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Below the horizon—the physics of extreme visual ranges

Michael Vollmer

Open Access Open Access

Abstract

Visual ranges of up to 440 km have recently been documented by photographs of ground-based observers. A report from 1948 claimed a record visual range from a plane of more than 530 km and a similar recent observation from 2017 was documented by a photo. Such extreme visual ranges can in principle be explained by the interplay of refraction and light scattering. However, they require optimal atmospheric conditions, and cleverly chosen locations and times.

© 2020 Optical Society of America

Full Article  |  PDF Article

Corrections

26 June 2020: Corrections were made to Figs. 8 and 9.

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References

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  • Year
  • Author
  • Publication
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    [Crossref]
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    [Crossref]
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    [Crossref]
  32. H. Özkayna, A. D. Schatz, G. D. Thurston, R. G. Isaacs, and R. B. Husar, “Relationships between aerosol extinction coefficients derived from airport visual range observations and alternative measures of airborne particle mass,” J. Air Pollut. Control Association 35, 1176–1185 (1985).
    [Crossref]
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    [Crossref]
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    [Crossref]
  36. M. Vollmer and J. A. Shaw, “Extended visual range during solar eclipses,” Appl. Opt. 57, 3250–3259 (2018).
    [Crossref]
  37. R. Penndorf, “Tables of the refractive index for standard air and the Rayleigh scattering coefficient for the spectral region between 0.2 and 20 µ and their application to atmospheric optics,” J. Opt. Soc. Am. 47, 176–182 (1957).
    [Crossref]
  38. B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Oceanic Technol. 16, 1854–1861 (1999).
    [Crossref]
  39. D. K. Lynch and S. Mazuk, “On the colors of distant objects,” Appl. Opt. 44, 5737–5745 (2005).
    [Crossref]
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  42. Planefinder App, https://planefinder.net .

2018 (1)

2014 (1)

S. Xiao, Q. Y. Wang, J. J. H. Cao, R.-J. Huang, W. D. Chen, Y. M. Han, H. M. Xu, S. X. Liu, Y. Q. Zhou, P. Wang, J. Q. Zhang, and C. L. Zhan, “Long-term trends in visibility and impacts of aerosol composition on visibility impairment in Baoji, China,” Atmos. Res. 149, 88–95 (2014).
[Crossref]

2006 (1)

A. T. Young, “Understanding astronomical refraction,” Observatory 126, 82–115 (2006).

2005 (1)

2003 (1)

B. B. Balsley, R. G. Frehlich, M. L. Jensen, Y. Meillier, and A. Muschinski, “Extreme gradients in the nocturnal boundary layer: structure, evolution, and potential causes,” J. Atmos. Sci. 60, 2496–2508 (2003).
[Crossref]

2001 (1)

R. D. Garreaud, J. Rutllant, J. Quintana, J. Carrasco, and P. Minnis, “CIMAR-5: a snapshot of the lower troposphere over the subtropical Southeast Pacific,” Bull. Am. Meteorol. Soc. 82, 2193–2207 (2001).
[Crossref]

1999 (1)

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Oceanic Technol. 16, 1854–1861 (1999).
[Crossref]

1998 (1)

S. Verhulst and F. W. Maes, “Scotopic vision in colour-blinds,” Vis. Res. 38, 3387–3390 (1998).
[Crossref]

1987 (1)

C. Owsley and M. E. Sloane, “Contrast sensitivity, acuity, and the perception of ‘real-world’ targets,” Br. J. Ophthalmol. 71, 791–796 (1987).
[Crossref]

1986 (1)

C. F. Bohren and A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
[Crossref]

1985 (1)

H. Özkayna, A. D. Schatz, G. D. Thurston, R. G. Isaacs, and R. B. Husar, “Relationships between aerosol extinction coefficients derived from airport visual range observations and alternative measures of airborne particle mass,” J. Air Pollut. Control Association 35, 1176–1185 (1985).
[Crossref]

1982 (1)

A. P. French, “How far is the horizon,” Am. J. Phys. 50, 795–799 (1982).
[Crossref]

1981 (2)

W. H. Lehn and B. A. German, “Novaya Zemlya effect: analysis of an observation,” Appl. Opt. 20, 2043–2047 (1981).
[Crossref]

M. A. Ferman, G. T. Wolff, and N. A. Kelly, “The nature and sources of haze in the Shenandoah Valley/Blue Ridge Mountains area,” J. Air Pollut. Control Association 31, 1074–1082 (1981).
[Crossref]

