(9)

Here $μ$ is used as a regularization parameter that controls the scaling of the data residuals. $M$ can be approximated by minimizing the L2 norm of the derivatives of J, with the minimum misfit value giving the Damped Least-Squares (DLS) solution for $M$ , as shown in Equation (10). This model Radon domain $M$ is an approximation of the “actual” Radon domain referred to in Equation (1)

M = (L^{H} L + μ I)^{- 1} M = L^{H} D .

(10)

This DLS solution is used in the forward ASRT, forming a pair with the corresponding inverse ASRT as shown in Equation (4). The introduction of the DLS approximation solution introduces some informational losses between the original signals and their transform.

Strategies for recovering events from the Radon domain

To use the ASRT as an extractive tool for SHM and NDE, we propose an algorithm that follows these steps:

Search: Conduct a multi-apex ASRT to locate the apexes of the target events in the Radon domain. The y-coordinates used in the multi-apex ASRT should be chosen according to where the existing and target events are expected. By default, we sample the entire spatial domain.

Target: Once the apex location of a target event is determined, conduct a single-apex ASRT at that y-coordinate.

Extract: To separate the target event from the rest of the data, first select an area containing the target event with as little other data as possible. In this paper, we use a rectangular selection for simplicity, and then reduce all the values outside the target area in the Radon domain to 0.

Return: Perform an inverse ASRT on the extracted information, which will give the successfully extracted target event in the $t - x$ domain.

Apart from the general outline of this algorithm, this paper proposes two additional signal processing techniques. The first is a process we have developed called the intercept-shifted LRT used to mute direct arrivals and recover linear events, and the second is a process we refer to as “crop-and-threshold,” which we use to minimize the clutter in the Radon domain so that target events are easy to find when the unwanted events are with high amplitude.

Intercept-shifted LRT

The “intercept-shifted LRT” is our proposed method for extracting linear events anywhere in the $t - x$ domain. One promising use for this process is for muting the direct arrivals of a linear sensor array (see Sections Elimination of the direct arrivals in a water-immersed scanning setup and Experimental verification). When the actuator is in line with a linear array, these direct arrivals will be received as a pair of linear events in the $t - x$ domain with a common intercept at the actuator’s location. In order to mute these direct arrivals, we propose a first order (linear) RT with its intercept shifted to the actuator’s position, such that the two linear events that make up the direct arrivals can be cleanly muted, making the rest of the signal events much more visible. The intercept-shifted RT can be used in a similar way to extract other target linear events as a first-order version of the normal ASRT algorithm.

Crop and threshold

This paper also proposes the following technique for event recovery. Using the ASRT, it can be difficult to extract relatively low-amplitude wave packets from surrounding higher-amplitude data. This is because the Radon domain will capture information from the higher-amplitude events, which can bury the target. To address this shortcoming, we propose a solution called the “crop-and-threshold method” that aims to limit the amount of undesirable data included in the RT. This technique is performed using the following steps during the proposed ASRT process: (1) If the apex of a target event can be identified, then the time axis of the $t - x$ data is cropped to begin at the apex . This is the “crop” part of the process. (2) A threshold is placed on the value of the data in the cropped section such that the values cannot exceed the apex’s amplitude at any point. This process reduces the influence of extraneous, large-amplitude events, limiting contamination in the Radon domain. This process is repeatedly used in our research (see Sections Abaqus/Explicit simulation of an L-shaped plate and Experimental verification), enabling the ability to target and extract very low-amplitude events. In summary, we focus our transform on a small part of the total data, and limit the amplitudes of the selected portion so the target is easy to locate.

Recognizing and extracting near-perfect synthetic events

The proposed step-by-step ASRT procedures are explained and illustrated by processing a synthetic $t - x$ domain signal (see Figure 2) that contains near-perfect parabolic events. The conceptual setup of the generation of this case is described as follows: A horizontal linear sensor array of 50 sensors with a spacing of $4.2 mm$ records information at a sampling interval of $d t = 0.25 μ s for 50 μ s$ . Two actuators displaced vertically away from the sensor array are located at two horizontal locations (i.e., $x = 21 mm$ and $189 mm$ ), each emits a tone-burst signal which begins to arrive at the sensor array at $25 μ s$ and $33 μ s$ , respectively. The excitation causes waves to propagate through the medium (which is assumed to have a single constant wavespeed, free from dispersion effects) in circular patterns, and as the waves arrive at the linear sensor array, they form a parabola in the $t - x$ domain. Both events arrive with a curvature of $0.961 ns / m m^{2}$ , and the pair overlap in the $t - x$ domain. Figure 2(a) show the two recorded overlapping parabolic events in the $t - x$ domain. The steps to separate the two events are summarized as follows.

