The following papers may be downloaded free, for personal use only.

“Isaac Newton and the astronomical refraction” Lehn 2008

Abstract: In a short interval toward the end of 1694, Isaac Newton developed two mathematical models for the theory of the astronomical refraction and calculated two refraction tables, but did not publish his theory. Much effort has been expended, starting with Biot in 1836, in the attempt to identify the methods and equations that Newton used. In contrast to previous work, a closed form solution is identified for the refraction integral that reproduces the table for his first model (in which density decays linearly with elevation). The parameters of his second model, which includes the exponential variation of pressure in an isothermal atmosphere, have also been identified by reproducing his results. The implication is clear that in each case Newton had derived exactly the correct equations for the astronomical refraction; furthermore, he was the first to do so.

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“Atmospheric refraction: a history” Lehn, van der Werf 2005

Abstract: We trace the history of atmospheric refraction from the ancient Greeks up to the time of Kepler. The concept that the atmosphere could refract light entered Western science in the second century B.C. Ptolemy, 300 years later, produced the first clearly defined atmospheric model, containing air of uniform density up to a sharp upper transition to the ether, at which the refraction occurred. Alhazen and Witelo transmitted his knowledge to medieval Europe. The first accurate measurements were made by Tycho Brahe in the 16th century. Finally, Kepler, who was aware of unusually strong refractions, used the Ptolemaic model to explain the first documented and recognized mirage (the Novaya Zemlya effect).

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“The hafstramb and margygr of the King’s Mirror: an analysis” Lehn, Schroeder 2004

Abstract: Greenland and Iceland are described with unusual scientific accuracy in the King’s Mirror. However this thirteenth-century manuscript contains a few ‘wonders’ that appear more mythological than rational. They include the hafstramb and the margygr, commonly translated respectively as merman and mermaid. The mermaid has a long history in western civilisation. The commonly accepted theory that it evolved from the classical Greek siren is critically examined here. The margygr is shown to be a distinct creature based on independent observation in northern Europe. The characteristics of these observations actually modified the siren of the Physiologus, a bird-woman, into the fish-woman known today. Observations of hafstramb and margygr are explained as superior mirages. These are caused by atmospheric refraction, which distorts and magnifies distant objects. Computer simulations and photographs show that mirages of an orca, a walrus, or even a boulder match almost point for point the descriptions in the King’s Mirror. Thus the apparently mythical components in the Greenland account are in fact careful scientific observations.

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“Hafgerdingar: a mystery from the King’s Mirror explained” Lehn, Schroeder 2003

Abstract: The medieval King’s Mirror describes Iceland and Greenland with a scientific accuracy that is remarkable. One of the very few exceptions is the hafgerdingar in the Greenland Sea. The term translates as ‘sea hedges,’ within which a mariner may become trapped at great peril. Many have believed that a real event was being described, although none of the proposed explanations has been totally satisfactory. The most common view currently is based on Steenstrup (1871), who explained the phenomenon as a tidal wave following a submarine earthquake. A simpler and more consistent theory is developed here: that the hafgerdingar are an optical phenomenon, specifically, a superior mirage. Such mirages, quite common in the polar regions, can produce an appearance fully consistent with the original description, as illustrated by several photographs and a computer simulation. Even the peril to seafarers has been corroborated, in the sense that such a mirage is frequently followed by a storm.

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“Gerrit de Veer’s true and perfect description of the Novaya Zemlya effect, 24 –27 January 1597” van der Werf, Können, Lehn, Steenhuisen, Davidson 2003

Abstract: The first recordings of the Novaya Zemlya NZ effect were made during Willem Barents’ third Arctic expedition. Ray-tracing analyses of the three key observations, on 24 –27 January 1597, show that all the reported details can be explained by adopting one common and realistic type of temperature inversion. In particular, the Moon–Jupiter conjunction could have been visible over the central mountain ridge of the island. We show that the NZ effect distorts the relative positions of Jupiter and the Moon in such a way that the looked-for fingerprint of the conjunction occurred almost 2 h after the true conjunction. The quoted direction for the apparent Moon–Jupiter conjunction is then found to be accurate to within 1°. This delay of the apparent conjunction largely explains the error of 29° in their longitude determination. The truthfulness of these observations, debated for four centuries, now appears to be beyond doubt.

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“Novaya Zemlya effect and sunsets” van der Werf, Können, Lehn 2003

Abstract: Systematics of the Novaya Zemlya NZ effect are discussed in the context of sunsets. We distinguish full mirages, exhibiting oscillatory light paths and their onsets, the subcritical mirages. Ray-tracing examples and sequences of solar images are shown. We discuss two historical observations by Fridtjof Nansen and by Vivian Fuchs, and we report a recent South Pole observation of the NZ effect for the Moon.

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“Bright superior mirages” Lehn 2003

Abstract: Superior mirages of unusual brightness are occasionally observed. Two such cases, photographed over the frozen surface of Lake Winnipeg, Canada, are documented. Visually, these mirages appear as featureless bright barriers far out on the lake. They are just images of the lake ice, yet the luminance in one case was 2.5 times in the other, 1.7 times the luminance of the ice surface in front of the mirage. The mirage itself can be modeled by means of a conduction inversion, but a proper explanation of the brightness is not yet available.

