{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T00:53:04Z","timestamp":1777942384203,"version":"3.51.4"},"reference-count":39,"publisher":"SAGE Publications","issue":"5","license":[{"start":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T00:00:00Z","timestamp":1722988800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"funder":[{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["306526\/2019-0"],"award-info":[{"award-number":["306526\/2019-0"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002322","name":"Coordena\u00e7\u00e3o de Aperfeioamento de Pessoal de N\u00edvel Superior","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002322","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["22\/12575-7"],"award-info":[{"award-number":["22\/12575-7"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["19\/19684-3"],"award-info":[{"award-number":["19\/19684-3"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Structural Health Monitoring"],"published-print":{"date-parts":[[2025,9]]},"abstract":"<jats:p>Computational models are crucial in applied science and engineering, offering valuable insights on the behavior of strutures and mechanical systems. However, their effectiveness is often hindered by complexity and substantial time required for execution. Metamodeling, or surrogate modeling, is a practical strategy to optimize computational time and resources. This approach involves substituting a complex model with a metamodel, that is, a simplified function that mimics the behavior of the original model, thereby significantly expediting the evaluation process. One widely utilized method is Gaussian Process Regression (GPR), also known as Kriging, which has demonstrated effectiveness in numerous structural health monitoring (SHM) applications. However, achieving accurate predictions for a target variable (e.g., damage-sensitive feature) often requires a significant amount of past data or well-calibrated models of the structure under analysis, presenting challenges and high costs. Therefore, the innovation presented in this article is applying a co-Kriging method, a multivariate extension of ordinary Kriging that leverages the covariance between two or more related datasets. This is an efficient decision-making process in various fields, especially when the co-variable is more cost-effective to measure than the target variable. Three distinct applications are presented here, showcasing the efficacy of the co-Kriging methodology. Two of these applications focus on generic mathematical functions. The third pertains to a real-world scenario involving the correlation of the natural frequencies of a concrete bridge under varying thermal conditions. Across all three scenarios, co-Kriging emerges as a robust method, consistently yielding superior results compared to ordinary Kriging.<\/jats:p>","DOI":"10.1177\/14759217241265375","type":"journal-article","created":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T01:55:07Z","timestamp":1723082107000},"page":"2927-2940","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Co-Kriging strategy for structural health monitoring of bridges"],"prefix":"10.1177","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9218-6858","authenticated-orcid":false,"given":"Henrique Cordeiro","family":"Novais","sequence":"first","affiliation":[{"name":"Departamento de Engenharia Mecnica, UNESP\u2014Universidade Estadual Paulista, Ilha Solteira, SP, Brasil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6430-3746","authenticated-orcid":false,"given":"Samuel","family":"da Silva","sequence":"additional","affiliation":[{"name":"Departamento de Engenharia Mecnica, UNESP\u2014Universidade Estadual Paulista, Ilha Solteira, SP, Brasil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9168-6903","authenticated-orcid":false,"given":"Eloi","family":"Figueiredo","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, Lusfona University, Lisboa, Portugal"},{"name":"CERIS, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Lisboa, Portugal"}]}],"member":"179","published-online":{"date-parts":[[2024,8,7]]},"reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/9780470740170"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1177\/14759217221075241"},{"key":"e_1_3_2_4_2","volume-title":"Structural health monitoring: a machine learning perspective","author":"Farrar CR","year":"2013","unstructured":"Farrar CR, Worden K. Structural health monitoring: a machine learning perspective. Hoboken, NJ: John Wiley & Sons, Ltd, 2013."