Symmetries in the reflection and transmission of elastic waves

Summary. The symmetry relations between the reflection and transmission coefficients for plane elastic waves incident upon an arbitrary horizontally stratified medium are derived by a novel approach. Previous results, particularly for a single interface, are obtained as special cases of this treatment.
In addition, for perfectly elastic media, projection operators for travelling and evanescent waves are introduced and used to derive a number of new relationships between the reflection and transmission coefficients.     

Guided wave propagation in laterally varying media - I. Theoretical development

Summary. The propagation of surface waves in a laterally varying medium can be described by representing the wavetrain as a superposition of modal contributions for a reference structure. As the guided waves propagate through a heterogeneous zone the modal coefficients needed to describe the wavetrain vary with position, leading to interconversions between modes and reflection into backward travelling modes. The evolution of the modal terms may be described by a set of first-order differential equations which allow for coupling to both forward and backward travelling waves; the coefficients in these equations depend on the differences between the actual structure and the reference structure. This system is established using the orthogonality properties of the modal eigenfunctions and is valid for SH -waves, P - SV -waves and full anisotropy.
The reflected and transmitted wavefields for a region of heterogeneity can be related to the incident wave by introducing reflection and transmission matrices which connect the modal coefficients in these fields to those in the incident wavetrain. By considering a sequence of models with increasing width of heterogeneity we are able to derive a set of Ricatti equations for the reflection and transmission matrices which may be solved by initial value techniques. This avoids an awkward two-point boundary value problem for a large number of coupled equations. The method is demonstrated for 1 Hz Lg - and Sn -waves in a multilayered model for which there are 19 coupled modes.
The method is applicable to three-dimensional heterogeneity, and we are able to show that the interconversion between Love and Rayleigh waves, in the presence of gradients in seismic properties transverse to the propagation path, leads to a net rate of increase of the transverse components of the seismogram at the expense of the other components.     

Asymptotic eigenfrequencies of the Earth's normal modes

We derive asymptotic formulae for the toroidal and spheroidal eigenfrequencies of a SNREI earth model with two discontinuities, by considering the constructive interference of propagating SH and P-SV body waves. For a model with a smooth solid inner core, fluid outer core and mantle, there are four SH and 10 P-SV ray parameters regimes, each of which must be examined separately. The asymptotic eigenfrequency equations in each of these regimes depend only on the intercept times of the propagating wave types and the reflection and transmission coefficients of the waves at the free surface and the two discontinuities. If the classical geometrical plane-wave reflection and transmission coefficients are used, the final eigenfrequency equations are all real. In general, the asymptotic eigenfrequencies agree extremely well with the exact numerical eigenfrequencies; to illustrate this, we present comparisons for a crustless version of earth model 1066A.     

Physical properties and structure of the lower crust revealed by one- and two-dimensional modelling

Summary. Data from a refraction and a reflection seismic survey in the Black Forest, southwest Germany, are used for extensive one- and two-dimensional modelling. The data are available along approximately the same line, and therefore the same piece of crust is probed by two seismic methods. We utilize this favorable circumstance for detailed model calculations concerning both data sets. Lower crustal properties vary on the scale of a wavelength and thus full solutions of the elastic equations are required: the Reflectivity Method for the evaluation of refraction seismograms and numerical solutions of the acoustic wave equation for the reflection response. Details of the geometry and physical properties of the lamination are derived. Vertical layering on a scale of 100 m is found; horizontal extent of reflecting elements is in the range of a few hundreds of meters; rocks with velocities between 5.6 and 7.2 km/s constitute the lower crust.     

Seismic reflection and transmission coefficients of a self-similar interface

The derivation of seismic reflection and transmission coefficients is generally based on the assumption that the medium parameters behave as step functions of depth, at least in a finite region around the interface. However, outliers observed in well logs generally behave quite differently from step functions. In this paper we represent an interface by a self-similar singularity, embedded between two homogeneous half-spaces, and we derive its frequency-dependent normal-incidence reflection and transmission coefficients. For ω  → 0 the expressions for the coefficients reduce to those for a discrete boundary between two homogeneous half-spaces; for ω → ∞ they become frequency-independent. These asymptotic expressions have a relatively simple form and depend on the singularity exponent α .
  The exact as well as the asymptotic expressions are used to evaluate the time-domain reflection and transmission responses of a self-similar interface. Finally, we use a numerical method to model the response of a smoothed version of a self-similar interface (note that the velocity of a smoothed singularity remains finite). It turns out that smoothing has hardly any effect on the response, provided that the smoothing does not affect the scales corresponding to the seismic frequency range.     

