3D Wave Propagation Research Articles

Numerical simulation of 2D wave propagation on unstructured grids using explicit differential operators

We present a numerical method of simulating seismic wave propagation on unstructured 2D grids. The algorithm is based on the velocity–stress formulation of the elastic wave equation and therefore uses a staggered grid approach. Unlike finite‐element or spectral‐element methods, which can also handle flexible unstructured grids, we use explicit differential operators for the calculation of spatial derivatives in each time step. As shown in previous work, three types of these operators are used, and their particular performance is analysed and compared with standard explicit finite‐difference operators on regular quadratic and hexagonal grids. Our investigations are especially focused on the influence of grid irregularity, sampling rate (i.e. gridpoints per wavelength) and numerical anisotropy on the accuracy of numerical seismograms. The results obtained from the various methods are therefore compared with analytical solutions. The algorithm is then applied to a number of models that are difficult to handle using (quasi‐)regular grid methods. Such alternative techniques may be useful in modelling the full wavefield of bodies with complex geometries (e.g. cylindrical bore‐hole samples, 2D earth models) and, because of their local character, they are well suited for parallelization.

One- and two-dimensional seismic analysis of solid-waste landfills

The results from one-dimensional (1D) and two-dimensional (2D) dynamic response analyses are compared to determine the reliability of the common practice of using 1D analysis to evaluate the seismic response of solid-waste landfills. Results indicate that 1D analysis generally provides a reasonably conservative estimate of the seismic loading and earthquake-induced permanent displacement for deep sliding surfaces, such as along the base liner. However, caution is warranted for shallower sliding surfaces, where 2D topographic amplification often results in larger values of seismic loading and permanent displacement than that predicted by 1D analysis. The 1D–2D analysis comparison is not just a function of geometry, but is also dependent on the different numerical formulations typically employed in each procedure (e.g., 1D wave propagation versus 2D finite element). Comparisons of accelerations along the landfill surface indicate that 1D analysis consistently under predicts the maximum horizontal acceleration (MHA) along the slope near the crest of the landfill. However, the seismic loading is shown to vary systematically with normalized cover slope length, with the maximum equivalent acceleration for the full cover slope being about 40% of the peak acceleration at the crest. Based on the results of this study, a simplified procedure is proposed for scaling 1D results to account for 2D topographic amplification and cover slope averaging when evaluating the seismic performance of landfill cover systems.Key words: solid-waste landfill, dynamic response, seismic loading, finite element analysis, Newmark sliding block analysis, damping.

3D Wave Propagation and an Ultra–Low-Frequency Instability in Magnetopause

A model set of equations for the low-frequency electromagnetic perturbations in a magnetized nonuniform plasma is presented. A more convenient and systematic procedure is suggested to treat the fluid equations in order to deduce electrostatic and electromagnetic limits. A general dispersion relation is derived for the waves propagating in 3D under local approximation in nonuniform plasmas, which includes almost all the known modes of cold ion magnetized plasmas in the limit ω < Ω i (where Ω i is the ion cyclotron frequency). Both the limits β ≪ 1 and β ∼ O(1) have been discussed briefly. The shear Alfven waves and electromagnetic ion acoustic waves near ultra low frequency (ULF) range are found unstable in the high β plasma of magnetopause. The results are in complete agreement with the satellite observations.

Live modeling of 1D wave propagation in layered soil media

AbstractThis paper presents the structure and use of Dr. Layer, a computer program designed to provide students a simulation/visualization environment for studying linear and nonlinear wave propagation behavior in layered soil media. To support student‐based exploration, Dr. Layer uses “live” modeling, i.e., modeling in which computation, interaction, and visualization occur simultaneously. © 2002 Wiley Periodicals, Inc. Comput Appl Eng Educ 9: 248–258, 2001; Published online in Wiley InterScience (www.interscience.wiley.com.); DOI 10.1002/cae.10003

