Gravity and Topography of the Terrestrial Planets
01 Jan 2015-Vol. 10, pp 153-193
TL;DR: A general review of the mathematical formalism that is used in describing gravity and topography of the terrestrial planets is given in this article, where the basic properties of Earth, Venus, Mars, Mercury, and the Moon are characterized.
Abstract: This chapter reviews our current knowledge of the gravity and topography of the terrestrial planets and describes the methods that are used to analyze these data. A general review of the mathematical formalism that is used in describing gravity and topography is first given. Next, the basic properties of Earth, Venus, Mars, Mercury, and the Moon are characterized. Following this, the relationship between gravity and topography is quantified, and techniques by which geophysical parameters can be constrained are detailed. Analysis methods include crustal thickness modeling, geoid/topography ratios, spectral admittance and correlation functions, and localized spectral analysis and wavelet techniques. Finally, the major results that have been obtained by modeling the gravity and topography of Earth, Venus, Mars, Mercury, and the Moon are summarized.
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01 Dec 2012
TL;DR: The Moon's gravity field shows that the lunar crust is less dense and more porous than was thought, and high-resolution gravity data obtained from the dual Gravity Recovery and Interior Laboratory (GRAIL) spacecraft show that the bulk density of the Moon's highlands crust is substantially lower than generally assumed.
Abstract: The Holy GRAIL? The gravity field of a planet provides a view of its interior and thermal history by revealing areas of different density. GRAIL, a pair of satellites that act as a highly sensitive gravimeter, began mapping the Moon's gravity in early 2012. Three papers highlight some of the results from the primary mission. Zuber et al. (p. 668, published online 6 December) discuss the overall gravity field, which reveals several new tectonic and geologic features of the Moon. Impacts have worked to homogenize the density structure of the Moon's upper crust while fracturing it extensively. Wieczorek et al. (p. 671, published online 6 December) show that the upper crust is 35 to 40 kilometers thick and less dense—and thus more porous—than previously thought. Finally, Andrews-Hanna et al. (p. 675, published online 6 December) show that the crust is cut by widespread magmatic dikes that may reflect a period of expansion early in the Moon's history. The Moon's gravity field shows that the lunar crust is less dense and more porous than was thought. High-resolution gravity data obtained from the dual Gravity Recovery and Interior Laboratory (GRAIL) spacecraft show that the bulk density of the Moon's highlands crust is 2550 kilograms per cubic meter, substantially lower than generally assumed. When combined with remote sensing and sample data, this density implies an average crustal porosity of 12% to depths of at least a few kilometers. Lateral variations in crustal porosity correlate with the largest impact basins, whereas lateral variations in crustal density correlate with crustal composition. The low-bulk crustal density allows construction of a global crustal thickness model that satisfies the Apollo seismic constraints, and with an average crustal thickness between 34 and 43 kilometers, the bulk refractory element composition of the Moon is not required to be enriched with respect to that of Earth.
470 citations
TL;DR: The Lunar Orbiter Laser Altimeter (LOLA) as discussed by the authors is an instrument on the payload of NASA's Lunar Reconnaissance Orbiter spacecraft (LRO) which is designed to measure the shape of the Moon by measuring precisely the range from the spacecraft to the lunar surface.
Abstract: The Lunar Orbiter Laser Altimeter (LOLA) is an instrument on the payload of NASA's Lunar Reconnaissance Orbiter spacecraft (LRO) (Chin et al., in Space Sci. Rev. 129:391-419, 2007). The instrument is designed to measure the shape of the Moon by mea- suring precisely the range from the spacecraft to the lunar surface, and incorporating preci- sion orbit determination of LRO, referencing surface ranges to the Moon's center of mass. LOLA has 5 beams and operates at 28 Hz, with a nominal accuracy of 10 cm. Its primary objective is to produce a global geodetic grid for the Moon to which all other observations can be precisely referenced.
431 citations
Massachusetts Institute of Technology1, Jet Propulsion Laboratory2, Goddard Space Flight Center3, Purdue University4, Southwest Research Institute5, Carnegie Institution for Science6, Lamont–Doherty Earth Observatory7, Institut de Physique du Globe de Paris8, University of Maryland, Baltimore County9
TL;DR: The Moon's gravity field reveals that impacts have homogenized the density of the crust and fractured it extensively, and GRAIL elucidates the role of impact bombardment in homogenizing the distribution of shallow density anomalies on terrestrial planetary bodies.
Abstract: Spacecraft-to-spacecraft tracking observations from the Gravity Recovery and Interior Laboratory (GRAIL) have been used to construct a gravitational field of the Moon to spherical harmonic degree and order 420. The GRAIL field reveals features not previously resolved, including tectonic structures, volcanic landforms, basin rings, crater central peaks, and numerous simple craters. From degrees 80 through 300, over 98% of the gravitational signature is associated with topography, a result that reflects the preservation of crater relief in highly fractured crust. The remaining 2% represents fine details of subsurface structure not previously resolved. GRAIL elucidates the role of impact bombardment in homogenizing the distribution of shallow density anomalies on terrestrial planetary bodies.
