MAP instruments measure radiances at different viewing angles, polarization angles, and spectral bands. The state of polarization of measured light can be represented by the Stokes vector, S, where the transpose of the vector is given as ST=[IQUV] (Schott, 2009). The elements in the Stokes vector are 1aIQUV=E∥E∥+E⊥E⊥E∥E∥*-E⊥E⊥*E∥E⊥*+E⊥E∥*i(E∥E⊥*-E⊥E∥*), where E⊥ and E∥ are the perpendicular and parallel components of the electric field E, respectively. The first element (I) represents the total radiance. The second and third elements (Q,U) represent the linear polarization of the radiance, and the fourth element (U) represents circular polarization. Passive remote sensors, like AirHARP, use the Sun as their light source. Therefore, sunlight incident on the atmosphere is defined as 1bSIncT=I000.

Since the light from the Sun is unpolarized, Q, U, and V of the Stokes vector are zero. The Stokes vector of the scattered light back into the instrument sensor is given by 1cSscaT=IscaQscaUsca0, where the light reaching the instrument sensor has now acquired some polarization but is assumed to be only linearly polarized, an assumption that holds well for the Earth's atmosphere and surface (Dubovik et al., 2011; Kokhanovsky et al., 2015). In this paper, we use reduced radiances RI=πIscaF0, RQ=πQscaF0, and RU=πUscaF0 to define the Stokes vector of the scattered light measured by the MAP with RI, RQ, and RU notation. F0 is the solar radiance (Wm-2µm-1), and hence RI, RQ, and RU are dimensionless variables. RI is the total radiation measured by MAP, the same that would be measured by a radiometer normalized by F0/π. RQ and RU define orthogonal states of linear polarization and, together, they form the polarized intensity, defined as (Schott, 2009) 1dRP=RQ2+RU2, and the degree of linear polarization (DoLP) is 1eDoLP=RP/RI.

The Hyper-Angular Rainbow Polarimeter (HARP) is a modern MAP concept capable of wide field-of-view (FOV), multi-angle, multi-wavelength polarimetric imagery of a ground scene even from a low-cost, CubeSat-size platform. The HARP program was initially funded by the NASA Earth Science and Technology Office (ESTO) InVEST program as a joint collaboration between the University of Maryland, Baltimore County (UMBC) in Baltimore, Maryland, and the Space Dynamics Laboratory at Utah State University in Logan, Utah. There are currently three instruments based on the original HARP concept: HARP CubeSat, a self-contained space technology demonstration mission launched to the International Space Station in November 2019 for a 1-year long mission beginning in February 2020; HARP2, a payload instrument for NASA's Plankton, Aerosols, Clouds, ocean Ecosystem (PACE) mission set to launch in the early 2023s (Werdell et al., 2019); and AirHARP, an airborne version of the HARP concept. In this paper, we focus on aerosol retrievals derived from measurements made from AirHARP as a proxy for these future space missions.

AirHARP's swath spans an angle of 114∘ along-track and 94∘ in cross-track; a simulated cross-section image of the AirHARP instrument is shown in Fig. 1a. It uses a Phillips prism system that splits the incoming beam of light into three components so that the radiation can be measured at three polarization angles simultaneously, with no moving parts. These polarization states are imaged on three CCD imaging sensors, denoted by A, B, and C, which measure the light at angles of linear polarization (AoLP) = 0∘, 45∘, and 90∘, respectively, which are hereby denoted as IA, IB, and IC radiances. AirHARP has three wavelengths (440, 550, and 870 nm) that measure at 20 along-track view angles plus a hyper-angle measurement at the 670 nm wavelength that measures at 60 along-track view angles (see Table 1). This capability allows AirHARP to view a single ground target from up to 60 different perspectives and measure the angular scattering response emanating from that location in both total and polarized radiances. These radiances are measured in all four channels, and each collocated detector pixel, which corresponds to a single channel and view angle, is calibrated independently for the radiometric and polarimetric measurements.

Using the calibration matrix C and measured IA, IB, and IC, the Isca, Qsca, and Usca elements of the Stokes vector are calculated using Eq. (2) (Fernandez-Borda et al., 2009) and subsequently reduced radiances RI, RQ, and RU for each collocated detector pixel. 2IscaQscaUsca=C11C12C13C21C22C23C31C32C33IAIBIC

The calibration matrix, C, is derived from a laboratory calibration scheme described in Fernandez-Borda et al. (2009). The AirHARP instrument was validated in the lab to perform at a 3 %–5 % radiometric and 0.5 % DoLP uncertainty across all spectral bands, though HARP2 may further reduce this uncertainty with improvements to onboard calibration and optical design (McBride et al., 2020). The study in this paper uses 3 % radiometric uncertainty for all the bands and 0.5 % DoLP uncertainty for 440, 550, and 670 nm and 1.5% for 870 nm as inputs to GRASP. The 870 nm polarimetric data have larger uncertainty due to a lower signal-to-noise ratio in the field data compared to 440, 550, and 670 nm; therefore, we give these data less relative weight in the retrievals. The total radiance (I) and DoLP are both useful for accuracy assessments and retrievals: they are not sensitive to a reference plane that defines electric field E. Q and U as measured by AirHARP, on the other hand, are defined based on a reference plane, and their absolute values depend directly on this chosen frame of reference. The details of this reference plane, including instrument scattering geometry, are described in the next section.

