Abstract

If digital images of clear daytime or twilight skies are acquired through a linear polarizing filter, they can be combined to produce high-resolution maps of skylight polarization. Here polarization P and normalized Stokes parameter Q are measured near sunset at one inland and two coastal sites. Maps that include the principal plane consistently show that the familiar Arago and Babinet neutral points are part of broader areas in which skylight polarization is often indistinguishably different from zero. A simple multiple-scattering model helps explain some of these polarization patterns.

© 1998 Optical Society of America

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References

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  1. K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (Deepak, Hampton, Va., 1988), pp. 375–391.
  2. J. W. Strutt, “On the light from the sky, its polarization and colour,” Philos. Mag. 41, 107–120, 274–279 (1871).
  3. Reference 1, pp. 269–270, lists skylight polarization studies by several of Rayleigh’s contemporaries.
  4. S. Chandrasekhar, D. Elbert, “The illumination and polarization of the sunlit sky on Rayleigh scattering,” Trans. Am. Philos. Soc. 44, Pt. 6, 643–728 (1954).
  5. H. Neuberger, Introduction to Physical Meteorology, Revised ed. (Pennsylvania State University, University Park, Pa., 1957), pp. 194–206.
  6. Z. Sekera, “Light scattering in the atmosphere and the polarization of sky light,” J. Opt. Soc. Am. 47, 484–490 (1957).
    [CrossRef]
  7. K. L. Coulson, “Effects of the El Chichon volcanic cloud in the stratosphere on the polarization of light from the sky,” Appl. Opt. 22, 1036–1050 (1983).
    [CrossRef] [PubMed]
  8. F. E. Volz, “Volcanic turbidity, skylight scattering functions, sky polarization, and twilights in New England during 1983,” Appl. Opt. 23, 2589–2593 (1984).
    [CrossRef] [PubMed]
  9. See Ref. 1, pp. 377–378.
  10. E. de Bary, “Influence of multiple scattering of the intensity and polarization of diffuse sky radiation,” Appl. Opt. 3, 1293–1303 (1964). The atmospheric principal plane is also called the Sun’s vertical.
  11. R. S. Fraser, “Atmospheric neutral points outside of the principal plane,” Contrib. Atmos. Phys. 54, 286–297 (1981).
  12. K. Bullrich, “Scattered radiation in the atmosphere and the natural aerosol,” Adv. Geophys. 10, 99–260 (1964). Polarization measurements that span half of the sky dome appear on pp. 212–215 (Figs. 49 and 50). Also see Ref. 1, pp. 216–218, 311, 325, 327.
    [CrossRef]
  13. T. Prosch, D. Hennings, E. Raschke, “Video polarimetry: a new imaging technique in atmospheric science,” Appl. Opt. 22, 1360–1363 (1983). Also see Ref. 1, p. 554.
  14. E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976), pp. 213, 268.
  15. D. J. Gambling, B. Billard, “A study of the polarization of skylight,” Aust. J. Phys. 20, 675–681 (1967).
    [CrossRef]
  16. F. S. Harris, “Calculated Mie scattering properties in the visible and infrared of measured Los Angeles aerosol size distributions,” Appl. Opt. 11, 2697–2705 (1972).
    [CrossRef] [PubMed]
  17. For examples, see Ref. 1, pp. 256–261.
  18. Ref. 5, p. 197.
  19. Ref. 1, p. 233. Also see E. Collett , Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), pp. 34–39.
  20. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 50–53.
  21. Ref. 20, p. 53.
  22. B. W. Fitch, R. L. Walraven, D. E. Bradley, “Polarization of light reflected from grain crops during the heading growth stage,” Remote Sensing Environ. 15, 263–268 (1984).
    [CrossRef]
  23. See Ref. 1, p. 254 and Ref. 20, p. 50.
