Deformation of the Earth by surface loads
TL;DR: In this article, the static deformation of an elastic half-space by surface pressure is reviewed and a brief mention is made of methods for solving the problem when the medium is plane-strategized, but the major emphasis is on the solution for spherical, radially stratified, gravitating earth models.
Abstract: The static deformation of an elastic half-space by surface pressure is reviewed. A brief mention is made of methods for solving the problem when the medium is plane stratified, but the major emphasis is on the solution for spherical, radially stratified, gravitating earth models. Love-number calculations are outlined, and from the Love numbers, Green's functions are formed for the surface mass-load boundary-value problem. Tables of mass-load Green's functions, computed for realistic earth models, are given, so that the displacements, tilts, accelerations, and strains at the earth's surface caused by any static load can be found by evaluating a convolution integral over the loaded region.
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TL;DR: In this paper, a relocatable system for generalized inverse (GI) modeling of barotropic ocean tides is described, where the GI penalty functional is minimized using a representer method, which requires repeated solution of the forward and adjoint linearized shallow water equations.
Abstract: A computationally efficient relocatable system for generalized inverse (GI) modeling of barotropic ocean tides is described. The GI penalty functional is minimized using a representer method, which requires repeated solution of the forward and adjoint linearized shallow water equations (SWEs). To make representer computations efficient, the SWEs are solved in the frequency domain by factoring the coefficient matrix for a finite-difference discretization of the second-order wave equation in elevation. Once this matrix is factored representers can be calculated rapidly. By retaining the first-order SWE system (defined in terms of both elevations and currents) in the definition of the discretized GI penalty functional, complete generality in the choice of dynamical error covariances is retained. This allows rational assumptions about errors in the SWE, with soft momentum balance constraints (e.g., to account for inaccurate parameterization of dissipation), but holds mass conservation constraints. Wh...
3,133 citations
Cites background or methods from "Deformation of the Earth by surface..."
...In fact zSAL should be computed by convolution of z with the Green’s function for loading and selfattraction (Farrell 1972; Ray 1998)....
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...Hatayama et al. (1996) review tidal studies, and present results from a new purely hydrodynamic model of the area....
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...Hatayama et al. (1996) review tidal studies, and present results from a new purely hydrodynamic model of the area. Tidal currents are large enough to be of great practical importance to navigation in many locations, and there is evidence that the tides may play a crucial role in interchange processes between the Pacific and Indian Oceans (Hatayama et al., 1996). Because the tides are greatly influenced by the complex topography of the area they are quite difficult to model accurately with a purely dynamical model (Mazzega and Berge, 1994). T/P altimeter data provide useful constraints on tidal elevations, at least in the larger seas within the Indonesian archipelago. Elevation maps for this area are included in most of the T/P based global tidal solutions discussed by Shum et al (1997). However, there are great differences between all of the various solutions, and as we shall show below for the TPXO....
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...In fact SAL ζ should be computed by convolution of ζ with the Green's function for loading and selfattraction (Farrell, 1972; Ray, 1998)....
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TL;DR: The impact of the changing surface ice load upon both Earth's shape and gravitational field, as well as upon sea-level history, have come to be measurable using a variety of geological and geophysical techniques.
Abstract: ▪ Abstract The 100 kyr quasiperiodic variation of continental ice cover, which has been a persistent feature of climate system evolution throughout the most recent 900 kyr of Earth history, has occurred as a consequence of changes in the seasonal insolation regime forced by the influence of gravitational n-body effects in the Solar System on the geometry of Earth's orbit around the Sun. The impacts of the changing surface ice load upon both Earth's shape and gravitational field, as well as upon sea-level history, have come to be measurable using a variety of geological and geophysical techniques. These observations are invertible to obtain useful information on both the internal viscoelastic structure of the solid Earth and on the detailed spatiotemporal characteristics of glaciation history. This review focuses upon the most recent advances that have been achieved in each of these areas, advances that have proven to be central to the construction of the refined model of the global process of glacial isos...
2,333 citations
01 Jan 2004
2,160 citations
TL;DR: In this article, the authors use output from hydrological, oceanographic, and atmospheric models to estimate the variability in the gravity field (i.e., in the geoid) due to those sources.
Abstract: The GRACE satellite mission, scheduled for launch in 2001, is designed to map out the Earth's gravity field to high accuracy every 2–4 weeks over a nominal lifetime of 5 years. Changes in the gravity field are caused by the redistribution of mass within the Earth and on or above its surface. GRACE will thus be able to constrain processes that involve mass redistribution. In this paper we use output from hydrological, oceanographic, and atmospheric models to estimate the variability in the gravity field (i.e., in the geoid) due to those sources. We develop a method for constructing surface mass estimates from the GRACE gravity coefficients. We show the results of simulations, where we use synthetic GRACE gravity data, constructed by combining estimated geophysical signals and simulated GRACE measurement errors, to attempt to recover hydrological and oceanographic signals. We show that GRACE may be able to recover changes in continental water storage and in seafloor pressure, at scales of a few hundred kilometers and larger and at timescales of a few weeks and longer, with accuracies approaching 2 mm in water thickness over land, and 0.1 mbar or better in seafloor pressure.
