Elastic Least-squares Gaussian beam migration

Yubo YueR&D Center, BGP, CNPC

ABSTRACT

Gaussian beam migration is an elegant imaging method which has the ability to imagemultiple arrivals while preserving the advantages of ray-based methods. In this paper,we extend this method to linearized least-squares imaging for elastic waves in isotropicmedia. We dynamically transform the multi-component data to the principal compo-nents of different wave modes using the polarization information available in the beammigration process, and then use Gaussian beams as wavefield propagator to constructthe forward modeling and adjoint migration operators. Based on the constructed oper-ators, we formulate a least-squares migration scheme which is iteratively solved usinga preconditioned conjugate gradient method. With this method, we can obtain multi-wave images with better subsurface illumination and higher resolution. It also allowsus to achieve a good balance between computational cost and imaging accuracy, whichare both important requirements for iterative least-squares migrations. Numerical testson two synthetic datasets demonstrate the validity and effectiveness of our proposedmethod.

1 INTRODUCTION

Depth migration is an important step of multi-componentdata processing. A common procedure for migrating multi-component data in practice is to treat the Z-component dataas P-waves and the radial-components as PS-waves, which arethen migrated separately using acoustic methods. However,this approach suffers from (1) crosstalk noise caused by di-rectly migrating the coupled elastic waves and (2) inaccurateimaging polarity for converted-wave imaging as polarity re-versal is no longer at zero-offset when the P/S velocity ratiovaries (Rosales and Rickett, 2001). In order to make full useof the vectorial wavefields and obtain high quality multi-waveimages, we approach this problem using elastic imaging meth-ods.

Elastic Kirchhoff migration (EKM) is typically used formigration of multi-component data (Kuo and Dai, 1984; Hok-stad, 2000). This technique computes traveltime and polariza-tion information for PP- and PS-waves via ray tracing andsums the vector data along traveltime trajectories. EKM offersa high level of efficiency, however, it suffers from the limita-tions of classical ray theory, including amplitude singularityin caustic regions and difficulties for imaging multiple arrivals(Gray et al., 2001). Another kind of elastic migration methodis elastic reverse-time migration (ERTM) (Yan and Sava, 2008;Du et al., 2012, 2017; Duan and Sava, 2015), in which a nu-merical solution to the elastic-wave equation is used to extrap-olate the P- and S-wave wavefields simultaneously, followedby wave mode separation during the migration process. Al-though ERTM has high imaging accuracy, its computationalefficiency is low, which significantly hinder its practical appli-cation. Gaussian beam migration (GBM) is an elegant imag-

ing method (Hill, 1990, 2001; Nowack et al., 2005; Gao et al.,2006; Gray, 2005; Gray and Bleistein, 2009; Yue et al., 2012),as it not only preserves the high efficiency of ray-based migra-tion methods, but also achieves high level of accuracy by mi-grating multipath arrivals. Recently, Protasov and Tcheverda(2012), Li et al. (2018) and Yang et al. (2018) have success-fully expanded GBM to elastic isotropic media, providing acompetitive alternative to imaging elastic waves.

Least-squares migration (LSM) seeks a reflectivity modelthat can predict data consistent with the observed data(Nemeth et al., 1999; Duquet et al., 2000; Zeng et al., 2014;Zhang et al., 2015). It is used to compensate for acquisitionnoise, poor sampling of sources and receivers on the surface,as well as poor illumination of the subsurface. The existingelastic LSM (ELSM) methods are mainly implemented withthe two-way wave propagator (Duan et al., 2017; Feng andSchuster, 2017). However, since the computational cost of oneiteration of ELSM is twice that of conventional ERTM, theexpense of all ELSM iterations is often prohibitively high inlarge scale application.

To provide an efficient approach of ELSM, we proposeelastic least-squares Gaussian beam migration (ELSGBM),which not only can produce better illuminated elastic imageswith enhanced resolution, but also can achieve a good bal-ance between computational cost and imaging accuracy. Wedynamically transform multi-component data to the princi-pal components of different wave modes using the polariza-tion vectors available in the migration process, and then useGaussian beams as wavefield propagator to construct the for-ward modeling and adjoint migration operators. Based on theconstructed operators, we formulate least-squares migration

Yubo Yue

for elastic waves and use preconditioned conjugate gradient(Nemeth et al., 1999) to iteratively solve for the optimal im-ages. To verify the validity and effectiveness of the method, weuse two synthetic datasets and show that ELSGBM is capableof producing elastic images with high resolution and balancedamplitudes.

