Consistent Fusion of MADOCA-PPP and PPP-B2b SSR Corrections for Robust Real-Time PPP

Highlights

What are the main findings?

• A correction-domain fusion framework based on bias-state-aware structured covariance intersection (SCI) consistently combines MADOCA-PPP and PPP-B2b SSR corrections for real-time precise point positioning (PPP).
• In offshore kinematic tests, the fused solution is the most accurate of the three schemes (0.23 m horizontal, 0.29 m 3D RMS) and maintains 100% availability under single-service outages of up to 600 s.

What are the implications of the main findings?

• Provides redundancy, continuity, and robustness for real-time PPP users in Asia-Pacific regions where network-based corrections are unavailable.
• Establishes a transferable, statistically consistent fusion architecture for combining heterogeneous open SSR services in next-generation multi-GNSS positioning.

Abstract

Real-time precise point positioning (PPP) is increasingly supported by open satellite-broadcast state-space representation (SSR) services, yet standalone operation with a single service remains vulnerable to limited constellation support, correction outages, latency variations, and service-dependent modeling inconsistencies. In the Asia-Pacific region, MADOCA-PPP and PPP-B2b provide two publicly accessible and complementary SSR sources, but their consistent fusion before user-level PPP estimation remains insufficiently investigated. This paper proposes a correction-domain fusion framework that combines MADOCA-PPP and PPP-B2b orbit and clock corrections before PPP estimation, rather than merging final positioning solutions. Inter-service discrepancies and unknown cross-correlations are handled by a bias-state-aware structured covariance intersection strategy, in which the relative weighting is derived from the respective correction information (inverse variance), preserving statistical consistency and avoiding overconfident fusion. A unified multi-GNSS PPP scheme further supports signal-priority harmonization, broadcast-ephemeris adaptation, correction-age control, and GLONASS inter-frequency and differential code bias handling. Static-station per-epoch (pseudo-kinematic) and offshore kinematic experiments validate the framework. In the static-station test, fusion raised the mean number of valid satellites from 21.98 and 14.98 to 26.56 and improved the horizontal RMS to 0.033 m—better than either standalone service (0.037 m, 0.079 m)—confirming a genuine combination rather than source selection, while the 3D RMS (0.068 m) matched the best standalone service (0.066 m). In the offshore test, fusion achieved the best overall accuracy (0.232 m horizontal, 0.290 m 3D, versus 0.332 m and 0.313 m for the standalone services) and the most satellites (25.4). It also degraded most slowly with increasing elevation cut-off, outperforming both services about threefold at 40°. A normalized-innovation-squared check confirmed the fused covariance is consistent and not overconfident (median ≈ 1.1; within the 99% bound in 100% of epochs). Under single-service outages from 30 s to 600 s, fusion maintained 100.0% availability, confirming its advantage in redundancy, continuity, and resilience.

1. Introduction

Precise point positioning (PPP) has become an important technique in high-accuracy global navigation satellite system (GNSS) positioning and navigation research [1,2,3]. Compared with Real-Time Kinematic (RTK), PPP does not rely on nearby reference stations and can therefore be more suitable for wide-area, infrastructure-limited, and dynamic environments [4,5]. Owing to these advantages, PPP has been widely applied in various scenarios, including marine navigation, offshore operations, and environmental monitoring [6,7]. Since the concept of PPP was established in the 1990s [8], continuous improvements in precise orbit and clock products, as well as the development of the International GNSS Service (IGS), have gradually promoted PPP from post-processing to real-time applications [9,10]. The accuracy of real-time PPP is fundamentally limited by the quality of the satellite orbit and clock corrections: it depends on the stability and noise of the on-board frequency standards [11], on the precision attainable when orbit and clock corrections are computed in an ultra-rapid/real-time regime [12], and on the product itself, whose accuracy varies appreciably among analysis centers and generation strategies [13].

With the growing demand for real-time high-precision positioning, open satellite-broadcast augmentation services have become increasingly attractive alternatives to network-delivered correction products. At present, BDS-3, Galileo, and Japan’s QZSS have launched public real-time PPP services, including PPP-B2b [14], high accuracy service (HAS), and MADOCA/CLAS, respectively. These services provide new opportunities for real-time PPP users in regions where network connectivity is limited or unstable. Among them, PPP-B2b and MADOCA-PPP are two representative and complementary correction sources in the Asia-Pacific region.

Table 1. Overview of public real-time PPP services.

Service/Product	Data Carrier	Data Format	Current Coverage	Supported System	Correction Messages
IGS Real-Time Service	Network	IGS-SSR	Global	GREC	Orbit/clock/biases */VTEC
CNES Real-time Product	Network	RTCM-SSR	Global	GREC	Orbit/clock/biases/VTEC
BDS PPP-B2b	PPP-B2b signal	SSR	Asia-Pacific	GC(BDS-3)	Orbit/clock/Differential Code Biases (DCBs)
MADOCA-PPP	L6E signal/ Network	CSSR	Asia-Pacific	GREJ	Orbit/clock/biases
HAS	E6 signal/ Network	CSSR-like	Global for SL1; Europe for SL2	GE	Orbit/clock/biases

* Biases include code and phase bias.

summarizes an overview of the services that are currently used primarily for real-time PPP.

As an important open augmentation service, PPP-B2b has been extensively studied in recent years. Existing works have investigated its correction precision [15], service availability, message characteristics, Issue of Data (IOD) matching behavior [16], positioning performance [17], and time-transfer capability [18,19,20,21]. These studies have provided a relatively complete understanding of PPP-B2b from the perspectives of signal structure, correction quality, and user-side PPP performance.

On the QZSS side, CLAS has demonstrated rapid centimeter-level regional augmentation capability within Japan [22]. Meanwhile, MADOCA-related studies have evaluated its usability and orbit/clock accuracy, assessed its multi-GNSS PPP and PPP-AR performance [23], and discussed its potential application in future real-time precise orbit determination scenarios [24,25]. These works indicate that MADOCA-PPP is also a promising open service for real-time PPP in the Asia-Pacific region.

Although PPP-B2b and MADOCA-PPP have both shown strong potential for real-time PPP, existing studies have mainly evaluated them separately or compared them as standalone services. In contrast, the necessity and methodology of combining these two open services for robust real-time PPP remain insufficiently investigated. In particular, practical multi-service use involves not only complementary satellite support and geometry enhancement, but also inter-service inconsistency, asynchronous updates, and possible interruption of a single correction stream. Therefore, the key issue is no longer simply which service performs better on average, but how to construct a consistent and robust multi-service fusion framework.

To address this problem, this paper proposes a consistent fusion framework for combining MADOCA-PPP and PPP-B2b corrections in real-time PPP. The proposed method performs correction-domain fusion before user-level PPP estimation and explicitly considers inter-service inconsistency and robustness under single-service degradation or interruption. Static-station and dynamic experiments are further conducted to evaluate the proposed framework in terms of positioning accuracy, convergence, satellite availability, and outage-recovery behavior.

The main contributions of this study are summarized as follows:

(1)

A unified correction-domain fusion framework is developed for combining MADOCA-PPP and PPP-B2b in real-time PPP.

(2)

A consistency-preserving fusion strategy is introduced to handle inter-service mismatch and improve robustness under single-service interruption.

(3)

Experiments using static-station and dynamic datasets demonstrate that the value of combining MADOCA-PPP and PPP-B2b lies not only in positioning accuracy, but also in solution continuity and service resilience.

2. Materials and Methods

This chapter first briefly introduces PPP-B2b and MADOCA real-time products, presents the PPP model of SSR correction used in the subsequent PPP performance comparison, and gives the recovery strategy of SSR correction.

