Elastic modelling on a grid with vertically varying spacing1

We present a discrete modelling scheme which solves the elastic wave equation on a grid with vertically varying grid spacings. Spatial derivatives are computed by finite-difference operators on a staggered grid. The time integration is performed by the rapid expansion method. The use of variable grid spacings adds flexibility and improves the efficiency since different spatial sampling intervals can be used in regions with different material properties. In the case of large velocity contrasts, the use of a non-uniform grid avoids spatial oversampling in regions with high velocities. The modelling scheme allows accurate modelling up to a spatial sampling rate of approximately 2.5 gridpoints per shortest wavelength. However, due to the staggering of the material parameters, a smoothing of the material parameters has to be applied at internal interfaces aligned with the numerical grid to avoid amplitude errors and timing inaccuracies. The best results are obtained by smoothing based on slowness averaging. To reduce errors in the implementation of the free-surface boundary condition introduced by the staggering of the stress components, we reduce the grid spacing in the vertical direction in the vicinity of the free surface to approximately 10 gridpoints per shortest wavelength. Using this technique we obtain accurate results for surface waves in transversely isotropic media.     

Comparison of the FORT approximation of the coupling ray theory with the Fourier pseudospectral method

The standard ray theory (RT) for inhomogeneous anisotropic media does not work properly or even fails when applied to S-wave propagation in inhomogeneous weakly anisotropic media or in the vicinity of shear-wave singularities. In both cases, the two shear waves propagate with similar phase velocities. The coupling ray theory was proposed to avoid this problem. In it, amplitudes of the two S waves are computed by solving two coupled, frequency-dependent differential equations along a common S-wave ray. In this paper, we test the recently developed approximation of coupling ray theory (CRT) based on the common S-wave rays obtained by first-order ray tracing (FORT). As a reference, we use the Fourier pseudospectral method (FM), which does not suffer from the limitations of the ray method and yields very accurate results. We study the behaviour of shear waves in weakly anisotropic media as well as in the vicinity of intersection, kiss or conical singularities. By comparing CRT and RT results with results of the FM, we demonstrate the clear superiority of CRT over RT in the mentioned regions as well as the dangers of using RT there.     

Cu- and Mn-bearing tourmalines from Brazil and Mozambique: crystal structures, chemistry and correlations

Cu- and Mn-bearing tourmalines from Brazil and Mozambique were characterised chemically (EMPA and LA-ICP-MS) and by X-ray single-crystal structure refinement. All these samples are rich in Al, Li and F (fluor-elbaite) and contain significant amounts of CuO (up to ~1.8 wt%) and MnO (up to ~3.5 wt%). Structurally investigated samples show a pronounced positive correlation between the <Y-O> distances and the (Li + Mn²⁺ + Cu + Fe²⁺) content (apfu) at this site with R ² = 0.90. An excellent negative correlation exists between the <Y-O> distances and the Al₂O₃ content (R ² = 0.94). The samples at each locality generally show a strong negative correlation between the X-site vacancies and the (MnO + FeO) content. The Mn content in these tourmalines depends on the availability of Mn, on the formation temperature, as well as on stereochemical constraints. Because of a very weak correlation between MnO and CuO we believe that the Cu content in tourmaline is essentially dependent on the availability of Cu and on stereochemical constraints.     

Li-bearing, disordered Mg-rich tourmaline from a pegmatite-marble contact in the Austroalpine basement units (Styria, Austria)

