Chapter 4 Optical Mirage William Viezee

1. Introduction

An optical mirage is a phenomenon associated with the refraction of light in the gaseous (cloud-free) atmosphere. During mirage a visible image of some distant object is made to appear displaced from the true position of the object. The image is produced when the light energy emanating from the distant source travels along a curvilinear instead of a rectilinear path, the curvilinear path, in turn, arises from abnormal spatial variations in density that are invariably associated with abnormal temperature gradients.

The visible image of the mirage can represent shape and color of the "mirrored" object either exactly or distorted. Distortions most commonly consist of an exaggerated elongation, an exaggerated broadening, or a complete or partial inversion of the object shape. Frequently, mirages involve multiple images of a single source. Under special conditions, refractive separation of the color components of white light can enhance the observation of a mirage. Atmospheric scintillation can introduce rapid variations in position, brightness, and color variations of the image.

When both the observer and the source are stationary, a mirage can be observed for several hours. However, when either one or both are in motion, a mirage image may appear for a duration of only seconds or minutes. Although men have observed mirages since the beginning of recorded history, extensive studies of the phenomenon did not begin till the last part of the 18th century. Since that time, however, a large volume of literature has become available from which emerges a clear picture of the nature of the mirage. The comprehensive body of information presented here is based on a survey of the literature, and constitutes the state-of-the-art knowledge on optical mirages. The report provides a ready source of up-to-date information that can be applied to problems involving optical mirages. [[987]]

No claim is made that all existing pertinent writings have been collected and read. The contents of many publications, especially of those dating back to the last part of the 18th Century and the beginning of the 19th Century are evaluated from available summaries and historical reviews. Also, when a particular aspect of the mirage phenomenon is considered, the collection of pertinent literature is discontinued at the point where the state-of-the-art knowledge appears clearly defined. The collected volume of literature covers the period 1796 to 1967. In essence, the literature survey yields the following principal characteristics of the mirage:

1.Mirages are associated with anomalous temperature gradients in the atmosphere. 2.Mirage images are observed almost exclusively at small angles above or below the horizontal plane of view; mirages, therefore, require terrain and meteorological conditions that provide extended horizontal visibility. 3.A mirage can involve the simultaneous occurrence of more than one image of the "mirrored" object; the images can have grossly distorted forms and unusual coloring. 4.Extreme brightening and apparent rapid movement of the mirage image in and near the horizontal plane can result from the effects of focussing and interference of wavefronts in selected areas of the refracting layer.

Only minor shortcomings appear to be evident in present knowledge of mirage phenomena. Ultimately, a unified theory is desirable that can deal with both the macroscopic and microscopic aspects. Currently, the behavior of light refraction on a large scale is represented by means of rays while the finer details are treated with the wave theory. More observations are needed that deal with the microscopic optical effects of the mirage. The finer details that arise mostly from focussing and interference are not commonly observed. They require close examination of areas that are highly selective in time and place. BACK TO TOP

2. Cross Section of Surveyed Literature The contents of this report are based on a survey of literature on atmospheric refraction in general and on optical mirages in particular. The survey began with the review of such basic sources of information on atmospheric optics as Meteorologische Optik, by Pernter and Exner, Physics of the Air, by Humphreys, The Nature of Light and Colour in the Open Air, [[988]]

by Minnaert, and The Compendium of Meteorology. These sources present historical summaries, and their contents are to a large extent based on literature surveys. Key references mentioned in these sources were examined and a large volume of literature was subsequently collected by following successive reference leads. Pertinent information on atmospheric scintillation was obtained from several sources, in particular from Optical Scintillation; A Survey of the Literature, by J. R. Meyer-Arendt. A cross section of the collected literature is listed below. Because of the wide range of aspects covered, the literature is listed in the following categories: 1.papers on optical mirage the contents of which are mostly descriptive, 2.papers that propose theoretical models of spheric refraction or optical mirage, 3.papers that compare theory and observation, 4.papers that are concerned with the application of terrestrial light refraction to meteorology, surveying, and hydrography, 5.papers that present average values of terrestrial refraction based on climatology, and 6.papers on atmospheric scintillation. Within each category, publications are arranged chronologically.

In category 1, descriptive accounts of mirages go back in time to 1796, when Joseph Huddart observed superior mirages near Macao. (Earlier accounts can be found in Meteorologische Optik.) Numerous recent observations of abnormal atmospheric refraction can be found in The Marine Observer. The two "classical" observations most frequently quoted as having "triggered" a long series of investigations on optical mirage are the observations of Vince and Scoresby. Vince (1798) from a position on the sea shore observed multiple images of ships, some upright and some inverted, above the ocean horizon; Scoresby (1820) observed elevated images of ships and coastal lines while navigating near Greenland. Both observations were carefully documented and results were read before bodies of the Royal Society. Proposed theories of the mirage (category 2) are basically of three types, that are best represented by the respective works of Tait (1883), Wegener (1918), and Sir C. V. Raman (1959). Tait (in his efforts to explain the observations by Vince and Scoresby) considers a vertically [[989]]

finite refracting layer having a continuous change in refractive index, and formulates the ray paths for a plane-stratified atmosphere. Wegener (motivated by mirage observations made during his stay in Greenland) replaces Tait's finite refraction layer with a "reflecting" surface - i.e. a surface of discontinuity in the refractive index - and formulates the ray paths for a spherically stratified atmosphere. Raman questions the use of geometric optics in the theory of the mirage and shows by means of physical optics that the upper boundary of the refracting layer resembles a caustic surface in the vicinity of which focussing and interference are the major mirage-producing effects. All three theories quite accurately describe various mirage observations.

