Heterogeneous Earth Models Research Articles

High-frequency seismic wave modelling of the deep Earth based on hybrid methods and spectral-element simulations: a conceptual study

SUMMARYOver the past few decades, seismic studies have revealed complex structural anomalies in the Earth’s deep interior at various scales, such as large low-shear-velocity provinces (LLSVPs) and ultra-low velocity zones (ULVZs) in the lowermost mantle, and small-scale scatterers in the mid-mantle. These structures which are critical for better understanding of the geodynamics and evolution of the deep Earth, need to be further resolved by high-resolution imaging techniques. The spectral-element method (SEM) can be used to accurately simulate seismic wave propagation in heterogeneous Earth models, and its application in full-waveform inversion (FWI) provides a promising high-resolution and high-fidelity imaging technique. But it can be computationally prohibitive when used to model small scale structures in the deep Earth based upon high-frequency seismic waves. The heavy computational cost can be circumvented by using hybrid methods, which restrict the main computation by SEM solver to only a small target region (e.g. above the CMB) encompassing possible 2-D/3-D anomalies, and apply efficient analytical or numerical methods to calculate the wavefield for 1-D background models. These forward modelling tools based on hybrid methods can be then used in the so-called ‘box tomography’ approach to resolve fine-structures in the deep Earth.In this study, we outline the theory of a hybrid method used to model small scale structures in the deep Earth and present its implementation based on SEM solvers in a three-step workflow. First, the wavefield generated by the source is computed for the 1-D background model with traction and velocity saved for the virtual boundary of the target region, which are then used as boundary inputs to simulate the wavefield in the target region based on absorbing boundary condition in SEM. In the final step, the total wavefield at receivers is reconstructed based upon the total wavefield on the virtual boundary computed in the previous step. As a proof-of-concept study, we demonstrate the workflow of the hybrid method based on a 2-D SEM solver. Examples of the hybrid method applied to a coupled fluid–solid model show that our workflow can accurately recover the scattered waves back to the surface. Furthermore, we benchmark the hybrid method on a realistic heterogeneous Earth model built from AK135-F and show how teleseismic scattered waves can be used to model deep Earth structures. By documenting the theory and SEM implementation of the hybrid method, our study lays the foundation for future two-way coupling of 3-D SEM solver with other efficient analytic or numerical 1-D solvers.

The 20 May 2016 Petermann Ranges earthquake: centroid location, magnitude and focal mechanism from full waveform modelling

Ground velocity records of the 20 May 2016 Petermann Ranges earthquake are used to calculate its centroid-moment-tensor in the 3 D heterogeneous Earth model AuSREM. The global-centroid-moment-tensor reported a depth of 12 km, which is the shallowest allowed depth in the algorithm. Solutions from other global and local agencies indicate that the event occurred within the top 12 km of the crust, but the locations vary laterally by up to 100 km. We perform a centroid-moment-tensor inversion through a spatiotemporal grid search in 3 D allowing for time shifts around the origin time. Our 3 D grid encompasses the locations of all proposed global solutions. The inversion produces an ensemble of solutions that constrain the depth, lateral location of the centroid, and strike, dip and rake of the fault. The centroid location stands out with a clear peak in the correlation between real and synthetic data for a depth of 1 km at longitude 129.8° and latitude –25.6°. A collection of acceptable solutions at this centroid location, produced by different time shifts, constrain the fault strike to be 304 ± 4° or 138 ± 1°. The two nodal planes have dip angles of 64 ± 5° and 26 ± 4° and rake angles of 96 ± 2° and 77 ± 5°, respectively. The southwest-dipping nodal plane with the dip angle of 64° could be seen as part of a near vertical splay fault system at the end of the Woodroffe Thrust. The other nodal plane could be interpreted as a conjugate fault rupturing perpendicular to the splay structure. We speculate that the latter is more likely, since the hypocentres reported by several agencies, including the Geoscience Australia, as well as the majority of aftershocks are all located to the northeast of our preferred centroid location. Our best estimate for the moment magnitude of this event is 5.9. The optimum centroid is located on the 20 km surface rupture caused by the earthquake. Given the estimated magnitude, the long surface rupture requires only ∼4 km of rupture down dip, which is in agreement with the shallow centroid depth we obtained.

