2D full-waveform modeling of seismic
waves in layered karstic media
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Zheng, Yingcai, Adel H. Malallah, Michael C. Fehler, and Hao Hu.
“2D Full-Waveform Modeling of Seismic Waves in Layered Karstic
Media.” GEOPHYSICS 81, no. 2 (March 2016): T115–T124. © 2016
Society of Exploration Geophysicists
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http://dx.doi.org/10.1190/GEO2015-0307.1
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Society of Exploration Geophysicists
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http://hdl.handle.net/1721.1/108503
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2D full-waveform modeling of seismic waves in layered karstic media
Yingcai Zheng
1, Adel H. Malallah
2, Michael C. Fehler
3, and Hao Hu
4ABSTRACT
We have developed a new propagator-matrix scheme to simulate seismic-wave propagation and scattering in a multi-layered medium containing karstic voids. The propagator matrices can be found using the boundary element method. The model can have irregular boundaries, including arbitrary free-surface topography. Any number of karsts can be in-cluded in the model, and each karst can be of arbitrary geo-metric shape. We have used the Burton-Miller formulation to tackle the numerical instability caused by the fictitious resonance due to the finite size of a karstic void. Our method was implemented in the frequency-space domain, so fre-quency-dependent Q can be readily incorporated. We have validated our calculation by comparing it with the analytical solution for a cylindrical void and to the spectral element method for a more complex model. This new modeling capability is useful in many important applications in seis-mic inverse theory, such as imaging karsts, caves, sinkholes, and clandestine tunnels.
INTRODUCTION
The subsurface of the earth contains voids. These voids may in-clude karsts, caves, sinkholes, and even clandestine tunnels. Proper imaging of these features using body or surface waves will be fa-cilitated if we have an approach for modeling the elastodynamical wave propagation and scattering in media containing voids.
Proper modeling of voids requires that the free-surface boundary condition at the void boundary must be explicitly enforced. Ana-lytical modeling methods are available for a rather limited number of cases if the void is of simple geometry (e.g., a cylindrical or
spherical void; seeLiu et al., 2000b). Finite-difference (FD) meth-ods (Gelis et al., 2005;Zeng et al., 2012) can be used to strictly implement the free-surface boundary condition for a simply shaped void. However, these methods usually suffer from staircase artifacts if the gridding does not conform to the geometry of the karst. Some-times, a collapsed void is approximated as a low-velocity/low-den-sity material (Xia et al., 2006a,2006b) so that FD methods can be used. The finite-element method is a numerical modeling method known for its flexibility in handling complicated geometry. Like FD methods, it decomposes the model into simple mesh domains. Within each mesh grid, interpolation functions (with coefficients to be determined) are used to represent the wavefield. The boundary conditions can be explicitly enforced between neighboring domains and, finally, a matrix inversion is needed to find all the coefficients for the interpolation functions. Its widely used variant is the spectral-element method (SEM) (Komatitsch and Vilotte, 1998;Komatitsch and Tromp, 2002). We will use the SEM as a benchmark tool for our boundary-element method (BEM).
Boundary integral equation (BIE) methods and the discretized implementation, the BEMs (Sanchez-Sesma and Campillo, 1991; Ge et al., 2005; Ge and Chen, 2007, 2008; Ge, 2010; Zheng, 2010), known for their flexibility in dealing with irregular geom-etries, have been used by many authors (Benites et al., 1997; Yo-mogida et al., 1997;Pointer et al., 1998;Liu et al., 2000a;Liu and Zhang, 2000,2001;Chen et al., 2012) to model seismic-wave scat-tering by cavities in a homogeneous medium. However, it is well known that BEMs (direct or indirect) have numerical difficulties for calculating scattering by an inclusion (Burton and Miller, 1971; Colton and Kress, 1983;Martin, 1990;Bielak et al., 1995;Martin, 2006). For direct or indirect BEM methods, at some frequencies (called irregular frequencies; see Martin, 1990) corresponding to the resonant frequencies of the cavity that would have been filled by the same exterior material with a rigid (or soft) boundary, the boundary integral operators are singular and noninvertible. The
Manuscript received by the Editor 10 June 2015; revised manuscript received 27 October 2015; published online 3 March 2016.
1Formerly Massachusetts Institute of Technology, Department of Earth, Atmospheric, and Planetary Sciences, Cambridge, Massachusetts, USA; presently University of Houston, Earth and Atmospheric Sciences, Houston, Texas, USA. E-mail: yzheng12@uh.edu.
