Abstract

Optical data containing the relative positions of an observer, a target, and an image of the target are inverted with a set of nonlinear polynomial equations to obtain a temperature profile near the earth’s surface. The temperature that is predicted at a specific height with the inversion of optical data is verified with a temperature that was measured with thermocouples when the optical data were collected. When the maximum uncertainty of ±0.04°C in the measured temperature is known, the largest difference between the measured temperature and the temperature obtained with the optical data is within ±0.02°C.

© 1979 Optical Society of America

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References

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  1. A. B. Fraser, W. H. Mach, Sci. Am. 234, 102 (1974).
    [CrossRef]
  2. N. K. Johnson, O. F. T. Roberts, Q. J. R. Meteorol. Soc. 51, 131 (1925).
    [CrossRef]
  3. F. J. W. Whipple, “Meteorological Optics” in Dictionary of Applied Physics, Vol. III, ed. R. Glazebrook (Macmillan, London, 1921), p. 519.
  4. D. Brunt, Q. J. R. Meteorol. Soc. 55, 335 (1929).
    [CrossRef]
  5. R. G. Fleagle, Bull. Am. Meteorol. Soc. 31, 51 (1950).
  6. R. G. Fleagle, J. Meteorol. 13, 160 (1956).
    [CrossRef]
  7. A. L. Friend, C. R. Stearns, Boundary-Layer Meteorol. 1, 227 (1970).
    [CrossRef]
  8. J. K. Sparkman, “A Remote Sensing Technique Using Terrestrial Refraction, for the Study of Low-Level Lapse Rate,” Ph.D. Thesis, University of Wisconsin (1971), 163 pp.
  9. C. R. Stearns, M. L. Wesely, Boundary-Layer Meteorol. 7, 441 (1974).
    [CrossRef]
  10. M. L. Wesely, J. Appl. Meteorol. 15, 1177 (1976).
    [CrossRef]
  11. A. B. Fraser, Appl. Opt. 16, 160 (1977).
    [CrossRef] [PubMed]
  12. J. C. Owens, Appl. Opt. 6, 51 (1967).
    [CrossRef] [PubMed]
  13. W. H. Mach, “Measurement of Micrometeorological Temperature Profiles by the Inversion of Optical Data,” Ph.D. Thesis, Pennsylvania State University (1978), 99 pp.
  14. A. B. Fraser, Appl. Opt. 18, 1724 (1979).
    [CrossRef] [PubMed]

1979 (1)

1977 (1)

1976 (1)

M. L. Wesely, J. Appl. Meteorol. 15, 1177 (1976).
[CrossRef]

1974 (2)

C. R. Stearns, M. L. Wesely, Boundary-Layer Meteorol. 7, 441 (1974).
[CrossRef]

A. B. Fraser, W. H. Mach, Sci. Am. 234, 102 (1974).
[CrossRef]

1970 (1)

A. L. Friend, C. R. Stearns, Boundary-Layer Meteorol. 1, 227 (1970).
[CrossRef]

1967 (1)

1956 (1)

R. G. Fleagle, J. Meteorol. 13, 160 (1956).
[CrossRef]

1950 (1)

R. G. Fleagle, Bull. Am. Meteorol. Soc. 31, 51 (1950).

1929 (1)

D. Brunt, Q. J. R. Meteorol. Soc. 55, 335 (1929).
[CrossRef]

1925 (1)

N. K. Johnson, O. F. T. Roberts, Q. J. R. Meteorol. Soc. 51, 131 (1925).
[CrossRef]

Brunt, D.

D. Brunt, Q. J. R. Meteorol. Soc. 55, 335 (1929).
[CrossRef]

Fleagle, R. G.

R. G. Fleagle, J. Meteorol. 13, 160 (1956).
[CrossRef]

R. G. Fleagle, Bull. Am. Meteorol. Soc. 31, 51 (1950).

Fraser, A. B.

Friend, A. L.

A. L. Friend, C. R. Stearns, Boundary-Layer Meteorol. 1, 227 (1970).
[CrossRef]

Johnson, N. K.

N. K. Johnson, O. F. T. Roberts, Q. J. R. Meteorol. Soc. 51, 131 (1925).
[CrossRef]

Mach, W. H.