1980 (1)

M. M. Meyer, J. E. Jiusto, and G. G. Lala, “Measurements of visual range and radiation-fog (haze) microphysics,” J. Atmos. Sci. 37, 622–629 (1980).
[Crossref]

1979 (1)

1977 (1)

K. Horváth, “Visibility in Geodesy,” Periodica Polytechnica Civil Eng. 21, 87–100 (1977).

1975 (1)

C. N. Davies, “Visual range and size of atmospheric particles,” J. Aerosol Sci. 6, 335–347 (1975).
[Crossref]

1973 (1)

E. Pöppel and L. O. Harvey, “Light-difference threshold and subjective brightness in the periphery of the visual field,” Psychol. Forsch. 36, 145–161 (1973).
[Crossref]

1957 (1)

1949 (1)

R. G. Beuttell and A. W. Brewer, “Instruments for the measurement of the visual range,” J. Sci. Instrum. 26, 357–359 (1949).
[Crossref]

1947 (1)

1946 (1)

1942 (1)

H. G. Houghton, “The transmission of light in the atmosphere with applications to aviation,” J. Aeronaut. Sci. 9, 103–107 (1942).
[Crossref]

1939 (1)

H. G. Houghton, “On the relation between visibility and the constitution of clouds and fog,” J. Aeronaut. Sci. 6, 408–411 (1939).
[Crossref]

1935 (1)

W. E. K. Middleton, “Unusually great visual range over Ontario,” Q. J. R. Meteorol. Soc. 61, 411–416 (1935).
[Crossref]

1924 (1)

H. Koschmieder, “Theorie der horizontalen Sichtweite,” Beiträge zur Physik der freien Atmosphäre, Vol. XII, 33–53 (1924), “Theorie der horizontalen Sichtweite II,” Beiträge zur Physik der freien Atmosphäre XII, 171–181 (1924); reprinted in Selected Papers on Scattering in the Atmosphere, C. F. Bohren, ed., SPIE milestone Series, Vol. MS7, SPIE, Bellingham (1989).

Balsley, B. B.

B. B. Balsley, R. G. Frehlich, M. L. Jensen, Y. Meillier, and A. Muschinski, “Extreme gradients in the nocturnal boundary layer: structure, evolution, and potential causes,” J. Atmos. Sci. 60, 2496–2508 (2003).
[Crossref]

Berg, H.

H. Berg, Allgemeine Meteorologie – Einführung in die Physik der Atmosphäre, (F. Dümmler Verlag, Bonn / Germany, 1948).

Beuttell, R. G.

R. G. Beuttell and A. W. Brewer, “Instruments for the measurement of the visual range,” J. Sci. Instrum. 26, 357–359 (1949).
[Crossref]

Blackwell, H. R.

Bodhaine, B. A.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Oceanic Technol. 16, 1854–1861 (1999).
[Crossref]

Bohren, C. F.

C. F. Bohren and A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
[Crossref]

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley, 2006).

Booker, R. L.

C. A. Douglas and R. L. Booker, Visual Range: Concepts, Instrumental Determination, and Aviation Applications (National Bureaus of Standards Monograph, 1977), p. 159.

Brewer, A. W.

R. G. Beuttell and A. W. Brewer, “Instruments for the measurement of the visual range,” J. Sci. Instrum. 26, 357–359 (1949).
[Crossref]

Cao, J. J. H.

S. Xiao, Q. Y. Wang, J. J. H. Cao, R.-J. Huang, W. D. Chen, Y. M. Han, H. M. Xu, S. X. Liu, Y. Q. Zhou, P. Wang, J. Q. Zhang, and C. L. Zhan, “Long-term trends in visibility and impacts of aerosol composition on visibility impairment in Baoji, China,” Atmos. Res. 149, 88–95 (2014).
[Crossref]

Carrasco, J.

R. D. Garreaud, J. Rutllant, J. Quintana, J. Carrasco, and P. Minnis, “CIMAR-5: a snapshot of the lower troposphere over the subtropical Southeast Pacific,” Bull. Am. Meteorol. Soc. 82, 2193–2207 (2001).
[Crossref]

Chen, W. D.

S. Xiao, Q. Y. Wang, J. J. H. Cao, R.-J. Huang, W. D. Chen, Y. M. Han, H. M. Xu, S. X. Liu, Y. Q. Zhou, P. Wang, J. Q. Zhang, and C. L. Zhan, “Long-term trends in visibility and impacts of aerosol composition on visibility impairment in Baoji, China,” Atmos. Res. 149, 88–95 (2014).
[Crossref]

Clothiaux, E. E.