Figure 2 defines the scattering geometry. Scattering angle (θsca) is defined as the angle between the Sun vector and the viewing direction. θs is the solar zenith angle, θv is the instrument viewing zenith angle, ϕsat is the satellite azimuthal angle and ϕsun is the solar azimuthal angle. The relative azimuthal angle is ϕ=ϕsun-ϕsat. For the calculation of θsca, we need to know θs, θv and ϕ.

The reference plane for the definition of E⊥ and E∥ is based on the local meridian plane, which is a standard reference frame used for reporting Q and U (Chandrasekhar, 1950; Emde et al., 2015; Hansen and Travis, 1974; Hovenier et al., 2004). For detailed information, please refer to the coordinate system as defined in the book by Hovenier et al. (2004). Q and U measured by AirHARP are based on the instrument reference frame, and these are rotated to the local meridian plane that is formed of the local nadir vector plus the viewing zenith vector (see Fig. 2). The electric field parallel to the local meridian plane is called E∥ and the electric field perpendicular to the local meridian plane is E⊥. Using the information from an aircraft's inertial measurement unit (IMU), the Q and U are rotated to the local meridian plane from the instrument reference frame.

The ACEPOL campaign (https://www-air.larc.nasa.gov/missions/acepol/index.html, last access: 8 September 2020) was a collaborative effort of NASA and SRON (Netherlands Institute of Space Research) based out of Armstrong Flight Research Center (AFRC) in Palmdale, California, USA. One primary aim of the campaign was to acquire data using airborne advanced passive and active remote sensing instruments and then use the expanded information content available from the new sensors, both individually and in synergy, to better characterize aerosol (Knobelspiesse et al., 2020). Multiple polarimeters and lidars were mounted on the NASA ER-2 aircraft. These included AirHARP as well as AirMSPI (Diner et al., 2013), RSP (Cairns et al., 1999, 2003), SPEX airborne (Smit et al., 2019), HSRL2 (Burton et al., 2018; Hair et al., 2008) and Cloud Physics Lidar (CPL) (McGill et al., 2002). ACEPOL also made use of ground-based instruments such as AERONET and the Micro-Pulse Lidar Network (MPLNET) for validation of aircraft measurements (Holben et al., 2001; Welton et al., 2001). The measurements and inversion algorithms used to analyze the ACEPOL data will be helpful in understanding the potential use of polarimeters in future satellite missions like the NASA PACE mission, the Aerosols, Clouds, Convection and Precipitation (A-CCP) Decadal Survey mission, and the European EarthCare mission. The ACEPOL campaign had nine flights over the period of 19 October to 9 November 2017, with a combined flight time of approximately 41.3 h. The main objectives of ACEPOL include the calibration of instruments over ocean and land with no clouds or aerosol, geolocation of images using coastlines, coordinated Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) or Cloud-Aerosol Transport System (CATS) under flights, validation with AERONET in low, medium and high aerosol loading, satellite intercomparison for aerosol and cloud retrievals, and calibration over a spatially uniform surface amongst other lower-priority goals, such as cirrus cloud observations (Knobelspiesse et al., 2020).

For the ACEPOL 2017 campaign, the AirHARP instrument was mounted on an ER-2 aircraft left-wing pod. It collected data over eight flights consisting of a total of 45 flight leg images. For this study, we have analyzed only 12 of these flight legs, listed in Table 2, including scenes over the ocean, dry lake, forest fire smoke, and AERONET sites. Along with the airborne polarimeters, the HSRL2 also flew aboard the ER-2 aircraft during ACEPOL. HSRL2 is a NASA high spectral lidar that has been used to measure clouds and aerosols (Burton et al., 2012; Hair et al., 2008). HSRL2 measures extinction coefficients at two wavelengths (2α at 355 and 532 nm) and backscattering coefficients at three wavelengths (3β at 355, 532, and 1064 nm) (Burton et al., 2018). These measurements allow the detection of the vertical distribution of aerosol extinction with a precision of about 0.02 km-1. The HSRL2 instrument points at nadir and measures the vertical profile of the aerosol backscatter coefficient at 0.1 Hz frequency with a vertical resolution of 15 m and an aerosol extinction coefficient at 1 min temporal resolution and 150 m vertical resolution. In some ACEPOL cases, due to atmospheric turbulence, interference in the HSRL2 measurement resulted in the inability to use the molecular channels at 355 and 532 nm to report the AOD and required assumed lidar ratios of 20 and 40 sr over the ocean and land, respectively. However, for all the comparisons shown in this study, those cases were avoided, and HSRL2 AOD reported here required no lidar ratio assumptions.

