Arbitrary Surface Topography Research Articles

Hardware, methodology and applications of 2+D backscatter Mössbauer spectroscopy with simultaneous x-ray and γ-ray detection

A unique method is presented for the acquisition and analysis of 57Fe backscatter Mössbauer spectra with simultaneous detection of the resonant 14.4 keV γ-rays and the characteristic 6.4 keV x-rays, using a custom-built multi-parameter analyser constructed on the basis of commercial analogue to digital converters and high-speed digital latches. The system allows for the simultaneous registration of Doppler-modulation velocities and photon energies, with up to 4096 and 8192 digital channels respectively. This arrangement is in contrast to most related systems, which detect at a single narrow energy window per detector. Samples of arbitrary atomic structure, morphology and surface topography can be studied without altering the setup or the analysis procedure, provided that the samples are at least micrometre sized. The hardware and software that are used to acquire data with minimal dead time are described and the custom and self-contained methods for post-measurement energy discrimination, background correction and velocity-axis folding are discussed. The data are fit using a general Hamiltonian model for the nuclear energy levels of 57Fe and a quantum mechanical description of the angular momentum coupling is utilised, with consideration of the crystalline and chemical disorder of the sample under examination. Three examples of distinct magnetic systems, with thicknesses ranging from m to 6 mm, that were studied using this method are presented, these are: an amorphous CoFeB-based ribbon with ultra-soft coercivity for high-frequency applications, magnetically hard Nd-Fe-B thick films on Si substrates, examined in both as-deposited and annealed states, and a sample from the nickel-rich iron meteorite NWA 6259 that contains the atomically ordered, elevated coercivity, phase of FeNi, tetrataenite. The wide applicability and usefulness of this method is thus demonstrated on three distinct sample morphologies that required little to no surface preparation prior to examination.

A parallel adaptive finite-element approach for 3-D realistic controlled-source electromagnetic problems using hierarchical tetrahedral grids

SUMMARY To effectively and efficiently interpret or invert controlled-source electromagnetic (CSEM) data which are recorded in areas with the kind of complex geological environments and arbitrary topography that are typical, 3-D CSEM forward modelling software that can quickly solve large-scale problems, provide accurate electromagnetic responses for complex geo-electrical models and can be easily incorporated into inversion algorithms are required. We have developed a parallel goal-oriented adaptive mesh refinement finite-element approach for frequency-domain 3-D CSEM forward modelling with hierarchical tetrahedral grids that can offer accurate electromagnetic responses for large-scale complex models and that can efficiently serve for inversion. The approach uses the goal-oriented adaptive vector finite element method to solve the total electric field vector equation. The geo-electrical model is discretized by unstructured tetrahedral grids which can deal with complex underground geological models with arbitrary surface topography. Different from previous adaptive finite element software working on unstructured tetrahedral grids, we have utilized a novel mesh refinement technique named the longest edge bisection method to generate hierarchically refined grids. As the refined grids are nested into the coarse grids, the refinement technique can precisely map the electrical parameters of inversion grids onto the forward modelling grids so that the extra numerical errors generated by the inconsistency of electrical parameters between inversion grids and forward modelling grids are eliminated. In addition, we use the parallel domain-decomposition technique to further accelerate the computations, and the flexible generalized minimum residual solver (FGMRES) with an auxiliary Maxwell solver pre-conditioner to solve the final large-scale system of linear equations. In the end, we validate the performance of the proposed scheme using two synthetic models and one realistic model. We demonstrate that accurate electromagnetic fields can be obtained by comparison with the analytic solutions and that the code is highly scalable for large-scale problems with millions or even hundreds of millions of unknowns. For the synthetic 3-D model and the realistic model with complex geometry, our solutions match well with the results calculated by an existing 3-D CSEM forward modelling code. Both synthetic and realistic examples demonstrate that our newly developed code is an effective, efficient forward modelling engine for interpreting CSEM field data acquired in areas of complex geology and topography.

