Abstract

The Monte Carlo method has been applied to numerical modeling of an integrating sphere designed for hemispherical-directional reflectance factor measurements. It is shown that a conventional algorithm of backward ray tracing used for estimation of characteristics of the radiation field at a given point has slow convergence for small source-to-sphere-diameter ratios. A newly developed algorithm that substantially improves the convergence by calculation of direct source-induced irradiation for every point of diffuse reflection of rays traced is described. The method developed is applied to an integrating sphere reflectometer for the visible and infrared spectral ranges. Parametric studies of hemispherical radiance distributions for radiation incident onto the sample center were performed. The deviations of measured sample reflectance from the actual reflectance as a result of various factors were computed. The accuracy of the results, adequacy of the reflectance model, and other important aspects of the algorithm implementation are discussed.

© 2003 Optical Society of America

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References

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  1. D. G. Goebel, “Generalized integrating-sphere theory,” Appl. Opt. 6, 125–128 (1967).
    [CrossRef]
  2. M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).
    [CrossRef]
  3. W. B. Fussell, “Approximate theory of the photometric integrating sphere,” Natl. Bur. Stand. (U.S.) Tech. Note 594-7 (1974).
  4. R. L. Brown, “A numerical solution of the integral equation describing a photometric integrating sphere,” J. Res. Natl. Bur. Stand. Sect. A 77, 343–351 (1973).
    [CrossRef]
  5. A. C. M. De Visser, M. Van der Woude, “Minimization of the screen effect in the integrating sphere by variation of the reflection factor,” Light. Res. Technol. 12, 42–49 (1980).
    [CrossRef]
  6. Y. Ohno, “Integrating sphere simulation: application to total flux scale realization,” Appl. Opt. 33, 2637–2646 (1994).
    [CrossRef] [PubMed]
  7. H. L. Tardy, “Matrix method for integrating sphere calculations,” J. Opt. Soc. Am. A 8, 1411–1418 (1991).
    [CrossRef]
  8. J. F. Clare, “Comparison of four analytic methods for the calculation of irradiance in integrating spheres,” J. Opt. Soc. Am. A 15, 3086–3096 (1998).
    [CrossRef]
  9. L. M. Hanssen, “Effects of non-Lambertian surfaces on integrating sphere measurements,” Appl. Opt. 35, 3597–3606 (1996).
    [CrossRef] [PubMed]
  10. A. V. Prokhorov, V. I. Sapritsky, S. N. Mekhontsev, “Modeling of integrating spheres for photometric and radiometric applications,” in Optical Radiation Measurements III, J. M. Palmer, ed., Proc. SPIE2815, 118–125 (1996).
    [CrossRef]
  11. B. G. Crowther, “Computer modeling of integrating spheres,” Appl. Opt. 35, 5880–5886 (1996).
    [CrossRef] [PubMed]
  12. A. Ziegler, H. Hess, H. Schimpl, “Rechnersimulation von Ulbrichtkugeln,” Optik (Stuttgart) 101, 130–136 (1996).
  13. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, 11–12 (1977).
  14. P. Shirley, C. Wang, K. Zimmerman, “Monte Carlo techniques for direct lighting calculations,” ACM Trans. Graphics 15, 1–36 (1966).
    [CrossRef]
  15. P. Shirley, Realistic Ray Tracing, (A K Peters, Natick, Mass., 2000).
  16. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Taylor & Francis, Washington, D.C., 1992).
  17. G. Marsaglia, “Choosing a point from the surface of a sphere,” Ann. Math. Stat. 43, 645–646 (1972).
    [CrossRef]
  18. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, San Diego, Calif., 2000).
  19. E. Lafortune, Y. Willems, “A theoretical framework for physically based rendering,” Comput. Graph. Forum 13, 97–107 (1994).
    [CrossRef]
  20. I. Manno, Introduction to the Monte-Carlo Method (Akademiai Kiado, Budapest, 1999).
  21. J. Spanier, E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, Reading, Mass., 1969).
  22. P. L’Ecuyer, “Uniform random number generators,” in Proceedings of 1998 Winter Simulation Conference (Association of Computing Machinery, New York, 1998), p. 104.
  23. K. Entacher, “Bad subsequences of well-known linear congruential pseudorandom number generators,” ACM Trans. Modeling Comput. Simulation 8, 61–70 (1998).
    [CrossRef]
  24. M. Matsumoto, T. Nishimura, “Mersenne Twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Modeling Comput. Simulation 8, 3–30 (1998).
    [CrossRef]
  25. A. V. Prokhorov, L. M. Hanssen, “Numerical modeling of an integrating sphere radiation source,” in Modeling and Characterization of Light Sources, C. B. Wooley, ed., Proc. SPIE4775, 106–118 (2002).
    [CrossRef]

