An optimization
approach has been developed for simultaneous retrieval of aerosol properties
and normalized water-leaving radiance (nLw) from multispectral,
multiangular, and polarimetric observations over ocean. The main features of
the method are (1) use of a simplified bio-optical model to estimate nLw,
followed by an empirical refinement within a specified range to improve its
accuracy; (2) improved algorithm convergence and stability by applying
constraints on the spatial smoothness of aerosol loading and Chlorophyll

Aerosols exist in the form of airborne suspensions of tiny particles that
scatter and absorb sunlight, leading to significant impacts on Earth's energy
and water cycles. Quantifying aerosol influences on climate requires accurate
determination of their abundances and optical/microphysical properties, which
are highly variable spatially and temporally. Aerosol characterization is
also crucial for ocean color remote sensing, as the spectral water-leaving
radiances account for only 10–15 % of the signal observed at the top of
the atmosphere (TOA) and most of the signal arises from atmospheric
scattering. Chlorophyll

When aerosol is a major target of retrieval, water-leaving radiance is often
empirically estimated or even neglected in operational algorithms employed by
current-generation satellite imagers due to their small contribution to TOA
signals. For examples, the MODIS Collection 5 algorithm uses zero
water-leaving radiance for all but the 550 nm band, where a value of
reflectance 0.005 is assumed (

As the main disadvantage of LUT approach, the solutions have to be selected from a finite number of aerosol models which might not be sufficiently representative in the relevant parameter space. New research efforts have been proposed to expand the LUT to cover more aerosol models (e.g., Limbacher and Kahn, 2014). An alternative to the LUT approach is optimization-based retrieval. It involves a direct inversion of the measurements within the context of a parametric description of the aerosol and surface characteristics that govern the radiation field observed at the TOA. The optimization-based retrieval is featured by a more compact and continuous representation of the relevant parameter space. A review of modern aerosol retrieval algorithms used by airborne and satellite-borne passive remote sensing instruments has been recently given by Kokhanovsky (2015).

When water-leaving radiance becomes the major target of retrieval, traditional retrievals decouple the atmosphere and surface using “atmospheric correction” procedures. The Ocean Biology Processing Group (OBPG) uses the atmospheric correction developed by Gordon and Wang (1994) and Gordon (1997) and refined by Ahmad et al. (2010). In this algorithm an aerosol optical property lookup table is built for ten aerosol models and eight relative humidity (RH) values based on the aerosol property statistics from Aerosol Robotic Network (AERONET) observations (Ahmad et al., 2010). Aerosol optical depth (AOD) and type are determined by fitting the observations in two near-infrared bands (e.g., 748 and 869 nm for MODIS), where water-leaving radiance is assumed negligible. The selected aerosol model is then extrapolated to shorter-wavelength visible bands and applied to the measured TOA radiances to retrieve normalized water-leaving radiance (nLw) (Gordon and Wang, 1994; Gordon, 1997). To reduce errors caused by this atmospheric correction procedure and instrumental radiometric uncertainties, empirical gain factors are derived by forcing agreement between retrieved nLw values and in situ measurements obtained at the Marine Optical Buoy (MOBY) site in Lanai, Hawaii (Franz et al., 2007).

For single-angle, nonpolarimetric instruments such as MODIS and the Sea-Viewing Wide Field-of-View Sensor (SeaWiFS), Franz et al. (2007) pointed out that, “the performance of satellite-based ocean color retrieval process is relatively insensitive to the aerosol model assumption …at least for open-ocean conditions where maritime aerosols dominate and aerosol concentrations are relatively low (i.e., aerosol optical thickness generally less than 0.3 at 500 nm)”. Therefore, the gain factors derived from conditions at the MOBY site can be applied globally to improve the agreement between satellite and in situ nLw over deep (Case 1) waters. In more challenging observing conditions, e.g., in the presence of absorbing aerosols or complex, spatially diverse (Case 2) waters, inaccurate knowledge of the absorbing aerosol optical properties or height distribution can lead to incorrect assumptions regarding CDOM and phytoplankton absorption coefficients (Moulin et al., 2001; Schollaert et al., 2003; Banzon et al., 2009). In addition, the vertical distribution of absorbing aerosols can affect the reflectance of the ocean–atmosphere system, resulting in errors in nLw (Duforêt et al., 2007). In coastal regions, where the traditional assumption of zero water-leaving radiance in the near-infrared (NIR) (Gordon, 1997; Siegel et al., 2000) breaks down, backscattering from suspended hydrosol particles (e.g., algae or sediment) can be misinterpreted as aerosols, leading to overestimation of AOD. The resulting overcorrection can lead to underestimated or even negative water-leaving radiances in the blue and green (e.g., Hu et al., 2000; Bailey et al., 2010; He et al., 2012).