Three-Dimensional Inversion of Borehole-Surface Resistivity Method Based on the Unstructured Finite Element

The electromagnetic wave signal from the electromagnetic field source generates induction signals after reaching the target geological body through the underground medium. The time and spatial distribution rules of the artificial or the natural electromagnetic fields are obtained for the exploration of mineral resources of the subsurface and determining the geological structure of the subsurface to solve the geological problems. The goal of electromagnetic data processing is to suppress the noise and improve the signal-to-noise ratio and the inversion of resistivity data. Inversion has always been the focus of research in the field of electromagnetic methods. In this paper, the three-dimensional borehole-surface resistivity method is explored based on the principle of geometric sounding, and the three-dimensional inversion algorithm of the borehole-surface resistivity method in arbitrary surface topography is proposed. The forward simulation and calculation start from the partial differential equation and the boundary conditions of the total potential of the three-dimensional point current source field are satisfied. Then the unstructured tetrahedral grids are used to discretely subdivide the calculation area that can well fit the complex structure of subsurface and undulating surface topography. The accuracy of the numerical solution is low due to the rapid attenuation of the electric field at the point current source and the nearby positions and sharply varying potential gradients. Therefore, the mesh density is defined at the local area, that is, the vicinity of the source electrode and the measuring electrode. The mesh refinement can effectively reduce the influence of the source point and its vicinity and improve the accuracy of the numerical solution. The stiffness matrix is stored with Compressed Row Storage (CSR) format, and the final large linear equations are solved using the Super Symmetric Over Relaxation Preconditioned Conjugate Gradient (SSOR-PCG) method. The quasi-Newton method with limited memory (L_BFGS) is used to optimize the objective function in the inversion calculation, and a double-loop recursive method is used to solve the normal equation obtained at each iteration in order to avoid computing and storing the sensitivity matrix explicitly and reduce the amount of calculation. The comprehensive application of the above methods makes the 3D inversion algorithm efficient, accurate, and stable. The three-dimensional inversion test is performed on the synthetic data of multiple theoretical geoelectric models with topography (a single anomaly model under valley and a single anomaly model under mountain) to verify the effectiveness of the proposed algorithm.

AstroSeis: A 3D Boundary Element Modeling Code for Seismic Wavefields in Irregular Asteroids and Bodies

Abstract We developed a 3D elastic boundary element method computer code, called AstroSeis, to model seismic wavefields in a body with an arbitrary shape, such as an asteroid. Besides the AstroSeis can handle arbitrary surface topography, it can deal with a liquid core in an asteroid model. Both the solid and liquid domains are homogenous in our current code. For seismic sources, we can use single forces or moment tensors. The AstroSeis is implemented in the frequency domain, and the frequency-dependent Q can be readily incorporated. The code is in MATLAB (see Data and Resources), and it is straightforward to set up the model to run the code. The frequency-domain calculation is advantageous to study the long-term elastic response of a celestial body due to a cyclic force, such as the tidal force, with no numerical dispersion issue suffered by many other methods requiring volume meshing. Our AstroSeis has been benchmarked with other methods such as normal-mode summation and the direct solution method. This open-source AstroSeis will be a useful tool to study the interior and surface processes of asteroids.

Micro hardness determination on a rough surface by using combined indentation and topography measurements

