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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
1. Introduction
At the North Sea in Germany near the border with Denmark there is a unique region of mud flats with ten small islands, called halligen. At a hallig there are one or a few hillocks with a few houses, called warfts. In summer, at low tide, inferior mirages often appear at the Halligen Sea.
Mirages at the Halligen Sea are fascinating phenomena. Mirages of warfts often look like airships at the horizon.1 In summer they appear at low tide and disappear at high tide. With good field glasses one can see a strange flickering of the mirages because of rapid air fluctuations. The flickering is well documented in Engler’s film.1
For simulation of inferior mirages a model of the temperature distribution near the ground is needed. Wegener2 chooses one spherical shell with a linear temperature decrease and a temperature jump at the ground. Lehn3 takes into account a few spherical shells with linear temperature gradients, which are adjusted to measured temperature profiles. An attractive two-parameter model for inferior mirages is proposed by Fraser.4 He developed a quadratic approximation to the thermodynamic equations that conventionally describe unstable stratification at the surface and found a convenient closed-form solution. A three-parameter model of exponential-plus-linear form is developed by Lehn and Morrish.5 This model reproduces observed inferior mirage images accurately over horizontal ranges up to 20 km above a flat ice surface.
In the Halligen Sea the topography is more complex. Besides extended flat regions of mud flats there are dry sandbanks and water channels. We extend the three-parameter model of Lehn to take into account effects of dry sandbanks and water channels.
The simulation and visualization method is introduced in Section 2 with three temperature profiles measured at the Halligen Sea. A mirage of the Ockenswarft at the hallig Hooge looks like a double inferior mirage. It is explained in Section 3. Two photographs of mirages of halligen show well-defined horizontal stripes in the mirage image. The appearance of a horizontal stripe in mirages is discussed in Section 4. Beautiful inferior mirages also appear at the Halligen Sea in the winter season at high tide.1 The simulation of a mirage of the hallig Südfall above the sea is presented in Section 5.
2. Typical Inferior Mirages
The refraction equation of a light ray with a continously variable refractive index n on a flat earth can be written as6
which depends on the temperature profile T(z, x) and a characteristic constant w. For the temperature profiles T(z, x) analytic functions with a few parameters are used. The refraction differential equation is integrated numerically.
This concept has a few advantages: There are no errors that are due to the shell approximation. For the numerical integration a Runge-Kutta method with an automatic control of the integration error is used. The algorithm works well for different kinds of temperature profile.
For typical inferior mirages we choose the temperature profile5
Table 1. Three Sets of Parameter Values of T1, from Measured Temperature Values on a Calm Day
The first term on the right models the fast temperature increase near the ground. The second term describes a linear temperature decrease with increasing height, while d is related to the surface temperature by d=T1(0)-a .
Using a simple thermometer with a sensor on a long line, I measured three temperature profiles consisting of ten datapoints above the mud flats and a sandbank on a calm day. By fit of the data points with T1, I obtained three sets of parameter values of T1, listed in Table 1.
It is a challenge to develop instructive computer graphics for mirage work. A collection of computer-generated pixel maps (pixmaps) of familiar objects undergoing inferior mirages was designed. They are used as objects for the simulation of mirages. Digitized photographs of the objects of mirages can be used, too.
As an application of the visualization algorithms developed, I show a few graphics of usual inferior mirages at the Halligen Sea. First, ray bending near the ground is visualized. Ray tracing with the profile P2 is shown in Fig. 1 for the pixmap hallig. On the left-hand side is the mirage of the hallig. The right-hand side shows the temperature profile, a sequence of rays for eye angles of constant step size and the pixmap hallig rotated in the drawing plane. The rays at low eye angles are bent upward. They produce an inverted image, because the observer mistakenly interprets the light path as a straight one and thus sees a displaced image. The body of the island is not reached by the rays. Therefore the warft appears not to be connected to the body of the hallig and seems to hover at the horizon. The eye angle of the lowest ray on the object rectangle is the separation line between the upright image and the
Table 2. Parameter Values for the Temperature Profiles used for the Simulations Shown
inverted image of the object. This line in the mirage image is called the discontinuity line, vanishing line, or optical horizon. I call it the vanishing line in what follows.
