Description of the solutions

The GRACE and GRACE-FO missions

The GRACE (Gravity Recovery and Climate Experiment; Tapley et al., 2004; Tapley et al., 2019) and GRACE-FO (GRACE Follow-On; Chen et al., 2022; Landerer et al., 2020) satellite missions have, since 2002, provided monthly estimates of variations in the Earth’s gravity field and of  mass redistributions on its surface or at depth. These redistributions include mass changes in ice sheets, movements of water masses linked to hydrological cycles, and displacements of mantle masses associated with glacial isostatic adjustment (GIA) and earthquakes. The GRACE and GRACE-FO missions consist of twin satellites equipped with a Star Camera Assembly (SCA), an accelerometer (ACC), GPS receivers, and K-band microwave transmitters (K-Band Ranging or KBR). The KBR data are processed as the relative velocity between the satellites, known as the K-Band Range-Rate (KBRR). These data are used to estimate Earth’s geopotential anomalies by inverting the normal equations generated by satellite orbit adjustments. This inversion yields the coefficients of the spherical harmonic decomposition of the geopotential, known as the Level 2A (L2A) solution. Within the framework of the unconstrained solution developed by CNES, classical inversion based on a Cholesky decomposition does not reduce the anisotropic noise in the solutions (see Lemoine et al., 2026 for more details). It is therefore necessary to apply Gaussian or DDK spatial filters (Kusche et al., 2009) to extract the geophysical signals of interest for Earth observation.

Files description

The time series of unconstrained GRACE CNES solutions is provided in the form of spherical harmonic coefficients up to degree and order 90, excluding degrees 0 and 1 (8,277 coefficients), as the GRACE and GRACE-FO satellites cannot estimate them accurately. These solutions are obtained through unconstrained inversion using a Cholesky decomposition (more details are available here). The solutions are distributed in ASCII files in the extended GRACE format, which is an extension of the original GRACE gravity model format (document GR-GFZ-FD-001 GRACE 327-732 (v1.1), November 27, 2003, available here). The coefficients are sorted in ascending order by degree and order. An example file is provided below.

Solution Quality Metrics

An initial assessment of the quality of the solutions can be found in the spectral amplitude of the coefficients. The GRACE/-FO satellites orbit the Earth approximately 15 times a day, so noise peaks are expected at degrees multiples of 15 (15, 30, 45, 60, …). This type of peak is an indicator of a noisier solution, as shown by the example of the 2025-11 solution illustrated in the figure below.

Due to the geometry of the GRACE/-FO satellites, the solutions have significant anisotropic noise that manifests as vertical stripes once converted to a grid (regardless of the unit), rendering the solution unusable in its current state. To reduce this noise, a filter must be applied to the harmonic coefficients. Generally, a DDK (Denoising Decorrelation Kernel) filter is applied to reduce noise, which decreases spatial resolution but makes the solution usable. You can find more details on the DDK filtering method here. Applying a DDK filter significantly reduces the amplitude of the spherical harmonic coefficients starting from a certain degree depending on the order of the DDK filter. Consequently, displaying the coefficients after applying the filter allows for the identification of abnormally high values, which are likely to generate anisotropic noise during the conversion of solutions to a grid, and provides a good estimate of the solution’s noise level. An example of a harmonic coefficient tree is shown below.

Using the solutions

In order to use these solutions, it is necessary to express the gravitational potential at every point on Earth. The Earth’s gravitational potential is expressed as follows:

Where r, φ and λ are, respectively, the radius, latitude, and longitude of the considered point, Cₙₘ and Sₙₘ re the harmonic coefficients of the solution, G es the universal gravitational constant, M eis the mass of the Earth, a is the Earth’s equatorial radius, P̄ₗₘ is the associated Legendre function of degree l and order m.

The solutions of the GRACE/-FO geopotential allow us to track variations in mass on the Earth’s surface, particularly water masses, since the solutions are generally expressed as variations in a thin layer of water on its surface. In this case, the previous equation becomes, when calculated up to a maximum degree Lₘₐₓ :

Où :

– ΔHᵂ(φ, λ) is the equivalent water height at latitude φ and longitude λ,
– a is the Earth’s average radius,
– ρᴱ is the average Earth density,
– ρʷ is the water density,
– kₗ is the Load Love Number (LLN) at any harmonic degree l,
– Ȳₗₘ are the Legendre polynomial functions for any degree l and order m, in complex notation
– ΔCₗₘ are the increments, relative to a reference gravitational solution, of the complex spherical harmonic coefficients of the solution.

More details about the method can be found in Ditmar (2018).

Dataset identifier

10.24400/170160/SAGSA_GGM_CHO_1MONTH_RL0005

Characteristics

	Product type	Stokes coefficients
	Format	ASCII files
	License	(CCBY)
	Start of production	GRACE: 01/04/2002 | GRACE-FO: 01/05/2018
	End of production	GRACE: 01/05/2017 | GRACE-FO: Still producing
	Coverage	Global
	Coverage type	Spherical harmonics
	Spatial resolution	Maximum degree: 90
	Time resolution	Monthly solutions
	Mission(s)	GRACE | GRACE-FO
	Instrument(s) / Captor(s)	Star Camera Assembly (SCA), Accelerometer (ACC), K-Band Ranging (KBR), GPS