Seismic imaging of the laterally varying D" region beneath the Cocos Plate

We use an axisymmetric, spherical Earth finite difference algorithm to model SH -wave propagation through cross-sections of laterally varying lower mantle models beneath the Cocos Plate derived from recent data analyses. Synthetic seismograms with dominant periods as short as 4 s are computed for several models: (1) a D" reflector 264 km above the core–mantle boundary with laterally varying S -wave velocity increases of 0.9–2.6 per cent, based on localized structures from a 1-D double-array stacking method; (2) an undulating D" reflector with large topography and uniform velocity increase obtained using a 3-D migration method and (3) cross-sections through the 3-D mantle S -wave velocity tomography model TXBW. We apply double-array stacking to assess model predictions of data. Of the models explored, the S -wave tomography model TXBW displays the best overall agreement with data. The undulating reflector produces a double Scd arrival that may be useful in future studies for distinguishing between D" volumetric heterogeneity and D" discontinuity topography. Synthetics for the laterally varying models show waveform variability not observed in 1-D model predictions. It is challenging to predict 3-D structure based on localized 1-D models when lateral structural variations are on the order of a few wavelengths of the energy used, particularly for the grazing geometry of our data. Iterative approaches of computing synthetic seismograms and adjusting model characteristics by considering path integral effects are necessary to accurately model fine-scale D" structure.     

P-SH conversions in a flat-layered medium with anisotropy of arbitrary orientation

P-SH conversion is commonly observed in teleseismic P waves, and is often attributed to dipping interfaces beneath the receiver. Our modelling suggests an alternative explanation in terms of flat-layered anisotropy. We use reflectivity techniques to compute three-component synthetic seismograms in a 1-D anisotropic layered medium. For each layer of the medium, we prescribe values of seismic velocities and hexagonally symmetric anisotropy about a common symmetry axis of arbitrary orientation. A compressional wave in an anisotropic velocity structure suffers conversion to both SV -and SH -polarized shear waves, unless the axis of symmetry is everywhere vertical or the wave travels parallel to all symmetry axes. The P-SV conversion forms the basis of the widely used 'receiver function' technique. The P-SH conversion occurs at interfaces where one or both layers are anisotropic. A tilted axis of symmetry and a dipping interface in isotropic media produce similar amplitudes of both direct ( P ) and converted ( Ps ) phases, leaving the backazimuth variation of the P-Ps delay as the main discriminant. Seismic anisotropy with a tilted symmetry axis leads to complex synthetic seismograms in velocity models composed of just a few flat homogeneous layers. It is possible therefore to model observations of P coda with prominent transverse components with relatively simple 1-D velocity structures. Successful retrieval of salient model characteristics appears possible using multiple realizations of a genetic-algorithm (GA) inversion of P coda from several backazimuths. Using GA inversion, we determine that six P coda recorded at station ARU in central Russia are consistent with models that possess strong (> 10 per cent) anisotropy in the top 5 km and between 30 and 43 km depth. The symmetry axes are tilted, and appear aligned with the seismic anisotropy orientation in the mantle under ARU suggested by SKS splitting.     

Seismic waves in stratified anisotropic media

Summary. The response of a structure composed of anisotropic strata can be built up from the reflection and transmission properties of individual interfaces using a slightly modified version of the recursion scheme of Kennett. This scheme is conveniently described in terms of scatterer operators and scatterer products. The effects of a free surface and the introduction of a simple point source at any depth can be accommodated in a manner directly analogous to the treatment for isotropic structures. As in the isotropic case the results so obtained are stable to arbitrary wavenumbers. For isotropic media, synthetic seismograms can be constructed by computing the structure response as a function of frequency and radial wavenumber, then performing the appropriate Fourier and Hankel transforms to obtain the wavefield in time-distance space. Such a scheme is convenient for any system with cylindrical symmétry (including transverse isotropy). Azimuthally anisotropic structures, however, do not display cylindrical symmétry; for these the transverse component of the wavenumber vector will, in general, be non-zero, with the result that phase, group, and energy velocities may all diverge. The problem is then much more conveniently addressed in Cartesian coordinates, with the frequency-wavenumber to time-distance transformation accomplished by 3-D Fourier transform.     