Second, two series of mirages for lowering of eye position and increasing distance between the object and the observer are presented. Figure 2 shows the dependence of the mirage of a hallig on the eye position for the temperature profiles P1, P2, and P3 (see Table 1). With lower eye position the lower part of the object disappears behind the vanishing line. In the case of a few warfts on a hallig, first the warfts become disconnected and seem to hover at the horizon. Then different parts of a warft, e.g., the flag-post and the house, become disconnected, too.
Figure 3 shows the dependence of the appearance of the mirage on the distance of walkers for the three profiles. With increasing distance the inverted mirage image becomes more complete. At the same time the lower part of the object disappears behind the vanishing line. In this case the dog, the child, and the woman, one after the other, vanish. In a few pictures the walkers seem to walk on the water.
Third, effects of fast air fluctuations are illustrated. In Engler’s film there are five scenes of a walker with a stick at different distances. From each scene a photograph and its best simulation are shown in Fig. 4 (see Table 2). As object for the simulation we use the photograph of the first scene. The parameters of T1 are adjusted by visual comparison of the graphics with the five photographs. As the comparison with the simulation shows, the defocusing of the mirage rapidly increases with distance. Structures of small width, such as the stick, may not be identifyable in mirages at large distances.
The main cause for the defocusing appears to be fast air fluctuations caused by unstable stratification at the surface. Effects of air fluctuation are not taken into account in the refraction equation and in the temperature profile T1. Nevertheless, the simulation helps us to recognize and quantify the effects of fast air fluctuations.
The typical inferior mirage has a simple appearance consisting of one upright image of the object on the top and one inverted image on the bottom. Inferior mirage images with another image structure are refered to as unusual inferior mirages. In the Halligen Sea we observed two kinds of unusual inferior mirage, two inferior mirages with four images and two inferior mirages with a horizontal stripe of constant width in the mirage image.
3. Double Inferior Mirage
The mirage of the Ockenswarft at the hallig Hooge (Fig. 5) has an unusual form. Clearly two upright images of the Ockenswarft can be seen. A thorough analysis of the photograph shows that below each upright image there is a strongly contracted inverted image of the warft. In this paper this form of mirage is refered to as a double inferior mirage.
Fig. 5. Double inferior mirage of the Ockenswarft at the hallig Hooge. The photograph is taken from the sandbank Japsand (see Fig. 6).
From structural arguments it follows that in this case the temperature profile function cannot be a smooth decreasing function like T1 but must have a turning point. Therefore an unusual temperature profile must have existed between the observer and the Ockenswarft. The position of the observer was at the sandbank Japsand, and the position of the Ockenswarft at the hallig Hooge (see Fig. 6). The mud flats between Japsand and Hooge lie 1.2 m lower than the sandbank Japsand, which normally stays dry at high tide. During the observation there was a light west wind, which may have carried warm air from above the sandbanks to the mud flats between the observer and the object.
Assuming the existence of a dynamically stable warm layer of some thickness at the ground, we may use the profile
where f is the thickness of the layer and e is the temperature change over the layer.
The structure of the double inferior mirage can be reproduced with reasonable values for all parameters (Fig. 7 and Table 2). Ray tracing shows that the upper inverted image is produced by upward bending of the rays at the surface of the warm layer, whereas the lower inverted image is created by upward bending of the rays at the ground.
The right-hand series of images in Fig. 8 shows the best simulation of the double inferior mirage of the Ockenswarft for four eye positions. The thickness of the warm layer is predicted to be 1.5 m. The photograph of the double inferior mirage can be compared with the simulation at h=2.5 m. The structure of the mirage image is the same. However, we cannot obtain the same form of the image in detail, because the model of the layer is very simple. For example, it is assumed that the thickness of the warm layer is constant between the observer and the object. The upper part of the mirage disappears if the eye position is lower than the thickness of the layer. Then the mirage has the structure of a typical inferior mirage, in agreement with another photograph of the same phenomenon taken at the same place, at h=1.5 m, above the mud flats.
On the left-hand side of Fig. 8 a simulation without the warm layer (e=0, with all other parameters fixed) is shown for comparison. For eye positions below the layer it is difficult for the observer to recognize whether the image is a single or double inferior mirage. The comparison of the two simulations at h=1 m shows that in the case of a double inferior mirage a large part of the object has disappeared behind the vanishing line, whereas in the case of a typical inferior mirage only a small part of the object is enveloped.