2.1. PPP-B2b and MADOCA-PPP Products

PPP-B2b and MADOCA-PPP are two representative open satellite-broadcast augmentation services for real-time precise point positioning in the Asia-Pacific region. Although both services provide state-space representation (SSR) corrections for user-side PPP, they differ significantly in constellation support, message organization, correction content, and update characteristics. These differences make them complementary in practical real-time PPP, while also introducing non-negligible challenges for direct combination.

PPP-B2b is a public augmentation service provided by the BDS-3 system through geostationary orbit (GEO) satellite broadcasting [26]. It mainly delivers satellite mask information, orbit corrections, user range accuracy index (URAI), DCB, and satellite clock corrections [27] in SSR form. Existing studies have shown that PPP-B2b can support real-time precise positioning and time transfer, while its correction characteristics, delay behavior, SISRE evaluation [28] and IOD matching properties have also been analyzed in detail. In practical PPP applications, PPP-B2b has the advantage of open access and direct satellite broadcasting, but its supported constellations and correction update characteristics remain service-specific.

MADOCA-PPP is developed within the QZSS augmentation framework and provides real-time precise corrections through L6 signals. Compared with PPP-B2b, MADOCA-PPP supports a broader multi-GNSS correction framework and adopts Compact SSR (CSSR), which offers a more compact message structure for real-time dissemination [29,30]. Previous studies have demonstrated the usability of MADOCA products in PPP and PPP ambiguity resolution (AR) applications, and have evaluated their orbit and clock quality as well as their potential in broader real-time precise positioning scenarios.

From the perspective of standalone service performance, the two services each possess their own strengths. However, in practical real-time PPP, relying on only one service may still lead to several limitations. First, the number of available satellites and the resulting observation geometry depend on service-specific constellation support. Second, the correction streams may experience temporary delay, degradation, or interruption, which can affect solution continuity and re-convergence behavior. Third, even when both services are available simultaneously, their corrections cannot be directly averaged in a statistically safe manner, because inter-service inconsistency may arise from different message conventions, reference realizations, update epochs, bias handling, and unknown common-mode correlation.

Therefore, the purpose of this study is not simply to compare PPP-B2b and MADOCA-PPP as two independent services, but to investigate how they can be consistently combined for robust real-time PPP. To achieve this goal, a unified correction-domain processing strategy is required before user-level PPP estimation, so that the complementary information from the two services can be utilized without introducing inconsistency or overconfident fusion.

2.2. Unified PPP Model and Correction-Domain Harmonization

In this study, an undifferenced ionosphere-free PPP model is adopted as the common user-side positioning model for all compared solutions. The same PPP estimator is used for MADOCA-PPP-only, PPP-B2b-only, and the fused solution. Therefore, the differences among the solutions mainly originate from the applied SSR correction products rather than from different user-side positioning models.

For a receiver $r$, satellite $s$, constellation $g$, and frequency $i$, the undifferenced pseudorange and carrier-phase observations can be expressed as follows:

$$\begin{array}{l}{P}_{r,i}^{s,g}={\rho }_r^{s,g}+c(d{t}_r-d{t}_s)+T_r^{s,g}+I_{r,i}^{s,g}+d_{r,i}^g-d_i^{s,g}+{\epsilon }_{P,i}^{s,g}\\ {\mathrm{\Phi }}_{r,i}^{s,g}={\rho }_r^{s,g}+c(d{t}_r-d{t}_s)+T_r^{s,g}-I_{r,i}^{s,g}+{\lambda }_i{N}_{r,i}^{s,g}+b_{r,i}^g-b_i^{s,g}+{\epsilon }_{\mathrm{\Phi },i}^{s,g}\end{array}$$

(1)

where ${\rho }_r^{s,g}$ is the geometric distance between the receiver and satellite, $c$ is the speed of light, $d{t}_r$ and $d{t}_s$ are the receiver and satellite clock offsets, $T_r^{s,g}$ is the tropospheric delay, $I_{r,i}^{s,g}$ is the slant ionospheric delay on frequency $i$, $d_{r,i}^g$ and $d_i^{s,g}$ are receiver and satellite-dependent code hardware delays, $b_{r,i}^g$ and $b_i^{s,g}$ are carrier-phase hardware delays, ${\lambda }_i$ is the carrier wavelength, $N_{r,i}^{s,g}$ is the integer ambiguity, and ${\epsilon }_{P,i}^{s,g}$ and ${\epsilon }_{\mathrm{\Phi },i}^{s,g}$ denote the remaining measurement noise and unmodeled errors.

For each constellation, two frequencies are selected according to the available receiver-independent exchange format (RINEX) observation types and the signal bias information supported by the corresponding SSR service. The ionosphere-free (IF) combinations are formed as follows:

$$\begin{array}{l}{P}_{\mathrm{IF}}^{s,g}=\alpha P_{r,1}^{s,g}-\beta P_{r,2}^{s,g}\\ {\mathrm{\Phi }}_{\mathrm{IF}}^{s,g}=\alpha {\mathrm{\Phi }}_{r,1}^{s,g}-\beta {\mathrm{\Phi }}_{r,2}^{s,g}\end{array}$$

(2)

with

$$\begin{array}{l}\alpha ={\displaystyle \frac{f_1^2}{f_1^2-f_2^2}},\\ \beta ={\displaystyle \frac{f_2^2}{f_1^2-f_2^2}}\end{array}$$

(3)

where $f_1$ and $f_2$ are the two selected carrier frequencies. This combination eliminates the first-order ionospheric delay. The frequency pair is constellation- and service-dependent.

After forming the ionosphere-free combinations and applying SSR orbit, clock, and signal bias corrections, the observation equations can be written as follows:

$$\begin{array}{l}{P}_{\mathrm{IF}}^{s,g}=\rho (\mathbf{r},{\mathbf{x}}_{\mathrm{SSR}}^s)+c(d{t}_r-d{t}_{\mathrm{SSR}}^s)+T_r^{s,g}+{\mathrm{ISB}}_g+B_{P,\mathrm{IF}}^{s,g}+{\epsilon }_{P,\mathrm{IF}}^{s,g}\\ {\mathrm{\Phi }}_{\mathrm{IF}}^{s,g}=\rho (\mathbf{r},{\mathbf{x}}_{\mathrm{SSR}}^s)+c(d{t}_r-d{t}_{\mathrm{SSR}}^s)+T_r^{s,g}+{\mathrm{ISB}}_g+N_{\mathrm{IF}}^{s,g}+B_{\mathrm{\Phi },\mathrm{IF}}^{s,g}+{\epsilon }_{\mathrm{\Phi },\mathrm{IF}}^{s,g}\end{array}$$

(4)

where ${\mathbf{x}}_{\mathrm{SSR}}^s$ and $d{t}_{\mathrm{SSR}}^s$ are the satellite position and clock corrected by SSR orbit and clock corrections. ${\mathrm{ISB}}_g$ is the inter-system bias of constellation $g$ with respect to the GPS time reference. $B_{P,\mathrm{IF}}^{s,g}$ and $B_{\mathrm{\Phi },\mathrm{IF}}^{s,g}$ denote the residual ionosphere-free code and phase bias terms after applying the available SSR signal bias corrections. If the corresponding signal bias correction is available, it is applied before PPP estimation; otherwise, the residual bias is absorbed by the receiver clock, inter-system bias, or ambiguity parameters depending on its physical property.

Since the MADOCA-PPP corrections include GLONASS, its FDMA inter-frequency biases (IFBs) are accounted for in the ionosphere-free PPP. The GLONASS-to-GPS inter-system bias (the frequency-independent hardware delay) is absorbed by a per-constellation receiver-clock parameter; the carrier-phase IFB is fully absorbed by the per-satellite float ambiguities and does not bias the float solution; and the pseudorange IFB is handled stochastically, by augmenting the GLONASS code variance with an IFB term (0.6 m, 1σ) rather than estimating explicit per-channel parameters. This is adequate in the float-PPP context used here, where GLONASS contributes to multi-constellation redundancy and geometry without integer ambiguity resolution.