Micro hardness determination on rough surfaces is a topic of interest e.g. in industrial applications where the component surface is of functional relevance. In most approaches, hardness on a rough surface is determined by including profile roughness parameters like Ra (e.g., [–]) to adjust measured hardness values or to get the minimum indentation depth value where the influence of the surface topography is assumed to become negligibly small. In the present study, local surface topography data were used instead to enable precise micro hardness measurements on a sample with arbitrary surface topography. Samples made of 1.457 1 stainless steel with different surface states were face milled by using an end mill with different feed rates. Instrumented indentation tests were performed on these samples as well as on comparative samples with polished surfaces. From the resulting load-indentation depth (F-d)-curves the averaged indentation hardness was calculated for all surface states. A method was applied to manipulate and to average the F-d-curves to eliminate deviations, occurring at the beginning of the indentation. The indentation hardness was calculated from these modified F-d-curves and compared to the indentation hardness from the actually measured F-d-curves of the polished samples with feasible results. Using surface topography measurements is considered to enable deriving more accurate indentation hardness values directly and to put the investigations to another level. The surface topography of the samples was evaluated by confocal microscope measurements before and after the indentation tests. From the surface topography data at the location of indentation, four parameters were calculated: volume, projected contact area and depth of the indentation mark, as well as the curvature of the surface topography before indentation. These parameters were correlated with the hardness value from the respective indentation and compared to the indentation hardness of the polished sample. The results of the present study are the basis for combining optical imaging techniques like confocal microscopy or white light interferometry and indentation testing equipment to broaden the field of application.

Comparing resistivity models from 2D and 1D inversion of frequency domain HEM data over rough terrains: cases study from Iran and Norway

ABSTRACTFrequency domain helicopter-borne electromagnetic (FHEM) surveys have been used as an effective tool for the exploration of underground resources for as long as airborne electromagnetics (AEM) has existed. Large FHEM data sets are commonly interpreted with very fast 1D inversion algorithms that often numerically adequately fit the data sets, even though they yield incorrect results near 2D/3D geological structures. The present study aims to compare 1D and 2D inversion algorithms when applied to the reconstruction of geologically complex regions. We have developed a 2D inversion algorithm incorporating the Levenberg–Marquardt least-squares approach regularised through spatial constraints to retrieve 2D electrical resistivity models associated with arbitrary surface topography. The approach uses a 2D finite element frequency domain solution and a tailored triangular meshing algorithm based on the Ruppert’s Delaunay refinement for the forward modelling. We illustrate how rough topographic effects obscure the FHEM response and affect recovered resistivity models through numerical experiments. We also demonstrate the influence of acquisition frequency and resistivity structure on the topographic effect. In the Appendix, we discuss the FHEM footprint concept from a 2D perspective to assess how 2D effects affect and bias 1D inversion results. Complex 2D synthetic scenarios are presented to compare 1D and 2D inversion in various settings. Two field cases from Norway and Iran are presented to show the model improvements with 2D inversion. For the Norwegian case, the 2D FHEM inversion aligns well with a model retrieved from ground-based electrical resistivity tomography. We show the bias imposed on the 2D inversion of the data set from Iran by improper system calibration.

An improved immersed boundary finite-difference method for seismic wave propagation modeling with arbitrary surface topography

To accurately simulate seismic wave propagation for the purpose of developing modern land data processing tools, especially full-waveform inversion (FWI), we have developed an efficient high-order finite-difference forward-modeling algorithm with the capability of handling arbitrarily shaped free-surface topography. Unlike most existing forward-modeling algorithms using curvilinear grids to fit irregular surface topography, this finite-difference algorithm, based on an improved immersed boundary method, uses a regular Cartesian grid system without suffering from staircasing error, which is inevitable in a conventional finite-difference method. In this improved immersed boundary finite-difference (IBFD) algorithm, arbitrarily curved surface topography is accounted for by imposing the free-surface boundary conditions at exact boundary locations instead of using body-conforming grids or refined grids near the boundaries, thus greatly reducing the complexity of its preprocessing procedures and the computational cost. Furthermore, local continuity, large curvatures, and subgrid curvatures are represented precisely through the employment of the so-called dual-coordinate system — a local cylindrical and a global Cartesian coordinate. To properly describe the wave behaviors near complex free-surface boundaries (e.g., overhanging structures and thin plates, or other fine geometry features), the wavefields in a ghost zone required for the boundary condition enforcement are reconstructed accurately by introducing a special recursive interpolation technique into the algorithm, which substantially simplifies the boundary treatment procedures and further improves the numerical performance of the algorithm, as demonstrated by the numerical experiments. Numerical examples revealed the performance of the IBFD method in comparison with a conventional finite-difference method.