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Fig. 2 Fig. 3 Fig. 5

“Exact temperature profile for the hillingar mirage” Lehn 2001

Abstract: In a hillingar mirage, the Earth’s surface appears flat, because nearly horizontal light rays have the same curvature as the Earth. A linear temperature profile is traditionally inferred; its gradient is calculated to give this curvature to the exact horizontal ray. To see an image, however, a bundle of rays is required. To ensure that each ray in the bundle have the same curvature, the temperature profile must contain a small positive quadratic term, the coefficient of which is derived.

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“Skerrylike mirages and the discovery of Greenland” Lehn 2000

Abstract: The Norse discovery of Greenland is associated with the sighting of low barren islands called Gunnbjörn’s Skerries, which have never been satisfactorily identified. Here the historical references that connect the skerries to Greenland are reviewed. A mirage of the Greenland coast, arising specifically from optical ducting under a sharp temperature inversion, is used to explain the vision of skerries seen by the Norse mariners. Images from both ducting and uniform inversions are calculated. Under the assumption of a clean Rayleigh atmosphere, sufficient visibility remains to see the skerry image at a distance of 220 km. There is significant circumstantial evidence to indicate that the Norse were familiar with the skerrylike mirage and that they used it to discover new lands.

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“Long-range superior mirages” Lehn, Legal 1998

Abstract: Superior mirages of simple appearance are occasionally observed over distances exceeding 70 km. These mirages cannot be explained in terms of standard textbook models; rather, they are shown to arise from fairly complex atmospheres. Two observations of different types, observed at Resolute Bay, Canada, are presented. The first is the basic three-image mirage in which one inverted and one erect image float above the object. The second is a single-image mirage in which the object is elevated but undistorted. For each, the most suitable atmospheric model contains several distinct atmospheres, and the first one requires sloped atmospheric layers as well.

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Fig. 1a Fig. 1b

“Analysis of an infrared mirage sequence” Lehn 1997

Abstract: Infrared observations of seaborne thermal sources are subject to the effects of atmospheric refraction. For low elevation angles at long ranges, out to the limit of visibility, the inevitable atmospheric temperature gradients frequently produce mirages. I present an analysis of a 22-min sequence of images recorded on 18 February 1994 at the U.S. Naval Surface Warfare Center at Wallops Island, Virginia. The infrared target is a heat source carried on a ship moving in a straight line toward the camera. The images show a quasi-periodic variation of the horizon elevation, as well as an extended range of visibility. A model that reasonably reproduces the observed features consists of a small temperature inversion in a slightly sloped atmosphere, with an atmospheric gravity wave moving across the line of sight.

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“Mirages with atmospheric gravity waves” Lehn, Silvester, Fraser 1994

Abstract: The temperature inversions that produce superior mirages are capable of supporting gravity (buoyancy) waves of very low frequency and long wavelength. This paper describes the optics of single mode gravity waves that propagate in a four-layer atmosphere. Images calculated by ray tracing show that (1) relatively short waves add a fine structure to the basic static mirage, and (2) long waves produce cyclic images, similar to those observed in the field, that display significant variation from a base image.

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Photographs appearing in the paper are available in full resolution:

Fig. 1a Fig. 1b Fig. 1c Fig. 1d Fig. 1e Fig. 1f Fig. 1g Fig. 1h Fig. 1i Fig. 2

“Differential geometric approach to atmospheric refraction” Kropla, Lehn 1992

Abstract: Differential geometric techniques are presented and used to model the optical properties of the atmosphere under conditions that produce superior mirages. Optical path length replaces the usual Euclidean metric as a distance-measuring function and is used to construct a surface on which the paths of light rays are geodesics. The geodesic equations are shown to be equivalent to the ray equation in the plane. A differential equation that relates the Gaussian curvature of the surface and the refractive index of the atmosphere is derived. This equation is solved for the cases in which the curvature vanishes or is constant. Illustrative observation demonstrate the use of geometric techniques in the analysis of mirage images.

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“Simulation of mirages” Lehn, Friesen 1992

Abstract: A mirage is seen when atmospheric refraction distorts or displaces an image. We describe a mirage simulator that uses digital imaging equipment to generate mirage images from normal photographs. The simulation program relocates horizonal image lines into positions that they appear to occupy, according to rays traced from observer to object. Image-brightness adjustments are not required; we show that, while the atmosphere can change the size or shape of an object, it does not change its apparent brightness. The realistic quality of the computed images makes this simulator a useful tool in mirage analysis.

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Photographs appearing in the paper are available in full resolution:

Fig. 2a Fig. 2b Fig. 5a Fig. 5b Fig. 5c

“The Scoresby ship mirage of 1822” Lehn, Rees 1990

Abstract: A very clear mirage observed by Scoresby in the Greenland Sea shows an inverted ship floating above the horizon. This mirage can be mathematically reconstructed using a linear image diagram. Scoresby’s description is here re-examined: a new set of essential assumptions is distilled from his report, and an ‘exact’ reproduction of the mirage is obtained to match these conditions.

