# American Institute of Mathematical Sciences

April  2018, 12(2): 261-280. doi: 10.3934/ipi.2018011

## Reconstruction of cloud geometry from high-resolution multi-angle images

 1 Departments of Statistics and Mathematics and CCAM, University of Chicago, Chicago, IL 60637, USA 2 Department of Mathematical Sciences, Rensselear Polytechnic Institute, Troy, NY 12180, USA 3 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA

Received  November 2015 Revised  October 2016 Published  February 2018

We consider the reconstruction of the interface of compact, connected "clouds" from satellite or airborne light intensity measurements. In a two-dimensional setting, the cloud is modeled by an interface, locally represented as a graph, and an outgoing radiation intensity that is consistent with a diffusion model for light propagation in the cloud. Light scattering inside the cloud and the internal optical parameters of the cloud are not modeled explicitly. The main objective is to understand what can or cannot be reconstructed in such a setting from intensity measurements in a finite (on the order of 10) number of directions along the path of a satellite or an aircraft. Numerical simulations illustrate the theoretical predictions. Finally, we explore a kinematic extension of the algorithm for retrieving cloud motion (wind) along with its geometry.

Citation: Guillaume Bal, Jiaming Chen, Anthony B. Davis. Reconstruction of cloud geometry from high-resolution multi-angle images. Inverse Problems and Imaging, 2018, 12 (2) : 261-280. doi: 10.3934/ipi.2018011
##### References:
 [1] M. D. Alexandrov et al., Derivation of cumulus cloud dimensions and shape from the airborne measurements by the Research Scanning Polarimeter, Remote Sensing of the Environment, 177 (2016), 144-152. [2] G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48pp. [3] B. Cairns, E. E. Russell and L. D. Travis, Research Scanning Polarimeter: Calibration and ground-based measurements, SPIE Proc., 3754 (1999), 186-197. [4] S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960. [5] C. Cornet and R. Davies, Use of MISR measurements to study the radiative transfer of an isolated convective cloud: Implications for cloud optical thickness retrieval, J. Geophys. Res.-Atmospheres, 113 (2008), D04202.  doi: 10.1029/2007JD008921. [6] A. B. Davis, Cloud remote sensing with sideways looks: Theory and first results using Multispectral Thermal Imager (MTI) data, SPIE Proc., 4725 (2002), 397-405.  doi: 10.1117/12.478772. [7] A. B. Davis and A. Marshak, Solar radiation transport in the cloudy atmosphere: A 3D perspective on observations and climate impacts, Reports on Progress in Physics, 73 (2010), 026801 (70pp). doi: 10.1088/0034-4885/73/2/026801. [8] D. J. Diner et al., Multi-angle Imaging SpectroRadiometer (MISR) instrument description and experiment overview, IEEE Transactions in Geoscience and Remote Sensing, 36 (1998), 1072-1087. [9] _____, The Airborne Multiangle SpectroPolarimetric Imager (AirMSPI): A new tool for aerosol and cloud remote sensing, Atmospheric