Forward Simulation and Complex Signal Analysis of Concrete Crack Depth Detection Using Tracer Electromagnetic Method

Abstract

Cracks are the most typical faults of concrete structures, and their extension can lead to structural fracture. However, when cracks develop inside a structure, the most important depth information is invisible and difficult to measure. The tracer electromagnetic method is an effective technique for detecting the depth of concrete cracks, but since concrete is a multiphase stochastic composite material, its complex internal structure often interferes with the radar detection results, making the conventional radar interpretation technique difficult. In this study, the detection results for concrete crack depth detection based on the tracer electromagnetic method were comprehensively analyzed by combining the complex signal analysis technique, using transient information such as amplitude, phase, and frequency in order to improve the precision and accuracy of radar signal interpretation. In this study, a numerical model was established to determine whether typical cracks such as vertical cracks and diagonal cracks contain indicators or not, and the ground-penetrating radar forward simulation software was used to perform forward simulation of the numerical model and analyze the forward results. The complex signal analysis technique was used to obtain the response characteristics of typical cracks when they did and did not contain the indicator, and the complex signal was finally analyzed by combining it with the actual crack depth detection data. The results show that the tracer electromagnetic method can significantly improve the crack bottom’s reflection ability for radar signals, and when the crack bottom contains an indicator, the amplitude of the reflected signal at the bottom of the crack is enhanced, the phase is reversed, and the frequency is reduced. The distribution of the crack morphology and the location of the crack bottom can be analyzed more conveniently by using the complex signal analysis technique.

1. Introduction

Concrete structures are subject to various factors that have adverse effects on the structure during the service period [1,2,3], such as material damage, aging, corrosion, and the possibility of natural disasters including typhoons and earthquakes [4,5,6]. These unfavorable effects can cause structural damage. When the damage accumulates to a certain extent, concrete will crack [7,8,9]. When cracks extend to the inside of the member, they may cross the critical location and seriously threaten structural safety [10,11], as shown in Figure 1. However, crack depth information is hidden, and crack depth detection is a technical challenge [12,13]. Traditional concrete crack depth detection technology often uses the damage detection method, which mainly includes drilling. These methods are inefficient and cause damage to the structure [14,15]. The commonly used non-destructive testing technologies are based on the stress wave theory, and mainly include the impact echo method, surface wave method, and ultrasonic method. The impact echo method sensor adopts fixed signal reception, which needs to be directly fixed on the surface of the concrete to be tested, and the test efficiency is low. In addition, the stability and state of the sensor will also affect the test results, especially on the surface of non-uniform and honeycomb surfaces [16]. The impact echo detector is suitable for detecting cracks in the range of 20~60 cm, and there is also the phenomenon of depth uncertainty caused by the approximate closure of the crack tip area. The surface wave method uses an electromagnetic or piezoelectric steady-state vibration source to excite a fixed frequency surface wave, and realizes layer-by-layer detection by adjusting the wavelength of the excitation wave. KCH and SM numerically simulate the detection of the surface wave method for crack depth, and propose that the cut-off frequency can be used to judge the depth of surface cracks. However, the deeper the crack, the more obvious the amplification effect of near-field diffraction on the surface wave amplitude, and the greater the impact on the detection accuracy [17]. Because the surface wave detection equipment is complex, the test speed is slow, and it is greatly affected by the boundary conditions (sidewall, corner, etc.), it is still difficult to apply in practical engineering detection [18,19]. The ultrasonic detector needs to fix the transducer on the surface of the structure, so the cleanliness, flatness, and shape of the detection surface must be high when used [20,21]. When the distance between the two transducers exceeds 60 cm, the acoustic energy of the transmitting transducer is greatly attenuated at the crack, and the ultrasonic signal transmitted to the receiving transducer by bypassing the crack is severely attenuated. The acoustic signal received by the receiving transducer is not a single sound wave, but a superposition of multiple waves [22]. Adams M’s test confirmed that the ultrasonic detection signal has fast energy attenuation in concrete [23]. In addition, a small crack tip size, variable crack development direction, and the presence of impurities in a crack gap will affect the detection accuracy of current conventional methods [24,25].

