Abstract

Aerosols affect climate, health and aviation. Currently, their retrieval assumes a plane-parallel atmosphere and solely vertical radiative transfer. We propose a principle to estimate the aerosol distribution as it really is: a three dimensional (3D) volume. The principle is a type of tomography. The process involves wide angle integral imaging of the sky on a very large scale. The imaging can use an array of cameras in visible light. We formulate an image formation model based on 3D radiative transfer. Model inversion is done using optimization methods, exploiting a closed-form gradient which we derive for the model-fit cost function. The tomography model is distinct, as the radiation source is unidirectional and uncontrolled, while off-axis scattering dominates the images.

© 2013 OSA

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H. Messer, A. Zinevich, and P. Alpert, “Environmental sensor networks using existing wireless communication systems for rainfall and wind velocity measurements,” IEEE Instrumentation & Measurement Magazine 15, 32–38 (2012).
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J. Gregson, M. Krimerman, M. B. Hullin, and W. Heidrich, “Stochastic tomography and its applications in 3D imaging of mixing fluids,” ACM Trans. Graph. 31,  52:1–52 (2012).
[Crossref]

2011 (2)

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multi-angle photo-polarimetry to vertical layering and mixing of absorbing aerosols: Quantifying measurement uncertainties,” Journal of Quantitative Spectroscopy and Radiative Transfer 112, 2149–2163 (2011).
[Crossref]

N. J. Pust, A. R. Dahlberg, M. J. Thomas, and J. a. Shaw, “Comparison of full-sky polarization and radiance observations to radiative transfer simulations which employ AERONET products,” Opt. Express 19, 18602–18613 (2011).
[Crossref] [PubMed]

2009 (2)

2008 (2)

U. Dayan, B. Ziv, T. Shoob, and Y. Enzel, “Suspended dust over southeastern Mediterranean and its relation to atmospheric circulations,” International Journal of Climatology 924, 915–924 (2008).
[Crossref]

I. Ihrke, K. N. Kutulakos, H. P. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in EUROGRAPHICS 2008 STAR–STATE OF THE ART REPORT, (2008).

2007 (1)

2006 (3)

H. Iwabuchi, “Efficient Monte Carlo Methods for Radiative Transfer Modeling,” Journal of the Atmospheric Sciences 63, 2324–2339 (2006).
[Crossref]

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proceedings of the IEEE 94.3 (2006).

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Transactions on Graphics (TOG) 25, 924–934 (2006).
[Crossref]

2004 (1)

2001 (2)

A. Bluestone, G. Abdoulaev, C. Schmitz, R. Barbour, and A. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express 9, 272–86 (2001).
[Crossref] [PubMed]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” Sig.Proc. Magazine 18, 57–75 (2001).
[Crossref]

1998 (1)

J. Martonchik, D. D. J. Diner, J. M. David, R. Kahn, T. P. Ackerman, M. M. Verstraete, B. Pinty, and H. R. Gordon, “Techniques for the Retrieval of Aerosol Properties over Land and Ocean Using Multi-angle Imaging,” IEEE Trans. on Geoscience and Remote Sensing 36, 1212–1227 (1998).
[Crossref]

1997 (1)

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550–560 (1997).
[Crossref]

1992 (1)

1986 (1)

L. Devroye, “Sample-based non-uniform random variate generation,” in Proceedings of the 18th conference on Winter simulation, (ACM, 1986), pp. 260–265.
[Crossref]

Abdoulaev, G.

Ackerman, T. P.

J. Martonchik, D. D. J. Diner, J. M. David, R. Kahn, T. P. Ackerman, M. M. Verstraete, B. Pinty, and H. R. Gordon, “Techniques for the Retrieval of Aerosol Properties over Land and Ocean Using Multi-angle Imaging,” IEEE Trans. on Geoscience and Remote Sensing 36, 1212–1227 (1998).
[Crossref]

Adams, A.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Transactions on Graphics (TOG) 25, 924–934 (2006).
[Crossref]

Alpert, P.

