Abstract

An entirely rigorous method for the solution of the equations for radiative transfer based on the matrix operator theory is reviewed. The advantages of the present method are: (1) all orders of the reflection and transmission matrices are calculated at once; (2) layers of any thickness may be combined, so that a realistic model of the atmosphere can be developed from any arbitrary number of layers, each with different properties and thicknesses; (3) calculations can readily be made for large optical depths and with highly anisotropic phase functions; (4) results are obtained for any desired value of the surface albedo including the value unity and for a large number of polar and azimuthal angles including the polar angle θ = 0°; (5) all fundamental equations can be interpreted immediately in terms of the physical interactions appropriate to the problem; (6) both upward and downward radiance can be calculated at interior points from relatively simple expressions. Both the general theory and its history together with the method of calculation are discussed. As a first example of the method numerous curves are given for both the reflected and transmitted radiance for Rayleigh scattering from a homogeneous layer for a range of optical thicknesses from 0.0019 to 4096, surface albedo A = 0, 0.2, and 1, and cosine of solar zenith angle μ = 1, 0.5397, and 0.1882. It is shown that the matrix operator approach contains the doubling method as a special case.

© 1973 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1950).
  2. Z. Sekera, Adv. Geophys. 3, 43 (1956).
    [Crossref]
  3. Z. Sekera, “Polarization of Skylight,” in Handbuch der Physik, S. Flugge, Ed. (Springer, Berlin, 1957), Vol. 48, p. 288–328.
    [Crossref]
  4. V. Kourganoff, Basic Methods in Transfer Problems (Clarendon, Oxford, 1952).
  5. K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960).
  6. B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1970).
    [Crossref]
  7. J. V. Dave, P. M. Furukawa, J. Opt. Soc. Am. 56, 394 (1966).
    [Crossref]
  8. A. B. Kahle, J. Geophy. Res. 73, 7511 (1968).
    [Crossref]
  9. A. B. Kahle, Astrophy. J. 151, 637 (1968).
    [Crossref]
  10. H. B. Howell, H. Jacobowitz, J. Atmos. Sci. 27, 1195 (1970).
    [Crossref]
  11. A. L. Fymat, K. D. Abhyankar, J. Geophy. Res. 76, 732 (1971).
    [Crossref]
  12. B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1970).
    [Crossref]
  13. S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
    [Crossref]
  14. J. V. Dave, J. Gazdag, Appl. Opt. 9, 1457 (1970).
    [Crossref] [PubMed]
  15. J. V. Dave, Appl. Opt. 9, 2673 (1970).
    [Crossref] [PubMed]
  16. J. E. Hansen, Astrophy. J. 155, 565 (1969).
    [Crossref]
  17. J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
    [Crossref]
  18. J. E. Hansen, J. Atmos. Sci. 28, 120 (1971).
    [Crossref]
  19. J. E. Hansen, J. Atmos. Sci. 28, 1400 (1971).
    [Crossref]
  20. J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
    [Crossref]
  21. G. G. Stokes, Proc. Roy. Soc. (London) 11, 545 (1862).
  22. J. A. McClelland, Royal Dublin Soc. Scientific Trans. (2) 9, 9 (1906).
  23. H. W. Schmidt, Ann. der Phys. 23, 671 (1907).
    [Crossref]
  24. R. Bellman, R. Kalaba, G. M. Wing, J. Math. Phys. 1, 280 (1960).
    [Crossref]
  25. V. A. Ambarzumian, Dokl. Akad. Nauk SSSR 38, 257 (1943).
  26. R. Redheffer, J. Math. Phys. 41, 1 (1962).
  27. R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, New York, 1965).
  28. H. C. Van de Hulst, New Look at Multiple Scattering, Tech. Rept. (Goddard Institute for Space Studies, NASA, New York, 1963), 81 pp.
  29. S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
    [Crossref]
  30. I. P. Grant, G. E. Hunt, Proc. Roy. Soc. (London) A313, 183 (1969).
  31. R. Bellman, Introduction to Matrix Analysis (McGraw Hill, New York, 1960).
  32. I. P. Grant, G. E. Hunt, Proc. Roy. Soc. (London) A313, 199 (1969).
  33. A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice Hill, Englewood Cliffs, N.J., 1966).
  34. H. C. Van de Hulst, J. Comput. Phys. 3, 291 (1968).
    [Crossref]

