## Abstract

We present an optimal-estimation-based retrieval framework, the microphysical aerosol properties from polarimetry (MAPP) algorithm, designed for simultaneous retrieval of aerosol microphysical properties and ocean color bio-optical parameters using multi-angular total and polarized radiances. Polarimetric measurements from the airborne NASA Research Scanning Polarimeter (RSP) were inverted by MAPP to produce atmosphere and ocean products. The RSP MAPP results are compared with co-incident lidar measurements made by the NASA High-Spectral-Resolution Lidar HSRL-1 and HSRL-2 instruments. Comparisons are made of the aerosol optical depth (AOD) at 355 and 532 nm, lidar column-averaged measurements of the aerosol lidar ratio and Ångstrøm exponent, and lidar ocean measurements of the particulate hemispherical backscatter coefficient and the diffuse attenuation coefficient. The measurements were collected during the 2012 Two-Column Aerosol Project (TCAP) campaign and the 2014 Ship-Aircraft Bio-Optical Research (SABOR) campaign. For the SABOR campaign, 73% RSP MAPP retrievals fall within $\pm 0.04$ AOD at 532 nm as measured by HSRL-1, with an $R$ value of 0.933 and root-mean-square deviation of 0.0372. For the TCAP campaign, 53% of RSP MAPP retrievals are within 0.04 AOD as measured by HSRL-2, with an $R$ value of 0.927 and root-mean-square deviation of 0.0673. Comparisons with HSRL-2 AOD at 355 nm during TCAP result in an $R$ value of 0.959 and a root-mean-square deviation of 0.0694. The RSP retrievals using the MAPP optimal estimation framework represent a key milestone on the path to a combined $\mathrm{lidar}+\mathrm{polarimeter}$ retrieval using both HSRL and RSP measurements.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

In this paper, we present a retrieval framework, microphysical aerosol properties from polarimetry (MAPP), designed to simultaneously retrieve aerosol microphysical properties and ocean color bio-optical parameters using multi-angular total and polarized radiances from the NASA GISS Research Scanning Polarimeter (RSP [1]). The atmosphere/ocean products are compared against NASA Langley High-Spectral-Resolution Lidar (HSRL-1 and HSRL-2 [2]) aerosol optical depths and lidar ocean measurements as well as lidar-intensive parameters (the lidar ratio and Ångstrøm exponent). The co-incident polarimeter and lidar measurements that are presented were collected during the 2012 Two-Column Aerosol Project (TCAP [3]) campaign and the 2014 Ship-Aircraft Bio-Optical Research (SABOR) campaign. TCAP was a Department of Energy campaign conducted off the coast of Cape Cod to address knowledge gaps in aerosol and cloud temporal evolution and uncertainties in direct and indirect aerosol forcing by improving remote sensing retrievals of aerosol optical and microphysical properties and studying aerosol–cloud interactions. Thus, the TCAP study sampled the atmosphere between and within two atmospheric columns: one fixed near the coast of North America (over Cape Cod, MA) and a second moveable column over the Atlantic Ocean several hundred kilometers from the coast. The NASA-led SABOR campaign was designed to assess the applicability of lidar and polarimetry for ocean biogeochemistry and involved ship-based *in situ* measurements as well as airborne remote sensors. The SABOR campaign provides a wide range of ocean optical properties from aircraft overflights of the ship, spanning ocean conditions from the eutrophic waters of the Gulf of Maine to oligotrophic waters near Bermuda.

The aircraft-based RSP instrument can retrieve aerosol/ocean products by measuring the upwelling total and polarized reflectance using six paired refractive telescopes, with each pair making measurements in three spectral bands. One telescope in each pair makes simultaneous measurements of the linear polarization components of the intensity in orthogonal planes at 0° and 90° to the meridional plane of the instrument, while the other telescope simultaneously measures equivalent intensities in orthogonal planes at 45° and 135°. The RSP instrument has nine spectral channels that are divided into two groups based on the type of detector used: visible/near infrared (VNIR) bands at approximate wavelengths (full width at half-maximum, FWHM, in parentheses) 410 (30), 469 (20), 555 (20), 670 (20), 864 (20), and 960 (20) nm and shortwave infrared (SWIR) bands at 1594 (60), 1880 (90), and 2264 (120) nm. The set of four measurements in a particular band are denoted ${S}_{1L}$ and ${S}_{1R}$ for the two orthogonal polarization states in the telescope observing at polarization azimuths of 0° and 90° and ${S}_{2L}$ and ${S}_{2R}$ for the two orthogonal polarizations in the second telescope of each pair. The Stokes parameters $I$, $Q$, and $U$ can be derived from these four measurements, and the corresponding orthogonal (parallel and perpendicular) components are obtained as ${I}_{L}=(I+Q)/2$ and ${I}_{R}=(I-Q)/2$, respectively. With appropriate normalization [see Eqs. (22), (23), and (24)], we can also define corresponding reflectances ${R}_{I}$, ${R}_{Q}$, ${R}_{U}$, ${R}_{{I}_{L}}$, and ${R}_{{I}_{R}}$.

Waquet *et al.* (2009) [4] used polarized reflectance (${R}_{Q}$) observations in seven spectral spectral bands (excluding the 960 and 1880 nm bands) obtained during the 2005 ALIVE campaign in the Southern Great Plains and in 2003 over the Simi Valley in the Mojave desert to retrieve aerosol microphysical parameters. The 2.26 μm polarized reflectances were primarily used to characterize the polarized surface reflectance as a function of scan angle, which to a good approximation is spectrally invariant. Knobelspiesse *et al.* (2011) [5] used polarized reflectances rotated into the scattering plane in the same bands used by Waquet *et al.* (2009) and two total reflectance measurements at 410 nm and 2264 nm from RSP to retrieve aerosol microphysical parameters of thick smoke over land during the 2008 ARCTAS campaign. Over water, Chowdhary *et al.* (C2006, C2012) [6,7] performed simultaneous aerosol and ocean retrievals using total and polarized reflectance observations for RSP observations collected off the coast of Veracruz, Mexico during the 2006 MILAGRO campaign, but focused mainly on development of an ocean bio-optical model for polarized reflectances. Wu *et al.* (2015) [8] used all seven RSP window channels for total reflectance (${R}_{I}$) and the degree of linear polarization, $\mathrm{DoLP}=\sqrt{{R}_{Q}^{2}+{R}_{U}^{2}}/{R}_{I}$, to retrieve properties of fine- and coarse-mode aerosol retrievals over land.

One of the main ideas in the “MAPP” algorithm is that the retrieval is “coupled,” in the sense that the atmosphere and ocean products are simultaneously retrieved, and that all seven RSP window channels are used, including polarization through both the reflectances ${R}_{{I}_{L}}$, ${R}_{{I}_{R}}$, and the degree of linear polarization. Similar “coupled” atmosphere and surface retrieval approaches have also been developed for the POLarization and Directionality of Earth Reflectances (POLDER) instrument on the PARASOL satellite, [9,10]. While MAPP may not work as well as “hand-tuned” retrievals one might conduct for a few cases, the goal is to produce an algorithm that is capable of automated processing of polarimeter (RSP) data. Also, MAPP was designed not only as a standalone polarimeter retrieval algorithm, but as the foundation for a combined $\mathrm{HSRL}+\mathrm{RSP}$ retrieval (or arbitrary $\mathrm{lidar}+\mathrm{polarimeter}$ configuration). The focus of this paper, and a key milestone on the path to the combined $\mathrm{HSRL}+\mathrm{RSP}$ retrieval, is a MAPP RSP (polarimeter-only) retrieval that provides the framework for processing polarimeter data, and a baseline retrieval that the combined MAPP $\mathrm{HSRL}+\mathrm{RSP}$ retrieval is expected to improve upon. Thus, for the RSP MAPP retrievals in this paper, we use lidar only to provide the aerosol top height and cloud screening to identify Aerosol-Above-Ocean (AAO) scenes. We present RSP MAPP results for all TCAP Phase 1 and SABOR AAO scenes, after screening for clouds and aircraft altitude/roll.

A unique feature of MAPP is the focus on coupled retrievals of aerosol microphysical properties and ocean color parameters using optimal estimation driven by on-the-fly vector radiative transfer and Mie calculations. The RSP instrument makes coupled retrievals easier in that, unlike an imager that has many scenes (pixels) but fewer measurements per scene, the RSP instrument scans along the track so that a single scene contains many measurements per pixel, with fewer scenes to process. Fewer scenes make it possible to perform Mie and vector radiative transfer calculations on the fly, rather than relying on lookup tables (LUTs) of pre-computed results. Performing the vector radiative transfer computations on the fly is advantageous for aircraft platforms where the sensor altitude varies and the aerosol location (top height, layering, etc.) is allowed to vary, whereas precomputed Mie LUTs would provide a speed benefit at the cost of reduced flexibility. Over 90% of the signal measured by a spectroradiometer or polarimeter is due to the molecular and aerosol scattering in the atmosphere and the Fresnel reflection of the ocean surface, so that less than 10% of the signal is due to subsurface ocean inherent optical properties (particulate and dissolved matter) [11]. RSP measurements of the polarized reflectance at multiple angles can lead to improved retrievals of underwater properties by enabling accurate characterization of the aerosol signal through retrieval of aerosol optical depth, microphysical aerosol properties including aerosol effective radius, single-scattering albedo, and the real part of the refractive index. Another advantage is that RSP does not saturate due to sunglint, which is a significant issue for instruments like MODIS.

