Code SHARM: fast and accurate radiative transfer over spatially variable anisotropic surfaces

  • Alexei LyapustinEmail author
  • Tolegen Muldashev
  • Yujie Wang
Part of the Springer Praxis Books book series (PRAXIS)


Numerous numerical methods have been developed to solve a plane-parallel radiative transfer problem, including discrete ordinates (Stamnes et al., 1988; Spurr et al., 2001), spherical harmonics (Dave, 1975; Benassi et al., 1984), adding-doubling (Hansen and Hovenier, 1971; Twomey, 1985), successive orders of scattering (Lenoble et al., 2007) etc. An extended reference to numerical methods and publicly available codes can be found in Lenoble (1985), Ricchiazzi et al. (1998), Mayer and Kylling (2005) and Cahalan (2005). In this chapter, we describe the method of spherical harmonics (MSH), in particular its very efficient implementation developed by Karp et al. (1980) and, later, Muldashev et al. (1999). From a numerical standpoint, here are several main components of the spherical harmonics solution: (1) obtaining the system of linear differential equations of MSH, (2) its reduction to the system of linear algebraic equations using singular value decomposition (SVD), (3) use of a system’s matrix symmetry to halve its size for SVD transformation with ~8 times gain in speed, and finally, (4) angular smoothing of the solution for radiance calculations in arbitrary directions. The detail of MSH for the 1-D radiative transfer problem with a uniform surface, and for the 3-D problem with a spatially variable surface, are presented in sections 6.1 and 6.5, respectively. Section 6.2 provides an overview of the 1-D radiative transfer code SHARM (Muldashev et al., 1999; Lyapustin and Wang, 2005) which is one of the most numerically efficient scalar codes. SHARM performs simultaneous monochromatic calculations for multiple sun-view geometries, and allows the user to make multi-wavelength calculations in one run. The code is user-friendly, featuring built-in aerosol models and the most popular models of the bi-directional reflectance factor (BRF) of land and wind-ruffled water surface. Comparisons of SHARM with the benchmark code DISORT showed agreement to 0.02%.


Singular Value Decomposition Radiative Transfer Spherical Harmonic Surface Albedo Path Radiance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, T. L., R. J. Charlson, D. M. Winker, J. A. Ogren, Kim Holmenm, 2003: Mesoscale variations of tropospheric aerosols, J. Atm. Sci., 60, 119–136.CrossRefGoogle Scholar
  2. Bell, G. I., and S. Glasstone, 1970: Nuclear Reactor Theory. Van Nostrand Reinhold, New York.Google Scholar
  3. Benassi, M., R. D. M. Garcia, A. Karp et al., 1984: A high-order spherical harmonics solution to the standard problem in radiative transfer, Astrophys. J., 280, 2, 853–864.Google Scholar
  4. Bodhaine, B. A., N. B. Wood, E. G. Dutton, J. R. Slusser, 1999: On Rayleigh Optical Depth Calculations, J. Atmos. Oceanic Technology, 16, 1854–1861.CrossRefGoogle Scholar
  5. Cahalan R. F., et al., 2005: The international intercomparison of 3D radiation codes (I3RC). Bringing together the most advanced radiative transfer tools for cloudy atmospheres, Bull. Amer. Meteor. Soc., 86, 1275–1293.Google Scholar
  6. Case, K. M., and P. F. Zweifel, 1967: Linear Transport Theory. Addison-Wesley Publishing Company, Reading, MS.Google Scholar
  7. Chandrasekhar, S., 1958: On the diffuse reflection of a pencil of radiation by a planeparallel atmosphere, Proc. Nat. Acad. Sci., 44, 933–940.CrossRefGoogle Scholar
  8. Chandrasekhar, S., 1960: Radiative Transfer. Dover, New York.Google Scholar
  9. Cox, C., and W. Munk, 1954: Measurements of the roughness of the sea surface from photographs of the Sun’s glitter, J. Opt. Society Am., 44, 838–850.CrossRefGoogle Scholar
  10. Dave, J. V., 1975: A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation, J. Atmos. Sci., 32, 790–798.CrossRefGoogle Scholar