Hamilton’s principle and normal mode coupling in an aspherical planet with a fluid core

Seismic free oscillations, or normal modes, provide a convenient tool to calculate low-frequency seismograms in heterogeneous Earth models. A procedure called ‘full mode coupling’ allows the seismic response of the Earth to be computed. However, in order to be theoretically exact, such calculations must involve an infinite set of modes. In practice, only a finite subset of modes can be used, introducing an error into the seismograms. By systematically increasing the number of modes beyond the highest frequency of interest in the seismograms, we investigate the convergence of full-coupling calculations. As a rule-of-thumb, it is necessary to couple modes 1–2 mHz above the highest frequency of interest, although results depend upon the details of the Earth model. This is significantly higher than has previously been assumed. Observations of free oscillations also provide important constraints on the heterogeneous structure of the Earth. Historically, this inference problem has been addressed by the measurement and interpretation of splitting functions. These can be seen as secondary data extracted from low frequency seismograms. The measurement step necessitates the calculation of synthetic seismograms, but current implementations rely on approximations referred to as self- or group-coupling and do not use fully accurate seismograms. We therefore also investigate whether a systematic error might be present in currently published splitting functions. We find no evidence for any systematic bias, but published uncertainties must be doubled to properly account for the errors due to theoretical omissions and regularization in the measurement process. Correspondingly, uncertainties in results derived from splitting functions must also be increased. As is well known, density has only a weak signal in low-frequency seismograms. Our results suggest this signal is of similar scale to the true uncertainties associated with currently published splitting functions. Thus, it seems that great care must be taken in any attempt to robustly infer details of Earth's density structure using current splitting functions.

Exact free oscillation spectra, splitting functions and the resolvability of Earth's density structure

Seismic free oscillations, or normal modes, provide a convenient tool to calculate low-frequency seismograms in heterogeneous Earth models. A procedure called ‘full mode coupling’ allows the seismic response of the Earth to be computed. However, in order to be theoretically exact, such calculations must involve an infinite set of modes. In practice, only a finite subset of modes can be used, introducing an error into the seismograms. By systematically increasing the number of modes beyond the highest frequency of interest in the seismograms, we investigate the convergence of full-coupling calculations. As a rule-of-thumb, it is necessary to couple modes 1–2 mHz above the highest frequency of interest, although results depend upon the details of the Earth model. This is significantly higher than has previously been assumed. Observations of free oscillations also provide important constraints on the heterogeneous structure of the Earth. Historically, this inference problem has been addressed by the measurement and interpretation of splitting functions. These can be seen as secondary data extracted from low frequency seismograms. The measurement step necessitates the calculation of synthetic seismograms, but current implementations rely on approximations referred to as self- or group-coupling and do not use fully accurate seismograms. We therefore also investigate whether a systematic error might be present in currently published splitting functions. We find no evidence for any systematic bias, but published uncertainties must be doubled to properly account for the errors due to theoretical omissions and regularization in the measurement process. Correspondingly, uncertainties in results derived from splitting functions must also be increased. As is well known, density has only a weak signal in low-frequency seismograms. Our results suggest this signal is of similar scale to the true uncertainties associated with currently published splitting functions. Thus, it seems that great care must be taken in any attempt to robustly infer details of Earth's density structure using current splitting functions.