Before the inversion of the AirHARP-measured I, Q and U components, each measured pixel must be prepared for aerosol retrievals. This involves first avoiding groups of pixels that are inappropriate for an aerosol retrieval, such as clouds, and correcting for gaseous absorption in the remaining signal. Automatic algorithm-level cloud masking can be challenging. In the work presented here, scenes were selected by eye, so that there was no need to develop an automatic cloud mask for AirHARP at this time. However, correction for gaseous absorption was necessary. I, Q, and U are corrected for the atmospheric gas absorption using the technique mentioned in Patadia et al. (2018). For AirHARP spectral bands, gas absorption is most significant at the 550 and 670 nm bands and is mainly due to the four atmospheric gasses O2, H2O, O3, and NO2. Columnar optical depths of 0.004, 0.032, 0.014, and 0.003 due to atmospheric gases are observed at the 440, 550, 670, and 870 nm spectral bands, respectively. The Unified Linearized Vector Radiative Transfer Model (UNL-VRTM) (Wang et al., 2014; Xu and Wang, 2019) is used to calculate transmission due to the total effect of all atmospheric gas absorption, which is translated to a multiplicative correction factor for each of the four AirHARP bands. These correction factors are a function of the path length through the atmosphere, which is a combination of solar and instrument viewing zenith. All the calculations are done for a mid-latitude summer US atmosphere assuming no variation in the four gases and for an AirHARP observation height of 20 km a.s.l. This correction is applied to each band, for each pixel, prior to the inversion.

GRASP is a versatile retrieval algorithm that can be used for a variety of remote sensing and in situ measurements to retrieve aerosol and surface properties (Dubovik et al., 2014). It is open-source software and is available free to non-commercial users for downloading from the website https://www.grasp-open.com/ (last access: 8 September 2020). GRASP first demonstrated its overall capability in an aerosol retrieval test study (Kokhanovsky et al., 2010). It has gone on to prove itself in a variety of real-world applications (Chen et al., 2018, 2019; Frouin et al., 2019; Li et al., 2019; Schuster et al., 2019; Torres et al., 2017). GRASP has been successfully applied to measurements from many different types of instruments (Benavent-Oltra et al., 2019; Espinosa et al., 2017, 2018; Román et al., 2018; Titos et al., 2019; Torres et al., 2017), but the most pertinent to AirHARP are the previous applications of GRASP to POLDER-3 on PARASOL (Chen et al., 2018, 2019; Dubovik et al., 2011; Frouin et al., 2019; Li et al., 2019) because of its familiar polarization, multi-wavelength, and multi-angle sampling characteristics.

GRASP consists of two modules: a forward model and an inversion module. For the case of aerosol measurements, the forward model consists of a polarized radiative transfer (RT) code to calculate the radiance measured by the instrument, and it uses a precalculated spheroidal kernel to calculate the contribution of single scattering by the aerosol particles following the strategy described by Dubovik et al. (2006, 2011). The kernel includes the pre-calculated full phase matrix elements, extinction and absorption for five log-normal size distributions with preselected size parameters for the ranges of the real refractive index 1.33 to 1.7 and 0.0005 to 0.5 for the imaginary part of the refractive index for both spherical and non-spherical aerosol approximated by a mixture of spheroids with a fixed particle shape distribution derived in Dubovik et al. (2006). This approach allows for very fast and accurate calculations of aerosol single-scattering properties in the wide range of refractive indices even for non-spherical aerosol. The details of the application of the kernels to satellite polarimetry are discussed in detail by Dubovik et al. (2011) (e.g., see Sect. 3.1 and Fig. 4 in Dubovik et al., 2011).

For the RT calculations, GRASP uses a successive order of scattering (SOS) scheme. The RT module consists of different surface bi-directional reflectance distribution function (BRDF) and bi-directional polarized distribution function (BPDF) models for land and ocean. These models are briefly discussed in Appendix A, and further information about the RT code and single-scattering database that is beyond the scope of this paper can be found in Dubovik et al. (2011) and Lenoble et al. (2007).

