Damage Index Analysis of RetainingWall Structures Based on the Impulse Response Function and Virtual Impulse Response Function

To identify the damage within retaining wall structures, the Hilbert–Huang Transforms of the impulse response function and virtual impulse response function were performed.,e Hilbert marginal energy ratio spectrums of the impulse response function and virtual impulse response function were acquired. To reflect damage information effectively, those bands with stronger damage sensitivity were extracted via the threshold value ε0. ,en, the Hilbert feature bands, which were more sensitive to damage within retaining walls, were selected by considering the contribution of the residual band to the damage identification. Based on the feature bands, the Hilbert damage feature vector, which reflects the variations of Hilbert marginal energy ratio caused by damage, was created. Based on the damage feature vector, two damage identification indexes (the energy ration standard deviation and Energy Ration Standard Deviation), which were based on the impulse response function and virtual impulse response function, respectively, were proposed to identify damage within retaining walls. To investigate the validity of the damage indexes, vibration tests on a pile plate retaining wall were done. ,e test results show that the damage feature vector is a zero vector or the value of damage index is zero when the wall is undamaged.,e damage feature vector is a nonzero vector or the value of the damage index is more than zero when the wall is damaged. ,us, the damage state of the wall can be detected sensitively via the damage feature vector or damage indexes. Partial damage causes greater fluctuation of trend surface of the damage index. ,e location of partial damage can be diagnosed validly via the coordinate of peak value in the trend surface. ,e quantitative relationship formula between the damage index and damage intensity is established.,e damage intensity of the wall can be calculated reversely, when the damage index is available. Either the energy ration standard deviation or Energy Ration Standard Deviation can be used to detect the damage state, diagnose the damage location, and identify the damage intensity. In comparison with the energy ration standard deviation, the stability and damage sensitivity of the Energy Ration Standard Deviation is much better.


Introduction
As a common type of retaining structure, the retaining walls (such as cantilever retaining walls, gravity retaining walls, and pile plate retaining walls) were often used to sustain foundation ditch, stabilize the cutting of highway, and prevent landslide or collapse.
us, retaining walls were widely used in building engineering, road engineering, slope engineering, and underground engineering, and so on. However, under the effects of many factors (such as environmental erosion, material degradation, and load variation), such damages as micro-cracks or holes might appear in the retaining wall. e wall might collapse when these damages accumulated to a certain extent, which caused serious accidents, as listed in Table 1 ( e data in Table 1 is from the Internet). To avoid or reduce these accidents in the future, it is necessary and significant to investigate the identification of damages of retaining wall structures.
Common damages within engineering structures mainly included internal faults, structural cracks, and holes. Changes of some structural parameters were often caused by these damages. us, the damaged state of structures could be detected, damage locations could be diagnosed, damage intensity could be identified, and residual lifespan could be evaluated by analyzing these structural parameters [1]. And, common methods for damage identification are listed here. ① e methods were based on such dynamic fingerprints as natural frequencies, mode shapes, and frequency response function, Hearn and Testa identified the damages within a steel frame and wire rope via analyzing the changes of natural frequencies [2]. Biswas et al. detected the damage intensity and location of the highway bridge via analysis of the mode shapes [3]. ② e methods based on the finite element model updating. Zhang et al. diagnosed the damages within structures via synchronically updating measured vibration modes and finite element model [4]. Zong et al. proposed a damage diagnosis method for bridges based on the response surface model updating and the element modal strain energy [5]. ③ e methods based on the genetic algorithm. Perera and Torres localized the damage within a beam via the analysis of natural frequencies and mode shapes based on the genetic algorithm [6]. Varmazyar et al. detected the damage within beams via analysis of the response power spectral density of the beam based the evolutionary algorithms [7]. ④ e method based on the neural network. Qu and Chen proposed a seismic damage diagnosis method for structures based