#### Abstract

Appropriate impact and sensor locations must be chosen in pile integrity tests to prevent three-dimensional effects caused by the torsional and flexural modes. The three-dimensional characteristics cause high-frequency interference, especially in bridge and wharf piles. A method is required to minimize the high-frequency interference without reducing the accuracy of the pile integrity test. A multivelocity integrity test method is proposed based on a sensor array and frequency-wavenumber (FK) domain analysis to eliminate high-frequency interference and reduce the errors in the output of integrity tests of platform-pile systems. FK filtering is performed to eliminate the spatial alias frequency and separate the upward and downward wavefield and the vibration modes in an integrity test of a platform-pile system. The optimum sensor location to minimize the influence of interference signals is at the bending plane relative to the impact location. Using a sensor array reduces the influence of the sensor location on the test results and minimizes the requirements for determining the location of the excitation point and sensors in the traditional low-strain integrity testing (LST) method, thereby improving the applicability of this method.

#### 1. Introduction

Assessing elastic wave propagation in structures is a highly promising technique for structural health monitoring (SHM) in civil engineering. The integrity assessments of pile foundations of bridges and wharves remain a significant challenge in superstructures with a lack of construction information [1, 2]. Pile integrity assessments are crucial for reinforcement and reconstruction [3]. The oxidation of steel rebars, chloride-induced steel corrosion, and the resulting degradation of concrete threaten the integrity of pile systems in marine environments [4, 5]. The coupled effects of current scour and wave load can cause deformation of the pile-soil system, causing damage to the structural components (such as piles and beams) and ultimately leading to structural instability of wharves, and the changes in environmental conditions (such as geological parameters) caused by wave and current scour make pile inspections vital [6, 7]. Due to the large number of overlapping reflected waves generated by complex structures, SHM with guided waves is complicated. A sparse array of transducers has been used for performance evaluations of structures in the presence of large noise [8, 9], but few studies focused on using sparse arrays for SHM of platform piles in soil.

Low-strain integrity testing (LST) has been utilized to assess the structural health of piles in soil for decades. The piles are subjected to an impact force at the pile head in an impulse response test, and the reflected wave signals are analyzed to evaluate the damage [10, 11]. In the traditional LST method, the optimal distance between the striking point and the receiver is 0.5R-0.7 R when the impact location is on the top surface of the pile [12–14]. However, wave interference occurs in superstructures, making it difficult to distinguish the damage from the pile toe signals in the reflected wave [15]. Three-dimensional (3D) wave effects caused by superstructures result in significant interference with the pile-soil vertical responses [16, 17]. When there is no impact location at the pile head, the results of a lateral impact are influenced by the reflected signals generated by the superstructure, complicating the use of LST for pile inspection.

Parallel seismic (PS) tests have provided excellent results [2, 18, 19] because the pile bottom can be identified by an inflection point in the stacked trace plots. A correction equation based on the wave velocity and pile-borehole distance has been proposed to improve the estimation accuracy [20]. An equation was also proposed to compensate for the borehole inclination [21]. A mathematical algorithm that considers the soil layers and borehole inclination was put forward [22], and the ray-tracing method was used to identify defects in a pile using the PS method [23]. Compared with the widely used LST method, the PS method has the advantage of being applicable to the integrity testing of superstructure piles. The sensors are placed by drilling a hole in the soil, but the complexity of layered soil may complicate the detection. Using a lateral horizontal impact has been suggested as a more feasible approach for assessing pile foundations of superstructures [24, 25]. Although PS tests have been used to determine the length and integrity of piles using the first arrival time and waveforms [19, 26], this method is limited by complex soil parameters, and a borehole must be drilled to place the sensors.

The frequency-wavenumber (FK) method has been used for wavefield transformations of vertical seismic profiles [27] and damage imaging [28–30]. Structural features and defects can be accurately imaged using frequency band-pass and wavenumber band-pass filters [31]. The traditional LST method typically uses only one or two sensors for the integrity testing of piles. Since the FK method requires large amounts of sensor data, few studies used this method for the LST of piles.

