Two-dimensional induction in a thin sheet of variable integrated conductivity at the surface of a uniformly conducting earth

Summary. The forward solution of the general two-dimensional problem of induction in a model earth comprising a uniformly conducting half-space covered by a thin sheet of variable integrated conductivity is obtained. Unlike some previous treatments of similar problems, the method presented here does not require the field to be separated into its normal and anomalous parts. Both the E - and B -polarization modes of induction are considered and in each case the solution is expressed in terms of the horizontal component of the electric field satisfying, on the surface of the conductor, a singular integral equation whose kernel is a well-known analytic function. A recently published solution of the coast effect is included as a special case. The numerical procedure for solving the integral equations is described and some illustrative calculations are presented.     

Induction in a thin sheet of variable conductance at the surface of a stratified earth – II. Three-dimensional theory

Summary. The algorithm of Dawson & Weaver for modelling electromagnetic induction effects in a thin sheet at the surface of a uniform earth is modified to permit the use of a layered earth model. The theory is developed in Fourier space in terms of the toroidal and poloidal transfer functions instead of with the Green's function approach which was used by Dawson & Weaver. The integral equation for the surface electric field and most of the integral formulae for the derived field components are the same as before, except for the inclusion of additional integral the kernel of which has to be calculated numerically with the aid of fast Hankel transforms. The accuracy of the results is tested by comparing solutions with those obtained from a related 2-D algorithm and finally an example of 3-D modelling is presented.     

Three-dimensional induction in a non-uniform thin sheet at the surface of a uniformly conducting earth

Summary. A new method for solving problems in three-dimensional electromagnetic induction in which the Earth is represented by a uniformly conducting half-space overlain by a surface layer of variable conductance is presented. Unlike previous treatments of this type of problem the method does not require the fields to be separated into their normal and anomalous parts, nor is it necessary to assume that the anomalous region is surrounded by a uniform structure; the model may approach either an E- or a B -polarization configuration at infinity. The solution is expressed as a vector integral equation in the horizontal electric field at the surface. The kernel of the integral is a Green's tensor which is expressed in terms of elementary functions that are independent of the conductance. The method is applied to an illustrative model representing an island near a bent coastline which extends to infinity in perpendicular directions.     

Green's functions for modelling the EM response of a target near a dipping contact

Summary. Although the integral equation method has shown promise for realistic modelling of electromagnetic exploration techniques, it has so far been limited to targets in horizontally layered hosts. This note shows how the Green's functions appropriate for horizontal layers can be used for targets near a dipping contact.     

Love waves excited by discontinuous propagation of a rupture front

Summary. Analytical results are presented for Love waves generated by sudden changes of the rate of advance of a curved rupture front in an inclined fault plane that is embedded in an elastic half-space. The boundary condition at the surface of the half-space approximates the presence of an overlying layer. The calculation consists of two parts. First, ray theory is used to calculate far-field approximations to the horizontally polarized wavefields which are emitted when the speed of the rupture front suddenly changes. These fields can be expressed as products of emission coefficients (which govern the angular dependence) and propagation terms. Secondly, a representation integral for the Love wave over a surface enclosing the rupture front is constructed, using the emitted signal and an appropriate Green's function. This integral is evaluated asymptotically. The resulting approximate Love-wave spectrum shows an explicit dependence on the nature of the rupture process, on the rupture-front and fault-plane geometry, and on the magnitude of a sudden change in the rate of advance of the rupture front.     

Teleseismic waveform modelling with a one-way wave equation

It is now widely accepted that elastic properties of the continental lithosphere and the underlying sublithospheric mantle are both anisotropic and laterally heterogeneous at a range of scales. To fully exploit modern three-component broad-band array data sets requires the use of comprehensive modelling tools. In this work, we investigate the use of a wide-angle, one-way wave equation to model variations in teleseismic 3-D waveforms due to 2-D elastic heterogeneity and anisotropy. The one-way operators are derived based on a high-frequency approximation of the square-root operator and include the effects of wave propagation as well as multiple scattering. Computational cost is reduced through a number of physically motivated approximations. We present synthetic results from simple 1-D (layer over a half-space) and 2-D (subduction zone) models that are compared with reference solutions. The algorithm is then used to model data from an array of broad-band seismograph stations deployed in northwestern Canada as part of the IRIS-PASSCAL/LITHOPROBE CANOE experiment. In this region radial-component receiver functions show a clear continental Moho and the presence of crustal material dipping into the mantle at the suture of two Palaeo-Proterozoic terranes. The geometry of the suture is better defined on the transverse component where subduction is associated with a ∼10 km thick layer exhibiting strong elastic anisotropy. The modelling reproduces the main features of the receiver functions, including the effects of anisotropy, heterogeneity and finite-frequency scattering.     

