# A Correlated Multi-Pixel Inversion Approach for Aerosol Remote Sensing

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## Abstract

**:**

## 1. Introduction

## 2. General Structure of the Algorithm

**v**), and spatial and temporal mean vector ${\overline{\mathbf{x}}}_{\mathrm{corr}}$, namely

**v**is made up of ${N}_{\mathrm{PC}}$ column vectors consisting of correlated aerosol fields, with ${N}_{\mathrm{PC}}$ denoting the number of retrieved principal components. By integrating all PCs into a matrix, we have

**w**that consists of N

_{pixel}columns, namely,

**v**,

**w**) are contained in the solution vector

**x**from the last iteration of an inversion process.

## 3. Inversion

#### 3.1. State Vector

#### 3.2. Constraints

- The first type of constraint, formulated in Section 3.2.1, consists of the observations provided directly by the remote sensing instrument(s).
- The second type of constraint consists of a priori values for the retrieval parameters, and is described in Section 3.2.2.
- For the uncorrelated fields (such as surface reflection properties), the pixel-resolved properties ${\mathbf{x}}_{\mathrm{uncorr}}$ can be subjected to both across-pixel and within-pixel constraints. Imposition of these types of constraints has been incorporated into the original multi-pixel inversion [6] and is repeated in Section 3.2.3 as well as in Appendix B as the third type of constraint.
- When a set of parameters (e.g., aerosol properties) are correlated with each other, their mean can be subjected to smoothness constraints. This is referred to as the fourth type of constraint in Section 3.2.3.
- Transformation of smooth variations of aerosol properties from regular aerosol parameters into the PC space forms the fifth type of constraint, discussed in Section 3.2.4.
- In PC space, the smoothness constraints can be applied to the PC weights
**w**and vectors**v**separately. Application to across-pixel weights**w**, discussed in Section 3.2.3, forms the sixth type of constraint. - Similarly, application of the smoothness constraints to certain type of parameters within a PC vector ${\mathbf{v}}_{k}$, also discussed in Section 3.2.3, forms the seventh type of constraint. Although it appears that the sixth and seventh types of constraints applied to “
**w**” and “**v**” separately are redundant with the fifth type of constraint that ensures a smooth variation of overall correlated field constructed from PCs; they are helpful when poor initial guesses of “**w**” and “**v**” are provided by a training dataset. - The eighth type of constraint, formulated in Section 3.2.5, imposes a zero sum of the PC weights.
- The ninth type of constraint, formulated in Section 3.2.6, imposes mutual orthogonality of the PC vectors.
- The tenth type of constraint, formulated in Section 3.2.7, imposes unit norm on all PC vectors.

**x**= ${\mathbf{x}}_{\mathrm{state}}$, as defined in Equation (6). Formally, the statistical independence of different sources of constraints means that the covariance matrix of joint constraint ${\mathbf{f}}^{\ast}=[{\mathbf{f}}_{1}^{\ast};{\mathbf{f}}_{2}^{\ast};\cdots ;{\mathbf{f}}_{M}^{\ast}]$ has the following structure

**W**) for multiple (M) types of constraints, the objective cost function to be minimized has a quadratic form, namely,

**x**approaches zero, such that

**x**by

**x**− Δ

**x**in Equation (22) and substitute

**K**and

**W**matrices for all ten types of constraints is discussed in the following subsections.

#### 3.2.1. Observational Constraints (i = 1)

**x**, namely,

#### 3.2.2. A Priori Constraints (i = 2)

**I**(identity matrix) and ${\mathbf{W}}_{i=2}=\frac{1}{{\epsilon}_{a\ast}^{2}}{\mathbf{C}}_{a\ast}$. More explicitly, ${\mathbf{W}}_{i=2}$ can be constructed from estimated range of each parameter relative to the first one,

#### 3.2.3. Smoothness Constraints in Regular Parameter Space (i = 3, 4, 6, and 7)

_{j}(namely, 1 ≤ j ≤ L) leads to

**W**has the following diagonal terms,

**W**is a unity matrix (namely

**W**=

**I**). To distinguish the smoothness constraints imposed on different types of parameters, we note here,

#### 3.2.4. Smoothness Constraints in PC Space (i = 5)

**x**(see Equation (57)), therefore ${\mathbf{K}}_{i}\ne {\mathbf{S}}_{i,m}$. Intuitively, one can think of ${\mathbf{f}}_{i=5}({\mathbf{x}}_{\mathrm{wv}})={\mathbf{S}}_{i,m=1}{\mathbf{x}}_{\mathrm{wv}}$ as a model prediction to fit the “observation” ${\mathbf{f}}_{i}^{\ast}$, which is a zero vector. Therefore, the above Jacobian matrix is evaluated in the same way as is done for observational constraints by use of finite difference methodology. The finite difference method is used for evaluating Equation (59).

