{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T13:13:49Z","timestamp":1775913229934,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2022,9,24]],"date-time":"2022-09-24T00:00:00Z","timestamp":1663977600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Space Agency","award":["4000125043\/18\/I-NB"],"award-info":[{"award-number":["4000125043\/18\/I-NB"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The analytical solutions of the radiative transfer equation are needed for the solution of various applied atmospheric and snow optics problems. In this paper, we propose a simple analytical equation for the top-of-atmosphere (TOA) spectral reflectance. To simplify the problem under study we consider the case of Antarctica, where both snow and atmosphere are almost free of pollutants. This work is focused on the simulation of the moderate spectral resolution TOA measurements (1 nm or so) and the spectral range 400\u20131000 nm. The values of the coefficient of variance (CV) between the measured by the Ocean and Land Colour Instrument (OLCI) on board Sentinel-3A and modelled spectra are smaller than 10% for most cases in Antarctica. There are regions in Eastern Antarctica, where the values of CV are smaller than 5%. The areas with larger deviations between measured and retrieved spectra could be due to the presence of clouds or structures on the snow surface not captured by the proposed model.<\/jats:p>","DOI":"10.3390\/rs14194778","type":"journal-article","created":{"date-parts":[[2022,9,26]],"date-time":"2022-09-26T03:34:17Z","timestamp":1664163257000},"page":"4778","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Approximate Analytical Solution for the Top-of-Atmosphere Spectral Reflectance of Atmosphere\u2014Underlying Snow System over Antarctica"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7110-223X","authenticated-orcid":false,"given":"Alexander","family":"Kokhanovsky","sequence":"first","affiliation":[{"name":"Max Planck Institute for Chemistry, Hahn-Meitner-Weg 1, 55128 Mainz, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,24]]},"reference":[{"key":"ref_1","unstructured":"Liou, K.-N. (2002). An Introduction to Atmopsheric Radiation, Academic Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1689","DOI":"10.5194\/gmd-15-1689-2022","article-title":"SSolar-GOA v1.0: A simple, fast, and accurate Spectral solar radiative transfer for clear skies","volume":"15","author":"Cachorro","year":"2022","journal-title":"Geosci. Model Dev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"107270","DOI":"10.1016\/j.jqsrt.2020.107270","article-title":"A fast and accurate radiative transfer model for aerosol remote sensing","volume":"256","author":"Mei","year":"2020","journal-title":"J. Quant. Spectrosc. Radiat. Transf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/j.isprsjprs.2022.04.010","article-title":"A new snow bidirectional reflectance distribution function model in spectral regions from UV to SWIR: Model development and application to ground-based, aircraft and satellite observations","volume":"188","author":"Mei","year":"2022","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kokhanovsky, A.A. (2021). Snow Optics, Springer Nature.","DOI":"10.1007\/978-3-030-86589-4"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jqsrt.2004.12.008","article-title":"Reflection of light from particulate media with irregularly shaped particles","volume":"96","author":"Kokhanovsky","year":"2005","journal-title":"J. Quant. Spectrosc. Radiat. Transf."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kokhanovsky, A., Box, J.E., Vandecrux, B., Mankoff, K.D., Lamare, M., Smirnov, A., and Kern, M. (2020). The determination of snow albedo from satellite measurements using fast atmospheric correction technique. Remote Sens., 12.","DOI":"10.3390\/rs12020234"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1007\/s10812-016-0357-3","article-title":"Accounting for atmospheric effects in the interpretation of satellite and ground-based optical measurements","volume":"83","author":"Malinka","year":"2016","journal-title":"J. Appl. Spectrosc."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Sobolev, V.V. (1975). Light Scattering in Planetary Atmospheres, Pergamon Press.","DOI":"10.1016\/B978-0-08-017934-6.50017-6"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.5194\/amt-3-1403-2010","article-title":"Speeding up the aerosol optical thickness retrieval using analytical solutions of radiative transfer theory","volume":"3","author":"Katsev","year":"2010","journal-title":"Atmos. Meas. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Katkovsky, L.V., Martinov, A.O., Siliuk, V.A., Ivanov, D.A., and Kokhanovsky, A.A. (2018). Fast atmospheric correction method for hyperspectral data. Remote Sens., 10.","DOI":"10.20944\/preprints201809.0119.v3"},{"key":"ref_12","first-page":"45","article-title":"The accuracy of the Sobolev approximation","volume":"3","author":"Avaste","year":"1960","journal-title":"Izv. Geophys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1086\/149294","article-title":"Reflection and transmission of light by a thick atmosphere according to a phase function 1+xcos\u03c5","volume":"149","author":"Busbridge","year":"1967","journal-title":"Astrophys. J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"9514","DOI":"10.1002\/2015JD023575","article-title":"Spectral calculations of Rayleigh-scattering optical depth at Arctic and Antarctic sites using a two-term algorithm","volume":"120","author":"Tomasi","year":"2015","journal-title":"J. Geophys. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5041","DOI":"10.1016\/j.atmosenv.2005.05.010","article-title":"First aerosol optical thickness measurements at Dome C (east Antarctica), summer season 2003\u20132004","volume":"39","author":"Six","year":"2005","journal-title":"Atmos. Env."},{"key":"ref_16","first-page":"156","article-title":"On the atmospheric transmission of Sun radiation and on dust in the air","volume":"11","year":"1929","journal-title":"Geogr. Ann."},{"key":"ref_17","unstructured":"Iqbal, M. (1983). An Introduction to Solar Radiation, Elsiever."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/BF00168069","article-title":"Light scattering in planetary atmospheres","volume":"16","author":"Hansen","year":"1974","journal-title":"Space Sci. Rev."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"D02205","DOI":"10.1029\/2009JD012852","article-title":"Characterizing polar atmospheres and their effect on Rayleigh-scattering optical depth","volume":"115","author":"Tomasi","year":"2010","journal-title":"J. Geophys. Res."},{"key":"ref_20","unstructured":"van de Hulst, H.C. (1980). Multiple Light Scattering, Academic Press."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1175\/1520-0469(1975)032<0409:TTSAIR>2.0.CO;2","article-title":"The two-stream approximation in radiative transfer: Including the angle of the incident radiation","volume":"32","author":"Coakley","year":"1975","journal-title":"J. Atmos. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2440","DOI":"10.1175\/1520-0469(1976)033<2440:TBFITS>2.0.CO;2","article-title":"The backscattered fraction in two-stream approximations","volume":"33","author":"Wiscombe","year":"1976","journal-title":"J. Atmos. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Abramowitz, M., and Stegun, I. (1964). Handbook of Mathematical Functions and Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series.","DOI":"10.1115\/1.3625776"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.atmosres.2004.07.004","article-title":"A parameterization of the diffuse transmittance and reflectance for aerosol remote sensing problems","volume":"73","author":"Kokhanovsky","year":"2005","journal-title":"Atmos. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1931","DOI":"10.1016\/j.jqsrt.2010.03.005","article-title":"Benchmark results in vector atmospheric radiative transfer","volume":"111","author":"Kokhanovsky","year":"2010","journal-title":"J. Quant. Spectrosc. Radiat. Transf."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.jqsrt.2017.04.035","article-title":"Vector radiative transfer code SORD: Performance analysis and quick start guide","volume":"200","author":"Korkin","year":"2017","journal-title":"J. Quant. Spectrosc. Radiat. Transf."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"107903","DOI":"10.1016\/j.jqsrt.2021.107903","article-title":"Retrieval of total ozone column using high spatial resolution top-of-atmosphere measurements by OLCI\/S-3 in the ozone Chappuis absorption band over bright underlying surfaces","volume":"276","author":"Kokhanovsky","year":"2021","journal-title":"J. Quant. Spectrosc. Radiat. Transf."