Step 1

Search

The multi-apex ASRT of the signals will be obtained first as shown in Figure 2(b). The transform is performed with respect to 9 (i.e., $n_{y} = 9$ ) evenly-spaced y locations, from an apex location of $21 mm$ to an apex location of $189 mm$ . In this representation, the Radon domain is shown in 2-D with nine adjacent “windows,” which are visually divided by a dashed line. We take only nine apexes in this transform to make the results easier to display, so that each “window” is clearly visible in this 2-D representation of the 3-D domain. Each of the nine windows in this domain corresponds to one of the nine evenly-spaced apex locations used in the ASRT algorithm. The result of the transform is that the amplitude energy from the two events are concentrated in the windows corresponding to each event’s apex, at $21 mm$ and $189 mm$ , respectively. These are taken as the apex locations of the target events. In windows that are far from any event’s apex, however (such as the window at $y = 105 mm$ ), there is only a low amount of unfocused and diffuse energy.

Figure 2(c) is the result after a full forward ASRT and then an inverse ASRT procedure were performed on the original signals. No extraction has been performed yet, and this figure illustrates the ability of the forward and inverse solution pair with to recover the $t - x$ domain information from the transform pairs. Figure 2(d) is a zoom-in view of the Radon domain window for the apex location at 21 mm, which corresponds to the first event. Figure 2(e) is the zoom-in view of the Radon domain window for the second event, located at an apex of 189 mm. In these windows, whose apex location corresponds to the apexes for each event, the events are well focused. This represents the result of the search for apex locations in the received signals.

Step 2

Target

Because the apexes of corresponding events have been identified in the last step, we can efficiently compress these two parabolic events shown in Figure 3(a) by directly targeting them using single-apex Radon transforms at each of the selected apex locations. This allows an easy separation in the Radon domain. Note that for the single-apex ASRT that we perform, the domain is now effectively 2-D, because we only have one y-coordinate. The single-apex RTs are performed for the apex locations of each target event at $21 mm$ and $189 mm$ , respectively (see Figure 3(b) and (c)). By limiting the ASRT to the apex location of each target event, extraction can be easily completed. The information in the single-apex ASRT shown in Figure 3(b) and (c) is different from Figure 2(d) and (e), because they contain all the transformed data from the entire domain, instead of solely the information from the windows corresponding to their apex locations.

Step 3

Extract

We then isolate each of the two parabolic events using a simple rectangular selection aimed at the well-focused point-like event. The rectangular selections in the $τ - q$ domain are shown in Figure 3(b) and (c) as the red, rectangular boxes. The data within the selection is left unchanged, and we set all the information outside window to be 0. After this process, the event is isolated in the Radon domain.

Step 4

Return

The inverse ASRT is performed on the selected Radon domain signals, returning each of the now-separated parabolic events back to the $t - x$ domain. The recovered event with an apex of 21 mm is shown in Figure 3(d), and the event with an apex location of $189 mm$ is shown in Figure 3(e). Through our algorithm, the two parabolic events have been well separated in the $t - x$ domain.

The forward RT, extraction, and inverse RT process cause information to be lost between the original $t - x$ signals and the recovered signals. This loss is seen in the recovered signals (Figure 3(d) and (e)) as a drop in amplitude in the recovered signals far from the apex. For instance, in Figure 3(c), the amplitudes of the signals after the sensor at $105 mm$ are noticeably lower. This is due to an effect called the “spatial truncation effect” which affects the extraction. The cause of this effect is that the Radon domain events are not 100% focused in the point-like region such that the extraction is not perfect. Even perfectly parabolic events that have been Radon transformed will have low-energy linear “tails” extending out from them where the energy is poorly focused. These “tails” are labeled in Figure 2(d). The extraction process used here completely cuts these tails off, which results in an amplitude loss in the recovered signals, especially in the regions far from the apex in the $t - x$ domain.³³

Error analysis

After the event has been separated in the Radon domain and the inverse RT is performed, the reconstructed wave packet will contain both amplitude error and phase error. Generally, the amplitude error will be more significant than the phase error. In this near-perfect synthetic case, the recovered amplitude is about 83% of the original amplitude near the apex. However, the amplitude error becomes more significant at greater distances from the apex. At the bottom of event 1, the amplitude drops to about 39%.