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley, 2006).

Davies, C. N.

C. N. Davies, “Visual range and size of atmospheric particles,” J. Aerosol Sci. 6, 335–347 (1975).
[Crossref]

Dietze, G.

G. Dietze, Einführung in die Optik der Atmosphäre (Akad. Verlagsges. Leipzig / Germany, 1957).

Douglas, C. A.

C. A. Douglas and R. L. Booker, Visual Range: Concepts, Instrumental Determination, and Aviation Applications (National Bureaus of Standards Monograph, 1977), p. 159.

Dutton, E. G.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Oceanic Technol. 16, 1854–1861 (1999).
[Crossref]

Ferman, M. A.

M. A. Ferman, G. T. Wolff, and N. A. Kelly, “The nature and sources of haze in the Shenandoah Valley/Blue Ridge Mountains area,” J. Air Pollut. Control Association 31, 1074–1082 (1981).
[Crossref]

Fraser, A. B.

C. F. Bohren and A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
[Crossref]

Frehlich, R. G.

B. B. Balsley, R. G. Frehlich, M. L. Jensen, Y. Meillier, and A. Muschinski, “Extreme gradients in the nocturnal boundary layer: structure, evolution, and potential causes,” J. Atmos. Sci. 60, 2496–2508 (2003).
[Crossref]

French, A. P.

A. P. French, “How far is the horizon,” Am. J. Phys. 50, 795–799 (1982).
[Crossref]

Garreaud, R. D.

R. D. Garreaud, J. Rutllant, J. Quintana, J. Carrasco, and P. Minnis, “CIMAR-5: a snapshot of the lower troposphere over the subtropical Southeast Pacific,” Bull. Am. Meteorol. Soc. 82, 2193–2207 (2001).
[Crossref]

German, B. A.

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge University, 1980).

Han, Y. M.

S. Xiao, Q. Y. Wang, J. J. H. Cao, R.-J. Huang, W. D. Chen, Y. M. Han, H. M. Xu, S. X. Liu, Y. Q. Zhou, P. Wang, J. Q. Zhang, and C. L. Zhan, “Long-term trends in visibility and impacts of aerosol composition on visibility impairment in Baoji, China,” Atmos. Res. 149, 88–95 (2014).
[Crossref]

Harvey, L. O.

E. Pöppel and L. O. Harvey, “Light-difference threshold and subjective brightness in the periphery of the visual field,” Psychol. Forsch. 36, 145–161 (1973).
[Crossref]

Hecht, S.

Hendley, C. D.

Horváth, K.

K. Horváth, “Visibility in Geodesy,” Periodica Polytechnica Civil Eng. 21, 87–100 (1977).

Houghton, H. G.

H. G. Houghton, “The transmission of light in the atmosphere with applications to aviation,” J. Aeronaut. Sci. 9, 103–107 (1942).
[Crossref]

H. G. Houghton, “On the relation between visibility and the constitution of clouds and fog,” J. Aeronaut. Sci. 6, 408–411 (1939).
[Crossref]

Huang, R.-J.

S. Xiao, Q. Y. Wang, J. J. H. Cao, R.-J. Huang, W. D. Chen, Y. M. Han, H. M. Xu, S. X. Liu, Y. Q. Zhou, P. Wang, J. Q. Zhang, and C. L. Zhan, “Long-term trends in visibility and impacts of aerosol composition on visibility impairment in Baoji, China,” Atmos. Res. 149, 88–95 (2014).
[Crossref]

Husar, R. B.

H. Özkayna, A. D. Schatz, G. D. Thurston, R. G. Isaacs, and R. B. Husar, “Relationships between aerosol extinction coefficients derived from airport visual range observations and alternative measures of airborne particle mass,” J. Air Pollut. Control Association 35, 1176–1185 (1985).
[Crossref]

Isaacs, R. G.

H. Özkayna, A. D. Schatz, G. D. Thurston, R. G. Isaacs, and R. B. Husar, “Relationships between aerosol extinction coefficients derived from airport visual range observations and alternative measures of airborne particle mass,” J. Air Pollut. Control Association 35, 1176–1185 (1985).
[Crossref]

Jacobsen, M. Z.

M. Z. Jacobsen, Atmospheric Pollution (Cambridge University, 2002).

Jensen, M. L.

