# Multi-View Polarimetric Scattering Cloud Tomography and Retrieval of Droplet Size

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## Abstract

**:**

## 1. Motivation, Context & Outline

#### 1.1. Why Polarized Light?

#### 1.2. Why Passive Tomography?

#### 1.3. Context

#### 1.4. Outline

## 2. Theoretical Background

#### 2.1. Scatterer Microphysical Properties

#### 2.2. Single Scattering of Polarized Light

#### Mie Scattering

#### 2.3. Multiple Scattering of Polarized Light

#### 2.4. Single-Scattering Separation

#### 2.5. Ray Tracing

## 3. Cloud Tomography

#### 3.1. Polarimetric Information

#### 3.2. Inverse Problem Formulation

#### 3.3. Iterative Solution Approach

`Step 1`uses vSHDOM to compute the forward (recursive) RT equations. This renders synthetic images according to the multi-view geometry, spectral bands and spatial samples of the cameras. Keeping ${\mathbf{I}}_{\mathrm{d}}$ fixed,

`Step 2`efficiently computes an approximate gradient with respect to $\Theta $. The approximate gradient is fed into an L-BFGS step to update the current estimate ${\Theta}_{b}$.

#### `Step 1: RTE Forward Model`

#### `Step 2: Approximate Jacobian Computation`

**is**impacted by ${\mathbf{I}}_{\mathrm{d}}$. This is because ${\mathbf{I}}_{\mathrm{d}}$ affects $\mathbf{I}$ through Equation (12), and $\mathbf{I}$ appears in the terms ${A}_{1},\dots ,{A}_{6}$. As the estimated medium properties evolve through iterations, so does ${\mathbf{I}}_{\mathrm{d}}$ (in

`Step 1`, above). We just assume during

`Step 2`that ${\partial}_{g}{\mathbf{I}}_{\mathrm{d}}$ is negligible compared to other terms in Equation (35).

## 4. Computational Efficiency

## 5. Simulations

`Scene A:`An atmospheric domain of dimensions 0.64 × 0.72 × 20 km${}^{3}$ with an isolated cloud (see synthetic AirMSPI nadir view in Figure 8).`Scene B:`An atmospheric domain of dimensions 2.42 × 2.1 × 8 km${}^{3}$ with several clouds of varying optical thickness (see synthetic AirMSPI nadir view in Figure 9).

`Scene A`are shown in Figure 10. Scatter plot of the recovered LWC and the recovery results of ${r}_{\mathrm{e}}$ for

`Scene A`are given in Figure 11. Qualitative volumetric results of the recovered LWC for

`Scene B`are shown in Figure 12. A scatter plot of the recovered LWC and the recovery results of ${r}_{\mathrm{e}}$ for

`Scene B`are given in Figure 13.

`Scene A:`${\u03f5}_{{r}_{\mathrm{e}}}\approx 11\%,\phantom{\rule{3.33333pt}{0ex}}{\u03f5}_{\mathrm{LWC}}\approx 30\%,\phantom{\rule{3.33333pt}{0ex}}{\vartheta}_{\mathrm{LWC}}\approx -4\%,\phantom{\rule{3.33333pt}{0ex}}\begin{array}{c}\hfill \mathrm{RMS}\approx 0.065\phantom{\rule{3.33333pt}{0ex}}g\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}{m}^{3},\phantom{\rule{3.33333pt}{0ex}}\mathrm{bias}\approx -0.002\phantom{\rule{3.33333pt}{0ex}}g\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}{m}^{3}\end{array}$,

`Scene B:`${\u03f5}_{{r}_{\mathrm{e}}}\approx 13\%,\phantom{\rule{3.33333pt}{0ex}}{\u03f5}_{\mathrm{LWC}}\approx 29\%,\phantom{\rule{3.33333pt}{0ex}}{\vartheta}_{\mathrm{LWC}}\approx -5\%,\phantom{\rule{3.33333pt}{0ex}}\begin{array}{c}\hfill \mathrm{RMS}\approx 0.085\phantom{\rule{3.33333pt}{0ex}}g\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}{m}^{3},\phantom{\rule{3.33333pt}{0ex}}\mathrm{bias}\approx -0.0035\phantom{\rule{3.33333pt}{0ex}}g\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}{m}^{3}\end{array}$.

`Scenes A,B`was $\sim 13$ h and $\sim 10$ days, respectively.