H. Messer, A. Zinevich, and P. Alpert, “Environmental sensor networks using existing wireless communication systems for rainfall and wind velocity measurements,” IEEE Instrumentation & Measurement Magazine 15, 32–38 (2012).
[Crossref]

Aviles, J. A.

J. A. Aviles, “The Development and Validation of a First Generation X-Ray Scatter Computed Tomography Algorithm for the Reconstruction of Electron Density Breast Images Using Monte Carlo Simulation,” Ph.D. thesis (2011).

Barbour, R.

Bluestone, A.

Boas, D. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” Sig.Proc. Magazine 18, 57–75 (2001).
[Crossref]

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” Sig.Proc. Magazine 18, 57–75 (2001).
[Crossref]

Byrd, R. H.

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550–560 (1997).
[Crossref]

Cornette, W. M.

Cosofret, B. R.

Dahlberg, A. R.

David, J. M.

J. Martonchik, D. D. J. Diner, J. M. David, R. Kahn, T. P. Ackerman, M. M. Verstraete, B. Pinty, and H. R. Gordon, “Techniques for the Retrieval of Aerosol Properties over Land and Ocean Using Multi-angle Imaging,” IEEE Trans. on Geoscience and Remote Sensing 36, 1212–1227 (1998).
[Crossref]

Davis, A.

A. Marshak and A. Davis, 3D Radiative Transfer in Cloudy Atmospheres, Physics of Earth and Space Environments (Springer, 2005).
[Crossref]

Davis, A. B.

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multi-angle photo-polarimetry to vertical layering and mixing of absorbing aerosols: Quantifying measurement uncertainties,” Journal of Quantitative Spectroscopy and Radiative Transfer 112, 2149–2163 (2011).
[Crossref]

Dayan, U.

U. Dayan, B. Ziv, T. Shoob, and Y. Enzel, “Suspended dust over southeastern Mediterranean and its relation to atmospheric circulations,” International Journal of Climatology 924, 915–924 (2008).
[Crossref]

Devroye, L.

L. Devroye, “Sample-based non-uniform random variate generation,” in Proceedings of the 18th conference on Winter simulation, (ACM, 1986), pp. 260–265.
[Crossref]

DiMarzio, C. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” Sig.Proc. Magazine 18, 57–75 (2001).
[Crossref]

Diner, D. D. J.

J. Martonchik, D. D. J. Diner, J. M. David, R. Kahn, T. P. Ackerman, M. M. Verstraete, B. Pinty, and H. R. Gordon, “Techniques for the Retrieval of Aerosol Properties over Land and Ocean Using Multi-angle Imaging,” IEEE Trans. on Geoscience and Remote Sensing 36, 1212–1227 (1998).
[Crossref]

Diner, D. J.

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multi-angle photo-polarimetry to vertical layering and mixing of absorbing aerosols: Quantifying measurement uncertainties,” Journal of Quantitative Spectroscopy and Radiative Transfer 112, 2149–2163 (2011).
[Crossref]

J. V. Martonchik, R. A. Kahn, and D. J. Diner, “Retrieval of aerosol properties over land using MISR observations,” in Satellite Aerosol Remote Sensing over Land,, A. A. Kokhanovsky and G. Leeuw, eds. (Springer BerlinHeidelberg, 2009), pp. 267–293.
[Crossref]

Enzel, Y.

U. Dayan, B. Ziv, T. Shoob, and Y. Enzel, “Suspended dust over southeastern Mediterranean and its relation to atmospheric circulations,” International Journal of Climatology 924, 915–924 (2008).
[Crossref]

Faghfouri, A.

Finson, M. L.

Footer, M.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Transactions on Graphics (TOG) 25, 924–934 (2006).
[Crossref]

Garay, M. J.

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multi-angle photo-polarimetry to vertical layering and mixing of absorbing aerosols: Quantifying measurement uncertainties,” Journal of Quantitative Spectroscopy and Radiative Transfer 112, 2149–2163 (2011).
[Crossref]

Gaudette, R. J.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” Sig.Proc. Magazine 18, 57–75 (2001).
[Crossref]

Geogdzhayev, I. V.

