| GNSS | |
GNSS Constellation Specific Monthly Analysis Summary: May 2025
The analysis performed in this report is solely his work and own opinion. State Program: U.S.A (G); EU (E); China (C) “Only MEO- SECM satellites”; Russia (R); Japan (J); India (I) |
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Introduction
This article continues the monthly performance analysis of the GNSS constellation. Readers are encouraged to refer to previous issues for foundational discussions and earlier results. As a complementary extension to last month’s article on the application of variational equations in parameter estimation, this issue includes a focused analysis on reduced-dynamic orbit determination. Specifically, it will explore how satellite orbits—such as those of LEO missions—can be estimated by combining GNSS observations with physical models of orbital dynamics, supported by the solution of variational equations. This approach bridges the gap between purely kinematic and fully dynamic orbit determination, offering a robust framework for high precision orbit estimation even in the presence of data gaps or modeling uncertainties.
Analyzed Parameters for April 2025
(Dhital et. al, 2024) provides a brief overview of the necessity and applicability of monitoring the satellite clock and orbit parameters.
a..Satellite Broadcast Accuracy, measured in terms of Signal-In Space Range Error (SISRE) (Montenbruck et. al, 2010).
b. SISRE-Orbit (only orbit impact on the range error), SISRE (both orbit and clock impact), and SISRE-PPP (as seen by the users of carrier phase signals, where the ambiguities absorb the unmodelled biases related to satellite clock and orbit estimations. Satellite specific clock bias is removed) (Hauschlid et.al, 2020)
c. Clock Discontinuity: The jump in the satellite clock offset between two consecutive batches of data uploads from the ground mission segment. It is indicative of the quality of the satellite atomic clock and associated clock model.
d. URA: User Range Accuracy as an indicator of the confidence on the accuracy of satellite ephemeris. It is mostly used in the integrity computation of RAIM.
e. GNSS-UTC offset: It shows stability of the timekeeping of each constellation w.r.t the UTC
f. Variational Equations and Satellite Orbit Estimation: The variational equations are a key set of differential equations to reliably capture the dynamics of the satellite motion in the orbit determination process. The solution to these equations is indicative of quality of the initial orbit determination and force models
Note:– for India’s IRNSS there are no precise satellite clocks and orbits as they broadcast only 1 frequency which does not allow the dual frequency combination required in precise clock and orbit estimation; as such, only URA and Clock Discontinuity is analyzed.
(f) Variational Equations and Satellite Orbit Estimation
Continuing from last month’s introduction to variational equations in dynamical parameter estimation, this section presents a practical example of satellite orbit determination using GNSS observables and dynamic modeling. The principles discussed here are equally applicable to GNSS network processing for both GNSS and LEO satellite orbit determination. However, for clarity and tractability, the focus is on the orbit determination of a Low Earth Orbit (LEO) satellite, which offers a more manageable framework compared to the complexity of a full GNSS network solution involving multiple ground stations and satellites.
In essence, LEO orbit determination is the inverse of GNSS satellite orbit determination from ground networks. In the LEO case, the positions of GNSS satellites are assumed known (from precise ephemerides), and the LEO satellite’s trajectory is estimated. Conversely, in GNSS network processing, the ground station positions are known, and the satellite orbits are estimated.
To support the analysis, the open-source GNSS processing software GROOPS (Mayer et.al, 2021) is used leveraging publicly available data from the International GNSS Service (IGS) and TU Graz for reference and auxiliary inputs. For rigorous review and testing purposes, (GROOPS et.al, 2025) and (Strasser et.al, 2022) are recommended.
Before jumping to the analysis, however, a short recap on the topic is provided. The variational equations give the partial derivatives of the satellite state vector with respect to its initial conditions. Simultaneously, the satellite’s orbit is propagated from the initial state using a dynamic model. This propagated trajectory represents a particular solution to the initial value problem defined by the satellite’s equations of motion.
Initial conditions for the orbit can be derived from either:
• A prior orbit solution (e.g., from orbit identification), or
• A purely kinematic solution based on GNSS observations from the LEO satellite’s onboard receiver.
In this analysis the example from GROOPS is taken (https://groops devs.github.io/groops/html/cookbook. kinematicOrbit.html) and adapted for reduced dynamic orbit estimation where the GRACE satellite’s dynamics are modeled using well-established force models and auxiliary data. The output of this process includes:
• The propagated orbit for 24 hours
• The corresponding partial derivatives (Jacobian matrix consisting of state transition and sensitivity matrices) evaluated at the epochs of the GPS observations (the GRACE FO satellite has onboard receiver providing only GPS observables).
Iterative Estimation and the Role of Jacobians
If the initial orbit is reasonably accurate, the variational equations can be used to linearize the system and apply least squares estimation to refine the orbit. The Jacobian matrix, derived from the variational equations, plays a central role in this process. A well-conditioned Jacobian provides strong dynamical constraints and enables accurate f itting of the GNSS observations.
However, if the initial orbit or Jacobians are poorly approximated, the residuals between observed and computed measurements will be large. In such cases, the least squares solution will yield significant corrections to the initial state, necessitating further iterations. These iterations continue until the propagated orbit becomes insensitive to small perturbations in the initial conditions, indicating convergence.
Failure to converge typically points to:
• Modeling errors in the satellite dynamics,
• Inaccuracies in the GNSS observation model, or
• Systematic biases in the measurements.