2 METHOD

We assume a 3D elastic and smoothly inhomogeneousisotropic medium with a free-space acquisition surface. Themulti-component data u(xr,xs, ω) recorded at receiver xr aregenerated by an omnidirectional point source with coordinatesxs.

2.1 Principal components of elastic waves

According to the zero-order ray theory (Cerveny , 2001), wecan use three mutually orthogonal, right-handed unit vectorse1, e2, e3 of the ray-centered coordinate system to denote thepolarization vectors of S1-waves, S2-waves, and P-waves, re-spectively. If we particularly chose e1 at reflection point tobe parallel to the reflection plane (Figure1), then e2 is per-pendicular to the plane (Figure 1) and therefore the S1- andS2-waves, which are also referred as SV- and SH-waves, arefully decoupled. With this particular choice, we can calculatethe polarization vectors eSV , eSH , eP at the receiver locationand the recorded multi-component data u(xr,xs, ω) can betransformed to the three principal components uSV , uSH , uP

of elementary SV-waves, SH-waves, and P-waves by uSV

uSH

uP

= HT

uxuyuz

, (1)

where

H = (eSV , eSH , eP ) =

eSVx eSHx ePx

eSVy eSHy ePy

eSVz eSHz ePz

, (2)

denotes the rotation matrix from the ray-centered coordinatesystem to the Cartesian coordinate system. Because the po-larization vector of each wave mode depends on the ray tra-jectory connecting source, reflection point and receiver, thistransformation can only be dynamically implemented duringthe modeling or migration process. After this transformation,each wave mode can be extracted from the elastic vector wave-fields and can be imaged using scalar wave operators (Jacksonet al., 1991; Takahashi, 1995; Goertz, 2002; Luth et al., 2005).

2.2 Elastic wave Born modeling formula

With the notated assumptions, we can apply the linearized sin-gle scattering theory (Aki and Richards, 2002; Stolt and We-

Figure 1. Schematic diagram showing the determination of po-larization vectors. The initial polarization vectors of SV-waveseSV0 and SH-waves eSH0 are determined at the reflection point,where eSV0 is parallel to the reflection plane defined by thesource ray slowness vector au and the receiver ray slownessvector av , and eSH0 is perpendicular to the plane. After wedetermine eSV0 and eSH0 , we can calculate the polarizationvectors eSV0 , eSH0 at the receiver point via ray tracing. Thepolarization vector of P-waves eP is always tangent to the ray.

glein, 2012) to derive the Born modeling formula of one prin-cipal wave component uwv(xr,xs, ω)

uwv(xr,xs, ω) = ew(x,xr) · u(xr,xs, ω)

= ω2

∫dxF (ω)Gw(x,xr, ω)Mwv(x)Gv(x,xs, ω), (3)

where superscripts w, v represent up-going and down-goingwave modes, respectively. F (ω) is the source wavelet, M(x)is the scattering potential at x, G(x,xr, ω) is the up-goingGreen’s function, and G(x,xs, ω) is the down-going Green’sfunction. After taking account all possible down-going and up-going wave modes, we can write the forward modeling for-mula in operator form

u(xr,xs, ω)

= ω2

∫dxF (ω)HG(x,xr, ω)M(x)G(x,xs, ω), (4)

where M(x) is the scattering potential matrix which has fivenon-zero components

M(x) =

MSV SV 0 MSV P

0 MSHSH 0MPSV 0 MPP

, (5)

G(x,xr, ω) is the 3× 3 matrix of the up-going Green’s func-tion

G(x,xr, ω) =

GSV 0 00 GSH 00 0 GP

, (6)

and G(x,xs, ω) is the 3×1 matrix of the down-going Green’sfunction

G(x,xr, ω) =(GSV , GSH , GP

)T. (7)

Elastic Least-squares GBM

2.3 Elastic wave Born modeling in terms of Gaussianbeams

The Green’s functionGu(x,x′, ω) of one wave mode u can berepresented by the summation of Gaussian beams connectingthe scattering point x and the surface point x′