B. B. Balsley, R. G. Frehlich, M. L. Jensen, Y. Meillier, and A. Muschinski, “Extreme gradients in the nocturnal boundary layer: structure, evolution, and potential causes,” J. Atmos. Sci. 60, 2496–2508 (2003).
[Crossref]

Jiusto, J. E.

M. M. Meyer, J. E. Jiusto, and G. G. Lala, “Measurements of visual range and radiation-fog (haze) microphysics,” J. Atmos. Sci. 37, 622–629 (1980).
[Crossref]

Kelly, N. A.

M. A. Ferman, G. T. Wolff, and N. A. Kelly, “The nature and sources of haze in the Shenandoah Valley/Blue Ridge Mountains area,” J. Air Pollut. Control Association 31, 1074–1082 (1981).
[Crossref]

Koschmieder, H.

H. Koschmieder, “Theorie der horizontalen Sichtweite,” Beiträge zur Physik der freien Atmosphäre, Vol. XII, 33–53 (1924), “Theorie der horizontalen Sichtweite II,” Beiträge zur Physik der freien Atmosphäre XII, 171–181 (1924); reprinted in Selected Papers on Scattering in the Atmosphere, C. F. Bohren, ed., SPIE milestone Series, Vol. MS7, SPIE, Bellingham (1989).

Lala, G. G.

M. M. Meyer, J. E. Jiusto, and G. G. Lala, “Measurements of visual range and radiation-fog (haze) microphysics,” J. Atmos. Sci. 37, 622–629 (1980).
[Crossref]

Lamar, E. S.

Lehn, W. H.

Liu, S. X.

S. Xiao, Q. Y. Wang, J. J. H. Cao, R.-J. Huang, W. D. Chen, Y. M. Han, H. M. Xu, S. X. Liu, Y. Q. Zhou, P. Wang, J. Q. Zhang, and C. L. Zhan, “Long-term trends in visibility and impacts of aerosol composition on visibility impairment in Baoji, China,” Atmos. Res. 149, 88–95 (2014).
[Crossref]

Livingston, W.

D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge University, 2001).

Lynch, D. K.

D. K. Lynch and S. Mazuk, “On the colors of distant objects,” Appl. Opt. 44, 5737–5745 (2005).
[Crossref]

D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. (Cambridge University, 2001).

Maes, F. W.

S. Verhulst and F. W. Maes, “Scotopic vision in colour-blinds,” Vis. Res. 38, 3387–3390 (1998).
[Crossref]

Mazuk, S.

Meillier, Y.

B. B. Balsley, R. G. Frehlich, M. L. Jensen, Y. Meillier, and A. Muschinski, “Extreme gradients in the nocturnal boundary layer: structure, evolution, and potential causes,” J. Atmos. Sci. 60, 2496–2508 (2003).
[Crossref]

Meyer, M. M.

M. M. Meyer, J. E. Jiusto, and G. G. Lala, “Measurements of visual range and radiation-fog (haze) microphysics,” J. Atmos. Sci. 37, 622–629 (1980).
[Crossref]

Middleton, W. E. K.

W. E. K. Middleton, “Unusually great visual range over Ontario,” Q. J. R. Meteorol. Soc. 61, 411–416 (1935).
[Crossref]

W. E. K. Middleton, Vision Through the Atmosphere (University of Toronto, 1952).

W. E. K. Middleton, “Vision through the atmosphere,” in Handbuch der Physik (Springer, 1957), Vol. 48 (Geophysics II), pp. 254–287.

Minnaert, M. G. J.

M. G. J. Minnaert, Light and Color in the Outdoors (Springer, 1993), older English ed., The Nature of Light and Colour in the Open Air (Dover, 1954).

Minnis, P.

R. D. Garreaud, J. Rutllant, J. Quintana, J. Carrasco, and P. Minnis, “CIMAR-5: a snapshot of the lower troposphere over the subtropical Southeast Pacific,” Bull. Am. Meteorol. Soc. 82, 2193–2207 (2001).
[Crossref]

Muschinski, A.

B. B. Balsley, R. G. Frehlich, M. L. Jensen, Y. Meillier, and A. Muschinski, “Extreme gradients in the nocturnal boundary layer: structure, evolution, and potential causes,” J. Atmos. Sci. 60, 2496–2508 (2003).
[Crossref]

Owsley, C.

C. Owsley and M. E. Sloane, “Contrast sensitivity, acuity, and the perception of ‘real-world’ targets,” Br. J. Ophthalmol. 71, 791–796 (1987).
[Crossref]

Özkayna, H.