Comparisons made between observation and theory (category 3) indicate that the two are compatible - i.e., abnormal light-refraction phenomena are associated with anomalous atmospheric-temperature structure. Many investigations (category 4) are concerned with determining the effects of light refraction on optical measurements made in such fields as surveying and hydrography. Corrections for refraction based on average atmospheric conditions have been computed (category 5). Of specific interest to meteorologists are the attempts to develop inversion techniques for obtaining low-level temperature structure from light-refraction measurements (category 4). The temperature profiles that can be obtained do not have the desired resolution and accuracy. During the last decade, literature on atmospheric scintillation has become extensive due to its importance to astronomy, optical communication, and optical ranging. A selected number of recent papers are presented in category 6.

The publications categorized below represent a cross section of the various endeavors that have resulted from the Earth's atmosphere having light-refraction properties. The body of information is fundamental to the contents of this report. In addition to the listed literature, many other sources of information on atmospheric optics were consulted in its production. They are referenced throughout the text, and are compiled in a bibliography at the end of the report.

CATEGORIES 1. Descriptive 2. Theoretical Models 3. Theory vs. Observation 4. Application 5. Average values 6. Atmospheric Scintillation BACK TO SECTION 2 [[990]]

Category 1 (Descriptive Accounts)

1.Huddart, Joseph, "Observations on Horizontal Refractions Which Affect the Appearance of Terrestrial Objects, and the Dip, or Depression of the Horizon of the Sea," Phil. Trans. Vol. 87, pp. 29-42 (1797).

2.Vince, S., "Observations on an Unusual Horizontal Refraction of the Air; with Remarks on the Variations to Which the Lower Parts of the Atmosphere are Sometimes Subject," London Phil. Trans., Part 1, pp. 436-441 (1799).

3.Wollaston, William Hyde, "On Double Images Caused by Atmospheric Refraction," Phil. Trans., Vol. 90, pp. 239-254 (1800).

4.Scoresby, William, "Description of Some Remarkable Atmospheric Reflections and Refractions, Observed in the Greenland Sea," Trans. Roy. Soc. Edinburgh, Vol. 9, pp. 299-305 (1823).

5.Parnell, John, "On a Mirage in the English Channel," Phil. Mag., Vol. 37, pp. 400-401 (1869).

6.Forel, F. A., "The Fata Morgana," Proc. Roy. Soc. Edinburgh, Vol. 32, pp. 175-182 (1911).

7.Hillers, Wilhelm, "Photographische Aufnahmen einer mehrfachen Luftspiegelung," Physik. Zeitschr., Vol. 14, pp. 718-719 (1913).

8.Hillers, Wilhelm, "Einige experimentelle Beitr~ge Zum Phanomen der dreifachen Luftspiegelung nach Vince," Physikalische Zeitschrift, Vol. 15, p. 304 (1914).

9.Visser, S. W. and J. Th. Verstelle, "Groene Straal en Kimduiking," Hemel en Dampkring, Vol. 32, No. 3, pp. 81-87 (March 1934).

10.Meyer, Rudolf, "Der grUne Strahl," Meteorologische Zeitschrift, Vol. 56, pp. 342-346 (September 1939).

11.Science Service, "Mirage in Desert May Explain How Israelites Crossed Red Sea Unharmed," Bull. Am. Met. Soc., Vol. 28, p. 186 (1947). [[991]]

12.Ives, Ronald L., "Meteorological Conditions Accompanying Mirages in the Salt Lake Desert," J. Franklin Institute, Vol. 245, No. 6, pp. 457-473 (June 1948).

13.St. Amand, Pierre and Harold Cronin, "Atmospheric Refraction at College, Alaska, During the Winter 1947-1948," Trans. Am. Geophys. Un., Vol. 31, No. 2, Part 1, (April 1950).

14.Ten Kate, H., "Luftspiegelungen," Hemel en Dampkring, Vol. 49, No. 5, pp. 91-94 (1951).

15.Ewan, D., "Abnormal Refraction of Coast of Portugal," The Marine Observer, Vol. 21, No. 152, p. 80 (April 1951).

16.Mitchell, G.E., "Mirage in Gulf of Cadiz," The Marine Observer, Vol. 21, No. 152, p. 81 (April 1951).

17.Illingsworth, J., "Abnormal Refraction in the Gulf of St. Lawrence," The Marine Observer, Vol. 22, No. 156, pp. 63-64 (April 1952).

18.Markgraf, H., "Fata Morgana an der Norseekuste," Wetterlotse, Vol. 47, pp. 200-204 (November 1952).

19.Ten Kate, H., "Fata Morgana," Hemel en Dampkring, Vol. 50, No. 2, pp. 32-34 (1952).

20.Heybrock, W., "Luftspiegelungen in Marokko," Meteorologische Rundschau, Vol. 6, No. 1/2, pp. 24-25 (January/February 1953).

21.Ruck, F.W.M., "Mirages at London Airport," Weather, Vol. 8, No. 1, pp. 31-32 (January 1953).

22.Ainsworth, P.P., "Abnormal Refraction in Cabot Strait, Gulf of St. Lawrence," The Marine Observer, Vol.

23, No. 160, pp. 77-78 (April 1953). 23.Kebblewhite, Alexander W. and W. J. Gibbs, "Unusual Phenomenon Observed from East Sale," Australian Meteorol. Mag., Melbourne, No. 4, pp. 32-34 (August 1953).