Measurement Techniques, 6 (2013), 2007-2025. [10] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996. [11] Á. Horváth and R. Davies, Feasibility and error analysis of cloud motion wind extraction from near-simultaneous multiangle MISR measurements, Journal of Atmospheric and Oceanic Technology, 18 (2001), 591-608. [12] _____, Simultaneous retrieval of cloud motion and height from polar-orbiter multiangle measurements, Geophysical Research Letters, 28 (2001), 2915-2918. [13] A. Levis, Y. Y. Schechner, A. Aides and A. B. Davis, Airborne three-dimensional cloud tomography, in Proceedings of the IEEE International Conference on Computer Vision 2015 (ICCV2015), (2015), 3379-3387. doi: 10.1109/ICCV.2015.386. [14] A. Levis, Y. Y. Schechner and A. B. Davis, Multiple-scattering microphysics tomography, in Proceedings of the 30th IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR17), (2017), http://openaccess.thecvf.com/content_cvpr_2017/papers/Levis_Multiple-Scattering_Microphysics_Tomography_CVPR_2017_paper.pdf. doi: 10.1109/CVPR.2017.614. [15] S. M. Lovejoy, The area-parameter relation for rain and clouds, Science, 216 (1982), 185-187. [16] B. B. Mandelbrot, Fractals: Form, Chance and Dimension, W. H. Freeman & Co., San Diego(CA), 1977. [17] R. Marchand and T. Ackerman, 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. - Atmospheres, 109 (2004), D18208. [18] A. Marshak and A. B. Davis, 3D Radiative Transfer in Cloudy Atmospheres, Springer, New York, 2005. doi: 10.1007/3-540-28519-9. [19] W. G. K. Martin, B. Cairns and G. Bal, Adjoint methods for adjusting three-dimensional atmosphere and surface properties to multi-angle polarimetric measurements, J. Quant. Spectroscopy Radiative Transfer, 144 (2014), 68-85.  doi: 10.1016/j.jqsrt.2014.03.030. [20] W. G. K. Martin and O. P. Hasekamp, A demonstration of adjoint methods for multi-dimensional remote sensing of the atmosphere and surface, J. Quant. Spectroscopy Radiative Transfer, 204 (2018), 215-231.  doi: 10.1016/j.jqsrt.2017.09.031. [21] C. Moroney, R. Davies and J. -P. Muller, Operational retrieval of cloud-top heights using MISR data, IEEE Transactions on Geoscience and Remote Sensing, 40 (2002), 1532-1540.  doi: 10.1109/TGRS.2002.801150. [22] J. -P. Muller et al., MISR stereoscopic image matchers: Techniques and results, IEEE Transactions on Geoscience and Remote Sensing, 40 (2002), 1547-1559. [23] T. Nakajima and M. D. King, Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. Part 1: Theory, Journal of the Atmospheric Sciences, 47 (1990), 1878-1893.  doi: 10.1175/1520-0469(1990)047<1878:DOTOTA>2.0.CO;2. [24] S. Platnick et al., The MODIS cloud products: Algorithms and examples from Terra, IEEE Transactions on Geoscience and Remote Sensing, 41 (2003), 459-473. [25] G. Seiz and R. Davies, Reconstruction of cloud geometry from multi-view satellite images, Remote Sensing of the Environment, 100 (2006), 143-149.  doi: 10.1016/j.rse.2005.09.016. [26] P. G. Weber, Multispectral Thermal Imager mission overview, SPIE Proc., 3753 (1999), 340-346.

