GRACE-FO Real-Time Precise Orbit Determination Using Onboard GPS and Inter-Satellite Ranging Measurements with Quality Control Strategy

Highlights

What are the main findings?

• Robustness of the quality control strategy: The adopted iterative quality control method based on post-fit residuals effectively suppresses GNSS observation noise and outliers, thereby significantly improving the stability of the filter and the orbit accuracy.
• Efficacy of inter-satellite range measurements in asymmetric conditions: The experiment verifies that the inclusion of inter-satellite range measurements yields substantial accuracy improvements for GRACE-FO under asymmetric observation conditions. This is particularly decisive for the satellite with poor data quality (GRACE-D), where orbit accuracy was improved by 39%, effectively recovering its performance to the level of GRACE-C.

What are the implications of the main findings?

• Stabilization of constellation geometry: The integration of inter-satellite range measurements effectively compensates for geometric deficiencies in LEO satellite observations, thereby guaranteeing the structural stability of formation flying or large-scale constellations even when individual nodes are degraded.
• Spatiotemporal reference transfer and formation stability: High-precision ISLs serve as a critical conduit for transferring spatiotemporal references within the constellation. By establishing rigid geometric constraints, precise state information from satellites with normal data quality is effectively propagated to constrain degraded satellites.

Abstract

Real-Time Precise Orbit Determination (RTPOD) of Low Earth Orbit (LEO) satellites relies primarily on onboard GNSS observations and may suffer from degraded performance when observation geometry weakens or tracking conditions deteriorate within satellite formations. To enhance the robustness and accuracy of RTPOD under such conditions, a cooperative Extended Kalman Filter (EKF) framework that fuses onboard GNSS and inter-satellite link (ISL) range measurements is established, integrated with an iterative Detection, Identification, and Adaptation (DIA) quality control algorithm. By introducing high-precision ISL range measurements, the strategy increases observation redundancy, improves the effective observation geometry, and provides strong relative position constraints among LEO satellites. This constraint strengthens solution stability and convergence, while simultaneously enhancing the sensitivity of the DIA-based quality control to observation outliers. The proposed strategy is validated in a simulated real-time environment using Centre National d’Etudes Spatiales (CNES) real-time products and onboard observations of the GRACE-FO mission. The results demonstrate comprehensive performance enhancements for both satellites over the experimental period. For the GRACE-D satellite, which suffers from about 17% data loss and a cycle slip ratio several times higher than that of GRACE-C, the mean orbit accuracy improves by 39% (from 13.1 cm to 8.0 cm), and the average convergence time is shortened by 44.3%. In comparison, the GRACE-C satellite achieves a 4.2% mean accuracy refinement and a 1.3% reduction in convergence time. These findings reveal a cooperative stabilization mechanism, where the high-precision spatiotemporal reference is transferred from the robust node to the degraded node via inter-satellite range measurements. This study demonstrates the effectiveness of the proposed method in enhancing the robustness and stability of formation orbit determination and provides algorithmic validation for future RTPOD of LEO satellite formations or large-scale constellations.

1. Introduction

LEO satellite constellations have evolved into essential infrastructures for gravity field recovery [1], ocean altimetry [2], LEO-enhanced precise point positioning [3] and so on. In recent years, the development of mega LEO constellations has surged, such as Starlink and OneWeb, becoming a focal point for both national strategies and commercial ventures. While orbit determination requirements vary across different missions, Real-Time Precise Orbit Determination (RTPOD) has become a critical strategy for specific advanced applications, such as autonomous formation flying and time-critical remote sensing [4,5].

Currently, GNSS-based precise orbit determination for LEO satellites has been extensively studied, including both post-processed and real-time solutions. Selvan et al. [6] conducted a comprehensive review that although centimeter-level accuracy is routinely achieved in GNSS-based LEO POD using post-processed least-squares solutions, real-time solutions generally deliver inferior accuracy due to limited redundancy, real-time GNSS products, and other constraints inherent to RTPOD. Mao et al. [7,8] demonstrated that high-precision GNSS-based LEO orbit determination is feasible when using refined modeling and high-quality GNSS observations, achieving centimeter-level accuracy for absolute orbits and millimeter-level precision for relative state estimation.

In the real-time domain, Montenbruck and Ramos-Bosch [9] utilized a Kalman filter to demonstrate real-time orbit accuracies of 0.5 m using broadcast ephemerides and 15 cm with TDRSS Augmentation Service for Satellites (TASS). More recently, Montenbruck et al. [10] achieved a real-time orbit determination accuracy of 10 cm for Sentinel-6A based on GPS and Galileo broadcast ephemerides. However, this performance was largely enabled by the superior quality of the Galileo broadcast ephemerides. Similarly, Shi et al. [11] achieved a real-time orbit accuracy of approximately 20 cm for the TJU-01 satellite using a Kalman filter based on GPS and BDS-3 broadcast ephemerides corrected by PPP-B2b signals. Li et al. [12] implemented RTPOD for Sentinel-6A using a Square Root Information Filter (SRIF) and Centre National d’Etudes Spatiales (CNES) real-time products, realizing an accuracy of 3.6 cm with ambiguity resolution.

However, the reliability of standalone GNSS-based RTPOD is fundamentally constrained by the performance of onboard instruments and the observation environment. Beyond hardware degradation due to long-term aging, LEO satellites equipped with single-constellation GNSS receivers typically observe only a limited number of GNSS satellites, often on the order of 4–10 per epoch. Under such conditions, the RTPOD solution becomes highly sensitive to unfavorable GNSS observation geometry and reduced redundancy due to its inherent reliance on epoch-wise measurements. Furthermore, in the context of large-scale constellations equipped with multiple payloads, challenges such as co-frequency interference [13,14] or partial equipment failures may arise, leading to significant asymmetry in observation quality among satellites. These limitations originate at the measurement level and are therefore difficult to fully mitigate through algorithmic refinement alone, potentially leading to degraded accuracy, slower convergence, or even estimator instability in real-time processing.

