ALBERT

Notes: We present a new method to calculate complete synthetic seismograms for a spherically symmetric earth model which uses neither eigenfrequencies and eigenfunctions nor an earth-flattening transformation. The response of the earth to a moment tensor point source is evaluated in the frequency domain for both spheroidal and toroidal motion by numerical integration of the appropriate system of ordinary differential equations with source term and summation over vector spherical harmonics. Attenuation is included by using complex elastic moduli. Owing to the discrete sampling of the response in the frequency domain, the numerical effort is proportional to the length of the desired time series for a fixed maximum frequency. This makes the method much more efficient than normal-mode calculations for higher frequency applications, where often seismogram lengths of 20 to 40 min are sufficient. Since the angular degree of the spherical harmonics provide a natural discretization in the wavenumber domain, spatial aliasing is unimportant. Time aliasing is suppressed by evaluating the response at complex frequencies with constant imaginary part.We have compared synthetic seismograms obtained by the new method with normal-mode seismograms up to a frequency of 20 mHz and achieve excellent agreement for all three components. The accuracy of the method is further corroborated by comparisons with real data up to a frequency of 200 mHz. We tested the numerical scheme up to frequencies of 1 Hz and harmonic degrees of 12 000 and did not find any numerical instabilities. Incidentally, the approach sheds some light on how normal modes make up body waves.

Notes: 3-D wave propagation in a waveguide composed of laterally homogeneous partitions separated by vertical interfaces is treated in an exact manner. In each laterally homogeneous subregion, the wavefield is represented by Love- and Rayleigh-type modes, combined in such a way that displacements and tractions are continuous at the vertical discontinuities. In order to achieve exact continuity at the interfaces, near-field modes which exponentially decay in the propagation direction and are associated to fully complex wavenumbers, must be included in the modal representation. The expansion coefficients of the mode series are computed directly by exploiting orthogonality relations between modes of different type, order and propagation direction. Since the boundary conditions at the discontinuities are satisfied exactly, the resulting expansion of the wavefield is valid even on the interfaces themselves. Numerical results are presented for a single vertical discontinuity and a lamella.

Notes: We present an exact treatment of wave propagation across a cylindrical inclusion embedded in a layered waveguide, which may serve as a test case allowing an estimation of the accuracy and validity range of approximation methods applied to surface wave propagation. Since, in contrast to a plane vertical interface, a cylindrical inclusion is of finite spatial extent, the test case presented here is especially well suited to assess the quality of approaches based on scattering theory.The wavefield is represented by Love and Rayleigh type modes. A 3-D orthonormality relation is derived, expressed as an integral over the cylinder surface which allows a direct and unique computation of basis solutions. After solving a linear system of equations these basis solutions are superimposed to form the complete wavefield both within and outside the cylinder. It is shown that exact continuity of displacements and tractions at the interface can not be achieved with propagating modes only. Non-propagating modes, which have complex wavenumbers and hence decay in the propagation direction, have to be included in the modal series. In contrast to propagating modes, complex modes have vanishing energy flux.Numerical results are presented for a layered halfspace, where only propagating modes were used, and for a layered waveguide for various sizes of the cylindrical inclusion. Astonishingly, vertical and horizontal displacements behave very differently. For instance, the scattered vertical displacement generated by an incoming Rayleigh fundamental mode is clearly dominated by the mode itself, while the scattered horizontal displacement is severely influenced by the excited higher Rayleigh modes. This might explain the difficulties met when interpreting horizontal components of surface wave data within a single-mode concept.

Notes: Teleseismic surface waves in general strongly deviate from plane waves as is evident from the analysis of surface-wave data recorded with dense networks. This causes conventional, ray-based tomographic techniques to break down if applied to network surface-wave data. We present a new inversion method based on the acoustic-wave equation and applicable to vertical-component surface-wave data which successfully deals with non-plane wavefield geometries.The basic idea of the method is a joint estimation of the incoming wavefield and heterogeneous structure within the network region. Crucial to the success of the method is an adequate parametrization of the incoming wavefield which is realized using Hermite-Gaussian basis functions. Additionally, we apply a constraint on the wavefield parameters that expresses the idea that the samples of the wavefield taken at the station locations should be representative for the wavefield in the whole network region. In this way, wavefields that show stronger fluctuations in spectral amplitude than observed at the stations are rejected. To represent heterogeneous structure within the network region we use an expansion into 2-D Hermite-Gaussian functions.Provided that the density of stations is sufficient, the proposed method retrieves heterogeneous structure in the network region very well. It is not sensitive to noise or non-uniform azimuthal coverage of earthquakes. Moreover, it yields smoothed versions of the true model if the roughness of the latter has been underestimated in the inversion. Conversely, if the true model is much smoother than anticipated, inspection of the trade-off between model smoothness and data misfit allows us to find the correct model.The limiting factor for the inversion is the density of stations, which must allow for a reliable interpolation of the observed wavefield within the network. Therefore, in order to perform regional surface-wave studies it is essential to deploy seismic stations in the region of interest itself.