Pale-blue to pale-green tourmalines from the contact zone of Permian pegmatites to mica schists and marbles from different localities of the Austroalpine basement units (Rappold Complex) in Styria, Austria, are characterized. All these Mg-rich tourmalines have small but significant Li contents, up to 0.29 wt% Li₂O, and can be characterized as dravite, with FeO contents of ?~?0.9–2.7 wt%. Their chemical composition varies from ^X (Na_0.67Ca_0.19?K_0.02?_0.12) ^Y (Mg_1.26Al_0.97Fe²⁺ _0.36Li_0.19Ti⁴⁺ _0.06Zn_0.01?_0.15) ^Z (Al_5.31?Mg_0.69) (BO₃)₃ Si₆O₁₈ ^V (OH)₃? ^W [F_0.66(OH)_0.34], with a?=?15.9220(3), c?=?7.1732(2) Å to ^X (Na_0.67Ca_0.24?K_0.02?_0.07) ^Y (Mg_1.83Al_0.88Fe²⁺ _0.20Li_0.08Zn_0.01Ti⁴⁺ _0.01?_0.09) ^Z (Al_5.25?Mg_0.75) (BO₃)₃ Si₆O₁₈ ^V (OH)₃? ^W [F_0.87(OH)_0.13], with a?=?15.9354(4), c?=?7.1934(4) Å, and they show a significant Al-Mg disorder between the Y and the Z sites (R1?=?0.013–0.015). There is a positive correlation between the Ca content and?<?Y-O?>?distance for all investigated tourmalines (r?≈?1.00), which may reflect short-range order configurations including Ca and Fe²⁺, Mg, and Li. The tourmalines have X_Mg (X_Mg?=?Mg/Mg?+?Fe_total) values in the range 0.84–0.95. The REE patterns show more or less pronounced negative Eu and positive Yb anomalies. In comparison to tourmalines from highly-evolved pegmatites, the tourmaline samples from the border zone of the pegmatites of the Rappold Complex contain relatively low amounts of total REE (~8–36 ppm) and Th (0.1–1.8 ppm) and have low La_N/Yb_N ratios. There is a positive correlation (r?≈?0.91) between MgO of the tourmalines and the MgO contents of the surrounding mica schists. We conclude that the pegmatites formed by anatectic melting of mica schists and paragneisses in Permian time. The tourmalines crystallized from the pegmatitic melt, influenced by the metacarbonate and metapelitic host rocks.     

Efficient finite-difference modelling of seismic waves using locally adjustable time steps

Conventional finite-difference modelling algorithms for seismic forward modelling are based on a time-stepping scheme with a constant (global) time step. Large contrasts in the velocity model or in the spatial sampling rate cause oversampling in time for some regions of the model. The use of locally adjustable time steps can save large amounts of computation time for certain modelling configurations. The computation of spatial derivatives across the transition zone between regions of the model with different temporal sampling requires the definition of the wavefield at corresponding time levels on both sides of the transition zone. This condition can be obtained by extrapolation in time, which is inaccurate, or by multiple time integration in the transition zone. The error in the latter solution is of the same order as the conventional time-stepping scheme because both methods are based on the same iteration formula. The technique of multiple time integration simply requires the use of different sizes of time step. It is applicable only for certain factors of variation of the time step.     

Effects of topography and crustal heterogeneities on the source estimation of LP event at Kilauea volcano

The main goal of this study is to improve the modelling of the source mechanism associated with the generation of long period (LP) signals in volcanic areas. Our intent is to evaluate the effects that detailed structural features of the volcanic models play in the generation of LP signal and the consequent retrieval of LP source characteristics. In particular, effects associated with the presence of topography and crustal heterogeneities are here studied in detail. We focus our study on a LP event observed at Kilauea volcano, Hawaii, in 2001 May. A detailed analysis of this event and its source modelling is accompanied by a set of synthetic tests, which aim to evaluate the effects of topography and the presence of low velocity shallow layers in the source region. The forward problem of Green's function generation is solved numerically following a pseudo-spectral approach, assuming different 3-D models. The inversion is done in the frequency domain and the resulting source mechanism is represented by the sum of two time-dependent terms: a full moment tensor and a single force. Synthetic tests show how characteristic velocity structures, associated with shallow sources, may be partially responsible for the generation of the observed long-lasting ringing waveforms. When applying the inversion technique to Kilauea LP data set, inversions carried out for different crustal models led to very similar source geometries, indicating a subhorizontal cracks. On the other hand, the source time function and its duration are significantly different for different models. These results support the indication of a strong influence of crustal layering on the generation of the LP signal, while the assumption of homogeneous velocity model may bring to misleading results.