},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1177\/1077546320966183"},{"key":"e_1_3_2_6_2","doi-asserted-by":"publisher","DOI":"10.1177\/14759217211007956"},{"key":"e_1_3_2_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.rineng.2021.100316"},{"key":"e_1_3_2_8_2","doi-asserted-by":"publisher","DOI":"10.1177\/13694332211038444"},{"key":"e_1_3_2_9_2","first-page":"1705","volume-title":"14th International symposium on process systems engineering, computer aided chemical engineering","volume":"49","author":"Franzoi RE","year":"2022","unstructured":"Franzoi RE, Menezes BC, Kelly JD, et al. Adaptive least-squares surrogate modeling for reaction systems. In: Yamashita Y, Kano M (eds.) 14th International symposium on process systems engineering, computer aided chemical engineering, vol. 49. Elsevier, 2022, pp. 1705\u20131710."},{"key":"e_1_3_2_10_2","doi-asserted-by":"publisher","DOI":"10.1615\/Int.J.UncertaintyQuantification.2014006914"},{"key":"e_1_3_2_11_2","volume-title":"Vibration based on inspections of civil engineering structures","author":"Rytter A.","year":"1993","unstructured":"Rytter A. Vibration based on inspections of civil engineering structures. PhD Thesis, Department of Building Technology and Structural Engineering. Aalborg University, Denmark, 1993."},{"key":"e_1_3_2_12_2","unstructured":"Fuentes R Cross E Halfpenny A et al. Autoregressive gaussian processes for structural damage detection. In: Proceedings of the international conference on noise and vibration engineering 26. Leuven: KU Leuven\u2014Department of Mechanical Engineering pp. 469\u2013483. http:\/\/past.isma-isaac.be\/downloads\/isma2014\/papers\/isma2014_0711.pdf"},{"key":"e_1_3_2_13_2","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)BE.1943-5592.0001949"},{"key":"e_1_3_2_14_2","doi-asserted-by":"publisher","DOI":"10.1007\/s13349-015-0118-7"},{"key":"e_1_3_2_15_2","doi-asserted-by":"publisher","DOI":"10.1061\/(ASCE)BE.1943-5592.0001432"},{"key":"e_1_3_2_16_2","doi-asserted-by":"publisher","DOI":"10.1177\/14759217221098998"},{"key":"e_1_3_2_17_2","doi-asserted-by":"publisher","DOI":"10.1002\/9780470770801"},{"key":"e_1_3_2_18_2","first-page":"120","article-title":"Extended co-kriging interpolation method based on multi-fidelity data","volume":"323","author":"Xiao M","year":"2018","unstructured":"Xiao M, Zhang G, Breitkopf P, et al. Extended co-kriging interpolation method based on multi-fidelity data. Appl Math Comput 2018; 323: 120\u2013131.","journal-title":"Appl Math Comput"},{"key":"e_1_3_2_19_2","doi-asserted-by":"publisher","DOI":"10.1615\/Int.J.UncertaintyQuantification.2014006914"},{"key":"e_1_3_2_20_2","volume-title":"A statistical approach to some mine valuation and allied problems on the Witwatersrand","author":"Krige DG.","year":"1951","unstructured":"Krige DG. A statistical approach to some mine valuation and allied problems on the Witwatersrand. Master\u2019s Thesis, University of the Witwatersrand, South Africa, 1951."},{"issue":"2","key":"e_1_3_2_21_2","first-page":"12461266","article-title":"Principles of geostatistics","volume":"58","author":"Matheron G.","year":"1963","unstructured":"Matheron G. Principles of geostatistics. Econo Geol 1963; 58(2): 12461266.","journal-title":"Econo Geol"},{"key":"e_1_3_2_22_2","first-page":"409435","article-title":"Design and analysis of computer experiments","volume":"4","author":"Sacks J","year":"1989","unstructured":"Sacks J, Welch WJ, Mitchell TJ, et al. Design and analysis of computer experiments. Statist Sci 1989; 4: 409435.","journal-title":"Statist Sci"},{"key":"e_1_3_2_23_2","volume-title":"Gaussian processes for machine learning","author":"Rasmussen C","year":"2006","unstructured":"Rasmussen C, Williams C. Gaussian processes for machine learning. Adaptive Computation and Machine Learning, Cambridge, MA: MIT Press, 2006."