Simulations of P-SV wave scattering due to cracks by the 2-D finite difference method
- Yuji Suzuki^{1},
- Takahiro Shiina^{2},
- Jun Kawahara^{3},
- Taro Okamoto^{4} and
- Kaoru Miyashita^{3}
https://doi.org/10.5047/eps.2013.06.008
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2013
Received: 3 September 2012
Accepted: 25 June 2013
Published: 6 December 2013
Abstract
We simulate P-SV wave scattering by 2-D parallel cracks using the finite difference method (FDM). Here, special emphasis is put on simplicity; we apply a standard FDM (second-order velocity-stress scheme with a staggered grid) to media including traction-free, infinitesimally thin cracks, which are expressed in a simple manner. As an accuracy test of the present method, we calculate the displacement discontinuity along an isolated crack caused by harmonic waves using the method, which is compared with the corresponding results based on a reliable boundary integral equation method. The test resultantly indicates that the present method yields sufficient accuracy. As an application of this method, we also simulate wave propagation in media with randomly distributed cracks. We experimentally determine the attenuation and velocity dispersion induced by scattering from the synthetic seismograms, using a waveform averaging technique. It is shown that the results are well explained by a theory based on the Foldy approximation, if the crack density is sufficiently low. The theory appears valid with a crack density up to at least 0.1 for SV wave incidence, whereas the validity limit appears lower for P wave incidence.
Key words
1. Introduction
Random inhomogeneities inside the Earth’s lithosphere scatter high-frequency seismic waves and thereby cause phenomena such as the attenuation, dispersion, and generation of coda waves (Sato et al., 2012). The modeling of such inhomogeneities can be categorized into two concepts: random spatial fluctuation of medium parameters (called random media), and discrete scatterers such as cracks, inclusions, or cavities. Although the concept of random media has been popularly adopted in theoretically modeling seismic scattering (Sato et al., 2012), it assumes quite often a continuous heterogeneity. In contrast, the real Earth’s interior obviously has a discontinuous nature. It includes a large amount of cracks or fractures, and it also has many irregular interfaces having a sharp material contrast. Such a discontinuous heterogeneity can be modeled using the concept of discrete scatterers, either directly or approximately (Benites et al., 1992).
One of the popular numerical methods to simulate seismic scattering due to discrete scatterers is the boundary integral equation method (BIEM) (e.g., Benites et al., 1992; Murai et al., 1995; Pointer et al., 1998; Kelner et al., 1999; Liu and Zhang, 2001; Yomogida and Benites, 2002). The BIEM has often been believed to have advantages over domain-type methods in computational accuracy when dealing with discrete inhomogeneities (e.g., Benites et al., 1992; Liu and Zhang, 2001), and in a great flexibility regarding the shapes of scatterers. However, it also has shortcomings: the difficulty in its application to the discrete scatterers embedded in matrices which themselves have a complex heterogeneity, computational costs that rapidly increase with the increasing numbers of scatterers, and the intractability for beginners to code their computer programs. Domain-type methods, such as the finite difference method (FDM) and the finite element method (FEM), are superior in these points.
A great advantage of the FDM is its high simplicity and tractability in general. Especially, the staggered-grid scheme (Virieux, 1984, 1986) has been known for its accuracy for media having large values of Poisson’s ratio (Moczo et al., 2002). In fact, with a proper assignment of medium parameters (Ohminato and Chouet, 1997; Okamoto and Takenaka, 2005; Takenaka et al., 2009), it is now possible to simulate wave propagation in media with large material contrasts such as irregular land topography (i.e., irregular free surface), irregular ocean-bottom topography (i.e., irregular liquid-solid interface), and three-dimensional heterogeneity including low-velocity soft sediments by using the staggered-grid scheme (e.g., Nakamura et al., 2012; Okamoto et al., 2012, 2013).
