Modelling the pole tide and its effect on the Earth's rotation

Summary. The pole tide is the response of the ocean to incremental centrifugal forces associated with the Chandler wobble. The tide has a potentially important effect on the period and damping of the wobble, but it is at present not well constrained by observations. Here, we construct both analytical and numerical models for the pole tide. The analytical models consider the tide first in a global ocean and then in an enclosed basin on a beta-plane. The results are found to approach equilibrium linearly with decreasing frequency and inversely with increasing basin depth. The numerical models solve Laplace's tidal equations over the world's oceans using realistic continental boundaries and bottom topography. The results indicate that the effects of the non-equilibrium portion of the deep ocean tide on the Chandler wobble period and damping are negligible.     

The effects of the atmosphere and oceans on the Earth's wobble — I. Theory

Summary The theory of wobble excitation for a non-rigid earth is extended to include the effects of the earth's fluid core and of the rotationally induced pole tide in the ocean. The response of the solid earth and oceans to atmospheric loading is also considered. The oceans are shown to be affected by changes in the gravitational potential which accompany atmospheric pressure disturbances and by the load-induced deformation of the solid earth. These various improvements affect the excitation equations by about 10 per cent. Atmospheric and oceanic excitation can be computed using either an angular momentum or a torque approach. We use the dynamical equations for a thin fluid to relate these two methods and to develop a more general, combined approach. Finally, geostrophic winds and currents are shown to be potentially important sources of wobble excitation, in contrast to what is generally believed.     

A search for the ratio between gravity variation and vertical displacement due to a surface load

A data space approach to magnetotelluric (MT) inversion reduces the size of the system of equations that must be solved from M × M , as required for a model space approach, to only N × N , where M is the number of model parameter and N is the number of data. This reduction makes 3-D MT inversion on a personal computer possible for modest values of M and N . However, the need to store the N × M sensitivity matrix J remains a serious limitation. Here, we consider application of conjugate gradient (CG) methods to solve the system of data space Gauss–Newton equations. With this approach J is not explicitly formed and stored, but instead the product of J with an arbitrary vector is computed by solving one forward problem. As a test of this data space conjugate gradient (DCG) algorithm, we consider the 2-D MT inverse problem. Computational efficiency is assessed and compared to the data space Occam's (DASOCC) inversion by counting the number of forward modelling calls. Experiments with synthetic data show that although DCG requires significantly less memory, it generally requires more forward problem solutions than a scheme such as DASOCC, which is based on a full computation of J .     

Non-linear inversion using general measures of data misfit and model structure

We investigate the use of general, non- l ₂ measures of data misfit and model structure in the solution of the non-linear inverse problem. Of particular interest are robust measures of data misfit, and measures of model structure which enable piecewise-constant models to be constructed. General measures can be incorporated into traditional linearized, iterative solutions to the non-linear problem through the use of an iteratively reweighted least-squares (IRLS) algorithm. We show how such an algorithm can be used to solve the linear inverse problem when general measures of misfit and structure are considered. The magnetic stripe example of Parker (1994 ) is used as an illustration. This example also emphasizes the benefits of using a robust measure of misfit when outliers are present in the data. We then show how the IRLS algorithm can be used within a linearized, iterative solution to the non-linear problem. The relevant procedure contains two iterative loops which can be combined in a number of ways. We present two possibilities. The first involves a line search to determine the most appropriate value of the trade-off parameter and the complete solution, via the IRLS algorithm, of the linearized inverse problem for each value of the trade-off parameter. In the second approach, a schedule of prescribed values for the trade-off parameter is used and the iterations required by the IRLS algorithm are combined with those for the linearized, iterative inversion procedure. These two variations are then applied to the 1-D inversion of both synthetic and field time-domain electromagnetic data.     

Spectral analysis using orthonormal functions with a case study on the sea surface topography

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Spherical harmonics are orthonormalized using the Gram-Schmidt process in a function space. The problem of linear dependence of spherical harmonics over the oceans is studied using the Gram matrices and consequently three sets of orthonormal (ON) functions have been constructed. For the process an efficient formula for computing inner products of spherical harmonics has been developed. Important spectral properties of the ON functions are addressed. The ON functions may be used for representing the sea surface topography (SST) in the analysis of satellite altimeter data. The geoid error can be transformed to a representation by the ON functions and hence the comparison of powers of the geoid error and the SST signal only over the oceans is possible, leading to a better way of determining the cut-off frequency of the SST in the simultaneous solution using satellite altimeter data. As a case study, the modified Levitus SST is expanded into the ON functions. The results show that 99.90 per cent of that signal's energy is contained within degree 24 of the orthonormal functions. Such expansions also render better spectral behaviour of oceanic signals as compared to that from spherical harmonic expansions. The study shows that these generalized Fourier functions are suitable for spectral analyses of oceanic signals and they can be applied to future altimetric mission where the geoid and the SST are to be recovered.     