24.Menzel, Donald H., "Lenses of Air," Chapter 16 of Flying Saucers (Harvard Univ. Press, Cambridge, Mass., 1953).

25.Nelson, Robert T., "Mirages and Chlorophyll." Better Farming, (Summer 1953). [[992]]

26.Richard, R., "Phenomene Optique Remarquable," La Meteorologie, 4th Ser., No. 32, pp. 301-302 (October/December 1953).

27.Vassy, E., "Quelques Remarques sur un Phenomene de Mirage du Disque Solaire," La Meteorologie, 4th Ser. No. 32, pp. 302-303,(October/December 1953).

28.Williams, A. E., "Abnormal Refraction in North Atlantic Ocean," The Marine Observer, No. 166, pp. 208-210 (October 1954).

29.Jezek, ___ and Milan Koldovsky, "Totalni reflexe na inversnich vrstvach pozorovana s Milesovsky dne 18, listopadu 1953," Meteorologicke Zpravy (Prague), Vol. 7, No. 1, pp. 11-12 (February 1954).

30.Baines, J. P. E., "Abnormal Refraction off Cape Town," The Marine Observer, Vol. 25, No. 167, pp. 31-34 (January 1955).

31.Ashinore, S. E., "A North Wales Road Mirage," Weather, Vol. 10, pp. 336-342 (1955).

32.Ballantyne, J., "Abnormal Refraction in North Atlantic Ocean," The Marine Observer, Vol. 26, No. 172, pp. 82-84 (April 1956).

33.Durst, C. S. and G. A. Bull, "An Unusual Refraction Phenomenon seen from a High-Flying Aircraft," Meteorological Magazine, Vol. 85, No. 1010, pp. 237-242 (August 1956).

34.Collin, P., "Abnormal Refraction in Gulf of Aden," The Marine Observer, Vol. 26, No. 174, pp. 201-202 (October 1956).

35.Baker, R. E., "Abnormal Refraction in Red Sea," The Marine Observer, Vol. 27, No. 175, pp. 12-15 (January 1957).

36.Gabler, Horst, "Beobachtung einer Luftspiegelung nach oben," Zeitschrift fur Meteorologie (Berlin), Vol. 12, No. 7, pp. 219-221 (1958).

37.Ives, Ronald L., "An Early Report of Abnormal Refraction over the Gulf of California," Bull. Am. Meteorol. Soc., Vol. 40, No. 4 (April 1959). [[993]]

38.Rossman, Fritz 0., "Banded Reflections from the Sea," Weather (London), Vol. 15, No. 12, pp. 409-414 (December 1960).

39.Gordon, James H., "Mirages," Report, Publication 4398, Smithsonian Institution, Washington, D.C., pp. 327-346, 1959 (Pub. 1960).

40.O'Connell, D.J.K., "The Green Flash and Kindred Phenomena," Endeavor, (July 1961).

41.Zamorskiy, A. D. , "Optical Phenomena in the Atmosphere," Priroda, (Nature), Moscow, No. 8, pp. 62-66 (Translation) , (1963)

42.Ives, Ronald L., "The Mirages of La Encantada," Weather, Vol. 23, No. 2 (February 1968). BACK to Categories

Category 2 (Proposed Theories)

1.Tait, Professor P.G., "On Mirage," Trans. Roy. Soc. Edinburgh, Vol. XXX (1883).

2.Forster, Gustav, "Beitrag zur Theorie der Seitenrefraction," Gerlands Beitrage sur Geophysik, Vol. 11, pp. 414-469 (1911).

3.Hillers, Wilhelm, "Bemerkung uber die Abhangigkeit der dreifachen Luftspiegelung nach Vince von der Temperaturverteilung," Physikalische Zeit schr., Vol. 14, pp. 719-723 (1913).

4.Hillers, Wilhelm, "Ueber eine leicht Beobachtbare Luftspiegelung bei Hamburg und die Erklarung solcher Erscheinungen," Unterrichtsblatter fur Mathematik und Naturwissenschaften, Vol. 19, No. 2, pp. 21-38 (1913).

5.Hillers, Wilhelm, "Nachtrag zu einer Bemerkung uber die Abhangigkeit der dreifachen Luftspiegelung nach Vince von der Temperaturverteilung," Physik. Zeitschr., Vol. 15, pp. 303-304 (1914). [[994]]

6.Nolke, Fr., "Zur Theorie der Luftspiegelungen," Physik, Zeitschr., Vol. 18, pp. 134-144 (1917).

7.Wegener, Alfred, "Elementare Theorie der Atmospharischen Spiegelungen," Annalen der Physik, Vol. 57, No. 4 pp. 203-230, (1918).

8.Wurschmidt,Joseph, "Elementare Theorie der Terrestrischen Refraction und der Atmospharischen Spiegelungen," Annalen der Physik, Vol. 60, pp. 149-180 (1919).

9.Hidaka, Koji, "On a Theory of Sinkiro or Japanese Fata Morgana," Geophys. Mag., Vol. 4, pp. 375-386 (1931)

10.Meyer, Rudolf, "Die Entstehung Optischer Bilder durch Brechung und Spiegelung in der Atmosphare," Meteorologische Zeitschrift, Vol. 52, pp. 405-408 (November 1935).

11.Schiele, Wolf-Egbert, "Zur Theorie der Luftspiegelungen," Spezialarbeiten aus dem Geophysikalischen Institut und Observatorium, Leipzig Universitat Veroffentlichungen Zweite Serie, Band VII, Heft 3 (1935).

