The effects of near-source heterogeneity on shear-wave evolution

Christopher S. Sherman1, James Rector1, and Steven Glaser1

ABSTRACT

The Born and Rytov approximation, radiative transfertheory, and other related techniques are commonly usedto model features of wave propagation through hetero-geneous geologic media such as scattering, attenuation,and pulse-broadening. However, due to the underlying as-sumptions about the scattering direction and the referenceGreen’s function, these methods overlook important featuresof the wavefield such as mode conversion and near-fieldterm coupling. These effects are particularly importantwithin the predicted S-wave nodes of a seismic source, sowe analyzed the problem of wave propagation beneath a ver-tical-point force on the surface of a heterogeneous, elastichalf space. To do this, we generated a suite of 3D syntheticheterogeneous geologic models using fractal statistics andsimulated the wave propagation using the finite-differencemethod. We derived an estimate for the effective source ra-diation patterns, and we used these to compare the results ofthe models. Our numerical results showed that, due to acombination of mode conversion and near-source couplingeffects, S-wave energy on the order of 10% of the P-waveenergy is generated within the shear-radiation node. In somecases, this S-wave energy may occur as a coherent pulse andmay be used to enhance seismic imaging.

INTRODUCTION

Many observations of geologic media show that they are hetero-geneous over a wide range of scales (Turcotte, 1997; Sato et al.,2012). In spite of this, many geophysical analyses assume that geo-logic media are effectively homogeneous for the scales of interest,or that a simple low-frequency heterogeneous model may character-ize them. These assumptions are often necessary to obtain usefulsolutions in seismic studies, especially while considering the elastic

wave equation. Over the past several decades, the seismic commu-nity has developed many powerful numerical tools and approximatemethods to evaluate higher order wave propagation effects intro-duced by heterogeneity.One common approach is to consider a version of the wave equa-

tion, in which velocity varies randomly as a function of position,commonly referred to as the stochastic wave equation. Becauseof their efficiency, approximate solutions to the stochastic waveequation such as Born and Rytov methods are commonly usedin areas dominated by low contrast and relatively high-frequencyheterogeneity. Both of these methods consider a reference Green’sfunction for an equivalent homogeneous method, which may be de-termined using classical geophysical techniques, and estimate thedeviations from the solution introduced by the heterogeneity. Inthe Born approximation, the secondary wavefield is modeled asa linear addition to the reference solution. This method is most ac-curate when the seismic sources and receivers are in the same lo-cation (i.e., the backscattering regime) and is often used in radaranalysis. In the Rytov approximation, the secondary wavefield ismodeled as an exponential function of amplitude and phase varia-tions around the reference Green’s function. This method is mostaccurate when the separation between the seismic sources andreceivers is large (i.e., the forward scattering regime) and is the mostcommon approximation in seismic analyses (Chapman and Coates,1994; Marks, 2006; Sato et al., 2012). Both of these approximationsare limited by the amplitude of the heterogeneity, the possibility ofmultiple scattering, and higher order wave propagation effects suchas mode conversion. A related, heuristic solution to the stochasticwave equation is radiative transfer theory (RTT), which was devel-oped for radar analysis and is useful for the analysis of seismographenvelopes. This method considers the transfer of energy through amedium neglecting phase information, and it uses scattering coef-ficients determined using the above approximations (Przybilla et al.,2006; Sato et al., 2012).Although these approximation methods may be used in a deter-

ministic analysis, it is more common to formulate them in terms of astatistical analysis. This approach improves the stability of the sol-utions, and it permits tractable characterization of large volumes of

Manuscript received by the Editor 24 May 2013; revised manuscript received 18 February 2014; published online 12 June 2014.1University of California Berkeley, Department of Civil and Environmental Engineering, Berkeley, California, USA. E-mail: cssherman@berkeley.edu;

glaser@berkeley.edu; jwrector@lbl.gov.© 2014 Society of Exploration Geophysicists. All rights reserved.

T233

GEOPHYSICS, VOL. 79, NO. 4 (JULY-AUGUST 2014); P. T233–T241, 13 FIGS.10.1190/GEO2013-0199.1

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heterogeneous media from measurements. For instance, by consid-ering the log-average values of the amplitude and phase fluctuationsof the wavefield in the Rytov approximation, researchers have de-veloped useful relationships for the most probable seismic pulsefrom a source. The statistical treatment of Born, Rytov, and RTTmethods has yielded many other important relationships for wavephenomena in heterogeneous media: seismic coda (Aki and Chouet,1975; Saito et al., 2003), scattering attenuation (Shapiro and Kneib,1993), scattering dispersion (Shapiro et al., 1996; Saito, 2006), trav-eltime anomalies (Baig et al., 2003), most probable seismic arrivals(Müller and Shapiro, 2001; Müller et al., 2002), etc. Moreover, be-cause of their efficiency, these approximate methods have been usedto develop inverse analyses for the statistical characteristics ofheterogeneity in a geologic system. Some examples of this includelaboratory scale (Nishizawa and Kitagawa, 2007; Nishizawa andFukushima, 2008), regional scale (Przybilla and Korn, 2008), andglobal scale analysis of heterogeneity (Shearer and Earle, 2008).The finite-difference (FD) and finite-element (FE) methods are

