Anelastic Media Research Articles

Imaging Attenuation From Array Analysis of Surface Waves

AbstractAnelastic attenuation provides key insight in our understanding of thermal and rheological structures and the associated deformation and dynamic mechanisms of the Earth's deep interior. Unfortunately, attenuation tomography is advanced far behind wave‐speed tomography due to the challenge in properly excluding the complex effects of elastic heterogeneities on seismic wave amplitude. By taking advantage of phase tracking in seismic array analysis, here we derive a new theory of Helmholtz tomography that well accounts for attenuation, source radiation, and scattering, etc., and present a technique called Helmholtz Multi‐Event Tomography (HelMET) to retrieve the attenuation properly. The effectiveness of this method is then validated by synthetic inversions. Our synthetic seismograms are calculated using a newly developed three‐dimensional finite‐difference algorithm that accounts for physical dispersion and dissipation in anelastic media and remains accurate and stable even if strong attenuation exists. Compared to the traditional method poorly performed in the synthetic inversion, the HelMET well recovers the input attenuation anomalies, suggesting that this method can be used to successfully isolate attenuation from the complicated effects of elastic heterogeneities. Our results underline the implication of the new theory and method in accurately imaging high‐resolution attenuation structures and unambiguously interpreting the anelastic heterogeneities of the Earth by array‐based earthquake and ambient noise data with inexpensive computation.

Fractional-order velocity-dilatation-rotation viscoelastic wave equation and numerical solution based on a constant-Q model

Viscoelastic wave equations based on the constant- Q (CQ) model can accurately describe the amplitude dissipation and phase distortion of waves in anelastic media. However, only three velocity or displacement components can be obtained directly by solving such equations. Starting from the time-domain second-order displacement viscoelastic wave equation, we derived the decoupled P- and S-wave displacement vector viscoelastic wave equation by using the polarization difference of P- and S-wave propagation in isotropic media. The equation can be transformed into the velocity-dilatation-rotation viscoelastic wave equation containing the first-order temporal derivative and fractional Laplacian operators, which can be solved directly by using the staggered-grid finite-difference and pseudospectral methods. We use the low-rank decomposition method to approximate the derived mixed space-wavenumber domain fractional Laplacian operators for modeling wave propagation in heterogeneous attenuating media. We also demonstrated the precision of our equation by comparing the numerical solutions with the analytical solutions. Furthermore, compared with the conventional velocity-stress viscoelastic wave equation, experimental results demonstrate that our equation can separate the pure P and S waves from the mixed wavefield during wavefield continuation. In addition, it can be separated into an equation containing predominantly an amplitude attenuation or a phase distortion term.

Complex spherical-wave seismic inversion in anelastic media for the P-wave minimum quality factor

Attenuation always exists when seismic waves propagate in underground anelastic media, especially in hydrocarbon-bearing reservoirs. Quality factor Q or attenuation factor 1/ Q can be used to quantify the seismic wave attenuation and has become an important hydrocarbon indicator. The relationship between the plane-wave reflection coefficient ([Formula: see text]) in anelastic media and P- and S-wave quality factors has been widely used in the plane-wave seismic inversion to estimate the quality factors. The [Formula: see text] provides an adequate approximation for the deeper subsurface. However, for the shallow subsurface and anelastic wavefields excited by point sources, the [Formula: see text] is inaccurate and its meaning involves some fundamental difficulties. In view of this, a Q-dependent P-P spherical-wave reflection coefficient ([Formula: see text]) in anelastic media is used here. Considering that having too many parameters to be inverted will lead to unstable and inaccurate inversion results, we further derive an approximate anelastic [Formula: see text] and anelastic spherical-wave impedance ([Formula: see text]), which are frequency dependent and are the functions of P- and S-wave velocities, density, and P-wave minimum quality factor ([Formula: see text]). Finally, a complex spherical-wave seismic inversion approach in anelastic media for the P-wave minimum quality factor is developed. Using the Bayesian inversion approach and complex convolution model, we first estimate the multilayer [Formula: see text] from the complex seismic traces with different frequencies and incidence angles. Based on the inverted angle- and frequency-dependent [Formula: see text], the P- and S-wave velocities, density, and P-wave minimum quality factor are further estimated using a nonlinear inversion tool. Synthetic examples verify the feasibility and robustness of the complex spherical-wave seismic inversion approach in anelastic media. In the shallow subsurface, the spherical-wave inversion is superior to plane-wave inversion. A field example further demonstrates the accuracy and great potential of our approach in hydrocarbon-bearing reservoir prediction.

