(i)

(1) 

The vector radiative code we employ is based on the Doubling-Adding formalism. Within assumptions intrinsic to every model, the code is exact to any arbitrary accuracy, and has been used to model Research Scanning Polarimeter (RSP) measurements over a variety of Earth’s scenes for 25 years, including scenes containing ice crystals in clouds [van Diedenhoven et al., 2013] and snow [Ottaviani et al. 2012; 2015]. The ice crystals Inherent Optical Properties are calculated with an advanced Geometric Optics code, as documented in many papers [Macke et al., 1996; van Diedenhoven et al., 2012]. These references are already reported in the draft, especially in the introduction.

Model errors include standard assumptions which, most importantly, do not invalidate the results of the information content analysis. Aerosols are treated as log-normal distributions of spherical particles, and so are the impurities in the snow that are also assumed to be externally mixed with the (hexagonal) snow grains. The code is plane-parallel.

Ottaviani et al. [2012; 2015] demonstrated the feasibility of retrievals of snow grain shape, and microscale roughness based on RSP observations. In these studies, we isolated the surface contribution to the total signal measured at sensor altitude via an iterative procedure that automatically includes a rigorous atmospheric correction. By fitting the surface signal to the database of hexagonal prisms it was established that, radiatively speaking, snow behaves as a collection of non-spherical crystals with extreme aspect ratios. The reason for choosing hexagonal prisms is described in detail by the papers of van Diedenhoven, and are amply commented on in the text. It should be noted that whether or not these shapes really correspond to those found in nature, the climatological relevance of snow resides in its radiative contribution (albedo) to the energy balance. In this respect, and for the purposes of atmospheric correction, a model capable of replicating the surface contribution can be considered as satisfactory as one giving the actual shape.

The retrieval of grain size exploits measurements at infrared wavelengths, following the same strategy as the MODIS [Stamnes et al., 2007, Painter et. al 2009] or AVIRIS [Painter et al., 2003; Nolin and Dozier 1993] teams. Such types of retrievals have been validated [Aoki et al., 2007; Painter et al., 2003].

To highlight these points, we have adjusted the text at the start of the new Sec. 2.1 to:

The RT code employed to generate the LUT is based on the general doubling-adding formalism described by De Haan et al. (1987), and features a consistent treatment of the radiative effects deriving from atmospheric molecular scattering, aerosols and clouds, and any surface whose reflectance is known in analytical form or in terms of its Bidirectional Reflectance Distribution Function (BRDF) properties. The code has been used for decades to model measurements from the RSP over a variety of Earth scenes, including those containing ice crystals in clouds (van Diedenhoven et al., 2013) and ground snow (Ottaviani et al. 2012; 2015). Ottaviani et al. (2012; 2015) retrieved snow grain shape and ice crystal roughness. The retrieval of grain size follows validated strategies already applied to sensors such as MODIS (Stamnes et al., 2007, Painter et. al 2009; Aoki et al., 2007) and the Airborne Visible/InfraRed Imaging Spectrometer (AVIRIS, Nolin and Dozier 1993; Painter et al., 2003).

(2)

This is a general problem in remote sensing. Uniqueness problems are always to be expected in systems of elevated complexity. However, we note that optimal estimation methods are designed to provide the optimal solution in a statistical sense. The rigorous application of inversion methods also requires the user to provide the measurement uncertainties at input (see e.g., Rodgers 2000), in order to obtain the uncertainties on the retrieved parameters. Note that we considered measurement uncertainties of different sensors. For the test scenes, the error bars requested by the reviewer are exactly what is reported in Figs. 6, 9 and 10.

In general, the inversions work as “optimally” as one is able to provide a good first guess, i.e. starting the inversion from a point reasonably close to the final solution, so as to avoid local minima and converge to the global minimum of the cost function. The simulated retrievals were repeated with different initial guesses. For the sample scenes discussed in the draft, if the initial guess is far from the true value the retrievals which use only VIS channels from MODIS-like or even RSP-like data (i.e., including multi-angular and polarimetric observations) fail. The situation is ameliorated by the inclusion of NIR and SWIR channels. The accuracy and precision of retrievals using MODIS+POLDER-like data remains the same when the initial guesses are far from the true value.

