{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:35:10Z","timestamp":1760146510459,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2024,11,8]],"date-time":"2024-11-08T00:00:00Z","timestamp":1731024000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62203349","201806280305"],"award-info":[{"award-number":["62203349","201806280305"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004543","name":"China Scholarship Council","doi-asserted-by":"publisher","award":["62203349","201806280305"],"award-info":[{"award-number":["62203349","201806280305"]}],"id":[{"id":"10.13039\/501100004543","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>This paper considers the problem of tracking a three-dimensional target under the condition that only a single two-dimensional radar is available. Since a two-dimensional radar can only measure the slant range and azimuth information relative to the target, an unobservability issue arises in this tracking application. Therefore, we first investigate the observability issue of tracking a three-dimensional target with a single two-dimensional radar from two perspectives, including intuitive illustration and quantitative analysis. From the perspective of intuitive illustration, we demonstrate \u201cWhat is the unobservability issue\u201d and \u201cHow does the relative target-radar geometry influence the observability of the tracking system\u201d. From the perspective of quantitative analysis, we construct a novel observability metric for this special tracking problem. Second, aiming at improving tracking performance under the unobservability of target height, we propose an observability-based Gaussian sum cubature Kalman filter. Built within the Gaussian sum framework and based on the height-parameterized strategy, this novel algorithm uses a set of independent fifth-degree cubature Kalman filters, each of which can detect the system observability variation and enhance the tracking accuracy by using a Gaussian splitting scheme under low-degree observability. Finally, the effectiveness of the presented filtering algorithm is validated through lots of simulation experiments.<\/jats:p>","DOI":"10.3390\/rs16224172","type":"journal-article","created":{"date-parts":[[2024,11,8]],"date-time":"2024-11-08T06:05:41Z","timestamp":1731045941000},"page":"4172","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Observability-Based Gaussian Sum Cubature Kalman Filter for Three-Dimensional Target Tracking Using a Single Two-Dimensional Radar"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5758-4372","authenticated-orcid":false,"given":"Haonan","family":"Jiang","sequence":"first","affiliation":[{"name":"Faculty of Electronic and Information Engineering, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yingjie","family":"Zhang","sequence":"additional","affiliation":[{"name":"Beijing Institute of Radio Measurement, Beijing 100854, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7340-9681","authenticated-orcid":false,"given":"Xiaotong","family":"Wang","sequence":"additional","affiliation":[{"name":"Faculty of Electronic and Information Engineering, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7364-3101","authenticated-orcid":false,"given":"Yuanli","family":"Cai","sequence":"additional","affiliation":[{"name":"Faculty of Electronic and Information Engineering, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,8]]},"reference":[{"key":"ref_1","unstructured":"Gai, M., Yi, X., He, Y., and Shi, B. (2005, January 9\u201312). An approach to tracking a 3D-target with 2D-radar. Proceedings of the IEEE International Radar Conference, Arlington, VA, USA."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1109\/TAES.2017.2761138","article-title":"Three-dimensional tracking of an aircraft using two-dimensional radars","volume":"54","author":"Mallick","year":"2017","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"107241","DOI":"10.1016\/j.sigpro.2019.107241","article-title":"3-D tracking of air targets using a single 2-D radar","volume":"166","author":"Li","year":"2020","journal-title":"Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1109\/TSMC.2020.3012485","article-title":"Observability metrics for single-target tracking with bearings-only measurements","volume":"52","author":"Jiang","year":"2022","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1109\/TAC.1977.1101601","article-title":"Nonlinear controllability and observability","volume":"22","author":"Hermann","year":"1977","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Yin, Z., Yang, J., Ma, Y., Wang, S., Chai, D., and Cui, H. (2023). A Robust Adaptive Extended Kalman Filter Based on an Improved Measurement Noise Covariance Matrix for the Monitoring and Isolation of Abnormal Disturbances in GNSS\/INS Vehicle Navigation. Remote Sens., 15.","DOI":"10.3390\/rs15174125"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Gao, N., Chen, X., Yan, Z., and Jiao, Z. (2024). Performance Enhancement and Evaluation of a Vector Tracking Receiver Using Adaptive Tracking Loops. Remote Sens., 16.","DOI":"10.3390\/rs16111836"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Cui, B., Chen, W., Weng, D., Wei, X., Sun, Z., Zhao, Y., and Liu, Y. (2023). Observability-Constrained Resampling-Free Cubature Kalman Filter for GNSS\/INS with Measurement Outliers. Remote Sens., 15.","DOI":"10.3390\/rs15184591"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1016\/j.automatica.2012.11.014","article-title":"High-degree cubature Kalman filter","volume":"49","author":"Jia","year":"2013","journal-title":"Automatica"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2589","DOI":"10.1109\/TSP.2007.914964","article-title":"Square-root quadrature Kalman