  24. Ref. 20, pp. 46, 50. χ has the same direction as skylight’s plane of polarization but avoids the conceptual difficulties that a plane of (partial) polarization entails.
  25. For examples, see Ref. 1, pp. 554–555 and 565–566.
  26. For example, see Ref. 5, pp. 194–197. Lines of zero Q are still called neutral lines (Ref. 1, pp. 254–258).
  27. ϕrel ranges between 0° and 360°, with values increasing clockwise from the Sun’s azimuth.
  28. Ref. 1, p. 254.
  29. Ref. 20, pp. 382–383. Equation (4) also defines polarization for specular reflection from planar surfaces. χ is horizontal for linear polarization by reflection from horizontal surfaces (e.g., calm water). To measure this polarization, once again set Eq. (4)’s 0° direction parallel to χ (i.e., horizontal).
  30. Ref. 20, p. 54.
  31. Ref. 14, pp. 198–199.
  32. Ref. 14, pp. 136–139. Note that 475 nm is a dominant wavelength typical of clear skies.
  33. R. L. Lee, “Colorimetric calibration of a video digitizing system: algorithm and applications,” Color Res. Appl. 13, 180–186 (1988).
    [CrossRef]
  34. For a remote-sensing application of P derived from photographs, see K. L. Coulson, V. S. Whitehead, C. Campbell, “Polarized views of the earth from orbital altitude,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 35–41 (1986). Narrow-FOV photographic polarimetry that uses a Savart plate is discussed in R. Gerharz, “Polarization of scattered horizon light in inclement weather,” Arch. Meteorol. Geophys. Bioklimatol. Ser. A 26, 265–273 (1977).
  35. For example, see Ref. 1, p. 261 (Fig. 4.36). θv is 0° at the astronomical horizon, except in Figs. 10–13, where θv is 0° at the slightly higher mean topographic horizon (see Fig. 3).
  36. My polarizer’s H90 is fairly uniform at visible wavelengths, although crossed pairs of such polarizers do transmit a dim violet from a white-light source. Because skylight dominant wavelengths at the Earth’s surface typically are 475 nm or more, the increase in photographic polarizers’ H90 at shorter wavelengths is unlikely to appreciably bias observations of skylight polarization.
  37. See Ref. 1, p. 582, for the general form of these Mueller matrix calculations.
  38. As noted above, P measured by the four-image technique depends only on a polarizer’s relative (rather than absolute) directions of 0°, 45°, 90°, and 135°. In other words, the four-image 0° direction can differ arbitrarily from χ.
  39. R. Gerharz, “Self polarization in refractive systems,” Optik 43, 471–485 (1975). Coulson calls self-polarization parasitic polarization (Ref. 1, p. 556).
  40. See Ref. 1, pp. 254–258.
  41. R. S. Fraser, “Atmospheric neutral points over water,” J. Opt. Soc. Am. 58, 1029–1031 (1968). Also see Ref. 1, pp. 381–382.
    [CrossRef]
  42. See Ref. 1, pp. 522–525, for a discussion of partial polarization on reflection by water.
  43. Ref. 1, p. 311 (Fig. 5.22). Large near-horizon pQ gradients at 90° from a low Sun appear consistently in my polarization maps.
  44. Usually red pixels in Figs. 2–4 are the result of identical 24-bit colors in the original digital images; so, in a limited sense, the maps do include points where pQ and P = 0.0 exactly, but this equality is just an artifact of the resolution with which the slide scanner quantized scene radiances.
  45. For example, K. F. Evans, G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
    [CrossRef]
  46. R. L. Lee, “Horizon brightness revisited: measurements and a model of clear-sky radiances,” Appl. Opt. 33, 4620–4628, 4959 (1994).
  47. Ref. 20, pp. 112–113.
  48. Ref. 1, pp. 391–393.

