N. Sneeuw

Band-limited functions on a bounded spherical domain : the Slepian problem on the sphere

Abstract. The Slepian problem consists of determining a sequence of functions that constitute an orthonormal basis of a subset of ℝ (or ℝ2) concentrating the maximum information in the subspace of square integrable functions with a band-limited spectrum. The same problem can be stated and solved on the sphere. The relation between the new basis and the ordinary spherical harmonic basis can be explicitly written and numerically studied. The new base functions are orthogonal on both the subspace and the whole sphere. Numerical tests show the applicability of the Slepian approach with regard to solvability and stability in the case of polar data gaps, even in the presence of aliasing. This tool turns out to be a natural solution to the polar gap problem in satellite geodesy. It enables capture of the maximum amount of information from non-polar gravity field missions.

A CHAMP‐only gravity field model from kinematic orbits using the energy integral

[1]xa0In this paper we present results of a global gravity field recovery using half a year of CHAMP data. We use the energy integral of the motion of a satellite to transform satellite velocities into values of gravitational potential. The feasibility of this approach has already been demonstrated by several groups, using CHAMP reduced-dynamic orbits. We show, that the potential recovered from this kind of orbits depends on the a priori gravity field used for orbit determination. Thus, it cannot be excluded that errors present in the prior field propagate into the new CHAMP gravity model. It is the intention of this paper to avoid this dependency through the use of kinematic orbits, which are free from prior information. The derived potential model, TUM-1S, is validated by comparison to ground data and by satellite orbit residuals. It is shown to be comparable in quality to other state-of-the-art gravity field models.

Global spherical harmonic computation by two-dimensional Fourier methods

A method is presented for performing global spherical harmonic computation by two-dimensional Fourier transformations. The method goes back to old literature (Schuster 1902) and tackles the problem of non-orthogonality of Legendre-functions, when discretized on an equi-angular grid. Both analysis and synthesis relations are presented, which link the spherical harmonic spectrum to a two-dimensional Fourier spectrum. As an alternative, certain functions of co-latitude are introduced, which are orthogonal to discretized Legendre functions. Several independent Fourier approaches for spherical harmonic computation fit into our general scheme.

Box inverse models, altimetry and the geoid: Problems with the omission error

[1]xa0When one combines satellite altimetry and a geoid model to improve estimates of the ocean general circulation from hydrographic data with a box inverse model, there arises a problem of different resolution and representation of the data types involved. Here we show how this problem can lead to an artificial leakage of the error estimates of short-scale (high degree) spherical harmonic functions into long wavelength (low wave number) Fourier functions. A similar paradox effect can be seen in an idealized box inverse model constrained by additional sea-surface topography data of low, medium, and high resolution: When more information is added in the form of additional smaller scales, the error of a transport estimate eventually increases. Consequently, including the large geoid omission errors associated with smaller scales in a box inverse model of the Southern Ocean increases the posterior errors of transport estimates over those of a model that does not include the geoid omission error. We do not claim that including or excluding the geoid omission error is correct. Instead, we juxtapose two different ways of estimating the geoid errors to demonstrate the effect that the omission error might have on the long, supposedly well-known, scales. How (or if) to properly account for the geoid omission error must be the topic of further research. A proper treatment of the geoid model errors is demanded when one evaluates the errors of absolute sea-surface topography data.

CHAMP Gravity Field Recovery with the Energy Balance Approach: First Results

Using the principle of energy conservation has been considered for gravity field determination from satellite observations since the early satellite era, see e.g. O’Keefe (1957), Bjerhammar (1968), Reigber (1969) or Ilk (1983). CHAMP is the first satellite to which the energy balance approach can be usefully applied, now that near-continuous orbit tracking by GPS is available, aided by accelerometry. Simulation studies show the feasibility of the approach. One concern is the sensitivity to velocity errors. As a next step CHAMP’s Rapid Science Orbits (RSO) are used. Their error level is sufficiently low to demonstrate the feasibility of the energy balance approach with real data. The accelerometer data (ACC), used for modeling the non-conservative forces, give rise to further concern.

Calibration and Validation of GOCE Gravity Gradients

GOCE will be the first satellite ever to measure the second derivatives of the Earth’s gravitational potential in space. With these measurements it is possible to derive a high accuracy and resolution gravitational field if systematic errors have been removed to the extent possible from the data and the accuracy of the gravity gradients has been assessed. It is therefore necessary to understand the instrument characteristics and to setup a valid calibration model. The calibration parameters of this model could be determined by using GOCE data themselves or by using independent gravity field information. Also the accuracy or error assessment relies on either GOCE or independent data. We will demonstrate how state-of-the-art global gravity field models, terrestrial gravity data and observations at satellite track crossovers can be used for calibration/validation. In addition we will show how high quality terrestrial data could play a role in error assessment.

Satellite Gravity Gradiometry with GOCE

During the past years the European Space Agency (ESA) has worked on the definition of a comprehensive Earth observation satellite programme for the era after ENVISAT. This “ESA Living Planet” (1998) programme has recently been published. It distinguishes three types of missions: the research oriented medium scale explorer missions, the smaller opportunity missions, and the monitoring type Earth watch missions. In preparation of a pre-selection of the first explorer missions a meeting of the European Earth science community had been organized at Granada/Spain in May 1996. The Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) was one of nine candidates that were presented there. Following a recommendation of the Earth Science Advisory Committee of ESA four of the nine were selected for a phase-A study. They were: the gravity field and steady-state ocean circulation mission, an atmospheric dynamics mission, an Earth radiation experiment, and a land surface processes and interaction mission.

One year of time-variable CHAMP-only gravity field models using kinematic orbits

A full year of champ gravity field solutions has been calculated using the energy integral approach. The monthly solutions in the time frame 03.2002–02.2003 were based solely on kinematic orbits from champ gps orbit tracking and accelerometry. These kinematic orbits have not been contaminated by a priori gravity field information.

Validation of fast pre-mission error analysis of the GOCE gradiometry mission by a full gravity field recovery simulation

Abstract Spherical harmonic error analysis fully relies on the validity of the a priori observational and stochastic models. In this paper we validate error analysis results of the gradiometer mission GOCE by a full-fledged spherical harmonic coefficient recovery. Both methods (least squares error analysis and full recovery) are based on a semi-analytical approach. The results compare very well in spectral and spatial domains. Thus, it is demonstrated that, besides being fast, the least squares error analysis is a reliable premission error assessment tool.

Simulation of the GOCE Gravity Field Mission

GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) is one of the four selected ESA Earth Explorer Missions. The main objective of GOCE is the determination of the Earth’s gravity field with high spatial resolution and with high homogeneous accuracy. For this purpose, two observation concepts will be realised. Satellite-to-Satellite Tracking (SST) in high-low mode will be used for the orbit determination and for the retrieval of the long-wavelength part of the gravity field. Satellite Gravity Gradiometry (SGG) will be employed for the derivation of the medium/short-wavelength parts of the gravity field..