1998 (3)

J. F. Clare, “Comparison of four analytic methods for the calculation of irradiance in integrating spheres,” J. Opt. Soc. Am. A 15, 3086–3096 (1998).
[CrossRef]

K. Entacher, “Bad subsequences of well-known linear congruential pseudorandom number generators,” ACM Trans. Modeling Comput. Simulation 8, 61–70 (1998).
[CrossRef]

M. Matsumoto, T. Nishimura, “Mersenne Twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Modeling Comput. Simulation 8, 3–30 (1998).
[CrossRef]

1996 (3)

1994 (2)

Y. Ohno, “Integrating sphere simulation: application to total flux scale realization,” Appl. Opt. 33, 2637–2646 (1994).
[CrossRef] [PubMed]

E. Lafortune, Y. Willems, “A theoretical framework for physically based rendering,” Comput. Graph. Forum 13, 97–107 (1994).
[CrossRef]

1991 (1)

1980 (1)

A. C. M. De Visser, M. Van der Woude, “Minimization of the screen effect in the integrating sphere by variation of the reflection factor,” Light. Res. Technol. 12, 42–49 (1980).
[CrossRef]

1977 (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, 11–12 (1977).

1973 (1)

R. L. Brown, “A numerical solution of the integral equation describing a photometric integrating sphere,” J. Res. Natl. Bur. Stand. Sect. A 77, 343–351 (1973).
[CrossRef]

1972 (1)

G. Marsaglia, “Choosing a point from the surface of a sphere,” Ann. Math. Stat. 43, 645–646 (1972).
[CrossRef]

1970 (1)

M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).
[CrossRef]

1967 (1)

1966 (1)

P. Shirley, C. Wang, K. Zimmerman, “Monte Carlo techniques for direct lighting calculations,” ACM Trans. Graphics 15, 1–36 (1966).
[CrossRef]

Brown, R. L.

R. L. Brown, “A numerical solution of the integral equation describing a photometric integrating sphere,” J. Res. Natl. Bur. Stand. Sect. A 77, 343–351 (1973).
[CrossRef]

Clare, J. F.

Crowther, B. G.

De Visser, A. C. M.

A. C. M. De Visser, M. Van der Woude, “Minimization of the screen effect in the integrating sphere by variation of the reflection factor,” Light. Res. Technol. 12, 42–49 (1980).
[CrossRef]

Entacher, K.

K. Entacher, “Bad subsequences of well-known linear congruential pseudorandom number generators,” ACM Trans. Modeling Comput. Simulation 8, 61–70 (1998).
[CrossRef]

Finkel, M. W.

M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).
[CrossRef]

Fussell, W. B.

W. B. Fussell, “Approximate theory of the photometric integrating sphere,” Natl. Bur. Stand. (U.S.) Tech. Note 594-7 (1974).

Gelbard, E. M.

J. Spanier, E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, Reading, Mass., 1969).

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, 11–12 (1977).

Goebel, D. G.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, San Diego, Calif., 2000).

Hanssen, L. M.

L. M. Hanssen, “Effects of non-Lambertian surfaces on integrating sphere measurements,” Appl. Opt. 35, 3597–3606 (1996).
[CrossRef] [PubMed]

A. V. Prokhorov, L. M. Hanssen, “Numerical modeling of an integrating sphere radiation source,” in Modeling and Characterization of Light Sources, C. B. Wooley, ed., Proc. SPIE4775, 106–118 (2002).
[CrossRef]

Hess, H.

A. Ziegler, H. Hess, H. Schimpl, “Rechnersimulation von Ulbrichtkugeln,” Optik (Stuttgart) 101, 130–136 (1996).

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Taylor & Francis, Washington, D.C., 1992).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, 11–12 (1977).

L’Ecuyer, P.

P. L’Ecuyer, “Uniform random number generators,” in Proceedings of 1998 Winter Simulation Conference (Association of Computing Machinery, New York, 1998), p. 104.

Lafortune, E.

E. Lafortune, Y. Willems, “A theoretical framework for physically based rendering,” Comput. Graph. Forum 13, 97–107 (1994).
[CrossRef]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, 11–12 (1977).

