Abstract

We were able to transform the usual refraction equations for an observer within a spherically symmetric atmosphere into an Abel integral equation that has a unique inverse. It is now possible to extract accurate temperature profiles of the atmosphere below an observer over the marine boundary layer with use of simple edge-finding software on solar images at sunset. Several examples are presented demonstrating the efficiency of the method.

© 1997 Optical Society of America

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References

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  1. A. B. Fraser, “Simple solution for obtaining a temperature profile from the inferior mirage,” Appl. Opt. 18, 1724–1731 (1979).
    [Crossref] [PubMed]
  2. W. H. Lehn, “Inversion of superior mirage data to compute temperature profiles,” J. Opt. Soc. Am. 73, 1622–1625 (1983).
    [Crossref]
  3. P. D. Sozou, “Inversion of mirage data: an optimization approach,” J. Opt. Soc. Am. A 11, 125–134 (1994).
    [Crossref]
  4. N. A. Vasilenko, K. P. Gaykovich, M. I. Sumin, “Determination of atmospheric temperature and pressure profiles from astronomical refraction measurements near the horizon,” Izv. Akad. Nauk SSSR, Atmos. Ocean. Phys. 22, 798–804 (1986).
  5. Concise Dictionary of Scientific Biography (Scribner, New York, 1981), p. 643. It is interesting to note that Willebrord Snel van Royen used only one l in his last name. In the English-speaking community, however, a convention of using two l’s has arisen, and it is difficult to find an English physics text that does not use this convention.
  6. A. B. Fraser, “Solutions of the refraction and extinction integrals for use in inversion and image formation,” Appl. Opt. 16, 160–165 (1977).
    [Crossref] [PubMed]
  7. D. H. Menzel, Fundamental Formulas of Physics (Prentice-Hall, New York, 1955).
  8. H. Bateman, “The solution of the integral equation connecting the velocity of propagation of an earthquake-wave in the interior of the Earth with the times which the disturbance takes to travel to the different stations on the Earth’s surface,” Philos. Mag. 6, 19, 576–19,587 (1910).
  9. A. Mallama, “Shoemaker-Levy 9 aftermath,” CCD Astron. 3, 4–5 (1996).
  10. U.S. Standard Atmosphere 1976 (U.S. Government Printing Office, Washington, D.C., 1976).
  11. P. Parviainen, “The setting Sun seen through two unusually well-defined inversion layers in the lower atmosphere,” Sky Telesc. 83, 708 (1992).
  12. A. T. Young, G. W. Kattawar, “Sunset science. I. The mock mirage,” Appl. Opt. 36, 2689–2700 (1997).
    [Crossref] [PubMed]
  13. W. H. Lehn, W. K. Silvester, D. M. Fraser, “Mirages with atmospheric gravity waves,” Appl. Opt. 33, 4639–4643 (1994).
    [Crossref] [PubMed]
  14. C. Floor, “The effect of waves on the image of the low sun and on a reflection in water,” Weather 37, 148–151 (1982).
    [Crossref]

1997 (1)

1996 (1)

A. Mallama, “Shoemaker-Levy 9 aftermath,” CCD Astron. 3, 4–5 (1996).

1994 (2)

1992 (1)

P. Parviainen, “The setting Sun seen through two unusually well-defined inversion layers in the lower atmosphere,” Sky Telesc. 83, 708 (1992).

1986 (1)

N. A. Vasilenko, K. P. Gaykovich, M. I. Sumin, “Determination of atmospheric temperature and pressure profiles from astronomical refraction measurements near the horizon,” Izv. Akad. Nauk SSSR, Atmos. Ocean. Phys. 22, 798–804 (1986).

1983 (1)

1982 (1)

C. Floor, “The effect of waves on the image of the low sun and on a reflection in water,” Weather 37, 148–151 (1982).
[Crossref]

1979 (1)

1977 (1)

1910 (1)

H. Bateman, “The solution of the integral equation connecting the velocity of propagation of an earthquake-wave in the interior of the Earth with the times which the disturbance takes to travel to the different stations on the Earth’s surface,” Philos. Mag. 6, 19, 576–19,587 (1910).

Bateman, H.

H. Bateman, “The solution of the integral equation connecting the velocity of propagation of an earthquake-wave in the interior of the Earth with the times which the disturbance takes to travel to the different stations on the Earth’s surface,” Philos. Mag. 6, 19, 576–19,587 (1910).

Floor, C.

C. Floor, “The effect of waves on the image of the low sun and on a reflection in water,” Weather 37, 148–151 (1982).
[Crossref]

Fraser, A. B.

Fraser, D. M.

Gaykovich, K. P.

N. A. Vasilenko, K. P. Gaykovich, M. I. Sumin, “Determination of atmospheric temperature and pressure profiles from astronomical refraction measurements near the horizon,” Izv. Akad. Nauk SSSR, Atmos. Ocean. Phys. 22, 798–804 (1986).

Kattawar, G. W.

Lehn, W. H.

Mallama, A.

A. Mallama, “Shoemaker-Levy 9 aftermath,” CCD Astron. 3, 4–5 (1996).

Menzel, D. H.

D. H. Menzel, Fundamental Formulas of Physics (Prentice-Hall, New York, 1955).

Parviainen, P.

P. Parviainen, “The setting Sun seen through two unusually well-defined inversion layers in the lower atmosphere,” Sky Telesc. 83, 708 (1992).

Silvester, W. K.

Sozou, P. D.

Sumin, M. I.

