GOCO06s is the latest satellite-only global gravity field model computed by the GOCO (Gravity Observation Combination) project. It is based on over a billion observations acquired over 15 years from 19 satellites with different complementary observation principles. This combination of different measurement techniques is key in providing consistently high accuracy and best possible spatial resolution of the Earth's gravity field.

The motivation for the new release was the availability of reprocessed observation data for the Gravity Recovery and Climate Experiment (GRACE) and Gravity field and steady-state Ocean Circulation Explorer (GOCE), updated background models, and substantial improvements in the processing chains of the individual contributions. Due to the long observation period, the model consists not only of a static gravity field, but comprises additionally modeled temporal variations. These are represented by time-variable spherical harmonic coefficients, using a deterministic model for a regularized trend and annual oscillation.

The main focus within the GOCO combination process is on the proper handling of the stochastic behavior of the input data. Appropriate noise modeling for the observations used results in realistic accuracy information for the derived gravity field solution. This accuracy information, represented by the full variance–covariance matrix, is extremely useful for further combination with, for example, terrestrial gravity data and is published together with the solution.

The primary model data consisting of potential coefficients
representing Earth's static gravity field, together with
secular and annual variations, are available on the International Centre for Global Earth Models (

Supplementary material consisting of the full
variance–covariance matrix of the static potential coefficients and estimated co-seismic mass changes is
available at

Global models of Earth's static gravity field are crucial for geophysical and geodetic
applications.
These include oceanography
(e.g.,

The era of dedicated gravity field satellite missions

Although SST-hl is sensitive to shorter wavelengths due to the generally lower
satellite altitude compared to SLR, the spatial resolution is still limited.
With the Gravity Recovery and Climate Experiment (GRACE) twin satellite mission launched in 2002

To increase the spatial resolution, tailored to the requirements of geodetic, geophysical and oceanographic applications,
the Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission

Within typical processing workflows, mission (or technique) specific gravity field models are derived by
technique-specific experts and published as mission/technique-only gravity field models.
These models are published in terms of a spherical harmonic expansion and reflect the state-of-the-art analysis.
Nowadays, these models are more often equipped with a full covariance matrix, which reflects the error structure
specific to the analysis and observation technique
(e.g.,

With a full covariance matrix in combination with a solution vector, the original
system of normal equations can be reconstructed.
Together with additional meta-information, such as the number of observations and the weighted
square sum of the observation vector, multiple gravity field solutions can be
used to derive

Primary focus of combined models is the static gravity field
(e.g.,

A specific and updated series of combined satellite-only models is provided by the EIGEN-*S series which started as a
pure CHAMP-only model

This paper continues the global satellite-only models of the GOCO*s series, which are produced by the Gravity Observation Combination (GOCO) consortium (

GOCO satellite-only models are widely used and accepted as state of the art by the community.
The models published so far (GOCO01s, GOCO02s, GOCO03s, and GOCO05s) were used in a wide range of applications,
for example, for global and local high-resolution models
(e.g.,

Within this contribution, the latest release – GOCO06S

The GOCO models make use of the normal equations of the time-wise gravity field models

Within GOCO06s, normal equations from the time-wise RL06 model (GO_CONS_EGM_TIM_RL06,

Due to the sensitivity of the gravity gradients and their spectral noise characteristics, normal equations are assembled only with respect to the static part of the Earth's gravity field.
To define a clear reference epoch and being consistent with the GRACE processing, the time-variable gravity signal was reduced from the along-track gravity gradient observations in advance.
For this purpose, the models as summarized in Table

Models used to reduce the temporal gravity changes.

It enters the combination procedure (see Sect.

The GRACE contribution to GOCO06s consists of a subset of the normal equations of ITSG-Grace2018s

For GOCO06s, we made use of kinematic orbit positions from nine satellites.
In addition to GRACE A/B (see Sect.

To stabilize the long-wavelength part of the spectrum, we added
SLR observations to the combination.
The observations used match the GRACE time span from April 2002 to August 2016 and
feature a total of 10 satellites,
including LAGEOS 1/2, Ajisai, Stella, Starlette, LARES, LARETS,
Etalon 1/2 and BLITS.
The SLR observations were processed in weekly batches consisting of three 7 d arcs and one arc of variable length to complete the month, resulting in systems of normal equations

Relative weights of SLR satellites determined through VCE.

