{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:08:40Z","timestamp":1760144920194,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,5,29]],"date-time":"2024-05-29T00:00:00Z","timestamp":1716940800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"ONR","award":["73-6C95025","DMS-1923221"],"award-info":[{"award-number":["73-6C95025","DMS-1923221"]}]},{"name":"NSF","award":["73-6C95025","DMS-1923221"],"award-info":[{"award-number":["73-6C95025","DMS-1923221"]}]},{"name":"ONR Summer Faculty Fellowship Program","award":["73-6C95025","DMS-1923221"],"award-info":[{"award-number":["73-6C95025","DMS-1923221"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The recently deployed Surface Water and Ocean Topography (SWOT) mission for the first time has observed the ocean surface at a spatial resolution of 1 km, thus giving an opportunity to directly monitor submesoscale sea surface height (SSH) variations that have a typical magnitude of a few centimeters. This progress comes at the expense of the necessity to take into account numerous uncertainties in calibration of the quality-controlled altimeter data. Of particular importance is the proper filtering of spatially correlated errors caused by the uncertainties in geometry and orientation of the on-board interferometer. These \u201csystematic\u201d errors dominate the SWOT error budget and are likely to have a notable signature in the SSH products available to the oceanographic community. In this study, we explore the utility of the block-circulant (BC) approximation of the SWOT precision matrix developed by the Jet Propulsion Laboratory for assessment of a mission\u2019s accuracy, including the possible impact of the systematic errors on the assimilation of the wide-swath altimeter data into numerical models. It is found that BC approximation of the precision matrix has sufficient (90\u201399%) accuracy for a wide range of significant wave heights of the ocean surface, and, therefore, could potentially serve as an efficient preconditioner for data assimilation problems involving altimetry observations by space-borne interferometers. An extensive set of variational data assimilation (DA) experiments demonstrates that BC approximation provides more accurate SSH retrievals compared to approximations, assuming a spatially uncorrelated observation error field as is currently adopted in operational DA systems.<\/jats:p>","DOI":"10.3390\/rs16111954","type":"journal-article","created":{"date-parts":[[2024,5,30]],"date-time":"2024-05-30T03:45:08Z","timestamp":1717040708000},"page":"1954","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Block-Circulant Approximation of the Precision Matrix for Assimilating SWOT Altimetry Data"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3280-5490","authenticated-orcid":false,"given":"Max","family":"Yaremchuk","sequence":"first","affiliation":[{"name":"US Naval Research Laboratory, Stennis Space Center, MS 39522, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3302-4845","authenticated-orcid":false,"given":"Christopher","family":"Beattie","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA"}]},{"given":"Gleb","family":"Panteleev","sequence":"additional","affiliation":[{"name":"US Naval Research Laboratory, Stennis Space Center, MS 39522, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2234-8425","authenticated-orcid":false,"given":"Joseph","family":"D\u2019Addezio","sequence":"additional","affiliation":[{"name":"US Naval Research Laboratory, Stennis Space Center, MS 39522, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1109\/JPROC.2010.2043031","article-title":"The surface water and ocean topography mission: Observing terrestrial surface water and oceanic submesoscale eddies","volume":"98","author":"Durand","year":"2010","journal-title":"Proc. IEEE"},{"key":"ref_2","unstructured":"Esteban-Fernandez, D. (2024, April 10). SWOT Project: Mission Performance and Error Budget. Revision A, NASA. \/JPL Tech. Rep. JPL D-79084, Available online: http:\/\/swot.jpl.nasa.gov\/files\/SWOT_D-79084_v5h6_SDT.pdf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"e2023GL107652","DOI":"10.1029\/2023GL107652","article-title":"The surface water and ocean topography mission: A breakthrough in radar remote sensing of the ocean and land surface water","volume":"51","author":"Fu","year":"2023","journal-title":"Geophys. Res. Lett."},{"key":"ref_4","first-page":"OS34B-05","article-title":"A Japanese new altimetry mission COMPIRA\u2014Towards high temporal and spatial sampling of sea surface height","volume":"2014","author":"Ito","year":"2014","journal-title":"Agu Fall Meet. Abstr."