## Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media

Applied Optics, Vol. 27, Issue 12, pp. 2502-2509 (1988)

http://dx.doi.org/10.1364/AO.27.002502

Acrobat PDF (936 KB)

### Abstract

We summarize an advanced, thoroughly documented, and quite general purpose discrete ordinate algorithm for time-independent transfer calculations in vertically inhomogeneous, nonisothermal, plane-parallel media. Atmospheric applications ranging from the UV to the radar region of the electromagnetic spectrum are possible. The physical processes included are thermal emission, scattering, absorption, and bidirectional reflection and emissionat the lower boundary. The medium may be forced at the top boundary by parallel or diffuse radiation and by internal and boundary thermal sources as well. We provide a brief account of the theoretical basis as well as a discussion of the numerical implementation of the theory. The recent advances made by ourselves and our collaborators—advances in both formulation and numerical solution—are all incorporated in the algorithm. Prominent among these advances are the complete conquest of two ill-conditioning problems which afflicted all previous discrete ordinate implementations: (1) the computation of eigenvalues and eigenvectors and (2) the inversion of the matrix determining the constants of integration. Copies of the fortran program on microcomputer diskettes are available for interested users.

© 1988 Optical Society of America

## I. Introduction

2. K.-N. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmosphere,” J. Atmos. Sci. **30**, 1303 (1973). [CrossRef]

3. K. Stamnes and R. A. Swanson, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculations in Anisotropically Scattering Atmospheres,” J. Atmos. Sci. **38**, 387 (1981). [CrossRef]

## II. Theory

### A. Basic Equations

11. J. E. Hansen and L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. **16**, 527 (1974). [CrossRef]

15. K. Stamnes, “The Theory of Multiple Scattering of Radiation in Plane Parallel Atmospheres,” Rev. Geophys. **24**, 299 (1986). [CrossRef]

*ν*through a plane-parallel medium is given by [1]

*u*

*(*

_{ν}*τ*

*,*

_{ν}*μ*,

*ϕ*) is the specific intensity along direction

*μ*,

*ϕ*at optical depth

*τ*

*measured perpendicular to the surface of the medium (*

_{ν}*ϕ*is the azimuthal angle, and

*μ*is the cosine of the polar angle).

*S*

*is the source function given by*

_{ν}*ω*

*(*

_{ν}*π*

*) is the single-scattering albedo, and*

_{ν}*P*

*(*

_{ν}*τ*

*,*

_{ν}*μ*,

*ϕ;μ*′,

*ϕ*′) is the phase function. For thermal emission in local thermodynamic equilibrium (LTE), the source term

*Q*

*is*

_{ν}*B*

*(*

_{ν}*T*) is the Planck function at frequency

*ν*and temperature

*T*. If the usual diffuse–direct distinction is made (Ref. [1], p. 22), so that

*u*

*in Eqs. (1) and (2) describes the diffuse radiation only, then for a parallel beam incident in direction*

_{ν}*μ*

_{0},

*ϕ*

_{0}on a nonemitting medium,

*μ*

_{0}

*I*

_{0}is the incident flux. We take

*ν*subscript in the following. By expanding the phase function

*P*(

*τ*, cos

*θ*) in a series of 2

*N*Legendre polynomials and the intensity in a Fourier cosine series, [1], [9]

*N*independent equations (one for each Fourier component)

*P*

*(*

_{l}*cosθ*) is the Legendre polynomial,

*θ*is the angle between the direction vectors before and after scattering. Solutions to Eq. (6) give the azimuthal components, and then Eq. (5) gives the complete azimuthal dependence of the intensity. [When there is no beam source, the sum in Eq. (5) reduces to the

*m*= 0 term, and

*μ*

_{0},

*ϕ*

_{0}are irrelevant.]

### B. Discrete Ordinate Approximation—Matrix Formulation

3. K. Stamnes and R. A. Swanson, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculations in Anisotropically Scattering Atmospheres,” J. Atmos. Sci. **38**, 387 (1981). [CrossRef]

4. K. Stamnes and H. Dale, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculation in Anisotropically Scattering Atmospheres, II: Intensity Computations,” J. Atmos. Sci. **38**, 2696 (1981). [CrossRef]

*μ*

*and*

_{i}*w*

*are quadrature points and weights.*

_{i}*ω*(

*τ*) and the phase function

*P*(

*τ*,

*μ*,

*ϕ;μ*′,

*ϕ*′) are functions of

*τ*in a vertically inhomogeneous medium, Eqs. (7) constitute (for each

*m*) a system of 2

*N*coupled differential equations with nonconstant coefficients for which analytic solutions do not exist. To obtain analytic solutions, the medium is assumed to consist of

*L*adjacent homogeneous layers in which the single-scattering albedo and the phase function are taken to be constant within each layer (but allowed to vary from layer to layer, see Fig. 1), and the thermal source term is approximated by a polynomial in

*τ*. For the time being, we shall consider a single homogeneous layer for which

*τ*

_{p}_{−1}≤

*τ*≤

*τ*

*. The*

_{p}*τ*arguments of

*D*and

*X*will be omitted since

*D*and

*X*are by assumption independent of

*τ*in any one layer. We will also omit the

*p*subscripts, which are implicit on all quantities, until later.

