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     

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.     

Numerical scheme for the inversion of acoustical impedance profile based on the Gelfand-Levitan method

Summary. An exact method for the solution of the inverse problem in plane wave propagation modelled after the Gelfand-Levitan technique is reviewed and refined. A numerical scheme for the solution of the integral equation that arises in the method is proposed. A discussion on the stability and an error analysis of the numerical approximation are presented. The applicability of the inversion algorithm is demonstrated in a numerical experiment.     

Elastic scattered waves from a continuous and heterogeneous layer

Elastic scattering from a continuous and laterally unbounded heterogeneous layer has been formulated using the Born approximation. A general solution of the scattered wave equation for the above-stated medium has been given in terms of a Fourier integral over plane waves. Far-field asymptotic expressions for weak elastic scattering by a finite, continuous and inhomogeneous layer have been presented which agree with earlier results. For perturbations of the two elastic parameters and the density having the same form of spatial variation, the spectrum of plane waves scattered from a heterogeneous layer is expressed as a product of an 'elastic scattering factor'and a 'distribution factor'. As in earlier results for small-scale heterogeneity, the scattering pattern depends on various combinations of perturbations of elastic parameters and density. In order to show the general characteristics of the elastic wave scattering, some scattering patterns have been given.     

An introduction to Maslov's asymptotic method

Summary. Familiar concepts such as asymptotic ray theory and geometrical spreading are now recognized as an asymptotic form of a more general asymptotic solution to the non-separable wave equation. In seismology, the name Maslov asymptotic theory has been attached to this solution. In its simplest form, it may be thought of as a justification of disc-ray theory and it can be reduced to the WKBJ seismogram. It is a uniformly valid asymptotic solution, though. The method involves properties of the wavefronts and ray paths of the wave equation which have been established for over a century. The integral operators which build on these properties have been investigated only comparatively recently. These operators are introduced very simply by appealing to the asymptotic Fourier transform of Ziolkowski & Deschamps. This leads quite naturally to the result that phase functions in different domains of the spatial Fourier transform are related by a Legendre transformation. The amplitude transformation can also be inferred by this method. Liouville's theorem (the incompressibility of a phase space of position and slowness) ensures that it is always possible to obtain a uniformly asymptotic solution. This theorem can be derived by methods familiar to seismologists and which do not rely on the traditional formalism of classical mechanics. It can also be derived from the sympletic property of the equations of geometrical spreading and canonical transformations in general. The symplectic property plays a central role in the theory of high-frequency beams in inhomogeneous media.     

The theory of finite frequency body wave synthetic seismograms in inhomogeneous elastic media

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A new method is presented by means of which one can compute finite frequency synthetic seismograms for media whose velocity and density are continuous functions of two or three spatial variables. Basically, the method is a generalization of the familiar phase integral method, to which it reduces in a stratified medium. For a given source location the travel-time and distance functions needed to compute synthetics are obtained by numerically tracing rays through the model. This information is then used to evaluate a double integral over frequency and take-off angle at the source. The solution obtained reduces to the geometrical optics solution wherever that is valid but it also works in shadows and at caustics without knowing explicitly where these may be located. The method can be used as a spectral method, in which the integral over take-off angle is evaluated first, or as a slowness method, in which the frequency integral is evaluated first.     

An Integral Equation and its Solution for some Two-and Three-Dimensional Problems in Resistivity and Induced Polarization

Summary. The potential, U , about a point electrode, at the surface of a layered ground in which there is an heterogeneity embedded, satisfies the integral equation:
Here, U * and σ* are the corresponding quantities for the potential and conductivity without the heterogeneity. The integral is taken over the surface of the heterogeneity, ∂ U /∂ n is the normal derivative (in the direction of the outward normal) of U , and G is a Green's function.
Solutions to this equation can readily be found by using the Galerkin method of solving integral equations. The solutions of this equation when the heterogeneity is a sphere or a cylinder in a uniform ground or beneath a conductive overburden are the most readily found.
When the solution of the integral has been found for the potential it is a simple matter to calculate the apparent resistivity or chargeability for any electrode configuration.     

Efficient global matrix approach to the computation of synthetic seismograms

Summary. A numerically efficient global matrix approach to the solution of the wave equation in horizontally stratified environments is presented. The field in each layer is expressed as a superposition of the field produced by the sources within the layer and an unknown field satisfying the homogeneous wave equations, both expressed as integral representations in the horizontal wavenumber. The boundary conditions to be satisfied at each interface then yield a linear system of equations in the unknown wavefield amplitudes, to be satisfied at each horizontal wavenumber. As an alternative to the traditional propagator matrix approaches, the solution technique presented here yields both improved efficiency and versatility. Its global nature makes it well suited to problems involving many receivers in range as well as depth and to calculations of both stresses and particle velocities. The global solution technique is developed in close analogy to the finite element method, thereby reducing the number of arithmetic operations to a minimum and making the resulting computer code very efficient in terms of computation time. These features are illustrated by a number of numerical examples from both crustal and exploration seismology.     

