Abstract
Satellite missions providing data for a continuous monitoring of the Earth gravity field and its changes are fundamental to study climate changes, hydrology, sea level changes, and solid Earth phenomena. GRACEFO (Gravity Recovery and Climate Experiment FollowOn) mission was launched in 2018 and NGGM (Next Generation Gravity Mission) studies are ongoing for the longterm monitoring of the timevariable gravity field. In recent years, an innovative mission concept for gravity measurements has also emerged, exploiting a spaceborne gravity gradiometer based on cold atom interferometers. In particular, a team of researchers from Italian universities and research institutions has proposed a mission concept called MOCASS (Mass Observation with Cold Atom Sensors in Space) and conducted the study to investigate the performance of a cold atom gradiometer on board a low Earth orbiter and its impact on the modeling of different geophysical phenomena. This paper presents the analysis of the gravity gradient data attainable by such a mission. Firstly, the mathematical model for the MOCASS data processing will be described. Then numerical simulations will be presented, considering different satellite orbital altitudes, pointing modes and instrument configurations (singlearm and doublearm); overall, data were simulated for twenty different observation scenarios. Finally, the simulation results will be illustrated, showing the applicability of the proposed concept and the improvement in modeling the static gravity field with respect to GOCE (Gravity Field and SteadyState Ocean Circulation Explorer).
1 Introduction
Successful satellite gravity missions such as GRACE (Tapley et al., 2004) and GOCE (ESA, 1999) have provided plenty of data for monitoring the Earth gravity field and its changes. More recently, the GRACEFO twin spacecrafts (Kornfeld et al., 2019) were launched on May 22^{nd}, 2018, in order to continue the monitoring of the Earth water transport with the aim of detecting changes in underground water storage, in the amount of water in large lakes and rivers, soil moisture, ice sheets and glaciers, and in sea level caused by the water transfer from continental areas to the ocean (Frappart & Ramillien, 2018).
Ongoing studies such as NGGM are also being carried out for the longterm monitoring of the timevariable gravity field with high temporal and spatial resolution (Haagmans et al., 2020). A deeper knowledge of several geophysical phenomena in many interesting areas can be made possible by further and more accurate satellitebased investigations of the Earth composition. As for the Solid Earth, knowledge improvements are expected for megathrust earthquake modeling, study of mass distribution and transport inside a volcano, mass transport at an increased spatial and temporal resolution, study of Moho discontinuity and upper mantle, and so on. Even more significant outcomes are expected in oceanography, hydrology, and cryosphere science. Just to mention an example, GOCE significantly contributed to the knowledge of the mean dynamic topography at global scale (Knudsen et al., 2019) thanks to a geoid estimation with an accuracy of 1–2 cm up to a maximum spherical harmonic degree 200–220 (ESA, 1999); however, space and time resolutions are still insufficient for small and closed or semiclosed basins, like the Mediterranean Sea, and an improved gravity mission could fill this gap. Besides, ice masses carry a significant gravity signal: timevariable gravity solutions allow to quantify the icesheet mass variations and their impact on climate. All these improvements can give great benefits to climate and climate change studies (Pail, 2015). The geophysical impact of future gravity missions with significantly improved performances with respect to GRACE and GOCE has been deeply studied, e.g., in the framework of the Global Geodetic Observing System (GGOS) Working Group for Satellite Missions (see again Pail, 2015).
While electrostatic sensors are approaching their ultimate performance, atomic quantum sensors have shown the potential to drastically increase the performance of inertial measurements (Peters et al., 2001; Sorrentino et al., 2014; Tino et al., 2013). For example, the GOCE gravity gradients showed poor performance in the lower frequency band, where the noise power spectral density (PSD) increases as the inverse of the frequency (Rummel et al., 2011). In order to improve the GOCE performance (of the order of 10 mE/√Hz in the 5–100 MHz band, where 1 E = 10^{−9} s^{−2}), future Earth gravity missions will require gradiometers with sensitivity of the order of 1 mE/√Hz over a wide spectral range. Indeed, the remarkable stability and accuracy that atom interferometers have reached for inertial measurements can play a crucial role for science and technology. The architecture of a quantum gradiometer, e.g., a gradiometer built on cold atom interferometry for the purpose of gravity field recovery, was proposed in Carraz et al. (2014). This new technology is expected to enable highsensitivity measurements of all diagonal elements of the gravity gradient tensor, as well as the full spacecraft angular velocity.
The technological development and performance of a quantum accelerometer were investigated in a more detailed way in several studies (Douch et al., 2018; Mottini & Anselmi, 2019; Trimeche et al., 2019) and supported by the European Space Agency (ESA) in the context of the “Study of a Cold Atom Interferometry (CAI) gradiometer sensor and mission concepts” (ESA, 2014).
In the years of 2014–2018, the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) started funding the “Collaborative Research Center 1128: Relativistic Geodesy and Gravimetry with Quantum Sensors (geoQ)”, consisting of 21 projects. The aim of this program was to explore new frontiers and techniques for estimating the Earth gravity field and monitoring the Earth mass redistribution at a new level of accuracy, providing improvements needed for the GRACE and GRACEFO data analysis. One of the three main research areas of this program focused on the design and development of new instruments and sensors for gravity missions. Particularly, the study of quantum gradiometer was also funded by ESA. During this research, the cold atom interferometry and the optical clock concepts were studied, the use of atomic clocks to directly observe gravity potential differences by measuring the relativistic redshift between clocks (“relativistic geodesy”) was proposed and the contribution of the quantum sensors to improve the Earth gravity field was evaluated (Müller & Wu, 2019).
Between 2016 and 2019, Thales Alenia Space Italia performed the study of the “Cold Atom Inertial Sensors (CAI): Mission Application” for ESA with the support of Marwan Technology, Politecnico di Milano and Technical University of Delft (Mottini & Anselmi, 2019). The purpose of this study was to investigate the possible implementation of a future generation of gravity missions using gradiometers based on the cold atom interferometry. In the scope of this study, the design of a mission with the cold atom gradiometer on board and the achievable systemlevel performances were investigated.
Finally, it should be remarked that at the IUGG General Assembly in Montreal in 2019, the International Association of Geodesy (IAG) decided to establish a project on Novel Sensors and Quantum Technology in Geodesy (QuGe); one of the main purposes of this project is to investigate the enhanced prospects for satellite geodesy that quantum technologies will open up (Poutanen & Rózsa, 2020).
In this paper, we will focus on the data analysis of the MOCASS mission proposal (Mass Observation with Cold Atom Sensors in Space), carried out by a team of researchers from Italian universities and research institutions in the period from March 2017 to June 2019 under a contract by the Italian Space Agency (ASI), in the framework of preparatory activities for future missions and payloads of Earth Observation. In Sect. 2 the mission concept will be explained, introducing the cold atom gradiometer noise characteristics and mission scenarios for which data simulations and analysis were performed by applying the socalled spacewise approach. In Sect. 3 the results of the numerical simulations will be described and compared to those of GRACE and GOCE missions. In Sect. 4 a discussion on the obtained results and some conclusions will be presented. Regarding the science applications of the results of this study, geophysical investigations on glaciers mass variations, tectonic deformations, mass movements and superficial hydrology in the Himalayan region and seamount growth in the area of the Indian Ocean can be found in Pivetta et al. (2021).
2 The MOCASS Mission Proposal and Data Analysis
The MOCASS study was carried out by a research team consisting of Politecnico di Milano, University of Trieste and AtomSensors srl (a spinoff of the University of Florence). The goal of this study was to determine:

the technological characteristics of the cold atom gradiometer capable of acquiring observations of gravity gradients at satellite altitude;

the estimation of the global signal characteristics as a result of the data analysis;

the requirements on the signal to be detected in order to identify the geophysical phenomena of interest.
An overview of the proposal and results of the study is presented in a paper by Migliaccio et al. (2019).
2.1 The Cold Atom Gradiometer
Future improvements of the GOCE mission concept can be achieved by going beyond the technology of electrostatic gradiometers and taking advantage of a new generation of quantum sensors. In recent years, a new and highly improved gradiometer concept has been proposed (Carraz et al., 2014), exploiting instruments that have been refined and tested over nearly three decades, based on the principle of Cold Atom Interferometry. Such instruments operate as inertial sensors and have been used for fundamental physics experiments. A gradiometer in one given direction (e.g., the radial one) can be obtained by measuring the acceleration of two clouds of cold atoms separated by a certain distance. At the same time, this instrument allows to measure the rotation angle around the axis perpendicular to the plane of motion of the atom clouds. Since this scheme can be repeated in the three directions, it will be possible to determine all the diagonal elements of the gravity gradient tensor and the angular rate vector.
For this type of instruments, it has been proved that a sensitivity of 3.5 mE/√Hz over a wide spectral range can be reached. An interesting feature is that its noise power spectral density is flat at low frequencies in contrast to the spectrally colored noise of the electrostatic accelerometers (see Fig. 1).
As a consequence of its spectral characteristics, such an instrument might allow to meet the requirements of a mission dedicated to the observation of both the timevariable gravity field (like GRACE) and the static field (like GOCE). Of course, in the former case a longer lifetime is required, and this would reflect in the mission characteristics.
2.2 SpaceWise Solution Strategy
In the case of the MOCASS study, the input data were the simulated second derivatives of the anomalous gravitational potential T at given points along the orbit; the output data were the spherical harmonic coefficients of the Earth potential, estimated from the cold atom gradiometer observations by applying the spacewise approach.
The main idea behind the spacewise approach (see e.g., Migliaccio et al., 2004a; Pail et al., 2011) is to estimate the spherical harmonic coefficients of the geopotential model by exploiting the spatial correlation of the Earth gravity field. For this purpose, a collocation solution can be devised, modeling the signal covariance as a function of spatial distance and not of time distance, as it is done in the case of the noise covariance. In this way, data which are close in space but far in time can be filtered together, thus overcoming the problems related to the strong time correlation of the observation noise.
Although a unique collocation solution would be theoretically clean and desirable, it is not computationally feasible due to the huge amount of data which usually must be processed for this type of satellite missions (Pail et al., 2011). The dimension of the system to be solved would be in fact equal to the number of input data. For this reason, the spacewise approach is implemented as a multistep collocation procedure (Reguzzoni & Tselfes, 2009), basically consisting of:

Wiener filtering (Albertella et al., 2004; Papoulis, 1984) applied to the data along the orbit in order to reduce the highly time correlated noise of the gradiometer;

spatial interpolation of filtered data to obtain values over spherical grids at mean satellite altitude, by applying collocation to local patches of data (Migliaccio et al., 2007);

