## 1 Introduction

[2] The Gravity Recovery and Climate Experiment (GRACE) has provided valuable information about the global integral effects of mass redistributions within the Earth system [*Kusche et al*., 2012and the references in it]. Yet, GRACE‒derived gravity field solutions display errors due to instrument noise (K‒band ranging system and GPS) [*Sheard et al*., 2012], anisotropic spatial sampling of the mission [*Schrama et al*., 2007; *Sneeuw et al*., 2004], and temporal aliasing caused by incomplete reduction of short‒term mass variations by models [*Flechtner et al*., 2010]. Increasing the spatial sampling isotropy has become a major issue in designing future missions dedicated to pursue the task of GRACE. The GRACE Follow‒On (GRACE‒FO) mission, scheduled for launch in 2017, still adopts the GRACE design of two satellites flying in one orbital plane. More sophisticated formation types are under discussion for a new generation of gravity missions beyond 2020. The options are investigated in a variety of studies, e.g., *Wiese et al*. [2009], *Elsaka* [2010], *Anselmi* [2010], and *NG2 Team* [2011]. These studies include single pair formations with a permanently changing link direction as well as parallel flights of two GRACE‒type missions, e.g., combining one single pair in a polar orbit with another pair flying in a relatively lower inclination of 63° as proposed by *Bender et al*. [2003]. Future gravity missions will thus have the potential to observe the temporal variations of the Earth's gravity field with a better accuracy, e.g., in *Visser et al*. [2010] and *Elsaka et al*. [2012].

[3] Isolating the long‒term and seasonal hydrological signals is one of the major applications of time‒variable satellite gravimetry products [*Tapley et al*., 2004]. Therefore, recovering monthly mean gravity field solutions, from the observations of satellite gravimetry missions, requires a careful reduction of the short‒term (e.g., subdaily to monthly) variations of the atmosphere and the oceans due to their dominant effects [*Han et al*., 2004]. Within the processing of satellite gravimetry data, it is common to reduce the high‒frequency nontidal oceanic and atmospheric mass redistributions from the level‒1 measurements by using “background” models [*Reigber et al*., 2005; *Flechtner*, 2007; *Mayer‒Gürr et al*., 2010a]. Otherwise, such high‒frequency mass changes will be aliased into long wavelength signals leading to misinterpretation of hydrological signals [*Velicogna et al*., 2001].

[4] GeoForschungsZentrum (GFZ) Potsdam routinely produces nontidal high‒frequency atmospheric and oceanic mass variation products called GRACE Atmosphere and Ocean Dealiasing level‒1B (GRACE‒AOD1B) products [*Flechtner*, 2007]. *Karbon et al*. [2011] showed that the current data‒processing strategy of the GRACE‒AOD1B [*Flechtner*, 2007] is sufficient for the current accuracy of GRACE monthly solutions. It should be mentioned here that, for the processing of the ITG‒GRACE2010 monthly gravity solutions at Bonn University, the daily Kalman filter‒based GRACE solutions [*Kurtenbach et al*., 2009] have been used to successfully improve the dealiasing procedure [*Mayer‒Gürr et al*., 2010b].

[5] The dealiasing process, however, represents still a concern for the quality of the gravity field solutions. For instance, several studies show that the accuracy of GRACE is above the simulated prelaunch baseline [see, e.g., *Meyer et al*., 2010], for which the errors within the dealiasing procedure are believed to be one of the potential causes. Comparing surface pressure fields derived from the National Centers for Environmental Prediction (NCEP) and the European Center for Medium‒Range Weather Forecast (ECMWF), *Salstein et al*. [2008] showed the high uncertainty of atmospheric products over most parts of Asia, Central Africa, high latitude oceans, and Antarctic. Through simulations, *Thompson et al*. [2004] showed that the impact of high‒frequency atmospheric and oceanic errors can reach up to 2 mm in terms of geoid heights at a spatial resolution of 500 km. Impacts of the atmospheric spatiotemporal variations and their errors on the observations of satellite gravimetry missions are discussed in *Gruber et al*. [2009], who stated that considering the uncertainties of computed atmospheric dealiasing products is crucial. *Zenner et al*. [2010, 2012] showed that including errors of the atmospheric and oceanic models in the procedure of computing dealiasing products has an impact that is strong enough to be sensed by the GRACE satellites.