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"194","DOI":"10.3389\/fmars.2019.00194","article-title":"Concept design of the Guanlan science mission: China\u2019s novel contribution to space oceanography","volume":"6","author":"Chen","year":"2019","journal-title":"Front. Mar. Sci."},{"key":"ref_6","unstructured":"(2024, May 26). SWOT Project: Product Description, Algorithm Theoretical Basis and Data, Level 2 KaRIn Low Rate Sea Surface Height Data Product, Version 1.1, 2023, Available online: https:\/\/podaac.jpl.nasa.gov\/dataset\/SWOT_L2_LR_SSH_1.1."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Dibarboure, G., Ubelmann, C., Flamant, B., Briol, F., Peral, E., Bracher, G., Vergara, O., Faug\u00e8re, Y., Soulat, F., and Picot, N. (2022). Data-driven calibration algorithm and pre-launch performance simulations for the SWOT mission. Remote Sens., 14.","DOI":"10.3390\/rs14236070"},{"key":"ref_8","unstructured":"Gaultier, L., Ubelmann, C., and Fu, L.-L. (2017). SWOT Simulator Documentation, CalTech. Tech. Rep. 2.3.0, Jet Propulsion Laboratory."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1175\/JTECH-D-15-0160.1","article-title":"The challenge of using future SWOT data for oceanic field reconstruction","volume":"33","author":"Gaultier","year":"2016","journal-title":"J. Atm. Oceanic Tech."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ma, C., Guo, X., Zhang, H., Di, J., and Chen, G. (2020). An investigation of the influences of SWOT sampling and errors on ocean eddy observation. Remote Sens., 12.","DOI":"10.3390\/rs12172682"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4838","DOI":"10.1029\/2018JC014869","article-title":"An observing system simulation experiment for ocean state estimation to assess the performance of the SWOT mission: Part 1\u2014A twin experiment","volume":"124","author":"Li","year":"2019","journal-title":"J. Geophys. Res. Oceans"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1175\/JTECH-D-17-0076.1","article-title":"An observing system simulation experiment for the calibration and validation of the SWOT sea surface height measurement using in situ platforms","volume":"35","author":"Wang","year":"2018","journal-title":"J. Ocean Atm. Tech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1701","DOI":"10.5194\/os-17-1791-2021","article-title":"Assimilating realistically simulated wide-swath altimeter observations in a high-resolution shelf-seas forecasting system","volume":"17","author":"King","year":"2021","journal-title":"Ocean Sci."},{"key":"ref_14","unstructured":"Gaultier, L., and Ubelmann, C. (2024, March 13). SWOT Science Ocean Simulator Open Source Repository. Available online: https:\/\/github.com\/SWOTsimulator\/swotsimulator."},{"key":"ref_15","unstructured":"(2024, May 26). SWOT Project: Release Note KaRIn Science Data Products, Version C, Available online: https:\/\/podaac.jpl.nasa.gov\/dataset\/SWOT_L2_LR_SSH_2.0."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1175\/JTECH-D-17-0061.1","article-title":"A cross-spectral approach to measure the error budget of the SWOT altimetry mission over the ocean","volume":"35","author":"Ubelmann","year":"2018","journal-title":"J. Ocean Atm. Tech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2119","DOI":"10.5194\/gmd-16-2119-2023","article-title":"4DVarNet-SSH: End-to-end learning of variational interpolation schemes for nadir and wide-swath satellite altimetry","volume":"16","author":"Beauchamp","year":"2023","journal-title":"Geosci. Model Dev."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Tr\u00e9boutte, A., Carli, E., Ballarotta, M., Carpentier, B., Faug\u00e8re, Y., and Dibarboure, G. (2023). KaRIn noise reduction using a convolutional neural network for the SWOT ocean products. Remote Sens., 15.","DOI":"10.3390\/rs15082183"},{"key":"ref_19","unstructured":"(2024, May 26). SWOT Project: Release Note KaRIn Science Data Products, Version C, Available online: https:\/\/podaac.jpl.nasa.gov\/dataset\/SWOT_L2_LR_SSH_EXPERT_2.0."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1111\/j.1600-0870.2006.00148.x","article-title":"Development of NAVDAS-AR: Non-linear formulation and outer loop nests","volume":"58A","author":"Rosmond","year":"2006","journal-title":"Tellus"},{"key":"ref_21","unstructured":"Fletcher, S.J. (2022). Data Assimilation Methods for the Geosciences, Elsevier."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Park, S.K., and Xu, L. (2013). Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications Vol. II, Springer.","DOI":"10.1007\/978-3-642-35088-7"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Saad, Y. (2003). Iterative Methods for Sparse Linear Systems, SIAM Press. Available online: https:\/\/epubs.siam.org\/doi\/book\/10.1137\/1.9780898718003.","DOI":"10.1137\/1.9780898718003"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2755","DOI":"10.1175\/JTECH-D-16-0048.1","article-title":"An efficient way to account for observation error correlations in the assimilation of data from the future SWOT high-resolution altimeter mission","volume":"33","author":"Ruggiero","year":"2016","journal-title":"J. Ocean Atm. Tech."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Yaremchuk, M., D\u2019Addezio, J., and Jacobs, G. (2020). Facilitating inversion of the error covariance models for the wide-swath altimeters. Remote Sens., 12.","DOI":"10.3390\/rs12111823"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"822","DOI":"10.3389\/fmars.2019.00822","article-title":"Wide-swath altimetric satellite data assimilation with correlated error reduction","volume":"6","author":"Metref","year":"2020","journal-title":"Front. Mar. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Yaremchuk, M. (2022). Sparse approximation of the precision matrices for the wide-swath altimeters. Remote Sens., 14.","DOI":"10.3390\/rs14122827"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4277","DOI":"10.3390\/rs15174277","article-title":"The effect of spatially correlated errors on sea surface height retrieval from SWOT altimetry","volume":"15","author":"Yaremchuk","year":"2023","journal-title":"Remote Sens."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1446","DOI":"10.1175\/JTECH-D-13-00179.1","article-title":"The effect of atmospheric water vapor content on the performance of future wide-swath ocean altimetry measurement","volume":"31","author":"Ubelmann","year":"2014","journal-title":"J. Atmos. Ocean Technol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5222011","DOI":"10.1109\/TGRS.2023.3334493","article-title":"The effect of differential tropospheric error on the measurement of wide-swath interferometric altimetry","volume":"61","author":"Gao","year":"2023","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3804","DOI":"10.1175\/MWR-D-14-00384.1","article-title":"A multi-scale variational data assimilation scheme: Formulation and illustration","volume":"143","author":"Li","year":"2015","journal-title":"Mon. Wea. Rev."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s10596-019-09839-2","article-title":"A multiscale method for data assimilation","volume":"24","author":"Hajibeygi","year":"2020","journal-title":"Comput. Geosci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1109\/TAP.1983.1143132","article-title":"The inverse of a block-circulant matrix","volume":"31","author":"Gerlic","year":"1983","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/j.ocemod.2005.01.004","article-title":"2006: Formulation, implementation and examination of vertical coordinate choices in the Global Navy Coastal Ocean Model (NCOM)","volume":"11","author":"Barron","year":"2006","journal-title":"Ocean. Model."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Helber, R.W., Smith, S.R., Panteleev, G., Shriver, J., and Pickard, R. (Deep Sea Res., 2023). Greenland Freshwater Stability in the East Greenland Current, Deep Sea Res., in press.","DOI":"10.1016\/j.dsr2.2024.105402"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"116","DOI":"10.5670\/oceanog.2014.73","article-title":"The Navy Global Environmental Model","volume":"27","author":"Hogan","year":"2014","journal-title":"Oceanography"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"13","DOI":"10.5670\/oceanog.2002.32","article-title":"Operational altimeter sea level products","volume":"15","author":"Jacobs","year":"2002","journal-title":"Oceanography"},{"key":"ref_38","unstructured":"Jacobs, G., Desai, S., D\u2019Addezio, J., and Bartels, B. (Geophys. Res. Lett., 2024). SWOT cross-track error characteristics estimated from observations, Geophys. Res. Lett., under review."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"101683","DOI":"10.1016\/j.ocemod.2020.101683","article-title":"Multi-scale assimilation of simulated SWOT observations","volume":"154","author":"Souopgui","year":"2020","journal-title":"Ocean. Model."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/16\/11\/1954\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:50:18Z","timestamp":1760107818000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/16\/11\/1954"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,29]]},"references-count":39,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["rs16111954"],"URL":"https:\/\/doi.org\/10.3390\/rs16111954","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2024,5,29]]}}}