The tracer electromagnetic method is an emerging technique in concrete crack depth detection. This new method is based on a combination of indicator and ground-penetrating radar, utilizing the indicator to diffuse to the bottom of the crack and then mapping out the position of the air–indicator interface through electromagnetic waves, thus visually displaying the crack depth information [26,27]. This paper proposes a new method based on the combination of tracer and hand-held radar to better complete this task. Its main procedures firstly include spreading tracer diffusion to the bottom of the crack propagation, and then mapping the air–indicator interface position using electromagnetic waves to visualize the crack depth information. Compared with the traditional crack detection technology, the tracer electromagnetic method is not limited by the detection distance and coupling agent viscosity, and does not require point-by-point detection; the equipment is easy to operate, and the detection results are highly accurate [28,29].

The accurate identification and interpretation of concrete cracks using ground-penetrating radar is an important prerequisite for the accurate determination of crack depth information; thus, it is important to correctly analyze ground-penetrating radar images [30]. Currently, inspectors mainly analyze the strong reflection coaxial axis in ground-penetrating radar profiles to deduce and decipher structural defects in concrete [31]. However, concrete is a typical multiphase composite system, and the phases are randomly mixed together to form an extremely complex internal structure. Therefore, crack depth detection lies within complex internal structural defect detection, and concrete’s complex internal structural environment often makes it difficult to interpret radar signals [32], which can easily lead to miscalculation and lack of proper judgment. Complex signal analysis technology, as a digital signal processing technology [33,34], is employed in real signal synthesis through the Hilbert transform, allowing for the characterization of the real signal’s instantaneous amplitude, instantaneous phase, and instantaneous frequency, among other parameters [35]. One can obtain more refined information and easily decipher geological anomalies. Liu Bin applied complex signal analysis technology to the prediction of karst fissure water and achieved good results [36]. However, there is no forward modeling and complex signal analysis for the depth of concrete cracks.

In this paper, crack depth detection and radar image interpretation are efficiently and accurately realized by the tracer electromagnetic method and complex signal analysis technique, aiming at solving the problem of difficult crack depth detection and effectively reducing the interference of noise signals inside the concrete structure. This paper establishes numerical models for whether typical cracks such as vertical cracks and diagonal cracks contain indicators or not, uses the ground-penetrating radar forward simulation software GprMax3.0 for forward simulation of the numerical model, and analyzes the forward simulation results using complex signal analysis to obtain the response characteristics of typical cracks. Finally, the results are combined with actual crack depth detection data to achieve complex signal analysis.

2. Principal Analysis

2.1. Tracer Electromagnetic Detection Principle

2.1.1. Electromagnetic Wave Propagation Characteristics

Electromagnetic waves, as a ground-penetrating radar detection information transfer medium, have their own inherent propagation law; with the help of its propagation law, we can understand the important information. Ground-penetrating radar electromagnetic pulse propagation speed in the medium is:

$$v={\displaystyle \frac{c}{\sqrt{\epsilon }}}$$

(1)

where, $c$ is the speed of light in a vacuum (c = 0.3 m/ns); $\epsilon$ is the relative dielectric constant of the measured object. The relative dielectric constant of common media is shown in

Table 1. Relative dielectric constants of common media.

Media	$\mathbf{Relative} \mathbf{Permittivity} {\mathit{\epsilon }}_{\mathit{r}}$	Media	$\mathbf{Relative} \mathbf{Permittivity} {\mathit{\epsilon }}_{\mathit{r}}$
Silty clay	6	Dry sand	3~5
Concrete	6~8	Water	80
Air	1	Metal	300

.

As can be seen from Formula (1), the propagation speed of an electromagnetic wave in concrete is mainly determined by the relative dielectric constant of the medium. During the propagation of electromagnetic waves, reflected and transmitted waves are generated on interfaces with different electromagnetic properties, as shown in Figure 2. The reflection and transmission follow the law of reflection and transmission, and the reflected energy of the electromagnetic wave signal is determined by the reflectivity.

$$R={\displaystyle \frac{\sqrt{{\epsilon }_1}-\sqrt{{\epsilon }_2}}{\sqrt{{\epsilon }_1}+\sqrt{{\epsilon }_2}}}$$

(2)

where ${\epsilon }_1$, ${ \epsilon }_2$ are the relative permittivity of the medium above and below the reflecting interface, respectively.

