4,312 research outputs found
How Ordinary Elimination Became Gaussian Elimination
Newton, in notes that he would rather not have seen published, described a
process for solving simultaneous equations that later authors applied
specifically to linear equations. This method that Euler did not recommend,
that Legendre called "ordinary," and that Gauss called "common" - is now named
after Gauss: "Gaussian" elimination. Gauss's name became associated with
elimination through the adoption, by professional computers, of a specialized
notation that Gauss devised for his own least squares calculations. The
notation allowed elimination to be viewed as a sequence of arithmetic
operations that were repeatedly optimized for hand computing and eventually
were described by matrices.Comment: 56 pages, 21 figures, 1 tabl
Normal gravity field in relativistic geodesy
Modern geodesy is subject to a dramatic change from the Newtonian paradigm to
Einstein's theory of general relativity. This is motivated by the ongoing
advance in development of quantum sensors for applications in geodesy including
quantum gravimeters and gradientometers, atomic clocks and fiber optics for
making ultra-precise measurements of the geoid and multipolar structure of the
Earth's gravitational field. At the same time, VLBI, SLR, and GNSS have
achieved an unprecedented level of accuracy in measuring coordinates of the
reference points of the ITRF and the world height system. The main geodetic
reference standard is a normal gravity field represented in the Newtonian
gravity by the field of a Maclaurin ellipsoid. The present paper extends the
concept of the normal gravity field to the realm of general relativity. We
focus our attention on the calculation of the first post-Newtonian
approximation of the normal field that is sufficient for applications. We show
that in general relativity the level surface of the uniformly rotating fluid is
no longer described by the Maclaurin ellipsoid but is an axisymmetric spheroid
of the forth order. We parametrize the mass density distribution and derive the
post-Newtonian normal gravity field of the rotating spheroid which is given in
a closed form by a finite number of the ellipsoidal harmonics. We employ
transformation from the ellipsoidal to spherical coordinates to deduce the
post-Newtonian multipolar expansion of the metric tensor given in terms of
scalar and vector gravitational potentials of the rotating spheroid. We compare
these expansions with that of the normal gravity field generated by the Kerr
metric and demonstrate that the Kerr metric has a fairly limited application in
relativistic geodesy. Finally, we derive the post-Newtonian generalization of
the Somigliana formula for the gravity field on the reference ellipsoid.Comment: 39 pages, no figures, accepted to Physical Review
The Stokes problem for the ellipsoid using ellipsoidal kernels
A brief review of Stokes' problem for the ellipsoid as a reference surface is given. Another solution of the problem using an ellipsoidal kernel, which represents an iterative form of Stokes' integral, is suggested with a relative error of the order of the flattening. On studying of Rapp's method in detail the procedures of improving its convergence are discussed
Applications of satellite and marine geodesy to operations in the ocean environment
The requirements for marine and satellite geodesy technology are assessed with emphasis on the development of marine geodesy. Various programs and missions for identification of the satellite geodesy technology applicable to marine geodesy are analyzed along with national and international marine programs to identify the roles of satellite/marine geodesy techniques for meeting the objectives of the programs and other objectives of national interest effectively. The case for marine geodesy is developed based on the extraction of requirements documented by authoritative technical industrial people, professional geodesists, government agency personnel, and applicable technology reports
Algorithms for geodesics
Algorithms for the computation of geodesics on an ellipsoid of revolution are
given. These provide accurate, robust, and fast solutions to the direct and
inverse geodesic problems and they allow differential and integral properties
of geodesics to be computed.Comment: LaTex, 12 pages, 8 figures. Version 2 corrects some errors and adds
numerical examples. Supplementary material is available at
http://geographiclib.sourceforge.net/geod.htm
LIBRA: An inexpensive geodetic network densification system