additional methods for solving the problem of wave propagationin heterogeneous media. These solve for the complete Green’s func-tion directly from the momentum equation, and they do not relyupon assumptions about the amplitude or distribution of the hetero-geneity (Aki and Richards, 2002). Their accuracy is not limited tothe forward or backward scattering regimes, and they naturally in-clude features such as multiple scattering, resonant scattering, andmode conversion (Levander and Hill, 1984; Frankel and Clayton,1986). In addition, these methods are commonly used to verify theaccuracy of the approximation methods discussed above (Shapiroand Kneib, 1993; Shapiro et al., 1996; Müller et al., 2002; Przybillaet al., 2006). The primary limitation of the FD/FE methods is thatthey require significant computational resources. To accurately dis-cretize a heterogeneous model and avoid numerical dispersion ef-fects, a very fine grid spacing must be used. These requirements arecompounded if we consider 3D models instead of 2D models. Dueto recent advances in the accessibility of supercomputing, thesenumerical models are becoming more popular in the seismiccommunity.The purpose of our research is to investigate the effects of near-

source heterogeneity on wave propagation effects. Developing theapproximate methods for modeling heterogeneous wave propaga-tion requires assumptions about the reference Green’s functionto make the mathematics reasonable. A common assumption isto omit the near- and intermediate-field terms of Green’s function(Sato et al., 2012). Although these terms are very small in the farfield, they may play an important role in the development of S-waves. In a recent study, Przybilla and Korn (2008) observe a sig-nificant amount of S-wave energy in their finite-difference modelsnot predicted by the RTT method. They attribute this increase to abreakdown of the Born approximation, which is used to calculatescattering coefficients near the source. In a recent field study ofnear-surface vertical-array data by B. Hardage (personal communi-cation, 2012), a significant amount of shear radiation was observeddirectly beneath the source in the shear node, some of which arrivedas a coherent pulse. This additional shear energy was not accountedfor by conventional approximation techniques, and it is attributed tothe interaction with near-source heterogeneity.To investigate the effect of near-source heterogeneity on the evo-

lution of S-waves, we begin by generating a set of synthetic 3Dheterogeneous isotropic velocity models using fractal statistics.

We use the elastodynamic finite-difference code E3D to calculatethe full wavefield for a vertical point source on the surface in thesemodels. By manipulating Green’s function for a general pointsource, we estimate the effective source radiation patterns foreach simulation. Finally, we evaluate the ratio of S- to P-waveenergy beneath the source for a range of fractal heterogeneitycharacteristics.

MODEL OF GEOLOGIC HETEROGENEITY

One of the outstanding features of heterogeneity in a geologicsystem is scale invariance — that, without an external reference,one cannot distinguish between images taken at very different scales(Turcotte, 1997). This behavior may be described using fractal sta-tistics, which is a common tool in geostatistics and scattering analy-ses. In our models, we assume that the near-source heterogeneity ischaracterized by a self-affine fractal, which permits us to use someuseful scaling relationships. This leads to three important dimen-sionless parameters:

• Scaled distance r∕λo, where r is the distance from the sourceand λo is the dominant wavelength.

• Fractal amplitude ε, which is the percent standard deviationof the heterogeneity from the mean.

• Fractal exponent β, which describes the correlation of theheterogeneity in space.

To create a synthetic fractal distribution, we begin by generating amatrix of normally distributed random values. We apply the fractalfilter S given by equation 1 in the frequency domain, where kx, ky,and kz are the wavenumber components and ax, ay, and az are scal-ing parameters (Turcotte, 1997; Browaeys and Fomel, 2009). Bar-ton and Le Pointe (1995) observe that in layered and isotropicmedia, β does not change significantly with direction; however,in layered media ε may change significantly with direction. There-fore, for an isotropic-fractal model, we choose a single value of β,set ax ¼ ay ¼ az ¼ 1, and apply the filter. To generate a layeredmodel, we set the scaling parameters to be ax ¼ ay ¼ b andaz ¼ 1, where b is greater than one. After applying the filter, wescale the results to have the desired fractal amplitude and mean:

S ¼ ½ðaxkxÞ2 þ ðaykyÞ2 þ ðazkzÞ2�−β∕4. (1)

For waves passing through this type of heterogeneous material,the choice of fractal amplitude governs the magnitude of the scat-tered wavefield, and the fractal exponent governs its characteristics.With an estimate of measurement noise, these values may be esti-mated directly from borehole logs, outcrop measurements, or othergeophysical observations (Klimeš, 2002). These values may also beestimated by considering the depositional environment in whichthey were formed. Browaeys and Fomel (2009) recommend thatquasicyclical deposition be characterized by 1 < β < 2 and that tran-sitional deposition be characterized by 2 < β < 3. In addition, inte-ger values of β yield interesting model behavior: For β ¼ 0, theresult is white noise; for β ¼ 1, the result is flicker noise (a commonfeature in electronic systems); and for β ¼ 2, the result is Browniannoise (Turcotte, 1997; Browaeys and Fomel, 2009).Another important consideration is the number of degrees of free-

dom allowed in the model. For each elastic parameter (P-wavevelocity VP, S-wave velocity VS, density ρ, etc.), it is necessaryto generate a synthetic model. Each of these parameters may have

