Abstract

The inversion of atmospheric mirage data to produce a refractive-index profile and thus a temperature profile is considered. The method presented goes further than previous methods by seeking the most plausible profile consistent with the data. The approach is to minimize a composite cost function; the cost function includes novel terms to penalize a refractive-index profile that is not monotonic and to penalize a refractive-index profile that is not fully convex or fully concave. An algorithm for minimizing the cost function is applied to five sets of mirage data, giving good results.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. G. Rees, “Mirages with linear image diagrams,” J. Opt. Soc. Am. A 7, 1351–1354 (1990).
    [CrossRef]
  2. W. H. Mach, A. B. Fraser, “Inversion of optical data to compute a micrometeorological temperature profile,” Appl. Opt. 18, 1715–1723 (1979).
    [CrossRef]
  3. W. H. Lehn, “Inversion of superior mirage data to compute temperature profiles,”J. Opt. Soc. Am. 73, 1622–1625 (1983).
    [CrossRef]
  4. W. G. Rees, C. M. Roach, C. H. F. Glover, “Inversion of atmospheric refraction data,” J. Opt. Soc. Am. A 8, 330–338 (1991).
    [CrossRef]
  5. M. Bertero, T. A. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
    [CrossRef]
  6. R. Fletcher, Practical Methods of Optimisation (Wiley, Chichester, UK, 1981), Vol. 2, Chap. 14.
  7. We define a function to be unicurved if the function is fully convex or fully concave. Thus a refractive-index profile n(z) will be unicurved if d2n/dz2is either nonnegative throughout the domain of zor nonpositive throughout the domain of z.
  8. J. B. Biot, “Recherches sur les réfractions extraordinaires qui s’observent très près de l’horizon,” Mém. Classe Sci. Math. Phys. Inst. France (Baudoin, Paris) 10, 1–266 (1809).
  9. F. J. W. Whipple, “Meteorological optics,” in Dictionary of Applied Physics, R. Glazebrook, ed. (Macmillan, London, 1921), Vol. 3.
  10. R. E. Scoresby-Jackson, The Life of William Scoresby (Nelson, London, 1861).
  11. W. H. Lehn, W. G. Rees, “The Scoresby ship mirage of 1822,” Polar Rec. 26, 181–186 (1990).
    [CrossRef]

1991 (1)

1990 (2)

W. H. Lehn, W. G. Rees, “The Scoresby ship mirage of 1822,” Polar Rec. 26, 181–186 (1990).
[CrossRef]

W. G. Rees, “Mirages with linear image diagrams,” J. Opt. Soc. Am. A 7, 1351–1354 (1990).
[CrossRef]

1988 (1)

M. Bertero, T. A. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
[CrossRef]

1983 (1)

1979 (1)

Bertero, M.

M. Bertero, T. A. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
[CrossRef]

Biot, J. B.

J. B. Biot, “Recherches sur les réfractions extraordinaires qui s’observent très près de l’horizon,” Mém. Classe Sci. Math. Phys. Inst. France (Baudoin, Paris) 10, 1–266 (1809).

Fletcher, R.

R. Fletcher, Practical Methods of Optimisation (Wiley, Chichester, UK, 1981), Vol. 2, Chap. 14.

Fraser, A. B.

Glover, C. H. F.

Lehn, W. H.

W. H. Lehn, W. G. Rees, “The Scoresby ship mirage of 1822,” Polar Rec. 26, 181–186 (1990).
[CrossRef]

W. H. Lehn, “Inversion of superior mirage data to compute temperature profiles,”J. Opt. Soc. Am. 73, 1622–1625 (1983).
[CrossRef]

Mach, W. H.

Poggio, T. A.

M. Bertero, T. A. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
[CrossRef]

Rees, W. G.

Roach, C. M.

Scoresby-Jackson, R. E.

R. E. Scoresby-Jackson, The Life of William Scoresby (Nelson, London, 1861).

Torre, V.

M. Bertero, T. A. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
[CrossRef]

Whipple, F. J. W.

F. J. W. Whipple, “Meteorological optics,” in Dictionary of Applied Physics, R. Glazebrook, ed. (Macmillan, London, 1921), Vol. 3.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Polar Rec. (1)

W. H. Lehn, W. G. Rees, “The Scoresby ship mirage of 1822,” Polar Rec. 26, 181–186 (1990).
[CrossRef]

Proc. IEEE (1)

M. Bertero, T. A. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
[CrossRef]

Other (5)

R. Fletcher, Practical Methods of Optimisation (Wiley, Chichester, UK, 1981), Vol. 2, Chap. 14.

We define a function to be unicurved if the function is fully convex or fully concave. Thus a refractive-index profile n(z) will be unicurved if d2n/dz2is either nonnegative throughout the domain of zor nonpositive throughout the domain of z.