on the neural network system of damage diagnosis [8]. Wang et al. proposed a new damage identification method based on the statistical neural network [9]. ⑤ e methods based on the dynamic responses. Hou et al. diagnosed damages within the Benchmark structure via the Wavelet Packet Decomposition of structural vibration signals [10]. Based on the extracted impulse response function, Li and Fan detected damages within a beam via optimizing sensor locations [11]. Ding et al. proposed a damage alarming method for structures based on the Wavelet Packet Decomposition of the virtual impulse response function [12]. Zheng et al. proposed a damage diagnosis method for bearings based on the Hilbert-Huang Transform of vibration signals to detect faults in bearings [13]. e damage identification methods based on structural dynamic fingerprints were widely used to diagnose damages, because these fingerprints could be easily acquired. However, it was easy to obtain the damage information of bands with lower frequencies but hard to measure the ones with higher frequencies via this method due to the limitation of existing measurement technology. Some damage information might be lost via this method. e method based on the finite element model updating required better environments (fewer interferences such as size effects, variation of temperature, and humidity) and excitations. However, the better environments and sufficient excitations were rare in practical engineering. us, this method might not be suitable for practical engineering with pool conditions and several interferences. e damage identification methods based on the genetic algorithm and neural network were affected greatly by sample functions, objective functions, and algorithms. e precision of damage identification was greatly dependent on these factors. e method based on the Wavelet Packet Transform had much better damage sensitivity and stronger noise immunity. However, the method lacked adaptivity to signals and was suitable only for linear signals. In addition, the wavelet packet analysis results were greatly dependent on the wavelet basis function. In comparison with the WPT (the Wavelet Packet Transform), the method based on HHT (the Hilbert-Huang Transform) had good adaptivity to signals and was suitable for either linear signals or nonlinear ones. us far, the attention of damage identification to engineering structures were mainly paid to such structural components as beams [14], plates [15], piles [16], building structures [17,18], bridge structures [19,20], and other structures [21,22]. However, the investigation of damage identification to retaining wall structures was much rarer. Consequently, the damage identification to retaining wall structures should be paid much attention. In this paper, the Hilbert marginal energy ratio spectrums of the impulse response function and virtual impulse response function are acquired via HHT. en, the Hilbert feature bands, which are more sensitive to damage within retaining walls, are selected by considering the contribution of the residual band to the damage diagnosis. Based on the feature bands, the Hilbert damage feature vector, which reflects the variations of Hilbert marginal energy ratio caused by damage, is created. Based on the damage feature vector, two damage

Damage Identification Indexes
In the light of literature [23,24], damages within the structure cannot be identified validly, if the dynamic responses to the structure are analyzed directly via WPT or HHT. However, damages within the structure can be identified validly based on IRF (the Impulse Response Function of the structural response and external excitation in a structure) or VIRF (the Virtual Impulse Response Function of the virtual response and virtual excitation in a structure).

e Impulse Response Function.
Under the effects of excitation, the motion equation of a multiple degree of freedom structural dynamic system can be expressed as where m ∈ R n×n is the mass matrix of the structural dynamic system, c ∈ R n×n is the damping matrix, k ∈ R n×n is the stiffness matrix, u ∈ R n×1 is the displacement matrix, and f ∈ R n×1 is the excitation matrix. e Fourier Transforms of the excitation f (t) and response u (t) are performed, respectively: where u (ω) and f (ω) are the Fourier Transforms of u (t) and f (t), respectively. en, the frequency response function h (ω) of u (t) and f (t) can be obtained and expressed as After the inverse Fourier Transforms of h (ω), IRF h (t) is obtained and expressed as From the analysis above, both IRF and VIRF are the inherent attributes of the structure. IRF can be acquired when both the external excitation and structural responses are available. In comparison with IRF, VIRF can be acquired when the virtual response and virtual excitation are available whether the external excitation is available or not. us, VIRF has no dependence on the external excitation.

Damage Identification Indexes.
