A comparison of Gaussian and Wiener filters to suppress GRACE data errors

The Gravity Recovery and Climate Experiment (GRACE) twin-satellite gravimetry mission has been monitoring time-varying changes of the Earth’s gravitational field on a near-global scale since 2002. GRACE has been producing monthly time series of Earth gravity models up to a degree and order of 120. Its major scientific objective is to obtain detailed information on global water storage changes via the recovery of gravity changes.
Filtering or smoothing of GRACE data is necessary to reduce the contribution of noisy short wavelength components of the geopotential models and, as a consequence, to obtain reliable estimates of time-varying gravity signals. Errors of GRACE data increase rapidly with the spherical harmonic degree and manifest themselves in maps of surface mass variability as long, linear features, generally oriented in north to south stripes.
The averaging operators or filters implemented on the GRACE data can be divided into two main categories: deterministic or stochastic. Deterministic filters are based on properly choosing an optimal averaging radius which leads to an optimal tradeoff between noise reduction and spatial resolution. In contrast, stochastic operators, or the so-called optimal filters, rely on the principal that external knowledge of the problem (such as desired signal structure and solution error estimates) can be used to set up the filter.
This study uses Gaussian averaging and Wiener optimal filters as examples of deterministic and stochastic operators, respectively. The Gaussian filter weighting coefficients can be computed by Jekeli’s recursion formula. Wiener optimal filtering is designed based on the minimum sum of squares of differences between the desired and corresponding filtered signals. It uses the power spectra information of the desired gravitational signal and the observation noise which is inferred from the averaged GRACE degree power spectrum. It was found that the power of signal decreases with increasing harmonic degree  with approximately , where  for   and  for  are estimated by a least squares adjustment of GRACE data. The degree power of the noise increases in the logarithmic scale, linearly with the increasing .We show that the Wiener optimal filter is a low-pass filter; that is, in general, it functions similarly to a Gaussian filter.
Moreover, these two filter coefficients have been applied to 55 monthly GRACE gravity models for the estimation of the monthly anomalies of total water storage over Iran. The results were compared with the output of the Global Land Data Assimilation System (GLDAS) hydrological model (snow cover plus soil moisture variations) and groundwater variations from borehole pizometer data for the estimation of monthly total water storage variations over Iran. It is shown that Wiener optimal filtering outcomes are nearly identical to those of Gaussian averaging. However, designing the optimal Wiener filter based on the observation is the main advantage of the Wiener filter over the Gaussian one.