A sunset's shape is a sounding of the air

Refraction near the horizon is usually treated as a correction. The setting Sun is lifted by about half a degree, so it is still visible when geometrically it has already gone, and the lift is tabulated as a function of apparent altitude for a standard atmosphere, with small adjustments for temperature and pressure. Over a cold sea or polar ice, in calm air, the correction stops being small and stops being a function. A layer of warm air resting on colder air bends near-horizontal rays so strongly that they can follow the curve of the Earth for hundreds of kilometres; distant coasts appear stacked into towers, ships float upside down above their own images, and the Sun can be seen for days after it has set.

Those effects are usually described one at a time, by the shapes they produce. They are all consequences of one object, and the shapes are pictures of it. For an observer at a given height, each apparent direction above the horizon corresponds, through the rays traced back through the air, to one true direction in space: the direction the light was travelling before the atmosphere touched it. The map from apparent to true elevation is the transfer curve, and a distant object’s appearance is its true shape pushed through that map. A mirage is the transfer curve made visible, and it can in principle be run backwards: from a photograph of a distorted sunset, whose true shape is known exactly in advance, to the curve, and from the curve to the temperature of the air that produced it.

Tracing the rays out #

The figures here trace rays from the observer outwards, through a spherical atmosphere with a stated temperature profile, until they are twenty kilometres up and effectively in space. A ray’s elevation above the local horizontal changes along its path at a rate set by two things: the horizontal itself tilts as the ray travels round the Earth, and the ray bends towards denser air in proportion to the vertical gradient of the refractive index. The index of air is fixed by its pressure and temperature, and the pressure follows from the temperature by hydrostatic balance, so a temperature profile determines everything. The profile is a standard lapse rate of 6.5 degrees per kilometre with one inversion layer added — a band of air in which the temperature rises with height instead of falling. A ray that reaches the sea ends there; one still inside the lowest twenty kilometres after twelve hundred kilometres of path is trapped in the layer and brings no light from outside.

The sequence is a sunset through a modest inversion — six degrees of warming over forty metres, starting thirty metres above the sea, seen from a ship’s deck at twenty. While the Sun is still well above the layer’s influence it is merely flattened and lifted. As it sinks, the lower limb is squeezed and the upper part swollen; then the disc separates into two pieces, the lower one an inverted copy of part of the Sun that is no longer visible directly; then the pieces join into a vase-shaped column and finally into a flat, rectangular slab resting on the sea horizon, a quarter of a degree below where a standard atmosphere would have put the last glimpse. Every one of those shapes has been photographed many times at sea and on polar coasts. The ray tracing shows that they are one phenomenon at five moments, and that the order in which they appear is fixed by the curve.

One curve behind every shape #

In a standard atmosphere the curve is a gentle arc, offset below the line of no refraction by about half a degree at the horizon and less above it; each apparent direction maps to one true one, and the Sun is flattened only slightly, because the curve’s slope changes little across the Sun’s half-degree. An inversion distorts it in two characteristic ways. A strong surface inversion traps the rays that leave the observer near the horizontal — they are bent back down into the layer and never escape — so a band of apparent elevations shows nothing of the outside at all, and the curve has a gap. An elevated inversion bends the near-horizontal rays that reach it back down past the observer’s height and then up again, so that rays leaving at slightly different angles reach the same true direction by different routes, and the curve folds back on itself.

A distant object’s appearance can be read off the curve directly. Where the curve is steep, many degrees of true elevation are packed into a small range of apparent elevation, and the image is compressed; where it is shallow, stretched; where it folds, one true elevation appears at three apparent ones, and a horizontal strip of the object appears three times, the middle copy inverted; where it has a gap, the image is cut. The sunset shapes are these rules applied to a disc whose true outline is a circle.

Why the Sun is the calibration target #

Any distant object seen through the layer is distorted by the same transfer curve, and a distant coastline or ship is distorted just as much. What makes the Sun the object to invert is that its true shape and position are known exactly. Its outline is a circle of known radius; its centre’s true elevation at any instant follows from the time, the observer’s position and an ephemeris, to a fraction of an arcminute. A photograph of the distorted disc therefore gives, for each horizontal strip of the image, both coordinates of the transfer curve at once: the strip’s apparent elevation, measured on the photograph, and its true elevation, computed from where on the circle a strip of that width must lie.

The Sun also moves, and that is the second gift. At middle latitudes it sinks through the lowest degree of the sky in about five minutes, near the poles in much longer, and it does so at a rate the ephemeris gives exactly. A sequence of photographs taken a few seconds apart therefore sweeps the Sun’s known disc through the transfer curve, and each feature — the moment the disc splits, the moment the neck forms, the moment the last strip vanishes — is the curve’s fold or gap passing a particular part of the disc. The timing of the features is itself a measurement of the curve, and a good one, because it depends only on the clock and not on the scale of the photograph. A single frame gives one pass through the curve; a sequence gives dozens, each through a different part of the disc.

