Abstract

A fast and accurate principal component-based radiative transfer model in the solar spectral region (PCRTM-SOLAR) has been developed. The algorithm is capable of simulating reflected solar spectra in both clear sky and cloudy atmospheric conditions. Multiple scattering of the solar beam by the multilayer clouds and aerosols are calculated using a discrete ordinate radiative transfer scheme. The PCRTM-SOLAR model can be trained to simulate top-of-atmosphere radiance or reflectance spectra with spectral resolution ranging from 1  cm1 resolution to a few nanometers. Broadband radiances or reflectance can also be calculated if desired. The current version of the PCRTM-SOLAR covers a spectral range from 300 to 2500 nm. The model is valid for solar zenith angles ranging from 0 to 80 deg, the instrument view zenith angles ranging from 0 to 70 deg, and the relative azimuthal angles ranging from 0 to 360 deg. Depending on the number of spectral channels, the speed of the current version of PCRTM-SOLAR is a few hundred to over one thousand times faster than the medium speed correlated-k option MODTRAN5. The absolute RMS error in channel radiance is smaller than 103  mW/cm2/sr/cm1 and the relative error is typically less than 0.2%.

© 2016 Optical Society of America

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2014 (4)

S. Kato, F. G. Rose, X. Liu, B. A. Wielicki, and M. G. Mlynczak, “Retrieval of atmospheric and cloud property anomalies and their trend from temporally and spatially averaged infrared spectra observed from space,” J. Climate 27, 4403–4420 (2014).
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C. M. Roithmayr, C. Lukashin, P. W. Speth, G. Kopp, K. Thome, B. A. Wielicki, and D. F. Young, “CLARREO approach for reference intercalibration of reflected solar sensors: on-orbit data matching and sampling,” IEEE Trans. Geosci. Remote Sens. 52, 6762–6774 (2014).
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2013 (6)

B. A. Wielicki, D. F. Young, M. G. Mlynczak, K. J. Thome, S. Leroy, J. Corliss, J. G. Anderson, C. O. Ao, R. Bantges, F. Best, K. Bowman, H. Brindley, J. Butler, W. Collins, J. A. Dykema, D. R. Doelling, D. R. Feldman, N. Fox, X. Huang, R. Holz, Y. Huang, Z. Jin, D. Jennings, D. G. Johnson, K. Jucks, S. Kato, D. B. Kirk-Davidoff, R. Knuteson, G. Kopp, D. P. Kratz, X. Liu, C. Lukashin, A. J. Mannucci, N. Phojanamongkolkij, P. Pilewskie, V. Ramaswamy, H. Revercomb, J. Rice, Y. Roberts, C. M. Roithmayr, F. Rose, S. Sandford, E. L. Shirley, W. L. Smith, B. Soden, P. W. Speth, W. Sun, P. C. Taylor, D. Tobin, and X. Xiong, “Achieving climate change absolute accuracy in orbit,” Bull. Amer. Meteorol. Soc. 94, 1519–1539 (2013).
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C. Wang, P. Yang, S. L. Nasiri, S. Platnick, B. A. Baum, A. K. Heidinger, and X. Liu, “A fast radiative transfer model for visible through shortwave infrared spectral reflectance’s in clear and cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 116, 122–131 (2013).
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P. Yang, L. Bi, B. A. Baum, K. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100  μm,” J. Atmos. Sci. 70, 330–347 (2013).
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X. Chen, X. L. Huang, and X. Liu, “Non-negligible effects of cloud vertical overlapping assumptions on longwave spectral fingerprinting studies,” J. Geophys. Res. 118, 7309–7320 (2013).
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W. Wu, X. Liu, H. Li, D. K. Zhou, and A. M. Larar, “Application study of principal component based physical retrieval algorithm for hyperspectral infrared sensors,” Proc. SPIE 8866, 88660G (2013).
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Q. Liu and F. Weng, “Using advanced matrix operator (AMOM) in community radiative transfer model,” IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens. 6, 1211–1218 (2013).
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2011 (1)

