GENERATION OF CORRECTIONS TO THE RPKNP TIME SCALE
Key Parameters
| Notation | Description |
|---|---|
| \( S^{[10]} \) | Vector of weighted observed pseudoranges (signal delay measurements corrected for relativistic and gravitational effects). |
| \( R \) | Vector of weighted computed geometric ranges (true distance between receiver and satellite). |
| \( A \) | Diagonal matrix of weighting coefficients (\( \alpha_{i,j,k} \)), reflecting the contribution of each PNS station. |
| \( \Delta \tau_1, \dots, \Delta \tau_L \) | Vector of clock offset estimates for \( L \) regional PNS stations. |
| \( \Delta \tau_{\text{GLONASS}} \) | Estimate of the "GLONASS-GPS" time scale divergence. |
| \( K_{\text{GLONASS}} \) | Vector including the coefficient \( k_{\text{GLONASS}} \) for applying the GLONASS-GPS time scale correction. |
| \( S_{i,j,k,3} \) | Observed signal delay (pseudorange) from satellite \( j \) of system \( k \) (GPS/GLONASS) to receiver \( i \). |
| \( \alpha_{i,j,k} \) | Weighting coefficient used to account for relativistic and gravitational effects in signal delay measurements. |
| \( \zeta \) | Small correction factor in the formula for \( \alpha_{i,j,k} \). |
| \( R_{i,j,k} \) | Geometric distance between receiver \( i \) and satellite \( j \) of system \( k \). |
| \( (X_i, Y_i, Z_i) \) | Known coordinates of receiver \( i \). |
| \( (x_{j,i,k}, y_{j,i,k}, z_{j,i,k}) \) | Coordinates of satellite \( j \) of system \( k \) relative to receiver \( i \). |
| \( k_{\text{GLONASS}} \) | Coefficient equal to 1 for GLONASS and 0 for GPS, determined by the NSA type. |
| \( c \) | Speed of light in vacuum (299,792,458 m/s). |
| \( V \) | Normal (variance-covariance) matrix for the least squares method, used to determine parameters. |
| \( \mathbf{W} \) | Diagonal matrix of weighting coefficients, with elements (\( 1/\sigma_k^2 \)) depending on measurement precision \( \sigma_k \). |
| \( \sigma_k \) | Standard deviation of measurements for satellites of system \( k \) (GPS or GLONASS). |
| \( I_4 \) | Identity matrix of dimension \( 4 \times 4 \). |
| \( V_{0,0}, V_{0,1}, V_{1,0}, V_{1,1} \) | Blocks of matrix \( V \) used for inversion via the Frobenius formula. |
| \( \mathbf{H} \) | Schur complement, a key component in the Frobenius formula for inverting block matrices. |
| \( \Delta \dot{\tau}_1, \dots, \Delta \dot{\tau}_L \) | Vector of frequency divergence estimates for \( L \) regional PNS stations. |
| \( \Delta \dot{\tau}_{\text{GLONASS}} \) | Estimate of the "GLONASS-GPS" generator frequency divergence. |
| \( \dot{\mathbf{S}}^{[10]} \) | Vector of weighted rates of change of observed pseudoranges (pseudorange rates or Doppler measurements). |
| \( \dot{\mathbf{R}} \) | Vector of weighted rates of change of computed geometric ranges. |
| \( \dot{R}_{i,j} \) | Rate of change of geometric distance between receiver \( i \) and satellite \( j \). |
| \( (\dot{x}_{j,i}, \dot{y}_{j,i}, \dot{z}_{j,i}) \) | Components of the relative velocity of satellite \( j \) with respect to receiver \( i \). |
GENERATION OF CORRECTIONS TO THE RPKNP TIME SCALE
1. Generation of frequency-time corrections is performed taking into account known coordinates and velocity vector components of the CCS.