The effect of phase error on the reconstruction can be seen in Figure 4. Here, we normalize the signal amplitudes between −1 and 1 so that the amplitude error is negated. Then we can see that the shape of the signals is largely preserved, but there are some distortions observable in the recovered signals. These distortions are due to the phase error. The distortion effect is worst near the tails of the event. They are also noticeably worse in the second event, which we postulate is due to a more contaminated Radon domain later in the trial (at higher values of $τ$ ). We suggest that this contamination is from the first event’s Radon domain presence.

Extraction of scattered waves in numerical simulations

The extraction of synthetic events detailed in the previous section has yielded promising results. This section and the following sections detail the testing results of the proposed methods for extracting signals acquired for SHM and NDE. To this end, the algorithm was applied to two numerical simulations of ultrasonic guided wave inspections.

The first simulation is a 2-D model of a scanning array with a water medium and an immersed steel specimen, where the goal is to extract the specimen-scattered waves. For sensors away from the actuator, the presence of the direct arrivals overlaps with specimen-scattered signals, causing difficulties of extraction of the scattered signals. This kind of $t - x$ domain distribution is frequently observed as the direct arrivals excited by actuators in a linear array. This might cause strong artifacts at the near field of the linear array during imaging.³⁴ The direct arrivals have high amplitude, which can pollute the Radon domain and negatively affect the signal extraction. To alleviate this challenge, we propose to develop an “intercept-shifted LRT” method to effectively mute the linear events with arbitrary starting spatial locations.

The second case is a 3-D multi-physics simulation of an L-shaped bent aluminum plate specimen, where the geometry causes many overlapping boundary reflections. The goal in this case is to separate each of these reflections for analysis as a demonstration of the algorithm’s extractive capability.

Elimination of the direct arrivals in a water-immersed scanning setup

Water-immersed scanning is common in ultrasonic NDE.³⁵ In this type of measurement setup, a movable transducer array is used to collect information about immersed objects by exciting waves in a water medium that then interact with the immersed specimen. Using SPECFEM2D, we generate a 2-D numerical simulation consisting of a water scattering region and an immersed square steel region.³⁶ The ASRT algorithm is then used to separate the scattered waves from the steel region.

The modeling setup is displayed in Figure 5. This model consists of a $20 mm long \times 18 mm$ wide region of liquid water with a square $1.5 mm \times 1.5 mm$ steel specimen beginning at 16 mm from the top of the region. An array of 64 sensors are located $1 mm$ from the top of the water region. The sensors start 1 mm from the left edge of the water medium and end $1 mm$ before, giving the array a length of $16 mm$ . The actuator is located at $x = 8.13 mm$ , the position of the 33rd sensor. From the actuator, a Ricker wavelet with a frequency of $1 MHz$ is excited. Some wavefield snapshots are included in Figure 6.

The recorded events, which are seen in Figure 7(a), are a combination of (1) the direct arrivals and (2) the parabolic scattered waves from the steel region. As seen in the $t - x$ domain representation in Figure 7(a), the direct arrivals (with two linear shapes) have much higher amplitude than the steel-scattered waves. These high-amplitude direct arrivals, if not properly processed, will overshadow with the low-amplitude scattered wave signals in the Radon domain. In this paper, we refer to this effect as “contamination” of the Radon domain, where a high-energy unwanted event is included in the RT, causing unfocused and high-amplitude energy that buries the target event, even if the target is well-compressed.

In order to recover the specimen-scattered waves, the direct arrivals must be muted. The steel-scattered waves are parabolic in shape and arrive directly after the direct arrivals. Because of their significantly lower amplitude, they are nearly invisible in this representation of the data. In order to mute the direct arrivals, we propose to develop an “intercept-shifted LRT.” This can be considered as a subtype of the ASRT algorithm that is useful in this case and is described as follows. As seen in Figure 7(a), the direct arrivals in the $t - x$ domain take the shape of a very high amplitude point-like event at a sensor position of $8.13 mm$ with two symmetrical linear events extending from it. These two events only share one intercept, which is at $x = 8.13 mm$ . By performing an LRT with respect to their shared intercept $x = 8.13 mm$ , the two events can be well focused in a single Radon domain, allowing for them to be isolated, and in this case muted. This “intercept-shifted” concept is the same as the ASRT concept we use for curved events, but since linear events do not technically have an apex, the terminology is altered.