Reflection coefficients for weak anisotropic media

The interaction of plane elastic waves with a plane boundary between two anisotropic elastic half-spaces is investigated. The anisotropy dealt with in this study is of a general type. Explicit expressions for energy-related reflection and transmission coefficients are derived. They represent an approximation which is valid for a small deviation of the elastic parameters from isotropy.
Classical perturbation theory is applied on a 6times6 non-symmetric real eigenvalue problem to calculate first-order corrections for the polarization and stress of the plane waves. The explicit solution of the isotropic problem is used as a reference case. Degenerate perturbation theory is used to consider the splitting of the isotropic S -wave into two anisotropic qS-waves. The boundary conditions for two half-spaces in welded contact lead to a 6times6 system of linear equations. A correction to the isotropic solution is calculated by linearization. The resultant coefficients are functions of horizontal slowness, Lamé parameters and densities of the reference media, and of the perturbation of the elasticity tensors from isotropy.     

Computation of high-frequency seismic wavefields in 3-D laterally inhomogeneous anisotropic media

Summary. An algorithm for the computation of travel times, ray amplitudes and ray synthetic seismograms in 3-D laterally inhomogeneous media composed of isotropic and anisotropic layers is described. All 21 independent elastic parameters may vary within the anisotropic layers. Rays and travel times are evaluated by numerical solution of the ray tracing equations. Ray amplitudes are determined by evaluating reflection/ transmission coefficients and the geometrical spreading along individual rays. The geometrical spreading is computed approximately by numerical measurement of the cross-sectional area of the ray tube formed by three neighbouring rays. A similar approximate procedure is used for the determination of the coefficients of the paraxial ray approximation. The ray paraxial approximation makes computation of synthetic seismograms on the surface of the model very efficient. Examples of ray synthetic seismograms computed with a program package based on the described algorithm are presented.     

Weak-contrast reflection/transmission coefficients in weakly anisotropic elastic media: P-wave incidence

We present approximate displacement and energy PP and PS reflection/transmission coefficients for weak-contrast interfaces in general weakly anisotropic elastic media. The coefficients were obtained by applying first-order perturbation theory and then expressed in a compact and relatively simple form. The formulae can be used for arbitrary orientations of the incidence plane and interface, without the need to transform the elasticity parameters to a local Cartesian coordinate system. The accuracy of the approximate formulae is illustrated for the PS reflection coefficient for two synthetic models. For these models, we also study the possibility of using the approximate PP reflection coefficient in the inverse problem.     

Experimental study of shear-waves from shots in anisotropic media

Summary. A complex array of vertical-seismic-profiles and reflection surveys in the Taman Peninsula, Krasnodar, using geophones with inclined axes, is used to investigate shear-wave splitting in thick Maikopian clays characterised by pronounced diapirism. The shear waves split into SH (faster) and SV (slower) components and display transverse isotropy with a vertical axis of symmetry with a maximum shear-wave delay of up to 0.5 s . Within the accuracy of the observations there is no azimuthal anisotropy.     

First-order P-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media

We propose approximate equations for P -wave ray theory Green's function for smooth inhomogeneous weakly anisotropic media. Equations are based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. For evaluation of the approximate Green's function, earlier derived first-order ray tracing equations and in this paper derived first-order dynamic ray tracing equations are used.
The first-order ray theory P -wave Green's function for inhomogeneous, weakly anisotropic media of arbitrary symmetry depends, at most, on 15 weak-anisotropy parameters. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to equations for computation of standard ray theory Green's function for isotropic media. These properties make the proposed approximate Green's function an easy and natural substitute of traditional Green's function for isotropic media.
Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of the approximate Green's function on inhomogeneity of the medium. Accuracy depends more strongly on strength of anisotropy in general and on angular variation of phase velocity due to anisotropy in particular. For example, for anisotropy of about 8 per cent, considered in the examples presented, the relative errors of the geometrical spreading are usually under 1 per cent; for anisotropy of about 20 per cent, however, they may locally reach as much as 20 per cent.     

