MEASUREMENT INFORMATION ANALYSIS AND NAVIGATION FIELD INTEGRITY CONTROL
Key Parameters
| Notation | Description |
|---|---|
| \(\Delta \hat{S}_{i,j,k}\) | Differential correction estimate for pseudorange measurements at i-th receiver, j-th satellite, k-th epoch. |
| \(\Delta \hat{\dot{S}}_{i,j,k}\) | Differential correction estimate for pseudovelocity measurements at i-th receiver, j-th satellite, k-th epoch. |
| \(R_{i,j,k}\) | Geometric distance between j-th NSA and phase center of receiving antenna of i-th NRS (meters). |
| \(\dot{R}_{i,j,k}\) | Rate of change of geometric distance \(R_{i,j}\) (m/s). |
| \(S_{i,j,k,3}\) | Corrected pseudorange measurement after atmospheric corrections (meters). |
| \(\dot{S}_{i,j,k,3}\) | Corrected pseudovelocity measurement after atmospheric corrections (m/s). |
| \(\Delta \tau_{\text{GLONASS}}\) | GLONASS-GPS time scale divergence correction (seconds). |
| \(\Delta \dot{\tau}_{\text{GLONASS}}\) | GLONASS-GPS frequency divergence correction (s/s). |
| \(x_{j,i,k}, y_{j,i,k}, z_{j,i,k}\) | Coordinates of j-th satellite relative to i-th receiver at epoch k (meters). |
| \(\dot{x}_{j,i,k}, \dot{y}_{j,i,k}, \dot{z}_{j,i,k}\) | Velocity components of j-th satellite relative to i-th receiver at epoch k (m/s). |
| \(X_i, Y_i, Z_i\) | Known coordinates of i-th receiver station (meters). |
| \(k\) | Coefficient equal to 1 for GLONASS spacecraft and 0 for other spacecraft (GPS). |
| \(c\) | Speed of light in vacuum, \(299,792,458\) m/s. |
| \(H_i\) | Integrity control matrix for pseudorange measurements, dimension \([J \times J]\). |
| \(G_i\) | Integrity control matrix for pseudovelocity measurements, dimension \([J \times J]\). |
| \(J\) | Number of detected NSA (Navigation Satellite Apparatus). |
| \(L\) | Number of NRS (Navigation Receiver Systems). |
| \(\Pi_{P-} S_{i,j,k}\) | Validity flag for pseudorange measurement: 0 = valid, 1 = invalid. |
| \(\Pi_{P-} \dot{S}_{i,j,k}\) | Validity flag for pseudovelocity measurement: 0 = valid, 1 = invalid. |
| \(\delta S_{m,n}\) | Absolute difference between differential correction estimates for pseudorange. |
| \(\delta \dot{S}_{m,n}\) | Absolute difference between differential correction estimates for pseudovelocity. |
| \(DS\) | A priori threshold parameter for pseudorange integrity control, set by operator. |
| \(D\dot{S}\) | A priori threshold parameter for pseudovelocity integrity control, set by operator. |
| \(\Sigma_{i,j_n}^{H}\) | Sum of elements in n-th column of matrix \(H_i\) for integrity analysis. |
| \(\Sigma_{i,j_n}^{G}\) | Sum of elements in n-th column of matrix \(G_i\) for integrity analysis. |
| \(\zeta^{S}_{i,j,k}\) | Reliability flag for TNPA parameters: 0 = reliable, 1 = unreliable. |
| \(\text{P}r\_\text{I}_{i,j,k}\) | Signal-to-noise ratio flag: 0 = S/N > 3, 1 = S/N ≤ 3. |
| \(\text{P}r\_chan_{ch}\) | Navigation channel health flag: 0 = operational, 1 = faulty. |