4. Inferior Mirages with Horizontal Stripes
Another unusual form of inferior mirages in the Halligen Sea are mirages with a horizontal stripe. In a photograph of the Schulwarft at the hallig Nordstrandischmoor, taken in the mud flats near Hamburger Hallig at an eye position of 1.4 m, a horizontal stripe appears in mirage image (Fig. 9). The stripe has the color of the mud flats. With the eye position lowered, the width of the stripe decreased, and it disappeared at ~h=1.2 m. A horizontal stripe also appeared in a mirage of the hallig Südfall, observed near the Dickehörn at the peninsula Nordstrand at h=1.2 m. A few mirages with a horizontal stripe also occur in Engler’s film.
The stripe in Fig. 10 looks like a local disturbance of a typical inferior mirage. This may be produced by a local disturbance of the temperature profile because of the complex seafloor topography in the Halligen Sea. As the map of the Halligen Sea (Fig. 6 shows that the line of sight between Hamburger Hallig and the Schulwarft at the hallig Nordstrandischmoor is crossed almost perpendicularly by a deep water channel of width 700 m at a distance of 3500 m. No channel is drawn in the map between the Dickehörn and the hallig Südfall. However, a water channel indeed occurs at the line of sight at a distance of 2500 m. In this case it is a shallow water channel of 10 m in width and 0.05–0.1 m in depth.
During sunshine in summer the water in the channels is colder than the mud flats. Therefore the rays above the channel can be bent downward and may reach the ground.
A dynamically stable temperature depression is assumed to occur over the water channel. With the temperature decrease near the channel modeled by a Gaussian, the profile can be expressed as
where e is the maximum temperature decrease above the channel and while f and g are the position and width of the channel.
Fig. 8. Dependence on eye position for the double inferior mirage of the Ockenswarft; T2 is adjusted.
Fig. 9. Inferior mirage of the Schulwarft at the hallig Nordstrandischmoor. The photograph was taken from the Hamburger Hallig.
Fig. 11. Dependence on eye position for an image with a stripe of a warft at the hallig Nordstrandischmoor; T3 is adjusted.
Fig. 12. Simulation of three photographs of a mirage of the hallig Südfall at high tide; T3 is adjusted.
In the simulation we use fixed values for f and g taken from the map and a fixed parameter value for the surface temperature at the ground. The four other parameters are adjusted by visual comparison of the graphics with the photographs.
The simulation of the mirage of the Schulwarft reproduces the stripe for an eye position of 1.35 m (Fig. 10). Ray tracing shows that for a small region of eye angles the rays are bent downward and indeed reach the ground. The stripe appears to be the image of the surface of the channel or the mud flats nearby. Moreover, ray tracing shows that the stripe vanishes at h<1.2 m because all rays are either bent upward or strike the ground a bit in front of the channel.
The right-hand side of Fig. 11 shows the simulation of the mirage of the Schulwarft for four values of the eye position. On the left-hand side a simulation without the heat depression (e=0, with all other parameters fixed) is shown for comparison. A stripe occurs for h>1.33 m. For h>1.55 m the stripe cannot be easily recognized because it is visually not separated from the mud flats in the foreground. As a reference the observer may use the straight upper border line of the stripe. Moreover, comparison of the right- and left-hand images at h=1.6 m shows that the stripe can take the place of the inverted image. In spite of favorable conditions, no inverted image is visible. In this case an observer should lower the eye position. The maximum temperature decrease above the deep water channel is predicted to be 0.2 °C.
By simulation of a photograph of the mirage with a horizontal stripe of the hallig Südfall a maximal temperature decrease above the shallow water channel of 0.01 °C is obtained. This small value is not unreasonable, because at low tide no cold water from the neighboring sea, but only water from the neighboring mud flats, flows in this shallow water channel.
Stripes and spots of the ground in mirages can be caused by the roughness of the ground. However, in that case the form of the stripe most likely would not be a horizontal stripe of constant width as in the case of water channels.
5. Inferior Mirages at High Tide
In the winter season, when the temperature of the air is <3 °C and the tide is high, beautiful inferior mirages appear at the Halligen Sea.1 Under these conditions the sea warms the colder air and produces a rapid temperature increase near the sea surface. In this case inferior mirages appear at high tide and disappear at low tide. Inferior mirages above the sea and above the mud flats may even appear on the same day.
