Abstract

The first-known, explicit, analytic optical inversion for a refractive-index profile with curvature is given. It enables a quasi-parabolic profile of height vs temperature to be calculated from observations of the inferior mirage of natural objects. Given sufficient fetch, an inferior mirage will occur anytime the heat flux is away from a horizontal surface such as a large body of warm water. All that is needed to obtain the data for a temperature profile over such a surface is a theodolite, a tape measure, and a topographic map. A total of four measurements and a pocket calculator are sufficient to determine the temperature profile on the spot. The resulting profile represents a weighted horizontal mean over the surface. Not all inferior mirages are amenable to the technique, but only those where the temperature gradient at the eye is no less than half the mean gradient, a situation that seems to require some minimum wind. The predictions of the theory are verified with measurements from thermocouples.

© 1979 Optical Society of America

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References

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  1. N. K. Johnson, O. F. T. Roberts, Q. J. R. Meteorol. Soc. 51, 131 (1925).
    [CrossRef]
  2. K. Brocks, Veroef. Meteorol. Inst. Univ. Berlin 3, No. 4, 1 (1939).
  3. R. G. Fleagle, Bull. Am. Meteorol. Soc. 31, 51 (1950).
  4. J. K. Sparkman, “A Remote Sensing Technique Using Terrestrial Refraction for the Study of Low-Level Lapse Rate,” Ph.D. Thesis, U. Wisconsin (1971).
  5. A. B. Fraser, Appl. Opt. 16, 160 (1977).
    [CrossRef] [PubMed]
  6. W. H. Mach, “Measurement of Micrometeorological Temperature Profiles by the Inversion of Optical Data,” Ph.D. Thesis, Penn State U. (1978).
  7. W. H. Mach, A. B. Fraser, Appl. Opt. 18, 1715 (1979).
    [CrossRef] [PubMed]

1979 (1)

1977 (1)

1950 (1)

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

1939 (1)

K. Brocks, Veroef. Meteorol. Inst. Univ. Berlin 3, No. 4, 1 (1939).

1925 (1)

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

Brocks, K.

K. Brocks, Veroef. Meteorol. Inst. Univ. Berlin 3, No. 4, 1 (1939).

Fleagle, R. G.

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

Fraser, A. B.

Johnson, N. K.

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

Mach, W. H.

W. H. Mach, A. B. Fraser, Appl. Opt. 18, 1715 (1979).
[CrossRef] [PubMed]

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

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, U. Wisconsin (1971).

Appl. Opt. (2)

Bull. Am. Meteorol. Soc. (1)

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

Q. J. R. Meteorol. Soc. (1)

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

Veroef. Meteorol. Inst. Univ. Berlin (1)

K. Brocks, Veroef. Meteorol. Inst. Univ. Berlin 3, No. 4, 1 (1939).

Other (2)

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

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

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Figures (5)

Fig. 1
Fig. 1

A map of a portion of Roskilde Fiord in Denmark. The observation point was on the shore at Risø. The three targets, Agernakke (1630 m), Sonderby (4940 m), and Gershoj (7220 m), are shown.

Fig. 2
Fig. 2

Optically deduced temperature profiles for Agernakke (solid line) and Gershoj (dashed line) along with the thermocouple measurements for the afternoon of 5 July 1978. The thermocouples have a maximum error of ±0.04°C, but the top (reference) point is assumed to have no error. The bottom scale shows the temperature departure from eye level (optical), while the top scale gives the mast temperatures (thermistor and thermocouples). The optical data are presented in Table I. The eye position is marked with a ×.

Fig. 3
Fig. 3

Optically deduced profiles for Gershoj deduced from two different heights (marked with ×’s) plotted on top of each other to give a single line.

Fig. 4
Fig. 4

Profiles obtained from viewing two different targets, Gershoj (G) and Sonderby (S) from two different heights (H = higher height, L = lower height). The profiles were positioned to match in the vicinity of the two observation heights. The discrepancies in the profiles near the surface may reveal horizontal inhomogeneities because each represents a different position in space. The lower scale shows the temperature departure from one of the surface values.

Fig. 5
Fig. 5

An optically deduced temperature profile (solid line) for 1600 h 6 July 1978 when the wind was calm. It has insufficient curvature near the surface. A more realistic profile (dashed line) is obtained with a free hand joining of the thermocouples (for 1550 h) and the optically deduced surface temperature. The data gave a value of F = 2.81, but the default value of F = 1.0 was used. Note the change of temperature scale from the other figures.