In Figure 7(b), the two symmetrical linear parts of the direct arrivals are compressed into two symmetrical point-like events in the Radon domain. These are removed using a single rectangular selection as displayed. The result, shown in Figure 7(c), contains the signals after subtracting these high-amplitude linear events. Figure 7(d) represents the recovered signals after muting the linear direct arrivals. Figure 7(d) contains the same data as Figure 7(a), except that the linear portions of the direct arrival have been significantly reduced.

After muting the direct arrivals, the steel-scattered portion of the signals can be recovered. We begin with the data that we have already processed to exclude the direct arrivals, as shown in Figure 7(d). There is a point-like remnant of the direct arrivals that still remains, so we simply crop out the data before $5 s$ to exclude it. This crop results in the $t - x$ domain data shown in Figure 8(a). We can now apply our algorithm as mentioned in Section Recognizing and Extracting Near-Perfect Synthetic Events. In order to isolate the steel-scattered waves, we perform an ASRT. Because the apex of the specimen-scattered signals can be easily seen at 8.13 mm, we select this number as the single-apex location to perform our ASRT. Figure 8(b) is the PRT of the signals in part (a), where the well-compressed parabolic event corresponding to the steel region is visible.

Our target event can be easily recognized since it is a high-energy region compressed at the same time-coordinate of the parabolic event’s apex. We extract the event using a straight range selection—denoted as the red box—and modify the outside signals to be zero in this domain. Here we are careful not to select any of the remaining energy from the direct arrivals, which can be seen directly above our selection in Figure 8(b). This remaining energy also bleeds over to the bottom of the Radon domain, since the Radon domain exhibits periodic boundaries. After performing the inverse ASRT to the selected signals, the recovered steel scattered waves are shown in Figure 8(c). Again, we observe some amplitude losses due to the spatial truncation effect but the scattered waves are isolated from the direct arrivals.

Comparison with baseline subtraction in a steel region with inclusion with temperature changes

In this section, we compare the ASRT algorithm against a simple baseline subtraction process with and without temperature changes. We use a SPECFEM2D simulation of a steel region with a small centrally located inclusion of a higher wavespeed region (as seen in Figure 9), and the resulting scatter is overlapped by the direct arrivals. The aim is to separate the scatter from the direct arrivals.

In order to perform this experiment, three total models are generated. The first is an empty steel region, which is the “baseline” case and features no inclusion. The second is a steel region with identical properties to case 1, but with a small centrally located inclusion. The third model is geometrically identical to the second, but the steel wavespeed has been altered to reflect a 15 °C temperature increase, from 10 °C to 25 °C. For comparison purposes, we perform a simple baseline subtraction as seen in Figure 10 for a constant temperature case and a 15 °C temperature change case. From Figure 10(f), it can be seen that in the presence of a temperature change, baseline subtraction fails fully to separate the scatter from the direct arrivals.

In Figure 11, we perform the ASRT extraction process, with a “crop and threshold” applied to the relevant event. This process is repeated for each temperature, with nearly identical results for the 10 °C and 25 °C cases. Note that the original event was not perfectly symmetrical, but the ASRT reconstruction is. This is a form of distortion associated with the single-apex ASRT, which causes a mirroring effect.

In Figure 12, we compare the results of ASRT to baseline subtraction. In the baseline subtraction case, the 10 ° case yields the scattered region exactly. This is expected, since the model is simulated. We can take the difference of these two signals to be the scattering effects exactly. In this case, the basline subtraction is superior to the ASRT in both amplitude and shape recovery. However, in the 25 °C case, the direct arrivals retain a very large amplitude due to the phase effects of the temperature change. The direct arrivals can no longer be separated from the scatter with simple baseline subtraction. It is also notable that the baseline subtraction also separates the additional scattering below the primary event (as seen in Figure 12(b)), whereas the ASRT only separates the curve of interest.

In practice, baseline subtraction has many compensation techniques that can adjust for environmental changes, but without knowledge of what changes have occurred, these compensations cannot be performed. This illustrates a clear use-case for the ASRT algorithm, when there is not enough environmental information available to perform baseline subtraction compensation methods.

Abaqus/Explicit simulation of an L-shaped plate

We apply the algorithm to a 3-D Abaqus/Explicit simulation of an L-shaped aluminum-1050 plate. The goal in this case was to separate and identify the boundary reflections, which will demonstrate the algorithm’s ability to separate wave packets. The geometry of an L-shaped plate is used here as a representation of a simple stiffened plate, with the bent section representing a stiffener. The overlapping and cluttered nature of these events in the $t - x$ domain is due to the complex shape of the specimen and this is a suitable dataset for the ASRT algorithm, which can easily separate all the events.