12.Brocks, Karl, "Die terrestrische Refraction in polytropen Atmospharen," Deutsche Hydrographische Zeitschrift, Vol. 2, No. 5, pp. 199-211 (1949).

13.Haug, Odd, "On the Theory of Superior Mirage," Norway Meteorologiske Institutt, Meteorologiske Annaler, Vol. 3, No. 12 (1953).

14.Ozorai, Zoltan, "Mirages on Wave Surfaces," Ido jaras, 58 (3):143-153 (May/June 1954).

15.Raman, Sir C. V. and S. Pancharatnam, " The Optics of Mirages," Proc. Indian Acad. Sci., pp. 251-261 (May 1959).

16.Raman, Sir C. V., "The Optics of Mirages," Current Science, Vol. 29, No. 8 (August 1959). [[995]]

17.Baldini, Angel A., "Formulas for computing Atmospheric Refraction for Objects Inside and Outside the Atmosphere," Research Note No. 8, Task 8T35-12-00l-0l, U. S. Army Engineer, Geodesy, Intelligence and Mapping Research and Development Agency (9 January 1963).

18.Kabelac, Josef, "Atmospharenmodelle und astronomische sowie parallaktische Refraction," Studia Geophysica et Geodaetica. Vol. 11, No. 1, pp. 1-20 (1967). BACK to Categories

Category 3 (Theory and Observation)

1.Fujiwhara, S., T. Oomari and K. Taguti, "Sinkiro or the Japanese Fata Morgana," Geophys. Mag., Vol. 4, pp. 317-374 (1931).

2.Futi, H., "On Road Mirage," Geophys. Mag., Vol. 4, pp. 387-396 (1931).

3.Wegener, K., "Bemerkungen zur Refraction," Gerlands Beitrage zur Geophysik, Vol. 47, pp. 400-408 (1936).

4.Kohl, G., "Erklarung einer Luftspiegelung nach oben aus Radiosondierungen," Zeitschrift fur Meteorologie, Vol. 6, No. 11, pp. 344-348 (November 1952).

5.Nakano, T., "Mirage in the Toyama Bay," J. Meteorol. Res. (Tokyo), Vol. 6, No. 1/3, pp. 67-70 (March 1954).

6.Hasse, Lutz, "Uber den Zusammenhang der Kimtiefe mit meteorologischen Grossen," Deutsche Hydrograph Zeitschrift (Hamburg), Vol. 13, No. 4, pp. 181-197 (August 1960)

7.Trautman, Ernst, "Uber Luftspiegelungen der Alpen, gesehen vom Bayerischen Wale," Mitteilungen des Deutschen Wetterdienstes, Vol. 3, No. 21 (1960).

8.Cameron, W.S., John H. Glenn, Scott M. Carpenter, and John A. O'Keefe, "Effect of Refraction on the Setting Sun as Seen from Space in Theory and Observation," The Astronomical J., Vol. 68, No. 5 (June 1963). [[996]] BACK to Categories

Category 4 (Application to Meteorology, Surveying, and Hydrography)

1.Maurer, Von J., "Beobachtungen ilber die irdische Strahlenbrechung bei typischen Formen der Luftdruckverteilung," Meteorologische Zeitschrift, pp. 49-63 (February 1905).

2.Johnson, N.K. and O.F.T. Roberts, "The Measurement of Lapse Rate of Temperature by an Optical Method," Quart. J. Roy. Meteorol. Soc., Vol. 51, pp. 131-138 (1925) 3.Brunt, D., "The Index of Refraction of Damp Air and Optical Determination of the Lapse Rate," Quart. J. Roy. Meteorol. Soc., Vol. 55, pp. 335-339 (1929).

4.Brocks, Karl, "Eine Methode zur Beobachtung des vertikalen Dichte und Temperaturegefalles in den bodenfernen Atmospharenschichten," Meteorologische Zeitschrift, Vol. 57, pp. 19-26 (1940).

5.Fleagle, Robert G., "The Optical Measurement of Lapse Rate," Bull. Am. Meteorol. Soc., Vol. 31, No. 2 (February 1950).

6.Freiesleben, H.C., "Die strahlenbrechung in geringer Hohe uber Wasseroberflachen," Deutsche Hydrographische Zeitschrift (Hamburg), Vol. 4, No. 1-2, pp. 29-44 (1951).

7.Freiesleben, H.C., "Refraction Occurring Immediately above the Water Surface," International Hydrographic Review, Vol. 28, No. 2, pp. 102-106 (1951).

8.Brocks, Karl, "Eine raumlich integrierende optische Methode fur Messung vertikaler Temperatur-und Wasserdampf gradienten in der untersten Atmosphare," Archiv fur Meteorologie, Geophysik und Bioklimatologie, Vol. 6, pp. 370-402 (1953). [[997]]

9.Bourgoin, Jean-Paul, "La refraction terrestre dans les basses couches de l'atmosphere sur l'inlandsis Groenlandais," Annales de Geophysique, Vol. 10, No. 168-174 (April/June 1954).

10.Yates, H. W., "Atmospheric Refraction over Water," Report 4786, Naval Research Laboratory, Washington, D.C. (July 1956).

11.Hradilek, Ludvik, "Untersuchung der Abhangigkeit der Lichtbrechung von den Meteorologischen Bedingungen auf dem Beobachtungstandpunkt," Studia Geoph. et Geod., Vol. 5, pp. 302-311 (1961).