1971 (3)

A. L. Fymat, K. D. Abhyankar, J. Geophy. Res. 76, 732 (1971).
[Crossref]

J. E. Hansen, J. Atmos. Sci. 28, 120 (1971).
[Crossref]

J. E. Hansen, J. Atmos. Sci. 28, 1400 (1971).
[Crossref]

1970 (6)

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[Crossref]

J. V. Dave, J. Gazdag, Appl. Opt. 9, 1457 (1970).
[Crossref] [PubMed]

J. V. Dave, Appl. Opt. 9, 2673 (1970).
[Crossref] [PubMed]

B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1970).
[Crossref]

H. B. Howell, H. Jacobowitz, J. Atmos. Sci. 27, 1195 (1970).
[Crossref]

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1970).
[Crossref]

1969 (4)

J. E. Hansen, Astrophy. J. 155, 565 (1969).
[Crossref]

J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
[Crossref]

I. P. Grant, G. E. Hunt, Proc. Roy. Soc. (London) A313, 183 (1969).

I. P. Grant, G. E. Hunt, Proc. Roy. Soc. (London) A313, 199 (1969).

1968 (3)

H. C. Van de Hulst, J. Comput. Phys. 3, 291 (1968).
[Crossref]

A. B. Kahle, J. Geophy. Res. 73, 7511 (1968).
[Crossref]

A. B. Kahle, Astrophy. J. 151, 637 (1968).
[Crossref]

1967 (1)

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[Crossref]

1966 (2)

J. V. Dave, P. M. Furukawa, J. Opt. Soc. Am. 56, 394 (1966).
[Crossref]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
[Crossref]

1962 (1)

R. Redheffer, J. Math. Phys. 41, 1 (1962).

1960 (1)

R. Bellman, R. Kalaba, G. M. Wing, J. Math. Phys. 1, 280 (1960).
[Crossref]

1956 (1)

Z. Sekera, Adv. Geophys. 3, 43 (1956).
[Crossref]

1943 (1)

V. A. Ambarzumian, Dokl. Akad. Nauk SSSR 38, 257 (1943).

1907 (1)

H. W. Schmidt, Ann. der Phys. 23, 671 (1907).
[Crossref]

1906 (1)

J. A. McClelland, Royal Dublin Soc. Scientific Trans. (2) 9, 9 (1906).

1862 (1)

G. G. Stokes, Proc. Roy. Soc. (London) 11, 545 (1862).

Abhyankar, K. D.

A. L. Fymat, K. D. Abhyankar, J. Geophy. Res. 76, 732 (1971).
[Crossref]

Ambarzumian, V. A.

V. A. Ambarzumian, Dokl. Akad. Nauk SSSR 38, 257 (1943).

Bellman, R.

R. Bellman, R. Kalaba, G. M. Wing, J. Math. Phys. 1, 280 (1960).
[Crossref]

R. Bellman, Introduction to Matrix Analysis (McGraw Hill, New York, 1960).

Browning, S. R.

B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1970).
[Crossref]

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1970).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1950).

Coulson, K. L.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960).

Curran, R. J.

B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1970).
[Crossref]

Dave, J. V.

J. V. Dave, J. Gazdag, Appl. Opt. 9, 1457 (1970).
[Crossref] [PubMed]

J. V. Dave, Appl. Opt. 9, 2673 (1970).
[Crossref] [PubMed]

J. V. Dave, P. M. Furukawa, J. Opt. Soc. Am. 56, 394 (1966).
[Crossref]

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960).

Furukawa, P. M.

Fymat, A. L.

A. L. Fymat, K. D. Abhyankar, J. Geophy. Res. 76, 732 (1971).
[Crossref]

Gazdag, J.

Grant, I. P.

I. P. Grant, G. E. Hunt, Proc. Roy. Soc. (London) A313, 199 (1969).

I. P. Grant, G. E. Hunt, Proc. Roy. Soc. (London) A313, 183 (1969).

Hansen, J. E.

J. E. Hansen, J. Atmos. Sci. 28, 120 (1971).
[Crossref]

J. E. Hansen, J. Atmos. Sci. 28, 1400 (1971).
[Crossref]

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[Crossref]

J. E. Hansen, Astrophy. J. 155, 565 (1969).
[Crossref]

J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
[Crossref]

Herman, B. M.