**x**(cases for 3, 4, 6, and 7), then

**K**can be analytically evaluated from Equation (59), namely,

#### 3.2.5. Zero-sum Constraint on PC Weights (i = 8)

**w**in the state vector by an

**O**matrix, such that,

#### 3.2.6. Mutual Orthogonality Constraint among PC Vectors (i = 9)

#### 3.2.7. Unity-norm Constraint PC Vectors (i = 10)

**U**is expressed as,

#### 3.2.8. Construction of Overall Equation System

_{f}, N

_{c}, N

_{a}, and N

_{a*}are the total number of observations, total number of constraints imposed on retrieval (including smoothness constraints in different dimensions, zero-sum constraint over PC weights and orthogonality and unity constraints over PC vectors), total number of retrieval parameters, and total number of a priori estimates of parameters, respectively; and ${\epsilon}_{\mathrm{f}}^{2}$ is the expected variance due to measurement errors. In practice, forward RT modeling error and other unmodeled effects can impede realization of the required cost function minimization. Therefore, the retrieval is also terminated when the relative difference of fitting residues with solutions from two successive iterations drops below a user-specified threshold value, ${\epsilon}_{\mathrm{c}}^{2}$, namely,

#### 3.2.9. Determination of Lagrange Multipliers

#### 3.3. Retrieval Error Estiamte

**A**and $\nabla {\Psi}_{\mathrm{total}}$ are calculated by substituting ${\mathbf{x}}^{\mathrm{true}}$ into Equations (88) and (89), respectively.

#### 3.4. Retrieval Options

**c**has ${N}_{\mathrm{corr}}$ elements and is introduced here to balance or weight the contribution of all types of retrieval parameters in the PC analysis. It is calculated from (a) the standard deviation of a correlated field x varying in spatial or temporal scales (${\sigma}_{\mathrm{spatio}-\mathrm{temporal},x}$), and (b) its uncertainty estimate (${\sigma}_{\mathrm{e},x}$), namely,

## 4. Radiative Transfer in a Coupled Atmosphere-Surface System

#### 4.1. Fast Multi-Pixel Polarized RT Modeling in the Atmosphere

#### 4.1.1. Fast Multiple-Pixel Radiative Transfer Modeling Utilizing Correlation

**Y**), including total AOD (${\tau}_{\mathrm{aer},\mathrm{tot}}$), absorption AOD (${\tau}_{\mathrm{aer},\mathrm{abs}}$), and reflection and transmission matrices (

**R**and

**T**) for the above-surface atmosphere, can be expanded into Taylor series for multi-variable vector-valued functions ${\mathbf{v}}_{k}$. Adopting the second order of approximation (justified later in this section) and finite difference method to calculate derivatives, we have,

**R**and

**T**matrices are calculated for a few view and azimuthal angular grids for each viewed image. Interpolation is then used to obtain the reflection and transmission matrices for an arbitrary pixel before coupling them with the surface reflection (as we assume no correlation between aerosol and surface reflection properties in this paper). Coupling of atmospheric radiation and surface reflection to get the TOA radiance for fitting observation is formulated in the next section.

