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Simulation of inferior mirages observed at the Halligen Sea

Eberhard Tränkle

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Abstract

Two unusual forms of inferior mirage are observed and photographed at the Halligen Sea. With heuristic analytic functions for the temperature profiles, numerical integration of the refraction differential equation on a flat earth is performed. The simulation shows that a double inferior mirage can appear if a light wind carries hot air from above dry sandbanks in the mud flats. Horizontal stripes can appear in the mirage image if a water channel crosses the line of sight between the observer and the object.

©1999 Optical Society of America

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Figures (12)
Fig. 1.
Fig. 1. Ray tracing for the pixmap hallig. Profile P2 is used.

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Fig. 2.
Fig. 2. Dependence on eye position for the pixmap hallig; P1, P2, and P3 are used.

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Fig. 3.
Fig. 3. Dependence on distance for the pixmap walker; P1, P2, and P3 are used.

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Fig. 4.
Fig. 4. Effects of fast air fluctuations for a walker from Engler’s film; T1 is adjusted.

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Fig. 5.
Fig. 5. Double inferior mirage of the Ockenswarft at the hallig Hooge. The photograph is taken from the sandbank Japsand (see Fig. 6).

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Fig. 6.
Fig. 6. Map of the Halligen Sea.

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Fig. 7.
Fig. 7. Ray tracing for the pixmap hallig; T2 is adjusted.

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Fig. 8.
Fig. 8. Dependence on eye position for the double inferior mirage of the Ockenswarft; T2 is adjusted.

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Fig. 9.
Fig. 9. Inferior mirage of the Schulwarft at the hallig Nordstrandischmoor. The photograph was taken from the Hamburger Hallig.

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Fig. 10.
Fig. 10. Ray tracing for the pixmap hallig; T3 is adjusted.

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Fig. 11.
Fig. 11. Dependence on eye position for an image with a stripe of a warft at the hallig Nordstrandischmoor; T3 is adjusted.

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Fig. 12.
Fig. 12. Simulation of three photographs of a mirage of the hallig Südfall at high tide; T3 is adjusted.

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Tables (2)

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Table 1. Three Sets of Parameter Values of T1, from Measured Temperature Values on a Calm Day

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Table 2. Parameter Values for the Temperature Profiles used for the Simulations Shown

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Equations (5)

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d z d x = ± 1 w n ( T ) 2 − w 2 with
n = 1 + 0.08 273.15 + T ( z , x ) ,
T 1 ( z ) = a exp ( − z b ) − cz + d .
T 2 ( z ) = T 1 ( z ) + e [ 1 π arctan ( z − f g ) + 0.5 ] ,
T 3 ( z , x ) = T 1 ( z ) − e exp [ − ( x − f ) 2 2 g 2 ] ,
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