B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1970).
[Crossref]

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1970).
[Crossref]

Howell, H. B.

H. B. Howell, H. Jacobowitz, J. Atmos. Sci. 27, 1195 (1970).
[Crossref]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[Crossref]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
[Crossref]

Hunt, G. E.

I. P. Grant, G. E. Hunt, Proc. Roy. Soc. (London) A313, 183 (1969).

I. P. Grant, G. E. Hunt, Proc. Roy. Soc. (London) A313, 199 (1969).

Jacobowitz, H.

H. B. Howell, H. Jacobowitz, J. Atmos. Sci. 27, 1195 (1970).
[Crossref]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[Crossref]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
[Crossref]

Kahle, A. B.

A. B. Kahle, J. Geophy. Res. 73, 7511 (1968).
[Crossref]

A. B. Kahle, Astrophy. J. 151, 637 (1968).
[Crossref]

Kalaba, R.

R. Bellman, R. Kalaba, G. M. Wing, J. Math. Phys. 1, 280 (1960).
[Crossref]

Kourganoff, V.

V. Kourganoff, Basic Methods in Transfer Problems (Clarendon, Oxford, 1952).

McClelland, J. A.

J. A. McClelland, Royal Dublin Soc. Scientific Trans. (2) 9, 9 (1906).

Pollack, J. B.

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[Crossref]

Preisendorfer, R. W.

R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, New York, 1965).

Redheffer, R.

R. Redheffer, J. Math. Phys. 41, 1 (1962).

Schmidt, H. W.

H. W. Schmidt, Ann. der Phys. 23, 671 (1907).
[Crossref]

Secrest, D.

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice Hill, Englewood Cliffs, N.J., 1966).

Sekera, Z.

Z. Sekera, Adv. Geophys. 3, 43 (1956).
[Crossref]

Z. Sekera, “Polarization of Skylight,” in Handbuch der Physik, S. Flugge, Ed. (Springer, Berlin, 1957), Vol. 48, p. 288–328.
[Crossref]

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960).

Stokes, G. G.

G. G. Stokes, Proc. Roy. Soc. (London) 11, 545 (1862).

Stroud, A. H.

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice Hill, Englewood Cliffs, N.J., 1966).

Twomey, S.

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[Crossref]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
[Crossref]

Van de Hulst, H. C.

H. C. Van de Hulst, J. Comput. Phys. 3, 291 (1968).
[Crossref]

H. C. Van de Hulst, New Look at Multiple Scattering, Tech. Rept. (Goddard Institute for Space Studies, NASA, New York, 1963), 81 pp.

Wing, G. M.

R. Bellman, R. Kalaba, G. M. Wing, J. Math. Phys. 1, 280 (1960).
[Crossref]

Adv. Geophys. (1)

Z. Sekera, Adv. Geophys. 3, 43 (1956).
[Crossref]

Ann. der Phys. (1)

H. W. Schmidt, Ann. der Phys. 23, 671 (1907).
[Crossref]

Appl. Opt. (2)

Astrophy. J. (2)

J. E. Hansen, Astrophy. J. 155, 565 (1969).
[Crossref]

A. B. Kahle, Astrophy. J. 151, 637 (1968).
[Crossref]

Dokl. Akad. Nauk SSSR (1)

V. A. Ambarzumian, Dokl. Akad. Nauk SSSR 38, 257 (1943).

J. Atmos. Sci. (9)

H. B. Howell, H. Jacobowitz, J. Atmos. Sci. 27, 1195 (1970).
[Crossref]

B. M. Herman, S. R. Browning, R. J. Curran, J. Atmos. Sci. 28, 419 (1970).
[Crossref]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[Crossref]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
[Crossref]

J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
[Crossref]

J. E. Hansen, J. Atmos. Sci. 28, 120 (1971).
[Crossref]

J. E. Hansen, J. Atmos. Sci. 28, 1400 (1971).
[Crossref]

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[Crossref]

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1970).
[Crossref]