S. Kato, F. G. Rose, B. A. Wielicki, X. Liu, P. Taylor, D. P. Kratz, M. Mlynczak, S. Sun-Mack, W. F. Miller, and Y. Chen, “Sensitivity of spatially and temporally averaged all-sky top of-atmosphere nadir-view longwave spectrum and application to detect atmospheric property changes,” J. Climate 24, 6392–6407 (2011).
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Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Trans. 111, 878–894 (2010).
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H. Li, X. Liu, P. Yang, and D. P. Kratz, “A principal component-based radiative transfer forward model (PCRTM) for vertically inhomogeneous cloud,” Proc. SPIE 7856, 785606 (2010).
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P. W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Trans. 111, 1025–1040 (2010).
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M. Duan, Q. Min, and D. Lu, “A polarized radiative transfer model based on successive order of scattering,” Adv. Atmos. Sci. 27, 891–900 (2010).
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C. Cornet, L. C. Labonnote, and F. Szczap, “Three-dimensional polarized Monte Carlo atmospheric radiative transfer model (3DM- CPOL): 3-D effects on polarized visible reflectances of a cirrus cloud,” J. Quant. Spectrosc. Radiat. Trans. 111, 174–186 (2010).
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2009 (2)

B. Mayer, “Radiative transfer in the cloudy atmosphere,” Eur. Phys. J. Conf. 1, 75–99 (2009).

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2008 (1)

X. Liu, D. K. Zhou, A. Larar, W. L. Smith, and P. Schluessel, “Atmospheric property retrievals from infrared atmospheric sounding interferometer (IASI),” Proc. SPIE 7107, 71070E (2008).
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2007 (3)

J. Lenoble, M. Herman, J. L. Deuze, B. Lafrance, R. Santer, and D. Tanre, “A successive order of scattering code for solving the vector equation of transfer in the earth’s atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Trans. 107, 479–507 (2007).
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R. Saunders, P. Rayer, P. Brunel, A. von Engeln, N. Bormann, L. Strow, S. Hannon, S. Heilliette, X. Liu, F. Miskolczi, Y. Han, G. Masiello, T. Scalo, J.-L. Moncet, G. Uymin, V. Sherlock, and D. S. Turner, “A comparison of radiative transfer models for simulating AIRS radiances,” J. Geophys. Res. 112, D01S90 (2007).
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X. Liu, D. K. Zhou, A. Larar, W. L. Smith, and S. A. Mango, “Case‐study of a principal‐component‐based radiative transfer forward model and retrieval algorithm using EAQUATE data,” Q. J. R. Meteorol. Soc. 133, 243–256. (2007).
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2006 (4)

Q. Liu and F. Weng, “Advanced doubling-adding method for radiative transfer in planetary atmospheres,” J. Atmos. Sci. 63, 3459–3465 (2006).
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V. V. Rozanov and A. A. Kokhanovsky, “The solution of the vector radiative transfer equation using the discrete ordinates technique: selected applications,” Atmos. Res. 79, 241–265 (2006).
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X. Liu, W. L. Smith, D. K. Zhou, and A. Larar, “Principal component-based radiative transfer model for hyperspectral sensors: theoretical concept,” Appl. Opt. 45, 201–209 (2006).
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2005 (1)

X. Liu, W. L. Smith, D. K. Zhou, and A. M. Larar, “A principal component-based radiative transfer forward model (PCRTM) for hyperspectral instruments,” Proc. SPIE 5655, 96 (2005).
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2004 (1)

Q. Min and M. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Trans. 87, 243–259 (2004).
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2003 (1)

L. Strow, S. E. Hannon, S. D. Souza-Machado, H. E. Motteler, and D. Tobin, “An overview of the AIRS radiative transfer model,” IEEE Trans Geosci. Remote Sens. 41, 379–389 (2003).
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2000 (2)

D. P. Edwards and G. L. Francis, “Improvements to the correlated-k radiative transfer method: application to satellite infrared sounding,” J. Geophys. Res. 105, 18135–18156 (2000).
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W. Lucht and J. L. Roujean, “Considerations in the parametric modeling of BRDF and albedo from multiangular satellite sensor observations,” Remote Sens. Rev. 18, 343–379 (2000).
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1999 (4)