The reflection coefficient is often used to describe the relationship between the phase and amplitude of the incident and reflected waves. At the interface of different media, the reflection coefficient is positive if the phase is the same as that of the transmitted pulse, and negative if the phase is the same as that of the transmitted pulse.

2.1.2. Analysis of the Detection Principle of Tracer Electromagnetic Method

The hand-held radar is based on detecting the dielectric difference between the target body and the surrounding medium, transmitting a certain center frequency high-frequency electromagnetic wave to the object under testing; the high-frequency electromagnetic wave enters the medium in the form of a broadband narrow pulse, and when the high-frequency electromagnetic wave encounters an interface or target body with electrical differences in the process of propagation, the electromagnetic wave will be reflected and scattered, and we analyze the reflected wave from the interface or target body, in which the positive and negative peaks of the wave are expressed in black, white, gray-scale, or color, respectively, so as to achieve the purpose of positioning. The positive and negative peaks of the waveforms are expressed in black, white, gray-scale, or color, respectively, and the same phase axis or equal gray line or equal color line can intuitively reflect the profile of the interface or target body, so as to achieve the purpose of positioning. Radar wave propagation in the measured object follows the theory of the fluctuation equation; the propagation path of the reflected wave, electromagnetic field strength, and waveform will change with the different electrical properties of the medium it passes through. Therefore, by analyzing the time, amplitude, frequency and phase characteristics of the received radar reflected waves, the location of the target body can be inferred.

The tracer electromagnetic method utilizes a radar device to track the electromagnetic signal of the indicator, thus mapping the location and depth of the crack development tip. Among them, the indicator is a kind of liquid with strong reflective properties with respect to electromagnetic waves, and its dielectric constant is very different from that of air, and its strong reflective property with respect to electromagnetic waves and its own mobility are utilized to detect the development depth of cracks. The detection principle is as follows. Under the action of filling pressure or gravity, the indicator enters into the crack and flows to the bottom of the crack, and when it reaches the bottom of the crack, radar is used to scan and detect the part of the crack, and the indicator at the bottom of the crack is obviously shown on the radar scanning image, and the real development position and depth of the crack are determined by the position of the indicator on the radar image. The schematic diagram of the detection principle of the tracer electromagnetic method is shown in Figure 3.

2.2. Principles of Complex Signal Analysis

The real signal received by the radar antenna can be expressed as:

$$x\left(t\right)=A\left(t\right)\mathrm{c}\mathrm{o}\mathrm{s} [{\omega }_0+\phi \left(t\right)]$$

(3)

where $t$ is the time variable; $A\left(t\right)$ is the amplitude function; ${\omega }_0$ is the center frequency; $\phi \left(t\right)$ is the center frequency; and $x\left(t\right)$ is the phase function. For the signal $x\left(t\right)$, the Hilbert transform is:

$$\widehat{x}\left(t\right)=x\left(t\right){\displaystyle \frac{1}{\pi t}}={\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{x\left(t\right)}{t-\tau }}{d}_{\tau }=A\left(t\right)\mathrm{s}\mathrm{i}\mathrm{n} [{\omega }_{0}t+\phi \left(t\right)]$$

(4)

Generalizing $x\left(t\right)$ and the Hilbert transform $\widehat{x}\left(t\right)$ combined, a complex signal can be obtained:

$$\begin{array}{ll}f\left(t\right)=x\left(t\right)+i\widehat{x}\left(t\right)& =A\left(t\right)\cos \left[{\omega }_{0}t+\phi \left(t\right)\right]+iA\left(t\right)\sin \left[{\omega }_{0}t+\phi \left(t\right)\right]\\ & =A\left(t\right)e^{i\left[{\omega }_{0}t+\phi \left(t\right)\right]}\end{array}$$

(5)

where $A\left(t\right)$ is $f\left(t\right)$, the instantaneous amplitude, which is obtained from Equation (5):

$$A\left(t\right)=\sqrt{x^2\left(t\right)+{\widehat{x}}^2\left(t\right)}$$

(6)

The instantaneous amplitude reflects the strength of the reflected signal and is proportional to the square root of the total energy of the ground-penetrating radar signal at that moment. The instantaneous amplitude changes strongly when there are significant undesirable geologic bodies in the formation.