J. Comput. Phys. (1)

H. C. Van de Hulst, J. Comput. Phys. 3, 291 (1968).
[Crossref]

J. Geophy. Res. (2)

A. L. Fymat, K. D. Abhyankar, J. Geophy. Res. 76, 732 (1971).
[Crossref]

A. B. Kahle, J. Geophy. Res. 73, 7511 (1968).
[Crossref]

J. Math. Phys. (2)

R. Redheffer, J. Math. Phys. 41, 1 (1962).

R. Bellman, R. Kalaba, G. M. Wing, J. Math. Phys. 1, 280 (1960).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. Roy. Soc. (London) (3)

G. G. Stokes, Proc. Roy. Soc. (London) 11, 545 (1862).

I. P. Grant, G. E. Hunt, Proc. Roy. Soc. (London) A313, 183 (1969).

I. P. Grant, G. E. Hunt, Proc. Roy. Soc. (London) A313, 199 (1969).

Royal Dublin Soc. Scientific Trans. (2) (1)

J. A. McClelland, Royal Dublin Soc. Scientific Trans. (2) 9, 9 (1906).

Other (8)

S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1950).

Z. Sekera, “Polarization of Skylight,” in Handbuch der Physik, S. Flugge, Ed. (Springer, Berlin, 1957), Vol. 48, p. 288–328.
[Crossref]

V. Kourganoff, Basic Methods in Transfer Problems (Clarendon, Oxford, 1952).

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Los Angeles, 1960).

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice Hill, Englewood Cliffs, N.J., 1966).

R. Bellman, Introduction to Matrix Analysis (McGraw Hill, New York, 1960).

R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, New York, 1965).

H. C. Van de Hulst, New Look at Multiple Scattering, Tech. Rept. (Goddard Institute for Space Studies, NASA, New York, 1963), 81 pp.

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Figures (13)

Fig. 1
Fig. 1

Graphic representation of matrix operator vectors.

Fig. 2
Fig. 2

Upward radiance at the top of atmosphere for Rayleigh scattering for μ0 = 1 and various values of the optical thickness τ as a function of the cosine of the nadir angle μ. (a) A = 0; (b) A = 0.2. The limiting curve for large τ is the same as the last plotted curve within the width of the symbols.

Fig. 3
Fig. 3

(a) Upward radiance for μ0 = 1 and A = 1; (b) μ0 = 0.1882, A = 0, and ϕ = 90°. See Fig. 2 for key.

Fig. 4
Fig. 4

Upward radiance for μ0 = 0.1882, A = 0 and ϕ = 0° and 180°. See Fig. 2 for key.

Fig. 5
Fig. 5

Upward radiance for μ0 = 0.1882, A = 0.2, and ϕ = 0° and 180°. See Fig. 2 for key.

Fig. 6
Fig. 6

(a) Upward radiance for μ0 = 0.1882, A = 0.2, and ϕ = 90°; (b) μ0 = 0.5379 and 0.1882, A = 1, and ϕ = 90°. See Fig. 2 for key.

Fig. 7
Fig. 7

Upward radiance for μ0 = 0.5379 and 0.1882, A = 1, and ϕ = 0° and 180°. See Fig. 2 for key.

Fig. 8
Fig. 8

(a) Downward radiance at bottom of atmosphere for μ0 = 1 and A = 0; (b) μ0 = 1 and A = 0.2. See Fig. 2 for key.

Fig. 9
Fig. 9

(a) Downward radiance for μ0 = 1 and A = 1; (b) μ0 = 0.1882, A = 0 and ϕ = 90°. See Fig. 2 for key.

Fig. 10
Fig. 10

Downward radiance for μ0 = 0.5379, A = 0, and ϕ = 0° and 180°. See Fig. 2 for key.

Fig. 11
Fig. 11

Downward radiance for μ0 = 0.1882, A = 0 and ϕ = 0° and 180°. See Fig. 2 for key.

Fig. 12
Fig. 12

(a) Downward radiance for μ0 = 0.1882, A = 0.2, and ϕ = 90°; (b) μ0 = 0.1882, A = 1, and ϕ = 0° and 180°. See Fig. 2 for key.

Fig. 13
Fig. 13

Downward radiance for μ0 = 0.1882, A = 1, and ϕ = 0° and 180°. See Fig. 2 for key.