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different values of β and ε and may be generated with different ran-dom seeds. In our models, we permit only a single degree of free-dom to simplify the analysis and to prevent excessive impedancecontrasts that may occur as the result of interference between multi-ple random models. The variables VP, VS, and ρ are generatedusing the same random seed; VP and VS have the same ε andρ has ε ¼ 0. In our reference geologic model, we choose anaverage VP ¼ 3.0 km∕s, an average VS ¼ 1.73 km∕s, and a con-stant ρ ¼ 2.7 g∕cm3. To facilitate the analysis of the secondarywavefield, we limit the fractal heterogeneity to the upper thirdof the synthetic models. In this study, we consider a range offractal dimensions 1.4 < β < 1.8, and a range of fractal amplitudes0 < ε < 3.3%. A cross section through a synthetic model realizationwith ε ¼ 1% and β ¼ 1.7 is given in Figure 1.Note that, although we consider statistical anisotropy in our

analysis using layered fractal models for simplicity, we do not con-sider the influence of intrinsic anisotropy in velocity. The effects ofintrinsic anisotropy on wave propagation are the subject of activeresearch in the literature (Tsvankin and Chesnokov, 1990; Tsvankinet al., 2010)

NUMERICAL ANALYSIS

We choose to use the elastodynamic finite-difference code E3Dto calculate Green’s functions in our analysis. E3D is developed atLawrence Livermore National Laboratory, is fourth-order accuratein space, second-order accurate in time, and uses a standard stag-gered-grid formulation. It has been applied to solve problems rang-ing from earthquake simulation to seismic exploration, and itwas notably used to generate an elastic portion of the SEG/EAGEacoustic numerical model data set (Larson and Grieger, 1998). Wegenerated approximately 100 distinct 3D heterogeneous velocitymodels, each 15λo × 15λo × 15λo in size, with a grid spacing ofλo∕24. A schematic of the model geometry is included in Figure 2.To promote numerical stability in the models, we require a time stepsize less than 1∕10th required by the Courant condition. We apply afree-surface boundary condition to the top of the models, absorbingboundary conditions to the sides and bottom of the models, andimpose a 25-grid-point layer of highly attenuating material alongthe quiet boundaries. During our analysis, we observed that it

was necessary to damp the fractal amplitude of the models withinfour grid points of the boundaries to avoid numerical instabilities.To simulate the seismic wavefield, we placed a vertical point

force along the top center of the model ðx; y; zÞ ¼ ð7.5λo; 7.5λo;0Þ. We chose the source input wavelet to be the integral of aRicker wavelet with a nominal center frequency of 0.5 Hz, whichcorresponds to a dominant wavelength of 6 km. Every 10 timesteps, we recorded the velocity, divergence, and curl of the wave-field within a vertical cross section at x ¼ 7.5λo. Each modelwas run two times the period required for the direct S-wave to tra-verse the model. The simulations were completed on a workstationcontaining two hex-core Xeon processors using the open MPI pro-tocol; total code runtime and postprocessing took approximately 4 hper model.

EFFECTIVE SOURCE RADIATION PATTERNS

In our analysis, we derive an effective radiation pattern Reff bymanipulating the general form of a Green’s function. For reference,the far-field Green’s function for a S-wave due to a point force in ahomogeneous, infinite elastic medium is given by equation 2 (Akiand Richards, 2002). Here, usi is the shear displacement, ρ is thedensity, VS is the S-wave velocity, r is the distance from the source,t is the time, δij is the Kronecker delta, γi is a direction cosine, and δis the Dirac delta function. A more general form for Green’s func-tion for a point source is given in equation 3, where S is the spread-ing function, R is the radiation pattern, θ is the polar angle, and T isthe source-time function (the contributions from equation 2 arenoted by the square brackets):

usi ¼�

1

4πρV2Sr

�½δij − γiγj�

�δ

�t −

rVS

��; (2)

ui ¼ SðrÞRðθÞT�t −

rV

�. (3)

We use equation 3 as the model for Green’s functions in ouranalysis. To relate this model to the output from E3D, we beginby calculating the S-wave potential for an arbitrary wavefield χ,which we define as the absolute value of the curl of the wavefieldparticle velocity. The value of χ in spherical coordinates is given inequation 4, where φ is the azimuth, the overdot indicates the time

x / λo

z / λ

o

0 5 10 15

0

5

10

15 2.85

2.9

2.95

3

3.05

3.1

3.15

VP (km/s)

Figure 1. Cross section through a synthetic fractal velocity modelwith VP ¼ 3 km∕s, ε ¼ 1%, and β ¼ 1.7. The heterogeneity is con-fined to the upper third of the model. Figure 2. Model geometry for the finite-difference simulations.