$\theta \left(t\right)={\omega }_{0}t+\phi \left(t\right)$ for $f\left(t\right)$.

The instantaneous phase is obtained from Equation (5) as:

$$\theta \left(t\right)={tan}^{-1}\left[{\displaystyle \frac{\widehat{x}\left(t\right)}{x\left(t\right)}}\right]$$

(7)

The instantaneous phase reflects the continuity of the co-phase axis on the ground-penetrating radar profile and shows the phase of the reflected wave regardless of its energy strength. When there are obvious cracks in the concrete structure, the same-phase axis will no longer be continuous; when the electromagnetic wave encounters the indicator, the phase of the reflected signal will be reversed by 180°.

$\theta \left(t\right)$, the derivation for time t, gives $f\left(t\right)$. The instantaneous frequency is:

$$S\left(t\right)={\displaystyle \frac{d\theta \left(t\right)}{dt}}={\displaystyle \frac{d}{dt}}{tan}^{-1}\left[{\displaystyle \frac{\widehat{x}\left(t\right)}{x\left(t\right)}}\right]$$

(8)

The instantaneous frequency is the time rate of change in a phase, and it reflects physical change in a concrete structure. When an electromagnetic wave passes through different media, the frequency of the reflected electromagnetic wave signal changes. If three kinds of transient information change significantly at the same location, it can reflect physical changes at that location. Among them, changes in instantaneous amplitude and instantaneous frequency are more intuitive, and the resolution of instantaneous phase is the highest. Usually, the approximate location of a crack is first determined based on the instantaneous amplitude and instantaneous frequency, and then the instantaneous phase is utilized to accurately determine the location and bottom boundary of the crack.

3. Numerical Examples

3.1. Example of Vertical Crack

A vertical crack model is shown in Figure 4. The media of the model in Figure 4a are air, vertical crack, and concrete, in order from top to bottom, where the crack is filled with air. The media of the model in Figure 4b are air, crack, and concrete, in order from top to bottom, where the upper part inside the crack is air and the lower part is the indicator. The model is 0.6 m long (x-axis direction), 0.5 m high (y-axis direction), and 0.001 m wide (z-axis direction). The crack is 0.2 m high and 0.01 m wide, and the indicator is 0.02 m high and 0.01 m wide. In order to attenuate or eliminate the influence of boundary reflections and at the same time to improve the computational efficiency, an electromagnetic wave absorbing layer (Perfectly Matched Layer, PML) is set up at the boundary, and the mesh has a side length of 0.001 m. The analog excitation source is selected as 1600 MHz Ricker, and the signal receiver is parallel to the emission source and located at the right side of the grid at 0.01 m. In the simulation process, the emission source and the signal receiver complete the detection scanning from left to right along the outer boundary of the air medium. The electrical parameters of the medium are shown in

Table 2. Electrical parameters.

Medium	$\mathbf{Relative} \mathbf{Permittivity} {\mathit{\epsilon }}_{\mathit{r}}$	Conductivity σ/(S/m)
atmosphere	1	0
concrete	8	1 × 10−3
indicator	270	1.6 × 104

.

When there is no indicator in the vertical concrete crack, the orthogonal result is as shown in Figure 5a. When the electromagnetic wave encounters the crack, it produces a hyperbolic reflection signal, and from the reflection homogeneous axis information, the boundary at the bottom of the crack can be deduced. If the size of the crack is small and the attenuation of the electromagnetic wave signal is large when it passes through the cavity of the crack, the electromagnetic wave signal produced at the bottom of the crack is not obvious and is difficult to recognize.

