Generation of complete theoretical seismograms for SH— II.

Summary. We report the results of our continuing efforts to compute theoretical seismograms for direct comparison with the experimental time series obtained with the long-period instruments of the WWSSN. The entire theoretical seismogram — body waves and surface waves — is generated for realistic sources buried in a radially heterogeneous, anelastic, spherical earth. The results described in Paper I (Nakanishi, Schwab & Knopoff) are extended to include the summation of 11 modes; for each, the dispersion, attenuation, and excitation are computed down to a minimum period of 1 s. Examples of the theoretical seismograms, and the comparison with experimental results are presented, The results of this comparison indicate that our first application of combined body- and surface-wave generation will concern the investigation of the intrinsic anelasticity in the upper mantle. The indicated technique for such an investigation is based on body waves simply crossing the region of high attenuation a few times in passing from focus to recording station, while a guided surface wave such as Sa , experiences this anelasticity over the entire propagation path.     

Generation of complete theoretical seismograms for SH-III

Summary. Earlier efforts to generate the entire theoretical seismograms, including both body and surface waves for realistic sources buried in a radially heterogeneous anelastic, spherical earth, are extended to include the summation of 16 modes. The comparison between a real seismogram and theoretical time series, relative to different attenuation models in the upper mantle, yields information concerning the anelasticity under the Pacific Ocean.     

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.     

Crust-mantle structure under South Pole and Ross Sea Beach in Antarctica

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On crustal corrections in surface wave tomography

Mantle models from surface waves rely on good crustal corrections. We investigated how far ray theoretical and finite frequency approximations can predict crustal corrections for fundamental mode surface waves. Using a spectral element method, we calculated synthetic seismograms in transversely isotropic PREM and in the 3-D crustal model Crust2.0 on top of PREM, and measured the corresponding time-shifts as a function of period. We then applied phase corrections to the PREM seismograms using ray theory and finite frequency theory with exact local phase velocity perturbations from Crust2.0 and looked at the residual time-shifts. After crustal corrections, residuals fall within the uncertainty of measured phase velocities for periods longer than 60 and 80 s for Rayleigh and Love waves, respectively. Rayleigh and Love waves are affected in a highly non-linear way by the crustal type. Oceanic crust affects Love waves stronger, while Rayleigh waves change most in continental crust. As a consequence, we find that the imperfect crustal corrections could have a large impact on our inferences of radial anisotropy. If we want to map anisotropy correctly, we should invert simultaneously for mantle and crust. The latter can only be achieved by using perturbation theory from a good 3-D starting model, or implementing full non-linearity from a 1-D starting model.     

Normal-mode representations of multiple-ray reflections in a spherical earth

Summary. In a spherically symmetric, isotropic earth model the duality between seismic rays and modes can be established completely by application of the principle of stationary phase to the summed normal-mode representation of the time signal. The requirement of stationary phase must be applied not only on the sum over the angular order but also over the radial-order summation.
This approach is illustrated by using asymptotic approximations to the equations for toroidal oscillations. In this way the travel-time formulae for rays from a surface source in a sufficiently smooth earth model are easily derived, and the distribution of modes between the different rays can be found. The result may be used as a selection criterion to reduce the number of modes that must be summed to construct synthetic seismograms at a certain distance and within a certain time window.     

Synthesis of seismic surface waves

Summary. The reflection method is used to produce complete sets of Rayleigh wave dispersion curves in a given phase velocity—frequency window for horizontally stratified media, including modes of very high numerical order, and theoretical surface wave seismograms are then synthesized.
The difficulties encountered when attempting to complete large numbers of dispersion curves are discussed. Particular problems arise from models with a low-velocity zone, when curves in a certain portion of the dispersion diagram split into two families, the crustal and the channel modes. The reflection method provides a convenient framework in which to examine this phenomenon heuristically and so devise a method to overcome the difficulties.
Seismograms are produced by mode summation and it is found that body-wave behaviour, as well as surface-wave features can be synthesized. The effect of truncating the mode series at a number comparable with the number of modes used in previous studies is examined. It is found that although the S -coda at longer ranges is not adversely affected, the arrivals attributable to body-waves are severely distorted. This must call into question the validity of synthetic seismograms generated by summation of only a few modes.     

