The spectrum that is a thermometer at every height

The height a planet is seen from found that the Earth, seen from space, is not glowing at the temperature of its surface but at the temperature of a level some five kilometres up, where the infrared it emits can finally escape. That one height was an average over the whole spectrum. It ended by pointing out that the average hides the most useful thing about the spectrum: each wavelength escapes from its own height, so the spectrum, taken apart, is a measurement of temperature at many heights at once.

That is the principle of infrared sounding, and it is one of the quiet foundations of modern life. Satellites carrying spectrometers that look down through the atmosphere in thousands of infrared channels supply more of the information that goes into weather forecasts than any other kind of observation. They work because Planck’s law ties the radiance at a wavelength to the temperature of whatever emits it, and because the atmosphere’s absorption decides, wavelength by wavelength, what that is. This essay follows the argument from a single weighting function to the point where it stops: a thermometer that cannot see anything thinner than itself.

Where each wavelength comes from #

Carbon dioxide is mixed evenly through the lower hundred kilometres of the atmosphere, so the amount of it above any height falls off with the air’s density, exponentially, with a scale height of about seven kilometres. At a wavelength it absorbs, the optical depth from space down to a height — the number of absorption lengths the light must cross to get out — falls off the same way. Radiation emitted by the air at a given height reaches space only if the optical depth above it is not much more than one.

The contribution each height makes to what a satellite sees at that wavelength is its emission weighted by the chance its emission escapes: the weighting function. Low down, there is plenty of gas to emit but its emission is absorbed on the way out. High up, the emission escapes freely but there is little gas to emit. In between, where the gas above amounts to about one optical depth, the product peaks. A strongly absorbed wavelength has its peak high, where the gas is thin; a weakly absorbed one has it low. Seven wavelengths of different absorption strength have seven weighting functions stacked up through the atmosphere, and the radiance each channel measures is the Planck emission of the air, averaged over its function.

The shape of the function is the same for every channel, because it depends only on how fast the absorber thins out with height. It is broad: for a scale height of seven kilometres it is about seventeen kilometres wide at half its peak. That breadth is the whole limitation of the method, and it comes from exponential physics that no instrument can change.

Why the peak sits at one optical depth #

The rule that each channel sees from the height where one optical depth of gas lies above can be read straight from the weighting function. The chance that radiation emitted at a height reaches space is $e^{-\tau}$, the fraction a beam keeps after crossing τ absorption lengths. The amount of emitting gas in a thin layer is proportional to how fast τ changes across it. Their product, the contribution of the layer, is proportional to $\tau e^{-\tau}$, and that is largest at exactly $\tau = 1$. Nothing about carbon dioxide enters except through the one number that says how strongly it absorbs at the wavelength in question.

A change in that number moves the peak without changing its shape. Double the absorption and every level’s optical depth doubles, so the height at which it equals one rises by the scale height times the logarithm of two — about five kilometres for an absorber falling off with a seven-kilometre scale height. The channels of a sounder are chosen at wavelengths whose absorptions differ by factors of a few, so that their peaks are spaced by a few kilometres up the atmosphere, and the whole design of such an instrument is a choice of where along an absorption band to sample. The same arithmetic describes a cloud that light must walk through: what a cloud looks like from above is the state of the layer about one optical depth below its top.

A spectrum that reads the air from the ground up #

Put the channels together and the spectrum itself becomes the thermometer. The drawing computes the radiance leaving the top of a standard atmosphere across a model carbon-dioxide band and expresses it as a brightness temperature, the temperature a black body would need to give that radiance. Far in the wings, where the gas hardly absorbs, the satellite sees through the air to the ground and reads its temperature. Moving towards the band centre, the absorption grows, each wavenumber’s weighting function rises, and the brightness temperature falls, because the air it reads is higher and colder. On the shoulders of the band the reading bottoms out at the temperature of the tropopause, the coldest part of the lower atmosphere.

At the very centre something striking happens. The band’s strongest absorption, in a narrow feature called the Q-branch, puts the weighting function up in the stratosphere — and the stratosphere is warmer than the tropopause, heated by ozone absorbing sunlight. The spectrum rises again in a narrow spike. Every infrared spectrum of the Earth taken from space shows this shape, a broad bowl with a spike in the middle, and it is the temperature profile of the atmosphere turned on its side: the ground at the edges, the tropopause in the bowl, the stratosphere at the spike. The shape is the same reasoning as a spectrum being a subtraction of what the absorbers take out, run in reverse — here the absorbers are also the emitters, and what each wavelength shows is where it was last emitted.