1994 (1)

R. L. Lee, “Horizon brightness revisited: measurements and a model of clear-sky radiances,” Appl. Opt. 33, 4620–4628, 4959 (1994).

1991 (1)

For example, K. F. Evans, G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[CrossRef]

1988 (1)

R. L. Lee, “Colorimetric calibration of a video digitizing system: algorithm and applications,” Color Res. Appl. 13, 180–186 (1988).
[CrossRef]

1984 (2)

B. W. Fitch, R. L. Walraven, D. E. Bradley, “Polarization of light reflected from grain crops during the heading growth stage,” Remote Sensing Environ. 15, 263–268 (1984).
[CrossRef]

F. E. Volz, “Volcanic turbidity, skylight scattering functions, sky polarization, and twilights in New England during 1983,” Appl. Opt. 23, 2589–2593 (1984).
[CrossRef] [PubMed]

1983 (2)

K. L. Coulson, “Effects of the El Chichon volcanic cloud in the stratosphere on the polarization of light from the sky,” Appl. Opt. 22, 1036–1050 (1983).
[CrossRef] [PubMed]

T. Prosch, D. Hennings, E. Raschke, “Video polarimetry: a new imaging technique in atmospheric science,” Appl. Opt. 22, 1360–1363 (1983). Also see Ref. 1, p. 554.

1981 (1)

R. S. Fraser, “Atmospheric neutral points outside of the principal plane,” Contrib. Atmos. Phys. 54, 286–297 (1981).

1975 (1)

R. Gerharz, “Self polarization in refractive systems,” Optik 43, 471–485 (1975). Coulson calls self-polarization parasitic polarization (Ref. 1, p. 556).

1972 (1)

1968 (1)

R. S. Fraser, “Atmospheric neutral points over water,” J. Opt. Soc. Am. 58, 1029–1031 (1968). Also see Ref. 1, pp. 381–382.
[CrossRef]

1967 (1)

D. J. Gambling, B. Billard, “A study of the polarization of skylight,” Aust. J. Phys. 20, 675–681 (1967).
[CrossRef]

1964 (2)

K. Bullrich, “Scattered radiation in the atmosphere and the natural aerosol,” Adv. Geophys. 10, 99–260 (1964). Polarization measurements that span half of the sky dome appear on pp. 212–215 (Figs. 49 and 50). Also see Ref. 1, pp. 216–218, 311, 325, 327.
[CrossRef]

E. de Bary, “Influence of multiple scattering of the intensity and polarization of diffuse sky radiation,” Appl. Opt. 3, 1293–1303 (1964). The atmospheric principal plane is also called the Sun’s vertical.

1957 (1)

1954 (1)

S. Chandrasekhar, D. Elbert, “The illumination and polarization of the sunlit sky on Rayleigh scattering,” Trans. Am. Philos. Soc. 44, Pt. 6, 643–728 (1954).

1871 (1)

J. W. Strutt, “On the light from the sky, its polarization and colour,” Philos. Mag. 41, 107–120, 274–279 (1871).

Billard, B.

D. J. Gambling, B. Billard, “A study of the polarization of skylight,” Aust. J. Phys. 20, 675–681 (1967).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 50–53.

Bradley, D. E.

B. W. Fitch, R. L. Walraven, D. E. Bradley, “Polarization of light reflected from grain crops during the heading growth stage,” Remote Sensing Environ. 15, 263–268 (1984).
[CrossRef]

Bullrich, K.

K. Bullrich, “Scattered radiation in the atmosphere and the natural aerosol,” Adv. Geophys. 10, 99–260 (1964). Polarization measurements that span half of the sky dome appear on pp. 212–215 (Figs. 49 and 50). Also see Ref. 1, pp. 216–218, 311, 325, 327.
[CrossRef]

Campbell, C.

For a remote-sensing application of P derived from photographs, see K. L. Coulson, V. S. Whitehead, C. Campbell, “Polarized views of the earth from orbital altitude,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 35–41 (1986). Narrow-FOV photographic polarimetry that uses a Savart plate is discussed in R. Gerharz, “Polarization of scattered horizon light in inclement weather,” Arch. Meteorol. Geophys. Bioklimatol. Ser. A 26, 265–273 (1977).

Chandrasekhar, S.

S. Chandrasekhar, D. Elbert, “The illumination and polarization of the sunlit sky on Rayleigh scattering,” Trans. Am. Philos. Soc. 44, Pt. 6, 643–728 (1954).

Collett, E.

Ref. 1, p. 233. Also see E. Collett , Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), pp. 34–39.

Coulson, K. L.

K. L. Coulson, “Effects of the El Chichon volcanic cloud in the stratosphere on the polarization of light from the sky,” Appl. Opt. 22, 1036–1050 (1983).
[CrossRef] [PubMed]

K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (Deepak, Hampton, Va., 1988), pp. 375–391.