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derivative, and the subscripts indicate the direction and spatialderivatives. This value naturally separates the shear energy, and al-lows us to obtain the effective shear radiation pattern. From sym-metry, we assume that the radial and azimuthal components of u arezero to obtain an approximation for χ:

χ ¼ 1

r sin θ

hðuϕ sin θÞ;θ þ ður − uϕÞ;ϕ

i

þ 1

r

hðruθ þ ruϕÞ;r − ur;θ

i

≈1

ruθ þ uθ;r. (4)

We then calculate the P-wave potential for an arbitrary wavefieldΦ, which we define as the divergence of the velocity wavefield. Thevalue of Φ in spherical coordinates is given by equation 5. Thisvalue naturally separates out the compressional energy, and allowsus to estimate the effective P-wave radiation pattern. Again, we as-sume that the tangential and azimuthal components of velocity arezero to obtain an approximation for Φ:

Φ ¼ 1

r2ðr2urÞ;r þ

1

r sin θ½ððuθ sin θÞ;θ þ uϕ;ϕÞ�

≈2

rur þ ur;r. (5)

Next, we insert the appropriate Green’s function (equation 3) intothe approximations for the S- and P-wave potentials (equations 4and 5), and we exchange the closed-form radiation patterns R, withthe effective radiation pattern estimate Reff . Because S and Reff donot vary in time, we are able to use the L2-norm to collapse the timedimension in T, χ, and Φ. The resulting expressions are given inequations 6 and 7, in which the double vertical brackets representthe norm:

kχk ¼�����1

rþ ∂

∂r

��SðrÞRs

effðθÞT�t −

rVS

������; (6)

kΦk ¼�����2

rþ ∂

∂r

��SðrÞRp

effðθÞT�t −

rVP

������. (7)

For our analysis, we insert the spreading function and calculatethe spatial derivatives. By rearranging these expressions and assum-ing that r is large, we obtain estimates for the effective S- andP-wave radiation patterns in equations 8 and 9. Although it is notincluded explicitly, we expect small variations in these expressionswith distance from the source:

RseffðθÞ ¼ �4πρV3

SrkχkkTk ; (8)

RpeffðθÞ ¼ �4πρV2

PrkΦk

k 1r T − 1

VPTk ≈ �4πρV3

PrkΦkkTk . (9)

To verify the accuracy of the effective radiation pattern estimates,it is useful to consider the closed form Green’s functions for pointforce on the surface of a homogeneous, elastic half-space. The

solution for these Green’s functions is called Lamb’s problem,and it has been extensively studied since its introduction in 1904(Lamb, 1904; Johnson, 1974). Assuming that the material is a Pois-son solid, the closed-form radiation patterns for the S- and P-wavesfor a vertical point force on the surface are given in equations 10 and11, respectively (White, 1983):

Rs ¼ sin θ cos θ½3 − sin2 θ�1∕2½1 − 2 sin2θ�2 þ 12 sin2 θ cos θ½3 − sin2 θ�1∕2 ; (10)

Rp ¼ cos θ½1 − 6 sin2 θ�1∕2½1 − 6 sin2 θ�2 þ 12 sin2 θ cos θ½3 − 9 sin2 θ�1∕2 .

(11)

RESULTS

Homogeneous velocity model

To evaluate the accuracy of our numerical models and the effec-tiveness of the radiation pattern estimates, we begin by consideringa homogeneous reference geologic model (ε ¼ 0%) in E3D. We useequations 8 and 9 to estimate Rs

eff and Rpeff at a constant distance of

5λo from the source, and compare the results to closed-form valuesof the radiation patterns (equations 10 and 11) in Figure 3. For an-gles greater than 2° from the horizontal, the amplitude and shape ofthe P-wave radiation patterns agree excellently. Outside of thisrange, the calculated effective radiation pattern is much larger thanexpected because of the contribution from the surface wave. Theamplitude of the S-wave radiation pattern also agrees well forangles greater than 5°, except that there is an issue with the phasereversal predicted approximately 55°. We believe that the disagree-ment between these results is most likely due to the implementationof the point force in E3D, rather than being an issue with the ef-fective radiation pattern estimate.In our analysis of the effective radiation patterns, we are particu-

larly interested in the results directly beneath the source. For theclosed-form Green’s function results, we see that this location cor-responds to the maximum P-wave radiation and a node in theS-wave radiation. The numerical results also agree with this obser-vation, except that we see a small amount of shear radiation withinthe node. This unaccounted for energy is due to a combination ofthe near-field terms, which are not included in the closed formGreen’s function, and numerical noise in the model. As the distanceto the source decreases, the ratio between Rs

eff and Rpeff in the sim-

ulation increases significantly. We use this ratio as the baseline forcomparison between the homogeneous reference and the hetero-geneous geologic models.

Isotropic fractal velocity models

In this section, we consider the results of several isotropic,heterogeneous geologic models within a range of β and ε. For eachof these models, we apply the same methodology in the previoussection to model the wavefield and calculate the effective radiationpatterns. The calculated Rs

eff and Rpeff for a single model realization,

with β ¼ 1.7, ε ¼ 1.5%, and r ¼ 5λo, is compared with the resultsof the homogeneous reference model in Figure 4.As expected, we observe significant variations from the homo-

geneous reference model, although the average shape remains

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relatively unchanged. Note that because we do not require the veloc-ity model to be axially symmetric, the effective radiation pattern isasymmetric. In addition, we see a significant increase in S-waveamplitude directly beneath the source. We also plot the value ofRseff∕R

peff , calculated throughout the model domain, in Figure 5.