Readings that round off the corners #

The simplest reading of a sounder is to plot each channel’s brightness temperature at the height of its weighting function’s peak. The drawing does that, and the result follows the true profile in outline — falling through the troposphere, flat through the tropopause, rising through the stratosphere — while missing it by several kelvin everywhere and by up to thirteen near the corners. Each reading is an average over seventeen kilometres of temperature, weighted towards the peak, and an average of a profile that bends is not the profile’s value at any one height. At the tropopause, where the profile turns sharply from cooling to warming with height, every channel whose function straddles the turn reads warmer than the coldest air, because it averages in the warmer air above and below.

There is a further asymmetry in the functions themselves. They have a long tail upwards and a sharper cut-off below, because the transmission drops off exponentially with increasing optical depth while the gas above thins more gently. A channel peaking at three kilometres therefore averages in a good deal of colder air above it and reads more than ten kelvin below the air at its peak. Correcting for that is not a matter of instrument quality. It needs a model of the whole profile.

Two atmospheres the channels cannot tell apart #

The breadth of the weighting functions has a consequence that is more than a bias. Two atmospheres that differ in fine detail give nearly the same readings. The drawing adds to the standard profile a sharp inversion — a layer eight kelvin warmer and two kilometres thick at five kilometres, the kind of warm lid that traps pollution over a city or caps the clouds over a cold ocean. Every channel’s reading changes by less than two thirds of a kelvin. Against measurement noise of a quarter of a kelvin the inversion is barely detectable, and there is no way to tell from the readings whether it is eight kelvin strong and two kilometres thick, four strong and four thick, or somewhere between five and seven kilometres up.

This is not a failing of the particular channels chosen. Any set of weighting functions seventeen kilometres wide averages away structure much thinner than that, and adding more channels adds functions of the same width at slightly different heights, which overlap heavily with the ones already there. The limit is set by the scale height of the absorber — the same exponential fall in density that makes the method work at all — and it is the reason a balloon carrying a thermometer, which measures the air it is actually in, is still flown twice a day from hundreds of stations around the world.

Turning readings back into a profile #

The practical question is how to get from seven readings to a profile at twenty-five levels. There are more unknowns than measurements, and the problem is underdetermined before noise is considered; with noise it is worse, because the weighting functions overlap so much that small differences between readings must be explained by large differences between layers. The drawing asks a straightforward inversion to fit the seven readings exactly and let the twenty-five temperatures fall where they may. The result swings wildly, tens of kelvin from the truth, because nothing stops it from putting large, alternating errors into layers the channels cannot distinguish.

Add a single extra requirement — that the profile be smooth, penalising sharp bends — and the retrieval settles onto something close to the truth, with an error under two kelvin, rounding off the tropopause’s corner exactly as the individual readings did. The readings are the same in both cases. The difference is information supplied from outside them: a statement of what profiles are plausible. In the language of inverse problems the first retrieval is ill-posed and the second regularised, and the choice of regulariser decides the answer where the data are silent. The same structure appears whenever a quantity is inferred from integrals over it — the field outside a body cannot find its core, and a gravity measurement is inverted for density only with assumptions about what densities are plausible.

In operational weather forecasting the outside information is a forecast. The atmosphere’s state six hours ago, carried forward by the equations of motion, gives a prior profile at every point, with estimated uncertainties; the satellite readings are used to correct it, each in proportion to how much it can say. The sounders do not so much measure the profile as tell the forecast where it is wrong, and they do so in the smooth, broad terms their weighting functions allow.

Why the method is quantum mechanics #

It is easy to forget where the thermometer’s calibration comes from. The radiance in each channel is converted to a temperature by Planck’s law, the curve that would not come down until the energy of light was counted in quanta. At fifteen micrometres and atmospheric temperatures the radiance changes by about two per cent per kelvin, steeply enough that a sounder calibrated against an onboard black body can read temperatures to a tenth of a kelvin. And the absorption that makes each wavelength see a different height comes from the quantised vibrations and rotations of the carbon dioxide molecule, whose lines crowd into the band and whose strengths are known to a per cent from laboratory spectroscopy. A sounder is a Planck thermometer reading through a quantum-mechanical filter whose transmission has been computed line by line.