Gittins, C. M.

Gordon, H. R.

J. Martonchik, D. D. J. Diner, J. M. David, R. Kahn, T. P. Ackerman, M. M. Verstraete, B. Pinty, and H. R. Gordon, “Techniques for the Retrieval of Aerosol Properties over Land and Ocean Using Multi-angle Imaging,” IEEE Trans. on Geoscience and Remote Sensing 36, 1212–1227 (1998).
[Crossref]

Gregson, J.

J. Gregson, M. Krimerman, M. B. Hullin, and W. Heidrich, “Stochastic tomography and its applications in 3D imaging of mixing fluids,” ACM Trans. Graph. 31,  52:1–52 (2012).
[Crossref]

Heidrich, W.

J. Gregson, M. Krimerman, M. B. Hullin, and W. Heidrich, “Stochastic tomography and its applications in 3D imaging of mixing fluids,” ACM Trans. Graph. 31,  52:1–52 (2012).
[Crossref]

I. Ihrke, K. N. Kutulakos, H. P. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in EUROGRAPHICS 2008 STAR–STATE OF THE ART REPORT, (2008).

Hielscher, A.

Hong, S.-H.

Horowitz, M.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Transactions on Graphics (TOG) 25, 924–934 (2006).
[Crossref]

Hullin, M. B.

J. Gregson, M. Krimerman, M. B. Hullin, and W. Heidrich, “Stochastic tomography and its applications in 3D imaging of mixing fluids,” ACM Trans. Graph. 31,  52:1–52 (2012).
[Crossref]

Ihrke, I.

I. Ihrke, K. N. Kutulakos, H. P. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in EUROGRAPHICS 2008 STAR–STATE OF THE ART REPORT, (2008).

Iwabuchi, H.

H. Iwabuchi, “Efficient Monte Carlo Methods for Radiative Transfer Modeling,” Journal of the Atmospheric Sciences 63, 2324–2339 (2006).
[Crossref]

Jang, J.-S.

Janov, T. E.

Javidi, B.

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proceedings of the IEEE 94.3 (2006).

S.-H. Hong, J.-S. Jang, and B. Javidi, “Three-dimensional volumetric object reconstruction using computational integral imaging,” Opt. Express 12, 483–491 (2004).
[Crossref] [PubMed]

Kahn, R.

J. Martonchik, D. D. J. Diner, J. M. David, R. Kahn, T. P. Ackerman, M. M. Verstraete, B. Pinty, and H. R. Gordon, “Techniques for the Retrieval of Aerosol Properties over Land and Ocean Using Multi-angle Imaging,” IEEE Trans. on Geoscience and Remote Sensing 36, 1212–1227 (1998).
[Crossref]

Kahn, R. A.

J. V. Martonchik, R. A. Kahn, and D. J. Diner, “Retrieval of aerosol properties over land using MISR observations,” in Satellite Aerosol Remote Sensing over Land,, A. A. Kokhanovsky and G. Leeuw, eds. (Springer BerlinHeidelberg, 2009), pp. 267–293.
[Crossref]

Kalashnikova, O. V.

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multi-angle photo-polarimetry to vertical layering and mixing of absorbing aerosols: Quantifying measurement uncertainties,” Journal of Quantitative Spectroscopy and Radiative Transfer 112, 2149–2163 (2011).
[Crossref]

Kilmer, M.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” Sig.Proc. Magazine 18, 57–75 (2001).
[Crossref]

Kim, J.

J. Kim, D. Lanman, Y. Mukaigawa, and R. Raskar, “Descattering transmission via angular filtering,” in Proc. ECCV’ 10, (Springer-Verlag, Berlin, Heidelberg, 2010), pp. 86–99.

Kindle, H. S.

Konno, D.

Krimerman, M.

J. Gregson, M. Krimerman, M. B. Hullin, and W. Heidrich, “Stochastic tomography and its applications in 3D imaging of mixing fluids,” ACM Trans. Graph. 31,  52:1–52 (2012).
[Crossref]

Kutulakos, K. N.