However, the incorporation of cracks into the grids is not so straightforward. Some authors modeled cracks in indirect manners, such as thin layers of equivalent homogeneous anisotropic materials (Vlastos et al., 2003) or arrayed point sources (van Antwerpen et al., 2002). In contrast, Saenger et al. (2000) modeled cracks as a thin inclusion with zero elastic constants and small mass density within the “rotated staggered grid” developed by them. An advantage of this method is the ability to treat cracks with arbitrary shapes and orientations and even with intersections. Using this method, Saenger and his coworkers have numerically investigated the effective elastic moduli of cracked media (Saenger and Shapiro, 2002; Orlowsky et al., 2003; Saenger et al., 2004, 2006). Krüger et al. (2005) also applied the method to the scattering and diffraction of an SH wave by a 2-D crack and demonstrated its high accuracy in comparison with an analytical solution.
The FEM has also been used for evaluating the effective elastic moduli of cracked media, though mostly adopted in simulations of elastostatic deformation rather than wave propagation (Dahm and Becker, 1998; Grechka and Kachanov, 2006; Grechka, 2007). Frehner and Schmalholz (2010) successfully applied the FEM to simulations of Stoneley guided waves around a fluid-filled fracture. Frehner et al. (2008) compared the FEM and the FDM applied to the same problem of P-SV wave scattering due to a 2-D circular inclusion. They concluded that the FEM is superior to the FDM in terms of computational time because the former provides the same numerical accuracy with much less numerical grid points for modeling the curved interface of an inhomogeneity. A disadvantage of the FEM is that it is somewhat more complex to implement than the FDM and often needs a third-party software for mesh generation.
In our previous paper (Suzuki et al., 2006; hereafter referred to as Paper I), we numerically simulated SH wave scattering due to traction-free, infinitesimally thin cracks by the 2-D FDM. Therein was proposed a new, simple and cost-effective simulation technique; we adopted a standard velocity-stress FDM (Virieux, 1984; Levander, 1988) and expressed cracks in a staggered grid just by arraying grid points with zero shear traction. This is a natural extension of the FDM-based simulations of crack propagation originated by Madariaga (1976), in which the boundary conditions on the crack planes can be directly given. It was much simpler than the FDM previously proposed to deal with cracked media, though its application was limited to cracks parallel to the grid lines. We proved that this technique successfully deals with SH wave scattering due to cracks with practically sufficient accuracy, through accuracy tests based on the comparison with reliable BIEM estimates (Murai et al., 1995). It remained, however, to be extended to 2-D P-SV or 3-D elastic wave scattering, which would be practically more important.
An interesting topic on cracked media is the attenuation and velocity dispersion of waves propagating therein. Theoretically, they can be predicted by a stochastic theory based on Foldy’s (1945) approximation. This approximation is based on the assumption of many scatterers distributed randomly and sparsely (Ishimaru, 1978), and it is expected to give results accurate to the first order in the distribution density (Keller, 1964). Although considerable efforts have been made to propose alternative approximations that are claimed to be valid for more densely distributed scatterers, the original Foldy approximation theory (hereafter, FAT) has maintained its popularity probably because of its mathematical simplicity (Kawahara, 2011) and the lack of consensus on which approximation is best as an alternative to the FAT (Kawahara et al., 2009). The FAT was applied to cracked media first by Kikuchi (1981a, b) and then by many authors in seismology and acoustics (Yamashita, 1990; Kawahara and Yamashita, 1992; Kawahara, 1992; Zhang and Gross, 1997; Caleap and Aristégui, 2010). However, its actual validity limit with respect to crack density remains unclear. Using their BIEM, Murai et al. (1995) simulated SH wave scattering by randomly distributed 2-D cracks and determined the scattering attenuation and dispersion experimentally, thus validating the FAT of Kawahara and Yamashita (1992) as long as the crack density is relatively low (≤ 0.02). Later, Murai (2007) revisited this problem and concluded that the FAT seems to be valid for a crack density up to at least 0.1. Unfortunately, the ordinary peak picking technique, as they used in their measurement, will not work for dense crack distributions, because the initial motions of propagating waves will often be distorted too much to pick the peaks properly. In Paper I, we also performed numerical experiments like Murai et al. (1995) and Murai (2007), though we adopted a waveform averaging method in measuring the attenuation and dispersion. We showed that these parameters are obtained stably even for rather dense crack distributions, and revealed that the FAT works well for a crack density up to about 0.1, but probably not beyond it. To the authors ’ knowledge, however, it has not been confirmed whether the same conclusion holds for P-SV wave scattering due to 2-D cracks.
