CovariancePod

Provides arc wise covariance matrices for precise orbit data. Temporal correlations are modeled in the orbit system (along, cross, radial). The inputfileCovarianceFunction provides temporal covariance functions for each axis. From the diagonal matrix for each time step \[ Cov_{3\times3}(t) = \text{diag}(cov_x(t), cov_y(t), cov_z(t)) \]the Toeplitz covariance matrix for an arc is constructed \[ \M C = \begin{pmatrix} Cov(t_0) & Cov(t_1) & \cdots & & & \\ Cov(t_1) & Cov(t_0) & Cov(t_1) & \cdots & & \\ \cdots & Cov(t_1) & Cov(t_0) & Cov(t_1) & \cdots & \\ & \cdots & \ddots & \ddots & \ddots & \cdots \\ \end{pmatrix} \] The epoch wise $3\times3$ covariance matrices given by inputfileCovariancePodEpoch are eigen value decomposed \[ \M C_{3\times3}(t_i) = \M Q \M\Lambda \M Q^T, \]where $\M Q$ is an orthgonal matrix and $\M\Lambda$ diagonal. This used to split the covariances matrices \[ \M C_{3\times3}(t_i) = \M D(t_i) \M D(t_i)^T = (\M Q \M\Lambda^{1/2} \M Q^T)(\M Q \M\Lambda^{1/2} \M Q^T)^T, \]and to compose a block diagonal matrix for an arc \[ \M D = \text{diag}(\M D(t_1), \M D(t_2), \ldots, \M D(t_2)). \] The complete covariance matrix of an arc is given by \[ \M C_{arc} = \sigma_0^2 \sigma_{arc}^2 \M D \M C \M D^T + \text{diag}(\sigma_1^2\M I_{3\times3}, \sigma_2^2\M I_{3\times3}, \ldots, \sigma_n^2\M I_{3\times3}) \]where sigma $\sigma_0$ is an overall factor and the arc specific factors $\sigma_{arc}$ can be provided with inputfileSigmasPerArc. The last matrix can be used to downweight outliers in single epochs and will be added if inputfileSigmasPerEpoch is provided.