Inter-satellite link (ISL) range measurements provide a promising complementary observation to mitigate these challenges. By introducing additional ranging information between satellites, ISL measurements increase observation redundancy, improve the effective geometric strength of the estimation problem, and impose strong relative constraints within a satellite formation that are largely independent of GNSS tracking conditions. ISL technology has matured at the hardware level, as evidenced by its deployment in modern GNSS constellations, such as BDS-3, where inter-satellite links support autonomous orbit determination and time synchronization [15,16,17,18]. However, due to data sensitivity, ISL measurements used within GNSS constellations are generally not publicly available, and related academic studies are largely limited to simulations.

For LEO satellite formations, ISL ranging is increasingly regarded as a key observation type for next-generation navigation systems (e.g., Kepler) and emerging LEO satellite constellations [19,20]. However, publicly accessible high-precision ISL ranging data for LEO satellites remain very limited. To date, only a few missions, most notably the GRACE or GRACE-FO earth gravity missions and the GRAIL lunar gravity mission, have provided such data [21,22]. In this context, the GRACE-FO mission constitutes an ideal real-data testbed, providing both onboard GNSS observations and precise ISL ranging measurements. Yang et al. [23] proposed a relative kinematic orbit determination scheme for GRACE-FO, showing that the integration of Laser Ranging Interferometer (LRI) observations improved the inter-satellite baseline accuracy by approximately 25.9% compared to the GPS-only solution. Based on BDS-3 PPP-B2b corrections and inter-satellite ranging measurements, Shi et al. [24] achieved sub-decimeter-level orbit accuracy for the GRACE-FO mission, while also verifying the positive impact of ISL constraints on maintaining orbital precision during observation interruptions. Chen et al. [25] utilized a single-ISL cooperative orbit determination strategy to recover the orbit of GNSS-denied satellites, achieving centimeter-level accuracy by applying tight dynamic constraints and precise ephemerides. Yang et al. [26] utilized ISL range measurements constraints to refine initial GNSS-derived orbits, demonstrating that ISL range measurements effectively correct orbital errors and rigorously stabilize constellation geometry, offering a scalable framework for large-scale LEO constellations. Nevertheless, most GNSS–ISL fusion studies for LEO satellites have focused on post-processing or simulation-based analyses. Openly documented investigations that validate the contribution of ISL measurements to real-time or near-real-time POD within a sequential filtering framework remain limited, particularly when using real flight data.

Motivated by this gap, this study proposes a robust RTPOD strategy based on iterative EKF integrated with post-fit residual QC. The proposed framework aims to enhance orbit determination robustness by jointly exploiting onboard GNSS observations and high-precision ISL range measurements. Using real observation data from the GRACE-FO mission, this study investigates the contribution of ISL range measurements to improving RTPOD performance for LEO satellite formations in a simulated real-time environment via data playback, from an algorithmic perspective. The following sections of this paper are organized in this way. Section 2 outlines the mathematical models and the filter design; Section 3 details the experimental datasets and parameter estimation strategies; Section 4 presents the experimental schemes and analyzes the results; and finally, Section 5 provides the discussion and conclusions.

2. Mathematical Models for the RTPOD Based on EKF

2.1. Observation Models

Since LEO satellites orbit well above the troposphere, the GNSS signals received by LEO satellites do not traverse this layer, making tropospheric delay negligible. The zero difference observation equations of spaceborne GNSS dual-frequency code and carrier phases can be written as

$$\left\{\begin{array}{l}{P}_f=\rho +d{t}_r-d{t}^s+I_f+b_{r,f}-b_f^s+e_f\hfill \\ L_f=\rho +d{t}_r-d{t}^s-I_f+{\lambda }_f{N}_f+B_{r,f}-B_f^s+{\epsilon }_f\hfill \end{array}\right.,$$

(1)

where superscript $s$ and subscript $r$ represent the satellite and receiver, respectively, and subscript $f$ denotes the carrier frequency, which typically takes two values in this study. $P_f$ and $L_f$ are the code and carrier phase observations, $\rho$ is the geometric distance between the satellite and receiver, $d{t}_r$ and $d{t}^s$ are the receiver and satellite clock offsets in meters, $I_f$ is the ionospheric delay, ${\lambda }_f$ and $N_f$ are the wavelength and corresponding integer carrier phase ambiguity, $b_{r,f}$ and $b_f^s$ are the receiver and satellite code biases, and $B_{r,f}$ and $B_f^s$ are the receiver and satellite phase biases. $e_f$ and ${\epsilon }_f$ are the combinations of measurement noise and multipath error for code and carrier phase observations. Other range errors caused by phase center offset/phase center variation (PCO/PCV), relativity effect, carrier phase wind-up, and so on, have been corrected using the existing models [27].

The dual-frequency ionosphere-free (IF) combination is widely employed to eliminate the first-order ionospheric delay, a primary error source in GNSS observations. Ignoring the negligible higher-order effects, the linearized IF observation equations are formulated as

$$\left\{\begin{array}{l}{p}_{IF}=u^T\left({\Phi }_{orb}\left(t_r,t\right)x_{orb}\left(t\right)\right)+d{t}_r-d{t}^s+b_{r,IF}-b_{IF}^s+e_{IF}\hfill \\ l_{IF}=u^T\left({\Phi }_{orb}\left(t_r,t\right)x_{orb}\left(t\right)\right)+d{t}_r-d{t}^s+B_{r,IF}-B_{IF}^s+{\lambda }_{IF}{N}_{IF}+{\epsilon }_{IF}\hfill \end{array}\right.,$$

(2)

where $p_{IF}$ and $l_{IF}$ denote the observed-minus-computed (OMC) values of the IF combination pseudorange and carrier phase observations. The term $u^T\left({\Phi }_{orb}\left(t_r,t\right)x_{orb}\left(t\right)\right)$ represents the projection of the orbit state correction onto the line-of-sight direction, and $u^T$ is the unit vector pointing from the receiver to the satellite, ${\Phi }_{orb}\left(t_r,t\right)$ is the orbit state transition matrix from the nearest integer second epoch $t$ to the signal reception epoch $t_r$ and $x_{orb}\left(t\right)$ is the orbit state correction vector at epoch $t$, $b_{r,IF}$ and $B_{r,IF}$ are the IF combination code and phase delay of the receiver, $b_{IF}^s$ and $B_{IF}^s$ are the IF combination code and phase delay of the satellite, ${\lambda }_{IF}$ and $N_{IF}$ are the wavelength and integer ambiguity of the IF combination, and $e_{IF}$ and ${\epsilon }_{IF}$ are the IF combination code and carrier phase measurement noise [12].