Possible mode conversion between Love and Rayleigh waves at a continental margin

Summary. Mode conversion at a continental margin between normal modes of surface waves is investigated by theoretical calculations for oblique incidence for periods longer than 15 s. It is suggested that significant conversion takes place between the various modes of Love waves in the period interval between 15 and 40 s, while there is negligible mode conversion for longer periods. The largest mode conversion involves the lowest modes. In addition the calculations have revealed a small but significant conversion between Love and Rayleigh fundamental modes around 20-s period. Reflections of Love waves are found to be significant only for the continental fundamental mode.     

A fault plane solution using theoretical P seismograms

We use the method of Hudson and Douglas, Hudson & Blarney to compute seismograms which simulate the codas of 10 short period P -wave seismograms from a shallow earthquake. The polarities and relative amplitudes of P and pP measured from seven of the observed seismograms are used to compute a fault plane solution with confidence limits, assuming that the source radiates as a double couple. This solution is in approximate agreement with that given for the same earthquake by Sykes & Sbar, who used only the onset polarities of short-period P waves. The small difference between the two solutions can be explained by interference between the true first motion of P and microseismic noise at two stations.
The results show that, for some shallow earthquakes, the relative amplitude method has the following advantages over the first motions method. First, a P/pP amplitude ratio (with appropriate confidence limits) can always be measured, even in seismograms which are so noisy that the first motion of P is uncertain. Second, the fault plane solutions obtained from relative amplitudes have known confidence limits. Finally, by using more information from each seismogram, the relative amplitude method requires considerably fewer seismograms than the first motions method.     

Surface wave dispersion and Earth structure in south-eastern China

Summary. A reconnaissance study of crust and mantle structure in southeastern China was made using surface waves confined to that region from recent earthquakes. Data from the WWSSN stations ANP arid SEO, and from the digital stations TATO and MAT, were used to measure fundamental-mode group velocities of Love and Rayleigh waves over nine paths in south-eastern China, an area which has been technically quiet since the early Cenozoic. Crustal structure in this region is typical of stable continents, but shear-wave velocities in the uppermost mantle are low for a continent, 4.45 km s⁻¹ or less. Other seismological data support this observation.     

Measurements and interpretation of normal mode attenuation

summary . Measurements of Q for modes of free oscillation provide the most accurate information about the anelastic properties of the whole Earth in the period range from 100 to 3000 s. We have obtained more than 230 Q measurements, by using two different techniques. Individual LaCoste—Romberg gravimeter recordings of three large earthquakes were used to observe the time rate of decay of spectral peaks corresponding to different modes. This method provided measurements of Q for 37 different modes. By stacking 211 WWSSN recordings of two deep earthquakes, we were able to measure Q for 197 modes, including many overtones which cannot be analysed using the spectra of individual recordings.     

Rayleigh Wave Spectra and Group Velocity Minima, and the resonance of P waves in layered structures

Summary. A connection is established between the group velocity of Rayleigh waves, the spectral amplitudes of surface waves generated by a source, and the resonance of vertically travelling P waves. It implies that a minimum in a group velocity curve is reflected in the spectral amplitudes as a maximum. That this is so, appears to have been first noticed by Longuet-Higgins in a study of microseisms. Also when a sharp impedance contrast occurs in a plane-layered model of the crust, the group velocity minimum in the fundamental mode occurs close to a period equal to four times the travel time of P -waves from the surface to the interface. More than one such contrast gives rise in general to more than one minimum. Similar relations hold for the higher modes.     