As mentioned before, damages within engineering structures can be identified based on IRF or VIRF via WPT or HHT. However, WPT lacks adaptivity to signals and is only suitable for linear signals. And, the wavelet packet analysis results are greatly dependent on the wavelet basis function. In comparison with WPT, HHT has good adaptivity to signals, and HHT is suitable for either linear signals or nonlinear ones [25]. us, the Hilbert-Huang Transform of IRF and VIRF are performed, respectively. en, the intrinsic mode functions of IRF and VIRF are acquired, respectively. On the basis of the Hilbert marginal energy ratio spectrum, the Hilbert feature vectors, which are based on IRF and VIRF, are created, respectively. Based on the Hilbert feature vectors, the Hilbert damage feature vectors are created, respectively. On the basis Shock and Vibration of the damage feature vectors, two damage identification indexes are proposed.
To obtain intrinsic mode functions, the empirical mode decompositions of IRF h (t) u and h (t) d are performed, respectively. And, the empirical mode decompositions of VIRF H (t) u and H (t) d are performed too, respectively. e h (t) u , h (t) d , H (t) u , and H (t) d can be expressed as where the h (t And, the n is the order of empirical mode decomposition [26]. Because the values of residual functions are much less, the residual functions can be neglected. us, the h (t) u , h (t) d , H (t) u , and H (t) d can be rewritten as: In the light of the empirical mode decomposition [26], the components of intrinsic mode functions imf ju , imf jd , IMF ju , and IMF jd (j � 1, 2, . . ., n) are expressed as: where the a j (t) and e i ω j (t)dt are the instantaneous am-  (13) and (14). where and H (ω, t) d , the Hilbert marginal energy spectrums e(ω) u , e(ω) d , E (ω) u , and E (ω) d can be expressed as [27,28]: 4 Shock and Vibration where the e(ω) u and e(ω) d are the Hilbert marginal energy spectrums of the h (t) u and h (t) d , the E (ω) u and E (ω) d are the Hilbert marginal energy spectrums of the H (t) u and H (t) d , the t and T are time lengths of the signal. e distribution characteristics of the signal energy can be reflected via the Hilbert marginal energy from an absolute energy perspective. However, the distribution characteristics of the signal energy are often reflected from such relative energy perspective as the Hilbert marginal energy ratio. en, the Hilbert marginal energy ratio er(ω) ju , er(ω) jd , ER (ω) ju , and ER (ω) ju can be defined as where the e(ω) ju and e(ω) jd are the Hilbert marginal energies of the h (t) u and h (t) d , the E (ω) ju and E (ω) jd are the Hilbert marginal energies of the H (t) u and H (t) d .
en, the Hilbert marginal energy ratio spectrums er(ω) u , er(ω) d , ER (ω) u , and ER (ω) d can be expressed as where er(ω) u and er(ω) d are the Hilbert marginal energy ratio spectrums of the h (t) u and h (t) d , respectively. e ER (ω) u and ER (ω) d are the Hilbert marginal energy ratio spectrums of the H (t) u and H (t) d , respectively. In the literature [29], due to the influence of inherent frequencies of the retaining wall, not all the sub-bands are useful for damage identification. Only those bands, whose frequencies are close to the ones of the retaining wall, are more useful for damage identification. In this paper, to sensitively identify damages within the wall, let Shock and Vibration ε IRF and ε VIRF are the ratios of marginal energy ratio deviations of er(ω) and ER (ω), respectively. To extract damage information, a threshold value ε 0 is defined, where ε (ε IRF or ε VIRF ) ≥ ε 0 . en, the p (p ≤ n) bands can be selected from the n bands via ε 0 . ese energy ratios of the p bands can be used to reflect damage information. However, the contribution of the residual (n − p) sub-bands to damage identification should not be ignored either. us, the (n − p) sub-bands are merged into one band, which is called the residual band. en, the energy ratios of the residual bands are defined as er(ω) p+1 and ER(ω) p+1.
