# Wave propagation

We embark on solving the equations of motion for seismic waves according to their desired, known, assumed or sufficient accuracy at scales ranging from the solar interior to the geotechnical layer beneath our feet. This scale-dependent, crucial issue in wave propagation is central to reaching those frequencies of interest that are resolved by both seismic data and structural heterogeneities.

## Axisymmetric spectral-element method for global waves

(T. Nissen-Meyer, A. Fournier, M. v. Driel, S. Staehler, A. Colombi)
We have developed a stand-alone spectral-element
method (AXISEM ) which simulates global seismic wave
propagation up to any desirable resolution on workstations or small
clusters. This huge gain in efficiency compared to fully 3D methods
relies on assuming spherical symmetry for the background medium, in
which case the equation system can be collapsed to a 2D numerical
domain while solving the full 3D, 3-component radiation. This code
is used at about 15 research institutions worldwide and will be
officially released open-source shortly. I also work with the IRIS
consortium to compute, store and make available a database of
waveforms that constitute a complete basis for instantaneous
computation of seismograms of any configuration, i.e. a
once-and-for-all solution to the forward problem upon spherically
symmetric Earth models. This is planned to be made available through
IRIS web services along with automated scripts to conduct
convolutions with any desired source time function, any
source-receiver configuration and freely chosen source radiation
patterns.

## Numerical error analysis

(J.-P. Ampuero, T. Nissen-Meyer)In light of convergent and scalable wave-propagation techniques, it is important to quantify their error in relation to the geophysical problem at hand. We derive compact relationships for numerical dispersion errors within spectral-element methods, quantitatively estimate their computational cost, and derive criteria to optimally choose numerical simulation parameters for a given error tolerance. In the context of ever-increasing resolution, careful consideration of dispersion errors which naturally grow with propagation distance will only become more crucial both in forward and inverse modeling.

## Time discretisation

(T. Nissen-Meyer, J.-P. Ampuero, M. Grote, O. Schenk)Since the second-order Newmark time discretization becomes rather inaccurate for large propagation distances, we adapted symplectic high-order schemes (as often used in celestial mechanics) to elastodynamics due to its preferable cost-performance character in the case of multiple-orbit surface waves or high-frequency body waves. Within this realm, it is feasible to derive an optimal set of symplectic time integration schemes at varying orders. Hexahedreal spectral- element meshes are generally not capable of alleviating the velocity structure in complex domains such that small layers control the global time step. I therefore collaborate with mathematicians at Basel on local timestepping in Newmark and symplectic time schemes. This will drastically reduce computational cost and only become more important as more complex small-scale structures (e.g., sedimentary basins, geotechnical layers, global crustal layers) are incorporated.

## Unstructured spectral elements on GPU

(M. Rietmann, and groups of D. Komatitsch, J. Tromp, O. Schenk)In a large collaborative effort, we have been involved in co-developing fully anisotropic visco-, poro-elastic and acoustic wave propagation upon the unstructured spectral-element method SPECFEM3D. This encapsulates meshing complex domains with hexahedral elements and offers an attractive tool to study scalable wave propagation in local settings. The benefit of this method relies on its capability to run and scale efficiently while tackling sufficiently complex structures. We have run successful tests on the scalability up to thousands of processors and recently reformulated the method to run on graphical processing units (GPU), observing a significant speedup.

## Scattering approaches to wave propagation

(M. van Driel, T. Nissen-Meyer, A. Kuvshinov)We use the axisymmetric spectral-element method as a basis to compute Green functions for spherically symmetric Earth models, and apply Born modeling for extremely efficient and flexible computation of seismograms due to 3D structure such as above-mentioned deep-mantle heterogeneities or tomographic models. We will use this to run a large number of simulations through existent and likely tomographic models and compare their data fit, thus allowing us to estimate uncertainties. We will extend this scattering theory approach to full 3D wave propagation by a modificiation to the Neumann series, allowing for accurate computation beyond weak and single scattering. This approach shall open new doors for global non-linear tomographic inversions of waveforms, in the longer context with a probabilistic tangent.