# GRACE gravity field recovery with background model uncertainties

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## Abstract

In this article, we present a computationally efficient method to incorporate background model uncertainties into the gravity field recovery process. While the geophysical models typically used during the processing of GRACE data, such as the atmosphere and ocean dealiasing product, have been greatly improved over the last years, they are still a limiting factor of the overall solution quality. Our idea is to use information about the uncertainty of these models to find a more appropriate stochastic model for the GRACE observations within the least squares adjustment, thus potentially improving the gravity field estimates. We used the ESA Earth System Model to derive uncertainty estimates for the atmosphere and ocean dealiasing product in the form of an autoregressive model. To assess our approach, we computed time series of monthly GRACE solutions from L1B data in the time span of 2005 to 2010 with and without the derived error model. Intercomparisons between these time series show that noise is reduced on all spatial scales, with up to 25% RMS reduction for Gaussian filter radii from 250 to 300 km, while preserving the monthly signal. We further observe a better agreement between formal and empirical errors, which supports our conclusion that used uncertainty information does improve the stochastic description of the GRACE observables.

## Keywords

GRACE GRACE follow-on Model uncertainties Stochastic modeling## 1 Introduction

The Gravity Recovery And Climate Experiment (GRACE, Tapley et al. 2004) satellite mission was in orbit for over 15 years and provided an invaluable data record for climate and Earth system sciences. Its primary data product was a time series of monthly gravity field snapshots—a proxy for mass distribution in the Earth system.

To recover the gravity field from GRACE data without prior information, a global, homogeneous data distribution is required (Weigelt et al. 2013). Typically, global coverage is reached after about one month. This means that any signal that cannot be represented by a monthly time series has to be reduced from the data before the gravity field is computed; otherwise, the sampling theorem (e.g., Shannon 1949) is not fulfilled. This signal reduction is performed by using geophysical models which provide high-frequency temporal gravity variations. As these models are not perfect, residual high-frequency signal is aliased into the monthly gravity field, deteriorating the solution (Han et al. 2004). This undesired effect is currently among the limiting factors for the overall solution quality for both GRACE and its successor GRACE-FO (Flechtner et al. 2016). There have already been efforts toward improving the dealiasing process which aims at directly improving the dealiasing model coefficients (Zenner et al. 2010, 2012).

The idea we present here is to use information about the uncertainty of these geophysical models within the gravity field recovery process. The overall goal of the approach is to improve the gravity field estimates by incorporating a more realistic stochastic observation model. Assuming random error behavior, a stochastic description of the model uncertainties provides a more appropriate weighting of the observations. Measurements, which are subjected are larger model error, will be downweighted in the adjustment process, which mitigates their impact on the estimated gravity field. Ideally, this reduces temporal aliasing effect, namely the north–south striping patterns in the GRACE solutions, while conserving the full monthly signal. We outline two approaches how this additional information can be efficiently introduced into the least squares adjustment and show their mathematical equivalence. We evaluate our approach using six years of GRACE data, from which two time series of monthly solutions, with and without the derived error model, are computed.

## 2 Least squares estimation with model errors

*f*, which relate the observations to the gravity field through Newton’s second law of motion, for example, integrated variational equations (e.g., Montenbruck and Gill 2000; Beutler and Mervart 2010) or integral equations (e.g., Mayer-Gürr 2006). As depicted in (1), this mapping will not only depend on the gravity field parameters \({\mathbf {x}}\), but also additional model parameters \({\mathbf {y}}\). In practice

*f*will be evaluated using fixed model output, denoted by \({\mathbf {y}}_m\), resulting in a functional model

*f*with respect to \({\mathbf {y}}\) evaluated at \(({\mathbf {x}}_0, {\mathbf {y}}_m)\). Note that in (6), no uncertainties of \({\mathbf {x}}_0\) and \({\mathbf {A}}\) are considered, as the Taylor series expansion point is purely deterministic. Solving the overdetermined system of observation equations (4) using the covariance matrix (6) leads to the well-known least squares estimate of \({\mathbf {x}}\),

## 3 Application to GRACE gravity field recovery

To investigate the impact of using model uncertainties, we estimate monthly gravity field solutions from GRACE data following the least squares adjustment (11). We restrict our study to errors of the atmosphere and ocean dealiasing (AOD) product used to remove high-frequency non-tidal variations in GRACE/GRACE-FO processing (AOD1B, Dobslaw et al. 2017). However, other background models, such as ocean tides, can be treated in a similar fashion. The current release of AOD1B is given as a three-hourly time series of potential coefficients up to a spherical harmonic degree and order of 180. The model is applied to the GRACE observations by linearly interpolating the corresponding nodal points to the observation times. If all model parameters are considered to be subject to uncertainties, \({\varDelta }{\mathbf {y}}\) would be represented by 248 nodal points with 32,757 parameters each, resulting in over 8 million additional parameters per month. To reduce this number and the accompanying computational burden, we restrict the spatial resolution of the co-estimated model corrections to maximum spherical harmonic degree of 40 and parameterize the temporal evolution as daily block means. This implies that higher frequencies in both space and time are treated as error free.