Here, er(ω) (p+1)u � 100 − p q�1 er(ω) qu and er(ω) (p+1)d � 100 − p q�1 er(ω) qd are the energy ratios of the residual band of the h (t) u and h (t) d , respectively. Similarly, qu and ER(ω) (p+1)d � 100 − p q�1 ER(ω) qd are the energy ratios of the residual bands of the H (t) u and H (t) d , respectively. e (p + 1) bands are defined as feature bands. e feature bands are sorted from the strongest damage sensitivity to the poorest one. en, the Hilbert feature vectorserv(ω) u , erv(ω) d , ERV (ω) u , and ERV (ω) d can be obtained and expressed as where the erv(ω) ku and erv(ω) kd (k � 1, 2, . . ., p, p + 1) are the energy ratios of the k th feature band in erv(ω) u and erv(ω) d , respectively. ERV (ω) ku and ERV (ω) kd are the energy ratios of the k th feature band in ERV (ω) u and ERV (ω) d , respectively. On the basis of the erv(ω) u , erv(ω) d , ERV (ω) u , and ERV (ω) d , the Hilbert marginal energy ratio deviation erd(ω) k and ERD(ω) k are calculated and expressed as If the energy ratios in the erv(ω) u can be regarded as baselines, the energy ratios in the erv(ω) d fluctuate up and down on the basis of the ones in erv(ω) u due to damages. Similarly, energy ratios in the ERV (ω) d also fluctuate up and down on the basis of the ones in ERV (ω) u .
To describe the fluctuation intensity of energy ratios, the Hilbert damage feature vectors erd(ω) d and ERD (ω) are acquired and expressed as e energy ratios in the erv(ω) d do not change in comparison with the ones in theerv(ω) u , when the erd(ω) � 0. us, the retaining wall is not damaged. e energy ratios in the erv(ω) d fluctuate up and down based on the ones in the erv(ω) u , when the erd(ω) ≠ 0. us, the wall is damaged. So, the damage state of the retaining wall can be detected qualitatively via the erd(ω). Similarly, the damage state of the retaining wall can also be detected qualitatively via the ERD (ω).
Based on the erd(ω) and ERD (ω), the damage identification indexes ersd (the energy ratio standard deviation) and ERSD (the Energy Ratio Standard Deviation) are defined as e damage index ersd � 0, when the erd(ω) � 0. us, the retaining wall is not damaged. e damage index ersd＞0, when the erd(ω) ≠ 0. us, the wall is damaged. e value of ersd becomes larger and larger, when the fluctuation intensity of erd(ω) becomes larger and larger. With the increase of the holes number, the values of the damage indexes become larger and larger. us, the either ersd or ERSD can be used to detect the damage state of the retaining wall quantitatively. In a structural system, the variation of structural stiffness caused by damage will give rise to the variations of structural responses, which will cause the variations of IRF and VIRF. Similarly, ersd and ERSD, which are based on IRF and VIRF, respectively, also change. Especially, the stiffness in partial damage changes the most. Similarly, the values of ersd and ERSD in partial damage also change the most. us, the partial damage location can be detected by variation characteristics of ersd and ERSD.
In addition, the damage intensity of partial damage within walls can be identified by establishing a quantitative relationship between the damage intensity and identification index. e damage intensity can be calculated reversely via ersd or ERSD, if the quantitative relationship between the damage intensity and damage index is known. Consequently, the damage (the damage state, damage location, and damage intensity) of the retaining wall can be detected via ersd and ERSD.

Comparison of ERSD and ERSD.
IRF is related to both the external excitation and structural responses. To calculate IRF, both the excitation and responses must be available. However, VIRF is only related to structural responses. VIRF can be calculated when virtual excitation and virtual response are available. To detect the damage state of the wall, the damage index ersd, which is based on IRF, requires that the external excitation and one structural response are available at least. e damage index ERSD, which is based on VIRF, requires that two structural responses are available at least. To diagnose the damage location, ersd requires m measuring points, but ERSD requires (m + 1) points, where m is the number of points. To acquire IRF and VIRF, the required sensors and signal transmission system are similar. e required excitation equipment for IRF must be the one that can record the excitation signal. In contrast, the excitation equipment for VIRF is just a common hammer. And, the excitation equipment for VIRF is not even required, when the excitations are such ambient ones as wind loads, vehicle loads. us, the equipment for ERSD are much simpler than the ones for ersd.