Paraxial ray approximations in the computation of seismic wavefields in inhomogeneous media

Summary. Several important applications of the paraxial ray approximation (PRA) to numerical modelling of high-frequency seismic body wavefields are discussed. The PRA can be used to evaluate the displacement vector not only directly on the ray, as in the standard ray method. but also approximately in the vicinity of this ray. The PRA also offers simple ways of approximate evaluation of paraxial rays, situated in the vicinity of the central ray, and of two-point ray tracing. A very important application of the PRA consists in a simple, fast and effective Computation of body-wave synthetic seismograms in general, 3-D, laterally inhomogeneous, layered structures. Examples of synthetic seismograms for 3-D structures, computed using the PRA, are presented.     

Computation of complete synthetic seismograms for laterally heterogeneous models using the Direct Solution Method

We use the Direct Solution Method (DSM) together with the modified operators derived by Geller & Takeuchi (1995) and Takeuchi, Geller & Cummins (1996) to compute complete synthetic seismograms and their partial derivatives for laterally heterogeneous models in spherical coordinates. The methods presented in this paper are well suited to conducting waveform inversion for 3-D Earth structure. No assumptions of weak perturbation are necessary, although such approximations greatly improve computational efficiency when their use is appropriate.
An example calculation is presented in which the toroidal wavefield is calculated for an axisymmetric model for which velocity is dependent on depth and latitude but not longitude. The wavefield calculated using the DSM agrees well with wavefronts calculated by tracing rays. To demonstrate that our algorithm is not limited to weak, aspherical perturbations to a spherically symmetric structure, we consider a model for which the latitude-dependent part of the velocity structure is very strong.     

A method for calculating synthetic seismograms in laterally varying media

Summary An effective algorithm for computing synthetic seismograms in laterally inhomogeneous media has been developed. The method, based on zero-order asymptotic ray theory, is primarily intended for use in refraction and reflection studies and provides an economical means of seismic modelling.
A given smoothed velocity-depth-distance model is divided into small squares with constant seismic parameters and first-order interfaces are represented by an arbitrary number of dipping linear segments. The computation of ray propagation and amplitudes through such a model does not involve complicated analytic expressions and therefore minimizes computer time.
Amplitudes are determined by geometrical spreading of spherical wave-fronts and energy partitioning at interfaces. Synthetic seismograms calculated for laterally homogeneous models are in good agreement with those obtained by the Reflectivity Method.     

Two efficient algorithms for iterative linearized inversion of seismic waveform data

We present two equivalent algorithms for iterative linearized waveform inversion for 3-D Earth structure with respect to an arbitrary 3-D starting model; one is a matrix formulation, and the second is a wavefield formulation. Both algorithms require the computation of accurate synthetic seismograms, but neither requires that any particular method be used to compute the synthetics. The matrix formulation is equivalent to our previously published algorithm (Hara, Tsuboi & Geller 1991), but requires less than 10 per cent of the CPU time of the previous algorithm. The wavefield algorithm is equivalent to that of Tarantola (1986) and Mora (1987), but appears to be substantially more efficient.     

Use of Kirchhoff s formula for body wave calculations in the Earth

Summary. Kirchhoff's time-dependent surface integral representation of a scalar wavefield is applied to the problem of computing synthetic seismograms for P -waves in the Earth. By means of an appropriate parameterization, the Kirchhoff integral is transformed into a convolution of a weight function with the derivative of the source function in the time domain. The weight function is calculated using simple ray theory. The method extends the applicability of simple ray theory to caustics and other diffraction phenomena and allows certain kinds of departures from spherical symmetry to be taken into account. The method is illustrated in detail by application to the PKP -wavefield in the Earth.     