The model setup for this $3 mm$ thick aluminum plate is shown in Figure 13. The square base has dimensions of $200 mm \times 200 mm$ and the bent section dimensions are $100 mm \times 200 mm$ . A $200 kHz$ five-peaked Hanning-windowed tone-burst signal was excited from the source (labeled as “Actuator”) and received at a series of laser data acquisition locations (the red dashed line, $65 mm$ from the actuator and parallel to the bend line). The plate is bent at a 90 °C, which is $85 mm$ from the actuator. In Figure 14, several wavefield snapshots in the bent plate are shown.

As seen in Figure 15(a), the received $t - x$ signals are a collection of overlapping linear and parabolic events along with some signal noise. In order to recover each wave packet effectively, we perform our proposed “crop and threshold” procedure. The full $t - x$ domain is first cropped to each relevant event, with the maximum value in the cropped section capped at the amplitude of the target event (which is the “threshold” part). This step is seen in Figure 15(b) to (e). The reason for this process is to prevent contamination of the Radon domain by other events. By cropping out undesired events and limiting undesirable high amplitudes before performing a RT, the Radon domain will be much more sparse, improving the extraction process.

The success of this algorithm is heavily dependent on these “crop and threshold” procedures. Since the ASRT does not perfectly collapse events to a point, especially when several event geometries are present, there will be significant diffusive energy from high amplitude events contaminating the Radon domain, which will obscure lower amplitude events, no matter how well these events are focused. The crop cuts out the nontarget events from being transformed, and the threshold prevents contamination from high amplitude events. These steps allow the algorithm to efficiently target and recover very low-amplitude events.

After the “crop and threshold” is performed, a single-apex ASRT is applied for each relevant event, as shown in Figure 15(f) to (i). To recover the parabolic events in Figure 15(f) and (h), we use PRTs. For the linear events in Figure 15(g) and (i), LRTs are used. The events are easily located in the Radon domain in each case because they are well-compressed regions of high energy. Since the well-focused events in the Radon domain have a recognizable shape, they are identifiable. As in the preceding cases, the events are extracted using a rectangular selection, and the values outside of the target area are set to zero in this domain.

After the events are isolated, an inverse ASRT is performed on each cropped section, and the events are recovered. Figure 15(j) to (m) show the result of the RTs, where the four sets of events are recovered, having been extracted from the original signals.

Figure 16 is a comparison of the original signals (a) and a recombination of the extracted events. This comparison allows us to see that each of the events in the original signal have been extracted with their shapes and positions maintained.

Experimental verification

To determine the method’s effectiveness on real data, the algorithm was applied to high-density polyethylene (HDPE), and the results are presented in this section.³⁴ The specimen was a $200 mm \times 50 mm$ rectangular prism with nine small identical and evenly-spaced through-holes aligned vertically down the middle of the specimen. The nine holes in this specimen are used as sources of reliable damage scattering, and the algorithm is used to extract as many of these scatters as possible.

HDPE is a challenging medium for ultrasonic testing. Due to its high density and structure, wave propagation is heavily attenuated, so collecting data using sparse sensors is made more difficult. Imaging algorithms such as delay-and-sum will create strong artifacts near the array.³⁴ The boundary reflected signals can also create artifacts if the defects are closer to the boundary and the reflected signals are not appropriately filtered out or utilized. In this data, the asymmetry in amplitudes between the left and right halves of the $t - x$ domain is due to a sensor error, rather than an intrinsic property of the specimen.

A transducer with 64 receivers is applied to the top of the specimen as shown in Figure 17. This type of testing setup is common in NDE. The actuator is located at the central 32nd receiver, positioned directly over the nine holes. From the actuator, a tone-burst signal with a central frequency of $2.5 MHz$ is excited in the HDPE, which has a longitudinal wave velocity of $2337 m / s$ . Information was sampled every $1.6 \times 10^{- 8} s$ , with a total test time of $4.8 \times 10^{- 5} s$ .