Tests Analyses
To identify the damage within the retaining wall via ersd and ERSD, vibration tests on a pile plate retaining wall (concrete wall) are performed, as shown in Figure 1. e geometrical parameters of the wall are listed in Table 2. Backfill behind the wall is sand, and soil in front of the wall is miscellaneous fill. Material parameters of the wall and backfill are shown in Table 3.
As mentioned above, damages may occur and accumulate within the retaining wall due to many causes. In the light of literature [29,30], the micro-cracks or holes are common damages in the structure. To simulate the actual damage within the retaining wall, holes are drilled in the wall, as shown in Figure 2. e parameters of the holes are listed in Table 4, where L is the length of the wall and H is the height of the wall.
No holes are drilled in the wall, when the value of damage intensity is zero. With the increase of the number of holes, the value of damage intensity increases continuously. To describe the partial damage intensity, DI (Damage Intensity) is defined as where N is the number of holes, V h is the volume of a single hole, and V is the total volume of the partial area (V � 0.25 m × 0.25 m × 0.1 m). Generally speaking, equipment for model tests should include exciting hammers, sensors, and a signal transmission system. In this paper, the exciting hammer is a DFC-2 hammer, which can record hammer impulse signals automatically, as shown in Figure 3(a). e sensors are 941B accelerometers ( e frequency range is 0.17-100 Hz, the sensitivity is 0.3 (V·s 2 /m). And, these sensors are fixed on the wall by binder, as shown in Figure 3(b). e signals recorded via accelerometers are collected by the signal collecting system, the JM3863A wireless vibration test system, as shown in Figure 3(c). e signals collected by the JM3863A are transmitted to a computer by the JM1802 gateway, as shown in Figure 3(d).
To localize the damage, 26 accelerometers are fixed on the wall, as shown in Figure 4. e IRF can be acquired via the excitation signal of DFC-2 hammer and response signals of point 1 to point 25. In the light of literature [22,23], the point with less response is selected as virtual excitation point, and other points are virtual responses points. Here, point 26, whose response is much less, is selected as the reference point. And, points 1 to 25 are calculating points. e VIRF can be acquired via the virtual excitation and virtual responses. e locations of hammer excitation are the points from I to VI. e locations of measuring points and  Shock and Vibration 7 excitation points are shown in Figure 4. In the light of literature [29], under the effects of hammer excitation (low strain excitation), both the wall and filled soil vibrate mildly without considering the additional dynamic effects of the filled soil on the wall. Here, eight damage cases are considered to identify the damage within the retaining wall, as listed in Table 5       e time of IRF curves is very short, because the duration of hammer excitation is about 0.001 s. us, the IRF curves within 0.06 s are extracted to identify damage, as shown in Figure 5. And, the effective duration of VIRF curves is within 6 s after signal processing, as shown in Figure 5. Obviously, the time history curves of IRF nearly overlap under different damage cases. Similarly, the curves of VIRF also nearly overlap. us, it is difficult to extract damage information via the IRF curves and VIRF ones.