Induction in arbitrarily shaped oceans—III. Oceans of finite conductivity

Summary. Approximate methods of solution for induction in arbitrarily shaped oceans, derived earlier for 'oceans of infinite conductivity, are extended to cover cases where the oceans are of finite conductivity. The method enables the magnetic potential to be evaluated without undue effort. It is further shown how these ideas link up with integro-differential formulations involving either the oceanic electric current density or its associated current function. The problem of uniqueness of solution when there are two or more land masses is finally considered.     

Geomagnetic induction in a heterogeneous sphere: fully three-dimensional test computations and the response of a realistic distribution of oceans and continents

Long-period geomagnetic data can resolve large-scale 3-D mantle electrical conductivity heterogeneities which are indicators of physiochemical variations found in the Earth's dynamic mantle. A prerequisite for mapping such heterogeneity is the ability to model accurately electromagnetic induction in a heterogeneous sphere. A previously developed finite element method solution to the geomagnetic induction problem is validated against an analytic solution for a fully 3-D geometry: an off-axis spherical inclusion embedded in a uniform sphere. Geomagnetic induction is then modelled in a uniform spherical mantle overlain by a realistic distribution of oceanic and continental conductances. Our results indicate that the contrast in electrical conductivity between oceans and continents is not primarily responsible for the observed geographic variability of long-period geomagnetic data. In the absence of persistent high-wavenumber magnetospheric disturbances, this argues strongly for the existence of large-scale, high-contrast electrical conductivity heterogeneities in the mid-mantle. Lastly, for several periods the geomagnetic anomaly associated with a mid-mantle spherical inclusion is calculated. A high-contrast inclusion can be readily detected beneath the outer shell of oceans and continents. A comparison between observed and computed c responses suggests that the mid-mantle contains more than one order of magnitude of lateral variability in electrical conductivity, while the upper mantle contains at least two orders of magnitude of lateral variability in electrical conductivity.     

Meteorological excitation of the annual polar motion

Summary. Numerous studies have indicated that the annual term in the polar motion cannot be explained in any detail by meteorological/hydrological excitation and no reasonable alternative excitations have been put forward. Part of the problem has been that the hydrostatic adjustment of the oceans to the atmospheric pressure changes has traditionally been computed using the inverse barometer approach. This approach does not properly model the gravitational interaction between the atmosphere and oceans, and the inverse barometer theory is modified in this paper to account for this properly. The information necessary to compute the ocean tide and polar excitation caused by any change in the atmospheric pressure pattern is presented. The results of the application of this theory to two global atmospheric pressure data sets are examined and compared to results of other workers.
It is concluded that the atmosphere is observed well enough to answer the question of the annual excitation of polar motion and it is argued that the ground water excitation is the component with the largest error and remains the chief obstacle to the successful solution of this problem.     

Some new ideas on induction in infinitely- conducting oceans of arbitrary shapes

summary . The problem of an infinitely-conducting hemispherical'ocean'above a perfectly-conducting concentric sphere is solved by using Legendre polynomial expansions. Some of the techniques used in manipulating the expansions are believed to be novel. The form of the solution leads to a new general method for oceans of arbitrary shapes.     

Geoelectromagnetic induction in a heterogeneous sphere:a new three-dimensional forward solver using a conservative staggered-grid finite difference method

A conservative staggered-grid finite difference method is presented for computing the electromagnetic induction response of an arbitrary heterogeneous conducting sphere by external current excitation. This method is appropriate as the forward solution for the problem of determining the electrical conductivity of the Earth's deep interior. This solution in spherical geometry is derived from that originally presented by Mackie et al. (1994 ) for Cartesian geometry. The difference equations that we solve are second order in the magnetic field H , and are derived from the integral form of Maxwell's equations on a staggered grid in spherical coordinates. The resulting matrix system of equations is sparse, symmetric, real everywhere except along the diagonal and ill-conditioned. The system is solved using the minimum residual conjugate gradient method with preconditioning by incomplete Cholesky decomposition of the diagonal sub-blocks of the coefficient matrix. In order to ensure there is zero H divergence in the solution, corrections are made to the H field every few iterations. In order to validate the code, we compare our results against an integral equation solution for an azimuthally symmetric, buried thin spherical shell model ( Kuvshinov & Pankratov 1994 ), and against a quasi-analytic solution for an azimuthally asymmetric configuration of eccentrically nested spheres ( Martinec 1998 ).     