W. L. Smith, A. M. Larar, D. K. Zhou, C. A. Sisko, J. Li, B. Huang, H. B. Howell, H. E. Revercomb, D. Cousins, M. J. Gazarik, D. Mooney, and S. A. Mango, “NAST-I: results from revolutionary aircraft sounding spectrometer,” Proc. SPIE 3756, 2–8 (1999).
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F. M. Schultz, K. Stamnes, and F. Weng, “VDISORT: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer,” J. Quant. Spectrosc. Radiat. Trans. 61, 105–122 (1999).
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R. Saunders, M. Matricardi, and P. Brunel, “An improved fast radiative transfer model for assimilation of satellite radiance observations,” Q. J. R. Meteorol. Soc. 125, 1407–1425 (1999).
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M. Matricardi and R. Saunders, “Fast radiative transfer model for simulation of infrared atmospheric sounding interferometer radiances,” Appl. Opt. 38, 5679–5691 (1999).
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1997 (5)

E. K. Mlawer, S. J. Taubman, P. D. Brown, M. J. Iacono, and S. A. Clough, “Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlated-k model for the long-wave,” J. Geophys. Res. 102, 16663–16682 (1997).
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W. J. Wiscombe and J. W. Evans, “Exponential-sum fitting of radiative transmission functions,” J. Comput Phys. 24, 416–444 (1997).
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S. A. Tjemkes and J. Schmetz, “Synthetic satellite radiances using the radiance sampling method,” J. Geophys. Res. 102, 1807–1818 (1997).
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W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. B. Schaaf, and M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
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J. L. Privette, T. F. Eck, and D. W. Deering, “Estimating spectral albedo and nadir reflectance through inversion of simple BRDF models with AVHRR/MODIS-like data,” J. Geophys. Res. 102, 29529–29542 (1997).
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1996 (3)

1995 (5)

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1994 (2)

R. Cahalan, W. Ridgway, and W. Wiscombe, “Independent pixel and Monte Carlo estimates of stratocumulus albedo,” J. Atmos. Sci. 51, 3776–3790 (1994).
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1993 (2)

M. F. Gerstell, “Obtaining the cumulative k-distribution of a gas mixture from those of its components,” J. Quant. Spectrosc. Radiat. Transfer 49, 15–38 (1993).
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1991 (2)

A. Lacis and V. Oinas, “A description of the correlated k-distribution method for modeling nongray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres,” J. Geophys. Res. 96, 9027–9063 (1991).
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1990 (2)

R. West, D. Crisp, and L. Chen, “Mapping transformations for broadband atmospheric radiation calculations,” J. Quant. Spectrosc. Radiat. Transfer 43, 191–199 (1990).
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1989 (2)

R. Goody, R. West, L. Chen, and D. Crisp, “The correlated-k method for radiation calculations in nonhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 42, 539–550 (1989).
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1988 (2)

1987 (1)

J. F. de Haan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).

1984 (1)

1983 (1)

J. Suskind, J. Rosenfield, and D. Reuter, “An accurate radiative transfer model for use in the direct physical inversion of HIRS2 and MSU temperature sounding data,” J. Geophys. Res. 88, 8550–8562 (1983).
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1981 (1)

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1974 (2)

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N. A. Scott, “A direct method of computation of the transmission function of an inhomogeneous gaseous medium. I: Description of the method,” J. Quant. Spectrosc. Radiat. Transfer 14, 691–704 (1974).
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1973 (2)

G. W. Kattawar, G. N. Plass, and J. A. Guinn, “Monte Carlo calculations of the polarization of radiation in the earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
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G. N. Plass, G. W. Kattawar, and F. E. Catchings, “Matrix operator theory of radiative transfer. 1. Rayleigh scattering,” Appl. Opt. 12, 314–329 (1973).
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1972 (1)

D. G. Collins, W. G. Blattner, M. B. Wells, and H. G. Horak, “Backward Monte Carlo calculations of the polarization characteristics of the radiation emerging from spherical-shell atmospheres,” Appl Opt. 11, 2684–2696 (1972).
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1971 (2)

J. W. Hovenier, “Multiple scattering of polarized light in planetary atmospheres,” Astron. Astrophys. 13, 7–29 (1971).

J. E. Hansen and J. W. Hovenier, “The doubling method applied to multiple scattering of polarized light,” J. Quant. Spectrosc. Radiat. Trans. 11, 809–812 (1971).
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1969 (1)

M. I. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. 17, 577–585 (1969).
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1968 (1)

1966 (1)