Centroid moment tensor catalogue using a 3‐D continental scale Earth model: Application to earthquakes in Papua New Guinea and the Solomon Islands

AbstractAlthough both earthquake mechanism and 3‐D Earth structure contribute to the seismic wavefield, the latter is usually assumed to be layered in source studies, which may limit the quality of the source estimate. To overcome this limitation, we implement a method that takes advantage of a 3‐D heterogeneous Earth model, recently developed for the Australasian region. We calculate centroid moment tensors (CMTs) for earthquakes in Papua New Guinea (PNG) and the Solomon Islands. Our method is based on a library of Green's functions for each source‐station pair for selected Geoscience Australia and Global Seismic Network stations in the region, and distributed on a 3‐D grid covering the seismicity down to 50 km depth. For the calculation of Green's functions, we utilize a spectral‐element method for the solution of the seismic wave equation. Seismic moment tensors were calculated using least squares inversion, and the 3‐D location of the centroid is found by grid search. Through several synthetic tests, we confirm a trade‐off between the location and the correct input moment tensor components when using a 1‐D Earth model to invert synthetics produced in a 3‐D heterogeneous Earth. Our CMT catalogue for PNG in comparison to the global CMT shows a meaningful increase in the double‐couple percentage (up to 70%). Another significant difference that we observe is in the mechanism of events with depth shallower then 15 km and Mw < 6, which contributes to accurate tectonic interpretation of the region.

High Order Accurate Finite Difference Modeling of Seismo-Acoustic Wave Propagation in a Moving Atmosphere and a Heterogeneous Earth Model Coupled Across a Realistic Topography

We develop a numerical method for simultaneously simulating acoustic waves in a realistic moving atmosphere and seismic waves in a heterogeneous earth model, where the motions are coupled across a realistic topography. We model acoustic wave propagation by solving the linearized Euler equations of compressible fluid mechanics. The seismic waves are modeled by the elastic wave equation in a heterogeneous anisotropic material. The motion is coupled by imposing continuity of normal velocity and normal stresses across the topographic interface. Realistic topography is resolved on a curvilinear grid that follows the interface. The governing equations are discretized using high order accurate finite difference methods that satisfy the principle of summation by parts. We apply the energy method to derive the discrete interface conditions and to show that the coupled discretization is stable. The implementation is verified by numerical experiments, and we demonstrate a simulation of coupled wave propagation in a windy atmosphere and a realistic earth model with non-planar topography.

Love-wave waveform inversion in time domain for shallow shear-wave velocity

High-frequency surface-wave techniques are widely used to estimate S-wave velocity of near-surface materials. Surface-wave methods based on inversions of dispersion curves are only suitable to laterally homogeneous or smoothly laterally varying heterogeneous earth models due to the layered-model assumption during calculation of dispersion curves. Waveform inversion directly fits the waveform of observed data, and it can be applied to any kinds of earth models. We have used the Love-wave waveform inversion in the time domain to estimate near-surface S-wave velocity. We used the finite-difference method as the forward modeling method. The source effect was removed by the deconvolution technique, which made our method independent of the source wavelet. We defined the difference between the deconvolved observed and calculated waveform as the misfit function. We divided the model into different sizes of blocks depending on the resolution of the Love waves, and we updated the S-wave velocity of each block via a conjugate gradient algorithm. We used two synthetic models to test the effectiveness of our method. A real-world case verified the validity of our method.

Varying dimensional Bayesian acoustic waveform inversion for 1D semi-infinite heterogeneous media

This paper introduces a methodology to infer the spatial variation of the acoustic characteristics of a 1D vertical elastic heterogeneous earth model via a Bayesian calibration approach, given a prescribed sequence of loading and the corresponding time history response registered at the ground level. This involves solving an inverse problem that maps the ground seismic response onto a random profile of the ground stratigraphy (i.e. a 1D continuous spatial random field). From a Bayesian point of view, the solution to an inverse problem is fully characterized by a posterior density function of the forward model random parameters, which explicitly overcomes the solution's non-uniqueness. This subsurface earth model is parameterized using a Bayesian partition model, where the number of soil layers, the location of the layers' interfaces, and their corresponding mechanical characteristics are defined as random variables. The partition model approach to an inverse problem is closely related to a Bayesian model selection problem, where the likely dimensionality of the inverse problem (number of unknowns) is inferred conditioned on the experimental observations. The main benefit of the proposed approach is that the explicit regularization of the inverted profile by global damping procedures is not required. A Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm is used to sample the target posterior of varying dimension, dependent on the number of layers. A synthetic case study is provided to indicate the applicability of the proposed methodology.

