{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T17:09:05Z","timestamp":1785431345409,"version":"3.56.0"},"reference-count":25,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T00:00:00Z","timestamp":1781740800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"State Administration for Market Regulation","award":["2022MK038"],"award-info":[{"award-number":["2022MK038"]}]},{"award":["2022MK038"],"award-info":[{"award-number":["2022MK038"]}],"id":[{"id":"https:\/\/ror.org\/00aaqtw69","id-type":"ROR","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100016355","name":"Xuzhou Science and Technology Bureau","doi-asserted-by":"publisher","award":["KC22310"],"award-info":[{"award-number":["KC22310"]}],"id":[{"id":"10.13039\/501100016355","id-type":"DOI","asserted-by":"publisher"}]},{"award":["KC22310"],"award-info":[{"award-number":["KC22310"]}],"id":[{"id":"https:\/\/ror.org\/016s74507","id-type":"ROR","asserted-by":"publisher"}]},{"name":"the General Program of Basic Science (Natural Science) Research in Higher Education Institutions of Jiangsu Province","award":["23KJD170007"],"award-info":[{"award-number":["23KJD170007"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>Ultrasonic inspection of welded steel components remains challenging due to weld-scale material gradients, local anisotropy, attenuation, and aperture limitations. These factors severely distort both the first-arrival wavefield and the late-arriving scattered wavefield. To address this, this study presents a numerical proof of concept for three-dimensional cross-weld virtual-array coda-wave tomography (VACWT). The \u201cvirtual array\u201d utilizes a synthetic aperture created by re-indexing sequential source\u2013receiver records from two opposing line scans into midpoint\u2013angle\u2013depth coordinates. This approach enables line-based data acquisition to achieve multi-angle volumetric coverage without requiring a two-dimensional matrix array. A parameterized welded-solid benchmark model was developed, incorporating effective longitudinal and shear wave velocities, attenuation, and out-of-plane tilt fields. Four defect scenarios were evaluated: a cylindrical void, a lack-of-fusion ribbon, a porosity cluster, and an interference case. For each source\u2013receiver path, four observables were extracted from the synthetic records: first-arrival travel time perturbations, coda wave stretching, coda decorrelation, and late-window energy ratios. These observables were then coupled into a volumetric inverse problem to separate smooth slowness variations, distributed scattering strength, and compact defect occupancy. Under the current simulation conditions, VACWT achieved smaller recovered support volumes and higher volumetric overlap compared to the delay-and-sum total focusing method (DAS-TFM), background-corrected TFM, and reverse time migration (RTM). In the interference case, applying a fixed defect-free calibration threshold yielded a centroid error of 0.48 mm, a volumetric intersection over union (IoU) of 0.856, and a false-positive volume fraction of 0.0%. While these findings serve as benchmark results rather than generalized experimental validation, the study demonstrates that late scattered wave observables provide valuable constraints for volumetric support recovery in a controlled welded-solid model. Future experimental verification on welded steel specimens with known defects remains necessary.<\/jats:p>","DOI":"10.3390\/ma19122633","type":"journal-article","created":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T09:18:43Z","timestamp":1781860723000},"page":"2633","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Numerical Study of Cross-Weld Virtual-Array Coda-Wave Tomography for Volumetric Imaging of Weld Defects in Steel Plates"],"prefix":"10.3390","volume":"19","author":[{"given":"Guiwu","family":"Chen","sequence":"first","affiliation":[{"name":"School of Civil Engineering, Xuzhou University of Technology, Xuzhou 221018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yan","family":"Li","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Xuzhou University of Technology, Xuzhou 221018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3953-8065","authenticated-orcid":false,"given":"Shaolei","family":"Song","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Xuzhou University of Technology, Xuzhou 221018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hao","family":"Wang","sequence":"additional","affiliation":[{"name":"Xuzhou Inspection and Testing Center, Xuzhou 221000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuxun","family":"Zhang","sequence":"additional","affiliation":[{"name":"Xuzhou Inspection and Testing Center, Xuzhou 221000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,6,18]]},"reference":[{"key":"ref_1","first-page":"677","article-title":"The post-processing of ultrasonic array data using the total focusing method","volume":"46","author":"Holmes","year":"2004","journal-title":"Insight\u2014Non-Destr. Test. Cond. Monit."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1016\/j.ndteint.2005.04.002","article-title":"Post-processing of the full matrix of ultrasonic transmit-receive array data for non-destructive evaluation","volume":"38","author":"Holmes","year":"2005","journal-title":"NDT E Int."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.ndteint.2015.12.009","article-title":"Ultrasonic array imaging through an anisotropic austenitic steel weld using an efficient ray-tracing algorithm","volume":"79","author":"Nowers","year":"2016","journal-title":"NDT E Int."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"M\u00e9nard, C., Dominguez, N., Richard, H., Lesselier, D., and Moreau, L. (2020). Optimization algorithms for ultrasonic array imaging in homogeneous anisotropic steel components with unknown properties. NDT E Int., 115.","DOI":"10.1016\/j.ndteint.2020.102327"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1840","DOI":"10.1080\/17415977.2020.1762596","article-title":"Effective grain orientation mapping of complex and locally anisotropic media","volume":"28","author":"Tant","year":"2020","journal-title":"Inverse Probl. Sci. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Kalkowski, M.K., Velichko, A., Wilcox, P.D., and Croxford, A.J. (2023). Weld-map tomography for determining local grain orientations in inhomogeneous and anisotropic welds. Proc. R. Soc. A, 479.","DOI":"10.1098\/rspa.2023.0236"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kalkowski, M.K., Velichko, A., Croxford, A.J., and Wilcox, P.D. (2023). Supporting imaging of austenitic welds with finite element simulations: A case study on model complexity. Appl. Sci., 13.","DOI":"10.3390\/app13137448"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Nicolson, E., Fan, Y., Mineo, C., Croxford, A.J., Drinkwater, B.W., and Wilcox, P.D. (2023). Towards an in-process ultrasonic phased array inspection of narrow-gap girth welds using a dual-tandem configuration. NDT E Int., 136.","DOI":"10.1016\/j.ndteint.2024.103074"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Gao, S., Zhang, R., Fan, Z., Li, N., and Xie, L. (2024). Online monitoring of fatigue damage in welded joints using diffuse ultrasound. Ultrasonics, 138.","DOI":"10.1016\/j.ultras.2023.107191"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Yang, X., Wang, K., Xu, Y., Xu, L., Hu, W., Wang, H., and Su, Z. (2020). A reverse time migration-based multistep angular spectrum approach for ultrasonic imaging of specimens with irregular surfaces. Ultrasonics, 108.","DOI":"10.1016\/j.ultras.2020.106233"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"B\u00fcrchner, T., Schmid, S., Bergbreiter, L., Rank, E., Kollmannsberger, S., and Grosse, C.U. (2025). Quantitative comparison of the total focusing method, reverse time migration, and full waveform inversion for ultrasonic imaging. Ultrasonics, 149.","DOI":"10.1016\/j.ultras.2025.107705"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Xu, X., Liu, Y., Li, X., Xiang, Y., and Xuan, F.-Z. (2025). Ultrasonic internal multiple migration: An imaging method for defect characterization in complex-structure welds. Mech. Syst. Signal Process., 231.","DOI":"10.1016\/j.ymssp.2025.112640"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Yin, X., Fan, Z., and Edwards, R.S. (2022). Hybrid geometrical full waveform inversion for ultrasonic defect characterization using phased arrays. J. Sound Vib., 542.","DOI":"10.1016\/j.jsv.2022.117099"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kumar, A., Balasubramaniam, K., and Krishnamurthy, C.V. (2020). Phased array ultrasonic imaging using angle beam virtual source full matrix capture-total focusing method. NDT E Int., 116.","DOI":"10.1016\/j.ndteint.2020.102324"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.ultras.2019.02.011","article-title":"Inspection of butt welds for complex surface parts using ultrasonic phased array","volume":"96","author":"Li","year":"2019","journal-title":"Ultrasonics"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Zhu, T., Zhou, Z., Zhou, W., and Ahmad, H.E. (2025). Precise internal geometric characterization of multilayer structures using ultrasonic array imaging technology. NDT E Int., 156.","DOI":"10.1016\/j.ndteint.2025.103468"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Yu, B., Jin, H., Mei, Y., Chen, J., Wu, E., and Yang, K. (2022). 3-D ultrasonic image reconstruction in frequency domain using a virtual transducer model. Ultrasonics, 118.","DOI":"10.1016\/j.ultras.2021.106573"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Wang, Z., Xiao, J., Li, D., Li, B., Zhang, J., and Ta, D. (2023). Full waveform inversion guided wave tomography with a recurrent neural network. Ultrasonics, 133.","DOI":"10.1016\/j.ultras.2023.107043"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2253","DOI":"10.1126\/science.1070015","article-title":"Coda wave interferometry for estimating nonlinear behavior in seismic velocity","volume":"295","author":"Snieder","year":"2002","journal-title":"Science"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Snieder, R., and Gr\u00eat, A. (2002). Coda wave interferometry and the equilibration of energy in elastic media. Phys. Rev. E, 66.","DOI":"10.1103\/PhysRevE.66.046615"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Lobkis, O.I., and Weaver, R.L. (2003). Coda-wave interferometry in finite solids: Recovery of P-to-S conversion rates in an elastodynamic billiard. Phys. Rev. Lett., 90.","DOI":"10.1103\/PhysRevLett.90.254302"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/j.cemconres.2013.07.009","article-title":"A review of ultrasonic coda wave interferometry in concrete","volume":"53","author":"Larose","year":"2013","journal-title":"Cem. Concr. Res."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Niederleithinger, E., Wang, X., Herbrand, M., and M\u00fcller, M. (2018). Processing ultrasonic data by coda wave interferometry to monitor load tests of concrete beams. Sensors, 18.","DOI":"10.3390\/s18061971"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Kim, J.Y. (2025). Velocity of ultrasound diffusion in heterogeneous media and its applications. Ultrasonics, 145.","DOI":"10.1016\/j.ultras.2024.107476"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Treeby, B.E., and Cox, B.T. (2010). k-Wave: MATLAB toolbox for the simulation and reconstruction of photoacoustic wave fields. J. Biomed. Opt., 15.","DOI":"10.1117\/1.3360308"}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/19\/12\/2633\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T04:48:01Z","timestamp":1782362881000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/19\/12\/2633"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,18]]},"references-count":25,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2026,6]]}},"alternative-id":["ma19122633"],"URL":"https:\/\/doi.org\/10.3390\/ma19122633","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,18]]}}}