spherical harmonic analysis of gridded data by numerical integration (Colombo, 1981) to retrieve the geopotential coefficients.
In the spacewise strategy as implemented in GOCE data analysis, the procedure was iterated until convergence to recover the signal frequencies cancelled by the Wiener filter along the orbit and to correct the estimated rotation from the gradiometer reference frame to the local orbital reference frame (LORF) (Migliaccio et al., 2004b). In the MOCASS simulations, no iterations were required (see Fig. 2) due to the prevailing white behavior of the noise spectrum of the cold atom gradiometer and to the assumption that the spacecraft (and therefore the gradiometer) is kept aligned with the LORF; the latter assumption would be justified by the use of an enhanced spacecraft attitude control system, like the one devised for NGGM (Bacchetta et al., 2017). The scheme in Fig. 2 without iterations allowed for higher speed in computations and hence the possibility to analyze several case studies.
Note that in the spacewise approach a prior model has to be used in order to reduce the spatial correlation of the signal, which is necessary when applying collocation gridding on local patches of data. In other words, after removing the prior model contribution from the observations, low frequency information in the residual signal is significantly reduced, leading to a signal covariance function with a correlation length comparable with the size of the local patches used for the gridding. For this purpose, the gravity field model derived by SST (SatellitetoSatellite Tracking) data was employed (Migliaccio et al., 2010), thus avoiding the use of external information.
2.3 Numerical Simulations for the MOCASS Study
The numerical simulations for the MOCASS study were performed considering as reference orbits the GOCE orbits from November 2009 to January 2010 (about 2 months of data along a “high” orbit at an altitude about 259 km) and from February 2013 to April 2013 (same time span of about 2 months, corresponding to a “low” orbit at an altitude about 239 km).
EIGEN_6C4 (GOCE/GRACE/ground data combined model) up to harmonic degree and order 360 was used as the reference model (Förste et al., 2014) in order to simulate the spherical harmonic coefficients as input.
Two different pointing modes of the instrument were considered: nadirpointing mode, which corresponds to the usual Earthpointing satellite attitude, and inertial mode (also denoted by IRF = Inertial reference frame) (see Fig. 3).
Data were simulated for two configurations of the gradiometer, namely: a singlearm gradiometer, and a doublearm gradiometer. For the singlearm case, gradients were simulated for each of the three possible directions: T_{xx} or T_{yy} or T_{zz}; for the doublearm case, gradients were simulated for T_{xx} and T_{zz} and for T_{yy} and T_{zz}. Note that in order to correctly compare two simulations, it was decided that the number of atoms in the cloud had to be kept unchanged in both cases (singlearm and doublearm gradiometer). Obviously, increasing the overall number of atoms would have generally implied a better performance for the doublearm instrument.
For all the simulated observation scenarios, a Monte Carlo (MC) approach was applied in order to compute the estimation errors of the results of the numerical simulations (grid errors and coefficient errors). The MC strategy consisted in simulating sample sets of spherical harmonic model coefficients \(T_{\ell m}\) and then computing error variances and covariances of these solutions. Twenty samples were generated for each scenario (Migliaccio et al., 2009), using the coefficient variances of the reference model EIGEN_6C4, which means that the sample data were equal to EIGEN_6C4 on average, but differ from it by a variation, related to the EIGEN_6C4 error variances. The error r.m.s. of the estimated coefficients \(\hat{T}_{\ell m}\) was thus computed as:
A higher number of samples would be recommendable to get more accurate error estimates; however, here the choice was driven by the need of limiting the computational burden.
2.4 The Cold Atom Gradiometer Noise and Wiener Deconvolution Filter
As described in Sect. 2.2, the first step of the spacewise approach consists in applying a Wiener filter to the data along the orbit to reduce the time correlated noise of the gradiometer. In the case of the MOCASS cold atom gradiometer, time correlation arises also from the atom interferometer integrator or transfer function (Tino & Kasevich, 2014). This means that the MOCASS gradiometer does not provide pointwise observations and, therefore, the Wiener filter was generalized to a Wiener deconvolution filter.
The basic assumption of this filtering technique is that we have a flow of data consisting of the observation signal and noise, which are both stationary processes (with their own respective covariances) and independent from each other. The observation equation in the time domain can be written as:
where \(y_{o} \left( t \right)\) is the vector of the observed values; \(h\left( t \right)\) is the atom interferometer integrator; \(y\left( t \right)\) is the signal and \(\nu \left( t \right)\) is the observation noise. The symbol \(*\) stands for the convolution operator.
In the MOCASS numerical simulations, the cold atom interferometer integrator parameters were taken into account (see Fig. 4 and Table 1). More technical details about the interferometer integrator can be found in the papers by Tino and Kasevich (2014) and Migliaccio et al. (2019).
The optimal filtering of the data along the orbit was obtained according to the WienerKolmogorov theory (Papoulis, 1984; Sansò, 1986). In the case of GOCE, a Wiener filter was applied (Reguzzoni, 2003), while in the case of MOCASS, a Wiener deconvolution filter was implemented to account for the cold atom interferometer integrator. The MOCASS filter is therefore given by:
where \(H\left( f \right)\) is the Fourier transform of the interferometer integrator; \(S_{y} \left( f \right)\) is the signal PSD and \(S_{\nu } \left( f \right)\) is the noise PSD. Figure 5 represents the shape of the resulting Wiener deconvolution filter \(W\left( f \right)\) for different observation signals, considering the nadirpointing mode and high orbit scenario in the case of a singlearm gradiometer. Once the Fourier transform of the observations \(y_{o} \left( t \right)\) is computed, the Wiener deconvolution filter is applied in a computationally efficient way in the frequency domain.
In the MOCASS numerical simulations, the shape of the signal and noise PSD depended on the simulation scenarios, so different PSD curves were used for the different cases. Examples of the analytical spectra of the noise for the case of the nadirpointing mode are shown in Fig. 6 for the high orbit (left) and low orbit cases (right). The corresponding curves for the case of the inertial mode are almost identical. As it can be seen, one of distinctive features of the noise PSD of the cold atom gradiometer is the presence of spikes at certain frequency interval (corresponding to about 10^{–4} Hz and 10^{–3} Hz). These spikes are related to a residual rotation rate at certain frequencies due to the fact that the satellite attitude cannot be perfectly controlled, so this effect was taken into account in the simulations (Migliaccio et al., 2019).
In the case of the singlearm gradiometer configuration (for T_{xx}, T_{yy} and T_{zz} signals), the noise PSD basically remains the same in each scenario. In the case of the doublearm gradiometer configuration, there is a slight degradation with respect to the singlearm one (the noise is about 1.5 times higher). This is due to the fact that the overall number of cold atoms does not change in the doublearm gradiometer case, but the available cold atoms are split into two directions.
In Fig. 7 the analytical spectrum of the noise is shown against the signal PSD for the cases of T_{xx}, T_{yy} and T_{zz} observations (for a singlearm gradiometer), both for the nadirpointing and inertial mode in high and low orbit.
At epoch t, the filtered observation is then represented by the signal plus the filtering error \(e\left( t \right)\):
Figure 8a shows the shape of the filtering error covariance function for T_{xx} observations in the case of a singlearm gradiometer, considering different orbit altitudes and pointing modes. As one can see in Fig. 8b for the case of nadirpointing mode in high orbit, these covariance functions show a quite short correlation that approaches zero within 100 s.
The parameters of the filtering error covariance function for the case of a singlearm gradiometer and a doublearm gradiometer are reported in Tables 2 and 3, respectively.
2.5 Gridding Procedure
After applying the Wiener deconvolution along orbit, a collocation gridding procedure was applied to local patches of data to obtain interpolated values over a boundary sphere Σ at mean satellite altitude. Moreover, a preliminary operation was carried out with the aim of numerically stabilizing the subsequent gridding solution by reducing the signal correlation in such a way that it is consistent with the size of local patches. This consisted in computing a satellitetosatellite tracking (SST) solution for the spherical harmonic coefficients of the gravity field. In this way, a “MOCASSonly” solution was obtained without combining data with any external apriori information.
The approach to compute the simulated SST solution for MOCASS was similar to the one used for the analysis of GOCE data (Pail et al., 2011) and consisted in:

