IMF-Slices for GPR Data Processing Using

remote sensing Article

IMF-Slices for GPR Data Processing Using Variational Mode Decomposition Method Xuebing Zhang 1 1 2 3

*

ID

, Enhedelihai Nilot 2 , Xuan Feng 2, *, Qianci Ren 3 and Zhijia Zhang 1

School of Geomatics and Prospecting Engineering, Jilin Jianzhu University, Changchun 130118, China; [email protected] (X.Z.); [email protected] (Z.Z.) College of Geo-Exploration Science and Technology, Jilin University, Changchun 130026, China; [email protected] College of Earth Sciences, Guilin University of Technology, Guilin 541006, China; [email protected] Correspondence: [email protected]; Tel.: +86-134-0435-3645

Received: 14 February 2018; Accepted: 15 March 2018; Published: 19 March 2018

Abstract: Using traditional time-frequency analysis methods, it is possible to delineate the time-frequency structures of ground-penetrating radar (GPR) data. A series of applications based on time-frequency analysis were proposed for the GPR data processing and imaging. With respect to signal processing, GPR data are typically non-stationary, which limits the applications of these methods moving forward. Empirical mode decomposition (EMD) provides alternative solutions with a fresh perspective. With EMD, GPR data are decomposed into a set of sub-components, i.e., the intrinsic mode functions (IMFs). However, the mode-mixing effect may also bring some negatives. To utilize the IMFs’ benefits, and avoid the negatives of the EMD, we introduce a new decomposition scheme termed variational mode decomposition (VMD) for GPR data processing for imaging. Based on the decomposition results of the VMD, we propose a new method which we refer as “the IMF-slice”. In the proposed method, the IMFs are generated by the VMD trace by trace, and then each IMF is sorted and recorded into different profiles (i.e., the IMF-slices) according to its center frequency. Using IMF-slices, the GPR data can be divided into several IMF-slices, each of which delineates a main vibration mode, and some subsurface layers and geophysical events can be identified more clearly. The effectiveness of the proposed method is tested using synthetic benchmark signals, laboratory data and the field dataset. Keywords: variational mode decomposition; empirical mode decomposition; IMF-slices; GPR data processing; GPR imaging; time-frequency analysis

1. Introduction Most of the geophysical data are non-stationary. Especially for the ground-penetrating radar (GPR) data, which can be treated as a set of time-series signals, such non-stationarity appears more obviously. Non-stationarity can be defined and measured by certain statistical methods [1], however, geophysicists are prone to using the time-frequency decomposition methods (e.g., wavelet transform [2,3], S-transform [4,5] and matching pursuit [6,7]) to delineate the non-stationary characteristics. This is because these methods provide a visualization and are convenient for further applications. Recently, empirical mode decomposition (EMD) has been providing alternative solutions with a fresh perspective. With EMD-based methods, the GPR data can be decomposed into a group of stationary sub-components (i.e., intrinsic mode functions, IMFs), and the subsequent processing for subsurface information is thus focused on the analysis of the generated IMFs instead [8]. The decomposed IMFs of GPR data highlight two important features. The first is that the IMFs are close to being stationary. This leads to a direct application of time-frequency analysis of GPR data [9], in which time-frequency structures can be computed steadily within each IMF. The other Remote Sens. 2018, 10, 476; doi:10.3390/rs10030476