S. Twomey, N. Jacobowitz, and H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–298 (1966).
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1965 (1)

W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 142, 1563–1575 (1965).
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1964 (1)

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1954 (1)

C. Cox and W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. A 44, 838–850 (1954).
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1951 (1)

G. H. Peebles and M. S. Plesset, “Transmission of gamma-rays through large thicknesses of heavy materials,” Phys. Rev. 81, 430–439 (1951).
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1948 (1)

H. C. van de Hulst, “Scattering in a planetary atmosphere,” Astrophys. J. 107, 220–246 (1948).
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1933 (1)

H. Hotelling, “Analysis of a complex of statistical variables into principal components,” J. Educ. Psychol. 24, 498–520 (1933).
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1862 (1)

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. London 11, 545–556 (1862).
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Anderson, J. G.

B. A. Wielicki, D. F. Young, M. G. Mlynczak, K. J. Thome, S. Leroy, J. Corliss, J. G. Anderson, C. O. Ao, R. Bantges, F. Best, K. Bowman, H. Brindley, J. Butler, W. Collins, J. A. Dykema, D. R. Doelling, D. R. Feldman, N. Fox, X. Huang, R. Holz, Y. Huang, Z. Jin, D. Jennings, D. G. Johnson, K. Jucks, S. Kato, D. B. Kirk-Davidoff, R. Knuteson, G. Kopp, D. P. Kratz, X. Liu, C. Lukashin, A. J. Mannucci, N. Phojanamongkolkij, P. Pilewskie, V. Ramaswamy, H. Revercomb, J. Rice, Y. Roberts, C. M. Roithmayr, F. Rose, S. Sandford, E. L. Shirley, W. L. Smith, B. Soden, P. W. Speth, W. Sun, P. C. Taylor, D. Tobin, and X. Xiong, “Achieving climate change absolute accuracy in orbit,” Bull. Amer. Meteorol. Soc. 94, 1519–1539 (2013).
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Ao, C. O.

B. A. Wielicki, D. F. Young, M. G. Mlynczak, K. J. Thome, S. Leroy, J. Corliss, J. G. Anderson, C. O. Ao, R. Bantges, F. Best, K. Bowman, H. Brindley, J. Butler, W. Collins, J. A. Dykema, D. R. Doelling, D. R. Feldman, N. Fox, X. Huang, R. Holz, Y. Huang, Z. Jin, D. Jennings, D. G. Johnson, K. Jucks, S. Kato, D. B. Kirk-Davidoff, R. Knuteson, G. Kopp, D. P. Kratz, X. Liu, C. Lukashin, A. J. Mannucci, N. Phojanamongkolkij, P. Pilewskie, V. Ramaswamy, H. Revercomb, J. Rice, Y. Roberts, C. M. Roithmayr, F. Rose, S. Sandford, E. L. Shirley, W. L. Smith, B. Soden, P. W. Speth, W. Sun, P. C. Taylor, D. Tobin, and X. Xiong, “Achieving climate change absolute accuracy in orbit,” Bull. Amer. Meteorol. Soc. 94, 1519–1539 (2013).
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Armbruster, W.

Bansemer, A.

B. A. Baum, P. Yang, A. J. Heymsfield, A. Bansemer, B. H. Cole, A. Merrelli, C. Schmitt, and C. Wang, “Ice cloud single-scattering property models with the full phase matrix at wavelengths from 0.2 to 100 μm,” J. Quant. Spectrosc. Raiat. Transfer 146, 123–139 (2014).
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Bantges, R.

B. A. Wielicki, D. F. Young, M. G. Mlynczak, K. J. Thome, S. Leroy, J. Corliss, J. G. Anderson, C. O. Ao, R. Bantges, F. Best, K. Bowman, H. Brindley, J. Butler, W. Collins, J. A. Dykema, D. R. Doelling, D. R. Feldman, N. Fox, X. Huang, R. Holz, Y. Huang, Z. Jin, D. Jennings, D. G. Johnson, K. Jucks, S. Kato, D. B. Kirk-Davidoff, R. Knuteson, G. Kopp, D. P. Kratz, X. Liu, C. Lukashin, A. J. Mannucci, N. Phojanamongkolkij, P. Pilewskie, V. Ramaswamy, H. Revercomb, J. Rice, Y. Roberts, C. M. Roithmayr, F. Rose, S. Sandford, E. L. Shirley, W. L. Smith, B. Soden, P. W. Speth, W. Sun, P. C. Taylor, D. Tobin, and X. Xiong, “Achieving climate change absolute accuracy in orbit,” Bull. Amer. Meteorol. Soc. 94, 1519–1539 (2013).
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Barnet, C.

