Abstract
The discrete ordinate method for the transfer of monochromatic unpolarized radiation in non-isothermal, vertically inhomogeneous media, as implemented in the computer code Discrete-Ordinate-Method Radiative Transfer, DISORT, is reviewed. Both the theoretical background and its algorithmic implementation are covered. Among others, described are the reduction of the order of the standard algebraic eigenvalue problem to increase efficiency in both the homogenous and particular solutions of the system of coupled ordinary differential equations, application of the scaling transformation to make the solution unconditionally stable for arbitrary large values of optical depth, application of the δ-M method to handle highly anisotropic scattering, the correction of intensities by the Nakajima-Tanaka method, and the implementation of a realistic bidirectional bottom boundary. Numerical considerations that make the implementation robust and efficient are also discussed. Examples of setting up DISORT runs are shown by using test cases with increasing complexity. Brief summaries of the versions released to date are provided, as well.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Anderson E, Bai Z, Bischof C, Demmel I, Dongarra J, Du Croz J, Greenbaum A, Hammarling S, McKenney A, Ostrouchov S, Sorensen D (1999) LAPACK user’s guide, 3rd ed. Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, Pennsylvania, ISBN: 978-0-89871-447-0, 19104-2688
Asano S (1975) On the discrete ordinates method for radiative transfer. J. Meteor Soc 53:92–95
Asheim A, Deano A, Huybrechs D, Wang HY (2014) A gaussian quadrature rule for oscillatory integrals on a bounded interval. Discret Contin Dyn Syst 34(3):883–901
Berk A, Bernstein LS, Anderson GP, Acharya PK, Robertson DC, Chetwynd JH, Adler-Golden SM (1998) MODTRAN cloud and multiple scattering upgrades with application to AVIRIS. Rem Sens Envir 65(3):367–375. doi:10.1016/S0034-4257(98)00045-5
Bohren C, Huffman D (1983) Absorption and scattering of light by small particles. Wiley, New York
Buras R, Dowling T, Emde C (2011) New secondary-scattering correction in DISORT with increased efficiency for forward scattering. J Quant Spectrosc Radiat Transfer 112(12):2028–2034. doi:10.1016/j.jqsrt.2011.03.019
Chandrasekhar S (1960) Radiative transfer. Dover Publications, New York 393 p
Clough SA, Shephard MW, Mlawer EJ, Delamere JS, Iacono MJ, Cady-Pereira K, Boukabara S, Brown PD (2005) Atmospheric radiative transfer modeling: a summary of the AER codes. J Quant Spectrosc Radiat Transfer 91(2):233–244. doi:10.1016/j.jqsrt.2004.05.058
Cowell WR (ed) (1980) Sources and developments of mathematical sofware. Prentice Hall, Englewood Cliffs
Cox C, Munk W (1954) Measurement of roughness of the sea surface from photographs of the sun’s glitter. J Opt Soc Am 44(11):838–850
Devaux C, Grandjean P, Ishiguro Y, Siewert CE (1979) On multi-region problems in radiative transfer. Astrophys Space Sci 62:225–233
Ding S, Yang P, Weng F, Liu Q, Han Y, van Delst P, Li J, Baum B (2011) Validation of the community radiative transfer model. J Quant Spectrosc Radiat Transfer 112(6):1050–1064. doi:10.1016/j.jqsrt.2010.11.009
Dongarra J, Moler C, Bunch J, Stewart GW (1979) LINPACK User’s guide. Society for Industrial and Applied Mathematics (SIAM) Press, Philadelphia, pp xx+344. ISBN 978-0-89871-172-1
Filon LNG (1928) On a quadrature formula for trigonometric integrals. Proc Roy Soc Edinburgh Sect A 49:38–47
Garcia RDM, Siewert CE (1985) Benchmark results in radiative transfer. Transp Theory Stat Phys 14:437–483
Godsalve C (1995) The Inclusion of Reflectances with Preferred Directions in Radiative-Transfer Calculations. J Quant Spectrosc Radiat Transfer 53(3):289–305
Godsalve C (1996) Accelerating the discrete ordinates method for the solution of the radiative transfer equation for planetary atmospheres. J Quant Spectrosc Radiat Transfer 56(4):609–616. doi:10.1016/0022-4073(96)00075-1
