Gravityfield2EmpiricalCovariance

This program estimates an spatial and temporal covariance matrix from a time series of gravity fields.

Firstly for every time step $t_i$ a spherical harmonics vector $\M x_i$ from the time variable gravity field is generated. The coefficients of the spherical harmonics expansion are in the sequence given by numbering. If set the expansion is limited in the range between minDegree and maxDegree inclusivly. The coefficients are related to the reference radius R and the Earth gravitational constant GM.

In the next step the empirical covariance matrix is estimated according to \[ \M\Sigma(\Delta i)_{full} = \frac{1}{N}\sum_{i=1}^N \M x_i \M x_{i+\Delta i}^T, \]where $\Delta i$ is given by differenceStep.

From the diagonal elements of $\M\Sigma(\Delta i)$ the isotropic accuracies are computed \[ \sigma_n^2 = \frac{1}{2n+1}\sum_{m=0}^n \sigma_{cnm}^2+\sigma_{snm}^2, \]and a diagonal matrix is constructed $\Sigma_{iso} = \text{diag}(\sigma_2^2,\ldots,\sigma_N^2)$. The result is computed: \[ \M\Sigma(\Delta i) = \alpha_{full}\M\Sigma(\Delta i)_{full}+\alpha_{iso}\M\Sigma(\Delta i)_{iso}. \]