The tropospheric delay is modeled as follows:

$$T_r^{s,g}=m_h(e)ZHD+m_w(e)ZWD$$

(5)

where $ZHD$ and $ZWD$ are the zenith hydrostatic and wet delays, and $m_h(e)$ and $m_w(e)$ are the corresponding mapping functions as a function of satellite elevation angle $e$. The hydrostatic component is corrected using an empirical model, while the wet component is estimated as an unknown parameter in the PPP filter.

The estimated state vector is defined as follows:

$$\mathbf{x}={[\mathbf{r},d{t}_r,IS{B}_{GLO},IS{B}_{GAL},IS{B}_{QZS},IS{B}_{BDS},ZWD,N_{IF}^1,\dots ,N_{IF}^m]}^T$$

(6)

where $\mathbf{r}$ is the receiver position vector, $d{t}_r$ is the receiver clock offset, $ZWD$ is the zenith wet delay, and $N_{IF}^1,\dots ,N_{IF}^m$ are the float ionosphere-free ambiguities for all tracked satellites. GPS is used as the reference system, and its system bias is absorbed into the receiver clock. For GLONASS, Galileo, QZSS, and BDS, one inter-system bias parameter is estimated when observations from the corresponding constellation are used.

The linearized ionosphere-free PPP observation model can be expressed as follows:

$$\mathbf{v}=\mathbf{H}\Delta \mathbf{x}-\mathbf{l}$$

(7)

where $\mathbf{v}$ is the post-fit residual vector, $\mathbf{H}$ is the design matrix, $\Delta \mathbf{x}$ is the state correction vector, and $\mathbf{l}$ is the observed-minus-computed vector after applying SSR and conventional geophysical corrections. The stochastic model is constructed according to satellite elevation angle and the quality indicators of the SSR corrections. In this way, satellites corrected by different services can be introduced into the same PPP estimator while retaining service-dependent correction uncertainty.

In the proposed correction-domain fusion strategy, MADOCA-PPP and PPP-B2b corrections are first converted into a unified SSR representation, including orbit, clock, and signal bias corrections. For overlapping satellites, especially GPS satellites corrected by both services, two correction estimates are available for the same satellite state and are fused before being introduced into the PPP filter. For non-overlapping satellites, such as GLONASS, Galileo, and QZSS from MADOCA-PPP or BDS-3 from PPP-B2b, the available single-service correction is used directly. This design allows the fused solution to benefit from the complementary constellation coverage of the two services while maintaining a consistent ionosphere-free PPP observation model.

The first harmonization step is satellite and constellation alignment. Since the two services do not support exactly the same constellation set, only those satellites with valid and correctly matched correction information are eligible for direct fusion, while service-exclusive satellites are retained in a single-service mode. In this way, the fusion framework can exploit common satellites for consistent combination and simultaneously preserve the geometry contribution of non-overlapping satellites.

The second step is broadcast-ephemeris consistency. Both PPP-B2b and MADOCA-PPP corrections are referenced to broadcast navigation information rather than directly to final precise products. Therefore, the corresponding broadcast ephemerides must be matched carefully in time and IOD space before correction application. This step is particularly important for avoiding artificial discrepancies caused by inconsistent ephemeris baselines rather than true service-level correction differences.

The third step is correction representation unification. Orbit corrections from different services may be transmitted in different representations or message structures, but they must ultimately be transformed into a common user-side form before PPP calculation. In this study, all orbit corrections are mapped into a unified satellite-related representation and then applied consistently with the associated broadcast ephemerides. Likewise, the clock corrections are converted into a common convention so that user-side clock modeling is not contaminated by service-dependent formulation differences.

The fourth step is bias and signal mapping harmonization. Because the two services may use different signal priorities and bias conventions, differential code bias handling must be made compatible with the actual observation types used in PPP processing. When service-native bias information is available and valid, it is applied directly; otherwise, external DCB information is used as a fallback to maintain consistency across services. This step is necessary to avoid introducing artificial inter-service offsets in the code observations.

After the above harmonization procedures, the two correction streams are transformed into a unified correction-domain representation, which provides the basis for subsequent consistent fusion. In other words, the purpose of harmonization is not to force the two services to become numerically identical, but to ensure that their differences primarily reflect true service-level characteristics rather than mismatched conventions or implementation artifacts. On this basis, the fused correction model can then be constructed in a statistically consistent manner.

2.3. Bias-State-Aware Structured Covariance Intersection

For a satellite corrected by both services, we collect the common-correction state into a four-dimensional vector $\mathbf{z}={\left[\delta O_R,\delta O_A,\delta O_{\mathrm{C}},c\delta t\right]}^T$, comprising the radial, along-track and cross-track orbit corrections and the clock correction (in range units) expressed in a unified frame. After the harmonization of Section 2.2, each service $s\in \left\{M,B\right\}$ (MADOCA-PPP and PPP-B2b) delivers an estimate that, in general, differs from the true state by both a slowly varying systematic offset and a random error,

$${\widehat{\mathbf{z}}}_s=\mathbf{z}+{\mathbf{b}}_s+{\mathsf{\epsilon }}_s,\hspace{1em}{\mathsf{\epsilon }}_s~(\mathbf{0},{\mathbf{R}}_s),\hspace{1em}\mathrm{E}[{\mathsf{\epsilon }}_M{\mathsf{\epsilon }}_B^{\top }]=\mathbf{C},$$

(8)

where the offset ${\mathbf{b}}_s$ originates from service-specific datum, broadcast-ephemeris baseline and antenna/EOP conventions, and the cross-covariance $\mathbf{C}$ is unknown: the two services rely on partially overlapping tracking networks and shared modeling assumptions, so their errors contain a common-mode component that cannot be recovered at the user side. This unknown correlation is precisely the condition under which a minimum-variance combination becomes overconfident and for which covariance intersection (CI) is appropriate. Stacking the two observations,

$$\left[\begin{array}{c}{\widehat{\mathbf{z}}}_M\\ {\widehat{\mathbf{z}}}_B\end{array}\right]=\mathbf{H}\left[\begin{array}{c}\mathbf{z}\\ {\mathbf{b}}_M\\ {\mathbf{b}}_B\end{array}\right]+\left[\begin{array}{c}{\mathsf{\epsilon }}_M\\ {\mathsf{\epsilon }}_B\end{array}\right],\hspace{1em}\mathbf{H}=\left[\begin{array}{ccc}\mathbf{I}& \mathbf{I}& \mathbf{0}\\ \mathbf{I}& \mathbf{0}& \mathbf{I}\end{array}\right],$$

(9)

reveals that the joint state $(\mathbf{z},{\mathbf{b}}_M,{\mathbf{b}}_B)$ is not observable: for any $\delta \in {ℝ}^4$ the triple $(\mathbf{z}+\mathbf{\delta },{\mathbf{b}}_M-\mathbf{\delta },{\mathbf{b}}_B-\mathbf{\delta })$ reproduces the same observations, so $\mathbf{H}$ has a rank deficiency equal to $\dim (\mathbf{z})=4$. This gauge freedom is removed by fixing one service as the bias datum. Taking MADOCA-PPP as reference, ${\mathbf{b}}_M=0$, leaves only the relative offset $\mathbf{\beta }\triangleq {\mathbf{b}}_B$ to be estimated and renders $\mathbf{z}$ identifiable,

$${\widehat{\mathbf{z}}}_M=\mathbf{z}+{\mathsf{\epsilon }}_M,\hspace{1em}{\widehat{\mathbf{z}}}_B=\mathbf{z}+\mathbf{\beta }+{\mathsf{\epsilon }}_B.$$

(10)

The relative offset is observable from the inter-service single difference $\mathbf{d}={\mathbf{Z}}_M-{\mathbf{Z}}_B=-\mathbf{\beta }+({\mathsf{\epsilon }}_M-{\mathsf{\epsilon }}_B)$ and is propagated as a slowly varying random-walk state shared across the overlapping satellites, which decouples the common state from the inter-service bias. The choice of datum does not affect identifiability: because only the inter-service difference $\mathbf{\beta }={\mathbf{b}}_B-{\mathbf{b}}_M$ is observable, fixing either service as the bias datum removes the same gauge freedom, leaving identifiability—and, up to this gauge, the positioning solution—unchanged. We fix the higher-accuracy service (MADOCA-PPP, Section 3.1.1) as the datum so that the random-walk bias state models the noisier PPP-B2b correction relative to a more stable reference, which yields better filter conditioning than the reverse assignment.