The same construction works for other gases and other purposes. Channels in the microwave oxygen band sound temperature through clouds, which are opaque in the infrared. Channels in water-vapour bands, whose absorber is not well mixed, read humidity at different heights once the temperature is known. And channels in the ozone band near 9.6 micrometres read ozone. Each is a family of weighting functions, and each has the same limit on vertical detail.

The same reading of the Sun #

The principle is older than satellites. An astronomer looking at the Sun sees, at each wavelength and at each point on the disc, the layer about one optical depth down along the line of sight, and a spectrum of the Sun is a stack of thermometers in exactly the same way. At the centre of the disc the line of sight goes straight down and reaches deeper, hotter gas; towards the edge it goes in obliquely and reaches one optical depth higher up, where the gas is cooler, so the disc darkens towards its limb. The relation that the emerging intensity roughly equals the Planck emission at one optical depth, used to read the temperature profile of the Sun’s atmosphere from its limb darkening, is the stellar version of the weighting function.

Absorption lines work the same way from the other side. At the centre of a strong line the gas is most opaque and the light comes from high in the solar atmosphere, which is cooler, so the line is dark; in the line’s wings it comes from deeper and hotter layers. A line’s profile is a temperature profile read across wavelength, which is why the Sun’s light, which takes an age to diffuse out from the core, carries in its last few hundred kilometres a detailed record of the layers it finally escaped from.

The greenhouse effect is written in the band #

The same weighting functions say how carbon dioxide warms a climate, which is where the height a planet is seen from left off. Adding carbon dioxide raises every channel’s peak by the scale height times the logarithm of the concentration ratio. On the shoulders of the band the peaks sit in the troposphere, where temperature falls with height at the rate a rising parcel cools; raising them means the planet emits at those wavelengths from colder air, emits less, and must warm until it balances again. At the band centre the peaks sit in the stratosphere, where temperature rises with height; raising them there means emitting from warmer air, and the stratosphere, losing more energy than before, cools. A warming surface under a cooling stratosphere is the fingerprint that distinguishes a greenhouse warming from a brighter Sun, which would warm both, and it has been measured.

Where the simple model stops #

Clouds. The drawings assume a clear sky. A cloud is nearly opaque in the infrared, and a channel whose weighting function reaches below the cloud top sees the cloud rather than the air beneath; much of the effort in using sounder data goes into detecting clouds in each field of view and either correcting for them or discarding the observation.

Line by line. The model band here is a smooth absorption falling away from its centre. A real band is thousands of individual rotational lines, each broadened by pressure, so that a line’s width — and hence the absorption between lines — depends on the height. That pressure broadening sharpens real weighting functions a little in the lower atmosphere and is why modern sounders, with thousands of narrow channels between the lines, do somewhat better than the smooth model suggests.

The surface. The wing channels assume the ground radiates as a black body. Deserts, ice and seas have emissivities a few per cent below one and varying with wavelength, which shifts the surface-sensing channels by a kelvin or more and must be modelled.

What the weighting functions leave out #

They are functions of height alone. A satellite’s field of view is ten or more kilometres across, so each reading also averages horizontally, and the lower atmosphere’s structure on scales of a few kilometres — sea breezes, convective cells, the edges of fronts — is smeared in both directions at once. They also assume the atmosphere is in local thermodynamic equilibrium, which holds below about sixty kilometres; above that, the carbon dioxide molecules are not excited in proportion to the local temperature, and the band centre stops being a thermometer.

Still open: how much vertical detail can be recovered #

The newest sounders have thousands of channels, and the information content of their spectra is not simply the number of channels: overlapping weighting functions carry largely redundant information, and analyses of how many independent pieces of temperature information a spectrum contains typically find a dozen to twenty in the vertical, not thousands. Whether sounders combined with other measurements — the bending of navigation-satellite signals passing through the atmosphere’s edge, which measures density along long horizontal paths with high vertical resolution — can resolve structures like boundary-layer inversions well enough for forecasting, and how best to combine two kinds of measurement with such different weighting, are active questions in data assimilation.

The habit worth carrying away is to ask what an averaged measurement has averaged over. A reading that is an integral of a quantity against a broad kernel can report the quantity only as smoothly as the kernel allows, and recovering detail finer than the kernel requires information that the reading does not contain — so a spectrum is a thermometer at every height, and each of those thermometers is seventeen kilometres long.