ALBERT

Description: The lattice Boltzmann (LB) method is a numerical method that has its origins in discrete mechanics. The method is based on propagating discrete density distributions across a fixed lattice and implementing conservation laws between the density distributions at lattice intersections. The method has been successfully applied to a wide variety of problems in fluid dynamics but has yet to be applied to elastic wave propagation. In this article we outline a new 2D and 3D LB solution to the elastic wave equation in a Poisson solid using a regular lattice, in 2D a square geometry and in 3D a cubic geometry. We outline the theory behind the method and derive the elastic wave equation from a Chapman–Enskog expansion about the Knudsen number. The scheme is shown to give rise to the elastic wave equation with a fixed Poisson ratio of 0.25 with a Knudsen number truncation error of order two. We have performed a von Neumann plane-wave analysis and found that the numerical dispersion is comparable to other discrete methods for modelling wave propagation. We have compared the numerical method to two problems, a 3D infinite homogeneous medium and a 2D heterogeneous block model. In both cases, we found the solutions agreed, thus showing that the LB method can be used to model elastic wave propagation. The scheme offers the potential to model the interaction of several continuum equations within one solver as the continuum equation is solely dependent on the equilibrium distribution.

Description: The representation of viscoelastic media in the time domain becomes more challenging with greater bandwidth of the propagating waves and number of travelled wavelengths. With the continuously increasing computational power, more extreme parameter regimes become accessible, which requires the reassessment and improvement of the standard ‘memory variable’ methods to implement attenuation in time-domain seismic wave-propagation methods. In this paper, we propose a method to minimize the error in the wavefield for a fixed complexity of the anelastic medium. This method consists of defining an appropriate misfit criterion based on a first-order analysis of how errors in the discretized medium propagate into errors in the wavefield and a simulated annealing optimization scheme to find the globally optimal parametrization. Furthermore, we derive an analytical time-stepping scheme for the memory variables that encode the strain history of the medium. Then we develop the coarse grained memory variable approach for the spectral element method (SEM) and benchmark it using the 2.5-D code AxiSEM for global body waves up to 1 Hz. Showing very good agreement with a reference solution, it also leads to a speedup of a factor of 5 in the anelastic part of the code (factor 2 in total) in this 2.5-D approach. A factor of 15 (3 in total) can be expected for the 3-D case compared to conventional implementations.

Description: We present a numerical method to compute 3-D elastic waves in fully anisotropic axisymmetric media. This method is based on a decomposition of the wave equation into a series of uncoupled 2-D equations for which the dependence of the wavefield on the azimuth can be solved analytically. Four independent equations up to quadrupole order appear as solutions for moment-tensor sources located on the symmetry axis while single forces can be accommodated by two separate solutions up to dipole order. This decomposition gives rise to an efficient solution of the 3-D wave equation in a 2-D axisymmetric medium. First, we prove the validity of the decomposition of the wavefield in the presence of general anisotropy. Then we use it to derive the reduced 2-D equations of motions and discretize them using the spectral element method. Finally, we benchmark the numerical implementation for global wave propagation at 1 Hz and consider inner core anisotropy as an application for high-frequency wave propagation in anisotropic media at frequencies up to 2 Hz.

Description: The topography of the core–mantle boundary (CMB) is directly linked to the dynamics of both the mantle and the outer core, although it is poorly constrained and understood. Recent studies have produced topography models with mutual agreement up to degree 2. A broad-band waveform inversion strategy is introduced and applied here, with relatively low computational cost and based on a first-order Born approximation. Its performance is validated using synthetic waveforms calculated in theoretical earth models that include different topography patterns with varying lateral wavelengths, from 600 to 2500 km, and magnitudes (~10 km peak-to-peak). The source–receiver geometry focuses mainly on the P diff , PKP , PcP and ScS phases. The results show that PKP branches, PcP and ScS generally perform well and in a similar fashion, while P diff yields unsatisfactory results. We investigate also how 3-D mantle correction influences the output models, and find that despite the disturbance introduced, the models recovered do not appear to be biased, provided that the 3-D model is correct. Using cross-correlated traveltimes, we derive new topography models from both P and S waves. The static corrections used to remove the mantle effect are likely to affect the inversion, compromising the agreement between models derived from P and S data. By modelling traveltime residuals starting from sensitivity kernels, we show how the simultaneous use of volumetric and boundary kernels can reduce the bias coming from mantle structures. The joint inversion approach should be the only reliable method to invert for CMB topography using absolute cross-correlation traveltimes.

Description: We present a new, computationally efficient numerical method to simulate global seismic wave propagation in realistic 3-D Earth models. We characterize the azimuthal dependence of 3-D wavefields in terms of Fourier series, such that the 3-D equations of motion reduce to an algebraic system of coupled 2-D meridian equations, which is then solved by a 2-D spectral element method (SEM). Computational efficiency of such a hybrid method stems from lateral smoothness of 3-D Earth models and axial singularity of seismic point sources, which jointly confine the Fourier modes of wavefields to a few lower orders. We show novel benchmarks for global wave solutions in 3-D structures between our method and an independent, fully discretized 3-D SEM with remarkable agreement. Performance comparisons are carried out on three state-of-the-art tomography models, with seismic period ranging from 34 s down to 11 s. It turns out that our method has run up to two orders of magnitude faster than the 3-D SEM, featured by a computational advantage expanding with seismic frequency.