The rotational feedback on linear-momentum balance in glacial isostatic adjustment

The influence of changes in surface ice-mass redistribution and associated viscoelastic response of the Earth, known as glacial isostatic adjustment (GIA), on the Earth's rotational dynamics has long been known. Equally important is the effect of the changes in the rotational dynamics on the viscoelastic deformation of the Earth. This signal, known as the rotational feedback, or more precisely, the rotational feedback on the sea level equation, has been mathematically described by the sea level equation extended for the term that is proportional to perturbation in the centrifugal potential and the second-degree tidal Love number. The perturbation in the centrifugal force due to changes in the Earth's rotational dynamics enters not only into the sea level equation, but also into the conservation law of linear momentum such that the internal viscoelastic force, the perturbation in the gravitational force and the perturbation in the centrifugal force are in balance. Adding the centrifugal-force perturbation to the linear-momentum balance creates an additional rotational feedback on the viscoelastic deformations of the Earth. We term this feedback mechanism, which is studied in this paper, as the rotational feedback on the linear-momentum balance. We extend both the time-domain method for modelling the GIA response of laterally heterogeneous earth models developed by Martinec and the traditional Laplace-domain method for modelling the GIA-induced rotational response to surface loading by considering the rotational feedback on linear-momentum balance. The correctness of the mathematical extensions of the methods is validated numerically by comparing the polar-motion response to the GIA process and the rotationally induced degree 2 and order 1 spherical harmonic component of the surface vertical displacement and gravity field. We present the difference between the case where the rotational feedback on linear-momentum balance is considered against that where it is not. Numerical simulations show that the resulting difference in radial displacement and sea level change between these situations since the Last Glacial Maximum reaches values of ±25 and ±1.8 m, respectively. Furthermore, the surface deformation pattern is modified by up to 10 per cent in areas of former or ongoing glaciation, but by up to 50 per cent at the bottom of the southern Indian ocean. This also results in the movement of coastlines during the last deglaciation to differ between the two cases due to the difference in the ocean loading, which is seen for instance in the area around Hudson Bay, Canada and along the Chinese, Australian or Argentinian coastlines.

The Impact of Including Heterogeneity into the Simulation Model for the Evaluation of CO2 Injectivity and Sequestration

Abstract We study the influence of including heterogeneity into earth models of a flow simulation for a prediction of the injected CO 2 behavior. Multiple heterogeneous earth models are generated from seismic and geological prior information in a stochastic manner. The flow simulations are performed with the heterogeneous models as well as a homogeneous model at one of the proposed storage site. The variation of the results for the heterogeneous models is significant and the large amount of CO 2 is migrated into the seal layer compared to the homogeneous case. Although CO 2 leakage from the seal layer is observed for a few models, the leakage mass ratio is less than 1 percent. The result suggests that even if a reservoir and a seal are not homogeneous, the both layers can work as a storage layer in the whole system.

Full waveform tomography of the upper mantle in the South Atlantic region: Imaging a westward fluxing shallow asthenosphere?