The overall methodology is described in Section 2 and is summarized by the retrieval parameters, which determine the state of the atmosphere (aerosol) and ocean models. In Section 3, we present the retrieval framework based on principles of optimal estimation, while Section 4 includes descriptions of the instruments, direct measurements, lidar ocean products, and noise profiles. Section 5 discusses simulated retrievals, and Sections 6 and 7 present results for measurements collected during the SABOR and TCAP campaigns, respectively. Section 8 provides an error analysis, and a conclusion is provided in Section 9. The RSP MAPP products are summarized on the NASA LaRC HSRL website (https://science.larc.nasa.gov/hsrl/) [12] and are available for download at the NASA GISS RSP website (https://data.giss.nasa.gov/pub/rsp) [13].

## 2. METHODOLOGY

The MAPP algorithm consists of an optimal-estimation-based solver, where the forward model consists of a vector radiative transfer code and a Mie scattering code. In optimal-estimation theory, the forward model input is specified by the state vector, and the forward model output matches the instrument, in this case, RSP’s total and polarized radiances. Each MAPP retrieval takes approximately 60 min to run on a single core of a Sandy Bridge Intel Xeon processor, although we expect to reduce processing (CPU) time by a factor of 2–3. The NASA LaRC K cluster was used to perform the retrievals in parallel, and since the CPU time scales linearly with the number of cores, then 1,000 cores enables processing of approximately 24,000 scenes per 24 h. Assuming an RSP-like along-track polarimeter was scanning continuously at 1 Hz, averaging to 10 s per measurement would result in 8,640 scenes per 24 h. Hence, MAPP is already fast enough to process 10-second-averaged RSP-like data within 60 min, given a dedicated resource of approximately 500 Sandy Bridge cores (allowing for some overhead).

#### A. Overview of MAPP Retrieval Parameters

The state vector of MAPP retrieval parameters is defined as

*a priori*to be equal to that of water to represent hydrated sea salt (marine) aerosols. The aerosol top heights for the coarse and fine mode were set to be the same (${z}_{f}={z}_{c}$), and were obtained by finding the altitude that represented an integral of 95% of the lidar backscatter or extinction at 532 nm. For the simulated retrieval, ${z}_{f}$ and ${z}_{c}$ were fixed and assumed to be known. While it is possible to use polarimetric observations to retrieve the aerosol layer height [14], here we assume that high-quality lidar observations are available to constrain this aspect of aerosol variability. The ocean surface roughness is described by the wind speed $v$ [m/s], and the subsurface absorption and scattering is parameterized in terms of the chlorophyll concentration CHL [$\mathrm{mg}/{\mathrm{m}}^{3}$]. The ranges allowed for each of the parameters in this 10-parameter retrieval are as follows:

The maximum allowed optical depth for the fine and coarse modes is up to 0.6 for the fine mode and up to 0.4 for the coarse mode, which are reasonable values for conditions encountered over the Atlantic Ocean off the East coast of the United States, but which need to be increased in areas with higher aerosol loading. The choices for all the aerosol parameters are explained in Section 2B.2. The allowed ranges for the wind speed and chlorophyll concentration are explained in Section 2C.

#### B. Atmosphere Models

### 1. Atmospheric Gases

The atmosphere is assumed to consist of five layers between 0 and 100 km to account for the vertical distribution of molecules that scatter and absorb sunlight. For retrievals of real data we used five layers with [0, ${z}_{\mathrm{aerosol}}$, ${z}_{\mathrm{aircraft}}$, 12, 22, 100] km, where ${z}_{\mathrm{aerosol}}$ is the aerosol top height and ${z}_{\mathrm{aircraft}}$ is the aircraft altitude, followed by three additional layers to allow for some vertical variation in molecular absorption. The molecular absorption is due mainly to trace gases plus ${\mathrm{O}}_{2}$, while molecular scattering is driven by ${\mathrm{N}}_{2}$ and ${\mathrm{O}}_{2}$ gas, which comprise the bulk of the atmosphere. For the calculation of absorption by gases in each window channel (band), MODTRAN [15] was used with the Mid-latitude Summer model atmosphere. In addition to the gases ${\mathrm{N}}_{2}$ and ${\mathrm{O}}_{2}$, the following trace gases were considered: ${\mathrm{H}}_{2}\mathrm{O}$, ${\mathrm{H}}_{2}\mathrm{O}$ continuum, ${\mathrm{CO}}_{2}$, ${\mathrm{O}}_{3}$, ${\mathrm{N}}_{2}\mathrm{O}$, CO, ${\mathrm{CH}}_{4}$, NO, ${\mathrm{SO}}_{2}$, ${\mathrm{NO}}_{2}$, ${\mathrm{NH}}_{3}$, ${\mathrm{HNO}}_{3}$. The Mid-latitude Summer model was created in the 1980s, and as a result the well-mixed gases that absorb in the RSP channels need to be scaled to current values. The default value of 330 ppmv of ${\mathrm{CO}}_{2}$ is scaled to a 2012–2014 value of 400 ppmv of ${\mathrm{CO}}_{2}$ using a scaling factor of 1.2. Absorption by water vapor is turned off in the MODTRAN model, and is instead derived using measurements from RSP’s 960 nm channel and the following parameterization for the water vapor column amount, ${w}_{\mathrm{col}}$ [cm] [16,17]:

### 2. Aerosol Models

Aerosols are assumed to be bimodally distributed, with a fine mode and a coarse mode. The complex refractive index of the fine mode is retrieved. The refractive index of the coarse mode is fixed to that of water, which is a reasonable assumption for hydrated sea-salt particles, particularly considering that the desired retrieval accuracy of the real refractive index is $\pm 0.02$.

The atmospheric layer that contains up to 95% of the lidar-backscattered signal above the ocean is assumed to contain all of the aerosol particles, distributed homogeneously. The bimodal lognormal volume-density size distribution is defined by

If we use the subscript $i=f$ to denote the fine mode and the subscript $i=c$ to denote the coarse mode, we have $V={V}_{f}+{V}_{c}$, and the fraction of fine mode particles in volume-density becomes ${f}_{v}={V}_{f}/V$. The definition of effective radius is

The effective radius [Eq. (9)] is a more useful parameter than the mode radius because the single-scattering properties depend on the geometrical cross section $\pi {r}^{2}$ [19,20], and for weak absorption the single-scattering albedo is proportional to the effective radius. Thus, aerosols with the same effective radius and effective variance tend to have similar scattering properties independent of size distribution. The aerosol scattering is therefore described more concisely by the effective radius, which is a function of both the mode radius and mode width, rather than by the mode radius alone.

The assumption of a single homogenous aerosol layer was applied to both the simulated data (Section 5) and real data (Sections 6 and 7). For real data we rely on lidar data to identify the aerosol location; otherwise, information about the aerosol location would also have to be retrieved. The inversions of real/simulated data are described in Section 3. For inversions of real RSP data, we determined the aerosol height in kilometers (km) from collocated HSRL data, by integrating the measured backscatter from the surface to 95% of the total measured aerosol backscatter, and used the height corresponding to the 95th percentile as the aerosol top height. Thus, the aerosols were approximated as being homogeneously distributed between 0 km and the aerosol top height, representing a column aerosol retrieval.

The MAPP aerosol model consists of a Lorenz–Mie [19,21–23] code that is run on the fly—at each step in the iterative retrieval—to calculate the scattering properties of spherical particles, i.e., the single-scattering albedo and scattering phase matrix needed for polarized radiative transfer calculations. The MAPP aerosol model incorporates fine-mode effective radii in the range [0.094 μm, 0.51 μm] and coarse-mode effective radii in the range [0.626 μm, 5.1 μm], which extends the size range used in the SeaDAS aerosol model for both modes (see Table 1). The goal is to capture most types of aerosols that can be modeled using spheres. Therefore, nonspherical dust particles, as well as aerosol particles with an effective radius above 5.1 μm, are not considered. Smoke particles are captured if the (ambient) refractive index is less than 1.65, and the (ambient) imaginary part is less than 0.03. Since the Mie code is run on the fly, MAPP retrieves aerosol optical depth and aerosol microphysical parameters directly. We found it useful to preserve a gap in the effective radius [24–26] between 0.51 to 0.626 μm for the fine and coarse modes to reduce crosstalk in the retrieval.

*Comparisons with the MODIS SeaDAS Aerosol Model:* The SeaDAS aerosol model [25] currently being used in the atmospheric correction for MODIS/SeaWiFS ocean color retrievals parameterizes the particle size of the coarse and fine modes, the real refractive index, and the single-scattering albedo, to vary with relative humidity. For example, at a relative humidity of 70% or $\mathrm{RH}=0.7$, ${r}_{vf}=0.158\text{\hspace{0.17em}}\mathrm{\mu m}$, and ${r}_{vc}=2.927\text{\hspace{0.17em}}\mathrm{\mu m}$. The standard deviations for both modes are kept fixed at all relative humidities: ${\sigma}_{gf}=0.437$, corresponding to an effective variance of ${v}_{\mathrm{eff},f}=0.21$, and ${\sigma}_{gc}=0.672$, corresponding to an effective variance of ${v}_{\mathrm{eff},c}=0.57$. For the atmospheric correction, one of 10 possible fine-mode fractions in volume-density, ${f}_{v}$, can be retrieved: 0, 0.01, 0.02, 0.05, 0.10, 0.20, 0.30, 0.50, 0.80, 0.95. Compared to using the SeaDAS aerosol model to retrieve a three-parameter set of {RH, ${\tau}_{f}$, ${\tau}_{c}$}, MAPP retrieves the two optical depths {${\tau}_{f}$, ${\tau}_{c}$} and directly retrieves six aerosol microphysical parameters: {${r}_{nf}$, ${r}_{nc}$, ${n}_{rf}$, ${n}_{\text{if}}$, ${\sigma}_{gf}$, ${\sigma}_{gc}$}.