A similar “crop and threshold” process to the one previously used (see Abaqus/Explicit simulation of an L-shaped plate) is applied to this data as shown in Figure 18(b) to (g). After this pre-processing, an ASRT is performed at a location of $32 mm$ (which corresponds to the hole location) for each relevant event, which is shown in Figure 18(h) to (m). In this data, all damage scattered events are caused by a small point-like defect, which results in scattering waves that arrive in a curved shape at the sensor array. The final curved event is not caused by damage scattering, but a boundary reflection. To compress each of the curved events, PRTs are used. Though the experimental data is significantly noisier than the simulated cases, many of the Radon domain events are still identifiable, thanks to the “crop-and-threshold” procedure. Following the algorithm, the events can then be extracted. The inverse ASRT is used on the extracted events in the $τ - q - y$ domain to return them to the $t - x$ domain. The recovered events are shown in Figure 18(n) to (s). As is most easily seen in the boundary reflection event, the energy imbalance between the sides of each event is balanced by the ASRT. We observe this effect for transforms about centrally-located apexes.

The comparison of the original data with the recombined signals can be found in Figure 19. In total, seven damage scattered events were isolated (with the eighth curved event corresponding to the bottom reflection of the specimen). Since we know the wavespeed of the medium is $2337 m / s$ , we know that the theoretical two-way travel time for a wave is $4.279 μ s \times n$ where n is the event number. We can then construct a table of the expected and actual arrival times to confirm which events we isolated correspond to which holes. This information is collected in Table 1. Due to amplitude attenuation effects, the amplitude of the damage scattered waves eventually becomes too small to be recoverable, and the events are buried by signal noise. Given the difficulties of the test material, these results are considered promising and can be combined with the amplitude-compensation methods proposed by Rao et al.³⁴

Table 1.

HDPE: Extraction using ASRT.

Hole #	t (Expected)	t (Actual)
1	$4.279 μ s$	$4.34 μ s$
2	$8.558 μ s$	$8.72 μ s$
3	$12.837 μ s$	$12.66 μ s$
4	$17.116 μ s$	$16.88 μ s$
5	$21.395 μ s$	$20.98 μ s$
6	$25.674 μ s$	$25.20 μ s$
7	$29.953 μ s$	$29.22 μ s$
8	$34.232 μ s$	N/A
9	$38.511 μ s$	N/A
Bottom	$42.790 μ s$	$41.790 μ s$

ASRT: Apex-shifted Radon transform.

Discussion

Several important considerations must be made when implementing the ASRT algorithm for NDE and SHM. For instance, something notable about the algorithm is the loss of amplitude and minor spatial distortions between the original signals and a recovered event, an effect that can be observed in the curved events, as in Figure 3(d) to (e) and Figure 8(d). This is due to a combination of two major effects: (1) there is some inherent loss associated with the DLS solution to the RT, being the approximate solution to an inverse problem; (2) the spatial truncation effect causes long energy tails to appear extending from events in the Radon domain.

Selection

The simple rectangle-shaped extraction technique being used in this research is straight-forward but exacerbates the losses from this effect by completely cutting off these tails. One promising way to mitigate this effect is the use of more advanced techniques such as the surgical mute technique presented by Wang.³³

Improving the RT

To mitigate the loss of amplitude and improve the results of extraction and removal compared to a conventional RT, a high-resolution variant of the Radon transform exists.^37,38 The high-resolution RT better focuses the information in the Radon domain, allowing for better event extraction. Given the limitations of the current algorithm presented here, implementing a high-resolution RT might significantly improve the results.

Because of the aforementioned distortions in the Radon domain, the algorithm’s ability to eliminate events from the original signals is still somewhat weak. In its current state, it is useful to reduce the amplitude of targeted high-amplitude events, but due to the spatial truncation effect, the full shape of an event is not perfectly captured so the deletion is not perfect and artifacts remain after its removal. If the Radon domain focusing could be improved (such as by using a high-resolution RT), the algorithm could be used to delete targeted events rather than isolate them, which would give better results for tasks like the linear mute strategy applied previously (see Section Elimination of the direct arrivals in a water-immersed scanning setup).

Locating events in the Radon domain

Another limitation of this algorithm is that it can be hard to identify target events in the Radon domain. As He et al.²¹ noted, this method still benefits from a baseline measurement, as the qualitative differences in the Radon domains of a damaged and undamaged pair of signals can help determine where the target events are located.

Selecting the appropriate range and spacing of apexes for the ASRT is heavily application dependent. The bigger $n_{y}$ becomes, the more computationally expensive the transform becomes. However, for very closely packed events, it may become necessary to use a high $n_{y}$ .