Additionally, we applied a Kaula constraint to stabilize the system of equations.
The final normal equation matrix and right-hand side used in the combination procedure are then computed as the weighted sum of all satellites,

We combined the individual contributions from GOCE, GRACE, kinematic LEO orbits, SLR and constraints on the basis
of normal equations using VCE.
VCE is a widely used technique in geodesy for
combining different observation groups

The accumulation of the individual contributions in Eq. (

Structure of the combined normal equation coefficient matrix, with all contributions overlaid. The different parameter groups for trend, annual and static potential coefficients are highlighted and plotted to scale.

We estimate 211 788 parameters in total with 90 597 to describe the static gravity field and
121 191 parameters for the co-estimated temporal variations.
Another 121 191 parameters representing co-seismic mass change caused by major earthquakes (see
Sect.

In order to properly decorrelate the co-estimated temporal variations from the long-term mean field, we constrained the corresponding parameters. While the previous solution, GOCO05s, featured a simple Kaula constraint, GOCO06s employs regionally varying prior information to account for the greatly different signal levels in individual regions. Using a globally uniform signal model, which a Kaula-type regularization provides, damps the secular signal in, for example, Greenland while overestimating the expected signal level in the ocean. To avoid this undesired behavior, but still introduce as little prior information as possible, we developed a tailored regularization strategy.

As a first step, the globe was subdivided into regions with similar temporal behavior
such as the ocean,
Greenland, Antarctica, the Caspian Sea and the remaining land masses.
For each region

Even though the observation contribution to the temporal variations is band-limited to degree
and order 120, as outlined in Sect.

For each region the unknown signal level

Co-estimated secular variation of GOCO05s

The impact of this novel approach compared to the Kaula-type regularization of the preceding
GOCO satellite-only model on the estimated secular variations can be seen in Fig.

When dealing with combined satellite models, the contribution of each component is of particular interest. It clearly reveals the strengths and weaknesses of the different techniques in determining specific parts of the spherical harmonic spectrum.

We compute the contribution of each component

The main diagonal of

Individual contribution to the static part of the estimated potential coefficients
for

Figure

SLR primarily contributes to degree 2 and even zonal coefficients up to degree 12, with a minor
contribution to the near-sectorials around degree 15.
These findings are consistent with

Contribution of

Contribution of

We find a similar picture when looking at the
contributions for the estimated trend and
annual oscillation shown in Figs.

The simple parametrization of Earth's gravity field with static, trend and annual signal basis functions
cannot capture instantaneous gravity changes caused by, for example, large earthquakes.
This mismodeling results in an apparent secular variation in the affected regions as the
co-seismic gravity change is mapped into the trend estimate.
To avoid this behavior, we estimate an additional step function in regions where co-seismic
mass change is expected, thus improving the description of the temporal evolution
of Earth's gravity field.
The methodology is exemplified on the basis of a single earthquake dividing
the whole observation time span into two intervals

Estimated co-seismic mass change for all modeled earthquakes (230 km Gaussian filter applied).

The estimated co-seismic mass changes can be seen in Fig.

Comparison of estimated secular variation from GOCO05s, GOCO06s (including estimated
co-seismic mass change) and filtered GRACE monthly solutions in terms of
equivalent water height (EWH). The gravity field solutions were evaluated
at 94.1

We can clearly observe that adding the co-estimation of co-seismic events greatly improves the accuracy of the secular variations. However, the monthly solutions show different rates before and after the event, which can obviously not be modeled by just a uniform trend over the whole observation time span. This is however a deliberate trade-off to retain redundancy in the trend estimates and simple usability of the data set.

The complete published data set of GOCO06s consists of a static gravity field solution
up to degree and order 300, an unconstrained system of normal equations of the static part for further combination, secular and annual gravity field variations up to degree and order 200, and
co-seismic mass changes for three major earthquakes.
All components are published in widely used data formats such as the ICGEM format for potential coefficients

The data product of primary interest for the community certainly is the estimated static gravity field together with its uncertainty information represented by the system of normal equations. Therefore, we focus our evaluations and discussions on these components.