The finite element method has been used to analyze PS test data, and a length correction method has been proposed [20]. The elastodynamic finite integration technique (EFIT) is a more accurate and stable time-domain method for wave propagation analysis in elastic media [32–35] than the boundary element method (BEM) method [36, 37] and finite element method [38]. The 3D characteristics of superstructures can be computed and analyzed using the boundary conditions of the piles in the soil. Therefore, the EFIT method is well suited for analyzing composite computational problems of platform piles in soil.

The objectives of this study were to develop a strategy to reduce the errors in the output of integrity tests of platform-pile systems using sensors, determine the effect of high-frequency modes, and improve the applicability of the traditional LST method for platform-pile systems. We propose an analysis method for platform-pile systems using a sensor array and wavenumber domain analysis for assessing the pile integrity. A filtering window function and FK analysis are used for wavefield separation after conducting integrity testing of piles in soil. The performance and reliability of the method are evaluated using experiments and numerical analysis. The results of the integrity analysis of different impaction locations and sensor locations indicate that the method can deal with the high-frequency modes caused by the pile cap. The proposed method minimizes the requirements for determining the location of the impact and sensors in traditional LST methods, improving their applicability.

#### 2. Methodology

##### 2.1. Sensor Array Configuration in Pile Integrity Testing

Few studies analyzed the use of LST of piles using a sensor array. However, it is necessary to understand the 3D effects caused by the torsional and flexural modes on stress wave propagation and develop a methodology to minimize the high-frequency interference without reducing the detection accuracy of pile integrity testing. The traditional LST method is not well suited for platform-pile systems due to the interference of the reflected wave of the upper structure with that of the lower structure. When the impact location is at the pile top of a superstructure, two wavefields propagate in different directions, i.e., the upward wavefield and the downward wavefield. The proposed method uses the FK filtering wavefield algorithm to separate the upward wavefield from the downward wavefield.

The sensors were placed on the lateral surface of the pile in the length direction, as shown in Figure 1. There were two groups of sensors, and , and three impact locations: (1) top impact, (2) lateral impact, and (3) impact on the concrete block. The concrete impact block can be quickly prefabricated and installed using epoxy resin and has strong applicability for on-site inspections.

The foundation pile with a cap was connected to the superstructure and underground rock mass, forming a vibration system. Structural cross-sectional differences and wave impedance changes occur at the connections, causing vibration reflection. The internal defects of the pile also change the wave impedance and cause wave scattering. The change in the wave impedance at the interface between a defect and the pile has the following characteristics: the greater the change in the section, the more significant the defect is, the stronger the scattered energy is, and the lower the frequency of the scattered wave is. The travel time of the scattered wave is related to the defect location. The farther the distance, the longer the travel time is. A multichannel receiving system was used to track the downward wavefield of the superstructure and the upward wavefield of the lower defects to determine their respective locations and filter out the downward wavefield. The damage locations can be determined according to the energy and travel time of the reflected wave signal.

##### 2.2. FK Wavefield Algorithm

The signals of a sensor array can be expressed as a binary function of the distance *z* and time *t*.

A Fourier transform is applied to equation (1) to obtain the two-dimensional spectrum [27], which is expressed as follows:where is the time-domain velocity data measured at or . The two-dimensional spectrum of can be obtained by a Fourier transform. When a sensor array is used, Eqs. (2) and (3) are discretized and expressed as follows:where *k* is the wavenumber relative to *z* and *f* is the frequency relative to time *t*. *M* is the number of sensors, and *N* is the size of time-domain velocity data; *p* is the index of the sensors, and *q* is the index of the time-domain velocity data. [−*A*, *A*] is the frequency range of *k*, and [−*B*, *B*] is the range of *f*.

