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

^{1}

^{2}

^{3}

^{*}

^{†}

## 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, ${\ell}_{g}\left(\right)open="("\; close=")">{x}_{0}\to {x}_{k}$ 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.

## References

- Trenberth, K.E.; Fasullo, J.T.; Kiehl, J. Earth’s global energy budget. Bull. Am. Meteorol. Soc.
**2009**, 90, 311–324. [Google Scholar] [CrossRef] - Boucher, O.; Randall, D.; Artaxo, P.; Bretherton, C.; Feingold, G.; Forster, P.; Kerminen, V.M.; Kondo, Y.; Liao, H.; Lohmann, U.; et al. Clouds and aerosols. In Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change; Cambridge University Press: Cambridge, UK, 2013; pp. 571–657. [Google Scholar]
- Rosenfeld, D.; Lensky, I.M. Satellite-based insights into precipitation formation processes in continental and maritime convective clouds. Bull. Am. Meteorol. Soc.
**1998**, 79, 2457–2476. [Google Scholar] [CrossRef][Green Version] - Platnick, S.; King, M.; Ackerman, S.; Menzel, W.; Baum, B.; Riedi, J.; Frey, R. The MODIS cloud products: Algorithms and examples from Terra. IEEE Trans. Geosci. Remote Sens.
**2003**, 41, 459–473. [Google Scholar] [CrossRef][Green Version] - Marshak, A.; Platnick, S.; Várnai, T.; Wen, G.; Cahalan, R.F. Impact of three-dimensional radiative effects on satellite retrievals of cloud droplet sizes. J. Geophys. Res. Atmos.
**2006**, 111. [Google Scholar] [CrossRef] - Cho, H.M.; Zhang, Z.; Meyer, K.; Lebsock, M.; Platnick, S.; Ackerman, A.S.; Di Girolamo, L.; C.-Labonnote, L.; Cornet, C.; Riedi, J.; et al. Frequency and causes of failed MODIS cloud property retrievals for liquid phase clouds over global oceans. J. Geophys. Res. Atmos.
**2015**, 120, 4132–4154. [Google Scholar] [CrossRef] - National Academies of Sciences, Engineering, and Medicine. Thriving on Our Changing Planet: A Decadal Strategy for Earth Observation from Space; The National Academies Press: Washington, DC, USA, 2018. [Google Scholar]
- Schilling, K.; Schechner, Y.Y.; Koren, I. CloudCT—computed tomography of clouds by a small satellite formation. In Proceedings of the IAA Symposium on Small Satellites for Earth Observation, Berlin, Germany, 6–10 May 2019. [Google Scholar]
- Nakajima, T.; King, M.D. Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. Part I: Theory. J. Atmos. Sci.
**1990**, 47, 1878–1893. [Google Scholar] [CrossRef][Green Version] - Deschamps, P.Y.; Bréon, F.M.; Leroy, M.; Podaire, A.; Bricaud, A.; Buriez, J.C.; Seze, G. The POLDER mission: Instrument characteristics and scientific objectives. IEEE Trans. Geosci. Remote. Sens.
**1994**, 32, 598–615. [Google Scholar] [CrossRef] - Bréon, F.M.; Goloub, P. Cloud droplet effective radius from spaceborne polarization measurements. Geophys. Res. Lett.
**1998**, 25, 1879–1882. [Google Scholar] [CrossRef] - Kalashnikova, O.V.; Garay, M.J.; Davis, A.B.; Diner, D.J.; Martonchik, J.V. Sensitivity of multi-angle photo-polarimetry to vertical layering and mixing of absorbing aerosols: Quantifying measurement uncertainties. J. Quant. Spectrosc. Radiat. Transf.
**2011**, 112, 2149–2163. [Google Scholar] [CrossRef] - Lukashin, C.; Wielicki, B.A.; Young, D.F.; Thome, K.; Jin, Z.; Sun, W. Uncertainty estimates for imager reference inter-calibration with CLARREO reflected solar spectrometer. IEEE Trans. Geosci. Remote. Sens.
**2013**, 51, 1425–1436. [Google Scholar] [CrossRef] - Diner, D.; Xu, F.; Garay, M.; Martonchik, J.; Rheingans, B.; Geier, S.; Davis, A.; Hancock, B.; Jovanovic, V.; Bull, M.; et al. The Airborne Multiangle SpectroPolarimetric Imager (AirMSPI): A new tool for aerosol and cloud remote sensing. Atmos. Meas. Tech.