The present article is a continuation of Paper I, and its purpose is two-fold. First, we extend the FDM of Paper I in order to simulate P-SV wave scattering due to 2-D traction-free, infinitesimally thin cracks (Section 2). We validate the method through an accuracy test, in which it is compared with a reliable BIEM in the calculation of the displacement discontinuity along a single crack (Section 3). Second, we examine the validity of the FAT concerning P-SV wave scattering due to many cracks on the basis of numerical experiments with the present simulation technique (Section 4). Discussion and conclusions are given in Section 5. As in Paper I, we treat only cracks parallel to the grid lines; the extension of the present method to the case of arbitrarily oriented cracks is discussed in Section 5.
2. Finite Difference Method
Then, in the vicinity of the crack, the normal stress and are calculated using instead of in Eqs. (1c) and (1d). Note that vertical cracks (parallel to the z-axis) can be also treated in almost the same manner. Although the above-mentioned manner to incorporate cracks into a grid may not be so simple as that for the case of SH wave scattering given in Paper I, it still seems to retain simplicity as compared with those of other previous studies (Section 1).
In any simulation performed below, a plane wavelet is assumed to propagate upward (in the negative z-direction) from below and be incident on cracks either normally or parallel. Then, velocity seismograms are synthesized at specified grid points, which are finally integrated with the trapezoidal rule to yield the displacement seismograms. As in Paper I, Δh and Δt are chosen to satisfy the stability condition and the sampling criterion Δh < β/ nf_{ ul } (Virieux, 1986; Moczo, 1998), in which n = 10 for the second-order scheme, and are the P and S wave velocity of the matrix, respectively, and f_{ ul } is the frequency up to which one needs to have accurate results without the grid dispersion. For convenience, β, ρ, and the crack half length a are set to be unity and the values of other parameters are normalized with them. We also assume a Poisson solid with λ = μ and, hence, . Concerning the artificial boundaries of the whole grid (model space), we impose again simple conditions. The cyclic boundary condition is applied to the left and right ends to express approximately an infinitely long cracked layer. A standard absorbing boundary condition of Clayton and Engquist (1977) is assumed along the top and bottom ends. Although artificial reflections from the absorbing boundaries cannot be perfectly erased, we actually analyze the portions of seismograms not contaminated with them.
3. Crack Displacement Discontinuity
As in Paper I, we compare the results of our FDM simulations with those based on a frequency-domain BIEM, thus trying to validate the present simulation method. For this purpose, we calculate the displacement discontinuity along a single crack caused by incident harmonic waves, using both methods. Note that the normal displacement discontinuity (NDD) along a traction-free, infinitesimally thin crack due to normally incident harmonic P waves was analytically solved by Mal (1970). Kawahara and Yamashita (1992) developed a BIEM algorithm for treating an isolated crack, which reproduces Mal’s results quite excellently. Hence, we infer their BIEM to be reliable and adopt it here. In order to treat the BIEM results as the correct answers, we make the discretization interval of the crack planes in the BIEM calculation to be much smaller than the crack length as well as the wavelength, as in Paper I.
