{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T19:02:55Z","timestamp":1773342175393,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T00:00:00Z","timestamp":1773273600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100014657","name":"University of Transport and Communications","doi-asserted-by":"publisher","award":["T2026-DT-002"],"award-info":[{"award-number":["T2026-DT-002"]}],"id":[{"id":"10.13039\/501100014657","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Micromachines"],"abstract":"<jats:p>Ultrasonic testing is a prevalent method for non-destructive evaluation of railway rails; however, conventional Time-of-Flight (ToF) approaches applied in practical dry-coupled inspections often rely on simplified assumptions regarding wave propagation velocity and neglect complex waveform characteristics. This paper presents a robust depth estimation framework for surface-breaking cracks that enhances sizing accuracy through effective velocity calibration and Hilbert envelope extraction. Unlike standard methods that assume the free-space speed of sound in air (343 m\/s) for wave propagation within the air-filled gap of a surface-breaking crack, we propose an effective velocity model derived from in situ calibration to account for the boundary layer viscosity and thermal conduction effects within narrow crack geometries. The signal processing chain incorporates spectral analysis, band-pass filtering, and Hilbert Transform-based envelope detection to mitigate noise and resolve phase ambiguities. Experimental validation on steel specimens with controlled defects (0.2\u201310.0 mm) demonstrates that the proposed method achieves an exceptional linear correlation (R2 \u2248 0.9976). The calibrated effective velocity was determined to be 289.3 m\/s, approximately 15.6% lower than the speed of sound in air, confirming the significant influence of confinement effects. Furthermore, excitation parameters were optimized, identifying that high-voltage excitation (\u2265110 V) and a tuned pulse width (\u2248150 ns) are critical for maximizing the signal-to-noise ratio. The results confirm that combining physical model calibration with advanced signal analysis significantly reduces systematic errors, paving the way for portable, high-precision rail inspection systems.<\/jats:p>","DOI":"10.3390\/mi17030346","type":"journal-article","created":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T09:23:03Z","timestamp":1773307383000},"page":"346","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Enhancing Ultrasonic Crack Sizing Accuracy in Rails: The Role of Effective Velocity and Hilbert Envelope Extraction"],"prefix":"10.3390","volume":"17","author":[{"given":"Trung Thanh","family":"Ho","sequence":"first","affiliation":[{"name":"Department of Electronic Engineering, University of Transport and Communications, No. 3, Cau Giay Street, Hanoi 100000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Toan Thanh","family":"Dao","sequence":"additional","affiliation":[{"name":"Department of Electronic Engineering, University of Transport and Communications, No. 3, Cau Giay Street, Hanoi 100000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"405","DOI":"10.47869\/tcsj.76.2025.4","article-title":"Overview of advances and practices in high-speed railway earth structures in China","volume":"76","author":"Tengfei","year":"2025","journal-title":"Transp. Commun. Sci. J."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Xiong, L., Jing, G., Wang, J., Liu, X., and Zhang, Y. (2023). Detection of rail defects using NDT methods. Sensors, 23.","DOI":"10.3390\/s23104627"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"108324","DOI":"10.1016\/j.triboint.2023.108324","article-title":"Experimental study on wheel\u2013rail rolling contact fatigue damage starting from surface defects under various operational conditions","volume":"181","author":"Zhang","year":"2023","journal-title":"Tribol. Int."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Zhao, Y., Liu, Z., Yi, D., Yu, X., Sha, X., Li, L., Sun, H., Zhan, Z., and Li, W.J. (2022). A review on rail defect detection systems based on wireless sensors. Sensors, 22.","DOI":"10.3390\/s22176409"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"117289","DOI":"10.1016\/j.measurement.2025.117289","article-title":"Development of an ultrasonic system for nondestructively detecting flaws in the rail foot of in-service railway","volume":"251","author":"Wang","year":"2025","journal-title":"Measurement"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Wang, Y., Miao, B., Zhang, Y., Huang, Z., and Xu, S. (2025). Review on rail damage detection technologies for high-speed trains. Appl. Sci., 15.","DOI":"10.3390\/app15147725"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"107014","DOI":"10.1016\/j.ultras.2023.107014","article-title":"A review on guided-ultrasonic-wave-based structural health monitoring: From fundamental theory to machine learning techniques","volume":"133","author":"Yang","year":"2023","journal-title":"Ultrasonics"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s10921-025-01190-0","article-title":"A review of ultrasonic testing and evaluation methods with applications in civil NDT\/E","volume":"44","author":"Alqurashi","year":"2025","journal-title":"J. Nondestruct. Eval."