Using an initial guess close to the true value improves the accuracy of retrievals from VIS and NIR channels of RSP-like data, although the retrievals still fail with poor precision. The uncertainty in LAI density retrieved from all wavelength combinations of RI+Rp RSP-like data also decreases, though the uncertainty is still greater than the optimal retrieval which uses DoLP. Retrievals using RSP-like and MODIS+POLDER-like perform similarly to those presented in Fig. 10, regardless of the initial guesses.

We make the following changes to Section 3.1:

Figure 5 summarizes the state parameters and their uncertainty obtained from the inversion. The solid lines represent the “true” values used in the forward simulations, and the dashed lines the initial guesses used to initialize the inversion algorithm. Initial guesses for each model parameter were randomly selected between their upper and lower bounds (see Table 1). The retrievals were repeated with different initial guesses to test the stability of the results.

… although the larger uncertainty of spaceborne measurements limits the retrieval quality compared to the RSP-like case. The accuracy and precision of retrievals from MODIS+POLDER-like and RSP-like data in Fig. 5 are insensitive to the choice of the initial guess.

And Section 3.2:

…using only VIS channels yields larger uncertainty in grain size than the pure-snow case, even though the total reflectance in the VIS is sensitive to grain size when impurities are present. Repeating these retrievals with an initial guess close to the true value improves their accuracy, but the precision remains poor. Improvements are observed when measurements in the SWIR are included, due to selective sensitivity to r_eff^T and 𝜏_C⁵⁵⁵. 

…confirming that polarimetric measurements in the SWIR are valuable for determining the vertical partitioning of impurities. Initializing the inversion with a guess close to the true value decreases the uncertainty on ⍴_c^T and ⍴_c^B retrieved from total and polarized reflectance in the VIS+NIR+SWIR, although using DoLP in place of the polarized reflectance is still optimal.

…facilitates a more accurate and precise determination of all parameters except for the apportioning of LAIs between the two layers. In this case, the uncertainty on the retrieved values of aerosol optical depth and grain size, shape, and microscale roughness decreases by an order of magnitude. 

The accuracy and precision of the retrievals from MODIS+POLDER-like and RSP-like data presented in Fig. 10 are not sensitive to the choice of initial data, a result that is promising for remote sensing applications. 

(ii)

We welcome this suggestion and have modified the Methods section so that it now contains the two subsections: 2.1 Radiative Transfer Simulations; and 2.2 Global Sensitivity Analysis Formalism.

(iii)

We have added the following paragraphs to Section 3.2 to emphasize precise numerical results:

Multi-angle polarimetric data (even at lower angular resolution than RSP’s) facilitates a more accurate and precise determination of all parameters except for the apportioning of LAIs between the two layers. In this case, the uncertainty on the retrieved values of aerosol optical depth and grain size, shape, and microscale roughness decreases by an order of magnitude. 

We also add recommendations for retrievals in Section 3.2:

    …retrievals from scenes with larger aerosol amounts (𝜏C555 = 1.0) are nearly identical to those in Figs. 9 and 10..

Our results indicate that retrievals with RSP-like and MODIS+POLDER-like data should use all available channels of total reflectance and DoLP. Simulated retrievals using this selection of data outperform those which include the polarized reflectance at the same wavelengths; the uncertainty on the retrieved values of LAI density is an order of magnitude lower when DoLP is used in place of the polarized reflectance and the microscale roughness is successfully retrieved. 

In the abstract:

…significantly decreasing the uncertainty in the derived impurity concentration and aerosol optical depth.

The results suggest that the degree of linear polarization (DoLP) should be used over the polarized reflectance in retrievals which use observations from instruments like the Research Scanning Polarimeter (RSP) and the Polarization and Directionality of the Earth’s Reflectances 3 (POLDER-3). These findings encourage the development of new remote sensing algorithms that fully leverage the multi-angle and polarimetric capabilities of modern remote sensors, like those onboard the PACE and 3MI spaceborne missions. The better characterization of surface and atmospheric parameters in the snow-covered regions of the cryosphere ultimately benefits the albedo estimates in climate models.