We set here Δt = 0.003a/β and Δh = 0.0125a. Choosing the same values of the parameters, the second-order FDM adopted in Paper I gave an SDD which was highly consistent with those by the BIEM, independently of the wavenumbers. Note that the fourth-order FDM was shown to systematically underestimate the SDD (by several percent), probably because of the nonlocality of the finite difference operator; see Paper I as to the detailed discussion. We use here the second-order FDM throughout. Note also that the present choice of Δh and the sampling criterion (Section 2) require an inequality f_{ ul } < 8.0 β/a whereas the incident waves considered here (and also in Section 4) are shown to have a negligibly small energy for a frequency beyond 8.0 β/a (corresponding to ka > 50). This implies that every simulation will not suffer grid dispersion.
In the following, we show only the results on non-zero NDD and SDD. Their values are normalized with h and k for P and S wave incidence, respectively, after Mal (1970). Note that the same normalization was also employed in figure 2 of Paper I, but without mention.
Figure 3(b) denotes the SDD due to the normal SV wave incidence for the normalized S wavenumber ka = 0, 1.0, 2.8, 4.2, and 6.0; the corresponding S wavelengths are ∞, 6.3a, 2.2a, 1.5a, and 1.0a, respectively. Note that the same values of ka were chosen in Paper I for the accuracy test of its FDM calculations after Mal (1970). In this case, the discontinuity by the FDM does not vanish outside the crack (at a distance larger than 1.0a in the figure). As mentioned in Paper I, this can be explained by the offsets of the measurement points from the crack plane, and, therefore, gets smaller for smaller Δh. Thus, it does not imply the failure of the present FDM. Except for this, the agreement between the FDM and BIEM estimates is satisfactory also in this case. Note that the agreement seems slightly worse than that seen in Fig. 3(a), especially for ka = 1. This may also be explained by the offsets of the measurement points stated above.
In summary, the present FDM produces the crack displacement discontinuity due to either P or SV wave incidence fairly accurately, independently of wavenumbers. We therefore conclude that the present method is applicable to the synthesis of P-SV waves scattered by cracks.
4. Validation of the Foldy Approximation Theory
In this section, we show an example of the application of the present FDM, whose accuracy has been confirmed above. Using the method, we investigate here the validity of the FAT of Kawahara (1992), which predicts the scattering attenuation and velocity dispersion of P-SV waves in 2-D media with identical parallel cracks. Here, the prediction is stochastic in the sense that the theory is based on the mean wave formalism; i.e., it treats the ensemble average of real wavefields. Note that the theory also adopts an approximation in which the waves effectively impinging on each scatterer (crack) are equal to the waves that would be observed without the scatterer. This approximation, first introduced by Foldy (1945), yields a closed equation for mean waves, called the Foldy-Twersky integral equation (Ishimaru, 1978). Kawahara and Yamashita (1992) solved it for P-SV as well as SH wave scattering due to cracks filled with an incompressible viscous fluid, and obtained the attenuation and phase velocity of the mean waves to the first order in the crack density. Kawahara (1992) then analyzed the case for P-SV wave scattering due to empty (traction-free) cracks. In Paper I, we validated in a numerical-experimental manner the FAT of Kawahara and Yamashita on SH wave scattering in the case of cracks filled with an inviscid fluid (or equivalently, shear stress-free cracks); the same experimental procedures are again adopted here, which are summarized below.
4.1 Outline of the simulations
Parameters of the crack distribution models used in the simulations. P and SV indicate the wave modes. H and V indicate the horizontal and vertical cracks (normal and parallel wave incidence), respectively.