Using the estimated offset $\widehat{\mathbf{\beta }}$ with covariance ${\mathbf{\Sigma }}_{\beta }$, the PPP-B2b state is aligned to the MADOCA datum,

$${\tilde{\mathbf{z}}}_B={\widehat{\mathbf{z}}}_B-\widehat{\mathbf{\beta }},\hspace{1em}{\tilde{\mathbf{R}}}_B={\mathbf{R}}_B+{\mathbf{\Sigma }}_{\beta },$$

(11)

so that ${\tilde{\mathbf{z}}}_M$ and ${\tilde{\mathbf{z}}}_B$ become unbiased estimates of the same state. The combination is termed structured because it is applied only within this common-correction subspace—block-diagonally, with a three-dimensional orbit block and a one-dimensional clock block—while the service bias is carried as a nuisance parameter outside the intersection; and because it is bias-state-aware, operating on the aligned quantities of (11). For the weight $\omega \in [0,1]$, the covariance-intersection estimate and its covariance are

$${\mathbf{P}}_f(\omega )={\left(\omega {\mathbf{R}}_M^{-1}+(1-\omega ){\tilde{\mathbf{R}}}_B^{-1}\right)}^{-1},\hspace{1em}\hspace{1em}\widehat{\mathbf{z}}(\omega )={\mathbf{P}}_f(\omega )\left(\omega {\mathbf{R}}_M^{-1}{\widehat{\mathbf{z}}}_M+(1-\omega ){\tilde{\mathbf{R}}}_B^{-1}{\tilde{\mathbf{z}}}_B\right).$$

(12)

The essential robustness property is that, for every $\omega \in [0,1]$ and every admissible cross-covariance $\mathbf{C}$, the result is consistent,

$${\mathbf{P}}_f(\omega )\succcurlyeq \mathrm{E}\left[(\widehat{\mathbf{z}}(\omega )-\mathbf{z}){(\widehat{\mathbf{z}}(\omega )-\mathbf{z})}^{\top }\right],$$

(13)

i.e., the fused covariance never underestimates the true error covariance irrespective of the unknown correlation. Property (13) is the formal basis of the consistency-preserving claim and, importantly, holds for any weight, not only the covariance-minimizing one; the correction injected into the user filter therefore cannot be overconfident.

MADOCA-PPP and PPP-B2b cannot be assumed mutually independent. Both are referenced to broadcast ephemerides rather than to independent final products; both are generated from globally distributed tracking networks whose stations partially overlap; and both estimate satellite orbits and clocks under broadly similar modeling conventions (reference-clock datum, terrestrial-frame realization, Earth-orientation and antenna models). Through these shared inputs and conventions, part of the orbit and clock errors of the two services is common-mode, so their cross-correlation is non-zero. At the user side, this correlation can neither be observed nor recovered, which is precisely why a minimum-variance combination—optimal only under independence—would be overconfident, whereas a covariance-intersection combination, consistent for any unknown cross-correlation, is appropriate. The two services additionally differ by service-specific systematic offsets (reference-frame and clock-datum conventions), which are represented by the bias state and removed prior to fusion so that the combination operates on quantities sharing a common reference.

Classical CI fixes the weight by ${\omega }^{*}=\arg {\min }_{\omega }\det {\mathbf{P}}_f(\omega )$ the tightest consistent bound. When one covariance is nested in the other in the Loewner order $({\mathbf{R}}_M\prec {\tilde{\mathbf{R}}}_B)$—the regime of the GPS overlap, where MADOCA-PPP uniformly dominates PPP-B2b (Section 3.1.1)—this minimizer is a vertex, ${\omega }^{*}\in \{0,1\}$, so CI degenerates to source selection and performs no blending. To realize a genuine combination while preserving (13), the weight is instead set from the relative information of the two sources, component-wise,

$${\omega }_i={\displaystyle \frac{{({\mathbf{R}}_M^{-1})}_{ii}}{{({\mathbf{R}}_M^{-1})}_{ii}+{({\tilde{\mathbf{R}}}_B^{-1})}_{ii}}},$$

(14)

which is the minimum-variance weight under independence and is interior whenever both sources carry finite information, so that the two services contribute in proportion to their accuracy. Although (14) is not the minimum-volume CI weight, it trades a marginally looser—yet, by (13), still consistent—covariance for a genuine, accuracy-proportional blend; because (13) holds for any $\omega$, this choice carries no risk of overconfidence.

Before combining, the statistical compatibility of the aligned sources is verified by the normalized innovation squared,

$$q={\tilde{\mathbf{d}}}^{\top }{\left({\mathbf{R}}_M+{\tilde{\mathbf{R}}}_B\right)}^{-1}\tilde{\mathbf{d}}\hspace{0.17em}~\hspace{0.17em}{\chi }^2(n_z),\hspace{1em}\tilde{\mathbf{d}}={\widehat{\mathbf{z}}}_M-{\tilde{\mathbf{z}}}_B,$$

(15)

with $n_z=4$; fusion proceeds only if $q\le \gamma$, where $\gamma$ is the $(1-\alpha )$ quantile of ${\chi }^2(4)$ ($\gamma =13.28$ at $\alpha =0.01$) and otherwise the datum correction is used. Because neither ${\mathbf{R}}_s$ nor $\mathbf{C}$ is recoverable at the user, ${\mathbf{R}}_s$ is built as a deliberately conservative correction-domain proxy from the service quality indicator (PPP-B2b URAI or the MADOCA variance field), an age-dependent inflation that treats orbit and clock separately, and a variance floor reflecting the measured product accuracy (Section 3.1.1); over-bounding the true precision, together with (13), guarantees a non-overconfident fused correction, which is confirmed empirically by the NIS statistic of (15).

Finally, the fused state $\tilde{\mathbf{z}}(\omega )$ replaces the satellite orbit and clock used in the user PPP observation equations; the correction covariances set the information weight (14) that forms this fused state, while ${\mathbf{P}}_f(\omega )$ additionally drives the consistency gate (15) and the satellite-screening test. The per-observation measurement variance follows the standard elevation/SNR model (with a URA-based term for PPP-B2b–corrected satellites); injecting ${\mathbf{P}}_f$ as an additional additive term in $\mathbf{R}$ is a natural extension and is left to future work. For satellites corrected by only one service—GLONASS, Galileo and QZSS from MADOCA-PPP, and BDS-3 from PPP-B2b—the available single-service correction and its covariance are directly used in (11), providing the multi-constellation geometry augmentation that complements the fused GPS overlap.

This scheme is distinguished from related fusion rules as summarized in

Table 2. Comparison with covariance-intersection-family fusion rules.

