Corrections and Auxiliary Products

A number of corrections and reductions are applied to the Level-2 spherical harmonics to generate Level-3 products that represent variations of the Earth's surface masses as accurately as possible. These post-processed Level-2 coefficients, denoted as Level-2B products, are provided as an additional data set for users who wish to undertake surface mass inversion starting from spherical harmonic coefficients by themselves.

As for Level-3 products, GravIS Level-2B products are provided for the most recent releases of both GFZ and COST-G. These products as well as individual data sets and models used during the post-processing steps mentioned below are available at ISDC for GFZ and COST-G.

Mean Field

GRACE/GRACE-FO Level-3 products represent mass anomalies, i.e., positive or negative variations about a long-term mean gravity field of the Earth. Essentially, the choice of this mean field is arbitrary, since using a different mean field only introduces a constant bias to the time series of mass anomalies. However, when comparing these Level-3 products to other data or models, all time series should refer to the same reference epoch.

All Level-2B/Level-3 products currently available at GravIS refer to a long-term mean field calculated as unweighted average of the 183 available GFZ RL06 Level-2 products in the period from 2002/04 through 2020/03. The only exception in this regard are the G3P prototype products which refer to the mean over the period 2002/04 through 2020/12 (considering months that are absent in the GRACE/GRACE-FO time series).

Anisotropic Filtering

In order to optimally separate signal and noise in the GRACE/GRACE-FO Level-2 data, filtering is necessary. Due to the observation geometry with its pure along-track ranging on polar orbits GRACE and GRACE-FO gravity fields reveal highly anisotropic error characteristics. An adequate filter technique to account for this is the decorrelation method by Kusche et al. (2009), named DDK, which is deduced from a regularization approach using signal and error information in terms of variance and covariance matrices. The filtering is applied in the spectral domain by multiplying the filter matrix to the unfiltered spherical harmonic (SH) coefficients (residual with respect to a mean field). This method has been adapted by Horvath et al. (2018) taking into account the temporal variations of the error variances and covariances, denoted by VDK filtering.

Hence, our monthly Level-2B products are optionally decorrelated and smoothed with an adaptive filter that explicitly takes into account the formal covariance information of the corresponding monthly GFZ RL06 Level-2 product. We provide the following variants of Level-2B products: filtered with VDK1, VDK2, VDK3, VDK4, VDK5, VDK6, VDK7, and VDK8 (where a larger number means weaker filtering), as well as unfiltered (NFIL) solutions. These variants are distinguishable by respective strings in the product file names.

Kusche, J., Schmidt, R., Petrovic, S., Rietbroek, R., 2009:
Decorrelated GRACE time-variable gravity solutions by GFZ, and their validation using a hydrological model
Journal of Geodesy, 83, 10, p. 903—913,

Horvath, A., Murböck, M., Pail, R., Horwath, M., 2018:
Decorrelation of GRACE Time Variable Gravity Field Solutions Using Full Covariance Information
Geosciences, 8, 323,

Replacement of Particular Low Degree Harmonics

The spherical harmonic coefficient of degree 2 and order 0 (C20) is related to the flattening of the Earth. Since it is known that monthly GRACE estimates of C20 are affected by spurious systematic effects (e.g. Cheng & Ries, 2017), it is recommended to replace the C20 coefficients by estimates derived from satellite laser ranging (SLR) observations that are regarded to be more reliable. The same recommendation is also made for GRACE-FO.

In addition, recent analysis revealed that also the C30 coefficient is poorly determined from GRACE/GRACE-FO when accelerometer observations for one of the two spacecraft are not available or degraded and need to be transplanted from the other spacecraft. Again, it is recommended to replace the C30 coefficients in such corresponding months by SLR-derived estimates. Note that due to its harmonic properties, C30 has a large impact in particular on Antarctic ice-mass change recovery (Loomis et al., 2020).

Furthermore, and solely affecting the GFZ RL06 solutions, an anomalous behavior of the C21 and S21 coefficients, particularly during the last seven months of the GRACE period, is observed (Dahle et al., 2019).

As replacement time series, we use here a combination of SLR and GRACE/GRACE-FO on the level of normal equations generated at GFZ. The SLR part includes the six geodetic satellites LAGEOS-1 and -2, AJISAI, Stella, Starlette, and LARES (starting from March 2012) and uses the same background models and standards as applied during GFZ's GRACE/GRACE-FO processing. Relative weighting of the individual SLR normals is done by means of variance component estimation whereas relative weighting of the SLR-combined and GRACE/GRACE-FO normals is based on empirical weights. Gravity field coefficients up to degree/order 6x6 are estimated independently for each month. From these estimates, the following coefficients as well as their formal standard deviations are used to replace the corresponding values of the GFZ and COST-G solutions: C20 for the complete time series of GFZ and COST-G, C30 for all GFZ and COST-G solutions starting from 2016/11 and later, and C21/S21 for the complete time series of GFZ only.

It has to be mentioned that other replacement time series are available and that their choice can significantly impact mass change results. Based on findings by Loomis et al. (2019), the operational C20 product by the GRACE/GRACE-FO Science Data System was recently changed and is now provided as Technical Note 14 (TN-14), which also includes a C30 time series. Another operational SLR-only product for C20, based on the same normals as used for the above-mentioned combination with GRACE/GRACE-FO, is provided by GFZ (König et al., 2019). Generally, the estimation and validation of low degree harmonics and their uncertainties from space-geodetic techniques as well as their impact on mass balance studies is an ongoing research topic.

