Combined satellite gravity field model GOCO01S derived from GOCE and GRACE



[1] The satellite-only gravity field model GOCO01S is a combination solution based on 61 days of GOCE gravity gradient data, and 7 years of GRACE GPS and K-band range rate data, resolved up to degree/order 224 of a harmonic series expansion. The combination was performed consistently by addition of full normal equations and stochastic modeling of GOCE and GRACE observations. The model has been validated against external global gravity models and regional GPS/leveling observations. While low to medium degrees are mainly determined by GRACE, significant contributions by the new measurement type of GOCE gradients can already be observed at degree 100. Beyond degree 150, GOCE becomes the dominant contributor. Correspondingly, with GOCO01S a global gravity field model with high performance for the complete spectral range up to degree/order 224 is now available. This new gravity model will be beneficial for many applications in geophysics, oceanography, and geodesy.

1. Introduction

[2] The launch of dedicated gravity satellite missions has revolutionized our knowledge of the Earth's gravity field. In 2002 the along-track satellite formation mission GRACE (Gravity Recovery and Climate Experiment) [Tapley et al., 2007] was launched into a near polar low Earth orbit. Apart from high-low satellite-to-satellite tracking by the GPS constellation and a 3-axis accelerometer measuring the non-conservative accelerations (e.g., air drag, solar radiation pressure) acting on each of the two satellites, the key element is a microwave ranging system observing the distance variation between the two satellites with micrometer accuracy. The most recent static models derived purely from GRACE data resolve the global gravity field up to degree 180, e.g., GGM03S [Tapley et al., 2007], EIGEN-5S [Förste et al., 2008], AIUB-GRACE02s [Jäggi et al., 2009], and ITG-Grace2010s [Mayer-Gürr et al., 2010].

[3] The GOCE satellite (Gravity Field and Steady-state Ocean Circulation Explorer) [Drinkwater et al., 2003] was successfully launched on 17 March 2009. GOCE is based on a sensor fusion concept: orbit analysis with a spaceborne GPS-receiver enabling continuous 3D tracking of the satellite with an accuracy 2–3 centimeters, plus on-board satellite gravity gradiometry (SGG). This completely new measurement concept is based on the observation of gravity gradients in space with accelerometers over short baselines within a platform flying in drag-free mode, i.e., by in-situ compensating the non-gravitational forces. Correspondingly, GOCE is the first mission that observes direct functionals of the Earth gravity field from space. First GOCE gravity field models based on 71 days of GOCE data from November 2009 to January 2010 have been computed in the frame of the ESA project “High-Level Processing Facility” by applying 3 independent and complementary processing strategies [Bruinsma et al., 2010; Pail et al., 2010; Migliaccio et al., 2010].

[4] One of these three solutions, the so-called time-wise gravity field model [Pail et al., 2010] (product ID: EGM_GOC_2__20091101T000000_20100111T000000_0002), will be used as a reference solution, and will be denoted as ESA GOCE-only model hereafter.

[5] Here, the first combined satellite-only gravity field model GOCO01S is presented, which is a combination of GRACE and GOCE measurements, to make optimal benefit of the individual strengths of these two satellite missions. (GOCO stands for “Combination of GOCE data with complementary gravity field information”, a project initiative with the objective to compute high-accuracy and high-resolution static global gravity field models based on satellite and terrestrial data. GOCO01S is the first step comprising a combined satellite-only model).

[6] The model is parameterized in terms of coefficients of a spherical harmonic series expansion of the gravitational potential V in spherical coordinates (with radius r, co-latitude ϑ, longitude λ):

equation image

with G the gravitational constant, M the mass of the Earth, R the mean Earth radius, equation imagelm the fully normalized Legendre polynomials of degree l and order m, and {equation imagelm, equation imagelm} the corresponding coefficients to be estimated.

2. Gravity Field Model GOCO01S

[7] The global gravity field model GOCO01S is composed of the following main components: (1) GOCE normal equations (representing GOCE gradients); (2) GRACE normal equations (representing GRACE GPS and range rate observations); (3) constraints.

2.1. GOCE

[8] The GOCE component is based on two months of GOCE gravity gradients (1 November 2010 to 31 December 2010), which are publicly available via at the ESA's Virtual Online Archive at The gravity gradients represent second order derivatives of the gravitational potential

equation image

defined in the gradiometer reference frame xi, with i, j = X, Y, Z [Drinkwater et al., 2003]. The method for assembling the normal equations based on SGG observations is described in detail by Pail and Plank [2002] and Pail et al. [2010]. Full normal equations complete to degree/order 224, corresponding to more than 50,000 parameters {equation imagelm, equation imagelm}, have been assembled on a PC cluster.