Dynamics in layer models of solid flame propagation

Self-propagating high-temperature synthesis (SHS) is a process in which combustion waves, e.g., “solid flames”, which are considered here, are employed to synthesize desired materials. Like many other systems, SHS is a pattern forming system. The problem of describing experimentally observed patterns and of predicting new, as yet unobserved, patterns continues to attract the attention of scientists and mathematicians due to the fundamental significance of the phenomena in combustion in particular, and in nonlinear science in general. Here, we analyze the dynamics of solid flame propagation in a 2D region by considering the region to be composed of parallel, identical layers aligned along the direction of propagation and having thermal contact. Each layer is then described by wave propagation in 1D, with the transverse Laplacian replaced by a term describing heat exchange between neighboring layers. This configuration is the simplest model of a 2D system because it accounts, in a simple way, for the principal feature of the problem, i.e., heat exchange between neighbors in the transverse direction. For simplicity, we describe the situation for two layers. Because the layers are identical, uniformly propagating waves in each layer must be identical, independent of the heat exchange rate α. When the Zeldovich number Z exceeds a critical value Z c, which depends on α, uniformly propagating waves become unstable. The stability diagram for the two coupled layers reproduces that for the full 2D problem after appropriate identification of parameters in the two problems. Depending on parameter values, we determine three different steady-state dynamical behaviors (though additional behaviors are also expected to occur). The three behaviors are: (i) waves in each layer which pulsate in phase as they propagate, so that together they form a single pulsating propagating wave; (ii) the waves in each layer are no longer identical, and antiphase pulsations occur, with the two waves alternately advancing and receding as they propagate. This wave is the analog of the spinning wave on the surface of a circular cylinder; (iii) finally, there is a region of bistability between the in phase and antiphase waves, with each having its own domain of attraction, so that which of the two behaviors occur depends on the condition of initiation of the wave. The results of our computations indicate a qualitative similarity in the behavior of combustion waves in the layer model and in the full 2D model. Specifically, due to the similarity between the small α wave behavior in the layer model and the large diameter behavior in the model of waves on the surface of a cylinder, we are able to predict the behavior of the mean velocity for the waves on the cylinder, where computations of the full problem can be rather difficult. We also compute solutions for three or more layers. The results of our computations prompt us to predict that, while planar uniformly propagating waves are unstable, the wave will be quasiplanar, i.e., the resulting spinning waves have very low amplitude hot spots, and travel with the velocity close to that of the uniformly propagating wave. Such waves may be difficult to distinguish from uniformly propagating waves in experiments. We also find that for both the layer model and full 2D problem, steady-state time-dependent waves, e.g., pulsating and spinning waves, have a conserved quantity H which characterizes the excess energy in the wave, just as in the case of uniformly propagating waves. The quantity H, which is generated by dissipation, does not vary in time and is proportional to the diffusivity and caloricity of the system, and inversely proportional to the mean wave velocity.

3D Viscoelastic Wave Propagation in the Upper Borrego Valley, California, Constrained by Borehole and Surface Data

We have simulated 2-Hz-wave propagation in a three-dimensional model of the upper Borrego Valley, southern California, for a M 4.9 earthquake with epicenter 5 km north of the valley. A 4th-order staggered-grid finite-difference method was used to calculate viscoelastic ground motion in a basin model (9 km by 5 km by 0.4 km) consisting of heterogeneous sediments surrounded by bedrock. We simulated the earthquake as a double-couple point source and computed the ground motions in the valley separately for the parts of the source incident from below and from the North. The earthquake was recorded by a surface array as well as a deep downhole array (0–238 m depth) in the center of the valley, all equipped with digital three-component seismic instruments. The simulation reproduces the overall pattern of ground motions at basin and borehole sites and shows a good correlation of observed to synthetic waveforms. In particular, the 3D simulation reproduces the recorded peak motions, cumulative kinetic energies, and Fourier spectral amplitudes within a factor of 2 for most components at the individual sites. The correlation between data and simulation allows us to identify the secondary arrivals in the records as Love and Rayleigh waves generated at the edges of the valley and the troughs of the basin. The peak velocities for the waves incident into the valley from below are generally more than an order of magnitude larger than those for the waves incident from the North. The success of the prediction requires the inclusion of anelastic attenuation in the simulation with Q values for P and S waves in the alluvium of about 30. We also used a profile of the 3D model and the soil parameters at the deep borehole to examine the ability of 2.5D and 1D model approximations to predict the data. The maximum peak velocities and total cumulative kinetic energies are reproduced at the recording sites within a factor of 2 for both 2.5D and 1D model approximations, but are underpredicted by up to an order of magnitude at some depths for individual components. In particular, the 2.5D and 1D simulations tend to underpredict the duration.