A ship or a coastline has neither property. Its true height above the horizon depends on its distance and its size, which are usually not known to the precision needed, and it does not move through the curve on a known schedule. That is why the few quantitative inversions of mirages have been made on sunsets and moonsets, and why the ordinary refraction tables were first calibrated against the Sun too — the table in every almanac is the transfer curve of a standard atmosphere, measured on the one object whose true position was known.

The standard correction as a special case #

It is worth seeing how the familiar correction sits inside this picture. The standard atmosphere’s transfer curve is smooth, and its slope is close to one at every elevation above a few degrees: every direction is lifted by nearly the same amount, which is why astrometry near the zenith needs only a small correction that barely varies across a field. Near the horizon the slope departs from one, the lift changes across a small field, and a square field is imaged as a squashed one — the same compression that flattens the setting Sun in an ordinary sunset, where the lower limb is lifted about six arcminutes more than the upper. The inversion’s transfer curve differs from the standard one only near the horizon and only in the lowest air, but there it differs completely: its slope changes sign at a fold, goes to zero at a stretched strip, and is undefined in a gap. Refraction as a correction and refraction as a mirage are the same curve in two regimes, and the second is what the first becomes when the lowest hundred metres stop being well mixed.

The lowest air, and nowhere else #

Before asking what the curve can say about the air, it is worth asking which air it says anything about.

The near-horizontal rays run almost parallel to the layer for tens of kilometres, and they collect most of their bending inside it. Rays leaving at half a degree cross the layer in a few kilometres and are bent mostly by the ordinary gradient of the air above it. So the part of the transfer curve that carries information about the layer is its lowest few tenths of a degree — which is exactly the part a setting Sun sweeps through in the last few minutes before it disappears. Above a few hundred metres the air enters the picture only through the standard refraction every ray collects, and the curve says almost nothing about it. A mirage is a sounding of the lowest hundred metres of air, and of nothing higher.

That is also its strength. The lowest hundred metres over a sea or an ice sheet are the part of the atmosphere least well measured by balloons, which ascend through them in seconds, and by satellites, which see them only through everything above. A photograph of a sunset over the sea records their structure along a path tens of kilometres long, averaged in a way that a single balloon profile cannot be.

The same air from three heights #

The observer’s height matters as much as the air. From ten metres, below the layer, the near-horizon rays run up into it and are bent back, and the curve is compressed but single-valued. From fifty metres, inside the layer, rays leaving near the horizontal are guided along it and trapped, and the curve has a gap and a turning point beside it. From ninety metres, above the layer, the near-horizon rays pass down through it, are bent, and come back up, and the curve folds in a different place. Three observers on the same ship — one at the rail, one on the bridge, one at the masthead — see three different mirages of one sunset.

The dependence on height is not a nuisance to be corrected; it is what makes the inversion possible. A single observer sees one curve, which is one set of constraints on the profile. Observers at several heights — or one observer who can move, which is why mirage enthusiasts have been known to photograph a sunset while climbing a cliff — see several curves through the same air, each weighting a different part of it. The profile that reproduces all of them is much better constrained than the profile that reproduces one.

What a single sunset can recover #

The inverse problem can be posed as a fit. Given a transfer curve measured from a sunset photograph — the true elevation of each strip of the Sun’s image, known because the Sun’s outline is a known circle moving at a known rate — search over temperature profiles for those whose computed curve matches.

The best fit lands on the true inversion, which is a check on the method rather than a result. The result is the shape of the region that fits almost as well: a curved valley from thin, strong layers to thick, gentle ones. A shallow layer with a steep gradient and a deep layer with a gentle one bend the near-horizontal rays by similar total amounts, because a ray running along the layer integrates the gradient over the layer’s depth, and what it responds to is close to the temperature jump across the layer. The fit therefore pins down a combination of the two — the depths allowed range over a factor of five, the gradients and jumps over factors of about two and a half — and not either one alone. The layer’s base height, fixed here, would widen the valley further if it were free.

That degeneracy is the practical limit of the method, and it is not special to mirages. Any measurement made with rays that run along a layer rather than across it integrates along the layer, and an integral determines a sum more readily than its parts. Separating them requires rays that cross the layer at different angles — a second observer height, or the same sunset followed as the Sun sinks through a range of elevations so that different rays are used at different moments — and a careful inversion uses both.