Method	Unknown Cross-Correlation	Inter-Service Offset	Fused State	Behavior
Inverse-variance	NO (assumes $\mathbf{C}=\mathbf{0}$)	NO	$\mathbf{z}$	overconfident
Classical CI	YES	NO	$\mathbf{z}$	degenerates to selection
Split-CI	YES	NO	$\mathbf{z}$	covariance only
Proposed (this work)	YES	YES	datum-aligned	consistent

: a naive inverse-variance combination assumes $\mathbf{C}=\mathbf{0}$ and is overconfident under common-mode errors; classical CI is consistent but, with the determinant-minimizing weight, reduces to source selection for nested covariances; and Split-CI and Inverse-CI address the covariance structure or the double-counting of common information but not the systematic inter-service offset. By combining a bias state with datum fixing—removing the offset that classical CI would otherwise propagate into a biased estimate—a subspace-restricted, block-wise intersection, and an information-based weight, the proposed method realizes a genuine, consistent blend of the two correction services.

2.4. Algorithm Summary

The overall framework of the proposed MADOCA-PPP/PPP-B2b fusion method is shown in Figure 1.

2.5. Dataset

To comprehensively evaluate the proposed fusion framework, this study used two complementary types of datasets: static-station datasets and a real offshore kinematic trajectory dataset in the Asia-Pacific region. The former provides stable, repeatable, and reference-controllable test conditions, while the latter represents a practical dynamic positioning scenario with time-varying satellite geometry, observation quality, and correction availability. This combination enables the performance of MADOCA-PPP, PPP-B2b, and the proposed fused solution to be assessed from both nominal accuracy and robustness perspectives.

The static-station experiments were carried out using IGS/Multi-GNSS experiment (MGEX) station observations located within the effective service coverage of both MADOCA-PPP and PPP-B2b; the station distribution is shown in Figure 2. In the experimental part, the WUH2 station is taken as the representative station. These datasets were used to evaluate nominal positioning accuracy, convergence behavior, valid-satellite number, constellation contribution, and geometric strength under relatively stable observation conditions. Because precise station coordinates are available from Solution (software/technique) Independent Exchange format (SINEX) products, the static-station experiments also provide reliable external references for assessing the consistency of different correction strategies.

To further investigate the behavior of the proposed method under realistic dynamic conditions, a real offshore kinematic dataset collected on day of year (DOY) 190, 2023 was used; the offshore scenario and navigation trajectory are shown in Figure 3. This dataset contains continuous multi-GNSS observations along a moving trajectory and therefore reflects practical challenges such as changing receiver environment, varying satellite visibility, fluctuating observation quality, and time-dependent correction support. It was mainly used to evaluate dynamic positioning accuracy, solution continuity, and post-outage recovery performance. Compared with the static-station datasets, the offshore trajectory is more representative of real-time maritime PPP applications, where service interruption and geometry variation can directly affect positioning reliability.

For all experiments, MADOCA-PPP corrections were decoded from QZSS L6 products, and PPP-B2b corrections were recovered from BDS-3 PPP-B2b messages [31]. The corresponding broadcast ephemerides were used in a unified user-side processing framework. The observation data are in RINEX 3.04 (offshore) and 3.05 (WUH2). The broadcast navigation messages are taken from the DLR multi-GNSS broadcast file (BRD400DLR), in RINEX 4.00 for the 2023 offshore dataset and RINEX 4.02 for the 2024 WUH2 dataset; the RINEX 4 format provides the BDS-3 and multi-GNSS navigation message types required for the PPP-B2b service. External auxiliary products, including antenna phase center models and other products, were applied when available to maintain consistency among different PPP solutions. By combining station-based experiments and a real offshore dynamic experiment, the dataset design covers both controlled reference conditions and practical dynamic positioning conditions, providing a sufficient basis for evaluating the accuracy, availability, continuity, and robustness of multi-service SSR fusion.

2.6. Compared Schemes

To demonstrate the necessity and effectiveness of the proposed multi-service fusion method, three solution schemes were considered.

The first scheme is MADOCA-only PPP, in which only MADOCA-PPP corrections are used for user-side PPP estimation. This scheme serves as one standalone reference and reflects the performance of the QZSS-based open correction service under the adopted processing strategy.

The second scheme is PPP-B2b-only PPP, in which only PPP-B2b corrections are used. This scheme provides the second standalone reference and reflects the positioning characteristics of the BDS-3-based open augmentation service under the same user-side PPP model.

The third scheme is the full proposed fusion scheme, namely the bias-state-aware structured covariance intersection framework described in Section 2. In this scheme, MADOCA-PPP and PPP-B2b corrections are first harmonized in the correction domain and then fused in a consistency-preserving manner before being injected into the PPP filter.

By comparing these schemes, the experiments are designed not only to show whether the fused solution is more accurate than the standalone solutions, but also to evaluate whether the proposed framework improves service continuity, geometry support, and robustness under correction interruption.

2.7. PPP Processing Strategy

To ensure that the comparison among different correction schemes is fair, all schemes were processed under a unified PPP strategy unless otherwise stated. An ionosphere-free multi-frequency PPP model was adopted for the user-side estimator, with consistent observation screening, tropospheric modeling, ambiguity treatment, and antenna correction settings across all experiments.

The basic observation model follows the framework introduced in Section 2. Dual- or multi-frequency observations from the supported constellations G, R, E, J and C were used whenever valid measurements and corresponding correction information were available. The cut-off elevation angle, observation interval, ambiguity treatment strategy, and atmospheric parameter estimation settings were kept consistent across the compared schemes. In this way, differences in positioning performance can be attributed primarily to the correction source and fusion strategy, rather than to changes in the PPP filter configuration.

In the fused solution, service-native corrections were first aligned in time and representation, and then the fused orbit and clock corrections were injected into the PPP filter. When a satellite was supported by only one service, the corresponding correction was retained in a single-service mode rather than discarded, so that the geometry benefit of non-overlapping constellations could be preserved.

For the dynamic experiments, the same PPP strategy was used, except that the observation interval and evaluation window followed the characteristics of the dynamic dataset. This unified processing configuration ensures that the static-station and dynamic results remain comparable at the methodological level. The detailed process strategies are shown in

Table 3. PPP process strategy.

Item	Process Strategy
GNSS system	MADOCA-PPP: GPS, GLONASS, Galileo, QZSS; PPP-B2b: GPS, BDS-3; fusion: all correction-supported satellites from both services.
Observation combinations	Ionosphere-free dual-frequency PPP
Observation Frequency	GPS/QZSS/GLONASS: L1 + L2; Galileo: E5a + E5b; BDS: B1C + B2a
Interval	Stations: 30s; offshore data: 1s
Atmosphere delay	Ionosphere: first-order ionospheric delay eliminated by ionosphere-free combination; troposphere: Saastamoinen model with zenith tropospheric delay estimated
Orbit and clock	SSR-derived orbit and clock corrections from MADOCA-PPP, PPP-B2b, or fused SSR products
Code/signal bias	SSR code-bias corrections applied
Ambiguity	Estimated as float
Phase Center Offset (PCO)/Phase Center Variation (PCV)	Igs20.atx

.

2.8. Evaluation Metrics

To comprehensively evaluate the performance of the proposed method, multiple metrics were adopted from the perspectives of positioning accuracy, convergence, geometry, continuity, and robustness.

The positioning accuracy was assessed using the root-mean-square (RMS) errors in the east, north, and up components, as well as the corresponding 2D and 3D RMS statistics. In addition, percentile-based metrics such as the 95th percentile of 3D error can also be reported to reflect tail behavior and occasional degradation more clearly than RMS alone.

