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High Resolution Gravity Field Modelling

Principal Investigator: Thomas Gruber, Institute of Astronomical and Physical Geodesy, Technische Universität München (Germany)
HPC Platform: SuperMUC of LRZ

Scientific challenge

The static gravity field of the Earth is one of the key parameters for the observation and measurement of a number of processes and flows in the dynamic system of the living planet Earth. Its knowledge is of importance for various scientific disciplines, such as geodesy, geophysics and oceanography. For geophysics the gravity field gives insight into the Earth’s interior, while by defining the physical shape of the Earth it provides an important reference surface for oceanographic applications, such as the determination of sea level rise or modelling of oceanographic currents. Moreover this reference surface is a key parameter on the way to a globally unified height system.

EXtreme PREcipitation and Hydrological Climate Scenario Simulations (EXPRESS-Hydro)High Resolution Gravity Field ModellingFig 1: The gravity field of the Earth in terms of gravity anomalies [mgal].
Copyright: Institute of Astronomical and Physical Geodesy, Technische Universität München

The scientific goal is to estimate the static gravity field as precise and detailed as possible. As the gravity field in general is represented by a spherical harmonic series, the parameters to be estimated in gravity field modelling are spherical harmonic coefficients. There exist various techniques to observe the gravity field, which have different advantages and complement each other. The observation of the Earth gravity field from dedicated satellite missions delivers highly accurate and globally homogenous gravity field information for the long to medium wavelengths of the spherical harmonic spectrum (corresponding to spatial resolutions down to roughly 100 km). However, due to the large distance between the satellite and the Earth’s surface, the gravity field signal is damped in satellite height. Therefore short wavelengths of the spherical harmonic spectrum (smaller than 100 km) cannot be observed from space.

To complement the satellite information, terrestrial gravity field measurements over land and satellite altimeter observations over the oceans, which need to be converted to gravity field quantities, are used as additional data. As these observations are taken at the Earth surface (land and ocean) they contain the full undamped signal.

The scientific challenge is to combine the different types of gravity field observations in a way in which all data types keep their specific strengths and are not degraded by the combination with other information in specific spherical harmonic wavelength regimes. As mentioned, this procedure shall result in a set of spherical harmonic coefficients representing the global Earth gravity field up to highest possible resolution.

Strict full normal equation approach and the need for supercomputing

The researchers’ approach to determine the spherical harmonic coefficients is based on a strict least squares adjustment with a Gauß-Markov model. By that, the different data types can be combined optimally on normal equation base. The approach enables the ideal relative weighting between different data sets as well as the individual weighting of every single observation, which is important as especially the quality of terrestrial measurement data differs significantly (observations are in general quite accurate in regions such as Europe or North America, whereas they are not as good in Africa or South America). Due to the high correlation of the unknown spherical harmonic coefficients, the corresponding normal equation system is a dense matrix. As the number of unknowns increases quadratic with the spherical harmonic degree, and again the number of elements of the full normal equation matrix is quadratic to the number of unknowns, full normal equation systems become quite large. This is a computational challenge, which can only be solved with supercomputers such as SuperMUC. With the available data the researchers currently estimate the coefficients of spherical harmonic expansion up to degree and order 720, which corresponds to more than 500,000 unknowns and a normal equation system of 2 TByte. With the availability of denser ground data coverage further extensions to even higher resolutions are planned for the near future.


In the images below, the benefit of the full normal equation approach is demonstrated exemplary for a gravity field solution in the area of South America. As observations data sets from the US-German satellite mission GRACE (Gravity and Climate Recovery Experiment), the satellite mission GOCE (Gravity Field and Steady State Ocean Circulation Explorer) of the European Space Agency (ESA), and terrestrial observations provided by the National Geospatial Intelligence Agency (NGA) are available. The quality of the terrestrial data differs significantly, e.g. it is quite high for example for coastal areas in Brazil and low for areas in the Amazon area. The major challenge is to find the optimal combination of the different data sets.

EXtreme PREcipitation and Hydrological Climate Scenario Simulations (EXPRESS-Hydro)High Resolution Gravity Field ModellingFig 2: Accuracy map of terrestrial observations in South America [mgal] (left); difference between the block-diagonal (center) and full normal (right) equation solution to the satellite information [mgal].
Copyright: Institute of Astronomical and Physical Geodesy, Technische Universität München

For that purpose, the scientists derive first an accuracy map of the terrestrial observations by comparison with the satellite information in the low to medium wavelengths (Fig 2 left). In the following, they calculate two different gravity field solutions. One is based on a reduced block-diagonal normal equation approach, which requires strong data constraints like equal data weights, but can be executed on a single node computer system. The second approach is based on full normal equations applying individual weights according to a predetermined accuracy map and using supercomputing facilities. One can investigate the difference between the two results and the original satellite information in the low to medium wavelengths, which delivers a measure of how good the estimation procedure is able to recover the field in this frequency range. The parameter estimation process can be regarded as good if the differences are small - which was proven in this case as the good performance of the satellite information was not degraded by the combination with terrestrial data. It is clearly visible, that the differences are quite high for the reduced block diagonal approach (Fig 2, center). Poor data from the Amazon area and from the Andes affected the combination solution. With full normal equations and supercomputing approach, nearly no differences are visible (Fig 2 right). The combination is ideal for that case, what demonstrates the power of using full normal equation systems and supercomputing facilities for this purpose.

Details about the procedure and results can be found in Fecher (2015).

Fecher, T.: Globale kombinierte Schwerefeldmodellierung auf Basis voller Normalgleichungssysteme; Dissertation, Ingenieurfakultät Bau Geo Umwelt (BGU), TU München, 2015

Scientific contact:
Dr.-Ing Thomas Gruber, Dr.-Ing. Thomas Fecher
Institute of Astronomical and Physical Geodesy
Technische Universität München
Arcisstr. 21, D-80333 München
e-mail: Thomas.Gruber @

November 2015