FORMATION OF DIFFERENTIAL CORRECTION DATA AND QUALITY CONTROL
Key Parameters
| Notation | Description |
|---|---|
| \(\Delta S_{i,j}\) | Differential correction estimate for pseudorange measurement at i-th receiver, j-th satellite. |
| \(\Delta \dot{S}_{i,j}\) | Differential correction estimate for pseudovelocity measurement at i-th receiver, j-th satellite. |
| \(\mathbf{S}^{[2]}\) | Vector of weighted observed pseudoranges after filtering and atmospheric corrections. |
| \(\dot{\mathbf{S}}^{[2]}\) | Vector of weighted observed pseudovelocities after filtering and atmospheric corrections. |
| \(\mathbf{R}\) | Vector of computed geometric distances between receivers and satellites. |
| \(\dot{\mathbf{R}}\) | Vector of computed rates of change of geometric distances. |
| \(\mathbf{A}\) | Diagonal matrix of weighting coefficients for regional stations. |
| \(\mathbf{K}_{\text{GLONASS}}\) | Vector of coefficients for GLONASS time scale correction application. |
| \(\Delta \tau_i\) | Clock offset estimate for i-th regional station (seconds). |
| \(\Delta \dot{\tau}_i\) | Frequency divergence estimate for i-th regional station (s/s). |
| \(\Delta \hat{S}_{i,j,k}\) | Differential correction estimate at registration time \(t_{z-count}\) for i-th receiver, j-th satellite, k-th epoch. |
| \(t_{z-count}\) | Registration time in GPS time scale, computed using integer part formula with 0.6 second intervals. |
| \(t_i\) | Measurement time at i-th receiver station. |
| \(R_{i,j,k}\) | Geometric distance between i-th receiver and j-th satellite at epoch k (meters). |
| \(c\) | Speed of light in vacuum, \(299,792,458\) m/s. |
| \(\sigma_{\Delta \hat{S}_{j,k}}\) | RMS (Root Mean Square) of differential correction estimates for j-th satellite at epoch k. |
| \(\Delta \bar{S}_{j,k}\) | Weighted mean value of differential corrections for j-th satellite at epoch k. |
| \(\Delta \overline{\dot{S}}_{j,k}\) | Weighted mean value of differential velocity corrections for j-th satellite at epoch k. |
| \(\zeta S_{i,j,k}\) | Reliability flag for TNPA parameters: 0 = reliable, 1 = unreliable/anomalous. |
| \(\sigma S_{i,j,k,2}\) | RMS error estimate for filtered pseudorange measurement. |
| \(\sigma_{\max \Delta S}\) | Maximum allowable RMS threshold for differential correction validation. |
| \(\sigma_{\max \Delta \dot{S}}\) | Maximum allowable RMS threshold for differential velocity correction validation. |
| \(Health_{j,k}\) | NSA status flag: 0 = healthy/usable, 1 = unhealthy/unusable. |
| \(S_{\text{CP},j,k}\) | Control receiver pseudorange measurement for j-th satellite at epoch k. |
| \(\dot{S}_{\text{CP},j,k}\) | Control receiver pseudovelocity measurement for j-th satellite at epoch k. |
| \(\vec{\Theta}\) | Vector of estimated parameters including coordinates, clock offsets, and time scale divergences. |
| \(\Delta \vec{\Theta}_n\) | Correction to the parameter vector at n-th iteration in Newton-Gauss method. |
| \(\vec{X}_{\text{CP}}\) | Control receiver coordinates \([X_{\text{CP}}, Y_{\text{CP}}, Z_{\text{CP}}]^T\) (meters). |
| \(\dot{\vec{X}}_{\text{CP}}\) | Control receiver velocity components \([\dot{X}_{\text{CP}}, \dot{Y}_{\text{CP}}, \dot{Z}_{\text{CP}}]^T\) (m/s). |
| \(\Delta \tau_{\text{CP}}\) | Control receiver clock offset estimate (seconds). |