Structure Guided Multiparameter Waveform Inversion With Attenuation Compensation in Viscoacoustic Medium

Full waveform inversion (FWI) has been proved as an effective method for subsurface parameter estimation by iteratively reducing the residual between the predictions and the observations. In recent years, FWI has been widely used due to the increasing computing power. However, seismic waves usually suffer from energy dissipation and phase distortion during their propagation in the anelastic medium, which will decelerate the convergence rate of FWI and make the inversion processing even more time-consuming. In this letter, we propose a structure guided FWI method with attenuation compensation to estimate velocity model and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> model simultaneously. We refer to the proposed method as SGQFWI. With the help of a fractional decoupled viscoacoustic equation, we introduce the attenuation compensation mechanism into full waveform inversion iterative algorithm, which can effectively compensate the deep weak amplitude, so as to obtain a more accurate inversion gradient and improve the accuracy and efficiency of inversion. Additionally, the structure regularization is added to model update, which helps to retrieve local detailed information more efficiently. Benefit from the incorporation of the priori structure constraint and attenuation compensation, this method has the ability to invert velocity and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> model with improved efficiency and resolution. Synthetic and field data examples are adopted to further verify the effectiveness of the proposed method.

A Spectral Solver for Solar Inertial Waves

Inertial waves, which are predominantly driven by the Coriolis force, likely play an important role in solar dynamics, and, additionally, they provide a window into the solar subsurface. The latter allows us to infer properties that are inaccessible to the traditional technique of acoustic wave helioseismology. Thus, a full characterization of these normal modes holds the promise of enabling investigations into solar subsurface dynamics. In this work, we develop a spectral eigenvalue solver to model the spectrum of inertial waves in the Sun. We model the solar convection zone as an anelastic medium, and solve for the normal modes of the momentum and energy equations. We demonstrate that the solver can well reproduce the observed mode frequencies and line widths, not only of sectoral Rossby modes, but also of recently observed high-frequency inertial modes. In addition, we believe that the spectral solver is a useful contribution to the numerical methods of modeling inertial modes on the Sun.

Viscoelastic and viscoacoustic modelling using the Lie product formula

ABSTRACTWe present efficient and accurate modelling of seismic wavefields in anelastic media. We use a first‐order viscoelastic wave equation based on the generalized Robertsson's formulation. In our work, viscoacoustic and viscoelastic wave equations use the standard linear solid mechanism. To numerically solve the first‐order wave equation, we employed a scheme derived from the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices, and each exponential matrix term is approximated by the Taylor series expansion. The accuracy of the proposed scheme is evaluated by comparison with the analytical solution for a homogeneous medium. We also present simulations of some geological models with different structural complexities, whose results confirm the accuracy of the proposed scheme and illustrate the attenuation effect on the seismic energy during its propagation in the medium. Our results demonstrate that the numerical scheme employed can be used to extrapolate wavefields stably for even larger time steps than those usually used by traditional finite‐difference schemes.

Modeling viscoacoustic wave propagation using a new spatial variable-order fractional Laplacian wave equation

We have developed a new time-domain viscoacoustic wave equation for simulating wave propagation in anelastic media. The new wave equation is derived by inserting the complex-valued phase velocity described by the Kjartansson attenuation model into the frequency-wavenumber domain acoustic wave equation. Our wave equation includes one second-order temporal derivative and two spatial variable-order fractional Laplacian operators. The two fractional Laplacian operators describe the phase dispersion and amplitude attenuation effects, respectively. To facilitate the numerical solution for our wave equation, we use the arbitrary-order Taylor series expansion (TSE) to approximate the mixed-domain fractional Laplacians and achieve the decoupling of the wavenumber and the fractional order. Then, our viscoacoustic wave equation can be directly solved using the pseudospectral method. We adopt a hybrid pseudospectral/finite-difference method (HPSFDM) to stably simulate wave propagation in arbitrarily complex media. We validate the high accuracy of our approximate dispersion term and approximate the dissipation term in comparison with the accurate dispersion term and accurate dissipation term. The accuracy of the numerical solutions is evaluated by comparison with the analytical solutions in homogeneous media. Theory analysis and simulation results indicate that our viscoacoustic wave equation has higher precision than the traditional fractional viscoacoustic wave equation in describing constant- Q attenuation. For a model with Q &lt; 10, the calculation cost for solving the new wave equation with TSE HPSFDM is lower than that for solving the traditional fractional-order wave equation with TSE HPSFDM under the high numerical simulation precision. Furthermore, we examine the accuracy of HPSFDM in heterogeneous media using several numerical examples.