#### C. Ocean Model

The ocean surface roughness is modeled by a one-dimensional Gaussian (Cox–Munk) distribution of surface slopes, parameterized by wind speed [27]. The underwater scattering and absorption properties are described by the one-parameter Chowdhary polarized ocean bio-optical model (C2012, C2006) [6,7], which is parameterized by the chlorophyll concentration, CHL [$\mathrm{mg}/{\mathrm{m}}^{3}$]. The one-parameter C2012 model was originally designed for chlorophyll concentrations $0\le \mathrm{CHL}\le 3$ [$\mathrm{mg}/{\mathrm{m}}^{3}$], but we found it useful to allow chlorophyll concentrations up to $10\text{\hspace{0.17em}}\mathrm{mg}/{\mathrm{m}}^{3}$ in order to represent more productive waters. Results from inversions of RSP polarimeter data collected during the SABOR campaign demonstrated the limitations of the original model and the improvement resulting from its extension to CHL values as large as 10 [$\mathrm{mg}/{\mathrm{m}}^{3}$]. These results are presented in Section 6. The C2012 model is summarized in Appendix B.

#### D. Vector Radiative Transfer Model

The vector radiative transfer code we used is based on doubling-adding principles first proposed by H.C. van de Hulst (1963) [28] and developed starting in the late 1960s by James Hansen and Joop W. Hovenier [19,29–31], which has been continuously upgraded [32,33]. Adding refers to combining two inhomogeneous layers, while doubling refers to combining two homogenous layers. Full polarization is available (all four Stokes parameters). The details are given in Appendix A.

## 3. OPTIMAL ESTIMATION

To test the MAPP algorithm, we generated simulated data for the seven RSP window channels at $\lambda =410$, 469, 555, 670, 864, 1594, and 2264 nm, and added noise as discussed in Section 4.B. The complete list of RSP channels is described in Section 4.A. The 960 nm channel to determine the water vapor absorption uses Eq. (3).

The cost function is defined as [34]

The vector radiative transfer model, described in Section 2.D, is our forward model, providing ${R}_{{I}_{L}}$, ${R}_{{I}_{R}}$, and DoLP, represented by $\mathit{f}$, which are a function of the state vector $\mathbf{x}$, Eq. (1), and which provide a suitable model for the measured ${R}_{{I}_{L}}$, ${R}_{{I}_{R}}$, and DoLP, represented by $\mathbf{y}$ (either real measurements from RSP or synthetic measurements generated by the forward model with noise added). The first term may be called the *data* term since it depends on residuals of the forward model and the measurement, taking into account measurement error through its covariance matrix ${\mathbf{S}}_{\epsilon}$. The second term may be considered the *a priori* term since it is the departure of the state vector $\mathbf{x}$ from the *a priori* state vector ${\mathbf{x}}_{a}$, with *a priori* uncertainty provided by the covariance matrix ${\mathbf{S}}_{a}$. The *a priori* state vector and covariance matrices for our problem are defined below. It is convenient to define the *normalized* cost function of the data term by dividing by the number of measurements, $m$,

Generally speaking, for a successful retrieval, $\frac{1}{m}\sqrt{\mathrm{\varphi}(\mathbf{x})}<1$, and if there happen to be redundant measurements, then $\frac{1}{m}\sqrt{\mathrm{\varphi}(\mathbf{x})}\ll 1$. For inversions of real RSP data in this paper, we consider retrievals successful if ${\chi}^{\prime}<0.1$.

The optimal-estimation method used in this paper follows a method that has been used for inverting atmospheric temperature and water vapor vertical profiles, cloud, and surface properties from hyperspectral infrared sounders [35]. It was found to be reliable and efficient in terms of the number of forward model evaluations required. The inversion results from real and simulated data presented in this paper used this method. The state vector $\mathbf{x}$ is transformed into fifth-root-space ${\mathbf{x}}_{\mathbf{1}}$ to smooth changes in the state parameters, analogous to a transformation into log-space,

In order to ensure that the state vector does not go out of bounds [described by ${\mathbf{x}}^{\mathrm{low}}$ and ${\mathbf{x}}^{\mathrm{high}}$ using the ranges in Eq. (2)], we found that the following transformation to $\mathbf{b}$-space works well:

If $\mathbf{x}$ is out of bounds, the argument of the log will be negative, and will evaluate to an imaginary number. In order to transform from $\mathbf{b}$-space to ${\mathbf{x}}_{\mathbf{1}}$-space, we can use the transformation

Then $\mathbf{x}$ is given by $\mathbf{x}={\mathbf{x}}_{\mathbf{1}}^{5}$. The retrieval is performed in $\mathbf{b}$-space. Therefore, by definition, ${\mathbf{x}}_{\mathbf{1}}\in {\mathbb{R}}^{+}$, and thus $\mathbf{x}$ will also be positive and real-valued. The $m\times n$ Jacobian matrix $\mathbf{K}\equiv \frac{\partial}{\partial \mathbf{x}}\mathit{f}$ is expressed in $\mathbf{b}$-space by

The *scaled* cost function is a scalar value ${\chi}_{s}^{2}$, defined by

*a priori*state vector in $\mathbf{b}$-space, and ${\mathbf{S}}_{\mathbf{1}a}$ is the

*a priori*covariance in $\mathbf{b}$-space proportional to the Jacobian. The following iteration over $i$ is performed until ${\mathbf{x}}_{\mathbf{1}}$ converges,

We used a conservative and simple way to specify the *a priori* state vector, ${\mathbf{x}}_{a}$, using the mean of the allowable range of each retrieval parameter, which is also used as the first guess ${\mathbf{x}}_{i}$,

The *a priori* covariance matrix, defined as a diagonal matrix, is given by

*a priori*values, which represent $1\sigma $ uncertainties, and is set equal to the

*a priori*state vector, so that ${\overrightarrow{\sigma}}_{a}={\mathbf{x}}_{a}$ and ${\overrightarrow{\sigma}}_{a}\circ {\overrightarrow{\sigma}}_{a}$ denotes the Schur product. [Also known as the Hadamard product, it is simply the element-by-element product. See https://en.wikipedia.org/wiki/Hadamard_product_(matrices).] The measurement error covariance matrix ${\mathbf{S}}_{\epsilon}$ is given in Section 4.B.

Once the retrieval has converged to $\widehat{\mathbf{x}}$, we calculate the posterior covariance matrix at $\widehat{\mathbf{x}}$, or the state error covariance matrix, given by

The square root of the diagonals of ${\widehat{\mathbf{S}}}^{-1}$ give the $1\sigma $ uncertainty estimates of the retrieval accuracy, and the non-diagonal elements indicate how correlated the retrieval parameters are to each other. A diagonal or nearly-diagonal ${\widehat{\mathbf{S}}}^{-1}(\widehat{\mathbf{x}})$ indicates that the retrieval parameters are orthogonal to each other, without overlapping impacts on the modeled measurements. The RSP MAPP data files with the retrievals provide the uncertainty estimates of the state vector given by Eq. (21), and the propagated uncertainty for the fine- and coarse-mode effective radii, effective variances, and the fine-mode single-scattering albedo.

#### A. Desired Uncertainty Targets

The desired [36] one-standard-deviation ($1\sigma $) target accuracies for each aerosol retrieval parameter are summarized in Table 2. As the aerosol optical depth becomes small, it becomes difficult to retrieve aerosol properties.

## 4. MEASUREMENTS

#### A. Polarimeter Measurements

The RSP instrument [1] is capable of measuring the $I$, $Q$, and $U$ Stokes parameters from multiple viewing angles and at multiple wavelengths. However, for this capability to be realized, the vector of aircraft motion must be parallel (or near-parallel) to the scanning direction. Data where the aircraft has substantial yaw is therefore excluded from this study. The exact nine RSP channel wavelengths, with FWHM in parentheses (nanometer, nm), are 410.27 (30), 469.13 (20), 554.96 (20), 670.01 (20), 863.51 (20), 960.0 (20), 1593.51 (60), 1880.0 (90), 2263.51 (120) nm, with angles measured between $\pm 65\xb0$ from zenith. These FWHM values are used to integrate the gaseous absorption in each channel, assuming a rectangular response function centered at the channel wavelength (see Section 2.B.1 and Chen *et al.*, 2017 [37]).