We omit the areas in close proximity to the absorbing boundariesbecause they may contain a significant amount of numerical noise.The radial variations in this ratio, which tend to stabilize forz > 5λo, reflect the complex interaction of the wavefield withthe model heterogeneity. Another important ob-servation is that Rs

eff∕Rpeff within the S-wave node

is increased significantly in this model.The increase in the S-wave energy beneath the

source is likely due to scattering of the shearwavefield, mode conversion of the compres-sional wavefield, and a near-field heterogeneitycoupling effect. The timing character of this en-ergy may give some insight as to its source. Weexpect that the contributions from scattering andmode conversion to be mostly incoherent and tonot focus back toward the source location. In ad-dition, we expect that as the source-receiver dis-tance increases, the relative contribution of modeconversion will decrease, due to the lowering ofthe scattering angle. On the other hand, the near-field coupling effect, which describes the interac-tion of the near-field energy with the hetero-geneity near the source, will likely produce acoherent S-wave arrival that will trace back to-ward the source. The horizontal wavefield, mea-sured directly beneath the source for the modelrealization shown in the previous figures and cor-rected for spreading, is included in Figure 6.In these results, we see incoherent S-wave en-

ergy arriving immediately following the arrivalof the direct P-wave, which is the result of S-P-S mode-converted energy. Around the pro-jected arrival time of a direct S-wave from thesource, we see a coherent arrival that we attributeto the near-field coupling term. In this case, theshape of the coherent arrival is the same as is ex-pected for the far-field arrivals. After this arrival,incoherent energy continues to arrive in what would normally beidentified as a coda wave, and is the result of the scattered S-wavefield.To explore the effect of the individual isotropic fractal parameters

on the model results, we begin by holding β constant at a value of1.7, while we consider values of ε from 0% to 3%. For each value ofε, we generate at least five independent, synthetic-heterogeneousmodels, simulate the wavefield using E3D, and calculate the effec-tive radiation patterns beneath the source. Because the variations inthe effective shear radiation may differ by orders of magnitude forsets of β and ε, we choose to consider the log-average values ofthese results in our analysis. We plot the ratio of the S- to P-waveradiation as a function of depth and ε in Figure 7. Note that becausewe are interested in contributions from the coherent and incoherentwavefield, we do not stack the individual realizations. Also note thatbecause of symmetry shear radiation directly beneath the sourcefor ε ¼ 0% is expected to be zero. However, due to the numerical

limitations of the models, the estimated shear radiation at thislocation is nonzero.We observed that, over the range of ε considered in this analysis,

the average value of Rseff∕R

peff exceeds that of the reference model

and increases with ε. As depth increases, this ratio tends to increasewith respect to the reference model until it reaches a steady value at5λo (the limit of the heterogeneous zone). In addition, the variationof these results decreases with depth, reflecting the self-averagingnature of this process (Müller and Shapiro, 2001). The deviations

0.2 0.4 0.6 0.8 1

30

60

90

120

150

180 0

φ (°)

R0.1 0.2 0.3 0.4 0.5

83

87

90

93

97

08001

φ (°)

Ra) b)

Figure 3. (a) Comparison of the effective radiation patterns Rseff and R

peff , for the homo-

geneous reference model (black) and the closed-form solution (red) and (b) close-upview of the shear radiation patterns.

0.2 0.4 0.6 0.8 1

30

60

90

120

150

180 0

φ (°)

R0.1 0.2 0.3 0.4 0.5

83

87

90

93

97

08001

φ (°)

Ra) b)

Figure 4. (a) Comparison of the effective radiation patterns, Rseff and Rp

eff , for an iso-tropic heterogeneous model with β ¼ 1.7, ε ¼ 1.5%, and r ¼ 5λo (black), and thehomogeneous reference model (red) and (b) close-up view of the shear-radiationpatterns.

x / λo

z / λ

o

–6 –4 –2 0 2 4 6

0

2

4

6

8

10 0.1

1

10

Rs/R

p

Figure 5. The Rseff∕R

peff for an isotropic-fractal model throughout

the domain (β ¼ 1.7 and ε ¼ 1.5%).

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seen for large depths and small ε are primarily due to boundarycondition issues.Next, we investigate the effect of fractal exponent on the effective

source radiation. We choose a range of β from 1.4 to 1.8, with a

constant value of ε ¼ 1.6%. For each value of β, we consider atleast five model realizations, and calculate the log-average values.The resulting values of Rs

eff∕Rpeff in this analysis are included in

Figure 8.For this range of β, the variations in Rs

eff∕Rpeff are very small, and

there are no trends apparent in the results. For larger changes in β,we expect the dominant regime of scattering to change and the char-acter of the results to change dramatically (Browaeys and Fo-mel, 2009).

Layered fractal velocity model

Although an isotropic fractal is a simple and useful model formany geologic media, a layered fractal model is often more appro-priate. This is especially the case for sedimentary structures, such asshale. To generate a layered model, we select values for fractal am-plitude ε, fractal dimension β, and a horizontal scaling factor b > 1

(see equation 1). Figures 9 and 10 show the effective radiation pat-tern and wavefield beneath the source for a model with ε ¼ 1.6%,β ¼ 1.7, and b ¼ 10. In this model, we see many of the same fea-tures as in the isotropic case — deviations from the radiation in thehomogeneous model and an increase in S-wave energy beneath thesource. However, compared with the isotropic model, the shear ra-diation beneath the source is decreased and the radiation at somelower angles is increased.To evaluate the expected effect of heterogeneity shape on the ef-

fective radiation patterns, we vary ε similar to the previous section.We hold β at a constant value of 1.7, b at 10, vary ε from 0% to3.3%, and calculate the log-average Rs

eff∕Rpeff for at least five model

realizations. The results of this analysis are given in Figure 11. Us-ing a layered fractal model, we see the same major trends as in theisotropic model. As ε increases, the expected value of Rs

eff∕Rpeff

tends to increase, and as depth increases the variance decreases.A significant difference between the results of the isotropic and lay-ered models is that, for a given value of ε, a layered velocity modelwill produce twice the amount of S-wave beneath the source onaverage. Because we do not see a strong dependence on β forthe isotropic model, we expect the same for the layered models.