Viscoelastic Wave Propagation Simulation Using New Spatial Variable-Order Fractional Laplacians

ABSTRACTTime-domain constant-Q (CQ) viscoelastic wave equations have been derived to efficiently model Q, but are known to break down in accuracy in describing CQ attenuation at low Q. In view of this, a new time-domain viscoelastic wave equation for modeling wave propagation in anelastic medium is evaluated based on Kjartansson’s CQ model to improve the accuracy in describing CQ attenuation at low Q. We use an approximate frequency-domain viscoelastic wave equation to replace the accurate frequency-domain viscoelastic wave equation. Then, a new time-domain wave equation is derived by converting the approximate viscoelastic wave equation from the frequency domain to the time domain. The newly derived viscoelastic wave equation consists of several Laplacian differential operators with variable fractional order. We use an arbitrary-order Taylor series expansion (TSE) to approximate the derived mixed domain fractional Laplacian operators, and realize the decoupling of the wavenumber and fractional order. Then, the proposed viscoelastic wave equation can be solved directly using the staggered-grid pseudospectral method (SGPSM). We evaluate the precision of the new viscoelastic wave equation by comparing the numerical solutions with the analytical solutions in homogeneous medium. Theoretical curve analysis and numerical results indicate that the proposed fractional viscoelastic wave equation has higher precision in describing CQ attenuation than that of the traditional fractional viscoelastic wave equation, especially for cases that P-wave quality factor QP is less than 10, and S-wave quality factor QS is less than 8. Furthermore, we use two numerical examples to verify the effectiveness of the TSE SGPSM in heterogeneous media. The discussion shows that the advantage of using our fractional viscoelastic wave equation over the traditional fractional viscoelastic wave equation is the higher precision in describing CQ attenuation at different frequency.

Mean-field model of interacting quasilocalized excitations in glasses

Structural glasses feature quasilocalized excitations whose frequencies \omegaω follow a universal density of states {D}(\omega)\!\sim\!\omega^4D(ω)∼ω4. Yet, the underlying physics behind this universality is not fully understood. Here we study a mean-field model of quasilocalized excitations in glasses, viewed as groups of particles embedded inside an elastic medium and described collectively as anharmonic oscillators. The oscillators, whose harmonic stiffness is taken from a rather featureless probability distribution (of upper cutoff \kappa_0κ0) in the absence of interactions, interact among themselves through random couplings (characterized by a strength JJ) and with the surrounding elastic medium (an interaction characterized by a constant force hh). We first show that the model gives rise to a gapless density of states {D}(\omega)\!=\!A_{g}\,\omega^4D(ω)=Agω4 for a broad range of model parameters, expressed in terms of the strength of the oscillators’ stabilizing anharmonicity, which plays a decisive role in the model. Then — using scaling theory and numerical simulations — we provide a complete understanding of the non-universal prefactor A_{g}(h,J,\kappa_0)Ag(h,J,κ0), of the oscillators’ interaction-induced mean square displacement and of an emerging characteristic frequency, all in terms of properly identified dimensionless quantities. In particular, we show that A_{g}(h,J,\kappa_0)Ag(h,J,κ0) is a non-monotonic function of JJ for a fixed hh, varying predominantly exponentially with -(\kappa_0 h^{2/3}\!/J^2)−(κ0h2/3/J2) in the weak interactions (small JJ) regime — reminiscent of recent observations in computer glasses — and predominantly decays as a power-law for larger JJ, in a regime where hh plays no role. We discuss the physical interpretation of the model and its possible relations to available observations in structural glasses, along with delineating some future research directions.

