Abstract

During a number of transmission experiments over littoral waters, quantitative measurements of atmospheric refraction phenomena were carried out to determine the range performance of optical-IR sensors. Examples of distortion and intensity gain generated by spatial variations of the atmospheric refractive index are shown. A high-precision ray-tracing model has been developed for better understanding of the phenomena and to satisfy the requirements for accuracy of the meteorological data used in refraction models. The output of the model includes the propagation function, the intensity gain, and details of the ray curvature and of the optical phase behavior along the path between the target and the observer. Examples of measured transmission data and their interpretation are presented.

© 2004 Optical Society of America

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References

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  1. G. de Leeuw, A. M. J. van Eijk, D. R. Jensen, “MAPTIP experiment, marine aerosol properties and thermal imager performance: an overview,” TNO-FEL Rep. FEL-94-A140 (Netherlands Organization for Applied Scientific Research, The Hague, 1994).
  2. D. R. Jensen, S. G. Gathman, C. R. Zeisse, C. P. McGrath, G. De Leeuw, M. H. Smith, P. A. Frederickson, K. L. Davidson, “Electro-optical propagation assessment in coastal environments (EOPACE): summary and accomplishments,” Opt. Eng. 40, 1486–1498 (2001).
    [CrossRef]
  3. A. N. de Jong, G. De Leeuw, P. J. Fritz, M. M. Moerman, “Long-range transmission at low elevations over the ocean,” in E-O Propagation, Signature and System Performance under Adverse Meteorological Conditions Considering Out-of-Area Operations, Tech. Rep. RTO-MP1 (Research and Technology Organization, Brussels, 1998), pp. 1501–1512.
  4. C. R. Zeisse, B. D. Nener, R. V. Dewees, “Measurement of low-altitude infrared propagation,” Appl. Opt. 39, 873–886 (2000).
    [CrossRef]
  5. S. M. Doss-Hammel, C. R. Zeisse, A. E. Barrios, G. De Leeuw, M. M. Moerman, A. N. de Jong, P. A. Frederickson, K. L. Davidson, “Low-altitude infrared propagation in a coastal zone: refraction and scattering,” Appl. Opt. 41, 3706–3724 (2002).
    [CrossRef] [PubMed]
  6. A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
    [CrossRef]
  7. J. S. Accetta, “Infrared Search and Track Systems,” in Infrared and Electro-Optical Systems Handbook, J. S. Acetta, ed. (SPIE Press, Bellingham, Wash., 1993), pp. 209–344.
  8. G. C. Holst, Electro-Optical Imaging System Performance, 2nd ed. (JCD, Winter Park, Fla., 2000).
  9. E. Tränkle, “Simulation of inferior mirages observed at the Halligen Sea,” Appl. Opt. 37, 1495–1505 (1998).
    [CrossRef]
  10. S. Y. van der Werf, “Ray tracing and refraction in the modified US1976 atmosphere,” Appl. Opt. 42, 354–366 (2003).
    [CrossRef] [PubMed]
  11. W. H. Lehn, “A simple parabolic model for the optics of the atmospheric surface layer,” Appl. Math. Model. 9, 447–453 (1985).
    [CrossRef]
  12. R. D. Sampson, E. P. Lozowski, A. E. Peterson, “Comparison of modeled and observed astronomical refraction of the setting Sun,” Appl. Opt. 42, 342–353 (2003).
    [CrossRef] [PubMed]
  13. W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sens. GE-24, 940–946 (1986).
    [CrossRef]
  14. A. N. de Jong, J. Winkel, “Enhanced IR point target detection by atmospheric effects,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 885–896 (2002).
    [CrossRef]

2003 (2)

2002 (1)

2001 (1)

D. R. Jensen, S. G. Gathman, C. R. Zeisse, C. P. McGrath, G. De Leeuw, M. H. Smith, P. A. Frederickson, K. L. Davidson, “Electro-optical propagation assessment in coastal environments (EOPACE): summary and accomplishments,” Opt. Eng. 40, 1486–1498 (2001).
[CrossRef]

2000 (1)

1998 (1)

1986 (1)

W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sens. GE-24, 940–946 (1986).
[CrossRef]

1985 (1)

W. H. Lehn, “A simple parabolic model for the optics of the atmospheric surface layer,” Appl. Math. Model. 9, 447–453 (1985).
[CrossRef]

Accetta, J. S.

