the reflection matrix just above the ocean surface, with the oceanic chlorophyll-a concentration as input. Finally, A(λ) in Eq. (B13) is a wavelength-dependent Lambertian albedo term that accounts for oceanic foam but may also correct for errors in Rul. Ws, Xchl, and A(λ) for the different spectral bands are included in the state vector.

## B3 GRASP ocean reflection

The reflection matrix over water surfaces (R$_s$) in the GRASP algorithm used in these studies is represented as follows:

$$ \mathbf{R}_s(\lambda, \theta_{in}, \theta_{out}, \Delta\varphi) = \mathbf{R}_{\text{frn}}(\theta_{in}, \theta_{out}, \Delta\varphi) \delta_{\text{frn}} \\ f_{\text{shad}} + r_0(\lambda)\mathbf{D}, \qquad (B14) $$

where R$_{frn}$ is the Fresnel reflection matrix from water surface facets with refractive index m, f$_{shad}$ is the shadowing function for the Gaussian random rough surface (Tsang et al., 1985; Mishchenko and Travis, 1997b), σ² is the mean square facet slope related to wind speed (Cox and Munk, 1954), δ$_{frn}$ is fraction of water surface providing the Fresnel reflection matrix, and r$_{0}$(λ) is the isotropic spectrally dependent water leaving reflectance. In the presented studies no a priori information about wind speed was used, and three parameters – σ², δ$_{frn}$, and r$_{0}$(λ) – were retrieved with application of proper spatial and temporal constraints Table A2.

**Data availability.** GRASP products are available at https://www.grasp-open.com/products/ (GRASP, 2023). RemoTAP products are available at https://public.spider.surfsara.nl/project/spexone/POLDER/GRIDDED/ (RemoTAP, 2023). The PARASOL level-1 data can be downloaded from https://www.icare.univ-lille.fr/data-access (ICARE, 2015). The AERONET data can be downloaded from https://aeronet.gsfc.nasa.gov/ (NASA and LOAPHOTONS, 1993). MODIS products can be downloaded from https://www.icare.univ-lille.fr/data-access (ICARE, 2020).

**Author contributions.** OH, PL, and OD designed the research. GF, CC, OH, and PL performed the research. OH wrote the first draft, and all authors contributed to the final version through comments and writing of subsections.

**Competing interests.** At least one of the (co-)authors is a member of the editorial board of Atmospheric Measurement Techniques. The peer-review process was guided by an independent editor, and the authors also have no other competing interests to declare.

**Disclaimer.** Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

**Acknowledgements.** The research of this paper has been performed in the framework of the project HARPOL (Harmonizing and advancing retrieval approaches for present and future polarimetric space-borne atmospheric missions), funded through the ESA program EO Science for Society. We acknowledge Christian Retscher (ESA) for his work as technical officer for HARPOL. We also acknowledge Daniele Gasbarra (ESA) and Alexandru Dandocsi (ESA) for important feedback during the course of the HARPOL project.

**Financial support.** This research has been supported by the European Space Agency (ESA contract no. 4000134035/21/I-DT-Ir).

**Review statement.** This paper was edited by Alexander Kokhanovsky and reviewed by six anonymous referees.

## References

Andreae, M., Jones, C., and Cox, P.: Strong present-day aerosol cooling implies a hot future, Nature, 435, 1187, 2005.

Bellouin, N., Quaas, J., Gryspeerdt, E., Kinne, S., Stier, P., Watson-Parris, D., Boucher, O., Carslaw, K. S., Christensen, M., Daniau, A.-L., Dufresne, J.-L., Feingold, G., Fiedler, S., Forster, P., Gettelman, A., Haywood, J. M., Lohmann, U., Malavelle, F., Mauritsen, T., McCoy, D. T., Myhre, G., Mülmenstädt, J., Neubauer, D., Possner, A., Rugenstein, M., Sato, Y., Schulz, M., Schwartz, S. E., Sourdeval, O., Storelvmo, T., Toll, V., Winker, D., and Stevens, B.: Bounding global aerosol radiative forcing of climate change, Rev. Geophys., 58, e2019RG000660, https://doi.org/10.1029/2019RG000660, 2020.