A 2-D spectral-element method for computing spherical-earth seismograms—II. Waves in solid–fluid media

We portray a dedicated spectral-element method to solve the elastodynamic wave equation upon spherically symmetric earth models at the expense of a 2-D domain. Using this method, 3-D wavefields of arbitrary resolution may be computed to obtain Fréchet sensitivity kernels, especially for diffracted arrivals. The meshing process is presented for varying frequencies in terms of its efficiency as measured by the total number of elements, their spacing variations and stability criteria. We assess the mesh quantitatively by defining these numerical parameters in a general non-dimensionalized form such that comparisons to other grid-based methods are straightforward. Efficient-mesh generation for the PREM example and a minimum-messaging domain decomposition and parallelization strategy lay foundations for waveforms up to frequencies of 1 Hz on moderate PC clusters. The discretization of fluid, solid and respective boundary regions is similar to previous spectral-element implementations, save for a fluid potential formulation that incorporates the density, thereby yielding identical boundary terms on fluid and solid sides. We compare the second-order Newmark time extrapolation scheme with a newly implemented fourth-order symplectic scheme and argue in favour of the latter in cases of propagation over many wavelengths due to drastic accuracy improvements. Various validation examples such as full moment-tensor seismograms, wavefield snapshots, and energy conservation illustrate the favourable behaviour and potential of the method.     

Simulating three-dimensional seismograms in 2.5-dimensional structures by combining two-dimensional finite difference modelling and ray tracing

Finite difference (FD) simulation of elastic wave propagation is an important tool in geophysical research. As large-scale 3-D simulations are only feasible on supercomputers or clusters, and even then the simulations are limited to long periods compared to the model size, 2-D FD simulations are widespread. Whereas in generally 3-D heterogeneous structures it is not possible to infer the correct amplitude and waveform from 2-D simulations, in 2.5-D heterogeneous structures some inferences are possible. In particular, Vidale & Helmberger developed an approach that simulates 3-D waveforms using 2-D FD experiments only. However, their method requires a special FD source implementation technique that is based on a source definition which is not any longer used in nowadays FD codes. In this paper, we derive a conversion between 2-D and 3-D Green tensors that allows us to simulate 3-D displacement seismograms using 2-D FD simulations and the actual ray path determined in the geometrical optic limit. We give the conversion for a source of a certain seismic moment that is implemented by incrementing the components of the stress tensor.
Therefore, we present a hybrid modelling procedure involving 2-D FD and kinematic ray-tracing techniques. The applicability is demonstrated by numerical experiments of elastic wave propagation for models of different complexity.     

The PWaves From the Amchitka Island Explosions

Summary. There is evidence that the equivalent seismic sources of the Amchitka Island explosions — Longshot, Milrow and Cannikin — depart significantly from the simple model of a point compressional-source in a layered elastic-medium. Consequently modelling the observed seismograms using standard source-models may not be the most efficient method of determining source properties. Here an alternative to modelling is used to obtain information on the seismic sources due to the explosions. Broad-band (BB) estimates of the P signals are obtained from the short-period (SP) seismograms, corrected for attenuation, and interpreted in terms of P, pP and radiation from secondary sources. the main conclusions are:
(i) BB estimates of the radiated displacement from the explosions can be obtained with only a small reduction in the signal-to-noise ratio seen on SP seismograms;
(ii) observations of differences in pulse amplitudes and spectra are not necessarily due to differences in anelastic attenuation;
(iii) P and pP at a given station may differ in shape so that notches in the signal spectrum may not be related to source depth;
(iv) there is evidence of arrivals that others have identified as due to slap-down but which could be interpreted as an overshoot to pP;
(v) direct interpretation of the estimated ground displacement is a better procedure for determining the seismic source properties of explosions than modelling SP seismograms using idealised models as a starting point.     

Three-dimensional geometrical ray theory and modelling of transmitted seismic energy of data from the Nevada Test Site

