The gravity Field - An important Parameter for Earth Observation

As the mass density in the Earth’s interior is varying and the shape of the Earth is non-uniform, gravity is differing at each observation point. This means, that the force of attraction acting on a test mass varies between the poles and the equator, mountains and valleys, or ocean and land areas. The challenge is to find a global representation of the gravity field of the Earth, as its knowledge is of primary interest for many applications and scientific disciplines in Earth sciences.

For instance, the gravity field is directly linked to the physical shape of the Earth. This physical shape does not correspond to a sphere or an ellipsoid, but to the so-called geoid, an irregular surface which is illustrated in Figure 1. Over the oceans the geoid coincides approximately with the mean sea surface assuming that only gravity and no other external forces are acting on it. Over the continents it can be regarded as a continuation of the mean sea surface below the topography. The gravity potential is constant on the geoid, which means that there is no water flowing between points located on the geoid. Therefore, it is appropriate to use the geoid as general reference and zero level for height systems. The deviation of the geoid from a rotational ellipsoid, which is used as best approximating geometric reference body for the Earth, is between -100 and 80 m. This quantity is called geoid height.

Also in oceanography the geoid serves as important reference surface for modelling of sea level change and ocean currents. The deviation of the ocean surface from the geoid is called dynamic ocean topography, and is one of the main drivers of ocean currents [1]. As warm and cold water is transported by these currents, the precise knowledge of the geoid is of great importance for studying climate change. Beside its importance as reference surfaces the gravity field provides insight into the Earth’s interior. Geophysicists gain information about the distribution of masses and density inside the Earth. The Earth’s gravity field is also sensitive to redistribution of masses due to continental water flows (hydrology), melting of continental ice sheets, or even large earthquakes. The loss of ice masses in Greenland and Antarctica as well as hydrological events like rainy seasons in the Amazon can be detected by observing the gravity field from space. Due to its importance for different scientific disciplines, the gravity field has to be determined as accurately as possible.

Since spherical harmonics, a generalization of the concept of Fourier series for the sphere, are global base functions, they are well suited to describe the Earth’s gravity field. Gravity field observations on ground, from airplanes or from satellites are used to determine the spherical harmonic coefficients by solving an inverse problem. There exists a multitude of methods to observe the gravity field. First, terrestrial or airborne gravity accelerations are observed with absolute or relative gravimeters. The gravity at a certain point can be determined through free fall experiments (absolute gravimeter) or by instruments measuring the tension of springs (relative gravimeter). The advantage of this method is that the full gravity signal content is observed, because the observations are taken on the Earth surface or close to it. However, each measurement campaign is restricted to a local area, whereas a global coverage of observations is essential for describing the global field. To achieve global coverage, the data of multiple campaigns must be collected, which is rather challenging, because in some Asian, African or South American areas only few and low-quality data sets exist. Often the access to these data is restricted. The result is an inconsistent data set with inhomogeneous coverage, which is not well suited for determining the long wavelengths of the gravity field. The second possibility to obtain gravity field observations is satellite altimetry. Altimetry is a method, which primarily determines the height of the ocean above the mean Earth ellipsoid. By subtracting the dynamic ocean topography (see above), geoid heights can be computed ocean-wide. With a further post-processing also altimetric gravity can be computed. The advantage of this procedure is that the signal content is not restricted and the dataset is consistent. Altimetric and terrestrial gravity measurements are generally combined to a more or less global set of observations. As a third method, satellites can be used to observe the Earth’s gravity field. The attraction of the Earth acting on a satellite causes orbit perturbations. While in the past orbits of arbitrary satellites were analyzed, nowadays data from dedicated gravity field satellite missions exist. The first mission was CHAMP (launched in 2000), where the satellite orbit was tracked by GPS. The second mission GRACE (launched in 2002) consists of two satellites following each other on the same orbital track, and observes distance changes between them. These distance changes are mainly caused by different gravitational attractions acting on the two satellites (separation distance about 220 km). The third and most recent mission GOCE (launched in 2009) is equipped with a space gradiometer, which enables the observation of gravity gradients along a baseline of 50 cm with very high performance. In general, data from these satellite missions are consistent, almost global (depending on the chosen orbit) and of very good and homogeneous quality in the low to medium frequency part (depending on the mission concept). However, the signal content of satellite observations is restricted due to the fact, that according to Newton’s law the gravity signal attenuates with increasing satellite height.