**2013**, 6, 2007–2025. [Google Scholar] [CrossRef][Green Version] - Diner, D.J.; Boland, S.W.; Brauer, M.; Bruegge, C.; Burke, K.A.; Chipman, R.; Di Girolamo, L.; Garay, M.J.; Hasheminassab, S.; Hyer, E.; et al. Advances in multiangle satellite remote sensing of speciated airborne particulate matter and association with adverse health effects: From MISR to MAIA. J. Appl. Remote. Sens.
**2018**, 12, 042603. [Google Scholar] [CrossRef][Green Version] - Martins, J.V.; Nielsen, T.; Fish, C.; Sparr, L.; Fernandez-Borda, R.; Schoeberl, M.; Remer, L. HARP CubeSat–An innovative hyperangular imaging polarimeter for earth science applications. In Proceedings of the Small Sat Pre-Conference Workshop, Logan, UT, USA, 3 August 2014; Volume 20. [Google Scholar]
- Emde, C.; Barlakas, V.; Cornet, C.; Evans, F.; Korkin, S.; Ota, Y.; Labonnote, L.C.; Lyapustin, A.; Macke, A.; Mayer, B.; et al. IPRT polarized radiative transfer model intercomparison project—Phase A. J. Quant. Spectrosc. Radiat. Transf.
**2015**, 164, 8–36. [Google Scholar] [CrossRef][Green Version] - Emde, C.; Barlakas, V.; Cornet, C.; Evans, F.; Wang, Z.; Labonotte, L.C.; Macke, A.; Mayer, B.; Wendisch, M. IPRT polarized radiative transfer model intercomparison project—Three-dimensional test cases (Phase B). J. Quant. Spectrosc. Radiat. Transf.
**2018**, 209, 19–44. [Google Scholar] [CrossRef][Green Version] - Kak, A.; Slaney, M. Principles of Computerized Tomographic Imaging IEEE Press; IEEE Press: Piscataway, NJ, USA, 1988. [Google Scholar]
- Gordon, R.; Bender, R.; Herman, G.T. Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography. J. Theor. Biol.
**1970**, 29, 471–481. [Google Scholar] [CrossRef] - Marshak, A.; Davis, A.; Cahalan, R.; Wiscombe, W. Nonlocal independent pixel approximation: Direct and inverse problems. IEEE Trans. Geosci. Remote. Sens.
**1998**, 36, 192–204. [Google Scholar] [CrossRef][Green Version] - Faure, T.; Isaka, H.; Guillemet, B. Neural network retrieval of cloud parameters of inhomogeneous and fractional clouds: Feasibility study. Remote Sens. Environ.
**2001**, 77, 123–138. [Google Scholar] [CrossRef] - Faure, T.; Isaka, H.; Guillemet, B. Neural network retrieval of cloud parameters from high-resolution multispectral radiometric data: A feasibility study. Remote Sens. Environ.
**2002**, 80, 285–296. [Google Scholar] [CrossRef] - Cornet, C.; Isaka, H.; Guillemet, B.; Szczap, F. Neural network retrieval of cloud parameters of inhomogeneous clouds from multispectral and multiscale radiance data: Feasibility study. J. Geophys. Res. Atmos.
**2004**, 109, D12203. [Google Scholar] [CrossRef] - Zinner, T.; Mayer, B.; Schröder, M. Determination of three-dimensional cloud structures from high-resolution radiance data. J. Geophys. Res. Atmos.
**2006**, 111, D08204. [Google Scholar] [CrossRef] - Iwabuchi, H.; Hayasaka, T. A multi-spectral non-local method for retrieval of boundary layer cloud properties from optical remote sensing data. Remote Sens. Environ.
**2003**, 88, 294–308. [Google Scholar] [CrossRef] - Diner, D.J.; Beckert, J.C.; Reilly, T.H.; Bruegge, C.J.; Conel, J.E.; Kahn, R.A.; Martonchik, J.V.; Ackerman, T.P.; Davies, R.; Gerstl, S.A.W.; et al. Multi-angle Imaging SpectroRadiometer (MISR) instrument description and experiment overview. IEEE Trans. Geosci. Remote Sens.