| \(\Delta \dot{\tau}_{\text{CP}}\) | Control receiver frequency offset estimate (s/s). |
| \(\Delta \vec{\tau}_{\text{NSA}}\) | Vector of time scale divergences for GLONASS and INMARSAT relative to GPS. |
| \(\Delta \dot{\vec{\tau}}_{\text{NSA}}\) | Vector of frequency divergences for GLONASS and INMARSAT relative to GPS. |
| \(\varepsilon_{\Theta}\) | Convergence tolerance for Newton-Gauss iteration, \(\varepsilon_{\Theta} = 10^{-5}\) km. |
| \(V\) | Weight matrix for Newton-Gauss method with identity blocks and scaling factors. |
| \(I_J\) | Identity matrix of dimension \(J \times J\), where J is number of tracked satellites. |
| \(M\) | Design matrix combining geometric and time-related partial derivatives for least squares. |
| \(H\) | Matrix of partial derivatives with respect to position coordinates. |
| \(G\) | Matrix of partial derivatives with respect to velocity components. |
| \(K\) | Matrix of coefficients for GLONASS and INMARSAT time scale corrections. |
| \(W\) | Measurement weight matrix based on inverse squared measurement uncertainties. |
| \(\alpha_j\) | Weighting coefficient for j-th satellite, \(\alpha_j = (1 - \text{Pr}_p - S_{\text{CP},j,k})\). |
| \(DS_T, DV_T\) | A priori thresholds for control receiver position and velocity validation. |
| \(DS_N, DV_N\) | A priori thresholds for checking previously issued DCI suitability. |
| \(Pr_{DCI}\) | DCI validity flag: 0 = valid/usable, 1 = invalid/unusable. |
| \(L\) | Number of NRS (Navigation Receiver Systems) in the network. |
| \(J\) | Number of NSA (Navigation Satellite Apparatus) tracked by receivers. |
| \(i\) | Receiver index (1 to L). |
| \(j\) | Satellite index (1 to J). |
| \(k\) | Measurement epoch index for time-series processing. |
| \(n\) | Iteration step number in Newton-Gauss method. |
1. Calculation of differential corrections in TNPA is performed according to the following algorithm.
2. Estimates of differential corrections are formed for all NSA-NRS combinations at the time of TNPA registration in the GPS time scale (each differential correction value refers to the time of signal transmission from the NSA).
\[ \left\| \begin{array}{c} \cdots \\ \Delta S_{1,j} \\ \cdots \\ \Delta S_{2,j} \\ \cdots \\ \Delta S_{L,j} \\ \cdots \end{array} \right\| = \left(\mathsf{S}^{[2]}-\mathbf{R}\right)_{-} \|\mathbf{A} \quad \mathbf{K}_{\mathrm{GLONASS}}\| \left\| \begin{array}{c} \Delta \tau_1 \\ \Delta \tau_2 \\ \vdots \\ \Delta \tau_L \end{array} \right\|, \]
Pseudorange Differential Correction: Estimates corrections for pseudorange by accounting for geometric distances and clock offsets. Critical for enhancing DGPS positioning accuracy.
\[ \left\| \begin{array}{c} \cdots \\ \Delta \dot{S}_{1,j} \\ \cdots \\ \Delta \dot{S}_{2,j} \\ \cdots \\ \Delta \dot{S}_{L,j} \\ \cdots \end{array} \right\| = \left(\dot{\mathbf{S}}^{\left[2\right]}-\dot{\mathbf{R}}\right) - \left\Vert\mathbf{A} \quad \mathbf{K}_{\mathrm{GLONASS}}\right\Vert \left\| \begin{array}{c} \Delta \dot{\tau}_1 \\ \Delta \dot{\tau}_2 \\ \vdots \\ \Delta \dot{\tau}_L \end{array} \right\|, \]Pseudovelocity Differential Correction: Adjusts pseudovelocity for geometric rates and frequency divergences. Important for precise DGPS velocity tracking.