Backscattering from spherical elastic inclusions and accuracy of the Kirchhoff approximation for curved interfaces

The Kirchhoff (or tangent plane) approximation, derived from the theoretically complete Kirchhoff–Helmholtz integral representation for the seismic wavefield, has been used extensively for the analysis of seismic-wave scattering from irregular interfaces; however, the accuracy of this method for curved interfaces has not been rigorously established. This paper describes an efficient Kirchhoff algorithm to simulate scattered waves from an arbitrarily curved interface in an elastic medium. Synthetic seismograms computed using this algorithm are compared with exact synthetics computed using analytical formulae for scattering of plane P waves by a spherical elastic inclusion. A windowing technique is used to remove strong internal reverberations from the analytical solution. Although the Kirchhoff method tends to underestimate the total scattering intensity, the accuracy of the approximation improves with increasing value of the wavenumber-radius product, kR . The arrival times and pulse shapes of primary reflections from the sphere are well approximated using the Kirchhoff approach regardless of curvature of the scattering surface, but the amplitudes are significantly underestimated for kR ≤ 5. The results of this work provide some new guidelines to assess the accuracy of Kirchhoff-synthetic seismograms for curved interfaces.     

Formalism for traveltime inversion with finite frequency effects

A simple modification of the waveform inversion formula, based on the normal mode perturbation theory, is shown to lead to a formula for traveltime anomalies. The kernel which is derived can be used for traveltime inversion with automatic inclusion of finite frequency effects. Inversion for Earth structure with such kernels will lead to better resolution estimates than ray-theoretical traveltime inversion. Examples of kernels for transverse component seismograms are shown for direct S waves, ScS , Love waves and diffracted S waves. A measure of finite frequency effects is also proposed by comparing our formula with the one from ray theory. A quantity which should be 1 in the case of ray theory is computed for the finite frequency kernels and is shown to have deviations up to about 30 per cent from 1. Therefore, the use of ray theory for long-period body waves applies incorrect weight along a ray path and may introduce a small bias to an earth model.     

A lower mantle S-wave triplication and the shear velocity structure of D"

Summary. A lower mantle S-wave triplication detected with short- and long-period WWSSN and CSN recordings indicates a substantial shear velocity discontinuity near 280 km above the core–mantle boundary. The triplication can be observed in rotated SH seismograms from intermediate and deep focus events throughout the distance range from 70° to 95°. Three distinct source region–receiver array combinations that have been investigated in detail demonstrate consistent travel time and relative amplitude behaviour of the triplication, with slight systematic shifts in the triplication indicating up to 40 km variations in the depth of the discontinuity. Modelling of the observations with synthetic seismograms produced with the Cagniard de Hoop and reflectivity methods constrains the shear velocity increase to be 235 ± 0.25 per cent, comparable to upper mantle discontinuities. Short-period observations indicate that the velocity increase may be a sharp first-order discontinuity, or may extend over a transition zone no more than 50 km thick. The shear velocity gradient below the discontinuity, within the D" layer, is not well-constrained by the SH data, but slightly positive or near zero velocity gradients are consistent with the long-period amplitude ratios of ScSH/SH .     

Simulation of SH- and P-SV-wave propagation in fault zones

Seismic fault-zone (FZ) trapped waves provide a potentially high-resolution means for investigating FZ and earthquake properties. Seismic waves emitted within and travelling along low-velocity FZ layers may propagate many kilometres within the low-velocity structure associated with the fault. Waveform observation of FZ trapped waves can be modelled in terms of FZ layer velocities, thicknesses and attenuation coefficients. This can greatly improve the resolution of imaged FZ structure and microearthquake locations. At present, broad-band theoretical seismograms are restricted to plane-parallel layers of uniform properties. However, it is not clear how realistic these models are compared with actual fault structures which could, for example, flare outwards near the surface, have irregular boundaries, interior heterogeneities, etc. To address these interpretational uncertainties, we perform finite-difference simulations for irregular FZ geometries and non-uniform material properties within the layers. The accuracy of the numerical solutions are verified by comparison with the analytical solution of Ben-Zion & Aki (1990) for plane-parallel structures. Our main findings are: (1) FZs can widen at the ctustal surface only slightly modifying the trapped waves; (2) velocity variations with depth destroy trapped wave propagation at all wavelengths; (3) FZ trapped waves can be obscured by the presence of a low-velocity surface layer; (4) models with short-scale random structures suggest that trapped waves average out irregular FZ geometries, and hence can be effectively modelled by average-property plane-layered media for the observed range of wavelengths.     