A prominent feature of the South Atlantic region is its strongly asymmetric residual bathymetry across the ocean basin. It has been suggested that the residual bathymetry is dynamic in nature, arising from the large slow velocity seismic anomaly located in the lower mantle beneath the African plate. Unfortunately, the pattern of mantle heterogeneity particularly in the upper mantle is not well known owing to the sparsity of seismic stations and the existence of large aseismic regions on the African and South American plates. Here we present a new seismic tomographic study of the South Atlantic upper mantle. Our model is based on a full seismic waveform inversion of ≈ 4000 high-quality seismograms for isotropic 3-D seismic structure using a powerful adjoint methodology capable of extracting maximum information from each seismogram. The theory requires simulation of seismic wave propagation in 3-D heterogeneous earth models computed with a spectral-element method where the differences between observed and synthetic seismograms are quantified using phase misfits obtained through a time–frequency transform. The model images a continuous channel of pronounced slow seismic velocity in the shallow sublithospheric mantle between ≈ 150 and ≈ 300 km depth that branches in between the cratonic roots under the African and South American continents. At greater depth, below 300–350 km, the slow anomalies are less pronounced and a change in the convective planform is indicated by isolated, round shaped patches in an overall faster mantle. It is possible that the depthwise change of the convective planform from vertical to horizontal advection of hot buoyant material in a low viscosity asthenosphere can reconcile the anomalous residual bathymetry in the region, and we present a simple fluid dynamic model of pressure driven flow to assess the feasibility of this scenario.

Calculation of normal mode spectra in laterally heterogeneous earth models using an iterative direct solution method

Normal mode observations play an important role in studying broad-scale lateral variations in the Earth. Such studies require the calculation of accurate synthetic spectra in realistic earth models, and this remains a computationally challenging problem. Here, we describe a new implementation of the direct solution method for calculating normal mode spectra in laterally heterogeneous earth models. In this iterative direct solution method, the mode-coupling equations are solved in the frequency-domain using the preconditioned biconjugate gradient algorithm, and the time-domain solution is recovered using a numerical inverse Fourier transform. A number of example calculations are presented to demonstrate the accuracy and efficiency of the method for performing large ‘full coupling’ calculations as compared to methods based on matrix diagonalization and the traditional direct solution method.

High-frequency Born synthetic seismograms based on coupled normal modes

SUMMARY High-frequency and full waveform synthetic seismograms on a 3-D laterally heterogeneous earth model are simulated using the theory of coupled normal modes. The set of coupled integral equations that describe the 3-D response are simplified into a set of uncoupled integral equations by using the Born approximation to calculate scattered wavefields and the pure-path approximation to modulate the phase of incident and scattered wavefields. This depends upon a decomposition of the aspherical structure into smooth and rough components. The uncoupled integral equations are discretized and solved in the frequency domain, and time domain results are obtained by inverse Fourier transform. Examples show the utility of the normal mode approach to synthesize the seismic wavefields resulting from interaction with a combination of rough and smooth structural heterogeneities. This approach is applied to an ∼4 Hz shallow crustal wave propagation around the site of the San Andreas Fault Observatory at Depth (SAFOD).

Application of the multiaxial perfectly matched layer (M-PML) to near-surface seismic modeling with Rayleigh waves

Perfectly matched layer (PML) absorbing boundaries are widely used to suppress spurious edge reflections in seismic modeling. When modeling Rayleigh waves with the existence of the free surface, the classical PML algorithm becomes unstable when the Poisson’s ratio of the medium is high. Numerical errors can accumulate exponentially and terminate the simulation due to computational overflows. Numerical tests show that the divergence speed of the classical PML has a nonlinear relationship with the Poisson’s ratio. Generally, the higher the Poisson’s ratio, the faster the classical PML diverges. The multiaxial PML (M-PML) attenuates the waves in PMLs using different damping profiles that are proportional to each other in orthogonal directions. The proportion coefficients of the damping profiles usually vary with the specific model settings. If they are set appropriately, the M-PML algorithm is stable for high Poisson’s ratio earth models. Through numerical tests of 40 models with Poisson’s ratios that varied from 0.10 to 0.49, we found that a constant proportion coefficient of 1.0 for the x- and z-directional damping profiles is sufficient to stabilize the M-PML for all 2D isotropic elastic cases. Wavefield simulations indicate that the instability of the classical PML is strongly related to the wave phenomena near the free surface. When applying the multiaxial technique only in the corners of the PML near the free surface, the original M-PML technique can be simplified without losing its stability. The simplified M-PML works efficiently for homogeneous and heterogeneous earth models with high Poisson’s ratios. The analysis in this paper is based on 2D finite difference modeling in the time domain that can easily be extended into the 3D domain with other numerical methods.

