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Altimeter data from a spacecraft borne altimeter was processed to estimate the fine structure of the marine geoid. Simulation studies show that, among several competing parameterizations, the mean free air gravity anomaly model exhibited promising geoid recovery characteristics. Using covariance analysis techniques, quantitative measures of the orthogonality properties are investigated.
An attempt is made to define the geoid on the basis of geodetic height systems determined from sea-surface topographic data. Existing height systems are corrected using sea-surface data. An equipotential surface is then chosen in such a way as to minimize the sum of the squares of the deviations from the corrected height systems. This equipotential surface is defined as the geoid. A least squares procedure is used to realize this definition.
A numerical method for the determination of gravity anomalies from geoid heights is described using the inverse Stokes formula. This discrete form of the inverse Stokes formula applies a numerical integration over the azimuth and an integration over a cubic interpolatory spline function which approximates the step function obtained from the numerical integration. The main disadvantage of the procedure is the lack of a reliable error measure. The method was applied on geoid heights derived from GEOS-3 altimeter measurements in the calibration area of the GEOS-3 satellite.
A revised 5 minute gravimetric geoid and its errors were computed for the North Atlantic calibration area using GEM-8 potential coefficients and the latest gravity data available from the Defense Mapping Agency. This effort was prompted by a number of inconsistencies and small errors found in previous calculations of this geoid. The computational method and constants used are given in detail to serve as a reference for future work.
Although geoid or surface gravity anomalies cannot be uniquely related to an interior distribution of mass, they can be related to a surface mass distribution. However, over horizontal distances greater than about 100 km, the condition of isostatic equilibrium above the asthenosphere is a good approximation and the total mass per unit column is zero. Thus the surface distribution of mass is also zero. For this case we show that the surface gravitational potential anomaly can be uniquely related to a surface dipole distribution of mass. Variations in the thickness of the crust and lithosphere can be expected to produce undulations in the geoid.
The regional geoid of the southwest Pacific is matched reasonably well by results from a model of the upper mantle density structure (including slabs) associated with subduction zones of the region. Estimates of the geoid are obtained from Geos-3 and Seasat radar altimeter data. These data are very well suited to the task of detecting intermediate wavelength (600-4000 km) geopotential variations. Actually, subducting slabs can be expected to produce primarily intermediate and longer wavelength variations. Gravimetric profiles across trench/island arc complexes resolve primarily short wavelengths. The model represents subducting slabs as thin surfaces of anomalous mass per unit area. These surfaces are positioned using published seismicity results which detail the configuration of the Benioff zones. Crustal effects are ignored. Effects due to the contrast between the young thermal lithosphere of the behind-arc regions (marginal basins) and the older lithosphere seaward of the trench are modelled. Results indicate that the New Hebrides slab possesses an average areal density anomaly of about 300,000 gm/sq cm. This is about three times that which is estimated for the Tonga-Kermadec slab. Additional modelling suggests that slabs worldwide may be an important source of large, long wavelength gravity highs; i.e., they may contribute substantially to geopotential power of harmonic degree as low as three or four up to twenty or more.
The truncation theory as it pertains to the calculation of geoid undulations based on Stokes' integral, but from limited gravity data, is reexamined. Specifically, the improved procedures of Molodenskii et al. are shown through numerical investigations to yield substantially smaller errors than the conventional method that is often applied in practice. In this improved method, as well as in a simpler alternative to the conventional approach, the Stokes' kernel is suitably modified in order to accelerate the rate of convergence of the error series. These modified methods, however, effect a reduction in the error only if a set of low-degree potential harmonic coefficients is utilized in the computation. Consider, for example, the situation in which gravity anomalies are given in a cap of radius 10 deg and the GEM 9 (20,20) potential field is used. Then, typically, the error in the computed undulation (aside from the spherical approximation and errors in the gravity anomaly data) according to the conventional truncation theory is 1.09 m; with Meissl's modification it reduces to 0.41m, while Molodenskii's improved method gives 0.45 m. A further alteration of Molodenskii's method is developed and yields an RMS error of 0.33 m. These values reflect the effect of the truncation, as well as the errors in the GEM 9 harmonic coefficients. The considerable improvement, suggested by these results, of the modified methods over the conventional procedure is verified with actual gravity anomaly data in two oceanic regions, where the GEOS-3 altimeter geoid serves as the basis for comparison. The optimal method of truncation, investigated by Colombo, is extremely ill-conditioned. It is shown that with no corresponding regularization, this procedure is inapplicable.
The computation of accurate geoid undulations is usually done combining potential coefficient information and terrestrial gravity data in a cap surrounding the computation point. In doing this a spherical approximation is made that can cause the errors that were investigated. The equations dealing with ellipsoidal corrections developed by Lelgemann and by Moritz were used to develop a computational procedure considering the ellipsoid as a reference surface. Terms in the resulting expression for the geoid undulation are identified as ellipsoidal correction terms. These equations were developed for the case where the Stokes function is used, and for the case where the modified Stokes function is used. For a cap of 20 deg the correction can reach -33 cm.
In the considered investigation, attention is given to the line source model, a surface source model, an application of the model, and a model of the thermal lithosphere associated with marginal basins. It is found that undulations in the altimetrically observed geoid of the southwest Pacific are strongly controlled by positive density anomalies in the subducting slabs of the region and the effects of elevation of the geotherm in behind arc lithosphere (corresponding to young marginal basins). Finer details of slab geometry do not obviously manifest themselves in the observed geoid. Such gravitational effects are quite attenuated at sea level and are apparently mixed with crustal effects, oceanographic noise, etc. It appears that slabs in global composite may contribute substantially to intermediate and long wavelength portions (down to spherical harmonic degree 3 or 4) of the earth's gravity field.