Model | Wave/Cracks | N | W ×L | ϵ | N _{ D } |
---|---|---|---|---|---|
1 | P/H, P/V, SV/H, SV/V | 19 | 40a × 38a | 0.0125 | 6 |
2a | P/H, SV/H, SV/V | 36 | 40a × 18a | 0.05 | 12 |
2b | P/V | 76 | 40a × 38a | 0.05 | 12 |
3a | P/H, SV/H, SV/V | 32 | 40a × 8a | 0.1 | 18 |
3b | P/V | 152 | 40a × 38a | 0.1 | 18 |
We notice that Shapiro and Kneib (1993) have stated that spatial averaging works well if the length of the receiver array is much larger than max(a_{ cl }, λ_{ wl }), in which acl is the correlation length of the medium heterogeneity and λ_{ wl } is a wavelength; in such cases, the incoherent part of a wavefield would interfere destructively during the spatial averaging process and only the coherent part (mean wave) would be retained. In the present simulation geometry, a_{ cl } nearly corresponds to the size of the scatterers (cracks) 2a (Kawahara, 2011), which is much smaller than the length of the receiver array W = 40a (Table 1). However, the wavelengths are not always sufficiently smaller than W. As seen in the following subsections, we will numerically evaluate Q^{−1} and v for 0.5 < ka < 10 (0.29 < ha < 5.77) in which the initialwavelet has a high energy. This wavenumber range corresponds to 21.8a > λ_{ P } > 1.1a and 12.6a > λ_{ S } > 0.6a, in which λ_{ P } and λ_{ S } are P and S wavelengths, respectively. Hence, W is only a few times larger than the maximum wavelength of either the P or the S wave corresponding to ka = 0.5. As a result, the spatial averaging of seismograms for only one realization of a crack distribution removes the incoherent waves only partially, and, hence, the ensemble averaging process among many realizations is required for the total removal of the incoherent waves. Actually, we chose N_{ D } in Table 1 to achieve this purpose by increasing N_{ D } until coda-like wave trains effectively vanish from the synthetic mean wave.
4.2 Results for sparsely distributed cracks
The values of Q^{−1} and Δv derived from the synthetic and predicted mean waves in Fig. 6(a) are plotted in Figs. 6(d) and 6(e) for 0.5 < ka < 10, respectively, in which Δv is the phase velocity reduction due to scattering; it is defined as either Δv = α − v or Δv = β − v, depending on the modes of the incident waves. The agreement between the experimental values (from the synthetic mean wave) and the theoretical values (from the predicted mean wave) are fairly good on the whole, even including the ripple pattern of δv for high wavenumbers, as is expected from the good agreement between the synthetic and predicted mean waves in Fig. 6(a).
4.3 Results for more densely distributed cracks
We next show the results for the models with denser distributions (Models 2 and 3), focusing on the validity limit of the FAT.
In contrast, the disagreement between the experimental and theoretical results still remains not very serious for SV wave incidence, even in the case of ϵ = 0.1 (Figs. 10(d) to 10(f) and 10(j) to 10(l)). One may therefore say that the FAT will be nearly valid at least up to ϵ = 0.1 concerning SV waves. This is consistent with the conclusion of Paper I as well as that of Murai (2007) on SH wave scattering.
We notice that the case of P wave incidence on vertical cracks with ϵ = 0.1 was examined on the basis of Model 3b with N = 152, many more cracks than assumed in Model 3a adopted for the other three cases, for the same reason as before (Model 2). In Paper I, however, we revealed that the numerical errors of the FDM seem to weakly increase with an increasing number of cracks. Hence, one might infer that the failure of the FAT, in that case, is partly because of the numerical errors. To examine this inference, we recalculate Q^{−1} and Δ v for this case by replacing Model 3b with Model 3a. The result (not shown) is effectively unchanged from that depicted in Figs. 10(h) and 10(i), indicating that the FDM calculations were sufficiently accurate and the failure of the FAT, in this case, is not due to the numerical errors caused by too many cracks.
One may also infer that strong multiple scattering among the cracks enhances the numerical errors. This effect would be significant for a high crack density and especially for high wavenumbers, even if the sampling criterion (Section 2) is satisfied. However, we confirmed that changing Δh by a factor of two does not effectively change the numerical results for Model 3a for all the considered wavenumbers (figure omitted). This again verifies the accuracy of the present FDM simulations, even up to ϵ = 0.1 and the considered wavenumbers.