The particle single-scattering calculations that we employ for the AirHARP retrieval use one of two possible retrieval setups: (1) five log-normal distribution modes as described in Table 3 or (2) the aerosol assumed to be an external mixture of five aerosol components as described in Table 4. Both approaches were extensively used in PARASOL/GRASP processing and, therefore, are considered here. For the first kernel possibility, the retrieval has 15 aerosol parameters to retrieve and is called a “GRASP/Five mode” kernel. Each of the log-normal modes has a fixed mode radius and width. The free parameter in the retrieval for particle size distribution is the concentration of particles in each bin. There are three log-normal modes in the fine mode (log-normal modes 1 to 3 in Table 3) and two in the coarse mode (log-normal modes 4 and 5 in Table 3). Other retrieved parameters related to aerosol properties include a complex refractive index, aerosol layer height, and the fraction of spherical particles (SF). The same kernel is used for all the retrievals in this paper, with an exception for the AERONET comparison mentioned in Sect. 5.2. For the AERONET comparison, we make use of the second GRASP kernel that has reduced the number of aerosol parameters from 15 to 6. This reduced parameter option is the “GRASP/Models” kernel, where particle properties are assumed for each aerosol component given in Table 4. Complex refractive index, SF, and particle size distribution of each aerosol components are fixed for this kernel. Only the concentration (weight) for each aerosol component is retrieved.

The inversion module in GRASP uses the multi-term least square method (LSM) to solve the following system of equations: 3af*=f(a)+Δf,0*=Ga+Δg,a*=a+Δa, where a is a vector of unknowns and is called a state vector. f* is the vector which contains the instrument observations, Δf is the uncertainty in the observations, and f(a) is the forward model simulated observations. For the AirHARP observations, f* is a vector containing information of RI, Q/I (same as RQ/RI), and U/I (same as RU/RI) or RI and DoLP for all the spectral bands and viewing angles. GRASP is able to accept different configurations of the input parameters to make its retrieval. We will use the following sets of input in this work in different situations: (RI, Q/I, U/I) or (RI, DoLP). The text will explicitly state the inputs in each instance. Given an ideal pixel, AirHARP measures 120 data points for each aforementioned variable. A priori smoothness constraints are imposed on the retrieved solution in order to suppress unrealistic oscillations in the retrieved characteristics. The second equation in Eq. (3a) represents such a smoothness constraint on the retrieved characteristics, and 0* is the zero vector and imposes the forced constraint that the derivatives of retrieved parameters be zero. The matrix G includes the coefficients for calculating derivatives of state vectors approximated by finite differences. For example, unrealistic oscillations in particle size distribution are eliminated using coefficients calculated from derivatives with respect to radius. Similarly, spectral dependencies of the refractive index are imposed using the coefficients calculated using wavelength. Uncertainties in the smoothness constraints are represented by the Δg term. GRASP can perform retrievals using multi-pixel information in both spatial and temporal dimensions; however, in this study, we are not utilizing this feature due to the limited availability of data over the same place in the temporal dimension. We use a priori constraints on the particle size distribution, and these constraints are represented by the third and last term in Eq. (3b). A priori estimates of state vector parameters are given by a* and Δa is the uncertainty in the a priori constraints of a*. The multi-term LSM in GRASP finds the statistically optimized solution of the set of equations mentioned in Eq. (3a) by minimizing the term: 3b2Ψ(a)=f(a)-f*TWf-1(f(a)-f*)+γgaTWg-1Ga+γa(a-a*)TWa-1(a-a*), where Wf=1ϵf2Cf is the weighting matrix calculated using the measurement covariance matrix Cf and the first diagonal element ϵf in Cf. Similarly, Wg=1ϵg2Cg, Wa=1ϵa2Ca are calculated using the covariance matrices of a priori smoothness constraints and a priori constraints on the retrieved parameters. γg=ϵf2ϵg2 and γa=ϵf2ϵa2 are the Lagrange multipliers (Phillips, 1962; Tikhonov, 1963) calculated using the information from the covariance matrices. In order to take into account the non-negative character of measured and retrieved physical values in the retrieval optimization, the log-normal error distributions are assumed for all positively defined measured characteristics, and the minimization is defined for logarithms of all positively defined retrieved parameters. The solutions to the set of equations in Eq. (3a) are found by minimizing the term Ψ(a) in Eq. (3b). Since the radiative transfer in the atmosphere has a pronounced nonlinear character, the Levenberg–Marquardt (Levenberg, 1944; Marquardt, 1963) algorithm is harmoniously adapted into the statistically optimized fitting to ensure the monotonic convergence of the iterative solution. These and other technical details of numerical inversion are described in Dubovik et al. (2011), and in-depth discussion of the above methodological aspects also can be found in Dubovik and King (2000) and Dubovik (2006).