The National Aeronautics and Space Administration's Pre-Aerosol, Clouds, and ocean Ecosystem (PACE) mission, with an anticipated launch date early in the next decade, is aimed at expanding upon current satellite ocean color measurements. The PACE payload is envisioned to include an ocean color spectrometer to measure ocean carbon storage and ecosystem function, and possibly a multiangle, multispectral polarimeter to provide advanced data records on clouds and aerosols and to assist with atmospheric correction of the ocean biology measurements. The capability of multiangle polarimetry in characterizing aerosols for the purposes of assessing their climatic or environmental impacts and improving nLw retrievals over turbid waters or in the presence of absorbing (dust or carbon-containing) aerosols motivates supplementing the vicarious calibration and LUT-based atmospheric correction procedures with one that permits simultaneous extraction of AOD, particle properties, and nLw. Inclusion of spectral bands covering the UV, visible, NIR, and shortwave infrared (SWIR), multiple view angles, and polarimetry in the retrieval enables retrieval of aerosol types that may be beyond the capabilities of the LUT and potentially improves accuracy of both the aerosol and ocean water properties. Given that measurements of atmospheric mineral dust and carbonaceous aerosols show a strong spectral dependence of absorption coefficient in the near-UV (e.g., Koven and Fung, 2006; Bergstrom et al., 2007; Russell et al., 2010) and have a spectral signature similar to those of CDOM, accurate modeling of radiative transfer (RT) in the coupled atmosphere–ocean system (CAOS) becomes necessary.

Without using bio-optical models, some RT models for CAOS consider specular reflection by assuming a flat ocean surface (Jin and Stamnes, 1994; Bulgarelli et al., 1999; Chami et al., 2001; Sommersten et al., 2009; Zhai et al., 2009) for simplicity. Better modeling fidelity and accuracy is then achieved by including sea surface roughness into the RT models (Nakajima and Tanaka, 1983; Fischer and Grassl, 1984; Masuda and Takashima, 1986; Kattawar and Adams, 1989; Mobley, 1994; Deuzé, 1989; Jin et al., 2006; Spurr, 2006) and including the water-leaving radiance and/or ocean foam reflection based on a Lambertian or a more general bidirectional reflectance distribution model (Koepke, 1984; Lyapustin and Muldashev, 2001; Mobley et al., 2003; Sayer et al., 2010; Sun and Lukashin, 2013; Gatebe et al., 2005). Though empirical parameterization of water-leaving radiance simplifies the radiative transfer, the relationship between water-leaving radiance and inherent optical properties (IOP) of dissolved or suspended ocean constituents is indirect. Such a weakness can be overcome by using bio-optical models to relate IOP directly to water-leaving radiance. The bio-optical model-based RT methods make it feasible to perform a one-step retrieval of IOP and aerosol optical properties from TOA measurements of radiance and polarization (e.g., Hasekamp et al., 2011), which is a complementary retrieval strategy to the prevailing two-step retrieval that obtains nLw from TOA via atmospheric correction and then determines IOP from nLw (IOCCG, 2006). Various RT solutions involving the use of bio-optical models have been developed and can be used for this purpose. These include the invariant imbedding method adopted by HydroLight (Mobley, 2008) and its faster version EcoLight (Mobley, 2011a) for scalar (intensity only) RT, in addition to the adding–doubling method (Chowdhary et al., 2006) and successive-order-of-scattering method (Zhai et al., 2010) for polarized RT in the CAOS.