2Kuwait University, Petroleum Engineering Department, Safat, Kuwait. E-mail: adel.malallah@ku.edu.kw.
3Massachusetts Institute of Technology, Department of Earth, Atmospheric, and Planetary Sciences, Earth Resources Laboratory, Cambridge, Massachusetts, USA. E-mail: fehler@mit.edu.
4University of Houston, Earth and Atmospheric Sciences, Houston, Texas, USA. E-mail: huhaoletitbe@gmail.com. © 2016 Society of Exploration Geophysicists. All rights reserved.
T25 GEOPHYSICS, VOL. 81, NO. 2 (MARCH-APRIL 2016); P. T25–T34, 11 FIGS.
10.1190/GEO2015-0307.1
singularity issue can be shown to exist using the Fredholm theory of integral operators (Colton and Kress, 1983). For a detailed exposi-tion on this subject matter, we refer to the book byMartin (2006). In seismology, BEMs have been used to study seismic scattering by voids or fluid-filled fractures (Benites et al., 1997;Yomogida et al., 1997;Pointer et al., 1998) but those authors do not mention the above stated singularity issue. If the voids are small in size, the seismic-frequency band is too low to overlap with the irregular frequencies for such voids, and we do not need to consider this sin-gularity issue. However, if the void size is comparable to the seismic wavelength, this issue must be considered and properly addressed. To make the integral operator invertible, an ad hoc solution is to make the frequency complex by adding a small imaginary part to the frequency, which corresponds to modeling the seismic wave-field that decays exponentially in time. When the imaginary part of the frequency is small (i.e., slow decay), the integral operator can still be close to singular. Another solution is the multiregion concept proposed byRodriguez-Castellanos et al. (2006), which splits an inclusion into three regions with overlapping boundaries extending to infinity. However, this technique may not be numerically straight-forward when dealing with multiple cavities. A third solution is called the Burton-Miller formulation. It is to complement direct BEMs with hypersingular BEMs (Burton and Miller, 1971; Liu and Rizzo, 1993;Martin, 2006) and is able to produce numerically stable results for all frequencies. In this paper, we adopt the third approach in our formulation. The BEM requires inverting a matrix whose size depends on the number of elements, and the matrix in-version can be computationally expensive.
A propagator-matrix approach (Bouchon and Campillo, 1989; Chen, 1990, 1995, 1996; Ge and Chen, 2008; Liu et al., 2008) is a scheme that can reduce the problem of inverting a large global matrix to one of inverting several small local matrices each of which is defined by the layer properties and its boundaries. To estimate the computational cost, if we have m boundaries and each boundary has n elements, to invert for a global matrix will need a computational complexity of Oðm3n3Þ, whereas the propagator-matrix approach needs a complexity of mOðn3Þ. This represents significant reduc-tion of computareduc-tional cost if m is large. Earlier propagator-matrix methods such as the discrete wavenumber method based on the indirect BIE (Bouchon and Campillo, 1989) using single-layer potentials can handle irregular-layered media with multiple boun-daries. However, because it uses a global Fourier transform between the space and wavenumber domains, the geometry of the internal boundary needs to be simple and single-valued with respect to one coordinate. Recently,Ge and Chen (2008)using direct BEM
and Liu et al. (2008)using indirect BEM, independently propose the frequency-space-domain propagator-matrix method to simulate waves in irregularly layered media with any number of layers. How-ever, the mathematical formulation of this important development may not be applicable if a layer contains karstic voids or inclusions. To include karstic voids in the elastic layers, a key step is to establish an integral relation between the displacement and the traction on each karst boundary. However, such a relation may not be possible when the frequency corresponds to one of the fictitious resonant frequen-cies of the cavity. We show that by using the Burton-Miller formu-lation (i.e., augmenting the direct BIE with a hypersingular BIE), such a relation can always be established for each cavity surface regardless of the frequencies. It is the aim of our paper to firmly establish this relation on the karst surface and propose a new propa-gator-matrix algorithm for simulating seismic waves in generally lay-ered media containing karstic voids or inclusions.