##### 2.3. FK Filtering with a Window Function

Low and high wavenumber passband cutoffs were used as a filter for wavefield separation to remove all other wavenumbers within the frequency bandwidth of the excitation signal [28, 39]. However, due to the reflection of the pile cap, reflected waves with different wave velocities are generated, and the interference wave signals cannot be separated effectively by this method. Thus, it is necessary to use a window function for wave velocity filtering to separate the wave signals with different wave velocities. The following filter function is applied to equation (4) to remove all other data outside the bandwidth of the signal:

The window function is expressed as follows:where *f* is the window function, such as a rectangular window and Hann window; is the slope of the band signal relative to the wave velocity. For example, the Hann window is defined as follows:

##### 2.4. Spatial Alias Frequency Elimination

During the FK domain calculation, a spatial alias frequency is generated due to the irregular sampling locations of the sensor array and sparse spatial sampling. The spurious frequency leads to frequency dispersion; thus, the upward wavefield cannot be obtained, and the damage cannot be identified. Several processing techniques have been used to eliminate spatial alias frequency and prevent irregular sampling, such as channel interpolation. Processing the wave velocity data with a spatial alias frequency filter ensures that the damage can be detected in the wavefield data. De-aliasing and normalization of the sensor array data are a challenging problem in damage identification. Curvelet transform has been used as a rapid calculation method to improve efficiency [40]. In addition, the curved wave transformation algorithm has been developed [41]. The Hilbert–Huang transform (HHT) algorithm has been used for signal filtering [42]. Due to the characteristics of the sensor array data, we use a modified algorithm based on the HHT to filter out the spatial alias frequency signal.

The process of removing the spatial alias frequency is as follows:(1)The local maximum and minimum points are detected in the input noise signal.(2)The maximum and minimum points are removed in the wavelength bandwidth.(3)The average and standard deviation of the amplitude and period of the spatial alias frequency signal are calculated. If the standard deviation is less than the set value, the maximum value and minimum value in the bandwidth are interpolated according to the adjacent feature points; otherwise, no interpolation is performed.(4)The curve is drawn using a cubic spline interpolation function.(5) is subtracted from the original data clock; the remaining signal is obtained.(6)Steps (1)-(5) are repeated until the root-mean-square (RMS) values of and are less than the preset value.

We remove the spatial alias frequency signal , and then,

An example of the raw data in the f-k domain before and after eliminating the spatial alias frequency signal example is shown in Figure 2.

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##### 2.5. Frequency-Wavenumber Analysis

The propagation direction of the upward wavefield is opposite to that of the downward wavefield; thus, this feature can be used for wavefield separation. Fourier transformation is used to transform the function of the distance *z* and time *t* into the function of the wavenumber *k* and frequency *f*. In the plane of the function, the slope represents the wave velocity. The velocity of the upward wavefield is positive (or negative), and the velocity of the downward wavefield is negative (or positive). If filtering is performed separately in the FK domain, wavefield separation can be performed by extracting positive or negative wavefield data and performing a two-dimensional inverse Fourier transform to obtain the upward wavefield .where and are the FK spectra of the upward and downward wavefields. Then,where and are the time-domain data of the upward and downward wavefields.

#### 3. Wavefield Simulation with EFIT

##### 3.1. Elastic Wave Equation

3D ultrasonic EFIT simulations aid in understanding unexpected features to provide insight into mode conversion as Lamb waves interact with damaged areas [43]. Many authors have used EFIT for ultrasonic damage detection [34, 44]. According to Hooke’s law and Cauchy’s equation of motion, the 3D elastic wave equations can be written as follows using the summation convention for repeated subscripts:where .

It is assumed in the computational model that the pile-soil system consists of linear elastic materials during LST. We use the (2M)th-order finite-difference scheme to calculate the temporal derivatives of the velocity and stress components and calculate the spatial derivatives with fourth-order staggered finite-difference (SFD) schemes to improve the accuracy [45, 46]. The heterogeneous finite-difference scheme is implemented following Moczo et al. (2002) [47].

##### 3.2. Initial Conditions

An impact force *p(t)* is applied to the impact surface as follows:where *I* is the unit impulse of the force and is the contact time. The stress ( or ) values in the impact area are determined by the pressure , and *r* is the radius of the impact surface.

##### 3.3. Free-Surface Boundary Condition

Zero stress must be satisfied on the free-surface boundaries. The stress imaging technique is used for the free-surface treatments [46, 48].