For a remote-sensing application of P derived from photographs, see K. L. Coulson, V. S. Whitehead, C. Campbell, “Polarized views of the earth from orbital altitude,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 35–41 (1986). Narrow-FOV photographic polarimetry that uses a Savart plate is discussed in R. Gerharz, “Polarization of scattered horizon light in inclement weather,” Arch. Meteorol. Geophys. Bioklimatol. Ser. A 26, 265–273 (1977).

de Bary, E.

Elbert, D.

S. Chandrasekhar, D. Elbert, “The illumination and polarization of the sunlit sky on Rayleigh scattering,” Trans. Am. Philos. Soc. 44, Pt. 6, 643–728 (1954).

Evans, K. F.

For example, K. F. Evans, G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[CrossRef]

Fitch, B. W.

B. W. Fitch, R. L. Walraven, D. E. Bradley, “Polarization of light reflected from grain crops during the heading growth stage,” Remote Sensing Environ. 15, 263–268 (1984).
[CrossRef]

Fraser, R. S.

R. S. Fraser, “Atmospheric neutral points outside of the principal plane,” Contrib. Atmos. Phys. 54, 286–297 (1981).

R. S. Fraser, “Atmospheric neutral points over water,” J. Opt. Soc. Am. 58, 1029–1031 (1968). Also see Ref. 1, pp. 381–382.
[CrossRef]

Gambling, D. J.

D. J. Gambling, B. Billard, “A study of the polarization of skylight,” Aust. J. Phys. 20, 675–681 (1967).
[CrossRef]

Gerharz, R.

R. Gerharz, “Self polarization in refractive systems,” Optik 43, 471–485 (1975). Coulson calls self-polarization parasitic polarization (Ref. 1, p. 556).

Harris, F. S.

Hennings, D.

T. Prosch, D. Hennings, E. Raschke, “Video polarimetry: a new imaging technique in atmospheric science,” Appl. Opt. 22, 1360–1363 (1983). Also see Ref. 1, p. 554.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 50–53.

Lee, R. L.

R. L. Lee, “Horizon brightness revisited: measurements and a model of clear-sky radiances,” Appl. Opt. 33, 4620–4628, 4959 (1994).

R. L. Lee, “Colorimetric calibration of a video digitizing system: algorithm and applications,” Color Res. Appl. 13, 180–186 (1988).
[CrossRef]

McCartney, E. J.

E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976), pp. 213, 268.

Neuberger, H.

H. Neuberger, Introduction to Physical Meteorology, Revised ed. (Pennsylvania State University, University Park, Pa., 1957), pp. 194–206.

Prosch, T.

T. Prosch, D. Hennings, E. Raschke, “Video polarimetry: a new imaging technique in atmospheric science,” Appl. Opt. 22, 1360–1363 (1983). Also see Ref. 1, p. 554.

Raschke, E.

T. Prosch, D. Hennings, E. Raschke, “Video polarimetry: a new imaging technique in atmospheric science,” Appl. Opt. 22, 1360–1363 (1983). Also see Ref. 1, p. 554.

Sekera, Z.

Stephens, G. L.

For example, K. F. Evans, G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[CrossRef]

Strutt, J. W.

J. W. Strutt, “On the light from the sky, its polarization and colour,” Philos. Mag. 41, 107–120, 274–279 (1871).

Volz, F. E.

Walraven, R. L.

B. W. Fitch, R. L. Walraven, D. E. Bradley, “Polarization of light reflected from grain crops during the heading growth stage,” Remote Sensing Environ. 15, 263–268 (1984).
[CrossRef]

Whitehead, V. S.

For a remote-sensing application of P derived from photographs, see K. L. Coulson, V. S. Whitehead, C. Campbell, “Polarized views of the earth from orbital altitude,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 35–41 (1986). Narrow-FOV photographic polarimetry that uses a Savart plate is discussed in R. Gerharz, “Polarization of scattered horizon light in inclement weather,” Arch. Meteorol. Geophys. Bioklimatol. Ser. A 26, 265–273 (1977).