J. B. Biot, “Recherches sur les réfractions extraordinaires qui s’observent très près de l’horizon,” Mém. Classe Sci. Math. Phys. Inst. France (Baudoin, Paris) 10, 1–266 (1809).

F. J. W. Whipple, “Meteorological optics,” in Dictionary of Applied Physics, R. Glazebrook, ed. (Macmillan, London, 1921), Vol. 3.

R. E. Scoresby-Jackson, The Life of William Scoresby (Nelson, London, 1861).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Exact-value error term (dashed curve) has a gradient discontinuity at S = EQ. We modified the gradient discontinuity to remove the discontinuity for the error term actually used (solid curve).

Fig. 2
Fig. 2

Temperature profiles deduced from mirage 1. The figure shows the solution given by the method of Rees et al.4 (dotted curve) and the least-cost-algorithm solutions starting from a uniform refractive index (solid curve) and from the solution of Rees et al.4 (dashed curve). The latter two curves are not distinguishable. The in situ temperature measurements are plotted as open circles.

Fig. 3
Fig. 3

Temperature profiles for mirage 2, showing the solution of Rees et al.4 (dotted curve) and the least-cost-algorithm solutions (solid and dashed curves, as in Fig. 2), and the in situ temperature measurements (open circles).

Fig. 4
Fig. 4

Temperature profiles for mirage 3. Definitions are as for Fig. 2.

Fig. 5
Fig. 5

Temperature profiles for mirage 4. Definitions are as for Fig. 2.

Fig. 6
Fig. 6

Temperature profiles for mirage 5. Definitions are as for Fig. 2.

Fig. 7
Fig. 7

Construction of m zones for computing the nonmonotonicity cost. (a) n(z) is monotonic, and there is just one negative m zone. (b)–(d) n(z) is nonmonotonic. In (b) there is a large zone of negative amplitude and a very small zone of positive amplitude; (c) is similar but with a larger positive zone; in (d) there are four m zones.

Fig. 8
Fig. 8

Effect of omitting terms from the cost function in the least-cost-algorithm inversion of mirage 3: (a) U3 term missing, (b) U4 term missing, (c) U3 and U4 terms missing. Solid curves show the computed temperature profiles. The in situ temperature measurements are plotted as open circles.

Tables (2)

Tables Icon

Table 2 Parameters of the Cost Function

Equations (22)

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

d 2 z d x 2 = 1 n d n d z + 1 R E ,
d 2 z d x 2 = 1 n c d n d z + 1 R E .
l ( z ) = 1 n c d n d z ,
d 2 z d x 2 = l ( z ) + 1 R E .
S = 0 if Δ Z r E Q , S = λ ( Δ Z r - E Q ) if Δ Z r > E Q .
S = λ { Δ Z r - E Q + t [ 1 - exp ( - E Q / t ) ] } if Δ Z r > E Q , S = λ t { exp [ - ( E Q - Δ Z r ) / t ] - exp ( - E Q / t ) } if Δ Z r E Q .
U = U 1 + U 2 + U 3 + U 4 .
K 1 D 4 ( z U - z L ) 2 z L z U [ l ( z ) ] 2 d z
U 1 = K 1 D 4 ( z U - z L ) 2 j = 1 , N l j 2 δ z .
K 2 D 4 z L z U ( d l d z ) 2 d z
U 2 = K 2 D 4 j = 1 , N - 1 l j 2 δ z ,
l j = l j + 1 - l j δ z             for 1 j N - 1.
d T d Z = - α - T n - 1 d n d z ,
T ( z + δ z ) = T ( z ) - α δ z - T n - 1 δ n .
ν m = m m zone j l j δ z ,
m m zone d n d z d z .
switching cost = A 3 D 2 z U - z L ω T log ( 1 + B 3 ν m ν T ) ,
ν T = m ν m = j = 1 , N l j δ z .
w m = m m zone j l j δ z ,
m c zone d l d z d z .
switching cost = A 4 D 2 w T log ( 1 + B 4 w m w T ) ,
w T = m w m = j = 1 , N - 1 l j δ z .

Metrics