Closely spaced satellite altimeter profiles collected during the Geosat Geodetic Mission (-6 km) and the ERS 1 Geodetic Phase (8 km) are easily converted to grids of vertical gravity gradient and gravity anomaly. The long-wavelength radial orbit error is suppressed below the noise level of the altimeter by taking the along-track derivative of each profile. Ascending and descending slope profiles are then interpolated onto separate uniform grids. These four grids are combined to form comparable grids of east and north vertical deflection using an iteration scheme that interpolates data gaps with minimum curvature. The vertical gravity gradient is calculated directly from the derivatives of the vertical deflection grids, while Fourier analysis is required to construct gravity anomalies from the two vertical deflection grids. These techniques are applied to a combination of high-density data from the dense mapping phases of Geosat and ERS 1 along with lower-density but higher-accuracy profiles from their repeat orbit phases. A comparison with shipboard gravity data shows the accuracy of the satellite- derived gravity anomaly is about 4-7 mGal for random ship tracks. The accuracy improves to 3 mGal when the ship track follows a Geosat Exact Repeat Mission track line. These data provide the first view of the ocean floor structures in many remote areas of the Earth. Some applications include inertial navigation, prediction of seafloor depth, planning shipboard surveys, plate tectonics, isostasy of volcanoes and spreading ridges, and petroleum exploration.

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Marine gravity anomaly from Geosat and ERS 1 satellite altimetry

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Regularization Of Inverse Problems

Green functions of the biharmonic operator, in one and two dimensions, are used for minimum curvature interpolation of irregularly spaced data points. The interpolating curve (or surface) is a linear combination of Green functions centered at each data point. The amplitudes of the Green functions are found by solving a linear system of equations. In one (or two) dimensions this technique is equivalent to cubic spline (or bicubic spline) interpolation while in three dimension it corresponds to multiquadric interpolation. Although this new technique is relatively slow, it is more flexible than the spline method since both slopes and values can be used to find a surface. Moreover, noisy data can be fit in a least squares sense by reducing the number of model parameters. These properties are well suited for interpolating irregularly spaced satellite altimeter profiles. The long wavelength radial orbit error is suppressed by differentiating each profile. The shorter wavelength noise is reduced by the least squares fit to nearby profiles. Using this technique with 0.5 million GEOS-3 and SEASAT data points, it was found that the marine geoid of the Caribbean area is highly correlated with the sea floor topography. This suggests that similar applications, in more remote, areas may reveal new features of the sea floor.

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Biharmonic spline interpolation of geos-3 and seasat altimeter data

Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

GOCE is the first gravitational gradiometry satellite mission. Gravitational gradiometry is the measurement of the second derivatives of the gravitational potential. The nine derivatives form a 3 × 3 matrix, which in geodesy is referred to as Marussi tensor. From the basic properties of the gravitational field, it follows that the matrix is symmetric and trace free. The latter property corresponds to Laplace equation, which gives the theoretical foundation of its representation in terms of spherical harmonic or Fourier series. At the same time, it provides the most powerful quality check of the actual measured gradients. GOCE gradiometry is based on the principle of differential accelerometry. As the satellite carries out a rotational motion in space, the accelerometer differences contain angular effects that must be removed. The GOCE gradiometer provides the components V
 xx
 , V
 yy
 , V
 zz
 and V
 xz
 with high precision, while the components V
 xy
 and V
 yz
 are of low precision, all expressed in the gradiometer reference frame. The best performance is achieved inside the measurement band from 5 × 10–3 to 0.1 Hz. At lower frequencies, the noise increases with 1/f and is superimposed by cyclic distortions, which are modulated from the orbit and attitude motion into the gradient measurements. Global maps with the individual components show typical patterns related to topographic and tectonic features. The maps are separated into those for ascending and those for descending tracks as the components are expressed in the instrument frame. All results are derived from the measurements of the period from November to December 2009. While the components V
 xx
 and V
 yy
 reach a noise level of about $${10\;\rm{\frac{mE}{\sqrt{Hz}}}}$$
 , that of V
 zz
 and V
 xz
 is about $${20\; \rm{\frac{mE}{\sqrt{Hz}}}}$$
 . The cause of the latter’s higher noise is not yet understood. This is also the reason why the deviation from the Laplace condition is at the $${20 \;\rm{\frac{mE}{\sqrt{Hz}}}}$$
 level instead of the originally planned $${11\;\rm{\frac{mE}{\sqrt{Hz}}}}$$
 . Each additional measurement cycle will improve the accuracy and to a smaller extent also the resolution of the spherical harmonic coefficients derived from the measured gradients.

GOCE gravitational gradiometry

Dedicated gravity field missions—principles and aims

Gravity Field and Steady-State Ocean Circulation Mission

/pdf/spherical-harmonic-analysis-of-satellite-gradiometry-3a4flef7s8.pdf

Spherical harmonic analysis of satellite gradiometry

Vertical datum definition is identical with the choice of a potential (or height) value for the fundamental bench mark. Also the connection of two adjacent vertical datums poses no principal problem as long as the potential (or height) value of two bench marks of the two systems is known and they can be connected by levelling. Only the unification of large vertical datums and the connection of vertical datums separated by an ocean remains difficult. Two vertical datums can be connected indirectly by means of a combination of precise geocentric positions of two points, as derived by space techniques, their potential (or height) value in the respective height datum and their geoid height difference. The latter requires the solution of the linear geodetic boundary value problem under the assumption that potential and gravity anomalies refer to a variety of height datums. The unknown off-sets between the various datums appear in the solution inside and outside the Stokes integral and can be estimated in a least squares adjustment, if geocentric positions, levelled heights and adequate gravity material are available for all datum zones. The problem can in principle also be solved involving only two datums, in case a precise global gravity field becomes available purely from satellite methods.

Reiner Rummel

Papers

GOCE gravitational gradiometry

Dedicated gravity field missions—principles and aims

Gravity Field and Steady-State Ocean Circulation Mission

Spherical harmonic analysis of satellite gradiometry

Height datum definition, height datum connection and the role of the geodetic boundary value problem