Description: We present a method to construct non-stationary and anisotropic second-order random model realizations that can be used for numerical wave propagation simulations in various geometries. Models are generated directly from a given covariance matrix using its eigenvector decomposition (principal component or Karhunen-Loève method). Because this decomposition is very expensive computationally in 3-D, we use model symmetries to reduce the size of the covariance matrix to its non-stationary components. Stationary components can then be described through their power spectrum, such that models with axisymmetric or spherically symmetric statistics can be generated from a 1-D covariance matrix. We focus in particular on models with spherically symmetric statistics that are important to study wave propagation in the Earth. We use this method to show the influence of hypothetical small-scale structure in the Earth's mantle on the elastic wavefield. To this end, we extend tomographic models beyond their spatial resolution limit with different distributions of small-scale scatterers that generate a coda and attenuate direct waves (scattering attenuation). We observe that scattering attenuation of fundamental mode Rayleigh waves is small (0.5–2 per cent of the total attenuation), if the elastic mantle structure does not become significantly stronger at smaller scales. At the examined heterogeneity strengths, scattering attenuation scales linearly with the model variance. The long-period fundamental mode Rayleigh wave coda is difficult to measure because it is weak and overlaps with other signals. However, it can be shown that its intensity also scales linearly with model power, and that it depends strongly on the spherical geometry of the Earth. It can therefore be used to distinguish between models with different small-scale power. We show qualitatively that the coda generated by the type of random models we consider can explain observed scattered energy at long periods (100 s).

Description: We analyse the lateral heterogeneity scales of recent upper mantle tomographic shear velocity ( Vs ) global and regional models. Our goal is to constrain the spherical harmonics power spectrum over the largest possible range of scales to get an estimate of the strength and statistical distribution of both long and small-scale structure. We use a spherical multitaper method to obtain high quality power spectral estimates from the regional models. After deconvolution of the employed taper functions, we combine global and regional spectral estimates from scales of 20 000 to around 200 km (degree 100). In contrast to previous studies that focus on linear power spectral densities, we interpret the logarithmic power per harmonic degree l as heterogeneity strength at a particular depth and horizontal scale. Throughout the mantle, we observe in recent global models, that their low degree spectrum is anisotropic with respect to Earth's rotation axis. We then constrain the uppermost mantle spectrum from global and regional models. Their power spectra transfer smoothly into each other in overlapping spectral bands, and model correlation is in general best in the uppermost 250 km (i.e. the ‘heterosphere’). In Europe, we see good correlation from the largest scales down to features of about 500 km. Detailed analysis and interpretation of spectral shape in this depth range shows that the heterosphere has several characteristic length scales and varying spectral decay rates. We interpret these as expressions of different physical processes. At larger depths, the correlation between different models drops, and the power spectrum exhibits strong small scale structure whose location and strength is not as well resolved at present. The spectrum also has bands with elevated power that likely correspond to length scales that are enhanced due to the inversion process.