The orthogonal results shown in Figure 6a illustrate that the indicator is present at the bottom of the vertical crack. Here, the amplitude of the reflected signal generated from the indicator level is enhanced and the phase is reversed, compared with that without the indicator in the crack. The bottom of the crack is difficult to recognize due to serious attenuation of the electromagnetic wave when crossing the indicator.

Interest in the instantaneous amplitude information discards the phase information in the radar detection results and retains only the amplitude information, which allows for a more intuitive observation of the strength of the reflected signal. The instantaneous amplitudes of vertical cracks without the indicator and vertical crack containing indicator are shown in Figure 5b and Figure 6b, respectively. As can be seen from the figures, the hyperbolic-like reflection produced by the indicator liquid surface is more obvious, but the reflection signal produced at the bottom of the crack cannot be distinguished. The air–indicator interface produces a stronger amplitude compared to the air–concrete interface.

The instantaneous phase is opposite to the instantaneous amplitude information, and only the phase information in the radar detection result is retained, while amplitude information is discarded, so that even weak signals can be displayed well. The instantaneous phases of vertical cracks without the indicator and vertical cracks containing the indicator are shown in Figure 5c and Figure 6c, respectively. As can be seen from the figures, the multiple reflected waves generated by the crack and the reflected waves generated at the bottom end of the crack are clearly shown, the location of the bottom end of the crack can be inferred, and there is a clear difference between the reflected signals generated in the indicator-containing crack and the waves generated in the open position of the crack, making them easier to differentiate. Compared with the unfilled crack, the reflected wave’s phase in the indicator-filled crack is reversed by 180°.

The instantaneous frequencies of unfilled vertical cracks and indicator-filled vertical cracks are shown in Figure 5d and Figure 6d, respectively. As can be seen in the figures, the reflected wave frequency of the indicator-containing crack is significantly lower than that of the indicator-free crack due to the fact that the indicator absorbs the high-frequency part of the electromagnetic wave. The frequency of the multiple-reflected signals generated at the bottom of the crack without the indicator is obviously higher than the frequency of the reflected wave generated at the crack opening, so the bottom of the crack’s position can be recognized; the frequency of the multiple reflected signals generated at the liquid surface of the indicator in the crack is obviously higher than the frequency of the reflected wave generated at the crack opening, and it is difficult to observe the frequency of the reflected signal at the bottom of the crack.

3.2. Example of Diagonal Crack

The diagonal crack model is shown in Figure 7. The media of the model in Figure 7a are air, diagonal crack, and concrete in order from top to bottom, where the crack is filled with air. The media of the model in Figure 7b are air, diagonal crack, and concrete in order from top to bottom, where the upper part of the crack is filled with air and the lower part is filled with the indicator. The model is 0.6 m long (x-axis direction), 0.5 m high (y-axis direction), and 0.001 m wide (z-axis direction). The crack is 0.2 m high and 0.01 m wide. The indicator is 0.2 m high and 0.01 m wide. The boundary setting, simulated excitation source, and detection method are the same as in the Chapter 3.1 calculations. The electrical parameters of the relevant medium are shown in Table 2.

The orthogonal detection results shown in Figure 8a illustrate that there is no indicator in the diagonal crack, and it can be seen that the strong reflection axis of the electromagnetic wave is more consistent with the crack morphology. The radar can acquire hyperbolic-like reflection signals at the upper and lower ends of the crack. Therefore, the morphology and distribution of the inclined crack and the angle between the crack and the radar detection direction can be obtained from the strong reflection co-axial information.

The forward detection results are shown in Figure 9a for when the indicator is present at the bottom of the crack. Compared with the crack without the indicator, the phase of the reflected signal of the electromagnetic wave is reversed and the amplitude is enhanced, and it is difficult to recognize the bottom of the crack below the indicator.

The instantaneous amplitudes of the diagonal crack without the indicator and the diagonal crack containing the indicator are shown in Figure 8b and Figure 9b, respectively; the strong reflection signals generated by the cracks are more easily observed, and the top position and morphological information of the crack without the indicator can be distinguished, while it is difficult to recognize the bottom information of the crack. The liquid level position in indicator-containing cracks can be effectively identified. Cracks containing the indicator produce a stronger-amplitude signal compared to cracks without the indicator.