**1998**, 36, 1072–1087. [Google Scholar] [CrossRef] - Marchand, R.; Ackerman, T. Evaluation of radiometric measurements from the NASA Multiangle Imaging SpectroRadiometer (MISR): Two-and three-dimensional radiative transfer modeling of an inhomogeneous stratocumulus cloud deck. J. Geophys. Res. Atmos.
**2004**, 109. [Google Scholar] [CrossRef] - Seiz, G.; Davies, R. Reconstruction of cloud geometry from multi-view satellite images. Remote. Sens. Environ.
**2006**, 100, 143–149. [Google Scholar] [CrossRef] - Cornet, C.; Davies, R. Use of MISR measurements to study the radiative transfer of an isolated convective cloud: Implications for cloud optical thickness retrieval. J. Geophys. Res. Atmos.
**2008**, 113. [Google Scholar] [CrossRef] - Evans, K.F.; Marshak, A.; Várnai, T. The Potential for Improved Boundary Layer Cloud Optical Depth Retrievals from the Multiple Directions of MISR. J. Atmos. Sci.
**2008**, 65, 3179–3196. [Google Scholar] [CrossRef] - Romps, D.M.; Öktem, R. Observing Clouds in 4D with Multiview Stereophotogrammetry. Bull. Am. Meteorol. Soc.
**2018**, 99, 2575–2586. [Google Scholar] [CrossRef][Green Version] - Castro, E.; Ishida, T.; Takahashi, Y.; Kubota, H.; Perez, G.J.; Marciano, J.S. Determination of cloud-top Height through three-dimensional cloud Reconstruction using DIWATA-1 Data. Sci. Rep.
**2020**, 10, 1–13. [Google Scholar] [CrossRef] - Alexandrov, M.D.; Cairns, B.; Emde, C.; Ackerman, A.S.; Ottaviani, M.; Wasilewski, A.P. Derivation of cumulus cloud dimensions and shape from the airborne measurements by the Research Scanning Polarimeter. Remote Sens. Environ.
**2016**, 177, 144–152. [Google Scholar] [CrossRef] - Lee, B.; Di Girolamo, L.; Zhao, G.; Zhan, Y. Three-Dimensional Cloud Volume Reconstruction from the Multi-angle Imaging SpectroRadiometer. Remote Sens.
**2018**, 10, 1858. [Google Scholar] [CrossRef][Green Version] - Yu, H.; Ma, J.; Ahmad, S.; Sun, E.; Li, C.; Li, Z.; Hong, J. Three-Dimensional Cloud Structure Reconstruction from the Directional Polarimetric Camera. Remote Sens.
**2019**, 11, 2894. [Google Scholar] [CrossRef][Green Version] - Veikherman, D.; Aides, A.; Schechner, Y.Y.; Levis, A. Clouds in The Cloud. In Proceedings of the Asian Conference on Computer Vision (ACCV), Singapore, 1–5 November 2014; pp. 659–674. [Google Scholar]
- Zinner, T.; Marshak, A.; Lang, S.; Martins, J.V.; Mayer, B. Remote sensing of cloud sides of deep convection: Towards a three-dimensional retrieval of cloud particle size profiles. Atmos. Chem. Phys.
**2008**, 8, 4741–4757. [Google Scholar] [CrossRef][Green Version] - Alexandrov, M.D.; Miller, D.J.; Rajapakshe, C.; Fridlind, A.; van Diedenhoven, B.; Cairns, B.; Ackerman, A.S.; Zhang, Z. Vertical profiles of droplet size distributions derived from cloud-side observations by the research scanning polarimeter: Tests on simulated data. Atmos. Res.
**2020**, 239, 104924. [Google Scholar] [CrossRef] [PubMed] - Okamura, R.; Iwabuchi, H.; Schmidt, K.S. Feasibility study of multi-pixel retrieval of optical thickness and droplet effective radius of inhomogeneous clouds using deep learning. Atmos. Meas. Tech.
**2017**, 10, 4747–4759. [Google Scholar] [CrossRef][Green Version] - Masuda, R.; Iwabuchi, H.; Schmidt, K.S.; Damiani, A.; Kudo, R. Retrieval of Cloud Optical Thickness from Sky-View Camera Images using a Deep Convolutional Neural Network based on Three-Dimensional Radiative Transfer. Remote Sens.