The inter-satellite link observation is modeled as the sum of the instantaneous geometric distance between the centers of mass of the two LEO satellites and various geometric effects [28,29]. The generalized observation equation can be written as

$$D_{ISL}=\rho +I_{ion}+\delta {\rho }_{LT}+\delta {\rho }_{ant}+R_{rel}+\delta {\rho }_{bias}+\epsilon ,$$

(3)

where $D_{ISL}$ represents the raw inter-satellite range observation, $\rho$ is the instantaneous geometric distance between the centers of mass of the two LEO satellites, $I_{ion}$ is the ionospheric path delay effect, which is significant for microwave links, e.g., the K-Band Ranging System (KBR), but is negligible for optical links, e.g., LRI, $\delta {\rho }_{LT}$ is the propagation time correction, $\delta {\rho }_{ant}$ denotes the antenna phase center offset correction [28], $R_{rel}$ is the correction of relativity effect, $\delta {\rho }_{bias}$ is systematic bias, typically modeled as a piecewise constant for each continuous observation arc, and $\epsilon$ is measurement noise.

The general orbit determination problem involves a non-linear dynamic system and non-linear measurement models. The state vector, $X={\left(r^T,{\dot{r}}^T,p^T\right)}^T$, which includes satellite position $r$, velocity $\dot{r}$, and dynamic parameters $p$, is governed by the equations of motion:

$$\dot{X}=F(X,t).$$

(4)

The relationship between the state vector and the observations $Y_i$ at epoch $t_i$ is described by the non-linear measurement equation:

$$Y_i=G(X_i,t_i)+{\epsilon }_i,$$

(5)

where ${\epsilon }_i$ is measurement noise.

Typically, a Taylor series expansion is employed to convert the non-linear dynamic and measurement functions into linear forms. This process requires a reference trajectory, $X^{\ast }(t)$, that is sufficiently close to the true trajectory, $X(t)$, in order to minimize the errors introduced by the linearization.

The true state vector can then be expressed as the sum of the reference state and a state deviation vector $x(t)$:

$$X(t)=X^{\ast }(t)+x(t).$$

(6)

By substituting Equation (6) into Equations (4) and (5) and applying a first-order Taylor expansion (truncating higher-order terms), the system is linearized. The dynamics of the state deviation vector $x(t)$ are described by the variational equation:

$$\dot{x}(t)=A(t)x(t).$$

(7)

And the linearized observation equation becomes

$$y_i=H_i{x}_i+{\epsilon }_i,$$

(8)

where $y_i$ is the OMC vector, $A(t)$ is the dynamics matrix, and $H_i$ is the design matrix, defined as

$$\left\{\begin{array}{c}\hspace{1em}{y}_i=Y_i-G(X_i^{\ast },t_i)\\ A(t)={\left.{\displaystyle \frac{\partial F(X,t)}{\partial X}}\right|}_{X^{\ast }}\\ \hspace{1em}{H}_i={\left.{\displaystyle \frac{\partial G(X,t_i)}{\partial X}}\right|}_{X^{\ast }}\end{array}\right..$$

(9)

The general solution of Equation (7) can be expressed as

$$x(t)=\Phi (t,t_k)x_k,$$

(10)

where $x_k$ is the value of $x$ at time $t_k$, and $\Phi (t,t_k)$ is the state transition matrix, which propagates the state correction vector from epoch $t_k$ to $t$.

Based on this linear system (Equations (8) and (10)), the state correction vector $x$ can be solved. For real-time applications, a sequential estimator, such as the EKF, is employed to update the state estimate at each epoch.

Unlike GNSS observations, which depend only on the state vector of a single LEO satellite, an ISL range observation simultaneously involves the states of both satellites. The linearization of the ISL observation therefore requires a common reference state for the two satellites.

In this study, the reference state is provided by the predicted state obtained through numerical integration from the previous epoch. Based on the predicted states $X_A^{\ast }$ and $X_B^{\ast }$, the ISL range observation is linearized using a first-order Taylor expansion:

$${\rho }_{ISL} (X_A ,X_B )\approx {\rho }_{ISL} (X_A^{\ast } ,X_B^{\ast } )+ {\left.{\displaystyle \frac{\partial {\rho }_{ISL}}{\partial X_A}}\right|}_{X_A^{\ast }}(X_A -X_A^{\ast } )+{\left.{\displaystyle \frac{\partial {\rho }_{ISL}}{\partial X_B}}\right|}_{X_B^{\ast }} (X_B -X_B^{\ast } ),$$

(11)

where ${\rho }_{ISL}$ denotes the geometric inter-satellite distance between the two LEO satellites and is a function of their positions only. $X_i$ denotes the real state of satellite $i$, and ${\left.\left(\cdot \right)\right|}_{X_i^{\ast }}$ denotes evaluation at the reference state $X_i^{\ast }$.

Since the ISL range represents the relative distance between the two satellites, it is a function of the satellite positions only, and the partial derivatives with respect to the remaining state parameters vanish. Accordingly, the position Jacobians are given by

$${\left.{\displaystyle \frac{\partial {\rho }_{ISL}}{\partial r_A}}\right|}_{X_A^{\ast }}=u_{AB}^T,{\left.{\displaystyle \frac{\partial {\rho }_{ISL}}{\partial r_B}}\right|}_{X_B^{\ast }}=-u_{AB}^T,u_{AB}={\displaystyle \frac{r_A^{\ast }-r_B^{\ast }}{\left|r_A^{\ast }-r_B^{\ast }\right|}},$$

(12)

where $r_i^{\ast }$ is the reference position of satellite $i$, and $u_{AB}$ denotes the line-of-sight unit vector pointing from satellite B to satellite A, evaluated at the reference state.

Consequently, each ISL observation introduces non-zero sensitivities to the state parameters of both LEO satellites, resulting in a coupled observation model when GNSS and ISL observations are jointly processed. In contrast, when only GNSS observations are used, each observation depends solely on the state parameters of the corresponding satellite, and no cross-coupling terms exist between the states of different satellites in the observation model. This transition from an uncoupled to a coupled observation model enables information transfer between the satellites within the EKF and constitutes the mathematical basis of the cooperative RTPOD strategy.