To detect the damage state of the wall, the Hilbert-Huang Transforms of IRF and VIRF are done. In the light of literature [31], the optimum value of EMD order is less than ten, so the value of EMD (the Empirical Mode Decomposition) order is ten. us, ten intrinsic mode functions can be acquired via EMDs of IRF and VIRF, respectively. And, ten energy ratios of sub-bands can be obtained via HT (the Hilbert Transform) of the intrinsic mode functions. e Hilbert marginal energy ratio spectrums of IRF and VIRF are shown in Figure 6. Due to the influence of the retaining wall with lower frequencies, the majority of sub-bands' energies are distributed over the minority of bands with lower frequencies. e total energy of the first five bands with lower frequencies is almost more than 90% of total energy, as shown in Figure 6. e energy ratio of the same band changes, when retaining wall changes from an undamaged state to damaged one. e energy ratio variations caused by damage are nearly undetectable in such bands with higher frequencies as 7 th band to 10 th one. us, it is unnecessary to use all of the bands to identify damage. Only these bands, which are more sensitive to damage, are more useful for identifying damage, such as 1 st band to 6 th one of IRF and 1 st band to 5 th one of VIRF are much more sensitive to damage. In contrast, 7 th band to 10 th one of IRF and 6 th band to 10 th one of VIRF are not sensitive to damage, as is shown in Figure 6 (point 12). ese bands, which are not sensitive to damage, are merged into the residual band. e values of the residual bands are listed in Table 6. Obviously, the values of the majority of residual bands are small. Nevertheless, the values of residual bands of IRF to point 1 are much larger; this is caused by the larger threshold value ε 0 . e number of feature bands is determined by threshold value ε 0 . e number of bands will become small, if the value of ε 0 is big. On the contrary, the number of bands will become large, if the value of ε 0 is small. Some bands, which are sensitive to damage, may be merged into the residual band due to the large ε 0 . is may result in damage information being missed. On the contrary, some bands, which are much less sensitive to damage, may be added into feature bands due to small ε 0 .
is may decrease the effect of feature bands on damage identification. Because the damage sensitivities to different points are different, the value of ε 0 is variable for different points, as shown in Figure 7. Generally speaking, the values of ε 0 should be smaller for those points whose damage sensitivities are much poorer. And, the values of ε 0 should be larger for those points whose damage sensitivities are much stronger. us, the threshold value ε 0 plays a prominent role in extracting feature bands. en, the Hilbert feature vector, which is sensitive to damage, is created via the threshold ε 0 to replace the Hilbert marginal energy ratio spectrum, as shown in Figure 7. Although the value of the band energy is the largest, the damage sensitivity of this band is not necessarily the strongest, as shown in Figure 7 (the energy of 3 rd band is the largest, but its damage sensitivity is not the strongest, as shown in Figures 7(c) and 7(d)).
Based on the Hilbert feature vector, the Hilbert damage feature vector is created, as shown in Figure 8. As mentioned above, the wall is undamaged when damage feature vector is a zero vector. e wall is damaged when damage feature vector is a nonzero vector. Obviously, all the damage feature vectors are nonzero vectors, when holes are drilled in the wall. us, the damage state of the pile plate retaining wall can be sensitively distinguished via damage feature vectors which are based on IRF or VIRF.
Under the influences of the excitation location and damage location, the damage sensitivities of different measuring points are different. For instance, the damage sensitivity of point 7 which is far away from the partial damage is much poorer, while the one of point 12 which is located in damage location is much stronger.

Diagnosing the Damage Location of the Wall.
As mentioned above, the damage indexes which are based on IRF or VIRF changes a lot in the location of partial damage. us, the damage location of the wall can be diagnosed via variation characteristics of the damage index. In the light of (21), ersd and ERSD are calculated under the different damage cases, as listed in Table 7. Firstly, the damage index values of these 25 measuring points are calculated under different damage cases. en, a damage index trend surface can be formed by these 25 damage index values via MATLAB program under a certain damage case. e damage location of the wall can be diagnosed via the variation properties of the trend surface, as shown in Figure 9.

Shock and Vibration
Obviously, the peak coordinate of the trend surface is located in the center of the scope {L ∈ (1.375 m, 1.625 m), H ∈ (0.875 m, 1.125 m)}, where the partial damage is simulated. us, the damage location of the wall can be diagnosed via the peak coordinate of the damage index trend surface. In comparison with ersd, the peak characteristics of ERSD trend surface caused by partial damage are much better than the ones of ersd trend surface. Especially, the damage location can be detected more effectively and accurately via ERSD trend surface, when DI is small (Figure 9(a)). What calls for special attention is that the values of the damage indexes are nonnegative. e negative values in trend surface are caused by the B-spline technique.

Identifying the Damage Intensity of the Wall.