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##### References:
 [1] M. D. Alexandrov et al., Derivation of cumulus cloud dimensions and shape from the airborne measurements by the Research Scanning Polarimeter, Remote Sensing of the Environment, 177 (2016), 144-152. [2] G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48pp. [3] B. Cairns, E. E. Russell and L. D. Travis, Research Scanning Polarimeter: Calibration and ground-based measurements, SPIE Proc., 3754 (1999), 186-197. [4] S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960. [5] C. Cornet and R. Davies, Use of MISR measurements to study the radiative transfer of an isolated convective cloud: Implications for cloud optical thickness retrieval, J. Geophys. Res.-Atmospheres, 113 (2008), D04202.  doi: 10.1029/2007JD008921. [6] A. B. Davis, Cloud remote sensing with sideways looks: Theory and first results using Multispectral Thermal Imager (MTI) data, SPIE Proc., 4725 (2002), 397-405.  doi: 10.1117/12.478772. [7] A. B. Davis and A. Marshak, Solar radiation transport in the cloudy atmosphere: A 3D perspective on observations and climate impacts, Reports on Progress in Physics, 73 (2010), 026801 (70pp). doi: 10.1088/0034-4885/73/2/026801. [8] D. J. Diner et al., Multi-angle Imaging SpectroRadiometer (MISR) instrument description and experiment overview, IEEE Transactions in Geoscience and Remote Sensing, 36 (1998), 1072-1087. [9] _____, The Airborne Multiangle SpectroPolarimetric Imager (AirMSPI): A new tool for aerosol and cloud remote sensing, Atmospheric Measurement Techniques, 6 (2013), 2007-2025. [10] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996. [11] Á. Horváth and R. Davies, Feasibility and error analysis of cloud motion wind extraction from near-simultaneous multiangle MISR measurements, Journal of Atmospheric and Oceanic Technology, 18 (2001), 591-608. [12] _____, Simultaneous retrieval of cloud motion and height from polar-orbiter multiangle measurements, Geophysical Research Letters, 28 (2001), 2915-2918. [13] A. Levis, Y. Y. Schechner, A. Aides and A. B. Davis, Airborne three-dimensional cloud tomography, in Proceedings of the IEEE International Conference on Computer Vision 2015 (ICCV2015), (2015), 3379-3387. doi: 10.1109/ICCV.2015.386. [14] A. Levis, Y. Y. Schechner and A. B. Davis, Multiple-scattering microphysics tomography, in Proceedings of the 30th IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR17), (2017), http://openaccess.thecvf.com/content_cvpr_2017/papers/Levis_Multiple-Scattering_Microphysics_Tomography_CVPR_2017_paper.pdf. doi: 10.1109/CVPR.2017.614. [15] S. M. Lovejoy, The area-parameter relation for rain and clouds, Science, 216 (1982), 185-187. [16] B. B. Mandelbrot, Fractals: Form, Chance and Dimension, W. H. Freeman & Co., San Diego(CA), 1977. [17] R. Marchand and T. Ackerman, 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. - Atmospheres, 109 (2004), D18208. [18] A. Marshak and A. B. Davis, 3D Radiative Transfer in Cloudy Atmospheres, Springer, New York, 2005. doi: 10.1007/3-540-28519-9. [19] W. G. K. Martin, B. Cairns and G. Bal, Adjoint methods for adjusting three-dimensional atmosphere and surface properties to multi-angle polarimetric measurements, J. Quant. Spectroscopy Radiative Transfer, 144 (2014), 68-85.  doi: 10.1016/j.jqsrt.2014.03.030. [20] W. G. K. Martin and O. P. Hasekamp, A demonstration of adjoint methods for multi-dimensional remote sensing of the atmosphere and surface, J. Quant. Spectroscopy Radiative Transfer, 204 (2018), 215-231.  doi: 10.1016/j.jqsrt.2017.09.031. [21] C. Moroney, R. Davies and J. -P. Muller, Operational retrieval of cloud-top heights using MISR data, IEEE Transactions on Geoscience and Remote Sensing, 40 (2002), 1532-1540.  doi: 10.1109/TGRS.2002.801150. [22] J. -P. Muller et al., MISR stereoscopic image matchers: Techniques and results, IEEE Transactions on Geoscience and Remote Sensing, 40 (2002), 1547-1559. [23] T. Nakajima and M. D. King, Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. Part 1: Theory, Journal of the Atmospheric Sciences, 47 (1990), 1878-1893.  doi: 10.1175/1520-0469(1990)047<1878:DOTOTA>2.0.CO;2. [24] S. Platnick et al., The MODIS cloud products: Algorithms and examples from Terra, IEEE Transactions on Geoscience and Remote Sensing, 41 (2003), 459-473. [25] G. Seiz and R. Davies, Reconstruction of cloud geometry from multi-view satellite images, Remote Sensing of the Environment, 100 (2006), 143-149.  doi: 10.1016/j.rse.2005.09.016. [26] P. G. Weber, Multispectral Thermal Imager mission overview, SPIE Proc., 3753 (1999), 340-346.
Geometry of cloud interface
Left: A cloud model. Right: Simulated radiances $u_j(X) : = u(X,Z,\theta_j)$ for that cloud using (7) with $j = 1,\dots,J = 5$ (specifically, $\theta \in \{90,90\pm26.1,90\pm45.6\}$ in degrees clockwise from the positive $x$ axis) for a uniform $\alpha$ and $\beta = \sin\phi$.
True angular radiation function $\beta(\phi) = \sin\phi$ (in green), reconstructed function (in blue), and initial guess (in red).
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