2.2. EKF Model Used in RTPOD

The EKF has been widely used in the adjustment of geodetic networks and GNSS POD [30,31]. EKF operates in a two-step process:

(1)

The Time Update (Prediction) step, which uses the linearized dynamic model (Equation (10)) to propagate the state vector and its covariance from the reference epoch to the current epoch.

(2)

The Measurement Update (Correction) step, which uses the linearized observation equation (Equation (8)) to correct the predicted state with new measurements.

2.2.1. RTPOD Procedure Based on EKF

In this study, we developed an EKF module based on the Positioning And Navigation Data Analyst (PANDA) software [32]. This module extends the software’s capabilities to support real-time POD for multiple LEO satellites simultaneously and incorporates the processing of inter-satellite range observations.

The workflow of the EKF-based LEO RTPOD is illustrated in Figure 1. The procedure is primarily composed of three phases: Initialization, Data Preprocessing, and Parameter Estimation.

The process begins with Initialization, where the state vector is constructed and assigned a priori values. For LEO satellites, initial orbital parameters are typically derived from Standard Point Positioning (SPP) [33] solutions using onboard pseudo-range observations or obtained from external ephemerides, while the initial state covariance matrix is determined empirically based on the expected system uncertainty (as detailed in

Table 1. Initialization and stochastic modeling strategy for the estimated states.

Parameter	Initial Uncertainty (${\mathit{\sigma }}_0$)	Process Noise (${\mathit{\sigma }}_{\mathit{n}\mathit{o}\mathit{i}\mathit{s}\mathit{e}}$)/Epoch
Position	$1.0\times {10}^3 \mathrm{m}$	$1.0\times {10}^{-3} \mathrm{m}$
Velocity	$1.0 \mathrm{m}/\mathrm{s}$	$1.0\times {10}^{-4} \mathrm{m}/\mathrm{s}$
Atmospheric drag factor	1.0	$1.0\times {10}^{-1}$
Solar radiation factor	1.0	$1.0\times {10}^{-1}$
Empirical accelerations	$1.0\times {10}^{-4} {\mathrm{m}/\mathrm{s}}^2$	$1.0\times {10}^{-6} {\mathrm{m}/\mathrm{s}}^2$
Receiver Clock	$10.0 \mathrm{m}$	$1.0\times {10}^{-1} \mathrm{m}$
Ambiguity	$7.0 \mathrm{m}$	$0.0 \mathrm{m}$

). Following Initialization, the Data Preprocessing phase performs rigorous screening on raw GNSS and inter-satellite range observations to detect cycle slips and outliers. Finally, the Parameter Estimation phase executes the recursive filter loop, propagating the state via dynamic models and updating it with validated observations to output precise orbits in real-time.

2.2.2. Estimator Formulation and Stochastic Modeling

• Filter Structure

The recursive estimation follows a standard predictor-corrector sequence. The state estimate $x$ and covariance $P$ are propagated and updated using the following algorithm:

$$\begin{array}{cc}\hfill \mathrm{Time} \mathrm{Update}:& x_k^{-}={\Phi }_{k,k-1}{x}_{k-1}^{+}+{\mathsf{\omega }}_k,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{P}_k^{-}={\Phi }_{k,k-1}{P}_{k-1}^{+}{\Phi }_{k,k-1}^T+Q_k\hfill \\ \hfill \mathrm{Measurement} \mathrm{Update}:& K_{\mathrm{K}}=P_k^{-}{H}_k^T{(H_k{P}_k^{-}{H}_k^T+R_k)}^{-1}\hfill \\ \hfill & x_k^{+}=x_k^{-}+K_K(y_k-H_k{x}_k^{-}),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{P}_k^{+}=(I-K_K{H}_k)P_k^{-}\hfill \end{array}$$

(13)

where superscripts $\left(-\right)$ and $\left(+\right)$ denote the a priori and a posteriori estimates, respectively; ${\Phi }_{k,k-1}$ is the state transition matrix, propagating the state from $t_{k-1}$ to $t_k$; ${\mathsf{\omega }}_k$ is process noise; $Q_k$ is the discrete process noise covariance matrix; $K_{\mathrm{K}}$ is the Kalman Gain; $R_k$ is the measurement noise covariance matrix; and $I$ is the identity matrix [31].

Several practical considerations are addressed in the software implementation. First, the predicted state vector $x_k^{-}$ and the state transition matrix ${\Phi }_{k,k-1}$ are computed by numerically integrating the variational equations (Equation (14) using a Runge–Kutta–Fehlberg 7(8) (RKF 7(8)) integrator [34]. Second, to ensure numerical stability and guarantee that the covariance matrix remains symmetric and positive-definite, the covariance update equation is replaced by the numerically robust Joseph stabilized form, Equation (15) [35]. Finally, to optimize computational efficiency, matrix operations are executed using the Linear Algebra PACKage (LAPACK) library.

$$\begin{array}{c}{\dot{x}}_k=A(t_k)x_k\\ {\dot{\Phi }}_{k,k-1}=A(t_k){\Phi }_{k,k-1}\end{array},$$

(14)

$$P_k^{+}=P_k^{-}-K_K\left[H_k{P}_k^{-}{H}_k^T+R_k\right]{K_K}^T.$$

(15)

2.

State Vector and Dynamic Compensation

Since deterministic perturbation models are often insufficient to fully capture complex dynamics, a reduced-dynamic method is employed [36]. The estimated state vector $x$ is augmented with empirical parameters to absorb these force model deficiencies. Consequently, the vector of estimated parameters is defined as

$$x={\left[r^T,v^T,C_d,C_r,a_{emp}^T,CL{K}_r,AMB,\dots \right]}^T,$$

(16)

where $r$ and $v$ are the position and velocity vectors, $C_d$ and $C_r$ are the drag and solar radiation pressure coefficients, and $a_{emp}^T={\left[a_R,a_T,a_N\right]}^T$ represents the empirical accelerations in the radial, along-track, and cross-track directions.

In this study, both GPS and ISL observations are referenced to the GPST time system. The onboard receiver clock offsets of the LEO satellites are estimated as state parameters, thereby unifying the onboard time scales with GPST within the EKF framework.

3.