12.Sparkman, James K., Jr., "Preliminary Report on an Optical Method for Low-Level Lapse Rate Determination" in "Studies of the Three Dimensional Structure of the Planetary Boundary Layer," Contract Report, Univ. of Wisconsin, Dept. Meteorology, 232 pp; see pp. 69-79 (1962) BACK to Categories

Category 5 (Average Values of Terrestrial Refraction)

1.Link, Frantisek and Zdenek Sekera, "Dioptric Tables of the Earth's Sphere," Publications of the Prague Observatory, No.l4 (Prometheus Press, Prague, 1940).

2.Brocks, K., "Die terrestrische Refraction," Annalen der Meteorologie, Vol. 1, pp. 329-336 (1948).

3.Brocks, K., "Die Lichtstrahlkrummung in den unteren 500 m der Atmosphare," Annalen der Meteorologie, Vol. 5, pp. 47-57 (1952).

4.Brocks, Karl, "Die Lichtstrahlkrummung in Bodennahe," Deutsche Hydrographische Zeitschrift (Hamburg), Vol. 13, No. 4, pp. 181-197 (August 1960).

5.Saunders, M.J., "Refraction Angles for Luminous Sources Within the Atmosphere," AIAA J., Vol. 1, No. 3, pp. 690-693 (March 1963). [[998]] BACK to Categories

Category 6 (Optical Scintillation)

1.Ellison, M . A., "Why Do Stars Twinkle?" J. Roy. Astron. Soc. Canada, Vol. 46, No. 5, pp. 191-194 (September/October 1952).

2."Transaction of the Conference of the Research on the Twinkling of Stars," English Translation, prepared by Translation Services Branch Foreign Technology Division, WP-AFB, Ohio.

3.Portman, Donald J., E. Ryznar, and F. C. Elder, "Visual Resolution and Optical Scintillation over Snow, Ice, and Frozen Ground," Research Report III, Part II, U.S. Army Materiel Command, Cold Regions Research ~ 'Engineering Laboratory, Hanover, N.H. (October 1965).

4.Carlon, Hugh R., "The Apparent Dependence of Terrestrial Scintillation Intensity upon Atmospheric Humidity," Technical Report CRDLR 3324, U.S. Army Edgewood Arsenal, Chemical Research Development Laboratories, Edgewood Arsenal, Maryland 21010 (November 1965).

5.Hudson, Craig, C., "Experimental Evidence of a Twinkling Layer in the Earth's Atmosphere," Nature, Vol. 207, No. 4994 (July 17, 1965)

6.Meyer-Arendt, Jurgen R., and Constantinos B. Emmanuel, "Optical Scintillation; A Survey of the Literature," Technical Note 225, National Bureau of Standards, U.S. Dept. of Commerce (April 1965).

7.Kucherov, N. I., ed., "Optical Instability of the Earth's Atmosphere," Translated form Russian, Israel Program for Scientific Translations, Jerusalem (1966).

8."Optical Effects of Atmospheric Turbulence," Compilation of the Results of Research Performed at the Electro-Optical Laboratory, Autonetics Division, North American Aviation, Inc. (March 1967). [[999]]

BACK to Categories BACK TO TOP 3. Basic Physical Concepts and Atmospheric Variables Involved in Light Refraction A. General B. Optical Refractive Index C. Snell's Law of Refraction D. Partial Reflections E. Spatial Variations F. Meteorological Conditions

A. General In a vacuum or in a medium of constant density, the energy from a light-emitting source travels along a straight line. Consequently, a distant observer sees the light source at its exact location. In a medium of variable density, such as the earth's atmosphere, the direction of energy propagation is deflected from a straight line; i.e., refracted. Refraction causes an observer to see a distant light source at an apparent position that differs from the true position by an angular distance the magnitude of which depends on the degree of refraction, i.e. on the degree of density variation between the observer and the light source. Changes in the direction of energy propagation arise principally from changes in the speed of energy propagation. The latter is directly related to density.

A clear picture of what causes refraction is obtained by means of Huygen's principle which states that each point on a wavefront may be regarded as the source or center of "secondary waves" or "secondary disturbances," At a given instant, the wavefront is the envelope of the centers of the secondary disturbances. In the case of a travelling wavefront the center of each secondary disturbance propagates in a direction perpendicular to the wavefront. When the velocity of propagation varies along the wavefront the disturbances travel different distances so that the orientation of their enveloping surface changes in time, i.e., the direction of propagation of the wavefront changes.

Practically all large-scale effects of atmospheric refraction can be explained by the use of geometrical optics, which is the method of tracing light rays -- i.e., of following directions of energy flow. The laws that form the basis of geometrical optics are the law of reflection (formulated by Fresnel) and the law of refraction (formulated by Snell). When a ray of light strikes a sharp boundary that separates two transparent media in which the velocity of light is appreciably different, [[1000]]

such as a glass plate or a water surface, the light ray is in general divided into a reflected and a refracted part. Such surfaces of dis- continuity in light velocity do not exist in the cloud-free atmosphere. Instead changes in the speed of energy propagation are continuous and are large only over layers that are thick compared to the optical wavelengths. It has been shown (J. Wallot, 1919) that, in this case, the reflected part of the incident radiation is negligible so that all the energy is contained in the refracted part. Since in the lower atmosphere, where mirages are most common, absorption of optical radiation in a layer of the thickness of one wavelength is negligible, Snell's law of refraction forms the basis of practically all investigations of large-scale optical phenomena that are due to atmospheric refraction (Paul S. Epstein, 1930).