3. K. Stamnes and R. A. Swanson, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculations in Anisotropically Scattering Atmospheres,” J. Atmos. Sci. **38**, 387 (1981). [CrossRef]

16. T. Nakajima and M. Tanaka, “Matrix Formulations for the Transfer of Solar Radiation in a Plane-Parallel Atmosphere,” J. Quant. Spectrosc. Radiat. Transfer **35**, 13 (1986). [CrossRef]

### C. Quadrature Rule

17. W. J. Wiscombe, “The Delta-M Method: Rapid Yet Accurate Radiative Flux Calculations for Strongly Asymmetric Phase Functions,” J. Atmos. Sci. **34**, 1408 (1977). [CrossRef]

*μ*

_{−}

*= −*

_{j}*μ*

*and*

_{j}*w*

_{−}

*=*

_{j}*w*

*. Double-Gauss simply refers to a quadrature rule suggested by Sykes [18] in which the Gaussian formula is applied separately to the half-ranges −1 <*

_{j}*μ*< 0 and 0 <

*μ*< 1. The main advantage of this double-Gauss scheme is that the quadrature points (in even orders) are distributed symmetrically around |

*μ*| = 0.5 and clustered both toward |

*μ*| = 1 and

*μ*= 0, whereas in the Gaussian scheme for the complete range, −1 <

*μ*< 1, they are clustered toward

*μ*= ±1. The clustering toward

*μ*= 0 will give superior results near the boundaries where the intensity varies rapidly around

*μ*= 0. A half-range scheme is also preferred since the intensity is discontinuous at the boundaries. Another advantage is that upward and downward fluxes and average intensities are obtained immediately without further approximations.

### D. Basic Solution

*N*-coupled ordinary differential equations with constant coefficients. These coupled equations are linear, and our goal is to uncouple them by using well-known methods of linear algebra.

*N*× 2

*N*determining the eigenvalues

*k*and the eigenvectors

**G**

^{±}.

*k*), and the order of the algebraic eigenvalue problem [Eq. (8b)] may be reduced by a factor of 2 as follows. [3]

**38**, 387 (1981). [CrossRef]

**38**, 387 (1981). [CrossRef]

**G**

^{+}+

**G**

^{−}). We then use Eq. (8d) to determine (

**G**

^{+}−

**G**

^{−}).

*Q*

^{(beam)}(

*τ*,

*μ*) =

*X*

_{0}(

*μ*) exp(−

*τ*/

*μ*

_{0}) [Eq. (6d)], it is easily verified that a particular solution of Eq. (7) is (omitting

*m*superscript)

*Q*

*= 0,*

^{m}*m*> 0,

*Q*(

*τ*) ≡

*Q*

^{0}(

*τ*) = (1 −

*ω*)

*B*(

*τ*). By approximating the Planck function for each layer by a polynomial in

*τ*, [3]

**38**, 387 (1981). [CrossRef]

19. W. J. Wiscombe, “Extension of the Doubling Method to Inhomogeneous Sources,” J. Quant. Spectrosc. Radiat. Transfer **16**, 477 (1976). [CrossRef]

*τ*:

*Z*

_{0}(

*μ*

*) and*

_{i}*Y*

*(*

_{l}*μ*

*) are readily determined by solving systems of linear algebraic equations (see Refs. [3]*

_{i}**38**, 387 (1981). [CrossRef]

*C*

*, of all the homogeneous solutions plus the particular solutions for beam and thermal emission sources (omitting the*

_{j}*m*superscript):

*k*

*and the*

_{j}*G*

*(*

_{j}*μ*

*) are the eigenvalues and eigenvectors obtained as described in Sec. III.A, the*

_{i}*μ*

*the quadrature angles, and the*

_{i}*C*

*the constants of integration. We have for convenience included the beam solution in the sum by defining for*

_{j}*j*= 0

### E. Boundary Conditions

*u*

_{∞}and

*u*

*are the intensities incident at the top and bottom boundaries, respectively, and*

