Page - 47 - 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

The linearised observations can be written as ∆ω˙=DACC∆α , (6.2.13) where∆ω˙= ω˙− ω˙0 and ω˙0 are the angular accelerations derived from q0 through polynomial differentiation.DACC is a double differentiation matrix based on a poly- nomial of degree d as described in section 2.7.DACC has a roughly block-diagonal structure, with each block of size 3×3 due to there being three angles per epoch. For a polynomial of degree d=2,DACC would be DACC=               D0,0 D0,1 D0,2 0 · · · 0 D1,0 D1,1 D1,2 0 0 D1,0 D1,1 D1,2 ... ... ... ... ... D1,0 D1,1 D1,2 0 0 D1,0 D1,1 D1,2 0 · · · 0 D2,0 D2,1 D2,2               3N×3N , (6.2.14) with Di,j=wi,jI3×3 (6.2.15) andwi,j the j-th weight factor of the differentiation polynomial computed to be evaluated at the support i. For the accelerometer observations, it is desirable to co- estimate calibration parameters such as biases or scales. These Pcal parameters are collected in the 3N×Pcal matrixAcal. The design matrix for the angular accelerations is then AACC= [ DACC Acal ] 3N×3N+Pcal . (6.2.16) The initial weights for the angular acceleration observations are set to the values given by (Klinger, 2018), 5 ·10−6rad/s2/√Hz for the x- and z-axis, and 2 ·10−7rad/s2/√Hz for the y-axis. 6.2.4 Combination The two observation groups are combined in an iterative least squares adjustment, with variance component estimation used to determine their relative weights. The initial Taylor point for the orientationq0 at each epoch is set to the nominal orientation of the satellite. The combined observation equation system for both SCA and ACC observations is[ ∆q[1:3] ∆ω˙ ] = [ ASCA 0 DACC Acal ][ ∆α ∆xcal ] . (6.2.17) 6.2 Sensor Fusion 47