A. B. Fraser, W. H. Mach, Sci. Am. 234, 102 (1974).
[CrossRef]

W. H. Mach, “Measurement of Micrometeorological Temperature Profiles by the Inversion of Optical Data,” Ph.D. Thesis, Pennsylvania State University (1978), 99 pp.

Owens, J. C.

Roberts, O. F. T.

N. K. Johnson, O. F. T. Roberts, Q. J. R. Meteorol. Soc. 51, 131 (1925).
[CrossRef]

Sparkman, J. K.

J. K. Sparkman, “A Remote Sensing Technique Using Terrestrial Refraction, for the Study of Low-Level Lapse Rate,” Ph.D. Thesis, University of Wisconsin (1971), 163 pp.

Stearns, C. R.

C. R. Stearns, M. L. Wesely, Boundary-Layer Meteorol. 7, 441 (1974).
[CrossRef]

A. L. Friend, C. R. Stearns, Boundary-Layer Meteorol. 1, 227 (1970).
[CrossRef]

Wesely, M. L.

M. L. Wesely, J. Appl. Meteorol. 15, 1177 (1976).
[CrossRef]

C. R. Stearns, M. L. Wesely, Boundary-Layer Meteorol. 7, 441 (1974).
[CrossRef]

Whipple, F. J. W.

F. J. W. Whipple, “Meteorological Optics” in Dictionary of Applied Physics, Vol. III, ed. R. Glazebrook (Macmillan, London, 1921), p. 519.

Appl. Opt. (3)

Boundary-Layer Meteorol. (2)

C. R. Stearns, M. L. Wesely, Boundary-Layer Meteorol. 7, 441 (1974).
[CrossRef]

A. L. Friend, C. R. Stearns, Boundary-Layer Meteorol. 1, 227 (1970).
[CrossRef]

Bull. Am. Meteorol. Soc. (1)

R. G. Fleagle, Bull. Am. Meteorol. Soc. 31, 51 (1950).

J. Appl. Meteorol. (1)

M. L. Wesely, J. Appl. Meteorol. 15, 1177 (1976).
[CrossRef]

J. Meteorol. (1)

R. G. Fleagle, J. Meteorol. 13, 160 (1956).
[CrossRef]

Q. J. R. Meteorol. Soc. (2)

N. K. Johnson, O. F. T. Roberts, Q. J. R. Meteorol. Soc. 51, 131 (1925).
[CrossRef]

D. Brunt, Q. J. R. Meteorol. Soc. 55, 335 (1929).
[CrossRef]

Sci. Am. (1)

A. B. Fraser, W. H. Mach, Sci. Am. 234, 102 (1974).
[CrossRef]

Other (3)

F. J. W. Whipple, “Meteorological Optics” in Dictionary of Applied Physics, Vol. III, ed. R. Glazebrook (Macmillan, London, 1921), p. 519.

J. K. Sparkman, “A Remote Sensing Technique Using Terrestrial Refraction, for the Study of Low-Level Lapse Rate,” Ph.D. Thesis, University of Wisconsin (1971), 163 pp.

W. H. Mach, “Measurement of Micrometeorological Temperature Profiles by the Inversion of Optical Data,” Ph.D. Thesis, Pennsylvania State University (1978), 99 pp.

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Figures (11)

Fig. 1
Fig. 1

Relationship among ξt, ζt, and ϕ0. A target is a distance ξt from the observer. A point, whose height is ζt, on this target is observed with a light ray that enters the eye at the angle ϕ0. The apparent height of this point is then ξtϕ0. If the light ray were not refracted, the actual elevation angle to the point would be ζt/ξt.

Fig. 2
Fig. 2

Image of the vertical target at 934 m, observed between 1450 and 1455 on 5 July 1978 at Roskilde Fjord. The observed optical data are noted with ⊙, and the predicted image is shown with the solid, curved line. The straight diagonal line, where ϕ0 = ζt/ξt, is the image of a diagonal target when the light rays are not refracted. This line is shown on all the image diagrams.