The state vector a includes the information on particle size distribution, which is the concentration for five log-normal modes of Table 3, the complex refractive index in the four spectral bands that are independent of particle size, the SF, aerosol layer height, and parameters characterizing the directional reflectance of the surface. AOD is derived from retrieved aerosol properties using the method mentioned in Appendix B1. Additionally, fine-mode AOD is calculated using modes 1–3 mentioned in Table 3 and coarse-mode AOD using modes 4 and 5. Single scattering albedo (SSA) and Angstrom exponent (AE) are also derived from the retrieved aerosol properties. For the GRASP/Models approach, the state vector a includes the concentration for each aerosol component mentioned in Table 4. State vector a does not contain information on the particle size distribution, SF, and complex refractive index. All this is embedded in the aerosol components which are close (with some modifications) to biomass burning, urban, urban polluted, maritime, and desert dust observed in AERONET climatology by Dubovik et al. (2002). Among these, only desert dust is considered completely non-spherical and, similarly to AERONET retrievals, uses a shape distribution mentioned in Dubovik et al. (2006). All the other types are treated as 100 % spherical particles. The details of the bi-modal size distribution parameters along with the fixed complex refractive index for each of the aerosol components are tabulated in Table 4 and are based on the work of Dubovik et al. (2002). Figure 3 shows the particle size distribution as a function of radii for the different aerosol components. The main differences between the GRASP/Five mode and GRASP/Models approaches are that (1) instead of retrieving the concentration of each log-normal mode, concentration (weight) for each of the aerosol components mentioned in Table 4 is retrieved. (2) RRI, IRI, and SF are not retrieved since these are fixed for each of the aerosol components. This simplified approach significantly drops the complexity of the aerosol model by reducing the number of parameters retrieved directly in the joint retrieval. It helps in reducing the nonlinearity of the inverse problem and makes the separation of the surface and aerosol signal much less complicated compared to the GRASP/Five mode approach. At the same time, all aerosol total properties, such as SSA, effective size distribution, and complex refractive index, can be obtained using an external aerosol mixture concept. The reduction of sought unknowns helps situations in lower information content (e.g., for low AOD) and makes the separation of the surface and aerosol signal much less complicated compared to a GRASP/Five mode kernel. This tendency is well identified in the in-depth analysis of PARASOL data processing using different retrieval setups by Chen et al. (2020). Like the GRASP/Five mode kernel, the state vector a includes the information on aerosol layer height. Even though aerosol layer height is retrieved during the retrieval process, the sensitivity to aerosol height for the AirHARP wavelengths is negligible for most of the low loading cases. Retrieved aerosol layer height is thus not discussed in this work.

Retrieval of aerosol properties from MAPs is highly sensitive to the accurate representation of the directional reflectance from the surface. For the ocean pixels, the surface model used is the NASA GISS model (Mishchenko and Travis, 1997) based on Cox and Munk (1954), in which the ocean surface reflectance is represented by three parameters: ocean surface albedo, the fraction of the Fresnel reflection surface, and wind speed denoted by a0, a1, and a2, respectively, with details given in Appendix A2. These three parameters are the surface components in the state vector a for the case of ocean pixels. For the case of land pixels, the Ross-thick Li-sparse linear BRDF model is used to represent the directional reflectance from the surface (see Appendix A1.1), which uses three parameters K0,K1,K2. K0 is a spectrally dependent parameter that represents the isotropic reflectance, and K1 and K2 normalized to K0 are the spectrally independent parameters which are the coefficients of geometric and volumetric scattering kernels, respectively (Maignan et al., 2004; Wanner et al., 1995). Polarized reflectance from the surface is modeled using the Maignan–Breon one-parameter model, and the retrieved parameter is a scaling factor (α) that is spectrally dependent (Maignan et al., 2009). Refer to Appendix A1.2 for detailed information on the surface models. The four parameters K0,K1/K0,K2/K0, and α are the surface components in the state vector a for the case of land pixels. In the next section, we discuss the results of applying GRASP to AirHARP measurements of RI, Q/I, and U/I for selected cases from the ACEPOL campaign.

We will focus on four specific cases from ACEPOL 2017 to illustrate the measurement characteristics of AirHARP and demonstrate GRASP retrieval. Figure 4 shows color composite imagery of the intensity and DoLP of each case, using the 440, 550, and 670 nm bands. Cases include a scene over the ocean with little aerosol loading (23 October 2017; T21:30 UTC), a scene over Rosamond Dry Lake (25 October 2017; T18:26 UTC), and two scenes of forest fire smoke (27 October 2017; T18:16 UTC, and 9 November 2017; T19:30 UTC).

The first of our analysis scenes is a cloud-free segment where the ER-2 flew over the USC SeaPRISM AERONET station located at 33.564∘ N, 118.118∘ W, on a platform off the coast of southern California. The AERONET station measured a low aerosol loading of AOD = 0.04 at 440 nm at the time of ER-2 overpass. The segment includes sunglint, and because of the low AOD, the sunglint and non-sunglint patterns are ideal for the intercomparison of different polarimeter measurements of I, Q, U, and DoLP. This flight track aligned with the solar principal plane so that the longer wavelength bands will be highly polarized for the sunglint viewing angles. For a scene with low aerosol loading above the ocean with no sunglint, the polarization follows the Rayleigh pattern and will peak at the 90∘ scattering angle. For the sunglint case, we expect the peak to be at scattering angles where the sunglint is observed. In this case, the maximum sunglint occurs for scattering angles 70 to 90∘. Figure 5 shows the measured RI, Q/I, U/I, and DoLP for 1 pixel (footprint of 55 m × 55 m) along the nadir track as a function of scattering and plotted as colored circles, for each of the four wavelengths. The measurements show that the maximum intensity occurs in the glint region (scattering angles 70 to 90∘) and confirms that the DoLP peak also occurs at the sunglint scattering angles. However, while the intensity falls to minimum levels outside of the glint region, the DoLP has a more gradual falloff, as the Rayleigh pattern with maximum DoLP at 90∘ is superimposed on the dark ocean scene.