J. S. Accetta, “Infrared Search and Track Systems,” in Infrared and Electro-Optical Systems Handbook, J. S. Acetta, ed. (SPIE Press, Bellingham, Wash., 1993), pp. 209–344.

Barrios, A. E.

Buss, J. R.

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

Cini, A.

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

Davidson, K. L.

S. M. Doss-Hammel, C. R. Zeisse, A. E. Barrios, G. De Leeuw, M. M. Moerman, A. N. de Jong, P. A. Frederickson, K. L. Davidson, “Low-altitude infrared propagation in a coastal zone: refraction and scattering,” Appl. Opt. 41, 3706–3724 (2002).
[CrossRef] [PubMed]

D. R. Jensen, S. G. Gathman, C. R. Zeisse, C. P. McGrath, G. De Leeuw, M. H. Smith, P. A. Frederickson, K. L. Davidson, “Electro-optical propagation assessment in coastal environments (EOPACE): summary and accomplishments,” Opt. Eng. 40, 1486–1498 (2001).
[CrossRef]

de Jong, A. N.

S. M. Doss-Hammel, C. R. Zeisse, A. E. Barrios, G. De Leeuw, M. M. Moerman, A. N. de Jong, P. A. Frederickson, K. L. Davidson, “Low-altitude infrared propagation in a coastal zone: refraction and scattering,” Appl. Opt. 41, 3706–3724 (2002).
[CrossRef] [PubMed]

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

A. N. de Jong, G. De Leeuw, P. J. Fritz, M. M. Moerman, “Long-range transmission at low elevations over the ocean,” in E-O Propagation, Signature and System Performance under Adverse Meteorological Conditions Considering Out-of-Area Operations, Tech. Rep. RTO-MP1 (Research and Technology Organization, Brussels, 1998), pp. 1501–1512.

A. N. de Jong, J. Winkel, “Enhanced IR point target detection by atmospheric effects,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 885–896 (2002).
[CrossRef]

De Leeuw, G.

S. M. Doss-Hammel, C. R. Zeisse, A. E. Barrios, G. De Leeuw, M. M. Moerman, A. N. de Jong, P. A. Frederickson, K. L. Davidson, “Low-altitude infrared propagation in a coastal zone: refraction and scattering,” Appl. Opt. 41, 3706–3724 (2002).
[CrossRef] [PubMed]

D. R. Jensen, S. G. Gathman, C. R. Zeisse, C. P. McGrath, G. De Leeuw, M. H. Smith, P. A. Frederickson, K. L. Davidson, “Electro-optical propagation assessment in coastal environments (EOPACE): summary and accomplishments,” Opt. Eng. 40, 1486–1498 (2001).
[CrossRef]

G. de Leeuw, A. M. J. van Eijk, D. R. Jensen, “MAPTIP experiment, marine aerosol properties and thermal imager performance: an overview,” TNO-FEL Rep. FEL-94-A140 (Netherlands Organization for Applied Scientific Research, The Hague, 1994).

A. N. de Jong, G. De Leeuw, P. J. Fritz, M. M. Moerman, “Long-range transmission at low elevations over the ocean,” in E-O Propagation, Signature and System Performance under Adverse Meteorological Conditions Considering Out-of-Area Operations, Tech. Rep. RTO-MP1 (Research and Technology Organization, Brussels, 1998), pp. 1501–1512.

Dewees, R. V.

Doss-Hammel, S. M.

Forand, J. L.

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

Frederickson, P. A.

S. M. Doss-Hammel, C. R. Zeisse, A. E. Barrios, G. De Leeuw, M. M. Moerman, A. N. de Jong, P. A. Frederickson, K. L. Davidson, “Low-altitude infrared propagation in a coastal zone: refraction and scattering,” Appl. Opt. 41, 3706–3724 (2002).
[CrossRef] [PubMed]

D. R. Jensen, S. G. Gathman, C. R. Zeisse, C. P. McGrath, G. De Leeuw, M. H. Smith, P. A. Frederickson, K. L. Davidson, “Electro-optical propagation assessment in coastal environments (EOPACE): summary and accomplishments,” Opt. Eng. 40, 1486–1498 (2001).
[CrossRef]

Fritz, P. J.