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feature is that the decomposed IMFs correspond to different frequency bands or ranges. This means that any operation performed on the IMFs (e.g., removing or extracting a few IMFs) amounts to filtering in the corresponding frequency domain. In this regard, the derived applications based on IMFs may include denoising [10], subsurface objects identification [11], etc. However, the mode-mixing effect and the lack of a mathematical foundation limit their further applications in GPR data processing and imaging. To utilize the benefits of the IMFs mentioned above, and avoid the negatives of the EMD, we introduce a new decomposition scheme called variational mode decomposition (VMD) [12] for GPR data processing as well as for imaging. The VMD decomposition is not “empirical” any more, while the mathematical framework is well-established. Specifically, though the decomposition results can still be seen as IMFs, the computation is implemented by solving an optimization problem. Some references have demonstrated that the VMD is able to avoid mode-mixing phenomena [13–15], and therefore the decomposition results match the input signal’s intrinsic vibration mode better. In this study, we test VMD decomposition on GPR data, and based on the VMD decomposition results, we propose a new method for GPR imaging which we refer as “the IMF-slice”. In the proposed method, the IMFs are generated by the VMD trace by trace, and then each IMF is sorted and recorded into different profiles (i.e., the IMF-slice) according to its center frequency. A GPR profile could be divided into several IMF-slices, each of which delineates a main vibration mode within certain frequency band. Using IMF-slices, some subsurface events are able to be identified more clearly. As parts of the long-term study of the mathematical transformations of GPR data, this work shows elementary research on the applications of the VMD decomposition for GPR imaging. The objectives of this work are to: (1) introduce the VMD decomposition and test its feasibility on GPR data; (2) compare the decomposition results of GPR data between the VMD and the classic EMD; (3) propose the concept of IMF-slices and discuss their applications in GPR imaging and interpretation. The paper starts with a methods section introducing the VMD decomposition and the IMF-slices for GPR data. We then show some results based on synthetic benchmark tests, laboratory data tests and a field dataset. Finally, the conclusions and discussions are provided in the last section. 2. Methods 2.1. The Variational Mode Decomposition The VMD algorithm was initially described as solving the following constrained optimization subject to equality constraints by Dragomiretskiy and Zosso [12],  min

{ u k },{ w k }

∑ k∂t

h

δ(t) +

k

j πt



∗ uk (t)

i

e− jωk t k

2



2

(1)

s.t. ∑ uk (t) = f (t) k

In the above form, δ(t) is the Dirac function, and {uk (t)}k=1,2,...N denotes the decomposed sub-components (i.e., the IMFs) embedded in the input signal f (t), where k identifies the number of IMFs. {ωk }k=1,2,...N denotes the corresponding center frequencies of each IMF. Within the Lagangian-multiplier framework, the penalty-term and the Lagangian-multiplier λ are introduced. Then the above optimization problem would be turned to solving the defined Lagrange function [16] as follows, L({uk }, {ωk }, λ) h = α∑ k∂t δ(t) + k

j πt



  i 2 2 ∗ uk (t) e− jωk t k + k f (t) − ∑ uk (t)k + λ(t), f (t) − ∑ uk (t) 2

k

2

k

(2)

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where L({uk }, {ωk }, λ) denotes the Lagrange function, and the parameter α is used for balancing the reconstruction error. Its solution can be derived by the well-known alternating direction method of multipliers (ADMM) [17]. Assuming that there are K IMFs decomposed, the frequency-domain expression of the kth IMF derived by the VMD is, k −1

uˆ nk +1 (ω ) =

K

fˆ(ω ) − ∑ uˆ in+1 (ω ) − ∑ uˆ in (ω ) + i =1

1 + 2α(ω

i = k +1 − ωkn )2

λˆ n (ω ) 2

(3)

in which fˆ(ω ), uˆ i (ω ), λˆ (ω ), and uˆ nk +1 (ω ) are the frequency-domain forms of f (t), ui (t), λ(t), unk +1 (t) using the Fourier transform, and n denotes iteration number. The criterion of iteration termination is 2 (4) ∑ kunk +1 (t) − unk (t)k2 /kunk (t)k22 ≤ ε, k

where ε denotes the tolerance of convergence. Based on the decomposed IMFs, the input signal can be represented by the decomposed IMFs and the residual signal r (t), K

f (t) =

∑ uin+1 (t) + r(t).

(5)

i =1

It should be noted that, though the final expansion form of the VMD decomposition (i.e., Equation (5)) is similar to that of the EMD, the computation of each IMF in VMD is totally different. Since the detailed deduction and comparisons with the EMD algorithm are not the main purposes of this paper, we will show the decomposition results in the Section 3. 2.2. IMF-Slices of GPR Data Compared with traditional common-frequency slices, the IMF-slices can be treated as a set of IMFs. In the VMD scheme, the center frequency ωk (t) of each decomposed IMF can be obtained simultaneously. Given a GPR profile (i.e., a B-scan), the IMFs can be derived by VMD trace by trace, where the “trace” here means the A-scan. If we pre-define the frequency intervals or ranges for the IMF-slices, the IMFs of each trace can be classified into different IMF-slices. It should be noted that the frequency ranges for each IMF-slice can be estimated empirically using the center frequencies obtained for each IMF of the first five or ten traces. For example, if the frequency range of a IMF-slice is defined as [ω1 , ω2 ], and combining the median filtering, each decomposed IMF of one trace are evaluated whether it belongs to this range. If yes, it is recorded in the corresponding trace of the IMF-slice, which can be represented as, K