_{g}*τ*

*is the total optical depth.*

_{L}*∊*(

*μ*) is the directional emissivity and

*T*

*is the temperature of the bottom boundary;*

_{g}*ρ*

_{d}(

*μ*,

*ϕ*;−

*μ*′,

*ϕ*′) is the bidirectional reflectivity, and

*I*

_{0}is the incident beam intensity at the upper boundary. Kirchhoff’s law

*θ*between the incident and the reflected radiation (i.e., there are no preferred directions at the lower boundary). Then it can be expanded in a series of 2

*N*Legendre polynomials:

7. K. Stamnes and P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer **31**, 273 (1984). [CrossRef]

*k*

*> 0 and*

_{jp}*k*

_{−}

*= −*

_{jp}*k*

*):*

_{jp}*C*

*) to be determined, and*

_{jp}*R*

*(*

_{p}*τ*,

*μ*

*) is the particular solution given by [Eq. (10a)]*

_{i}*N*×

*L*) × (2

*N*×

*L*) system of linear algebraic equations from which the 2

*N*×

*L*unknown coefficients, the

*C*

*(*

_{jp}*j*= ±1,…,±

*N; p*= 1,…,

*L*) are determined. To solve this system of equations, we take advantage of the fact that the coefficient matrix is a (6

*N*− 1) diagonal band matrix. [7]

7. K. Stamnes and P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer **31**, 273 (1984). [CrossRef]

## III. Numerical Implementation

### A. Computation of Eigenvalues and Eigenvectors

**38**, 387 (1981). [CrossRef]

*et al*., [8]

8. K. Stamnes, S.-C. Tsay, and T. Nakajima, “Computation of Eigenvaluesand Eigenvectors for Discrete Ordinate and Matrix Operator Method Radiative Transfer,” J. Quant. Spectrosc. Radiat. Transfer in press (1988). [CrossRef]

*α*−

*β*) (

*α*+

*β*) in Eq. (8e) is that only one matrix multiplication is necessary. Nakajima and Tanaka [16]

16. T. Nakajima and M. Tanaka, “Matrix Formulations for the Transfer of Solar Radiation in a Plane-Parallel Atmosphere,” J. Quant. Spectrosc. Radiat. Transfer **35**, 13 (1986). [CrossRef]

*et al*. [8]

8. K. Stamnes, S.-C. Tsay, and T. Nakajima, “Computation of Eigenvaluesand Eigenvectors for Discrete Ordinate and Matrix Operator Method Radiative Transfer,” J. Quant. Spectrosc. Radiat. Transfer in press (1988). [CrossRef]

8. K. Stamnes, S.-C. Tsay, and T. Nakajima, “Computation of Eigenvaluesand Eigenvectors for Discrete Ordinate and Matrix Operator Method Radiative Transfer,” J. Quant. Spectrosc. Radiat. Transfer in press (1988). [CrossRef]

16. T. Nakajima and M. Tanaka, “Matrix Formulations for the Transfer of Solar Radiation in a Plane-Parallel Atmosphere,” J. Quant. Spectrosc. Radiat. Transfer **35**, 13 (1986). [CrossRef]

*et al*. [8]

- In single precision the Stamnes/Swanson procedure is more accurate than the Cholesky procedure, which in turn is slightly more accurate than the Nakajima/Tanaka procedure.
- The Cholesky procedure is slightly faster than Nakajima/Tanaka’s, which is slightly faster than Stamnes/Swanson’s.

### B. Scaling Transformation

7. K. Stamnes and P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer **31**, 273 (1984). [CrossRef]

*k*

*> 0 by convention). This is achieved by the scaling transformation [7]*

_{jp}**31**, 273 (1984). [CrossRef]

*C*

*, we find that all the exponential terms in the coefficient matrix have the form exp[−*

_{jp}*k*

*(*

_{jp}*τ*

*−*

_{p}*τ*

_{p}_{−1})];i.e., all the exponentials in the coefficient matrix have negative arguments (

*k*

*> 0,*

_{jp}*τ*

*>*

_{p}*τ*

_{p}_{−1}). Consequently, numerical ill-conditioning is avoided implying that our numerical scheme for finding the

### C. Scaled Solutions—Angular Distributions

*k*

*> 0 and*

_{jp}*τ*

_{p}_{−1}≤

*τ*≤

*τ*

*, all exponentials in Eq. (23) have negative arguments as they should to avoid fatal overflows.*