#### 4.1.2. Coupling Atmospheric Radiation with Surface Reflection

**Q**and

**S**are defined to account for the interaction between the surface and atmosphere, respectively,

#### 4.2. Jacobian Evaluation

_{f}measurements (${\mathbf{f}}_{1}^{\ast}=[{y}_{1};{y}_{2};\cdots ;{y}_{{N}_{f}}]$) and a state vector of ${N}_{a}$ components ($\mathbf{x}=[{x}_{1};{x}_{2};\cdots ;{x}_{{N}_{a}}]$), the Jacobian matrix has the following structure,

**R**, and

**T**with respect to the mean aerosol properties and pixel-resolved PC weights. Then, without applying the finite difference method again, the derivative of quantity

**Y**= {${\tau}_{\mathrm{aer},\mathrm{tot}}$, ${\tau}_{\mathrm{aer},\mathrm{abs}}$,

**R**,

**T**} at pixel p with respect to an element in k-th PC vector can be derived in a fast manner, namely, from the derivatives with respect to the mean and to the PC weights associated with p-th pixel,

## 5. Inversion of Aerosol and Surface Properties

#### 5.1. AirMSPI Datasets

^{4}RS) (August to September 2013), CalWater-2 (January to March 2015), and Imaging Polarimetric Assessment and Characterization of Tropospheric Particulate Matter (ImPACT-PM) (July 2016) campaigns. From these, 27 AirMSPI step-and-stare data collection sequences were identified to be cloud-free and collocated with AERONET sun photometers for retrieval validation. Locations of these AERONET sites and AirMSPI/AERONET measurement times can be found in a previous study [19]. To control the strength of multiple types of constraints, the initial values of Lagrange multipliers for the two PCs used in our retrieval are provided in Table 3, Table 4 and Table 5. Note that the difference between Table 3 and Table 4 and Table 5 is that Table 3 and Table 4 are for constructing the smoothness constraints over the correlated fields constructed from the combined set of PCs and PC weights. Table 5 is for constructing the smoothness constraints over PC weights and PC vectors separately. Table 3, Table 4 and Table 5 also list the first guesses of the relevant state vector components and the order of difference for imposing the smoothness constraints on these components.

#### 5.2. Retrieval Validation against AERONET Products

_{surf}) at 555 nm in the left, middle, and right panels, respectively. The retrieved AOD, SSA, and volume weighted aerosol size distribution for the atmosphere above the super-pixel closest to the Baskin site is compared to the AERONET reference data in the left, middle, and right panels of Figure 8d. Good agreement (quantified below) is obtained for all of these quantities, except for the coarse particle size distribution, likely due to the lack of bands longward of 1000 nm in AirMSPI. Generally, the difference between AirMSPI retrievals of AOD, SSA, and size distribution and AERONET reference data is within their retrieval uncertainties. The AirMSPI uncertainties plotted in Figure 8d are estimated as the root mean square of the retrieval uncertainties of these aerosol quantities and the standard deviation of their variations over the whole image. The AERONET uncertainties consist of two parts: temporal variation within the ±~1-h window centered on the AirMSPI nadir overpass time, and aerosol measurement and retrieval error [34]. A temporally closest AERONET reference data was identified compare to AirMSPI retrieval at the spatially closest pixel. To account for airmass change during the measurements, the ±~1-h window centered on AirMSPI nadir overpass time is used to calculate the AERONET uncertainty from temporal variation.

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^{2}) from the correlated multi-pixel inversion with those retrieved using original multi-pixel algorithm adapted for AirMSPI [19]. The AOD results for seven AirMSPI spectral bands are plotted in different colors: pink (355 nm), purple (380 nm), dark blue (445 nm), light blue (470 nm), green (555 nm), red (660 nm), and brown (865 nm). Linear regression is performed to obtain slope a, intercept b, as well as the coefficient of determination R

^{2}. The mean absolute difference (MAD) in AERONET and AirMSPI results is also calculated to measure the overall deviation. Strong correlation and low bias (R

^{2}≥ 0.88, a ~ 0.90, b ≤ 0.05, and MAD < 0.02) are observed. It can also be observed that a variation of aerosol loading by ~30% around the mean (namely in the range 0.28 ≤ ${\mathrm{AOD}}_{445nm}$ ≤ 0.40 with mean value 0.32) across the retrieval area is captured by the correlated multi-pixel inversion. Implementation of our approach using several datasets with even higher (~90%) variation of aerosol loading over several smoke scenes acquired by AirMSPI during the recent Aerosol Characterization from Polarimeter and Lidar (ACEPOL) campaign showed the algorithm to be capable of capturing this variation. The regression in Figure 9b shows correlations and low bias of SSAs (${\mathrm{R}}^{2}$ > 0.40, a > 0.60, b < 0.030, and MAD ≤ 0.004) from the two inversions as well. Figure 9c shows basic consistency in the retrieved aerosol size distributions: both algorithms find the peaks of fine and coarse mode aerosol size to be around ~0.15 μm and ~2 μm, respectively. Due to the lower sensitivity of AirMSPI’s longest wavelength 865 nm to coarse mode aerosols, some differences in coarse mode aerosol size can be observed. This indicates the impact of insufficient observational information about certain aerosol properties. Comparisons of pixel-scale AOD, SSA, and size distribution at other retrieval cases show a similar quality of agreement.