estimating the anomalous potential from kinematic orbits by the energy integral approach;

estimating the spherical harmonic coefficients by least squares adjustment.
The anomalous potential was synthesized along the orbit from the EIGEN_6C4 reference model, adding white noise with 1.35 m^{2}/s^{2} standard deviation (see Migliaccio et al., 2010). Moreover, since the velocity estimation from kinematic orbit coordinates was based on a moving window of 31s lag, an undersampling of 1:31 was applied to comply with the hypothesis that the observation noise is uncorrelated in time. Spherical harmonic coefficients were estimated up to degree 100, also applying a standard Kaula regularization. This maximum spherical harmonic degree was chosen for computational reasons and because the error degree variances of the coefficients estimated from the SST solutions approach the signal degree variances close to degree 100, as shown in Fig. 9.
Coming to the gridding, the local collocation applied for the estimate of interpolated values over the boundary sphere Σ at equiangular grid nodes is represented by:
where \(\hat{y}_{o}\) is the vector of observations forming the local patch of data around the grid node \(P\left( {\vartheta , \lambda } \right)\); \(\hat{z}\left( {\vartheta , \lambda } \right)\) is the predicted functional of the potential at the grid node P; \(C_{zy}\) is the crosscovariance vector between observed and predicted signal; \(C_{yy}\) is the covariance matrix of the signal; \(C_{ee}\) is the covariance matrix of the filtering error; correlations between signal and filtering error are neglected.
In all scenarios, a collocation procedure was applied to the simulated T_{xx}, T_{yy} and T_{zz} observations in order to predict values of other functionals of the anomalous potential such as T_{rr}, T_{λλ} and T on the grid. Basically, these changes of functionals are possible by exploiting the crosscovariance information between observed and predicted quantities. Note that the gridding procedure was implemented by leastsquares collocation on local data patches because the full dataset of observations could not be processed as a whole in a unique collocation gridding procedure, due to the huge amount of data.
A particular issue in the gridding procedure regarded the data selection criteria, which is fundamental to ensure the good quality of the results. Some preliminary tests were performed during the study, considering either fixed size equiangular “patches” of data or adaptable “clouds” of data. In the latter case (which was the one applied in the MOCASS study), the idea is to consider clouds of observation points to compute the values of the grid nodes: the closer to the nodes to be estimated, the more observations are used and vice versa, the far from the nodes to be estimated, the less observations are used. In this way the local information is enhanced and possibly all data are used at least once with suitable overlapping clouds. To this aim, the points inside each cloud were randomly chosen by also applying the following conditions: use the largest possible set of input data, and not use the same input data to estimate the values for different grid nodes. The idea of the “clouds” of data is represented in Fig. 10. An equiangular grid with a resolution of 0.5° was devised in the case of MOCASS data processing. Each cloud of data points around the grid nodes is basically divided into three areas with different radius size and different data density.
The three areas are:

a “full density” (FD) area with the nearest points to the grid nodes to be estimated; no undersampling is here applied;

a “high density” (HD) area with points at intermediate distance to the grid nodes to be estimated;

a “low density” (LD) area with the farthest points to the grid nodes to be estimated.
The size of each area depends on the available data, which means that it is not constant for all clouds, but ranges between a minimum and a maximum value, thus adapting to the local density of the data, which in turn mainly depends on the latitude. In the case of MOCASS, Table 4 shows the minimum and maximum size of these areas and the maximum number of points that they could contain. Overall, 8325 clouds of data (containing a maximum of 5000 points) without undersampling were used (Reguzzoni et al., 2014).
The global signal covariance function, i.e. the signal degree variances, was adapted to the local signal variances of the “clouds” by applying suitable scale factors. They were computed as the ratio between the local and global variance of T_{rr} signal, even if the observation functional was different. The scale factors for the signal covariance adaptation are shown in Fig. 11; obviously, high values of the scale factors are related to areas where the T_{rr} local signal is much stronger than the global one and, therefore, the collocation filtering is less pronounced.
The error r.m.s. of the predicted grid of T_{rr} values is reported in Tables 5 and 6 for the cases of a singlearm and of a doublearm gradiometer.

for the singlearm gradiometer configuration, observations of T_{zz} in nadirpointing mode and observations of T_{yy} in inertial mode represent the optimal cases, i.e. in these cases the error r.m.s. remains below 0.50 mE;