On the parametrization of equilibrium stress fields in the Earth

SUMMARY A new method for parametrizing the possible equilibrium stress fields of a laterally heterogeneous earth model is described. In this method a solution of the equilibrium equations is first found that satisfies some desirable physical property. For example, we show that the equilibrium stress field with smallest norm relative to a given inner product can be obtained by solving a static linear elastic boundary value problem. We also show that the equilibrium stress field whose deviatoric component has smallest norm with respect to a given inner product can be obtained by solving a steady-state incompressible viscous flow problem. Having found such a solution of the equilibrium equations, all other solutions can be written as the sum of this equilibrium stress field and a divergence-free stress tensor field whose boundary tractions vanish. Given n divergence-free and traction-free tensor fields, we then obtain a simple n-dimensional parametrization of equilibrium stress fields in the earth model. The practical construction of such divergence- and traction-free tensor fields in the mantle of a spherically symmetric reference earth model is described using generalized spherical harmonics.

Modelling broad-band poroelastic propagation using an asymptotic approach

An asymptotic method, valid in the presence of smoothly varying heterogeneity, is used to derive a semi-analytic solution to the equations for fluid and solid displacements in a poroelastic medium. The solution is defined along trajectories through the porous medium model, in the manner of ray theory. The lowest order expression in the asymptotic expansion provides an eikonal equation for the phase. There are three modes of propagation, two modes of longitudinal displacement and a single mode of transverse displacement. The two longitudinal modes define the Biot fast and slow waves which have very different propagation characteristics. In the limit of low frequency, the Biot slow wave propagates as a diffusive disturbance, in essence a transient pressure pulse. Conversely, at low frequencies the Biot fast wave and the transverse mode are modified elastic waves. At intermediate frequencies the wave characteristics of the longitudinal modes are mixed. A comparison of the asymptotic solution with analytic and numerical solutions shows reasonably good agreement for both homogeneous and heterogeneous earth models. Journal compilation © 2009 RAS No claim to original US government works.

Full seismic waveform tomography for upper-mantle structure in the Australasian region using adjoint methods

SUMMARY We present a full seismic waveform tomography for upper-mantle structure in the Australasian region. Our method is based on spectral-element simulations of seismic wave propagation in 3-D heterogeneous earth models. The accurate solution of the forward problem ensures that waveform misfits are solely due to as yet undiscovered Earth structure and imprecise source descriptions, thus leading to more realistic tomographic images and source parameter estimates. To reduce the computational costs, we implement a long-wavelength equivalent crustal model. We quantify differences between the observed and the synthetic waveforms using time–frequency (TF) misfits. Their principal advantages are the separation of phase and amplitude misfits, the exploitation of complete waveform information and a quasi-linear relation to 3-D Earth structure. Fr´ echet kernels for the TF misfits are computed via the adjoint method. We propose a simple data compression scheme and an accuracy-adaptive time integration of the wavefields that allows us to reduce the storage requirements of the adjoint method by almost two orders of magnitude. To minimize the waveform phase misfit, we implement a pre-conditioned conjugate gradient algorithm. Amplitude information is incorporated indirectly by a restricted line search. This ensures that the cumulative envelope misfit does not increase during the inversion. An efficient pre-conditioner is found empirically through numerical experiments. It prevents the concentration of structural heterogeneity near the sources and receivers. We apply our waveform tomographic method to ≈1000 high-quality vertical-component seismograms, recorded in the Australasian region between 1993 and 2008. The waveforms comprise fundamental- and higher-mode surface and long-period S body waves in the period range from 50 to 200 s. To improve the convergence of the algorithm, we implement a 3-D initial model that contains the long-wavelength features of the Australasian region. Resolution tests indicate that our algorithm converges after around 10 iterations and that both long- and short-wavelength features in the uppermost mantle are well resolved. There is evidence for effects related to the non-linearity in the inversion procedure. After 11 iterations we fit the data waveforms acceptably well; with no significant further improvements to be expected. During the inversion the total fitted seismogram length increases by 46 per cent, providing a clear indication of the efficiency and consistency of the iterative optimization algorithm. The resulting SV -wave velocity model reveals structural features of the Australasian upper mantle with great detail. We confirm the existence of a pronounced low-velocity band along the eastern margin of the continent that can be clearly distinguished against Precambrian Australia and the microcontinental Lord Howe Rise. The transition from Precambrian to Phanerozoic Australia (the Tasman Line) appears to be sharp down to at least 200 km depth. It mostly occurs further east of where it is inferred from gravity and magnetic anomalies. Also clearly visible are the Archean and Proterozoic cratons, the northward continuation of the continent and anomalously low S-wave velocities in the upper mantle in central Australia.

