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
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.
Fig. 1. Ray tracing for the pixmap hallig. Profile P2 is used.
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
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
P1||Sandbank at Japsand||3.7||0.06||0.51||25|
P2||Mud flats at Japsand||2.8||0.11||0.13||23|
P3||Mud flats at Ockholm||1.5||0.17||0.12||24|
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
Fig. 2. Dependence on eye position for the pixmap hallig; P1, P2, and P3 are used.
Fig. 3. Dependence on distance for the pixmap walker; P1, P2, and P3 are used.
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 T
1, I obtained three sets of
parameter values of T
1, 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 P
2 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
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.
Fig. 4. Effects of fast air fluctuations for a walker from Engler’s film;
T1 is adjusted.
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 P
2, and P
(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
, 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.
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 T
1 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.
Double inferior mirage of the Ockenswarft at the hallig Hooge. The photograph is taken from
the sandbank Japsand (see Fig. 6
Fig. 6. Map of the Halligen Sea.
From structural arguments it follows that in this case the temperature profile function cannot
be a smooth decreasing function like T
1 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
=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
=1.5 m, above the mud flats.
Fig. 7. Ray tracing for the pixmap hallig; T2 is adjusted.
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
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
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
observed near the Dickehörn at the peninsula Nordstrand at h
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. 10. Ray tracing for the pixmap hallig; T3 is adjusted.
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
<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
=0, with all other parameters fixed) is shown for comparison. A stripe occurs
>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.
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
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.