IMF-slice( x, t) =

∑ θ (M(ωi (t)))ui (t)

(6)

i =1

where M denotes the median filtering operator, and the θ is the Heaviside function, ( θ (ω ) =

1, for ω ∈ [ω1 , ω2 ], 0, else.

(7)

Equation (6) appears to be a band-pass filter for the IMFs. A schematic diagram based on the VMD decomposition is shown in Figure 1.

1, for   1 , 2  , 0, else.

    

(7)

Equation (6) appears to be a band-pass filter for the IMFs. A schematic diagram based on the 4 of 13 VMD decomposition is shown in Figure 1.

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Figure Figure 1. 1. Schematic diagram of IMF-slices. IMF-slices.

Similarly, the the IMF-slices IMF-slices can can be be also also generated generated based based on on EMD. EMD. However, However, since since the the center center Similarly, frequencies for each IMF, accompanied by the VMD decomposition results, cannot be obtained in the the frequencies for each IMF, accompanied by the VMD decomposition results, cannot be obtained in EMD scheme, scheme, we we select select the the Taner’s Taner’s instantaneous instantaneous frequency frequency[18] [18]to toreplace replaceit, it, EMD IMF dfI MF  t  − f * ∗ (df t 1 f I MF I MF dt f (tt) IMF  f IMF  t  t)IMF dt , ω (t) = 1 IMF   dt dt 2 ∗ 2  t   , 2π f I MF 2(t) + f I MF (t) 2 f IMF  t   f IMF *2  t 

df

*∗ ( t )

df

(t)

(8) (8)

where f I MF (t) and f I MF ∗ (*t) here denotes the IMF and its conjugate form. where f IMF  t  and f IMF  t  here denotes the IMF and its conjugate form. 3. Results 3. Results 3.1. Synthetic Benchmark Tests 3.1. Synthetic Benchmark Tests test a synthetic benchmark signal formulated by Herrera et al. [19,20], In this section, we firstly which characterized as test a non-stationary and multi-component signal. signalet(Figure 2a) Incan thisbe section, we firstly a synthetic benchmark signal formulated byThe Herrera al. [19,20], is comprised of four sub-components, which are relatively stationary certain ranges, which can be characterized as a non-stationary and multi-component signal.in The signal time (Figure 2a) is see Figure 2b–e. Figure 3 shows the corresponding instantaneous frequencies (IFs).ranges, see Figure comprised of four sub-components, which are relatively stationary in certain time benchmark can be decomposed into afrequencies set of IMFs(IFs). by using both EMD and VMD, 2b–e.The Figure 3 shows signal the corresponding instantaneous as shown in Figures 4 and 5, respectively. It is obvious that the decomposition results using VMD The benchmark signal can beusing decomposed into a set of IMFs by usingsimilar both EMD and VMD, as are sparser than those obtained the EMD method. Additionally, vibration modes shownsimilar in Figures 4 and 5, respectively. It isalways obvious that the decomposition using is VMD are (with time-frequency structures) are decomposed into differentresults IMFs, which known sparser than those obtained using the EMD method. Additionally, similar vibration modes (with as the mode-mixing effect). For example, the third sub-component signal (Figure 2d) is mainly located between 6–10 s and its IF is around 10 Hz. However, the EMD scheme separates them and records them in IMF1 and IMF2, in Figure 4. By contrast, the IMFs derived by the VMD scheme match the benchmark better (see Figure 5). More specifically, in Figure 5, IMF1 refers to Figure 2c, IMF2 refers to Figure 2e, and IMF3 refers to both Figure 2b,d. We remark here that the correspondence can be judged not only by the proximity of frequency ranges, but also by the proximity of amplitude. Although there still exist some “mode-mixing” phenomena in the decomposition results by VMD, this defect has been extremely reduced compared with EMD. For example in Figure 5, there are some vibrations with low amplitude or energy remaining at 6~10 s of IMF1, but the sub-component signals (Figure 2d) are mainly reflected in the IMF3. From the above benchmark tests, we observe that the