Tables (1)

Tables Icon

Table I Upward Radiance

Equations (60)

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I + ( τ ) = [ I + ( τ , μ 1 ) I + ( τ , μ 2 ) I + ( τ , μ m ) ] ,
I 1 + = t 01 I 0 + + r 10 I 1 - + J 01 + ,
I 0 - = r 01 I 0 + + t 10 I 1 - + J 10 - ,
I 2 + = t 12 I 1 + + r 21 I 2 - + J 12 + ,
I 1 - = r 12 I 1 + + t 21 I 2 - + J 21 - ,
I 2 + = t 02 I 0 + + r 20 I 2 - + J 02 + ,
I 0 - = r 02 I 0 + + t 20 I 2 - + J 20 - ,
I 1 - = ( E - r 12 r 10 ) - 1 ( t 21 I 2 - + r 12 t 01 I 01 I 0 + + J 21 - + r 12 J 01 + ) .
I 1 + = ( E - r 10 r 12 ) - 1 ( t 01 I 0 + + r 10 t 21 I 2 - + J 01 + + r 10 J 21 - ) .
t 02 = t 12 ( E - r 10 r 12 ) - 1 t 01 ,
r 20 = r 21 + t 12 ( E - r 10 r 12 ) - 1 r 10 t 21 ,
J 02 + = J 12 + + t 12 ( E - r 10 r 12 ) - 1 ( J 01 + + r 10 J 21 - ) .
t 20 = t 10 ( E - r 12 r 10 ) - 1 t 21 ,
r 02 = r 01 + t 10 ( E - r 12 r 10 ) - 1 r 12 t 01 ,
J 20 - = J 10 - + t 10 ( E - r 12 r 10 ) - 1 ( J 21 - r 12 J 01 + ) .
( E - r 10 r 12 ) - 1 = k = 0 ( r 10 r 12 ) k
r 20 = r 21 + t 12 k = 0 ( r 10 r 12 ) k r 10 t 21 .
I 1 - = r G I 1 + .
( μ d d τ + 1 ) I ( τ ; μ , ϕ ) = F exp ( - τ / μ 0 ) ω ( τ ) p ( τ ; μ , ϕ ; μ 0 , ϕ 0 ) + - 1 1 0 2 π p ( τ ; μ , ϕ ; μ , ϕ ) I ( τ ; μ , ϕ ) d μ d ϕ ,
- 1 1 0 2 π p ( τ ; μ , ϕ ; μ , ϕ ) d μ d ϕ = 1.
p ( μ , ϕ ; μ , ϕ ) = p ( μ , ϕ ; μ , ϕ ) p ( - μ , ϕ ; - μ , ϕ ) = p ( μ , ϕ ; μ , ϕ ) p ( μ , ϕ ; - μ , ϕ ) = p ( - μ , ϕ ; μ , ϕ ) = p ( μ , ϕ ; - μ , ϕ ) .
p ( cos θ ) = l = 0 N p l ( μ , μ ) cos l ( ϕ - ϕ ) .
- 1 1 p 0 ( μ , μ ) d μ = ( 2 π ) - 1
I ( τ , μ , ϕ ) = l = 0 N I l ( τ , μ ) cos l ( ϕ - ϕ 0 )
( μ d d τ + 1 ) I l ( τ , μ ) = ω ( τ ) F exp ( - τ / μ 0 ) p l ( μ , μ 0 ) + π ( 1 + δ 0 I ) - 1 1 p l ( μ , μ ) I l ( τ , μ ) d μ
( μ d d τ + 1 ) I l + ( τ , μ ) = ω ( τ ) F exp ( - τ / μ 0 ) p l ( μ , μ 0 ) + ω ( τ ) π ( 1 + δ 0 l ) [ 0 1 p l ( μ , μ ) I l + ( τ , μ ) d μ + 0 1 p l ( μ , - μ ) I l - ( τ , μ ) d μ ] , ( - μ d d τ + 1 ) I l - ( τ , μ ) = ω ( τ ) F exp ( - τ / μ 0 ) p l ( - μ , μ 0 ) + ω ( τ ) π ( 1 + δ 0 l ) [ 0 1 p l ( - μ , μ ) I l + ( τ , μ ) d μ + 0 1 p l ( - μ , - μ ) I l - ( τ , μ ) d μ ] ,