Conclusion

In this paper, we introduce and evaluate a BSF damage scattered wave extraction method based on an ASRT. We conclude from the results of the studies that this algorithm shows promise as an extractive tool for damage detection in complex structures. The proposed ASRT method has potential to become a robust signal processing tool in both SHM and NDE.

To introduce our algorithm, we first outline the derivation of a DLS solution to the discrete forward ASRT and the inverse ASRT with which it forms a pair. This allows us to perform the ASRT algorithm on digital data from ultrasonic testing. Next, we introduced the steps to our extraction algorithm. We then illustrated the working principle of the proposed algorithm by separating a pair of parabolic events in a simple synthetic example. Next, the algorithm was applied to a numerically simulated water-immersed steel scanning setup, where the direct arrivals were muted and the steel specimen-scattered waves were extracted. Then, the algorithm was used to separate the boundary reflections in a 3D numerical simulation of the wave propagation in a bent-plate specimen. Finally, the algorithm was used on real data from a specimen of HDPE with nine holes, and the damage scattered waves were successfully separated for five holes. From these studies, we find the algorithm to be effective in targeting and extracting wave packets of a selected shape. This process can be conducted without the use of baseline subtraction.

Future work related to this research can focus on improving the recovered amplitude and preserving the original event shapes by investigating related technologies such as the high-resolution RT. Another way to expand on this research is through automation, by integrating machine learning algorithms to automatically detect damage-related signals in the Radon domain.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors thank the financial support from The National Aeronautics and Space Administration with the grant number 80NSSC22M0223. This work used computational resources provided by the Extreme Science and Engineering Discovery Environment (XSEDE) (grant number MSS190025). The authors also greatly appreciate assistance from Dr. Paul Rodriguez at the University of California, San Diego on cluster usage, under XSEDE’s Extended Collaborative Support Service (ECSS).

ORCID iDs

Jiaze He

Jing Rao

References

Rose

. Ultrasonic guided waves in structural health monitoring. Key Eng Mater 2004; 270: 14–21.

Rose

. Successes and challenges in ultrasonic guided waves for ndt and shm. In: Proceedings of the national seminar & exhibition on non-destructive evaluation, 2009, pp. 10–12.

Rose

. The upcoming revolution in ultrasonic guided waves. In: Nondestructive characterization for composite materials, aerospace engineering, civil infrastructure, and homeland security 2011, San Diego, CA, USA, vol. 7983, p. 798302. International Society for Optics and Photonics.

Cho

Lissenden

. Structural health monitoring of fatigue crack growth in plate structures with ultrasonic guided waves. Struct Health Monit 2012; 11(4): 393–404.

Dawson

Michaels

. Isolation of ultrasonic scattering by wavefield baseline subtraction. Mech Syst Signal Process 2016; 70: 891–903.

Abbas

Shafiee

. Structural health monitoring (SHM) and determination of surface defects in large metallic structures using ultrasonic guided waves. Sensors 2018; 18(11): 3958.

Cawley

Cegla

Galvagni

. Guided waves for ndt and permanently-installed monitoring. Insight-Non-Destr Test Cond Monit 2012; 54(11): 594–601.

Chukwuemeka Olisa

Khan

Starr

. Review of current guided wave ultrasonic testing (GWUT) limitations and future directions. Sensors 2021; 21(3): 811.

Capineri

Bulletti

. Ultrasonic guided-waves sensors and integrated structural health monitoring systems for impact detection and localization: a review. Sensors 2021; 21(9): 2929.

10.

Clarke

Cawley

David Wilcox

, et al. Evaluation of the damage detection capability of a sparse-array guided-wave shm system applied to a complex structure under varying thermal conditions. IEEE Trans Ultrason Ferroelectr Freq Control 2009; 56(12): 2666–2678.

11.

Croxford

Moll

Wilcox

, et al. Efficient temperature compensation strategies for guided wave structural health monitoring. Ultrasonics 2010; 50(4): 517–528.

12.

Jun Lee

Michaels

, et al. On the optimization of temperature compensation for guided wave structural health monitoring. In: AIP conference proceedings, Kingston, RI, USA, vol. 1211, pp. 1860–1867, 2010. American Institute of Physics.

13.

Michaels

. A methodology for structural health monitoring with diffuse ultrasonic waves in the presence of temperature variations. Ultrasonics 2005; 43(9): 717–731.

14.