**2019**, 11, 1962. [Google Scholar] [CrossRef][Green Version] - Liou, K.N.; Ou, S.C.; Takano, Y.; Roskovensky, J.; Mace, G.G.; Sassen, K.; Poellot, M. Remote sensing of three-dimensional inhomogeneous cirrus clouds using satellite and mm-wave cloud radar data. Geophys. Res. Lett.
**2002**, 29, 1360. [Google Scholar] [CrossRef][Green Version] - Barker, H.; Jerg, M.; Wehr, T.; Kato, S.; Donovan, D.; Hogan, R. A 3D cloud-construction algorithm for the EarthCARE satellite mission. Q. J. R. Meteorol. Soc.
**2011**, 137, 1042–1058. [Google Scholar] [CrossRef][Green Version] - Fielding, M.D.; Chiu, J.C.; Hogan, R.J.; Feingold, G. A novel ensemble method for retrieving properties of warm cloud in 3-D using ground-based scanning radar and zenith radiances. J. Geophys. Res. Atmos.
**2014**, 119, 10–912. [Google Scholar] [CrossRef][Green Version] - Hasmonay, R.A.; Yost, M.G.; Wu, C.F. Computed tomography of air pollutants using radial scanning path-integrated optical remote sensing. Atmos. Environ.
**1999**, 33, 267–274. [Google Scholar] [CrossRef] - Todd, L.A.; Ramanathan, M.; Mottus, K.; Katz, R.; Dodson, A.; Mihlan, G. Measuring chemical emissions using an open-path Fourier transform infrared (OP-FTIR) spectroscopy and computer-assisted tomography. Atmos. Environ.
**2001**, 35, 1937–1947. [Google Scholar] [CrossRef] - Kazahaya, R.; Mori, T.; Kazahaya, K.; Hirabayashi, J. Computed tomography reconstruction of SO
_{2}concentration distribution in the volcanic plume of Miyakejima, Japan, by airborne traverse technique using three UV spectrometers. Geophys. Res. Lett.**2008**, 35. [Google Scholar] [CrossRef] - Wright, T.E.; Burton, M.; Pyle, D.M.; Caltabiano, T. Scanning tomography of SO
_{2}distribution in a volcanic gas plume. Geophys. Res. Lett.**2008**, 35. [Google Scholar] [CrossRef] - Warner, J.; Drake, J.; Snider, J. Liquid water distribution obtained from coplanar scanning radiometers. J. Atmos. Ocean. Technol.
**1986**, 3, 542–546. [Google Scholar] [CrossRef][Green Version] - Huang, D.; Liu, Y.; Wiscombe, W. Determination of cloud liquid water distribution using 3D cloud tomography. J. Geophys. Res. Atmos.
**2008**, 113. [Google Scholar] [CrossRef] - Huang, D.; Liu, Y.; Wiscombe, W. Cloud tomography: Role of constraints and a new algorithm. J. Geophys. Res. Atmos.
**2008**, 113. [Google Scholar] [CrossRef] - Garay, M.J.; Davis, A.B.; Diner, D.J. Tomographic reconstruction of an aerosol plume using passive multiangle observations from the MISR satellite instrument. Geophys. Res. Lett.
**2016**, 43, 12–590. [Google Scholar] [CrossRef] - Aides, A.; Schechner, Y.Y.; Holodovsky, V.; Garay, M.J.; Davis, A.B. Multi-sky-view 3D aerosol distribution recovery. Opt. Express
**2013**, 21, 25820–25833. [Google Scholar] [CrossRef][Green Version] - Geva, A.; Schechner, Y.Y.; Chernyak, Y.; Gupta, R. X-ray computed tomography through scatter. In Proceedings of the European Conference on Computer Vision (ECCV), Munich, Germany, 8–14 September 2018; pp. 34–50. [Google Scholar]
- Arridge, S.R. Optical tomography in medical imaging. Inverse Probl.
**1999**, 15, R41. [Google Scholar] [CrossRef][Green Version] - Boas, D.A.; Brooks, D.H.; Miller, E.L.; DiMarzio, C.A.; Kilmer, M.; Gaudette, R.J.; Zhang, Q. Imaging the body with diffuse optical tomography. IEEE Signal Process. Mag.
**2001**, 18, 57–75. [Google Scholar] [CrossRef] - Arridge, S.R.; Schotland, J.C. Optical tomography: Forward and inverse problems. Inverse Probl.