Butz, A., Hasekamp, O. P., Frankenberg, C., and Aben, I.: Retrievals of atmospheric CO_2 from simulated space-borne measurements of backscattered near-infrared sunlight: accounting for aerosol effects, Appl. Opt., 48, 3322, https://doi.org/10.1364/AO.48.003322, 2009.

Butz, A., Guerlet, S., Hasekamp, O., Schepers, D., Galli, A., Aben, I., Frankenberg, C., Hartmann, J.-M., Tran, H., Kuze, A., Keppel-Aleks, G., Toon, G., Wunch, D., Wennberg, P., Deutscher, N., Griffith, D., Macatangay, R., Messerschmidt, J., Notholt, J., and Warneke, T.: Toward accurate CO_2 and CH_4 observations from GOSAT, Geophys. Res. Lett., 38, L14812, https://doi.org/10.1029/2011GL047888, 2011.

Chen, C., Dubovik, O., Henze, D. K., Lapyonak, T., Chin, M., Ducos, F., Litvinov, P., Huang, X., and Li, L.: Retrieval of desert dust and carbonaceous aerosol emissions over Africa from POLDER/PARASOL products generated by the GRASP algorithm, Atmos. Chem. Phys., 18, 12551–12580, https://doi.org/10.5194/acp-18-12551-2018, 2018.

Chen, C., Dubovik, O., Henze, D. K., Chin, M., Lapyonok, T., Schuster, G. L., Ducos, F., Fuertes, D., Litvinov, P., Li, L., Lopatin, A., Hu, Q., and Torres, B.: Constraining global aerosol emissions using POLDER/PARASOL satellite remote sensing observations, Atmos. Chem. Phys., 19, 14585–14606, https://doi.org/10.5194/acp-19-14585-2019, 2019.

Chen, C., Dubovik, O., Fuertes, D., Litvinov, P., Lapyonok, T., Lopatin, A., Ducos, F., Derimian, Y., Herman, M., Tanré, D., Re-
the reflection matrix just above the ocean surface, with the oceanic chlorophyll-a concentration as input. Finally, A(λ) in Eq. (B13) is a wavelength-dependent Lambertian albedo term that accounts for oceanic foam but may also correct for errors in Rul. Ws, Xchl, and A(λ) for the different spectral bands are included in the state vector.

## B3 GRASP ocean reflection

The reflection matrix over water surfaces (R_s) in the GRASP algorithm used in these studies is represented as follows:

$$ \mathbf{R}_s(\lambda, \theta_{in}, \theta_{out}, \Delta\varphi) = \mathbf{R}_{\text{frn}}(\theta_{in}, \theta_{out}, \Delta\varphi) \delta_{\text{frn}} \\ f_{\text{shad}} + r_0(\lambda)\mathbf{D}, \qquad (B14) $$

where R_frn is the Fresnel reflection matrix from water surface facets with refractive index m, f_shad is the shadowing function for the Gaussian random rough surface (Tsang et al., 1985; Mishchenko and Travis, 1997b), σ² is the mean square facet slope related to wind speed (Cox and Munk, 1954), δ_frn is fraction of water surface providing the Fresnel reflection matrix, and r₀(λ) is the isotropic spectrally dependent water leaving reflectance. In the presented studies no a priori information about wind speed was used, and three parameters – σ², δ_frn, and r₀(λ) – were retrieved with application of proper spatial and temporal constraints Table A2.

**Data availability.** GRASP products are available at https://www.grasp-open.com/products/ (GRASP, 2023). RemoTAP products are available at https://public.spider.surfsara.nl/project/spexone/POLDER/GRIDDED/ (RemoTAP, 2023). The PARASOL level-1 data can be downloaded from https://www.icare.univ-lille.fr/data-access (ICARE, 2015). The AERONET data can be downloaded from https://aeronet.gsfc.nasa.gov/ (NASA and LOAPHOTONS, 1993). MODIS products can be downloaded from https://www.icare.univ-lille.fr/data-access (ICARE, 2020).

**Author contributions.** OH, PL, and OD designed the research. GF, CC, OH, and PL performed the research. OH wrote the first draft, and all authors contributed to the final version through comments and writing of subsections.