  11. Dave, J. V., and B. H. Armstrong, 1974: Smoothing of intensity curve obtained from a solution of spherical harmonics approximation to transfer equation, J. Atmos. Sci., 31, 1934–1937.CrossRefGoogle Scholar
  12. Davis, A. B., I. N. Polonski, and A. Marshak, 2009: Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing. In Light Scattering Reviews, 4. Single Light Scattering and Radiative Transfer, A. A. Kokhanovsky (ed.), Springer, 169–292.Google Scholar
  13. Diner, D. J., and J. V. Martonchik, 1984a: Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground – I. Theory, J. Quant. Spectrosc. Radiat. Transfer, 31, 97–125.Google Scholar
  14. Diner, D. J., and J. V. Martonchik, 1984b: Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground – II. Computational considerations and results, J. Quant. Spectrosc. Radiat. Transfer, 32, 279–304.Google Scholar
  15. Diner, D. J., and J. V. Martonchik, 1985: Influence of aerosol scattering on atmospheric blurring of surface features, IEEE Trans. Geosci. Remote Sens., GE-23, 618–624.Google Scholar
  16. Dubovik, O., A. Sinyuk, T. Lapyonok, B. N. Holben, M. Mishchenko, P. Yang, T. F. Eck, H. Volten, O. Munoz, B. Veihelmann, van der Zander, J.-F. Leon, M Sorokin, and I. Slutsker, 2006: Application of spheroid models to account for aerosol particle non-sphericity in remote sensing of desert dust, J. Geophys. Res., 111, D11208, doi:10.1029/2005JD006619.CrossRefGoogle Scholar
  17. Evans, K. F., 1998: The spherical harmonics discrete ordinate method for threedimensional atmospheric radiative transfer, J. Atm. Sci., 55, 429–446.CrossRefGoogle Scholar
  18. Gatebe, C., M. D. King, A. I. Lyapustin, G. T. Arnold, and J. Redermann, 2005: Airborne spectral measurements of ocean directional reflectance, J. Atmos. Sci., 62, 1071–1091.CrossRefGoogle Scholar
  19. Germogenova, T. A., 1986: The Local Properties of the Solution of the Transport Equation (in Russian). Nauka, Moscow, Russia.Google Scholar
  20. Gerstl, S. A. W., 1982: Application of the adjoint method in atmospheric radiative transfer calculations. in Atmospheric Aerosols: Their Formation, Optical Properties and Effects. A. Deepak (ed.). Spectrum Press, Hampton VA, 241–254.Google Scholar
  21. Hansen, J. E., and J. W. Hovenier, 1971: The doubling method applied to multiple scattering of polarized light, J. Quant. Spectrosc. Radiat. Transfer, 11, 809–812.CrossRefGoogle Scholar
  22. Holben, B. N., T. F. Eck, I. Slutsker, D. Tanr´e, J. P. Buis, A. Setzer, E. Vermote, J. A. Reagan, Y. J. Kaufman, T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, 1998: AERONET-A Federated instrument network and data archive for aerosol characterization, Rem. Sens. Environ., 66, 1–16.Google Scholar
  23. Ioltukhovskii, A. A., 1999: Radiative transfer over the surface with an arbitrary reflection: Green’s functions method, Transport Theory and Statistical Physics, 28 (4), 349–368.CrossRefGoogle Scholar
  24. Karp, A. H., 1981: Computing the angular dependence of the radiation of a planetary atmosphere, J. Quant. Spectrosc. Radiat. Transfer, 25, 403–412.CrossRefGoogle Scholar
  25. Karp, A. H., J. Greenstadt and J. A. Fillmore, 1980: Radiative transfer through an arbitrarily thick scattering atmosphere, J. Quant. Spectrosc. Radiat. Transfer, 24, 391–406.CrossRefGoogle Scholar
  26. King, M., and R. Greenstone (eds), 1999: EOS Reference Handbook: A Guide to Earth Science Enterprise and the Earth Observation System (p. 355). Greenbelt, MD: EOS Project Science Office, NASA/Goddard Space Flight Center.Google Scholar
  27. Kneizys, F. X., L. W. Abreu, G. P. Anderson, J. H. Chetwynd, E. P. Shettle, A. Berk, L. S. Bernstein, D. C. Robertson, P. Acharya, L. S. Rothman, J. E. A. Selby, W. O. Gallery, S. A. Clough, 1996: The MODTRAN 2/3 Report and LOWTRAN 7 Model, Modtran Report. Ontar Corporation, North Andover, MA.Google Scholar
  28. Landgraf J., Hasekamp O. P., Trautmann T., 2002: Linearization of radiative transfer with respect to surface properties, J. Quant. Spectrosc. Radiat. Transfer, 72 (4), 327–339.CrossRefGoogle Scholar