This paper presents a geometrically based algorithm for computing synthetic seismograms for energy transmitted through a 3-D velocity distribution. 3-D ray tracing is performed to compute the traveltimes and geometrical spreading (amplitude). The formulations of both kinematic and dynamic ray-tracing systems are presented. The two-point ray-tracing problem is solved by systematically updating the initial conditions and adjusting the ray direction until the ray intersects the specified endpoint. The amount of adjustment required depends on the derivatives of the position with respect to the given starting angles between consecutive rays. The algorithm uses derivatives to define the steepest-descent direction and to update the initial directions. The convergence rate depends on the complexity of the model.
Test seismograms compare favourably with those from a 2-D asymptotic ray theory algorithm and a 3-D Gaussian-beam algorithm. The algorithm is flexible in modelling arbitrary source and recorder geometries for various smoothly varying 3-D velocity distributions. The algorithm is further tested by simulating surface-to-tunnel vibroseis field data. Shear waves as well as compressional waves may be approximately included. Application of the algorithm to a data set from the Rainier Mesa of the Nevada Test Site produced a good fit to the transmitted (first arrival) traveltimes and amplitudes, with approximately 15 per cent variation in the local 3-D velocity.     

Synthetic body wave seismograms for three-dimensional laterally varying media

Summary. Two methods of computing body wave synthetic seismograms in three-dimensional laterally varying media are discussed. Both these methods are based on the summation of Gaussian beams. In the first, the initial beam parameters are chosen at the source, in the second at the beam endpoints. Both these variants eliminate the ray method singularities. The expansion of the wavefield into plane waves may be considered as the limiting case of the first approach and the Chapman–Maslov method as the limiting case of the second approach. Computer algorithms are briefly described and numerical examples presented. In the first numerical example, the comparisons of the two approaches, based on summing Gaussian beams, with the reflectivity method indicate that the computed synthetic seismograms are satisfactorily accurate even in the caustic region. The next example suggests that the two methods discussed can be simply and effectively applied to 3-D laterally inhomogeneous structures.     

Generation of complete theoretical seismograms for SH—

Summary. We report the initial results of our attempts to obtain theoretical seismograms for direct comparison with the experimental time series obtained with the long-period instruments of the WWSSN. The entire theoretical seismogram, including both body waves and surface waves, can be generated for a spherical, anelastic earth by simple inverse Fourier transformation of the sum of the propagating fundamental and higher-mode surface waves. The key to success in reproducing the WWSSN records involves the number of modes, and the minimum period used in these computations; here we use eight modes and a minimum period of 2 s. Efficient computational algorithms make it possible to handle up to 2000 frequency points for each mode; approximately 200 layers are used to model the radial heterogeneity of the earth; attenuation is treated exactly. Examples are given of the SH theoretical seismograms resulting from dislocation sources buried at various depths in the Earth.     

Exact synthetic seismograms for an inhomogeneity in a layered elastic half-space

Summary. The propagation of a pulsed elastic wave in the following geometry is considered. An elastic half-space has a surface layer of a different material and the layer furthermore contains a bounded 3-D inhomogeneity. The exciting source is an explosion, modelled as an isotropic pressure point source with Gaussian behaviour in time.
The time-harmonic problem is solved using the null field approach (the T matrix method), and a frequency integral then gives the time-domain response. The main tools of the null field approach are integral representations containing the free space Green's dyadic, expansions in plane and spherical vector wave functions, and transformations between plane and spherical vector wave functions. It should be noted that the null field approach gives the solution to the full elastodynamic equations with, in principle, an arbitrarily high accuracy. Thus no ray approximations or the like are used. The main numerical limitation is that only low and intermediate frequencies, in the sense that the diameter of the inhomogeneity can only be a few wavelengths, can be considered.
The numerical examples show synthetic seismograms consisting of data from 15 observation points at increasing distances from the source. The normal component of the velocity field is computed and the anomalous field due to the inhomogeneity is sometimes shown separately. The shape of the inhomogeneity, the location and depth of the source, and the material parameters are all varied to illustrate the relative importance of the various parameters. Several specific wave types can be identified in the seismograms: Rayleigh waves, direct and reflected P -waves, and head waves.     