^{2}) and MAD are indicated in all panels. The AOD regression shows a spectral means of coefficient of determination 0.91, slope 0.93, and intercept 0.03, reflecting high retrieval quality. While SSA and refractive index in Figure 11, Figure 12 and Figure 13 show relatively larger differences between the AirMSPI and AERONET retrievals, the differences are generally within their respective uncertainties, which in turn depend on AirMSPI and AERONET observation errors and the sensitivities of the respective retrieval algorithms. Figure 14 shows a maximum difference of 30% between AirMSPI and AERONET retrieved fine mode aerosol size, whereas larger differences (up to 80%) are observed in coarse mode aerosol size. As noted above, shortwave infrared spectral bands, which AirMSPI lacks, are necessary to constrain the coarse mode aerosol size.

## 6. Summary and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Symbols and Abbreviations

Symbol/Abbreviation | Description |
---|---|

a | Slope of a regression line |

$\overline{a}$ | Mean slope of a set of regression lines |

${a}_{\lambda}$ | Spectral weight of surface BRDF |

A | Coefficient for a basic smooth function shape, e.g., “g(z) = Az ^{2} + Bz + C” for a parabola |

A | Fischer matrix |

AERONET | Aerosol Robotic Network |

AirMSPI | Airborne Multiangle SpectroPolarimetric Imager |

AOD | Aerosol optical depth |

ASTER | Advanced Spaceborne Thermal Emission and Reflection Radiometer |

A_{surf} | Surface albedo |

b | Intercept of a regression line |

$\overline{b}$ | Mean intercept of a set of regression lines |

B | Coefficient for a basic smooth function shape, e.g., “g(z) = Az ^{2} + Bz + C” for a parabola |

BRDF | Bidirectional reflectance distribution function |

BRF | Bidirectional reflectance factor (BRF) associated with Stokes vector component I |

c | Constant column vector that weight correlated fields in PC analysis |

C | Coefficient for a basic smooth function shape, e.g., “g(z) = Az ^{2} + Bz + C” for a parabola |

${\mathbf{C}}_{i}$ | Covariance matrix of i-th type of constraint |

${\mathbf{C}}_{\Delta \mathbf{x},\mathrm{rand}}$ | Covariance matrix of random errors in the measurements |

C_{v,sphere} | Column volume concentration of spherical aerosols |

C_{v,tot} | Total column volume concentration of aerosols |

d | Differentiation array |

${d}_{\mathrm{ES}}$ | Earth-Sun distance |

DOLP | Degree of linear polarization |

dV/dln(r) | Volume weighted aerosol size distribution |

${E}_{0}$ | Exo-atmospheric solar irradiance |

${\mathbf{f}}_{i}^{\ast}$ | Column vector of i-th type of constraint |

${\mathbf{f}}_{i}$ | Column vector that contains model prediction to fit i-th type of constraint |

$\Delta {\mathbf{f}}_{i}^{\ast}$ | Column vector that contains the errors of i-th type of constraint |

f_{fine} | Fine mode aerosol fraction |

f_{v,sphere} | Volume fraction of spherical aerosols |

g(z) | Smooth function with variable z |

${g}_{\lambda}$ | Spectral anisotropy parameter of surface BRDF |

GRASP | Generalized Retrieval of Aerosol and Surface ProperEes |

h | Cartesian coordinate in the direction h |

h_{a} | Central height of aerosol vertical profile (constrained by Gaussian profile) |

I | First Stokes vector component |

I | Identity matrix |

I_{meas} | Measured radiance |

${k}_{\lambda}$ | Spectral anisotropy parameter of surface BRDF |

${k}_{\gamma}$ | Shadowing width of polarized BRDF |

$\mathbf{K}$ | Jacobian matrix |

${\mathbf{K}}_{i}$ | The Jacobian matrix associated with i-th type of constraint |