filtering","volume":"56","author":"Arasaratnam","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wu, Q., Chen, L., Li, Y., Wang, Z., Yao, S., and Li, H. (2022). Reweighted robust particle filtering approach for target tracking in automotive radar application. Remote Sens., 14.","DOI":"10.3390\/rs14215477"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Huang, Z., Chen, S., Hao, C., and Orlando, D. (2021). Bearings-only target tracking with an unbiased pseudo-linear Kalman filter. Remote Sens., 13.","DOI":"10.3390\/rs13152915"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/s40815-020-00956-0","article-title":"Maximum fuzzy correntropy Kalman filter and its application to bearings-only maneuvering target tracking","volume":"23","author":"Li","year":"2021","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"78","DOI":"10.2514\/1.G002696","article-title":"Multiple maneuvering target tracking using a single unmanned aerial vehicle","volume":"42","author":"Zhang","year":"2019","journal-title":"J. Guid. Control Dyn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1109\/TAES.2009.5089543","article-title":"Multi-target\/multi-sensor tracking using only range and doppler measurements","volume":"45","author":"Deming","year":"2009","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"8504","DOI":"10.1109\/JSEN.2023.3243039","article-title":"Event-triggered distributed bias-compensated pseudolinear information filter for bearings-only tracking under measurement uncertainty","volume":"23","author":"Jiang","year":"2023","journal-title":"IEEE Sens. J."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Aoki, E.H. (2010, January 26\u201329). A general approach for altitude estimation and mitigation of slant range errors on target tracking using 2D radars. Proceedings of the 13th International Conference on Information Fusion, Edinburgh, UK.","DOI":"10.1109\/ICIF.2010.5711932"},{"key":"ref_18","unstructured":"Rakvongthai, Y., Ru, J., Sivananthan, S., and Oraintara, S. (2011, January 5\u20138). Altitude estimation for 3-D tracking with two 2-D radars. Proceedings of the 14th International Conference on Information Fusion, Chicago, IL, USA."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1109\/7.570737","article-title":"Discrete-time observability and estimability analysis for bearings-only target motion analysis","volume":"33","author":"Jauffret","year":"1997","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1290","DOI":"10.1109\/7.543850","article-title":"Observability in passive target motion analysis","volume":"32","author":"Jauffret","year":"1996","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1109\/TAES.1985.310617","article-title":"Observability requirements for three dimensional tracking via angle measurements","volume":"21","author":"Hammel","year":"1985","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1049\/ip-cta:19951614","article-title":"Bearings-only tracking using a set of range-parameterised extended Kalman filters","volume":"142","author":"Peach","year":"1995","journal-title":"IEE Proc. Control Theory Appl."},{"key":"ref_23","unstructured":"Luo, Z., and He, J. (2007, January 9\u201312). ML estimation of true height in 2-D radar network. Proceedings of the 10th International Conference on Information Fusion, Quebec, QC, Canada."},{"key":"ref_24","first-page":"408","article-title":"Technology of height estimation in distributed 2D radar network","volume":"39","author":"Xiong","year":"2010","journal-title":"Inf. Control"},{"key":"ref_25","first-page":"1055","article-title":"A 3D Target Tracking Algorithm in Distributed 2D Radar Network","volume":"36","author":"Yan","year":"2014","journal-title":"J. Electron. Inf. Technol."},{"key":"ref_26","first-page":"285","article-title":"BLUE Tracking Algorithm with Predicted Slant Range and Altitude","volume":"35","author":"Sheng","year":"2019","journal-title":"J. Signal Process."},{"key":"ref_27","first-page":"801","article-title":"Overview of deterministic sampling filtering algorithms for nonlinear system","volume":"27","author":"Wang","year":"2012","journal-title":"Control Decis."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.1109\/TAES.2014.130136","article-title":"Random-point-based filters: Analysis and comparison in target tracking","volume":"51","author":"Dunik","year":"2015","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1431","DOI":"10.1109\/TAES.2014.120563","article-title":"Decorrelated unbiased converted measurement Kalman filter","volume":"50","author":"Bordonaro","year":"2014","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Leutnant, V., Krueger, A., and Haeb-Umbach, R. (2011, January 27\u201331). A versatile Gaussian splitting approach to non-linear state estimation and its application to noise-robust ASR. Proceedings of the 12th Annual Conference on the International Speech Communication Association, Florence, Italy.","DOI":"10.21437\/Interspeech.2011-208"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1386","DOI":"10.1109\/78.668800","article-title":"Posterior Cram\u00e9r-Rao bounds for discrete-time nonlinear filtering","volume":"46","author":"Muravchik","year":"1998","journal-title":"IEEE Trans. Signal Process."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/16\/22\/4172\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:28:48Z","timestamp":1760113728000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/16\/22\/4172"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,8]]},"references-count":31,"journal-issue":{"issue":"22","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["rs16224172"],"URL":"https:\/\/doi.org\/10.3390\/rs16224172","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2024,11,8]]}}}