Adv. Geophys. (1)

K. Bullrich, “Scattered radiation in the atmosphere and the natural aerosol,” Adv. Geophys. 10, 99–260 (1964). Polarization measurements that span half of the sky dome appear on pp. 212–215 (Figs. 49 and 50). Also see Ref. 1, pp. 216–218, 311, 325, 327.
[CrossRef]

Appl. Opt. (6)

Aust. J. Phys. (1)

D. J. Gambling, B. Billard, “A study of the polarization of skylight,” Aust. J. Phys. 20, 675–681 (1967).
[CrossRef]

Color Res. Appl. (1)

R. L. Lee, “Colorimetric calibration of a video digitizing system: algorithm and applications,” Color Res. Appl. 13, 180–186 (1988).
[CrossRef]

Contrib. Atmos. Phys. (1)

R. S. Fraser, “Atmospheric neutral points outside of the principal plane,” Contrib. Atmos. Phys. 54, 286–297 (1981).

J. Opt. Soc. Am. (2)

Z. Sekera, “Light scattering in the atmosphere and the polarization of sky light,” J. Opt. Soc. Am. 47, 484–490 (1957).
[CrossRef]

R. S. Fraser, “Atmospheric neutral points over water,” J. Opt. Soc. Am. 58, 1029–1031 (1968). Also see Ref. 1, pp. 381–382.
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

For example, K. F. Evans, G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[CrossRef]

Optik (1)

R. Gerharz, “Self polarization in refractive systems,” Optik 43, 471–485 (1975). Coulson calls self-polarization parasitic polarization (Ref. 1, p. 556).

Philos. Mag. (1)

J. W. Strutt, “On the light from the sky, its polarization and colour,” Philos. Mag. 41, 107–120, 274–279 (1871).

Remote Sensing Environ. (1)

B. W. Fitch, R. L. Walraven, D. E. Bradley, “Polarization of light reflected from grain crops during the heading growth stage,” Remote Sensing Environ. 15, 263–268 (1984).
[CrossRef]

Trans. Am. Philos. Soc. (1)

S. Chandrasekhar, D. Elbert, “The illumination and polarization of the sunlit sky on Rayleigh scattering,” Trans. Am. Philos. Soc. 44, Pt. 6, 643–728 (1954).

Other (31)

H. Neuberger, Introduction to Physical Meteorology, Revised ed. (Pennsylvania State University, University Park, Pa., 1957), pp. 194–206.

Reference 1, pp. 269–270, lists skylight polarization studies by several of Rayleigh’s contemporaries.

K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (Deepak, Hampton, Va., 1988), pp. 375–391.

See Ref. 1, pp. 377–378.

E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976), pp. 213, 268.

For examples, see Ref. 1, pp. 256–261.

Ref. 5, p. 197.

Ref. 1, p. 233. Also see E. Collett , Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), pp. 34–39.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 50–53.

Ref. 20, p. 53.

See Ref. 1, p. 254 and Ref. 20, p. 50.

Ref. 20, pp. 46, 50. χ has the same direction as skylight’s plane of polarization but avoids the conceptual difficulties that a plane of (partial) polarization entails.

For examples, see Ref. 1, pp. 554–555 and 565–566.

For example, see Ref. 5, pp. 194–197. Lines of zero Q are still called neutral lines (Ref. 1, pp. 254–258).

ϕrel ranges between 0° and 360°, with values increasing clockwise from the Sun’s azimuth.

Ref. 1, p. 254.

Ref. 20, pp. 382–383. Equation (4) also defines polarization for specular reflection from planar surfaces. χ is horizontal for linear polarization by reflection from horizontal surfaces (e.g., calm water). To measure this polarization, once again set Eq. (4)’s 0° direction parallel to χ (i.e., horizontal).

Ref. 20, p. 54.

Ref. 14, pp. 198–199.

Ref. 14, pp. 136–139. Note that 475 nm is a dominant wavelength typical of clear skies.

See Ref. 1, pp. 522–525, for a discussion of partial polarization on reflection by water.