稳定渠道几何形态解析

本文基于临界理论,同时引入合理的水流切应力表达式,建立起稳定渠道断面形态微分方程,据此获得断面曲线、过水面积及湿周的全部分析解。在此基础上,结合现有基本力学方程,严格从理论上导出稳定渠道的河相关系,并用实测资料作初步验证,结果较为满意。还运用积分中值定理给出断面宽度的另一形式的分析解,以便于水力计算中应用。     

Propagation of harmonic plane waves in a general anisotropic porous solid

Wave propagation is studied in a general anisotropic poroelastic solid. The presence of dissipation due to fluid-viscosity as well as hydraulic anisotropy of pore permeability are also considered. Biot's theory is used to derive a system of modified Christoffel equations for the propagation of plane harmonic waves in porous media. A non-trivial solution of this system is ensured by a determinantal equation. This equation is separated into two different polynomial equations. One is the quartic equation whose roots represent the complex velocities of four attenuating waves in the medium. The other is a eighth-degree polynomial whose roots represent the vertical slowness values for the four waves propagating upward and downward in a finite porous medium. Procedure is explained to associate the numerically obtained roots with the waves propagating in the medium. The slowness surfaces of waves reflected at the boundary of the medium are computed for a realistic numerical model. The behaviours of phase velocity surfaces are analysed with the help of numerical examples.     

Generalized Born scattering of elastic waves in 3-D media

It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansätze. In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution. The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach. The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed.     

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.     

Dynamic behaviour of a phase boundary under non-uniform surface loads

Summary. A method is outlined to determine the dynamic behaviour of a phase boundary in the Earth when non-uniform time-varying pressure and temperature conditions are assumed at the Earth's surface. An integral equation describing the phase boundary motion is derived and it is solved under a linearizing assumption. The solution is obtained in the form of a double integral transform. Short and long time-expansions of the solution can be obtained from series expansion and integration of the Laplace transform along a branch cut. The method is illustrated by considering a stepwise change in surface pressure conditions.
For short times, the solution exhibits the same type of time dependence (i.e. the first-order term is in t ^1/2) as the one obtained in the one-dimensional case (i.e. uniform pressure perturbation at the Earth's surface).
For long times, it is shown that the time dependence of the phase boundary motion is almost identical to the one derived for the one- dimensional case if the wavenumber k _L of the surface excitation is such that κ k ²_Lτ≤ 1 (where τ is the relaxation time associated with the one-dimensional phase boundary motion and κ is the thermal diffusivity). If κ k ²_Lτ > 1, then the relaxation time for the phase boundary motion in two dimensions is of the order of κ⁻¹ k ⁻²_L.
When considering parameters that would be appropriate for a basalt to eclogite phase transition at Moho depth, the latter situation is met only when the load wavelength is smaller than 35 km.     

On the estimation of the vertical distribution of radiogenic heat production in the crust

Summary. A differential-difference equation governing the distribution of radiogenic heat in the crust has been obtained. The solution of this equation gives the exponential model of the heat production distribution with the logarithmic decrement as determined in Singh & Negi.     

Elastic wave propagation in model sediments – II

Summary. A fluid-saturated cubic packing of like elastic spheres is taken to be in equilibrium under the effect of gravity and the effects of a superimposed low-frequency elastic wave are considered. In the first place, expressions for the wave velocity, dispersion and attenuation are derived for the dry packing. This dynamic theory leads to the result that, for very low frequencies, the wave velocity is proportional to the third root of the depth and not the sixth root as is obtained by using the effective elastostatic modulus of the packing. For the fluid-saturated packing, two waves, termed respectively the 'solid wave' and the 'fluid wave', are found to propagate. The 'solid wave' has the characteristics of a wave propagating within a dry packing whose parameters differ in a specified way from those of the original packing, whereas the 'fluid wave' has those of a wave within a homogeneous fluid with similarly modified parameters.     

Gaussian beams for surface waves in laterally slowly-varying media

Summary. Asymptotic ray theory is applied to surface waves in a medium where the lateral variations of structure are very smooth. Using ray-centred coordinates, parabolic equations are obtained for lateral variations while vertical structural variations at a given point are specified by eigenfunctions of normal mode theory as for the laterally homogeneous case. Final results on wavefields close to a ray can be expressed by formulations similar to those for elastic body waves in 2-D laterally heterogeneous media, except that the vertical dependence is described by eigenfunctions of 'local' Love or Rayleigh waves. The transport equation is written in terms of geometrical-ray spreading, group velocity and an energy integral. For the horizontal components there are both principal and additional components to describe the curvature of rays along the surface, as in the case of elastic body waves. The vertical component is decoupled from the horizontal components. With complex parameters the solutions for the dynamic ray tracing system correspond to Gaussian beams: the amplitude distribution is bell-shaped along the direction perpendicular to the ray and the solution is regular everywhere, even at caustics. Most of the characteristics of Gaussian beams for 2-D elastic body waves are also applicable to the surface wave case. At each frequency the solution may be regarded as a set of eigenfunctions propagating over a 2-D surface according to the phase velocity mapping.     

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.     

Split-step complex Padé-Fourier depth migration

We present a split-step complex Padé-Fourier migration method based on the one-way wave equation. The downward-continuation operator is split into two downward-continuation operators: one operator is a phase-shift operator and the other operator is a finite-difference operator. A complex treatment of the propagation operator is applied to mitigate inaccuracies and instabilities due to evanescent waves. It produces high-quality images of complex structures with fewer numerical artefacts than those obtained using a real approximation of a square-root operator in the one-way wave equation. Tests on zero-offset data from the SEG/EAGE salt data show that the method improves the image quality at the cost of an additional 10 per cent computational time compared to the conventional Fourier finite-difference method.     

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.     

Computation of wave fields in inhomogeneous media — Gaussian beam approach

Summary. An asymptotic procedure for the computation of wave fields in two-dimensional laterally inhomogeneous media is proposed. It is based on the simulation of the wave field by a system of Gaussian beams. Each beam is continued independently through an arbitrary inhomogeneous structure. The complete wave field at a receiver is then obtained as an integral superposition of all Gaussian beams arriving in some neighbourhood of the receiver. The corresponding integral formula is valid even in various singular regions where the ray method fails (the vicinity of caustic, critical point, etc.). Numerical examples are given.