Invariant expressions for elastic radiation from a fault with prescribed slip

Summary New formulae are obtained for the displacement potentials and displacements due to a point source with moment tensor, and for a fault with prescribed slip. These formulae, unlike previous formulae, are invariant, i.e. they are valid in any coordinate system, not just Cartesian coordinates or orthogonal curvilinear coordinates.
Apart from their invariance, these formulae have other advantages: they are exceedingly simple; the expressions for P -motion are nearly the same as those for S -motion; and the separation of far field motion from final static displacement is automatic.     

Ray tracing and inhomogeneous dynamic ray tracing for anisotropy specified in curvilinear coordinates

Ray tracing has recently been expressed for anisotropy specified in a local Cartesian coordinate system, which may vary continuously in a model specified by elastic parameters. It takes advantage of the fact that anisotropy is often of a simpler nature locally (and is thus specified by a smaller number of elastic parameters) and that the orientation of its symmetry elements may vary. Here we extend this approach by replacing the local Cartesian coordinate system with a curvilinear coordinate system of global extent and by applying the new approach to ray tracing and inhomogeneous dynamic ray tracing. The curvilinear coordinate system is orthogonal and is constructed so that the coordinate axes are consistent with the considered anisotropy of the medium. Our formulation allows for computation of ray attributes (e.g. ray velocity vector and paraxial ray attributes) in the curvilinear coordinate system, while rays are computed in global Cartesian coordinates. Compared to the classic formulation in terms of 21 elastic moduli in global Cartesian coordinates, the main advantages are improved efficiency, lower computer-memory requirements, and conservation of anisotropic symmetry throughout the model.     

Spherical prism magnetic effects by Gauss–Legendre quadrature integration

Regional spherical coordinate observations of the Earth's crustal magnetic field components are becoming increasingly available from shipborne, airborne, and satellite surveys. In assessing the geological significance of these data, theoretical anomalous magnetic fields from geologic models in spherical coordinates need to be evaluated. This study explicitly develops the elegant Gauss–Legendre quadrature formulation for numerically modelling the complete magnetic effects (i.e. potential, vector and tensor gradient fields) of the spherical prism. We also use these results to demonstrate the magnetic effects for the crustal prism and to investigate the crustal magnetic effects at satellite altitudes for a large region of the Middle East centred on Iran.     

Cluster analysis of seismic moment tensor orientations

This paper demonstrates that well-known methods of cluster analysis and multivariate data analysis are useful for geodynamic interpretation of seismic moment tensors. To use these methods, moment tensors are expressed as vectors in a 6-D space. These are vectors in a rigorous sense, rather than an arbitrary set of ordered numbers, because a dot product can be defined that is independent of the coordinate system. In this vector space, non-isotropic moment tensors are a 5-D linear subspace and normalized moment tensors are unit vectors, or points on a unit sphere. Distance along a great circle of the unit sphere satisfies reasonable requirements for any measure of the difference between normalized moment tensors. In regions with a few isolated sets of orientations, cluster analysis based on the great circle distance identifies the same groups of earthquakes that a seismologist would. Figures based on principal component analysis and discriminant analysis illustrate orientation clustering better than equal area projections of moment tensor principal axes. In one case where clusters have been claimed to exist, orientations appear to be continuously distributed and no evidence is found for separate populations of moment tensors.     

On covariance propagation of eigenparameters of symmetric n-D tensors

Symmetric tensors are typically encountered during investigations associated with stress and strain analysis and, thus, they are of particular interest to geophysicists and geodesists. Furthermore, symmetric tensors are studied using eigentheory analysis which provides the decomposition of the tensor on its principal components (n independent eigenvalues and the corresponding eigenvectors). In this paper, an analytical expression of the covariance matrix of the eigenvalues and eigenvectors of an n-D symmetric tensor is derived based on the principles of linear algebra and differential calculus. Through numerical tests, the proposed formulation is proven to give realistic uncertainty estimates of the determined eigenparameters. The methodology also reveals the significant impact on uncertainty assessments when the parameter dependencies between principal components are neglected.     

On Some implications of the Poisson relation

The potentialities of the so-called 'Poisson relation', which holds for uniformly magnetized bodies of constant density, for showing the connections between the gravitational and the magnetic fields of such bodies are considered. In particular, it is seen that the same characteristic ratios occur among the components of the magnetic field intensity and the components of the gradient tensors of the two fields, both locally and non-locally. It is also shown that along the magnetization axis of the body the gradient tensor of the magnetic field displays a 'tidal' structure.     