1,821 citations
TL;DR: A review of the state-of-the-art in the field of finite element solutions (FES) atlases can be found in this paper, where the authors introduce the FES2004 tidal atlas and validate the model against in situ and satellite data.
Abstract: During the 1990s, a large number of new tidal atlases were developed, primarily to provide accurate tidal corrections for satellite altimetry applications. During this decade, the French tidal group (FTG), led by C. Le Provost, produced a series of finite element solutions (FES) tidal atlases, among which FES2004 is the latest release, computed from the tidal hydrodynamic equations and data assimilation. The aim of this paper is to review the state of the art of tidal modelling and the progress achieved during this past decade. The first sections summarise the general FTG approach to modelling the global tides. In the following sections, we introduce the FES2004 tidal atlas and validate the model against in situ and satellite data. We demonstrate the higher accuracy of the FES2004 release compared to earlier FES tidal atlases, and we recommend its use in tidal applications. The final section focuses on the new dissipation term added to the equations, which aims to account for the conversion of barotropic energy into internal tidal energy. There is a huge improvement in the hydrodynamic tidal solution and energy budget obtained when this term is taken into account.
1,553 citations
Additional excerpts
...…displacement: Π λ;ϕð Þ ¼ 1þ k2ð ÞΠa λ;ϕð Þ þ Z Gπ λ;ϕ;λ 0 ;ϕ 0 α λ;ϕð Þds (5) Here, R Gδ λ;ϕ;λ 0 ;ϕ 0 α λ;ϕð Þds and R Gπ λ;ϕ;λ0; ϕ 0 Þα λ;ϕð Þds are the convolution integrals between a preexisting tidal solution and the appropriate Green’s functions (see Farrell 1972; Francis and Mazzega 1990)....
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References
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Book•
01 Jan 1944
TL;DR: The tabulation of Bessel functions can be found in this paper, where the authors present a comprehensive survey of the Bessel coefficients before and after 1826, as well as their extensions.
Abstract: 1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel functions 7. Asymptotic expansions of Bessel functions 8. Bessel functions of large order 9. Polynomials associated with Bessel functions 10. Functions associated with Bessel functions 11. Addition theorems 12. Definite integrals 13. Infinitive integrals 14. Multiple integrals 15. The zeros of Bessel functions 16. Neumann series and Lommel's functions of two variables 17. Kapteyn series 18. Series of Fourier-Bessel and Dini 19. Schlomlich series 20. The tabulation of Bessel functions Tables of Bessel functions Bibliography Indices.
9,584 citations
Book•
01 Jan 1971
TL;DR: In this paper, the authors describe how people search numerous times for their favorite books like this the finite element method in engineering science, but end up in malicious downloads, and instead they cope with some infectious bugs inside their computer.
Abstract: Thank you very much for downloading the finite element method in engineering science. Maybe you have knowledge that, people have search numerous times for their favorite books like this the finite element method in engineering science, but end up in malicious downloads. Rather than enjoying a good book with a cup of tea in the afternoon, instead they cope with some infectious bugs inside their computer.
3,688 citations
Book•
01 Jan 1955
TL;DR: The transformation of Laplace's equation in polar coordinates and the Legendres associated functions can be found in this article, where the authors also give approximate values of the generalized Legendres functions.
Abstract: Preface 1. The transformation of Laplaces's equation 2. The solution of Laplace's equation in polar coordinates 3. The Legendres associated functions 4. Spherical harmonics 5. Spherical harmonics of general type 6. Approximate values of the generalized Legendres functions 7. Representation of functions by series 8. The addition theorems for general Legendres functions 9. The zeros of Legendres functions and associated functions 10. Harmonics for spaces bounded by surfaces of revolution 11. Ellipsoidal harmonics List of authors quoted General index.
1,678 citations
TL;DR: In this article, it was shown that a given set G of measured gross Earth data permits such a construction of localized averages, and if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount.
Abstract: A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We suppose that a finite set G of gross Earth data has been measured, that the measurements are inaccurate, and that the variance matrix of the errors of measurement can be estimated. We show that some such sets G of measurements determine the structure of the Earth within certain limits of error except for fine-scale detail. That is, from some setsG it is possible to compute localized averages of the Earth structure at various depths. These localized averages will be slightly in error, and their errors will be larger as their resolving lengths are shortened. We show how to determine whether a given set G of measured gross Earth data permits such a construction of localized averages, and, if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount. We apply the general theory to the linear problem of finding the depth variation of a frequency-independent local elastic dissipation ( Q ) from the observed damping rates of a finite number of normal modes. We also apply the theory to the nonlinear problem of finding density against depth from the total mass, moment and normal-mode frequencies, in case the compressional and shear velocities are known.
1,291 citations