I. Ihrke, K. N. Kutulakos, H. P. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in EUROGRAPHICS 2008 STAR–STATE OF THE ART REPORT, (2008).

Lanman, D.

J. Kim, D. Lanman, Y. Mukaigawa, and R. Raskar, “Descattering transmission via angular filtering,” in Proc. ECCV’ 10, (Springer-Verlag, Berlin, Heidelberg, 2010), pp. 86–99.

Lensch, H. P.

I. Ihrke, K. N. Kutulakos, H. P. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in EUROGRAPHICS 2008 STAR–STATE OF THE ART REPORT, (2008).

Levi, L.

L. Levi, Applied Optics (John Wiley & Sons, Inc., 1980).

Levoy, M.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Transactions on Graphics (TOG) 25, 924–934 (2006).
[Crossref]

Levreault, M. J.

Lu, P.

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550–560 (1997).
[Crossref]

Magnor, M.

I. Ihrke, K. N. Kutulakos, H. P. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in EUROGRAPHICS 2008 STAR–STATE OF THE ART REPORT, (2008).

Marinelli, W. J.

Marshak, A.

A. Marshak and A. Davis, 3D Radiative Transfer in Cloudy Atmospheres, Physics of Earth and Space Environments (Springer, 2005).
[Crossref]

Martonchik, J.

J. Martonchik, D. D. J. Diner, J. M. David, R. Kahn, T. P. Ackerman, M. M. Verstraete, B. Pinty, and H. R. Gordon, “Techniques for the Retrieval of Aerosol Properties over Land and Ocean Using Multi-angle Imaging,” IEEE Trans. on Geoscience and Remote Sensing 36, 1212–1227 (1998).
[Crossref]

Martonchik, J. V.

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multi-angle photo-polarimetry to vertical layering and mixing of absorbing aerosols: Quantifying measurement uncertainties,” Journal of Quantitative Spectroscopy and Radiative Transfer 112, 2149–2163 (2011).
[Crossref]

J. V. Martonchik, R. A. Kahn, and D. J. Diner, “Retrieval of aerosol properties over land using MISR observations,” in Satellite Aerosol Remote Sensing over Land,, A. A. Kokhanovsky and G. Leeuw, eds. (Springer BerlinHeidelberg, 2009), pp. 267–293.
[Crossref]

Messer, H.

H. Messer, A. Zinevich, and P. Alpert, “Environmental sensor networks using existing wireless communication systems for rainfall and wind velocity measurements,” IEEE Instrumentation & Measurement Magazine 15, 32–38 (2012).
[Crossref]

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” Sig.Proc. Magazine 18, 57–75 (2001).
[Crossref]

Mishchenko, M. I.

Miyashiro, R. K.

Mukaigawa, Y.

J. Kim, D. Lanman, Y. Mukaigawa, and R. Raskar, “Descattering transmission via angular filtering,” in Proc. ECCV’ 10, (Springer-Verlag, Berlin, Heidelberg, 2010), pp. 86–99.

Namer, E.

Ng, R.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Transactions on Graphics (TOG) 25, 924–934 (2006).
[Crossref]

Nocedal, J.

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550–560 (1997).
[Crossref]

Pinty, B.

J. Martonchik, D. D. J. Diner, J. M. David, R. Kahn, T. P. Ackerman, M. M. Verstraete, B. Pinty, and H. R. Gordon, “Techniques for the Retrieval of Aerosol Properties over Land and Ocean Using Multi-angle Imaging,” IEEE Trans. on Geoscience and Remote Sensing 36, 1212–1227 (1998).
[Crossref]

Pust, N. J.

Raskar, R.

J. Kim, D. Lanman, Y. Mukaigawa, and R. Raskar, “Descattering transmission via angular filtering,” in Proc. ECCV’ 10, (Springer-Verlag, Berlin, Heidelberg, 2010), pp. 86–99.

Schechner, Y. Y.

Schmitz, C.

Shanks, J. G.

Shaw, J. a.

Shoob, T.