Reflecting velocity model

To evaluate whether the coherent S-wave pulse is useful for seis-mic imaging, we consider an isotropic fractal velocity model similarto those used in the previous sections, but with a 30% lower averagevelocity in the upper layer. We use E3D to simulate the seismicwavefield from a vertical point force on the surface of the model,and record the results for a vertical trace beneath the source. Themeasured horizontal velocity for a model with ε ¼ 1.6% and β ¼1.7 is included in Figure 12.For a model with stacked homogeneous layers, we still expect a

S-wave node to exist beneath the vertical point source. However, aswe observed for the previous heterogeneous geologic models, shearenergy is introduced into this area from scattering, mode conver-sion, and near-source coupling effects. The coherent S-wave pulsein this model, and the incoherent arrivals around it, reflect off theinterface at a depth of 5λo and may be traced back to the surface. Inaddition to determining the location of the reflecting interface, geo-physicists often attempt to determine the statistical characteristics ofthe geologic medium through a statistical analysis of the traveltimevariations seen in reflection or refraction data (Iooss, 1998; Kaslilar

t (s)

z / λ

o

0 10 20 30 400

2

4

6

8

10

Figure 6. Horizontal wavefield directly beneath the point source foran isotropic-fractal model (β ¼ 1.7 and ε ¼ 1.5%).

0 2 4 6 8 10

10–2

10–1

100

z / λo

Rs /

Rp

0.0%0.3%0.8%1.7%2.5%

Figure 7. Effect of fractal amplitude (ε) on Rseff∕R

peff beneath the

source.

0 2 4 6 8 10

10–2

10–1

100

z / λo

Rs /

Rp

1.41.51.61.71.8

Figure 8. Effect of fractal dimension (β) on Rseff∕R

peff beneath the

source.

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et al., 2008). These additional S-waves contain useful informationconcerning the statistics of the source medium, as discussed in theprevious sections, and may be used to supplement the seismic im-aging applications.

DISCUSSION

Approximate methods for solving the problemof the elastic wave equation in heterogeneousmedia, such as the Born approximation andRTT, are very efficient and effective at model-ing the effect of scattering, attenuation, anddispersion of waves. However, previous numeri-cal investigations have discovered that S-waveenergy, on the order of 10% of the P-wave en-ergy, may occur in these media that is not ac-counted for by the approximations (Przybillaand Korn, 2008). Our numerical analysis showsthat, even for a small value of ε, a synthetic frac-tal geologic model is capable of producing thisenergy. We believe that because of the characterand the timing of this additional shear energy, itis the result of S-P-S mode conversion and anear-source coupling effect. Two of the fundamental assumptionsof the Born and Rytov approximations prevent them from modelingthese phenomena: These approximations assume that the angle ofscattering is either very high (backward scattering) or very low (for-ward scattering), which implies that mode conversion of the inci-dent wavefield is negligible. In addition because each of thesemodels rely on the far-field Green’s functions as a reference, theinteraction of the near-field term with the near-source heterogeneitythrough scattering and higher order mechanical deformation (i.e.,the near-field coupling effect) is overlooked.The results of our analysis suggest that we may develop some

correlations between this additional shear energy introduced bythese higher order terms and the fractal characteristics of themedium. We plot the log-average value of Rs

eff∕Rpeff , measured in

the far field, as a function of ε and fractal shape in Figure 13.We observe a strong linear trend in the results for the isotropic frac-tal (R2 ¼ 0.99) and layered fractal (R2 ¼ 0.98) models. We mayalso draw some correlations between the characteristics of the co-herent S-wave pulse and the medium characteristics. We observethat the shape of this pulse does not vary significantly with εand β. However, as ε increases, the amplitude of the pulse tendsto increase as is expected.The drop in the expected value of Rs

eff∕Rpeff for the layered veloc-

ity models is related to the horizontal wavenumber scaling factor, b,in equation 1. As b increases, the synthetic models are biased to-ward lower frequencies in the horizontal direction, and the averageshape of the heterogeneity changes from a sphere to an oblate sphe-roid. In effect, this reduces the density of vertically oriented velocitydiscontinuities in the model, which are important for the horizontalscattering potential and the effectiveness of mode conversion of theprimary wavefield.Although we do not consider intrinsic anisotropy in our numeri-

cal models, we may draw some conclusions about its expected ef-fect on the S-wave radiation beneath the source. In anisotropicmedia, we expect that the wavefield energy will tend to focus indirections where velocity is maximum and will defocus wherevelocity is minimum (Tsvankin and Chesnokov, 1990; Tsvankin,

1995). In our results, we expect the same trends for a VTI mediumwith minimum velocity in the vertical direction and we expect adecrease in the shear energy generated beneath the source.By comparing the trends in the results for ε and β (Figures 7 and

