Stochastic Modeling in GRACE-FO Gravity Field Estimation Using Two Types of Satellite-to-Satellite Tracking Observations

When determining monthly gravity fields, as commonly done for the operational ITSG solutions, an adequate stochastic model is crucial. By setting up a realistic stochastic model within the least-squares adjustment, the observations are weighted properly and consequently an optimal solution with reasonable formal errors can be obtained. 

GRACE-FO carries two independent inter-satellite ranging systems: the K/Ka-band ranging instrument (KBR) and the laser ranging interferometer (LRI). The LRI, originally conceived as a technology demonstrator, exhibits significantly higher measurement precision and is particularly beneficial for the determination of high-degree spherical harmonic coefficients. However, the combined use of both ranging instruments enables the determination of the best achievable gravity field, provided that the stochastic modeling is properly taken into account. 

The non-linearity of the functional model relating the observations to the estimated gravity field parameters entails that forward-modeled observations are subtracted from the actual measurements. Consequently, the reduced observations are contaminated not only by the noise of the ranging observables, but also by noise contributions from the accelerometer, the star camera, and uncertainties in the background models. This noise component is inherent to all observation types and induces correlations between the LRI and KBR measurements. To ensure proper stochastic modeling, this cross-correlation must be considered. 

The stochastic modeling presented here is realized by determining each amplitude of the power spectrum via a frequency-wise variance component estimation. This procedure involves estimating the covariance function for each observation type and down-weighting flawed observation data.  In addition, the common noise component can be separated from the instrument-specific noise of the KBR and LRI. The resulting formal errors of the gravity field solutions derived from the combined use of both ranging systems show good agreement with the empirical estimates, which is particularly important for subsequent combinations with other data types. This stochastic modeling approach was therefore also applied in the determination of the static gravity field model GOCO2025s, in which monthly GRACE-FO solutions were combined with GRACE, GOCE, kinematic orbit, and satellite laser ranging data.