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Thus, the spatial variation in the refractive index, i.e., light refraction, depends primarily on the vertical temperature gradient. When ðn/ðz is negative and the direction of energy propagation is from dense to rare, the curvature of light rays in the earth's atmosphere is in the same sense as that of the earth's surface. Equation (4) shows that ðn/ðz is negative for all vertical gradients of temperature except those for which the temperature decreases with height 3.4°C/l00 m. No light refraction takes place when ðn/ðz = 0; in this case ðn/ðz = -3.4°C/100 m. which is the autoconvective lapse rate, i.e., the vertical temperature gradient in an atmosphere of constant density. Table 2 gives the curvature of a light ray in seconds of arc per kilometer for various values of ðn/ðz near the surface of the earth (standard P and T). When ray curvature is positive, it is in the same sense as an earth's curvature. *When horizontal gradients in the refractive index are present, the complex mirage images that occur are often referred to as Fata Morgana. It is believed, however, that the vertical gradient is the determining factor in the formation of most images.

3. Effects from Focussing and Interference A recent theoretical and experimental investigation of the optical mirage is presented by Sir C. V. Raman (1959). Sir C. V. Raman demonstrates that multiple, inverted images of a single object can arise from interference and focussing of the incident and reflected wavefronts near the boundary of total reflection. Raman's work, which is entirely based on wave theory, suggests the interaction of wavefronts within a refracting layer as a mechanism in mirage formation. The occurrence of focussing and interference in situations that give rise to mirage, examined specifically by Raman, is also evident from various investigations based on geometrical optics. For example, the crossing of light rays mentioned in connection with image inversion implies interference of wavefronts at the points of intersection.

The visual effects from focussing and interference must be considered in particular when plane-parallel radiation (radiation from a very distant source) is incident on a layer of total reflection. In this case, there is a constant crossing of light rays within a relatively narrow region of the refracting layer, as illustrated in Fig. 12 (for the sake of clarity, height and elevation angles are exaggerated). In Fig. 12, a circular collimated light-beam of diameter A is incident on the lower boundary of a temperature-inversion layer at angle equal to or exceeding the critical angle for total reflection. Interference of the incident and reflected wavefronts occurs in a selected layer near the level of total reflection. This layer, shaded in Fig. 12, has a maximum thickness B, which is dependent on A. In the absence of absorption, the amount of radiant energy, flowing per unit time through Pi·A2 equals that flowing through Pi·B2. When B is less than A, the energy density at B is larger than at A, so that the brightness of the refracted light beam increases in the layer of interference.

An example of the ratio of A to B can be Given with the aid of Eq. (3). It is assumed that the optical refractive index through the inversion layer varies from no = 1.00029 to n = 1.00026 according to n2 = no2 - z. When the angle of incidence is near the critical angle for total reflection (Thetao ~ 89.5°), the light rays within the inversion layer are parabolas and [[1033]] Figure 12: Energy in Inversion Click on thumbnail to see full-size image. [[1034]]

the level of total reflection coincides with the upper boundary of the inversion layer. Under these conditions, it can be shown that B/A = A/16H where H is the thickness of the temperature-inversion layer. When the diameter A of the incident light beam is less than 16H, B is less than A and a brightening or focussing occurs near the top of the inversion. When the angle of incidence of the light beam is larger than the critical angle, ~89.5°, the level of total reflection lies below the upper boundary of the inversion layer. In this case, brightening can still occur near the level of total reflection, but the restrictions on the required beam-diameter become rather severe. The above example, based on a special case, demonstrates that sudden brightening can be encountered near the upper boundary of a refracting layer when optical mirages are associated with a refracting layer that is thick with respect to the diameter of the incident light beam from a distant source and when the angle of incidence is near the critical angle.

Observations of the brightening phenomenon must be considered rare in view of the selective location of its occurrence within the temperature-inversion layer and the requirement of plane-parallel incident radiation. Upper-level inversions seem most likely to produce the phenomenon. Some photographs showing apparent brightening of "spike" reflections on the edge of the setting sun are shown in O'Connell (1958, c.f., p. 158).

Microscopic effects due to interference of wavefronts within the area of brightening are illustrated in Fig. 13. Wavefronts are indicated rather than light rays. Unless absorption is extremely large, light rays are normal to the wavefront. A train of plane-parallel waves is assumed incident on the lower boundary of a refracting layer in which the refractive-index decreases with height. When the angle of incidence equals the critical angle, the incident waves are refracted upon entering the refracting layer and are totally reflected at the upper boundary The crests and troughs of the waves are indicated by solid lines and dashed lines, respectively. At the upper boundary, the wavefronts of the incident and reflected waves converge to a focus. The focus is called a cusp. The upper boundary of the refracting layer resembles a caustic, [[1035]]

Figure 13: Wavefront Diagram Click on thumbnail to see full-size image. [[1036]] i.e., an envelope of the moving cusps of the propagating wavefronts. Because of the focussing of wavefronts, a large concentration of radiant energy is usually found along the caustic (see Raman, 1959). In the area where the incoming and outgoing wavefronts interact, destructive interference is found along AA' and CC' (troughs meeting crests), while constructive interference is found along BB' (incident and reflected waves have similar phase). Hence, brightness variations can be expected in the interference layer, as demonstrated by Sir C. V. Raman (1959). To what extent the microscopic effects from interference and focussing can be observed under actual atmospheric conditions of mirage is not known. Undoubtedly, the proper relation between refracting layer and distant light source must be combined with an observer's position near the upper boundary of the refracting layer. If the dark and bright bands in the area of interference can be observed, the observer could easily get the impression that he is viewing a rapidly oscillating light or a light that is drawing near and moving away at rapid intervals. Nighttime observations by airplane are most likely to provide proper evidence of this effect. Currently, the focussing and interference effects are the least explored and consequently the least discussed of the various aspects associated with optical mirage.