\[ \begin{align*} \dot{\mathbf{S}}^{\scriptstyle[2]} &= \left\| \begin{array}{c} \cdots \\ \dot{S}_{1,j,k,2}\cdot \boldsymbol{\alpha}_{1,j} \\ \cdots \\ \dot{S}_{2,j,k,2}\cdot \boldsymbol{\alpha}_{2,j} \\ \cdots \\ \dot{S}_{L,j,k,2}\cdot \boldsymbol{\alpha}_{L,j} \\ \cdots \end{array} \right\|; \end{align*} \]Weighted Pseudovelocity Vector: Combines filtered pseudovelocities with weighting coefficients for each receiver-satellite pair. Necessary for accurate differential correction calculations.
3. Formation of differential correction estimates in registration sequence at time tz-count.
\[ \Delta \hat{S}_{i,j,k} = \Delta S_{i,j,k} + \Delta \dot{S}_{i,j,k} \cdot (t_{z-count} - (t_i - \Delta \tau_i - \frac{R_{i,j,k}}{c})) \]
Differential Correction Estimate: Combines pseudorange and pseudovelocity corrections with time adjustments. Fundamental to synchronizing GNSS measurements.
where
\[ t_{\text{z-count}} = \text{int.part}\,\biggl[ \Bigl(\frac{t_{1} - 3600\,\cdot\text{int.part}\,\bigl[\tfrac{t_{1}}{3600}\bigr]}{0.6}\Bigr)\,\cdot\,0.6 \;+\; 3600\,\text{int.part}\,\bigl[\tfrac{t_{1}}{3600}\bigr] \biggr]. \]
GPS Time Registration: Calculates registration time in GPS time scale with 0.6-second intervals. Critical for aligning differential corrections.
4. Analysis of differential correction estimates is performed for each j-th NSA.
If for all (L) NRS there is only one estimate of differential corrections, then analysis (beyond that performed at previous processing stages) is not conducted, differential corrections are placed in the output frame, the NSA status flag is set to zero (\(Health_{i,j} = 0\)), the RMS of the differential correction is taken as \(\sigma_{i,j,k}\) and entered into the output frame.
If for all (L) NRS there are two or more estimates of differential corrections, then the RMS of these estimates is calculated:
\[ \sigma_{\Delta \hat{S}_{j,k}} = \sqrt{ \frac{ \displaystyle \sum_{i=1}^{L} \bigl(\Delta \hat{S}_{i,j,k} - \Delta \bar{S}_{i,j,k}\bigr)^{2} \,\cdot \bigl(1 - \zeta \cdot S_{i,j,k}\bigr) }{ \Bigl(\displaystyle \sum_{i=1}^{L} \bigl(1 - \zeta \cdot S_{i,j,k}\bigr)\Bigr) - 1 } }. \]
RMS of Differential Corrections: Quantifies the variability in correction estimates, excluding unreliable data. Important for assessing GNSS data quality.
\[ \Delta \overline{{S}}_{j,k} = \frac{ \displaystyle \sum_{i=1}^L \Delta\hat{S}_{i,j,k} \cdot \frac{(1 - \zeta S_{i,j,k})}{(\sigma S_{i,j,k,2} + 10^{-10})^2} }{ \displaystyle \sum_{i=1}^L \frac{(1 - \zeta S_{i,j,k})}{(\sigma S_{i,j,k,2} + 10^{-10})^2} }. \]
Weighted Mean Pseudorange Correction: Computes the average correction weighted by reliability and precision. Essential for robust RTK positioning accuracy.