404 citations
Cites background from "Gravity and Topography of the Terre..."
...(40), RM is the lunar reference radius of 1737....
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...Following the notation of (40) the power spectrum of the function f is...
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TL;DR: In this paper, an improved digital elevation model (DEM) covering latitudes within ±60°, at a horizontal resolution of 512 pixels per degree (∼60 m at the equator) and a typical vertical accuracy ∼3 to 4 m.
316 citations
TL;DR: In this paper, the Earth's topography harmonic expansion was used to obtain a spherical harmonic model of the topography of the Earth, which was then used for the estimation of induced gravity perturbations.
Abstract: The availability of high-resolution global digital elevation data sets has raised a growing interest in the feasibility of obtaining their spherical harmonic representation at matching resolution, and from there in the modelling of induced gravity perturbations. We have therefore estimated spherical Bouguer and Airy isostatic anomalies whose spherical harmonic models are derived from the Earth’s topography harmonic expansion. These spherical anomalies differ from the classical planar ones and may be used in the context of new applications. We succeeded in meeting a number of challenges to build spherical harmonic models with no theoretical limitation on the resolution. A specific algorithm was developed to enable the computation of associated Legendre functions to any degree and order. It was successfully tested up to degree 32,400. All analyses and syntheses were performed, in 64 bits arithmetic and with semi-empirical control of the significant terms to prevent from calculus underflows and overflows, according to IEEE limitations, also in preserving the speed of a specific regular grid processing scheme. Finally, the continuation from the reference ellipsoid’s surface to the Earth’s surface was performed by high-order Taylor expansion with all grids of required partial derivatives being computed in parallel. The main application was the production of a 1′ × 1′ equiangular global Bouguer anomaly grid which was computed by spherical harmonic analysis of the Earth’s topography–bathymetry ETOPO1 data set up to degree and order 10,800, taking into account the precise boundaries and densities of major lakes and inner seas, with their own altitude, polar caps with bedrock information, and land areas below sea level. The harmonic coefficients for each entity were derived by analyzing the corresponding ETOPO1 part, and free surface data when required, at one arc minute resolution. The following approximations were made: the land, ocean and ice cap gravity spherical harmonic coefficients were computed up to the third degree of the altitude, and the harmonics of the other, smaller parts up to the second degree. Their sum constitutes what we call ETOPG1, the Earth’s TOPography derived Gravity model at 1′ resolution (half-wavelength). The EGM2008 gravity field model and ETOPG1 were then used to rigorously compute 1′ × 1′ point values of surface gravity anomalies and disturbances, respectively, worldwide, at the real Earth’s surface, i.e. at the lower limit of the atmosphere. The disturbance grid is the most interesting product of this study and can be used in various contexts. The surface gravity anomaly grid is an accurate product associated with EGM2008 and ETOPO1, but its gravity information contents are those of EGM2008. Our method was validated by comparison with a direct numerical integration approach applied to a test area in Morocco–South of Spain (Kuhn, private communication 2011) and the agreement was satisfactory. Finally isostatic corrections according to the Airy model, but in spherical geometry, with harmonic coefficients derived from the sets of the ETOPO1 different parts, were computed with a uniform depth of compensation of 30 km. The new world Bouguer and isostatic gravity maps and grids here produced will be made available through the Commission for the Geological Map of the World. Since gravity values are those of the EGM2008 model, geophysical interpretation from these products should not be done for spatial scales below 5 arc minutes (half-wavelength).
284 citations
References
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Book•
31 Jan 1986TL;DR: Numerical Recipes: The Art of Scientific Computing as discussed by the authors is a complete text and reference book on scientific computing with over 100 new routines (now well over 300 in all), plus upgraded versions of many of the original routines, with many new topics presented at the same accessible level.
Abstract: From the Publisher:
This is the revised and greatly expanded Second Edition of the hugely popular Numerical Recipes: The Art of Scientific Computing. The product of a unique collaboration among four leading scientists in academic research and industry, Numerical Recipes is a complete text and reference book on scientific computing. In a self-contained manner it proceeds from mathematical and theoretical considerations to actual practical computer routines. With over 100 new routines (now well over 300 in all), plus upgraded versions of many of the original routines, this book is more than ever the most practical, comprehensive handbook of scientific computing available today. The book retains the informal, easy-to-read style that made the first edition so popular, with many new topics presented at the same accessible level. In addition, some sections of more advanced material have been introduced, set off in small type from the main body of the text. Numerical Recipes is an ideal textbook for scientists and engineers and an indispensable reference for anyone who works in scientific computing. Highlights of the new material include a new chapter on integral equations and inverse methods; multigrid methods for solving partial differential equations; improved random number routines; wavelet transforms; the statistical bootstrap method; a new chapter on "less-numerical" algorithms including compression coding and arbitrary precision arithmetic; band diagonal linear systems; linear algebra on sparse matrices; Cholesky and QR decomposition; calculation of numerical derivatives; Pade approximants, and rational Chebyshev approximation; new special functions; Monte Carlo integration in high-dimensional spaces; globally convergent methods for sets of nonlinear equations; an expanded chapter on fast Fourier methods; spectral analysis on unevenly sampled data; Savitzky-Golay smoothing filters; and two-dimensional Kolmogorov-Smirnoff tests. All this is in addition to material on such basic top
12,662 citations
TL;DR: In this paper, a digital bathymetric map of the oceans with a horizontal resolution of 1 to 12 kilometers was derived by combining available depth soundings with high-resolution marine gravity information from the Geosat and ERS-1 spacecraft.
