Abstract

In a recent paper by Di Girolamo et al. [J. Geophys. Res. D 103, 8795 (1998)] a heuristic argument was used to derive a reciprocity principle applicable to reflected solar radiation measurements. Here a formal derivation of this reciprocity principle is presented. It is also demonstrated that a purely spatial reciprocal relationship exists between one-dimensional radiative transfer theory and the three-dimensional searchlight problem for horizontally homogeneous media.

© 1999 Optical Society of America

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References

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  1. L. Di Girolamo, T. Várnai, R. Davies, “Apparent breakdown of reciprocity in reflected solar radiances,” J. Geophys. Res. D 103, 8795–8803 (1998).
    [CrossRef]
  2. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  3. K. M. Case, “Transfer problems and the reciprocity principle,” Rev. Mod. Phys. 29, 651–663 (1957).
    [CrossRef]
  4. R. Aronson, “Radiative transfer implies a modified reciprocity relation,” J. Opt. Soc. Am. A 14, 486–490 (1997).
    [CrossRef]
  5. H. Yang, H. G. Gordon, “Remote sensing of ocean color: assessment of water-leaving radiance bidirectional effects on atmospheric diffuse transmittance,” Appl. Opt. 36, 7887–7897 (1997).
    [CrossRef]
  6. R. Davies, “Spatial autocorrelation of radiation measured by the Earth Radiation Budget Experiment: scene inhomogeneity and reciprocity violation,” J. Geophys. Res. D 99, 20,879–20,887 (1994).
    [CrossRef]
  7. H. von Helmholtz, “Theorie der Luftschwingungen in Rohren mit offenen Enden,” Crelle LVII, 1 (1859).
  8. D. E. Kornreich, B. D. Ganapol, “Numerical evaluation of the three-dimensional searchlight problem in half-space,” Nucl. Sci. Eng. 127, 317–337 (1997).
  9. A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B. 24(3), 177–185 (1999).
    [CrossRef]

1999 (1)

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B. 24(3), 177–185 (1999).
[CrossRef]

1998 (1)

L. Di Girolamo, T. Várnai, R. Davies, “Apparent breakdown of reciprocity in reflected solar radiances,” J. Geophys. Res. D 103, 8795–8803 (1998).
[CrossRef]

1997 (3)

1994 (1)

R. Davies, “Spatial autocorrelation of radiation measured by the Earth Radiation Budget Experiment: scene inhomogeneity and reciprocity violation,” J. Geophys. Res. D 99, 20,879–20,887 (1994).
[CrossRef]

1957 (1)

K. M. Case, “Transfer problems and the reciprocity principle,” Rev. Mod. Phys. 29, 651–663 (1957).
[CrossRef]

1859 (1)

H. von Helmholtz, “Theorie der Luftschwingungen in Rohren mit offenen Enden,” Crelle LVII, 1 (1859).

Aronson, R.

Cahalan, R. F.

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B. 24(3), 177–185 (1999).
[CrossRef]

Case, K. M.

K. M. Case, “Transfer problems and the reciprocity principle,” Rev. Mod. Phys. 29, 651–663 (1957).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Davies, R.

L. Di Girolamo, T. Várnai, R. Davies, “Apparent breakdown of reciprocity in reflected solar radiances,” J. Geophys. Res. D 103, 8795–8803 (1998).
[CrossRef]

R. Davies, “Spatial autocorrelation of radiation measured by the Earth Radiation Budget Experiment: scene inhomogeneity and reciprocity violation,” J. Geophys. Res. D 99, 20,879–20,887 (1994).
[CrossRef]

Davis, A. B.

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B. 24(3), 177–185 (1999).
[CrossRef]

Di Girolamo, L.

L. Di Girolamo, T. Várnai, R. Davies, “Apparent breakdown of reciprocity in reflected solar radiances,” J. Geophys. Res. D 103, 8795–8803 (1998).
[CrossRef]

Ganapol, B. D.

D. E. Kornreich, B. D. Ganapol, “Numerical evaluation of the three-dimensional searchlight problem in half-space,” Nucl. Sci. Eng. 127, 317–337 (1997).

Gordon, H. G.

Kornreich, D. E.

D. E. Kornreich, B. D. Ganapol, “Numerical evaluation of the three-dimensional searchlight problem in half-space,” Nucl. Sci. Eng. 127, 317–337 (1997).

Love, S. P.

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B. 24(3), 177–185 (1999).
[CrossRef]

McGill, M. J.