Three-dimensional structure influences on the strong-motion wavefield of the 1989 Loma Prieta earthquake

AbstractThe accurate calculation of ground motions for future large earthquakes depends on detailed knowledge of three-dimensional (3D) geologic structure and the earthquake source process, as well as sufficient computational resources. In this article we describe the results of finite-difference simulations for the 1989 Loma Prieta, CA, earthquake, with a 3D seismic velocity model for the San Francisco Bay region and a heterogeneous slip model of the source. Additionally, we explore the sensitivity of the synthetics to the major geologic structures in the velocity model. The San Francisco Bay region (particularly the Loma Prieta region) is a unique area for the study of 3D wave propagation because of the pronounced lateral velocity contrasts across the strike-slip faults of the region. Understanding the effects of such long wavelength structure is doubly important when considering a source located close to such a contrast. Our simulations show that the lateral velocity contrast across the San Andreas Fault (SAF) would be expected to substantially affect the propagation of elastic waves radiated from a source in the Loma Prieta region. Indeed, we find that the refraction of energy by the SAF serves to reduce ground motions at stations located along the San Francisco Peninsula, and the Quaternary and Tertiary alluvial basins of the San Francisco Bay region are found to amplify and extend the duration of ground motions in Santa Clara Valley, Livermore Valley, and San Pablo Bay. We find that the 3D model as currently defined accurately describes the spatial variation of peak ground velocity for frequencies less than 0.5 Hz, which suggests that this model may be used to estimate ground motions for future earthquake scenarios.

Earthquake Ground Motion and Structural Response in Alluvial Valleys

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 for 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.

EURO-SEISTEST strong-motion array near Thessaloniki, Northern Greece: A study of site effects

AbstractThe effects of local geology on the characteristics of strong ground motion have been, and continue to be, a field of active research. Despite the considerable efforts made so far, there are still several unresolved and controversial issues remaining. In particular, debates still continue over the limits of applicability of one-dimensional (1D) wave-propagation models. There are also unresolved questions related to the implementation and reliability of site-response estimation techniques such as the standard spectral ratio (SSR) and the horizontal-to-vertical spectral ratio (HVSR). This study addresses these issues on the basis of data from the EURO-SEISTEST strong-motion array at Volvi, near Thessaloniki, Greece. The data set used consists of accelerograms of 32 earthquakes, almost exclusively local, covering a magnitude range from ML 2.0 to 6.1. The range of recorded accelerations is 0.001 − 0.042 g.We implement two 1D wave-propagation models (the Haskell-Thompson matrix method and Kennett's reflectivity-coefficient method) and the HVSR site-response assessment technique. We test the applicability of the 1D models in a basin environment. We use both Fourier-amplitude and response spectra to compute HVSR and investigate how HVSR is influenced by the choice of the time window and smoothing procedure.We found the HVSR technique and 1D SH-wave modeling to perform reasonably well in a sediment-valley environment (the modeling was performed for a site in the center of the valley).

Principe d'absorption limite pour l'opérateur de Maxwell dans un milieu bihomogène

We consider the Maxwell operator A that describes 3D electromagnetic wave propagation in an infinite cylindrical wave guide with a rectangular section, where the medium is made of two homogeneous parts separated by a vertical interface. The spectral analysis of this operator points out a countable family of real numbers in the spectrum called thresholds, and the resolvent operator R A (z) = ( A − z) −1, Im (z)) ≠ 0, can be extended (‘limiting absorption principle’) continuously to the lower or upper half-planes (origin excepted) in a suitable weighted L 2-topology. In particular, this continuity holds at the thresholds.

Intrinsic anisotropy and thin multilayering-two anisotropy effects combined

SUMMARY Elastic wave propagation in1-D randomly layered media is characterized by frequencydependent anisotropy. We study the transmission of obliquely incident plane qP and qSV waves propagating through a thick 1-D randomly multilayered stack embedded between two identical homogeneous half-spaces and allow each individual layer to be transversely isotropic. Theoretical and numerical investigations show that the presence of intrinsic anisotropy in addition to thin multilayering causes a shift of the phase velocities towards higher values when compared with a stack of isotropic layers with the same vertical-incidence velocity (provided that the background velocity normal to the symmetry axis exceeds that along the symmetry axis). The magnitude of the shift depends on the magnitude of the intrinsic anisotropy and increases with increasing angle of incidence. The frequency dependence of the phase velocities is only changed if the intrinsic anisotropy is not constant but varies along the symmetry axis.The in£uence of the intrinsic anisotropy on the phase velocities is quite strong in comparison with multilayering eiects, even if the velocity and density £uctuations in the thin-layer stack are much larger. In the small £uctuation approximation the three eiects, background anisotropy, static anisotropy due to multilayering, and frequency-dependent anisotropy due to multilayering, provide additive contributions to the vertical slowness. In this approximation the eiects are also additive in terms of the phase velocity. The attenuation coe⁄cients for both qP and qSV waves are controlled exclusively by frequency-dependent multilayering eiects. However, the intrinsic anisotropy in£uences the attenuation coe⁄cients in the case that it is £uctuating, since £uctuations of the intrinsic anisotropy aiect the phase velocities in the single layers, leading to an increase in internal scattering. This eiect can be rather pronounced for large incidence angles.