Corrections accounting for the divergence between PNS and GPS time scales \(\left| \begin{matrix} \Delta \tau_1 & \dots & \Delta \tau_L \end{matrix} \right|^T\), as well as estimates of the "GLONASS-GPS" time scale divergence (\(\Delta \tau_{\text{GLONASS}}\)), are determined by solving the following system of equations:
\[ S^{[10]} = R + A \cdot \left| \begin{matrix} \Delta \tau_1 \\ \vdots \\ \Delta \tau_L \end{matrix} \right| + K_{\text{GLONASS}} \cdot \Delta \tau_{\text{GLONASS}}. \]
Where
\[ S^{[10]} = \left| \begin{matrix} \dots \\ S_{1,j,k,3} \cdot \alpha_{1,j,k} \\ \dots \\ S_{2,j,k,3} \cdot \alpha_{2,j,k} \\ \dots \\ S_{L,j,k,3} \cdot \alpha_{L,j,k} \\ \dots \end{matrix} \right|; \quad R = \left| \begin{matrix} \dots \\ R_{1,j,k} \cdot \alpha_{1,j,k} \\ \dots \\ R_{2,j,k} \cdot \alpha_{2,j,k} \\ \dots \\ R_{L,j,k} \cdot \alpha_{L,j,k} \\ \dots \end{matrix} \right|; \quad A = \left| \begin{matrix} \alpha_{1,j,k} & 0 & \cdots & 0 \\ 0 & \alpha_{2,j,k} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \alpha_{L,j,k} \end{matrix} \right|; \]
\[ \alpha_{i,j,k} = \left( 1 - \zeta S_{i,j,k} \right); \]
\[ R_{i,j,k} = \sqrt{(x_{j,i,k} - X_i)^2 + (y_{j,i,k} - Y_i)^2 + (z_{j,i,k} - Z_i)^2}; \]
\[ \mathbf{K}_{\text{GLONASS}} = \left| \begin{array}{c} \dots \\ k_{\text{GLONASS}} \cdot \alpha_{1,j,k} \\ \dots \\ k_{\text{GLONASS}} \cdot \alpha_{2,j,k} \\ \dots \\ k_{\text{GLONASS}} \cdot \alpha_{L,j,k} \\ \dots \end{array} \right|; \]
\( k_{\text{GLONASS}} \) — coefficient equal to one for GLONASS and zero for GPS, determined by the NSA type \((k)\).
The system of equations is solved in the form:
\[ \begin{vmatrix} \Delta \tau_1 \\ \Delta \tau_2 \\ \vdots \\ \Delta \tau_L \\ \Delta \tau_{\text{GLONASS}} \end{vmatrix} = \frac{1}{c} \cdot V^{-1} \cdot \left( \begin{vmatrix} A \quad \mathbf{K}_{\text{GLONASS}} \end{vmatrix}^T \cdot \mathbf{W} \cdot \left( \mathbf{S}^{[10]} - \mathbf{R} \right) \right). \]
where
\[ c = 299792458 \text{ m/s} \text{ — speed of light}; \]
\[ V = \left| A \quad \mathbf{K}_{\text{GLONASS}} \right|^T \cdot \mathbf{W} \cdot \left| A \quad \mathbf{K}_{\text{GLONASS}} \right| + 10^{-10} \cdot I_4 = \]
\[ = \left| \begin{array}{ccccc} 10^{-10} + \sum\limits_{j,k} \frac{\alpha_{1,j,k}^2}{\sigma_k^2} & 0 & \cdots & 0 & \sum\limits_{j,k} \frac{k_{\text{GLONASS}} \cdot \alpha_{1,j,k}^2}{\sigma_k^2} \\ 0 & 10^{-10} + \sum\limits_{j,k} \frac{\alpha_{2,j,k}^2}{\sigma_k^2} & \cdots & 0 & \sum\limits_{j,k} \frac{k_{\text{GLONASS}} \cdot \alpha_{2,j,k}^2}{\sigma_k^2} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 10^{-10} + \sum\limits_{j,k} \frac{\alpha_{L,j,k}^2}{\sigma_k^2} & \sum\limits_{j,k} \frac{k_{\text{GLONASS}} \cdot \alpha_{L,j,k}^2}{\sigma_k^2} \\ \sum\limits_{j,k} \frac{k_{\text{GLONASS}} \cdot \alpha_{1,j,k}^2}{\sigma_k^2} & \sum\limits_{j,k} \frac{k_{\text{GLONASS}} \cdot \alpha_{2,j,k}^2}{\sigma_k^2} & \cdots & \sum\limits_{j,k} \frac{k_{\text{GLONASS}} \cdot \alpha_{L,j,k}^2}{\sigma_k^2} & 10^{-10} + \sum\limits_{i=1}^{L} \sum\limits_{j,k} \frac{k_{\text{GLONASS}} \cdot \alpha_{i,j,k}^2}{\sigma_k^2} \end{array} \right| ; \]
\( W \) — diagonal matrix of weighting coefficients, whose elements \( \left( \frac{1}{\sigma_k^2} \right) \) depending on which NSA
(\( k = 0 \) or \( k = 1 \)) they correspond to, equal \( \sigma_{GPS}^{-2} \) or \( \sigma_{\text{GLONASS}}^{-2} \).
\( I_4 \) — identity matrix of dimension \( 4 \times 4 \).