[9] In this framework, one of the key elements is the correct stochastic modeling of the gradiometer measurement errors. Figure 1a shows power spectral densities (PSDs) of the gradiometer error (estimated from the residuals of a gravity field adjustment) for the main diagonal components VXX, VYY and VZZ. The optimum instrument performance is achieved only in the measurement bandwidth (MBW) from 5 to 100 mHz. Digital auto-regressive moving average (ARMA) filters are used to set-up the variance-covariance information of the gradient observations [Schuh, 2003]. Technically, this is done by applying these filters to the full observation equation, i.e., both to the observations and the columns of the design matrix. Thus, the gradiometer error information is introduced as the metric of the normal equation system. By this strategy, the full spectral range of the gravity gradients enters the gravity field solution, but they are properly weighted according to their spectral behavior. Exemplarily, Figure 1b shows the filter model for the gradiometer component VZZ. A more detailed discussion on the refinement of the stochastic model for this GOCE solution is given by Schuh et al. [2010], and its application to GOCE normal equations by Pail et al. [2010].

Figure 1.

(a) GOCE gradiometer error PSD of the main diagonal components of the gravity gradiometry tensor VXX, VYY and VZZ; gradiometer MBW indicated by black dashed lines; (b) ARMA filter model for the gradiometer component VZZ.

[10] Because of the combination with GRACE, no gravity field information derived from GOCE GPS orbits was included. It is important to emphasize that no a priori gravity field information enters the SGG normal equations. Thus, they are completely independent of GRACE and terrestrial gravity field data, and reflect the pure contribution by the GOCE mission.

2.2. GRACE

[11] GRACE normal equations of ITG-Grace2010s available with full variance/covariance matrix at up to degree/order 180, which are based on GRACE data covering the time span from August 2002 to August 2009, have been used for the combination. The ITG-Grace2010s gravity field solution is calculated with the integral equation approach using short arcs with a maximum length of 60 min [Mayer-Gürr et al., 2010]. K-band range rates and kinematic orbits are used as observations. As the observations are strongly correlated, an appropriate stochastic model has been introduced for each short arc.

2.3. Constraints

[12] In order to improve the signal-to-noise ratio in the high degrees, Kaula regularization was applied to coefficients with degrees larger than 170 [cf. Pail et al., 2010]. Since we do not use any reference gravity field model for our solution, but estimate the gravity field coefficients from the scratch, the solution is Kaula constrained towards a zero model avoiding high energy in less accurately estimated high degree coefficients.

2.4. Combination Procedure

[13] The combination has been performed by addition of full normal equations. Due to the adequate stochastic modeling of the GOCE and GRACE components, they entered the solution with unit weight, while a single weight factor was determined for the Kaula constraints as a kind of regularization parameter in the frame of a variance component estimation procedure [Koch and Kusche, 2002; Brockmann et al., 2010]. This weight scales the relative spectral weights provided by Kaula's rule so that the final spherical harmonic parameters have minimal variance. The resulting normal equations are then solved for the unknown spherical harmonic coefficients {equation imagelm, equation imagelm} using a massive parallel implementation of the Cholesky factorization. The covariance information is determined by the inversion of the joint normal equation matrix.

3. Results

[14] Figure 2 shows degree median errors, i.e., the median deviation derived from all coefficients of a specific harmonic degree, of the individual components as well as the combined solution. The fact that only the SGG component was used for GOCE results in a bad performance of the 61-days GOCE-only solution (magenta curve) in the very low degrees, which is due to the error characteristic of the GOCE gradiometer [Drinkwater et al., 2003], while this spectral region is dominated by GRACE (red curve). As a reference, also the ESA GOCE-only model, which is based on 71 days of data, is displayed in green color. In addition to SGG, it is based on kinematic precise science orbits in the low degrees. The blue curve shows the combined solution GOCO01S. Evidently, GOCE starts to contribute already below degree/order 100, shown by the divergent error curves (red vs. blue). The GRACE-GOCE performance cross-over is in the spectral range of degrees 130 to 150.

Figure 2.

Degree error medians of the GOCE component (magenta), the GRACE component (red), and the combined GOCO01S solution (blue dashed). As a reference, the ESA GOCE-only solution (green), and the absolute gravity field signal based on EGM2008 (black) are displayed.