Long-period seismic source inversions using global tomographic models

We investigate the effect of laterally varying earth structure on centroid moment tensor inversions using fundamental mode mantle waves. Theoretical seismograms are calculated using a full formulation of surface wave ray theory. Calculations are made using a variety of global tomographic earth models. Results are compared with those obtained using the so-called great-circle approximation, which assumes that phase corrections are given in terms of mean phase slowness along the great circle, and which neglects amplitude effects of heterogeneity. Synthetic tests suggest that even source parameters which fit the data very well may have large errors due to incomplete knowledge of lateral heterogeneity. The method is applied to 31 shallow, large earthquakes. For a given earthquake, the focal mechanisms calculated using different earth models and different forward modelling techniques can significantly vary. We provide a range of selected solutions based on the fit to the data, rather than one single solution. Difficulties in constraining the dip-slip components of the seismic moment tensor often produce overestimates of seismic moment, leading to near vertical dip-slip mechanisms. This happens more commonly for earth models not fitting the data well, confirming that more accurate modelling of lateral heterogeneity can help to constrain the dip-slip components of the seismic moment tensor.     

Extension of the Thomson-Haskell method to non-homogeneous spherical layers

Summary. Amplitude spectra of Rayleigh and Love waves in a layered non-gravitating spherical earth have been obtained using as a source, displacement and stress discontinuities. In each layer elastic parameters and density follow specified functions of radial distance and the solutions of the equations of motion are obtained in terms of exponential functions. The Thomson—Haskell method is extended to this case. The problem reduces to simple calculations as in a plane-layered medium. Numerical results of phase and group velocities up to periods of 300 s in various earth models when compared with earlier results (obtained by numerical integration) show that the present method can be used with sufficient accuracy. The differences in phase velocity, group velocity and amplitude (also surface ellipticity in the case of Rayleigh waves) between spherical- and flat-earth models have been investigated in the range 20–300–s period and expressed in polynomials in the period.     

Shear-wave polarizations on a curved wavefront at an isotropic free surface

Summary. We present polarization diagrams of the particle motions at the free surface of an isotropic half-space generated by incident shear waves from a local buried point source. The reflectivity technique is used to calculate synthetic seismograms from which the particle motions are plotted. The particle motions are examined over a range of epicentral distances in a uniform isotropic half-space for different source frequencies and polarization angles, and for different Poisson's ratios. The particle motions due to a curved wavefront possess different characteristics from those generated by plane wavefronts at corresponding incidence angles. A curved wavefront generates a local SP -phase: a P -headwave which propagates along the free surface, and arrives shortly before the direct S -wave. These two arrivals give rise to cruciform particle motions in the sagittal and horizontal planes, which could be misinterpreted as anisotropy-induced shear-wave splitting. An examination of the particle motion in the transverse plane, mutually orthogonal to the sagittal and horizontal planes, can be used to discriminate between isotropic and anisotropic interpretations. The amplitude of the SP -phase is enhanced when it propagates in a low-velocity surface layer overlying the source layer, and may then become the dominant phase on radial-component seismograms. The presence of even a single surface layer may introduce considerable complexity into the seismogram, and we examine the effects of layer thickness, velocity contrast, and source depth on the corresponding polarization diagrams. Reliable information on the source and propagation path characteristics of shear waves from a buried local point source can only be obtained from free-surface records if they are recorded within a very limited epicentral distance range.     

Seismic body waves in anisotropic media: synthetic seismograms

Summary. Synthetic seismograms and particle motion diagrams are computed for simple, layered Earth models containing an anisotropic layer. The presence of anisotropy couples the P, SV and SH wave motion so that P waves incident on the anisotropic layer from below produce P, SV and small-amplitude SH waves at the surface both the P velocity and the amplitudes of the converted phases vary with azimuth. Significant SH amplitudes may be generated even when the wavelength of the P wave is much greater than the thickness of the anisotropic layer. Incident SV or SH waves may each generate large amplitudes of both SV and SH motion. This strong coupling is largely independent of the degree of velocity anisotropy of the medium. The arrivals from short-period S waves exhibit S-wave splitting, but arrivals from longer period S waves superpose into a modified waveform. This strong coupling does not allow the arrival of separate phases with pure SV and SH polarization except along directions of symmetry where the motion decouples.