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S U M M A R YWe 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.

S U M M A R YSeismic 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 reducedvariance 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.

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.

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