A. N. de Jong, G. De Leeuw, P. J. Fritz, M. M. Moerman, “Long-range transmission at low elevations over the ocean,” in E-O Propagation, Signature and System Performance under Adverse Meteorological Conditions Considering Out-of-Area Operations, Tech. Rep. RTO-MP1 (Research and Technology Organization, Brussels, 1998), pp. 1501–1512.

Gathman, S. G.

D. R. Jensen, S. G. Gathman, C. R. Zeisse, C. P. McGrath, G. De Leeuw, M. H. Smith, P. A. Frederickson, K. L. Davidson, “Electro-optical propagation assessment in coastal environments (EOPACE): summary and accomplishments,” Opt. Eng. 40, 1486–1498 (2001).
[CrossRef]

Holst, G. C.

G. C. Holst, Electro-Optical Imaging System Performance, 2nd ed. (JCD, Winter Park, Fla., 2000).

Jensen, D. R.

D. R. Jensen, S. G. Gathman, C. R. Zeisse, C. P. McGrath, G. De Leeuw, M. H. Smith, P. A. Frederickson, K. L. Davidson, “Electro-optical propagation assessment in coastal environments (EOPACE): summary and accomplishments,” Opt. Eng. 40, 1486–1498 (2001).
[CrossRef]

G. de Leeuw, A. M. J. van Eijk, D. R. Jensen, “MAPTIP experiment, marine aerosol properties and thermal imager performance: an overview,” TNO-FEL Rep. FEL-94-A140 (Netherlands Organization for Applied Scientific Research, The Hague, 1994).

Lehn, W. H.

W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sens. GE-24, 940–946 (1986).
[CrossRef]

W. H. Lehn, “A simple parabolic model for the optics of the atmospheric surface layer,” Appl. Math. Model. 9, 447–453 (1985).
[CrossRef]

Lozowski, E. P.

McGrath, C. P.

D. R. Jensen, S. G. Gathman, C. R. Zeisse, C. P. McGrath, G. De Leeuw, M. H. Smith, P. A. Frederickson, K. L. Davidson, “Electro-optical propagation assessment in coastal environments (EOPACE): summary and accomplishments,” Opt. Eng. 40, 1486–1498 (2001).
[CrossRef]

Moerman, M. M.

S. M. Doss-Hammel, C. R. Zeisse, A. E. Barrios, G. De Leeuw, M. M. Moerman, A. N. de Jong, P. A. Frederickson, K. L. Davidson, “Low-altitude infrared propagation in a coastal zone: refraction and scattering,” Appl. Opt. 41, 3706–3724 (2002).
[CrossRef] [PubMed]

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

A. N. de Jong, G. De Leeuw, P. J. Fritz, M. M. Moerman, “Long-range transmission at low elevations over the ocean,” in E-O Propagation, Signature and System Performance under Adverse Meteorological Conditions Considering Out-of-Area Operations, Tech. Rep. RTO-MP1 (Research and Technology Organization, Brussels, 1998), pp. 1501–1512.

Morrish, J. S.

W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sens. GE-24, 940–946 (1986).
[CrossRef]

Nener, B. D.

Peterson, A. E.

Potvin, G.

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

Sampson, R. D.

Smith, M. H.

D. R. Jensen, S. G. Gathman, C. R. Zeisse, C. P. McGrath, G. De Leeuw, M. H. Smith, P. A. Frederickson, K. L. Davidson, “Electro-optical propagation assessment in coastal environments (EOPACE): summary and accomplishments,” Opt. Eng. 40, 1486–1498 (2001).
[CrossRef]

Stark, E.

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

Stein, K.

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

Tränkle, E.

van der Werf, S. Y.

van Eijk, A. M. J.

G. de Leeuw, A. M. J. van Eijk, D. R. Jensen, “MAPTIP experiment, marine aerosol properties and thermal imager performance: an overview,” TNO-FEL Rep. FEL-94-A140 (Netherlands Organization for Applied Scientific Research, The Hague, 1994).

Vogel, H. H.

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

Weiss-Wrana, K.

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

Winkel, J.

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

A. N. de Jong, J. Winkel, “Enhanced IR point target detection by atmospheric effects,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 885–896 (2002).
[CrossRef]

Zeisse, C. R.