[15] Figure 3 shows the redundancy factors of the 3 contributing components GRACE, GOCE, and constraints, indicating the relative contributions of these components to the combined solution. The solution is based mostly on ITG-Grace2010s up to degree/order 100 (redundancy factors close to 1), but there are sectorial coefficients which are determined by GOCE by more than 10% below degree 100. This is due to the fact that GRACE is less sensitive to measure the sectorial coefficients, because of the along-track measurement principle. Beyond degree 100, GOCE starts to contribute significantly. In the polar areas (opening angle 6.5 degrees), corresponding to zonal and near-zonal coefficients, the solution is exclusively based on GRACE. The Kaula constraint starts to act at degree 170, and is gradually increasing with growing degrees.

Figure 3.

Redundancy factors expressing the relative contributions per coefficient of (a) GRACE, (b) GOCE, and (c) the Kaula constraint to the combined model GOCE01S.

[16] Figure 4 shows geoid height differences of the combined model with respect to the GRACE component ITG-Grace2010s for different maximum degrees. Already at degree/order 100 (Figure 4a) there appear characteristic features which are typical for GRACE errors, and correspondingly, it can be concluded that there are significant contributions by GOCE with amplitudes of about 1 cm (global rms of 1 mm). At degree 180 (Figure 4b), GOCE is the dominant data source, and typical GRACE errors of more than 1 m (global rms of 0.23 m) appear.

Figure 4.

Geoid height differences [m] between GOCO01S and ITG-Grace2010s at degree/order (a) 100 (max. absolute deviation: 8 mm, rms: 1 mm); (b) 180 (max. abs. dev.: 1.20 m, rms: 0.23 m).

4. Validation Against External Gravity Field Data

[17] A comparison of GOCO01S with independent global gravity field models such as EGM2008 [Pavlis et al., 2008] or EIGEN-5C [Förste et al., 2008], which are combined solutions of GRACE, terrestrial gravimetry and satellite altimetry, leads to similar conclusions as for the ESA GOCE-only model [Pail et al., 2010]. Large deviations appear in regions where the terrestrial gravity data are known to be of low accuracy, e.g., South America, Africa, or Himalaya (not shown).

[18] GOCO01S was also externally validated by means of comparisons with independent geoid height observations determined from GPS and leveling in several regions (for a description of the methodology see Gruber [2009]). Models were truncated and analyzed at various degrees. Table 1 shows the standard deviations computed from the geoid height differences for the three regions Germany, Japan, and Canada.

Table 1. RMS of Geoid Height Differences σN [cm] Between Gravity Field Models and GPS/Leveling Observations in Selected Regionsa
ModelGermany σN [cm]Japan σN [cm]Canada σN [cm]
  • a

    Germany (675 points), Japan (873 points), Canada (430 points), truncated at degree/order Nmax.

Nmax = 100
ESA GOCE-only4.210.814.6
Nmax = 150
ESA GOCE-only5.311.214.7
Nmax = 170
ESA GOCE-only5.710.715.4
Nmax = 200
ESA GOCE-only15.113.918.2

[19] Evidently, combining GOCE and GRACE information in a consistent way further improves the performance of the single mission gravity field models in all spectral ranges. The multi-year GRACE normal equations dominate the combined solution up to degree and order 100, while the 2 months of GOCE data significantly contribute from degree and order 150 onwards. Between degree 100 and 150 both missions overlap. Compared to the ESA GOCE-only model [Pail et al., 2010], the combination with GRACE results in a significant improvement of the low to medium degrees.

5. Conclusions

[20] The gravity field model GOCO01S represents a consistent combination of GRACE and GOCE information at normal equation level. Due to the fact that neither the GRACE nor the GOCE normal equations contain gravity prior information, it is a pure satellite-only model using data from two complementary and independent satellite missions. The strength of such a pure model lies in the fact that it can be used for an independent comparison with terrestrial gravity field data, and subsequently for a consistent combination of satellite and terrestrial data. In ocean regions it provides a pure high-resolution geoid, which is independent of altimetry, and which will be beneficial for oceanography and the derivation of dynamic ocean topography. In geophysics, it can be applied for an improved modelling of 3D density structures and lithospheric mass distribution and transport, also in regions where terrestrial gravity data are sparse and of low quality.

[21] The low to medium degrees of GOCO01S are mainly determined by ITG-Grace2010s, due to the superior performance of the GRACE low-low inter-satellite tracking concept. However, in spite of the fact that only 2 months of GOCE data have been used, significant contributions to the combined solution can already be observed at degree 100, and GOCE becomes dominant beyond degree 150. Correspondingly, with GOCO01S a global gravity field model with utmost performance for the complete spectral range up to degree/order 224 is now available. The model is available via, or at