Manno, I.

I. Manno, Introduction to the Monte-Carlo Method (Akademiai Kiado, Budapest, 1999).

Marsaglia, G.

G. Marsaglia, “Choosing a point from the surface of a sphere,” Ann. Math. Stat. 43, 645–646 (1972).
[CrossRef]

Matsumoto, M.

M. Matsumoto, T. Nishimura, “Mersenne Twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Modeling Comput. Simulation 8, 3–30 (1998).
[CrossRef]

Mekhontsev, S. N.

A. V. Prokhorov, V. I. Sapritsky, S. N. Mekhontsev, “Modeling of integrating spheres for photometric and radiometric applications,” in Optical Radiation Measurements III, J. M. Palmer, ed., Proc. SPIE2815, 118–125 (1996).
[CrossRef]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, 11–12 (1977).

Nishimura, T.

M. Matsumoto, T. Nishimura, “Mersenne Twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Modeling Comput. Simulation 8, 3–30 (1998).
[CrossRef]

Ohno, Y.

Prokhorov, A. V.

A. V. Prokhorov, L. M. Hanssen, “Numerical modeling of an integrating sphere radiation source,” in Modeling and Characterization of Light Sources, C. B. Wooley, ed., Proc. SPIE4775, 106–118 (2002).
[CrossRef]

A. V. Prokhorov, V. I. Sapritsky, S. N. Mekhontsev, “Modeling of integrating spheres for photometric and radiometric applications,” in Optical Radiation Measurements III, J. M. Palmer, ed., Proc. SPIE2815, 118–125 (1996).
[CrossRef]

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, 11–12 (1977).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, San Diego, Calif., 2000).

Sapritsky, V. I.

A. V. Prokhorov, V. I. Sapritsky, S. N. Mekhontsev, “Modeling of integrating spheres for photometric and radiometric applications,” in Optical Radiation Measurements III, J. M. Palmer, ed., Proc. SPIE2815, 118–125 (1996).
[CrossRef]

Schimpl, H.

A. Ziegler, H. Hess, H. Schimpl, “Rechnersimulation von Ulbrichtkugeln,” Optik (Stuttgart) 101, 130–136 (1996).

Shirley, P.

P. Shirley, C. Wang, K. Zimmerman, “Monte Carlo techniques for direct lighting calculations,” ACM Trans. Graphics 15, 1–36 (1966).
[CrossRef]

P. Shirley, Realistic Ray Tracing, (A K Peters, Natick, Mass., 2000).

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Taylor & Francis, Washington, D.C., 1992).

Spanier, J.

J. Spanier, E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, Reading, Mass., 1969).

Tardy, H. L.

Van der Woude, M.

A. C. M. De Visser, M. Van der Woude, “Minimization of the screen effect in the integrating sphere by variation of the reflection factor,” Light. Res. Technol. 12, 42–49 (1980).
[CrossRef]

Wang, C.

P. Shirley, C. Wang, K. Zimmerman, “Monte Carlo techniques for direct lighting calculations,” ACM Trans. Graphics 15, 1–36 (1966).
[CrossRef]

Willems, Y.

E. Lafortune, Y. Willems, “A theoretical framework for physically based rendering,” Comput. Graph. Forum 13, 97–107 (1994).
[CrossRef]

Ziegler, A.

A. Ziegler, H. Hess, H. Schimpl, “Rechnersimulation von Ulbrichtkugeln,” Optik (Stuttgart) 101, 130–136 (1996).

Zimmerman, K.

P. Shirley, C. Wang, K. Zimmerman, “Monte Carlo techniques for direct lighting calculations,” ACM Trans. Graphics 15, 1–36 (1966).
[CrossRef]

ACM Trans. Graphics (1)

P. Shirley, C. Wang, K. Zimmerman, “Monte Carlo techniques for direct lighting calculations,” ACM Trans. Graphics 15, 1–36 (1966).
[CrossRef]

ACM Trans. Modeling Comput. Simulation (2)

K. Entacher, “Bad subsequences of well-known linear congruential pseudorandom number generators,” ACM Trans. Modeling Comput. Simulation 8, 61–70 (1998).
[CrossRef]

M. Matsumoto, T. Nishimura, “Mersenne Twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Modeling Comput. Simulation 8, 3–30 (1998).
[CrossRef]

Ann. Math. Stat. (1)