Appl. Math. Model. (1)

W. H. Lehn, “A simple parabolic model for the optics of the atmospheric surface layer,” Appl. Math. Model. 9, 447–453 (1985).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Geosci. Remote Sens. (1)

W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sens. GE-24, 940–946 (1986).
[CrossRef]

Opt. Eng. (1)

D. R. Jensen, S. G. Gathman, C. R. Zeisse, C. P. McGrath, G. De Leeuw, M. H. Smith, P. A. Frederickson, K. L. Davidson, “Electro-optical propagation assessment in coastal environments (EOPACE): summary and accomplishments,” Opt. Eng. 40, 1486–1498 (2001).
[CrossRef]

Other (6)

A. N. de Jong, G. De Leeuw, P. J. Fritz, M. M. Moerman, “Long-range transmission at low elevations over the ocean,” in E-O Propagation, Signature and System Performance under Adverse Meteorological Conditions Considering Out-of-Area Operations, Tech. Rep. RTO-MP1 (Research and Technology Organization, Brussels, 1998), pp. 1501–1512.

A. N. de Jong, J. Winkel, M. M. Moerman, K. Stein, K. Weiss-Wrana, J. L. Forand, G. Potvin, J. R. Buss, A. Cini, H. H. Vogel, E. Stark, “TG16 point target detection experiment POLLEX, Livorno 2001,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 849–860 (2002).
[CrossRef]

J. S. Accetta, “Infrared Search and Track Systems,” in Infrared and Electro-Optical Systems Handbook, J. S. Acetta, ed. (SPIE Press, Bellingham, Wash., 1993), pp. 209–344.

G. C. Holst, Electro-Optical Imaging System Performance, 2nd ed. (JCD, Winter Park, Fla., 2000).

A. N. de Jong, J. Winkel, “Enhanced IR point target detection by atmospheric effects,” in Infrared Technology and Applications XXVIII, B. F. Andresen, G. F. Fulop, M. Strojnik, eds., Proc. SPIE4820, 885–896 (2002).
[CrossRef]

G. de Leeuw, A. M. J. van Eijk, D. R. Jensen, “MAPTIP experiment, marine aerosol properties and thermal imager performance: an overview,” TNO-FEL Rep. FEL-94-A140 (Netherlands Organization for Applied Scientific Research, The Hague, 1994).

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Figures (14)

Fig. 1
Fig. 1

Images of an array of lamps on a platform in the North Sea taken in 1994 from a pier at Scheveningen, The Netherlands, on 1 June [10.03 (UTC+2 h); Fig. 1(a)], 6 July [10.57 (UTC+2 h); Fig. 1(b)], and 6 July [12.36 (UTC+2 h), Fig. 1(c)], showing the variability of the geometrical distortion caused by changes in the temperature profile. The pictures were taken with a CCD camera with a 1.25-m lens (frame time, ∼10 ms) from the Defense Research Establishment Valcartier (Canada).

Fig. 2
Fig. 2

The signal increase with range for a point target near the horizon. Data were collected with a point source on a boat at a height of 10 m above the water level. The receiver was located on a seacoast at a height of 16 m. The source emitted 12.8 W/sr in the 3.6–4.0-μm spectral band. The camera was a Radiance HS with a 250-mm lens and a 0.12-mrad instantaneous field of view. The average difference of sums (DOS) value (pupil irradiance) was taken for a series of 100 frames and multiplied by the square of the range, providing a kind of apparent source intensity contrast (not corrected for atmospheric transmission loss).

Fig. 3
Fig. 3

(a) Transmission versus time (hours:minutes:seconds). The source at 18.5-km distance was modulated with a frequency of 800 Hz. The 0.9- and 10-μm channels used an aperture of 20 cm; the 4-μm channel used a 5-cm aperture. The integration time was set at 1 s. (b) Air and water temperature measured at mid-path for the data set of (a).

Fig. 4
Fig. 4

Mid-wave (3.60–4.04) transmission plot over the San Diego Bay on 14 November 1996, showing an irregular signal boost owing to changing temperature profiles (wind direction) in the morning (UTC time is local time + 8 h). The data were averaged over a period of 10 s. In this experimental period the receiver height was 6.4 m. Occasional mirage bars were discernable.