Gu(x,x′, ω) =iω

2π

∫ ∫dpuxdp

uy

puzAu(x,x′)

× exp[iωτu(x,x′)], (8)

where pu = (px, py, pz)T is the slowness vector of the central

ray at the surface. Au(x,x′) and τu(x,x′) are complex am-plitude and traveltime (Cerveny, 2001) in the following forms

Au(x,x′) = AuR(x,x′) + iAuI (x,x′), (9)

τu(x,x′) = τuR(x,x′) + iτuI (x,x′), (10)

where the subscripts R, I represent the real and imaginaryparts of complex quantities, respectively. By inserting equa-tion (8) into equation (4), we now obtain the elastic wave Bornmodeling formula in terms of Gaussian beams

um(x,x′, ω)

= − ω4

4π2

∑w

∑v

∫ ∫dpwx dp

wy

pwz

∫ ∫dpvxdp

vy

pvz

∫dxF (ω)

× ewm(x,xr)Mwv(x)[AwvR (xr,xs) + iAwvI (xr,xs)]

× exp[iωτwvR (xr,xs)− ωτwvI (xr,xs)], (11)

where m = 1, 2, 3 represents the X-, Y-, and Z-componentof modeled multi-component data, respectively.AwvR (xr,xs) ,AwvI (xr,xs) are the real and imaginary parts of the product ofsource and receiver beam complex amplitudes, τwvR (xr,xs),τwvR (xr,xs) are the real and imaginary parts of the sum ofsource and receiver beam complex traveltimes. These quanti-ties have the following expressions:

AwvR (xr,xs) = AwR(x,xr)AvR(x,xs)−AwI (x,xr)A

vI (x,xs),

AwvI (xr,xs) = AwR(x,xr)AvI (x,xs)−AwI (x,xr)A

vR(x,xs),

τwvR (xr,xs) = τwR (x,xr) + τvR(x,xs),

τwvI (xr,xs) = τwI (x,xr) + τvI (x,xs).

(12)

In order to accelerate equation (11), we trace beams atcoarse beam center locations L instead of the dense receiverlocations (Hill, 2001; Gray and Bleistein, 2009), and use themto approximately calculate the receiver wavefields by incorpo-rating a phase shift exp[−iωpw · (xr − L)]:

Gw(x,xr, ω) =iω

2π

∫ ∫dpwx dp

wy

pwzAw(x,L)

× exp[iωτw(x,L)− iωpw · (xr − L)]. (13)

To reduce the error of this process, we apply partition of unity(Bleistein, 2010; Gray and Bleistein, 2009) to equation (11)√

3

4π| ωωr|(∆L

ω0)∑L

exp[−| ωωr| |xr − L|2

2ω20

] ≈ 1, (14)

and arrive at the final Born modeling formula

um(xr,xs, ω) = −Φ∑w

∑v

∑L

∫ ∫dpwLxdp

wLy

pwLz

× Uwvm (L,xs,pw, ω)

× exp[−iωpw · (xr − L)− | ωωr| |xr − L|2

2ω20

], (15)

where Φ =√

3ω4

16π3 | ωωr|( ∆Lω0

), ∆L is the beam center spacing,and ωr and ω0 are the selected reference frequency and initialhalf-width of Gaussian beams. Uwvm (L,xs,p

w, ω) representsthe modeled plane-waves of the m-component for the down-going wave mode w and up-going wave mode v:

Uwvm (L,xs,pw, ω) =

∫ ∫dpvxdp

vy

pvz

∫dxF (ω)

× ewm(x,L)Mwv(x)[AwvR (L,xs) + iAwvI (L,xs)]

× exp[iωτwvR (L,xs)− ωτwvI (L,xs)]. (16)

The summation in equation (15) accumulates the mod-eled plane-waves of each beam center to the receiver wave-field. It implies an inverse slant stack operation which can beefficiently implemented in FK domain. Equation (16) repre-sents a combined smearing and convolution process which iscomputed in the time domain to ensure efficiency.