Rigorous velocity bounds from soft τ (p) and X(p) data

Summary. The convergence of two methods of inferring bounds on seismic velocity in the Earth from finite sets of inexact observations of τ ( p ) and X( p ) are examined: the linear programming (LP) method of Garmany, Orcutt & Parker and the quadratic programming (QP) method of Stark & Parker. The LP method uses strict limits on the observations of τ and X as its data, while QP uses estimated means and variances of τ and X. The approaches are quite similar and involve only one inherent approximation: they use a finite-dimensional representation of seismic velocity within the Earth. Clearly, not every Earth model can be written this way. It is proved that this does not hinder the methods - they may be made as accurate as desired by increasing the number of dimensions in a specified way. It is shown how to get the highest accuracy with a given number of dimensions.     

Seismic waves in a stratified half space

Summary. The response of a stratified elastic half space to a general source may be represented in terms of the reflection and transmission properties of the regions above and below the source. For P-SV and SH waves and both buried sources and receivers, convenient forms of the response may be found in which no loss of precision problems arise from growing exponential terms in the evanescent regime. These expressions have a ready physical interpretation and enable useful approximations to the response to be developed. The reflection representation leads to efficient computational procedures for models composed of uniform layers, which may be extended in an asymptotic development to piecewise smooth models.     

Identification of symmetry planes in weakly anisotropic elastic media

A procedure is proposed to obtain symmetry properties of weakly anisotropic (WA) elastic media by giving 15 WA parameters of qP wave in an arbitrary Cartesian coordinate system. The 15 WA parameters, which linearly depend on 21 elastic elements, form a complete set to determine the symmetry planes in WA materials. The procedure is based on the eigenvalue problems of two matrices. One of the matrices consists of the Voigt and dilatational matrices, and the other is an acoustic tensor defined by an irreducible, completely symmetric and traceless, fourth-rank tensor resulting from decomposition of the fourth-rank elastic tensor. If the eigenvectors are taken as the axes of a new coordinate system (called symmetry Cartesian coordinate system), the transformation of WA parameters from an arbitrary Cartesian coordinate system to a symmetry Cartesian coordinate system can reduce the number of distinct WA parameters of elastic materials except in triclinic medium.     

The explicit crustal transfer function for SH-waves by ray theory — the case of one and two layers

Summary. Previous theoretical studies on the transfer function of crustal plane surface layers have been primarily based on applying recursion formulae to the solutions of wave equations. In this way the detailed physical insight of the transfer function is often obscure and it is very difficult to express the transfer function in an explicit form. To show that this situation can be improved by an alternative approach, we demonstrate in this paper with models of one and two surface layers that the explicit transfer function can be derived by summing multiple-reflected rays between interfaces. Since the derived transfer function is related explicitly to some physical parameters, such as wave travel times between interfaces, reflection and transmission coefficients at the interfaces, and attenuation and dispersion of waves in each layer of the model, it is more flexible than the traditional recursion form when applied to different models. This suggests that the ray theory can play an important role in problems of layered media.     

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.     

Kirchhoff–Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium – II. Computations

Summary. High-frequency reflection and refraction seismograms for laterally variable multi-layered elastic media are computed by using the frequency domain elastic Kirchhoff–Helmholtz (KH) theory of Frazer and Sen. Both source and receiver wavefields are expanded in series of generalized rays and then elastic (KH) theory is applied to determine the coupling between each source ray and each receiver ray at each interface. The motion at the receiver is given as a series of integrals, one for each generalized ray. We use geometrical optics and plane wave reflection and transmission coefficients for rapid evaluation of the integrand. When the source or the receiver ray field has caustics on the surface of integration geometrical ray theory breaks down and this gives rise to singularities in the KH integrand. We repair this using methods suggested by Frazer and Sen.
Examples of reflection seismograms for 2-D structures computed by elastic KH theory are shown. Those for a vertical fault scarp structure are compared with the seismograms obtained by physical modelling. Then OBS data obtained from the mid-America trench offshore Guatemala area are analysed by computing KH synthetics for a velocity model that has been proposed for that area. Our analysis indicates the existence of a small low-velocity zone off the trench axis.
No head wave arrivals are obtained in our KH synthetics since we do not consider multiple interactions of a ray with an interface. The nearly discontinuous behaviour of elastic R/T coefficients near the critical angle causes small spurious phases which arrive later than the correct arrivals.     