As mentioned above, DI can be calculated reversely via the damage index, when the quantitative relationship between the damage intensity and damage index is known. Here, DI is the one of partial damages (measuring point 12) that is diagnosed via the damage index trend surface. To establish the quantitative relationship between the damage intensity and damage index, the quantitative relationship between damage index and DI of point 12 is fitted by polynomial fitting. Firstly, the mean value of the damage index of point 12 is calculated under the same damage intensity at different exciting locations. en, the quantitative relationship between the damage index and damage intensity of point 12 can be acquired. e polynomial coefficient will become much larger, if the relationship between the damage index and DI which is much less than the value of damage index is fitted directly. us, the quantitative relationship between di (the relative damage intensity) and damage index is fitted. Finally, the quantitative relationship between damage index and di (di � DI × 100) can be expressed as:     Residual band Y � −0.24x 5 + 2.57x 4 − 9.28x 3 + 12.89x 2 − 0.88x + 0.14, where Y is the relative damage intensity di, x is the mean value of ersd for three tests at different excitation points, and z is the mean value of ERSD for three tests at different excitation points, too. (23) and (24) can be expressed as two data-fitting curves. e data-fitting curves and test curves are shown in Figure 10(a). It is not difficult to verify whether the test curves and fitting curves are nearly overlapped. It is believed that (23) and (24) have a better fitting degree to the quantitative relationship between di and the damage index. Once the damage index is known, di can be calculated reversely.
us, (23) and (24) can be used to identify the damage intensity of partial damage within the retaining wall. For instance, di � 5.2, when ersd � 1.
us, DI � di/100 � 4.27%. us, the partial damage intensity of the wall can be identified via this quantitative relationship.
Under the same case, the values of ersd and ERSD are close, when DI is much less (early damage). e value of ERSD becomes much larger than the one of ersd when DI is large, as shown in Figure 10(a). It is predicted that the gap between ERSD and ersd will become larger and larger with continuous increase of DI. us, the fluctuation intensity of ERD (ω) is much larger than the one of er d(ω). To some extent, the damage sensitivity of ERSD is better than that of ersd. e location of excitation point has certain influence on the damage indexes. e values of damage index will fluctuate when the excitation location changes under the same damage intensity, as shown in Figures 10(b) and 10(c), where the ersd and ERSD are mean values of damage indexes at each excitation point for three vibration tests. To describe this fluctuation intensity of the damage indexes, the variance analyses of the damage indexes are performed under each damage intensity. e variance curves of ersd and ERSD are plotted, as shown in Figures 10(b) and 10(c). With variations of excitation position, the fluctuation intensity of ERSD is lower, but the fluctuation intensity of ersd is much higher. Consequently, the location of excitation point has larger influence on the ersd, but has smaller influence on the ERSD. us, the stability of ERSD is better than that of ersd.
eoretically speaking, both IRF and VIRF are the inherent attributes to retaining wall structures. Both ersd and ERSD, which are based on IRF and VIRF, respectively, should be valid for damages' diagnosis for the retaining wall. However, the validity of ERSD is better than that of ersd.
is outcome may be caused by the following factors. (1) e frequency ranges of IRF and VIRF, and the frequency ranges of IRF, are determined by the external excitation and structural responses, and the frequency ranges of VIRF are determined by the structural responses. Especially, the frequency ranges of IRF may be smaller, if the frequency range of the external excitation is not matching well with those of the responses. Due to this, the value of ersd may become smaller. In comparison with IRF, changes of the frequency range of VIRF are smaller, because the frequency ranges of the virtual excitation and virtual response are close. So, the value of the ERSD may be larger. (2) Mode aliasing. e mode aliasing phenomena is caused by such factors as EMD algorithm, frequency components of the original signal, the amplitude of the original signal, and sampling frequency of the original signal. In this paper, the sampling frequency of the original signal and algorithm for EMD are constant. e mode aliasing may be mainly caused by the frequency components and the amplitude of the original signal. Due to the mode aliasing, the IMF component may contain two or more frequency components. e damage sensitivity of the IMF component becomes poor due to the mode aliasing, and the value of the damage index may become smaller. us, the mode aliasing should be eliminated to improve the damage sensitivities of IMF components. (3) e order of EMD. e number of bands will become smaller and the value of damage index will become larger, if the order of EMD is smaller. However, the damage information may be lost. e number of bands will become larger, if the order of EMD is larger. With an increase of the band numbers, the number of feature bands may become larger too. is may make the value of the damage index smaller.