for the doublearm gradiometer configuration, observations of the pair T_{yy} and T_{zz} represent the optimal case in both pointing modes, i.e. in this case the error r.m.s. remains below 0.55 mE; we recall that in the case of the doublearm gradiometer, the available cold atoms are split into two directions, and this is why the overall accuracy is not improved with respect to the singlearm gradiometer.
For the sake of comparison, the error r.m.s. of the predicted grids of T_{rr} and T_{λλ}, computed considering T_{zz} observations in nadirpointing mode and high orbit, are displayed in Figs. 12 and 13, respectively. As it can be seen, the T_{λλ} component is less sensitive to the polar gaps effect than T_{rr}. Note that T_{λλ} is the second derivative of the anomalous potential T with respect to the longitude. Since the longitude is an angle, the units are not changed, and the result is not formally a “gravity gradient”.
Finally, Fig. 14 shows the grid errors along parallels, in terms of r.m.s. of the T_{rr} predicted data, highlighting the difference of the results when predictions in the polar gaps are also considered. In this figure, the simulated observations are T_{zz} in nadirpointing mode and high orbit.
3 Spherical Harmonic Analysis
Once the T, T_{λλ} and T_{rr} grids have been predicted by the local collocation procedure applied to the simulated and filtered data, spherical harmonic coefficients can be computed by numerical integration (Migliaccio et al., 2004a) for each grid, and then combined according to the MC solution errors to obtain a final estimate of the global geopotential model. Figure 15 shows the complete scheme of the MOCASS data simulations and spacewise analysis.
The spherical harmonic analysis is applied to the gridded observations:
where \(\eta \left( {\vartheta , \lambda } \right)\) is the gridding error. The estimate of the geopotential coefficients is computed by the following equation:
where \(T_{\ell m}\) is the spherical harmonic coefficient of degree \(\ell\) and order m and \(Y_{\ell m} \left( {\vartheta ,\lambda } \right)\) is the corresponding spherical harmonic function. The parameter \(a_{\ell }\) depends on the type of data. For example, in the case of the second radial derivative of the potential:
where the constant GM is the product between the gravitational constant G and the mass of the Earth M; R is the reference radius for the Earth and r is the radius of the spherical grid, which in our case is the mean orbital radius R_{sat}.
4 Gravity Field Recovery Results
Considering 2month solutions and excluding nearzonal coefficients that are strongly degraded by polar gaps (Sneeuw & van Gelderen, 1997), the results of the MOCASS numerical simulations are shown in Figs. 16 and 17 in terms of error degree variances, for the high orbit case and for the low orbit case, respectively. In these figures, the MOCASS results are presented against those of GOCE and GRACE based on the same amount of data, i.e. considering the error of the GOCE_TIM_R1 model (Pail et al., 2011) and the error of the average of two GRACE monthly solutions (MayerGürr et al., 2016), corresponding to the same data period of the GOCE_TIM_R1 model. The improvement of MOCASS is clearly visible. In fact, in the “optimal” scenarios MOCASS observations allow for an improvement with respect to GOCE at all spherical harmonic degrees and with respect to GRACE at higher harmonic degrees (approximately above 35–40). At lower harmonic degrees, the GRACE estimates remain better than the MOCASS ones.
It has to be underlined that this comparison is a general indication of how well the MOCASS mission would perform with respect to GOCE and GRACE; in fact, the quality of real solutions such as GOCE_TIM_R1 and ITSGGrace2016 also depends on other factors, e.g., the processing method and the version of the model, which have not been considered in the present study.
For further comparison, the r.m.s. of individual spherical harmonic coefficients estimated in the optimal scenarios from data of a singlearm gradiometer are displayed in logarithmic scale in Fig. 18 for the case of high orbit. The spherical harmonic order is displayed along the horizontal axis, whereas the degree is along the vertical axis. A negative value of the order indicates the sine coefficients, a positive value the cosine ones.
