Narayan Dhital Actively involved to support international collaboration in GNSSrelated activities. He has regularly supported and contributed to different workshops of the International Committee on GNSS (ICG), and the United Nations Office for Outer Space Affairs (UNOOSA). As a professional employee, the author is working as GNSS expert at the Galileo Control Center, DLR GfR mbH, Germany

Introduction

The article is a continuation of monthly performance analysis of the GNSS constellation. Please refer to previous issues for past analysis. As an addition, there is a section that explores mathematical concepts of variational equations used in the estimation of satellite clocks and orbits. This is in turn linked to the challenges of estimating orbits of inclined geosynchronous and geo-stationary satellites in QZSS and Beidou constellations.

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.

Satellite orbit and clock parameters form the backbone of broadcast navigation messages, providing essential data for users worldwide. Since January 2024, monthly performance reports in mycoordinates have analyzed their characteristics and quality. This series of articles now shifts focus to the computational aspects of these parameters—not to rehash existing mathematical methods, but to distill key concepts in a simplified, yet insightful manner. By doing so, this analysis aims to enhance comprehension of the performance variations observed in different GNSS constellations. In this issue, a brief linkage to the performance characteristics of IGOS and GEO satellites is provided.

Variational equations are fundamental to modern orbit determination and adjustment algorithms. They enable efficient and accurate refinement of estimated parameters by providing sensitivity information, which is crucial for adjusting the satellite’s trajectory to match observations.

Linearization and the Role of the Jacobian

In the first step of orbit determination, the highly nonlinear GNSS observation equations are linearized around an initial state. This yields a Jacobian matrix that contains the partial derivatives of the measurements with respect to the estimated parameters—primarily the satellite’s position and velocity.

However, the satellite’s state itself evolves according to nonlinear equations of motion. Therefore, to understand how changes in the initial state affect the satellite’s position at later times, we must also linearize the dynamics. This is where the state transition matrix and sensitivity matrix come into play.

Dynamics and Variational Equations

The satellite’s trajectory is governed by a set of differential equations where acceleration is a function of position, velocity, and dynamical parameters (e.g., drag coefficient, solar radiation pressure). A particular orbit solution is uniquely determined by its initial state and dynamical parameters.

Numerical integration of the equations of motion yields a predicted orbit. However, this orbit is only accurate if the initial conditions and force models are close to the truth. To refine this orbit using observations, we need to understand how small changes in these inputs affect the output trajectory.

Why Variational Equations?

Using finite differences to compute these sensitivities of the orbit trajectory to small changes in initial values is computationally expensive and numerically unstable. Instead, variational equations provide a continuous and efficient way to compute:

Quantifies how uncertainties in dynamical parameters affect the trajectory.

These matrices are integrated alongside the equations of motion using the same force models. At each observation epoch, they provide the necessary partial derivatives for the Jacobian matrix used in batch least squares estimation, for example.

Constructing the Jacobian Matrix

The full Jacobian matrix (J) used in orbit determination is built using the chain rule:

This formulation ensures that the measurement sensitivity accounts for how errors in the initial state propagate to the observation epoch.

What If We Ignore Variational Equations?

• Loss of sensitivity to initial conditions
• Poor convergence in least squares estimation
• Inaccurate orbit corrections and higher residuals

Practical Implications

• Large values in Φ indicate strong dependence on initial conditions—small errors grow significantly over time.
• Stable orbit solutions are achieved when the sensitivity to initial conditions and dynamical parameters becomes small.
• Variational equations reduce the number of iterations needed and improve convergence speed and accuracy.

Taking examples of IGSO and GEO satellites of Beidou and QZSS, the changes in observation geometry for GEO and IGSO satellites are
much smaller compared to MEO satellites. As a result, strong correlations often arise between:
• State parameters (e.g., orbital elements),
• Dynamical parameters (e.g., solar radiation pressure coefficients),
• And measurement model parameters (e.g., ambiguities, differential code biases).

The reduced orbit quality of IGSO satellites can also be attributed to:
• Limited and regional ground station coverage,
• Radiation pressure modeling issues, especially due to large communication antennas not accounted for in standard box-wing models.

For GEO, additional SRP modeling difficulties from large communication antennas with undisclosed shape and material properties impact the orbit quality.

Moreover, delta-V maneuvers (for orbit maintenance) and attitude control maneuvers (for yaw and orbit normal attitudes) are hard to model precisely. Both GEO and IGSO satellites need frequent maneuvers in comparison to MEO satellites. All above mentioned characteristics introduce discontinuities or biases that are not easily captured by variational equations.

In fact, analysis of the sensitivity matrix often reveals large values associated with dynamical parameters—an indicator of modeling deficiencies. Ideally, a well converged solution should show:
• Diagonal elements close to 1 (indicating stable propagation),
• Off-diagonal elements that are small (indicating low parameter correlation).

To dig deeper into these challenges, future work will involve using advanced tools for detailed analysis of the sensitivity and covariance matrices in reduced dynamics orbit determination. In this regard, this section serves as an introductory overview, laying the foundation for deeper investigations into these modeling and estimation challenges.

Monthly Performance Remarks:

1. Satellite Clock and Orbit Accuracy:
• The performance of all constellations is relatively stable with some minor changes from previous month.
• The satellite clock jumps identified a couple of issues in Galileo satellites. E11 and E12 had large jumps on DOY 91 and DOY 105, respectively. Satellite E19 continues to have the noisiest clock (95 percentile is 0.36 ns vs 95 percentile of 0.2 ns of the whole constellation).
• The improvement in the QZSS satellite clock and orbit accuracy is visible. It is improved by 5 cm. Further investigation (once the QZSS operational history is available) is needed to correlate to the Sun angle and the switches between the yaw attitude and orbit normal modes of the satellites.
• The URA for I02 showed a little more scatter in comparison to previous months. It suggests a degraded confidence in its satellite orbit.

2. UTC Prediction (GNSS-UTC):
• GPS showed some variations as in previous months. Glonass UTC prediction started to deviate significantly in the second half of the month.