\[ \Delta \overline{\dot{S}}_{j,k} = \frac{ \displaystyle \sum_{i=1}^L \Delta\dot{S}_{i,j,k} \cdot \frac{(1 - \zeta S_{i,j,k})}{(\sigma S_{i,j,k,2} + 10^{-10})^2} }{ \displaystyle \sum_{i=1}^L \frac{(1 - \zeta S_{i,j,k})}{(\sigma S_{i,j,k,2} + 10^{-10})^2} }. \]
Weighted Mean Pseudovelocity Correction: Calculates the average velocity correction using reliability weights. Vital for precise RTK velocity estimates.
\[\text{If } \sigma_{\Delta \hat{S}_{j,k}} < \sigma_{\max \Delta S} \;\;\text{or}\;\; \sigma_{\Delta \dot{S}_{j,k}} < \sigma_{\max \Delta \dot{S}}, \quad \text{then the output frame contains} \]
mean values of differential corrections and estimates of their (mean values)
RMS \( \sigma_{\Delta \bar{S}_{j,k}} \), the NSA status flag is set to zero \(( Health_{j,k} = 0 )\).
Values \( \sigma_{\Delta \bar{S}_{j,k}} \) are calculated by formulas: \[ \sigma_{\Delta \hat{S}_{j,k}} = \left( \displaystyle \sum_{i=1}^L \frac{(1 - \zeta S_{i,j,k})}{(\sigma S_{i,j,k,2} + 10^{-10})^2} \right)^{-1}. \]
RMS of Weighted Corrections: Estimates the precision of weighted differential corrections. Fundamental to validating GNSS correction reliability.
If \( \sigma_{\Delta \dot{S}_{j,k}} > \sigma_{\max \Delta S} \) or \( \sigma_{\Delta \dot{S}_{j,k}} > \sigma_{\max \Delta \dot{S}} \), then anomaly search is performed using the condition:
\[ \left| \Delta \hat{S}_{i,j,k} - \Delta \bar{S}_{j,k} \right| > 2 \cdot \sigma_{\Delta \hat{S}_{j,k}} \quad \text{or} \quad \left| \Delta \dot{S}_{i,j,k} - \Delta \overline{\dot{S}}_{j,k} \right| > 2 \cdot \sigma_{\Delta \dot{S}_{j,k}} \]
Anomaly Detection Condition: Identifies outliers in differential corrections using 2-sigma rule. Important for ensuring GNSS data integrity.
When anomalous measurements are detected, a flag \( \zeta S_{i,j,k} \) is formed, set to one, and differential correction analysis is repeated.
If no anomalous values are found, i.e. \[ \left| \Delta \hat{S}_{i,j,k} - \Delta \bar{S}_{j,k} \right| < 2 \cdot \sigma_{\Delta \hat{S}_{j,k}} \]
Pseudorange Correction Validation: Verifies corrections align with 2-sigma statistical bounds. Critical for reliable DGPS positioning.
and \[ \left| \Delta \dot{S}_{i,j,k} - \Delta \overline{\dot{S}}_{j,k} \right| < 2 \cdot \sigma_{\Delta \dot{S}_{j,k}}, \] then differential corrections are considered invalid, the NSA status flag is set to zero \( (Health_{j,k} = 0) \), and TNPA validity flags \( \zeta S_{i,j,k} \) are set to one.Pseudovelocity Correction Validation: Confirms velocity corrections meet 2-sigma reliability criteria. Essential for accurate DGPS velocity tracking.
5. Formed differential correction values are checked for compliance with conditions:
\[ \left| \Delta \bar{S}_{j,k} \right| < 10485.44\,m, \]
Pseudorange Correction Threshold: Checks if mean corrections are within allowable limits. Fundamental to ensuring GNSS correction validity.
\[ \left| \Delta \overline{\dot{S}}_{j,k} \right| < 4.064\, \frac{m}{s}. \]
Pseudovelocity Correction Threshold: Verifies velocity corrections are within acceptable bounds. Necessary for reliable GNSS velocity estimates.
If any condition is not satisfied, the NSA status flag is set to one \( (Health_{j,k} = 1) \).