ANALYSIS OF AMPLIFICATION CHARACTERISTICS OF GROUND MOTIONS IN THE HEAVILY DAMAGED BELT ZONE DURING THE 1995 HYOGO-KEN NANBU EARTHQUAKE

To estimate the amplification characteristics of ground motions in the heavily damaged belt zone in Kobe City during the 1995 Hyogo-ken Nanbu earthquake, 3D wave propagation analyses of a 2D deep irregular underground structure model with a vertical discontinuity were performed at an early stage as a preliminary and qualitative study. The hyperelement method was applied to the analyses for incident plane waves expected from the wavefields due to the source mechanism. The observation records at Kobe University of the rock site were used as control motions. The ground motions on the engineering bedrock (assumed to be on the free surface of the Osaka group layers having a shear velocity of 500 m/s) and at ground surface were calculated. The effects of the deep irregular underground structure and shallow surface layers on the ground motion amplification are discussed. Although there are qualifications due to the uncertain characteristics of the input rock motion and shear wave velocities of the underground structure, the analytical results show that the ground motion in the heavily damaged belt zone were amplified due to the focusing effect of the deep irregular underground structure as well as the shallow surface layers, and that the calculated peak ground acceleration (PGA) distribution coincided closely with the distributions of structural damage.

2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method

A new numerical method is presented for propagating elastic waves in heterogeneous earth media, based on spectral approximations of the wavefield combined with domain decomposition techniques. The flexibility of finite element techniques in dealing with irregular geologic structures is preserved, together with the high accuracy of spectral methods. High computational efficiency can be achieved especially in 3D calculations, where the commonly used finite-difference approaches are limited both in the frequency range and in handling strongly irregular geometries. The treatment of the seismic source, introduced via a moment tensor distribution, is thoroughly discussed together with the aspects associated with its numerical implementation. The numerical results of the present method are successfully compared with analytical and numerical solutions, both in 2D and 3D.

Parallel interactive visualization of 3D mantle convection

In 3D simulation of thermal convective flows that change with time, each run can generate gigabytes of data. A client server visualization tool called pV3, running on a parallel computer, allows data exploration during computation instead of by postprocessing. While pV3 has been primarily used in the area of flow visualization designed with aerospace applications in mind, it has easily been applied in other flow visualization areas, such as geoscience. It has also been used for 3D wave propagation problems, including seismic waves and electromagnetics. More importantly, many of the concepts of pV3's client server interactive parallel visualization model could be used as a prototype for other application areas, such as enhanced MRI in medicine. The key to this kind of interactive visualization lies in being able to scale the compute part of the visualization as well as the application is scaled, while still minimizing network traffic. The article discusses the use of pV3 for investigating thermal convection in the interior of the Earth. pV3 was used to help understand details of the physics of variable viscosity and phase transitions in high Rayleigh number convection.

Evaluation of local site effects and their removal from borehole records observed in the Sendai region, Japan

AbstractWe present results from a borehole observation network composed of 12 borehole sites deployed in a 20 × 20 km area in the Sendai region, Japan. The subsurface Quaternary layers vary with each site, but a Pliocene layer is commonly underlying at a depth of 0 to 80 m throughout the Sendai region. We define this Pliocene layer, whose S-wave velocity is larger than 500 m/sec and whose N value of the standard penetration test is greater than 50, as an engineering bedrock in the region. Once we characterize ground motions at the engineering bedrock, we can either use these motions directly in the response analysis of whole soil-building systems or use them in the nonlinear analysis of soft soil layers as an input. The purpose of this study is to evaluate the local site effects due to surface layers overlying the engineering bedrock and to remove them by using one-dimensional (1D) soil models whose properties are determined by weak-motion records of 18 events. First we identify S-wave velocities and frequency-dependent quality factors Q from amplification factors between surface records and borehole records observed at 10 sites at depths of several tens of meters. The identified S-wave velocities are very close to S-wave logging values. The identified Q values show strong frequency dependence, proportional to f(0.46 to 1.16), where f is frequency. We confirm that the observed amplification factors in the frequency range from about 0.1 to 20 Hz can be explained well by the theoretical ones based on the 1D wave propagation theory with the identified S-wave velocities and Q. Then we estimate the so-called engineering bedrock waves, which are supposed to be observed on the outcrop of the engineering bedrock, from borehole records by using the 1D models with these identified soil constants. We confirm that local site effects due to surface layers overlying the engineering bedrock are properly removed so that we have similar characteristics in the resultant engineering bedrock waves from records at different sites.