2.4 Adjoint elastic Gaussian beam migration

Elastic migration is defined as the adjoint operator of forwardmodeling. From equation (3), we derive the migration formula:

Mwv(x) =∑xs

∫dω

∫dxrω

2F ∗(ω)Gw∗(x,xr, ω)

× [ew(x,xr) · u(xr,xs, ω)]Gv∗(x,xs, ω), (17)

where superscript ∗ denotes complex conjugate. Here we alsouse the summation of Gaussian beams to calculate the Greensfunction. By inserting equation (8) into equation (17), we have

Mwv(x) = − ω4

4π2

∑xs

∫dω

∫d(x)ω2F ∗(ω)

×∫ ∫

dpwx dpwy

pwz

∫ ∫dpvxdp

vy

pvz[AwvR (xr,xs)− iAwvI (xr,xs)]

× exp[iωτwvR (xr,xs)− ωτwvI (xr,xs)]

× [ew(x,xr) · u(xr,xs, ω)].

(18)

To accelerate migrations, we apply the same strategy aswe did in equations (13) and (14), and arrive at the final ex-pression of elastic Gaussian beam migration (EGBM)

Mwv(x) =−∑xs

∑L

∫dω

∫ ∫dpwLxdp

wLy

pwLz

∫ ∫dpvxdp

vy

pvz

× [AwvR (L,xs)− iAwvI (L,xs)]Dw(L,pw, ω)

× exp[−iωτwvR (L,xs)− ωτwvI (L,xs)],

(19)

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whereDw(L,pw, ω) is the tapered slant stack of the principalcomponent of the selected wave mode w

Dw(L,pw, ω) = ΦF ∗(ω)

×∫dxre

w(L,xr) · u(xr,xs, ω)

× exp[iωpw · (xr − L)− | ωωr| |xr − L|2

2ω20

].

(20)

The dot-product operation in equation (20) represents a model-driven elastic wave mode separation that projects the multi-component data onto the polarization vector calculated fromray tracing. This operation extracts the wavefield correspond-ing to a selected wave mode and simultaneously suppressescrosstalk with other wave modes. Moreover, this operationalso ensures correct imaging polarity for converted waves, as-suming that the velocity model is accurate enough.

2.5 Least-squares Gaussian beam migration

Elastic least-squares Gaussian beam migration (ELSGBM)aims to find the multi-wave images that best predicts, ina least-squares sense, the recorded multi-component seismicdata. The misfit function in LSM is built using the l2 norm as

J =1

2‖Lm− d‖2, (21)

where m is the multi-wave image defined in equation (5), dis the multi-component data, and L is the beam-based elasticforward modeling whose adjoint operator is defined as LT .We use a preconditioned conjugate gradient method (Nemethet al., 1999) to iteratively solve the optimization problem

m(k+1) = m(k) − αdk(k), (22)

where dk(k) is the revised conjugate gradient direction of thekth iteration and α is the corresponding step length. They canbe calculated with the following equations

dk(k) = Cg(k) + βdk(k−1), (23)

α =(dk(k))Tg(k)

(Lk(k−1))T (Ldk(k−1)), (24)

where C is a preconditioning operator which is the source sideillumination and used to accelerate the convergence. g(k) andβ are the gradient and step length of the kth iteration using thesteepest descent method, which are expressed as

g(k) = LT (Lm(k)−d), (25)

β =(g(k))TCg(k)

(g(k−1))T (Cg(k−1)). (26)

3 EXAMPLES

The application of ELSGBM is now demonstrated with twosynthetic examples: (1) a multi-layered model and (2) a com-plex structural model. We compute multi-wave images us-

Figure 2. A synthetic multi-layered model. (a) P-wave velocity model,(b) S-wave velocity model.

ing different beam migration algorithms and compare themto demonstrate the improvements in image resolution andillumination of ELSGBM. We also extracted angle-domaincommon-image gathers (ADCIGs), which are easily imple-mented using ray information available in beam migrations,to show the suppression of crosstalk noise after ELSGBM.

3.1 Multi-layered model

We first demonstrate the advantages of ELSGBM using amulti-layered model. The P- and S-wave velocities are shownin Figures 2a and 2b, respectively, and the density is constant.We use finite-difference (FD) modeling to generate 200 multi-component shot gathers using a 20 Hz Ricker wavelet. We alsocompute the same shot gathers using EGB modeling to com-pare the accuracy and efficiency with FD-modeled results. Themodeled X- and Z-component data are shown in Figures 3a-d, respectively. Kinematically, the multi-component data pro-duced using EGB modeling are very similar to that of FD mod-eling, except the direct waves, multiples and wide-angle reflec-

(a)

(b)

Elastic Least-squares GBM

tions which are not accounted for in EGB modeling. Whilecomparing the computational time, EGB modeling is almost50 times faster than FD modeling. The comparisons show thatEGB modeling is an accurate and efficient method to simulatethe elastic reflection waves.