In summary, we have confirmed that the FAT is actually valid for low crack density (say, ϵ ~ 0.01) for all the cases considered. Its validity limit with regard to ϵ, however, seems to depend on the wave modes and, sometimes, the incident angles of waves to cracks. The FAT would be nearly valid with ϵ up to at least 0.1 for the case of SV wave incidence. In contrast, the validity limit of the FAT seems to be lower for P wave incidence. It may be, say, 0.05 for normal incidence, whereas even a lower value of ϵ may be preferable for a grazing incidence.
5. Discussion and Conclusions
In this paper, we have implemented a simple, but effective, finite-difference scheme to simulate P-SV wave scattering due to 2-D parallel cracks. We expressed traction-free, infinitesimally thin cracks in a finite-difference staggered grid in probably the simplest manner: we assumed arrays of grid points with zero shear traction. We split the grid points for vertical particle velocity on the planes into pairs of very close points, thus succeeding in expressing the NDD along the planes. As an accuracy test, we calculated the displacement discontinuity along an isolated crack acted on by harmonic waves using the present method and compared this with the corresponding results based on a reliable BIEM of Kawahara and Yamashita (1992). The test resultantly indicated that the present method yielded a sufficient accuracy for any wavenumber. As an application of the method, we next performed simulations of wave propagation in media with randomly distributed parallel cracks of the same length. We determined experimentally the attenuation and velocity dispersion induced by scattering from the synthetic seismograms, using a waveform averaging technique. It was shown that the results are well described by the FAT of Kawahara (1992), if the crack density ϵ is sufficiently low, say, 0.01. The FAT apparently remains to be always valid with ϵ up to at least 0.1 for SV wave incidence, whereas the validity limit seems to be lower for P wave incidence; it may be, say, 0.05 for normal incidence and even lower for grazing incidence.
To the authors’ knowledge, the present study is the first one to have successfully validated the FAT for cracked media, even if there exists mode conversion between P and S waves. Although we only dealt with the simplest type of cracks (i.e., parallel equal-sized traction-free cracks), some previous studies based similarly on numerical experiments showed that the FAT works well also for others kinds of cracks concerning SH wave scattering. The examples are cracks filled with a viscous fluid (Murai et al., 1995), randomly oriented cracks (Yoshida et al., 2003), and cracks of unequal lengths (Paper I). On the basis of the same approach as the present study, Kawahara et al. (2009) also proved that the FAT is nearly valid for SH wave scattering due to 2-D round cavities—possibly identified with high-aspect-ratio cracks—with a volume concentration up to 0.1. All these results suggests that the validity of the FAT would be universal as long as the scatterers are sparsely distributed, irrespective of the geometry and physical properties (boundary conditions) of the cracks, the wave modes, and, possibly, the spatial dimensions.
In contrast, the validity limit of the FAT for cracked media seems to depend on the wave modes. Moreover, this is as well as the incident angle in the case of P wave incidence. The details of the dependence of the validity limit on the incident angle, however, remain unrevealed since we considered here only two kinds of incident angles (0° and 90°); the case of oblique wave incidence should be investigated in future. It also remains to be clarified why the validity limit of the FAT seems to be lower for P wave incidence than for SV wave incidence.
