Mirages are caused by the curvilinear propagation of light rays as a result of variations in the atmospheric refractive index. The refractive index is most strongly determined by the air temperature, so mirages carry information about the atmospheric temperature profile. It is straightforward to calculate the appearance of a mirage (defined by its image diagram) given the refractive-index profile (the forward problem); it is much less so to solve the inverse problem of deducing the refractive-index profile from the appearance of the mirage. The methods that have been employed to tackle this inverse problem are reviewed briefly; then a semirigorous treatment of mirages having linear image diagrams is developed. This class of mirages, while rare meteorologically, is interesting mathematically because it is possible to classify all the possible forms of refractive-index profile that can give rise to such mirages, and it includes the important phenomenon of the ducting of light rays. The treatment presented yields the refractive-index profile from the image diagram, except for certain well-defined cases for which no solution is possible. It also shows, by counterexample, that there can be no uniqueness theorem stating that a given image diagram can arise from only one refractive-index profile.
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