Torsten Mayer-Guerr, Annette Eicker, Karl-Heinz Ilk

The static gravity field
The time variable gravity field
Variance-Covariance Information
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The Static Gravity Field

Figure 1: ITG-Grace03s as of gravity anomalies (n=150) (left) and in terms of degree amplitudes (right)

The static gravity field is computed from accumulated normal equations over a time span from September 2002 to April 2007 using GRACE data only and without applying any apriori information or regularization. The solution is represented by a spherical harmonics expansion up to degree n=180. We used the short arc integration method as described in (Mayer-Guerr 2006, [1]). The method is also described in the paper dealing with the preceding model ITG-Grace02s (Mayer-Guerr et al. 2006, [2]).


Further details:

Earth rotation:  
IERS 2003 
Moon, sun and planets ephem.:  
JPL DE405 
Earth tide:  
IERS 2003 
Ocean tide:  
FES 2004 
Pole tide:  
IERS 2003 
Ocean pole tide:  
Deasi 2003 
Atmosphere and Ocean Dealiasing:  
Static gravity field:  
IERS 2003 
Reference Epoch:  
Permanent tidal deformation:  
included (zero tide) 

The Time Variable Gravity Field

Figure 2: Equivalent water heights at 2005-03-15 12:00 (left) computed from the evaluation of quadratic spline basis functions (right)

As the gravity field changes smoothly in time, we believe that the estimation of the time variable gravity field in terms of block mean values in time domain (e.g. monthly mean) is not optimal. A better approximation can be achieved when applying a set of smooth basis functions. Therefore in our solution the time variable gravity field is parametrized by quadratic splines in time domain. In space domain the common sphercial harmonic expansion is used from degree n=2 to n=40. The quadratic splines have a nodal point distance of half a month (about 15 days) to each nodal point there belongs a set of spherical harmonic coefficients. In the estimation process the variations are filtered by applying a regularization matrix for each set of spherical harmonic coefficients. The regularization matrices were choosen by analyzing hydrologycal models and have a Kaula type form. For each regularization matrix an indiviudally weight is determined by the variance component estimation method.

To get a set of spherical harmonic coefficients at a given time t the following steps are required:

    1. t has to be given in Modified Julian Date (mjd)
    2. Use the file indexTime.txt to determine the index of the time interval, so that t is between index(i) and index(i+1)
    3. Compute the normalized time tau according to

tau = (t - time.index(i)) / (time.index(i+1) - time.index(i))

4. Use the three files ITG-Grace03_quadraticSplines_xxx.coeff with xxx being index(i-1), index(i) and index(i+1) and multiply the respective coefficients with the following factors:

(cnm(t),snm(t)) = ( 1/2*tau^2)) * (cnm,snm).index(i+1)                

+ ( -tau^2 + tau + 1/2)) * (cnm,snm).index(i)       

+ ( 1/2*tau^2 - tau + 1/2)) * (cnm,snm).index(i-1)

The result is a set of spherical harmonic coefficients representing differences to the static gravity field ITG-Grace03s at time t.

Additionally we provide monthly means of this solution for those users being more comfortable with this way of representing the time variablities.



Variance-Covariance Information

Figure 4: Accuracies of spherical harmonic coefficients (left), full variance-covariance matrix (right)

During the estimation process of the static gravity field ITG-Grace03s 32,757 unknown gravity field parameters were estimated. The coefficients are ordered by order, within each order they are sorted by degree with c_nm and s_nm alternating. The position of each unknown parameter in the parameter vector can be found in varianceIndex.txt.


The full variance-covariance matrix has a size of about 8 GB memory. To handle this huge data set the matrix is split up into smaller files and only the upper triangle of the matrix is stored. Each line of the files contains row number column number (counting from 0) and value of the matrix. The individual files are denoted by covariance_xx_yy.txt.gz and they each contain a block of the covariance matrix according to:


Be carefull, the matrix contains only the formal errors and does not have to reflect the true accuracies.




Download Data

Format description: 
ITG-Grace03s static gravity field: 
ITG-Grace03 time variable field (quadratic splines): 
ITG-Grace03 time variable field (monthly means): 
Full variance-covariance matrix: 


[1] Mayer-Guerr, T. (2006): Gravitationsfeldbestimmung aus der Analyse kurzer Bahnbögen am Beispiel der Satellitenmissionen CHAMP und GRACE, Dissertation, University of Bonn (pdf, 5.5 MB)


[2] Mayer-Guerr, T., A. Eicker, K.-H. Ilk (2007): ITG-Grace02s: A GRACE Gravity Field Derived from Short Arcs of the Satellites Orbit, Procedings of the 1st International Symposium of the International Gravity Field Sevice "Grvaity Field of the Earth", Istanbul


[3] Mayer-Guerr, T. (2007): ITG-Grace03s: The latest GRACE gravity field solution computed in Bonn, presentation at GSTM+SPP, 15-17 Oct 2007, Potsdam (pdf, 7.9 MB)