Content

Abstract:

An efficiency of the singular value decomposition (SVD) method and ordinary differential equation (ODE) solvers in finding the reflection matrix are compared. A reflection matrix can be found by solving the one-dimensional radiative transfer equation. The latter’s solution based on the discrete ordinate method leads to the singular value decomposition (SVD) method. Alternative approach consists in transforming the original problem into a matrix Riccati equation written specifically for the reflection matrix. The matrix Riccati equation is solved using numerical integration techniques for ordinary differential equations (ODEs). It is found that for a single layer case, the SVD approach is faster than the ODE solvers by an order of magnitude. Yet as the number of layers increases, the ODE solvers become more efficient than the SVD approach. In addition, they outperform the SVD method when a solution for a set of optical thicknesses of the medium should be found or when retrieval of optical thickness should be performed. The comparison between different ODE solvers is performed, as well.
References:

1. Twomey, S., Jacobowitz, H. and Howell, H.B. Matrix Methods for Multiple-Scattering Problems // J. Atmos. Sci. 1966, 23(3), pp. 289–298.2. Sobolev, V. V. and ter Haar, D. Light Scattering in Planetary Atmospheres / Pergamon Press, 1975.

3. Hansen, James E., Travis Larry D. Light scattering in planetary atmospheres / Space Science Reviews, 1974, 16(4), pp. 527–610.

4. Irvine, W.M. Multiple scattering in planetary atmospheres // Icarus, 1975, 25(2), pp. 175–204.

5. Lenoble, J. Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures of Studies in geophysical optics and remote sensing. A. Deepak, 1985.

6. Budak, V.P. and Klyuykov, D.A. and Korkin, S.V. Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering. In Kokhanovsky, A A, editors, Light scattering reviews, pages 147–203. Springer Berlin Heidelberg, 2010.

7. Afanas’ev, V. P. and Efremenko, D. S. and Lubenchenko, A.V. On the application of the invariant embedding method and the radiative transfer equation codes for surface state analysis. In Kokhanovsky, A.A., editors, Light Scattering Reviews 8, pages 363–423. Springer Science + Business Media, 2013.

8. Rozanov, V.V. and Rozanov, A.V. and Kokhanovsky, A.A. and Burrows, J.P. Radiative transfer through terrestrial atmosphere and ocean: Software package SCIATRAN. Journal of Quantitative Spectroscopy and Radiative Transfer, 133:13–71, 2014.

9. Rogovtsov, N.N. and Borovik, F. Application of general invariance relations reduction method to solution of radiation transfer problems. Journal of Quantitative Spectroscopy and Radiative Transfer, 183:128–153, 2016.

10. Ganapol, Barry D. and Patel, J. Matrix Riccati Equation Solution of the 1D Radiative Transfer Equation //Journal of Computational and Theoretical Transport, 2020, 50(5), pp. 297–327.

11. Wick, G.C. Über ebene Diffusionsprobleme // Zeitschrift für Physik, 1943, 121(11–12), pp. 702–718.

12. Chandrasekhar, S. Radiative Transfer / Dover Publications, Inc., New York, 1950.

13. Stamnes, K. The theory of multiple scattering of radiation in plane parallel atmospheres // Rev. Geophys. 1986, 24(2):299.

14. Waterman, P.C. Matrix-exponential description of radiative transfer // Journal of the Optical Society of America,1981, 71(4):410.

15. Efremenko, Dmitry and Doicu, Adrian and Loyola, Diego and Trautmann, Thomas. Acceleration techniques for the discrete ordinate method // Journal of Quantitative Spectroscopy and Radiative Transfer, 2013, 114, pp. 73–81.

16. Efremenko, Dmitry S. and Loyola, Diego G. and Doicu, Adrian and Spurr, Robert J D. Multi-core-CPU and GPU-accelerated radiative transfer models based on the discrete ordinate method // Computer Physics Communications, 2014, 185(12), pp. 3079–3089.