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“A Three-Parameter Inferior Mirage Model for Optical Sensing of Surface Layer Temperature Profiles” Lehn, Morrish 1986

Abstract: Inferior mirages provide a sensitive and fairly accurate probe for determining vertical temperature distributions in the atmospheric surface layer. Optical measurements on the image can be used to calculate the parameters in a temperature profile model, in this case a function with three adjustable parameters. The function contains an exponential term (two parameters) and an additive linear term (one parameter). The optical observations, for which a known target is required, consist of the elevation angles of the apparent peak, caustic, and horizon. Analytic expressions that must be simultaneously satisfied are derived for all three conditions. The parameter values are extracted numerically by minimizing a positive definite function of the three conditions. The model is tested on a set of images for which nearly simultaneous photographs, theodolite readings, and temperature profiles were available. For each image the three calculated elevations matched the measured values very closely. The complete images also match well in most of the cases. The results, a distinct improvement over previous two-parameter models, also provide a more accurate reconstruction than is obtained from the thermodynamic model for unstable stratification.

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Inversion of superior mirage data to compute temperature profiles Lehn 1983

Abstract: Information derived from the superior mirage is used to compute the average vertical temperature profile in the atmosphere between the observer and a known object. The image is described by a plot of ray-elevation angle at the eye against elevation at which that ray intersects the object. The computational algorithm, based on the tracing of rays that have at most one vertex, iteratively adjusts the temperature profile until the observed image characteristics are reproduced. An example based on an observation made on the Beaufort Sea illustrates the process.

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Fig. 5a Fig. 5b

“The Norse merman as an optical phenomenon” Lehn, Schroeder 1981

Abstract: Mediaeval Norse writings that describe the merman are considered accurate observations of a natural phenomenon. Images of common sea mammals, severely distorted by strong, non-uniform atmospheric refraction, fit the mediaeval descriptions remarkably well. Three examples are presented: computer-generated images of a killer whale and a walrus, and a photograph of a suitably distended boulder. The mediaeval correlation of a merman sighting with the advent of storms is also verified.

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“Novaya Zemlya effect: analysis of an observation” Lehn, German 1981

Abstract: The Novaya Zemlya effect, historically identified with the premature rebirth of the sun during the polar night, is a long range optical ducting phenomenon in the lower atmosphere. An occurrence of the effect was observed at Tuktoyaktuk, Canada (69˚26'N, 133˚02'W) on 16 May 1979, when the minimum solar altitude was -1°34'. The sun's image remained above the horizon, within a gray horizontal band, and assumed the various expected shapes, ranging from a bright rectangle filling the band, to three flat suns stacked one over the other, to several thin vertically separated strips. A model for the corresponding atmospheric conditions was identified by matching the observations with images calculated from a computer simulation study.

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Fig. 4a Fig. 5a Fig. 6a Fig. 8a Fig. 9

“Polar Mirages as Aids to Norse Navigation” Lehn, Schroeder 1979

Abstract: The possibility is examined that the Norse may have gleaned information from polar mirages for their westward expansion across the North Atlantic. Two types of superior mirages, the hillingar and the Novaya Zemlya effects, are explained briefly. Examples reported by early explorers are used to familiarize the reader with the effects and to illustrate both their informative and confusing natures. Specifically, optical theory is applied in an attempt to establish the Gunnbjorn Skerries as images of the Greenland coast, transmitted by mirage across the Denmark Strait. This hypothesis is supported by an examination of available historical evidence.

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“The Novaya Zemlya effect: An arctic mirage” Lehn 1979

Abstract: The arctic mirage is an atmospheric refraction phenomenon caused by a temperature inversion in the lower atmosphere. It is classified into three basic types, two of which (hillingar and hafgerdingar effects) occur fairly frequently. The third is the Novaya Zemlya effect, reported by polar explorers on several occasions as an anomalous sunrise during the polar winter, when the position of the sun was below the horizon. The Novaya Zemlya effect consists of the trapping of light rays beneath a thermocline of large horizontal extent. Within the thermocline layers, the coefficient of refraction must exceed 1, while above and below it the coefficient must be less than 1. The certain upward rays repeatedly bounce back from the thermocline and are transmitted for long distances around the earth’s curvature. The anomalous sunrise is a special case of this generalized definition. The properties of the Novaya Zemlya effect, analyzed using a laterally uniform stratified-atmosphere model, agree with those reported by polar expeditions. A narrow strip or window appears near the horizon, with or without an image of the sun in the window. An observation sketched by Liljequist in Antarctica is reconstructed to demonstrate the model’s accuracy.

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“Atmospheric Refraction and Lake Monsters” Lehn 1979

Abstract: A survey of reported sightings of lake monster phenomena suggests that many of them may be attributable to atmospheric image distortion. The existence of the necessary conditions (surface temperature inversion and hence strong atmospheric refraction) can be inferred from most of the reports. Under such conditions familiar objects can easily take on unrecognizable form. Two photographs demonstrate the extent of the distortion that can occur.

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