Hapke B (1993) Theory of reflectance and emittance spectroscopy. Cambridge University Press, Cambridge, p 455
Iserles A, Nørsett SP (2004) On quadrature methods for highly oscillatory integrals and their implementation. BIT 44(4):755–772
Iserles A, Nørsett SP, Olver S (2006) Highly oscillatory quadrature: the story so far. In: de Castro A, Gómez D, Quintela P, Salgado P (eds) Numerical mathematics and advanced applications. Springer, Berlin, pp 97–118
Jerg M, Trautmann T (2007) One-dimensional solar radiative transfer: Perturbation approach and its application to independent-pixel calculations for realistic cloud fields. J Quant Spectrosc Radiat Transfer 105(1):32–54. doi:10.1016/j.jqsrt.2006.09.014
Key J, Schweiger AJ (1998) Tools for atmospheric radiative transfer: Streamer and FluxNet. Comput Geosci 24(5):443–451
Kokhanovsky AA et al (2010) Benchmark results in vector atmospheric radiative transfer. J Quant Spectrosc Radiat Transfer 111(12–13):1931–1946
Kylling A, Stamnes K (1992) Efficient yet accurate solution of the linear transport equation in the presence of internal sources: the exponential-linear-in-depth approximation. J Comp Phys 102:265–276
Laszlo I, Stamnes K, Wiscombe WJ, Tsay S-C (2010) Towards generalized boundary conditions in DISORT. In: Extended abstracts of the AMS 13th conference on atmospheric radiation, 28 June–2 July 2010, Portland, Oregon. https://ams.confex.com/ams/13CldPhy13AtRad/techprogram/paper_171283.htm
Levin D (1996) Fast integration of rapidly oscillatory functions. J Comput Appl Math 67(1):95–101
Lin Z, Stamnes S, Jin Z, Laszlo I, Tsay SC, Wiscombe WJ, Stamnes K (2015) Improved discrete ordinate solutions in the presence of an anisotropically reflecting lower boundary: upgrades of the DISORT computational tool. J Quant Spectrosc Radiat Transfer 157:119–134. doi:10.1016/j.jqsrt.2015.02.014
Lindner B (1988) Ozone on mars: the effects of clouds and airborne dust. Planet Space Sci 36:125–144
Liou K-N (1973) A numerical experiment on Chandrasekhar’s discrete-ordinate method for radiative transfer: applications to cloudy and hazy atmospheres. J Atmos Sci 30(7):1303–1326
Nakajima T (2010) A retrospective view on “algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation” by Teruyuki Nakajima and Masayuki Tanaka (1988). J Quant Spectrosc Radiat Transfer 111(11):1651–1652. doi:http://dx.doi.org/10.1016/j.jqsrt.2010.01.002
Nakajima T, Tanaka M (1986) Matrix formulations for the transfer of solar radiation in a plane-parallel atmosphere. J Quant Spectrosc Radiat Transfer 35:13–21
Nakajima T, Tanaka M (1988) Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation. J Quant Spectrosc Radiat Transfer 40:51–69
Ozisik M, Shouman S (1980) Source function expansion method for radiative transfer in a two-layer slab. J Quant Spectrosc Radiat Transfer 24:441–449
Parlett and Reinsch (1969) Balancing a matrix for calculation of eigenvalues and eigenvectors. Num Math 13:293–304
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in Fortran: the art of scientific computing, 2nd edn. Cambridge University Press, Cambridge 963 p
Rahman H, Pinty B, Verstraete MM (1993) Coupled surface-atmosphere reflectance (CSAR) model 2. Semiempirical surface model usable with NOAA advanced very high resolution radiometer data. J Geophys Res 98(D11):20791–20801
Ricchiazzi P, Yang S, Gautier C, Sowle D (1998) SBDART: a research and teaching software tool for plane-parallel radiative transfer in the earth’s atmosphere. Bull Am Meteorol Soc 79(10):2101–2114. doi:10.1175/1520-0477(1998)079<2101:SARATS>2.0.CO;2
Roujean J-L, Leroy M, Deschamps P-Y (1992) A bidirectional reflectance model of the Earth’s surface for the correction of remote sensing data. J Geophys Res 97(D18):20455–20468
Siewert CE (2000) A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer. J Quant Spectrosc Radiat Transfer 64(2):109–130. doi:10.1016/S0022-4073(98)00144-7
Stamnes K (1982a) On the computation of angular distributions of radiation in planetary atmospheres. J Quant Spectrosc Radiat Transfer 28:47–51