The convergence performance was evaluated by the time required for the PPP solution first to satisfy a predefined accuracy threshold and remain within that threshold for a continuous time interval. Following the convention used in the original study, horizontal and vertical thresholds can be jointly defined to characterize practical convergence performance in real-time PPP. This metric is particularly important for comparing the impact of different correction schemes on initialization and post-recovery behavior.

To reflect the geometric contribution of the fused solution, the number of valid satellites and the positional dilution of precision (PDOP) were also analyzed. These metrics are useful for showing whether multi-service fusion improves the observation geometry even when the best single-service accuracy is already strong in nominal periods.

2.9. Correction-Outage Simulation Design

To verify the robustness of the proposed fusion method under practical service disturbances, controlled correction-outage experiments were designed. In these experiments, one correction stream was intentionally interrupted for a specified time window, while the other stream remained available. This setup was used to simulate temporary signal blockage, decoding failure, message interruption, or service degradation in practical real-time PPP applications.

Specifically, outage scenarios were designed for both MADOCA-PPP-only interruption and PPP-B2b-only interruption. During the outage interval, the standalone solutions relying on the interrupted service naturally experienced correction loss, while the fused solution was expected to continue operating by relying on the remaining available service and the proposed consistency-preserving fusion mechanism. After service restoration, the post-outage error evolution and re-convergence time were analyzed for all compared schemes.

Different outage durations can be considered to evaluate the sensitivity of the proposed method to short and medium interruption events. Through this design, the experiments directly address a key motivation of this study, namely that the value of combining MADOCA-PPP and PPP-B2b is not limited to nominal positioning accuracy, but also includes improved continuity and resilience against temporary single-service failure.

3. Results

3.1. Baseline Positioning Performance

3.1.1. Correction-Domain Assessment Against a Precise Reference Product

To characterize the two services at the correction level, the satellite orbit and clock corrections recovered from MADOCA-PPP and PPP-B2b were assessed against the WUM multi-GNSS final products (WUM0MGXFIN) for DOY 341, 2024. For each epoch, the broadcast-plus-SSR satellite position and clock were compared with the precise product interpolated at the same signal-transmission time (10-point Lagrange for the 5 min orbits, linear for the 30 s clocks), which avoids the receiver-clock-induced along-track artifact incurred when the nominal epoch is used. Reference frames were handled per service: MADOCA-PPP is referenced to the ionosphere-free antenna phase center (APC), so the WUM center-of-mass (CoM) orbit was converted to the IF APC using the IGS20 phase-center offsets (L1/L2 for GPS, B1I/B3I for BDS); PPP-B2b, as recovered here, is in the CoM frame and was compared with the WUM CoM orbit directly. Orbit errors were projected onto the radial/along-track/cross-track (R/A/C) axes. Clock errors, after removing the relativistic periodic term, were single-differenced against a continuously tracked reference satellite of the same system to eliminate the inter-product datum, and the residual standard deviation (STD) is reported after constant-bias removal; a 3σ iterative edit discarded stale-correction epochs (1–6% of samples). Results are given in

Table 4. Orbit (R/A/C/3D, cm) and clock (STD, ns) accuracy of the recovered SSR corrections versus WUM-FIN at WUH2 (DOY 341, 2024).

Service/Type	Radial	Along-Track	Cross-Track	3D	Clock STD (ns)
MADOCA-PPP, GPS	3.1	9.2	2.5	10.1	0.09
PPP-B2b, GPS	6.8	28.2	18.2	34.3	0.23
PPP-B2b, BDS-3 (all)	7.6	25.6	19.0	32.7	0.15
BDS-3 (MEO)	6.5	23.6	19.0	31.0
BDS-3 (IGSO)	11.8	32.9	18.8	39.7

.

3.1.2. Static-Station PPP at WUH2

At the WUH2 station on DOY 341, 2024, the three correction schemes were evaluated under an identical processing configuration, with the static-antenna observations processed epoch-by-epoch (pseudo-kinematic) and the SINEX coordinates taken as reference; the steady-state RMS errors are summarized in

Table 5. Positioning performance at WUH2.

Solution	RMS E (m)	RMS N (m)	RMS U (m)	RMS2D (m)	RMS3D (m)	Mean NS	PDOP
MADOCA-PPP	0.029	0.022	0.055	0.036	0.066	21.98	1.10
PPP-B2b	0.066	0.043	0.147	0.079	0.167	14.98	1.48
Proposed fusion	0.029	0.015	0.060	0.033	0.068	26.56	0.97

. The proposed fusion achieved the best horizontal accuracy of the three schemes (east/north/2-D RMS of 0.029/0.015/0.033 m), outperforming MADOCA-PPP (0.029/0.022/0.037 m) and PPP-B2b (0.066/0.043/0.079 m; the least accurate, 3D RMS 0.167 m)—a genuine combination rather than a selection of one source. After correction-domain fusion, the mean number of valid satellites rose from 21.98 (MADOCA-PPP) and 14.98 (PPP-B2b) to 26.56, and the geometry strengthened accordingly, the mean PDOP decreasing from 1.10 and 1.48 to 0.97. In the 3D sense, the fused solution (0.068 m) was essentially on par with MADOCA-PPP (0.066 m), the marginal difference (+0.002 m) arising entirely from a slightly larger vertical component (0.060 m vs. 0.055 m).

This residual is concentrated in the vertical for well-understood reasons. The vertical (up) component of GNSS PPP is typically two to three times less accurate than the horizontal components, because satellites are observed only above the horizon, giving a weaker vertical geometry (VDOP > HDOP) in which the receiver height and clock are strongly correlated [32,33]. Moreover, residual satellite orbit and clock errors reach the user as a line-of-sight range error dominated by the radial orbit component and the clock offset—the signal-in-space range error [34]—and the radial orbit error is preferentially assimilated into the satellite clock and the user height rather than the horizontal components [35]; residual orbit/clock errors therefore manifest primarily in the vertical. At WUH2, a strong open-sky station where MADOCA-PPP alone is already near-optimal, the satellites added by fusion are the BDS-3 satellites corrected by PPP-B2b, whose corrections are quantifiably noisier than MADOCA-PPP on GPS (3D orbit RMS ~31 cm and clock STD ~0.15 ns versus ~10 cm and ~0.09 ns); their inclusion thus provides geometric redundancy together with a marginally higher correction-noise level that is absorbed predominantly in the vertical. The slightly larger fused 3D RMS is therefore the expected behavior of constellation augmentation at a station where a single service is already near-optimal, not a deficiency of the fusion algorithm—as confirmed by the dynamic offshore experiment, where the same augmentation yields the best accuracy among all schemes.

3.1.3. Kinematic PPP of 0709 Offshore Dataset

The offshore dataset represents the dynamic scenario in this study, with a continuously moving platform, time-varying satellite visibility, and fluctuating observation quality. Because real-time maritime users cannot discard the convergence transient, the positioning accuracy is reported as the full-window RMS against the WUM forward/backward-smoothed reference trajectory. The east/north/up, horizontal (2-D) and three-dimensional (3D) RMS errors, together with the mean number of valid satellites and the mean PDOP, are summarized in

Table 6. Positioning performance of 0709 offshore dataset.

Solution	RMS E (m)	RMS N (m)	RMS U (m)	RMS2D (m)	RMS3D (m)	Mean NS	PDOP
MADOCA-PPP	0.234	0.132	0.195	0.269	0.332	21.4	0.93
PPP-B2b	0.236	0.084	0.188	0.251	0.313	16.9	0.87
Proposed fusion	0.206	0.106	0.173	0.232	0.290	25.4	0.67

; the corresponding ENU error time series are shown in Figure 4.

