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NASA NTRS1982

The inverse problem of constructing a gravimetric geoid

Computation of a single geoidal height from gravity acceleration data formally requires that the latter be known everywhere on the earth. A computational procedure based on linear inverse theory for estimating geoidal heights from incomplete sets of data is presented. The same scheme can be used to estimate gravity accelerations from altimetry-derived geoids. The systematic error owing to lack of data and the choice of a particular inverse operator is described by using resolution functions and their spherical harmonic expansions. An rms value of this error is also estimated by assuming a spectrum for the unknown geoid. The influence of the size of the data region, the spacing between data, the filtering applied to the data, and the model weighting function chosen are all quantified in a spherical geometry. The examples presented show that when low degree spherical harmonic coefficients are available - from satellite orbit analysis - a band-passed version of the geoid can be constructed from local gravity data, even with a relatively restricted data set.

Zlotnicki, V.↗

NASA NTRS1982

Isostatic geoid anomalies on a sphere

Long-wavelength geoid anomalies due to lateral variations in the density or thickness of a thin layer in local isostatic equilibrium on the surface of a sphere depend sensitively on the assumed state of stress within the layer. A number of common intuitive definitions of local isostasy generally believed to be essentially identical are associated with quite different states of isostatic stress, and the corresponding theoretical geoid anomalies can vary by more than a factor of 2. This sensitivity of the theoretical anomaly to the exact definition of local isostasy constitutes an obstacle to any proposed program of interpreting or exploiting observed global isostatic geoid anomalies such as those associated with the aging of the oceanic lithosphere.

Dahlen, F. A.↗

NASA NTRS1983

Subducted slabs and the geoid: Constraints on mantle rheology and flow

The total geoid anomaly which is the result of a given density contrast in a convecting viscous earth is affected by the mass anomalies associated with the flow induced deformation of the upper surface and internal compositional boundaries, as well as by the density contrast itself is discussed. If the internal density contrasts can be estimated, the depth and variation of viscosity with depth of the convecting system can be constrained. The observed long wavelength geoid is highly correlated with that predicted by a density model for seismically active subducted slabs. The amplitude of the correlation is explained if the density contrasts associated with subduction extend into the lower mantle or if subducted slabs exceeding 350 km in thickness are piled up over horizontal distances of thousands of km at the base of the upper mantle. Mantle wide convection in a mantle that has a viscosity increasing with depth provides the explanation of the long-wavelength geoid anomalies over subduction zones.

Hager, B. H.↗

NASA NTRS1984

MAGSAT correlations with geoid anomalies

The MAGSAT data of the Gulf of Mexico were analyzed to define better the possible relation of the negative MAGSAT anomaly there to the negative residual geoid anomaly in the western Gulf of Mexico. The estimated magnetic crystal anomaly pattern has a magnetic low in the region of the residual geoid low, but the shape of the anomalies are different. Since the shape and location of the negative magnetic anomaly are variable depending upon the particular polynomial and curve orders used, the degree of correspondance between the residual geoid and MAGSAT lithosphere anomalies was not established definitively.

Bowin, C. O.↗

NASA NTRS1984

Subducted slabs and the geoid - Constraints on mantle rheology and flow

Hager, B. H.↗

NASA NTRS1985

Does the geoid drift west?

In 1970 Hide and Malin noted a correlation of about 0.8 between the geoid and the geomagnetic potential at the Earth's surface when the latter is rotated eastward in longitude by about 160 degrees and the spherical harmonic expansions of both functions are truncated at degree 4. From a century of magnetic observatory data, Hide and Malin inferred an average magnetic westward drift rate of about 0.27 degrees/year. They attributed the magnetic-gravitational correlation to a core event at about 1350 A.D. which impressed the mantle's gravity pattern at long wavelengths onto the core motion and the resulting magnetic field. The impressed pattern was then carried westward 160 degrees by the nsuing magnetic westward drift. An alternative possibility is some sort of steady physical coupling between the magnetic and gravitational fields (perhaps migration of Hide's bumps on the core-mantle interface). This model predicts that the geoid will drift west at the magnetic rate. On a rigid earth, the resulting changes in sea level would be easily observed, but they could be masked by adjustment of the mantle if it has a shell with viscosity considerably less than 10 to the 21 poise. However, steady westward drift of the geoid also predicts secular changes in g, the local acceleration of gravity, at land stations. These changes are now ruled out by recent independent high-accuracy absolute measurements of g made by several workers at various locations in the Northern Hemisphere.