Numerical simulation of seismic wave propagation produced by earthquake by using a particle method

SUMMARY We propose a forward wavefield simulation based on a particle continuum model to simulate seismic waves travelling through a complex subsurface structure with arbitrary topography. The inclusion of arbitrary topography in the numerical simulation is a key issue not only for scientific interests but also for disaster prediction and mitigation purposes. In this study, a Hamiltonianparticlemethod(HPM)isemployed.Itiseasytointroducetraction-freeboundary conditionsinHPMandtorefinetheparticledensityinspace.Anymodelwithcomplexgeometry and velocity structure can be simulated by HPM because the connectivity between particles is easily calculated based on their relative positions and the free surfaces are automatically introduced. In addition, the spatial resolution of the simulation could be refined in a simple manner even in a relatively complex velocity structure with arbitrary surface topography. For these reasons, the present method possesses great potential for the simulation of strong ground motions. In this paper, we first investigate the dispersion property of HPM through a plane wave analysis. Next, we simulate surface wave propagation in an elastic half space, and compare the numerical results with analytical solutions. HPM is more dispersive than FDM, however, our local refinement technique shows accuracy improvements in a simple and effective manner. Next, we introduce an earthquake double-couple source in HPM and compare a simulated seismic waveform obtained with HPM with that computed with FDM to demonstrate the performance of the method. Furthermore, we simulate the surface wave propagation in a model with a surface of arbitrary topographical shape and compare with results computed with FEM. In each simulation, HPM shows good agreement with the reference solutions. Finally, we discuss the calculation costs of HPM including its accuracy.

An easy-to-implement numerical simulation method for adhesive contact problems involving asymmetric adhesive contact

A novel numerical method to solve asymmetric adhesive contact problems in rectangular coordinates has been developed. Surface interaction is modelled using an interface potential, deformation is coupled using Green's functions for a half space, and the resulting system of equations is solved by a relaxation technique. The method can handle arbitrary surface topography and properties. Compared with previous methods, this numerical scheme is much easier to implement and is just as accurate. Here, it is applied to two adhesive contact problems: one between a sphere and a cylinder; and the other between two identical cylinders in oblique contact. The numerical results reveal inaccuracies in elliptical contact theory when the skew angles between the two cylinders are small and the resulting contact is highly eccentric. The pull-off forces show an indiscernible decrease with decreasing value of the skew angle, which is quite different from the elliptical JKR theory. This technique can be used to solve adhesive contact problems that involve partial contact or complex geometry, such as rippled or rough surfaces.

DC resistivity modelling in anisotropic media with Gaussian quadrature grids

It has been shown that the spectral method (Trefethen 2000) and the spectral element method (Komatitsch and Tromp 1999) have more attractive features than the traditional finite difference (FD) and the finite element (FE) numerical methods used in resistivity modelling. The main advantages lie in the capability to simulate complex physical models and the exponential power convergence. They have been successfully applied to fluid flow dynamic modelling (Boyd 1989), seismic wave simulations (Komatitsch and Tromp 1999) and electromagnetic computations (Martinec 1999). The spectral method uses some global series of orthogonal functions to represent the unknown solution at the irregular collocation points, subject to boundary conditions. The resulting linear system matrix is full. The spectral element method combines the spectral method and the finite element method, and it possesses the main advantages of each. This includes the capability to handle various model shapes, the sparse matrix format of the FEM and the exponential power convergence of the spectral method. In a recent paper (Zhou et al., 2008) we presented the theory for a new resistivity modelling method based on Gaussian Quadrature Grids (GQG). It readily enables calculation of the electric potential in 2.5-D / 3-D heterogeneous, anisotropic models having arbitrary surface topography. The method co-operatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it transforms the 2.5-D / 3-D resistivity modelling problem into a sparse and symmetric linear equation system, which can be solved by an iterative or matrix inversion method.