Stochastic Modeling

To maintain stable responsiveness to incoming observations in EKF-based LEO orbit determination, a stochastic modeling strategy is introduced for selected estimated parameters. Specifically, a Random Walk (RW) process is adopted for state elements, and temporal evolution is modeled as being driven by zero-mean white noise processes. The corresponding process noise levels, expressed in terms of standard deviations, are summarized in Table 1.

Process noise ${\omega }_k\sim N(0,Q_k)$ includes the kinematic states (position and velocity) and the dynamic force parameters. For dynamic parameters, the process noise is tuned to absorb the uncertainty in atmospheric drag and other non-conservative forces. For kinematic states, small process noise terms are explicitly added. This technique, known as state noise compensation, prevents the error covariance matrix $P$ from becoming overly optimistic (vanishing gain) over long arcs. It effectively relaxes the rigid dynamic constraints, allowing the filter to absorb linearization errors and remain sensitive to new observations. The measurement noise covariance $R_k$ is determined based on an elevation-dependent weighting model for GNSS observations and fixed noise characteristics for inter-satellite ranging.

2.2.3. Real-Time Quality Control Strategy

DIA is a well-established quality control framework originally introduced by Teunissen and has been widely applied in navigation and orbit determination to ensure estimation reliability under imperfect observation conditions [37]. Owing to its solid statistical foundation and suitability for sequential estimation, DIA has been adopted in real-time and near-real-time filtering applications [38,39].

The DIA-based quality control strategy adopted in this study is founded on the assumption that post-fit residuals follow a zero-mean Gaussian distribution under nominal observation conditions. Within the EKF framework, a chi-square test applied to the normalized post-fit residuals provides an effective mechanism for detecting global inconsistencies arising from observation outliers or model deficiencies.

Compared to conventional least-squares-based quality control approaches, the DIA framework remains applicable in real-time filtering scenarios where the number of available observations per epoch is limited and may even be smaller than the number of estimated parameters. In such cases, the estimation of an epoch-wise variance factor becomes infeasible. Moreover, the statistical optimality of DIA-based estimators and decision rules has been rigorously established within a hypothesis-testing framework, further justifying its use in real-time filtering applications [40].

To ensure the statistical consistency of the orbit solution, we implement a robust quality control strategy integrated directly into the filtering loop. This strategy adopts the DIA framework, implemented here as an iterative procedure based on standardized post-fit residuals, as illustrated in Figure 2. Unlike methods that screen outliers before the update, this approach evaluates the a posteriori consistency of the measurements to refine the state estimate. The detailed procedure is as follows:

• Standardization

The EKF measurement update is first executed to generate the post-fit residual vector $v_k$. To assess the statistical significance of the residuals, the post-fit residuals are standardized. The standardized residual ${\tilde{v}}_i$ for the $i$-th observation is computed as

$${\tilde{v}}_i={\displaystyle \frac{v_{k,i}}{{\sigma }_{v,i}}},$$

(17)

where ${\sigma }_{v,i}$ is the theoretical standard deviation of the post-fit residual derived from the posterior covariance matrix.

2.

Detection (Global Test)

A right-tailed chi-square (${\chi }^2$) test is applied to the weighted sum of squared residuals. The test statistic is defined as

$$T={\tilde{v}}^T\tilde{v}={\displaystyle \sum _{i=1}^m{\tilde{v}}_i^2}.$$

(18)

The null hypothesis is rejected if $T>{\chi }_{\alpha ,m}^2$, where ${\chi }_{\alpha ,m}^2$ is the upper critical value for a significance level $\alpha$ (e.g., 0.05) and $m$ degrees of freedom.

3.

Identification (Local Test)

Regardless of the global test outcome, local screening is performed to pinpoint the most likely outlier candidate. The observation corresponding to the maximum absolute standardized residual is identified:

$$|\tilde{v}{|}_{max}=\max |{\tilde{v}}_i|.$$

(19)

4.

Adaptation (Iterative Rejection)

The system adapts by rejecting the identified candidate if either of the following conditions is met:

(1)

Global Anomaly: The detection step failed ($T>{\chi }_{\alpha ,m}^2$), indicating overall inconsistency.

(2)

Local Anomaly: The candidate’s standardized residual exceeds the empirical threshold (e.g., $|\tilde{v}{|}_{max}>3.0$), indicating a significant single-point outlier.

If a rejection occurs, the filter re-executes the measurement update with the remaining clean subset of observations. This DIA cycle iterates until the system stabilizes.

In the proposed iterative implementation, only the single most suspicious observation is removed at each DIA iteration. This conservative strategy is motivated by the fact that secondary suspicious residuals may be influenced by the dominant outlier, and their direct removal may not reflect the true error source. Moreover, for LEO missions with limited GNSS visibility, each observation carries significant geometric and informational value, and excessive observation rejection may degrade the overall solution robustness.

3. Dataset and Processing Strategy

3.1. Experimental Data

To validate the proposed RTPOD strategy and to rigorously assess the impact of inter-satellite range observations on the orbit determination of multi-LEO satellite formations, we utilize onboard observations from the GRACE-FO mission, as illustrated in Figure 3. This mission consists of two twin satellites, GRACE-C and GRACE-D, co-orbiting at an altitude of approximately 490 km. The experimental dataset covers the period of the first two weeks of June 2021 (DOY 152–165).

Three primary types of data are employed in this study:

• Onboard GNSS Observations:

Dual-frequency GPS code and carrier phase observations were extracted from Level-1B products (GPS1B). The sampling interval was standardized to 10 s.

• ISL Range Measurements:

The biased inter-satellite ranges were obtained from KBR1B (or LRI1B) products. The raw data were resampled to align with the GPS observations.

• Real-time Orbit and Clock Products:

The GPS orbit, clock, and observable specific bias (OSB) products were provided by the CNES real-time service (RTS). In addition, the CNES PPP-WIZAD project contributed satellite code and phase biases.

For accuracy assessment, the post-processed precise science orbits (PSOs) provided by JPL served as the reference.

3.2. ISL Range Data Pre-Processing and Bias Calibration

In this study, to enable the utilization of inter-satellite range observations as geometric ranges within the real-time filter, the inherent instrument systematic bias is explicitly pre-determined. This bias, which encompasses the carrier phase integer ambiguity and uncalibrated hardware delays, is calibrated using post-processed precise science orbits as a high-precision reference.