Due to the wavelength dependence of the optical refractive index, systematic refraction of white light leads to a separation of the composing colors. Visible effects of color separation are most frequently associated with astronomical refraction. In this case, the light enters the atmosphere at an upper boundary where n approaches unity for all wavelengths. At an observation site near sea level n is wavelength-dependent, so that from the upper boundary of the atmosphere to the observation site the individual color components are refracted at different angles. The basic composing color of white light may be assumed to be red (24%), green (38%), and blue-violet 38%); the red is refracted less than the green, while the green is refracted less than the blue-violet. The visual effects of color separation depend on the zenith [[1037]]

angle of the extraterrestrial light source. When a white light source is more than 50° above the horizon, the color separation is simply too small to be resolved by the eye. Close to the horizon it can be observed only in the case of very small light sources. The principle of color separation in astronomical refraction is illustrated in Fig. 14. The light from an extended source enters the top of the atmosphere and is separated with respect to color in the order red, green, blue, and violet. A bundle of light rays of diameter D can be selected for which all colors, upon refraction, converge at O. Hence, an observer at O sees the entire color mixture as white. When the extended source has a diameter larger than D, an area rather than a single point of color blending is formed. However, when the diameter of the source becomes less than D, the point of color convergence, O, recedes from the location of the observer. Now the observer begins to see a gradual refractive separating of color such that red tends to lie below green, and green tends to lie below blue-violet (see Fig. 14).

The diameter of the light beam from a given extraterrestrial source decreases with respect to an earth-bound observer, with increasing distance from the zenith, as illustrated in Fig. 14. Thus, when the zenith angle increases, the apparent diameter D of the light source decreases rapidly to a minimum value on the horizon. Hence, the chance of having a light source of diameter less than D is greatest on the horizon. Therefore, color separation is observed most frequently on the horizon, when the light source is reduced to a bright point like a star or a minute portion of the solar or lunar disc. A prominent example of the visible effects of color separation is the so-called Green Flash. This phenomenon is sometimes observed when the sun disappears in a clear sky below a distant horizon. The last star-like point can then be seen to change rapidly from pale yellow or orange, to green, and finally, blue, or at least a bluish-green. The vividness of the green, when the sky is exceptionally clear, together with its almost instant appearance and extremely short duration, has given rise to the name "green flash" for this phenomenon. [[1038]] Figure 14: Refractive Color Separation Click on thumbnail to see full-size image. [[1039]]

The same gamut of colors, only in reverse order, occasionally is seen at sunrise. The observations of the Green Flash require an unusually clear atmosphere such that the sun is yellowish, and not red, as it begins to sink below the horizon. A red setting sun means that the blue and green portions of the spectrum are relatively strongly attenuated by the atmosphere and hence indicates that conditions are not favorable for seeing the greenish segment. Thus, the meteorological conditions required for observing color separation are even more stringent than those required for observing optical mirages. Examples of color separation associated with astronomical refraction are given on the following page in excerpts from The Marine Observer. In terrestrial refraction the composing colors of white light are very seldom separated to the extent that the effects can be observed with the naked eye. When the wavelength dependence of the refractive index is put back into Eq. (4),

Hence, for a given temperature inversion, the refractive index (n) decreases somewhat faster with height (z) for Lambda = 0.4µ (blue) than for Lambda = 0.7µ (red), so that the blue rays are refracted more than the red rays. However, the difference is generally too small to be resolved by the eye. Only under very special conditions can a visible effect be imagined.

5. Evaluation of the State-of-the-Art Knowledge During the last decade, active interest in optical mirage appears to have waned. The reasons for the apparent decline are believed to be two-fold. Firstly, on the basis of simple ray-tracing techniques, the mirage theories satisfactorily explain the various large-scale aspects of observations. Thus no disturbing contradictions between theory and observation have been found. Secondly, although atmospheric refraction remains of great interest to astronomy, optical communication, and optical ranging, the phenomenon of the mirage has so far failed to demonstrate a major use.

At the present time, there is no single theoretical model that explains all the aspects, both macroscopic and microscopic, of the mirage phenomenon. The absence of such a model must stand as evidence that shortcomings remain in current knowledge. These shortcomings are most eloquently discussed by Sir. C. V. Raman (1959), who suggests and actually demonstrates that any approach to explain the phenomenon must be based on wave-optics rather than ray-optics. The theory of wave-optics, as applied by Sir. C. V. Raman, suggests the presence of some intriguing aspects of the mirage that arise from the interference and focussing of wavefronts in selected regions of the refracting layer. Raman's experimental studies reveal that when a collimated pencil of light is incident obliquely on a heated plate in contact with air, the field of observation exhibits a dark region adjacent to the plate into which the incident radiation does not penetrate, followed by a layer in which there is an intense concentration of light and then again by a series of dark and bright bands of progressively diminishing intensity.

Further theoretical and experimental investigations are warranted in order to determine to what extent the brightening and brightness variations that arise from interference and focussing can add unusual effects to observations of phenomenon associated with abnormal refraction in the atmosphere. [[1052]] BACK TO TOP

When an unusual optical phenomenon is observed in the atmosphere, its positive identification as a mirage cannot be made without a physically meaningful description of what is seen and a complete set of meteorological and astronomical data. The required "hard" data are practically never available for the specific place and time of observation, so that the descriptive account remains the only basis for identification; in this case, successful identification depends on a process of education. Thus, the casual observer of an optical phenomenon can establish the likelihood that his observation is a mirage only by being aware of the basic characteristics of mirage and the physical principles that govern its appearance and behavior.