THEORY AND METHODS
We first review some basics of the BIE formulation. Later, we will show that these BIEs can be used to build propagator matrices. In the frequencyω domain, the direct BIE for the interior domain VI(Figure1) reads as follows:
1 2 unðx0Þ ¼ u0nðx0Þ − ∯ S ½uiðxÞTνinðx; x0; ωÞ − Ginðx; x0; ωÞtiðxÞdSðxÞ; x; x0∈ S; (1)
where Gijðxjx0Þ is the interior elastic Green’s function for a receiver at x and a point force f at x0, f ¼ ejδðx − x0Þ. The force direction is ej the unit vector along jth axis, and the receiver polarization is along the ith axis. The Green’s traction tensor Tνinðxjx0Þ in VI is the ith component traction for a surface element with outward nor-malν at x due to a point force at x0in theω domain. Here, u0nðx0Þ is the nth component of the incident displacement field at location x0 due to internal sources. The surface integral in equation1is under-stood in the Cauchy principal value sense. In the operator form, equation1reads as follows:
1
2 u ¼ u0− Au þ Bt; (2)
which expresses a relation between the surface displacement u and the surface traction t as follows:
u ¼ ½I∕2 þ A−1Bt þ ½I∕2 þ A−1u0: (3) However,½I∕2 þ A−1does not exist if the frequencyω is an ei-genfrequency of the interior problem with a rigid boundary condi-tion on S (Martin, 2006;Rodriguez-Castellanos et al., 2006). In the eigenstate, there are nonzero eigenfunctions vks that satisfy the fol-lowing equation:
½I∕2 þ ATv
k¼ 0; k ¼ 1; 2; : : : : (4) The eigenfunction vkis called a normal mode corresponding to a rigid boundary. The value ATis the transpose of A. By the Fredholm alternative, if there is nontrivial solution to equation4, the solution u to equation3is nonunique. To remedy this, one needs to use the V0
VI
S
x' x
ν
Figure 1. A finite domain VIbounded by surface S with infinite exterior domain V0. The outward surface normal at a surface point x ∈ S is ν ¼ νðxÞ.
hypersingular BIE, which is constructed by taking spatial derivative of equation1with respect to x0and then converting∇x0uðx0Þ to
traction vector tðx0Þ with the surface normal ν0¼ νðx0Þ using Hooke’s law (Liu and Rizzo, 1993) as follows:
1 2 tnðx0Þ ¼ t0nðx0Þ − ∯ S ½uiðxÞHν 0ν inðx; x0; ωÞ − Kν0 inðx; x0; ωÞtiðxÞdSðxÞ; x; x0∈ S; (5)
where again the surface integral is in the sense of Cauchy principal value. The kernel H can be found as Hννin0ðxjx0; ωÞ ¼ cnmpqðx0Þ Tνip0qðxjx0; ωÞνm0ðx0Þ, where the fourth-rank tensor cmnpq is the generalized Hooke’s constants. Similarly, the kernel K is Kννin0ðxjx0; ωÞ ¼ cnmpqðx0ÞGνip0qðxjx0; ωÞνm0ðx0Þ. We can write equa-tion5in the following operator form:
1
2 t ¼ t0− Hu þ Kt: (6)
Burton and Miller (1971)show that if we linearly combine equa-tions2and6, we get the following equation:
1
2 ðu þ iγtÞ ¼ u0þ iγt0− ðA þ iγHÞu þ ðB þ iγKÞt; (7) whereγ ∈ R and γ ≠ 0, there is a unique relation between u and t on S for the exterior BIE:
u¼½I∕2þAþiγH−1ðBþiγKÞtþ½I∕2þAþiγH−1ðu0þiγt0Þ: (8) Althoughγ can be any nonzero value, in practice, we choose γ to balance the magnitude of the kernels in equations 1and 5. For example,jHj ∼ μj∇0Tj ∼ μVωSjTj ¼ ρVSωjTj, where μ is the shear rigidity, ρ is the density, and VS is the S-wave velocity; there-fore,γ ¼ ðωρVSÞ−1.
Next we build propagator matrices using the BIEs. We consider a multilayered medium (Figure2). The first interface can be a free surface. The wavefield below the last interface satisfies the Som-merfeld radiation condition (SomSom-merfeld, 1949). On the ith inter-face, we use uiand tito denote displacement and traction field on the interface, respectively. Our objective here is to show the fact that ui and tiare related uniquely to each other by a matrix. This has been observed for general multilayered media without inclusions (Ge and Chen, 2007,2008). With inclusions, this relation still holds provided that one can solve the difficulty caused by resonance due to the finite inclusion.
We consider a multilayered medium (Figure2) containing karstic voids. The medium properties in each layer are constant, but the layer interfaces can be irregular and of any shape. We first construct the propagator matrix upward starting from the bottom layer up-ward to the first layer.