Brownian Dynamics Simulations of Rotational Diffusion Using the Cartesian Components of the Rotation Vector as Generalized Coordinates

AbstractHere, we report on the first Brownian dynamics (BD) simulations of rotational diffusion using the Cartesian components of the rotation vector as the generalized coordinates. The model system employed in this study consists of freely rotating and non‐interacting rigid particles with arbitrary surface topography. The numerical BD algorithm contains no singularities and yields numerical results that are in full agreement with known theoretical results. Because of the absence of singularities, this new algorithm is several orders of magnitude more efficient than a simple BD algorithm employing the Euler angles as the generalized coordinates. The general theory for using generalized coordinates in studies of more complex systems involving both translation, rotation, and fluid dynamic interactions is well known. Consequently, the benefits reported here can readily be extended to such systems. Important examples are segmented polymer chains, with and without holonomic constraints, and liquid crystals.magnified image

Free Rotational Diffusion of Rigid Particles with Arbitrary Surface Topography: A Brownian Dynamics Study Using Eulerian Angles

AbstractRotational diffusion of rigid bodies is an important topic that has attracted sustained interest for many decades, but most existing studies are limited to particles with simple symmetries. Here, we present a simple Brownian dynamics algorithm that can be used to study the free rotational diffusion of rigid particles with arbitrary surface topography. The main difference between the new algorithm and previous algorithms is how the numerical values of the mobility tensor are calculated. The only parameters in the numerical algorithm that depend on particle shape are the principal values of the particle rotational mobility tensor. These three scalars contain all information about the surface topography that is relevant for the particle rotational diffusion. Because these principal values only need to be pre‐calculated once, the resulting general algorithm is highly efficient. The algorithm is valid for arbitrary mass density distribution throughout the rigid body. In this paper, we use Eulerian angles as the generalized coordinates describing the particle angular orientation.magnified image

Adaptive unstructured grid finite element simulation of two-dimensional magnetotelluric fields for arbitrary surface and seafloor topography

We present an adaptive unstructured triangular grid finite element approach for effectively simulating plane-wave diffusive electromagnetic fields in 2-D conductivity structures. The most striking advantage of irregular grids is their potential to incorporate arbitrary geometries including surface and seafloor topography. Adaptive mesh refinement strategies using an a posteriori error estimator yield most efficient numerical solutions since meshes are only refined where required. We demonstrate the robustness of this approach by comparison with analytical solutions and previously published numerical simulations. Maximum errors may systematically be reduced to, for example, 0.8 per cent for the apparent resistivity and 0.2° in the phase. An additional accuracy study of the thickness of the air layer in E-polarization suggests to keep a minimum thickness depending on lateral conductivity contrasts within the earth. Furthermore, we point out the new quality and flexibility of our simulation technique by addressing two marine magnetotelluric applications. In the first case, we discuss topographic effects associated with a synthetic sinusoidal sea bottom model and in the second case, we show a close-to-reality scenario using real bathymetry data from the East Pacific Rise at 17°S.

Common reflection surface stack using dip decomposition for rugged surface topography

We present an extension of the Common Reflection Surface (CRS) stack that provides support for an arbitrary top surface topography. CRS stacking can be applied to the original prestack data without the need for any elevation statics. The CRS-stacked zero-offset section can be corrected (redatumed) to a given planar level by kinematic wave field attributes. The seismic processing results indicate that the CRS stacked section for rugged surface topography is better than the conventional stacked section for S/N ratio and better continuity of reflection events. Considering the multiple paths of zero-offset rays, the method deals with reflection information coming from different dips and performs the stack using the method of dip decomposition, which improves the kinematic and dynamic character of CRS stacked sections.