X. Liu, A. M. Larar, D. K. Zhou, S. H. Kizer, W. Wu, C. Barnet, M. Divakarla, G. Guo, W. Blackwell, W. L. Smith, P. Yang, and D. Gu, “Retrieving atmospheric profiles data in the presence of clouds from hyperspectral remote sensing data,” in Hyperspectral Imaging and Sounding of the Environment, Toronto, Canada (2011).

Barnsley, M. J.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. B. Schaaf, and M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
[Crossref]

Baum, B. A.

B. A. Baum, P. Yang, A. J. Heymsfield, A. Bansemer, B. H. Cole, A. Merrelli, C. Schmitt, and C. Wang, “Ice cloud single-scattering property models with the full phase matrix at wavelengths from 0.2 to 100 μm,” J. Quant. Spectrosc. Raiat. Transfer 146, 123–139 (2014).
[Crossref]

P. Yang, L. Bi, B. A. Baum, K. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100  μm,” J. Atmos. Sci. 70, 330–347 (2013).
[Crossref]

C. Wang, P. Yang, S. L. Nasiri, S. Platnick, B. A. Baum, A. K. Heidinger, and X. Liu, “A fast radiative transfer model for visible through shortwave infrared spectral reflectance’s in clear and cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 116, 122–131 (2013).
[Crossref]

Ben-David, A.

Best, F.

B. A. Wielicki, D. F. Young, M. G. Mlynczak, K. J. Thome, S. Leroy, J. Corliss, J. G. Anderson, C. O. Ao, R. Bantges, F. Best, K. Bowman, H. Brindley, J. Butler, W. Collins, J. A. Dykema, D. R. Doelling, D. R. Feldman, N. Fox, X. Huang, R. Holz, Y. Huang, Z. Jin, D. Jennings, D. G. Johnson, K. Jucks, S. Kato, D. B. Kirk-Davidoff, R. Knuteson, G. Kopp, D. P. Kratz, X. Liu, C. Lukashin, A. J. Mannucci, N. Phojanamongkolkij, P. Pilewskie, V. Ramaswamy, H. Revercomb, J. Rice, Y. Roberts, C. M. Roithmayr, F. Rose, S. Sandford, E. L. Shirley, W. L. Smith, B. Soden, P. W. Speth, W. Sun, P. C. Taylor, D. Tobin, and X. Xiong, “Achieving climate change absolute accuracy in orbit,” Bull. Amer. Meteorol. Soc. 94, 1519–1539 (2013).
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Bi, L.

P. Yang, L. Bi, B. A. Baum, K. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100  μm,” J. Atmos. Sci. 70, 330–347 (2013).
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Figures (11)