Instantaneous phases of a diagonal crack without the indicator and a diagonal crack containing the indicator are shown in Figure 8c and Figure 9c, respectively, and the multiple reflections generated by the crack can be clearly seen. The bottom end of the indicatorless diagonal crack and the indicator liquid level of the indicator-containing diagonal crack are clearly identified. The phase of the reflected wave from the indicator-containing crack is reversed by 180°, compared to the crack without the indicator.

The instantaneous frequencies of the diagonal crack without the indicator and the diagonal crack containing the indicator are shown in Figure 8d and Figure 9d, respectively, and the reflected wave frequency of the diagonal crack containing the indicator is significantly lower than that of the crack without the indicator. For diagonal cracks containing the indicator, there is a clear signal anomaly at the indicator level.

4. Example Exploration

Plain concrete beams with prefabricated cracks were selected to carry out crack depth detection tests, and vertical and diagonal crack depth detection based on the tracer electromagnetic method, respectively; the actual detection results were analyzed via the complex signal.

4.1. Test Materials and Equipment

(1)

Indicator

The indicator consists of two parts, the mother liquor and the dispersant, and it needs to have the properties of permeability and electromagnetic wave reflectivity. The permeability of the indicator describes the ability of the fracture to adsorb the indicator, which is realized by the mother liquor, and the mother liquor transports the dispersant to the bottom of the fracture. The electromagnetic wave reflectivity of the indicator is determined by the dispersant.

In this test, alkali metal silicate solution was chosen as the mother liquor of the indicator, and iron powder with a particle size of 50 nm was chosen as the dispersant of the indicator, as shown in Figure 10. The indicator was made by mixing the two, as shown in Figure 11. The physical parameters of the indicator are detailed in

Table 3. Physical and chemical parameters of indicators.

Physical and Chemical Projects	Parameter
Appearance	Black transparent liquid
Density (g/cm3)	≥1.10
pH value	11 ± 1
Viscosity (s)	11.0 ± 1.0
Surface tension (mN/m)	≤26.0
Dielectric constant	270
Conductivity (s/m)	1.25 × 104

.

(2)

Indicator filling equipment

The indicator injector used in this test is shown in Figure 12. The filling process was automatic low-pressure filling. The indicator injector consisted of two parts: the injector and the base. The role of the injector was to extract, store, and output the indicator, and the base contained an inlet hole connecting the crack and the output port of the injector.

(3)

Ground-penetrating radar

The ground-penetrating radar used in this test was a three-dimensional perspective scanning test system, which had a working surface 22 cm long and 14 cm wide, with a probe spacing of 14 cm and a center frequency of 2.6 GHz, as shown in Figure 13.

4.2. Test Methods and Procedures

The method and steps for detecting the depth of concrete cracks using tracer electromagnetic method are as follows.

(1)

Clean the structure’s surface of dust and impurities in the cracks before testing, place two bases on the cracks on the structure’s top surface, seal the top surface of the structure’s cracks with epoxy cement on both sides, and fix the grouting bases.

(2)

Fill the indicator into one of the bases using a filler, using 20 mL of indicator for this test.

(3)

After indicator filling, take the centerline of the concrete specimen perpendicular to the crack as the direction of ground-penetrating radar detection, and overlap the front and rear laser marking lines of the ground-penetrating radar with the centerline when detecting.

(4)

Start the detection at 40 cm from the crack opening of the concrete member, put the hand-held radar close to the detection surface and keep a uniform speed, travel through the crack until the detection ends at 40 cm from the other end of the crack opening, and record and save the data. When detecting diagonal cracks, extend the detection length in the direction of crack inclination.

In accordance with the above inspection method, a depth inspection test was carried out on a vertical crack on a plain concrete member with an actual depth of 12.5 cm and a width of 0.3 mm, as shown in Figure 14a; at the same time, a depth inspection test was carried out on a diagonal crack with an actual depth of 12.5 cm, a width of 0.3 mm, and a horizontal direction of 60° on the plain concrete member, as shown in Figure 14b.