6. If for any NSA all TNPA validity flags \( \zeta S_{i,j,k} \) are set to one, then differential corrections are not formed for it, and the NSA status flag is set to one \( (Health_{j,k} = 1) \).
7. If a control receiver of navigation signals is available, additional quality check of DCI is performed. In this case, pseudoranges and pseudovelocities measured by the control receiver are corrected, smoothed, and compared with its actual coordinates. Based on comparison results, a conclusion is formed about the possibility (necessity) of issuing DCI to users.
Differential corrections are entered into TNPA only for those NSA for which the validity flag \(Pr\_S \) is zero, according to the following formulas:
\[ S_{\text{CP},j,k} = S_{\text{CP},j,k,2} + \Delta S_j, \]
Control Receiver Pseudorange Correction: Applies differential correction to control receiver pseudorange. Critical for validating DGPS positioning accuracy.
\[ \dot{S}_{\text{CP},j,k} = \dot{S}_{\text{CP},j,k,2} + \Delta \dot{S}_j; \]
Control Receiver Pseudovelocity Correction: Adjusts control receiver pseudovelocity with differential correction. Important for ensuring DGPS velocity precision.
where CP — control receiver identifier.
Calculation of control receiver coordinates is performed using the Newton-Gauss method:
\[ \overset{\rightharpoonup}{\widetilde{\Theta}}_n = \overset{\rightharpoonup}{\widetilde{\Theta}}_{n-1} + \Delta \overset{\rightharpoonup}{\widetilde{\Theta}}_n, \]
Newton-Gauss Parameter Update: Iteratively refines coordinates, clock, and time scale estimates. Fundamental to achieving precise GNSS positioning.
where
\[ \overset{\rightharpoonup}{\widetilde{\Theta}} = \lVert \overset{\rightharpoonup}{\widetilde{X}}_{\text{CP }}^T \hspace{10pt} C \cdot \Delta \tau_{CP} \hspace{10pt} \dot{\overrightarrow{\widetilde{X}}}^{\,T}_{CP} \hspace{10pt} C \cdot \Delta \dot{\tau}_{CP} \hspace{10pt} C \cdot \Delta \overrightarrow{\tau}^{\,T}_{NSA} \hspace{10pt} C \cdot \Delta \dot{\overrightarrow{\tau}}^{\,T}_{NSA} \lVert^{T} \]
— vector of estimated parameters;
\[ \Delta \vec{\tau}_{\text{NSA}} = \begin{Vmatrix} \Delta \tau_{\text{GLONASS}} \\ \Delta \tau_{\text{INMARSAT}} \end{Vmatrix} \]
— estimates of time scale divergence "GLONASS-GPS" and "INMARSAT-GPS";
\[ \Delta \dot{\vec{\tau}}_{\text{NSA}} = \begin{Vmatrix} \Delta \dot{\tau}_{\text{GLONASS}} \\ \Delta \dot{\tau}_{\text{INMARSAT}} \end{Vmatrix} \]
— estimates of generator frequency divergence "GLONASS-GPS" and "INMARSAT-GPS";
\( \Delta \overrightarrow{\widetilde{\Theta}} \) — correction to the vector of estimated parameters.
C — speed of light \((C = 299792.458 \text{ m/s})\);
\( n \) — iteration step number.
The iterative process terminates when \( k \) reaches 5 or when the condition is satisfied:
\[ \sqrt{\Delta \overrightarrow{\widetilde{\Theta}}_n^T \cdot V \cdot \Delta \overrightarrow{\widetilde{\Theta}}_n} \leq \varepsilon_{\Theta}, \]
Convergence Criterion: Tests if Newton-Gauss iteration has reached sufficient precision. Critical for ensuring GNSS solution accuracy.
where \( \varepsilon_{\Theta} \) — tolerance defining computation errors, \( \varepsilon_{\Theta} = 10^{-5} \) km;
\[ V = \begin{Vmatrix} I_J & 0 \\ 0 & 10^6 \cdot I_J \end{Vmatrix} \quad \text{— weight matrix;} \]
\( I_J \) — diagonal matrix of dimension \( J \times J \), \( J \) — number of NSA registered by NRS.