About the influence of pre-stress upon adiabatic perturbations of the Earth

Summary. In this paper we examine the influence of the state of stress in the equilibrium configuration of the Earth (i.e. the pre-stress) upon its adiabatic perturbations. The equations governing these perturbations to the first order (Woodhouse & Dahlen; Dahlen) are re-derived using a Lagrangian approach. Different expressions of the sesquilinear form associated to the elastic-gravitational operator are given. One of these provides a way to extend to hydrostatically pre-stressed solids the criterion of local stability given by Friedman & Schutz for uniformly rotating fluids. Then the propagation in the Earth of seismic wavefronts is considered. It is shown that the nature of these different wavefronts is entirely determined by the quadratic coefficients of the development of the specific internal energy variation, corresponding to isentropic evolution, with respect to the Lagrangian finite deformation tensor. Expressions for the velocities of the various waves are given as functions of incidence angle and pre-stress for orthotropic elastic material. In the particular case where the elastic parameters depend only on one coordinate of a curvilinear system and the axis of orthotropy of the material coincides with the corresponding natural base vector, the elastodynamic equations are reduced to a simple system for a displacement stress vector, using surface operators. In particular for spherical geometry, equations are obtained which generalize to orthotropic pre-stress those given by Alterman et al. and Takeuchi & Saito.     

Stable inversions for complete moment tensors

The seismic moment tensors for certain types of sources, such as volcanic earthquakes and nuclear explosions are expected to contain an isotropic component. Some earlier efforts to calculate the isotropic component of these sources are flawed due to an error in the method of Jost & Herrmann. We corrected the method after Herrmann & Hutchensen and found great improvement in the recovery of non-double-couple moment tensors that include an isotropic component. Tests with synthetic data demonstrate the stability of the corrected linear inversion method, and we recalculate the moment tensor solutions reported in Dreger et al. for Long Valley caldera events and Dreger & Woods for Nevada Test Site nuclear explosions. We confirm the findings of Dreger et al. that the Long Valley volcanic sources contain large statistically significant isotropic components. The nuclear explosions have strikingly anomalous source mechanisms, which contain very large isotropic components, making it evident that these events are not tectonic in origin. This indicates that moment tensor inversions could be an important tool for nuclear monitoring.     

Statistics and geometry of the eigenspectra of three-dimensional second-rank symmetric random tensors

Geoscientists have undertaken mapping of the Earth's crustal strain (or stress) fields using a great variety of field data. The output can be represented by a 3-D second-rank symmetric random strain tensor. The random principal strains-land rotations of the random tensor are frequently computed. The accuracy is calculated using a first-order approximation. The distribution aspects of the random principal strains and rotations have received almost no attention in Earth Sciences. A first-order approximation of accuracy may not be sufficient if the signal-to-noise ratio is small, as is often the case for geodetically derived random strain tensors. Therefore, the purpose of this paper is to investigate the distribution and estimation problems of the general 3-D second-rank tensor equation GΛG ^T= T , where T is a given 3-D second-rank symmetric random tensor, Λ a diagonal (3 × 3) random eigenvalue matrix, and G a (3 × 3) random orientation matrix, which is also orthogonal. Λ and G are to be estimated (or solved) from T . If some eigenvalues coincide, additional conditions are imposed on the eigenvectors so that they can be chosen uniquely. The joint probability density function (pdf) of the random eigenvalues and rotations will be worked out, given a joint pdf of the elements of random tensors T. Because the rotations are of special interest in Earth Sciences, we shall also derive the joint marginal pdf of random rotations. The geometry of eigenspectra will be studied. The biases of random eigenvalues and rotations will be derived, which have been neglected in the past. They can be very crucial in interpreting the pattern of a derived strain field, however, when applied to a real Earth Science problem. The variance-covariance matrices will be computed using a second-order approximation.     

Point Sources of Elastic Deformation: Elementary Sources, Static Displacements

Summary. We construct a catalogue of all possible elementary point sources for static deformation in an elastic solid. the familiar double-couples, CLVD's, centres of compression and dilatation, etc., are all members of the complete catalogue. the sources are classified according to the rank of the seismic moment tensor, and according to the weight (or order) of the irreducible tensor representation of the 3-D rotation group. These sources can be classified as belonging to one of three general classes. the static excitation functions are calculated for an infinite, homogeneous, isotropic medium for all these sources. We show that, except for sources belonging to these three general classes, all other sources — which are numerous for the tensors of high rank — are null static sources. That is, sources that do not produce any static displacement outside of the source region. Due to the presence of null sources, an inversion of the static deformation data is non-unique. the expansion of the equivalent-force tensors and the stress glut tensors (or seismic moment tensors) into a set of the symmetric trace-free source tensors is proposed. the sources corresponding to seismic moment tensors of the second, third and fourth ranks are considered in more detail. We identify the third-rank sources with rotational dislocations or disclinations.     