Bottom layer
In the bottom layer N that extends to infinity (Figure2a), we have two interfaces: boundary N and karstic boundary Γ, and the BIE (equation2) reads as the following equation:
1
2 uN ¼ −ANuNþ BNtN− ANΓuΓþ BNΓtΓþ u0N; (9) 1
2 uΓ¼ −AΓNuNþ BΓNtN− AΓΓuΓþ BΓΓtΓþ u0Γ; (10) where uNand tN(or uΓand tΓ) are displacement and traction on the boundary N (or Γ). The value u0Nand u0Γare the incident displace-ments on the boundary N and Γ, respectively, and they should van-ish if there is no source in the layer. As and Bs with different subscripts are integral operators defined in equation1. The hyper-singular BIE (equation6) in the last layer is given as
1
2 tΓ¼ −HΓNuNþ KΓNtN− HΓΓuΓþ KΓΓtΓþ t0Γ; (11) where Hs and Ks with different subscripts are integral operators, and t0Γ is the incident traction on the boundaryΓ due to a source in layer N. We can linearly combine equations10 and11 to get the following equation:
1 2 ðuΓþ iγtΓÞ ¼ − ~AΓNuNþ ~BΓNtN− ~AΓΓuΓþ ~BΓΓtΓþ ~u0Γ; (12) where ~AΓN¼ AΓNþ iγ ~HΓN; ~BΓN¼ BΓNþ iγ ~BΓN; ~AΓΓ¼ AΓΓþ iγ ~HΓΓ; ~BΓΓ¼ BΓΓþ iγ ~BΓΓ; ~u0 Γ¼ u0Γþ iγt0Γ; (13)
andγ is a real number. If the boundary Γ is a free surface, which corresponds to a karstic void, we have tΓ¼ 0. Equation12reduces to the following equation:
a) b)
ν
ν
ν
ν
ν
Figure 2. A multilayered medium containing karstic voids: (a) the bottom layer bounded on the top by the Nth boundary. This region may contain voids and (b) an intermediate layer bounded by Mth and Nth boundaries. The karst boundary is Γ; ν is the boundary normal direction, and it depends on boundary point location.
2D full-waveform modeling of seismic waves T27
1
2 uΓ¼ − ~AΓNuNþ ~BΓNtN− ~AΓΓuΓþ u0Γ; (14) with which we can eliminate uΓfrom equation9to obtain a matrix relation between boundary values uN and tN as follows:
uN ¼ CNNtN þ b0N: (15) It has been proven that ifγ is nonzero, the relation15is unique and stable (Burton and Miller, 1971;Colton and Kress, 1983). We will show that for an intermediate boundary i, we also have the fol-lowing equation:
ui¼ Ciitiþ b0i: (16)
Intermediate Layer m
For layer m (we use M to denote the mth interface above the karst and N for the (m þ 1)th interface below the karst) (Figure2b), the direct BIE (equation2) reads as the following equations:
1 2 uM ¼ −AMMuMþ BMMtM− AMNuN þ BMNtN− AMΓuΓþ u0M; (17) 1 2 uN ¼ −ANMuMþ BNMtM− ANNuN þ BNNtN− ANΓuΓþ u0N; (18) where u0M and u0N are the incident field on boundaries M and N, respectively, due to a source in layer m. For the karsts in layer m, we can combine its direct BIE and the hypersingular BIE and obtain the following equation:
1
2 uΓ¼ − ~AΓMuMþ ~BΓMtM− ~AΓNuN
þ ~BΓNtN− ~AΓΓuΓþ ~u0Γ; (19) from which we can eliminate uΓin equations17and18. Together with the known relation15for interface N, we can get the relation between uM and tM for the upper interface as follows:
uM¼ CMMtMþ b0M: (20) We continue the propagator matrix upward to the first layer (M ¼ 1) where t1¼ 0 for the free surface. From equation 20, we immediately obtain the displacement u1 on the free surface. If we know u1and t1, we can obtain u2and t2and the displacement uΓ on the karstic surfaces from equations17 to19. By recursion, displacement and traction on all boundaries can be obtained. Then, one can use these boundary values to compute the wavefield at any point within the model using the Kirchhoff integral.
The propagator-matrix approach may have important applica-tions in seismic inversion because if we update (insert, delete, or modify) the elastic properties or boundary geometries for one particular layer, we only need to update the propagator matrix in this layer and the matrices in all other layers will not be changed. To image near-surface karstic features using seismic waves, our approach can be used to understand the image artifacts. As our im-plementation is in the frequency-space domain, quality Q factors, QP(for the P wave) and QS(for the S wave) can be incorporated by making the substitution V−1P;S→ V−1P;S½1 þ i∕ð2QP;SÞ in the traction tensor and the Green’s tensor. Here, VP and VS are the P- and S-wave velocities, respectively.