Joint retrieval of aerosol and nLw properties requires supplementing the forward RT calculations with a sophisticated and computationally efficient inverse model to disentangle their contributions to TOA radiometry and polarimetry. Motivated by the development of a multiangle imaging polarimeter at JPL – the Airborne Multiangle SpectroPolarimetric Imager (AirMSPI) (Diner et al., 2013) – this paper describes the development of a coupled aerosol-ocean retrieval methodology. Our method (1) employs a simplified bio-optical model to obtain a reasonable estimate of nLw in the first retrieval step, followed by an empirical refinement in the subsequent step; (2) applies constraints on the spatial smoothness of aerosol and Chl loadings across neighboring image patches and spectral constraints on aerosol optical properties and on nLw across relevant bands to improve the convergence and stability of the algorithm; and (3) models and stores the RT fields in the aerosol/Rayleigh mixed layer, the pure Rayleigh-scattering layers, and the ocean medium separately, then couples them to obtain the radiative field at the sensor – thereby enhancing the Jacobian evaluations by reusing RT fields in the unperturbed layers. The Markov chain and doubling methods are applied to the mixed and uniform layers, respectively, to gain computational efficiency.

The parameters of our retrieval include spectrally dependent real and
imaginary parts of aerosol refractive index, aerosol concentrations of
different size components, mean height and width of aerosol distribution,
nonspherical particle fraction, wind speed over ocean surface, and normalized
water-leaving radiance. As auxiliary product, aerosol phase matrix is
obtained from the retrieved refractive index and normalized size
distribution. Throughout the paper, we use the definition of “exact”
normalized water-leaving radiance (nLw) given by Morel et al. (2002). It is
consistent with the definition adopted by Franz et al. (2007) and Zibordi et
al. (2009) and is related to the remote sensing reflectance (

The paper is organized as follows. In Sect. 2, we introduce our development
of the RT model that integrates the Markov chain, doubling and adding methods
for CAOS. The multipatch retrieval algorithm is described in Sect. 3. In
Sect. 4, a truth-in/truth-out test is performed to assess the retrieval
uncertainties for a variety of synthetic scenarios combined from three types
of aerosols, five aerosol loadings, three Chl

A five-layer model is established for a CAOS system, which consists (from the bottom up) of the ocean medium, the air–water interface, a pure Rayleigh layer, an aerosol/Rayleigh mixed layer, and a second pure Rayleigh layer (see Fig. 1). All layers are vertically homogeneous except for the mixed layer, where the aerosol has its own vertical distribution profile, different than that of the Rayleigh-scattering molecular atmosphere. Parameterizations of distribution profile, size, and single scattering properties of aerosols in the mixed layer are demonstrated in Appendix A.

Depiction of the 5-layer CAOS model. A Gaussian vertical
distribution profile for aerosols in the mixed layer is assumed and the
Markov chain model is used for RT in this optically inhomogeneous layer. The
ocean medium and the two Rayleigh layers (below and above the mixed layer,
respectively) are treated as optically homogeneous and the doubling method is
used for the RT computations. Coupling of these layers and inclusion of the
air–water interface are completed by use of the adding strategy. The Sun
illuminates the top-of-atmosphere with solar zenith angle

The five-layer CAOS model allows the use of different RT methods to model radiative transfer in different layers based on their computational strength. As an example, the Markov chain method (Esposito and House, 1978; Xu et al., 2010, 2011, 2012), which exhibits high computational efficiency for modeling RT in vertically inhomogeneous media (Esposito, 1979), is adopted in this work for the aerosol/Rayleigh mixed layer (see Appendix B for details). The doubling method (Stokes, 1862; van de Hulst, 1963; Hansen, 1971; de Haan et al., 1987; Evans and Stephens, 1991; among others), which exhibits high efficiency for modeling RT in optically homogeneous media (Esposito, 1979) is used for the two pure Rayleigh layers and the ocean medium (assumed to be homogeneous throughout the paper). Appendix C gives an example of using the doubling method for modeling RT in the ocean medium. The radiative fields from all layers are then coupled using an adding strategy to obtain the TOA fields.