The calibration procedure involves the following steps:

• Break Detection

The continuity of the raw biased range time series is first verified. By computing the first-order temporal differences in the biased range, potential discontinuities caused by instrument resets or cycle slips can be identified. If significant jumps are detected, the observation arc is segmented, and the bias is solved independently for each continuous segment.

2.

Bias Estimation

Leveraging the high stability of the instrument bias over short durations. The bias is calculated as the statistical average (e.g., mean or median) of the difference between the PSO-derived geometric distance and the KBR1B/LRI1B corrected range, which accounts for light–time delays and, in the case of KBR, antenna phase center offsets.

3.

Observation Calibration

Once the bias is determined for each arc, it is applied to the real-time biased range. This recovers the absolute inter-satellite geometric distances, which are then ingested into the EKF for orbit estimation.

Figure 4 visualizes this calibration process using a representative daily arc. The top panel displays the temporal evolution of the raw biased range. The middle panel presents the first-order epoch differences in the biased range. The smoothness of this sequences confirms that the phase measurements is continuous and free of cycle slips for this arc. The bottom panel illustrates the residuals derived by differencing the PSO-computed geometric distance and the KBR1B-corrected range. Given that the absence of cycle slips in this arc has been verified, this residual is utilized to calculate the constant instrument bias.

3.3. Models and Estimation Strategy

The specific dynamic and observation models used in the RTPOD estimator are summarized in

Table 2. Models and the parameter estimation strategy for RTPOD of GRACE-FO.

Category	Item	Models/Strategies
Measurement Model	GPS observations	Dual-frequency IF combination; Elevation mask is 1°.
	Sampling interval	10 s.
	GPS observation weight	A priori precision of 0.02 cycles and 1.0 m for raw phase and code, respectively.
	ISL range observations	KBR, resample to 10 s.
	GPS orbit and clocks	CNES RTS products.
	GPS satellite biases	Real-time OSB products from PPP-WIZARD project of CNES.
	GPS satellite antenna	igs14.atx [41]
	Receiver antenna	PCO + PCV
	LEO satellite attitude	Quaternions (measured), LEVEL-B SCA1B files.
	Carrier phase wind-up	Applied
Dynamic Models	Earth orientation parameters	IERS Bulletin A
	Gravitational forces	EIGEN-6c (80 × 80) [42]; N-body gravity
	Solid Earth tide	IERS conventions 2010 [43]
	Ocean tide	FES2004 20 × 20 [44]
	Pole tides	IERS conventions 2010 [43]
	General relativity	IERS conventions 2010 [43]
	Solar radiation pressure	Macro-model [45]
	Atmospheric drag	NRLMSIS-00 [46]
	Empirical accelerations	Consider the parameters of Ca, Sa, Cc, Sc, Cr and Sr for the along-, cross-rack and radial
Estimation	Parameter estimator	EKF
	Integrator	(Runge–Kutta-Fehlberg) RKF 7(8)
	Integral step	5 s
	Ambiguity-fixed	Ambiguity-float solutions

. The processing generally follows the IERS conventions 2010. A reduced-dynamic strategy is adopted, where the state vector is augmented with empirical accelerations to compensate for force model deficiencies.

4. Experiment and Results Analysis

4.1. Experimental Design

To systematically evaluate the contribution of inter-satellite range observations to the RTPOD performance, two processing schemes are defined:

• Scheme A (GPS-only): The orbits of GRACE-C and GRACE-D are estimated using only onboard GPS observations. This configuration is designed not only to verify the fundamental capability of the RTPOD strategy, but also to serve as a rigorous control group. By comparing against this baseline, the specific contributions of inter-satellite range observations can be isolated and assessed.
• Scheme B (GPS + ISL): The orbits are estimated using both onboard GPS and inter-satellite range observations. In this scheme, the inter-satellite range measurements are ingested as absolute geometric ranges with a weighting sigma of 1.0 cm, serving as a strong geometric constraint to tightly couple the relative states of the satellite formation.

The accuracy of the real-time orbits is assessed by comparing them with the post-processed precise science orbits provided by JPL. In this paper, we define orbit convergence as the first time the 3D orbital error is better than 10 cm for 5 consecutive minutes.

4.2. Assessment of GPS Observation Quality

Before conducting the RTPOD, an analysis of the observation quality is performed. Figure 5 illustrate the data availability and cycle slip characteristics (based on GF and MW combinations, computed using RINGO v0.9.4 [47]) for both GRACE-C and GRACE-D over the experiment period.

The statistics reveal a distinct disparity between the two satellites. The average number of GPS observations for GRACE-C and GRACE-D are 30,469 and 25,289, respectively, with GRACE-D having 17% fewer observations than GRACE-C. Furthermore, the average GF cycle slip ratios are 0.03% and 0.26%, while the average MW cycle slip ratios are 3.44% and 4.14%, respectively. This implies that the occurrence of GF and MW cycle slips on GRACE-D is 8.4 times and 1.2 times higher, respectively, than on GRACE-C. These indicators reflect that the GPS observation quality of GRACE-D is significantly inferior to that of GRACE-C, which is expected to compromise the orbit determination performance in the GPS-only mode.

4.3. Impact of QC-with-DIA on RTPOD

Given the sensitivity of the EKF to measurement anomalies, the efficacy of the quality control strategy is a prerequisite for high-precision orbit determination. To quantify the contribution of the proposed iterative DIA procedure (Section 2.2.3), we conducted a comparative analysis between solutions obtained with and without the QC module.

Figure 6 illustrates the 3D orbit error sequences for GRACE-C and GRACE-D on a representative day (1 June 2021). In the absence of the QC module (denoted as ‘No-QC’), the orbit solutions for both satellites exhibit frequent and significant fluctuations. These excursions are directly triggered by undetected cycle slips or gross errors, which propagate into the state estimate and degrade the accuracy. Conversely, when the iterative DIA strategy is activated (‘With-QC’), these spikes are effectively suppressed, especially for GRACE-D. The filter successfully identifies and rejects the contaminated observations, resulting in a significantly more stable difference profile with mitigated fluctuations. This improvement is particularly critical for GRACE-D, where the raw data quality is inherently poorer.