us, only the order of EMD is optimum, the number of the feature bands is the most appropriate. (4) e value of ε 0 . e ε 0 has bigger influences on the value of damage index. e number of feature bands is determined by the ε 0 . e number of feature bands will become larger, if the value of ε 0 is smaller. e value of damage index may become smaller due to the smaller ε 0 . e number of feature bands will become smaller, if the value of ε 0 is larger. Although the value of damage index may become larger, the damage information may be lost. us, the gap between ersd  and ERSD may be related to the ε 0 . (5) e coherence of data. e mutual interference of structural responses exists in the structural system. Either IRF or VIRF is affected by the mutual interference of responses. e value of the damage index is also affected by the coherence of data.

Conclusions
In this paper, two damage indexes ersd and ERSD are proposed via the Hilbert-Huang Transforms of IRF and VIRF. Damages within the pile plate retaining wall are identified via ersd and ERSD. e damage state of the wall can be detected sensitively, the damage location can be diagnosed validly, and the damage intensity can also be identified quantitatively via ersd and ERSD. Similarly, ersd and ERSD can also be used to identify the damages within other types of retaining walls (such as gravity retaining walls, cantilever retaining walls). Although only the single damage within the wall is diagnosed via ersd and ERSD, the multidamages within the wall can also be diagnosed similarly. In addition, ERSD has much better stability, but ersd has much poorer stability. Besides, the damage sensitivity of ERSD is better than that of ersd.
us, the outcomes of damage identification of ERSD are much better than the ones of ersd.
Due to the difference of damage sensitivity to different measuring points and the requirements for locating damage, enough measuring points should be fixed on the wall to diagnose damages sensitively and validly. In addition, the location of partial damages can be precisely identified when sensors are very close to partial damages. However, the damage location is unknown in practice. us, it is required that sensor locations should be adjusted repeatedly to diagnose the location of partial damage precisely. In addition, the identification of damage intensity is related to tests data. e tests data are much more, and the identification accuracy of damage intensity is much better. us, the precise identification of damage intensity requires the collection of a great deal of tests data. Besides, the threshold value ε 0 has great influence on extracting feature bands. us, the investigation of ε 0 should be performed to select reasonable feature bands in the future.
In the literature [29], a damage identification method for retaining walls is proposed based on the Wavelet Packet frequency band energy spectrum. e feature bands, which are used to detect damage, are the ones with much larger energy. In the literature [24], a damage diagnosis method for retaining walls is also proposed based on the Hilbert energy spectrum. e feature bands are the ones with much larger energy too. In these two methods, it is thought that the bands with larger energy are useful for damage identification without considering damage sensitivity of bands. However, the damage sensitivities of those bands with larger energy may be not strong, and the sensitivities of those bands with smaller energy may not be poor. In this paper, the feature bands are determined by the damage sensitivities of bands. ose bands with stronger damage sensitivities are extracted to identify damage. It may be much more reasonable to select feature bands via damage sensitivity of bands.
Due to some technology issue, the Hilbert-Huang Transforms of IRF and VIRF were performed without considering aliasing and leakage phenomena. And, the analysis of coherence in experimental data is also ignored. However, the aliasing, leakage phenomena, and coherence in data should be performed to get more precise results. In addition, all the order of EMD is ten in this paper. However, the optimum order of EMD to IRF or VIRF may be variable.
us, the optimum order of EMD needs to be investigated. Besides, the value of ε 0 is also variable. To select the valid feature bands, the investigation of the optimum value of ε 0 needs to be done. us, these works should be considered in the future.
Data Availability e data used in this paper are available within this paper. Based on the tables and figures in this paper, the readers can verify the results of this paper, replicate the analysis, and conduct secondary analyses.

Conflicts of Interest
e author of this paper does not have any conflicts of interest regarding the publication of this paper.