Ref. 1, p. 311 (Fig. 5.22). Large near-horizon pQ gradients at 90° from a low Sun appear consistently in my polarization maps.

Usually red pixels in Figs. 2–4 are the result of identical 24-bit colors in the original digital images; so, in a limited sense, the maps do include points where pQ and P = 0.0 exactly, but this equality is just an artifact of the resolution with which the slide scanner quantized scene radiances.

For a remote-sensing application of P derived from photographs, see K. L. Coulson, V. S. Whitehead, C. Campbell, “Polarized views of the earth from orbital altitude,” in Ocean Optics VIII, M. A. Blizard, ed., Proc. SPIE637, 35–41 (1986). Narrow-FOV photographic polarimetry that uses a Savart plate is discussed in R. Gerharz, “Polarization of scattered horizon light in inclement weather,” Arch. Meteorol. Geophys. Bioklimatol. Ser. A 26, 265–273 (1977).

For example, see Ref. 1, p. 261 (Fig. 4.36). θv is 0° at the astronomical horizon, except in Figs. 10–13, where θv is 0° at the slightly higher mean topographic horizon (see Fig. 3).

My polarizer’s H90 is fairly uniform at visible wavelengths, although crossed pairs of such polarizers do transmit a dim violet from a white-light source. Because skylight dominant wavelengths at the Earth’s surface typically are 475 nm or more, the increase in photographic polarizers’ H90 at shorter wavelengths is unlikely to appreciably bias observations of skylight polarization.

See Ref. 1, p. 582, for the general form of these Mueller matrix calculations.

As noted above, P measured by the four-image technique depends only on a polarizer’s relative (rather than absolute) directions of 0°, 45°, 90°, and 135°. In other words, the four-image 0° direction can differ arbitrarily from χ.

See Ref. 1, pp. 254–258.

Ref. 20, pp. 112–113.

Ref. 1, pp. 391–393.

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Figures (14)

Fig. 1
Fig. 1

Signed polarization as a function of scattering angle Ψ for scattering by a small sphere (approximately molecular scattering) and by an ensemble of spherical haze droplets whose size spectrum follows the Deirmendjian haze-M distribution at λ = 475 nm. The haze droplets’ real and imaginary refractive indices are 1.5 and 0.01, respectively.

Fig. 2
Fig. 2

Maps of clear-sky p Q at the coastal site of Chesapeake Beach, Md., on 6 February 1996. In map (a), the Sun elevation = 1.8° at 2220 UTC and azimuth relative to the Sun ϕrel = 180° at map center. In map (b), the Sun elevation = -0.46° at 2234 UTC and the p Q maximum occurs at ϕrel = 270°. Each map’s angular size is ∼23.5° × 36°, and every map gray level spans Δp Q = 0.05; pixels with |p Q | < 0.002 are colored red.

Fig. 3
Fig. 3

Maps of clear-sky polarization p Q at Marion Center, Pa., on 31 August 1996. This inland site is ∼425 km from the Atlantic Ocean, and its elevation is 450 m above mean sea level. In map (a), the Sun elevation = -0.7° at 2352 UTC and the center ϕrel = 0°; in map (b), the Sun elevation = 8.7° at 2259 UTC and the center ϕrel = 104°. Each map’s angular size is ∼24.3° × 35.7°.

Fig. 4
Fig. 4

Comparison of p Q [map (a)] and P [map (b)] for the antisolar clear sky at Annapolis, Md., on 12 March 1997 at 2255 UTC. In both maps, ϕrel = 180° is marked and the Sun elevation = 2.4°. In map (a), pixels are colored red for |p Q | < 0.002; in map (b) the criterion is P < 0.005. Each map’s angular size is ∼36.2° × 23.8°.

Fig. 5
Fig. 5

Depolarization factor d(ϑ) as a function of the misalignment ϑ of my polarizing filter’s transmission axis relative to χ. If a light source were completely linearly polarized, these d(ϑ) would equal its P [Eq. (5)] as measured for polarizer misalignments of 0° ≤ ϑ ≤ 5°.