In addition to the benefit of enabling a combination of the strengths of different RT methods, the strategy of separate RT modeling in five layers also makes for an efficient optimization-based retrieval. During the iterative optimization process, Jacobians are calculated to represent how the radiation fields vary as a function of the model parameters. When they are evaluated by perturbing a model parameter within one of the layers, the diffuse RT fields for all other layers are unchanged from the values obtained from the forward RT simulation and thus can be reused. For example, calculation of the Jacobians with respect to surface or ocean bio-optical parameters does not require recomputation of RT in the atmospheric layer because it has already been derived from the previous forward model calculation. Similarly, when evaluating Jacobians with respect to the aerosol parameters, it is unnecessary to repeat the RT computation of the Rayleigh layers and in the ocean or at the air–water interface. Because optimization-based retrievals involve Jacobian evaluations for a large number of parameters at all iterative steps, this strategy significantly improves the retrieval efficiency.

For the atmosphere, the diffuse radiative fields for the aerosol/Rayleigh mixed layer and the pure Rayleigh layers are computed by Markov chain and doubling methods, respectively, then coupled to get the diffuse reflection and transmission matrices for the whole atmosphere (see Appendix B).

For the ocean, the radiative field in the bulk medium is computed by the doubling
method with the optics of ocean constituents evaluated by use of a simplified
bio-optical model (see Appendix D), then coupled with the reflection and
transmission across the air–water interface, and finally corrected to account
for Raman scattering (see Appendix C). In addition to the contribution by
water-leaving radiance from the simplified bio-optical model, total light
leaving ocean surface also includes polarized specular reflection
(

Once the diffuse reflection and transmission matrices of the atmosphere and
reflection from ocean system are individually known, their coupling to get RT
field for the full CAOS is implemented by using the adding method. Two
operators

Note that the above formalism for modeling RT in a CAOS assumes a
horizontally homogeneous atmosphere above a uniform surface, which is known
as the independent pixel/patch approximation (IPA) in RT theory (Cahalan et
al., 1994). In reality, however, aerosol properties and surface reflection
vary across the pixels/patches. To reduce the IPA errors, the single-scattering contribution to the total field evaluated by Eq. (4) is replaced
by an exact evaluation of radiance along the line of sight. Moreover, for
simplicity of model demonstration, our five-layer model assumes the sensor to
be located at the TOA. For real airborne measurements, however, the sensor is
located inside the atmosphere. Therefore to improve the modeling accuracy,
the radiative field is actually computed at the sensor location. This is
realized by adding an extra Rayleigh layer above the sensor altitude (e.g.,

The integrated RT model established in the current section will be used as the forward model in retrieval, which is to be introduced in the next section.

Within the framework of optimization-based retrievals for nonlinear
problems, various approaches have been proposed to invert passive remote
sensing data for aerosol, ocean, and surface properties. Ideally, the solution
vector

To maximize the use of information provided by different remote sensing
instruments on aerosol and surface properties, various algorithms have been
applied to inverse radiance and polarimetric signals (Kokhanovsky, 2015;
Kokhanovsky et al., 2015). For the particular application of AirMSPI aerosol
and water-leaving radiance retrievals, an adaptation of the inversion
approach of Dubovik (2004) and Dubovik et al. (2008, 2011) is used. This
approach considers inversion as a multiterm least square
fitting with multiple a priori constraints. Particularly, as suggested by Dubovik et al. (2008,
2011), additional constraints on temporal or spatial variability of the
retrieved characteristics can be used if the retrieval is performed for a
group of observed pixels/patches. In the present application, a smoothness
constraint is imposed to constrain spatial variation of aerosol properties
and Chl

Median radius (

Note that though accurate forwarding RT modeling with multiple aerosol
species is possible, the increased number of free parameters challenges the
ability to retrieve a globally optimized solution in an efficient way.
Therefore, as described in Appendix A, a single aerosol species is assumed to
represent an effective set of aerosol optical properties, size
distribution (which may be multimodal), and vertical profile. Five log-normal
size distribution components (

In the next three subsections, we will give some details on the design of a multipatch retrieval algorithm for joint aerosol and water-leaving radiance retrieval. Readers not interested in it could skip over them.