### 3.1 Derivation of \({\varvec{\Sigma }}_{{\mathbf {y}}_m}\)

To derive the AOD model uncertainty covariance matrix \({\varvec{\Sigma }}_{{\mathbf {y}}_m}\), we make use of the ESA Earth System Model (ESA ESM, Dobslaw et al. 2015, 2016) for satellite gravity mission simulations. The dataset provides six-hourly time series of potential changes from atmosphere (A), ocean (O), hydrosphere (H), cryosphere (I), and solid Earth (S) as well as a time series of estimated atmosphere and ocean model errors (aoErr) up to a spherical harmonic degree of 180. These error estimates correspond to release 05 (RL05) of the AOD1B product; however, the currently employed AOD1B RL06 is a notable improvement over RL05 both in terms of high-frequency variability and also long-term consistency (e.g., Figures 1 to 3 in Dobslaw et al. 2017). The major error characteristics as reflected by spatial and temporal correlations and regional distributions, however, remain largely unchanged. This includes in particular the much higher accuracy over the continents, in particular over the densely populated regions of the world; higher errors in some open-ocean places that are resonant at daily-to-weekly periods (Bellingshausen Basin, North Pacific, Arctic Ocean); and the highest errors at the shelf or in semi-enclosed seas. Error characteristics as represented in ESA ESM are thus perceived as still representative even for RL06, albeit those are now somewhat more conservative than in the case of RL05 (Henryk Dobslaw, personal communication).

*p*of \(\varvec{\Phi }^{(p)}_k\) and \({\varvec{\Sigma }}^{(p)}_{\mathbf {w}}\) denotes that these coefficients and white noise covariance are part of an AR(

*p*) model, where

*p*is the model order. The model coefficients and white noise covariance were determined using Yule–Walker equations for multivariate AR processes (e.g., Lütkepohl 2005; Brockwell and Davis 2010) with empirical covariance estimates from the sample \({\tilde{{\mathbf {y}}}}_e\). The maximum order

*p*for which a stable AR model could be estimated from the given AOD error time series was \(p_{\text {max}}=3\). We obtained the required empirical covariance matrices for the lags \(k = \{0,\ldots ,p_{\text {max}}\}\) through the unbiased estimator

*j*and

*N*is the total number of months. After the coefficients were determined, we used the AR model equation (23) to transform the pseudo-observations (10). By separating \({\mathbf {y}}_e\) and \({\mathbf {w}}\) in (23) to the left- and right-hand side, we get

### 3.2 Parameterization of \({\varDelta }{\mathbf {y}}\) and \({\varDelta }{\mathbf {x}}\)

*M*is the number of days in that specific month. Correspondingly, the observation equation coefficient matrix \({\mathbf {B}}\) is block-diagonal. The monthly mean \({\varDelta }{\mathbf {x}}\) is also parameterized as potential coefficients, though from degree 2 to 120. This choice of parametrization introduces a linear dependency between \({\varDelta }{\mathbf {y}}\) and \({\varDelta }{\mathbf {x}}\). The zero-observations (10) however imply that the daily co-estimates \({\varDelta }{\hat{{\mathbf {y}}}}_k\) have zero mean over one month with respect to \({\varvec{\Sigma }}_{{\mathbf {y}}_m}\). We can verify that \({\varDelta }{\hat{{\mathbf {y}}}}\) is centered by computing the sample mean \({\mathbf {m}}\) through the least squares adjustment

## 4 Evaluation of estimated solutions

We now evaluate the impact of incorporating the AOD model uncertainties into the gravity field recovery. As stated in the introduction, the aim of introducing these AOD model uncertainties is to reduce the effects of temporal aliasing without losing monthly gravity field signal. To verify if this goal was achieved, we gauge the high-frequency noise content and the signal level of a series of monthly solutions.

Basis of the evaluation are two time series, “OBS” and “OBS+AOD”, of GRACE monthly solutions in the time span of 2005-01 to 2010-12. We chose this period because we observe a nearly homogeneous observation quality due to the still-active thermal management of the satellites (Tapley et al. 2015), minimal orbit decay, and an extended solar flux quiet period (e.g., Agee et al. 2010). Moreover, this time span contains no large data gaps or orbit repeat cycles. Given these circumstances, we expect a consistent set of solutions without large monthly outliers. For both time series, the monthly gravity field is parameterized up to a spherical harmonic degree of 120 and the processing scheme follows ITSG-Grace2018 (Mayer-Gürr et al. 2018). Whereas OBS only considers a stochastic model for the GRACE observations, OBS+AOD additionally incorporates AOD error information as outlined in the previous section. All other processing steps are identical. The estimated solutions follow the signal definition of the official GRACE products, and hence contain only non-tidal variations in hydrosphere, cryosphere, and solid Earth as well as residual atmosphere and ocean signal.