The upwelling (reflected) total and polarized radiances (Stokes parameters) measured in the Earth’s atmosphere are defined by

*reflectance*of the $I$, $Q$, or $U$ Stokes parameter, respectively. The RSP L1B files contain measurements of calibrated

*normalized radiances*, which are defined by $(\frac{\pi}{{F}_{0}}\times \text{Total Radiance})$,

Therefore, the RSP measurement is related to the reflectance calculated from vector radiative transfer programs by

To retrieve aerosol microphysical and ocean color parameters, MAPP used all seven of RSP’s window channels, and the 960 nm water vapor detection channel was used to correct the measurements for water vapor absorption as described in Section 2.B. RSP also has a channel at 1880 nm for cirrus detection using water vapor absorption, but that channel was used only for cloud screening. The RSP field of view (FOV) is 14 mrad corresponding to a horizontal resolution of about 100 m with measurements occurring about every 0.8 s corresponding to 72 revolutions per minute. The RSP measurements were horizontally averaged $\pm 30\text{\hspace{0.17em}}\mathrm{s}$, or 6.6 km assuming an aircraft cruising speed of about 110 m/s, since aerosol properties would not be normally expected to vary significantly over this distance. The measurements are then mapped to the same point on the ocean surface. As a pre-processing step, a cloud mask was constructed using HSRL/RSP cloud products to screen for clouds within $\pm 160\text{\hspace{0.17em}}\mathrm{s}$ of a particular scene, to mitigate contamination from cloud 3D side illumination effects. Thus, no RSP MAPP retrievals should have been attempted within 160 s of a cloud, or about 17 km, but clouds to the side of the aircraft could currently go undetected.

#### B. Polarimeter Measurement Noise

Optimal estimation fits the forward model to the measurements while accounting for measurement errors via the measurement covariance matrix ${\mathbf{S}}_{\epsilon}$. The form of ${\mathbf{S}}_{\epsilon}$ depends upon the measurements and the instrument calibration. We adopted the commonly used approximation that the measurements are independent [38,39], so that the measurements are uncorrelated and ${\mathbf{S}}_{\epsilon}$ is a diagonal matrix, which is valid to the first order for RSP. For all channels, RSP has an accuracy in the Stokes parameters {${R}_{I}$, ${R}_{Q}$, ${R}_{U}$} of 3.5%, an accuracy of 0.2% for the degree of linear polarization, and polarization azimuth within 0.5°.

**Simulated data RSP error model:** For simulated data, we assumed an error of 2% normally distributed for each Stokes parameter in the set {${R}_{I}$, ${R}_{Q}$, ${R}_{U}$}. Thus, for each simulated measurement {${R}_{I}$, ${R}_{Q}$, ${R}_{U}$}, there is a 99.7% probability that the measurement will be found within $\pm 6\%$ of the measured value. The simulated RSP measurement error covariance thus used ${C}_{I}^{\mathrm{sim}}={C}_{Q}^{\mathrm{sim}}={C}_{U}^{\mathrm{sim}}={(0.02{R}_{I,Q,U})}^{2}$, and which was then propagated to ${R}_{{I}_{L}}=({R}_{I}+{R}_{Q})/2$, ${R}_{{I}_{R}}=({R}_{I}-{R}_{Q})/2$, and the degree of linear polarization, $\mathrm{DoLP}=\frac{\sqrt{{R}_{Q}^{2}+{R}_{U}^{2}}}{{R}_{I}}$: ${\mathbf{S}}_{\u03f5}^{\mathrm{sim}}\equiv {C}_{\mathrm{RSP}}^{\mathrm{sim}}=\mathrm{diag}({C}_{{I}_{L}}^{\mathrm{sim}},{C}_{{I}_{R}}^{\mathrm{sim}},{C}_{\mathrm{DoLP}}^{\mathrm{sim}})$.

**Real data RSP error model:** The actual RSP measurement error covariances on ${R}_{I}$, ${R}_{Q}$, ${R}_{U}$, ${R}_{{I}_{L}}$, ${R}_{{I}_{R}}$, and ${R}_{\mathrm{DoLP}}$ are given in Appendix C.

#### C. Lidar Measurements

The NASA Langley airborne High-Spectral-Resolution Lidar (HSRL-1 and HSRL-2) beam is linearly h-polarized (i.e., horizontal or ∥) and both the h-polarized and v-polarized (i.e., vertical polarization or ⊥) components of the returned signal are measured, the latter of which provides the “depolarization” of the returned transmitted signal.

The lidar equation [2] for the measured total scattering (TS) signal $\mathrm{[W}/{\mathrm{m}}^{2}/\mathrm{sr}]$ for each channel is

High-spectral-resolution lidars like HSRL-1 and HSRL-2 have an additional 532-nm channel which is sensitive to only the molecular ($m$) backscatter and thereby provides a means to retrieve the profile of extinction [2]:

Here $F$ is the transmission through the iodine filter employed in the molecular channel to optically remove aerosol backscatter, and ${C}_{m,532}^{\parallel}$ is a system calibration constant. The iodine is kept at high temperature so that it is always in the gas phase. Then the aerosol extinction coefficient can be obtained by differentiating Eq. (26) [2]. HSRL-2 has an additional high-spectral-resolution channel at 355 nm. Thus HSRL-2 can measure five aerosol parameters in each vertically resolved bin: three backscatter values at 355, 532, and 1064 nm, denoted $\beta $, and two extinction values at the high-spectral-resolution channels, 355 and 532 nm, denoted $\alpha $. These five HSRL-2 measurements ($3\beta +2\alpha $) allow us to make comparisons of AOD at 532 and 355 nm, the 355/532 Ångstrøm exponent, the aerosol lidar ratio at 532 and 355 nm, and can also be used to retrieve microphysical aerosol parameters directly [40]. The three HSRL-1 measurements ($2\beta +1\alpha $) allow us to compare AOD and the lidar ratio at 532 nm and contain significant information about aerosol vertical distribution and type. In 2012, HSRL-1 completed upgrades to make underwater measurements of the diffuse attenuation coefficient, ${K}_{d}$, and the hemispherical backscatter coefficient, ${b}_{\mathrm{bp}}$.

### 1. HSRL AOD

HSRL lidars can measure aerosol optical depth by calculating the transmittance at a point near the aircraft (${r}_{1}$) and near the surface (${r}_{2}$) using the molecular lidar return ${P}_{m,532}^{\parallel}(r)$ and the theoretical molecular lidar backscattering coefficient ${\beta}_{m}^{\parallel}$ at the two points [2]

### 2. HSRL Column-Averaged Intensive and Extensive Parameters

The HSRL-intensive parameters, such as the lidar ratio [sr] and the lidar aerosol depolarization ratio (in the single-scattering limit) do not depend on aerosol concentration, but only the aerosol scattering and absorption cross sections [41]. By contrast, extensive parameters, such as the aerosol backscatter [1/km/sr] and aerosol extinction ${\mathrm{km}}^{-1}$, depend on the aerosol concentration. Note that both lidar-intensive and lidar-extensive parameters are inherent optical properties, since they depend on the aerosol concentration and cross sections, but not on the light field itself. We found it useful to compare the HSRL “column-averaged” lidar ratio $\u27e8{S}_{a}\u27e9$ and Ångstrøm exponent at 355/532 (available on HSRL-2) to the RSP MAPP retrieved column-average values of lidar ratio and Ångstrøm exponent. The HSRL “column-averaged” lidar ratio is calculated as follows:

### 3. MAPP Aerosol Top Height and Lidar Parameter Mixing Rules

The aerosol top height, ${h}_{t}$, is defined as the height at which

This definition of aerosol top height has the advantage of being simple and powerful, although complex, multi-layered aerosol scenes may require a more sophisticated approach.

The corresponding RSP MAPP column-averaged values for backscattering and extinction coefficients and lidar ratio were calculated using the same equations for each mode separately. Since the MAPP retrieval assumes the aerosols to be distributed homogeneously between the surface $h=0$ and the aerosol top height $h={h}_{t}$, the integration is from 0 to ${h}_{t}$, and the backscattering and extinction coefficients are assumed to be constant with height. Thus, we have $\u27e8\beta \u27e9=\beta $, $\u27e8\alpha \u27e9=\alpha $, $\u27e8{S}_{a}\u27e9=\frac{\alpha}{\beta}$ for each aerosol mode, fine and coarse. The following mixing rules are applied to calculate the RSP estimate of the column lidar ratio, which is the ratio of the total extinction to the total backscatter,

### 4. HSRL Ocean Products

The HSRL-1 instrument was upgraded [42] with detectors that quickly recover from the strong Fresnel surface reflection and detect the underwater signal [43]. The lidar beam diffuse attenuation coefficient, ${K}_{\mathrm{beam}}$, is measured along the direction of the lidar beam by using the molecular backscatter channel, normalized to the molecular signal directly above the water. The slope of the molecular backscatter channel with depth, $z$, is directly related to the diffuse attenuation coefficient [42,44],

*in situ*measurements of ${b}_{\mathrm{bp}}$ ([44,42]) so that

## 5. SIMULATED RESULTS

Simulations were performed using the 10-parameter state vector defined by Eq. (1), and the MAPP retrieved results using a simulated dataset are depicted in Fig. 1. The truth values were selected by using a random uniform distribution to select each retrieval parameter, independently. The ranges for the 10 parameters are given by Eq. (2). The aerosols were distributed homogeneously in a single layer between [0, 4 km]. The sensor geometry was fixed to a solar cosine zenith angle of 0.936 and relative azimuth of 105.27°. For each scene of simulated data, we randomly added Gaussian noise (one standard $\mathrm{deviation}=2\%$ of signal) to the Stokes parameters {${R}_{I}$, ${R}_{IQ}$, ${R}_{U}$} and propagated the noise to the set of Stokes parameters {${R}_{{I}_{L}}$, ${R}_{{I}_{R}}$, DoLP}. The simulated retrieval then used all seven RSP window channels simultaneously using the set {${R}_{{I}_{L}}$, ${R}_{{I}_{R}}$, DoLP} and the optimal estimation method described in Section 3.