judged not only by the proximity of frequency ranges, but also by the proximity of amplitude. Although there still exist some “mode-mixing” phenomena in the decomposition results by VMD, this defect has been extremely reduced compared with EMD. For example in Figure 5, there are some vibrations with low amplitude or energy remaining at 6~10 s of IMF1, but the 5 subRemote Sens. 2018, 10, 476 of 13 component signals (Figure 2d) are mainly reflected in the IMF3. From the above benchmark tests, we observe that the IMFs by VMD are able to reveal the intrinsic properties of the non-stationary signal, IMFs by VMD able to reveal intrinsic properties of the non-stationary signal, whereasby EMD may whereas EMDare may mislead us.the Thus, we can draw the conclusion that decomposition VMD is mislead us. Thus, we can draw the conclusion that decomposition by VMD is always more reasonable. always more reasonable.

synthetic data (a) composed of four sub-components (b–e). 2. The Remote Sens. 2018, 10, Figure x FOR PEER REVIEW

Figure 3. 3. The The corresponding corresponding instantaneous instantaneous frequencies frequencies of of the the sub-components sub-components in in Figure Figure 1. 1. Figure

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Figure 3. The corresponding instantaneous frequencies of the sub-components in Figure 1. Remote Sens. 2018, 10, 476

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Figure 4. The EMD decomposition results and the residual.

Figure 4. The EMD decomposition results and the residual.

Figure Figure5.5. The The VMD VMD decomposition decomposition results results and and the the residual. residual. IMF1 IMF1 and and IMF2 IMF2 refer refer to to Figure Figure 2c,e, 2c,e, respectively, respectively,and andIMF3 IMF3refers referstotoboth bothFigure Figure2b,d. 2b,d.

Thecorresponding correspondingIFsIFs can computed by Taner’s equation the IMFs decomposed by The can be be computed by Taner’s equation usingusing the IMFs decomposed by EMD EMD andThen VMD. the instantaneous can beasconstructed as shown in Figure with 6a,b. and VMD. theThen instantaneous spectra canspectra be constructed shown in Figure 6a,b. Compared Compared with the time-frequency structures in Figure 3, the instantaneous spectrum by VMD the time-frequency structures in Figure 3, the instantaneous spectrum by VMD is relatively moreis relativelyThis more Thisofisthe also because of the fact that VMDisdecomposition is more accurate. is accurate. also because fact that VMD decomposition more reasonable. Thesereasonable. synthetic These synthetic tests illustrate how the VMDcan decomposition can conducted ondata, nonbenchmark tests benchmark illustrate how the VMD decomposition be conducted onbe non-stationary stationary data, avoiding the mode-mixing phenomenon. avoiding the mode-mixing phenomenon.

The corresponding IFs can be computed by Taner’s equation using the IMFs decomposed by EMD and VMD. Then the instantaneous spectra can be constructed as shown in Figure 6a,b. Compared with the time-frequency structures in Figure 3, the instantaneous spectrum by VMD is relatively more accurate. This is also because of the fact that VMD decomposition is more reasonable. synthetic RemoteThese Sens. 2018, 10, 476benchmark tests illustrate how the VMD decomposition can be conducted on non-7 of 13 stationary data, avoiding the mode-mixing phenomenon.

Figure 6. The instantaneousspectrum spectrum based and (b)(b) VMD. Figure 6. The instantaneous basedon on(a)(a)EMD EMD and VMD.