G. Marsaglia, “Choosing a point from the surface of a sphere,” Ann. Math. Stat. 43, 645–646 (1972).
[CrossRef]

Appl. Opt. (4)

Comput. Graph. Forum (1)

E. Lafortune, Y. Willems, “A theoretical framework for physically based rendering,” Comput. Graph. Forum 13, 97–107 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Res. Natl. Bur. Stand. Sect. A (1)

R. L. Brown, “A numerical solution of the integral equation describing a photometric integrating sphere,” J. Res. Natl. Bur. Stand. Sect. A 77, 343–351 (1973).
[CrossRef]

Light. Res. Technol. (1)

A. C. M. De Visser, M. Van der Woude, “Minimization of the screen effect in the integrating sphere by variation of the reflection factor,” Light. Res. Technol. 12, 42–49 (1980).
[CrossRef]

Natl. Bur. Stand. (U.S.) Monogr. (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, 11–12 (1977).

Opt. Commun. (1)

M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).
[CrossRef]

Optik (Stuttgart) (1)

A. Ziegler, H. Hess, H. Schimpl, “Rechnersimulation von Ulbrichtkugeln,” Optik (Stuttgart) 101, 130–136 (1996).

Other (9)

P. Shirley, Realistic Ray Tracing, (A K Peters, Natick, Mass., 2000).

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Taylor & Francis, Washington, D.C., 1992).

I. Manno, Introduction to the Monte-Carlo Method (Akademiai Kiado, Budapest, 1999).

J. Spanier, E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, Reading, Mass., 1969).

P. L’Ecuyer, “Uniform random number generators,” in Proceedings of 1998 Winter Simulation Conference (Association of Computing Machinery, New York, 1998), p. 104.

W. B. Fussell, “Approximate theory of the photometric integrating sphere,” Natl. Bur. Stand. (U.S.) Tech. Note 594-7 (1974).

A. V. Prokhorov, V. I. Sapritsky, S. N. Mekhontsev, “Modeling of integrating spheres for photometric and radiometric applications,” in Optical Radiation Measurements III, J. M. Palmer, ed., Proc. SPIE2815, 118–125 (1996).
[CrossRef]

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, San Diego, Calif., 2000).

A. V. Prokhorov, L. M. Hanssen, “Numerical modeling of an integrating sphere radiation source,” in Modeling and Characterization of Light Sources, C. B. Wooley, ed., Proc. SPIE4775, 106–118 (2002).
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

Sectional view of the integrating sphere reflectometer: 1, integrating sphere; 2, elliptical opening; 3, radiation source; 4, sample; 5, sample holder; 6, sample thermostat; 7, reference; 8, reference holder; 9, reference thermostat; 10, 11, baffles; 12, folding mirror.

Fig. 2
Fig. 2

Local spherical coordinate system.

Fig. 3
Fig. 3

Angular distributions of radiation source radiant intensity for ν = 0.5, 1, 2.

Fig. 4
Fig. 4

Schematic of inverse ray tracing.

Fig. 5
Fig. 5

Schematic for calculation of irradiance produced by source S at point P.

Fig. 6
Fig. 6

Schematic of the shadow rays algorithm.

Fig. 7
Fig. 7

Flow chart of the algorithm for computing radiance.

Fig. 8
Fig. 8

Artifactual pattern in a color map of the hemispherical distribution of radiance incident onto a sample center (a standard commercial PRNG was used).

Fig. 9
Fig. 9

Same distribution as in Fig. 8 but computed with the use of a Mersenne-Twister PRNG.

Fig. 10
Fig. 10

Hemispherical distribution of radiance incident onto the sample center; 10,000 rays were traced for every N θ × N ϕ = 181 × 361 points of a uniform grid. Radiation source with ν = 1. All specularities are 0; reflectances from Variant 1 of Table 2.

Fig. 11
Fig. 11

Hemispherical distribution of radiance incident onto the sample center; 10,000 rays were traced for every of N θ × N ϕ = 181 × 361 points of a uniform grid. Radiation source with ν = 1. All specularities are 0; reflectances from Variant 2 of Table 2.

Fig. 12
Fig. 12

Hemispherical distribution of radiance incident onto the sample center; 10,000 rays have been traced for every of N θ × N ϕ = 181 × 361 points of a uniform grid. Radiation source with ν = 1. All specularities are 0; reflectances from Variant 3 of Table 2.

