[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85048-en":3,"doc-seo-85048-105":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},85048,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","A Non-Decoupled Time-Domain Direct Sampling Method for Inverse Elastic Medium Scattering","Inverse elastic wave scattering focuses on reconstructing unknown inhomogeneities from time-resolved boundary measurements. The work proposes a time-domain direct sampling strategy for locating scatterers from a single incident source, avoiding assumptions on the excitation’s temporal profile. A time-shifted correlation imaging functional replaces traditional P–S wave decomposition via travel-time alignment in the coupled elastic field. Parseval-based analysis with Fourier–Laplace transforms yields a frequency-domain reformulation and asymptotic characterization using modified Bessel functions.","arXiv :2607 .08067v1 [math .NA] 9 Jul 2026  \nA Non-Decoupled Time-Domain Direct Sampling Method for Inverse  \nElastic Medium Scattering  \nLefu Cai∗ Hongjie Li† Xianchao Wang‡  \nAbstract  \nThis work is concerned with an inverse medium problem for elastic waves, in which unknown inhomogeneities are reconstructed from time-resolved boundary measurements. We propose a novel time-domain direct sampling method for locating scatterers from a single incident source, without imposing specific assumptions on the temporal profile of the excitation. In particular, the imaging functional introduces a time-shifted correlation strategy that replaces the traditional P-S wave decomposition with a travel-time alignment mechanism, thereby enabling direct imaging from the coupled elastic wave field. To analyze the proposed time-domain imaging functional, we employ Parseval’s identity for the Fourier–Laplace transform and reformulate the functional in the frequency domain. By exploiting properties of modified Bessel functions, we characterize the asymptotic behavior of the imaging functional and show that it attainsits maximum at the target location, which enables reliable identification of the scatterer. Rigorous theoretical justifications are provided to substantiate the effectiveness of the proposed method. Numerical experiments are also presented to demonstrate its performance and applicability.  \nKeywords: inverse medium problem, elastic waves, time-domain direct sampling method, modified Bessel functions  \n2020 Mathematics Subject Classification:  \n1 Introduction  \nIn this paper, we investigate an inverse medium scattering problem in time-domain elasticity, where the objective is to reconstruct unknown scatterers from boundary measurements of coupled elastic waves. Typically, an incident elastic wave is emitted toward the targets of interest, and an array of receivers is placed on a closed or open measurement boundary located away from the scatterers. The scattered elastic wave fields recorded by these receivers are then used to determine the locations and geometric shapes of the unknown objects. This class of inverse problems arises in numerous scientific and engineering applications, including seismic exploration in geophysics [9], nondestructive evaluation of engineering structures [1], and medical ultrasound elastography [3] .  \nWe next present the mathematical formulation of the inverse medium problem for time-dependent elastic waves. Let D ⊂ Rd , d = 2 , 3, denote the inhomogeneous scatterers with Lam´e parameters (λ1 ,µ 1 ) and density ρ1 (x) . The background medium Rd \\D is characterized by the constants (λ2 ,µ2 ,ρ2 ), and we assume that λ2 /λ 1 = µ2 /µ 1 . Then the global parameters (λ,µ,ρ) admit the representation  \nλ (x) = λ 1 χ (D) + λ2 χ (Rd \\D  ) ,  \nµ (x) = µ1 χ (D) + µ2 χ (Rd \\D  ) , (1 . 1)  \nρ (x) = ρ1 (x)χ (D) + ρ2 χ (Rd \\D  ) .  \n∗ School of Mathematics, Harbin Institute of Technology, Harbin, People’s Republic of China. ([25B312009@stu.hit.edu.cn](25B312009@stu.hit.edu.cn)).  \n†Yau Mathematical Sciences Center, Tsinghua University, Beijing, China. The work of this author was substantially supported by NSFC grant (12401561). ([hongjieli@tsinghua.edu.cn](hongjieli@tsinghua.edu.cn) ; hongjie [li@yeah.net](li@yeah.net)).  \n‡School of Mathematics, Harbin Institute of Technology, Harbin, People’s Republic of China. The work of this author was supported by NSFC grant 12471397 and Heilongjiang Provincial Natural Science Foundation grant YQ2024A003 .([xcwang90@gmail.com](xcwang90@gmail.com)).  \nMoreover, the Lam´e constants in the two regions are assumed to satisfy the following strong convexity condition:  \ni) . µi > 0 and ii) . 3λi + 2µi > 0 ,  \nwith i = 1 , 2. Given a causal incident wave ui , namely ui ≡ 0 for t ≤ 0, the propagation of the elastic scattered wave us(x, t) is governed by the following initial-value problem:  \nLλ,µus(x, t) − ρ(x)∂2~~ ~~t(x2,~~ ~~t) = 􀀒ρ (x) − λλ(x2) ρ2 􀀓 ∂2~~ ~~u∂it(x2,~~ ~~t) , us(x, 0) = ∂tus(","cbCaid6WLDYoaFrA","https://ap.wps.com/l/cbCaid6WLDYoaFrA","pdf",1306508,1,23,"English","en",105,"# 1 Introduction\n## Inverse medium scattering in time-domain elasticity\n## Mathematical formulation and governing elastic wave equation\n## Review of frequency-domain and sampling-type methods","[{\"question\":\"What is the main objective of the proposed method?\",\"answer\":\"To locate scatterers in an inhomogeneous elastic medium by reconstructing unknown support regions from time-resolved boundary measurements.\"},{\"question\":\"How does the method form the imaging functional?\",\"answer\":\"It uses a time-shifted correlation strategy that aligns travel times, avoiding traditional P–S wave decomposition and enabling direct imaging from the coupled elastic wave field.\"},{\"question\":\"What theoretical tools are used to analyze the imaging functional?\",\"answer\":\"Parseval’s identity for the Fourier–Laplace transform is used to reformulate the functional in the frequency domain, and properties of modified Bessel functions characterize its asymptotic behavior and show it peaks at the target location.\"}]",1784200625,58,{"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},"a-non-decoupled-time-domain-direct-sampling-method-for-inverse-elastic-medium-scattering","",{"@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/a-non-decoupled-time-domain-direct-sampling-method-for-inverse-elastic-medium-scattering/85048/",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 is the main objective of the proposed method?","Question",{"text":75,"@type":76},"To locate scatterers in an inhomogeneous elastic medium by reconstructing unknown support regions from time-resolved boundary measurements.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method form the imaging functional?",{"text":80,"@type":76},"It uses a time-shifted correlation strategy that aligns travel times, avoiding traditional P–S wave decomposition and enabling direct imaging from the coupled elastic wave field.",{"name":82,"@type":73,"acceptedAnswer":83},"What theoretical tools are used to analyze the imaging functional?",{"text":84,"@type":76},"Parseval’s identity for the Fourier–Laplace transform is used to reformulate the functional in the frequency domain, and properties of modified Bessel functions characterize its asymptotic behavior and show it peaks at the target 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