**2009**, 25, 123010. [Google Scholar] [CrossRef] - Che, C.; Luan, F.; Zhao, S.; Bala, K.; Gkioulekas, I. Inverse transport networks. arXiv
**2018**, arXiv:1809.10820. [Google Scholar] - Evans, K.F. The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer. J. Atmos. Sci.
**1998**, 55, 429–446. [Google Scholar] [CrossRef] - Doicu, A.; Efremenko, D.; Trautmann, T. A multi-dimensional vector spherical harmonics discrete ordinate method for atmospheric radiative transfer. J. Quant. Spectrosc. Radiat. Transf.
**2013**, 118, 121–131. [Google Scholar] [CrossRef] - Levis, A.; Schechner, Y.Y.; Aides, A.; Davis, A.B. Airborne three-dimensional cloud tomography. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), Santiago, Chile, 7–13 December 2015; pp. 3379–3387. [Google Scholar]
- Holodovsky, V.; Schechner, Y.Y.; Levin, A.; Levis, A.; Aides, A. In-situ multi-view multi-scattering stochastic tomography. In Proceedings of the IEEE International Conference on Computational Photography (ICCP), Evanston, IL, USA, 13–14 May 2016; pp. 1–12. [Google Scholar]
- Levis, A.; Schechner, Y.Y.; Davis, A.B. Multiple-scattering microphysics tomography. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Honolulu, HI, USA, 21–26 July 2017; pp. 6740–6749. [Google Scholar]
- Aides, A.; Levis, A.; Holodovsky, V.; Schechner, Y.Y.; Althausen, D.; Vainiger, A. Distributed Sky Imaging Radiometry and Tomography. In Proceedings of the IEEE International Conference on Computational Photography (ICCP), Saint Louis, MO, USA, 24–26 April 2020; pp. 1–12. [Google Scholar]
- Loeub, T.; Levis, A.; Holodovsky, V.; Schechner, Y.Y.; Chernyak, Y.; Gupta, R. Monotonicity Prior for Cloud Tomography. In Proceedings of the European Conference on Computer Vision (ECCV), Glasgow, Scotlang, 24–29 August 2020. [Google Scholar]
- Hansen, J.E. Multiple scattering of polarized light in planetary atmospheres, Part II. Sunlight reflected by terrestrial water clouds. J. Atmos. Sci.
**1971**, 28, 1400–1426. [Google Scholar] [CrossRef][Green Version] - Marshak, A.; Davis, A. 3D Radiative Transfer in Cloudy Atmospheres; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Chylek, P. Extinction and liquid water content of fogs and clouds. J. Atmos. Sci.
**1978**, 35, 296–300. [Google Scholar] - Bohren, C.F.; Huffman, D.R. Absorption and Scattering of Light by Small Particles; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Chandrasekhar, S. Radiative Transfer; Oxford University Press: Oxford, UK, 1950. [Google Scholar]
- Mayer, B. Radiative transfer in the cloudy atmosphere. Eur. Phys. J. Conf.
**2009**, 1, 75–99. [Google Scholar] [CrossRef] - Nakajima, T.; Tanaka, M. Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation. J. Quant. Spectrosc. Radiat. Transf.
**1988**, 40, 51–69. [Google Scholar] [CrossRef] - Zhu, C.; Byrd, R.H.; Lu, P.; Nocedal, J. Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization. ACM Trans. Math. Softw.
**1997**, 23, 550–560. [Google Scholar] [CrossRef] - Doicu, A.; Efremenko, D.S. Linearizations of the Spherical Harmonic Discrete Ordinate Method (SHDOM). Atmosphere
**2019**, 10, 292. [Google Scholar] [CrossRef][Green Version] - Martin, W.; Cairns, B.; Bal, G. Adjoint methods for adjusting three-dimensional atmosphere and surface properties to fit multi-angle/multi-pixel polarimetric measurements. J. Quant. Spectrosc. Radiat. Transf.
**2014**, 144, 68–85. [Google Scholar] [CrossRef][Green Version] - Martin, W.G.; Hasekamp, O.P. A demonstration of adjoint methods for multi-dimensional remote sensing of the atmosphere and surface. J. Quant. Spectrosc. Radiat. Transf.