As initial approximation for estimated parameters \( \vec{\Theta}_0 \), actual coordinates of the control NRS are used.
\[ \left( \overset{\rightharpoonup}{X}_{\text{CP}} = \begin{bmatrix} X_{\text{CP}} & Y_{\text{CP}} & Z_{\text{CP}} \end{bmatrix}^T \right) \text{ and zero values of time scale and frequency divergence.} \]
Correction to the vector of estimated parameters \( \Delta \overset{\rightharpoonup}{\widetilde{\Theta}} \) is calculated by solving the following system of equations:
\[ \begin{vmatrix} \dots & \dots \\ S_{CP,j,k} \cdot \alpha_j & \tilde{S}_{CP,j} \cdot \alpha_j \\ \dots & \dots \\ \hat{S}_{CP,j,k} \cdot \alpha_j & \tilde{\hat{S}}_{CP,j} \cdot \alpha_j \\ \dots & \dots \end{vmatrix}_{k-1} = \begin{vmatrix} H & E & 0 & 0 & K & 0 \\ G & 0 & H & E & 0 & K \end{vmatrix}_{k-1} \cdot \Delta \tilde{\Theta}_n = M \cdot \Delta \tilde{\Theta}_n, \]Newton-Gauss System of Equations: Solves for parameter corrections using design matrix M and weighted measurements. Essential for iterative GNSS positioning refinement.
\[ \left\lVert \begin{array}{c} \dots \\ S_{\text{CP}, j, k} \cdot \alpha_j \\ \dots \\ \dot{S}_{\text{CP}, j, k} \cdot \alpha_j \\ \dots \end{array} \right\rVert - \left\lVert \begin{array}{c} \dots \\ \widetilde{S}_{\text{CP}, j} \cdot \alpha_j \\ \dots \\ \widetilde{\!\dot{S}}_{\text{CP}, j} \cdot \alpha_j \\ \dots \end{array} \right\rVert _{k-1} = \left\lVert \begin{array}{cccccc} H & E & 0 & 0 & K & 0 \\ G & 0 & H & E & 0 & K \end{array} \right\rVert \cdot \Delta \overset{\rightharpoonup}{\widetilde{\Theta}}_n = M \cdot \Delta \overset{\rightharpoonup}{\widetilde{\Theta}}_n, \]
Newton-Gauss Measurement Equations: Relates observed and computed measurements in matrix form for parameter estimation. Critical for GNSS solution convergence.
where
\[ \alpha_j = \left( 1 - \text{Pr}_p - S_{\text{CP}, j, k} \right); \]
Satellite Weighting Coefficient: Defines weights for satellite measurements based on reliability and pseudorange validity. Essential for accurate GNSS data processing.
\[ \widetilde{S}_{\text{CP}, j} = \widetilde{R}_{\text{CP}, j} + C \cdot \Delta \widetilde{\tau}_{\text{CP}} + C \cdot k_{\text{GLONASS}} \cdot \Delta \widetilde{\tau}_{\text{GLONASS}} + C \cdot \Delta_{\text{INMARSAT}} \cdot \widetilde{\tau}_{\text{INMARSAT}}; \]
Control Receiver Pseudorange Estimate: Computes pseudorange for the control receiver, including clock and time scale corrections. Critical for validating DGPS measurements.
