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. Those coefficients are provided as 'Level-2b' products for users who wish to undertake surface mass inversion starting from spherical harmonic coefficients by themselves.

GRACE 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 the GRACE Level-3 products to other data or models, all time series should refer to the same reference epoch.

All Level-3 products currently presented at GravIS refer to the mean over the period from 2002/04 up to and including 2016/08.

The spherical harmonic coefficient of degree 2 and order 0 (C_{20}) is related
to the flattening of the Earth.
Since it is known that monthly GRACE estimates of C_{20} are affected by spurious
systematic effects (e.g. Cheng & Ries, 2017), the C_{20} coefficients and their
formal errors are replaced by estimates derived from satellite laser ranging (SLR)
observations that are regarded to be more reliable.

Here, we use a C_{20} time series based on six SLR satellites (LAGEOS-1 and -2,
AJISAI, Stella, Starlette, LARES) that is processed at GFZ using the same background
models and standards as applied during GFZ GRACE processing, including the Atmosphere
and Ocean De-aliasing model AOD1B.
The relative weighting of the six individual satellites is based on a variance component
estimation.

Cheng, M., Ries, J., 2017:

**The unexpected signal in GRACE estimates of C _{20}**

Journal of Geodesy, 91, 8, p. 897—914 , https://doi.org/10.1007/s00190-016-0995-5

Cheng, M., Tapley, B., Ries, J., 2013:

**Deceleration in the Earth's oblateness**

Journal of Geophysical Research: Solid Earth, 118, p. 740—747
,
https://doi.org/10.1002/jgrb.50058

The response of the solid Earth and oceans to the Earth's polar motion, also referred to
as pole tide, causes a gravitational effect which is mainly reflected by the spherical
harmonic coefficients of degree 2 and order 1 (C_{21}, S_{21}).
During GRACE Level-2 processing, the pole tide is regarded as background model and its
effect should thus be removed from monthly gravity field solutions.
However, Wahr et al. (2015) show that the pole tide correction is incompletely modelled
during GRACE RL05 processing and recommend an additional pole tide correction to be applied
to the monthly GRACE RL05 Level-2 products.

Here, we follow this recommendation and correct the C_{21} and S_{21}
coefficients according to equation (22) in Wahr et al. (2015).

Wahr, J., Nerem, R.S., Bettadpur, S., 2015:

**The pole tide and its effect on GRACE time-variable gravity measurements: Implications for estimates of surface mass variations**

Journal of Geophysical Research: Solid Earth, 120, p. 4597—4615
,
https://doi.org/10.1002/2015JB011986

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, degree 1 and C_{20} coefficients are also affected.

The Level-2b/Level-3 products provided here are corrected for a GIA model based on ICE-5G ice load history (Peltier, 2004) and applied in the 3D-Viscoelastic Lithosphere and Mantle Model VILMA (Martinec, 2000; Klemann et al., 2008).

Peltier, W.R., 2004:

**Global Glacial Isostasy and the Surface of the Ice-Age Earth: The ICE-5G(VM2) Model and GRACE**

Annual Review of Earth and Planetary Sciences, 32, p. 111—149
,
https://doi.org/10.1146/annurev.earth.32.082503.144359

Martinec, Z., 2000:

**Spectral-Finite Element Approach for Three-Dimensional Viscoelastic Relaxation in a Spherical Earth**

Geophysical Journal International, 142, p. 117—141
,
https://doi.org/10.1046/j.1365-246x.2000.00138.x

Klemann, V., Martinec, Z., Ivins, E.R., 2008:

**Glacial Isostasy and Plate Motions**

Journal of Geodynamics, 46, p. 95—103
,
https://doi.org/10.1016/j.jog.2008.04.005

The spherical harmonic coefficients of degree 1 (C_{10}, C_{11}, S_{11})
are related to the distance between the Earth's centre of mass (CM) and centre of figure (CF).
However, GRACE is insensitive to variations in the offset between CM and CF (denoted as geocentre
motion), i.e., the coefficients C_{10}, C_{11} and S_{11} are not estimated
and thus are set to zero by definition.

To add information about geocentre motion to our Level-2b/Level-3 products, we have estimated degree 1 coefficients according to the method of Swenson et al. (2008) using only information of monthly GRACE gravity fields in an iterative procedure as outlined in Bergmann-Wolf et al. (2014).

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
,
https://doi.org/10.1029/2007JB005338

Bergmann-Wolf, I., Zhang, L., Dobslaw, H., 2014:

**Global eustatic sea-level variations for the approximation of geocenter motion from GRACE**

Journal of Geodetic Science, 4, p. 37—48
,
https://doi.org/10.2478/jogs-2014-0006

A global ocean tide model is used as a background model during GRACE 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 typically related to errors in the S2 tide of both ocean and atmosphere. A harmonic signal at this frequency is estimated over the whole GRACE mission period and subsequently subtracted from each of the monthly Level-2 products.

Stammer, D. et al., 2014:

**Accuracy assessment of global barotropic ocean tide models**

Reviews of Geophysics, 52, 3, p. 243—282
,
https://doi.org/10.1002/2014RG000450

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
,
https://doi.org/10.1007/s10712-015-9338-y

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
,
https://doi.org/10.1007/s00190-013-0671-y