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2266","DOI":"10.1364\/AO.27.002266","article-title":"Analytic spectral functions for atmospheric transmittance calculations","volume":"27","author":"Green","year":"1988","journal-title":"Appl. Opt."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"609","DOI":"10.5194\/amt-7-609-2014","article-title":"High spectral resolution ozone absorption cross-sections\u2014Part 1: Measurements, data analysis and comparison with previous measurements around 293 K","volume":"7","author":"Gorshelev","year":"2014","journal-title":"Atmos. Meas. Technol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2458","DOI":"10.1364\/AO.25.002458","article-title":"Molecular transmission band model for oxygen in the visible","volume":"25","author":"Pierluissi","year":"1986","journal-title":"Appl. Opt."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Zege, E.P., Ivanov, A.P., and Katsev, I.L. (1991). Image Transfer through Light Scattering Media, Springer.","DOI":"10.1007\/978-3-642-75286-5"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1589","DOI":"10.1364\/AO.43.001589","article-title":"Scattering optics of snow","volume":"43","author":"Kokhanovsky","year":"2004","journal-title":"Appl. Opt."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2371","DOI":"10.5194\/tc-12-2371-2018","article-title":"On the reflectance spectroscopy of snow","volume":"12","author":"Kokhanovsky","year":"2018","journal-title":"Cryosphere"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Kokhanovsky, A., Lamare, M., Danne, O., Brockmann, C., Dumont, M., Picard, G., Arnaud, L., Favier, V., Jourdain, B., and Meur, E.L. (2019). Retrieval of snow properties from the Sentinel-3 Ocean and Land Colour Instrument. Remote Sens., 11.","DOI":"10.20944\/preprints201906.0162.v1"},{"key":"ref_35","first-page":"644551","article-title":"Retrieval of dust properties from spectral snow reflectance measurements","volume":"9","author":"Kokhanovsky","year":"2021","journal-title":"Front. Environ. Sci. Inform. Remote Sens."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2655","DOI":"10.5194\/tc-10-2655-2016","article-title":"Refinement of the ice absorption spectrum in the visible using radiance profile measurements in Antarctic snow","volume":"10","author":"Picard","year":"2016","journal-title":"Cryosphere"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"D14","DOI":"10.1029\/2007JD009744","article-title":"Optical constants of ice from the ultraviolet to the microwave: A revised compilation","volume":"113","author":"Warren","year":"2008","journal-title":"J. Geophysical Research"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Preusker, R., Carbajal Henken, C., and Fischer, J. (2021). Retrieval of daytime total column water vapor from OLCI measurements over land surfaces. Remote Sens., 13.","DOI":"10.3390\/rs13050932"},{"key":"ref_39","unstructured":"Mazeran, C., and Rueskas, A. (2020). Ocean Colour System Vicarious Calibration Tool Documentation, EUMETSAT."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1109\/TGRS.2009.2029345","article-title":"HAMSTRAD-Tropo, A 183-GHz Radiometer Dedicated to Sound Tropospheric Water vapor Over Concordia Station, Antarctica","volume":"48","author":"Ricaud","year":"2010","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"904585","DOI":"10.3389\/fenvs.2022.904585","article-title":"Snow surface properties derived from PRISMA satellite data over the Nansen Ice Sheet (East Antarctica)","volume":"10","author":"Kokhanovsky","year":"2022","journal-title":"Front. Environ. Sci"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"527","DOI":"10.3189\/172756502781831016","article-title":"Snow grain-size measurements in Antarctica","volume":"48","author":"Gay","year":"2002","journal-title":"J. Glaciol."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Kokhanovsky, A.A. (2006). Cloud Optics, Springer.","DOI":"10.1007\/1-4020-4020-2"},{"key":"ref_44","unstructured":"van de Hulst, H.C. (1981). Light Scattering by Small Particles, Dover."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/14\/19\/4778\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:38:47Z","timestamp":1760143127000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/14\/19\/4778"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,24]]},"references-count":44,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["rs14194778"],"URL":"https:\/\/doi.org\/10.3390\/rs14194778","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,24]]}}}