Implementation of perfectly matched layers in an arbitrary geometrical boundary for elastic wave modelling

The perfectly matched layer (PML) absorbing boundary condition is incorporated into an irregular-grid elastic-wave modelling scheme, thus resulting in an irregular-grid PML method. We develop the irregular-grid PML method using the local coordinate system based PML splitting equations and integral formulation of the PML equations. The irregular-grid PML method is implemented under a discretization of triangular grid cells, which has the ability to absorb incident waves in arbitrary directions. This allows the PML absorbing layer to be imposed along arbitrary geometrical boundaries. As a result, the computational domain can be constructed with smaller nodes, for instance, to represent the 2-D half-space by a semi-circle rather than a rectangle. By using a smooth artificial boundary, the irregular-grid PML method can also avoid the special treatments to the corners, which lead to complex computer implementations in the conventional PML method. We implement the irregular-grid PML method in both 2-D elastic isotropic and anisotropic media. The numerical simulations of a VTI lamb's problem, wave propagation in an isotropic elastic medium with curved surface and in a TTI medium demonstrate the good behaviour of the irregular-grid PML method.     

Three-dimensional massively parallel electromagnetic inversion—I. Theory

An iterative solution to the non-linear 3-D electromagnetic inverse problem is obtained by successive linearized model updates using the method of conjugate gradients. Full wave equation modelling for controlled sources is employed to compute model sensitivities and predicted data in the frequency domain with an efficient 3-D finite-difference algorithm. Necessity dictates that the inverse be underdetermined, since realistic reconstructions require the solution for tens of thousands of parameters. In addition, large-scale 3-D forward modelling is required and this can easily involve the solution of over several million electric field unknowns per solve. A massively parallel computing platform has therefore been utilized to obtain reasonable execution times, and results are given for the 1840-node Intel Paragon. The solution is demonstrated with a synthetic example with added Gaussian noise, where the data were produced from an integral equation forward-modelling code, and is different from the finite difference code embedded in the inversion algorithm     

A hybrid calculation technique of the Indirect Boundary Element Method and the analytical solutions for three-dimensional problems of topography

We present a new approach of the Indirect Boundary Element Method (IBEM) for 3-D topographic problems which can be used to deal with an infinitely spread free surface owing to the introduction of a reference solution, that is the analytical solution for the half-space with a flat free surface. This approach is an efficient countermeasure for the non-physical waves owing to the domain truncation which contaminates the computed results in the ordinary approach. Theoretical consideration shows that this newly proposed approach is a higher-grade approximation than some existing ones and achieves a higher efficacy and accuracy than those of existing ones. The discretization of the resulting boundary integral equation for this formulation is carried out with triangular elements. Their contributions to the solution are calculated by Gaussian numerical integration except in the case where the wavefield is evaluated on the source element itself. For this case, we present an analytical formula based on the reasonable assumption that the elements are much smaller than the wavelengths appearing in the calculation. Several numerical examples used for validation show acceptably precise results.     

Inversion of a normally incident reflection seismogram by the Gopinath–Sondhi integral equation

Summary. The one-dimensional acoustic wave equation has been transformed to two coupled first-order equations whose inverse solution is obtained through application of the Gopinath and Sondhi integral equation. A scattering solution of the Schrödinger wave equation for an explosive source leads us to express the kernel of the Gopinath–Sondhi integral equation in terms of a seismic reflection response. A numerical solution of the integral equation obtained by a trapezoidal rule yields a continuous impedance profile whose derivative has step-like discontinuities. The method is illustrated with computer model studies.     