The Passive Influence of the Oceans upon the Rotation of the Earth

A general theory is developed which allows the exact numerical computation of the static equilibrium response of a non-rotating spherically symmetric Earth model covered by thin oceans with geometrically irregular coastlines to the action of an imposed static tidal or centrifugal potential. The theory is self-consistent, and takes into account the gravitational self-attraction of the oceans and the elastic-gravitational response of the Earth model to both the applied potential and the equilibrium oceanic tidal load on the surface. The results are used to determine the influence of an equilibrium pole tide on the free period and the associated rotational eigenfunction of the Chandler wobble. If the pole is globally well represented by this equilibrium approximation, its effect is to increase the Chandler wobble period by 27·6 days. It is shown that a fully self-consistent theory of the rotation of an Earth model with oceans predicts that changes in spin and wobble will be coupled, and that the Chandler wobble should, as a result, be accompanied by an associated periodic change in the length of day. The consequences of spin-wobble coupling are explored quantitatively, and found to be slight.     

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 center-based modeling approach to solve the districting problem

This article deals with the districting problem arising in applications such as political districting, police patrol area delineation and sales territory design. The aim of districting is to group basic areal units into geographic districts such that some set of criteria are satisfied, with basic criteria being district balance, compactness and contiguity. This article proposes a center-based mixed-integer linear programming model to solve the districting problem. Given the central units of districts, the model optimizes weighted objectives of district balance and compactness while satisfying contiguous constraints on districts. The performance of the model was tested using three study areas with 297, 324 and 1297 areal units, respectively. Experimentation shows that, using the district centers identified by a multistart weighted K-medoids algorithm, the model instances can be solved optimally or near-optimally. Compared with local search-based algorithms, the center-based approach outperforms metaheuristics such as simulated annealing, variable neighborhood descent, iterative local search and old bachelor acceptance search in terms of solution quality.     

地理学多视角研究方法——Braess网络车流分配过程的理论分析与数值计算

对复杂的地理系统采用多种方法从不同的视角开展分析,可以降低错误结论的概率。本文以Braess交通网络为例,提出一个地理系统多视角分析的研究案例。首先借助La氏乘数法预测奇对称Braess网络的车流优化分配的结果。然后采用数值计算和模拟方法论证,在该网络中,车流会通过自组织过程自动向着优化分配的方向演化,并且利用Markov链预测各个阶段的车流分配数值。最后借助最大熵原理从理论上证明,上述最优化过程的本质是地理系统的熵最大化;运用对偶规划和对称思想揭示,熵最大化的实质是车流运行的耗时总量最小。不同的方法给出的结果殊途同归、互相印证。这一套研究方法可以推广到多维不对称的交通网络,进而推广应用于地理学其他方面的理论分析和应用研究。     

Analysis of positional uncertainty of road networks in volunteered geographic information with a statistically defined buffer-zone method

Volunteered geographic information (VGI) is crowdsourced information that can enrich and enhance research and applications based on geo-referenced data. However, the quality of VGI is of great concern, and positional accuracy is a fundamental basis for the VGI quality assurance. A buffer-zone method can be used for its assessment, but the buffer radius in this technique is subjectively specified; as result, different selections of the buffer radius lead to different positional accuracies. To solve this problem, a statistically defined buffer zone for the positional accuracy assessment in VGI is proposed in this study. To facilitate practical applications, we have also developed an iterative method to obtain a theoretically defined buffer zone. In addition to the positional accuracy assessment, we have derived a measure of positional quality, which comprises the assessment of positional accuracy and the level of confidence in such assessment determined with respect to a statistically defined buffer zone. To illustrate and substantiate the theoretical arguments, both numerical simulations and real-life experiments are performed using OpenStreetMap. The experimental results confirm the high significance of the proposed statistical approach to the buffer zone-based assessment of the positional uncertainty in VGI.     

Inversion of time domain three-dimensional electromagnetic data

We present a general formulation for inverting time domain electromagnetic data to recover a 3-D distribution of electrical conductivity. The forward problem is solved using finite volume methods in the spatial domain and an implicit method (Backward Euler) in the time domain. A modified Gauss–Newton strategy is employed to solve the inverse problem. The modifications include the use of a quasi-Newton method to generate a pre-conditioner for the perturbed system, and implementing an iterative Tikhonov approach in the solution to the inverse problem. In addition, we show how the size of the inverse problem can be reduced through a corrective source procedure. The same procedure can correct for discretization errors that inevidably arise. We also show how the inverse problem can be efficiently carried out even when the decay time for the conductor is significantly larger than the repetition time of the transmitter wave form. This requires a second processor to carry an additional forward modelling. Our inversion algorithm is general and is applicable for any electromagnetic field  ( E , H , d B / dt )  measured in the air, on the ground, or in boreholes, and from an arbitrary grounded or ungrounded source. Three synthetic examples illustrate the basic functionality of the algorithm, and a result from a field example shows applicability in a larger-scale field example.     