And in the conclusions: 

The development of a robust inversion scheme to be applied to the collocated, multi-year observations of POLDER and MODIS over the study region of Greenland is underway. The augmented results favor the inclusion of the DoLP over the polarized reflectance in retrievals using data from instruments like the RSP and POLDER. This perspective is particularly exciting considering the higher accuracies enabled by recent technological progress for spaceborne observations, like those of the polarimeters on PACE and 3MI.

Note that in our retrievals we subsampled the measurements of DoLP to only positive viewing angles. This inference came from examining the angular variations of the Sobol indices. To highlight the angular information contained in the Sobol Indices, we initially had included in the manuscript heatmaps of this kind (see attached  heatmap.pdf).

The number in each grid location indicates the viewing zenith angle at which the Sobol index is maximum for each parameter (rows) and wavelength (columns). These heatmaps depend on solar zenith angle (the one above is for 65°). Despite their attractive look, such diagrams do not contain further information than that already included in the plots of the Sobol indices (Figs 4, 7, 8). Since the paper contains a significant number of figures, we ultimately decided to exclude them. While we’d prefer not to burden the paper with an excessive number of plots, we would like to collect the editor’s opinion on the possible inclusion of these maps. 

(iv)

The external labels in Fig. 1 have been moved inside the layers to make the figure slimmer and decrease its overall size. All labels have been reformatted.

The plots of the Sobol indices (Figs. 4, 7, 8) should be given in a consistent format, so that toggling among them aids their intercomparison. We have tried several options to produce such figures, and we actually think that the current version contains considerable information and is visually appealing, without being excessively cluttered. Note that the panels are arranged so that the legend fits without overlapping the curves or making the entire figure larger. Removing the empty panels in Fig. 4 defeats the purpose of highlighting the complex system of correlations stemming from the inclusion of impurities (Figs. 7 and 8). 

Figs. 5, 6, 9, 10: We have restricted the space between the vertical bars in Figs. 6 and 10 to make the figures more compact, as the reviewer suggests. The visual appearance of where the retrievals land (given the initial guesses) and the relative comparison of the error bars obtained using different instrument configurations is just simpler to perceive graphically, rather than having the reader parsing and comparing numbers in a table.

(1)

Thank you for catching this! It was a mistake coming from an earlier version of the draft. We corrected the relevant sentence to: 

Zhang et. al. (2023) have recently evaluated the performance of introducing a snow kernel in an inverse algorithm to retrieve the microphysics of aerosols above snow based on observations of the Polarization and Directionality of the Earth’s Reflectances 3 (POLDER-3) instrument, that flew aboard the Polarization & Anisotropy of Reflectances for Atmospheric Sciences coupled with Observations from a Lidar (PARASOL) spacecraft.

(2)

The range of AOD was chosen to target impurity amounts which are difficult to detect via remote sensing, as explained at the beginning of Sec. 3.2. For completeness, attached below are the results of the GSA when the AOD is allowed to vary up to 1.2 (attached ST_larger_AOD.pdf). We also duplicate Figs. 9 and 10 in Section 3.1, but now with 𝜏_C⁵⁵⁵ = 1.0 (attached PM_like_retrieval_larger_AOD.pdf and RSP_like_retrieval_larger_AOD.pdf).

The absolute Sobol indices are proportional to the square root of the variance (see Eq. 17). If the range of AOD is extended, the absolute Sobol indices for the visible and NIR wavelengths increase because black carbon absorption causes large decreases in the total reflectance. Despite these changes, retrievals which use only VIS channels still fail due to simultaneous sensitivity to many parameters (as mentioned on lines 368-369).

Similarly, increased AOD leads to increased polarized reflectance (Ottaviani, 2022) in the NIR and SWIR, especially at positive viewing zenith angles. The Sobol indices and the uncertainty threshold reflect this increase in variance at larger viewing (smaller scattering) angles. Larger variance in the total and polarized reflectance also causes an increase in variance in the DoLP, because DoLP = Rp / RI (line 152). Larger values of DoLP (corresponding to the largest values of AOD) also lead to a minor increase in the uncertainty threshold. The retrievals are nearly identical to those in Figs. 9 and 10, indicating that the conclusions drawn in the manuscript remain valid for scenes with larger aerosol concentrations. 