Errors in computations of the geoid undulation, which determines the position of the geoid with respect to a reference ellipsoid, introduced by various modifications of the Stokes' function in the Stokes integral formula for the undulation are examined in light of currently available harmonic coefficients. The variants considered include the conventional Molodenskii truncation theory, in which integration is limited to a spherical cap, and the variations suggested by Molodenskii (1958) and Meissl (1971). In computations of the undulation from gravity anomalies in a spherical cap and terrestrial and satellite harmonic potential coefficients (GEM 9 and GEM 10B), it is shown that either of the modifications to the conventional Molodenskii theory results in substantial reductions in truncation errors, with the simpler modification of Meissl even providing superior results in certain circumstances.
A digital data library of MAGSAT data consisting of 1,615,636 measurements from the quiet data set, is geographically sorted, and allows rapid analysis and processing of all the quiet magnetic data about any selected location. Because this library of MAGSAT data is compatible with existing gravity and geoid data library processing and display system software, correlations between MAGSAT, surface gravity, GEOS-3 radar altimeter geoid and bathymetric data sets can be conveniently detected and analyzed. Polynomial trends from each half-orbit were removed as an effective way of estimating and removing ring current effects following estimation of the core field contribution. It was found that a third order polynomial is the lowest polynomial order that appears to provide the best consistency of residual anomalies between coincident orbits.
Density contrasts in the lower mantle, recently imaged using seismic tomography, drive convective flow which results in kilometers of dynamically maintained topography at the core-mantle boundary and at the Earth's surface. The total gravity field due to interior density contrasts and boundary topography predicts the largest wavelength components of the geoid remarkably well. Neglecting dynamic surface deformation leads to geoid anomalies of opposite sign than are observed.
A digital data library of MAGSAT data is described and its applications and capabilities are reviewed. Polynomial trends were removed from each half-orbit in order to estimate and remove ring current effects from the data. The MAGSAT data in the Gulf of Mexico region was 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. Since the shape and location of the negative magnetic anomaly are variable depending upon the particular polynomial surface and curve orders used, no definitive conclusion as to the degree of correspondance between the residual geoid and MAGSAT lithosphere anomalies is offered.
Ellipsoidal height differences have been determined for 13 station pairs in the central Ohio region using measurements made with the Global Positioning System. This information was used to compute geoid undulation differences based on known orthometric heights. These differences were compared to gravimetrically-computed undulations (using a Stokes integration procedure, and least squares collocation having an internal r.m.s. agreement of plus or minus 1 cm in undulation differences). The two sets of undulation differences have an r.m.s. discrepancy of plus or minus 5 cm while the average station separation is of the order of 14 km. This good agreement suggests that gravimetric data can be used to compute accurate geoid undulation differences that can be used to convert ellipsoidal height differences obtained from GPS to orthometric height differences.
Density contrasts in the lower mantle, recently imaged using seismic tomography, drive convective flow which results in kilometers of dynamically maintained topography at the core-mantle boundary and at the earth's surface. The total gravity field due to interior density constrasts and boundary topography predicts the largest wavelength components of the geoid remarkably well. Neglecting dynamic surface deformation leads to geoid anomalies of opposite sign than are observed.
Another type of geophysical observation has proved to be very useful in the study of thermal cooling of the oceanic lithosphere. It is the geoid height derivative with respect to plate age, a quantity computed from the short wavelength geoid step across fracture zones measured by altimeter satellites. Two categories of simples models are proposed to describe cooling and contraction of the oceanic lithosphere with age. Both plate model and half space model, give almost similar results up to ages of 50 to 70 ma, but predict quite distinct behavior of seafloor depth, heat flow and other parameters in old basins. Tests of thermal models are based on heat flow and topography data. However, heat flow is not very sensitive to the form of the thermal model. Large areas of the ocean floor are particularly shallow, and as a result topography data may not be very appropriate to discriminate between plate and half space models, and no clear concensus on a preferred model yet exists.
The high degree and order SEASAT geoid in the central Pacific correlates closely with the structure of the cooling lithosphere. Relative changes in plate age across major fracture zones in relatively young seafloor frame the east-west trending pattern formed by the geoid anomalies. The field removal in bathymetry corresponds to removal of some of the low degree and order geoidal components, the step like structure across fracture zones is also removed. The regional thermal subsidence was removed from the bathymetry by subtracting a mean subsidence surface from the observed bathymetry. This produces a residual bathymetry map analogous to the usual residual depth anomaly maps. The residual bathymetry obtained in this way contains shallow depths for young seafloor, and larger depths for older seafloor, thus retaining the structure of the lithosphere while removing the subsidence of the lithosphere.
The long wavelength components of the Earth's gravity field result mainly from density contrasts associated with convection in the mantle. Direct interpretation of the geoid for mantle convection is complicated by the fact that convective flow results in dynamically maintained deformation of the surface of the Earth, the core mantle boundary (CMB), and any interior chemical boundaries which might exist. These boundary deformations effect the geoid opposite in sign and are comparable in magnitude to those of the interior density contrasts driving the flow. The total difference of two relatively large quantities.