Backus, G. E.↗

NASA NTRS1985

Surface deformation and geoid anomalies over single and double-layered convective systems

Using a primitive variable formulation of the finite-element method, the differences in the surface observables, such as topography and geoid, produced by single- and double-layered thermal convection, were compared. Both constant and depth-dependent viscosities have been considered. For the same Rayleigh number, larger surface perturbations are produced by single-cell convection. For the same Nusselt number, the magnitudes of the surface observables are greater for double-layered convection. For the same surface heat-flux, surface topographies have similar magnitudes, but the relative amplitudes of geoid anomalies depend greatly on the style of viscosity stratification. This difference in the geoid between the two systems increases with greater surface heat-flow, regardless of viscosity structure.

Koch, M.↗

NASA NTRS1987

Surface deformation, gravity and the geoid from a three-dimensional convection model at low Rayleigh numbers

The two-dimensional surface deformation, gravity field, and geoid are calculated from the temperature fields of a number of numerical models of constant-viscosity three-dimensional convective flows, heated from within and from below. The surface deformation and the geoid are insensitive to the short-wavelength features of the temperature variation; the gravity field is less smooth, although it still does not clearly indicate the narrowness of the upwelling and downwelling regions at large Rayleigh numbers. The results indicate that the convective upwelling beneath regions showing positive geoid and residual depth anomalies is more localized than the horizontal extent of these features would suggest. The results are relevant to attempts to understand the behavior of the earth mantle.

Craig, Claire Harvey↗

NASA NTRS1988

The earth's geoid and the large-scale structure of mantle convection

It is shown that most of the earth's low-degree geoid power is derived from density heterogeneity in the lower mantle. Much of the remaining geoid power is due to high-density slabs in active subduction zones. The compensated topography and lithospheric or crustal thickness variations contribute significantly to the observed geoid only for harmonic degrees greater than or equal to 6.

Richards, Mark A.↗

NASA NTRS1989

Geoid height versus topography for oceanic plateaus and swells

Gridded geoid height data (Marsh et al.l, 1986) and gridded bathymetry data (Van Wykhouse, 1973) are used to estimate the average compensation depths of 53 oceanic swells and plateaus. The relationship between geoid height and topography is examined using Airy and thermal compensation models. It is shown that geoid height is linearly related to topography between wavelengths of 400 and 4000 m as predicted by isostatic compensation models. The geoid/topography ratio is dependent on the average depth of compensation. The intermediate geoid/topography ratios of most thermal swells are interpreted as a linear combination of the decaying thermal swell signature and that of the persisting Airy-compensated volcanic edifice.

Sandwell, David T.↗

NASA NTRS1989

Effects of lateral viscosity variations on long-wavelength geoid anomalies and topography

The effects of lateral variations in the earth mantle viscosity, due to temperature- or stress-dependent rheology, on the long-wavelength geoid anomalies are examined. Results from simple perturbation theory combined with findings from numerical models for convective flow led to a conclusion that the geoid due to the very longest wavelength convective patterns (l = 2,3) on earth is probably not seriously contaminated by lateral variations due either to temperature or stress dependence. Considerable contamination of the higher-degree geoid (l value of no less than 4) is to be expected due to lateral viscosity variations in phase with the fundamental convection scale length.

Richards, Mark A.↗

NASA NTRS1990

A preliminary estimate of geoid-induced variations in repeat orbit satellite altimeter observations

Altimeter satellites are often maintained in a repeating orbit to facilitate the separation of sea-height variations from the geoid. However, atmospheric drag and solar radiation pressure cause a satellite orbit to drift. For Geosat this drift causes the ground track to vary by + or - 1 km about the nominal repeat path. This misalignment leads to an error in the estimates of sea surface height variations because of the local slope in the geoid. This error has been estimated globally for the Geosat Exact Repeat Mission using a mean sea surface constructed from Geos 3 and Seasat altimeter data. Over most of the ocean the geoid gradient is small, and the repeat-track misalignment leads to errors of only 1 to 2 cm. However, in the vicinity of trenches, continental shelves, islands, and seamounts, errors can exceed 20 cm. The estimated error is compared with direct estimates from Geosat altimetry, and a strong correlation is found in the vicinity of the Tonga and Aleutian trenches. This correlation increases as the orbit error is reduced because of the increased signal-to-noise ratio.