Fig. 1.
Fig. 1. Typical results of the optical properties of the cloud water droplets. (a) Water droplet extinction coefficient (WDEC, km 1 m 3 / g ). (b) Coalbedo ( 1 ω ) . (c) Water droplet asymmetry factor (WDAF). The legends represent the water droplet effective diameter (μm).
Fig. 2.
Fig. 2. Typical results of the optical properties of the cloud ice particles. (a) Ice particle extinction coefficient (IPEC, km 1 m 3 / g ). (b) Coalbedo ( 1 ω ) . (c) Ice particle asymmetry factor (IPAF). The legends represent the ice effective diameter (μm).
Fig. 3.
Fig. 3. Typical TOA radiances obtained using MODTRAN5 for both land and ocean surfaces, respectively. There are many fine features in the spectrum. The high-resolution channel spectrum makes the channel-based RTMs inefficient. Each color represents a randomly selected case.
Fig. 4.
Fig. 4. Singular value for each principal component (upper) and normalized cumulative singular value as function of the selected number of PCs (lower) in the band range of 625 nm to 980 nm for ocean surface case.
Fig. 5.
Fig. 5. Typical TOA radiance and the reconstructed radiance using 299 PCs for ocean surface case in wavelength range from 625 to 980 nm (upper). The RMS errors of the reconstructed TOA radiances compared to the original radiances (lower).
Fig. 6.
Fig. 6. TOA radiances obtained using MODTRAN5 (blue) and fast PCRTM-SOLAR (green) for an ocean surface case with 1    cm 1 resolution (upper). The difference between the TOA channel radiances obtained using MODTRAN5 and those obtained using PCRTM-SOLAR (lower). This case was included in training.
Fig. 7.
Fig. 7. Typical TOA radiances obtained using MODTRAN5 (blue) and fast PCRTM-SOLAR (green) for an ocean surface case with 1    cm 1 resolution (upper). The difference between the TOA channel radiances obtained using MODTRAN5 and those obtained using PCRTM-SOLAR (lower). This case was not included in training.
Fig. 8.
Fig. 8. Mean absolute errors in PCRTM-SOLAR calculated TOA channel radiances with respect to those simulated using MODTRAN5. The blue line is for the validation data set while the green one is for the training data set.
Fig. 9.
Fig. 9. RMS errors in PCRTM-SOLAR calculated TOA channel radiances with respect to those simulated using MODTRAN5. The blue line is for the validation data set while the green one is for the training data set.
Fig. 10.
Fig. 10. RMS errors in PCRTM-SOLAR calculated TOA channel radiances with respect to those simulated using MODTRAN5. The blue lines are for the validation data set while the green ones are for the training data set. (a) Land surface with 1    cm 1 resolution; (b) land surface with 8 nm resolution; (c) ocean surface with 8 nm resolution.
Fig. 11.
Fig. 11. Typical systematic model biases of PCRTM-SOLAR model. The data shown are for land surface with 8 nm resolution. The blue line is for the validation data set while the green one is for the training data set.

Tables (2)

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Table 1. Number of Monofrequencies (nmo), Number of Channel Frequencies (nch), Number of Significant PCs (npc), and Number of Preselected Monofrequencies (nsmo), for Both Land and Ocean Surfaces with Both 1    cm 1 and 8 nm Resolutions

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Table 2. Increase in Speed of PCRTM-SOLAR Compared to MODTRAN5

Equations (20)

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β ext LWC = a 1 r e b 1 + c 1 ,
1 ω = a 2 r e b 2 + c 2 ,
g = a 3 r e b 3 c 3 ,
ρ ( θ v , θ S , φ ) = P 1 + P 2 + K LSR ( θ v , θ S , φ ) + P 3 K RT ( θ v , θ S , φ ) .
K RT = ( π / 2 ξ ) cos ξ + sin ξ cos θ v + cos θ s π 4 ,
K LSR = 1 + sec θ v sec θ s + tan θ v tan θ s cos φ 2 + ( t sin t cos t π 1 ) ( sec θ v + sec θ s ) ,
cos 2 t = ( 2 sec θ v + sec θ s ) 2 × [ G ( θ v θ s φ ) 2 + ( tan θ v tan θ s sin φ ) 2 ] ,
G ( θ v , θ s , φ ) + tan 2 θ v + tan 2 θ s 2 tan θ v tan θ s cos φ ,
tan θ x = P 5 tan θ x ; x = v or S ,
P ( μ n ) = 1 π σ 2 exp ( 1 μ n 2 σ 2 μ n 2 ) ,
σ 2 = 0.003 + 0.00512 V ,
R 0 ( μ , φ , μ , φ , n ) = r × p × s ,
R nch × ns chan = U nch × nch S nch × nch V ns × nch = U nch × nch Y nch × ns ,
NCSV ( npc ) = i = 1 npc SV ( i ) i = 1 nch SV ( i ) ,
R nch × ns chan = U nch × npc Y nch × ns ( npc nch ) .
Y npc × ns = U npc × nch R nch × ns chan .
R nch × ns chan = k = 1 N Φ k R k × ns mono k = 1 N Φ k .
Y npc × ns = A npc × nsmo R nsmo × ns mono .
MA Error = 1 N i = 1 N | R i chan_PCRTM R i chan_MODRAN | ,
RMS Error = 1 N i = 1 N ( R i chan_PCRTM R i chan_MODRAN ) 2 ,

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