Stable Iterative Methods for the Inversion of Geophysical Data

Summary. Interpretation of earth electrical measurements can often be assisted by inversion, which is a non-linear model-fitting problem in these cases. Iterative methods are normally used, and the solution is defined by' best fit'in the sense of generalized least-squares.
The inverse problems we describe are ill-posed. That is, small changes in the data can lead to large changes in both the solution and in the iterative process that finds the solution. Through an analysis of the problem, based on local linearization, we define a class of methods that stabilize the iteration, and provide a robust solution. These methods are seen as generalizations of the well-known Singular Value Truncation and Marquardt Methods of iterative inversion.
Here, and in a companion paper, we give examples illustrating the successful application of the method to ill-posed problems relating to the resistivity of the Earth.     

Ray asymptotic propagation of the spectra of elastic surface waves in a waveguide with laterally smooth inhomogeneities

Summary. A technique based on ray asymptotics has been developed to propagate complex spectra of elastic normal mode surface waves in a waveguide with material and geometrical properties varying smoothly in the lateral directions. In the technique, the original problem defined in the unstretched coordinates has been transformed into an eiconal equation as well as into a certain number of transport equations defined in stretched coordinates.
The solution of the eiconal equation is equal to the solution of the eigenproblem of the eiconal operator A₀. Due to the self-adjointness of A₀, in each of the relevant local inner product spaces, LIPS, the solution of the eigenproblem, A₀ψ= v ψ results in the set { v _t} of real local eigenvalues and in the orthonormal system {ψ_t} of local eigenvectors.
As the Hamiltonian function of an initial value problem, each eigenvalues gives birth to a bicharacteristic curve as well as to the related ray. The introduction of the rays induces connections between the vertical cross-sections of the waveguide.
Finally, for each asymptotic order j , the LIPS-valued transport equations are reduced to a set of matricial propagation equations in the local spectral amplitude vectors, LSAVs. Consequently, a knowledge of the initial conditions at a vertical cross-section makes it possible to propagate the LSAVs along the rays of the relevant modes. However, to complete the propagation one needs, in addition to the initial values, information about certain additional quantities, non-diagonal terms of order j , diagonal terms of orders lower than j and the auxiliary boundary terms of orders from 1 to j . The treatment has been completed by the propagation of the modal phases along the relevant rays.     

A parallel algorithm for coverage optimization on multi-core architectures

Location siting is an important part of service provision, with much potential to impact operational efficiency, safety, security, system reliability, etc. A class of location models seeks to optimize coverage of demand for service that is continuously distributed across space. Decision-making and planning contexts include police/fire resource allocation for a community, siting cellular towers to support cell phone signal transmission, locating emergency warning sirens to alert the public of severe weather and other related dangers, and many others as well. When facilities can be sited anywhere in continuous space to provide coverage to an entire region, this is a very computationally challenging problem to solve because potential demand for service is everywhere and there are an infinite number of potential facility sites to consider. This article develops a new parallel solution approach for this location coverage optimization problem through an iterative bounding scheme on multi-core architectures. The developed approach is applied to site emergency warning sirens in Dublin, Ohio, and fire stations in Elk Grove, California. Results demonstrate the effectiveness and efficiency of the proposed approach, enabling real-time analysis and planning. This work illustrates that the integration of cyberinfrastructure can significantly improve computational efficiency in solving challenging spatial optimization problems, fitting the themes of this special issue: cyberinfrastructure, GIS, and spatial optimization.     

Plane wave propagation in nonlinear-elastic anisotropic media

Summary. Kelvin-Christoffel equations describing plane wave propagation in anisotropic media are generalized to account for the effects of nonlinear elasticity. The polarization and waveform of nonlinear distortions of a transient plane wave are investigated by means of perturbation theory. Detailed analysis for an anisotropic medium with hexagonal symmetry shows that for "pure" shear-waves the polarization vector of the nonlinear component is always perpendicular to that of the linear wave. In the case of a high-amplitude excitation (for instance, in the vicinity of large earthquakes) the influence of nonlinearity may cause distortions of shear-wave polarization, which contains the most reliable information on the presence and characteristics of anisotropy. The solutions presented in this paper make it possible to solve reflection-transmission problems in nonlinear-elastic anisotropic media.