3.2. Laboratory Data Tests

3.2. Laboratory Data Tests

In order to test the VMD method for GPR data decomposition, we employ the laboratory data

In orderintoFigure test the VMD method forThe GPR data wetrough employ the laboratory (shown 7) for further analysis. data wasdecomposition, collected in a sand as shown in Figure data (shown in Figure 7) for further analysis. The data was collected in a sand trough as shown in Figure 8a, 8a, where the target body (Figure 8b) was set at a certain depth. whereRemote the target body (Figure 8b) was set at a certain depth. Sens. 2018, 10, x FOR PEER REVIEW 8 of 14 Remote Sens. 2018, 10, x FOR PEER REVIEW

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Figure 7.7.The datacollected. collected. Figure Thelaboratory laboratory data Figure 7. The laboratory data collected.

Figure 8. The sand trough (a) and the target body (b). Figure 8. The sand trough (a) and the target body (b).

Figure 8. The sand trough (a) and the target body (b). Typically, each trace of the GPR data is non-stationary. We apply the VMD method to Typically, each trace of the GPR data is non-stationary. We apply the VMD method to decompose the trace GPR data trace by trace.isThen we sort the main components and collect to them into Typically, of the GPR non-stationary. the VMD decompose decomposeeach the GPR data trace bydata trace. Then we sort the We mainapply components andmethod collect them into several profiles (i.e., the IMF-slices), see Figure 9a–d. For comparison, we adopt a similar way of several (i.e., the IMF-slices), seethe Figure For comparison, we adopt similar wayprofiles of the GPR dataprofiles trace by trace. Then we sort main9a–d. components and collect themainto several processing the decomposed IMFs using EMD decomposition, and the generated IMF-slices are processing the decomposed decomposition, and athe generated are the (i.e., the IMF-slices), see FigureIMFs 9a–d.using For EMD comparison, we adopt similar way IMF-slices of processing shown in Figure 10. In Figure 9, the GPR data are divided into four main components, each of which shown inIMFs Figure 10. In EMD Figuredecomposition, 9, the GPR data are divided into four main components, each of which decomposed and the of generated are shown Figure 10. demonstrates a using different vibration mode. The reflection the buriedIMF-slices target is identified in theinsecond demonstrates a different vibration mode. The reflection of the buried target is identified in the second In Figure 9, the data arepointing dividedto into main each in ofFigure which9d, demonstrates IMF-slice by GPR the red arrow it (seefour Figure 9b).components, The noise is shown without IMF-slice by the red arrow pointing to it (see Figure 9b). The noise is shown in Figure 9d, without a different vibrationinformation mode. Thefrom reflection of the buried targetmixed. is identified in 10, thethe second IMF-slice by any important the subsurface layers being In Figure first IMF-slice any important information from the subsurface layers being mixed. In Figure 10, the first IMF-slice is meaningless, and the yellow arrows illustrate the so-called mode-mixing effect. Additionally, the is meaningless, and the yellow arrows illustrate the so-called mode-mixing effect. Additionally, the reflection indicated by the red arrow in Figure 10b is not as distinct as that achieved using VMD. reflection indicated by the red arrow in Figure 10b is not as distinct as that achieved using VMD.

Figure 8. The sand trough (a) and the target body (b).

Typically, each trace of the GPR data is non-stationary. We apply the VMD method to decompose the GPR data trace by trace. Then we sort the main components and collect them into several profiles (i.e., the IMF-slices), see Figure 9a–d. For comparison, we adopt a similar way of Remote Sens. 2018,the 10, 476 8 of 13 processing decomposed IMFs using EMD decomposition, and the generated IMF-slices are shown in Figure 10. In Figure 9, the GPR data are divided into four main components, each of which demonstrates a different vibration mode. The reflection of the buried target is identified in the second theIMF-slice red arrow to it (see Figureto9b). TheFigure noise 9b). is shown in Figure 9d, without bypointing the red arrow pointing it (see The noise is shown in Figureany 9d, important without information from the subsurface layers being mixed. In Figure 10, the first IMF-slice is any important information from the subsurface layers being mixed. In Figure 10, the firstmeaningless, IMF-slice andisthe yellow arrows illustrate so-called mode-mixing effect. Additionally, the reflection indicated meaningless, and the yellowthe arrows illustrate the so-called mode-mixing effect. Additionally, the by reflection the red arrow in Figure 10b is not as distinct as that achieved using VMD. indicated by the red arrow in Figure 10b is not as distinct as that achieved using VMD.