For the simulated study, 1024 scenes were randomly constructed, and then the retrieval was performed on each of them. Out of the 1024 scenes, 902 fit the criteria where ${\tau}_{555f}\ge 0.05$ and ${\tau}_{555c}\ge 0.02$. Of these retrievals, 87% of the scenes were successful in that the eight retrieved aerosol and wind speed parameters were all within $3\sigma $ (relative to the $1\sigma $ uncertainty goals listed in Table 2) of the true values. And 67% of the cases successfully retrieved chlorophyll concentration to $3\sigma $, or $2.1\text{\hspace{0.17em}}\mathrm{mg}/{\mathrm{m}}^{3}$. Note that for some scenes we would not expect to be able to retrieve all 10 parameters within 3 standard deviations (for example, it is difficult to retrieve coarse-mode aerosol properties when the coarse-mode optical depth is small, especially if the fine-mode optical depth is large).

## 6. RESULTS FROM SABOR, 31 JULY 2014

In this section we present the results from SABOR 31 July 2014, which included both shallow- (less than 30 m ocean depth) and deep-water ocean measurements, and the flight track is shown in green in Fig. 2 together with the MODIS Aqua true color image. The other flight tracks and the ship track are also shown for reference. The HSRL extinction and aerosol typing for this day are given in Figs. 3(a) and 3(b), respectively. The resulting RSP MAPP retrievals of aerosol optical depth, $\tau $, ${b}_{bp}$, and ${K}_{d}$, are depicted in Fig. 3(c). There was heavy cirrus reported as the aircraft ascended until its cruising altitude at around 14 UTC, and became thinner, broken cirrus from 14.03 UTC onwards, with clear skies reported at 14.52 UTC. The influence of the cirrus (which could also be super-cooled liquid water droplets or mixed-phase cloud) appears to erroneously increase the aerosol optical depth. For this day and the TCAP case presented in the next section, all successful retrievals based on the magnitude of the cost function are displayed, without the altitude mask or clouds-above-aircraft mask used for the correlation plots. To help illustrate possible factors for discrepancies, the following masks are also displayed: clouds-above-aircraft (cyan), aircraft not at cruising altitude (magenta), aircraft turns/course corrections (yellow), and clouds below the aircraft (blue). According to the flight logs, there should have been clear sky conditions without significant clouds above the aircraft, as indicated by the lack of a cyan mask in Fig. 3(c). The variability of the ocean color measurements on this day, in addition to the complex, absorbing aerosols, make it a challenging and interesting case to study. Over the ocean, there was a tiny cloud spotted at 15:29 UTC below the aircraft, and a few small, low-level cumulus clouds were detected at around 15:38 through 15:42 UTC, followed by tiny, rogue clouds sprinkled here and there. Starting at around 16:02 UTC, the aircraft flew into a low-level field of cumulus clouds, and it can be seen that the AOD begins to drop significantly and the aerosol type changes from predominantly smoke to polluted marine. (Where clouds are detected by the cloud mask based on the HSRL-1 and RSP products, RSP MAPP retrievals were not attempted.) There were also some very thin clouds near the aircraft at around 16:03 and 16:06 UTC. Another thin cloud was near the aircraft, which grew thicker starting at 16:20 through 16:28 UTC, with the aircraft brocken spectre and surrounding glory clearly visible in the aircraft’s nadir camera, followed by another thin cloud at 16:29 through 16:34 UTC. The RSP retrievals for these thin, opaque clouds show optical depths between 1 and 3 with droplet sizes of 5 μm or smaller [46], and the cloud top is at 8 km, so they are significantly super-cooled.

The atmosphere (aerosols and molecules) is responsible for the major part of the signal measured by RSP. Therefore, when the aerosol retrieval is successful, the aerosol optical depth in Fig. 3(c) should match within 0.02 between RSP and HSRL, but not necessarily for all cases (see Section 8 for a discussion of discrepancies, error sources, and retrieval comparisons for the entire SABOR campaign). HSRL-1 can measure plankton and underwater particulate matter, but does not have the extra 355 nm channel available on HSRL-2. It is difficult to compare to the column-averaged HSRL-1 532 nm lidar ratio since the scenes on this day consisted of multi-layered aerosols.

#### A. SABOR 532 nm AOD Comparison

The HSRL ocean profiling capability allows us to test whether the extended one-parameter ocean model is helpful in improving our AOD retrievals, and we can compare the RSP MAPP retrieved AOD at 555 nm to the HSRL-1 AOD at 532 nm [see Fig. 3(c)] in order to validate our aerosol retrievals. We note that in most cases the agreement is within 0.02–0.04. Aerosols that happen to be near or above the airplane can explain larger RSP MAPP retrieved values compared to HSRL-1 but not smaller values.

#### B. SABOR Ocean Product Comparison

The HSRL-1 ocean measurements indicate that a significant amount of complex, coastal-type water was encountered, in that ${K}_{d}$ and ${b}_{\mathrm{bp}}$ do not covary closely with chlorophyll, i.e., they do not lie along a curve in Fig. 4, even for deeper waters (defined here as having an ocean bottom greater than 30 m). However, we can also see that a curve is not an entirely unreasonable approximation, where we have used the HSRL column-averaged values for $\u27e8{K}_{d}\u27e9$ and $\u27e8{b}_{\mathrm{bp}}\u27e9$, suggesting that the one-to-one relationship between ${K}_{d}$ and ${b}_{\mathrm{bp}}$ in our simplified one-parameter bio-optical model is not without merit.

#### C. Bio-optical Model Impact on Aerosol Microphysics

Figure 5 illustrates how the aerosol microphysics products are “coupled” to the ocean bio-optical model, since changing the bio-optical model changes the aerosol microphysics retrievals. The set of retrievals labeled “v1.21” was performed using the $[\mathrm{0,3}]\text{\hspace{0.17em}}\mathrm{mg}/{\mathrm{m}}^{3}$ chlorophyll concentration ocean model (colored in cyan), while the set of retrievals labeled “v1.22” was performed using the extended $[0,10]\text{\hspace{0.17em}}\mathrm{mg}/{\mathrm{m}}^{3}$ ocean model (colored in red). The $[0,3]\text{\hspace{0.17em}}\mathrm{mg}/{\mathrm{m}}^{3}$ chlorophyll concentration was saturated in high-sediment/mesotrophic waters (${K}_{d}>0.15$), since this situation corresponds to the maximum allowable chlorophyll concentration value of $3.0\text{\hspace{0.17em}}\mathrm{mg}/{\mathrm{m}}^{3}$. The saturation shows how bio-optical modeling errors can propagate into the aerosol retrievals (not just vice versa). We found that the extended $[0,10]\text{\hspace{0.17em}}\mathrm{mg}/{\mathrm{m}}^{3}$ one-parameter ocean model helped to smooth out some of the oscillations in the aerosol single-scattering albedo and real part of the refractive index, although it had less impact on the aerosol optical depth and aerosol effective radius, which tend to be easier to retrieve. Bio-optical models designed with three (or more) parameters can address the limitations in the one-parameter model, and potentially improve match-ups with the HSRL-1 ocean measurements and the quality of the aerosol microphysical retrievals.

## 7. RESULTS FROM TCAP, 17 JULY 2012

The NASA Langley HSRL-2 instrument flew together with the NASA GISS RSP during the 2012 TCAP mission. In this section we present the results from the TCAP flight on 17 July 2012, which is highlighted in green, together with the MODIS Aqua true color image, in Fig. 6.

As the aircraft flew out over the ocean, and before turning point back to Cape Cod, there was a low-altitude fog observed near the ocean surface at 14.75 (and again 16.25 UTC) in the RSP TCAP cloud retrievals [46], which is also apparent in the lidar backscatter in Fig. 7(a). The fog had an optical depth generally between 2 and 3 with a multi-modal size distribution dominated by large drops of 10–17 μm [47] in effective radius, so it is more like a surface cloud than a classic radiation fog [48]. Above the fog, clouds at around 2 km are also apparent in the RSP cloud retrievals and in the HSRL-1 backscatter. The HSRL-1 AOD increases near the clouds, which could be due to hydrated aerosol or small cloud droplets. RSP also detected clouds near 15.5 UTC when the aircraft flew over Cape Cod as it was turning back for its return leg out over the ocean.

#### A. TCAP 532 nm AOD Comparison

Both HSRL-1 and HSRL-2 are able to directly and accurately measure extinction ${\mathrm{km}}^{-1}$ at 532 nm, and thus optical depth, which is extremely useful for validation and improvement of atmospheric correction algorithms for passive instruments like radiometers and polarimeters. The HSRL-2 extinction product is plotted in Fig. 7(a). The HSRL aerosol typing product [41] is depicted in Fig. 7(b), which uses the wavelength-dependent lidar-intensive parameters to discriminate between aerosol types: on this day, the aerosol was characterized by a homogenous layer of urban aerosol outflow, but the plume or lofted aerosol layer overflown at 15.5 UTC and again at 17 UTC may have different properties than the aerosol below it, despite being categorized as the same type.