Imposing smoothness constraints on both the spatial variations of aerosol
loading and Chl

Parameters in ocean retrieval and Lagrange multipliers for smoothness constraints.

The optimal solution is approached in an iterative way so that after

As listed in Table 2, the parameters of the retrieval include spectrally
dependent real (

In our retrieval test, an a priori estimate is assumed unavailable so we set

Ideally, the retrieval is deemed successful when the minimization of the cost
function is achieved, such that

Following Dubovik and King (2000), the Lagrange multipliers reflecting the
strength of smoothness constraints are defined as

There are two differences between

The multipliers

The multiplier

As water-leaving radiance is a small contribution to TOA signals, opening a
large number of parameters for its retrieval increases the risk of obtaining
solutions at local minima of the fitting metric and a significant slowdown of
the retrieval. To improve retrieval efficiency and reliability, we use a
two-step retrieval strategy: obtaining a reasonable estimate of
water-leaving radiance (i.e., close to the truth) by using a bio-optical
model constrained by a single parameter (namely Chlorophyll

Cases for truth-in/truth-out retrieval tests.

Technologies to extend the observational capabilities of JPL's Multi-angle
Imaging SpectroRadiometer (MISR, Diner et al., 1998) have been developed over
the past decade for the purpose of providing additional observational
constraints on aerosol and surface properties. These have been incorporated
into AirMSPI, as described in Diner et al. (2013). AirMSPI is an
ultraviolet-visible-near-infrared imager that has been flying aboard the NASA
ER-2 high-altitude aircraft since October 2010. At the heart of the
instrument is an 8-band (355, 380, 445, 470, 555, 660, 865, and 935 nm)
pushbroom camera mounted on a gimbal to acquire multiangle observations over
a

Prior to performing retrievals with actual AirMSPI data, truth-in/truth-out
tests with simulated data were conducted to assess the accuracy and stability
of our optimization approach. The simulation generates modeled TOA radiance
and polarization fields based on AirMSPI observations over the USC SeaPRISM
AERONET-OC site (118.12

Using the AirMSPI observational characteristics described above,
simulated measurements were generated for five different aerosol loadings,
three aerosol types, three Chl

To cover a wide range of observing geometries, a total of nine scenarios
based on the AirMSPI USC_SeaPRISM viewing geometry is used, as illustrated
in Fig. 2: the Sun is placed at the original incidence angle

Random noise was added to the simulated radiance and DoLP values.
This is a commonly adopted measure to test the impact of measurement errors
on retrieval algorithm performance (Dubovik et al., 2011; Hasekamp and
Landgraf, 2005, 2007). We added a relative measurement uncertainty of

Retrieved aerosol properties and Chl

Simulation geometries based on AirMSPI observations over the AERONET
OC-site USC_SeaPRISM on 6 February 2013. The three red dots indicate the
Sun's location

As an example, we use one of the simulated scenarios of AirMSPI observation
over USC_SeaPRISM AERONET-OC site (

For all aerosol types, the shapes of AOD, SSA, and nLw, as a function of
wavelength and PSD as a function of particle radius, are similar to their
true values. Due to the limited contribution of nLw to TOA radiance, the
aerosol retrieval accuracy is not significantly affected by the Chl

Simulated true and retrieved spectral AOD for different scene
conditions. Left column of three panels: weakly absorbing aerosol. Middle
column of three panels: moderately absorbing aerosol. Right column of three
panels: dust aerosol. AOD is retrieved for three values of Chl

Panel layout as in Fig. 3 but for retrieved single-scattering albedo. The black line with dots placed at the AirMSPI wavelengths represents the true SSA. The colored symbols represent retrieved SSA for various values of AOD.

Panel layout as in Fig. 3 but for retrieved normalized aerosol size distribution. The black lines correspond to the true size distribution, with dots at discrete values of particle radius. The colored lines represent retrieved size distributions for various values of AOD.