As stated in Section 4, the overestimation by the FAT of Q^{−1} and Δ v observed for P waves in densely cracked media (Fig. 10) may be partly attributed to the imperfect randomness of distributions enhanced with an increasing ϵ. This implies that the validity of the FAT would be limited not only by ϵ but also by the spatial correlation among cracks. Murai (2007) demonstrated the importance of the spatial correlation by evaluating Q^{−1} and Δ v of SH waves in media with periodically arrayed cracks, and with ϵ = 0.1, through numerical experiments. The obtained Q^{−1} and Δ v were compared with those for (nearly) random crack distributions with the same ϵ. It was revealed that the periodicity (or strong correlation) of crack locations largely reduces Q^{−1} for ka < 1 and caused the FAT to fail. This result is qualitatively consistent with the present results. However, his result also indicated that the periodicity hardly affects Δ v, in contrast to the present result that Δ v is also reduced for ka < 1 with increasing ϵ. Note that Murai adopted a peak picking technique in evaluating Q^{−1} and Δ v, unlike us; how the choice of the evaluation methods affects the results would be of interest. Future works should also reveal the effect of the crack spatial correlation on Q^{−1} and Δ v concerning P-SV wave scattering.
We notice that Orlowsky et al. (2003) numerically evaluated the effective velocity (phase velocity in the long-wavelength limit), v_{ eff }, of elastic waves normally impinging on 2-D parallel empty cracks. Their results indicate that v_{ eff } of P waves for ϵ = 0.1 is, in our notation, about 0.7α (i.e., Δv/α ≈ 0.3) and that of SV waves is about 0.87β (i.e., Δ v/β ≈ 0.13). These values are roughly consistent with the theoretical values of Δv at ka = 0.1 for the same ϵ shown in Figs. 10(c) and 10(f). Unfortunately, a comparison of the experimental values shown in these figures with v_{ eff } by Orlowsky et al. (2003) may not be straightforward because the former were measured only for ka > 0.5. Figure 10(c, i, f, l), as well as Kawahara (1992), also suggests that a medium with parallel empty cracks is strongly anisotropic for P waves but nearly isotropic for SV waves; for example, Fig. 10 shows that the theoretical value of Av/± at ka = 0.1 for P waves for normal incidence is nearly ten times larger than that for parallel incidence. Note also that Saenger and Shapiro (2002) estimated v_{ eff } in media with 2-D randomly orientated empty cracks. Comparison of their results with ours implies that v_{ eff } of P waves for randomly oriented cracks lies between those for a normal and parallel incidence on parallel cracks, whereas v_{ eff } of S waves is nearly independent of the crack orientations. These features are shared by media with penny-shaped empty cracks (Crampin, 1984).
Although the simplicity of the present FDM is a great advantage, this method, as it stands, has some limitations on the geometry of cracks. One of these is that cracks must be parallel to the grid lines so that one can only choose, for 2-D grids, one or two orientations of cracks simultaneously. For a wider application of the present FDM (e.g., a medium with randomly oriented cracks), it is crucial to be able to incorporate arbitrarily oriented cracks into the standard staggered grid used in the present FDM. We infer that this will be achieved on the basis of the staircase approximation as has been successfully applied to irregular free surfaces and irregular liquid-solid boundaries (Ohminato and Chouet, 1997; Okamoto and Takenaka, 2005; Takenaka et al., 2009). It is worthwhile stating, however, that there are no limitations, even in the present FDM, on the incident angle of elastic waves to a crack, although we considered here only two incident angles. For example, one could simulate scattering by parallel cracks of a plane wave with an oblique incidence if the source is appropriately given in the grid.
Another limitation of the present FDM may be that it can treat non-intersecting empty cracks only. Relaxation of these limitations should be attempted in future works. A recent development has been achieved by Shiina et al. (2011), who succeeded in modeling 2-D cracks filled with a viscous fluid in the standard staggered grid and simulated the consequent scattering of SH waves with high accuracy. Extension of their method to P-SV wave scattering would be of value.
Notes
Declarations
Acknowledgments
The authors would like to thank Satoshi Nozawa and the members of his laboratory for using their computing environment and fruitful discussion. The critical comments of Marcel Frehner and an anonymous reviewer helped to largely improve the manuscript. This study was partly supported by the Earthquake Research Institute Cooperative Research Program (2000-B-07, 2003-B-04). All the figures in this paper were generated using the GMT software of Wessel and Smith (1998).
Authors’ Affiliations
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