A description is given of the Libra (Locations Interposed by Ranging Aircraft) system, by which geodesy and earth strain measurements can be performed rapidly and inexpensively to several hundred auxiliary points with respect to a few fundamental control points established by any other technique, such as radio interferometry or satellite ranging. This low-cost means of extending the accuracy of space age geodesy to local surveys provides speed and spatial resolution useful, for example, for earthquake hazards estimation. Libra may be combined with an existing system, Aries (Astronomical Radio Interferometric Earth Surveying) to provide a balanced system adequate to meet the geophysical needs, and applicable to conventional surveying. The basic hardware design was outlined and specifications were defined. Then need for network densification was described. The following activities required to implement the proposed Libra system are also described: hardware development, data reduction, tropospheric calibrations, schedule of development and estimated costs
Computational science and re-discovery: open-source implementations of ellipsoidal harmonics for problems in potential theory
We present two open-source (BSD) implementations of ellipsoidal harmonic
expansions for solving problems of potential theory using separation of
variables. Ellipsoidal harmonics are used surprisingly infrequently,
considering their substantial value for problems ranging in scale from
molecules to the entire solar system. In this article, we suggest two possible
reasons for the paucity relative to spherical harmonics. The first is
essentially historical---ellipsoidal harmonics developed during the late 19th
century and early 20th, when it was found that only the lowest-order harmonics
are expressible in closed form. Each higher-order term requires the solution of
an eigenvalue problem, and tedious manual computation seems to have discouraged
applications and theoretical studies. The second explanation is practical: even
with modern computers and accurate eigenvalue algorithms, expansions in
ellipsoidal harmonics are significantly more challenging to compute than those
in Cartesian or spherical coordinates. The present implementations reduce the
"barrier to entry" by providing an easy and free way for the community to begin
using ellipsoidal harmonics in actual research. We demonstrate our
implementation using the specific and physiologically crucial problem of how
charged proteins interact with their environment, and ask: what other
analytical tools await re-discovery in an era of inexpensive computation?Comment: 25 pages, 3 figure
Interaction of marine geodesy, satellite technology and ocean physics
The possible applications of satellite technology in marine geodesy and geodetic related ocean physics were investigated. Four major problems were identified in the areas of geodesy and ocean physics: (1) geodetic positioning and control establishment; (2) sea surface topography and geoid determination; (3) geodetic applications to ocean physics; and (4) ground truth establishment. It was found that satellite technology can play a major role in their solution. For solution of the first problem, the use of satellite geodetic techniques, such as Doppler and C-band radar ranging, is demonstrated to fix the three-dimensional coordinates of marine geodetic control if multi-satellite passes are used. The second problem is shown to require the use of satellite altimetry, along with accurate knowledge of ocean-dynamics parameters such as sea state, ocean tides, and mean sea level. The use of both conventional and advanced satellite techniques appeared to be necessary to solve the third and fourth problems
A General Approach to Regularizing Inverse Problems with Regional Data using Slepian Wavelets
Slepian functions are orthogonal function systems that live on subdomains
(for example, geographical regions on the Earth's surface, or bandlimited
portions of the entire spectrum). They have been firmly established as a useful
tool for the synthesis and analysis of localized (concentrated or confined)
signals, and for the modeling and inversion of noise-contaminated data that are
only regionally available or only of regional interest. In this paper, we
consider a general abstract setup for inverse problems represented by a linear
and compact operator between Hilbert spaces with a known singular-value
decomposition (svd). In practice, such an svd is often only given for the case
of a global expansion of the data (e.g. on the whole sphere) but not for
regional data distributions. We show that, in either case, Slepian functions
(associated to an arbitrarily prescribed region and the given compact operator)
can be determined and applied to construct a regularization for the ill-posed
regional inverse problem. Moreover, we describe an algorithm for constructing
the Slepian basis via an algebraic eigenvalue problem. The obtained Slepian
functions can be used to derive an svd for the combination of the regionalizing
projection and the compact operator. As a result, standard regularization
techniques relying on a known svd become applicable also to those inverse
problems where the data are regionally given only. In particular, wavelet-based
multiscale techniques can be used. An example for the latter case is elaborated
theoretically and tested on two synthetic numerical examples
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