||Application of the Wiener optimal filter to GRACE gravity solutions
We present a spatial-averaging method based on the Wiener optimal filter, and describe its application to the Gravity Recovery and Climate Experiment (GRACE) gravity solutions. In contrast to the more commonly used Gaussian filter, the spatial width of the Wiener filter does not need to be specified. Instead, it is designed directly from a least-squares minimization of the difference between the desired and filtered signals. This requires information about the power spectrum of the desired gravitational signal and the contaminating noise, both of which are inferred from the average GRACE degree-power spectrum. The lower-degree part describes the spectrum of the useful gravitational signals. We show that this decreases as 1/j^p, where p > 1, with increasing spherical-harmonic degree j. We term this dependency the Second Kaula rule for temporal variations in the Earth's gravity field. The higher-degree part of the average GRACE degree-power spectrum describes the noise. In a logarithmic scale, this increases linearly with increasing j. Extrapolating the noise spectrum to the lower-degree part, and subtracting it from the average GRACE spectrum results in the spectrum of the desired gravitational signal. We show that the filtered GRACE gravity signal is relatively insensitive to the exponential p. The Wiener optimal filter determined for the GRACE gravity-field solutions closely resembles a Gaussian filter with a spatial half width of 440 km. This is demonstrated using the GFZ-GRACE solution from April 2004.