**2018**, 204, 215–231. [Google Scholar] [CrossRef] - Forster, L.; Davis, A.B.; Diner, D.J.; Mayer, B. Toward Cloud Tomography from Space using MISR and MODIS: Locating the “Veiled Core” in Opaque Convective Clouds. arXiv
**2019**, arXiv:1910.00077. [Google Scholar] - Zhao, M.; Austin, P.H. Life cycle of numerically simulated shallow cumulus clouds. Part II: Mixing dynamics. J. Atmos. Sci.
**2005**, 62, 1291–1310. [Google Scholar] [CrossRef] - Anderson, G.P.; Clough, S.A.; Kneizys, F.; Chetwynd, J.H.; Shettle, E.P. AFGL Atmospheric Constituent Profiles (0.120 km); Technical Report; Air Force Geophysics Lab: Hanscom AFB, MA, USA, 1986. [Google Scholar]
- Matheou, G.; Chung, D. Large-eddy simulation of stratified turbulence. Part 2: Application of the stretched-vortex model to the atmospheric boundary layer. J. Atmos. Sci.
**2014**, 71, 4439–4460. [Google Scholar] [CrossRef] - Yau, M.K.; Rogers, R.R. A Short Course in Cloud Physics; Elsevier: Amsterdam, The Netherlands, 1996. [Google Scholar]
- Seethala, C. Evaluating the State-Of-The-Art of and Errors in 1D Satellite Cloud Liquid Water Path Retrievals with Large Eddy Simulations and Realistic Radiative Transfer Models. Ph.D. Thesis, University of Hamburg, Hamburg, Germany, 2012. [Google Scholar]
- Ewald, F.; Zinner, T.; Kölling, T.; Mayer, B. Remote sensing of cloud droplet radius profiles using solar reflectance from cloud sides – Part 1: Retrieval development and characterization. Atmos. Meas. Tech.
**2019**, 12, 1183–1206. [Google Scholar] [CrossRef][Green Version] - Alexandrov, M.D.; Cairns, B.; Emde, C.; Ackerman, A.S.; van Diedenhoven, B. Accuracy assessments of cloud droplet size retrievals from polarized reflectance measurements by the research scanning polarimeter. Remote Sens. Environ.
**2012**, 125, 92–111. [Google Scholar] [CrossRef][Green Version] - Blyth, A.M.; Latham, J. A Climatological Parameterization for Cumulus Clouds. J. Atmos. Sci.
**1991**, 48, 2367–2371. [Google Scholar] [CrossRef] - French, J.R.; Vali, G.; Kelly, R.D. Observations of microphysics pertaining to the development of drizzle in warm, shallow cumulus clouds. Q. J. R. Meteorol. Soc.
**2000**, 126, 415–443. [Google Scholar] [CrossRef] - Gerber, H.E.; Frick, G.M.; Jensen, J.B.; Hudson, J.G. Entrainment, mixing, and microphysics in trade-wind cumulus. J. Meteorol. Soc. Jpn. Ser. II
**2008**, 86, 87–106. [Google Scholar] [CrossRef][Green Version] - Khain, P.; Heiblum, R.; Blahak, U.; Levi, Y.; Muskatel, H.; Vadislavsky, E.; Altaratz, O.; Koren, I.; Dagan, G.; Shpund, J.; et al. Parameterization of Vertical Profiles of Governing Microphysical Parameters of Shallow Cumulus Cloud Ensembles Using LES with Bin Microphysics. J. Atmos. Sci.
**2019**, 76, 533–560. [Google Scholar] [CrossRef] - Pinsky, M.; Khain, A. Theoretical Analysis of the Entrainment–Mixing Process at Cloud Boundaries. Part I: Droplet Size Distributions and Humidity within the Interface Zone. J. Atmos. Sci.
**2018**, 75, 2049–2064. [Google Scholar] [CrossRef] - Bera, S.; Prabha, T.V.; Grabowski, W.W. Observations of monsoon convective cloud microphysics over India and role of entrainment-mixing. J. Geophys. Res. Atmos.
**2016**, 121, 9767–9788. [Google Scholar] [CrossRef] - Costa, A.A.; de Oliveira, C.J.; de Oliveira, J.C.P.; da Costa Sampaio, A.J. Microphysical observations of warm cumulus clouds in Ceara, Brazil. Atmos. Res.
**2000**, 54, 167–199. [Google Scholar] [CrossRef] - Lu, M.L.; Feingold, G.; Jonsson, H.H.; Chuang, P.Y.; Gates, H.; Flagan, R.C.; Seinfeld, J.H. Aerosol-cloud relationships in continental shallow cumulus. J. Geophys. Res. Atmos.