Attenuation and dispersion phenomena of shear waves in anelastic and elastic porous strips

PurposeThis paper aims to study the attenuation and dispersion phenomena of shear waves in anelastic and elastic porous strips. Numerical investigations are performed for the phase and damped velocity profiles of the wave. For numerical computation purposes, water-saturated limestone and kerosene oil saturated sandstone for the first and second porous strips, respectively. Some other peculiarities have been observed and discussed.Design/methodology/approachDispersion and attenuation characteristic of the shear wave propagations have been studied in an inhomogeneous poro-anelastic strip of finite thickness, which is clamped between an inhomogeneous poroelastic strip of finite thickness and an elastic half-space. Both the strips are initially stressed and the half-space is self-weighted. Analytical methods are used to calculate the interior deformations of the model with the involvement of special functions. The determination of the frequency equation, which includes the Bessel’s and Whittaker functions, has been obtained using the prescribed boundary conditions.FindingsImpacts of attenuation coefficient, dissipation factor, inhomogeneities, initial stresses, Biot’s gravity, porosity and thickness ratio parameters on the velocity profile of the wave have been demonstrated through the graphical visuals. These parameters are playing an important role and working as a catalyst in affecting the propagation behaviour of the wave.Originality/valueInclusion of the concept of doubly layered initially stressed inhomogeneous porous structure of elastic and anelastic medium bedded over a self-weighted half-space medium brings a novelty to the existing literature related to the study of shear wave. It may be helpful to geologists, seismologists and structural engineers in the development of theoretical and practical studies.

Estimating elastic properties and attenuation factor from different frequency components of observed seismic data

SUMMARY Based on an attenuation model, we first express frequency-dependent P- and S-wave attenuation factors as a function of P-wave maximum attenuation factor, and then we re-express P- and S-wave velocities in anelastic media and derive frequency-dependent stiffness parameters in terms of P-wave maximum attenuation factor. Using the derived stiffness parameters, we propose frequency-dependent reflection coefficient in terms of P- and S-wave moduli at critical frequency and P-wave maximum attenuation factor for the case of an interface separating two attenuating media. Based on the derived reflection coefficient, we establish an approach to utilize different frequency components of observed seismic data to estimate elastic properties (P- and S-wave moduli and density) and attenuation factor, and following a Bayesian framework, we construct the objective function and an iterative method is employed to solve the inversion problem. Tests on synthetic data confirm that the proposed approach makes a stable and robust estimation of unknown parameters in the case of seismic data containing a moderate noise/error. Applying the proposed approach to a real data set illustrates that a reliable attenuation factor is obtained from observed seismic data, and the ability of distinguishing oil-bearing reservoirs is improved combining the estimated elastic properties and P-wave attenuation factor.

Modeling the wave propagation in viscoacoustic media: An efficient spectral approach in time and space domain

We present an efficient and accurate modeling approach for wave propagation in anelastic media, based on a fractional spatial differential operator. The problem is solved with the Fourier pseudo-spectral method in the spatial domain and the REM (rapid expansion method) in the time domain, which, unlike the finite-difference and pseudo-spectral methods, offers spectral accuracy. To show the accuracy of the scheme, an analytical solution in a homogeneous anelastic medium is computed and compared with the numerical solution. We present an example of wave propagation at a reservoir scale and show the efficiency of the algorithm against the conventional finite-difference scheme. The new method, being spectral in the time and space simultaneously, offers a highly accurate and efficient solution for wave propagation in attenuating media.

Q-factor inversion using reconstructed source spectral consistency

Seismic waves propagating in an anelastic medium undergo phase and amplitude distortions. Although these effects may be compensated for during imaging processes, a background [Formula: see text]-model is generally required for their successful application. We have developed a new approach to the [Formula: see text]-estimation problem, which is fundamentally related to the basic physical principle of time reversal. It is based on back-propagating recorded traces to their known source location using the reverse tomographic equation. This equation is a ray approximation of viscoelastic wave propagation. It is applied assuming a known and correct velocity model. We subsequently measure consistency between spectral shapes of traces that were back-propagated using the tomographic equation. We formulate an inverse problem using this consistency as an objective function. In conventional inversion, on the contrary, the discrepancy between modeled and recorded data, or some data characteristics, is minimized. The inverse problem is solved by ant-colony optimization, a global optimization approach, to avoid local minima present in the objective function. This method does not require knowledge of the source function and uses the full spectrum rather than its parametric reduction. Through synthetic and field cross-hole examples, we illustrate its accuracy and sensitivity in inverting for complex attenuation models. In the synthetic case, we also compare reconstructed source consistency with the conventional centroid frequency shift objective function. The latter displays poor resolution when recovering complex [Formula: see text] structures. We determine that the reconstructed source-consistency approach should be used as a part of an iterative workflow, possibly yielding initial models for a joint velocity and [Formula: see text] inversion.