L(j) | Length of j-th type of correlated parameters (fields) |

m | Order of difference in constructing smoothness matrix |

M | Number of types of constraints imposed on retrieval |

MAD | Mean absolute difference |

MAIAC | MultiAngle Implementation of Atmospheric Correction |

MISR | Multi-angle Imaging SpectroRadiometer |

MODIS | Moderate Resolution Imaging Spectroradiometer |

n_{r, j} | Real part of refractive index at j-th wavelength |

n_{i, j} | Imaginary part refractive index at j-th wavelength |

${N}_{\mathrm{a}}$ | Total number of retrieval parameters |

${N}_{{\mathrm{a}}^{\ast}}$ | Total number of a priori estimate of parameters |

${N}_{\mathrm{c}}$ | Total number of constraints imposed on retrieval |

${N}_{\mathrm{corr}}$ | Number of correlated parameters (fields) |

${N}_{\mathrm{f}}$ | Total number of observations (all pixels are accounted) |

${N}_{i}$ | Total number of i-th type of constraint |

${N}_{\mathrm{PC}}$ | Total number of principle components |

${N}_{\mathrm{pixel}}$ | Total number of pixels |

${N}_{\mathrm{SC}}$ | Number of aerosol size components |

${N}_{\mathrm{TP},\mathrm{uncorr}}$ | Total number of types of uncorrelated parameters |

O | Constraining matrix that reflects zero-sum of PC weights |

pBRDF | Polarized BRDF |

P | Probability distribution function (PDF) |

PC | Principal component |

PCA | Principal component analysis |

POLDER | Polarization and Directionality of Earth’s Reflectance |

q | Ratio of Stokes components Q and I |

q | Iterative step during optimization |

qBRF_{TOA} | Top-of-atmosphere BRF associated with Stokes vector component Q |

Q | Second Stokes vector component |

r | Radius of aerosol |

r_{eff,coarse} | Effective radius of coarse mode aerosols |

r_{eff,fine} | Effective radius of fine mode aerosols |

R | Reflection matrix |

R^{2} | Coefficient of determination |

${\mathbf{R}}_{\mathrm{atmos}}$ | Reflection matrix for atmosphere associated with light illumination from top of the atmosphere |

${\mathbf{R}}_{\mathrm{atmos}}^{\ast}$ | Reflection matrix for atmosphere associated with light illumination from bottom of the atmosphere |

${\mathbf{R}}_{\mathrm{CAS}}$ | Reflection matrix for the coupled atmosphere-surface system (CAS) |

RPV | Rahman-Pinty-Verstraete (surface BRDF model) |

${\mathbf{R}}_{\mathrm{surf}}$ | Surface reflection matrix |

${\mathbf{R}}_{\mathrm{surf},\mathrm{BRDF}}$ | Depolarizing part of surface reflection matrix |

${\mathbf{R}}_{\mathrm{surf},\mathrm{pBRDF}}$ | Polarizing part of surface reflection matrix |

RT | Radiative transfer |

s_{a} | Standard deviation of aerosol vertical profile (constrained by Gaussian profile) |

${\mathbf{S}}_{i,m}$ | Differentiation matrix of m-th order for i-th type of constraint |

SSA | Single scattering albedo |

t | Temporal coordinate |

T | Transmission matrix |

${\mathbf{T}}_{\mathrm{atmos}}$ | Transmission matrix for atmosphere associated with light illumination from top of the atmosphere |

${\mathbf{T}}_{\mathrm{atmos}}^{\ast}$ | Transmission matrix for atmosphere associated with light illumination from bottom of the atmosphere |