Modelling landscape evolution on geological time scales: a new method based on irregular spatial discretization

We present simulations of large-scale landscape evolution on tectonic time scales obtained from a new numerical model which allows for arbitrary spatial discretization. The new method makes use of efficient algorithms from the field of computational geometry to compute the set of natural neighbours of any irregular distribution of points in a plane. The natural neighbours are used to solve geomorphic equations that include erosion/deposition by channelled flow and diffusion. The algorithm has great geometrical flexibility, which makes it possible to solve problems involving complex boundaries, radially symmetrical uplift functions and horizontal tectonic transport across strike-slip faults. The algorithm is also ideally suited for problems which require large variations in spatial discretization and/or self-adaptive meshing. We present a number of examples to illustrate the power of the new approach and its advantages over more 'classical' models based on regular (rectangular) discretization. We also demonstrate that the synthetic river networks and landscapes generated by the model obey the laws of network composition and have scaling properties similar to those of natural landscapes. Finally we explain how orographically controlled precipitation and flexural isostasy may be easily incorporated in the model without sacrificing efficiency.
     

潜在蒸散发公式2种修正方法及其在闽江流域的应用研究

各类潜在蒸散发经验估算法具有区域性的特点,实际运用时需要进行修正。常用的修正方法是通过线性回归法对经验公式的计算结果进行修正,并没有将修正反映到经验公式中的系数上。本研究提出一种系数修正的方法,通过逐月误差比例校正来修正经验公式中的参数,称之为误差比例校正法。以FAO推荐的Penman-Monteith法为标准,比较经线性回归法、误差比例校正法2种方法修正前后6种经验估算法（Blaney-Criddle、Hargreaves、Hamon、Makkink、Priestley and Taylor、Rohwer）在闽江流域的计算精度,从而对误差比例校正法的修正效果及稳健性进行探讨。结果表明：逐月误差比例校正法整体上具有较好的修正效果,可以较大程度提高6种经验方法在该区域的计算精度;常用的线性回归法虽计算较误差比例校正法稍复杂,但其修正效果及稳健性略优于误差比例校正法。     

Surface wave tomography for azimuthal anisotropy in a strongly reduced parameter space

Large scale seismic anisotropy in the Earth's mantle is likely dynamically supported by the mantle's deformation; therefore, tomographic imaging of 3-D anisotropic mantle seismic velocity structure is an important tool to understand the dynamics of the mantle. While many previous studies have focused on special cases of symmetry of the elastic properties, it would be desirable for evaluation of dynamic models to allow more general axis orientation. In this study, we derive 3-D finite-frequency surface wave sensitivity kernels based on the Born approximation using a general expression for a hexagonal medium with an arbitrarily oriented symmetry axis. This results in kernels for two isotropic elastic coefficients, three coefficients that define the strength of anisotropy, and two angles that define the symmetry axis. The particular parametrization is chosen to allow for a physically meaningful method for reducing the number of parameters considered in an inversion, while allowing for straightforward integration with existing approaches for modelling body wave splitting intensity measurements. Example kernels calculated with this method reveal physical interpretations of how surface waveforms are affected by 3-D velocity perturbations, while also demonstrating the non-linearity of the problem as a function of symmetry axis orientation. The expressions are numerically validated using the spectral element method. While challenges remain in determining the best inversion scheme to appropriately handle the non-linearity, the approach derived here has great promise in allowing large scale models with resolution of both the strength and orientation of anisotropy.     

Electromagnetic 2.5-dimensional forward modelling with a boundary integral formulation

An effective and accurate technique for the numerical solution of 2-D electromagnetic scattering problems with 3-D sources is presented. This solution introduces a set of the usual boundary integral equations and uses a scalar Green's function. In this scalar version, the unknowns of the problem are the boundary values of the longitudinal fields and their normal derivatives in the Fourier domain. A generalization of the usual boundary integral formulation enables us to handle a large class of models composed of piecewise homogeneous domains, including contiguous domains, multiply-connected domains and unbounded domains. This formulation involves the solution of a system of linear equations, and results in a significant saving in computation time in comparison with other rigorous methods.
  The requirements for the numerical implementation of this solution are described in detail. Numerical tests were carried out using the important example of electromagnetic tomography. The specific symmetry properties of the response function in this case are illustrated. Numerical accuracy is verified over a large frequency range, up to 1  MHz.