## 6. Discussion

## 7. Summary & Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Extended Theoretical Background

#### Appendix A.1. Scatterer Microphysical Properties

#### Appendix A.2. Polarized Light

#### Appendix A.3. Single Scattering of Polarized Light

#### Appendix A.3.1. Rayleigh Scattering

#### Appendix A.3.2. Mie Scattering

## Appendix B. Jacobian Derivation

`Step 1`and are therefor ready for use when computing ${A}_{1},{A}_{2},{A}_{3},{A}_{5}$ and ${A}_{6}$. Furthermore, ${\ell}_{g}\left({x}_{0}\to {x}_{k}\right)=0$ for any voxel that is not on the LOS of pixel k. Therefore, the terms ${A}_{1},{A}_{2},{A}_{3},{A}_{5},{A}_{6}$ are computed using a single path tracing ${x}_{k}\to {x}_{0}$.

## Appendix C. Measurement Noise

**Figure A1.**A normalized frame spans the interval $[-0.5,0.5]$, evenly divided into ${N}_{\mathrm{sub}}$ subframes.

**Table A1.**Modulation parameters [105] used for synthesis of AirMSPI measurements.

${\mathit{\gamma}}_{0}\left(470\phantom{\rule{3.33333pt}{0ex}}\mathbf{nm}\right)$ | ${\mathit{\gamma}}_{0}\left(660\phantom{\rule{3.33333pt}{0ex}}\mathbf{nm}\right)$ | ${\mathit{\gamma}}_{0}\left(865\phantom{\rule{3.33333pt}{0ex}}\mathbf{nm}\right)$ | $\mathit{\xi}\left(470\phantom{\rule{3.33333pt}{0ex}}\mathbf{nm}\right)$ | $\mathit{\xi}\left(660\phantom{\rule{3.33333pt}{0ex}}\mathbf{nm}\right)$ | $\mathit{\xi}\left(865\phantom{\rule{3.33333pt}{0ex}}\mathbf{nm}\right)$ | $\mathit{\eta}$ |
---|---|---|---|---|---|---|

4.472 | 3.081 | 2.284 | 1.0 | 0.27 | 0.03 | 0.009 |

## Appendix D. Numerical Considerations

#### Appendix D.1. Hyper-Parameters

`Step 1`and optimization with scipy L-BFGS [73,107] in

`Step 2`. Table A2 summarizes the numerical parameters used in our simulations.

vSHDOM | L-BFGS | ||||
---|---|---|---|---|---|

${N}_{\mu}$ | ${N}_{\varphi}$ | splitting accuracy | gtol | gtol | maxls |

8 | 16 | 0.1 | $1\times {10}^{-16}$ | $1\times {10}^{-16}$ | 30 |

#### Appendix D.2. Preconditioning

#### Appendix D.3. Initialization

- Each image is segmented into potentially cloudy and non-cloudy pixels (we use a simple radiance threshold).
- From each camera viewpoint, each potentially cloudy pixel back-projects a ray into the 3D domain. Voxels that this ray crosses are voted as potentially cloudy.
- Voxels which accumulate “cloudy” votes in at least 8 out of the 9 AirMSPI viewpoints are marked as cloudy.

#### Appendix D.4. Convergence

`Step 1`(RTE rendering) and

`Step 2`(approximate gradient) until convergence (Figure 7). The convergence criteria are dictated by the L-BFGS step: at each iteration, the relative change to the forward model and its gradient are compared to the ftol and gtol parameters (see Table A2 for values used). See SciPy documentation [107] for exact description of the L-BFGS stopping criteria.

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**Figure 1.**Artist’s illustration of the CloudCT [8] mission: a distributed multi-view system of 10 nano-satellites orbiting the Earth in formation. Measurements acquired by the formation will enable tomographic retrievals of cloud properties. Courtesy Mark Sheinin.

**Figure 2.**(

**Left**) A Normalized Gamma droplet size distribution. The effective radius and variance dictate the centroid and width of the droplet size distribution. The limit of very low ${v}_{\mathrm{e}}$ approaches a mono-disperse distribution. (

**Center**) Log-polar plot of the Mie phase-function ${p}_{11}$ induced by a single water sphere of radius r. (

**Right**) Log-polar plot of the effective phase-function ${\langle {s}_{\mathrm{s}}{p}_{11}\rangle}_{r}/{\sigma}_{\mathrm{s}}$ induced by a small volume that includes particles of different sizes.