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"106922","DOI":"10.1016\/j.measurement.2019.106922","article-title":"Rail foot flaw detection based on a laser induced ultrasonic guided wave method","volume":"148","author":"Pathak","year":"2019","journal-title":"Measurement"},{"key":"ref_10","first-page":"437","article-title":"Ultrasonic detection of transversal cracks in rail heads\u2014Theoretical approach","volume":"69","author":"Katz","year":"2021","journal-title":"Eng. Trans."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Lian, Y., Du, F., Xie, L., Qi, X., Jin, P., Wang, Y., and Lu, Z. (2024). Simulation and experimental research of V-crack testing of rail surfaces based on laser ultrasound. Photonics, 11.","DOI":"10.3390\/photonics11100920"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"166732","DOI":"10.1016\/j.ijleo.2021.166732","article-title":"Visual quantitative detection of rail surface crack based on laser ultrasonic technology","volume":"237","author":"Jiang","year":"2021","journal-title":"Optik"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"103026","DOI":"10.1016\/j.ndteint.2023.103026","article-title":"Guided waves in long range nondestructive testing and structural health monitoring: Principles, history of applications and prospects","volume":"142","author":"Cawley","year":"2024","journal-title":"NDT E Int."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"102737","DOI":"10.1016\/j.ndteint.2022.102737","article-title":"Ultrasonic imaging of buried defects in rails","volume":"133","author":"Rodriguez","year":"2023","journal-title":"NDT E Int."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Gong, W., Akbar, M.F., Jawad, G.N., Mohamed, M.F.P., and Wahab, M.N.A. (2022). Nondestructive Testing Technologies for Rail Inspection: A Review. Coatings, 12.","DOI":"10.3390\/coatings12111790"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Yang, F., Shi, D., Lo, L.Y., Mao, Q., Zhang, J., and Lam, K.H. (2023). Auto-diagnosis of time-of-flight for ultrasonic signal based on defect peaks tracking model. Remote Sens., 15.","DOI":"10.3390\/rs15030599"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"115467","DOI":"10.1016\/j.sna.2024.115467","article-title":"A review of the state-of-the-art approaches in detecting time-of-flight in room impulse responses","volume":"374","author":"Othmani","year":"2024","journal-title":"Sens. Actuators A Phys."},{"key":"ref_18","first-page":"851","article-title":"A concrete surface measurement system using SRF05 ultrasonic sensor and TOF method","volume":"76","author":"Ho","year":"2025","journal-title":"Transp. Commun. Sci. J."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Rose, J.L. (2014). Ultrasonic Guided Waves in Solid Media, Cambridge University Press.","DOI":"10.1017\/CBO9781107273610"},{"key":"ref_20","unstructured":"Royer, D., and Dieulesaint, E. (2000). Elastic Waves in Solids II: Generation, Acousto-Optic Interaction, Applications, Springer."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"B\u00fchling, B., K\u00fcttenbaum, S., Maack, S., and Strangfeld, C. (2022). Development of an accurate and robust air-coupled ultrasonic time-of-flight measurement technique. Sensors, 22.","DOI":"10.3390\/s22062135"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Tum\u0161ys, O. (2025). A new signal processing method for time-of-flight and center frequency estimation. Appl. Sci., 15.","DOI":"10.3390\/app15105721"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Chiu, Y., Wang, C., Gong, D., Li, N., Ma, S., and Jin, Y. (2021). A novel ultrasonic ToF ranging system using AlN-based PMUTs. Micromachines, 12.","DOI":"10.3390\/mi12030284"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ono, K. (2020). A comprehensive report on ultrasonic attenuation of engineering materials, including metals, ceramics, Polymers, Fiber-Reinforced Composites, Wood, and Rocks. Appl. Sci., 10.","DOI":"10.3390\/app10072230"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Yu, M.G., and Kim, D.S. (2025). Low-complexity ultrasonic flowmeter signal processor using peak detector-based envelope detection. J. Sens. Actuator Netw., 14.","DOI":"10.3390\/jsan14010012"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Wang, J., Zhang, C., Zhao, M., Zou, H., Qi, L., and Wang, Z. (2024). A Composite Pulse Excitation Technique for Air-Coupled Ultrasonic Detection of Defects in Wood. Sensors, 24.","DOI":"10.3390\/s24237550"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1109\/TUFFC.2016.2574399","article-title":"Tissue attenuation estimation by mean frequency downshift and bandwidth limitation","volume":"63","author":"Klimonda","year":"2016","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1109\/TUFFC.2016.2538719","article-title":"Optimum diffraction-corrected frequency-shift estimator of the ultrasonic attenuation coefficient","volume":"63","author":"Samimi","year":"2016","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Xu, W., Bai, X., Sun, Z., Meng, X., and Guo, Z. (2021). Correlation between laser-ultrasound and microstructural properties of laser melting deposited Ti6Al4V\/B4C composites. Metals, 11.","DOI":"10.3390\/met11121951"}],"container-title":["Micromachines"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-666X\/17\/3\/346\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T09:50:20Z","timestamp":1773309020000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-666X\/17\/3\/346"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,12]]},"references-count":29,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["mi17030346"],"URL":"https:\/\/doi.org\/10.3390\/mi17030346","relation":{},"ISSN":["2072-666X"],"issn-type":[{"value":"2072-666X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,12]]}}}