benchmarks:benchmarks

The results in this section are well established benchmark results to be used for validation of newly developed codes.

- Comprehensive tables for Rayleigh scattering:

*K. L. Coulson, J. V. Dave, and Z. Sekera. Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering. University of California Press, 1960.* - Correction for the Coulson's et al. Rayleigh tables:

*Nataraj V., Li K.-F., and Young Y.I., 2009: Rayleigh scattering in planetary atmospheres: corrected tables through accurate computation of X and Y functions. Astrophysical Journal, 691, pp. 1909-1920.*

Tables can be downloaded at http://web.gps.caltech.edu/~vijay/Rayleigh_Scattering_Tables/. - Rayleigh scattering (single scattering albedo as a function of optical depth):

*Mishchenko, M.I., 1990: The fast invariant imbedding method for polarized light: computational aspects and numerical results for Rayleigh scattering. J. Quant. Spectrosc. Radiat. Transfer 43, No 2, 163-171.* - Internal radiation field of a planetary atmosphere:

*Stammes P., de Haan JF, and Hovenier JW, 1989: The polarized internal radiation field of a planetary atmosphere. Astron.Astroph, 225, pp.239-259.*

- Results for aerosol scattering (aspherical particles) can be found in the following publication:

*Wauben, W. M. F. and Hovenier, J. W.: Polarized radiation of an atmosphere containing randomly-oriented spheroids, J. Quant. Spectrosc. Radiat. Transfer, 47, 491–504, doi:10.1016/0022-4073(92)90108-G, 1992.* - Two homogeneous layers above a Lambertian surface. Non-zero depolarization factor is assumed for the top (Rayleigh) layer:

*Wauben, W.M.F., J.F. de Haan, and J.W. Hovenier, 1994: A method for computing visible and infrared polarized monochromatic radiation in planetary atmospheres. Astron. Astrophys. 282, 277-290.* - Some results for I and Q for low orders of scattering, as well as sum of all scatterings are given in:

*Wauben WMF, de Haan JF, and Hovenier JW, 1993: Low orders of scattering in plane-parallel homogeneous atmosphere. Astron.Astroph., 276, pp.589-602.* - Two homegeneous layers (aerosol+Rayleigh)

*Haan, J.F. de, P.B. Bosma, and J.W. Hovenier, 1987: The adding method for multiple scattering calcualtions of polarized light. Astron. Astrophys. 183, 371-391.* - Reflected, transmitted and internal light field: Two types of aerosols with L=13 and 60 moments are considered in

*Garcia, R.D.M., C.E. Siewert, 1989: The FN method for radiative transfer models that include polarization effects, . J. Quant. Spectrosc. Radiat. Transfer 41, No 2, 117-145.* - Results for similar scenarios but with smaller number of significant digits are tabulated in:

*Garcia, R.D.M., C.E. Siewert, 1986: A generalized spherical harmonics solution for radiative transfer models that include polarization effects, . J. Quant. Spectrosc. Radiat. Transfer 36, No 5, 401-423.* - One of the two scenarios mentioned above, the L=13 case, is tabulated in:

*Siewert, C. E., 2000: A discrete-ordinates solution for radiative-transfer models that include polarization effects. J. Quant. Spectrosc. Radiat. Transfer 64, 227-254.* - Those working with Fourier decomposition of the light field, might find useful the following paper where results are given for several first Fourier moments separately:

*Vestrucci, P., C.E. Siewert, 1984: A numerical evaluation of analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light. J. Quant. Spectrosc. Radiat. Transfer, 31, No 2, 177-183.*

- Realistic phase matrices with sharp forward scattering peak:

*A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege. Benchmark results in vector atmospheric radiative transfer. J. Quant. Spectrosc. Radiat. Transfer, 111(12-13):1931 - 1946, 2010.*

Benchmark results available at A. Kokhanovsky's homepage.

- Scattering of a completely linear polarized beam, Fo=[1 1 0 0]:

*Mishchenko M.I., 1991: Reflection of polarized light by plane-parallel slabs containig randomly-oriented, nonspherical particles. JQSRT, 46(3), pp.171-181.* - Results for different total optical thicknesses of a layer, different phase matrices, and two types of incident beams, natural Fo=[1 0 0 0] and elliptically polarized Fo=[1 0 0 1]:

*Garcia RDM, Siewert CE, 2011: A simplified implementation of the discrete -ordinates method for a class of problems in radiative transfer with polarization. JQSRT, 112, pp.2801-2813.* *Garcia R, Siewert CE, 1985: Benchmark results in raditive transfer. Transport Theory and Statistical Physics, 14, pp.437-484.**Maiorino J.R., Siewert C.E., 1980: The FN method for poalrization studies - II.Numerical Results. JQSRT, 24, pp.159-165.*

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