Three-dimensional modelling and inversion of dc resistivity data incorporating topography - II. Inversion

We present a novel technique for the determination of resistivity structures associated with arbitrary surface topography. The approach represents a triple-grid inversion technique that is based on unstructured tetrahedral meshes and finite-element forward calculation. The three grids are characterized as follows: A relatively coarse parameter grid defines the elements whose resistivities are to be determined. On the secondary field grid the forward calculations in each inversion step are carried out using a secondary potential (SP) approach. The primary fields are provided by a one-time simulation on the highly refined primary field grid at the beginning of the inversion process. We use a Gauss—Newton method with inexact line search to fit the data within error bounds. A global regularization scheme using special smoothness constraints is applied. The regularization parameter compromising data misfit and model roughness is determined by an L-curve method and finally evaluated by the discrepancy principle. To solve the inverse subproblem efficiently, a least-squares solver is presented. We apply our technique to synthetic data from a burial mound to demonstrate its effectiveness. A resolution-dependent parametrization helps to keep the inverse problem small to cope with memory limitations of today's standard PCs. Furthermore, the SP calculation reduces the computation time significantly. This is a crucial issue since the forward calculation is generally very time consuming. Thus, the approach can be applied to large-scale 3-D problems as encountered in practice, which is finally proved on field data. As a by-product of the primary potential calculation we obtain a quantification of the topography effect and the corresponding geometric factors. The latter are used for calculation of apparent resistivities to prevent the reconstruction process from topography induced artefacts.

Parallel 3-D simulation of seismic wave propagation in heterogeneous anisotropic media: a grid method approach

SUMMARY This paper presents a parallel numerical technique for modelling wave propagation in 3-D heterogeneous anisotropic media. The scheme is developed by following a so-called 3-D grid method of the elastic-isotropic case. The proposed parallel algorithm needs small data exchanges between subdomains in contrast to that developed based on other numerical techniques; therefore, it is more suitable for a PC-Cluster. The algorithm is implemented on a mesh of mixed tetrahedrons and parallelepipedons, thus providing an accurate description of arbitrary 3-D surface and interface topographies and an easy generation of a non-uniform, unstructured mesh. The unstructured mesh means that the proposed algorithm can reduce the memory requirement by flexibly assigning small grid spacing in regions with low velocities and larger grid spacing in regions with higher velocities. Like the 3-D grid method, the resulting anisotropic scheme naturally satisfies the free-surface boundary conditions of arbitrary surface topography. As a result, the near-surface scattering effects can be more accurately modelled. The proposed scheme can handle a general anisotropy without any interpolations. In this paper, the transversely isotropic medium with a tilted symmetry axis, as typically caused by a system of parallel cracks or fine layers, is discussed in detail. A paraxial absorbing boundary condition in a 3-D general anisotropic case is also proposed. Comparisons with analytical solutions demonstrate the accuracy of the parallel algorithm. Computed 3-D radiation patterns illustrate shear-wave splitting, as predicted by the theory. We show the generality and flexibility of the algorithm by modelling wave propagation in an anisotropic half-space with a hemispherical crater on the surface and in mixed isotropic/anisotropic models with horizontal and inclined interfaces.

Exact three‐dimensional spectral solution to surface‐groundwater interactions with arbitrary surface topography

It has been long known that land surface topography governs both groundwater flow patterns at the regional‐to‐continental scale and on smaller scales such as in the hyporheic zone of streams. Here we show that the surface topography can be separated in a Fourier‐series spectrum that provides an exact solution of the underlying three‐dimensional groundwater flows. The new spectral solution offers a practical tool for fast calculation of subsurface flows in different hydrological applications and provides a theoretical platform for advancing conceptual understanding of the effect of landscape topography on subsurface flows. We also show how the spectrum of surface topography influences the residence time distribution for subsurface flows. The study indicates that the subsurface head variation decays exponentially with depth faster than it would with equivalent two‐dimensional features, resulting in a shallower flow interaction.