When the geometry is favourable #

There is a measurement that inverts atmospheric bending exactly, and the contrast shows what a sunset lacks. When a satellite passes behind the Earth’s limb as seen from another satellite, the radio signal between them is bent by the atmosphere by an amount that can be measured to a fraction of a microradian from the Doppler shift of the carrier. As the ray descends through the atmosphere its bending angle is measured at each tangent height, and in a spherically symmetric atmosphere the bending angle as a function of the ray’s closest approach is an Abel transform of the refractive-index gradient — which can be inverted uniquely. Radio occultation of this kind delivers temperature profiles from the stratosphere down to the lower troposphere, thousands a day, and they are among the most accurate temperatures assimilated into weather forecasts.

What makes it invertible is that the geometry does the scanning. The tangent height sweeps smoothly through the atmosphere as the satellites move, so every layer is visited once, from outside, by a ray that crosses it symmetrically; and the source is a point whose true direction is known to arbitrary precision. An observer on the ground watching a sunset is inside the medium, at one height, seeing rays that graze a layer at every angle at once, and the source is a disc whose position is known only as well as the clock and the ephemeris allow. The same physics is being inverted, but with a single fixed tangent point instead of a sweep, and that is why one sunset gives a valley rather than a point. An occultation of a star by a planet is the astronomical cousin of radio occultation, and its central flash — starlight focused by the planet’s whole atmosphere onto the centre of its shadow — is the same fold in a transfer curve that stacks the setting Sun, seen from the other side.

A fold that is everywhere #

Folds in a map from source to image are not peculiar to air. A gravitational lens has the same structure: a microlensing light curve has a fold in it where the map from the source plane to the image plane doubles back, and a source crossing the fold brightens enormously because images are being created or destroyed in pairs. The superior mirage is the same catastrophe in one dimension. Where the transfer curve turns back, two new images of the Sun appear at the turning point — an upright one and an inverted one — and they are stretched without limit at the moment they appear, which is why the neck of a vase-shaped sunset and the edges of a stacked one are bright, thin and stretched. The mathematics of those folds, and of the cusps where two folds meet, is the same in every medium that maps one surface smoothly onto another, and the shapes it produces are the same too.

Reading the old records #

Mirages have been described for as long as sailors have written. Barentsz’s crew, wintering on Novaya Zemlya in 1597, saw the Sun return two weeks before it should have, and the observation was dismissed as an error of their calendar for three centuries before it was recognised as ducting. Medieval Norse sagas describe hafgerdingar — sea hedges, walls of water rising round a ship — and the merman seen off Greenland, a figure rising from the sea with a flat head and shoulders; both have been argued to be superior mirages of a distant shoreline or iceberg seen through a strong inversion, which the waters off Greenland produce readily in calm spring weather. Such records carry, in principle, information about the air over polar seas long before instruments measured it.

The figures here say how much. A qualitative description — the Sun appeared early, a wall rose from the sea — constrains the transfer curve only to the extent of saying it folded or ducted, which the valley already shows can happen for a range of layers differing by factors of several in depth and strength. Without the observer’s height, which a ship’s log rarely gives to better than the deck or the mast, even that constraint loosens, since the same air makes a different mirage from a few tens of metres higher. What survives is the jump: that a layer of air at least a few degrees warmer than the surface lay within some tens of metres of it, over a path of tens of kilometres, on that day. For a climate record that is not nothing — the frequency of strong inversions over polar seas is a quantity of interest in its own right — but it is a count of events rather than a profile, and it inherits every uncertainty in how often such events were written down.

What the rays leave out #

The calculation assumes a horizontally uniform atmosphere, the same profile along the whole path. Real inversions over the sea vary over tens of kilometres, and the rays that make a sunset mirage sample exactly that range; a profile that changes along the path produces transfer curves no uniform profile can match, and a uniform-profile fit to them returns an average of uncertain meaning. It ignores the Sun’s limb darkening and colour, which make the thin strips of a stacked sunset fainter and redder than the figure’s uniform fill — the reddening being the same scattering that colours the twilight sky, acting over the enormous air paths of the near-horizontal rays — and the dispersion of the air, which separates the green and red images near the folds and is the origin of the green flash. It treats the sea as absorbing every ray that reaches it, when a calm sea returns a faint reflection that adds a dim inverted image below the horizon. And it takes the pressure to follow hydrostatically from the temperature, which is right on the scale of the layer and wrong only in fast, turbulent air, where a mirage would not be stable enough to photograph anyway.

Still open: how much of the old record can be read #

The method works in the present, where the observer’s height, the time and the Sun’s true position are all known and a sequence of photographs samples the transfer curve over minutes; inversions from such sequences have recovered temperature profiles over lakes and polar seas that are consistent with measurements made alongside them. Whether the historical descriptions can be turned into anything more quantitative than a count of strong inversions depends on how much the observers recorded that they did not know was important — the height they were standing at, how long the effect lasted, whether the image was stacked or merely lifted — and on whether enough such records survive from the same waters to give a frequency rather than an anecdote. The mirage that brought the Sun back early to a stranded crew is one data point about one layer of air on one day, and the question is whether there are enough of them to be a record.