Cheng, M., Ries, J., 2017:
The unexpected signal in GRACE estimates of C20
Journal of Geodesy, 91, 8, p. 897—914,

Loomis, B. D., Rachlin, K. E., Wiese, D. N., Landerer, F. W., Luthcke, S. B., 2020:
Replacing GRACE/GRACE-FO C30 with satellite laser ranging: Impacts on Antarctic Ice Sheet mass change
Geophysical Research Letters, 47, e2019GL085488,

Dahle, C., Murböck, M., Flechtner, F., Dobslaw, H., Michalak, G., Neumayer, K.H., Abrykosov, O., Reinhold, A., König, R., Sulzbach, R., Förste, C., 2019:
The GFZ GRACE RL06 Monthly Gravity Field Time Series: Processing Details and Quality Assessment
Remote Sensing, 11, 2116,

Loomis, B. D., Rachlin, K. E., Luthcke, S. B., 2019:
Improved Earth oblateness rate reveals increased ice sheet losses and mass-driven sea level rise
Geophysical Research Letters, 46, 6910—6917,

Loomis, B. D., Rachlin, K. E., Technical Note 14:
NASA GSFC SLR C20 and C30 solutions
available at the GRACE/GRACE-FO archives ISDC and PODAAC,

König, R., Schreiner, P., Dahle, C., 2019:
Monthly estimates of C(2,0) generated by GFZ from SLR satellites based on GFZ GRACE/GRACE-FO RL06 background models. V. 1.0.
GFZ Data Services,

GIA Correction

Glacial Isostatic Adjustment (GIA) denotes the surface deformation of the solid Earth (lithosphere and mantle) caused by ice-mass redistribution over the last 100,000 years, dominated by the termination of the last glacial cycle. Due to the Earth's viscoelastic response to mass redistribution between the ice sheets and the ocean, the Earth's gravity field is affected by long term secular trends mainly in previously glaciated regions such as North America, Fennoscandia and Antarctica. Moreover, also coefficients of low degrees and orders are affected.

The Level-2B/Level-3 products provided here are corrected using the GIA model ICE-6G_D (VM5a) (Peltier et al., 2018).

Peltier, W.R., Argus, D.F., Drummond, R., 2018:
Comment on 'An Assessment of the ICE-6G_C (VM5a) Glacial Isostatic Adjustment Model' by Purcell et al.
Journal of Geophysical Research: Solid Earth, 123, p. 2019—2028,

Geocenter Coefficients

The spherical harmonic coefficients of degree 1 (C10, C11, S11) are related to the distance between the Earth's centre of mass (CM) and centre of figure (CF), which is commonly denoted as geocenter motion. However, a GRACE-like mission as sensor system is insensitive to CF so that the coefficients C10, C11 and S11 are not estimated and thus set to zero by definition.

To add information about geocenter motion to our Level-2B/Level-3 products, which is essential to correctly quantify both oceanic and terrestrial mass distributions, we approximate those coefficients according to the method of Swenson et al. (2008) and insert them into the Level-2B products. Apart from the effect of self-attraction and loading (SAL), which is not implemented in our approximation, the recommended setup as outlined by Sun et al. (2016) is used. Note that the geocenter approximation is done individually for the GFZ and COST-G products.

Swenson, S., Chambers, D., Wahr, J., 2008:
Estimating geocenter variations from a combination of GRACE and ocean model output
Journal of Geophysical Research: Solid Earth, 113, B08410,

Sun, Y., Riva, R., Ditmar, P., 2016:
Optimizing estimates of annual variations and trends in geocenter motion and J2 from a combination of GRACE data and geophysical models
Journal of Geophysical Research: Solid Earth, 121, p. 8352—8370,

Aliased Signal of the S2 Tide

A global ocean tide model is used as a background model during GRACE/GRACE-FO Level-2 gravity field processing to remove ocean tide signals. However, errors are present in ocean tide models (Stammer et al., 2014), and these errors are known to be amongst the largest error sources in GRACE-like gravity field recovery (Flechtner et al., 2016). Apart from model errors, additional gravity field errors are caused by temporal aliasing of ocean tide signals (e.g. Murböck et al., 2014).

A prominent alias frequency in the GRACE gravity fields has a period of 161 days which is likely caused by model errors of the semi-diurnal solar tide S2 present in both ocean and atmosphere. A harmonic signal at this frequency is fitted (together with bias, linear trend, annual, and semi-annual components) to the time series and subtracted from each monthly Level-2B product. This is done here individually for the GFZ and COST-G products. Note that when fitting the S2 tidal alias frequency to a combined GRACE and GRACE-FO time series, a phase offset of 100 degrees should be applied between the two missions as the nodal planes of both are not specifically aligned to each other (Landerer et al., 2020).

Stammer, D. et al., 2014:
Accuracy assessment of global barotropic ocean tide models
Reviews of Geophysics, 52, 3, p. 243—282,

Flechtner, F., Neumayer, K.H., Dahle, C., Dobslaw, H., Fagiolini, E., Raimondo, J.-C., Güntner, A., 2016:
What Can be Expected from the GRACE-FO Laser Ranging Interferometer for Earth Science Applications?
Surveys in Geophysics, 37, 2, p. 453—470,

Murböck, M., Pail,. R., Daras, I., Gruber, T., 2014:
Optimal orbits for temporal gravity recovery regarding temporal aliasing
Journal of Geodesy, 88, p. 113—126,

Landerer, F., Flechtner, F., Save, H., Webb, F., et al., 2020:
Extending the global mass change data record: GRACE Follow-On instrument and science data performance
Geophysical Research Letters, 47, e2020GL088306,

Contact: Christoph Dahle (dahle (at)