Diffraction of P, S and Rayleigh waves by three-dimensional topographies

The diffraction of P, S and Rayleigh waves by 3-D topographies in an elastic half-space is studied using a simplified indirect boundary element method (IBEM). This technique is based on the integral representation of the diffracted elastic fields in terms of single-layer boundary sources. It can be seen as a numerical realization of Huygens principle because diffracted waves are constructed at the boundaries from where they are radiated by means of boundary sources. A Fredholm integral equation of the second kind for such sources is obtained from the stress-free boundary conditions. A simplified discretization scheme for the numerical and analytical integration of the exact Green's functions, which employs circles of various sizes to cover most of the boundary surface, is used.
The incidence of elastic waves on 3-D topographical profiles is studied. We analyse the displacement amplitudes in the frequency, space and time domains. The results show that the vertical walls of a cylindrical cavity are strong diffractors producing emission of energy in all directions. In the case of a mountain and incident P, SV and SH waves the results show a great variability of the surface ground motion. These spatial variations are due to the interference between locally generated diffracted waves. A polarization analysis of the surface displacement at different locations shows that the diffracted waves are mostly surface and creeping waves.     

Surface-wave propagation in a cracked poroelastic half-space lying under a uniform layer of fluid

The effect of cracks on the elastic properties of an isotropic elastic solid is studied when the cracks are saturated with a soft fluid. A polynomial equation in effective Poisson's ratio is obtained, whose coefficients are functions of Poisson's ratio of the uncracked solid, crack density and saturating fluid parameter. Elastic and dynamical constants used in Blot's theory of wave propagation in poroelastic solids are modified for the introduction of cracks. The effects of cracks on the velocities of three types of waves are observed numerically. The frequency equation is derived for the propagation of Rayleigh-type surface waves in a saturated poroelastic half-space lying under a uniform layer of liquid. Dispersion curves for a particular model of oceanic crust containing cracks are plotted. The effects of variations in crack density and saturation on the phase and group velocity are also analysed.     

Effects of the free surface on shear wavetrains

Summary. The behaviour of shear-waves is of great importance in identifying and investigating seismic anisotropy in the Earth. However, shear wavetrains recorded at the Earth's surface do not always reflect the motion at depth, introducing practical problems of interpretation. Shear wavetrains incident on the surface of an isotropic half-space at angles less than critical (about 35°) are broadly preserved, but at greater angles substantial distortions can occur. For stations situated close to the source, as in local earthquake studies, the local SP phase, a radially polarized precursor to S , may occur. The behaviour at the surface of an anisotropic half-space is further complicated by the divergence of phase and energy propagation vectors. All of these complications suggest that detailed seismogram modelling is essential to any study of shear wave propagation in the Earth, and in particular to investigations of anisotropy-induced shear-wave splitting.     

Ground deformation in a viscoelastic medium composed of a layer overlying a half-space: a comparison between point and extended sources

We obtain and compare analytical and numerical solutions for ground displacement caused by an overpressurized magma chamber placed in a linear viscoelastic medium composed of a layer over a half-space. Different parameters such as size, depth and shape of the chamber, crustal rheology and topography are considered and discussed. Numerical solutions for an axisymmetric extended source are computed using a finite element method (FEM). Analytical solutions for a point source are obtained using the dislocation theory and the propagator matrix technique. In both cases, the elastic solutions are used together with the correspondence principle of linear viscoelasticity to obtain the solution in the Laplace transform domain. Viscoelastic solutions in the time domain are derived inverting the Laplace transform using the Prony series method. The differences between the results allow us to constrain the applicability of the point source and the flat surface hypothesis, which are usually implicitly assumed when analytical solutions are derived. The effect of the topography is also considered. The results obtained show that neglecting the topographic effects may, in some cases, introduce an error greater than that implicit in the point-source hypothesis. Therefore, for an adequate modelling and interpretation of the time-dependent displacements, topography must be considered.     

Electromagnetic response of an arbitrarily shaped three-dimensional conductor in a layered earth—numerical results

Summary. In an earlier work, mathematical formulation on computing the electromagnetic response of an arbitrarily shaped three-dimensional inhomogeneity in a layered earth had been worked out using an integral equation technique. The method has been used to show its efficacy by computing numerical results. Introducing suitable changes of variables the secondary contributions to Green's dyadic are put in the form of convolution integrals and are computed using a digital linear filtering scheme. The matrix equation is solved for the unknown electric fields in the inhomogeneity. The scattered fields are then calculated at the surface of the Earth using the appropriate Green's dyadic. The performance of the computations has been shown by comparing the numerical results with those obtained by analogue modelling as well as by other numerical schemes. The use of digital linear filtering saves an enormous amount of computer time.
The effects of varying excitation-frequency, conductivity of the host medium and that of the overburden have been studied in detail for a horizontal loop system traversing over a two-layered earth with a prismatic inhomogeneity situated in the lower conducting half space.     