Fig. 13
Fig. 13

Hemispherical distribution of radiance incident onto the sample center; 10,000 rays have been traced for every N θ × N ϕ = 181 × 361 points of a uniform grid. Radiation source with ν = 1. All specularities are 0; reflectances from Variant 4 of Table 2.

Fig. 14
Fig. 14

Hemispherical distribution of radiance incident onto the sample center; 10,000 rays have been traced for every N θ × N ϕ = 181 × 361 points of a uniform grid. Radiation source with ν = 0.5. All specularities are 0; reflectances from Variant 1 of Table 2.

Fig. 15
Fig. 15

Hemispherical distribution of radiance incident onto the sample center; 10,000 rays have been traced for every N θ × N ϕ = 181 × 361 points of a uniform grid. Radiation source with ν = 2. All specularities are 0; reflectances from Variant 1 of Table 2.

Fig. 16
Fig. 16

Hemispherical distribution of radiance incident onto the sample center; 50,000 rays have been traced for every N θ × N ϕ = 181 × 361 points of a uniform grid. Radiation source with ν = 1. Sphere and baffles specularities are 0.2; all other specularities are 0; reflectances from Variant 1 of Table 2.

Fig. 17
Fig. 17

Hemispherical distribution of radiance incident onto the sample center; 80,000 rays have been traced for every N θ × N ϕ = 181 × 361 points of a uniform grid. Radiation source with ν = 1. Sphere and baffles specularities are 0.4; all other specularities are 0; reflectances from Variant 1 of Table 2.

Fig. 18
Fig. 18

Measured sample reflectance relative to sphere and baffle specularity for several values of sample specularity. Reference specularity is 0; reflectances from Variant 1 of Table 2; ν = 1. Sample specularity: ●, 0; ○, 0.2; ◆, 0.4; ◇, 0.6; ▼, 0.8; and ▽, 1.

Fig. 19
Fig. 19

Measured reflectance of a purely specular sample relative to sphere and baffle specularity. Reference specularity is 0; ν = 1. For four variants of reflectance from Table 2. Variant: ●, 1; ○, 2; ◆, 3; and ◇, 4.

Tables (2)

Tables Icon

Table 1 Numerical Values for IS Reflectometer Geometrical Parameters

Tables Icon

Table 2 Reflectances of IS Reflectometer Components

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

R1λ, 2πΩ, T1R2λ, 2πΩ, T2=V1V2,
ρ1λ, ω2π, T1=ρ2λ, ω2π, T2V1λV2λ.
x2+y2+z2=r2,
x/rx2+y/ry2=1.
S=ρs/ρs+ρd=ρs/ρ,
Iθ=ν+12π I0cosν θ,
ωr=ωi-2n · ωin,
θ=arcsinηθ, ϕ=2πηϕ,
x=x+ωrxt, y=y+ωryt, z=z+ωrzt,
ux=2ηx-1, uy=2ηy-1.
x=2ux1-s, y=2uy1-s, z=1-2s.
Lsθ=ν+12π I0cosν-1 θ.
Lθ=ρ1ρ2L32=ρ1ρ2Lsθ3s+ρ3sρ4L54=ρ1ρ2Lsθ3s+ρ3sρ4ρ5L65=ρ1ρ2Lsθ3s+ρ3sρ4ρ5Lsθ6s.
Lθ=1ni=1nj=1mi Lsθijk=1lij ρijk,
ρmeas=ρ0i=1n Li cos θii=1n L0,i cos θ0,i,
nγ=logργ=ln γln ρ
np=n=1 npn,
np=q n=1 n1-qn=1-qq.
EP=S VP, dSLsθScos θS cos θPπd2dS,
Ei=Smj=1m VijLSθjcos θi cos θjπdij2,
Li,0θ=Ls0θ0Vs0+ρ11π E1+ρ21π E2+ρ31π E3+ρ41π E4+=Ls0θ0Vs0+1πρ1E1+ρ1ρ2E2+ρ1ρ2ρ3E3+ρ1ρ2ρ3ρ4E4+=Ls0θ0Vs0+1πj=1m Ejk=1lj ρk
Ej=SVssjLsθsjcos θsj cos θjπdsj,j2,
L0θ=1ni=1nLSi0θ0VSi0+Sπ2j=1mi LSθSij×cos θSij cos θijdSijj2 VSijjk=1lij ρk.
wr,i=ρiwi, i=1, 2,, M,
L¯=i=1Nθsin θij=1Nϕ Lijnji=1Nθsin 2θi,

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