Kirchhoff–Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium – II. Computations

Summary. High-frequency reflection and refraction seismograms for laterally variable multi-layered elastic media are computed by using the frequency domain elastic Kirchhoff–Helmholtz (KH) theory of Frazer and Sen. Both source and receiver wavefields are expanded in series of generalized rays and then elastic (KH) theory is applied to determine the coupling between each source ray and each receiver ray at each interface. The motion at the receiver is given as a series of integrals, one for each generalized ray. We use geometrical optics and plane wave reflection and transmission coefficients for rapid evaluation of the integrand. When the source or the receiver ray field has caustics on the surface of integration geometrical ray theory breaks down and this gives rise to singularities in the KH integrand. We repair this using methods suggested by Frazer and Sen.
Examples of reflection seismograms for 2-D structures computed by elastic KH theory are shown. Those for a vertical fault scarp structure are compared with the seismograms obtained by physical modelling. Then OBS data obtained from the mid-America trench offshore Guatemala area are analysed by computing KH synthetics for a velocity model that has been proposed for that area. Our analysis indicates the existence of a small low-velocity zone off the trench axis.
No head wave arrivals are obtained in our KH synthetics since we do not consider multiple interactions of a ray with an interface. The nearly discontinuous behaviour of elastic R/T coefficients near the critical angle causes small spurious phases which arrive later than the correct arrivals.     

Comment on: 'Can horizontal P waves be trapped and resonate in a shallow sedimentary basin?' by F. J. Sácnhez-Sesma and F. LuzÓ

We show that most of the arguments in the above paper are either incorrect or irrelevant to the point the authors are trying to make. We show that their results have no bearing on the model proposed by our group, as they claim. They discuss the seismic response of a valley with a 2-D trapezoidal cross-section in a vertical plane, whereas we dealt with a closed basin with a 2-D cross-section but of arbitrary geometry and in the horizontal plane. Even more significantly, the width of the valley they use is much smaller than the wavelength of the horizontal P waves that can resonate, thereby precluding any possibility of them being trapped. Therefore, their arguments do not clarify the issue posed in the title of their article.     

Surface coseismic gravity changes caused by dislocations in a 3-D heterogeneous earth

This study describes an examination of surface gravity changes caused by dislocations within a 3-D heterogeneous earth. This new theory is described using six independent dislocations: a vertical strike-slip, two vertical dip-slips perpendicular to each other, and three tensile openings on three perpendicular planes. A combination of the six independent dislocations is useful to compute coseismic gravity changes resulting from an arbitrary seismic source at an arbitrary position. Based on the 3-D lateral inhomogeneous P -wave velocity model, we deduce the 3-D density and S -wave velocity models using the relation of Karato. Finally, numerical computations are performed for a location south of Japan (30°N, 135°E). We calculate the coseismic gravity changes resulting from the six independent dislocations for source depths of 100, 300 and 637 km, respectively. Numerical results show that the maximum 3-D effect varies concomitantly with the dislocation type and the source depth. For seismic problems, the effect of elastic parameter  μ  is dominant.     

Kirchhoff synthetics for seismic reflections and diving waves in two dimensions and application to the D" layer

The Kirchhoff-Helmholtz (KH) integration has been used to model the reflected and the diving waves from an interface with a positive velocity gradient. The modelling is carried out for a spherical boundary and for a sinusoidal topography with a long-scale wavelength.
An artefact, which is a major problem in modelling the seismic response using the KH integration, has been reduced by introducing a Hilbert transform sign manipulation. Cleaner synthetic seismograms with correct amplitudes have been produced by this method. A discretization in larger surface elements has been made possible by introducing a smoothing factor that suppresses the noise that normally follows the constructed signal if a large element size is taken.     

Seismogram synthesis using normal mode superposition: the locked mode approximation

Summary. A normal mode superposition approach is used to synthesize complete seismic codas for flat layered earth models and the P-SV phases. Only modes which have real eigenwavenumbers are used so that the search for eigenvalues in the complex wavenumber plane is confined to the real axis. In order to synthesize early P -wave arrivals by summing a number of'trapped'modes, an anomalously high velocity cap layer is added to the bottom of the structure so that most of the seismic energy is contained in the upper layers as high-order surface waves. Causality arguments are used to define time windows for which the resulting synthetic seismograms are close approximations to the exact solutions without the cap layer. The traditional Thomson—Haskell matrix approach to computing the normal modes is reformulated so that numerical problems encountered at high frequencies are avoided and numerical results of the locked mode approximation are given.