Thus, the stability and conditioning of the Jacobian matrix are critical. A well conditioned Jacobian ensures that the linearized system accurately reflects the underlying nonlinear dynamics, enabling robust estimation even in the presence of measurement gaps or noise.
Mathematical Formulation
Where:
• H is the Jacobian matrix, consisting of state transition matrix, sensitivity matrix and GNSS measurement sensitivity matrix (please refer to previous month’s issue on this topic)
• W is the weight matrix (inverse of measurement variances),
• r is the residual vector (observed minus computed measurements).
The condition number of the Jacobian matrix, defined as the ratio of its largest to smallest singular values, quantifies the sensitivity of the solution to perturbations. A high condition number indicates an ill-conditioned system, which can lead to unstable or unreliable estimates.
The effect of the solutions of the variational equations can also be seen through the fitting residuals captured by the a-posterior sigma value of the GNSS observations fitting after solving the normal equations. This is given by:
Where:
• nn is the number of observations,
• mm is the number of estimated parameters.
Illustrative Results
The following plots (Figure F1-F4) illustrate the impact of variational equation solutions on orbit accuracy for the GRACE satellite in the Celestial Reference Frame, compared against the f inal precise orbit product from TU Graz:
1. Top Plot (Figure F1): Orbit derived from a purely kinematic solution (epoch-wise), without using dynamic models. Note the presence of gaps due to missing GNSS measurements.
2. Middle Plot (Figure F1): Orbit obtained by fitting GNSS observations to a dynamically propagated orbit, using an initial state offset by several hundred meters. The variational equations are solved once, without iteration. The a posteriori sigma (Equation 2) for the f itting is a bit higher. The residuals (r) (Equation 1) on the observations f itting are also high for first few iterations as the system attempts to correct the initial approximation.
3. Bottom Plot (Figure F1): Final orbit after multiple iterations in solving the variational equations and least squares fitting with GPS observations. and r get smaller with each adjustment in the Jacobians and each iteration in the fitting. Despite gaps in GNSS measurements, the dynamic model f ills in the trajectory, resulting in a continuous and accurate orbit as shown in Figure F2. For broader overview on the reduced-dynamic approach for the orbit determination of GRACE-FO satellite, readers are recommended (Jiabo et.al, 2021).
These results underscore the importance of accurate initial conditions, well conditioned Jacobians derived from the solution of variational equations, and iterative refinement in achieving high precision orbit determination. As the estimate improves, the Jacobian is evaluated closer to the true trajectory, where the system behaves more linearly. This makes the elements of Jacobians derived from solving the variational equations more stable and smaller in variation. In Figure F3, the plots illustrate the behavior of the satellite orbit propagated from two different initial conditions: one based on an initial state offset by several hundred meters from the true orbit, and the other derived from an iteratively adjusted solution. The latter, obtained through successive corrections using the variational equations, yields an initial state that closely approximates the true orbit. This comparison clearly demonstrates that, when the initial conditions are sufficiently accurate, the physics-based dynamical model can propagate the orbit with high fidelity. It also demonstrates that even though the orbit trajectory is a particular solution of the initial conditions, the rate at which the trajectory diverges from a nearby depends on sensitivity information of the Jacobin. The iterative refinement of the Jacobian significantly enhances the accuracy and smoothness of the orbit solution, highlighting the importance of both precise initial values and well-conditioned variational solutions (Jacobian) in the orbit determination.
Similarly, the final analysis presented in Figure F4 demonstrates how an initially approximated orbit can be progressively refined through dynamic fitting and iterative updates of the Jacobian matrix. As the propagated orbit is adjusted to better fit the GNSS observations, the corrections to the initial state bring the trajectory closer to the true orbit. Consequently, the Jacobian matrix—comprising the state transition and sensitivity components—is evaluated along a trajectory that more accurately reflects the system›s true dynamics. In this region, the system exhibits more linear behavior, resulting in Jacobian elements that are more stable and exhibit reduced variability. This improved conditioning enhances the reliability of subsequent corrections and contributes to the convergence of the orbit determination process.
Monthly Performance Remarks:
1. Satellite Clock and Orbit Accuracy:
▪ The performance of Beidou (improved), QZSS (slightly degraded) and GLONASS (slightly degraded) constellations showed a small change. For Galileo and GPS, the overall performances remain the same. However, for Galileo there was a noticeable performance degradation between 12th and 19th May due to testing activities.
▪ The satellite clock jump on E11 (discontinuity > 250 ns) is reported for the 2nd of May. No NAGU and health issues are found so far. This event will be analyzed and reported again next month.
▪ The URA and satellite clock discontinuity for IRNSS showed some improvement. The URA values are less scattered than previous month. 2. UTC Prediction (GNSS-UTC):
▪ All constellations reported relatively stable and consistent UTC predictions.
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Note: References in this list might also include references provided to previous issues.
Data sources and Tools:
https://cddis.nasa.gov (Daily BRDC); http://ftp.aiub. unibe.ch/CODE_MGEX/CODE/ (Precise Products); BKG “SSRC00BKG” stream; IERS C04 ERP files
(The monitoring is based on following signals- GPS: LNAV, GAL: FNAV, BDS: CNAV-1, QZSS:LNAV IRNSS:LNAV GLO:LNAV (FDMA))
Time Transfer Through GNSS Pseudorange Measurements: https://e-learning.bipm.org/login/index.php
Allan Tools, https://pypi.org/project/AllanTools
gLAB GNSS, https://gage.upc.edu/en/learning materials/software-tools/glab-tool-suite.
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