Panel layout as in Fig. 3 but for retrieved values of nLw
(mW cm

Retrieval errors of

Comparison of single-patch- and multipatch-based retrievals of

AOD, SSA, aerosol size distribution, and nLw retrieved using the bio-optical model compared to retrievals in which water-leaving radiance is modeled simply as Lambertian with arbitrary albedo.

A more comprehensive view of aerosol retrieval errors is displayed in Fig. 7a–d.
Though the absolute error of retrieved AOD increases as the aerosol
loading increases (see Fig. 7a), the relative error of AOD (

Figure 7d shows that for weakly and moderately absorbing aerosols the effective radius for coarse-mode aerosols has larger retrieval errors than the fine mode aerosol. We attribute this to the fact that the longest spectral band of AirMSPI used in the retrievals (865 nm) is insufficient to fully constrain the coarse-mode aerosol PSD.

In Fig. 7e–f, which correspond to Chl

Figure 7 shows that for all aerosol types, even though the retrieval errors
of SSA and AOD at low aerosol loading (

Taking the case of median loading (

For the same scene, parameters used to compare the single- and
multipatch-based retrievals, Fig. 9 compares a retrieval using the
bio-optical model and one in which nLw is modeled using unconstrained
Lambertian reflectance factors at each wavelength. Using the bio-optical
model reduces the parameter space for the water-leaving radiance from seven
independent spectral values to a single parameter (Chl

The above truth-in/truth-out tests were performed assuming instrumental
errors are completely random. Such an assumption, however, is not applicable
to radiometric errors and their band-to-band variations, which represent
systematic deviations from the true values due to calibration errors. For a
satellite instrument such as MISR, the radiometric uncertainty is 4 % and
the band-to-band variations are about 1.5 % (Bruegge et al., 2002).
Because the absolute error is larger in magnitude than band-to-band error and
represents a systematic bias that applies to all measurements, it can
potentially have greater impact on retrieval accuracy than band-to-band
errors and random noise. To model its effect, we keep the random noise levels
used in the previous analysis and add a

Comparison of Figs. 7 and 10 shows that systematic errors have a larger
impact on retrieval accuracy than random errors, as the latter are suppressed
by using a lot of patches for retrieval while the former are not. For AOD and
SSA, a negative radiance bias causes larger retrieval errors than a positive
bias. Comparison of Figs. 10e and 7e shows that errors in nLw due to an
intensity bias increase at all AODs: at low aerosol loading the errors
propagate to nLw while at high loading the contribution of nLw to the TOA
signal is weak, exacerbating errors. On the other hand, comparison of
Figs. 10f with 7h shows a much smaller effect of systematic errors on

Similar to Fig. 7a–e and h for [Chl_

Following algorithm validation using the truth-in/truth-out tests, we applied the algorithm to actual AirMSPI observations acquired over the USC_SeaPRISM AERONET-OC site and near the AERONET site in La Jolla. The USC_SeaPRISM and La_Jolla scenes were chosen from a larger set of AirMSPI field campaign images to ensure cloud free conditions. The data were processed with the recently upgraded data processing pipeline, which includes vicarious radiometric calibrations and improved polarimetric calibration making use of onboard polarization sources. Nadir intensity and DoLP images from combinations of different spectral bands for these two target areas are shown in Figs. 11a and 12a. Maps of retrieved AOD and SSA at 555 nm, nLw at 445 and 555 nm spectral bands are displayed in Figs. 11b and 12b.

Selecting the image patch that is closest to the AERONET site, our retrieved
AOD, SSA, size distribution, and nLw are compared to the independent AERONET
results, as shown in Figs. 11c and 12c. We first discuss results from the
USC_SeaPRISM retrievals. The AERONET site reported a relatively high 550 nm
AOD of 0.30 and 0.26 at 19:08 and 20:08 UTC, respectively, and our
retrieval returns an intermediate value of 0.27 from the AirMSPI data
acquired at 19:40 UTC. The differences between the AirMSPI and AERONET AOD
and SSA retrievals are within the AERONET SSA retrieval uncertainties (e.g.,
0.015 for

As illustrated in the bottom right panel of Fig. 11c, the retrieved nLw also
compares favorably to AERONET reported values. After interpolating AERONET
nLw in logarithmic space to obtain nLw in the AirMSPI bands, the differences
are found to be 0.0396, 0.0118, 0.0198, and 0.0077 mW cm

Similar to Fig. 11 but corresponding to the AirMSPI observations near AERONET La Jolla site on 14 January 2013 at 21:09 UTC. The bluish part at the bottom right part of the DoLP image indicates the shallow water area which was not captured by all images and hence excluded in retrieval.