GRASP is applied to invert the measured RI, Q/I and U/I for the pixel represented in Fig. 5 for aerosol retrievals. Because the aerosol loading is very low for this scene, there is insufficient aerosol loading to retrieve the real (RRI) and imaginary (IRI) parts of the complex refractive index, and instead they are constrained (RRI = 1.4 and IRI = 0.0001) in GRASP using the values of the oceanic aerosol model mentioned in Hasekamp et al. (2008). This will reduce the number of retrieved parameters and thus reduce the complexity of the inversion problem by reducing the nonlinearity of the forward model. The GRASP/Five mode kernel is used in the retrieval, with the concentrations of each of the five modes unconstrained. Therefore, the retrieved parameters include the five concentrations for the five log-normal modes shown in Table 3, aerosol spherical fraction, aerosol layer height, and the ocean model parameters a0, a1, and a2. AOD is derived from the retrieved and modeled parameters. The solid black lines plotted in all panels of Fig. 5 are the GRASP fits using the AirHARP-measured RI, Q/I, and U/I as input. The DoLP is also calculated from the fitted variables and plotted in the same figure. The sunglint registers in RI as a sharp peak, with the width of that peak dependent on surface roughness primarily caused by surface wind. The retrieval of aerosol properties is highly sensitive to wind speed. An inappropriate wind speed estimate can result in high uncertainty in the aerosol properties retrieved. The goodness of fit in Fig. 5 suggests that retrieval of the ocean parameters, including wind speed, is very good for this sampled pixel.

To achieve a better understanding of how well the GRASP retrieval can fit the measurements, we apply the retrieval to 3600 pixels (60 × 60) of this ocean scene. Here, a pixel footprint of 55 m × 55 m is used, since the variability due to the geolocation is negligible for an open-ocean pixel as compared to a land pixel. Figure 6 shows 2-D density scatter plots of the AirHARP-measured variables RI, Q/I, and U/I for the four spectral bands vs. the GRASP fit, and the histogram of the number of points used for each bin is plotted on the respective axes. A dashed magenta line represents the ordinary least square (OLS) fit between retrieved and measured parameters; a black solid line denotes the 1:1 line. The goodness of fit, χnorm2 for each pixel, is calculated using the mathematical expression mentioned in Table 5. A list of the statistical parameters used in this study is formulated in Table 5, where xi is the measured value and yi is the GRASP fit. N is the total number of observations for the pixel; Sy is the error covariance matrix for the observations. Sy includes only diagonal elements, and off-diagonal elements are assumed to be zero since we do not consider the cross-correlation between the different viewing angles for the same spectral band. For the reduced radiance RI, all spectral bands show a good comparison at lower RI values, within the χnorm2 confidence interval, and show a slight deviation from the 1:1 line for the sunglint angle data points. The GRASP fit in blue band yields an underestimated RI when its values are greater than 0.2, giving an overall OLS slope of 0.967. However, the range where the underestimation occurs represents only a small fraction of the total analyzed samples. This underestimation is also observed for the 550 nm band, with a slightly better R2 value of 0.995 compared to 0.986 for the 440 nm band. For the case of the 670 and 870 nm bands, we observed a slight overestimation with the GRASP fitting having R2 values 0.997 and 0.996 for red and near-infrared (NIR) bands, respectively. Some of this underestimation and overestimation is because the isotropic wind model has trouble simulating the multi-angle views in the sunglint region. In terms of the spread of points around the regression line, we expected much higher noise for the 870 nm band due to the lower signal-to-noise ratio (SNR) in this band. Surprisingly, in terms of fitting the RI component, the 870 nm band does not display any repercussion of the lower SNR. For the polarization components Q/I and U/I, all the spectral bands demonstrate a good correlation between the GRASP fit and AirHARP measurements. This demonstrates that the polarization variables are less affected by the discrepancies in sunglint pixels for the extreme viewing angles. The average AODs retrieved for these 60 pixel by 60 pixel regions are 0.07 ± 0.03, 0.04 ± 0.02, 0.03 ± 0.01, and 0.02 ± 0.01 at 440, 550, 670, and 870 nm, respectively. In the following section, we detail several case studies of AirHARP land surface and aerosol plume data applied to GRASP for retrieval of aerosol microphysical and optical properties.