A 3D perfectly matched medium from modified maxwell's equations with stretched coordinates

AbstractA modified set of Maxwell's equations is presented that includes complex coordinate stretching along the three Cartesian coordinates. The added degrees of freedom in the modified Maxwell's equations allow the specification of absorbing boundaries with zero reflection at all angles of incidence and all frequencies. The modified equations are also related to the perfectly matched layer that was presented recently for 2D wave propagation. Absorbing‐material boundary conditions are of particular interest for finite‐difference time‐domain (FDTD) computations on a single‐instruction multiple‐data (SIMD) massively parallel supercomputer. A 3D FDTD algorithm has been developed on a connection machine CM‐5 based on the modified Maxwell's equations and simulation results are presented to validate the approach. © 1994 John Wiley &amp; Sons, Inc.

Fluid–foundation interaction in the seismic response of gravity dams

AbstractThe seismic response of a dam is strongly influenced by its interaction with the water reservoir and the foundation. The hydrodynamic forces in the reservoir are in turn affected by radiation of waves towards infinity, wave absorption at the reservoir bottom, and cross‐coupling between the foundation below the dam and the reservoir bottom. The fluid–foundation interaction effect, i.e. the wave absorption along the reservoir bottom, can be accounted for by using either an approximate one‐dimensional (1D) wave propagation model or a rigorous analysis of interaction between the flexible soil along the base and the water. The rigorous approach requires enormous computational effort because of (a) cross‐coupling between the foundation of the dam and the soil below the reservoir and (b) frequency dependence of the boundary condition along the fluid‐foundation interface. The analysis can be simplified by ignoring the cross‐coupling and by using the approximate 1D wave propagation model. The effects of each of these two simplifications on the accuracy and computational efficiency of the procedure used for the seismic response analysis of a dam are examined. Analytical results are presented for the complex frequency‐response functions as well as the time histories of the response of Pine Flat dam to Taft and E1 Centro ground motions.

Localization of waves in one-, two-, and three-dimensional random media with large-scale inhomogeneities

Path integrals and a complex Monte Carlo method are used to describe wave propagation and localization in 1D, 2D, and 3D media with small-scale random and large-scale regular inhomogeneities. The probability of wave passage over distance\(|\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} '|\) averaged over an ensemble of scatterers is calculated. A comparison with analytical results is performed. Unexpected sharp peaks interrupting the exponential decay of the mean square absolute value of Green's function are detected numerically together with the possibility of wave delocalization in the presence of regular inhomogeneities.

Concepts and results for 3D digital terrain-based wave propagation models: an overview

Mobile communication links are severely influenced by propagation effects. Wave propagation in the VHF/UHF frequency range over natural and man-made terrain is strongly dependent on topography and morphography. Propagation modeling is based on a ray-optical approach. Wave interactions, like diffraction and scattering, over the propagation path are described by the uniform theory of diffraction (UTD) and physical optics (PO). Propagation models for rural and urban areas are presented for 2-D and 3-D ray tracing. Near-range models apply to the corresponding areas in forest and urban sites. The field-strength delay spectrum describes ray contributions with deterministic amplitudes but statistical phases are used to derive time-and frequency-domain channel characteristics. Comparisons between measured and predicted data are presented. >

The simplified FitzHugh-Nagumo model with action potential duration restitution: Effects on 2D wave propagation

A modification of the simplified FitzHugh-Nagumo (FN) equations is proposed for introducing a residual component of the slow variable, which determines the restitution of action potential duration (APD) also known as the interval-excitation duration relationship. The three-step-wise approximation of ϵ( E) which is widely used in current publications is replaced in a new model by a four-step approximation. This change is used for studying by computer simulation the effects of APD restitution properties independently of the APD and refractory period on 2D wave propagation in an isotropic matrix (made by 128 × 128 nodes). The method for fitting the model to the given experimental restitution data (obtained from myocardial cells) is presented. The computer simulations implemented on a massively parallel computer (Connection Machine) showed at least three important qualitative distinctions in behavior which demonstrate the effect of APD restitution: changes in the speed and wavelength of propagated waves with the period of stimulation, non-stationary propagation of spiral waves, and site-specific induction of spiral waves with premature stimulation not on the tail of the previous wave. Quantitative effects of differing restitution properties are expressed in the size and location of a window of vulnerability in 2D excitable media. These windows are characterized by the appearance of single and double spiral waves in response to premature stimulation applied inside the window. Thus the APD restitution incorporated in the FN model produces a significant effect on the formation and propagation of spiral waves.