The daily 3D Root Mean Square (RMS) statistics over the entire 14-day period are summarized in Figure 7. The average 3D RMS for GRACE-C decreases from 8.2 cm (No-QC) to 7.3 cm (With-QC), representing an improvement of approximately 8.7%, while GRACE-D exhibits an even more pronounced reduction from 21.7 cm to 13.1 cm (an improvement of 29.4%). The different improvement levels observed for GRACE-C and GRACE-D are primarily due to asymmetric GNSS observation quality. As shown in Figure 5, GRACE-D suffers from reduced tracking capability and higher cycle slip rates. Under such degraded conditions, the DIA strategy effectively suppresses abnormal observations and mitigates error propagation into the state estimates, resulting in a more pronounced performance improvement. In contrast, the GNSS observations of GRACE-C are generally of higher quality, and therefore the improvement introduced by DIA is more limited.

However, it is important to note that while the DIA strategy significantly improves performance, a performance gap remains between the two satellites. Even with rigorous QC, the average RMS of GRACE-D (13.1 cm) is still inferior to that of GRACE-C (7.3 cm). This orbit difference degradation is attributed to the weak geometric redundancy caused by data loss and worse GPS observation quality (as analyzed in Section 4.2). This limitation underscores the necessity of introducing external constraints, such as inter-satellite range observations, to further stabilize the solution.

4.4. Analysis of Inter-Satellite Range Enhancement Mechanism

To explicitly characterize the impact of observation quality and the corrective capability of inter-satellite range augmentation, we first examine the temporal evolution of orbit errors for a representative day (1 June 2021).

Figure 8 shows the orbit errors in the radial (R), along-track (T), and cross-track (N) directions and 3D at each epoch. The amplitude of the orbit error curve of the GPS + ISL solution is lower than that of the GPS-only solution, and the convergence period is shorter, especially for GRACE-D.

The different performance gains observed for the two satellites mainly reflect the distinct roles played by ISL range observations in the orbit determination process. For GRACE-C, the ISL observations primarily increase measurement redundancy and improve the observation geometry, resulting in a moderate accuracy enhancement. For GRACE-D, ISL observations provide not only geometric improvement but also strong relative constraints that tightly couple its orbit with the more accurately determined GRACE-C orbit. This coupling effectively enhances solution stability and leads to a substantially larger improvement in orbit accuracy and convergence behavior. A minor degradation for GRACE-C is observed on DOY 152, which coincides with severely degraded GNSS observation conditions for GRACE-D. This suggests that under highly unbalanced observation quality, the strong ISL-induced coupling may slightly influence the solution of the better-observed satellite.

The GRACE-D orbit error sequences show some spikes, primarily due to the GPS observation quality and the Position Dilution of Precision (PDOP). Figure 9 illustrates the correlation between the GRACE-D orbital error and the PDOP. It can be observed that the PDOP is larger at the time of the spikes, indicating geometric instability, where small measurement errors are amplified into larger position errors. Furthermore, analysis of the original observations reveals that spikes are also prone to occur when the number of visible satellites is less than three. In such scenarios of weak geometry, the system lacks the necessary redundancy for the iterative quality control to reliably identify outliers, allowing measurement errors to propagate into the state estimate.

The GPS + ISL solution (Top, orange) is far less sensitive to these high-PDOP events. While minor fluctuations persist, the magnitude of the error spikes is largely mitigates compared to the GPS-only case. This stability confirms that the inter-satellite range observations can introduce a rigid inter-satellite geometric constraint.

Mathematically, the assimilation of high-precision inter-satellite range observations sharply reduces the predicted state error covariance. In the EKF update mechanism, the Kalman Gain ($K_K$) serves as a weighting factor that balances the prediction uncertainty ($P_k^{-}$) against the measurement noise ($R_k$). Since the Kalman Gain is proportional to $P_k^{-}$, this reduction naturally decreases the gain assigned to the noisier GPS observations. Consequently, the filter effectively down-weights the GPS measurement errors during the update, thereby mitigating their impact on the final orbit solution.

To verify the rigorous implementation of the inter-satellite constraint, the post-fit residuals of the inter-satellite range observations were analyzed. Figure 10 presents the inter-satellite range residual sequence and frequency distribution on 1 June 2021.

The statistical analysis reveals a negligible mean residual of 0.01 cm, validating the efficacy of the bias pre-determination strategy (Section 3.2). Furthermore, the residuals exhibit a distinct Gaussian distribution with an RMS of only 0.17 cm. This millimetric level of internal consistency—significantly tighter than the conservative a priori weighting of 1.0 cm—demonstrates that the EKF tightly constrains the relative distance between the two satellites to the high-precision inter-satellite measurements.

The results highlight the algorithmic synergy between inter-satellite constraints and the post-fit residual-based quality control. From a mathematical perspective, this constraint significantly alters the error propagation mechanism within the filter. In a loosely constrained system, e.g., GPS-only, the filter tends to minimize cost by assimilating measurement anomalies into the state estimate, which artificially shrinks the residuals and leads to the masking effect. Conversely, the rigid inter-satellite range constraint prevents the filter from absorbing GPS errors into the state estimate. Instead, these errors are more prominently exposed in the post-fit residuals. This amplification significantly boosts the test statistics, enabling the QC procedure to detect outliers that might otherwise remain hidden.

4.5. RTPOD Improvements Contributed by Inter-Satellite Range Measurements

Figure 11 shows the daily RMS values of the orbit errors in 3D directions for different scheme solutions, and the corresponding average RMS and some other statistical information are shown in

Table 3. Mean RMS (unit cm) in R, T, N, and 3D directions, and the convergence time (unit min) of the two schemes.

Scheme	GRACE-C					GRACE-D
	R	T	N	3D	Convergence Time	R	T	N	3D	Convergence Time
GPS-only	4.6	4.3	4.0	7.3	46.9	8.3	7.4	7.8	13.1	77.8
GPS + ISL	4.6	3.7	3.7	7.0	46.3	4.8	4.8	4.1	8.0	43.4

. It should be noted that this RMS refers to the accuracy after orbit convergence. For the GPS-only scheme, the average 3D RMS values for GRACE-C and GRACE-D are 7.3 cm and 13.1 cm, respectively. This indicates that the orbit determination accuracy of GRACE-D is significantly worse than that of GRACE-C during this period. For the GPS + ISL scheme, the average orbit determination accuracies of GRACE-C and GRACE-D are 7.0 cm and 7.8 cm, respectively, representing improvements of 4.2% and 39%. Simultaneously, the satellite orbit convergence time is also reduced by 1.3% and 44.3%, respectively. For GRACE-D, the magnitude of ISL-induced improvement on a daily basis shows a consistency with the corresponding GNSS observation quality as shown in Figure 5. Compared to the GPS-only scheme, the GPS + ISL scheme significantly improves the consistency of the orbit determination results for the two satellites, providing a substantial improvement in accuracy for the satellite with degraded tracking performance. This demonstrates that inter-satellite range observations have a positive impact on maintaining stable satellite formation.