Wave propagation across fluid-solid interfaces: a grid method approach

SUMMARY This paper presents a new numerical technique for modelling wave propagation in media with both fluid (acoustic) and solid (elastic) regions, as found for instance in a marine seismic case. The scheme can correctly satisfy the fluid‐solid interface conditions and accurately models the arbitrary interface topography. This work is an extension of the grid method, which is able to model wave propagation in heterogeneous solid (elastic) media with arbitrary surface topography and irregular interfaces. The scheme is developed by formulating the problem in terms of displacements in elastic regions and pressure in acoustic regions with an explicit boundary between them. The fluid‐solid interface conditions on this explicit boundary are implemented by introducing an integral approach to the fluid‐solid interface conditions. The solution in terms of pressure in acoustic regions together with this integral approach means that no extra computational cost is needed to implement the fluid‐solid interface conditions for a complex geometry, instead of resulting in additional computations as with the spectralelement, finite-element or finite-difference methods. In this paper an acoustic grid method is developed to solve the acoustic problem inside the fluid, and the (elastic) grid method, which is improved based on a parsimonious staggered-grid scheme, is used to solve the elastic problem inside the solid. The numerical dispersion and stability criteria for the acoustic grid method are discussed in detail. Comparison with an analytical solution demonstrates that the proposed scheme handles the fluid‐solid interface conditions correctly. Examples of wave propagation in a mixed fluid/solid model with both weak and strong sinusoidal fluid‐solid interfaces illustrate the suitability of the proposed scheme for modelling wave propagation across fluid‐solid interfaces with complex geometries.

Elastic wave propagation in heterogeneous anisotropic media using the lumped finite‐element method

A numerical technique for wave‐propagation simulation in 2‐D heterogeneous anisotropic structures is presented. The scheme is flexible in incorporating arbitrary surface topography, inner openings, liquid/solid boundaries, and irregular interfaces, and it naturally satisfies the free‐surface conditions of complex geometrical boundaries. The algorithm, based on a discretization mesh of triangles and quadrilaterals, solves the problem using integral equilibrium around each node instead of satisfying elastodynamic differential equations at each node as in the finite‐difference method. This study is an extension of previous work for the elastic‐isotropic case. Besides accounting for anisotropy, a simplified quadrilateral grid cell with low computational cost is introduced. The transversely isotropic medium with a symmetry axis on the horizontal or vertical plane, as typically caused by a system of parallel cracks or fine layers, is discussed in detail. A 2‐D algorithm is presented that can handle the situation where the symmetry axis of the anisotropy does not lie in the 2‐D plane. The proposed scheme is successfully tested against an analytical solution for Lamb's problem with a symmetry axis normal to the surface and agrees well with a numerical solution of the reflectivity method for a plane‐layered model in the isotropic case. Computed radiation patterns show characteristics such as shear‐wave splitting and triplications of quasi‐SV wavefronts, as predicted by the theory. Examples of surface‐wave propagation in an anisotropic half‐space with a semicylindrical pit on the surface and mixed liquid/(anisotropic) solid model with an inclined liquid/solid interface are presented. Moreover, seismograms are modeled for dome‐layered and plane‐layered anisotropic structures.

Earthquake location in strongly heterogeneous media

Traveltime computation methods for strongly heterogeneous 3-D media developed during recent years are well suited for earthquake location. We present here a new method based on the traveltime algorithm of Podvin-Lecomte, related to the inverse problem formulation of Tarantola & Valette. The Podvin-Lecomte method, based on the Huygens principle, is very robust and allows arbitrary surface topography and station placement even for borehole instruments. First arrival traveltimes are computed for each of the recording stations using a fine 3-D velocity mesh (up to 106 cells on a workstation). The traveltime grid allows the use of the Tarantola & Valette formulation, which enables a full non-linear approach. The solution is given as a 3-D probability density function of hypocentre coordinates, which accounts for the arrival time measurements as well as a priori information for the location, the accuracy of both the arrival time readings and the computation of the theoretical traveltimes. This powerful method called 3dgridloc gives the location of the induced seismicity of the gas field of Lacq (France) using 443 520 cells of a 3-D velocity mesh and the observations from nine recording stations, one of which is located at the bottom of a 3880 m deep borehole. Location of synthetic foci as well as more than 500 actual earthquakes shows the real advantages of this new method versus the classical hypo71. A new insight into the induced seismicity is now possible: induced seismicity may occur as far away as 10 km from the gas reservoir and involve a much greater volume of rock than expected using earlier locations.