Sohn

Bum Kim

. Development of dual PZT transducers for reference-free crack detection in thin plate structures. IEEE Trans Ultrason Ferroelectr Freq Control 2009; 57(1): 229–240.

15.

Anton

Inman

Park

. Reference-free damage detection using instantaneous baseline measurements. AIAA J 2009; 47(8): 1952–1964.

16.

Overly

Park

Farinholt

, et al. Piezoelectric active-sensor diagnostics and validation using instantaneous baseline data. IEEE Sens J 2009; 9(11): 1414–1421.

17.

Rocha

Sava

. Guided wave tomography based on least-squares reverse-time migration. Struct Health Monit 2020; 19(4): 1237–1249.

18.

Yoon

Marfurt

. Reverse-time migration using the poynting vector. Exploration Geophysics 2006; 37(1): 102–107.

19.

Rao

Ratassepp

Fan

. Guided wave tomography based on full waveform inversion. IEEE Trans Ultrason Ferroelectr Freq Control 2016; 63(5): 737–745.

20.

Rao

Fleming

, et al. Numerical ultrasonic full waveform inversion (FWI) for complex structures in coupled 2d solid/fluid media. Smart Mater Struct 2021; 30(8): 085044.

21.

Leser

, et al. IWSHM 2017: damage-scattered wave extraction in an integral stiffened isotropic plate: a baseline-subtraction-free approach. Struct Health Monit 2018; 17: 1365–1376.

22.

Hampson

. Inverse velocity stacking for multiple elimination. In: SEG technical program expanded abstracts 1986, Houston, TX, USA, pp. 422–424. Society of Exploration Geophysicists.

23.

Schonewille

Duijndam

AJW

. Parabolic Radon transform, sampling and efficiency. Geophysics 2001; 66(2): 667–678.

24.

Trad

Ulrych

Sacchi

. Latest views of the sparse Radon transform. Geophysics 2003; 68(1): 386–399.

25.

Ibrahim

Sacchi

. Simultaneous source separation using a robust Radon transform. Geophysics 2014; 79(1): V1–V11.

26.

Yong

Zheng-ou

Jia-li

. Clutter reduction based on apex shifted Radon transform in subsurface forward-looking ground penetrating radar. In: 2006 CIE international conference on radar, Shanghai, China, 2006, pp. 1–3. IEEE.

27.

Trad

Siliqi

Poole

, et al. Fast and robust deblending using apex shifted Radon transform. In: SEG technical program expanded abstracts 2012, 2012, pp. 1–5. Society of Exploration Geophysicists.

28.

Sabbione

Sacchi

Velis

. Radon transform-based microseismic event detection and signal-to-noise ratio enhancement. J Appl Geophys 2015; 113: 51–63.

29.

Seher

. A high-resolution apex-shifted hyperbolic Radon transform and its application to multiple attenuation. In: 2017 SEG international exposition and annual meeting, Houston, TX, USA, 2017. OnePetro.

30.

Peng

Zhao

, et al. Polarity-preserved diffraction extracting method using modified apex-shifted Radon transform and double-branch Radon transform. J Geophys Eng 2018; 15(5): 1991–2000.

31.

Huthwaite

Simonetti

. High-resolution guided wave tomography. Wave Motion 2013; 50(5): 979–993.

32.

Rocha

Leser

, et al. Least-squares reverse time migration (lsrtm) for damage imaging using Lamb waves. Smart Mater Struct 2019; 28(6): 065010.

33.

Wang

. Multiple attenuation: coping with the spatial truncation effect in the Radon transform domain. Geophys Prospect 2003; 51(1): 75–87.

34.

Rao

Qiu

Teng

, et al. Ultrasonic array imaging of highly attenuative materials with spatio-temporal singular value decomposition. Ultrasonics 2022; 124: 106764.

35.

Chen

Xiao

, et al. A study on the inclusion sizing using immersion ultrasonic c-scan imaging. J Phys: Conf Ser 2017; 842: 012003.

36.

Tromp

Komattisch

Liu

. Spectral-element and adjoint methods in seismology. Commun Comput Phys 2008; 3(1): 1–32.

37.

Sacchi

Ulrych

. High-resolution velocity gathers and offset space reconstruction. Geophysics 1995; 60(4): 1169–1177.

38.

Herrmann

Mojesky

Magesan

, et al. De-aliased, high-resolution Radon transforms. In: SEG technical program expanded abstracts 2000, pages 1953–1956, Calgary, AL, Canada, 2000. Society of Exploration Geophysicists.