8), we see that the expected value of Rseff∕R

peff is much more sen-

sitive to the choice of ε than β. This result is expected because β

0.2 0.4 0.6 0.8 1

30

60

90

120

150

180 0

φ (°)

R0.1 0.2 0.3 0.4 0.5

83

87

90

93

97

08001

φ (°)

Ra) b)

Figure 9. (a) Comparison of the effective radiation patterns, Rseff and Rp

eff , for a layeredfractal model with ε ¼ 1.6%, β ¼ 1.7, and b ¼ 10 (black), and the homogeneous refer-ence model (red) and (b) close-up view of the shear-radiation patterns.

t (s)z

/ λo

0 10 20 30 400

2

4

6

8

10

Figure 10. Horizontal wavefield directly beneath the point sourcefor a layered fractal model (ε ¼ 1.6%, β ¼ 1.7, and b ¼ 10).

0 2 4 6 8 10

10–2

10–1

100

z / λo

Rs /

Rp

0.0%0.3%0.8%1.7%2.5%

Figure 11. Effect of fractal amplitude (ε) on Rseff∕R

peff for a layered

fractal model directly beneath the source.

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influences the correlation of the heterogeneity within the medium,and therefore the character of the scattered wavefield. On the otherhand, ε influences the amplitude of the heterogeneities, which willdirectly affect the amplitude of scattering. These effects are evidentwhen attempting to invert for the fractal characteristics of an object.While attempting to invert regional seismic data in the western Bo-hemia region, Klimeš (2002) finds that the inversion for the regionalε is stable and reliable. This makes sense, considering the linearrelationship we have observed between the scattered energy andε. Klimeš (2002) also finds that the inversion for the regional Hurstexponent, which is related to β, is difficult and nonlinear. This ob-servation is also supported by our analysis, as we did not see anystraightforward trends in our results with β.

CONCLUSION

The interaction of the seismic wavefield with geologic hetero-geneity is commonly modeled using the Born or Rytov approxima-tions. However, due to basic assumptions about the heterogeneitycharacteristics, mode conversion, and the reference Green’s func-tions, some aspects of the secondary wavefield are overlooked.We generate synthetic fractal models and use finite differencesto investigate the generation of additional S-waves directly beneath

a point source on the surface. Our results show that this shear energyincreases linearly with fractal amplitude and is not a strong functionof fractal exponent. In addition, the part of the additional shear en-ergy that is coherent is due to a near-source coupling effect. Theseresults have important implications for the predicted coverage ofS-S seismic imaging because we have demonstrated that a signifi-cant amount of S-wave energy may be produced directly beneaththe source without the need for a traditional shear source.

ACKNOWLEDGMENTS

We would like to thank S. Larson at Lawrence Livermore Na-tional Laboratory for the use of the finite-difference code E3D. Thisresearch was made possible by grants from the National ScienceFoundation, CMMI-0919595, and CMMI-0727726.

REFERENCES

Aki, K., and B. Chouet, 1975, Origin of coda waves: Source, attenuation,and scattering effects: Journal of Geophysical Research, 80, 3322–3342,doi: 10.1029/JB080i023p03322.

Aki, K., and P. G. Richards, 2002, Quantitative seismology: University Sci-ence Books.

Baig, A. M., F. A. Dahlen, and S. H. Hung, 2003, Traveltimes of waves inthree-dimensional randommedia: Geophysical Journal International, 153,467–482, doi: 10.1046/j.1365-246X.2003.01905.x.

Barton, C. C., and P. R. Le Pointe, 1995, Fractals in petroleum geology andearth processes: Springer.

Browaeys, T. J., and S. Fomel, 2009, Fractal heterogeneities in sonic logsand low-frequency scattering attenuation: Geophysics, 74, no. 2, WA77–WA92, doi: 10.1190/1.3062859.

Chapman, C. H., and R. T. Coates, 1994, Generalized Born scattering inanisotropic media: Wave Motion, 19, 309–341, doi: 10.1016/0165-2125(94)90001-9.

Frankel, A., and R. W. Clayton, 1986, Finite difference simulations of seis-mic scattering: Implications for the propagation of short-period seismicwaves in the crust and models of crustal heterogeneity: Journal ofGeophysical Research, 91, 6465–6489, doi: 10.1029/JB091iB06p06465.

Iooss, B., 1998, Seismic reflection traveltimes in two-dimensional sta-tistically anisotropic random media: Geophysical Journal International,135, 999–1010, doi: 10.1046/j.1365-246X.1998.00690.x.

Johnson, L. R., 1974, Green’s function for Lamb’s problem: GeophysicalJournal of the Royal Astronomical Society, 37, 99–131, doi: 10.1111/j.1365-246X.1974.tb02446.x.

Kaslilar, A., Y. A. Kravtsov, and S. A. Shapiro, 2008, Geometrical optics ofacoustic media with anisometric random heterogeneities: Travel-time sta-tistics of reflected and refracted waves, in R. Dmowska, ed., Earth hetero-geneity and scattering effects on seismic waves: Elsevier, Advances inGeophysics, vol. 50, 95–121.

Klimeš, L., 2002, Estimating the correlation function of a self-affine randommedium: Pure and Applied Geophysics, 159, 1833–1853, doi: 10.1007/s00024-002-8711-1.