Difference degree amplitudes of various state-of-the-art satellite-only
models compared to the combined model XGM2016 (polar cap with 8

Figure

Difference degree amplitudes compared to GOCO06s (solid lines) and corresponding formal errors (dashed lines) of the individual GOCO06s components (polar cap with
8

Figure

Over-land geoid heights (height anomalies or geoid undulations) computed from global gravity models can be validated against independent geoid observations as they are determined by on ground GNSS leveling.
By subtracting physical heights determined by spirit leveling (orthometric or normal heights) from ellipsoidal heights as they are observed with GNSS, one can compute such independent geoid heights.
The challenge when comparing geoid heights derived from satellite-only gravity field models to observed geoid heights mainly is to eliminate the so-called omission error.
This error, which represents the higher-resolution geoid signal not observable by satellites due to the gravitational signal attenuation at satellite altitude, causes the major part of the differences leading to unrealistic quality estimates for the satellite-only gravity model.
Therefore, the omission error needs to be estimated from other sources and taken into account prior to computing the differences.
For this purpose, high-resolution gravity field models such as EGM2008

For validating the GOCO06s model a number of global gravity field models and a number of GNSS-leveling data sets is applied.
For gravity field models the previous satellite-only models of the GOCO series are used (GOCO01s, GOCO02s, GOCO03s and GOCO05s).
These models are characterized by combining GRACE and GOCE satellite data with an increasing number of observations and some additional satellite data (see Sect.

Root mean square of geoid height differences between global gravity models and independently observed geoid heights from GNSS leveling. Global models are truncated with steps of 10

Figure

Regarding this it becomes obvious that the German data set seems to be the best ground data set, while the Brazilian data set on average suffers from some inaccuracies, which are due to the size of the country and other geodetic infrastructure weaknesses. For Japan and the United States the rms of differences is at a level of 7.5 and 9.5 cm respectively. The general shape of the curves is completely different for Brazil and the other data sets under investigation. The reason behind this is that the more information from the EGM2008 model (without GOCE data) is used for computing the omission error the worse the result of the differences is. This indicates that GOCE data can significantly improve the overall performance of the global gravity field models in case no high-quality ground data are available. In the other areas we consider here high-quality ground data were used in EGM2008, and therefore it performs very well when most of the signal is computed from this model (i.e., for low truncation degrees). Looking to the different satellite-only solutions one can identify the limit of a pure GRACE solution (ITSG-Grace2018s) as the rms of the differences starts to increase at lower degrees than for the other models. Including GOCE (and more GRACE) data significantly improves the higher-resolution terms of the satellite-only models as it can be seen from the series from GOCO01s to GOCO06s, where the higher release number means more satellite information included. When comparing the satellite-only models to the combined XGM2019e model, the degree where they start to diverge somehow represents the maximum signal content of the satellite-only models, or in other words up to what degree and order such a model contains the full gravity field signal. Depending on the area, we can identify that around degree 200 the models start to diverge. For Germany, where we probably have the best ground data set, the satellite contribution is superior and may be up to degree 180. Regarding the GOCO06s model we can state from these comparisons that it is a state-of-the-art satellite-only model and that it performs best together with the GOCE-DIR6 and GOCE-TIM6 models, which on the higher end of the spectrum are all based on the same complete GOCE data set.

To evaluate the long-wavelength part of GOCO06s, we analyze orbit residuals between integrated
dynamic orbits and GPS-derived kinematic orbits of four LEO satellites of various altitudes and
inclinations.
The missions used with corresponding altitude and inclination are GOCE (

First, a dynamic orbit for each satellite is integrated based on a fixed set of geophysical models, where only the static gravity field is substituted.

Background models used in the dynamic orbit integration.

The models used can be found in Table

Since satellites are not only affected by gravitational forces, but also non-conservative forces
such as atmospheric drag, solar radiation pressure and Earth radiation pressure, both GOCE and GRACE are equipped with accelerometers which measure the impact of these forces.
The other satellites (Jason-2, TerraSAR-X) do not feature such an instrument; therefore, we make use of models to compute the effect.
Specifically, we model the impact of atmospheric drag

We compare GOCO06s with its predecessor GOCO05s and other recent static gravity field models (see Fig.

Root mean square of differences between integrated dynamic and GPS-derived kinematic orbits in centimeters for four selected months during the GOCE measurement time span for satellites of different altitude and inclination.

From the computed rms values we can conclude that the most recent combination models (GOCO06s and DIR6) perform nearly equally well, with GOCO06s having a slight edge. Furthermore, we can see that there is a quality jump from the previous release, GOCO05s. The GOCE-only model TIM6 performs the worst for all satellites, which highlights the importance of GRACE for stabilizing the long to medium wavelengths and the polar gap. The overall larger rms values for TerraSAR-X and Jason-2 reflect the challenges in modeling non-conservative forces for these satellites. Combined with the higher altitude, the contrast between the individual static gravity field solutions becomes very small. Still, we observe the same tendencies as for GOCE and GRACE.