Fig. 5
Fig. 5

Geometry for the ray-tracing calculations: C is the circle of curvature; M e is the center of the Earth; a ray departs O at jth step and ends at P to start the j + 1st step, running over path t, while the height above the Earth’s surface increases from h j to h j+1.

Fig. 6
Fig. 6

Propagation functions for a T profile with constant slope a and variable ASTD (z/ a).

Fig. 7
Fig. 7

Example of ray tracing through a focusing atmosphere. The departure angles for the 14 rays were u = -0.0008 - 1 × 0.00012. Horizontal step size s was 10 m. The focusing starts beyond a range of 15 km.

Fig. 8
Fig. 8

Arrival angle versus arrival height at three ranges for the same profile as the ray tracing shown in Fig. 7.

Fig. 9
Fig. 9

Effect of an incoming cold front. The range to the edge varies from 15.6 km (j 2 = 1560) to 3.6 km (j 2 = 360). The width of the edge g 4 is ∼2 km.

Fig. 10
Fig. 10

(a) Ray curvature K c and (b) path difference e over a 19.2-km path with z = 0.5 K/m and a = 0.5 m-1. The sea surface has waves with an amplitude of 0.5 m and a wavelength of 1000 m. Curves show the first departure angle u = -0.001 r; u increases further in steps of -0.0002 r.

Fig. 11
Fig. 11

Variation of minimum arrival height (a) with the phase of the surface wave and (b) with wave amplitude w. In (a) w = 0.5 and L = 400 m; in both figures the source height and s are 10 m. The path length is again 19.2 km. In (b) L = 400 m, z = 0.25 K/m, and a = 0.25 m-1. The calculations were made for the condition that L > s.

Fig. 12
Fig. 12

Propagation functions for three heights h 0 at the source location. Other conditions are range, 25.6 km; z = 0.5 K/m; a = 0.5 m-1; s = 10 m. The center of the source is at 10.0 m, and a standard temperature profile [Eqs. (6)] is used. The departure angle for the minimum arrival height is ∼-0.00133 r.

Fig. 13
Fig. 13

Calculated apparent intensity of an extended source (0.3 m) at a 25.6-km range to a receiver with finite size (0.1 m), showing the intensity gain in subrefractive conditions (ASTD, -1 K) when the receiver is close to or at the minimum arrival height.

Fig. 14
Fig. 14

Variation of arrival angle and total range-integrated optical path difference e t with arrival height. Range, 25.6 km; ASTD, -1 K; z = 1 K/m; a = 1 m-1; source height, 10.0 m.

Equations (14)

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

R2=ΔW×τ/NEI×S/Nd.
τ×ΔT=IETD×S/Ni/MTF6R/D,
pj=hj+uj×s/2+s2/8R+qj+1-qj/2.
n=1+b/T×786×10-9,dn/dh=-dT/dh+0.0348×786×10-9×b/T2.
n=1.000271,dn/dh=-9.35×10-7×dT/dh+0.0348m-1.
Th=z/a×exp-a×h-1-0.006×h,dT/dh=-z×exp-a×h-0.006,
Example 1: Th=g1×h m1+1×exp-h;Example 2: Th=a2/g2+hm2,
zj=z0-z1×arctanj-j2×s/g4,aj=a0-a1×arctanj-j2×s/g4,
hj+1=hj+s×1+hj/R×v/2+uj×1+hj/R+uj2/3+s×1+hj×2/R+uj2/2+uj+s/2Rc2/2Rc+qj+1-qj,
uj+1=uj+v+s×1+hj/R+uj2/2+v×uj+v+uj×s/2Rc+s/Rc2/6/Rc.
e=s×1.000271×-9.35×10-7×Th+hj/R+uj2/2+v×uj+v+uj×s/2Rc+s/Rc2/6.
Mi=hc0-Ds/2hc0+Ds/2 4×Ds/22-h0-hc020.5×dh0hac-Dr/2hac+Dr/2 mh0, ha×Dr/22-ha-hac20.5×dha,
qj=w×sin2πj×s/L+phase,qj+1=w×sin2πj+1×s/L+phase,
Mi=1.223×10-4 Rphc+D0/2+gDs/21.5-hc-D0/2+gDs/21.5-hc+D0/2-gDs/21.5+hc-D0/2-gDs/21.5/gD0Ds.

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