Brenner, Anita C.↗

NASA NTRS1991

Mantle viscosity - A comparison of models from postglacial rebound and from the geoid, plate driving forces, and advected heat flux

Models of the radial variation of effective viscosity inferred from the earth's response to surface loads associated with Pleistocene deglaciation are compared to structures inferred from models of geodynamic phenomena associated with convection: the geoid, plate-driving forces, and advected heat flux. While observations of the earth's response to surface loads do not have sufficient resolution to justify more than two viscous layers, adequately matching the observed long-wavelength geoid anomalies associated with density contrasts in the lower mantle (inferred from seismic tomography) and in the upper mantle (inferred from a model of subducted slabs) requires more structure. It is possible to explain the geoid, observed plate velocities, the advected heat flux in the lower mantle, and relative sea-level variations in oceanic regions, all with a mantle with a high-viscosity/elastic lid, an asthenospheric channel of 2 x 10 exp 19 Pa s from 100 to 400-km depth, a 6 x 10 exp 20 Pa s transition zone, and a lower mantle of 6 x 10 exp 21 Pa s. The uplift history of Australia, Fennoscandia, and Laurentia can be explained with an asthenospheric viscosity less than a factor of 10 higher. Lateral variations in lower mantle viscosity are not required. Transient creep appears to be unimportant for the recent response-to-surface loads from Pleistocene deglaciation.

Hager, Bradford H.↗

NASA NTRS1992

Geoid anomalies and dynamic topography from convection in cylindrical geometry - Applications to mantle plumes on earth and Venus

A variety of evidence suggests that at least some hotspots are formed by quasi-cylindrical mantle plumes upwelling from deep in the mantle. Such plumes are modeled in cylindrical, axisymmetric geometry with depth-dependent, Newtonian viscosity. Cylindrical and sheet-like, Cartesian upwellings have significantly different geoid and topography signatures. However, Rayleigh number-Nusselt number systematics in the two geometries are quite similar. The geoid anomaly and topographic uplift over a plume are insensitive to the viscosity of the surface layer, provided that it is at least 1000 times the interior viscosity. Increasing the Rayleigh number or including a low-viscosity asthenosphere decreases the geoid anomaly and the topographic uplift associated with an upwelling plume.

Kiefer, Walter S.↗

NASA NTRS1991

Venusian highlands - Geoid to topography ratios and their implications

Geoid-to-topography ratios (GTRs) are estimated for 12 Venusian highland features to allow comparison with convection calculations and with terrestrial data of oceanic hot spots, swells, and plateaus. The geoid is estimated in the wavenumber domain from the isostatic formula, using the topography and the apparent depths isostatic compensation (ADC) for each region. In the space domain, the GTR is equal to the least squares slope of the linear fit of the geoid to the topograpy. The resulting GTR range is 7-31 m/km, which is much higher than terrestrial oceanic values (-1 to 5 m/km). The features fall into two distinct groups, one with a GTR range of 7-13 m/km, and one with a range of 19-25 m/km. A model for thermal thinning of a 100-km-thick lithosphere fits all values in the lower GTR group to within one standard deviation.

Smrekar, Suzanne E.↗

NASA NTRS1993

Spectral study of Venus global topography and geoid from Magellan and PVO data

An analysis of newly available global harmonic models for topography and geoid on Venus was conducted. It was found that the power spectral density for Venus topography has a power law dependence on wave-number characteristic of Brown Noise, similar to what is found for the Earth. However, the Venus topography spectrum presents a rollover at lower degree (l = 3) than is observed for the Earth spectrum and has smaller amplitudes than that of the Earth's. The Venus geoid also obeys a power law relationship, at least for small values of l, but with a smaller slope and more power (for l greater than 3) than the Earth geoid.

Kucinskas, A. B.↗

NASA NTRS2000

Geoid Recovery Using Geophysical Inverse Theory Applied to Satellite to Satellite Tracking Data

This report describes a new method for determination of the geopotential, or the equivalent geoid. It is based on Satellite-to-Satellite Tracking (SST) of two co-orbiting low earth satellites separated by a few hundred kilometers. The analysis is aimed at the GRACE Mission, though it is generally applicable to any SST data. It is proposed that the SST be viewed as a mapping mission. That is, the result will be maps of the geoid or gravity, as contrasted with determination of spherical harmonics or Fourier coefficients. A method has been developed, based on Geophysical Inverse Theory (GIT), that can provide maps at a prescribed (desired) resolution and the corresponding error map from the SST data. This computation can be done area by area avoiding simultaneous recovery of all the geopotential information. The necessary elements of potential theory, celestial mechanics, and Geophysical Inverse Theory are described, a computation architecture is described, and the results of several simulations presented. Centimeter accuracy geoids with 50 to 100 km resolution can be recovered with a 30 to 60 day mission.

Gaposchkin, E. M.↗

NASA NTRS1972

A simplified model for the gravitational potential of the atmosphere and its effect on the geoid

The earth's atmosphere is considered as made up of oblate spheroidal layers of variable density lying over an oblate spheroidal earth. The gravitational attraction of the atmosphere at exterior points is computed and its contribution to the usual spherical harmonic gravitational expansion is assessed. The potential is also found for points at the bottom of the model atmosphere. This latter result is of interest for determination of the potential at the surface of the geoid. The atmospheric correction to the geoid determination from satellite coefficients is given.

Madden, S. J., Jr.↗