##### 3.4. Perfectly Matched Layer Technique

The velocity stress of a perfectly matched layer (PML) is used to eliminate the reflection of the waves at the boundaries [49], and the maximum damping value is calculated as follows:where is the reflection coefficient and is the shear velocity.

#### 4. Numerical Model and Experimental Setup

##### 4.1. Numerical Model

The integrity coefficient in the pile length direction was used for damage definition, where is the cross-sectional impedance, is the density, is the elastic wave velocity, and is the cross-sectional area of the pile. Numerical models of two damaged piles with a cap and a damaged pile in soil were established to assess the 3D high-frequency interference effects caused by the cap. The impact locations on the top surface are shown in Figure 3.

The two-pile model consists of two square piles and a cap. The computational models are shown in Figure 4. The EFIT numerical model of the two-pile model was constructed with the following parameters: m, m, and , and the impact parameters of equation (10) are as follows: , ms, and m. The other parameters are listed in Tables 1 and 2.

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The pile-soil model has the following soil parameters: S wave velocity = 100 m/s, Poisson’s ratio = 0.35, and density = . The soil location was at *x* = −1 m, 1 m, *y* = −1 m, 1 m, and *z* = 4–10 m. The computational domain is exhibited in Figure 4(b). The EFIT numerical model of the pile-soil model was constructed with the following parameters: the step size was m, m, and . The thickness of the PML was 0.4 m in the *x-* and *y*-directions and 0.8 m in the *z*-direction to meet the grid requirements of the PMLs. The other parameters are listed in Table 3 and Table 4.

##### 4.2. Experimental Setup

A test model of the two-pile model was created for verification with the same parameters as the two-pile model. The lateral impact location was = 1.0 m (Figure 1), and the first sensor was located at = 3.0 m. The size of the middle hole is cm, and the theoretical integrity coefficient is 0.36. The number of sensors was 16, and the spacing of the sensors was 0.25 m. The sensor array configuration is shown in Figure 1, and the test setup is shown in Figure 5. The acquisition instrument was a multichannel high-frequency tester with a maximum sampling frequency of 192 kHz.

#### 5. Results and Discussion

##### 5.1. Verification of EFIT

The EFIT results of the two-pile model were compared with the results of the Abaqus software to verify the accuracy and reliability of the numerical simulation [50]. Figures 6 and 7 show the Abaqus and EFIT models, respectively. The Abaqus and EFIT results are consistent. Figure 8 reveals that the peak values and phases of the wavelet are similar, indicating that the velocities of the Abaqus and EFIT models are consistent. These results verify the accuracy of the EFIT algorithm.

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##### 5.2. Verification of the FK Method

During the test, 16 sensors were used. The field detection layout is shown in Figure 5. When the pile is subjected to a lateral impact, bending and longitudinal waves are generated and propagated downward. The damage signals of these waves are difficult to detect using the traditional LST method, as shown in Figure 9. Figure 10 shows the results of the frequency-wavenumber domain analysis of the test data. Figure 11 shows the upward wavefield after wavefield separation. When the sensor is located close to the damage position ( = 4.5 m), the amplitude of the damage signal is smaller due to the 3D effect. The proposed method can distinguish between the damage signal and the pile toe signal, as shown in Figure 11. The average arrival time of the damage signals indicates that the average location = 4.46 m is consistent with the actual location = 4.5 m.

##### 5.3. Results of the Impact on the Top Surface

When the impact location is on the axis of symmetry (*x* = −0.1 m), one vibration mode occurs at sensors , and two vibration modes occur at sensors , as shown in Figure 12. In contrast, when the impaction location is not in the symmetrical location (*y* = 0.1 m), there are two vibration modes at sensors and , as shown in Figure 13. The reason is that the sensors are not in the center of the bending plane. The hammering signal continues to reflect and propagate downward inside of the pile cap, and the asymmetric bending wave at the pile cap causes a slow-wave propagation speed. It is difficult to identify the reflected wave signals of the damage using the raw velocity data by the traditional LST method, as shown in Figure 14. Figure 15 and Figure 16 show that the reflected wave signals of the damage and pile toe are detected, indicating that using a sensor array can reduce the influences of the sensor location and impact location on the test results when the impact locations are on the top surface of the pile cap.