#### B. TCAP Aerosol Microphysics

In Fig. 7(c), top panel, we compare the RSP-MAPP-retrieved aerosol optical depth at 532 nm to the HSRL-2 directly measured optical depth at 532 nm. The aerosol optical depths for this day generally agree within 0.02, despite thin cirrus (or super-cooled liquid clouds) reported above the aircraft during the beginning of the flight. In the second panel from top in Fig. 7(c), we plot the aerosol RSP-MAPP-retrieved single-scattering albedo (SSA), which quantifies how much the aerosols absorb. Defined as the scattering coefficient divided by the extinction coefficient, an SSA value of 1.0 represents a completely non-absorbing aerosol, values close to 1 represent weakly absorbing aerosols, while values smaller than 0.95 represent absorbing aerosols. Thus, the aerosols were found to be weakly to moderately absorbing, with retrieved aerosol SSA values between 0.9 and 1.0. Note that for a fine mode optical depth of 0.2, an SSA value of 0.95 corresponds to an aerosol absorption optical depth of 0.01, which makes it challenging to detect. The RSP-MAPP-retrieved aerosol effective radius (third panel from top) is about 0.15–0.17 μm for these urban aerosols. The fourth panel depicts the fine-mode real part of the refractive index, which is found to vary between 1.5 and 1.6, although becoming closer to 1.4 later in the day.

#### C. TCAP Intensive/Extensive Product Comparison

In the first panel of Fig. 8, the optical depths measured by the HSRL-2 at 355 and 532 nm are plotted, compared to the RSP MAPP retrieved aerosol product at 532 nm and 355 nm. In the second panel, the 532 nm HSRL-2 directly measured aerosol lidar ratio, or the extinction-to-backscatter ratio, is plotted in black. The HSRL-2 column aerosol lidar ratio varies from about 50 to 70 sr on this day. The RSP-MAPP-retrieved lidar ratio is plotted for comparison at 555 nm (red) and at 532 nm (cyan). The RSP aerosol lidar ratio is generally within 20% of the HSRL-2 extinction-weighted aerosol lidar ratio, and although in most cases it is within 5–15 sr, it can vary up to 25 sr. Note that the polarimeter will be influenced by aerosols or ice/liquid clouds above the aircraft (see Section 8 for a discussion of discrepancies, sources of error, and retrieval comparisons for the entire TCAP campaign). In the third panel, the Ångstrøm exponent (355/532) measured by the HSRL-2 is compared to the RSP MAPP retrieval. We note that it is important to calculate it at 355/532, since when a size distribution is bimodal, the Ångstrøm exponent can change substantially, depending on which wavelength pair is used. The HSRL-2 Ångstrøm exponent is generally quite stable (1.3–1.6), while the RSP Ångstrøm exponent is generally lower (0.9–1.46), which would correspond to larger particles. Note that the HSRL-1 and HSRL-2 column aerosol lidar ratio will not include a contribution from aerosols near the surface, e.g. sea-salt aerosols in the (low) marine boundary layer, but the HSRL-2 Ångstrøm exponent represents all aerosols between the lidar and the surface.

## 8. STATISTICAL OVERVIEW OF RESULTS AND ERROR ANALYSIS

In the following correlation plots, a clouds-above-aircraft mask was applied, and an altitude mask was applied to select only cases when the aircraft was flying a remote sensing leg at cruising altitude. For the SABOR comparisons, only the “deep-water” scenes were selected, defined as having an ocean bottom of 30 m or greater, to avoid complications from ocean bottom reflectance and/or very turbid coastal waters. Although there was no instrument to specifically detect clouds above the aircraft during TCAP or SABOR, the flight logs facilitated construction of a clouds-above-aircraft mask, particularly for SABOR.

#### A. SABOR and TCAP AOD Correlations

Correlations of the SABOR and TCAP Aerosol-Above-Ocean (AAO) RSP MAPP 532 nm AOD results compared to the HSRL AOD measurements are given in Figs. 9 and 10. The one-to-one line is colored in black, and the dashed black lines represent the desired $\pm 1\sigma $ accuracy of the RSP MAPP algorithm, which is $\pm 0.02$ for the aerosol optical depth (AOD) at 532 nm. The HSRL-1 and HSRL-2 can measure AOD below the aircraft within $\pm 0.02$ at 532 nm. The red line represents the least-squares bisector.

The majority of RSP MAPP retrievals fall within $\pm 0.04$ HSRL-1 or HSRL-2 AOD at 532 nm. For the SABOR campaign, which had better reporting of clouds above the aircraft, of the 532 nm AOD comparisons in Fig. 9, 73% are within 0.04, 88% are within 0.06, and 98% are within 0.08, and $R$ is 0.933 with a root-mean-square deviation of about 0.0372. For the TCAP 532 nm AOD comparisons in Fig. 10(a), 53% are within 0.04, 74% are within 0.06, and 81% are within 0.08, and $R$ is 0.927 with a root-mean-square deviation of about 0.0673.

Correlations of the TCAP RSP MAPP 355 nm AOD results compared to the HSRL-2 355 nm AOD measurements are given in Fig. 10(b). The RSP MAPP retrievals are generally within $\pm 0.04$ AOD at 355 nm (red lines), and $R$ is 0.959 and the root-mean-square deviation of about 0.0694.

#### B. SABOR ${\mathsf{K}}_{\mathsf{d}}$ and ${\mathsf{b}}_{\mathsf{bp}}$ Ocean Product Correlation

Correlations of SABOR RSP MAPP ${K}_{d}$ and ${b}_{\mathrm{bp}}$ at 532 nm compared to HSRL-1 ocean measurements are given in Fig. 11. The one-to-one line is colored in black, and the dashed black lines represent the desired $\pm 1\sigma $ accuracies of the RSP MAPP algorithm, which are $\pm 0.1\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$ for ${K}_{d}$ and $\pm 0.0005\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$ for ${b}_{\mathrm{bp}}$. The ${K}_{d}$ root-mean-square deviation is 0.0221 and R is 0.867. At the lower limit for clearer waters, the RSP ${K}_{d}$ is $\sim 0.05{\mathrm{m}}^{-1}$ when the HSRL ${K}_{d}$ $\sim 0.06-0.075$, a 20%–45% bias. At the lower limit, the RSP ${b}_{\mathrm{bp}}$ approaches $0\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$, whereas the HSRL ${b}_{\mathrm{bp}}$ is $0.001\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$. At RSP MAPP retrieved ${b}_{\mathrm{bp}}$ values of $0.0005\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$, the HSRL ${b}_{\mathrm{bp}}$ generally varies between $0.001-0.0015\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$, which is a bias of 200%–300%, or in absolute terms, $0.0005-0.001\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$, which is close to the root-mean-square of about 0.000102. $R$ for ${b}_{\mathrm{bp}}$ is 0.759. There is a discrepancy in ${K}_{d}$ and ${b}_{\mathrm{bp}}$ for clearer waters that could be due to simplifications in the RSP MAPP ocean bio-optical model, which currently has only one parameter (chlorophyll concentration). The RSP MAPP passive ocean color retrieval uses multiple wavelengths to retrieve the bulk properties of the water assuming a single homogenous layer, whereas the HSRL measurement is at a single wavelength and is capable of providing vertical profiles in the water. We simply stress that the RSP ocean color retrieval correlates well with the average of the HSRL measurements, and therefore shows significant potential. In the future we plan to use a more advanced ocean bio-optical model to reconcile the differences at the low end (clear/oligotrophic waters) and at the high end (coastal and mesotrophic/eutrophic waters) between the ocean RSP retrieval and the HSRL measurement.

#### C. Active/Passive Discrepancies

The following conditions will lead to discrepancies between RSP passive retrievals and HSRL active measurements: (i) aerosols near the aircraft, (ii) aerosols within 100 m of the surface, (iii) two or more layers of different aerosol types in the column, (iv) lofted or multi-layered absorbing aerosols, (v) non-spherical aerosols, (vi) three-dimensional effects caused by clouds above or in the vicinity of the aircraft, including cloud shadowing and cloud brightening [49,50], (vii) aerosol above the aircraft, (viii) cloud above the aircraft, and (ix) inherent retrieval and measurement uncertainties including those for viewing geometry caused by variations in the aircraft pitch, yaw and roll, and ocean modeling errors.

