3D atmospheric dealiasing product (ERA5 global reanalysis, 1980-2023, 1h sampling)


The Atmospheric Time-Variable Gravitational Potential (ATVGP) is generated by the time variable distribution of the atmospheric density field. The density field depends on the time, the latitude, the longitude and the radial distance. It constantly changes with the dynamics of Atmosphere. Hence the hybrid coordinates of the atmospheric levels in which the quantities are given change not only in time but also in the 3D space. At each epoch the spatial structure of the density field has to be rebuilt. And the density has to be infered from the atmospheric state. The density depends on the thermodynamic conditions but also on the constituents which are the dry air and the water content in different phases. The hydrometeors are the cloud rain water content, the cloud snow water content, the cloud liquid water content and the cloud ice water content.

The Atmospheric Time-Variable Gravitational Potential (ATVGP) includes the direct attraction of masses on the satellite above the atmosphere. The direct attraction is simply given by an upward continuation of the gravitational potential above the atmospheric masses. The atmosphere deforms the Earth since the atmospheric surface pressure stress the Earth surface and the gravitational potential generated at the Earth surface change. These two boundary conditions exerted on the Earth’s surface deform the Earth, which in turn changes its shape and its gravitational potential. A second downward continuation of the gravitational potential generated by the heterogenous density field is required to apply this boundary condition. The Earth’s gravitational potential due to atmospheric loading is simply given by Love numbers, solutions of the gravito-elasticity problem. The Love numbers are the proportionaly factors between the applied boundary conditions and its consequence in terms of gravitational potential.
The gravitational potential due to the Earth’s deformation induced by atmospheric loading is added to the direct attraction. Direct attraction and Earth’s atmospheric loading are added to give the (total) Atmospheric Time-Variable Gravitational Potential (ATVGP).

The time-variable gravitational potential is realized from the ERA5 global reanalysis at the ECMWF by the Copernicus Climate Change Service (C3S). ERA5 is the fifth generation ECMWF atmospheric reanalysis of the global climate covering the period from January 1940 to present. ERA5 is produced by the Copernicus Climate Change Service (C3S) at ECMWF. ERA5 provides hourly estimates of a large number of atmospheric, land and oceanic climate variables. The data covers the Earth on a 30km grid and resolve the atmosphere using 137 levels from the surface up to a height of 80km. ERA5 includes information about uncertainties for all variables at reduced spatial and temporal resolutions. Daily and monthly aggregates of the hourly fields are available as well.

The hourly time-variable global gravitational potential is defined relatively to a reference gravitational potential. The reference (mean) gravitational potential of the atmosphere is realized by the mean field generated by the mean of monthly mean density structure. This mean structure is calculated over a 23 years span (from 2000 to 2022) from atmospheric monthly means provided by the ERA5 global reanalysis (MODA). The state of the atmosphere is provided hourly by the model levels of the ERA5 global reanalysis fields. The density field is recovered from the temperature, the surface pressure, the geopotential, the specific humidity, and the specific cloud rain/snow/liquid/ice water contents, at the native vertical and horizontal resolutions i.e. on the native reduced gaussian grid of the model.

The daily ATVGP files contain:

  • a header report of the autosensed ECMWF inputs (class, stream, expver, type, levtype), the MPI tasks and OpenMP threads per task of the partitioning, the specifications of the reduced gaussian grids and the spherical harmonics and their domain partitioning on the available cores and nodes;
  • twenty four hourly spherical harmonics expansions of the global atmospheric time-variable gravitational potential up to the degree 180 (AOD1B format).

The size of one daily (text) file is 17 MB. The size of one yearly compressed (gzip) archive (tar) is 2.3 GB. The size of one yearly uncompressed folder is 6 GB. The size of the fourty four years archived dataset is 100 GB. The size of the fourty four years uncompressed dataset is 264 GB.
The command « tar xvfz ATVGP_YEAR.tar.gz » uncompress and extract files from the (tar) archive.

Data download: ERA5 atmospheric dealiasing


The ERA5 atmospheric model outputs are not free from jumps every 12 hours. As stated by the authors of the article « The ERA5 global reanalysis » (DOI 10.1002/qj.3803, Hans Hersbach et al.) : « Although within the 12-hr assimilation windows the model constraint ensures a smooth hourly product, analysis increments can introduce systematic jumps at the transition points between the windows. »


The source code atvgp.f90 computes the Atmospheric Time-Variable Gravitational Potential as described in ”Background Models used in Geodetic Data Processing”. It is an hybrid MPI/OpenMP code designed for the multi-nodes and multi-cores architecture of the High Performance Computing facilities.

It relies on the Fast Spherical Harmonics Transform for Global Numerical Weather Prediction and Climate Models (Nils P. Wedi, Mats Hamrud, and George Mozdzynski) operationally used in the Integrated Forecast System (IFS) at the European Center for Medium-range Weather Forecast (ECMWF).

• References: IFS DOCUMENTATION – Cy47r3 – Operational implementation 12 Oct 2021 – PART VI: TECHNICAL AND COMPUTATIONAL PROCEDURES
• NEWSLETTER ARTICLE: An open-source Integrated Forecasting System, Michael Sleigh, Willem Deconinck, Michael Lange, Olivier Marsden, Balthasar Reuter
• ARTICLE: A Fast Spherical Harmonics Transform for Global NWP and Climate Models, Nils P. Wedi, Mats Hamrud, and George Mozdzynski Monthly Weather
Review, pp. 3450-3461, 2013. https://journals.ametsoc.org/view/journals/mwre/141/10/mwr-d-13-00016.1.xml
• ECMWF REPOSITORIES: . https://smex-ctp.trendmicro.com:443/wis/clicktime/v1/query?url=https%3a%2f%2fgithub.com%2fecmwf%2difs%2fectrans&umid=9b4bf7ad-eea9-49e6-8713-a4c99190ae90&auth=ca8f4c8bd45f1ec8d56a660a0cef9357eaf5c28d-c5d5ae7dab09dd2e63b0234dd5216c1eed9a994a . https://smex-ctp.trendmicro.com:443/wis/clicktime/v1/query?url=https%3a%2f%2fgithub.com%2fecmwf%2difs%2ffiat&umid=9b4bf7ad-eea9-49e6-8713-a4c99190ae90&auth=ca8f4c8bd45f1ec8d56a660a0cef9357eaf5c28d-1427b602831250b3e55011bdd2dc68e13f4e98bf . https://github.com/ecmwf
• The Atmospheric Time-Variable Gravitational Potential, P. Gegout, H. Legros, J. Hinderer, 2002. http://pnaf.oca.eu/Site_old/Agret02_gegout.ppt.gz
• Background Models used in Geodetic Data Processing, P. Gegout, F. Lyard, J.-M. Lemoine, IAG/GGOS Workshop: Towards a Roadmap for Future Satellite Gravity Missions – Sept. 30, 2009 – Graz – Austria http://igcp565.org/workshops/Graz/pdfs/B3_04_Gegout_BackgroundModels_IAG2009.pdf
• Dealiasing Products: The Time-variable Atmospheric and Oceanic Gravitational Potential from 1980 to 2017, Gegout, Pascal. (2020). Dealiasing Products: Time-variable Atmospheric and Oceanic Gravitational Potential from 1980 to 2017.
• Yang, F., Forootan, E., Wang, C., Kusche, J., & Luo, Z. (2021). A new 1-hourly ERA5-based atmosphere de-aliasing product for GRACE, GRACE-FO, and future gravity missions. Journal of Geophysical Research: Solid Earth, 126, e2021JB021926. https://doi.org/10.1029/2021JB021926