U. Dayan, B. Ziv, T. Shoob, and Y. Enzel, “Suspended dust over southeastern Mediterranean and its relation to atmospheric circulations,” International Journal of Climatology 924, 915–924 (2008).
[Crossref]

Shwartz, S.

Stern, A.

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proceedings of the IEEE 94.3 (2006).

Thomas, M. J.

Verstraete, M. M.

J. Martonchik, D. D. J. Diner, J. M. David, R. Kahn, T. P. Ackerman, M. M. Verstraete, B. Pinty, and H. R. Gordon, “Techniques for the Retrieval of Aerosol Properties over Land and Ocean Using Multi-angle Imaging,” IEEE Trans. on Geoscience and Remote Sensing 36, 1212–1227 (1998).
[Crossref]

Zhang, Q.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” Sig.Proc. Magazine 18, 57–75 (2001).
[Crossref]

Zhu, C.

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550–560 (1997).
[Crossref]

Zinevich, A.

H. Messer, A. Zinevich, and P. Alpert, “Environmental sensor networks using existing wireless communication systems for rainfall and wind velocity measurements,” IEEE Instrumentation & Measurement Magazine 15, 32–38 (2012).
[Crossref]

Ziv, B.

U. Dayan, B. Ziv, T. Shoob, and Y. Enzel, “Suspended dust over southeastern Mediterranean and its relation to atmospheric circulations,” International Journal of Climatology 924, 915–924 (2008).
[Crossref]

ACM Trans. Graph. (1)

J. Gregson, M. Krimerman, M. B. Hullin, and W. Heidrich, “Stochastic tomography and its applications in 3D imaging of mixing fluids,” ACM Trans. Graph. 31,  52:1–52 (2012).
[Crossref]

ACM Trans. Math. Softw. (1)

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Figures (10)

Fig. 1
Fig. 1

Integral (lightfield) imaging through a volumetric distribution in the atmosphere, using ground-based cameras.

Fig. 2
Fig. 2

Simulated 3D aerosol distributions: [a] Haze blobs and [b] Haze Front. Here, an aerosol density unit is 106 particles/m3. The atmopsheric domain has area of 50 × 50km2, extending from the ground up to altitude of 10km. The domain is divided into a rectilinear grid having Nvoxels voxels.

Fig. 3
Fig. 3

Different viewpoints of a sky that includes Haze Blobs. Images are rendered using the single-scattering model. A yellow dot marks the Sun location. Dashed white circles mark zenith angles.

Fig. 4
Fig. 4

[a] Multiple-scattering forward model by MC. [b] Photograph of the same sky and view point as in Fig. 3(a), rendered using MC.

Fig. 5
Fig. 5

Images simulated by MC, from the same viewpoint, but different aerosol characteristics. (a) Haze blobs, characterized in Sec. 5. (b) Partly absorbing aerosol. (c) Aerosol having an isotropic phase function. (d) High aerosol density. Details of the scenarios used in creating (b) and (c) are given in Sec. 8.

Fig. 6
Fig. 6

Separation of the image shown in Fig. 4(b) to contributions by successive orders of scattering: (a) first, (b) second, (c) third and (d) forth scattering order.

Fig. 7
Fig. 7

Cross-sections along the X-axis of photographs rendered by the single-scattering (dotted) and MC (solid line) forward models. (a) cross sections of the Haze Blobs scene (described in Sec. 5). The difference between the models is much more pronounced (b) when the scatterers are ten times denser (see Sec. 8).

Fig. 8
Fig. 8

Recoveries of scenes shown in Fig. 2. Color represents aerosol density, in units of 106 particles/m3. In (a) and (b) images simulated by single-scattering are used as input for the recovery. In (c) and (d), MC simulated images are used as input for the recovery.

Fig. 9
Fig. 9

The relative error ε measure decreases with the number of cameras. Bars represent the standard deviation of ε, over our random tests.

Fig. 10
Fig. 10

Recovered distributions when the aerosol is either party absorbing (a), has isotropic phase function (b) or has high density (c).

Tables (1)

Tables Icon

Table 1 Relative errors in various simulations. Here �� denotes order of magnitude.