4.3. Test Results and Analysis

(1)

Vertical crack depth test results

The results of the vertical crack depth detection are shown in Figure 15a. From the figure, it can be seen that there is an obvious reflection signal at a depth of 12 cm, which is inferred to be the liquid surface of the indicator stored at the bottom of the crack.

In order to further identify the precise positional reflection at the bottom of the crack, the radar detection results were analyzed by complex signal analysis, and the results are as follows.

The instantaneous amplitude is shown in Figure 15b, and the hyperbolic reflection signal is clearly visible at a depth of 12 cm. The instantaneous phase is shown in Figure 15c, and there is obviously an anomaly in the same-phase axis at a position of 12 cm. The range of the anomaly can be clearly inferred from the anomalous region of the same-phase axis, which is reversed by 180° and is inferred to contain the indicator in this position. The instantaneous frequency is shown in Figure 15d, and there is an obvious frequency division interface at the position of 12 cm depth. Additionally, there is a low-frequency reflection region at this location, which is consistent with the presence of indicator at this location. Due to the small amount of indicator in the upper part of the crack bottom, the actual detection height is slightly larger than the actual crack depth.

(2)

Oblique crack depth detection results

The results of the oblique crack depth detection are shown in Figure 16a. From the figure, it can be seen that the strong reflection axis of the electromagnetic wave is consistent with the crack pattern, and there is an obvious hyperbolic reflection signal at a position of 12 cm depth, which is inferred to be the liquid surface of the indicator stored at the bottom of the crack.

In order to further precisely identify the positional reflection at the bottom of the crack, the radar detection results were analyzed by complex signal analysis. The results are as follows.

The instantaneous amplitude is shown in Figure 16b, and the hyperbolic reflection signal generated at 12 cm depth is very obvious. The instantaneous phase is shown in Figure 16c, and there is an obvious anomaly in the same-phase axis at a depth of 12 cm. The range of the anomaly can be clearly inferred from the anomalous region of the same-phase axis, which has been reversed by 180° and is inferred to contain the indicator in this location. The instantaneous frequency is shown in Figure 16d, and there is an obvious frequency division interface at the position of 12 cm depth. Additionally, there is a low-frequency reflection region at this location, which is consistent with the presence of indicator at this location. Due to the small amount of indicator in the upper part of the crack bottom, the actual detection height is slightly larger than the actual crack depth.

As can be seen from the above tests, when the crack width is small, the reflection of electromagnetic waves at the air–concrete interface is weak, and it is difficult to directly scan and identify the bottom of the crack by ground-penetrating radar. The air–indicator interface can produce effective reflection of electromagnetic waves.

5. Conclusions and Outlook

5.1. Conclusions

(1)

The homogeneous axis of the crack reflection signal is hyperbolic. The location of the crack can be inferred from the homogeneous axis, and it is difficult to distinguish the bottom of the crack. Meanwhile, the homogeneous axis of the crack reflection signal is consistent with its shape. After injecting the indicator into the crack, the reflection signal at the bottom of the crack is more obvious. The height of the indicator affects the detection accuracy of the tracer electromagnetic method. The detected crack depth is smaller than the actual crack depth.

(2)

Compared with an indicator-free crack, the bottom amplitude of an indicator-containing crack is significantly enhanced. The instantaneous phase can be acquired with higher resolution, and even very weak signals can be displayed, so the location of the bottom end of the crack can be distinguished more easily. When the bottom of a crack contains an indicator, the phase is reversed by 180, and the instantaneous frequency reflects the physical change in the concrete structure and is significantly lower when the bottom of the crack contains an indicator.

(3)

Both the forward simulation and the actual physical test show that the crack morphology distribution and the location of the crack bottom can be analyzed more conveniently using the complex signal analysis technique, which improves the precision and accuracy of the crack depth detection and interpretation.

5.2. Outlook

In this test, the indicator height deposited at the bottom of the crack affects the actual crack depth. At present, the effective geometric center position of the indicator-reflected radar signal has not been evaluated and quantified. It is hoped that in the follow-up study, the position of the reflection center can be determined, and the relationship between the position and the actual crack bottom can be inferred, so as to determine the crack depth more accurately.