**Competing interests.** At least one of the (co-)authors is a member of the editorial board of Atmospheric Measurement Techniques. The peer-review process was guided by an independent editor, and the authors also have no other competing interests to declare.

**Disclaimer.** Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

**Acknowledgements.** The research of this paper has been performed in the framework of the project HARPOL (Harmonizing and advancing retrieval approaches for present and future polarimetric space-borne atmospheric missions), funded through the ESA program EO Science for Society. We acknowledge Christian Retscher (ESA) for his work as technical officer for HARPOL. We also acknowledge Daniele Gasbarra (ESA) and Alexandru Dandocsi (ESA) for important feedback during the course of the HARPOL project.

**Financial support.** This research has been supported by the European Space Agency (ESA contract no. 4000134035/21/I-DT-Ir).

**Review statement.** This paper was edited by Alexander Kokhanovsky and reviewed by six anonymous referees.

## References

Andreae, M., Jones, C., and Cox, P.: Strong present-day aerosol cooling implies a hot future, Nature, 435, 1187, 2005.

Bellouin, N., Quaas, J., Gryspeerdt, E., Kinne, S., Stier, P., Watson-Parris, D., Boucher, O., Carslaw, K. S., Christensen, M., Daniau, A.-L., Dufresne, J.-L., Feingold, G., Fiedler, S., Forster, P., Gettelman, A., Haywood, J. M., Lohmann, U., Malavelle, F., Mauritsen, T., McCoy, D. T., Myhre, G., Mülmenstädt, J., Neubauer, D., Possner, A., Rugenstein, M., Sato, Y., Schulz, M., Schwartz, S. E., Sourdeval, O., Storelvmo, T., Toll, V., Winker, D., and Stevens, B.: Bounding global aerosol radiative forcing of climate change, Rev. Geophys., 58, e2019RG000660, https://doi.org/10.1029/2019RG000660, 2020.

Butz, A., Hasekamp, O. P., Frankenberg, C., and Aben, I.: Retrievals of atmospheric CO_2 from simulated space-borne measurements of backscattered near-infrared sunlight: accounting for aerosol effects, Appl. Opt., 48, 3322, https://doi.org/10.1364/AO.48.003322, 2009.

Butz, A., Guerlet, S., Hasekamp, O., Schepers, D., Galli, A., Aben, I., Frankenberg, C., Hartmann, J.-M., Tran, H., Kuze, A., Keppel-Aleks, G., Toon, G., Wunch, D., Wennberg, P., Deutscher, N., Griffith, D., Macatangay, R., Messerschmidt, J., Notholt, J., and Warneke, T.: Toward accurate CO_2 and CH_4 observations from GOSAT, Geophys. Res. Lett., 38, L14812, https://doi.org/10.1029/2011GL047888, 2011.

Chen, C., Dubovik, O., Henze, D. K., Lapyonak, T., Chin, M., Ducos, F., Litvinov, P., Huang, X., and Li, L.: Retrieval of desert dust and carbonaceous aerosol emissions over Africa from POLDER/PARASOL products generated by the GRASP algorithm, Atmos. Chem. Phys., 18, 12551–12580, https://doi.org/10.5194/acp-18-12551-2018, 2018.

Chen, C., Dubovik, O., Henze, D. K., Chin, M., Lapyonok, T., Schuster, G. L., Ducos, F., Fuertes, D., Litvinov, P., Li, L., Lopatin, A., Hu, Q., and Torres, B.: Constraining global aerosol emissions using POLDER/PARASOL satellite remote sensing observations, Atmos. Chem. Phys., 19, 14585–14606, https://doi.org/10.5194/acp-19-14585-2019, 2019.