The HSRL is able to detect the vertical extinction and backscatter profile of aerosols, but has larger uncertainties when aerosols are in the near field of the lidar or if the aerosols are very close to the surface, since contamination from the surface return becomes a concern (i and ii). But we note that the HSRL AOD product uses the lidar two-way transmittance to accurately measure aerosol AOD from the aircraft all the way down to the surface. The HSRL typing algorithm can potentially determine how many aerosol types are present, and the HSRL depolarization measurement can be used to identify the presence of non-spherical aerosols. Thus items (iii) through (vi) are largely avoidable from co-incident HSRL measurements, although three-dimensional effects from nearby clouds can be difficult to detect and will potentially require ancillary data and improved screening techniques. Aerosols and clouds above the aircraft will both hamper the algorithm’s performance, but we would not typically expect significant amounts of aerosol above the airplane. Therefore we focus on (viii) liquid or ice clouds above the aircraft as a probable cause of the horizontal “streamers” in the correlation plots for RSP AOD and (ix) retrieval uncertainties. For the latter, simplifications or unrealistic assumptions can cause the retrieval to diverge or converge to an erroneous solution. The current MAPP polarimeter retrieval is approximate in that it assumes that the aerosols are located in a single layer and modeled as a bimodal aerosol distribution, whereas in reality we know that aerosols can be layered in complex structures with different aerosol types. Also, the ocean bio-optical model is simplistic in that it depends on only a single parameter. The RSP versus HSRL correlation plots provide important campaign-wide aerosol and ocean product statistics, and the discrepancies indicate where the RSP MAPP retrievals can be further improved, such as enhanced above-aircraft-cloud and nearby cloud detection and ocean bio-optical modeling.

## 9. CONCLUSION

A new algorithm called MAPP has been created that can invert polarimetric data to retrieve detailed aerosol microphysical parameters such as aerosol single-scattering albedo and complex refractive index, in addition to ocean color products such as the diffuse attenuation coefficient. The MAPP algorithm is fully automated and capable of bulk processing entire field campaigns of data (thousands of scenes), and the entire framework is described in this paper. The MAPP algorithm has some unique advantages, since it does not rely on look-up tables and simultaneously retrieves aerosol and ocean products. For the first time, hundreds of aerosol microphysics retrievals from the RSP polarimeter have been compared to HSRL-1/HSRL-2 AOD at multiple wavelengths, and also comparisons have been made to the HSRL-1 ocean measurements.

The MAPP algorithm was tested on simulated data and used to analyze field data from AAO (Aerosol-Above-Ocean) scenes. These high-density retrievals from polarimeter measurements were collected during two aircraft campaigns, SABOR and TCAP. RSP MAPP retrievals were validated by comparing them to co-incident high-spectral-resolution lidar measurements obtained by the Langley airborne High-Spectral-Resolution Lidars HSRL-1 and HSRL-2. The AOD at 532 and 355 nm (when available) are compared, as are the 532-nm ocean measurements of the diffuse attenuation coefficient, ${K}_{d}$, and the hemispherical particulate backscattering coefficient, ${b}_{\mathrm{bp}}$.

The co-incident HSRL-1/HSRL-2 measurements of AOD and column-averaged intensive parameters (lidar ratio and the Ångstrøm exponent) were useful for evaluating the MAPP algorithm, and they show significant promise as additional validation tools for atmospheric correction algorithms for passive ocean color remote sensing. We found that the HSRL aerosol optical depth at 532 nm is an extremely useful and accurate check on the aerosol retrieval, i.e., the atmospheric correction, which is important, because accurate atmospheric correction is a prerequisite for retrieval of marine parameters with reliable accuracy from passive ocean color sensors. In addition, HSRL-1’s underwater measurement capability of the diffuse attenuation coefficient, the hemispherical backscattering coefficient, and potentially the hydrosol depolarization ratio are groundbreaking measurements for evaluating polarimeter- (e.g., RSP) and radiometer-retrieved (e.g. MODIS) ocean products. Taken together, these measurements have significant potential to advance the next stage of ocean color retrievals using data obtained by passive ocean color sensors. The agreement between the polarimeter and lidar observations, despite being very different measurement systems, underscores the importance of developing joint polarimeter/HSRL observations of the atmosphere/ocean system, since the two types of instruments are in many ways complementary. The RSP MAPP products are summarized on the NASA LaRC website [12] and are available for download at the NASA GISS RSP website [13].

Future work involves implementing a more advanced three- or four-parameter ocean bio-optical model into MAPP that has, for example, not only a chlorophyll concentration parameter (CHL), but also an independently varying color dissolved organic matter parameter (CDOM]), together with its spectral slope, ${S}_{\mathrm{CDOM}}$, and a hemispherical backscatter parameter, (${b}_{\mathrm{bp}}$) to allow for the effects of scattering by non-algal particles. A more comprehensive ocean model would allow us to model the two-dimensional ocean complexities detected by lidar as depicted in Fig. 4 and to potentially retrieve the CDOM and/or hemispherical backscatter wavelength-dependent slope, which can then be also compared against ship-based and buoy *in situ* ocean and underwater radiance-based measurements. It is also of interest to compare the MAPP aerosol retrievals to *in situ* measurements of aerosols made during SABOR and TCAP. We plan to add the capability to retrieve the aerosol layer height independently from RSP, since collocated lidar measurements are not always available. Another task is to include scattering by non-spherical aerosols, such as dust.

Lastly, the MAPP framework was designed with the ultimate goal of being able to use HSRL-2 measurements directly in a joint polarimeter+lidar retrieval. The joint HSRL+RSP retrieval algorithm is expected to improve upon the column-averaged microphysics of the polarimeter-only version presented in this paper, while simultaneously improving the vertically resolved microphysics retrievals of the HSRL-2-only retrievals [40,51].

## APPENDIX A: VECTOR RADIATIVE TRANSFER

The solution to the vector radiative transfer equation is given by the following recipe:

${\tau}_{a}$ is the optical thickness of the first layer, and ${\tau}_{b}$ is the optical thickness of the second layer. The exponential terms represent direct beam attenuation at solar zenith cosine ${\mu}_{0}$, while $\mathbf{T}$ is the diffuse transmission matrix. The bold-faced uppercase letters are $4\times 4$ matrices that represent diffuse scattering matrices, the reflection matrix, $\mathbf{R}$, the transmission matrix, $\mathbf{T}$, and the downwelling and upwelling ($\mathbf{D}$ and $\mathbf{U}$) matrices between the two layers. By multiplying the above reflection, transmission, upwelling, and downwelling matrices by the incident beam flux $\pi {\mu}_{0}{\mathit{f}}_{0}$, and by including the direct beam exponential term, we can find the Stokes vector components representing the reflection at the top, ${\mu}_{0}\pi {\mathit{f}}_{0}\mathbf{R}$, the transmission at the bottom, ${\mu}_{0}\pi {\mathit{f}}_{0}\mathbf{T}({\tau}_{a}+{\tau}_{b})+{\mu}_{0}\pi {\mathit{f}}_{0}{e}^{-({\tau}_{a}+{\tau}_{b}){\mu}_{0}}$, and the downwelling and upwelling (${\mu}_{0}\pi {\mathit{f}}_{0}\mathbf{D}$ and ${\mu}_{0}\pi {\mathit{f}}_{0}\mathbf{U}$) between the two layers. The superscript * denotes reflection (or transmission) matrices when illuminated from below, and for homogenous layers, ${\mathbf{R}}^{*}(\mu ,{\mu}_{0},\mathrm{\Delta}\varphi )={\mathbf{R}}^{*}(\mu ,{\mu}_{0},-\mathrm{\Delta}\varphi )$, where $\mu $ is the polar angle cosine and $\mathrm{\Delta}\varphi =\varphi -{\varphi}_{0}$ is the difference between the azimuthal angle $\varphi $ and the azimuthal solar angle ${\varphi}_{0}$.

## APPENDIX B: C2012 ONE-PARAMETER POLARIZED OCEAN BIO-OPTICAL MODEL

## 1. Scattering Coefficient of the Ocean

The total scattering coefficient for the ocean as a function of wavelength, $\lambda $ [nm], is given by

The hemispherical backscattering coefficient ${b}_{\mathrm{bp}}\equiv {b}_{\mathrm{p}}{q}_{\mathrm{p}}$ for ocean particles is parameterized as [53]

The normalization of ${F}_{\mathrm{p}}$ differs by a factor $\frac{1}{4\pi}$ to that commonly used by the ocean color community. The C2012 ocean model was constructed by constraining Eq. (B5) using the following empirical relation for ${q}_{\mathrm{p}}$ (C2006, MM2001), which has been modified so that there is no ${(\frac{\lambda}{660})}^{k}$ wavelength dependence (C2012) for the particulate backscattering ratio,

Thus, in this model, the hemispherical backscattering coefficient ${b}_{\mathrm{p}}\equiv {b}_{\mathrm{p}}(\lambda ,\mathrm{CHL})$ has wavelength dependence, but not the hemispherical backscattering ratio, ${q}_{\mathrm{p}}\equiv {q}_{\mathrm{p}}(\mathrm{CHL})$. Then the particulate hemispherical backscattering coefficient becomes

## 2. Hydrosol Mie Computations and Scattering Matrix

The ocean particles are modeled as a bimodal mixture (D–P) of detritus (D) and plankton (P) spherical particles. Each mode is assumed to have a Junge (power-law) size distribution with radii between 0.01 and 100 μm and Junge-exponents ${\gamma}_{\mathrm{det}}$ and ${\gamma}_{\mathrm{plk}}$, respectively. The real part of the refractive indices relative to water, ${n}_{r,\mathrm{det}}$ and ${n}_{r,\mathrm{plk}}$, are assumed to be wavelength independent. Mie computations are used to calculate the resulting scattering matrices ${\mathit{f}}_{\mathrm{det}}(\mathrm{\Theta})$, ${\mathit{f}}_{\mathrm{plk}}(\mathrm{\Theta})$, scattering cross sections ${\sigma}_{\mathrm{det}}$, ${\sigma}_{\mathrm{plk}}$, and the backscattering ratios ${q}_{\mathrm{det}}$, ${q}_{\mathrm{plk}}$. The imaginary part of the refractive index is set to zero, since its impact on the backscattering ratio is negligible (C2012). Note that we use an empirical relation, Eq. (B17), to determine the total D–P particle absorption and also use an empirical relation to calculate the particulate scattering coefficient from Eq. (B2), so that the hydrosol particulate single-scattering albedo is