Lamb, H., 1904, On the propagation of tremors over the surface of an elasticsolid: Philosophical Transactions of the Royal Society of London, 203, 1–42, doi: 10.1098/rsta.1904.0013.

Larsen, S., and J. Grieger, 1998, Elastic modeling initiative, Part III: 3-Dcomputational modeling: 68th Annual International Meeting, SEG, Ex-panded Abstracts, 1803–1806.

Levander, A. R., and N. R. Hill, 1984, Finite-difference simulations of noiseproblems caused by near-surface heterogeneity: 54th Annual InternationalMeeting, SEG, Expanded Abstracts, 392–395.

Marks, D. L., 2006, A family of approximations spanning the Born and Ry-tov scattering series: Optics Express, 14, 8837–8848, doi: 10.1364/OE.14.008837.

Müller, T. M., and S. A. Shapiro, 2001, Most probable seismic pulses insingle realizations of two- and three-dimensional random media: Geo-physical Journal International, 144, 83–95, doi: 10.1046/j.1365-246x.2001.00320.x.

Müller, T. M., S. A. Shapiro, and C. M. A. Sick, 2002, Most probable bal-listic waves in random media: A weak-fluctuation approximation andnumerical results: Waves in Random Media, 12, 223–245, doi: 10.1088/0959-7174/12/2/305.

t (s)z

/ λo

0 10 20 30 400

2

4

6

8

10

Figure 12. Horizontal wavefield directly beneath the point force fora reflecting fractal model.

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

ε (%)

Rs /

Rp

IsotropicR

s / R

p = 1.2 ε

LayeredR

s / R

p = 0.5 ε

Figure 13. Correlations of ε and Rseff∕R

peff for the isotropic fractal

models (R2 ¼ 0.98) and layered fractal models (R2 ¼ 0.99).

T240 Sherman et al.

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to 1

28.3

2.44

.129

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

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Nishizawa, O., and Y. Fukushima, 2008, Laboratory experiments of seismicwave propagation in random heterogeneous media, in R. Dmowska, ed.,Earth heterogeneity and scattering effects on seismic waves, Elsevier,Advances in geophysics, vol. 50, 219–246.

Nishizawa, O., and G. Kitagawa, 2007, An experimental study of phase an-gle fluctuation in seismic waves in random heterogeneous media: Time-series analysis based on multivariate AR model: Geophysical JournalInternational, 169, 149–160, doi: 10.1111/j.1365-246X.2006.03270.x.

Przybilla, J., and M. Korn, 2008, Monte Carlo simulation of radiative energytransfer in continuous elastic random media — Three-component enve-lopes and numerical validation: Geophysical Journal International, 173,566–576, doi: 10.1111/j.1365-246X.2008.03747.x.

Przybilla, J., M. Korn, and U. Wegler, 2006, Radiative transfer of elasticwaves versus finite difference simulations in two-dimensional randommedia: Journal of Geophysical Research, Solid Earth, and Planets,111, B04305, doi: 10.1029/2005JB003952.

Saito, T., 2006, Velocity shift in two-dimensional anisotropic random mediausing the Rytov method: Geophysical Journal International, 166, 293–308, doi: 10.1111/j.1365-246X.2006.02976.x.

Saito, T., H. Sato, M. Fehler, and M. Ohtake, 2003, Simulating the envelopeof scalar waves in 2D randommedia having power-law spectra of velocityfluctuation: Bulletin of the Seismological Society of America, 93, 240–252, doi: 10.1785/0120020105.

Sato, H., M. Fehler, and T. Maeda, 2012, Seismic wave propagation andscattering in the heterogeneous earth, 2nd ed.: Springer.

Shapiro, S. A., and G. Kneib, 1993, Seismic attenuation by scattering:Theory and numerical results: Geophysical Journal International, 114,373–391, doi: 10.1111/j.1365-246X.1993.tb03925.x.

Shapiro, S. A., R. Schwarz, and N. Gold, 1996, The effect of random iso-tropic inhomogeneities on the phase velocity of seismic waves: Geophysi-cal Journal International, 127, 783–794, doi: 10.1111/j.1365-246X.1996.tb04057.x.

Shearer, P. M., and P. S. Earle, 2008, Observing and modeling elastic scat-tering in the deep earth, in R. Dmowska, ed., Advances in geophysics,vol. 50: Elsevier, 167–193.

Tsvankin, I., 1995, Body-wave radiation patterns and AVO in transverselyisotropic media: Geophysics, 60, 1409–1425, doi: 10.1190/1.1443876.

Tsvankin, I., J. Gaiser, V. Grechika, M. van der Baan, and L. Thomsen,2010, Seismic anisotropy in exploration and reservoir characterization:An overview: Geophysics, 75, no. 5, 75A15–75A29, doi: 10.1190/1.3481775.

Tsvankin, I. D., and E. M. Chesnokov, 1990, Synthesis of body wave seis-mograms from point sources in anisotropic media: Journal of GeophysicalResearch: Solid Earth, 95, 11317–11331.

Turcotte, D. L., 1997, Fractals and chaos in geology and geophysics:Cambridge University Press.

White, J. E., 1983, Underground sound: Application of seismic waves:Elsevier.

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