For the second study site, the AirMSPI target area was about 13 km away from
the La Jolla AERONET station. In spite of the distance, the differences
between the AirMSPI and AERONET AOD and SSA values are both within AERONET's
uncertainty, as observed from the upper two plots of Fig. 12c. Though the
difference in PSD in some size bins falls outside the AERONET uncertainty
range, the bimodality of the size distribution is identified even at the low
aerosol loading for this case (

Accurate retrieval of both aerosol properties and water-leaving radiance is challenging as the latter only accounts for a small fraction of TOA signals and can be easily contaminated by Rayleigh and/or aerosol scattering. To ensure high-quality retrievals of the aerosol properties, traditional atmospheric correction schemes, which are focused primarily on retrieval of surface characteristics, may not be sufficient. In light of the additional information provided by multiangular, multispectral, and polarimetric measurements, we tested the concept of simultaneous aerosol and water-leaving radiance retrieval which include spectrally dependent real and imaginary parts of aerosol refractive index, aerosol concentrations of different size components, mean height and width of aerosol distribution, nonspherical particle fraction, wind speed over ocean surface, and normalized water-leaving radiance. An efficient RT modeling strategy has been developed that couples separate runs for modeling RT in two Rayleigh layers, an aerosol/Rayleigh mixed layer, and an ocean medium. Repeated, time-consuming RT computations for layers whose properties are not perturbed during Jacobian evaluations are avoided. The Markov chain method is used for modeling RT in the mixed layer and the doubling method is used to model RT in the pure Rayleigh layer and ocean medium. These features are implemented to enhance computational efficiency.

Next, an optimization approach has been developed for joint aerosol and
water-leaving radiance retrieval. The algorithm involves a two-step retrieval
strategy, first relying on a bio-optical model to retrieve a single parameter
(Chl

In future work, the influence of modeling errors on nLw retrievals will be
investigated. Since water-leaving radiance accounts for a small fraction of
the TOA signals, small forward modeling errors can translate into large nLw
retrieval errors. The modeling error can arise from various sources, e.g.,
neglect of cirrus cloud contamination, approximate treatment of trace-gas
absorption and the atmosphere profile, salinity of sea-water, assumption of
plane-parallel atmosphere, retrieval of effective aerosol optical properties
from assuming single aerosol species and size-independent refractive index,

The aerosol/Rayleigh mixed layer is defined to have the minimum altitude

Breaking the aerosol volumetric size distribution

Moreover, the total aerosol size distribution is constituted as

Using a log-normal volume weighted size distribution for all size components,

The mixed layer is subdivided into

As functions of aerosol refractive index, shape, and size distribution, the
elements of

The light propagation direction in the mixed layer is discretized into a
finite number of angles over the range

Equation (B1) is the basic form of the Markov chain method. The majority of
computational time is spent in computing the matrix inverse
[

Setting 3–4 sublayers for each subgroup, fast convergence and accuracy of
matrix inverse computation is usually achieved by using the first 3–4 series
terms of Eq. (B4) (namely

The reflection and transmission matrices of the two Rayleigh-scattering
layers above and below the mixed layer, (

Comparison of top-of-ocean radiance (

In the five-layer CAOS system illustrated in Fig. 1, the ocean system is composed of the ocean medium and the air–water interface. The diffuse reflection matrix of the ocean medium and the reflection and transmission matrices of the air–water interface need to be known before they are coupled to evaluate the diffuse field at the top of ocean.

