2 FastMAPOL Retrieval Framework
FastMAPOL is an efficient coupled aerosol and ocean color retrieval algorithm that combines neural network (NN) forward models with iterative optimization (Gao et al. 2021, 2023; Gao et al. 2021, 2026). The NN forward models emulate vector radiative transfer calculations for a coupled atmosphere-ocean system based on the PACE simulator (Zhai et al. 2009, 2010; Zhai and Hu 2022). Please find more details in the NN, automatic differentiaton and aerosol and surface model details in the technical section.
2.1 Radiative transfer parameterization
The radiative transfer (RT) model used in this work is the PACE simulator (Zhai and Hu 2022), which solves the polarized radiance field in a coupled atmosphere-ocean system with flexible solar and viewing geometries, aerosol microphysical properties, and ocean bio-optical models (Gao et al. 2018). The RT model is used to generate the training datasets for the neural network forward models (Gao et al. 2021, 2023).
The forward radiative transfer simulations use atmospheric molecular vertical distributions from the U.S. Standard Atmosphere profile (Anderson et al. 1986), scaled by surface pressure. Absorption by oxygen, water vapor, methane, carbon dioxide, ozone, and nitrogen dioxide is accounted for using line-by-line calculations and the double-k distribution method (Duan et al. 2005; Zhai and Hu 2022). Surface pressure \(P_s\) and ozone column density \(n_{\mathrm{O}_3}\) are considered as input parameters, with values obtained from the MERRA-2 reanalysis (Gelaro et al. 2017).
More details are provided in Appendix B and Appendix C.
Aerosol model
Aerosols are represented by a multi-modal framework that allows independent characterization of fine- and coarse-mode size distributions, each with its own complex refractive index defined by real (\(m_r\)) and imaginary (\(m_i\)) components, assuming flat spectral dependence. The fine mode is represented by three submodes, while the coarse mode is represented by two submodes. Each submode is described by a log-normal size distribution with mean radii of 0.1, 0.1732, 0.3, 1.0, and 2.9 \(\mu\mathrm{m}\) and geometric standard deviations of 0.35, 0.35, 0.35, 0.5, and 0.5, respectively, consistent with Dubovik et al. (2006) and Xu et al. (2016). The volume density of each submode is denoted by \(V_i\).
Both fine and coarse modes allow mixtures of spherical and spheroidal particles to account for particle non-sphericity (Dubovik et al. 2006). The corresponding spherical fractions are represented by \(\mathrm{sph}_f\) and \(\mathrm{sph}_c\). The aerosol vertical distribution is parameterized by the aerosol layer height (\(z_c\)), defined as the center of a Gaussian vertical profile describing aerosol number density.
More details are provided in Appendix K, Appendix M and Appendix L.
Ocean model
Ocean bio-optical properties, including contributions from pure seawater, phytoplankton, and colored dissolved organic matter (CDOM), are parameterized using chlorophyll-a concentration (Chl-a). The scattering phase functions of phytoplankton are represented by the Fournier-Forand (FF) phase function (Fournier and Forand 1994), which is mixed with the pure seawater phase function to describe total scattering in seawater. The Mueller matrix is represented by the product of the mixed phase function and the average normalized Mueller matrix measured by Voss and Fry (1984) and parameterized by Kokhanovsky (2003).
Ocean surface roughness is described using the Cox-Munk model, parameterized by surface wind speed (\(w_s\)). The whitecap fraction is parameterized in terms of wind speed, with whitecap reflectance based on Koepke (1984).
The current bio-optical model is best suited for waters optically dominated by phytoplankton and performs well over most global ocean waters (Gao et al. 2026), but it faces challenges in accurately representing optically complex coastal waters, particularly those influenced by non-algal particles and strong CDOM absorption. Ongoing development of more advanced bio-optical models capable of representing complex water types will enable more realistic retrievals in coastal waters, particularly in regions influenced by both complex aerosol properties and bright ocean constituents Aryal et al. (2026).
More details are provided in Appendix Q.