\[ \widetilde{\dot{S}}_{\text{CP}, j} = \frac{ (\dot{x}_j - \widetilde{\dot{X}}_{\text{CP}}) \cdot (x_j - \widetilde{X}_{\text{CP}}) + (\dot{y}_j - \widetilde{\dot{Y}}_{\text{CP}}) \cdot (y_j - \widetilde{Y}_{\text{CP}}) + (\dot{z}_j - \widetilde{\dot{Z}}_{\text{CP}}) \cdot (z_j - \dot{Z}_{\text{CP}}) }{ \widetilde{R}_{\text{CP}, j} } + \]
\[ + C \cdot \Delta \widetilde{\dot{\tau}}_{\text{CP}} + C \cdot k_{\text{GLONASS}} \cdot \Delta \widetilde{\dot{\tau}}_{\text{GLONASS}} + C \cdot \Delta_{\text{INMARSAT}} \cdot \widetilde{\dot{\tau}}_{\text{INMARSAT}}; \]
Control Receiver Pseudovelocity Estimate: Calculates pseudovelocity using relative satellite-receiver velocities and time corrections. Vital for precise DGPS velocity estimation.
\[ \widetilde{R}_{\text{CP}, j} = \sqrt{ (x_j - \widetilde{X}_{\text{CP}})^2 + (y_j - \widetilde{Y}_{\text{CP}})^2 + (z_j - \widetilde{Z}_{\text{CP}})^2 }; \]
Geometric Distance Estimate: Computes the distance between the control receiver and satellite using estimated coordinates. Fundamental to GNSS positioning accuracy.
\[ H = - \left\| \frac{x_j - \widetilde{X}_{\text{CP}}}{\widetilde{R}_{\text{CP}, j}} \cdot \alpha_j \quad \frac{y_j - \widetilde{Y}_{\text{CP}}}{\widetilde{R}_{\text{CP}, j}} \cdot \alpha_j \quad \frac{z_j - \dot{Z}_{\text{CP}}}{\widetilde{R}_{\text{CP}, j}} \cdot \alpha_j \right\|; \]
Position Partial Derivatives: Calculates partial derivatives of pseudorange with respect to receiver coordinates. Necessary for Newton-Gauss positioning solutions.
\[ G = \left\| -\left( \frac{\dot{x}_j - \widetilde{\dot{X}}_{\text{CP}}}{\widetilde{R}_{\text{CP}, j}} + \frac{\widetilde{\dot{S}}_{\text{CP}, j} \cdot (x_j - \widetilde{X}_{\text{CP}})}{\widetilde{R}_{\text{CP}, j}} \right) \alpha_j \quad -\left( \frac{\dot{y}_j - \widetilde{\dot{Y}}_{\text{CP}}}{\widetilde{R}_{\text{CP}, j}} + \frac{\widetilde{\dot{S}}_{\text{CP}, j} \cdot (y_j - \widetilde{Y}_{\text{CP}})}{\widetilde{R}_{\text{CP}, j}} \right) \alpha_j \quad -\left( \frac{\dot{z}_j - \widetilde{\dot{Z}}_{\text{CP}}}{\widetilde{R}_{\text{CP}, j}} + \frac{\widetilde{\dot{S}}_{\text{CP}, j} \cdot (z_j - \widetilde{Z}_{\text{CP}})}{\widetilde{R}_{\text{CP}, j}} \right) \alpha_j \right\|; \]
Velocity Partial Derivatives: Computes partial derivatives of pseudovelocity with respect to receiver velocities. Critical for accurate GNSS velocity solutions.
\( K = \left\|\begin{array}{cc} k_{\text{GLONASS}} \cdot \alpha_j & k_{\text{INMARSAT}} \cdot \alpha_j \end{array}\right\| \)
Time Scale Correction Coefficients: Defines coefficients for GLONASS and INMARSAT time scale adjustments. Important for synchronizing multi-GNSS systems.