Equations (33)

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τ = d τ = ( β aerosol + β air ) d l = ( σ aerosol n + β air ) d l = τ air + σ aerosol n d l ,
t = exp ( τ ) .
α aerosol = ϖ aerosol β aerosol = ϖ aerosol σ aerosol n .
α ˜ aerosol ( Φ scatter ) = α aerosol P aerosol ( Φ scatter ) = ϖ aerosol σ aerosol n P aerosol ( Φ scatter ) .
P g aerosol ( Φ scatter ) 3 8 π ( 1 g 2 ) [ 1 + ( cos Φ scatter ) 2 ] ( 2 + g 2 ) ( 1 + g 2 2 g cos Φ scatter ) 3 2
P air ( Φ scatter ) 3 16 π ( 1 + cos 2 Φ scatter )
α air ( h , λ ) = β air ( h , λ ) 1.09 × 10 3 λ 4 exp ( h / H air ) ,
F ( τ ) = 0 τ exp ( τ ) d τ = 1 exp ( τ ) .
τ random = F 1 ( u ) = ln ( 1 u ) .
l random = ( β aerosol + β air ) 1 τ random = ( β aerosol + β air ) 1 ln ( 1 u ) ,
D Sun voxel ( k , q ) = { l SR ( q ) if q [ SR , k ] 0 otherwise
D c voxel cam ( k , m ) = { l LOS c ( m ) if m [ LOS c , k ] 0 otherwise .
τ SR ( k ) = q [ SR , k ] l SR ( q ) [ β air ( q ) + σ aerosol n ( q ) ] τ LOS c ( k ) = m [ SR , k ] l LOS c ( m ) [ β air ( m ) + σ aerosol n ( m ) ] .
τ SR = D Sun voxel β air + σ aerosol D Sun voxel n τ LOS c = D c voxel cam β air + σ aerosol D c voxel cam n .
τ c = τ c air + σ aerosol D c n ,
α ˜ c air = β air P air ( Φ c scatter ) , α ˜ c aerosol = ϖ aerosol σ aerosol n P g aerosol ( Φ c scatter ) .
p c ( k ) = L TOA [ α ˜ c air ( k ) + α ˜ c aerosol ( k ) ] exp [ τ c ( k ) ] ,
p c = L TOA ( α ˜ c air + α ˜ c aerosol ) exp ( τ c ) .
i c = Π c p c ,
i c = L TOA Π c { [ α ˜ c air + ϖ aerosol σ aerosol P g aerosol ( Φ c scatter ) n ] exp [ ( τ c air + σ aerosol D c n ) ] } .
f 1 [ h ( k ) ] = n sealevel exp [ h ( k ) / H aerosol ] ,
f 2 ( k ) = { 1 if k any blob cluster 0 otherwise .
0 l random ( σ aerosol n + β air ) d l = τ random
n ^ = arg min n 𝒞 E ( n )
E ( n ) = c = 1 N views [ i c measured i c ( n ) ] 2 2 + η Ψ ( n ) .
Ψ ( n ) = W n 2 2
n E = 2 c = 1 N views [ J i c ( n ) ] [ i c measured i c ( n ) ] + 2 η W W n .
C ( a u ) n = [ a n 𝔻 { u } + u n 𝔻 { a } ] C .
a n = C 𝔻 { exp ( Cn ) } .
exp [ ( τ c air + σ aerosol D c n ) ] n = σ aerosol D c 𝔻 { exp [ ( τ c air + σ aerosol D c n ) ] } ,
[ ϖ aerosol σ aerosol P g aerosol ( Φ c scatter ) n ] n = ϖ aerosol σ aerosol 𝔻 { P g aerosol ( Φ c scatter ) } .
J i c ( n ) = L TOA σ aerosol ( A B ) 𝔻 { exp [ ( τ c air + σ aerosol D c n ) ] } Π c ,
A = ϖ aerosol 𝔻 { P g aerosol ( Φ c scatter ) } , B = D c 𝔻 { [ α ˜ c air + ϖ aerosol σ aerosol P g aerosol ( Φ c scatter ) n ] }

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