_{p}*m*superscript. These two equations show that if we know the source function

*S*(

*t*,

*±μ*), we can find the intensity at arbitrary angles by integrating the source function. The discrete ordinate solutions are used to derive explicit expressions for the source function, which can be integrated analytically. [3]

**38**, 387 (1981). [CrossRef]

5. K. Stamnes, “On the Computation of Angular Distributions of Radiation in Planetary Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer **28**, 47 (1982a). [CrossRef]

5. K. Stamnes, “On the Computation of Angular Distributions of Radiation in Planetary Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer **28**, 47 (1982a). [CrossRef]

5. K. Stamnes, “On the Computation of Angular Distributions of Radiation in Planetary Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer **28**, 47 (1982a). [CrossRef]

*τ*and

*μ*:

*n*>

*p*and

*n*<

*p*and

*G*

*terms and*

_{jn}*n*subscript):

*b*

_{0}is defined in Eq. (9b). Formulas (27a)–(27c) clearly reveal the interpolatory nature of Eqs. (25) and (26). Although we have generally omitted the

*m*superscript above, we have explicitly written

*D*

^{0}in Eq. (27c) to indicate that thermal radiation contributes only to the azimuth-independent part of the intensity.

### D. Computational Shortcut for Highly Absorbing Media

### E. Simplified Albedo and Transmissivity Computations

*a*(

*μ*) and the transmissivity

*t*(

*μ*) of a vertically inhomogeneous plane-parallel medium lacking thermal sources. Thus

- the plane albedo for a given angle of parallel incidence equals the azimuthally averaged reflected intensity
*u*^{0}(0,*μ*) for isotropic illumination of unit intensity incident at the top boundary; - the transmissivity equals the azimuthally averaged transmitted intensity
*u*^{0}*(0,*μ*) for isotropic illumination of unit intensity incident at the bottom boundary.

6. K. Stamnes, “Reflection and Transmission by a Vertically Inhomogeneous Planetary Atmosphere,” Planet. Space Sci. **30**, 727 (1982b). [CrossRef]

*τ*

*is to the optical thickness of the inhomogeneous medium, and the asterisk refers to illumination from below.*

_{L}6. K. Stamnes, “Reflection and Transmission by a Vertically Inhomogeneous Planetary Atmosphere,” Planet. Space Sci. **30**, 727 (1982b). [CrossRef]

## IV. Summary

- It is unconditionally stable for an arbitrarily large number of quadrature angles (streams) and arbitrarily large optical depths.
- It can be forced by any combination of parallel beam or diffuse incidence and thermal sources in the medium and at the boundaries.
- It allows for an arbitrary bidirectional reflectivity at the lower boundary; directional emissivity is computed from this reflectivity.
- It offers rapid computation of bulk albedo and transmissivity when thermal sources are absent.
- Unlike the popular doubling method, computing time for individual layers is independent of optical thickness.
- Because the solution is analytic, and by using the iteration-of-the-source-function method, intensities can be returned at arbitrary angles and optical depths, unrelated to the computational meshes for these quantities.
- It has been thoroughly tested against a wide variety of published solutions.
- It is thoroughly documented both in Ref. [9] and in the code itself (with extensive references to published equation numbers).

## References

1. | S. Chandrasekhar, |

2. | K.-N. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmosphere,” J. Atmos. Sci. |

3. | K. Stamnes and R. A. Swanson, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculations in Anisotropically Scattering Atmospheres,” J. Atmos. Sci. |

4. | K. Stamnes and H. Dale, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculation in Anisotropically Scattering Atmospheres, II: Intensity Computations,” J. Atmos. Sci. |

5. | K. Stamnes, “On the Computation of Angular Distributions of Radiation in Planetary Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer |

6. | K. Stamnes, “Reflection and Transmission by a Vertically Inhomogeneous Planetary Atmosphere,” Planet. Space Sci. |

7. | K. Stamnes and P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer |

8. | K. Stamnes, S.-C. Tsay, and T. Nakajima, “Computation of Eigenvaluesand Eigenvectors for Discrete Ordinate and Matrix Operator Method Radiative Transfer,” J. Quant. Spectrosc. Radiat. Transfer in press (1988). [CrossRef] |

9. | K. Stamnes, S.-C. Tsay, W. J. Wiscombe, and K. Jayaweera, “An Improved, Numerically Stable Computer Code for Discrete-Ordinate-Method Radiative Transfer in Scattering and Emitting Layered Meda,” NASA Report, in press (1988). |

