Page - 63 - 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

6.5.2 Covariance Function and Power Spectral Density For a wide-sense stationary process such as the assumed observation noise, the co- variance function of the process and the power spectral density (PSD) of the process form a Fourier pair (Etten, 2006). As the covariance function describing the observation noise is only estimated and known in its discretised form, a discrete transform to the spectral domain is used to compute the PSD. For ITSG-Grace2016, the type I discrete cosine transform (DCT) is used (Rao and Yip, 1990). This transform implies that the covariance function is even at the origin, with C−nxx =Cnxx (6.5.9) and also even at the upper boundary of the domain, with CNmax−nxx =CNmax+nxx . (6.5.10) As the DCT is a linear operator, it can be written as x˜=Xx , (6.5.11) withX the N×N matrix of DCT coefficients,xan N×1 vector of equidistant data points in the time domain, and x˜ the N×1 DCT of x. The elements of the DCT matrixX are (see e.g. Rao and Yip, 1990, p. 11) Xmn= √ 2 N−1 ( kmkncos ( mnpi N−1 )) , m,n∈ [0,N) , (6.5.12) with ki= { 1 if i 6=0 1√ 2 otherwise. (6.5.13) The PSD of the observation noise is then, in terms of the covariance function, Sxx=XCxx . (6.5.14) The elements of the PSD are Sjxx=Sxx(fj) , j∈ [0,N) . (6.5.15) with f0 = 0Hz and fN−1 the Nyquist frequency of the signal. Each entry gives the amplitude of the observation noise at that specific frequency. 6.5.3 Estimation of Covariance Function The Toeplitz cofactor matrix for each observation type can be written as a sum of Nmax individual cofactor matrices, each only dependent on the covariance function of one 6.5 Fit of Stochastic Model 63