At high tide the topography of the ground is simple; it is a large-scale flat surface. The small-scale roughness of the sea contributes to the defocusing of the image; however, it should not determine the main structure of the image. Therefore the temperature profile over the sea can be approximated by formula (2).
On a cold day in the spring of 1997 an inferior mirage of the hallig Südfall at high tide was observed. With a changing eye position, a series of photographs was taken near the Dickehörn at Nordstrand during a rather strong wind.
For the simulation of the mirage a photograph of the hallig Südfall taken on a day without mirages at high tide was selected as object. The three parameters a, b, and c were adjusted by visual comparison of the graphics with three photographs of the mirage at h=3.0, 1.8, 1.0 m. A graphic of the best simulation is shown in Fig. 12. The white foam spots of the sea surface give an impression of the roughness of the sea. Nevertheless the defocusing of the mirage images was less than in the case of the mirages of the hallig Südfall observed in summer at low tide. The rapid temperature increase near the water surface is predicted to be 0.3 °C. The temperature difference between the sea and the cold air, measured at the coast of Nordstrand, was 1 °C.
6. Conclusions
Inferior mirages of halligen, lighthouses, boats, walkers, horsemen and the Sun can be studied in the Halligen Sea. Typically the halligen and islands are at a distance of 5–10 km. Therefore, a hallig can be observed from another hallig or island in the neighborhood. In the Halligen Sea inferior mirages occur throughout the year. In summer they appear at low tide above the mud flats, and in the winter season at high tide above the sea. The warfts at the halligen are photogenic objects for mirages. When an observer varies the eye position by climbing up and down a dike, the change of the form of the appearance of the mirages can be easily observed with good binoculars and can be photographed with a strong tele-photo lens. From a hallig one often observes mirages of a few halligen at different distances. At low eye positions or large distances the warfts seem to hover like airships at the horizon.
Because of the complex topography of the Halligen Sea, two unusual forms of inferior mirage can occur. If a light wind carries warm air from the dry sand-banks over the mud flats between the observer and the object, a double inferior mirage can appear. If a water channel crosses the line of sight between the observer and the object, a horizontal stripe of constant width, of the ground, can occur in the mirage image.
The mirages were analysed by computer simulations by use of heuristic analytical functions with a few parameters for the temperature profiles. The simulations can provide instructive computer graphics. These can used to help explain the mirage by ray tracing and for the adjustment of the parameters by visual comparison with photographs. Moreover, the simulation can help the observer visualize the changing form of the mirage with eye position and distance. Although the simulation does not describe effects of fast air fluctuations, the graphics help one recognize and quantify air-fluctuation effects in mirage images.
The temperature profiles T2 and T3 are heuristic functions with three additional parameters. They are not based on calculated or measured meteorological data. Moreover, no attempt was made to determine the temperature profiles T2 and T3 from photographs of the mirages by use of an inversion of the computation algorithm.
The numerical integration algorithm with adaptive step size works well for the temperature profiles T1, T2, and T3 for double-precision arithmetics. In the case of T3 no algorithm with fixed step size size can be used, because for water channels of small width the resulting horizontal stripes in the mirage image might be missed.
The author thanks M. Engler for sending a video tape of his film and for additional information on the content of the film. He thanks W. Lehn for help in the interpretation of a photograph and for many discussions on the simulation of mirages.
References
1. M. Engler, “Luftspiegelungen WATT’N DAT?” Geo5, 144–162 (1992); Television film “Fata Morganen—Zauberspiegel am Horizont,” first broadcast 8 March 1996, M. Engler, producer, ARTE (1996).
2. A. Wegener, “Elementare Theorie der atmosphärischen Spiegelungen,” Annalen der Phys. 4. Folge, Band 57, 203–230 (1918).
3. W. H. Lehn, “A simple parabolic model for the optics of the atmospheric surface layer,” Appl. Math. Model. 9, 447–453 (1985). [CrossRef]
4. A. B. Fraser, “Simple solution for obtaining a temperature profile from the inferior mirage,” Appl. Opt. 18, 1724–1731 (1979). [CrossRef] [PubMed]
5. W. H. Lehn and J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. on Geosci. Remote Sensing , GE-24, 940–964 (1986). [CrossRef]
6. J. M. Pertner and F. M. Exner, Meteorologische Optik, 2nd ed. (Braumuller, Vienna, 1922).