We test different migration algorithms on the FD-modeled data to show the advantages of ELSGBM. First, weuse acoustic Gaussian-beam migration (AGBM) to migrate theZ-component data to produce a PP-image and the polarity-reversed X-component data to produce a PS-image, shown inFigure 4 and Figure 7, respectively. Although AGBM correctlyrecover subsurface structures in both PP- and PS-images, mi-grating single-component data using AGBM causes strongcrosstalk (marked by the red arrows) in the extracted ADCIGs.The crosstalk may be partially attenuated in the stacked image(Figures 4a and 7a), but its non-flat appearance in ADCIGsinevitably complicates the velocity updating. We also use theproposed ELSGBM to migrate both X- and Z-component data;the images after 1 iteration (also known as the adjoint EGBM)are shown in Figures 5 and 8. We can see that EGBM pro-duces similar images in the stacked images and ADCIGs asAGBM, but the crosstalk noise is significantly attenuated inADCIGs due to the wavefiled separation operation (equation20) embedded in EGBM. Figures 6 and 9 show the PP- andPS-images after 15 iterations of ELSGBM. ELSGBM gener-ates improved resolution and more balanced illumination, andthe extracted ADCIGs are almost free of crosstalk noise.

For better comparison, we calculate the spectra of theEGBM and ELSGBM images shown in Figure 10. We find thatELSGBM images (the red curves) have broader bandwidth atboth high and low wavenumbers, indicating higher resolutionthan the EGBM images (the blue curves). Figures 11 and 12show the multi-component data residuals after 15 iterationsand normalized objective function versus number of iterations,respectively. Compared with the original FD modeled multi-component data shown in Figures 3a and 3c, most of the multi-component data residuals are eliminated after ELSGBM, il-lustrating our algorithm accurately predict the recorded elasticwave reflections.

3.2 Marmousi2 model

We use the synthetic Marmousi2 model (Martin et al., 2006) toverify the effectiveness of our proposed ELSGBM with com-plex structures. Figure 13 shows the P-wave velocity model;the S-wave velocity is a scaled version of the P-wave veloc-ity with a Vs/Vp ratio of 0.6. A total of 240 multi-componentshot gathers are synthesized using a FD wave equation solverwith 40m source spacing. The explosive P-wave source uses aRicker wavelet with 20Hz peak frequency. For each shot, thereare 351 receivers evenly distributed around the source posi-tion with 20m spacing. The EGBM results after 1 iteration areshown in Figures 15 and 17, and the ELSGBM results after20 iterations are shown in Figures 16 and 18, respectively. Al-though EGB is capable of accurately recovering the complexMarmousi2 structures in both PP- (Figure 15a) and PS- (Fig-ure 17a) images, the illumination and resolution of the central

(a) (b)

(c) (d)

Figure 3. Synthetic shot gathers at x=2.0km generated using differentelastic modeling methods. (a) FD-modeled X-component data, (b) FD-modeled Z-component data, (c) GB-modeled X-component data, and(d) GB-modeled Z-component data. For better comparison, we calcu-late the spectra of the EGBM and ELSGBM images shown in Figure10. We find that ELSGBM images (the red curves) have broader band

steep reflectors are poor. In contrast, ELSGBM produce im-ages and ADCIGs with more balanced illumination and higherresolution (Figures 16 and 18).

We compare EGBM and ELSGBM stacked images inFigure 19 for the dotted red box shown in Figure 13. Com-pared with EGBM, ELSGBM improves the image amplitudesand continuity of steeply dipping structures (black arrows) andalso reveals the thin layered structures. As shown in Figure 20,the spectra of ELSGBM images (red curves) are clearly widerthan that of EGBM images, confirming the higher resolutionof ELSGBM images. The multi-component data residuals af-ter ELSGBM are shown in Figure 21 and the correspondingconvergence curve is in Figure 22. After 20 iterations of ELS-GBM, most recorded PP- and PS- residuals are eliminated,demonstrating the effectiveness of our proposed ELSGBM.