Description: We present a new webservice, Syngine, running at the Incorporated Research Institutions for Seismology Data Management Center (IRIS-DMC), that offers on-demand and custom-tailored seismograms served over HTTP. The free service produces full seismic waveforms, including effects like attenuation and anisotropy, that are calculated in commonly used spherically symmetric Earth models (preliminary reference Earth model [PREM], ak135-f, IASP91). Users can freely adjust sources and receivers, retrieve seismograms from finite sources, convolve with arbitrary source time functions, and download Green’s functions suitable for moment tensor inversions. Syngine extracts and processes seismograms in as fast as fractions of a second, making it suitable for applications demanding short iteration times and a large number of waveforms. For the first time, researchers without large computational resources or specialized knowledge can easily access high-quality, custom, broadband seismograms. In this article, we present the rational and basic principles of our method, including its limitations. Additionally, we demonstrate the features of Syngine and the included Earth models, showcase several applications, and discuss future possibilities.

Description: An important goal of computational seismology is to simulate dynamic earthquake rupture and strong ground motion in realistic models that include crustal heterogeneities and complex fault geometries. To accomplish this, we incorporate dynamic rupture modelling capabilities in a spectral element solver on unstructured meshes, the 3-D open source code SPECFEM3D, and employ state-of-the-art software for the generation of unstructured meshes of hexahedral elements. These tools provide high flexibility in representing fault systems with complex geometries, including faults with branches and non-planar faults. The domain size is extended with progressive mesh coarsening to maintain an accurate resolution of the static field. Our implementation of dynamic rupture does not affect the parallel scalability of the code. We verify our implementation by comparing our results to those of two finite element codes on benchmark problems including branched faults. Finally, we present a preliminary dynamic rupture model of the 2011 M w 9.0 Tohoku earthquake including a non-planar plate interface with heterogeneous frictional properties and initial stresses. Our simulation reproduces qualitatively the depth-dependent frequency content of the source and the large slip close to the trench observed for this earthquake.

Description: A supervirtual interferometry (SVI) method is presented that can enhance the signal-to-noise ratio (SNR) of core diffracted waveforms by as much as $O(\sqrt{N})$ , where N is the number of inline receivers that record the core–mantle boundary (CMB) diffractions from more than one event. Here, the events are chosen to be approximately inline with the receivers along the same great circle. Results with synthetic and teleseismic data recorded by USArray stations demonstrate that formerly unusable records with low SNR can be transformed to high SNR records with clearly visible CMB diffractions. Another benefit is that SVI allows for the recording of a virtual earthquake at stations not deployed during the time of the earthquake. This means that portable arrays such as USArray can extend the aperture of one recorded earthquake from the West coast to the East coast, even though the teleseism might have only been recorded during the West coast deployment. In summary, SVI applied to teleseismic data can significantly enlarge the catalogue of usable records both in SNR and available aperture for analysing CMB diffractions. A potential drawback of this method is that it generally provides the correct kinematics of CMB diffractions, but does not necessarily preserve correct amplitude information.

Description: SUMMARY Small-scale heterogeneities in the Earth’s mantle, the origin of which is likely compositional anomalies, can provide critical clues on the evolution of mantle convection. Seismological investigation of such small-scale heterogeneities can be facilitated by forward modelling of elastic wave scattering at high frequencies, but doing so with conventional 3-D numerical methods has been computationally prohibitive. We develop an efficient approach for computing high-frequency synthetic wavefields originating from small-scale mantle heterogeneities. Our approach delivers the exact elastodynamic wavefield and does not restrict the geometry or physical properties of the local heterogeneity and the background medium. It combines the technique of wavefield injection and a numerical method called AxiSEM3D. Wavefield injection can decompose the total wavefield into an incident and a scattered part. Both these two parts naturally have low azimuthal complexity and can thus be solved efficiently using AxiSEM3D under two different coordinate systems. With modern high-performance computing (on an order of magnitude of 105 CPU-hr), we have achieved a 1 Hz dominant frequency for global-scale problems with strong deep Earth scattering. Compared with previous global injection approaches, ours allows for a 3-D background medium and yields the exact solution without ignoring any higher-order scattering by the background medium. Technically, we develop a traction-free scheme for realizing wavefield injection in a spectral element method, which brings in several flexibilities and simplifies the implementation by avoiding stress or traction computation on the injection boundary. For a spherical heterogeneity in the mid-lower mantle, we compare the 3-D full-wave solution with two approximate ones obtained, respectively, by the perturbation theory and in-plane (axisymmetric) modelling. As a comprehensive application, we study S-wave scattering by a 3-D ultra-low velocity zone, incorporating 3-D crustal structures on the receiver side as part of the background model.