Figure 4 shows a comparison of the monthly block means of OBS+AOD, the sum of monthly block means and daily co-estimates, and an independently computed daily GRACE time series in terms of basin average time series. We briefly note that the basin average of the unconstrained monthly solutions and therefore the long wavelength variations of the time series will depend on the applied smoothing, which in this case was performed through a 300 km Gaussian filter. Both daily time series exhibit a similar short-wavelength behavior, from which we conclude that the daily co-estimates indeed pick up sub-monthly geophysical signals. This in turn means that since these high-frequency gravity field variations are absorbed by the additional parameters, their effect on the monthly estimates through temporal aliasing is also reduced.

In order to ensure that the approach does not cause signal attenuation, we examine regions where strong geophysical signals are present. Specifically, we look at river basins with a high signal-to-noise ratio (SNR). To identify river basins which match this criterion, we first derive noise estimates for 405 catchments for which the outlines were kindly provided by the Global Runoff Data Centre (2007). We follow a similar approach to Bonin et al. (2012) and move all basin polygons into quiet ocean areas while conserving their latitude. Leaving the basin polygons on their respective parallel is motivated by the latitude dependency of GRACE errors (Wahr et al. 2006). Following the same argumentation as for the total ocean RMS, the basin average time series for these shifted catchments will be dominated by random errors and can therefore be used to derive basin-specific noise estimates. To inhibit high-frequency spatial noise in smaller catchments and to reduce residual signal, we apply a 300 km Gaussian filter and subtract a climatology before computing basin average time series for all 405 shifted river basins. The temporal RMS over each time series then yields an upper bound for the noise in the corresponding catchment.

## 5 Summary and outlook

We presented a computationally efficient method to incorporate background model uncertainties into the gravity field recovery process. Assuming random error behavior, the model uncertainties can either be considered by augmenting the covariance matrix of observations, or by co-estimating constrained model corrections. Since both approaches lead to identical monthly gravity field estimates, which one is chosen mainly depends on implementation aspects. We used the ESA Earth System Model to derive uncertainty estimates for the atmosphere and ocean dealiasing product (AOD) in the form of an autoregressive model. This parametric method allowed us to approximate the spatiotemporal correlations of the AOD errors as a stationary process. Representing the error process as an autoregressive model also has benefits concerning the required computational resources. The resulting system of normal equations is block-banded, compared to the dense block Toeplitz structure of the covariance matrix of a stationary process.

To evaluate our approach, we computed time series of GRACE monthly solutions from L1B data with and without incorporating the derived AOD error model. Variability over the ocean shows a noise reduction over the whole spatial spectrum, with a maximum of 25% for Gaussian filter radii from 250 to 300 km. Also, month-to-month variations of the variability are lower, resulting in a more consistent time series. We confirmed that the lower noise level is not accompanied by signal loss by examining time series of drainage basin averages in regions with high signal-to-noise ratio. These evaluations showed consistent signal levels for both solutions in medium-to-large river basins, where we expect noise to play a minor role. We further found a better agreement between empirical and formal errors when considering AOD model uncertainties. This supports our initial assumption that the use of this additional information does improve the stochastic modeling of the GRACE observables.

We have shown that the current parameterization of the AOD model corrections as daily sets of spherical harmonic coefficients from degree 2 to 40 does significantly improve the estimated monthly gravity field solutions. The approach has been employed in this form for the processing of ITSG-Grace2018 (Mayer-Gürr et al. 2018). The next steps are to increase both spatial and temporal resolution of the error model, and to possibly include the uncertainties of the used global ocean tide model.

The theoretical framework of the approach was developed by Andreas Kvas and Torsten Mayer-Gürr, Andreas Kvas analyzed the GRACE L1B data and wrote this manuscript. The computed gravity field time series are publicly available through PANGEA.

## Notes

### Acknowledgements

Open access funding provided by Graz University of Technology. We thank three anonymous reviewers for insightful comments which helped to greatly improve the manuscript. We would further like to thank the German Space Operations Center (GSOC) of the German Aerospace Center (DLR) for providing continuously and nearly 100% of the raw telemetry data of the twin GRACE satellites. The presented work was funded by the Austrian Research Promotion Agency (FFG) in the frame of the Austrian Space Applications Programme Phase 13 (Project 859736).

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