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(a) (b)

Figure 4. Acoustic Gaussian beam migration. (a) PP- image, (b) PP-ADCIG at x=6.0km.

(a) (b)

Figure 5. Elastic Gaussian beam migration (the adjoint operator). (a)PP-image, (b) PP-ADCIG at x=6.0km.

(a) (b)

Figure 6. Elastic least-squares Gaussian beam migration. (a) PP-image, (b) PP-ADCIG at x=6.0km.

(a) (b)

Figure 7. Acoustic Gaussian beam migration. (a) PS-image, (b) PS-ADCIG at x=6.0km.

(a) (b)

Figure 8. Elastic Gaussian beam migration. (a) PS-image, (b) PS-ADCIG at x=6.0km.

(a) (b)

Figure 9. Elastic least-squares Gaussian beam migration. (a) PS-image, (b) PS-ADCIG at x=6.0km.

Elastic Least-squares GBM

(a) (b)

Figure 10. Spectrum comparisons between EGBM and ELSGBM. (a)PP-image comparison, (b) PS-image comparison.

(a) (b)

Figure 11. Data residual after 15 iterations of ELSGBM (Plottedon the same scale as the original data shown in Figure 3). (a) X-component data residual, (b) Z-component data residual.

Figure 12. Normalized objective function versus iterations for ELS-GBM on the multi-layered example.

Figure 13. Marmousi2 P-wave velocity.

(a) (b)

Figure 14. Synthetic multi-component shot gathers at x=7.0km gen-erated using FD modeling. (a) X-component data, (b) Z-componentdata. Note that we have muted the first arrivals. whatever it is it isalways going to be fine. keep you feet on the ground.

(a) (b)

Figure 15. Elastic Gaussian beam migration. (a) PP-image, (b) PP-ADCIG at x=9.0km.

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(a) (b)

Figure 16. Elastic least-squares Gaussian beam migration. (a) PP-image, (b) PP-ADCIG at x=9.0km.For better comparison, we calcu-late the spectra of the EGBM and ELSGBM images shown in Figure10. We find that ELSGBM images (the red curves) have broader band

(a) (b)

Figure 17. Elastic Gaussian beam migration. (a) PS-image, (b) PS-ADCIG at x=9.0km.

(a) (b)

Figure 18. Elastic least-squares Gaussian beam migration. (a) PS-image, (b) PS-ADCIG at x=9.0km.

(a) (b)

(c) (d)

Figure 19. Magnified views of the structures marked by the dottedred box shown in Figure (13). (a) PP-EGBM image, (b) PP-ELSGBMimage, (c) PS-EGBM image, and (d) PS-ELSGBM image.For bettercomparison, we calculate the spectra of the EGBM and ELSGBMimages shown in Figure 10. We find that ELSGBM images (the redcurves) have broader band

(a) (b)

Figure 20. Spectrum comparisons between EGBM and ELSGBM im-ages. (a) PP-image comparison, (b) PS-image comparison.For bettercomparison, we calculate the spectra of the EGBM and ELSGBM im-ages shown in Figure 10. We find that

(a) (b)

Figure 21. Data residual after 20 iterations of ELSGBM (Plottedon the same scale as the original data shown in Figure 14). (a) X-component data residual, (b) Z-component data residual.

Figure 22. Normalized objective function versus iterations for ELS-GBM on the Marmousi2 example.

Elastic Least-squares GBM

4 CONCLUSIONS

We derive an ELSGBM method for elastic waves in isotropic media. By using Gaussian beams as wave propagator to for-mulate the least-squares migration scheme, we achieve high resolution images, while preserving computational efficiency and accuracy. Our method operates by extracting the princi-pal components of the multi-component data, which enable us to suppress crosstalk and ensure accurate polarity for con-verted wave imaging. Trials on two synthetic datasets show that our proposed method produces multi-wave images with higher resolution and better illumination compared with con-ventional acoustic or elastic Gaussian beam migrations.

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I acknowledge my company, the Bureau of Geophysical Prospecting, for supporting my visit to the Center for Wave Phenomena (CWP), and CWP people for their generous help of this research. I am grateful to Prof. Sava for revising the manuscript and Qifan Liu for helping me to modify the for-mat of this paper. This work was supported by the Young Elite Scientists Sponsorship Program by CAST (2016QNRC001).

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