Notes: Seismic surface waves exhibit much more complicated wavefields than is commonly assumed. We are led to this conclusion after analysing 90 teleseismic events recorded at on average eight broad-band stations in Southern Germany. Large amplitude and phase fluctuations across the network are observed which, as we show, are definitely not due to instrument response or calibration problems. In order to give an impression of how surface wavefields may look in reality, we fit wavefields to the observed network data using basis wavefields derived from Hermite-Gaussian functions. We show both amplitude and phase distributions of several events.Synthetic wavefields, generated by acoustic finite-difference computations in a random medium with realistic correlation length and rms velocity fluctuations, support the interpretation that most of the observed wavefield anomalies have accumulated on the path of the wave from the source to the network. Even if no lateral heterogeneities existed in the region of the network, the large-scale features of the observed wavefields would remain nearly unaltered. We also model the synthetic wavefields using Hermite-Gaussian wavefields and achieve extremely good fits.In view of the large anomalies of the wavefields, we investigate their implications for the regional surface-wave tomography. In this method, one infers regional structure from a set of measurements of phase traveltime between pairs of stations. the basic assumption of the method is that the wavefield incident on the network be plane. Thus, all observed phase traveltime anomalies are interpreted in terms of regional heterogeneous structure. the consequence is a seemingly highly inconsistent data set. We perform several tomographic experiments with realistic synthetic data sets. Our experiments indicate that regional surface-wave tomography may work, if there is an excellent coverage of paths crossing the region and if each path is sampled several times. Under realistic conditions, that is on a sparse network with not much more than one velocity value per path, the imaging power of surface-wave tomography is very poor. Owing to the seeming inconsistency of the phase traveltimes, variance reduction is generally low. With an increasing number of traveltime data it further decreases, while at the same time the imaging power increases. the opposite behaviour is observed if the number of data is reduced—variance reduction increases, but imaging power decreases. Hence, in connection with regional surface-wave tomography, high variance reduction indicates a lack of data rather than the goodness of reconstruction.

Notes: We present a 2-D reformulation of surface wave scattering theory in terms of potentials, which allows an extension of the Born single-scattering approach to include multiple forward scattering. No additional numerical effort compared to single scattering is required for a computation of the wavefield over the whole heterogeneous region. Born single scattering for elastic surface waves and both multiple and single scattering for acoustic waves are also covered by the formulation. It is valid for fully anisotropic perturbations of the reference medium. We use the flexibility of our formulation to compare the different approximations with each other and, additionally, test all of them against an exact solution for the particular case of a cylindrical inclusion in a layered waveguide.Our numerical results, obtained for shear velocity contrasts of about 6 per cent, show that the method which includes multiple scattering is superior to the single-scattering methods if the scattering region extends over more than one wavelength. If coupling to higher modes is suppressed, the multiple-scattering method still yields nearly exact results for the vertical displacement. the influence of mode coupling and type conversion leads to only small errors in vertical displacement. Moreover, as we show for a cylinder with a diameter of two wavelengths, even an acoustic treatment of surface waves including multiple forward scattering may be more accurate than single scattering within an elastic treatment. For scatterer sizes below one wavelength the single-scattering approaches are accurate enough, while elastic and acoustic treatments of surface waves may differ considerably.The proposed multiple-scattering method is numerically very efficient, because the numerical effort mainly depends on the degrees of smoothness of the wavefield and the heterogeneity, and is not directly coupled to the wavelength.

Notes: The subject of this paper is the application of the Gaussian beam method to the propagation of surface waves on a laterally heterogeneous, spherical earth taking fully into account the curvature of the Earth's surface. This is achieved by constructing a ray centred coordinate system on general curved surfaces. The application of this method to a spherical shell leads to a ray centred coordinate system adjusted to the spherical geometry. Solving the elastodynamic equation in this coordinate system by a perturbation ansatz yields two interesting results. (1) The well-known parabolic differential equation for the zeroth order displacement follows directly from a solvability condition imposed on the right-hand side of an inhomogeneous differential equation arising from the perturbation ansatz. (2) The dynamic ray-tracing system which is derived from the parabolic equation is a special case of a general dynamic ray-tracing system valid for arbitrary curved surfaces which can be derived by perturbing the ray equations. It is shown that the geometry of the surface enters only by its intrinsic curvature into the dynamic ray-tracing system. Numerical computations of rays and synthetic seismograms based on this new approach for the model M84C of Woodhouse & Dziewonski show that there can occur considerable deviations of the ray-paths from great circles leading to visible amplitude anomalies in the synthetic seismograms. Comparison of synthetic seismograms with selected data recorded at the Black Forest Observatory (BFO) indicate that the model M84C is still too smooth to yield satisfactory agreement of the synthetics with the data.