Effects of Earth’s Lateral Inhomogeneous Structures on Coseismic Gravity Changes

Based on the perturbation method, we present a new method to study the effects of Earth’s laterally inhomogeneous structures on coseismic gravity changes caused by dislocations within a 3-D heterogeneous spherical earth model. We describe this method by six independent dislocations: A vertical strike-slip, two vertical dip-slips perpendicular to each other, and three tensile openings on three perpendicular planes. We derived the calculation formulae for the six independent dislocations. A combination of the six independent dislocations is useful to compute the effects on coseismic gravity changes resulting from an arbitrary seismic source at an arbitrary position.

Rotational Motions of Seismic Surface Waves in a Laterally Heterogeneous Earth

A number of recent studies have analyzed seismic rotational data using a relationship between transverse acceleration and rotation rate derived for a homogeneous full space. In this study we explore this relationship further theoretically by presenting a full ray theory (FRT) method to simulate rotational motions of fundamental mode seismic surface waves in smooth, laterally heterogeneous Earth models. In the ray picture of wave propagation the vertical component of the rotational rate motion of fundamental mode Love waves is obtained by dividing the transverse component of ground acceleration by the Love-wave local phase velocity beneath the seismic recording station. We illustrate the method with examples of theoretical calculations of T ≈40 sec rotational rate ground motions of fundamental Love waves using the crust model CRUST2.0 combined with the mantle model S20RTS for the 25 September 2003 M 8.1 Tokachi-oki earthquake, Japan. FRT rotation synthetics match complete calculations using the spectral-element method very well and fit real data reasonably well. Furthermore, we show that the effect of realistic local structure beneath receivers on rotational motions is strong enough to be observable. FRT calculations could potentially help to determine Love-wave local dispersion curves and, thus, to estimate the 1D local shear velocity structure beneath seismic stations from point measurements of rotational rate and acceleration ground motions.

Introduction to the supplement on seismic modeling with applications to acquisition, processing, and interpretation

Modeling of seismic wave propagation plays a key role in almost every aspect of exploration seismology. Fundamentally, it provides a means of understanding the character of recorded seismic data. Although analytical or semianalytical solutions exist for several canonical models, those are often insufficient to explain the full plethora of phenomena that arise in complex heterogeneous earth models governed by, e.g., anisotropic, viscoelastic, or poroelastic rheologies. Typical manifestations of such complex phenomena that are observed widely in surface seismic data include scattering, generation of multiples, and interface waves. Another application area of seismic modeling is survey evaluation and design, in which different acquisition geometries and subsurface model hypotheses are assessed to choose an optimal acquisition and processing strategy. A recent example in which seismic modeling is having a significant impact and proving to be a valuable tool is 3D wide-azimuth acquisition design in high-risk, high-reward ultra-deepwater exploration scenarios (see, for instance, contributions in this supplement).