Land Model
We use the land surface reflectance model implemented in the PACE simulator, with key components:
- Ross–Li kernel-driven BRDF
- Polarized bidirectional reflectance distribution function (BPDF)
- Fresnel reflection physics
Details are provided in Appendix S.
Parameter range
The forward-model parameters and their ranges are summarized in Table 2.1 using coupled atmosphere-ocean system as example. Solar and viewing geometries are specified by the solar zenith angle (\(\theta_0\)), viewing zenith angle (\(\theta_v\)), and relative azimuth angle (\(\phi_v\)), which are obtained from PACE Level-1C data. Surface pressure and ozone are obtained from ancillary data, while the aerosol, ocean, and surface parameters are retrieved by FastMAPOL.
| Parameter | Unit | Min | Max |
|---|---|---|---|
| \(\theta_0\) | \(^\circ\) | 0 | 80 |
| \(\theta_v\) | \(^\circ\) | 0 | 80 |
| \(\phi_v\) | \(^\circ\) | 0 | 180 |
| \(n_{\mathrm{O}_3}\) | Dobson | 150 | 450 |
| \(P_s\) | mb | 950 | 1050 |
| \(m_{r,f}\) | – | 1.33 | 1.55 |
| \(m_{r,c}\) | – | 1.33 | 1.55 |
| \(m_{i,f}\) | – | 0 | 0.03 |
| \(m_{i,c}\) | – | 0 | 0.001 |
| \(V_1\) | \(\mu\mathrm{m}^3\,\mu\mathrm{m}^{-2}\) | 0 | 0.14 |
| \(V_2\) | \(\mu\mathrm{m}^3\,\mu\mathrm{m}^{-2}\) | 0 | 0.11 |
| \(V_3\) | \(\mu\mathrm{m}^3\,\mu\mathrm{m}^{-2}\) | 0 | 0.07 |
| \(V_4\) | \(\mu\mathrm{m}^3\,\mu\mathrm{m}^{-2}\) | 0 | 0.20 |
| \(V_5\) | \(\mu\mathrm{m}^3\,\mu\mathrm{m}^{-2}\) | 0 | 0.62 |
| \(\mathrm{sph}_f\) | – | 0 | 1 |
| \(\mathrm{sph}_c\) | – | 0 | 1 |
| \(z_c\) | km | 0.1 | 6.0 |
| \(w_s\) | \(\mathrm{m}\,\mathrm{s}^{-1}\) | 0.5 | 15 |
| Chl-a | \(\mathrm{mg}\,\mathrm{m}^{-3}\) | 0.01 | 10 |
Note that aerosol retrieval over land has also been implemented for HARP2, with the corresponding product available as MAPOL_LAND. The retrieval follows the same coupled atmosphere-surface radiative transfer framework. Additional technical details are provided in later sections.
2.2 Neural network forward models
Neural network models are employed to represent TOA reflectance (\(\rho_t\)) and degree of linear polarization (DoLP) with two levels of accuracy. A smaller, computationally efficient neural network is used to generate an initial first guess, while a larger and more accurate neural network is applied in subsequent refinement steps of the retrieval. This two-stage strategy balances computational efficiency with retrieval accuracy (Gao et al. 2023).
In addition to the TOA forward models, separate neural network models are developed to support atmospheric correction. These include NNs trained to represent atmospheric and surface path reflectance (\(\rho_{\mathrm{atm+surface}}\)), total transmittance (\(1/(T_d t_u)\)), and a combined transmittance and BRDF correction term (\(C_{\mathrm{BRDF}}/(T_d t_u)\)). These components enable efficient decomposition of the TOA signal and facilitate the retrieval of water-leaving reflectance.