Spectral theory of constrained second-rank symmetric random tensors

The random principal eigenvalues and random eigenvector parameters have been routinely estimated from second-rank symmetric (SRS) random tensors and geophysically interpreted in the Earth Sciences. Statistical inference of random eigen-values and random eigenvector parameters has almost always been made as if they were normally distributed. The practical validity and applicability of the assumption of normal distributions for random eigenvalues and random eigenvector parameters has not yet been checked, however. Statistical inference of random eigenvalues and random eigenvector parameters should be based on their joint probability density function (pdf) derived from that of the original random tensor. We shall extend the work of Xu & Grafarend (1996a , b ) to the case of constrained SRS random tensors in this paper. All the relevant Jacobians for n -D unconstrained and 3-D constrained SRS tensors have been obtained. We then propose three pdf models for original SRS random tensors, which cover the commonly used Gaussian and Laplace pdfs and include pdf models for positive definite random material tensors. The pdfs of the random eigenvalues and random eigenvector parameters have been worked out. It is shown that the pdfs of the random eigenvalues and random eigenvector parameters are significantly different from the commonly used Gaussian pdf model. Deviatoric stress tensors and double-couple seismic moment tensors have been simulated to show the applications of the developed theory. The simulations have additionally indicated that Fisher's pdf model for directional data is not representative of the random rotations of constrained SRS random tensors.     

A matrix representation of the potential second-rank gradient tensor for local modelling

Summary. This paper introduces a new matrix computational approach to the local determination of gravity gradients, convenient for comparing with gradient signals from moving base gradiometer systems or calculating topographic effects at instrument heights. The method represents a practical alternative to the more conventional spherical harmonics formulation, primarily global in nature, and it may be considered as an extension to other previously used local representations, such as point masses. Important characteristics of the analytical development outlined herein are its conceptual simplicity and the possibility of obtaining at once, up to a certain order n , and in an arbitrary Cartesian coordinate system, the symmetric point gradient tensor of second rank.     

Seismic moment tensors and kinematic source parameters

Summary. Parameters pertaining to the kinematics of a finite source are usually estimated by fitting specific fault models to the data. On the other hand, these parameters, including source location, are also contained in the moment tensors of higher degree. In this paper, the seismic response is represented in terms of 20 source parameters which are related to components of the moment tensors; they are also related to the parameters of fault models, as will be demonstrated for a number of 'classical' models. A linearized inversion for the moment tensor shows that with real data, or with realistic synthetic data, the results are not necessarily physically meaningful, unless constraints are imposed. The constraints are precisely those appearing as a priori assumptions in the conventional methods of source analysis; it is thus possible to investigate the impact of these assumptions. We will discuss in particular the assumption of a general deviatoric point source (not necessarily a double couple) versus that of a plane fault in finite sources. Although at this stage experience with practical performance of the new method is limited, it is suggested that in the appropriate circumstances constrained inversion for the seismic moment tensors offers a viable alternative to estimate kinematic source parameters.     

Ray amplitudes of seismic body waves in laterally inhomogeneous media

Summary . The most complicated part in the computation of ray amplitudes of seismic body waves in laterally inhomogeneous media with curved interfaces lies in the evaluation of the geometrical spreading. Geometrical spreading can be simply expressed in terms of the Jacobian J of the transformation from the Cartesian into ray coordinates. Several systems of ordinary differential equations to compute the function J are suggested. For general three-dimensional media, in which the velocity changes with all the three spatial coordinates, a system of three non-linear ordinary differential equations of the first order is derived. If the velocity does not depend on one coordinate, the system of equations reduces to only one non-linear differential equation. The initial conditions for these differential equations at point (or line) source and at points of intersection of the ray with curved interfaces are presented.     

SRG剖分编码与地理坐标的转换算法研究

提出一种基于SRG剖分方法的剖分编码与经纬度坐标之间的转换算法,不涉及任何投影变换,计算过程只需加、减、乘、除简单算术运算,计算速度快,且SRG剖分编码本身就具有固定的方向性,利于邻近搜索。算法中采用坐标系辅助区分一些难以区分的菱形块,降低了转化误码率,进而提高了转换的精度。     

Three-dimensional numerical modelling of thermal transfer through the Vittel aquifer (Vosges, France), including approximated topographical effects

Summary. A geothermal and hydrochemical anomaly was observed in the Lower Triassic sandstone aquifer in the Vittel region (Vosges, France). This anomaly was attributed to a hydrothermal spring under sedimentary cover. In order to localize the hot spring and to quantify more precisely its flow rate and temperature, it is necessary to consider the 3-D thermal problem by taking into account the complex geometry of the domain and the flow rate in the aquifer. A 3-D numerical model of thermal conductive and forced convective transfer, developed for hydrological problems including approximate geometrical and topographical effects, is used which (i) can be directly applied to geologic strata (aquifers) with varying thickness and top and base slope; and (ii) allows calculation of heat flux anomalies associated with fluid flow in such geologically realistic aquifers.
The heat transfer equation is formulated in an orthogonal curvilinear coordinate system. As most geometries dealing with geothermal phenomena in sedimentary basins are nearly horizontal, this formulation can be simplified, leading readily to numerical solution with a finite difference method. The application of the 3-D model to the Vittel aquifer gives temperature results in agreement with measurements. These results provide evidence for the importance of associated forced convection and topographical effects for ground temperature distribution, and show clearly that heat flow in many basins is interpretable only if careful hydrological and thermal studies are made.     