Exact synthetic seismograms for an inhomogeneity in a layered elastic half-space

Summary. The propagation of a pulsed elastic wave in the following geometry is considered. An elastic half-space has a surface layer of a different material and the layer furthermore contains a bounded 3-D inhomogeneity. The exciting source is an explosion, modelled as an isotropic pressure point source with Gaussian behaviour in time.
The time-harmonic problem is solved using the null field approach (the T matrix method), and a frequency integral then gives the time-domain response. The main tools of the null field approach are integral representations containing the free space Green's dyadic, expansions in plane and spherical vector wave functions, and transformations between plane and spherical vector wave functions. It should be noted that the null field approach gives the solution to the full elastodynamic equations with, in principle, an arbitrarily high accuracy. Thus no ray approximations or the like are used. The main numerical limitation is that only low and intermediate frequencies, in the sense that the diameter of the inhomogeneity can only be a few wavelengths, can be considered.
The numerical examples show synthetic seismograms consisting of data from 15 observation points at increasing distances from the source. The normal component of the velocity field is computed and the anomalous field due to the inhomogeneity is sometimes shown separately. The shape of the inhomogeneity, the location and depth of the source, and the material parameters are all varied to illustrate the relative importance of the various parameters. Several specific wave types can be identified in the seismograms: Rayleigh waves, direct and reflected P -waves, and head waves.     

Scattering of horizontally-polarized shear waves by surface irregularities

Summary. The problem of the scattering of harmonic SH waves by an arbitrary surface irregularity in an otherwise semi-infinite elastic, homogeneous, isotropic two-dimensional half-space is examined in this study in order to ascertain the effect of topography on this type of seismic ground motion and to develop a useful scheme which can realistically handle arbitrary two-dimensional topography. Three geometric models are considered: a semicircular hill which is of academic interest; a mountain with a Gaussian shape which utilizes realistic dimensions and the combination of a ridge and a depression that models a region in Sylmar, California.
A singular Fredholm integral equation of the second kind for the displacement at the free surface is developed and solved numerically. In the case of the semicircular hill, horizontal ground motion can be more than twice that occurring in the case of smooth topography. The mountain simulated by a Gaussian profile experiences at its crest amplifications for certain angles of incidence and de-amplifications for other angles of incidence, as well as displacements whose amplitudes vary slowly with frequency on the side of the mountain which is in the same direction as the incident waves. The ridge-depression combination which is approximated by a sixth-order polynomial actually experienced shattered earth at its ridge crest during the San Fernando, California earthquake of 1971. This amplification is also exhibited by the results of the analysis which, predicts amplifications of over 75 per cent at the top of the ridge for waves arriving on the same side as the ridge.     

Displacements and stresses due to a single force in a half-space in welded contact with another half-space

Closed-form analytical expressions for the displacements and stresses induced by a single force of arbitrary orientation located in an elastic half-space in welded contact with another elastic half-space are obtained. These expressions are valid for arbitrary values of the Poisson's ratio and for arbitrary source and observer locations. The final results are given in a form that makes numerical computation straightforward and accurate.     

Some useful approximations to generalized ray theory for regional distance seismograms

Summary . Seismograms recorded at regional distances (2°–12°) are quite complicated due to the waveguide nature of the crust. Generalized ray theory can be used to model the body waves in this distance range but a very large number of rays is required. Here I present a series of approximations to streamline generalized ray theory for the waveguide problem. If a layer over a half-space is used for the structure, then the de Hoop contour for a given ray is most strongly dependent on the fastest velocity of any leg of the ray. This results in analytic approximations to locate the contour. Each ray has two body wave arrivals (a headwave and a reflected arrival) so the displacement response of the ray need only be evaluated at a few points in time about the two arrival times and interpolated in between. A change in structure (increasing crustal thickness or Pn velocity) most strongly affects the relative timing of the headwave and the reflected arrival, so it is possible to 'stretch' or 'squeeze' the waveform of a representative model to simulate a whole suite of models.
Also discussed is the applicability of a single layer over a half-space structure for modelling the observed regional distance waveforms for shallow earthquakes. At periods greater than a few seconds crustal layering can be replaced by a single layer having the appropriate average velocities. Lateral variations in crustal thickness with scale lengths of less than about 100 km can also be modelled with a simple horizontal layer of appropriate average thickness.