We totally agree that these points should be mentioned in the manuscript, so we have updated the Methods section:

The complete list of the descriptive parameters can be found in Table 1. Note that the ranges of LAI density and aerosol optical depth target amounts which are difficult to detect via remote sensing. As mentioned below, the analysis presented in this study was repeated for a range of aerosol optical depth up to 𝜏_C⁵⁵⁵ = 1.2. The results are nearly identical to those in Section 3.2, indicating that the conclusions remain valid for scenes with larger aerosol concentrations.

And Section 3.2:

…and even just 5 angles are sufficient to adequately sample the signals (Wu et. al. 2015).

When the range of impurities is extended to larger aerosol amounts, the absolute Sobol indices for the visible and NIR wavelengths increase because BC absorption causes large decreases in the total reflectance. Despite these changes, retrievals which use only VIS channels still fail due to simultaneous sensitivity to many parameters. Similarly, increased AOD leads to increased polarized reflectance (Ottaviani, 2022) in the NIR and SWIR, especially at positive viewing zenith angles. The Sobol indices and the uncertainty threshold reflect this increase in variance at larger viewing (smaller scattering) angles. Larger variance in the total and polarized reflectance also causes an increase in variance in the DoLP. Larger values of DoLP (corresponding to the largest values of AOD) also lead to a minor increase in the uncertainty threshold. These changes have minimal impact on the findings of this study, and retrievals from scenes with larger aerosol amounts (𝜏_C⁵⁵⁵ = 1.0) are nearly identical to those in Figs. 9 and 10.

(3)

To address the reviewer’s concern, we have added the following sentence to the introduction: 

In this paper, these details are examined via a Global Sensitivity Analysis (GSA) of simulated top-of-the-atmosphere (TOA) polarimetric observations, generalizing the studies presented in Ottaviani (2022). The focus is on scenes consisting of optically semi-infinite snow in homogeneous “pixels”, where both the atmosphere and the snowpack itself can host light-absorbing impurities.

To Section 3.1:

As explained in Sec. 2, the snowpack consists of a mixture of crystals (f^T=f^B=0.5). Fresher snow (smaller grains) is simulated in the top layer (r_eff^T=150 𝜇m, Z^T=3 cm, ⍴^T=0.2 g/cm3, AR^T=0.05 for plates and corresponding 1/AR^T=20 for columns, D^T=0.3 (Ottaviani, 2015)). More compact, larger and rounder grains are located in the bottom layer (r_eff^B=250 𝜇m, ⍴^B=0.30 g/cm3, AR^B=0.15 for plates and 6.67 for columns, D^B=0.40). The snowpack is assumed to be optically semi-infinite, so contributions from the underlying surface are ignored. For the reasons given at the end of Sec. 3…

And to the Conclusions:

The GSA presented in this study can be extended to LUTs that consider a whole suite of aerosol optical properties, region-specific impurity amounts, or optically thin snowpacks with different underlying land cover types.

(4)

As we mentioned at the end of the Conclusions (see lines 428-429), such work is underway with the only extensive spaceborne polarimetric dataset available (POLDER). Does the reviewer mean any other specific dataset? The results of the GSA are independent of validation, although we have made clear that Sobol indices surpassing the uncertainties threshold are not an absolute guarantee of retrievability (lines 227-229, 367-370, and 305-308 in section 3): full and precise measurement covariance matrices including off-diagonal elements that express cross-channel (spectral, angular) intercorrelations are notoriously hard to assess, and are rarely provided by the instrument teams.

Gao et al. [2023] evaluated the impact of uncertainty due to angular correlation on retrievals of aerosol properties, wind speed, and Chl-a from AirHARP and HARP2 data. Although accounting for off-diagonal terms can decrease retrieval uncertainty [Gao et al., 2023], the results of our best-performing retrieval (VIS+NIR+SWIR with RI+DoLP in Fig. 9) are anyway very accurate. The results of other simulated retrievals that use RSP-like data (VIS with RI in Fig. 9 for instance) might benefit from incorporating off-diagonal terms, but retrievals from real RSP data would anyway utilize all available channels (VIS+NIR+SWIR with RI+DoLP).

The angular spacing of MODIS and POLDER data is greater than that of the RSP, so the amount of angle-to-angle correlation [Knobelspiesse et al., 2012] and its effect on retrieval results is smaller.