Tables (1)

Tables Icon

Table I Tabulation of the Data and Some Calculations used in this Paper

Equations (47)

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T - T 0 = T * k ( ln z z 0 - ψ ) ,
T - T e = T * k ln [ ( z + z e ) / z e ] .
1 + z z e = exp [ - ( T - T e ) ( - T * / k ) ] exp ( - T ) ,
½ T s - T e z e - ( d T d z ) z e T s - T e z e ,
F = ( 4 β h 2 z s ) / ( 3 β c 3 x c ) ,
τ s = 2 3 { 2 cos [ 4 π + arccos ( 1 - 27 F / 16 ) 3 ] + 1 } ;
τ * = β h 2 / τ s ,             where 0 < τ * ,
ζ * = z s τ s / F ,             where ζ * < 0.
T - T e = A τ * [ 1 - ( 1 - z s / ζ * - h / ζ * ) 1 / 2 ] + B ( h + z s ) ,
A = T ¯ 2 / ( 1.58 × 10 - 4 p ¯ ) ,
B = 2 A / r e - γ a .
tan β = d ln r / d θ ,
d ( n r cos β ) / d s = 0 ,
τ = 1 - ( n r / n e r e ) 2 ϕ = tan β cos β e ξ = r e θ sec β e ζ = r e ln ( r / r e ) } ,
ζ z
ζ x ϕ β } .
τ = A - 1 [ ( T - T e ) - B z ] ,
ξ t = - 2 ϕ e ϕ t ζ d ϕ ,
ζ t = - 2 ϕ e ϕ t ζ ϕ d ϕ ,
ζ d ζ / d τ
ζ = 0 τ ζ d τ .
τ = ϕ e 2 - ϕ 2 ,             τ t = ϕ e 2 - ϕ t 2 ,
τ = τ / τ * ϕ = ϕ / ϕ * ξ = ξ / ξ * ζ = ζ / ζ * } ,
ζ * = ξ * ϕ * τ * = ϕ * 2 } .
ζ = 2 τ - τ 2 .
ζ = 2 ( 1 - τ ) = 2 ( 1 - ϕ e 2 + ϕ 2 ) .
ξ t = - 4 ϕ e ϕ t ( 1 - ϕ e 2 + ϕ 2 ) d ϕ ,
ξ t / 4 = ( ϕ e - ϕ t ) ( 1 - ϕ e 2 ) + ( ϕ e 3 - ϕ t 3 ) .
ζ t = 2 ( ϕ e 2 - ϕ t 2 ) - ( ϕ e 2 - ϕ t 2 ) 2 .
0 = ¼ d ξ t / d ϕ e = 1 - ϕ e / ϕ t - 2 ϕ e 2 + ϕ e ϕ t + ϕ e 3 / ϕ t
ϕ e - ϕ t = ϕ e ( ϕ e - ϕ t ) 2 ,
ϕ c 3 ξ c = 4 / 3 ,
¾ ϕ c 3 ξ c = ζ * τ * ,
ζ s = 2 τ s - τ s 2 ,
τ s 3 - 2 τ s 2 + F = 0 ,
τ s ζ s = τ s ζ s τ * ζ * = 4 τ s ζ s 3 ϕ c 3 ζ c ,
τ s = ϕ h 2 ,
F = ( 4 ϕ h 2 ζ s ) / ( 3 ϕ c 3 ξ c ) .
τ s = 2 3 { 2 cos { 4 π + arcson ( 1 - 27 F / 16 ) 3 } + 1 } .
τ * = ϕ h 2 / τ s             where τ * > 0.
ζ * = ζ s τ s / F             where ζ * < 0.
τ = τ * [ 1 - ( 1 - ζ / ζ * ) 1 / 2 ] .
T - T c = A τ * [ 1 - ( 1 - z / ζ * ) 1 / 2 ] + B z ,
T - T e = A τ * [ 1 - ( 1 - ζ s / ζ * - h / ζ * ) 1 / 2 ] + B ( h + ζ s ) .
T - T s = A τ * [ ( 1 - ζ s - ζ * ) 1 / 2 - ( 1 - ζ s / ζ * - h / ζ * ) 1 / 2 ] + B h ,
ξ h = 4 ξ * ( τ s ) 1 / 2 ( 1 - 2 τ s / 3 ) .
ϕ c < ϕ * / ( 2 ) 1 / 2 , ξ c < ξ * 8 ( 2 / 3 ) 1 / 2 ,

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