A bundle of FORTRAN90 and Matlab programs handles the pre-processing of different data sets with more than 200 million observations in case of terrestrial/altimetric observations, resulting in a single data file which is input to the main processing chain. The way from these observations to the gravity field coefficients is computationally challenging. As spherical harmonic coefficients are obtained by least-squares adjustment, the corresponding normal equation system can be very large. Up to now it was standard to estimate coefficients based on full normal equations up to spherical harmonic degree 360, which corresponds to a matrix size of 126 GByte. The coefficients of higher degrees were generally obtained by using block-diagonal techniques. At LRZ’s HLRB II an approach was implemented dealing with full normal equations up to significantly higher degrees. This is useful in order to preserve the full variance-covariance information of the normal equation systems. Calculations on the HLRB II were performed up to spherical harmonic degree 600, corresponding to a normal equation size of 972 GByte [2]. The main work is done by two FORTRAN90 programs using

BLACS, MPI and SCALAPACK. The first one is organized in a one-dimensional process grid, where each process assembles a certain part of the normal equation matrix (for all observations). Each of these parts handles a certain group of spherical harmonic coefficients belonging together. The second program is organized in a two-dimensional grid. The normal equation is block-cyclic distributed, and the program solves for the spherical harmonic coefficients and inverts the normal equation system for computing error estimates of the coefficients. Further routines were implemented, which handle observations of the GOCE gradiometer. The processing chain is different compared to the case of terrestrial data. Six gravity gradient observations and one 3D-position are measured per second since the start of the mission in 2009. The pre-processing requires data editing and filtering in order to cope with correlated noise. Solving of the normal equations for GOCE data is not as demanding as for terrestrial data, because, due to the satellite height, the signal is damped and the normal equation systems are restricted to degree and order 250 (corresponding to about 30 GByte normal equation file size).

The best gravity field solution is obtained by combination of all possible data sources. Figure 2 shows the gravity field signal per spherical harmonic degree (grey line) and the corresponding errors of different gravity field solutions. It can be identified, that the GRACE mission delivers the best result in the very low degrees, whereas GOCE is performing better in the low to medium degrees. Both missions together provide an excellent satellite-only gravity field solution [3]. Furthermore it is shown, that only by including terrestrial/altimeter data a large spherical harmonic expansion can be obtained. The percental contribution of the satellite data to the combination solution is displayed in the upper right of the figure. The quota depends on the relative weights for the observation groups. In areas of poor terrestrial data quality the weight for the satellite data is higher.

There is also a large impact of GOCE to combined gravity field solutions (due to the fact that the GOCE mission is the newest data source with the highest spatial resolution). Figure 3 illustrates the improvements of the gravity field due to the inclusion of GOCE [4]. Shown is the difference between geoid heights of a global gravity field with and without GOCE data. New and so far unknown gravity field signal detected by GOCE can be especially seen in areas where the quality of terrestrial data is poor, such as South America, Africa and the Himalaya region.

But improved gravity solution based on GOCE result also in large improvements of dynamic ocean topography estimates, because they represent a more accurate geoid solution and thus a more accurate reference surface. This was proven by an oceanographic application. The Gulf current was modelled with and without GOCE data (Fig. 4). By including GOCE data the noise is reduced significantly, and more detailed structures of the ocean current become visible.

These results show that the gravity field is an important parameter for various Earth related disciplines. As millions of data have to be analyzed and hundred of thousands of parameters have to be estimated, high performance computer systems are an essential component to achieve such global gravity field models.

Acknowledgement

The authors gratefully acknowledge computational resources granted by LRZ. Special thanks for all the help and patience concerning our questions and incidents.

References

[1] Bingham, R. J., Knudsen, P., Anderson, O., Pail, R. An initial estimate of the North Atlantic steady-state geostrophic circulation from GOCE, Vol. 38, American Geophysical Union, DOI: 10.1029/2010GL045633, 2011

[2] Fecher, T., Pail, R., Gruber, T. Global gravity field determination from terrestrial data, Amercian Geophysical Union Fall Meeting, San Francisco, 2010

[3] Pail, R., Goiginger, H., Schuh, W. D., Höck, E., Brockmann, J. M., Fecher, T., Gruber, T., Mayer-Gürr, T., Kusche, J., Jäggi, A., Rieser, D. Combined satellite gravity field model GOCO01S derived from GOCE and GRACE, Geophysical Research Letters, Vol 37, American Geophysical Union, DOI: 10.1029/2010GL044906, 2010

[4] Pail, R., Bruinsma, S., Migliaccio, F., Förste, C., Goiginger, H., Schuh, W. D., Höck, E., Reguzzoni, M., Brockmann, J. M., Abrikosov, O., Veicherts, M., Fecher, T., Mayrhofer, R., Krasbutter, I., Sansò, F., Tscherning, C. C. First goce gravity field models derived by three different approaches, Journal of Geodesy, DOI: 10.1007/s00190-011-0467-x

• Thomas Fecher
Institut für Astronomische und Physikalische Geodäsie, TU München

• Thomas Gruber
Institut für Astronomische und Physikalische Geodäsie, TU München

• Roland Pail
Institut für Astronomische und Physikalische Geodäsie, TU München

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