In contrast to the strong inland station, where MADOCA-PPP alone is already near-optimal, the two services are of comparable accuracy in the offshore scenario (MADOCA-PPP 3D RMS 0.332 m; PPP-B2b 0.313 m), and the fused solution achieves the best overall accuracy of the three schemes, with a horizontal RMS of 0.232 m and a 3D RMS of 0.290 m. This corresponds to a 3D RMS reduction of 12.7% relative to MADOCA-PPP and 7.3% relative to PPP-B2b. The fusion is the most accurate in the east (0.206 m), up (0.173 m), and in the 2-D and 3D components, and second only to PPP-B2b in the north; the lowest vertical error is particularly notable given that the vertical is the component most sensitive to orbit/clock correction errors. At the same time, the fusion tracks the largest mean number of valid satellites (25.4, versus 21.4 for MADOCA-PPP and 16.9 for PPP-B2b) and yields the lowest mean PDOP (0.67, versus 0.93 and 0.87); the corresponding satellite availability over the entire period is compared in Figure 5, confirming that the additional satellites strengthen the geometry rather than merely increasing the count.

These results demonstrate that, when the two correction sources are of comparable quality and contribute complementary constellations (G/R/E/J from MADOCA-PPP and G/C from PPP-B2b), the consistency-preserving information-weighted fusion produces a genuine accuracy gain over either standalone service—not a selection of one source—while simultaneously improving satellite availability and geometric strength. Together with the static-station result (Section 3.1.2), this shows that the value of the fusion is realized most clearly under challenging dynamic conditions, where neither service is uniformly dominant.

3.1.4. Sensitivity to Reduced Satellite Visibility

To address robustness under partially obstructed and low-elevation-visibility conditions, a controlled occlusion experiment was performed on the offshore dataset by progressively raising the cut-off elevation angle from 10° to 40°, which removes low-elevation satellites and emulates increasingly obstructed sky conditions. All three schemes were evaluated against the same precise reference trajectory (computed with full visibility). The results are summarized in

Table 7. Positioning performance under elevation-mask occlusion (offshore vs. precise reference).

Cut-Off Elev.	MADOCA-PPP 3D (m)/NS	PPP-B2b 3D (m)/NS	Fusion 3D (m)/NS
10°	0.332/21	0.313/16	0.290/25
20°	0.386/18	0.502/14	0.339/22
30°	0.478/14	0.659/12	0.417/18
40°	1.579/10	1.403/8	0.493/13

.

Under nominal visibility (10°), the fused solution already provides the best 3D accuracy and the largest satellite count. As the cut-off elevation increases, the two standalone services degrade rapidly: the MADOCA-PPP 3D RMS grows from 0.332 m to 1.579 m and the PPP-B2b from 0.313 m to 1.403 m, while their satellite counts fall to 10 and 8, respectively. In contrast, the fused solution degrades far more gracefully, from 0.290 m to only 0.493 m, retaining 13 satellites at a 40° cut-off. At this severe-occlusion level, the fused solution is about three times more accurate than either standalone service, and all schemes maintain 100% solution availability. This demonstrates that the multi-constellation geometry provided by correction-domain fusion translates directly into robustness under reduced satellite visibility, complementing the single-service outage results of Section 3.2.

3.1.5. Consistency of the Fused Covariance

To verify that the fused solution is statistically consistent—neither optimistic nor overconfident—we evaluate the normalized innovation squared (NIS) of the inter-service innovation on the overlapping GPS satellites, for both the static-station (WUH2) and the dynamic offshore datasets. Under a consistent combination, the NIS follows a chi-square distribution with four degrees of freedom (the dimension of the common-correction subspace), whose 99% bound is 13.28. As summarized in

Table 8. Consistency of the fused solution: NIS statistics of the inter-service innovation on overlapping GPS satellites (4 degrees of freedom; 99% Chi-square bound = 13.28).

Dataset	Epochs	Median	Mean	95th Pct.	99th Pct.	Max
WUH2	22,160	1.12	1.29	3.08	3.88	6.08
Offshore	77,465	1.10	1.30	3.97	3.97	4.24

for WUH2 (n = 22,160 satellite-epochs), the NIS has a median of 1.12, a 95th percentile of 3.08 and a maximum of 6.08; for the offshore kinematic dataset (n = 77,465), the median is 1.10, the 95th percentile 3.33 and the maximum 4.24. In both cases, the NIS lies within the 99% bound in 100% of epochs and sits well inside—indeed somewhat more conservative than—the theoretical chi-square envelope. This confirms that the covariance bound produced by the combination is consistent and does not understate the fused uncertainty across both static and dynamic conditions, the intended behavior of the covariance-intersection design under unknown cross-correlation.

3.2. Outage-Recovery Performance

Controlled single-service outage experiments were conducted to evaluate robustness under practical correction interruption. For both a MADOCA-PPP-only interruption and a PPP-B2b-only interruption, outages of 30 s, 60 s, 120 s, 300 s and 600 s were injected into the 0709 offshore dataset and the standalone and fused solutions were compared in terms of availability, three-dimensional (3D) error during the outage and post-outage re-convergence. The two services exhibit clearly different failure modes, which provides direct evidence that they are complementary rather than redundant.

When the MADOCA-PPP stream was interrupted, the standalone MADOCA-PPP solution was strongly affected: no valid solution was obtained during the 30 s and 60 s outages, and the availability recovered only to 50.0%, 80.0% and 90.0% for the 120 s, 300 s and 600 s outages, with the mean 3D error rising to 1.679 m, 1.850 m and 2.848 m and re-convergence times of 203 s, 230 s and 157 s, respectively (Figure 6,

Table 9. Single-service correction outage results for the 0709 offshore experiment.

Service	Outage (s)	Availability (%)	Mean 3D During (m)	Peak 3D During (m)	Mean 3D Post (m)	Recovery Time (s)
MADOCA-PPP	30	0.0	No solution		0.001	0
MADOCA-PPP	60	0.0	No solution		0.139	300
MADOCA-PPP	120	50.0	1.679	1.974	0.112	203
MADOCA-PPP	300	80.0	1.850	2.725	0.104	230
MADOCA-PPP	600	90.0	2.848	4.352	0.140	157
PPP-B2b	30	100.0	1.994	2.026	0.012	0
PPP-B2b	60	100.0	2.017	2.079	0.023	0
PPP-B2b	120	100.0	2.082	2.192	0.046	0
PPP-B2b	300	100.0	2.290	2.586	0.088	92
PPP-B2b	600	100.0	2.565	3.092	0.214	336

). When the PPP-B2b stream was interrupted, the standalone PPP-B2b solution retained 100% availability but at a degraded accuracy level (mean 3D error 1.994–2.565 m), with re-convergence required only for the longer 300 s and 600 s outages (92 s and 336 s). Thus, MADOCA-PPP offers higher nominal accuracy but is more vulnerable to interruption, whereas PPP-B2b is more continuity-tolerant but less accurate. The re-convergence times and availabilities of the two standalone services across all tested outage durations are summarized in Figure 7, which shows the contrasting failure modes: MADOCA-PPP recovers only after a relatively long re-convergence (up to 230 s) once availability is restored, whereas PPP-B2b retains availability throughout but re-converges more slowly for the longest (600 s) outage (336 s).

The 3-D deviation of the fused solution from its own outage-free baseline is shown in Figure 8: the deviation grows with outage duration but stays bounded (peak below 0.10 m for a MADOCA-PPP outage and below 0.23 m for a PPP-B2b outage) and decays after restoration, while the solution remains 100% available (Figure 8,

Table 10. Outage-induced 3D deviation and availability of the fused solution under single-service interruption (0709 offshore).