| \(i\) | Receiver index in measurement processing sequence (1 to L). |
| \(j\) | Satellite index (NSA - Navigation Satellite Apparatus) (1 to J). |
| \(k\) | Measurement epoch index for time-series processing. |
| \(m, n\) | Matrix indices for integrity control calculations (1 to J). |
1. Differential corrections estimates
Estimates of differential corrections \( \Delta \hat{S}_{i,j,k} \) and \( \Delta \hat{\dot{S}}_{i,j,k} \) in TNPA are calculated by the following formulas (initial values of quantities \( \Delta \tau_{\text{GLONASS}} \) and \( \Delta \dot{\tau}_{\text{GLONASS}} \) at CCS SW startup are equal to zero):
\[ \Delta \hat{S}_{i,j,k} = R_{i,j,k} - \left( S_{i,j,k,3} - k \cdot c \cdot \Delta \tau_{\text{GLONASS}} \right), \]
\[ \Delta \hat{\dot{S}}_{i,j,k} = \dot{R}_{i,j,k} - \left( \dot{S}_{i,j,k,3} - k \cdot c \cdot \Delta \dot{\tau}_{\text{GLONASS}} \right), \]
where
\[ R_{i,j,k} = \left[ (x_{j,i,k} - X_i)^2 + (y_{j,i,k} - Y_i)^2 + (z_{j,i,k} - Z_i)^2 \right]^{\frac{1}{2}} \]
— geometric distance between (j-th) NSA and phase center of receiving antenna of (i-th) NRS;
\[ \dot{R}_{i,j,k} = \frac{1}{R_{i,j,k}} \left[ (x_{j,i,k} - X_i) \cdot \dot{x}_{j,i,k} + (y_{j,i,k} - Y_i) \cdot \dot{y}_{j,i,k} + (z_{j,i,k} - Z_i) \cdot \dot{z}_{j,i,k} \right] \]
— rate of change of parameter \( R_{i,j} \),
\( k \) — coefficient equal to unity for GLONASS spacecraft and zero for other spacecraft.
\( c = 299792458 \) m/s — speed of light.
2. Integrity control of observed NSA constellation
Integrity control of observed NSA constellation is performed based on separate processing results of pseudoranges and pseudovelocities in the following sequence.
Matrices \( H_i \) and \( G_i \) of dimension \( [J \times J] \) each are formed with elements
\[ \{ H_i \}_{m,n} = 0, \text{ if } \Pi_{P-} S_{i,j_m,k_m} = \Pi_{P-} S_{i,j_n,k_n} = 0 \text{ and } \delta S_{m,n} \leq DS \]
\[ \text{or} \quad \Pi_{P-} S_{i,j_m,k_m} = \Pi_{P-} S_{i,j_n,k_n} = 0, \quad \Delta \tau_{\text{GLONASS}} = 0 \text{ and } k_m \neq k_n, \]
\( DS \) — a priori parameter set by operator,
\[ \{ H_i \}_{m,n} = 1 \quad \text{in other cases}, \]
\[ \delta S_{m,n} = \left| \Delta \hat{S}_{i,j_m,k_m} - \Delta \hat{S}_{i,j_n,k_n} \right|, \]
\[ \{ G_i \}_{m,n} = 0, \text{ if } \Pi_{P-} \dot{S}_{i,j_m,k_m} = \Pi_{P-} \dot{S}_{i,j_n,k_n} = 0 \text{ and } \delta \dot{S}_{m,n} \leq D\dot{S} \]
\[ \text{or} \quad \Pi_{P-} \dot{S}_{i,j_m,k_m} = \Pi_{P-} \dot{S}_{i,j_n,k_n} = 0, \quad \Delta \dot{\tau}_{\text{GLONASS}} = 0 \text{ and } k_m \neq k_n, \]
\( D\dot{S} \) — a priori parameter set by operator,
\[ \{ G_i \}_{m,n} = 1 \quad \text{in other cases}, \]
\[ \delta \dot{S}_{m,n} = \left| \Delta \hat{\dot{S}}_{i,j_m,k_m} - \Delta \hat{\dot{S}}_{i,j_n,k_n} \right|, \]
where \( m, n = 1, \dots, J \);
\( J \) — number of detected NSA,
\( i = 1, \dots, L \);
\( L \) — number of NRS.
