Scale-dependent truncated Lévy modeling of Gamma-ray log increments for characterizing stratigraphic heterogeneity in IODP well logs

Scale-dependent truncated Lévy modeling of Gamma-ray log increments for characterizing stratigraphic heterogeneity in IODP well logs | 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 Scale-dependent truncated Lévy modeling of Gamma-ray log increments for characterizing stratigraphic heterogeneity in IODP well logs Zhicheng Lei This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9337867/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 7 You are reading this latest preprint version Abstract Gamma-ray (GR) well logs record lithologic variability across multiple scales, but their increment statistics are often analyzed under stationary Lévy assumptions. In this study, we introduce a scale-dependent framework based on the gradually truncated Lévy flight model to characterize nonstationary increment behavior in 23 International Ocean Discovery Program (IODP) wells. This approach treats the stability index \(\:\alpha\:\) and truncation scale \(\:{l}_{c}\) as functions of lag scale, combining Lévy-stable core distributions with exponential truncation of large deviations. A robust estimation pipeline is developed that integrates core-distribution fitting, tail-sensitive truncation detection, and profile-likelihood optimization. Mean squared displacement (MSD) analysis reveals a crossover from super-diffusive (β > 1) to sub-diffusive (β < 1) regimes at scales of approximately 0.8–1.2 m, reflecting changes in dominant stratigraphic controls. α and \(\:{l}_{c}\) exhibit scale-dependent evolution, from interface-dominated increments at small scales to bounded variability in larger stratigraphic units. The proposed framework enhances stochastic modeling by extending traditional Gaussian assumptions and provides a versatile tool for quantifying anomalous diffusion and stratigraphic heterogeneity in well-log data. The Python implementation is open-source, ensuring reproducibility and flexibility in broader geological contexts. Truncated Lévy processes Nonstationary stochastic modeling Multiscale analysis Stratigraphic heterogeneity Full Text Additional Declarations No competing interests reported. Supplementary Files TableS1.csv Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 12 May, 2026 Reviews received at journal 11 May, 2026 Reviewers agreed at journal 21 Apr, 2026 Reviewers invited by journal 10 Apr, 2026 Editor assigned by journal 09 Apr, 2026 Submission checks completed at journal 08 Apr, 2026 First submitted to journal 06 Apr, 2026 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. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-9337867","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":620809562,"identity":"43f44195-a5c6-43b1-afe7-9f5fe6dd4b31","order_by":0,"name":"Zhicheng Lei","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVRIie3PMQrCMBTG8VcK7fJoHRMqeoVIQRwCXsUiZHJw7BgJOOUGgmdpCejSA1Scai/QURex6Cpt3RzyX97yfsMHYLP9YQE4EiAlGPrthS0A6SPemxR8THUGkLFBpM3ZC87K1VDiK9XcPYPsUp+ihvEJlW51K7sI5nKm0SC9CkEyJuIIvDjedBGSyDUQg8F1M2+JSY6AXtRJppU0wAzCpRhKiLNTsBI4KvFDDr0EE+XojCPVYr0o2i1U9WwJ/XPdPJ5kGfomL9OUT8hZVXUX+ZL727vNZrPZvvQCBoVGU6jDosUAAAAASUVORK5CYII=","orcid":"","institution":"East China University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Zhicheng","middleName":"","lastName":"Lei","suffix":""}],"badges":[],"createdAt":"2026-04-07 00:53:25","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9337867/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9337867/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107483550,"identity":"081fa49b-1040-4f2c-9125-6d90944e4d43","added_by":"auto","created_at":"2026-04-22 02:28:14","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":858974,"visible":true,"origin":"","legend":"","description":"","filename":"Manuscript2.0.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9337867/v1_covered_deabbf07-e973-4733-992e-0c1eebfb3db3.pdf"},{"id":107183220,"identity":"c961035e-4374-4151-bd5e-23ce82558792","added_by":"auto","created_at":"2026-04-17 18:07:50","extension":"csv","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":14952,"visible":true,"origin":"","legend":"","description":"","filename":"TableS1.csv","url":"https://assets-eu.researchsquare.com/files/rs-9337867/v1/023521d8be1ffe4eab0d2cd5.csv"}],"financialInterests":"No competing interests reported.","formattedTitle":"Scale-dependent truncated Lévy modeling of Gamma-ray log increments for characterizing stratigraphic heterogeneity in IODP well