[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82332-en":3,"doc-seo-82332-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82332,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Full Cross-Correlation Inversion for Quantitative Passive Imaging with Time-Harmonic Acoustic Waves","Inverse reconstruction of physical properties in passive imaging uses ambient wavefields to infer a medium, modeled as a superposition of waves from stochastic sources. For time-harmonic acoustic propagation with zero-mean, spatially uncorrelated sources, the expected cross-correlation between two receiver signals links to the deterministic Green’s function and the source covariance. A first-order wave formulation supports correlations across wavefield types, enabling a nonlinear inverse framework solved by iterative minimization. Gradients follow the adjoint-state method, validated by 2D/3D numerical experiments and compared with active-source measurements.","arXiv :2607 .09392v1 [math .NA] 10 Jul 2026  \nFull cross-correlation inversion for quantitative passive imaging with time-harmonic acoustic waves  \nJean Dutheil∗ and Florian Faucher∗  \nJuly 13, 2026  \nAbstract  \nWe consider the inverse problem for the quantitative reconstruction of physical properties in the context of passive imaging, where ambient wavefields are used to infer a medium. The data are modeled as a superposition of waves generated by stochastic sources. In this work, we focus on time-harmonic acoustic wave propagation and assume that the stochastic sources exciting the medium are zero-mean and spatially uncorrelated. Under these assumptions, the expected value of the cross-correlation between signals recorded at two locations can be related to the deterministic Green’s function and the covariance of the source terms. We follow a firstorder formulation of the wave equation, which enables the treatment of correlations between different types of wavefields. A numerical framework is developed for the resulting nonlinear inverse problem. The quantitative reconstruction is carried out using an iterative minimization scheme, in which the gradient of the misfit functional is computed via the adjoint-state method. Numerical experiments in two and three dimensions are performed using synthetic data, and inversions based on the expected value of cross-correlations are compared with those relying on direct wavefield measurements from active-source acquisitions.  \nContents  \n1 Introduction 2  \n2 Time-harmonic cross-correlation of acoustic waves 3  \n2.1 First-order wave equations ................................ 4  \n2.2 Cross-correlation of ambient signals ........................... 5  \n2.2.1 Source covariance ................................. 5  \n2.2.2 Expected value of the cross-correlation ..................... 5  \n2.3 Computation of the cross-correlation .......................... 6  \n3 Hybridizable Discontinuous Galerkin discretization 7  \n3.1 Notations ......................................... 7  \n3.2 Hybridizable Discontinuous Galerkin discretization .................. 8  \n3.3 Computation of the components of the cross-correlation ............... 10  \n4 Quantitative nonlinear inverse problem 11  \n4.1 Iterative minimization .................................. 12  \n4.2 Gradient computation with the adjoint-state method ................. 13  \n4.2.1 Lagrangian function ............................... 13  \n4.2.2 Selection of the adjoint states .......................... 14  \n4.2.3 Expression of the gradients ........................... 16  \n4.3 Minimization strategy .................................. 17  \n∗ Project-Team Makutu, Inria Bordeaux, University of Pau and Pays de l’Adour, Pau, France.  \n5 Numerical experiments 17  \n5.1 Experiment 1: Reconstruction of contrasting objects with full data ......... 18  \n5.2 Experiment 2: Seismic configuration with partial data ................ 20  \n5.2.1 Wave speed reconstruction ........................... 20  \n5.2.2 Source covariance reconstruction ........................ 21  \n5.3 Experiment 3: 3D test-case with high-contrast inclusions ............... 22  \n6 Conclusion 23  \n1 Introduction  \nIn passive imaging, background ambient signals are exploited to probe a medium of interest. Passive imaging is in particular used in asteroseismology and planetary seismology, where continuous vibrations of the bodies are recorded, for example in helioseismology [28] and terrestrial seismology [38] . For instance, in helioseismology, the observed solar oscillations are stochastically driven by turbulent convective motions [11, 28], and the resulting signals are used to probe the solar interior [28, 6 , 42 , 37] . On Earth, ambient noise can result from microseismic activity (and other natural or anthropogenic sources) [38] . Ambient signals provide vast amounts of data because they only require sensors to monitor background vibrations. Consequently, passive imaging has seen a growing number of ","cbCaifXUWJfzbmIj","https://ap.wps.com/l/cbCaifXUWJfzbmIj","pdf",5325927,1,26,"English","en",105,"# Introduction\n# Time-harmonic cross-correlation of acoustic waves\n## First-order wave equations\n## Cross-correlation of ambient signals\n# Hybridizable Discontinuous Galerkin discretization\n# Quantitative nonlinear inverse problem\n## Iterative minimization\n## Gradient computation with the adjoint-state method\n# Numerical experiments\n## Experiment 1\n## Experiment 2\n## Experiment 3\n# Conclusion","[{\"question\":\"What inverse problem does the paper address in passive imaging?\",\"answer\":\"The paper addresses reconstructing physical properties of a medium using ambient wavefields by formulating an inverse problem driven by stochastic sources and their recorded signals.\"},{\"question\":\"How is the expected cross-correlation connected to deterministic quantities?\",\"answer\":\"Under zero-mean, spatially uncorrelated source assumptions for time-harmonic acoustic waves, the expected cross-correlation between two receiver signals is related to the deterministic Green’s function and the covariance of the source terms.\"},{\"question\":\"How are gradients of the misfit functional computed during inversion?\",\"answer\":\"The inversion uses an iterative minimization scheme where the gradient of the misfit functional is computed via the adjoint-state method.\"}]",1784179701,66,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"full-cross-correlation-inversion-for-quantitative-passive-imaging-with-time-harmonic-acoustic-waves","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/full-cross-correlation-inversion-for-quantitative-passive-imaging-with-time-harmonic-acoustic-waves/82332/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What inverse problem does the paper address in passive imaging?","Question",{"text":75,"@type":76},"The paper addresses reconstructing physical properties of a medium using ambient wavefields by formulating an inverse problem driven by stochastic sources and their recorded signals.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the expected cross-correlation connected to deterministic quantities?",{"text":80,"@type":76},"Under zero-mean, spatially uncorrelated source assumptions for time-harmonic acoustic waves, the expected cross-correlation between two receiver signals is related to the deterministic Green’s function and the covariance of the source terms.",{"name":82,"@type":73,"acceptedAnswer":83},"How are gradients of the misfit functional computed during inversion?",{"text":84,"@type":76},"The inversion uses an iterative minimization scheme where the gradient of the misfit functional is computed via the adjoint-state 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