**2008**, 113. [Google Scholar] [CrossRef][Green Version] - Martins, J.A.; Dias, M.A.F.S. The impact of smoke from forest fires on the spectral dispersion of cloud droplet size distributions in the Amazonian region. Environ. Res. Lett.
**2009**, 4, 015002. [Google Scholar] [CrossRef][Green Version] - Hudson, J.G.; Noble, S.; Jha, V. Cloud droplet spectral width relationship to CCN spectra and vertical velocity. J. Geophys. Res. Atmos.
**2012**, 117. [Google Scholar] [CrossRef] - Pandithurai, G.; Dipu, S.; Prabha, T.V.; Maheskumar, R.S.; Kulkarni, J.R.; Goswami, B.N. Aerosol effect on droplet spectral dispersion in warm continental cumuli. J. Geophys. Res. Atmos.
**2012**, 117. [Google Scholar] [CrossRef] - Igel, A.L.; van den Heever, S.C. The Importance of the Shape of Cloud Droplet Size Distributions in Shallow Cumulus Clouds. Part I: Bin Microphysics Simulations. J. Atmos. Sci.
**2017**, 74, 249–258. [Google Scholar] [CrossRef][Green Version] - Lu, M.L.; Seinfeld, J.H. Effect of aerosol number concentration on cloud droplet dispersion: A large-eddy simulation study and implications for aerosol indirect forcing. J. Geophys. Res. Atmos.
**2006**, 111, D02207. [Google Scholar] [CrossRef] - Wang, X.; Xue, H.; Fang, W.; Zheng, G. A study of shallow cumulus cloud droplet dispersion by large eddy simulations. Acta Meteorol. Sin.
**2011**, 25, 166–175. [Google Scholar] [CrossRef] - Milbrandt, J.A.; Yau, M.K. A Multimoment Bulk Microphysics Parameterization. Part I: Analysis of the Role of the Spectral Shape Parameter. J. Atmos. Sci.
**2005**, 62, 3051–3064. [Google Scholar] [CrossRef][Green Version] - Cairns, B.; Russell, E.E.; Travis, L.D. Research Scanning Polarimeter: Calibration and ground-based measurements. In Polarization: Measurement, Analysis, and Remote Sensing II; International Society for Optics and Photonics: Bellingham, WA, USA, 1999; Volume 3754, pp. 186–196. [Google Scholar]
- Levis, A.; Loveridge, J.; Aides, A. Pyshdom. 2020. Available online: https://github.com/aviadlevis/pyshdom (accessed on 1 January 2020).
- Sanghavi, S.; Davis, A.B.; Eldering, A. vSmartMOM: A vector matrix operator method-based radiative transfer model linearized with respect to aerosol properties. J. Quant. Spectrosc. Radiat. Transf.
**2014**, 133, 412–433. [Google Scholar] [CrossRef] - Xu, F.; Davis, A.B. Derivatives of light scattering properties of a nonspherical particle computed with the T-matrix method. Opt. Lett.
**2011**, 36, 4464–4466. [Google Scholar] [CrossRef] - Florescu, L.; Markel, V.A.; Schotland, J.C. Inversion formulas for the broken-ray Radon transform. Inverse Probl.
**2011**, 27, 025002. [Google Scholar] [CrossRef][Green Version] - Van Harten, G.; Diner, D.J.; Daugherty, B.J.; Rheingans, B.E.; Bull, M.A.; Seidel, F.C.; Chipman, R.A.; Cairns, B.; Wasilewski, A.P.; Knobelspiesse, K.D. Calibration and validation of Airborne Multiangle SpectroPolarimetric Imager (AirMSPI) polarization measurements. Appl. Opt.
**2018**, 57, 4499–4513. [Google Scholar] [CrossRef] - Pincus, R.; Evans, K.F. Computational cost and accuracy in calculating three-dimensional radiative transfer: Results for new implementations of Monte Carlo and SHDOM. J. Atmos. Sci.
**2009**, 66, 3131–3146. [Google Scholar] [CrossRef][Green Version] - Scipy. L-BFGS-B. 2020. Available online: https://docs.scipy.org/doc/scipy/reference/optimize.minimize-lbfgsb.html (accessed on 1 January 2020).

**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