logs","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"stochastic-environmental-research-and-risk-assessment","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"serr","sideBox":"Learn more about [Stochastic Environmental Research and Risk Assessment](https://www.springer.com/journal/477)","snPcode":"477","submissionUrl":"https://submission.nature.com/new-submission/477/3","title":"Stochastic Environmental Research and Risk Assessment","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Truncated Lévy processes, Nonstationary stochastic modeling, Multiscale analysis, Stratigraphic heterogeneity","lastPublishedDoi":"10.21203/rs.3.rs-9337867/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9337867/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eGamma-ray (GR) well logs record lithologic variability across multiple scales, but their increment statistics are often analyzed under stationary L\u0026eacute;vy assumptions. In this study, we introduce a scale-dependent framework based on the gradually truncated L\u0026eacute;vy flight model to characterize nonstationary increment behavior in 23 International Ocean Discovery Program (IODP) wells. This approach treats the stability index \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\alpha\\:\\)\u003c/span\u003e\u003c/span\u003e and truncation scale \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{l}_{c}\\)\u003c/span\u003e\u003c/span\u003e as functions of lag scale, combining L\u0026eacute;vy-stable core distributions with exponential truncation of large deviations. A robust estimation pipeline is developed that integrates core-distribution fitting, tail-sensitive truncation detection, and profile-likelihood optimization. Mean squared displacement (MSD) analysis reveals a crossover from super-diffusive (β\u0026thinsp;\u0026gt;\u0026thinsp;1) to sub-diffusive (β\u0026thinsp;\u0026lt;\u0026thinsp;1) regimes at scales of approximately 0.8\u0026ndash;1.2 m, reflecting changes in dominant stratigraphic controls. α and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{l}_{c}\\)\u003c/span\u003e\u003c/span\u003eexhibit scale-dependent evolution, from interface-dominated increments at small scales to bounded variability in larger stratigraphic units. The proposed framework enhances stochastic modeling by extending traditional Gaussian assumptions and provides a versatile tool for quantifying anomalous diffusion and stratigraphic heterogeneity in well-log data. The Python implementation is open-source, ensuring reproducibility and flexibility in broader geological contexts.\u003c/p\u003e","manuscriptTitle":"Scale-dependent truncated Lévy modeling of Gamma-ray log increments for characterizing stratigraphic heterogeneity in IODP well logs","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-17 18:07:47","doi":"10.21203/rs.3.rs-9337867/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-05-12T09:31:06+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-11T14:32:31+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"142030757625262929009848857777761925096","date":"2026-04-21T08:13:15+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-04-10T08:31:44+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-04-09T21:01:19+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-04-08T10:17:48+00:00","index":"","fulltext":""},{"type":"submitted","content":"Stochastic Environmental Research and Risk Assessment","date":"2026-04-07T00:43:15+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"stochastic-environmental-research-and-risk-assessment","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"serr","sideBox":"Learn more about [Stochastic Environmental Research and Risk Assessment](https://www.springer.com/journal/477)","snPcode":"477","submissionUrl":"https://submission.nature.com/new-submission/477/3","title":"Stochastic Environmental Research and Risk Assessment","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"1ec4dd43-d672-4a8e-a489-5e3014988e16","owner":[],"postedDate":"April 17th, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Revision requested","date":"2026-05-12T09:31:06+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-11T14:32:31+00:00","index":18,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[],"tags":[],"updatedAt":"2026-05-12T10:24:16+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-17 18:07:47","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9337867","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9337867","identity":"rs-9337867","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}