The conditions required for mirage formation and the principal characteristics of mirage images, as described in this report, are summarized below. The summary presents a set of standards by which to interpret the nature of an optical observation in terms of a specific natural atmospheric phenomenon.

A. Meteorological Conditions Optical mirages arise from abnormal temperature gradients in the atmosphere. A temperature decrease with height (temperature lapse) exceeding 3.4°C per 100 m or a temperature increase with height (temperature inversion) is most commonly responsible for a mirage sighting.

Large temperature lapses are found in the first 10 meters above the ground during daytime. They occur when ground surfaces are heated by solar radiation, while during nighttime they can occur when cool air flows over a relatively warm surface such as a lake. When the temperature decreases with height more than 3.4° per 100 m over a horizontal distance of 1 kilometer or more, an observer located within the area of temperature lapse can sight an inferior mirage near the ground (e.g., road mirage, "water" on the desert)

Layers of temperature inversion ranging in thickness from a few meters to several hundred meters may be located on the ground or at various levels above it. In areas where they are horizontally extensive, an observer can sight a superior mirage that usually appears far away (beyond 1 kilometer) and "low in the sky." The strength of the inversion determines the degree of image-elevation; the stronger the inversion, the higher the image appears above the horizon. Layers of maximum temperature inversion (30°C) are usually found adjacent to the ground. [[1053]]

Calm, clear-weather conditions (no precipitation or high winds) and good horizontal visibility are favorable for mirage formation. Warm days or warm nights during the summer are most likely to produce the required temperature gradients.

B. Geometry of Illumination and Viewing The geometry of illumination and viewing in the case of optical mirage is determined by the spatial variations of refraction index that occur in the cloud-free atmosphere, and by Snell's law of refraction, which relates these variations to changes in the direction of propagating wavefronts. The spatial variations in refractive index are associated with layers of temperature inversion or temperature lapse. Variations of 3x10-5, corresponding to temperature changes of 30°C, are considered near maximum. As a consequence of Snell's law and the small changes in the atmospheric refractive index, an optical mirage develops only when a temperature inversion layer or a layer of large temperature lapse is illuminated at grazing incidence. The requirement of grazing incidence implies that the source of illumination must be either far away, i.e., near the horizon, or very close to or within the layer of temperature gradient. Therefore, both terrestrial and extraterrestrial sources can be involved. Because of the distance factor, the actual source of illumination may not be visible. Its location, however, must always be in the direction in which the mirage image is observed, i.e., observer, image and "mirrored" source are located in the same vertical plane.

Another consequence of Snell's law and the small spatial changes in refractive index is that noticeable refractive effects are not likely beyond an angular distance of approximately 14 degrees above the horizon and that a superior mirage image is not likely beyond an angular distance of 1 to 2 degrees above the horizon. Hence, mirages appear "low in the sky" and near the horizontal plane of view. An optical image seen near the zenith is not attributable to mirage. Because of the restricted geometry between observer, mirage image, and source of illumination, the observed image can often be made to disappear abruptly by moving to higher or lower ground. Furthermore, when mirage [[1054]]

observations are made from a continuously moving position, the image can move also, or can move for a while and then abruptly disappear.

C. Shape and Color A mirage can involve more than one image of a single object. Observations of up to four separate images, some inverted and some upright, are encountered in the literature. When multiple images occur they all lie in a single vertical plane or very close to it. The apparent shape of a mirage can vary from clearly outlined images of an identifiable object such as a distant ship, landscape, or the sun or moon, to distorted images that defy any description in terms of known objects (e.g., Fata Morgana). Apparent stretching either in the vertical or in the horizontal plane is common.

During daytime, a mirage can appear silvery white ("water" on the ground), or dark when projected against a bright sky background, or it can reflect the general color of the land or seascape. Distinctly colored images ranging from red and yellow to green and blue are observed when unusual conditions of mirage occur near sunrise or sunset (e.g., Red and Green Flash) or, at night, during rising or setting of the moon or of a planet such as Venus.

In the presence of atmospheric turbulance and convection, the effects of scintillation become superimposed on the large-scale mirage image. When scintillation occurs, extended mirage images appear in constant motion by changing their shape and brightness. When the image is small and bright, as may be the case at night, large fluctuations in brightness and under unusual conditions in color can give an illusion of blinking, flashing, side to side oscillation, or motion toward and away from the observer. The effects associated with scintillation can dominate the visual appearance of any bright point-object in the area between the horizon and approximately 14 degrees above the horizon.

D. Present Uncertainties The theory of ray optics adequately explains such observed large-scale aspects of the mirage as the number of images, image inversion, and apparent vertical stretching and shrinking. However, if the interference and focussing of wavefronts within the refracting layer are as fundamental [[1055]] in mirage formation as purported by Sir C.V. Raman, the ray-tracing technique may have to be replaced by the theory of wave-optics.

Sir C. V. Raman's application of wave-optics to mirage suggests that under special conditions of illumination, the upper boundary of an atmospheric temperature inversion could exhibit a large concentration of radiant energy due to focussing of wavefronts. Also, interference of wavefronts could produce alternating layers of high and low brightness. Under what conditions and to what extent these brightness effects can be observed in the atmosphere is not known. Relevant observations have not been encountered in the literature, although some unusual observations of the green flash made under mirage conditions (O'Connel, 1958) could possibly have been caused by the enhancement of brightness in an inversion. The visual effects from focussing and interference of wavefronts must be considered as the least explored aspect of mirage. [[1056]]

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