Next to the validation of static gravity fields, orbit residuals are also very useful in
evaluating the temporal constituents of gravity field solutions.
We gauge the quality of the co-estimated temporal constituents of GOCO06s by
integrating dynamic orbit arcs for GRACE Follow-On 1 (GRACE-C).
We compare GOCO06s to EIGEN-GRGS RL04, a recent gravity field model which also features a time-variable part; the previous solution, GOCO05s; and a GRACE-FO monthly solution.
Instead of the trend and annual variation estimated from CSR RL06, we use the temporal constituents
of the gravity fields to be compared.
Both GOCO06s and GOCO05s provide secular and annual variations, while EIGEN-GRGS RL04
has secular, annual and semi-annual variations estimated for shorter intervals.
We perform the evaluation for January 2020, which is well beyond the GRACE measurement time
span.
This choice is deliberate to assess the extrapolation capabilities of the
compared gravity fields.
The computed rms values can be found in Table

Root mean square of differences between integrated dynamic orbit and GPS-derived kinematic positions for GRACE-C in January 2020 in centimeters.

We can see that GOCO06s outperforms both its predecessor and EIGEN-GRGS RL04 while, unsurprisingly, the GRACE-FO monthly solution performs best.

The primary model data consisting of potential coefficients
representing Earth's static gravity field, together with
secular and annual variations, are available on ICGEM

Supplementary material consisting of the full
variance–covariance matrix of the static potential coefficients and estimated co-seismic mass changes is
available at

The satellite-only gravity field model GOCO06s provides a consistent combination of spaceborne gravity observations from a variety of satellite missions and measurement techniques. Each component of the solution was processed using state-of-the-art methodology which results in a clear increase in the solution quality compared to the preceding GOCO solutions and the individual input models. All contributing data sources were combined on the basis of full normal equations, with the individual weights being determined by VCE. The long to medium spatial wavelengths covered by GOCO06s are mainly determined by GRACE due to the high sensitivity of the inter-satellite ranging observation to this frequency band. SLR primarily contributes to degree 2, while the kinematic LEO orbits mainly contribute to the sectorial coefficients up to degree and order 150. Finally, the medium to short wavelengths of the solution, starting from degree 120, are dominated by the GOCE gradiometer observations. To reduce the energy in the higher spherical harmonic degrees, a Kaula-type regularization was applied from degree 151 to 300. The complementary nature of the data that were used mitigates weaknesses of the individual observation types, thus providing the highest accuracy throughout the spatial frequency band covered by the solution.

Since an appropriate stochastic observation model is used during processing, GOCO06s exhibits
realistic formal errors, which alleviates further combination with additional, for example,
terrestrial gravity field observations

AK computed the GRACE and LEO normal equations, developed and implemented the regularization strategies for the temporal constituents and earthquakes, performed the combination of all individual data contributions, performed the orbit validation, and partially wrote the manuscript. JMB computed the GOCE SGG system of normal equations and partially wrote the manuscript. TS and WDS contributed to the GOCE processing. SK computed and provided the systems of normal equations for all SLR satellites. TG performed the model validation with in situ GNSS-leveling data. UM provided the kinematic orbits of GRACE and GOCE used in the validation. TMG, AJ, WDS and RP acted as scientific advisors. All authors commented on the draft of the manuscript and on the discussion of the results.

The authors declare that they have no conflict of interest.

Jan Martin Brockmann and Wolf-Dieter Schuh gratefully acknowledge the Gauss Centre for Supercomputing e.V. (

Precise orbit determination of SLR satellites was accomplished with the software package GEODYN, kindly provided by the NASA Goddard Space Flight Center.

GNSS-leveling data sets applied in this study have been kindly provided by the Brazilian Institute of Geography and Statistics – IBGE, 2019 (Brazil); by the GeoBasis-DE/Geobasis NRW, 2018 (Germany); by the Japanese Geographical Survey Institute, 2003 (Japan); and by the National Geodetic Survey, 2012 (USA). The authors are grateful to the institutions who made available the GNSS-leveling data as they provide a unique reference for validating global gravity field models.

The authors gratefully acknowledge the ISDC at the German Research Centre for Geosciences for providing Champ, TerraSAR-X and TanDEM-X data; the European Space Agency for providing GOCE and SWARM data; Aviso for providing Jason data; and the International Laser Ranging Service for providing satellite laser ranging measurements.

This research has been supported by the ESA GOCE HPF (grant no. 18308/04/NL/MM).

This paper was edited by Christian Voigt and reviewed by two anonymous referees.