  29. Lenoble, J. (ed.), 1985: Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. A. Deepak Publishing, Hampton, VA.Google Scholar
  30. Lenoble, J., M. Herman, J.L. Deuz´e, B. Lafrance, R. Santer, D. Tanre, 2007: A successive order of scattering code for solving the vector equation of transfer in the earth’s atmosphere with aerosols, J. Quant. Spectrosc. Radiat. Transfer, 107, 479–507.Google Scholar
  31. Lucht W., Schaaf C. B., Strahler A. H., 2000: An algorithm for the retrieval of albedo from space using semiempirical BRDF models, IEEE Trans. Geosci. Remote Sens., 38, 977–998.CrossRefGoogle Scholar
  32. Lyapustin, A. I., 2001: 3-D effects in the remote sensing of surface albedo, IEEE Trans. Geosci. Remote Sensing, 39, 254–263.CrossRefGoogle Scholar
  33. Lyapustin, A., 2002: Radiative transfer code SHARM-3D for radiance simulations over a non-Lambertian nonhomogeneous surface: intercomparison study, Appl. Optics, 41, 5607–5615.Google Scholar
  34. Lyapustin, A., 2005: Radiative transfer code SHARM for atmospheric and terrestrial applications, Appl. Optics, 44, 7764–7772.Google Scholar
  35. Lyapustin, A. I., and Y. J. Kaufman, 2001: Role of adjacency effect in the remote sensing of aerosol. J. Geophys. Res., 106, D11, 11,909–11,916.Google Scholar
  36. Lyapustin, A., and Yu. Knyazikhin, 2001: Green’s function method in the radiative transfer problem. I: Homogeneous non-Lambertian surface, Appl. Optics, 40, 3495–3501.Google Scholar
  37. Lyapustin, A., and Yu. Knyazikhin, 2002: Green’s function method in the radiative transfer problem. II: Spatially heterogeneous anisotropic surface, Appl. Optics, 41, 5600–5606.Google Scholar
  38. Lyapustin, A. I., and T. Z. Muldashev, 1999: Method of spherical harmonics in the radiative transfer problem with non-Lambertian surface, J. Quant. Spectrosc. Radiat. Transfer, 61, 545–555.CrossRefGoogle Scholar
  39. Lyapustin, A. I., and T. Z. Muldashev, 2000: Generalization of Marshak boundary condition for non-Lambert reflection, J. Quant. Spectrosc. Radiat. Transfer, 67, 457–464.CrossRefGoogle Scholar
  40. Lyapustin, A. I., and T. Z. Muldashev, 2001: Solution for atmospheric optical transfer function using spherical harmonics method. J. Quant. Spectrosc. Radiat. Transfer, 68, 43–56.CrossRefGoogle Scholar
  41. Lyapustin, A. I., and J. L. Privette, 1999: A new algorithm for retrieving surface BRDF from ground measurements: atmospheric sensitivity study, J. Geophys. Res., 104, 6257–6268.CrossRefGoogle Scholar
  42. Lyapustin, A., and Y. Wang, 2005: Parameterized code Sharm-3D for radiative transfer over inhomogeneous surfaces, Appl. Optics, 44, 7602–7610.Google Scholar
  43. Lyapustin, A., Wang, Y., 2009: The time series technique for aerosol retrievals over land from MODIS, pp. 69–99, in Satellite Aerosol Remote Sensing over Land, A. Kokhanovky and G. de Leeuw (eds). Springer Praxis Books, IBSN: 978-3-540-69396-3.Google Scholar
  44. Marchuk, G., G. Mikhailov, N. Nazaraliev, R. Darbinjan, B. Kargin, and B. Elepov, 1980: The Monte Carlo Methods in Atmospheric Optics. Springer-Verlag, New York.Google Scholar
  45. Marshak R. E., 1947: Note on the spherical harmonics method as applied to the Milne problem for a sphere, Phys. Rev., 71, 443–446.CrossRefGoogle Scholar
  46. Martonchik, J. V., D. J. Diner, B. Pinty, M. M. Verstratete, R. B. Myneni, Yu. Knyazikhin, and H. R. Gordon, 1998: Determination of land and ocean reflective, radiative and biophysical properties using multiangle imaging, IEEE Trans. Geosci. Remote Sens., 36, 1266–1281.CrossRefGoogle Scholar
  47. Mayer, B., and A. Kylling, 2005: Technical note: The libRadtran software package for radiative transfer calculations: Description and examples of use. ACPD, 5, 1319–1381.Google Scholar
  48. Mekler, Yu., and Y. J. Kaufman, 1980: The effect of Earth’s atmosphere on contrast reduction for a non-uniform surface albedo and ‘two-halves’ field, J. Geophys. Res., 85, 4067–4083.CrossRefGoogle Scholar