[6] Based on the mentioned studies, therefore, improving the dealiasing products in order to reduce the temporal aliasing and obtain more accurate gravity fields is essential. The quest is even more critical for next‒generation gravity missions which aim to determine the geoid with an accuracy of 1 mm [*Anselmi*, 2010; *NG2 Team*, 2011]. *Nerem et al*. [2006], *Pierce et al*. [2008], and *Dehne et al*. [2009] stated that, as a result of using more advanced laser‒ranging devices in a GRACE‒FO mission, the precision of the range rate measurements may be in the range of ∼ 0.6 nms ^{−1}instead of the current ∼ 0.2 *μ**ms*^{−1}precision of the GRACE microwave system [see also *Loomis et al*., 2012]. Provided that all other short‒term mass variations are perfectly known, a future four‒satellite mission of the Bender‒type configuration will even be sufficiently sensitive to detect 1 *μ*m degree variance error in a background model, up to degree 50 [*NG2 Team*, 2011]. In section 4, therefore, we will discuss the effects of geometrical, numerical, and physical approximations for computing the atmospheric dealiasing products which may pose a limitation for exploiting the full accuracy of the satellite gravimetry measuring systems.

[7] To improve dealiasing products, *Flechtner et al*. [2010] refer to experiments performed with the AOD RL04 baroclinic Ocean Model for Circulation and Tides (OMCT) model and three‒hourly ECMWF forecasts. Although results appeared promising from theory, improvements were not clearly visible in the final gravity fields. Therefore, AOD RL05 is only based on a revised version of the OMCT model but still relies on six‒hourly ECMWF atmospheric models (see, www.gfz‒potsdam.de/aod1b).

[8] This paper, specifically, focuses on the atmospheric part of the dealiasing products, in which surface pressure, geopotential, temperature, and specific humidity fields from the ECMWF operational analysis (ECMWFop) are extracted and converted to potential coefficients. This conversion has been realized using a three‒dimensional (3‒D) integration approach including various approximations [*Swenson and Wahr*, 2002; *Boy and Chao*, 2005; *Flechtner*, 2007]. To investigate this procedure, first, the previous 3‒D formulations of the atmospheric loading effects described in *Boy and Chao* [2005] and *Flechtner* [2007] were revisited (section 3.2). Then, possible modifications of the 3‒D integration, with considering a more realistic physical and geometrical Earth's shape, were discussed (section 3.3). Numerical aspects of the computations were also investigated and improved (section 3.4). All physical and geometrical assumptions within the modified 3‒D model were compared to the prelaunch baseline and the current error‒curve of GRACE monthly fields (section 4). In addition, a predicted error‒curve of a Bender‒type satellite configuration was used to assess the effect of atmospheric dealiasing products on the scenario of future gravimetry missions [*NG2 Team*, 2011].

[9] The main changes within the new 3‒D integration method, called “ITG‒3D” in this paper, over the common approach used for computing the atmospheric part of GRACE‒AOD1B are threefold: (1) geometrical modification including ellipsoidal radius *r*_{e}(*θ*) instead of a constant radius, incorporating geoid heights from the ITG‒GRACE2010s static solution instead of using the surface geopotential from ECMWF, as well as using a more accurate transformation for computing radial coordinates [*Office of the Federal Coordinator for Meteorological Services and Supporting Research (OFCM)*, 1997]; (2) physical modification including the more accurate latitude‒ and altitude‒dependent gravity acceleration formula by *Heiskanen and Moritz* [1967] within the vertical integration instead of a simple linear approximation of the latitude‒dependent gravity acceleration; and finally, (3) numerical improvements, i.e., considering subintervals between each model level for better performing the vertical integration, as well as using the Gauss‒Legendre Quadrature (GLQ) method [*Krylov*, 1962] for improving the computation of the desired atmospheric dealiasing spherical harmonics by horizontal integration.

[10] The modified 3‒D integration approach was then used to compute new sets of atmospheric dealiasing products based on ECMWFop and ERA‒Interim, covering the period 2001 to 2009 (sections 4.2 and 4.3). The impacts of input atmospheric parameters on the new products were numerically assessed by comparing the atmospheric dealiasing products derived from ECMWFop to those of ERA‒Interim (section 4.3).

[11] The remaining part of the manuscript is organized as follows: Atmospheric data derived from the ECMWFop along with ERA‒Interim reanalysis data are described in section 2. In section 3, first, the previous formulations of the atmospheric loading effects described in *Boy and Chao* [2005] and *Flechtner*[2007] based on 3‒D approaches are revisited (section 3.2), and their possible modifications are discussed (sections 3.3and 3.4). Section 4 is devoted to the numerical results of the study, including the impact of assumptions on radial coordinates and gravity acceleration, the influence of each physical and geometrical assumptions on the modified ITG‒3‒D method as well as the impacts of atmospheric parameters on the new computed atmospheric dealiasing products. Section 5 discusses our findings and concludes the study.