17. Efremenko, D S and Loyola, D and Spurr, R J D and Doicu, A. Acceleration of radiative transfer model calculations for the retrieval of trace gases under cloudy conditions // J. Quant Spectrosc Radiat Transfer, 2014, 35, pp. 58–65.

18. Ambarzumian, V.A. Diffuse reflection of light by a foggy medium // Dokl. Akad. Nauk SSSR, 1943, 38, pp. 229–232.

19. Bellman, R. and Kalaba, R. and Wing, G.M. Invariant imbedding and mathematical physics-I: Particle processes // J. Math. Phys. 1960, 1, pp. 280–308.

20. Flatau, Piotr J. and Stephens, Graeme L. On the fundamental solution of the radiative transfer equation // J. Geophys. Res. 1988, 93(D9):11037.

21. Nakajima, T and Tanaka, M. Matrix formulations for the transfer of solar radiation in a plane-parallel scattering atmosphere // J. Quant Spectrosc Radiat Transfer, 1986, 35(1), pp. 13–21.

22. Efremenko, D.S. and Kokhanovsky, A. Foundations of Atmospheric Remote Sensing / Springer, 2021.

23. Budak, V.P., Efremenko, D.S, and Shagalov, O.V. Efficiency of algorithm for solution of vector radiative transfer equation in turbid medium slab // J. Phys.: Conf. Ser., 369:012021, 2012.

24. Afanas’ev, Viktor P., Basov, Alexander Yu., Budak, V.P., Efremenko D.S., and Kokhanovsky, A.A. Analysis of the Discrete Theory of Radiative Transfer in the Coupled “Ocean–Atmosphere” System: Current Status, Problems and Development Prospects // Journal of Marine Science and Engineering, 8(3):202, 2020.

25. Dormand, J.R. and Prince, P.J. A family of embedded Runge-Kutta formulae // Journal of Computational and Applied Mathematics, 1980, 6, pp. 19–26.

26. Bogacki, P. and Shampine, L.F. A 3(2) pair of Runge – Kutta formulas // Applied Mathematics Letters, 1989, 2(4), pp. 321–325.

27. E. Hairer, G. Wanner and Norsett, S.P. Solving Ordinary Differential Equations I: Nonstiff problems / Springer Berlin Heidelberg, 1993.

28. Shampine, L. F., Reichelt, M. W. THE MATLAB ODE SUITE // SIAM J. SCI. COMPUTE, 18:1–22, 1997.

29. Petzold, L. Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations // SIAM Journal on Scientific and Statistical Computing, 4(1), pp. 136–148, 1983.

30. Afanas’ev, V.P. and Efremenko, D.S. and Kaplya, P.S. Analytical and numerical methods for computing electron partial intensities in the case of multilayer systems // Journal of Electron Spectroscopy and Related Phenomena, 2016, 210, pp. 16–29.

31. Wiscombe, W.J. The Delta–M Method: Rapid Yet Accurate Radiative Flux Calculations for Strongly Asymmetric Phase Functions // Journal of the Atmospheric Sciences, 1977, 34, pp. 1408–1422.

32 Minzheng Duan, Qilong Min, and Stamnes, K. Impact of vertical stratification of inherent optical properties on radiative transfer in a plane-parallel turbid medium // Opt. Express, 2010, 18, pp. 5629–5638.

33. Yi-Ning Shi, Feng Zhang, Ka Lok Chan, Thomas Trautmann, and Jiangnan Li. Multi-layer solar radiative transfer considering the vertical variation of inherent microphysical properties of clouds // Opt. Express, 2019, 27, pp. A1569-A1590.

Keywords

Recommended articles

Discrete Ordinate Radiative Transfer Model With the Neural Network Based Eigenvalue Solver: proof Of Concept Light & Engineering Vol. 29, No. 1

Illumination Correction of Multi-Time RGB Images Obtained with an Unmanned Aerial Vehicle L&E, Vol. 29, No. 2, 2021

In-Terrestrial Aquaculture Fields Mapping from High Resolution Remote Sensing Images L&E, Vol.31, No.5, 2023