The results can also be considered in terms of gravity anomaly (Δg) cumulative error at ground level, again for the 2month solutions. Figure 19 shows the performances of MOCASS for the low orbit case (which of course delivers more informative observations on the gravity field) both in nadirpointing mode and in inertial mode. The “optimal” results are obtained from observations of the T_{zz} component in nadirpointing mode, with an error of 3 mGal at harmonic degree 250, and from observations of the T_{yy} component in inertial mode, with an error of about 3.5 mGal at harmonic degree 250. The cumulative error was also computed for T_{rr} at satellite altitude and is shown in Fig. 20. Again, the comparison was extended to GOCE and GRACE results, considering the same models used in Figs. 16 and 17. Of course, if the aim is to obtain a static gravity field model, the quality of the solution improves by exploiting a much longer observation period, at least of the duration of 1 year.
Figure 21 shows the expected improvement in the estimation of Δg at ground level and T_{rr} at satellite altitude for different mission periods (1, 2 and 5year data), considering T_{zz} observations from a nadirpointing mission in high orbit. These curves have been obtained by properly rescaling the corresponding ones in Figs. 19 and 20, assuming independent 2month solutions. For the sake of comparison, the curves for the GOCE and GRACE missions represent results based on 5year data, obtained from GOCE_TIM_R5 (Brockmann et al., 2014) and ITSGGrace2014k (MayerGürr et al., 2014) models, respectively.
Further tests were also performed to assess the level of accuracy attainable with MOCASS if the aim is to estimate the timevariable gravity signal; the results are represented in Fig. 22, which shows the cumulative errors of gravity anomalies Δg at ground level and T_{rr} at satellite altitude. The curves are obtained by propagating the MOCASS monthly solution error to the estimation of a possible linear trend, for a 1, 2 and 5year mission, again considering T_{zz} observations from a nadirpointing mission in high orbit.
In Fig. 22, the GOCE and GRACE curves represent the linear trend of monthly solutions for a 5year mission, rescaling the whole mission error of the GOCE_TIM_R5 and ITSGGrace2014k models on 1 month only. As it can be seen from Fig. 22a, in this case the MOCASS commission error for gravity anomalies is below 1 mGal/month at harmonic degree 300 (and below 0.6 mGal/month at degree 250) even for a 1year mission; for a 5year mission it remains below 0.1 mGal/month at harmonic degree 300 (and below 0.05 mGal/month at degree 250).
Figure 23a shows another comparison between MOCASS and the GRACE and GOCE missions based on 2month data in terms of the error degree variances, including and excluding the effect of polar gaps, i.e. including or excluding error variances of nearzonal coefficients in the computation. As one can see, if polar gaps are included (+ PG), MOCASS gives worse estimates than GRACE, but slightly better ones than GOCE at higher harmonic degrees; however, if polar gaps are excluded (− PG), MOCASS results are better than GOCE at all harmonic degrees, but worse than GRACE at lower harmonic degrees. Therefore, MOCASS estimates at lower harmonic degrees are not satisfactory to detect the timevariable gravity field for geophysical applications; in fact, as an example, in Fig. 23a the MOCASS error is seen to be above the timevariable gravity induced by continental hydrology processes (Iran Pour et al., 2015). Furthermore, a comparison was made with the GOCO05S combined model (MayerGürr et al., 2015), obtained from both GOCE and GRACE data. MOCASS estimates are better than GOCO05S at high harmonic degrees (e.g., over degree 50 for a 5year mission), as displayed in Fig. 23b.
Finally, Table 7 summarizes the results for the “optimal” MOCASS solutions (in terms of degree variances) for the static and timevariable gravity field, for a mission in high orbit; the GOCE TIM and GRACE solutions corresponding to the MOCASS static and timevariable gravity field solutions are also shown in the table, for the sake of comparison. In the case of static gravity field, the maximum harmonic degree \(\ell\) = 200 is considered, while in the case of timevariable gravity field the maximum degree \(\ell\) = 45 is taken into account, since the signal power is much lower and a higher accuracy is required for the signal detection, at the cost of a lower spatial resolution.
5 Discussion and Conclusions
In the frame of the MOCASS mission proposal, a research was conducted to investigate the impact of a cold atom gradiometer on board a low Earth orbiter acquiring data for the estimation of gravity field models. Several mission scenarios were numerically simulated and the corresponding gravity field models were estimated in order to test the mission performances, using orbit data of about 2 months of the GOCE mission as a “reference orbit”. Important aspects that were considered are:

the type of data which could be measured by the MOCASS payload (e.g., second spatial derivatives of the potential T_{xx}, T_{yy}, T_{zz});

the characteristics of the MOCASS payload (the cold atom gradiometer) and its noise PSD, from which the alongorbit filtering procedure was defined;

the local gridding procedure required to manage the large amount of data by collocation, in turn requiring a smart data partitioning as well as a local estimation of the signal covariance.
The simulations were set up under different configurations of the satellite orbit and pointing mode of the instrument: namely high (259 km) or low (239 km) altitude, and nadirpointing or inertial mode. The simulated data were T_{xx}, T_{yy} and T_{zz} observations for the case of a singlearm gradiometer and a pair of observations (T_{xx} and T_{zz}) and (T_{yy} and T_{zz}) for the case of a doublearm gradiometer.
In terms of reconstruction of a global gravity model, considering all the simulations, the results showed that:

in the case of the singlearm gradiometer configuration, T_{zz} observations in nadirpointing mode and T_{yy} observations in inertial mode provided the best accuracy level;

in the case of the doublearm gradiometer configuration, T_{yy} and T_{zz} observations (in both operation modes) provided a good accuracy level;

the results of the simulations of T_{zz} observations in nadirpointing mode were also directly compared with GOCE and GRACE results, showing that the MOCASS mission profile could significantly improve the results of the GOCE mission over all the harmonic spectrum, especially at high degrees, with a commission error of about of 1.4 mGal at degree 300 (0.9 mGal at degree 250) for a 5year mission, but would still be weaker than GRACE at low harmonic degrees, thus limiting its applicability for the timevariable gravity field investigations.
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Acknowledgements
The MOCASS study has been funded under Italian Space Agency contract N. 20169U0 “Proposal of a satellite mission and sensor concept based on advanced atom interferometry accelerometers for high resolution monitoring of mass variations on and below the Earth surface”.
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Reguzzoni, M., Migliaccio, F. & Batsukh, K. Gravity Field Recovery and Error Analysis for the MOCASS Mission Proposal Based on Cold Atom Interferometry. Pure Appl. Geophys. 178, 2201–2222 (2021). https://doi.org/10.1007/s00024021027565
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DOI: https://doi.org/10.1007/s00024021027565
Keywords
 Earth gravity field
 gradiometry
 cold atom interferometry
 spherical harmonic analysis