VALIDATION OF THE METHODOLOGY To validate our method, we compare it with the analytical solu-tion for a cylindrical inclusion in an otherwise infinite homo-geneous medium (Figure 3). The P- and S-wave speeds are, VP¼ 1800 m∕s and VS¼ 1000 m∕s, respectively, and the density
−2 −1 0 1 2 −1.5 −1 −0.5 0 0.5 1 1.5 X (km) Z (km) Source Receivers R = 1 .0 k m −3 −2 −1 0 1 2 3 2 4 6 8 10 X (km) Time (s) ux BEM Analytical −3 −2 −1 0 1 2 3 2 4 6 8 10 uz X (km) Time (s) BEM Analytical a) b) c)
Figure 3. Cavity model (a) with source (star) and receivers (trian-gles); and waveform comparison between our BEM method and the analytical solution for (b) x- and (c) z-component.
ρ ¼ 2000 kg∕m3. The cavity radius is R ¼ 1000 m. The seismic source (explosion) is a line source at ðxs; zsÞ ¼ ð0; −1200Þ m. The linear geophone array is located at depth z ¼ zr¼ 1200 m. The source wavelet is a Ricker pulse with a center frequency f0¼ 1.0 Hz. We use 400 elements of equal length for the cylindri-cal cavity boundary. For our BEM method, the coupling constantγ at frequency f is γ ¼ 1∕ð2πρVSfÞ. We compute the multi-component seismic wavefield using our approach and compare it with the analytical approach described in Appendix A, and the rms error is <1% (Figure 3). As the BEM method is a fre-quency-domain method, our highest frequency fmaxin computation is fmax¼ 3f0¼ 3.0 Hz. In this example, the irregular frequencies
obtained by solving equation4(ksR ¼ 2πfR∕VS¼ 3.457, 3.832, 5.334, 5.391, : : : ) are within the seismic bandwidth for our BEM modeling.
To test our ability to compute seismic wavefield in a layered kars-tic medium, we used the model in Figure4, which consists of three elastic layers and two karstic voids of triangular and rectangular shapes. The model is bounded on the top by a free surface. The source is an explosion source with a Ricker source wavelet with a 10 Hz central frequency. The source is located at 1000 and 150 m. We use an SEM (Komatitsch and Vilotte, 1998;Komatitsch and Tromp, 2002) to verify our results and they agree with each other for the x- and z-component displacements (Figure4aand4b).
−800 −600 −400 −200 0 200 400 600 800 0 100 200 300 400 500 600 X (m) Z (m) VP=2500 m/s, VS=1250 m/s, density = 2030 kg/m3 VP=3500 m/s, VS = 1809 m/s, density = 2277 kg/m3 VP=4000 m/s, VS = 2009 m/s, density = 2477kg/m3 V P=4500 m/s, VS = 2209 m/s, density = 2577 kg/m3
Receiver 1 Receiver 101 Receiver 201 Figure 5. Layered karstic model with free-surface
topography. The P-wave velocity VP, S-wave velocity VS, and density for each layer are labeled. Karsts are denoted as the white ellipses with ran-dom orientations. Each ellipse has a semimajor axis length of 20 m and semiminor axis length of 8 m. A total of 201 receivers (black triangles) are placed along the surface from x ¼ −500 to 500 m, spaced equally along the horizontal axis at 5 m. Source positions are shown as stars for Fig-ures6–8. The attenuation factors QPand QSin all layers are the same, QP¼ 9000 and QS¼ 4000.
0 500 1000 1500 2000 0 0.2 0.4 0.6 0.8 1 1.2 Xr (m) Time (s) U z BEM SEM 0 500 1000 1500 2000 0 0.2 0.4 0.6 0.8 1 1.2 Xr (m) Time (s) U x BEM SEM 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 100 200 300 400 500 600 700 800 900 1000 Source: 10 Hz Ricker Receivers VP=1500 m/s; VS=866 m/s ; ρ = 2700 kg/m 3 VP=3000 m/s; VS=1732 m/s ; ρ = 2700 kg/m3 VP=4500 m/s; VS=2598 m/s ; ρ = 2700 kg/m3 Distance (m) Depth (m) a) c)
b) Figure 4. Comparison between BEM and SEM
synthetic seismograms for x- and z-component displacements (a) Ux and (b) Uz, respectively, along with (c) the model and source-receiver ac-quisition geometry.