  49. Muldashev, T. Z., A. I. Lyapustin, and U. M. Sultangazin, 1999: Spherical harmonics method in the problem of radiative transfer in the atmosphere-surface system, J. Quant. Spectrosc. Radiat. Transfer, 61, 393–404.CrossRefGoogle Scholar
  50. Nakajima, T., and M. Tanaka, 1983: Effect of wind-generated waves on the transfer of solar radiation in the atmosphere–ocean system, J. Quant. Spectrosc. Radiat. Transfer, 29, 521–537.CrossRefGoogle Scholar
  51. Otterman, J., and R. S. Fraser, 1979: Adjacency effects on imaging by surface reflection and atmospheric scattering: Cross radiance to zenith, Appl. Opt., 18, 2852–2860.CrossRefGoogle Scholar
  52. Pearce, W. A., 1977: A study of the effect of atmosphere on the Thematic Mapper observations. Rep. 004-77, EG&G/Washington Anal. Serv. Center, Appl. Syst. Dept., Riverdale, MD. Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992: Numerical Recipes in C, 2nd edn, Cambridge University Press, New York.Google Scholar
  53. Qin Y., Box M. A., 2005: Analytic green’s function for radiative transfer in plane-parallel atmospheres, J. Quant. Spectrosc. Radiat. Transfer, 62 (8), 2910–2924.Google Scholar
  54. Rahman, H., B. Pinty, and M. M. Verstraete, 1993: Coupled surface–atmosphere reflectance (CSAR) model. 2. Semiempirical surface model usable with NOAA advanced very high resolution radiometer data, J. Geophys. Res., 98, 20,791–20,801.Google Scholar
  55. Ricchiazzi, P., S. Yang, C. Gautier, and D. Sowle, 1998: SBDART: A research and teaching software tool for plane-parallel radiative transfer in the Earth’s atmosphere, BAMS, 79, 2101–2114.CrossRefGoogle Scholar
  56. Riesz, F., and B. Sz.-Nagy, 1990: Functional Analysis. Dover, New York.Google Scholar
  57. Roujean, J.-L., M. Leroy, and P. Y. Deschamps, 1992: A bidirectional reflectance model of the Earth’s surface for the correction of the remote sensing data, J. Geophys. Res., 97, 20,455–20,468.Google Scholar
  58. Schaaf C. B., Gao F., Strahler A.H., W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette and D. Roy, 2002: First operational BRDF, albedo nadir reflectance products from MODIS, Rem. Sens. Environ., 83, 135–148.CrossRefGoogle Scholar
  59. Spurr, R. J. D., Kurosu T. P., Chance K. V., 2001: A linearized discrete ordinate radiative transfer model for atmospheric remote sensing retrieval, J. Quant. Spectrosc. Radiat. Transfer, 68, 689–735.CrossRefGoogle Scholar
  60. Stamnes, K., S. C. Tsay, W. Wiscombe and K. Jayaweera, 1988: Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media, Appl. Opt., 27, 2502–2509.CrossRefGoogle Scholar
  61. Sushkevich, T. A., S. A. Strelkov, and A. A. Ioltuhovskii, 1990: Method of Path Integration in the Problems of Atmospheric Optics (in Russian). Nauka, Moscow, Russia.Google Scholar
  62. Twomey, S., 1985: Green’s function formulae for the internal intensity in radiative transfer computations by matrix-vector methods, J. Quant. Spectrosc. Radiat. Transfer, 33, 575–579.CrossRefGoogle Scholar
  63. Vladimirov, V. S., 1963: Mathematical problems in the one-velocity theory of particle transport. Tech. Rep. AECL-1661, At. Energy of Can. Ltd., Chalk River, Ontario.Google Scholar
  64. Wang, Y., A. Lyapustin, J. L. Privette, J. T. Morisette, B. Holben, 2009: Atmospheric correction at AERONET locations: A new science and validation data set. IEEE Trans. Geosci. Remote Sens., in press.Google Scholar
  65. Wiscombe, W. J., 1977: The Delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions. J. Atm. Sci., 34, 1408–1422.CrossRefGoogle Scholar
  66. Zege, E. P., I. L. Katsev, A. P. Ivanov, 1991: Image Transfer through a Scattering Medium. Berlin: Springer.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Alexei Lyapustin
    • 1
    Email author
  • Tolegen Muldashev
    • 2
  • Yujie Wang
    • 1
  1. 1.NASA Goddard Space Flight CenterGreenbeltUSA
  2. 2.AlmatyKazakhstan

Personalised recommendations