Then the sums of elements of each column of matrices \( H_i \) and \( G_i \) are calculated.
\[ \Sigma_{i,j_n}^{H} = \sum_{m=1}^{M_i} \{H_i\}_{m,j_n}, \quad \Sigma_{i,j_n}^{G} = \sum_{m=1}^{M_i} \{G_i\}_{m,j_n}. \]
Based on analysis of parameter values \( \Sigma_{i,j_n}^{H} \) and \( \Sigma_{i,j_n}^{G} \) for each NRS, the numbers of NSA whose information cannot be used for synchronization of NRS and GPS time scales are determined.
Analysis of the matrices \( H_i \) and \( G_i \) is performed for each NRS individually. If \( \Sigma^{H}_{i,j_n} = \Sigma^{G}_{i,j_n} = 0 \), then, among the NSAs tracked by the \( i \)-th NRS, there are no unreliable parameters and the flags \( \zeta^{S}_{i,j,k} \) are equal to zero.
If \( \Sigma^{H}_{i,j_n} \) or \( \Sigma^{G}_{i,j_n} \) differs from zero, unreliable TNPA parameters are present. Their indices (\( m \)) are determined by the positions of the unity elements in the first columns with the minimum values of \( \Sigma^{H}_{i,j_n} \) or \( \Sigma^{G}_{i,j_n} \). The flags \( \zeta^{S}_{i,j,k} \) corresponding to these unreliable TNPA values are set to unity.
Finally, for each NSA, analysis is performed by comparing the expressions:
\[ \sum_i \left( 1 - \zeta S_{i,j,k} \right) \cdot \left( 1 - \text{P}r\_\text{I}_{i,j,k} \right) \quad \text{AND} \quad \sum_i \left( 1 - \text{P}r\_\text{I}_{i,j,k} \right) \]
\( (\text{P}r\_\text{I}_{i,j,k} = 0 \text{ if signal-to-noise ratio is greater than 3 and } (\text{P}r\_\text{I}_{i,j,k} =1 \text{ otherwise}) \text{ according to the following algorithm.} \)
If \[ 0 < \sum_i \left( 1 - \zeta S_{i,j,k} \right) \cdot \left( 1 - \text{P}r\_\text{I}_{i,j,k} \right) < \sum_i \left( 1 - \text{P}r\_\text{I}_{i,j,k} \right), \]
then NRS channels for which \( \zeta S_{i,j,k} \) equals unity and \( \text{P}r\_\text{I}_{i,j,k} \) equals zero are considered faulty \((\text{P}r\_chan_{ch} = 1)\).
In other cases, it is considered that the corresponding NRS channels function normally \((\text{P}r\_chan_{ch} = 0)\).
Table of Flags (Features)
| # | Flag name | Meaning | 0 — OK | 1 — Bad / Rejected |
|---|---|---|---|---|
| 1 | Pr_Si,j,k | Pseudorange validity | accepted | rejected |
| 2 | Pr_Sdoti,j,k | Pseudorange‑rate (Doppler) validity | accepted | rejected |
| 3 | Pr_Ii,j,k | S / I quality check (signal‑to‑interference) | S/I > 3 | S/I ≤ 3 |
| 4 | Pr_chan_ch | Composite navigation‑channel health | operational | faulty |
Implementation Result: The described algorithms perform differential correction estimation and navigation field integrity monitoring for GNSS measurement data. These formulas reflect the natural flow of algorithms without which no navigation system can function. This implementation applies fundamental mathematical operations (geometric distance calculations, statistical threshold analysis, binary flag logic) essential for any reliable positioning system operation. If you intend to write and defend your thesis project on GNSS topics, it is recommended to analyze and evaluate navigation field integrity as presented on this page.