In summary, while the inter-satellite range measurements augmentation provides slight refinement for the satellite with better GPS observations (GRACE-C), it yields a decisive recovery for the satellite with degraded tracking conditions (GRACE-D). This phenomenon suggests that the inter-satellite range measurements do not merely add observations but impose fundamental constraints to stabilize the constellation geometry, even under asymmetric data quality conditions.

5. Discussion

The results presented in Section 4 demonstrate that the proposed cooperative GNSS–ISL RTPOD strategy yields asymmetric yet systematic performance improvements across the satellite formation. To better interpret these findings, this section discusses the underlying mechanisms from an estimation perspective. Specifically, we first analyze how inter-satellite range measurements contribute to cooperative stabilization within the EKF framework, then examine their interaction with the adopted quality control strategy, and finally address practical limitations and implications for future operational LEO constellations.

5.1. Cooperative Stabilization Mechanism Enabled by ISL Ranging

The asymmetric performance enhancement observed in the previous sections provides insight into the role of inter-satellite range measurements in cooperative orbit determination.

GNSS observations primarily constrain the absolute state of individual satellites and are therefore sensitive to local observation geometry and tracking quality. In contrast, ISL range measurements are inherently relative observables, providing strong geometric constraints between satellites within a formation. From a mathematical standpoint, ISL ranges do not directly increase the absolute observability of the system. Instead, they enhance the internal consistency of the estimation problem by tightly coupling the relative positions of the satellites.

In this study, GRACE-D experiences degraded GNSS observation conditions, characterized by reduced satellite visibility and frequent cycle slips, which weakens its local geometric strength and increases the risk of solution instability. GRACE-C, on the other hand, maintains stable and high-quality GNSS tracking, resulting in a well-determined absolute orbit. When inter-satellite range measurements are introduced, the EKF jointly estimates the states of both satellites under rigid relative constraints. As a result, reliable positional information from GRACE-C is indirectly transferred to GRACE-D through the constrained relative geometry, thereby stabilizing the estimation of the degraded satellite.

This mechanism explains why the inclusion of inter-satellite ranging yields a substantial accuracy recovery for GRACE-D, while only marginal refinement is observed for GRACE-C. The observed performance improvement arises not from a direct enhancement of absolute positioning capability, but from improved estimator conditioning and reduced sensitivity to unfavorable GNSS observation geometry. Consequently, orbit determination performance within a connected satellite formation becomes mutually reinforced rather than independent.

5.2. Interaction Between Iterative Quality Control and ISL Range Constraints

The results further indicate that inter-satellite range augmentation complements the adopted quality control strategy by improving the overall robustness of the estimation process.

In GNSS-only configurations, the effectiveness of post-fit residual-based quality control is inherently limited by observation redundancy and geometric strength. Under degraded tracking conditions, limited redundancy may force the estimator to rely on weakly supported measurements, increasing the risk of error propagation into the state vector.

To mitigate this vulnerability, the integration of inter-satellite range observations reinforces the system through a threefold mechanism. (1) Geometry Optimization: High-precision ranging strengthens the formation structure, effectively compensating for instability under poor GNSS visibility; (2) Redundancy Augmentation: Independent measurements increase degrees of freedom, strengthening the statistical basis for hypothesis testing; and (3) Masking Effect Suppression: Inter-satellite range rigid constraints prevent the assimilation of GNSS measurement errors into the state, forcing outliers to be exposed in post-fit residuals and thereby sharpening quality control sensitivity.

5.3. Limitations and Future Perspectives

While the proposed strategy has demonstrated significant performance gains, certain limitations remain to be addressed, primarily regarding the handling of the ISL instrument bias. In this study, the systematic bias is pre-determined using post-processed precise orbits and treated as a constant over each processing arc. While this approach is reasonable for short-term stability, it relies on external products and is therefore compatible with algorithmic validation in a simulated real-time environment. For a fully autonomous real-time system, future work should investigate estimating the systematic bias directly within the filter. Given its high stability over short durations, this bias is physically best modeled as a constant parameter for each continuous arc—analogous to the treatment of GNSS carrier phase ambiguities. However, implementing this strategy requires robust methods to decouple the bias from orbital parameters to prevent potential solution instability.

Furthermore, the findings highlight the potential of inter-satellite range measurements as a key augmentation for future LEO formations. By stabilizing orbit determination under degraded observation conditions, cooperative GNSS–ISL strategies can enhance robustness, maintain service continuity, and reduce sensitivity to individual satellite performance degradation. The GRACE-FO mission provides a valuable real-data testbed for validating these concepts, offering practical insights that may inform the design of future autonomous LEO constellations.

6. Conclusions

This study proposed and validated a cooperative RTPOD strategy for LEO satellite formations by jointly processing onboard GNSS and ISL ranging observations within an EKF framework integrated with DIA-based quality control. Using real data of GRACE-FO in a simulated real-time environment, the experiments demonstrate that introducing ISL observations leads to improvements in orbit accuracy and convergence performance in formation-based RTPOD.

The results further reveal that ISL ranging provides strong relative geometric constraints between LEO satellites, stabilizing orbit estimation when local GNSS observation geometry weakens. Consequently, the degraded satellite achieves significant accuracy recovery and faster convergence, while the well-observed satellite obtains moderate refinement. This confirms that cooperative GNSS–ISL processing enhances estimator robustness and mitigates performance degradation caused by asymmetric observation quality within a satellite formation.

It should be noted that the present study is conducted in a simulated real-time processing environment based on data playback. For future fully onboard real-time implementation, further investigation is required on real-time estimation of ISL instrumental biases and on engineering aspects related to onboard operational robustness.