10. | K. N. Liou, |

11. | J. E. Hansen and L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. |

12. | W. M. Irvine, “Multiple Scattering in Planetary Atmospheres,” Icarus |

13. | J. Lenoble, Ed., |

14. | W. J. Wiscombe, “Atmospheric Radiation: 1975–1983,” Rev. Geophys. |

15. | K. Stamnes, “The Theory of Multiple Scattering of Radiation in Plane Parallel Atmospheres,” Rev. Geophys. |

16. | T. Nakajima and M. Tanaka, “Matrix Formulations for the Transfer of Solar Radiation in a Plane-Parallel Atmosphere,” J. Quant. Spectrosc. Radiat. Transfer |

17. | W. J. Wiscombe, “The Delta-M Method: Rapid Yet Accurate Radiative Flux Calculations for Strongly Asymmetric Phase Functions,” J. Atmos. Sci. |

18. | J. B. Sykes, “Approximate Integration of the Equation of Transfer,” Mon. Not. Roy. Astron. Soc. |

19. | W. J. Wiscombe, “Extension of the Doubling Method to Inhomogeneous Sources,” J. Quant. Spectrosc. Radiat. Transfer |

20. | W. R. Cowell, Ed., |

21. | J. J. Dongarra, C. B. Moler, J. R. Bunch, and G. W. Stewart, |

22. | V. Kourganoff, |

**History**

Original Manuscript: August 7, 1987

Published: June 15, 1988

**Citation**

Knut Stamnes, S-Chee Tsay, Warren Wiscombe, and Kolf Jayaweera, "Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media," Appl. Opt. **27**, 2502-2509 (1988)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-27-12-2502

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### References

- S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
- K.-N. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmosphere,” J. Atmos. Sci. 30, 1303 (1973). [CrossRef]
- K. Stamnes, R. A. Swanson, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculations in Anisotropically Scattering Atmospheres,” J. Atmos. Sci. 38, 387 (1981). [CrossRef]
- K. Stamnes, H. Dale, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculation in Anisotropically Scattering Atmospheres, II: Intensity Computations,” J. Atmos. Sci. 38, 2696 (1981). [CrossRef]
- K. Stamnes, “On the Computation of Angular Distributions of Radiation in Planetary Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 28, 47 (1982a). [CrossRef]
- K. Stamnes, “Reflection and Transmission by a Vertically Inhomogeneous Planetary Atmosphere,” Planet. Space Sci. 30, 727 (1982b). [CrossRef]
- K. Stamnes, P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 31, 273 (1984). [CrossRef]
- K. Stamnes, S.-C. Tsay, T. Nakajima, “Computation of Eigenvaluesand Eigenvectors for Discrete Ordinate and Matrix Operator Method Radiative Transfer,” J. Quant. Spectrosc. Radiat. Transfer in press (1988). [CrossRef]
- K. Stamnes, S.-C. Tsay, W. J. Wiscombe, K. Jayaweera, “An Improved, Numerically Stable Computer Code for Discrete-Ordinate-Method Radiative Transfer in Scattering and Emitting Layered Meda,” NASA Report, in press (1988).
- K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Orlando, FL, 1980).
- J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527 (1974). [CrossRef]
- W. M. Irvine, “Multiple Scattering in Planetary Atmospheres,” Icarus 25, 175 (1975). [CrossRef]
- J. Lenoble, Ed., Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (A. Deepak, Hampton, VA, 1985).
- W. J. Wiscombe, “Atmospheric Radiation: 1975–1983,” Rev. Geophys. 21, 957 (1983). [CrossRef]
- K. Stamnes, “The Theory of Multiple Scattering of Radiation in Plane Parallel Atmospheres,” Rev. Geophys. 24, 299 (1986). [CrossRef]
- T. Nakajima, M. Tanaka, “Matrix Formulations for the Transfer of Solar Radiation in a Plane-Parallel Atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 35, 13 (1986). [CrossRef]
- W. J. Wiscombe, “The Delta-M Method: Rapid Yet Accurate Radiative Flux Calculations for Strongly Asymmetric Phase Functions,” J. Atmos. Sci. 34, 1408 (1977). [CrossRef]
- J. B. Sykes, “Approximate Integration of the Equation of Transfer,” Mon. Not. Roy. Astron. Soc. 11, 377 (1951).
- W. J. Wiscombe, “Extension of the Doubling Method to Inhomogeneous Sources,” J. Quant. Spectrosc. Radiat. Transfer 16, 477 (1976). [CrossRef]
- W. R. Cowell, Ed., Sources and Developments of Mathematical Software (Prentice Hall, Englewood Cliffs, NJ, 1980).
- J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, linpack User’s Guide (SIAM, Philadelphia, 1979).
- V. Kourganoff, Basic Methods in Transfer Problems (Dover, New York, 1963).

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