TOA | Top-of-atmosphere |

u | Cartesian coordinate in the direction u |

u | Ratio of Stokes components U and I |

uBRF_{TOA} | Top-of-atmosphere BRF associated with Stokes vector component U |

U | Third Stokes vector component |

U | Constraining matrix that reflects the unit length of a PC |

USGS | U.S. Geological Survey |

v | Cartesian coordinate in the direction v |

$\mathbf{v}$ | PC matrix containing N_{PC} columns PC vectors |

vBRF_{TOA} | Top-of-atmosphere BRF associated with Stokes vector component V |

${\mathbf{v}}_{k}$ | The k-th PC vector |

${\mathbf{v}}_{\mathrm{state}}$ | Column vector containing all PC vectors |

w | PC weight matrix containing N_{pixel} column vectors containing PC weights |

${\mathbf{w}}_{i}$ | Weight matrix for i-th type of constraint |

${\mathbf{w}}_{p}$ | Column vector containing PC weights for p-th pixel |

${\mathbf{w}}_{\mathrm{state}}$ | Column vector containing all PC weights |

$\mathbf{x}$ | Column state vector including all retrieval parameters |

${\mathbf{x}}^{apriori}$ | a priori of state vector |

${\overline{\mathbf{x}}}_{\mathrm{corr}}$ | Column vector containing spatial and temporal mean of correlated parameters (fields) |

${\mathbf{x}}_{\mathrm{corr},p}$ | Column vector containing correlated parameters (fields) for p-th pixel |

${\mathbf{x}}_{q,\mathrm{aer}}$ | The vector consisting of correlated aerosol properties – calculated from the solution at q-th iteration |

${\mathbf{x}}_{q,\mathrm{surf}}$ | The vector consisting of uncorrelated surface reflection properties – containing in the solution at q-th iteration |

${\mathbf{x}}^{\mathrm{retrieved}}$ | Retrieved column state vector |

$\Delta {\mathbf{x}}_{\mathrm{syst}}$ | Systematic error in retrieval |

${\mathbf{x}}_{\mathrm{uncorr},p}$ | Column vector containing uncorrelated parameters (fields) for p-th pixel |

${\mathbf{x}}^{\mathrm{true}}$ | Column state vector associated with true solution |

${\mathbf{x}}_{\mathrm{wv}}$ | Column vector including PC weights and vectors |

${(\Delta x)}_{j}$ | The retrieval error in j-th parameter |

${y}_{i}$ | i-th observational signal |

${\mathbf{Y}}^{\mathrm{HS}}$ | Output of RT calculation with high stream approximation |

${\mathbf{Y}}^{\mathrm{LS}}$ | Output of RT calculation with low stream approximation |

z | Variable of a smooth function |

z_{min} | Lower bound of z |

z_{max} | Upper bound of z |

Z | Number of observations per pixel |

$\delta $ | Kronecker delta |

${\delta}_{\mathrm{s}}$ | Scale factor that perturbs a PC vector |

${\epsilon}_{\mathrm{rand}}$ | Random error in measurements |

${\epsilon}_{\lambda}$ | Spectral weight of pBRDF |

${\epsilon}_{i}^{2}$ | First diagonal element of C_{i} |

${\epsilon}_{\mathrm{c}}^{2}$ | User-specified threshold value to diagnose the convergence of optimization |

${\epsilon}_{\mathrm{f}}^{2}$ | Expected variance due to measurement errors |

${\theta}_{0}$ | Solar zenith angle |

${\theta}_{\mathrm{v}}$ | View zenith angle |

$\lambda $ | Wavelength |

${\mu}_{0}$ | Cosine of solar zenith angle |

${\gamma}_{i}$ | Lagrange factor for i-th type of constraint |

${\varphi}_{0}$ | Solar azimuthal angle |

${\Psi}_{i}$ | Objective cost function for i-th type of constraint |

${\Psi}_{\mathrm{total}}$ | Overall objective cost function |

$\nabla {\Psi}_{i}$ | Gradient of the objective cost function for i-th type of constraint |

$\nabla {\Psi}_{\mathrm{total}}$ | Gradient of the overall objective cost function |

${\mathbf{\Omega}}_{i}$ | The smoothness matrix associated with i-th type of constraints |

${\mathbf{\Omega}}_{\dots}^{\mathrm{Ra}}$ | The rearranged smoothness matrix from ${\mathbf{\Omega}}_{\dots}$ |

${\mathbf{\Omega}}_{\mathrm{uncorr}}$ | The smoothness matrix imposed on uncorrelated parameters (fields) |

${\mathbf{\Omega}}_{\mathrm{corr}}^{{\displaystyle \overline{\mathrm{x}}}}$ | The smoothness matrix imposed on spatial and temporal mean mean of correlated parameters (fields) |

