SUMMARY We present an introduction to the spectral element method, which provides an innovative numerical approach to the calculation of synthetic seismograms in 3-D earth models. The method combines the £exibility of a ¢nite element method with the accuracy of a spectral method. One uses a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements that is adapted to the free surface and to the main internal discontinuities of the model. The wave¢eld on the elements is discretized using high-degree Lagrange interpolants, and integration over an element is accomplished based upon the Gauss^Lobatto^Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix, which greatly simpli¢es the algorithm. We illustrate the great potential of the method by comparing it to a discrete wavenumber/re£ectivity method for layer-cake models. Both body and surface waves are accurately represented, and the method can handle point force as well as moment tensor sources. For a model with very steep surface topography we successfully benchmark the method against an approximate boundary technique. For a homogeneous medium with strong attenuation we obtain excellent agreement with the analytical solution for a point force.

/pdf/introduction-to-the-spectral-element-method-for-three-2x6bedmxj4.pdf

Introduction to the spectral element method for three-dimensional seismic wave propagation

SUMMARY We use a spectral-element method to simulate seismic wave propagation throughout the entire globe. The method is based upon a weak formulation of the equations of motion and combines the flexibility of a finite-element method with the accuracy of a global pseudospectral method. The finite-element mesh honours all first- and second-order discontinuities in the earth model. To maintain a relatively constant resolution throughout the model in terms of the number of grid points per wavelength, the size of the elements is increased with depth in a conforming fashion, thus retaining a diagonal mass matrix. In the Earth’s mantle and inner core we solve the wave equation in terms of displacement, whereas in the liquid outer core we use a formulation based upon a scalar potential. The three domains are matched at the inner core and core‐ mantle boundaries, honouring the continuity of traction and the normal component of velocity. The effects of attenuation and anisotropy are fully incorporated. The method is implemented on a parallel computer using a message passing technique. We benchmark spectral-element synthetic seismograms against normal-mode synthetics for a spherically symmetric reference model. The two methods are in excellent agreement for all body- and surface-wave arrivals with periods greater than about 20 s.

/pdf/spectral-element-simulations-of-global-seismic-wave-3pgbn1ckn8.pdf

Spectral-element simulations of global seismic wave propagation—I. Validation

The perfectly matched layer (PML) absorbing boundary condition has proven to be very efficient from a numerical point of view for the elastic wave equation to absorb both body waves with nongrazing incidence and surface waves. However, at grazing incidence the classical discrete PML method suffers from large spurious reflections that make it less efficient for instance in the case of very thin mesh slices, in the case of sources located close to the edge of the mesh, and/or in the case of receivers located at very large offset. We demonstrate how to improve the PML at grazing incidence for the differential seismic wave equation based on an unsplit convolution technique. The improved PML has a cost that is similar in terms of memory storage to that of the classical PML. We illustrate the efficiency of this improved convolutional PML based on numerical benchmarks using a finite-difference method on a thin mesh slice for an isotropic material and show that results are significantly improved compared with the classical PML technique. We also show that, as the classical PML, the convolutional technique is intrinsically unstable in the case of some anisotropic materials.

An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation

Introduction.- Numerical Solution of the Elastic Wave Equation.- Computing Sensitivity Kernels.- Seismological Data Functionals and their Associated Adjoint Sources.- Iterative Optimisation.- Full Waveform Tomography for Upper-mantle Structure in Australasian Region.- A Comparative Study of Local-scale full Waveform Tomographies.- Source Staking and Data Reduction in Global full Waveform Tomography.

Full Seismic Waveform Modelling and Inversion

With power dissipation becoming an increasingly vexing problem across many classes of computer systems, measuring power dissipation of real, running systems has become crucial for hardware and software system research and design. Live power measurements are imperative for studies requiring execution times too long for simulation, such as thermal analysis. Furthermore, as processors become more complex and include a host of aggressive dynamic power management techniques, per-component estimates of power dissipation have become both more challenging as well as more important. In this paper we describe our technique for a coordinated measurement approach that combines real total power measurement with performance-counter-based, per-unit power estimation. The resulting tool offers live total power measurements for Intel Pentium 4 processors, and also provides power breakdowns for 22 of the major CPU subunits over minutes of SPEC2000 and desktop workload execution. As an example application, we use the generated component power breakdowns to identify program power phase behaviour. Overall, this paper demonstrates a processor power measurement and estimation methodology and also gives experiences and empirical application results that can provide a basis for future power-aware research.

/pdf/runtime-power-monitoring-in-high-end-processors-methodology-4i6fufrrl9.pdf

Runtime power monitoring in high-end processors: methodology and empirical data

https://people.eecs.berkeley.edu/~jrs/papers/cmame.j.pdf

Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers

This paper is concerned with the problem of soil amplification and structural damage due to local site conditions in sedimentary valleys during earthquakes. It focuses on a small valley in Kirovakan, for which 1D wave propagation analyses have failed to provide adequate answers for the large extent and spatial distribution of damage during the 1988 Armenia earthquake. A more realistic 2D finite-element analysis is performed herein in search of an explanation of the observed behavior. Using as input an inferred rock accelerogram, the response of the valley is calculated for a vertically incident SH-wave. Synthetic accelerograms of the surface ground motion are presented for different sites; these accelerograms are then used to determine the amplification ratios of the surface response with respect to that of the free-field motion of the rock outcrop, for different frequencies and for a continuous set of sites. In addition, response spectra are evaluated for simple oscillators representing structures located at various sites. Results of the 2D simulations show striking differences with respect to those from 1D analyses. In particular, (1) while the resonant frequencies exhibited by a 1D model for a given site also appear in the 2D model, the peak ground response and structural response are almost twice as large for the 2D as for the 1D model; (2) the 2D model exhibits, in addition, a new set of resonant frequencies and concomitant “mode shapes” across the valley, which are directly related to its finite width; and (3) due to these additional resonances the ground amplification ratio tends to oscillate very rapidly, both spatially and with frequency, leading to the observation that two identical structures located in the same vicinity or two slightly different structures located essentially on the same site can be subjected to significantly different seismic forces, even if the underlying soils have very similar characteristics. These results provide a meaningful explanation for the observed damage, and thus serve to exemplify a situation in which site effects caused by the finite lateral extent of a valley must be taken into consideration in order to model satisfactorily seismic behavior.

Earthquake Ground Motion and Structural Response in Alluvial Valleys

We describe the design and discuss the performance of a parallel elastic wave propagation simulator that is being used to model and study earthquake-induced ground motion in large sedimentary basins. The components of the system include mesh generators, a mesh partitioner, a parceler, and a parallel code generator, as well as parallel numerical methods for applying seismic forces, incorporating absorbing boundaries, and solving the discretized wave propagation problem. We discuss performance on the Cray T3D for unstructured mesh wave propagation problems of up to 14 million tetrahedra. By paying careful attention to each step of the process, we obtain excellent performance despite the highly irregular structure of the coefficient matrices of the problem. The mesh generator, partitioner, parceler, and code generator have been incorporated into an integrated toolset/compiler. This system, called Archimedes, automates the solution of unstructured mesh PDE problems on parallel computers, and is being used for other unstructured mesh applications beyond ground motion modeling.

Earthquake ground motion modeling on parallel computers

Three-dimensional nonlinear seismic ground motion modeling in basins

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Papers

Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers

Earthquake Ground Motion and Structural Response in Alluvial Valleys

Earthquake ground motion modeling on parallel computers

Three-dimensional nonlinear seismic ground motion modeling in basins