**Figure 3.**Normalized phase matrix element $-{p}_{12}^{\mathrm{Mie}}/{p}_{11}^{\mathrm{Mie}}$ around the cloud-bow and glory regions. For highly disperse droplet distributions (large ${v}_{e}$) the secondary lobes of the cloud-bow ($\theta \sim {140}^{\xb0}$) and glory ($\theta \sim {180}^{\xb0}$) diminish. The main cloud-bow peak is slightly sensitive to $\lambda $ or ${v}_{\mathrm{e}}$. The side-lobe angles are more sensitive to $\lambda $ and ${r}_{\mathrm{e}}$. The side-lobe amplitude is sensitive to ${v}_{\mathrm{e}}$. This cloud-bow signal is helpful for retrievals of ${r}_{\mathrm{e}}$. [Right plot] Solid lines indicate monochromatic light. Dashed lines indicate spectral averaging over a 100 nm bandwidth, which is more than double any of the spectral bands considered further on.

**Figure 4.**(

**Left**) Light scatters in the medium, generally multiple times, creating a partially polarized (vector) scatter field $\mathbf{J}$ (9). Integration yields the partially polarized (vector) light field $\mathbf{I}$ (8). Here $I({x}_{k},{\mathbf{\omega}}_{k})$ is a pixel measurement at the top of the atmosphere (TOA) and ${\mathbf{I}}_{\mathrm{Single}}$ is the single-scattered contribution from ${x}^{\prime}$. (

**Right**) Ray tracing of a line-integral over a discretized voxel field $h\left[g\right]$ (zero-order interpolation).

**Figure 5.**A homogeneous cubic cloud illuminated with solar radiation at a zenith angle of ${15}^{\xb0}$ off-nadir. The solar azimuth angles are ${\varphi}_{0}=[{0.0}^{\xb0},{67.5}^{\xb0}]$. The outgoing Stokes vector $\mathbf{I}$ is simulated at AirMSPI resolution and wavelengths, with AirMSPI measuring along a North-bound track.

**Figure 7.**A block diagram of the iterative algorithm. Red marks hyper-parameters. Numerical parameters of vector Spherical Harmonics Discrete Ordinates Method (vSHDOM) and L-BFGS are summarized in Appendix D (Table A2).

**Figure 8.**

`Scene A`synthesized Stokes image using vSHDOM, before and after the application of a realistic AirMSPI noise model. We show here the Bidrectional Reflectance Factor (BRF) of the nadir view at $\lambda =0.67\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$.

**Figure 9.**

`Scene B`synthesized Stokes using vSHDOM. We show here the BRF of the nadir view at $\lambda =0.67\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$.

**Figure 10.**

`Scene A`recovery results. (

**Left**) Slices of the true cloud generated by Large Eddy Simulation (LES). (

**Right**) Slices of the cloud estimated tomographically using AirMSPI polarized bands.

**Figure 11.**

`Scene A`recovery results. (

**Left**) Scatter plot of estimated vs. true Liquid Water Content (LWC). The correlation coefficient is 0.94, the Root Mean Square (RMS) is 0.065 g/m${}^{3}$ and the bias is −0.002 g/m${}^{3}$. (

**Right**) recovery results of the 1D effective radius.

**Figure 12.**

`Scene B`recovery results. (

**Left**) Slices of the true LES generated region. (

**Right**) Slices of the estimated region.

**Figure 13.**

`Scene B`(

**Left**) Scatter plot of the estimated vs. true LWC. The fit correlation is 0.96, the RMS is 0.085 g/m${}^{3}$ and the bias is −0.0035 g/m${}^{3}$. (

**Right**) Recovery results of the 1D effective radius.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Levis, A.; Schechner, Y.Y.; Davis, A.B.; Loveridge, J.
Multi-View Polarimetric Scattering Cloud Tomography and Retrieval of Droplet Size. *Remote Sens.* **2020**, *12*, 2831.
https://doi.org/10.3390/rs12172831

**AMA Style**

Levis A, Schechner YY, Davis AB, Loveridge J.
Multi-View Polarimetric Scattering Cloud Tomography and Retrieval of Droplet Size. *Remote Sensing*. 2020; 12(17):2831.
https://doi.org/10.3390/rs12172831

**Chicago/Turabian Style**

Levis, Aviad, Yoav Y. Schechner, Anthony B. Davis, and Jesse Loveridge.
2020. "Multi-View Polarimetric Scattering Cloud Tomography and Retrieval of Droplet Size" *Remote Sensing* 12, no. 17: 2831.
https://doi.org/10.3390/rs12172831