${\mathbf{\Omega}}_{\mathrm{corr}}^{\mathrm{w}}$ | Smoothness matrix imposed on pixel resolved PC weights |

${\mathbf{\Omega}}_{\mathrm{corr}}^{\mathrm{v}}$ | Smoothness matrix imposed on a PC vector |

${\mathbf{\Omega}}_{\mathrm{corr}}^{\mathrm{wv}}$ | Smoothness matrix imposed on correlated parameters (fields) |

${\sigma}_{\mathrm{spatio}-\mathrm{temporal},x}$ | Standard deviation of a correlated field x |

${\sigma}_{\mathrm{e},x}$ | Uncertainty estimate of a correlated field x |

${\sigma}_{\mathrm{s}}$ | Slope variance of polarized BRDF |

${\tau}_{\mathrm{aer},\mathrm{tot}}$ | Total aerosol optical depth |

${\tau}_{\mathrm{aer},\mathrm{abs}}$ | Total absorption aerosol optical depth |

${\tau}_{\mathrm{atmos}}$ | Atmospheric optical depth |

$\mathbf{\Gamma}$ | Constraining matrix that reflects the mutual orthogonality in PCs |

## Appendix B. Smoothness Matrix to Constrain Uncorrelated Parameter Retrieval

**0**is the zero block matrix. Pixel resolved matrices ${\gamma}_{\mathrm{uncorr},\diamond}{\mathbf{\Omega}}_{\mathrm{uncorr},\diamond ,p}$ do not interfere with each other, so they are arranged along the diagonal axis of the large matrix on the right-hand-side of the above equation. To facilitate the use of ${\gamma}_{\mathrm{uncorr}}{\mathbf{\Omega}}_{\mathrm{uncorr}}$ calculated via Equation (A1), the uncorrelated parameters are grouped together in the order of ${\mathbf{x}}_{\mathrm{uncorr}}=\left[\mathbf{x}({t}_{1});\mathbf{x}({t}_{2});\cdots ;\mathbf{x}({t}_{{N}_{t}})\right]$, where $\mathbf{x}({t}_{j})=\left[\mathbf{x}({v}_{1};{t}_{j});\mathbf{x}({v}_{2};{t}_{j});\cdots ;\mathbf{x}({v}_{{N}_{v}};{t}_{j})\right]$, $\mathbf{x}({v}_{i};{t}_{j})=\left[\mathbf{x}({u}_{1};{v}_{i};{t}_{j});\mathbf{x}({u}_{2};{v}_{i};{t}_{j});\cdots ;\mathbf{x}({u}_{{N}_{u}};{v}_{i};{t}_{j})\right]$, and $\mathbf{x}({u}_{k};{v}_{i};{t}_{j})$ is a vector that contains uncorrelated parameters for the pixel $({u}_{k},{v}_{i})$ observed at temporal point ${t}_{j}$. Evaluations of ${\gamma}_{\mathrm{uncorr},\Delta}{\mathbf{\Omega}}_{\mathrm{uncorr},\Delta}$ and ${\gamma}_{\mathrm{uncorr},\diamond}{\mathbf{\Omega}}_{\mathrm{uncorr},\diamond ,p}$ are discussed in Appendix B.1 and Appendix B.2, respectively.

#### Appendix B.1. Across-Pixel Smoothness Matrix

**d**accounts for the distance between neighboring pixels in the direction v and is evaluated by Equation (40).

**d**accounts for the temporal gap between successive measurements and is evaluated by Equation (40).

#### Appendix B.2. Within-Pixel Smoothness Matrix

**d**accounts for the distance between neighboring wavelengths and proceeds in the way as expressed in Equations (A7) and (A8).

## Appendix C. Smoothness Matrix for Correlated Parameters

#### Appendix C.1. Across-Pixel Smoothness Constraints

**B**and

**C**that account for the contributions by PCs and PC weights, respectively, in the following form,

**B**matrix is expressed as,

**C**is a null matrix except its i-th row has fill-in values, namely

#### Appendix C.2. Within-Pixel Smoothness Matrix

**W**that has a similar structure as Equations (A16) and (A17) (but for the i-th type of correlated parameters and is expanded to account for the number of pixels). To account for the smooth variation of a type of correlated parameters, the contributions of PC vectors and weights are coupled in the following way,