N. A. Vasilenko, K. P. Gaykovich, M. I. Sumin, “Determination of atmospheric temperature and pressure profiles from astronomical refraction measurements near the horizon,” Izv. Akad. Nauk SSSR, Atmos. Ocean. Phys. 22, 798–804 (1986).

Vasilenko, N. A.

N. A. Vasilenko, K. P. Gaykovich, M. I. Sumin, “Determination of atmospheric temperature and pressure profiles from astronomical refraction measurements near the horizon,” Izv. Akad. Nauk SSSR, Atmos. Ocean. Phys. 22, 798–804 (1986).

Young, A. T.

Appl. Opt. (4)

CCD Astron. (1)

A. Mallama, “Shoemaker-Levy 9 aftermath,” CCD Astron. 3, 4–5 (1996).

Izv. Akad. Nauk SSSR, Atmos. Ocean. Phys. (1)

N. A. Vasilenko, K. P. Gaykovich, M. I. Sumin, “Determination of atmospheric temperature and pressure profiles from astronomical refraction measurements near the horizon,” Izv. Akad. Nauk SSSR, Atmos. Ocean. Phys. 22, 798–804 (1986).

J. Opt. Soc. Am. (1)

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

Philos. Mag. (1)

H. Bateman, “The solution of the integral equation connecting the velocity of propagation of an earthquake-wave in the interior of the Earth with the times which the disturbance takes to travel to the different stations on the Earth’s surface,” Philos. Mag. 6, 19, 576–19,587 (1910).

Sky Telesc. (1)

P. Parviainen, “The setting Sun seen through two unusually well-defined inversion layers in the lower atmosphere,” Sky Telesc. 83, 708 (1992).

Weather (1)

C. Floor, “The effect of waves on the image of the low sun and on a reflection in water,” Weather 37, 148–151 (1982).
[Crossref]

Other (3)

Concise Dictionary of Scientific Biography (Scribner, New York, 1981), p. 643. It is interesting to note that Willebrord Snel van Royen used only one l in his last name. In the English-speaking community, however, a convention of using two l’s has arisen, and it is difficult to find an English physics text that does not use this convention.

U.S. Standard Atmosphere 1976 (U.S. Government Printing Office, Washington, D.C., 1976).

D. H. Menzel, Fundamental Formulas of Physics (Prentice-Hall, New York, 1955).

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Figures (6)

Fig. 1
Fig. 1

Refraction geometry for a spherically symmetric atmosphere. The path of any light ray can be described by the coordinates r and θ for spherically symmetric atmospheres.

Fig. 2
Fig. 2

Geometry for probing the atmosphere below the observer. In this case the observer is considered to be within the atmosphere. The ray begins at the source considered to be at infinity and travels to a height ro, then to the ray vertex at rv, and then back out to ro. The angle βo is labeled with use of symmetry.

Fig. 3
Fig. 3

Refraction angles and the solution to the inverse problem for an atmosphere with two temperature inversions below the observer. The agreement between the true and computed profiles demonstrates the effectiveness of the integral inversion method even when discrete data are used. The related features A and B are labeled on each plot.

Fig. 4
Fig. 4

Example sunset and the corresponding disk contours. The photographs (a) were taken on 29 August 1995 by Andrew Young on the coast near San Diego, California. The observation site has a latitude of 32.72 °N, a longitude of 117.17 °W, and an elevation of 100 m. From left to right the images were taken at 7:13:59.5, 7:16:10, 7:16:24, 7:17:21, and 7:18:08.5 p.m. PDT. The disk contours (b) were found with edge-finding software. The tick marks show the location of the astronomical horizon.

Fig. 5
Fig. 5

Refraction angles from a sunset. These refraction angles were found with the disk contours in Fig. 4. An example distorted disk is shown on its side for reference of the major features.

Fig. 6
Fig. 6

The temperature profile computed from the refraction angles shown in Fig. 5.

Equations (19)

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dθ=cot βrdr
dnrr cos βds=0,
dR=-cot βndn,
Rno, ro, βo=-nons±noro cos βonn2r2-no2ro2 cos2 βo1/2dn,
Rno, ro, βo=-nonv-noro cos βonn2r2-no2ro2 cos2 βo1/2dn-nvno+noro cos βonn2r2-no2ro2 cos2 βo1/2dn-nons+noro cos βonn2r2-no2ro2 cos2 βo1/2dn,
Rno, ro, βo-Rno, ro, βo=nonv2noro cos βonn2r2-no2ro2 cos2 βo1/2dn=rorvdndr2noro cos βonn2r2-no2ro2 cos2 βo1/2dr=noronvrvd lnndnr2noro cos βon2r2-no2ro2 cos2 βo1/2dnr.
Fs=asϕts-tq dt,
ϕt=sin qππddtatFst-s1-qds.
q=12; ω=nr; α=noro cos βo=nvrv;  b=noro; t=1ω2; s=1α2; a=1b2.
Fα=ba-2αϕωω2ω2-α21/2dω
ϕω=ω3πddωbωωFαα2α2-ω21/2dα=ω2πddωbωFαα2-ω21/2dα,
ϕω=-ω2d lnndω=ω2πddωbωRno, ro, βo-Rno, ro, βoα2-ω21/2dα.
nr=no×exp1πnrnoroRno, ro, βo-Rno, ro, βoα2-n2r21/2dα.
τno, ro, βo-τno, ro, βo=rvrv2crnrn2r2-no2ro2 cos2 βo1/2dr,
cr=-1π×ddrnrnoronrτno, ro, βo-τno, ro, βoαα2-n2r21/2dα.
dpdr=-GMEr2ρ,
p=ρRTM,
n=1+ερ,
n=1+εMpRT.

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