Scattering of torsional surface waves in a three layered model structure

In this article, a comparative study has been made to investigate the scattering behaviour of three layered structure model on torsional surface wave. For such model intermediate layer is taken as fiber reinforced composite, resting over a dry sandy Gibson substratum and underlying by different anelastic media. We consider two distinct mediums for topmost layer. In the first case, topmost layer has been taken as fluid saturated homogeneous porous layer, while in the second case the fluid saturated porous layer has been replaced by a transversely isotropic layer. Simple form expression for the secular equation of torsional surface wave has been worked out in both the cases by executing specific boundary conditions, which comprises Whittaker\'s function and its derivative, for imminent result that have been elaborated asymptotically. Some special cases have been constituted which are in excellent compliance with recorded literatures. For the sake of comparative study, numerical estimation and graphical illustration have been accomplished to identify the effects of the width ratio of the layers, Biot\'s gravity parameter, sandy parameter, porosity parameter and other heterogeneity parameters corresponding to the layers and half spaces, horizontal compressive and tensile initial stress on the phase velocity of torsional surface wave.

Fractional integration of seismic wavelets in anelastic media to recover multiscale properties of impedance discontinuities

Multiscale seismic attributes based on wavelet transform properties have recently been introduced and successfully applied to identify the geometry of a complex seismic reflector in an elastic medium. We extend this quantitative approach to anelastic media where intrinsic attenuation modifies the seismic attributes and thus requires a specific processing to retrieve them properly. The method assumes an attenuation linearly dependent with the seismic wave frequency and a seismic source wavelet approximated with a Gaussian derivative function (GDF). We highlight a quasi-conservation of the Gaussian character of the wavelet during its propagation. We found that this shape can be accurately modeled by a GDF characterized by a fractional integration and a frequency shift of the seismic source, and we establish the relationship between these wavelet parameters and [Formula: see text]. Based on this seismic wavelet modeling, we design a time-varying shaping filter that enables making constant the shape of the wavelet allowing retrieval of the wavelet transform properties. Introduced with a homogeneous step-like reflector, the method is first applied on a thin-bed reflector and then on a more realistic synthetic data set based on an in situ acoustic impedance sequence and a high-resolution seismic source. The results clearly highlight the efficiency of the method in accurately restoring the multiscale seismic attributes of complex seismic reflectors in anelastic media by the use of broadband seismic sources.

Extended reflectivity method for modelling the propagation of diffusive–viscous wave in dip‐layered media

ABSTRACTThe reflectivity method plays an important role in seismic modelling. It has been used to model different types of waves propagating in elastic and anelastic media. The diffusive–viscous wave equation was proposed to investigate the relationship between frequency dependence of reflections and fluid saturation. It is also used to describe the attenuation property of seismic wave in a fluid‐saturated medium. The attenuation of diffusive–viscous wave is mainly characterised by the effective attenuation parameters in the equation. Thus, it is essential to obtain those parameters and further characterise the features of the diffusive–viscous wave. In this work, we use inversion method to obtain the effective attenuation parameters through quality factor to investigate the characteristics of diffusive–viscous wave by comparing with those of the viscoacoustic wave. Then, the reflection/transmission coefficients in a dip plane‐layered medium are studied through coordinate transform and plane‐wave theory. Consequently, the reflectivity method is extended to compute seismograms of diffusive–viscous wave in a dip plane multi‐layered medium. Finally, we present two models to simulate the propagation of diffusive–viscous wave in a dip plane multi‐layered medium by comparing the results with those in a viscoacoustic medium. The numerical results demonstrate the validity of our extension of reflectivity method to the diffusive–viscous medium. The numerical examples in both time domain and time–frequency domain show that the reflections from a dip plane interface have significant phase shift and amplitude change compared with the results of horizontal plane interface due to the differences in reflection/transmission coefficients. Moreover, the modelling results show strong attenuation and phase shift in the diffusive–viscous wave compared to those of the viscoacoustic wave.