Stamnes K (1982b) Reflection and transmission by a vertically inhomogeneous planetary atmosphere. Planet Space Sci 30:727–732
Stamnes K (1986) The theory of multiple scattering of radiation in plane parallel atmospheres. Rev Geophys 24:299–310
Stamnes K, Conklin P (1984) A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres. J Quant Spectrosc Radiat Transfer 31:273–282
Stamnes K, Dale H (1981) A new look at the discrete ordinate method for radiative transfer calculation in anisotropically scattering atmospheres. II: Intensity computations. J Atmos Sci 38:2696–2706
Stamnes K, Swanson RA (1981) A New look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. J Atmos Sci 38(2):387–399. doi:10.1175/1520-0469(1981)038<0387:ANLATD>2.0.CO;2
Stamnes K, Tsay S-C, Wiscombe W, Jayaweera K (1988a) Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media. Appl Opt 27:2502–2509
Stamnes K, Tsay S-C, Nakajima T (1988b) Computation of eigenvalues and eigenvectors for discrete ordinate and matrix operator method radiative transfer. J Quant Spectrosc Radiat Transfer 39:415–419
Stamnes K, Tsay S-C, Wiscombe WJ, Laszlo I (2000) DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media: documentation of methodology. NASA Technical Report, version 1.1
Stamnes S (2011) Calculations of the bidirectional reflectance distribution function (BRDF) of a planetary surface. MS thesis, Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey, USA
Sweigart A (1970) Radiative transfer in atmospheres scattering according to the Rayleigh phase function with absorption. Astrophys J 22:1–80
Sykes JB (1951) Approximate integration of the equation of transfer. Mon Not R Astron Soc 111(4):377–386
Thomas GE, Stamnes K (1999) Radiative transfer in the atmosphere and ocean. Cambridge University Press, Cambridge
Tsay S-C (1986) Numerical study of the atmospheric radiative transfer process with application to the Arctic energy balance. Ph.D. thesis, Alaska University, Fairbanks
Tsay S-C, Stamnes K, Jayaweera K (1989) Radiative energy budget in the cloudy and hazy Arctic. J Atmos Sci 46:1002–1018
Tsay S-C, Stamnes K, Jayaweera K (1990) Radiative transfer in stratified atmospheres: development and verification of a unified model. J Quant Spectrosc Radiat Transfer 43:133–148
Tsay S-C, Stamnes K (1992) Ultraviolet radiation in the Arctic: the impact of potential ozone depletions and cloud effects. J Geophy Res 97:7829–7840
Van de Hulst HC (1980) Multiple light scattering, tables, formulas and applications, vol 1 and 2. Academic Press, New York
Wilkinson J (1965) The algebraic eigenvalue problem. Clarendon Press, Oxford
Wiscombe WJ (1976) Extension of the doubling method to inhomogeneous sources. J Quant Spectrosc Radiat Transfer 16:477–489
Wiscombe WJ (1977) The delta–M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions. J Atmos Sci 34(9):1408–1422
Acknowledgments
The authors thank Z. Lin, S. Stamnes, L. Rokke, M. Zhou and H. Liu for their critical reading of and useful comments on an earlier version of the manuscript, and A. Kokhanovsky for inviting us to write this review and for his patience. IL acknowledges the assistance of K. Laszlo with typing in many of the equations.
Disclaimer
The contents of this paper are solely the opinions of the authors and do not constitute a statement of policy, decision, or position on behalf of the U.S. National Oceanic and Atmospheric Administration (NOAA) or the U.S. Government.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Laszlo, I., Stamnes, K., Wiscombe, W.J., Tsay, SC. (2016). The Discrete Ordinate Algorithm, DISORT for Radiative Transfer. In: Kokhanovsky, A. (eds) Light Scattering Reviews, Volume 11. Springer Praxis Books. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49538-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-662-49538-4_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-49536-0
Online ISBN: 978-3-662-49538-4
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)