A Fast Fractional Cycle Bias Estimation Method Based on Matrix Dimensionality Reduction

A Fast Fractional Cycle Bias Estimation Method Based on Matrix Dimensionality Reduction | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Fast Fractional Cycle Bias Estimation Method Based on Matrix Dimensionality Reduction Ke Qi, Shuqiang Xue, Bo Wang, Tong Hu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8907433/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Efficient fractional cycle bias (FCB) estimation is critical for real-time precise point positioning with ambiguity resolution (PPP-AR). However, conventional undifferenced models simultaneously estimate receiver and satellite parameters, causing the dimension of normal equations to expand rapidly in large-scale networks. To address this computational bottleneck, this paper proposes a fast FCB estimation algorithm based on matrix dimensionality reduction (MDR). By leveraging the theoretical equivalence between differenced and undifferenced estimators, the algorithm analytically eliminates receiver parameters from the observation equations. This reduces the unknowns from m + n to m (satellites), drastically decreasing computational complexity without compromising precision. Validation using 305 global IGS stations (GPS/Galileo/BDS) confirms that the MDR method is numerically equivalent to the classical solution within double-precision limits. Computational tests demonstrate an acceleration factor of 2.5-4.0× per epoch and a cumulative time reduction of over 90% for continuous processing. Independent PPP-AR experiments verify significantly improved positioning accuracy and stability compared to float solutions. Furthermore, a maritime buoy experiment in the South China Sea confirms the method's robustness under dynamic offshore conditions. The proposed MDR method significantly enhances efficiency while preserving rigor, making it ideal for large-scale real-time multi-GNSS applications. Geology fractional cycle bias (FCB) precise point positioning (PPP) ambiguity resolution (PPP-AR) matrix dimensionality reduction (MDR) undifferenced FCB estimation. Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8907433","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":593194555,"identity":"2b1c5b4c-1cee-4478-8c59-19bd242f6875","order_by":0,"name":"Ke Qi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVRIie3NMQrCMBSA4ZRCXQJdXxHrFQqFIuhhEoR2KUVwcXAIFNpFcO0xFC8QeaBL3V27O9QLiFFwECTWzSE/CbzhfQkhJtMfBupKfgPfJbZQs92RtM4k9IT1A7EqJ+aB7Eq8Mt8ipcjCIy+ALMZc9E5SS/r0MEMKmEV1o0idcEEzpiU+pAHSAOfRWf1iFcgF0EBPhhdFGPJd9SC3DqSvFvaVjPkGHkR0IN4qnsmrmIRQN/mIHZKwoKmewBF3rXrZd8vp/twux4N1r9YT1WuBPQ9xvu2/E5PJZDJ96g6fd0iIJSATsAAAAABJRU5ErkJggg==","orcid":"","institution":"State Key Laboratory of Physical Oceanography, Institute of Oceanographic Instrumentation, Qilu University of Technology (Shandong Academy of Sciences)","correspondingAuthor":true,"prefix":"","firstName":"Ke","middleName":"","lastName":"Qi","suffix":""},{"id":593199159,"identity":"e6c263d1-d53e-4c24-be29-51ba4eb26562","order_by":1,"name":"Shuqiang Xue","email":"","orcid":"","institution":"State Key Laboratory of Spatial Datum, Chinese Academy of Surveying and Mapping","correspondingAuthor":false,"prefix":"","firstName":"Shuqiang","middleName":"","lastName":"Xue","suffix":""},{"id":593199160,"identity":"9f829844-ffe3-4d3d-aae6-42214b65b3c0","order_by":2,"name":"Bo Wang","email":"","orcid":"","institution":"State Key Laboratory of Physical Oceanography, Institute of Oceanographic Instrumentation, Qilu University of Technology (Shandong Academy of Sciences)","correspondingAuthor":false,"prefix":"","firstName":"Bo","middleName":"","lastName":"Wang","suffix":""},{"id":593199161,"identity":"37b4ce0d-acbb-4d79-8103-d1159d75928d","order_by":3,"name":"Tong Hu","email":"","orcid":"","institution":"State Key Laboratory of Physical Oceanography, Institute of Oceanographic Instrumentation, Qilu University of Technology (Shandong Academy of Sciences)","correspondingAuthor":false,"prefix":"","firstName":"Tong","middleName":"","lastName":"Hu","suffix":""}],"badges":[],"createdAt":"2026-02-18 08:46:54","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8907433/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8907433/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eA Fast Fractional Cycle Bias Estimation Method Based on Matrix Dimensionality Reduction\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Qilu University of Technology","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"fractional cycle bias (FCB), precise point positioning (PPP), ambiguity resolution (PPP-AR), matrix dimensionality reduction (MDR), undifferenced FCB estimation.","lastPublishedDoi":"10.21203/rs.3.rs-8907433/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8907433/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eEfficient fractional cycle bias (FCB) estimation is critical for real-time precise point positioning with ambiguity resolution (PPP-AR). However, conventional undifferenced models simultaneously estimate receiver and satellite parameters, causing the dimension of normal equations to expand rapidly in large-scale networks. To address this computational bottleneck, this paper proposes a fast FCB estimation algorithm based on matrix dimensionality reduction (MDR). By leveraging the theoretical equivalence between differenced and undifferenced estimators, the algorithm analytically eliminates receiver parameters from the observation equations. This reduces the unknowns from \u003cem\u003em\u003c/em\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003en\u003c/em\u003e to \u003cem\u003em\u003c/em\u003e (satellites), drastically decreasing computational complexity without compromising precision. Validation using 305 global IGS stations (GPS/Galileo/BDS) confirms that the MDR method is numerically equivalent to the classical solution within double-precision limits. Computational tests demonstrate an acceleration factor of 2.5-4.0\u0026times; per epoch and a cumulative time reduction of over 90% for continuous processing. Independent PPP-AR experiments verify significantly improved positioning accuracy and stability compared to float solutions. Furthermore, a maritime buoy experiment in the South China Sea confirms the method's robustness under dynamic offshore conditions. 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