Equally important over the land for a multi-angle instrument is the need to co-register each along-track view angle of the same target. Over the flat ocean, co-registration is straightforward and is based on a projection of the measurements onto a representation of a smooth geoid Earth. Over the land, topography introduces a challenging situation in which forward and aft views of the same target might image different slopes of a ridge. Topographically corrected projections need to be made either to a digital elevation model at a resolution comparable to the measurements (this is operational for AirHARP Level 1B data) or the measurements need to be projected to a specific altitude in the atmosphere, perhaps cloud top height or an aerosol layer. Figure 7 shows the measured RI, Q/I, U/I, and DoLP from the 550 nm wavelength for selected pixels in each of the following three flight legs under analysis. In this figure, unlike Fig. 5, the ocean scene is from an offglint pixel. The other pixels represent a dry lake surface, vegetation, and smoke, respectively. Also plotted as the black curves are the GRASP fit to each of these targets. The ocean pixel appears the easiest to fit, and then the smooth dry lake pixels. The other land surface types, with their variable topography, present a greater challenge for GRASP. In the next section, we detail these three flight segments over land that include one on 25 October 2017 over Rosamond Dry Lake at 18:26 UTC, a second one that is a forest fire smoke scene near the Kaibab National Forest and Grand Canyon National Park in Arizona, USA, on 27 October 2017 at 18:16 UTC, and a third scene of fresh smoke on 9 November 2017 at 19:30 UTC.

HSRL2, flying on the same aircraft with AirHARP during ACEPOL, provides the opportunity to compare the GRASP retrievals of AOD with an independent and collocated measurement. AirHARP lacks a wavelength channel identical to the wavelengths measured by HSRL2; therefore, for this study, we make use of the 440 and 550 nm channels on AirHARP to calculate the Angstrom exponent and then use that information to interpolate the AOD to HSRL2's wavelength of 532 nm. We collocate HSRL2 and AirHARP measures of AOD for the smoke plume shown in Fig. 4d (9 November 2017; 19:30 UTC). The smoke plume in this image is a controlled fire started in the Kaibab National Forest, Arizona, USA, and is highly non-homogenous near the source fire. The extreme non-homogeneity of the smoke in this scene introduces additional uncertainty to the particle property retrievals, and only AOD will be shown here. To match the HSRL2 ground pixel in the AirHARP image, the latitude and the longitude are matched to a tolerance level of about 200 m on the ground. For this flight leg, HSRL2 reports the aerosol extinction at 10 s intervals, which translates to 2 km in the ground distance for an ER-2 aircraft flying at 20 km altitude with a speed of ∼ 200 ms-1 but with a narrow cross-track footprint of only 15 m. Thus the 2000 m × 15 m footprints of the HSRL2 measurements are inherently mismatched with AirHARP's 275 m × 275 m pixels. Given the inhomogeneity of this aerosol, we do not expect perfect agreement between the two sensors' retrieved AODs, simply because of the mismatch in spatial sampling.

For this study, we make use of the HSRL2 AOD at 532 nm, where an assumption of the lidar ratio is not required. All the GRASP retrievals for this comparison are done using the RI and DoLP measurements from AirHARP. Aerosol optical depth at 532 nm as a function of the collocated along-track pixels is plotted in Fig. 13a, and the scatter plot of the comparison is shown in Fig. 13b. In Fig. 13a HSRL2-measured AOD is denoted by the green diamond markers and AirHARP AOD by grey squares. Each square represents the mean of 28 pixels around the collocated HSRL2 ground pixel in the AirHARP image. We used seven pixels along-track (∼ 1.93 km) and four pixels cross-track (∼ 1.1 km) to find those mean values. The error bar in the AirHARP data points is the SD of AOD of all pixels within the ∼ 1.93 km × 1.1 km region around the HSRL2 ground pixel, representing the spatial variability of the smoke plume within the averaging rectangular box. For this heterogenous smoke plume case, we had to apply χnorm2<20, to filter out bad pixels/fits. Non-homogeneity of the smoke makes the retrieval complicated since we see different parts of the smoke plume when we scan through the different viewing angles of AirHARP data. So, using a higher χnorm2 value for filtration helps to catch the higher AOD values. A scatter plot of these two data sets is shown in Fig. 13b and the solid black line is the 1:1 line, whereas the yellow error bar represents the spatial variability of AirHARP AOD, similar to that in Fig. 13a. A comparison of HSRL2-measured AOD at 532 nm with the AirHARP AOD retrievals at 532 nm shows a strong, positive correlation and only deviates when the plume is thick and heterogeneous. A Pearson correlation coefficient (ρ) of 0.940, BIAS=-0.062, and mean absolute error (MAE)=0.122 is obtained for this comparison. Matching the HSRL2 AOD in regions of heterogeneity is challenging due to spatial mismatch between AirHARP and HSRL2 pixels. This will create issues where there is a sharp variation in the AOD, like close to the source and in the boundary of the smoke plume. The different cross-track pixel size between the HSRL2 and AirHARP measurements makes the intercomparison difficult to interpret in some cases. For points near the plume source, higher pixel variability may also bias AirHARP AOD retrievals performed at the same general location as the HSRL2 measurement. In a scene with this much complexity, there is additional uncertainty in matching multi-angle views for the AirHARP retrieval, because each viewing angle of the instrument will be looking at a different plume thickness, and this violates the plane-parallel assumption.