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Dubovik, O., Fuertes, D., Litvinov, P., Lopatin, A., Lapyonok, T., Doubovik, I., Xu, F., Ducos, F., Chen, C., Torres, B., Derimian, Y., Li, L., Herrera-Giralda, M., Herrera, M., Karol, Y., Matar, C., Schuster, G. L., Espinosa, R., Puthukkudy, A., Li, Z., Fischer, J., Preusker, R., Cuesta, J., Kreuter, A., Cede, A., Aspetsberger, M., Marth, D., Bindreiter, L., Hangler, A., Lanzinger, V., Holter, C., and Federspiel, C.: A Comprehensive Description of Multi-Term LSM for Applying Multiple a Priori Constraints in Problems of Atmospheric Remote Sensing: GRASP Algorithm, Concept, and Applications, Front. Remote Sens., 2, 706851, https://doi.org/10.3389/frsen.2021.706851, 2021.

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Chen, C., Dubovik, O., Litvinov, P., Fuertes, D., Lopatin, A., Lapyonok, T., Matar, C., Karol, Y., Fischer, J., Preusker, R., Hangler, A., Aspetsberger, M., Bindreiter, L., Marth, D., Chimo- mot, J., Fougnie, B., Marbach, T., and Bojkov, B.: Prop- erties of aerosol and surface derived from OLCI/Sentinel- 3A using GRASP approach: Retrieval development and pre- liminary validation, Remote Sens. Environ., 280, 113142, https://doi.org/10.1016/j.rse.2022.113142, 2022a.

Chen, C., Dubovik, O., Schuster, G. L., Chin, M., Henze, D. K., Lapyonok, T., Li, Z., Derimian, Y., and Zhang, Y.: Multi- angular polarimetric remote sensing to pinpoint global aerosol absorption and direct radiative forcing, Nat. Commun., 13, 7459, https://doi.org/10.1038/s41467-022-35147-y, 2022b.

Chowdhary, J., Cairns, B., Waquet, F., Knobelspiesse, K., Ottaviani, M., Redemann, J., Travis, L., and Mishchenko, M.: Sensitivity of multiangle, multispectral polarimetric remote sensing over open oceans to water-leaving radiance: Analyses of RSP data acquired during the MILAGRO campaign, Remote Sens. Environ., 118, 284–308, https://doi.org/10.1016/j.rse.2011.11.003, 2012.

Cox, C. and Munk, W.: Statistics of the sea surface derived from sun glitter, J. Mar. Res., 13, 198–227, 1954.

d'Almeida, G., Koepke, P., and Shettle, E.: Atmospheric Aerosols: Global Climatology and Radiative Characteristics, A. Deepak Publishing, Hampton, VA, ISA, 1991.

Deuzé, J. L., Bréon, F. M., Devaux, C., Goloub, P., Herman, M., Lafrance, B., Maignan, F., Marchand, A., Nadal, F., Perry, G., and Tanré, D.: Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements, J. Geophys. Res., 106, 4913–4926, 2001.

Di Noia, A., Hasekamp, O. P., Wu, L., van Diedenhoven, B., Cairns, B., and Yorks, J. E.: Combined neural network/Phillips-Tikhonov approach to aerosol retrievals over land from the NASA Research Scanning Polarimeter, Atmos. Meas. Tech., 10, 4235–4252, https://doi.org/10.5194/amt-10-4235-2017, 2017.

Dubovik, O., Smirnov, A., Holben, B. N., King, M. D., Kaufman, Y. J., Eck, T. F., and Slutsker, I.: Accuracy assessments of aerosol optical properties retrieved from Aerosol Robotic Network (AERONET) Sun and sky radiance measurements, J. Geophys. Res.-Atmos., 105, 9791–9806, https://doi.org/10.1029/2000JD900040, 2000.

Dubovik, O., Holben, B., Eck, T. F., Smirnov, A., Kauf- man, Y. J., King, M. D., Tanré, D., and Slutsker, I.: Variability of Absorption and Optical Properties of Key Aerosol Types Observed in Worldwide Locations, J. Atmos. Sci., 59, 590–608, https://doi.org/10.1175/1520- 0469(2002)059<0590:VOAAOP>2.0.CO;2, 2002.

Dubovik, O., Sinyuk, A., Lapyonok, T., Holben, B. N., Mishchenko, M., Yang, P., Eck, T. F., Volten, H., Muñoz, O., Veihelmann, B., van der Zande, W. J., Leon, J., Sorokin, M., and Slutsker, I.: Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust, J. Geophys. Res.-

Atmos., 111, D11208, https://doi.org/10.1029/2005JD006619, 2006.