A physical solution for plane SH waves in anelastic media

A physical solution for plane SH waves in anelastic media

Travelling wave modes of a plane layered anelastic earth

Abstract : Incorporation of attenuation into the normal mode sum representations of seismic signals is commonlyeffected by applying perturbation theory. This is fine for weak attenuation, but problematic forstronger attenuation. In this work modes of the anelastic medium are represented as complexsuperpositions of elastic eigenfunctions. For the P-SV system a generalized eigenvalue equation for thecomplex eigenwavenumbers and complex coefficients used to construct the anelastic eigenfunctions isderived. The generalized eigenvalue problem for the P-SV problem is exactly linear in theeigenwavenumber at the expense of doubling the dimension. The SH problem is exactly linear in thesquare of the eigenwavenumber. This is in contrast to a similar standing wave problem for the earthfree oscillations (Tromp and Dahlen,1990). Attenuation is commonly incorporated into syntheticseismogram calculations by introduction of complex frequency dependent elastic moduli. The modulidepend nonlinearly on the frequency. The independent variable in the standing wave free oscillationproblem is the frequency, which makes the eigenvalue problem nonlinear. The choice of the wavenumber asthe independent variable for the traveling wave problem leads to a linear problem.

On Fermat’s principle and Snell’s law in lossy anisotropic media

Fermat’s principle of least action is one of the methods used to trace rays in inhomogeneous media. Its form is the same in anisotropic elastic and anelastic media, with the difference that the velocity depends on frequency in the latter case. Moreover, the ray, envelope, and energy velocities replace the group velocity because this concept has no physical meaning in anelastic media. We have first considered a lossy (anelastic) anisotropic medium and established the equivalence between Fermat’s principle and Snell’s law in homogeneous media. Then, we found that the different ray velocities defined in the literature were the same for stationary rays in homogeneous media, with phase and inhomogeneity angles satisfying the principle and the law. We considered an example of a transversely isotropic medium with a vertical symmetry axis and wavelike and diffusionlike properties. In the first case, the differences were negligible, which was the case of real rocks having a quality factor greater than five. Strictly, ray tracing should be based on the so-called stationary complex slowness vector to obtain correct results, although the use of homogeneous viscoelastic waves (zero inhomogeneity angle) is acceptable as an approximation for earth materials. However, from a rigorous point of view, the three velocities introduced in the literature to define the rays present discrepancies in heterogeneous media, although the differences are too small to be measured in earth materials. The findings are also valid for electromagnetic waves by virtue of the acoustic-electromagnetic analogy.

Importance of crustal structure and anelastic attenuation for estimating ground motion parameters by finite difference simulation

We carry out simulations of seismic wave propagation in anelastic media in order to study the relative importance of source parameters as well as viscoelastic structures in affecting the decay of the ground motions. First, we verify the efficiency of the implementation in a finite difference code of two coarse-grain memory variables methods to model the anelastic behavior of the soil. We find that both methods are sufficiently consistent for a quality factor (Q) larger than 20. Then, we study the relative importance of the focal mechanism, the magnitude, the source depth, the crustal structure and the quality factor in affecting the decay of the ground motions. We verify that the magnitude and the focal mechanism of the source do not have a significant effect on the decay, whereas the focal depth is more important in explaining the variations in decay. The variations of the decay depending on the crustal structure are more difficult to assess. For the shorter distances (up to about 20 km), the velocity structure does not have a significant effect on the decay of the ground motions. The effect of the quality factor is perceptible but remains less important than the effect of the focal depth. However, for epicentral distances larger than about 20 km, both the velocity structure and the quality factor begin to affect significantly the decay. The effect of the quality factor on the decay becomes then even more important than the effect of the focal depth. Therefore, a reduction in the standard deviation of GMPEs could possibly be achieved through taking into account appropriate anelastic attenuation for each region considered. The effect of a 3D velocity structure is studied introducing a typical basin structure. The presence of a sedimentary basin can affect the decay even outside the basin. The spatial difference in ground motion is more pronounced in the elastic case than in the anelastic case.