Validation of AirHARP–GRASP retrievals using AERONET-measured aerosol optical depth for the collocated AirHARP pixels during the ACEPOL campaign are discussed in this section. A list of the collocated AERONET stations and measurements used for this analysis is given in Table 2. Only AERONET stations with a data quality of Level 2.0 are used for the comparison. For the AERONET validation, AirHARP pixels with a resolution of 550 m × 550 m are used for the GRASP retrievals to avoid the issues due to the small pixel shifting during the reprojection to a common latitude–longitude grid as well as to avoid strong fine-resolution surface features that appear over the urban area. To further protect the algorithm from subpixel inhomogeneity and other features inappropriate for retrieval, a χnorm2<1.5 filter is used to remove the bad pixels/fits which may be caused by the presence of thin clouds (Stap et al., 2015) or due to the inability of surface reflection models to represent the directional reflectance from a complicated surface. To collocate the AERONET station (a single pixel) within the AirHARP image, the latitude and longitude of the AERONET location are matched to the AirHARP latitude and longitude with a tolerance of 2×10-3∘, which is approximately equivalent to 200 m on the ground. An area of 5.5 km × 5.5 km (10 × 10 retrieval pixels) around this collocated pixel is used for the calculation of the area mean AOD from the AirHARP retrievals, and this is matched to a 1 h temporal mean from AERONET measurements. Each of the 5.5 km × 5.5 km averaging boxes includes 100 pixels; however, many of them are removed after the χnorm2 filtering. AERONET-measured AOD is interpolated linearly in log–log space using the AE-to-AirHARP spectral bands for 1:1 comparison. The Ross–Li BRDF surface model along with the Maignan–Breon BPDF models are used for representing the directional reflectance from the land surface for all retrievals used in the validation.

AirHARP observations in this validation exercise are retrieved with two versions of the GRASP aerosol kernels, one using the GRASP/Five mode kernel with 15 free parameters (Table 3) that allows for retrieval of particle properties and the other using the GRASP/Models kernel with only 6 free parameters (Table 4) that restricts the particle properties to focus on the AOD retrieval. When aerosol loading is low, there is insufficient signal to retrieve particle properties. Allowing for additional free parameters without having sufficient signal will degrade the accuracy of the AOD retrieval. The maximum AOD measured by a collocated AERONET station during the ACEPOL campaign is 0.158 at 440 nm, suggesting that in this exercise the simplified aerosol component model would be preferred over using the option with a greater number of free parameters. This is evident in the two figures, Fig. 14a and b. Figure 14 shows the scatter plot with AODAERONET on the X axis and AODAirHARP on the Y axis. The spatial SD of AOD within this 5.5 km × 5.5 km box is indicated using the error bars in Fig. 14. Statistical parameters that represent the correlation between these two data sets are shown in the table inside the plots. In Fig. 14a, for the case of AOD retrievals based on the GRASP/Five mode kernel with the greater number of free parameters, MAEs of 0.041, 0.039, 0.037, and 0.035 are obtained for the 440, 550, 670, and 870 nm bands, respectively. The bias ranges from 0.022 to 0.038, with the 440 nm band having the largest bias and the 870 nm band with the least. However, in Fig. 14b, for the case using the GRASP/Models kernel with the reduced number of free parameters, mean absolute errors of 0.010, 0.010, 0.011, and 0.015 are obtained for the spectral bands 440, 550, 670, and 870 nm, respectively, with the 870 nm band having a slightly higher spread than the other bands. Also, a similar trend is seen for 870 nm in the case of BIAS, where the 440, 550, and 670 nm bands have a BIAS of -0.002, -0.003, and -0.004, respectively, whereas 870 nm has a BIAS of -0.009. Figure 14 demonstrates the need to match the appropriate kernel to the available information in the scene and to not attempt the retrieval of more free parameters than the aerosol loading permits. Overall, the performance of the AirHARP observations plus the GRASP retrieval algorithm gives a good agreement with the collocated AERONET observations, especially when the GRASP/Models kernel is used. The above tendency is well identified in the in-depth analysis of PARASOL data processing using different retrieval setups by Chen et al. (2020).