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In the traditional LST method, the optimal distance between the striking point and the receiver is 0.5R–0.7 R when the impact location is on the top surface of the pile [12–14]. When two receivers are used, the impact location should be on the top surface of the pile cap, and the sensors should be placed on the bending plane relative to the impact location [46, 51]. Figure 17 exhibits different results of the upward velocities of different impact locations. The damage and toe signals can be distinguished, and there are no influences of the impact location and sensor location. There are slight differences in the damage and the toe signals when sensors are not on the symmetry axis of the two-pile model, and the upward velocities of sensors are almost the same. This result demonstrates that the use of a sensor array can reduce the influences of the sensor location and impact location.

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##### 5.4. Results of the Impact on the Concrete Block

When the impact occurs on the concrete block, only one vibration mode occurs at sensor when the sensors are located on the bending plane relative to the impact location, and two vibration modes occur at sensor . It is difficult to identify the reflected wave signals from the raw velocity data by the traditional LST method, as shown in Figure 18. Figures 19(a) and 18(a) show that the high-frequency interference is caused by the concrete block and is detected at location . Figures 20 and 21 show that the high-frequency interference can be eliminated by the FK filtering method. When the impact is on the concrete block, the asymmetry of the pile cap has a negligible influence on the results. The upward velocity results of the different sensor locations (Figure 22) show that although the upward waves of are affected by the high-frequency wave, the damage and toe signals can still be identified, in contrast to the results of . It can be concluded that the sensors do not have to be placed on the bending plane relative to the impact location in the sensor array method. These findings indicate that using a sensor array can reduce the influence of the sensor location on the test results when the impact location is at the concrete block on the lateral surface of the pile.

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##### 5.5. Pile-Soil Model Results of the Impact on the Concrete Block

The results of the pile-soil model show the following. When the sensor is located on the bending plane relative to the impact location, only one vibration mode occurs at sensors . The FK domain results show that the interference signal is more pronounced, and the frequency bandwidth of the upward wave is wider at sensors than at sensors . It is difficult to identify the meaningful reflected wave signals from the raw velocity data by the traditional LST method, as shown in Figure 23. Figures 24–26 show that the high-frequency interference can be eliminated by the FK filtering method. The interference of the time-domain signal is caused by the wider bandwidth of the upward wave. Since the reflected wave of the interference signal is received later at sensors , the interference signal does not affect the identification of the defect signal. Due to the scattering effect of the soil and the influence of high-frequency waves (Figure 27), the upward wave of shows high-frequency oscillations caused by the concrete block, and the toe signal is affected by the high-frequency wave, in contrast to the results of . Since the upward waves of occur on the bending plane relative to the impact location, the influence of high-frequency oscillations is small. These results indicate that the use of a sensor array reduces the influence of the sensor location on the results.

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#### 6. Conclusions

An analysis method based on FK analysis, a filtering window function, and a sensor array was proposed for integrity assessments of platform-pile systems. Different impact locations and sensor configurations were considered. The following conclusions were obtained:(1)A comparison of the results of numerical simulations and experiments indicated the accuracy and reliability of the proposed analysis method for integrity testing.(2)Two wave vibration modes were observed because of the interference signal. Since the sensors were not located at the center of the bending plane, the asymmetric bending waves at the pile cap resulted in slow-wave propagation speed. Placing the sensors at the bending plane relative to the impact location is the optimum choice to reduce the influence of the interference signal.(3)The FK filtering method eliminated the propagation of the high-frequency interference. When the impact was on the concrete block, the asymmetry of the pile cap had a negligible influence on the results.(4)Using a sensor array can reduce the influence of the sensor location on the test results and minimize the requirements for determining the location of the impact point and sensors in traditional LST methods, improving their applicability.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors are grateful for the financial support from the National Natural Science Foundation of China (grant number: 11672338) and the Science and Technology Program of Guangzhou, China (grant number: 201904010332).