Fig. 3
Fig. 3

Temperature profile obtained with the inversion of the optical data collected between 1450 and 1455 on 5 July 1978 at Roskilde Fjord. The profile is shown with the solid line. Two temperatures that were measured with thermocouples are shown with error bars that indicate the maximum uncertainty in the measured temperatures.

Fig. 4
Fig. 4

Image of the target at 1694 m, observed around 1410 on 5 July 1978 at Roskilde Fjord. The target appeared as a two-image inferior mirage. The six observed optical data are noted with ⊙. The image predicted with four coefficients is shown with the solid, curved line; the image predicted with three coefficients is shown with the dashed line. The measured angle to the caustic is denoted with ϕcm.

Fig. 5
Fig. 5

Temperature profile obtained with the inversion of the optical data that were collected around 1410 on 5 July 1978 at Roskilde Fjord. The profile is shown with the solid line. The dashed line shows the different profile when only three coefficients were used to calculate the profile. Three temperatures that were measured with thermocouples are shown with error bars that indicated the maximum uncertainty in the measured temperatures.

Fig. 6
Fig. 6

Image of the target at 1694 m, observed at 1540 on 6 July 1978 at Roskilde Fjord. The optical data are noted with ⊙, and the predicted image is shown with the solid, curved line.

Fig. 7
Fig. 7

Temperature profile obtained with the inversion of the optical data that were collected on 6 July 1978 at Roskilde Fjord. The profile is shown with the solid line. The temperatures measured with thermocouples are shown with error bars that indicate the maximum uncertainty in the measured temperatures.

Fig. 8
Fig. 8

Image of the target at 701 m from the observer at Black Moshannon Lake at 1628 on 11 May 1978. The target appeared as a single-image superior mirage. The optical data are noted with ⊙; the solid, curved line represents the predicted image.

Fig. 9
Fig. 9

Temperature profile obtained with the inversion of optical data collected at 1628 on 11 May 1978 at Black Moshannon Lake. The profile is shown with a solid line. The three temperatures obtained with thermocouples are shown without closed error bars because the amount of error is unknown for these measurements of temperatures.

Fig. 10
Fig. 10

Temperature profiles obtained with the inversion, by two different methods, of the optical data that were collected on 5 July 1978 at 1450 to 1455 at Roskilde Fjord. The profile obtained with an analytic inversion is shown with a dashed line. The profile obtained with nonlinear least squares is shown with a dashed and dotted line. The estimate of the actual profile is shown with a solid line. The temperatures measured with thermocouples are shown with error bars that indicate the maximum uncertainty in the measured temperature. The optical data for the analytic inversion were collected by an observer whose height above the surface is denoted by E1.

Fig. 11
Fig. 11

Temperature profiles obtained with the inversion, by two different methods, of optical data that were collected on 6 July 1978 at Roskilde Fjord. The profile obtained with the analytic inversion is shown with a dashed line; the profile obtained with nonlinear least squares is shown with a dashed and dotted line; the estimate of the actual profile is shown with a solid line. The bottom scale shows the difference between the temperature at any height and the temperature at the surface. The temperatures measured with thermocouples are shown with error bars that indicate the maximum uncertainty in the measured temperature.

Equations (10)

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ξ = r 0 θ sec β 0 ,
ζ = r 0 ln ( r / r 0 ) ,
ϕ = tan β cos β 0 ,
τ = 1 - ( n r / n 0 r 0 ) 2 .
τ 2 × 10 - 6 ( T - T 0 ) - 2.6 × 10 - 7 ( r - r 0 ) ,
ξ t = m = 1 P ζ 0 ( m ) [ 2 2 m - 1 ( m - 1 ) ! ( 2 m - 1 ) ! ϕ 0 2 m - 1 - s = 1 m 2 2 s - 1 ( s - 1 ) ! ( m - s ) ! ( 2 s - 1 ) ! ϕ t 2 s - 1 ( ϕ 0 2 - ϕ t 2 ) m - s ] ,
ζ t = m = 1 p ζ 0 ( m ) ( ϕ 0 2 - ϕ t 2 ) m / m ! .
ζ = m = 1 p ζ 0 ( m ) τ m / m ! .
ζ * = ξ * ϕ * .
τ * = ϕ * 2 ,

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