The instantaneous apparent resistivity tensor: a visualization scheme for LOTEM electric field measurements

Long-offset transient electromagnetic (LOTEM) data have traditionally been represented as early- and late-time apparent resistivities. Time-varying electric field data recorded in a LOTEM survey made with multiple sources can be represented by an 'instantaneous apparent resistivity tensor'. Three independent, coordinate-invariant, time-varying apparent resistivities can be derived from this tensor. For dipolar sources, the invariants are also independent of source orientation. In a uniform-resistivity half-space, the invariant given by the square root of the tensor determinant remains almost constant with time, deviating from the half-space resistivity by a maximum of 6 per cent. For a layered half-space, a distance–time pseudo-section of the determinant apparent resistivity produces an image of the layering beneath the measurement profile. As time increases, the instantaneous apparent resistivity tensor approaches the direct current apparent resistivity tensor. An approximate time-to-depth conversion can be achieved by integrating the diffusion depth formula with time, using the determinant apparent resistivity at each instant to represent the resistivity of the conductive medium. Localized near-surface inhomogeneities produce shifts in the time-domain apparent resistivity sounding curves that preserve the gradient, analogous to static shifts seen in magnetotelluric soundings. Instantaneous apparent resistivity tensors calculated for 3-D resistivity models suggest that profiles of LOTEM measurements across a simple 3-D structure can be used to create an image that reproduces the main features of the subsurface resistivity. Where measurements are distributed over an area, maps of the tensor invariants can be made into a sequence of images, which provides a way of 'time slicing' down through the target structure.     

Using Vertical Aerial Photography to Estimate Mass Balance at a Point

The mass balance distribution over a 0.5 km² area of the lower part of South Cascade Glacier is obtained from remotely sensed measurements of its geometry and velocity field over two periods, 1992–93 and 1993–94. Vertical aerial photography from late summer 1992, 1993, and 1994 is analyzed photogrammetrically to get surface topography of South Cascade Glacier on a 100-meter square grid. The known bed topography is subtracted from the surface topography to get the ice thickness, and the surface topographies are subtracted from each other to get the thickness change. Annual displacement vectors, determined at points where natural features could be tracked from one year to the next, are contoured by hand and interpolated to the grid. Assuming that the ice follows Glen's flow law with exponent n = 3, and that 10% of the ice flow is due to sliding at the bed, the surface velocity is scaled by 0.82 to get the average velocity in the vertical ice column. The average velocities are combined with the thicknesses to calculate the flux divergence at each of 46 gridpoints on a 100-m square grid, where it is subtracted from the thickness change to get the mass balance.
Use of the same control points from year to year makes any systematic error in photogrammetric coordinates temporally constant, so such error has no effect on the mass balance estimate. Random error in coordinates is assumed to be uncorrelated from coordinate to coordinate, from point to point, from year to year; the standard error is estimated to be 1 m, resulting in a standard error in coordinate differences of about 1.5 m. A 1 m error in a vertical coordinate has nearly twice the effect on the estimated balance that one in a horizontal coordinate has and more than ten times the effect that one in ice thickness has. Compared with measurements at a stake, the estimated balances are about 1 m too negative.     

Efficient encoding and spatial operation scheme for aperture 4 hexagonal discrete global grid system

Discrete global grid systems (DGGSs) are considered to be promising structures for global geospatial information representation. Square and triangular DGGSs have had the advantage over hexagonal ones in geospatial data processing over the past few decades. Despite a significant body of research supporting hexagonal grids as the superior alternative, the application thereof has been hindered partly owing to the lack of a hierarchy. This study presents an original perspective to combine two types of aperture 4 hexagonal discrete grid systems into a hierarchy. Each cell of the hierarchy is assigned a unique code using a linear quadtree that constructs the hexagonal quaternary balanced structure (HQBS). The mathematical system described by HQBS addressing and the vector operations, including addition, subtraction, multiplication, and division, are defined. Essential spatial operations for HQBS cell retrieval, transformation between HQBS codes and other coordinate systems, and arrangement of HQBS cells on spherical surfaces were studied and implemented. The accuracy and efficiency of algorithms were validated through experiments. The results indicate that the average efficiency of cell retrieval using the HQBS is higher than that using other schemes, thus proving it to be more efficient.