Fig. 6
Fig. 6

Clear-sky polarization component p Q (≈P here) at Chesapeake Beach, Md., on 6 February 1996 at 2220 UTC for ϕrel = 180° [see Fig. 2(a)]. Each error bar spans two standard deviations of p Q at the given θ v and is calculated from the standard deviations of Fig. 7’s radiances. The above-horizon rms difference between observed polarizations and those predicted by a double-scattering model = 0.0359. The model wavelength is 475 nm, and its molecular and aerosol normal optical depths are 0.15 and 0.05, respectively.

Fig. 7
Fig. 7

Polarized clear-sky radiances looking ENE over the Chesapeake Bay (Chesapeake Beach, Md.) on 6 February 1996 at 2220 UTC. The Sun’s elevation = 1.8°, and the measurement meridian’s azimuth relative to it (ϕrel) is 180°. Radiances were averaged 2° on either side of the principal plane, and the L(0°) were measured with the polarizing filter’s transmission axis perpendicular to that plane (i.e., oriented horizontal). Multiplying the scaled radiances by L scale approximates the absolute radiances at the film plane, including the effects of the filter’s average spectral transmittance T.

Fig. 8
Fig. 8

Polarized clear-sky radiances looking SSE at Chesapeake Beach, Md., on 6 February 1996 at 2234 UTC. The Sun elevation = -0.46°, and ϕrel = 270°. Radiances were averaged 2° on either side of ϕrel = 270°, and the L(0°) were measured at the filter orientation that produced the brightest image of this scene.

Fig. 9
Fig. 9

Clear-sky p Q (∼P) at Chesapeake Beach, Md., on 6 February 1996 at 2234 UTC for ϕrel = 270° [see Fig. 2(b)]. The above-horizon rms difference between observed polarizations and those predicted by a double-scattering model = 0.0309. The model wavelength is 475 nm, and its molecular and aerosol normal optical depths are 0.15 and 0.05, respectively.

Fig. 10
Fig. 10

Polarized clear-sky radiances in the sunset sky near Marion Center, Pa., on 31 August 1996 at 2352 UTC. The Sun elevation = -0.7°, and ϕrel = 0°. Radiances were averaged 2.5° on either side of the principal plane, and the L(0°) were measured with the polarizing filter’s transmission axis perpendicular to that plane (i.e., oriented horizontally).

Fig. 11
Fig. 11

Clear-sky p Q (≈P here) near Marion Center, Pa., on 31 August 1996 at 2352 UTC for ϕrel = 0° [see Fig. 3(a)]. Each error bar spans two standard deviations of p Q at the given θ v and is calculated from the standard deviations of Fig. 10’s radiances.

Fig. 12
Fig. 12

Polarized clear-sky radiances near Marion Center, Pa., on 31 August 1996 at 2259 UTC. The Sun elevation = 8.7°, and ϕrel = 104°. Radiances were averaged 2.5° on either side of ϕrel = 104°, and the L(0°) were measured at the filter orientation that produced the brightest image of this scene. Here small cumulus usually are brighter than their surroundings, with the notable exception of cloud D’s L(0°).

Fig. 13
Fig. 13

Clear-sky p Q (∼P) near Marion Center, Pa., on 31 August 1996 at 2259 UTC for ϕrel = 104° [see Fig. 3(b)]. Although most small cumulus here are local p Q minima, note that cloud B is both a local p Q minimum and maximum.

Fig. 14
Fig. 14

Clear-sky polarization P and its component |p Q | at Annapolis, Md., on 12 March 1997 at 2255 UTC for ϕrel = 180° (see Fig. 4). Both P and p Q were averaged 2.5° on either side of the principal plane. Each error bar spans two standard deviations of P at the given θ v . Topography that extends slightly above the astronomical horizon produces the large variations in P and |p Q | below θ v = 2°.

Equations (5)

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P T = Q 2 + U 2 + V 2 1 / 2 I ,
P = Q 2 + U 2 1 / 2 I ,
I = 0.5 L 0 ° + L 45 ° + L 90 ° + L 135 ° , Q = L 0 ° - L 90 ° , U = L 45 ° - L 135 ° .
p Q = L 0 ° - L 90 ° L 0 ° + L 90 ° ,
P = L max - L min L max + L min .

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