Evaluation of the reflection matrix of the ocean system follows a similar
methodology as the atmosphere system. However, instead of considering the
contributions by molecules and aerosols, RT in the ocean involves scattering
and absorption by sea water, CDOM, and phytoplankton and their covariant
particles. Evaluation of the IOPs of these components relies on a simplified
bio-optical model described in Appendix D, which determines absorption and
scattering of CDOM and phytoplankton particles, then bulk optical depth

As described in Appendix E, reflection of light from ocean surface and its
transmission through an air–ocean interface are evaluated using the model of
Cox and Munk (1954a, b) for a wind-roughened ocean surface. The set of
reflection and transmission matrices (

However, unlike a real atmospheric layer that attenuates light during its
transmission, the air–water interface is a pseudolayer without any thickness,
so all attenuation-related terms should be removed. This leads to a
modification of the classical adding–doubling scheme (named the “extended
adding–doubling method” in the remainder of the paper) for coupling the
transfer of radiation between the ocean bulk medium and the air–water
interface: the matrices describing the downwelling and upwelling of diffuse
light at the top of the ocean now become

As a numerical validation, Fig. C1 compares top-of-ocean radiance and DoLP
computed with the extended adding–doubling method via Eq. (C1) and an
independent successive-orders-of-scattering code (Zhai et al., 2010). Chl

The RT modeling formulated in Section C1 does not account for inelastic
scattering processes including Raman scattering by water and fluorescence by
chlorophyll and CDOM. Accurate modeling of these processes is necessary
(Mobley, 2008; Zhai et al., 2015) but requires additional inputs and
computations that can significantly slow down the retrievals (Mobley, 2011b).
To optimize the trade-off between computational efficiency and numerical
accuracy, the correction scheme proposed by Lee et al. (2013) is used to
quantify the contribution by Raman scattering, namely,

Since the two reference spectral bands at 440 and 550 nm in Eq. (C2) are
close to the AirMSPI bands at 445 and 555 nm,

Left panel: refractive index (

As indicated in the last two terms of Eq. (1), our water-leaving radiance
model consists of two parts. The first part
(

Pure sea water, CDOM, and phytoplankton and their covariant particles are considered to be the primary contributors to the oceanic absorption and scattering.

The absorption coefficients of water (

The depolarization factor of sea water is currently set to zero.

Invoking the Einstein–Smoluchowski theory of fluctuation scattering provides

Due to the symmetry of scattering function of seawater around 90

Phytoplankton and their covariant particles are assumed to conform to the
hyperbolic (Junge) size-distribution, namely,

Knowing the real refractive index of particles (

Mobley (2002) found that the detailed shape of particle-scattering function
is not critical if a correct backscatter fraction

The spectrally neutral assumption for the backscattering efficiency also
indicates that the refractive index and slope parameter are not independent
to each other. Knowing

Thus, given Chl

The absorption coefficients of phytoplankton and their covariant particles
for

Integrated with Vasilkov et al.'s (2005)

The particle-scattering coefficients are evaluated based on the model by
Morel and Maritorena (2001):

Absorption of CDOM (

The scattering coefficient for CDOM is treated as zero in the present study.

Summarizing the contribution of all components gives the total absorption
coefficient of ocean bulk (

Polarized radiative transfer computations require the full phase matrix of
bulk ocean scattering. To this purpose, we construct other phase matrix
entries (

Taking the geometric thickness of ocean as

With

With the micro-facet
assumption of oceanic wave structure, the polarized ocean surface reflectance
(Tsang, 1985; Mishchenko, 1997) is modeled as

For the downwelling light, the transmission matrix (Zhai et al., 2010) is

The Eqs. (E1) and (E5) also apply to the evaluation of reflection and
transmission matrices

The authors are grateful to Zia Ahmad at NASA Goddard Space Flight Center for providing the information on aerosol models used in MODIS ocean color retrieval and Jianwei Wei at Optical Oceanography Laboratory of University of Massachusetts Boston for discussing the AERONET Ocean Color product of normalized water-leaving radiance. This work was performed at the Jet Propulsion Laboratory, California Institute of Technology under contract with the National Aeronautics and Space Administration. Edited by: A. Kokhanovsky Reviewed by: two anonymous referees