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Figure 9. The derivedIMF-slices IMF-slices using VMD. VMD. (a–d) (a–d) show the IMF-slice 1–4 respectively. Figure 9. 9. The showthe theIMF-slice IMF-slice1–4 1–4respectively. respectively. Figure Thederived derived IMF-slicesusing using VMD. (a–d) show

Figure 10. The derived IMF-slices using EMD. (a–d) show the IMF-slice 1–4 respectively. Figure 10.10.The (a–d) show showthe theIMF-slice IMF-slice1–4 1–4respectively. respectively. Figure Thederived derivedIMF-slices IMF-slicesusing using EMD. EMD. (a–d)

4. The Use of VMD with Field Dataset 4. The Use of VMD with Field Dataset In this section, we use variational mode decomposition to process a field dataset and show the In this section, we use variational mode decomposition to process a field dataset and show the results of the proposed IMF-slices. Subsequently, the corresponding interpretations are drawn. results of the proposed IMF-slices. Subsequently, the corresponding interpretations are drawn. The dataset is shown in Figure 11. It was collected in Yan’an, in the Shanxi Province of China. The dataset is shown in Figure 11. It was collected in Yan’an, in the Shanxi Province of China.

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so there exists discontinuity of neighboring traces in gray scale. The second and third IMF-slices (Figure 15b–c) demonstrate the subsurface structures at different scales. In the last IMF-slice, the noise 4. Use of VMD with Field Dataset of The the dataset is extracted, especially in the deep part of the profile.

Remote Sens. 2018, 10,use x FORvariational PEER REVIEW mode decomposition to process a field dataset10and of 14 show the In section, In this Figure 16, wewe also provide comparisons with the IMF-slices based on EMD. In some locations results of the proposed IMF-slices. Subsequently, the corresponding interpretations are drawn. of these IMF-slices, the interpretations are greatly influenced by the mode-mixing of IMFs for so there exists discontinuity of neighboring traces in gray scale. The second and third IMF-slices The dataset is shown in Figure 11. It was in Yan’an, inthe the Province China. different traces. For instance, inthe Figure 16d thecollected important information inShanxi the shallow partofand the (Figure 15b–c) demonstrate subsurface structures at different scales. In last IMF-slice, the noise The had initially processed; however, theofnoise still remains. firstly use and of the dataset extracted, especially thebe deep part thethat, profile. noisedata in the deep partis been are mixed up. It in can observed although theWe IMF-slices by VMD EMD may EMD to decompose one trace respectively (Figure 12), and the decomposed results are shown in refer to different frequency ranges, the continuity between neighboring traces is not as good as that In Figure 16, we also provide comparisons with the IMF-slices based on EMD. In some locations Figures 14,IMF-slices, respectively. There are four IMFs generated Figure 13, of which ofand these the interpretations are greatly influenced byinthe mode-mixing of IMFsthe for first three achieved13with VMD. different For instance,The in Figure 16dIMF the important information the shallow part and the IMFs are the maintraces. components. fourth can be seen as theinnoise components. In contrast, noise in the deep are mixed up. It can observed that, although theinto IMF-slices by EMD the EMD generates morepart IMFs in Figure 14, be and the noise is mixed the first IMF.may In this case, refer to different frequency ranges, the continuity between neighboring traces is not as good as that the mode-mixing phenomenon that accompanies EMD decomposition can be lessened by the VMD. achieved with VMD.

Figure The field field GPR dataset. Figure 11. GPR dataset. Figure 11.11.The The field GPR dataset.

Figure 12. A single trace of the GPR dataset. Remote Sens. 2018, 10, x FOR PEER REVIEW

Figure 12. 12. A A single single trace trace of of the the GPR GPR dataset. dataset. Figure

13. The decomposed IMFs IMFs using VMD. FigureFigure 13. The decomposed using VMD.

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Figure 13. The decomposed IMFs using VMD.

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Figure 14. The decomposed IMFs using EMD. Figure 14. The decomposed IMFs using EMD.