2D full-waveform modeling of seismic waves T29
For our 2D examples, for the BEM, we only need to discretize the 1D boundaries. However, for the SEM, we need to discretize the whole 2D computational domain. This represents a domain dimen-sion reduction. If there are multiple karsts of complex geometric shapes, then SEM meshing can be challenging. Our BEM is imple-mented in the space-frequency domain, whereas the SEM is in the time-space domain. The BEM can be more efficient when we need to simulate wavefields for a large number of shots in the exploration seismology setting. This is because the matrices inverted in the modeling can be repeatedly used for all other shots. For a a 3D space, the BEM scheme presented in this paper may not be numeri-cally feasible due to significant increase of the matrix size and memory storage in the computer. In this case, the fast multipole methods may be used (Liu, 2009). The limitation for the BEM
is well known, especially when the medium is heterogeneous. In such a case, the SEM has the advantage in computation.
EXAMPLES
Two examples are used to demonstrate the capability of our mod-eling approach and to show some challenges in imaging karsts in the subsurface.
The first example is to show the effect of karsts on seismic wave-fields. The model is an irregularly layered model with karstic voids and a free-surface topography, including foothill and valley features (Figure 5). For each void, its boundary is discretized into equal-length elements. The element equal-length is 2.3 m. The source wavelet is a Ricker pulse of a central frequency of 50 Hz. The minimum S
0 0.1 0.2 0.3 0.4 0.5 Time (s) 0 0.1 0.2 0.3 0.4 0.5 Time (s) 0 0.1 0.2 0.3 0.4 0.5 Time (s) 0 0.1 0.2 0.3 0.4 0.5 Time (s) 50 100 150 200 Receiver Number 50 100 150 200 Receiver Number 50 100 150 200 Receiver Number 50 100 150 200 Receiver Number a) c) b) d) karsts no Ux Uznokarsts karsts with Uz karsts with Ux
Figure 7. Common shot gathers for the model in Figure5(c and d) with or (a and b) without the karsts. The source is an explosion at x ¼ 0 and 10 m below the earth surface. The source wavelet is a Ricker time function of 50 Hz. The values Ux and Uzare for x- and z-component of the displace-ment wavefields. 0 0.1 0.2 0.3 0.4 0.5 Time (s) 0 0.1 0.2 0.3 0.4 0.5 Time (s) 50 100 150 200 Receiver Number 50 100 150 200 Receiver Number 50 100 150 200 Receiver Number 50 100 150 200 Receiver Number 0 0.1 0.2 0.3 0.4 0.5 Time (s) 0 0.1 0.2 0.3 0.4 0.5 Time (s) s t s r a k o n Uz karsts no Ux karsts with Uz karsts with Ux a) b) c) d)
Figure 6. Common shot gathers for the model in Figure5(c and d) with or (a and b) without the karsts. The source is an explosion at x ¼ −320 and 10 m below the earth surface. The source wavelet is a Ricker time function of 50 Hz. The values Ux and Uz are for x- and z-component of the displacement wavefields.
wavelength is 25 m. A general rule of thumb for element size is that we should have 10 elements per wavelength. Within each layer, the elastic properties are uniform. Because we are interested in the ef-fect of karsts, we also considered a second model, a no-karsts model, in which we fill the voids in Figure5with the same material of the layer that contains the voids. We computed the horizontal and vertical components of the displacements for receivers placed on the free surface for three shot locations (Figure 5). The presence of karsts disrupted the moveout continuity of the seismic reflection events (see the events indicated by black arrows in Figures6–8), in particular those associated with the deeper boundaries below the karsts. This potentially causes challenges for imaging below the karstic layer.
Our second example is a near-surface karst of an irregular shape (Figure9), which mimics the dissolution feature found in many car-bonate or dolomite geologic settings. These types of subsurface fea-tures (e.g., sinkholes) pose significant hazards to construction and to the existing buildings. It is beneficial to see if we can image such a karst using seismic migration. To investigate whether the free sur-face can be used to identify subsursur-face karsts, we considered two model cases: with and without the free-surface boundary condition for the model in Figure9. If the seismic acquisition is right above the karst, we expect to see significant reverberations of the waves between the free surface and the subsurface karst (Figure10aand 10b). The strong reverberations in seismograms can be indicative of the presence of the near-surface karst. In our numerical synthetics, we used a Ricker source wavelet of 100 Hz (up to 300 Hz in cal-culation), which can be generated by a hammer source in practice. We calculated 41 shot gathers and performed Kirchhoff migration (Figure11) for the two model cases. The migration velocity model is the P-wave velocity model in Figure9but with no karst in it (i.e., it is filled with ambient background material). In our migration, we used only vertical components of the displacement. We did not per-form the P- and S-mode separation because it is not the focus of this paper to discuss details on how to image karsts. As expected, strong image multiples below the karst are produced (Figure11a). Three strong seismic image amplitude events labeled 1, 2, and 3 in
Figure 11bcan be associated with the three vertically protruded karstic features. Several strong seismic image features below the karst, labeled as 4, are produced by wave interaction with the karst (not by the presence of the free surface). This shows great chal-lenges for imaging karsts and for imaging below karsts using point sources. It can be speculated that localized beam migration (Zheng et al., 2013) may be one approach for avoiding interaction with karsts and may allow imaging below karsts.