Additional neural network models are trained to compute AOD and SSA for both fine and coarse modes, allowing consistent treatment of aerosol optical and microphysical properties within the coupled retrieval framework. All NN models, architectures, and associated uncertainties are summarized in Table 2.2.
| Quantity | Neural network architecture | Uncertainty |
|---|---|---|
| \(\rho_t\) | \(19\times256\times256\times256\times34\) | 0.5% |
| \(\rho_t\) (first guess) | \(19\times128\times128\times128\times34\) | 0.6% |
| \(\rho_{\mathrm{atm+surface}}\) | \(18\times256\times256\times256\times34\) | 0.5% |
| DoLP | \(19\times512\times512\times512\times34\) | 0.0007 |
| DoLP (first guess) | \(19\times256\times256\times256\times34\) | 0.0012 |
| \(C_{\mathrm{BRDF}}/(T_d t_u)\) | \(19\times256\times256\times34\) | 0.006 |
| \(1/(T_d t_u)\) | \(19\times256\times256\times34\) | 0.005 |
| AOD (fine) | \(6\times64\times64\times34\) | 0.003 |
| AOD (coarse) | \(5\times64\times64\times34\) | 0.001 |
| SSA (fine) | \(6\times64\times64\times34\) | 0.003 |
| SSA (coarse) | \(5\times64\times64\times34\) | 0.01 |
More details are provided in Appendix G and Appendix H.
2.3 Retrieval and Optimization
FastMAPOL determines the aerosol and surface state parameters by minimizing the differences between multi-angle polarimetric measurements and NN forward-model predictions. The measurements used in the retrieval are reflectance and degree of linear polarization (DoLP), defined as
\[ \rho_t = \frac{\pi L_t}{\mu_0 F_0}, \tag{2.1}\]
and
\[ P_t = \frac{\sqrt{Q_t^2+U_t^2}}{L_t}, \tag{2.2}\]
where \(L_t\), \(Q_t\), and \(U_t\) are the Stokes parameters at the sensor altitude, \(F_0\) is the extraterrestrial solar irradiance, and \(\mu_0\) is the cosine of the solar zenith angle.
The retrieval minimizes the normalized cost function
\[ \chi^2({\mathbf x}) = \frac{1}{N} \sum_i \left[ \frac{[\rho_t(i)-\rho_t^f({\mathbf x};i)]^2}{\sigma_\rho^2(i)} + \frac{[P_t(i)-P_t^f({\mathbf x};i)]^2}{\sigma_P^2(i)} \right], \tag{2.3}\]
where \(\rho_t^f\) and \(P_t^f\) are the NN forward-model predictions, \({\mathbf x}\) is the retrieval state vector, and \(\sigma_\rho\) and \(\sigma_P\) are the corresponding uncertainties. The state vector contains the retrieved aerosol and surface parameters described in Table 2.1.
FastMAPOL solves this nonlinear optimization problem iteratively using the NN forward model and its Jacobian calculated through automatic differentiation. Details of the inversion, uncertainty weighting, convergence criteria, and optimization algorithm are provided in Appendix D.
2.4 Input uncertainty model
The input uncertainty model used in the retrieval cost function accounts for instrument measurement uncertainty, NN forward-model approximation, and numerical uncertainty in the radiative transfer (RT) simulations used for NN training. The RT and NN contributions are characterized based on the RT configuration and NN training performance.
Instrument uncertainties are based on instrument characteristics and calibration information provided by the instrument teams. All FastMAPOL retrievals adopt the following values with discussions in Appendix E:
| Instrument | Reflectance uncertainty | DoLP uncertainty |
|---|---|---|
| HARP2 v1 | 3% | 0.005 |
| SPEXone v1 | 2% | 0.003 |
2.5 Data screening and uncertainty quantifization
FastMAPOL incorporates adaptive multi-angle data screening and pixel-level uncertainty quantification to improve retrieval robustness and characterize retrieval confidence. The data-screening approach identifies and removes individual viewing angles that cannot be adequately represented by the forward model, while retaining valid observations from the same pixel whenever sufficient information remains for retrieval. This is particularly important for multi-angle polarimetric observations, where clouds, sunglint, surface heterogeneity, and other scene-dependent effects may affect only a subset of viewing angles. The screening methodology is described in detail in Appendix F.
FastMAPOL also provides a framework for propagating measurement and forward-model uncertainties into the retrieved state parameters and derived geophysical products using the forward-model Jacobian and retrieval covariance matrix. Automatic differentiation enables efficient calculation of the derivatives required for pixel-level uncertainty propagation. The methodology and approaches for evaluating the resulting uncertainty estimates are described in Appendix I and Appendix J when considering data correlations.