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B. 24(3), 177–185 (1999).
[CrossRef]

Spinhirne, J. D.

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B. 24(3), 177–185 (1999).
[CrossRef]

Várnai, T.

L. Di Girolamo, T. Várnai, R. Davies, “Apparent breakdown of reciprocity in reflected solar radiances,” J. Geophys. Res. D 103, 8795–8803 (1998).
[CrossRef]

von Helmholtz, H.

H. von Helmholtz, “Theorie der Luftschwingungen in Rohren mit offenen Enden,” Crelle LVII, 1 (1859).

Yang, H.

Appl. Opt. (1)

Crelle (1)

H. von Helmholtz, “Theorie der Luftschwingungen in Rohren mit offenen Enden,” Crelle LVII, 1 (1859).

J. Geophys. Res. D (2)

L. Di Girolamo, T. Várnai, R. Davies, “Apparent breakdown of reciprocity in reflected solar radiances,” J. Geophys. Res. D 103, 8795–8803 (1998).
[CrossRef]

R. Davies, “Spatial autocorrelation of radiation measured by the Earth Radiation Budget Experiment: scene inhomogeneity and reciprocity violation,” J. Geophys. Res. D 99, 20,879–20,887 (1994).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nucl. Sci. Eng. (1)

D. E. Kornreich, B. D. Ganapol, “Numerical evaluation of the three-dimensional searchlight problem in half-space,” Nucl. Sci. Eng. 127, 317–337 (1997).

Phys. Chem. Earth B. (1)

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: an emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth B. 24(3), 177–185 (1999).
[CrossRef]

Rev. Mod. Phys. (1)

K. M. Case, “Transfer problems and the reciprocity principle,” Rev. Mod. Phys. 29, 651–663 (1957).
[CrossRef]

Other (1)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

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Figures (1)

Fig. 1
Fig. 1

Arbitrarily shaped enclosure with surface S, bounding a volume V, used to describe the geometry of the derivation.

Equations (17)

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Ω·Ir, Ω=-σerIr, Ω+σsr4π4π Pr, Ω, Ω×Ir, ΩdΩ+qr, Ω,
Pr, Ω, Ω=Pr, -Ω, -Ω,
SΩ·n<0 |Ω·n|I1r, -ΩI2incr, Ω-I1incr, Ω×I2r, -ΩdΩdr=V4πI2r, -Ωq1r, Ω-I1r, Ωq2r, -ΩdΩdr,
I1r, Ω=I1incr, Ω,  rS, n·Ω<0.
-Ω·I2r, -Ω=-σerI2r, -Ω+σsr4π×4π Pr, -Ω, -ΩI2r, -ΩdΩ+q2r, -Ω
I2r, -Ω=I2incr, -Ω,  rS, -n·Ω<0.
SΩ·n<0 Ω·nI1r, -ΩI2incr, ΩdΩdr=SΩ·n<0 Ω·nI1incr, ΩI2r, -ΩdΩdr.
I1r, Ω=Ir, Ω; A, Ω1,  r, AS
I1incr, Ω=δr-rAδ˜Ω·Ω1F1r, Ω,  rAA,
I2r, Ω=Ir, Ω; B, Ω2,  r, BS
I2incr, Ω=δr-rBδ˜Ω·Ω2F2r, Ω,  rBB.
SΩ·n<0 Ω·nδr-rBδ˜Ω·Ω2F2r, Ω×Ir, -Ω; A, Ω1dΩdr=SΩ·n<0 Ω·nδr-rA×δ˜Ω·Ω1F1r, ΩIr, -Ω; B, Ω2dΩdr,
Ω2·nB F2r, Ω2Ir, -Ω2; A, Ω1dr=Ω1·nA F1r, Ω1Ir, -Ω1; B, Ω2dr.
I-Ω2; Ω1Ω1·nF1Ω1=I-Ω1; Ω2Ω2·nF2Ω2.
Ω2·n F1DΩ2ISLr, -Ω2; A, Ω1dr=Ω1·nA FSLr, Ω1I1D-Ω1; Ω2dr.
 ISLr, -Ω2; A, Ω1drΩ1·nA FSLr, Ω1dr=I1D-Ω1; Ω2Ω2·nF1DΩ2.
 ISLr, -Ω2; A, Ω1drΩ1·nA FSLr, Ω1dr=I1D-Ω2; Ω1Ω1·nF1DΩ1.

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