Page - 20 - in Contributions to GRACE Gravity Field Recovery - Improvements in Dynamic Orbit Integration, Stochastic Modelling of the Antenna Offset Correction, and Co-Estimation of Satellite Orientations

3.4.3 Coefficient Triangles The Stokes coefficients and their formal errors can be displayed as coefficient triangles. In fig. 3.2a, the deviation of the Stokes coefficients of a monthly GRACE gravity field from a reference static field is shown. The cosine coefficients cnm are shown in the right half of the triangle, with their order increasing with growing distance from the centreline. The sine coefficients snm are displayed in the left half of the triangle. The degree of the coefficients is zero at the very top of the triangle, and increases in the downward direction. The deviation from the reference field is shown as the base-10 logarithm of the absolute difference from the reference field. Figure 3.2b shows the base-10 logarithm of the formal errors of the monthly solution. The formal errors clearly show some features which are due to the specific GRACE observation and orbit geometry, such as the bands of increased variance at orders around multiples of≈16. 120 90 60 30 0 30 60 90 120 0 30 60 90 120 Ordermsnm cnm 10−13 10−12 10−11 10−10 Difference amplitude [unitless] (a) Deviation from a reference field. 120 90 60 30 0 30 60 90 120 0 30 60 90 120 Ordermsnm cnm 10−13 10−12 10−11 10−10 Formal error [unitless] (b) Formal errors of Stokes coefficients. Figure 3.2: GRACE monthly solution Stokes coefficients displayed as coefficient triangles. 3.4.4 Filters Gravity fields determined through GRACE suffer from distinct correlated errors manifesting as a north-south striping pattern in the spatial domain. These errors are mainly due to the satellites’ ground track geometry. There exist many algorithms to filter a gravity field to at least partially remove such aliasing effects, ranging from simple methods of smoothing using a spherical cap to very sophisticated methods considering the complete stochastic information of GRACE gravity fields (Swenson and Wahr, 2006). One implementation of the first approach, smoothing with a spherical cap, is known as a Gaussian filter (Jekeli, 1981). A gravity field is filtered using a Gaussian filter by damping the Stokes coefficients with a progressively smaller factor for progressively higher degrees. The damping factor is determined by the filter radius Chapter3 Gravity and Other Signals20