Inversion of matrix \( V \) is performed using the Frobenius formula (often referred to as the block matrix inversion formula using the Schur complement; see MathWorld or Wikipedia).
\[ V^{-1} = \left| \begin{array}{cc} V_{0,0}^{-1} + V_{0,0}^{-1} \cdot V_{0,1} \cdot H^{-1} \cdot V_{1,0} \cdot V_{0,0}^{-1} & -V_{0,0}^{-1} \cdot V_{0,1} \cdot H^{-1} \\ H^{-1} \cdot V_{1,0} \cdot V_{0,0}^{-1} & H^{-1} \end{array} \right|, \]
\[ \text{where} \quad \textbf{H} = \textbf{V}_{1,1} - \textbf{V}_{1,0} \cdot \textbf{V}_{0,0}^{-1} \cdot \textbf{V}_{0,1}; \]
\[ \textbf{V}_{0,0}, \textbf{V}_{0,1}, \textbf{V}_{1,0}, \textbf{V}_{1,1} \text{ — blocks of matrix } \textbf{V}: \]
\[ \mathbf{V} \;=\; \left|\, \begin{array}{c|cc} & L & 1 \\ \hline L & \mathbf{V}_{0,0} & \mathbf{V}_{0,1} \\ 1 & \mathbf{V}_{1,0} & \mathbf{V}_{1,1} \end{array} \right|. \]
If any diagonal element of matrix \( \mathbf{V}^{-1} \) exceeds \( 10^8 \), then the corresponding parameter \( (\tau_i, \dot{\tau}_i \text{ or } \tau_{\text{GLONASS}}, \dot{\tau}_{\text{GLONASS}}) \) cannot be determined, and its previously formed value remains unchanged.
The values of divergence between PNS and GPS generator frequencies \( (|\Delta \dot{\tau}_1 \dots \Delta \dot{\tau}_L|^T) \), as well as the estimate of "GLONASS-GPS" generator frequency divergence \( (\Delta \dot{\tau}_{\text{GLONASS}}) \), are determined by the formula:
\[ \begin{vmatrix} \Delta \dot{\tau}_1 \\ \Delta \dot{\tau}_2 \\ \vdots \\ \Delta \dot{\tau}_L \\ \Delta \dot{\tau}_{\text{GLONASS}} \end{vmatrix} = \frac{1}{c} \cdot V^{-1} \cdot \left( \begin{vmatrix} A \quad \mathbf{K}_{\text{GLONASS}} \end{vmatrix}^T \cdot \mathbf{W} \cdot \left( \dot{\mathbf{S}}^{[10]} - \dot{\mathbf{R}} \right) \right). \]
where
\[ \dot{S}^{[10]} = \left| \begin{matrix} \dots \\ \dot{S}_{1,j,k,3} \cdot \alpha_{1,j,k} \\ \dots \\ \dot{S}_{2,j,k,3} \cdot \alpha_{2,j,k} \\ \dots \\ \dot{S}_{L,j,k,3} \cdot \alpha_{L,j,k} \\ \dots \end{matrix} \right| \quad \quad \dot{R} = \left| \begin{matrix} \dots \\ \dot{R}_{1,j} \cdot \alpha_{1,j} \\ \dots \\ \dot{R}_{2,j} \cdot \alpha_{2,j} \\ \dots \\ \dot{R}_{m,j} \cdot \alpha_{m,j} \\ \dots \end{matrix} \right|; \]
\[ \dot{R}_{i,j} = \frac{ \dot{x}_{j,i} \cdot (x_{j,i} - X_{i}) + \dot{y}_{j,i} \cdot (y_{j,i} - Y_{i}) + \dot{z}_{j,i} \cdot (z_{j,i} - Z_{i}) }{R_{i,j}}. \]
Implementation Result: The described algorithm generates time corrections \( \Delta \tau_i \) and frequency divergences \( \Delta \dot{\tau}_i \) for the RPKNP regional positioning system. This implementation applies weighted matrix operations and block inversion techniques to ensure synchronization between regional PNS stations and GPS/GLONASS time scales.
This equation calculates time corrections for regional stations and the GLONASS-GPS time offset by modeling the observed signal delays \( S^{[10]} \).
Purpose: Align the clocks of regional (PNS) stations with GPS and GLONASS systems by expressing the observed signal delay \( S^{[10]} \) as a sum of the geometric distance \( R \), station clock offsets \( \Delta \tau_i \), and the GLONASS-GPS time offset \( \Delta \tau_{\text{GLONASS}} \).
Components: The matrix \( A \) assigns weights to each station’s time offset contribution based on measurement reliability. The vector \( K_{\text{GLONASS}} \) enables the GLONASS correction only when the signal is from a GLONASS satellite (i.e., \( K_{\text{GLONASS}} = 1 \)).