},{"key":"e_1_3_2_24_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3799-8"},{"key":"e_1_3_2_25_2","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2007.1900"},{"key":"e_1_3_2_26_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-91641-5_18"},{"key":"e_1_3_2_27_2","volume-title":"Interpolation of spatial data: a theoretical approach to Kriging","author":"Stein ML.","year":"2012","unstructured":"Stein ML. Interpolation of spatial data: a theoretical approach to Kriging. Heidelberg: Springer Science & Business Media, 2012."},{"key":"e_1_3_2_28_2","volume-title":"UQLab user manual\u2014Kriging (Gaussian process modeling). Technical report, Chair of Risk, Safety and Uncertainty Quantification, Report UQLab-V2.0-105","author":"Lataniotis C","year":"2022","unstructured":"Lataniotis C, Wicaksono D, Marelli S, et al. UQLab user manual\u2014Kriging (Gaussian process modeling). Technical report, Chair of Risk, Safety and Uncertainty Quantification, Report UQLab-V2.0-105. Switzerland: ETH Zurich, 2022."},{"key":"e_1_3_2_29_2","volume-title":"UQLab user manual \u2014 Kriging (Gaussian process modeling)","author":"Lataniotis C","year":"2024","unstructured":"Lataniotis C, Wicaksono D, Marelli S, et al. UQLab user manual \u2014 Kriging (Gaussian process modeling). Technical report, Chair of Risk, Safety and Uncertainty Quantification. Report UQLab-V2.1-105. Switzerland: ETH Zurich, 2024."},{"key":"e_1_3_2_30_2","first-page":"2554","volume-title":"Proceedings of the 2nd international conference on vulnerability and risk analysis and management\u2014ICVRAM","author":"Marelli S","unstructured":"Marelli S, Sudret B. UQLab: a framework for uncertainty quantification in Matlab. In: Proceedings of the 2nd international conference on vulnerability and risk analysis and management\u2014ICVRAM. Liverpool: University of Liverpool, pp. 2554\u20132563."},{"key":"e_1_3_2_31_2","first-page":"2554","volume-title":"UQLab: a framework for uncertainty quantification in Matlab, chapter 257","author":"Marelli S","year":"2014","unstructured":"Marelli S, Sudret B. UQLab: a framework for uncertainty quantification in Matlab, chapter 257. Reston, VA: American Society of Civil Engineers, 2014. pp. 2554\u20132563."},{"key":"e_1_3_2_32_2","doi-asserted-by":"publisher","DOI":"10.2307\/2532147"},{"key":"e_1_3_2_33_2","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/87.1.1"},{"key":"e_1_3_2_34_2","doi-asserted-by":"publisher","DOI":"10.1006\/mssp.2002.1548"},{"key":"e_1_3_2_35_2","doi-asserted-by":"publisher","DOI":"10.1002\/1096-9845(200102)30:2<149::AID-EQE1>3.0.CO;2-Z"},{"key":"e_1_3_2_36_2","doi-asserted-by":"publisher","DOI":"10.1002\/9780470061626.shm165"},{"key":"e_1_3_2_37_2","doi-asserted-by":"publisher","DOI":"10.1006\/mssp.2002.1548"},{"key":"e_1_3_2_38_2","doi-asserted-by":"publisher","DOI":"10.1177\/1475921710388971"},{"key":"e_1_3_2_39_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.patrec.2005.10.010"},{"key":"e_1_3_2_40_2","doi-asserted-by":"publisher","DOI":"10.1186\/s12880-015-0068-x"}],"container-title":["Structural Health Monitoring"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/14759217241265375","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.1177\/14759217241265375","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/14759217241265375","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:48:20Z","timestamp":1777704500000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/14759217241265375"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,7]]},"references-count":39,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["10.1177\/14759217241265375"],"URL":"https:\/\/doi.org\/10.1177\/14759217241265375","relation":{},"ISSN":["1475-9217","1741-3168"],"issn-type":[{"value":"1475-9217","type":"print"},{"value":"1741-3168","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,7]]}}}