Abstract: A digital bathymetric map of the oceans with a horizontal resolution of 1 to 12 kilometers was derived by combining available depth soundings with high-resolution marine gravity information from the Geosat and ERS-1 spacecraft. Previous global bathymetric maps lacked features such as the 1600-kilometer-long Foundation Seamounts chain in the South Pacific. This map shows relations among the distributions of depth, sea floor area, and sea floor age that do not fit the predictions of deterministic models of subsidence due to lithosphere cooling but may be explained by a stochastic model in which randomly distributed reheating events warm the lithosphere and raise the ocean floor.
4,433 citations
TL;DR: The Generic Mapping Tools (GMT) is introduced, which is a free, public domain software package that can be used to manipulate columns of tabular data, time series, and gridded data sets and to display these data in a variety of forms ranging from simple x-y plots to maps and color, perspective, and shaded-relief illustrations.
Abstract: When creating camera-ready figures, most scientists are familiar with the sequence of raw data → processing → final illustration and with the spending of large sums of money to finalize papers for submission to scientific journals, prepare proposals, and create overheads and slides for various presentations. This process can be tedious and is often done manually, since available commercial or in-house software usually can do only part of the job.
To expedite this process, we introduce the Generic Mapping Tools (GMT), which is a free, public domain software package that can be used to manipulate columns of tabular data, time series, and gridded data sets and to display these data in a variety of forms ranging from simple x-y plots to maps and color, perspective, and shaded-relief illustrations. GMT uses the PostScript page description language, which can create arbitrarily complex images in gray tones or 24-bit true color by superimposing multiple plot files. Line drawings, bitmapped images, and text can be easily combined in one illustration. PostScript plot files are device-independent, meaning the same file can be printed at 300 dots per inch (dpi) on an ordinary laserwriter or at 2470 dpi on a phototypesetter when ultimate quality is needed. GMT software is written as a set of UNIX tools and is totally self contained and fully documented. The system is offered free of charge to federal agencies and nonprofit educational organizations worldwide and is distributed over the computer network Internet.
4,128 citations
"Gravity and Topography of the Terre..." refers methods in this paper
...Most figures were created using the Generic Mapping Tools of Wessel and Smith (1991)....
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...Missing nodes were filled by data obtained by the Pioneer Venus and Venera 15/16 missions; the remaining gaps were filled by interpolation using the Generic Mapping Tools (GMT) (Wessel and Smith, 1991) command <pr>surface</pr> with a tension parameter of 0.35, and the spherical harmonic expansion was performed using the sampling theorem of Driscoll and Healy (1994)....
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...Missing nodes were filled by data obtained by the Pioneer Venus and Venera 15/16 missions, the remaining gaps were filled by interpolation using the GMT (Wessel and Smith 1991) command “surface” with a tension parameter of 0....
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...…data obtained by the Pioneer Venus and Venera 15/16 missions; the remaining gaps were filled by interpolation using the Generic Mapping Tools (GMT) (Wessel and Smith, 1991) command <pr>surface</pr> with a tension parameter of 0.35, and the spherical harmonic expansion was performed using the…...
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Book•
01 Oct 1988
TL;DR: In this article, the authors present a collection of useful formulas besides those related to angular momentum, and compare different notations used by previous authors, and present results relating to different aspects of the angular momentum theory.
Abstract: Containing basic definitions and theorems as well as relations, tables of formulas and numerical tables which are essential for applications to many physical problems, the book is useful for specialists in nuclear and particle physics, atomic and molecular spectroscopy, plasma physics, collision and reaction theory, quantum chemistry, etc. The authors write many formulas in different coordinate systems. Each chapter opens with a list of its contents. New results relating to different aspects of the angular momentum theory are included. This book gathers together many useful formulas besides those related to angular momentum, and compares different notations used by previous authors.
3,987 citations
"Gravity and Topography of the Terre..." refers background or result in this paper
...…window bandlimited to degree L, F, and G are the windowed fields hf and hg, respectively, and the last symbol in parentheses is a Clebsch–Gordan coefficient (these are related to the Wigner 3 j symbols and are proportional to the integral of three Legendre functions; see Varshalovich et al., 1988)....
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...In contrast to the ‘geodesy’ or ‘4p’ normalization of eqn [7] (cf. Kaula, 2000), the seismology (e.g., Dahlen and Tromp, 1998) and physics (e.g., Varshalovich et al., 1988) communities often use orthonormal harmonics, whose squared integral is equal to unity....
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