H. Özkayna, A. D. Schatz, G. D. Thurston, R. G. Isaacs, and R. B. Husar, “Relationships between aerosol extinction coefficients derived from airport visual range observations and alternative measures of airborne particle mass,” J. Air Pollut. Control Association 35, 1176–1185 (1985).
[Crossref]

Penndorf, R.

Pöppel, E.

E. Pöppel and L. O. Harvey, “Light-difference threshold and subjective brightness in the periphery of the visual field,” Psychol. Forsch. 36, 145–161 (1973).
[Crossref]

Quintana, J.

R. D. Garreaud, J. Rutllant, J. Quintana, J. Carrasco, and P. Minnis, “CIMAR-5: a snapshot of the lower troposphere over the subtropical Southeast Pacific,” Bull. Am. Meteorol. Soc. 82, 2193–2207 (2001).
[Crossref]

Rutllant, J.

R. D. Garreaud, J. Rutllant, J. Quintana, J. Carrasco, and P. Minnis, “CIMAR-5: a snapshot of the lower troposphere over the subtropical Southeast Pacific,” Bull. Am. Meteorol. Soc. 82, 2193–2207 (2001).
[Crossref]

Schatz, A. D.

H. Özkayna, A. D. Schatz, G. D. Thurston, R. G. Isaacs, and R. B. Husar, “Relationships between aerosol extinction coefficients derived from airport visual range observations and alternative measures of airborne particle mass,” J. Air Pollut. Control Association 35, 1176–1185 (1985).
[Crossref]

Shaw, J. A.

Shlaer, S.

Sloane, M. E.

C. Owsley and M. E. Sloane, “Contrast sensitivity, acuity, and the perception of ‘real-world’ targets,” Br. J. Ophthalmol. 71, 791–796 (1987).
[Crossref]

Slusser, J. R.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Oceanic Technol. 16, 1854–1861 (1999).
[Crossref]

Thurston, G. D.

H. Özkayna, A. D. Schatz, G. D. Thurston, R. G. Isaacs, and R. B. Husar, “Relationships between aerosol extinction coefficients derived from airport visual range observations and alternative measures of airborne particle mass,” J. Air Pollut. Control Association 35, 1176–1185 (1985).
[Crossref]

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[Crossref]

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[Crossref]

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Figures (9)
Fig. 1.
Fig. 1. Geometry of finding the distance to an elevated object.

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Fig. 2.
Fig. 2. Schematic curvature of light rays in a concentric atmosphere with given lapse rates. An observer sees rays coming from the various directions indicated by the arrows.

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Fig. 3.
Fig. 3. Schematic ray paths due to rectilinear geometry, normal refraction, and abnormal refraction due to inversion layers.

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Fig. 4.
Fig. 4. (a) Geometry for curved light ray in spherical geometry and (b) most simple approximation of its left part in flat Earth geometry.

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Fig. 5.
Fig. 5. (a) Decrease of effective scattering coefficient as a function of minimum height of the light ray (for ${H} = {8}\;{\rm km}$, ${{h}_{\max}} = {4}\;{\rm km}$) and (b) corresponding visual range for a given wavelength averaged scattering coefficient ${\beta _0}$ at sea level.

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Fig. 6.
Fig. 6. Geometry for a symmetrical light ray with given minimum height.

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Fig. 7.
Fig. 7. Map including two observation sites in the Pyrenees and two observed objects in the Alps. Details, see text (image with some added distances and lines based on Google Maps).

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Fig. 8.
Fig. 8. Record long-distance observation of an Alps mountain peak in a distance of 443 km from the Pyrenees. (a) Scenery, (b) enlarged part with identified peaks, (c) original photo, courtesy photographer Marc Bret.

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Fig. 9.
Fig. 9. Record airplane observation; (a) wider panorama; (b) Mount Blanc enlarged, from a distance of about 538 km (details, see text). Courtesy photographer Ramon Ibarz.

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Equations (9)

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

x ≈ 2 R E h o b s .
x ≈ 2 R E h o b s + 2 R E h o b j .
k = 35 − α 150 .
L ( x ) = L 0 e − β ( λ ) x ,
− ln ⁡ ( C ) = β ⋅ d .
d = 3.9 β .
β ( h ) = β 0 e − h H ,
β e f f ≈ β 0 H h max − h min [ e − h min H − e − h max H ] .
C = e − β d 2 ( 1 − a + a e − β d 1 ) .

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