Interrupted Service	Outage (s)	Mean 3D During (m)	Peak 3D During (m)	Mean NS During Outage	Availability %
PPP-B2b	30	0.044	≤0.45	16.83	100
	60	0.132		16.7
	120	0.190		18.67
	300	0.277		20.07
	600	0.340		20.16
MADOCA-PPP	30	0.044	≤1.13	12.0	100
	60	0.043		12.0
	120	0.291		14.5
	300	0.539		15.99
	600	0.616		16.07

). When MADOCA-PPP was interrupted, the fused solution continued on the surviving PPP-B2b corrections, with the mean 3D error growing from 0.044 m (30 s) to 0.616 m (600 s) and a peak below 1.13 m. When PPP-B2b was interrupted, the fused solution continued on the more accurate MADOCA-PPP corrections, with the mean 3D error increasing from 0.044 m to 0.340 m and a peak below 0.45 m. In both cases, the fused solution degraded gracefully to the accuracy level of the surviving service, in sharp contrast to the standalone service whose stream was interrupted and which lost its solution entirely during the short outages. This continuity—rather than any accuracy gain during the outage itself—is the central robustness benefit of the fusion: the fused solution never experiences a complete loss, because the enlarged, multi-constellation corrected-satellite set always retains a consistently corrected sub-constellation.

4. Discussion

The results clarify that the role of PPP-B2b in the proposed framework should not be judged by its standalone positioning accuracy alone. Although PPP-B2b is less accurate than MADOCA-PPP in the tested datasets—an ordering confirmed at the correction level against precise WUM products, where the PPP-B2b GPS orbit/clock errors (3D RMS ~34 cm, ~0.23 ns) and BDS-3 errors (~31 cm, ~0.15 ns) exceed those of MADOCA-PPP on GPS (~10 cm, ~0.09 ns)—it provides an independent broadcast correction source, additional GPS and BDS-3 correction support, and a different outage response. The information-based weighting reflects this accuracy ordering automatically, assigning the larger share to the more accurate service while still forming a genuine blend on the overlapping GPS satellites.

The benefit of the fusion is scenario-dependent. At the strong open-sky WUH2 station, where MADOCA-PPP alone is already near-optimal, the fusion increased the mean number of valid satellites from 21.98 to 26.56 and improved the horizontal RMS to 0.033 m—better than either standalone service (0.037 m and 0.079 m)—while the 3D RMS (0.068 m) remained on par with the best standalone service (0.066 m), the small residual difference confined to the vertical. In the offshore kinematic experiment, the fusion achieved the best overall accuracy among the three schemes (2-D 0.232 m, 3D 0.290 m), reducing the 3D RMS by 12.7% relative to MADOCA-PPP and 7.3% relative to PPP-B2b while raising the mean satellite count to 25.4. Correction-domain fusion therefore improves nominal accuracy where the additional correction information is consistent and strengthens the geometry, but it does not guarantee uniform improvement in every component at an already-optimal station—behavior that is the expected outcome of constellation augmentation rather than a deficiency of the algorithm.

The robustness experiments demonstrate the framework’s main practical value. Under increasing elevation cut-off, the fused solution degraded most slowly and remained the most accurate at every cut-off, outperforming both standalone services by about a factor of three at 40°. Under single-service outages, the standalone services showed complementary failure modes—MADOCA-PPP losing the solution for short interruptions, PPP-B2b retaining continuity at degraded accuracy—whereas the fused solution maintained 100% availability for all 30–600 s outages of either service, degrading gracefully to the surviving service’s accuracy level rather than suffering a complete loss. The statistical consistency of the fused covariance was confirmed by the normalized-innovation-squared test on both datasets (median ≈ 1.1; within the 99% bound in 100% of epochs), indicating that the consistency-preserving design is not overconfident.

The scope of the per-satellite blend deserves emphasis. The covariance-intersection blend operates on satellites corrected by both services, which at present are the GPS satellites; this reflects the current service configuration—MADOCA-PPP correcting GPS/GLONASS/Galileo/QZSS and PPP-B2b correcting GPS/BDS-3—rather than a limitation of the method, which would blend any constellation observed by both services. As the services expand their commonly corrected constellations (MADOCA-PPP with BeiDou support under development, and the PPP-B2b service designed to augment the four core constellations), the shared overlap will grow, and the same consistency-preserving blend will extend to the newly shared constellations without modification. The framework thus provides a theory and method already prepared for the deeper multi-service fusion anticipated as these open SSR services evolve.

The current implementation still has limitations. The fused correction covariance is used to form the blend, to drive the inter-service consistency gate, and to screen satellites, but it is not yet injected as an additive term into the measurement stochastic model; a direct mapping of the fused covariance into the observation weights is a natural extension. The residual inter-service inconsistency, particularly the time-varying clock-datum offset, is presently absorbed on GPS by the bias state and elsewhere by the receiver inter-system bias, and could be modeled more explicitly in multi-datum scenarios. Future work will therefore address the propagation of the fused covariance into the estimator, adaptive correlation bounding, refined constellation- and frequency-dependent bias handling, ambiguity resolution, and tight integration with inertial navigation.

5. Conclusions

This paper investigated the fusion of MADOCA-PPP and PPP-B2b SSR corrections for robust real-time precise point positioning. A bias-state-augmented structured covariance intersection framework was proposed to combine heterogeneous SSR products in the correction domain. The proposed method explicitly considers inter-service discrepancies and unknown cross-correlation, deriving the relative weighting of the two streams from their respective correction information, so that overlapping corrections are fused in a consistency-preserving manner while non-overlapping constellations still contribute to the user-side ionosphere-free PPP solution.

The experimental results show that MADOCA-PPP and PPP-B2b are complementary in satellite coverage and service behavior, but not equivalent in correction accuracy. A correction-domain assessment against precise WUM products confirmed that MADOCA-PPP is markedly more accurate than PPP-B2b on the overlapping GPS satellites, which the information-based weighting reflects automatically. In the WUH2 static-station per-epoch (pseudo-kinematic) experiment, the fusion increased the average number of valid satellites to 26.56 and improved the horizontal RMS to 0.033 m—better than either standalone service (0.037 m and 0.079 m)—while the 3D RMS (0.068 m) remained on par with the best standalone service (MADOCA-PPP, 0.066 m). This demonstrates that the fusion genuinely combines complementary information rather than merely selecting the better source. A more pronounced benefit was observed in the offshore kinematic experiment, where the fusion achieved the best overall horizontal and three-dimensional accuracy among the three schemes, with RMS2D and RMS3D of 0.232 m and 0.290 m. Compared with MADOCA-PPP, the fused solution reduced the 3D RMS error by 12.7% and increased the mean number of valid satellites from 21.40 to 25.4; compared with PPP-B2b, the 3D RMS improvement was 7.3%. Under increasing elevation cut-off angles, the fusion degraded most slowly and remained the most accurate at every cut-off, outperforming both standalone services by about a factor of three at 40°.

The main value of the proposed framework lies in its ability to jointly improve correction-source redundancy, satellite availability, positioning continuity, and robustness, while preserving statistical consistency. A normalized-innovation-squared check confirmed that the fused covariance is consistent and not overconfident across both the static and dynamic datasets. The outage experiments further demonstrated that the fused solution maintained 100.0% positioning availability when either MADOCA-PPP or PPP-B2b was temporarily interrupted; during the tested 30–600 s outages, the mean 3D error degraded to the surviving-source level with no significant divergence. Therefore, the fusion of MADOCA-PPP and PPP-B2b should be understood as a continuity-oriented and consistency-preserving PPP architecture that improves robustness and satellite availability—and can also improve nominal accuracy in favorable scenarios—rather than as a simple accuracy-maximization strategy that always dominates every standalone service.