- 1 1 g ( μ ) d μ = j = 0 n g ( μ j ) c j + R n ,
I l + ( τ , μ i , μ j ) = [ I 11 + l I 12 + l I l m + l I m 1 + l I m 2 + l I m m + l ] , p l + + ( τ ; μ i , μ j ) = [ p l 11 + + p l 12 + + p l 1 m + + p l m 1 + + p l m 2 + + p l m m + + ] , p l + - ( τ ; μ i , μ j ) = [ p l 11 + - p l 12 + - p l 1 m + - p l m 1 + - p l m 2 + - p l m m + - ] , J l + ( τ , μ i , μ j ) = F ω ( τ ) [ p l 11 + + exp ( - τ / μ 1 ) p l 1 m + + exp ( - τ / μ m ) p l m 1 + + exp ( - τ / μ 1 ) p l m m + + exp ( - τ / μ m ) ] , J l - ( τ , μ i , μ j ) = F ω ( τ ) [ p l 11 + - exp ( - τ / μ 1 ) p l 1 m + - exp ( - τ / μ m ) p l m 1 + - exp ( - τ / μ 1 ) p l m m + - exp ( - τ / μ m ) ] , c = [ c j δ j k ] , M = [ μ j δ j k ] ,
M d I l + ( τ ) d τ + I l + ( τ ) = J l + ( τ ) + ω ( τ ) π ( 1 + δ 0 l ) × [ p l + + ( τ ) c I l + ( τ ) + p l + - ( τ ) c I l - ( τ ) ] , - M d I l - ( τ ) d τ + I l - ( τ ) = J l - ( τ ) + ω ( τ ) π ( 1 + δ 0 l ) × [ p l - + ( τ ) c I l + ( τ ) + p l - - ( τ ) c I l - ( τ ) ] .
Γ l + + ( τ ) = M - 1 [ E - ω ( τ ) π ( 1 + δ 0 l ) p l + + ( τ ) c ] , Γ l + - ( τ ) = M - 1 ω ( τ ) π ( 1 + δ 0 l ) p l + - ( τ ) c , Γ l - + ( τ ) = M - 1 ω ( τ ) π ( 1 + δ 0 l ) p l - + ( τ ) c , Γ l - - ( τ ) = M - 1 [ E - ω ( τ ) π ( 1 + δ 0 l ) p l - - ( τ ) c ] , Σ l + ( τ ) = M - 1 J l + ( τ ) , Σ l - ( τ ) = M - 1 J l - ( τ ) ,
t 10 = E - Γ l + + ( ξ ) Δ 10 + o ( Δ 10 ) , t 01 = E - Γ l - - ( ξ ) Δ 10 + o ( Δ 10 ) , r 01 = Γ l + - ( ξ ) Δ 10 + o ( Δ 10 ) , r 10 = Γ l - + ( ξ ) Δ 10 + o ( Δ 10 ) ,
Σ 01 + = Σ l + ( ξ ) Δ 10 + o ( Δ 10 ) , Σ 01 - = Σ l - ( ξ ) Δ 10 + o ( Δ 10 ) .
0 ( τ 1 - τ 0 ) < min μ i { μ i / [ 1 - ½ ω ( τ ) ] } .
p 0 ( μ i , μ j ) = 1 N k = 0 N - 1 p ( μ i , μ j , k ϕ ) , p l ( μ i , μ j ) = 2 N k = 0 N - 1 p ( μ i , μ j , k ϕ ) cos ( l k ϕ ) , l 0 ,
j = i = 1 n p 0 ( μ i , μ j ) c i - ( 2 π ) - 1 , j = 1 , 2 , , m
r G = A i = 1 m c i μ i [ μ l c l μ m c m μ l c l μ m c m ]
Σ G - = A 2 π i = 1 m c i μ i [ μ 1 exp ( - τ / μ 1 ) μ m exp ( - τ / μ m ) μ 1 exp ( - τ / μ 1 ) μ m exp ( - τ / μ m ) ]
r 01 = M - 1 p + - c α Δ , t 01 = E - M - 1 Δ + M - 1 p + + c α Δ ,