CONCLUSIONS
We have developed a propagator-matrix approach for simulating seismic waves in the layered karstic media. We have shown that in previous direct BIE, there was a singular integral operator (not invertible) for calculating scattering by a finite-sized inclusion (or karstic void) at some irregular frequencies (fictitious resonance frequencies). As such, the Burton-Miller formulation, which aug-ments the direct BIE formulation by a hypersingular BIE, is used
200 300 400 500 600 0 20 40 60 80 100 120 Depth (m) Distance (m) Source: 100 Hz Ricker Receivers Layer 1: VP=1800 m/s; VS= 800 m/s ; ρ = 2030 kg/m3 Layer 2: VP=2000 m/s; VS=1000 m/s ; ρ = 2277 kg/m3 Layer 3: VP=2400 m/s; VS=1200 m/s ; ρ = 2477 kg/m3 Karst
Figure 9. A layered model with a near-surface irregularly shaped karstic void (white color). Receivers are on the free surface. For each layer, the P-wave velocity VP, S-wave velocity VS, and density ρ are labeled. Receivers are placed on the free surface from 300 to 500 m in the horizontal direction, spaced at every 2 m.
0 0.1 0.2 0.3 0.4 0.5 Time (s) 0 0.1 0.2 0.3 0.4 0.5 Time (s) 0 0.1 0.2 0.3 0.4 0.5 Time (s) 0 0.1 0.2 0.3 0.4 0.5 Time (s) 50 100 150 200 Receiver Number 50 100 150 200 Receiver Number 50 100 150 200 Receiver Number 50 100 150 200 Receiver Number a) b) c) d) karsts no Ux Uznokarsts karsts with Ux Uzwithkarsts
Figure 8. Common shot gathers for the model in Figure5(c and d) with or (a and b) without the karsts. The source is an explosion at x ¼ 280 and 10 m below the earth surface. The source wavelet is a Ricker time function of 50 Hz. The values Ux and Uz are for x- and z-component of the displacement wavefields.
2D full-waveform modeling of seismic waves T31
to overcome this difficulty and can obtain stable results for all frequencies. We benchmarked our BEM calculations with an analytical solution and with the SEM.
ACKNOWLEDGMENTS We are grateful to editor D. Komatitsch for handling our paper, who also gave us useful com-ments. We also thank three anonymous reviewers for their useful reviews. We thank Z. Ge from Peking University for fruitful discussions. We thank the Kuwait Foundation for the Advance-ment of Sciences and the Kuwait-MIT Center for Natural Resources and the Environment for support to carry out most part of this research. We are also grateful to the University of Houston for the funding provided to Y. Zheng and H. Hu. Computational facilities from the Earth Resour-ces Lab and UH are acknowledged. We thank the authors of the Specfem2D package (https:// geodynamics.org/cig/software/specfem2d/, last accessed April 2015) for their laudable effort to put their code in the CIG repository, so we are able to compare numerical results.
APPENDIX A
ELASTIC WAVE SCATTERING BY A CYLINDRICAL VOID For a cylindrical void extending to infinity along the y-direction, embedded in a homo-geneous medium, the scattering can be readily computed by modal summation. The basic mate-rials in this appendix can also be found elsewhere (Morse and Feshbach, 1953). However, there can be numerical issues for high-frequency waves. It is our purpose to present a technique here to solve these numerical issues.