Estimate of the increment \( \overset{\rightharpoonup}{\widetilde{\Theta}} \) is calculated by the formula:
\[ \Delta \overset{\rightharpoonup}{\widetilde{\Theta}}_k = \left( M^T \cdot W \cdot M \right)^{-1} \cdot M^T \cdot W \cdot \left\| \begin{array}{c} \dots \\ S_{\text{CP}, j, k} \cdot \alpha_j \\ \dots \\ \dot{S}_{\text{CP}, j, k} \cdot \alpha_j \\ \dots \end{array} \right\| - \left\| \begin{array}{c} \dots \\ \widetilde{S}_{\text{CP}, j, k} \cdot \alpha_j \\ \dots \\ \widetilde{\dot{S}}_{\text{CP}, j, k} \cdot \alpha_j \\ \dots \end{array} \right\|_{k-1}, \]
Parameter Increment Estimate: Updates estimated parameters using weighted least squares in Newton-Gauss iteration. Essential for refining GNSS solutions.
\[ \text{where } W = \left\| \begin{array}{cc} \left( \sigma S_{\text{CP}, j, k} \right)^{-2} & 0 \\ 0 & \left( \sigma \dot{S}_{\text{CP}, j, k} \right)^{-2} \end{array} \right\| \text{ — weight matrix.} \]
Measurement Weight Matrix: Assigns weights to measurements based on inverse squared uncertainties. Fundamental to accurate GNSS parameter estimation.
The obtained estimates of coordinates and velocity vector components of the control NRS are compared with their actual values using conditions:
\[ \sqrt{ (X_{\text{CP}} - \widetilde{X}_{\text{CP}})^2 + (Y_{\text{CP}} - \widetilde{Y}_{\text{CP}})^2 + (Z_{\text{CP}} - \widetilde{Z}_{\text{CP}})^2 } \leq DS_T, \]
Coordinate Validation Condition: Verifies estimated control receiver coordinates against actual values within a threshold. Critical for validating DGPS accuracy.
\[ \sqrt{ \widetilde{\dot{X}}_{\text{CP}}^2 + \widetilde{\dot{Y}}_{\text{CP}}^2 + \widetilde{\dot{Z}}_{\text{CP}}^2 } \leq DV_T, \]
where DST and DVT are a priori parameters specified by the operator.
If both conditions are satisfied, then the DCI prepared by CCS is considered valid and the flag PrDCI is set to zero. Otherwise, PrDCI is set to one.
The need to refine previously issued DCI to users is checked using this DCI, and TNPA measured by the control receiver.
Estimates of coordinates and velocity vector components of the control NRS, obtained using previously issued DCI, are compared with their actual values. The following condition is used:
\[ \sqrt{(X_{\text{CP}} - \widetilde{X}_{\text{CP}})^2 + (Y_{\text{CP}} - \widetilde{Y}_{\text{CP}})^2 + (Z_{\text{CP}} - \widetilde{Z}_{\text{CP}})^2} \leq DS_{N}, \]
Previous DCI Validation Condition: Validates previously issued differential corrections by comparing estimated with actual coordinates. Essential for DCI refinement decisions.
\[ \sqrt{\widetilde{\dot{X}}_{\text{CP}}^2 + \widetilde{\dot{Y}}_{\text{CP}}^2 + \widetilde{\dot{Z}}_{\text{CP}}^2} \leq DV_{N}, \]
Velocity Validation Condition: Confirms estimated control receiver velocities meet predefined accuracy thresholds. Essential for reliable DGPS velocity estimates.
---------------Previous DCI Velocity Validation: Validates estimated velocities from prior DCI against accuracy thresholds. Critical for deciding DCI suitability.
=========where DSN and DVN are a priori parameters specified by the operator.
If both conditions are satisfied, then DCI previously issued to users is considered suitable for use and does not require refinement, and DCI is not issued to users. Otherwise, DCI is issued to users.
Implementation Result: The described algorithms perform differential correction formation and quality control for GNSS positioning systems. These formulas reflect the natural computational flow essential for any differential navigation system operation. This implementation applies fundamental statistical methods established in classical mathematics: weighted least squares (Legendre, 1805), anomaly detection via statistical thresholds (Gauss, 1809), and Newton-Gauss iteration (Newton, 1669; Gauss, 1809) along with validation procedures necessary for reliable differential correction generation and distribution to navigation users.