We then compute VMD decomposition for the whole dataset; the derived IMF-slices are shown in Figure 13. The low-frequency components are mainly collected in the first IMF-slice (Figure 15a). We remark here that in this IMF-slice, the DC and the slowly-varying trend term [1] are also collected, so there exists discontinuity of neighboring traces in gray scale. The second and third IMF-slices (Figure 15b–c) demonstrate the subsurface structures at different scales. In the last IMF-slice, the noise of theRemote dataset extracted, especially Sens.is 2018, 10, x FOR PEER REVIEWin the deep part of the profile. 12 of 14

Figure 15.derived The derived IMF-slices thedataset field dataset using (a–d) VMD.show (a–d)the show the IMF-slice 1–4 Figure 15. The IMF-slices of theof field using VMD. IMF-slice 1–4 respectively. respectively.

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In Figure 16, we also provide comparisons with the IMF-slices based on EMD. In some locations of these IMF-slices, the interpretations are greatly influenced by the mode-mixing of IMFs for different traces. For instance, in Figure 16d the important information in the shallow part and the noise in the deep part are mixed up. It can be observed that, although the IMF-slices by EMD may refer to different Figure frequency ranges, continuity between neighboring traces not as as that 15. The derivedthe IMF-slices of the field dataset using VMD. (a–d)isshow thegood IMF-slice 1–4 achieved with VMD. respectively.

Figure 16. The derived IMF-slices the dataset field dataset (a–d) the show the IMF-slice 1–4 Figure 16. The derived IMF-slices of theoffield using using EMD. EMD. (a–d) show IMF-slice 1–4 respectively. respectively.

5. Conclusions

5. Conclusions

EMD-based methods for GPR imaging provide a novel prospect for interpreting GPR data. Using EMD, the GPR data are decomposed, and the relative stationary IMFs are derived and analyzed for subsequent application. In this paper, variational mode decomposition was introduced. Using VMD, the derived IMFs are sparser and match well with the signal’s intrinsic properties. Based on the decomposition results, the IMFs are sorted and recorded into the proposed IMF-slices for GPR data imaging. We showed its utility on a group of examples, including synthetic benchmark signals, laboratory data and a field dataset. Using IMF-slices, GPR data could be divided into several IMF-slices, each of which delineates a main vibration mode and some subsurface layer and geophysical events can be identified more clearly. However, the proposed IMF-slices method still requires empirically pre-defined parameters. If the frequency ranges for each IMF-slice are estimated wrongly, or the time-frequency structures of the traces in the GPR profile vary sharply, it is difficult to find a unified standard to separate the different IMF-slices. In our future work, we need to rethink and seek for more intelligent schemes or more automatic procedures for classifying the derived IMFs for the IMF-slices. Additionally, for the accurate detection of the subsurface targets, some figures of merit could also be employed to assess the generated IMF-slices, such as receiver operating characteristics (ROC) [21]. In this case, VMD decomposition may be extended and applied to more aspects for GPR data processing and imaging. Acknowledgments: The authors thank Jing Li and the three reviewers for meaningful and inspiring discussions. This work is supported in part by the National Key Research and Development Program of China under grant

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2016YFC0600505 and the Scientific Research Program of the Education Department of Jilin Province (the grant number will be issued soon). The language proofreading of this paper during the revision stage is supported by the Provincial Training Program of Innovation and Entrepreneurship for Undergraduates under grant 2017S1027 and 2017S1030. Author Contributions: “X.Z. and X.F. conceived and designed the experiments; E.N. performed the experiments; X.Z. and Z.Z. analyzed the data and wrote the code; E.N. contributed the field data materials; X.Z. wrote the paper.” Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations The following abbreviations are used in this paper: GPR ground-penetrating radar EMD empirical mean decomposition IMF intrinsic mode function VMD variational mode decomposition ADMM alternate direction method of multipliers

References 1.

2. 3. 4. 5. 6. 7.

8. 9.

10. 11.

12. 13. 14. 15. 16. 17.

Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.; Yen, N.-C.; Tung, C.C.; Liu, H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. Seri. A Math. Phys. Eng. Sci. 1998, 454, 903–995. [CrossRef] Rioul, O.; Vetterli, M. Wavelets and signal processing. IEEE Signal Process. Mag. 1991, 8, 14–38. [CrossRef] Sinha, S.; Routh, P.; Anno, P.; Castagna, J. Spectral Decomposition of Seismic Data with Continuous Wavelet Transform. Geophysics 2005, 70, P19–P25. [CrossRef] Stockwell, R.G.; Mansinha, L.; Lowe, R.P. Localization of the complex spectrum: The S transform. IEEE Trans. Signal Process. 1996, 44, 998–1001. [CrossRef] Cheng, Z.; Chen, W.; Chen, Y.; Liu, Y.; Liu, W.; Li, H.; Yang, R. Application of bi-Gaussian S-transform in high-resolution seismic time-frequency analysis. Interpretation 2017, 5, SC1–SC7. [CrossRef] Feng, X.; Zhang, X.; Liu, C.; Lu, Q. Single-channel and multi-channel orthogonal matching pursuit for seismic trace decomposition. J. Geophys. Eng. 2017, 14, 90–99. [CrossRef] Zhang, X.; Feng, X.; Liu, C.; Chen, C.; Li, X.; Zhang, Y. Seismic matching pursuit decomposition based on the attenuated Ricker wavelet dictionary. In Proceedings of the 79th EAGE Conference and Exhibition 2017, Paris, France, 12–15 June 2017. Li, J.; Liu, C.; Zeng, Z.; Chen, L. GPR Signal Denoising and Target Extraction With the CEEMD Method. IEEE Geosci. Remote Sens. Lett. 2015, 12, 1615–1619. Lu, Q.; Liu, C.; Zeng, Z.; Li, J.; Zhang, X. Detection of human’s motions through a wall using UWB radar. In Proceedings of the 2016 16th International Conference on Ground Penetrating Radar (GPR), Hong Kong, China, 13–16 June 2016; pp. 1–4. Chen, Y.; Ma, J. Random noise attenuation by f-x empirical-mode decomposition predictive filtering. Geophysics 2014, 79, V81–V91. [CrossRef] Qin, Y.; Qiao, L.H.; Ren, X.Z.; Wang, Q.F. Using bidimensional empirical mode decomposition method to identification buried objects from GPR B-scan image. In Proceedings of the 2016 16th International Conference on Ground Penetrating Radar (GPR), Hong Kong, China, 13–16 June 2016; pp. 1–5. Dragomiretskiy, K.; Zosso, D. Variational Mode Decomposition. IEEE Trans. Signal Process. 2014, 62, 531–544. [CrossRef] Liu, W.; Cao, S.; Chen, Y. Applications of variational mode decomposition in seismic time-frequency analysis. Geophysics 2016, 81, V365–V378. [CrossRef] Liu, W.; Cao, S.; Wang, Z.; Kong, X.; Chen, Y. Spectral Decomposition for Hydrocarbon Detection Based on VMD and Teager-Kaiser Energy. IEEE Geosci. Remote Sens. Lett. 2017, 14, 539–543. [CrossRef] Liu, W.; Cao, S.; Wang, Z. Application of variational mode decomposition to seismic random noise reduction. J. Geophys. Eng. 2017, 14, 888–899. [CrossRef] Boyd, S.; Vandenberghe, L. Convex Optimization; Cambridge University Press: New York, NY, USA, 2004. Hestenes, M.R. Multiplier and gradient methods. J. Optim. Theory Appl. 1969, 4, 303–320. [CrossRef]

Remote Sens. 2018, 10, 476

18. 19.

20. 21.

13 of 13

Taner, M.; Koehler, F.; Sheriff, R. Complex seismic trace analysis. Geophysics 1979, 44, 1041–1063. [CrossRef] Herrera, R.H.; Tary, J.B.; van der Baan, M. Time-frequency representation of microseismic signals using the synchrosqueezing transform. In Proceedings of the GeoConvention 2013, Integration, Calgary, AB, Canada, 6–12 May 2013. Fomel, S. Seismic data decomposition into spectral components using regularized nonstationary autoregression. Geophysics 2013, 78, O69–O76. [CrossRef] Rodríguez, A.; Salazar, A.; Vergara, L. Analysis of split-spectrum algorithms in an automatic detection framework. Signal Process. 2012, 92, 2293–2307. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).