Page - 52 - 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

Monthly mean Time Sum of daily and monthly mean Signal Figure 6.5: Daily total potential as piecewise constant function. The monthly coefficients are estimated from degree 2 and order 0 to degree and order (D/O) 120, while the daily coefficients are only estimated to D/O 40. Schematically, the observation equations for the Stokes coefficients are         l1 l2 ... lK−1 lK         =           Ad1 0 · · · · · · 0 0 Ad2 ... ... ... ... ... ... ... Am ... ...AdK−1 0 0 · · · · · · 0 AdK           ·            xd1 xd2 ... xdK−1 xdK xm            , (6.4.5) whereAd1 . . .A d K are the observations equations for the daily Stokes coefficients xdk = [ c2,0 · · · s40,40 ]T , (6.4.6) andAm are the observation equations for the monthly Stokes coefficients xm= [ c2,0 · · · s120,120 ]T . (6.4.7) Equivalently, l1 . . .lK are the observations for the individual days of the month. For the case of piecewise constant daily potential coefficients, the observation equations for the daily coefficients are identical to those of the monthly coefficients of the same degree and order. This means that the respective partials only need to be computed once, and can be re-used for both design matrix blocks, as illustrated in fig. 6.6. The partials of the gravity field w.r.t. the Stokes coefficients are computed once for GRACE-A and once for GRACE-B (see e.g. Ilk, 1983) Jgrav,A(τ)= [ ∂g(τ,rA(τ)) ∂c2,0 · · · ∂g(τ,rA(τ)) ∂s120,120 ] , (6.4.8) Jgrav,B(τ)= [ ∂g(τ,rB(τ)) ∂c2,0 · · · ∂g(τ,rB(τ)) ∂s120,120 ] . (6.4.9) Chapter6 ITSG-Grace201652