Assume the void of radius R is centered at ðx; zÞ ¼ ð0; 0Þ and the point source (line source in 2D) is located atðxs; zsÞ in Cartesian and polar coordinatesðr0; ϕ0Þ. We use α, β, and ρ to denote the P-, S-wave speeds, and density, respectively. We use a standard potential method to find the scattered wavefield (Pao and Mow, 1973; Liu et al., 2000b). Liu et al. (2000b) consider plane-wave incidence. Here, we give results for the point source excitation. The P-wave po-tential can be expanded into summation of differ-ent angular orders m as follows:
Φðr; ϕÞ ¼ X∞ m¼−∞
½Φinc
m ðr; ϕÞ þ Φscattm ðr; ϕÞ: (A-1) Similarly for the S-wave potential, we have the following expansion: 0 0.1 0.2 0.3 0.4 Time (s) 50 100 Receiver Number 0 0.1 0.2 0.3 0.4 Time (s) 50 100 Receiver Number 0 0.1 0.2 0.3 0.4 Time (s) 50 100 Receiver Number 0 0.1 0.2 0.3 0.4 Time (s) 50 100 Receiver Number Ux Ux Uz Uz
with free surface with free surface
no free surface no free surface
a)
c)
b)
d)
Figure 10. Common shot gathers calculated for the model in Figure9(a and b) with or (c and d) without the free-surface boundary condition. The source is an explosion at x ¼ 280 and 5 m below the earth’s surface. The source wavelet is a Ricker time function of 100 Hz. The values Uxand Uzare for x- and z-component of the displacement wavefields.
50 100 Depth (m) 0 0 5 0 0 4 0 0 3 Distance (m) –2 –1 0 1 2 x10 –8 50 100 Depth (m) 0 0 5 0 0 4 0 0 3 Distance (m) –1.0 –0.5 0 0.5 x10 –8 1 2 3 4 4 4 4
With free surface
Without free surface a)
b)
Figure 11. Kirchhoff depth migration images (a) with and (b) without the effect of the free surface. Receivers are fixed in position, but source locations are from 300 to 500 m, every 5 m. All sources are at 5 m depth and are of the explosion type.
Ψðr; ϕÞ ¼ X∞ m¼−∞
Ψscatt
m ðr; ϕÞ: (A-2)
Because the wavefields for different orders are independent, we consider the case for order m. At order m, the incident P-wave po-tential is as follows (Morse and Feshbach, 1953):
Φinc mðr; ϕÞ ¼
i
4 JmðkαrÞHmðkαr0Þeimðϕ−ϕ0Þ; (A-3) the scattered P-wave potential is
Φscatt
m ðr; ϕÞ ¼ BmHmðkαrÞeimϕ; (A-4) and the scattered SV-wave potential is
Ψscatt
m ðr; ϕÞ ¼ CmHmðkβrÞeimϕ; (A-5) where kα¼ ω∕α is the background P-wave wavenumber, and kβ¼ ω∕β is the background S-wave wavenumber. The displacement field UðmÞcan be computed as gradient of the potential fields
UðmÞðr; ϕÞ ¼ ∇ðΦincm þ Φscattm Þ þ ∇ × ð0;0; Ψscattm Þ (A-6) in cylindrical coordinates. With this we can compute traction TðmÞðR; ϕÞ along the karst boundary. The traction should be zero, i.e., TðmÞðR; ϕÞ ¼ 0, in the radial and tangential directions, which gives two equations from which we can solve for Bm and Cm as follows: " −ðρω2R2− 2μm2Þ − 2μχ mðkαRÞ −2iμm½1 − χmðkβRÞ −2iμm½1 − χmðkαRÞ ðρω2r2− 2μm2Þ þ 2μχmðkβRÞ # × " BmHmðkαRÞ CmHmðkβRÞ # ¼4 Ji mðkαRÞ " −ðρω2R2− 2μm2Þ − 2μσ mðkαRÞ −2iμm½1 − σmðkαRÞ # ; (A-7) where χmðzÞ ¼ zHm0ðzÞ∕HmðzÞ (A-8) and σmðzÞ ¼ zJm0ðzÞ∕JmðzÞ: (A-9) From equationA-7we can solve for Bmand Cmfor order m. The displacement UðmÞfor order m can then be computed using equa-tionsA-3–A-6. The total displacement field u is the sum of all or-ders UðmÞas follows:
uðr; ϕÞ ¼ X ∞ m¼−∞
UðmÞðr; ϕÞ: (A-10) We note that the direct computation of equationsA-8andA-9for high-frequency waves at large m may not be numerically stable. However, using basic properties of the Bessel (including Hankel) functions, we found the following recursive calculation forχmðzÞ (also true forσmðzÞ):
χmþ1ðzÞ ¼ z2 m − χmðzÞ
− ðm þ 1Þ; (A-11)
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