But the combined D–P Stokes scattering matrix is given by the Mie-computed scattering cross sections and scattering Stokes matrices,

For each mode, the values are determined to be (${n}_{r,\mathrm{det}}=1.15$, ${\gamma}_{\mathrm{det}}=4.4$) and (${n}_{r,\mathrm{plk}}=1.04$, ${\gamma}_{\mathrm{plk}}=3.7$). The detritus fraction of the D–P mixture, ${f}_{\mathrm{det}}$, then depends on the chlorophyll concentration, and can be fitted by the following polynomial to within $\pm 0.003$:

The values above were determined as follows (C2012):

- (1) For a given refractive index ${n}_{r}$, the Junge exponent $\gamma $ was varied to minimize $d=\sqrt{\frac{1}{N}\sum _{i=1}^{N}h{({p}_{\mathrm{hyd}}({\mathrm{\Theta}}_{i})-{p}_{\mathrm{w}}({\mathrm{\Theta}}_{i}))}^{2}}$, where $p=-\frac{F21(\mathrm{\Theta})}{F11(\mathrm{\Theta})}$, ${p}_{\mathrm{hyd}}$ is for the plankton or detritus, ${p}_{\mathrm{w}}$ is for water using a depolarization ratio of $\delta =0.09$, and the scaling factor $h=4$ when ${p}_{\mathrm{hyd}}\ge 0.85$ and 1 otherwise.
- (2) A linear fit formula was derived from the results of (1): ${\gamma}_{\mathrm{hyd}}=6.63{n}_{\mathrm{hyd}}-3.25$.
- (3) The minimum and maximum values, ${q}_{p,\mathrm{min}}$ and ${q}_{p,\mathrm{max}}$, are obtained by using $\mathrm{CHL}=0.03$ and $3.0\text{\hspace{0.17em}}\mathrm{mg}/{\mathrm{m}}^{3}$ in Eq. (B6), respectively. Thus ${q}_{\mathrm{p},\mathrm{min}}=0.01081$ and ${q}_{\mathrm{p},\mathrm{max}}=0.00581$ for the four RSP ocean color channels at 410, 469, 555, and 670 nm.
- (4) Based on applying the constraint in (3), the real refractive indices are found to be ${n}_{r,\mathrm{det}}=1.15$ and ${n}_{r,\mathrm{plk}}=1.04$. Then Junge exponents ${\gamma}_{\mathrm{det}}=4.4$ and ${\gamma}_{\mathrm{plk}}=3.7$ are obtained from the fit in (2). The resulting effective radii are small, at 0.35 μm (plankton) and 0.034 μm (detritus), since the emphasis is to reproduce variations in ${q}_{p}$ corresponding to the MM2001 bio-optical model, and smaller particles contribute more to backscattering than larger particles.
- (5) Using the refractive index and Junge exponent values in (4), we can calculate the scattering matrices ${\mathit{f}}_{\mathrm{det}}(\mathrm{\Theta}),{\mathit{f}}_{\mathrm{plk}}(\mathrm{\Theta})$, scattering cross sections ${\sigma}_{\mathrm{det}}$ and ${\sigma}_{\mathrm{plk}}$, and backscattering ratios ${q}_{\mathrm{det}}$ and ${q}_{\mathrm{plk}}$.
- (6) The empirical relationship between ${q}_{p}$ as a function of CHL, Eq. (B6), is used with Eq. (B11) given values in (5) for ${\sigma}_{\mathrm{det}}$, ${\sigma}_{\mathrm{plk}}$, ${q}_{\mathrm{det}}$, ${q}_{\mathrm{plk}}$, to provide ${f}_{\mathrm{det}}$ as a function of CHL [Eq. (B12)]. The resulting D–P scattering matrix is consistent with a wavelength-dependent Fournier–Forand phase function (C2012).

## 3. Absorption Coefficient of the Ocean

In order to calculate the absorption coefficient of the ocean, C2012 uses a model of the diffuse attenuation coefficient ${K}_{d}(\lambda ,\mathrm{CHL})$. The diffuse attenuation coefficient is a *quasi* inherent optical property defined as [55]

*quasi*inherent optical property, the error resulting from the addition in Eq. (B14) is assumed to be small (MM2001).

The total absorption coefficient for the ocean is then given by solving the following equation for ${a}_{\mathrm{ocn}}$ (C2006):

The quantity $\alpha (\lambda ,\mathrm{CHL},{\theta}_{0})\frac{{b}_{\mathrm{b}}(\lambda ,\mathrm{CHL})}{{a}_{\mathrm{ocn}}}\equiv \frac{{E}_{u}(\lambda )}{{E}_{d}(\lambda )}$ is the ratio of the upwelling to downwelling irradiance measured just below the ocean surface and is equal to $\alpha (1+\frac{{b}_{\mathrm{b}}}{{a}_{\mathrm{ocn}}})(\frac{{b}_{\mathrm{b}}}{{a}_{\mathrm{ocn}}+{b}_{b}})$ when ${b}_{\mathrm{b}}$ is large compared to ${a}_{\mathrm{ocn}}$ [57].

Since we know the absorption for pure water, we can then find the particulate absorption (by plankton and implicitly CDOM),

Thus the C2012 bio-optical model depends only on chlorophyll concentration (one parameter), and is consistent with (i) the commonly used Fournier–Forand phase function for scattering by plankton particles, (ii) empirical measurements of scattering and absorption coefficients, and (iii) provides the full $4\times 4$ scattering matrix for polarized light calculations. The goal of a bio-optical model should be to capture the relevant changes in the inherent optical properties in the ultraviolet (UV) and visible with as few parameters as possible, and ideally to be as general and realistic as possible to allow application to as many different regions as possible. In that context, the C2012 model works well for having only a single parameter. Future work would reduce assumptions and increase the number of parameters to allow more detailed retrieval of in-water inherent optical properties.

## 4. Example C2012 IOP Calculation ($\mathsf{\lambda}=\mathsf{555}\text{\hspace{0.17em}}\mathsf{nm}$, $\mathsf{CHL}=\mathsf{1}\text{\hspace{0.17em}}\mathsf{mg}/{\mathsf{m}}^{\mathsf{3}}$)

From Morel and Mueller (2002) [57], $\alpha (\lambda ,\mathrm{CHL},{\theta}_{0})\approx 0.38$. ${K}_{d}(\lambda ,\mathrm{CHL})=0.09857$ (MM2001). ${b}_{\mathrm{b}}(\lambda ,\mathrm{CHL})=0.00431$ since ${b}_{\mathrm{b},\mathrm{sw}}(\lambda )=0.0019$ [52] and ${b}_{\mathrm{bp}}(\lambda ,\mathrm{CHL})=0.00236$ (MM2001). At $\mathrm{SZA}=30\xb0$, ${\mu}_{d}(\mathrm{SZA},\lambda ,\mathrm{CHL})=0.832$. ${\mu}_{u}=0.4$ for all wavelengths and chlorophyll concentrations (MM2001). Thus, ${a}_{\mathrm{ocn}}(\lambda ,\mathrm{CHL})=0.077515\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$.

## APPENDIX C: RSP UNCERTAINTY MODEL

The actual RSP measurement error covariances are given as follows, substituting in the values ${\sigma}_{\mathrm{floor}}^{\prime}=0.00002$, ${a}^{\prime}=1\times {10}^{-7}$, ${\sigma}_{\mathrm{ln}\text{\hspace{0.17em}}{K}_{1}}=0.005$ (0.5%), ${\sigma}_{\mathrm{ln}\text{\hspace{0.17em}}{\alpha}_{c}}=0.015$ (1.5%), ${\sigma}_{\mathrm{ln}\text{\hspace{0.17em}}{\alpha}_{1}}=0.002$ (0.2%):

## Funding

Langley Research Center; Goddard Space Flight Center (GSFC); National Aeronautics and Space Administration (NASA); U.S. Department of Energy (DOE) (DE-A06-76RLO 1830, DE-SC0006730, KP1701000/57131).

## Acknowledgment

This paper was funded by the NASA Aerosols-Clouds-Ecosystems (ACE) project, NASA ROSES Remote Sensing Theory for Earth Science, the NASA Ocean Biology and Biogeochemistry Program, and the NASA Science Innovation Fund 2015-2016. The SABOR campaign was funded by the NASA Ocean Biology and Biogeochemistry Program. Support for the RSP and HSRL-2 flight operations during TCAP was provided by the Department of Energy Atmospheric Radiation Measurement Climate Research Facility (DOE ARM): Interagency Agreement. Larry Berg was supported by the U.S. DOE’s Atmospheric System Research (ASR) Program, and TCAP was supported by ARM. The Pacific Northwest National Laboratory is operated for DOE by the Battelle Memorial Institute.

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