S Δ p + - α Δ ; T Δ p + + α Δ ,
F Δ [ exp ( - Δ / μ 1 ) 0 0 0 exp ( - Δ / μ 2 ) 0 0 0 exp ( - Δ / μ m ) ] .
r 01 = M - 1 S Δ c , t 01 = F Δ + M - 1 T Δ c ,
r o 2 = M - 1 S Δ c + ( F Δ + M - 1 T Δ c ) [ E - ( M - 1 S Δ c ) 2 ] - 1 × M - 1 S Δ c ( F Δ + M - 1 T Δ c ) .
r o 2 = M - 1 { S Δ + ( F Δ + T Δ c M - 1 ) S Δ [ E - ( c M - 1 S Δ ) 2 ] - 1 ( F Δ + M - 1 c T Δ ) } c .
S 2 Δ S Δ + ( F Δ + T Δ c M - 1 ) S Δ [ E - ( c M - 1 S Δ ) 2 ] - 1 [ F Δ M - 1 cT Δ ] .
S ( 2 Δ ; μ , μ 0 ) = S ( Δ ; μ , μ 0 ) + exp ( - τ / μ ) Σ 0 ( τ ; μ , μ 0 ) exp ( - τ / μ 0 ) + exp ( - τ / μ 0 ) 0 1 T ( Δ ; μ , μ ) Σ 0 ( Δ ; μ , μ 0 ) d μ μ + exp ( - τ / μ ) 0 1 Σ 0 ( Δ ; μ , μ ) T ( Δ ; μ , μ 0 ) d μ μ + 0 1 0 1 T ( Δ ; μ , μ ) Σ 0 ( Δ ; μ , μ ) T ( Δ , μ , μ 0 ) d μ μ d μ μ .
Σ 0 ( Δ ; μ , μ 0 ) = i = 2 n + 1 α S i ( Δ ; μ , μ 0 ) , n = 0 , 1 , 2 , ,
S 1 ( Δ ; μ , μ 0 ) = S ( Δ ; μ , μ 0 )
S i ( Δ ; μ , μ 0 ) = 0 1 S ( Δ ; μ , μ ) S i - 1 ( Δ ; μ , μ 0 ) d μ μ .
S 2 Δ = S Δ + ( F Δ + T Δ c M - 1 ) Σ 0 Δ ( F Δ + M - 1 c T Δ ) ,
Σ Δ = S Δ [ E + ( c M - 1 S Δ ) 2 + ( c M - 1 S Δ ) 4 + ] .
S 2 Δ = S Δ + ( F Δ + T Δ c M - 1 ) [ E - ( c M - 1 S Δ ) 2 ] - 1 ( F Δ + M - 1 c T Δ ) .
t 02 = ( F Δ + M - 1 T Δ c ) [ E - ( M - 1 Sc ) 2 ] - 1 [ F Δ + M - 1 T Δ c ] .
[ E - ( M - 1 Sc ) 2 ] - 1 = E + ( M - 1 Sc ) 2 [ E - ( M - 1 Sc ) 2 ] - 1 ,
t 02 = F 2 Δ + F Δ M - 1 T Δ c + M - 1 T Δ c F Δ + ( M - 1 T Δ c ) 2 + ( F Δ + M - 1 T Δ c ) ( M - 1 S Δ c ) 2 [ E - ( M - 1 S Δ c ) 2 ] - 1 ( F Δ + M - 1 T Δ c ) .
t 02 = F 2 Δ + M - 1 T 2 Δ c .
T 2 Δ = F Δ T Δ + T Δ F Δ + T Δ c M - 1 T Δ + ( F Δ + T Δ c M - 1 ) × S Δ c M - 1 S Δ [ E - ( c M - 1 S Δ ) 2 ] - 1 × [ F Δ + M - 1 cT Δ ] .
T 2 Δ = T Δ F Δ + F Δ T Δ + T Δ c M - 1 T Δ ( F Δ + T Δ c M - 1 ) Σ Δ [ F Δ + c M - 1 T Δ ] ,
Σ Δ = i = 2 n S i             n = 1 , 2 , 3 , .
Σ Δ = S Δ c M - 1 S Δ ( E - ( c M - 1 S Δ ) 2 ] - 1 .
T 2 Δ = T Δ F Δ + F Δ T Δ + T Δ c M - 1 T Δ + ( F Δ + T Δ c M - 1 ) S Δ c M - 1 S Δ [ E - ( c M - 1 T Δ ) 2 ] - 1 × ( F Δ + c M - 1 T Δ ) .

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