Dubovik, O., Herman, M., Holdak, A., Lapyonok, T., Tanré, D., Deuzé, J. L., Ducos, F., Sinyuk, A., and Lopatin, A.: Statistically optimized inversion algorithm for enhanced retrieval of aerosol properties from spectral multi-angle polarimetric satellite observations, Atmos. Meas. Tech., 4, 975–1018, https://doi.org/10.5194/amt-4-975-2011, 2011.

Dubovik, O., Lapyonok, T., Litvinov, P., Herman, M., Fuertes, D., Ducos, F., Lopatin, A., Chaikovsky, A., Torres, B., Derimian, Y., Huang, X., Aspetsberger, M., and Federspiel, C.: GRASP: a versatile algorithm for characterizing the atmosphere, SPIE Newsroom, https://doi.org/10.1117/2.1201408.005558, 2014.

Dubovik, O., Li, Z., Mishchenko, M. I., Tanré, D., Karol, Y., Bojkov, B., Cairns, B., Diner, D. J., Espinosa, W. R., Goloub, P., Gu, X., Hasekamp, O., Hong, J., Hou, W., Knobelspiesse, K. D., Landgraf, J., Li, L., Litvinov, P., Liu, Y., Lopatin, A., Marbach, T., Maring, H., Martins, V., Meijer, Y., Milinevsky, G., Mukai, S., Parol, F., Qiao, Y., Remer, L., Rietjens, J., Sano, I., Stammes, P., Stamnes, S., Sun, X., Tabary, P., Travis, L. D., Waquet, F., Xu, F., Yan, C., and Yin, D.: Polarimetric remote sensing of atmospheric aerosols: instruments, methodologies, results, and perspectives, J. Quant. Spectrosc. Ra., 224, 474–511, https://doi.org/10.1016/j.jqsrt.2018.11.024, 2019.

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ble fraction using multi-angle, multi-spectral polarimetry, Atmos. Meas. Tech., 15, 7411–7434, https://doi.org/10.5194/amt-15-7411-2022, 2022.

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Xu, F., van Harten, G., Diner, D. J., Kalashnikova, O. V., Seidel, F. C., Bruegge, C. J., and Dubovik, O.: Coupled retrieval of aerosol properties and land surface reflection using the Airborne Multiangle SpectroPolarimetric Imager, J. Geophys. Res.-Atmos., 122, 7004–7026, https://doi.org/10.1002/2017JD026776, 2017.

Xu, F., Harten, G., Diner, D. J., Davis, A. B., Seidel, F. C., Rheingans, B., Tosca, M., Alexandrov, M. D., Cairns, B., Ferrare, R. A., Burton, S. P., Fenn, M. A., Hostetler, C. A., Wood, R., and Redemann, J.: Coupled Retrieval of Liquid Water Cloud and Above-Cloud Aerosol Properties Using the Airborne Multiangle SpectroPolarimetric Imager (AirMSPI), J. Geophys. Res.-Atmos., 123, 3175–3204, https://doi.org/10.1002/2017JD027926, 2018.

Zeng, S., Parol, F., Riedi, J., Cornet, C., and Thieuleux, F.: Examination of POLDER/PARASOL and MODIS/Aqua cloud fractions and properties representativeness, B. Am. Meteorol. Soc., 24, 4435–4450, https://doi.org/10.1175/2011JCLI3857.1, 2011.

Zhang, H. and Wang, M.: Evaluation of sun glint models using MODIS measurements, J. Quant. Spectrosc. Ra., 111, 492–506, https://doi.org/10.1016/j.jqsrt.2009.10.001, 2010.

Zhang, X., Li, L., Chen, C., Chen, X., Dubovik, O., Derimian, Y., Gui, K., Zheng, Y., Zhao, H., Zhang, L., Guo, B., Wang, Y., Holben, B., Che, H., and Zhang, X.: Validation of the aerosol optical property products derived by the GRASP/Component approach from multi-angular polarimetric observations, Atmos. Res., 263, 105802, https://doi.org/10.1016/j.atmosres.2021.105802, 2021.