5.1 Radiative Transfer

Both remote sensing and ground-based radiometric techniques measure radiances or irradiances, at various wavelengths. Since the concentrations of particular chemical components in the atmosphere determine the radiances, we can exploit our knowledge of how matter changes electromagnetic radiation as it passes through the atmosphere to understand the transfer of radiation from one place (say, the top of the atmosphere) to another (say, Earth's surface or the location of a satellite). In this section, we describe this process of radiative transfer.

As light travels through a vacuum, the photons travel in straight-line paths. If the light travels through any kind of material, as the photons encounter atoms and molecules, each encounter is an opportunity for a photon to either be absorbed or be scattered. The probability that either of these will occur depends on the energy (or wavelength) of the photon and the type of atom, moleucule or other particle involved. So, at least in principle, we could simulate the trajectory of a photon through the atmosphere on a computer.

If we followed the paths of a large number of photons, we could construct the radiances at any place we wanted (particularly at the top and bottom of the atmosphere). But on its way from the top of the atmosphere to the bottom, a photon passes by billions of atoms and molecules. A good job of constructing radiances would require you to compute several million photon paths. This basic method of computation by direct simulation of the physical process (called a Monte Carlo method)is sometimes used, but it does take a lot of computer time.

The more convenient way to treat radiative transfer is to think of dividing up the atmosphere into a large number of layers, and keep track of the irradiances entering and leaving the surfaces of those layers. Figure 4.10 shows a simple, flat-atmosphere model of the radiative transfer problem for Earth's atmosphere. So, for a layer somewhere in the middle of the atmosphere, the irradiance that enters the top is the same as the irradiance that exited from the bottom of the layer just above. Figure 4.11 shows two of the atmospheric layers from Figure 4.10. Just for illustration, we put a little "empty" space between them.

Figure 4.12 shows the model we use for each layer. When light enters the layer from some direction, some of that light is transmitted through (going in the same direction), some is absorbed inside the layer, and some is scattered into other directions leaving the layer, entering the layers above or below. Note that the principle of conservation of energy requres that, in each layer, the amount of energy entering must equal the amount leaving (regardless of direction) plus the amount absorbed.

For computational efficiency, one would like to have as few of these layers in the model as possible. The principal factor that determines the number of layers necessary is the rate at which chemical concentrations, temperature, and pressure change as you travel vertically in the atmosphere. Each layer is treated as if the physical and chemical state of the atmosphere is constant throughout the layer. So the layers are usually chosen in such a way that this is approximately true.

Earth, of course, is almost spherical, and the atmosphere forms a spherical shell around it. It turns out that creating a mathematical or computational model of the atmosphere that takes its spherical shape into account is very difficult. Treating the atmosphere as a "cake" of flat layers, rather than as a set of nested spherical shells, turns out to give computed radiances that are very close to the correct answers, except when the sun is very low in the sky. This is the most common method for treating the problem of radiative transfer in the atmosphere.

In this section we will discuss how the simple flat, layered model works, and show some of the results of this kind of calculation. 

5.1.1 Radiative Transfer Model -- The atmosphere isn't quite flat. Its curvature follows that of the Earth. If you walk a few miles, you don't perceive any effect of the fact the Earth is round. In the same way, by thinking of the atmosphere as being flat, we won't get into trouble until we consider light that has to travel through a great horizontal distance. That will happen, for example, when we want to consider the light that falls on us when the Sun is near the horizon. When we happen to want to solve that problem, we will have to abandon this flat earth/flat atmosphere model.

Remember that, in general, the atmosphere has a lot of vertical structure in its physical and chemical properties: Temperature, pressure, and chemical composition can change quite a lot as you ascend through the atmosphere. That's one of the reasons why, to measure the properties of the atmosphere from the Earth, weather balloons and aircraft are often used: You just can't get all the useful information by measuring close to the surface. So, because of this inhomogeneity, it is desirable to divide our flat atmosphere into many layers, where each layer is small enough that its chemical and physical state can be considered to be constant throughout the layer.

In the simplest possible radiative transfer model, we ignore all the different directions the light can go, and consider each photon, at any given time, simply to be either going up or going down. A downward photon that enters a layer has a certain probability of being absorbed in the layer, and a certain probability of being scattered into the upward direction, and a certain probability of exiting the layer going downward. In Figure 4.13, a basic radiative transfer model of the atmosphere, we illustrate all the possible trajectories for photons entering a 7-layer "atmosphere," in a layer-by-layer fashion. Light entering the top of the atmosphere (big arrow) goes into the topmost layer. Some of this light proceeds in the downward direction, and some of it is scattered into the upward direction. As light proceeds through each layer, some of it is transmitted, and some is reflected.

Using a model like this, if we have the probabilities of scattering and absorption for each layer (and they may be different in each layer of the atmosphere), we can easily calculate the amount of light that makes it to the surface, and the amount of light that is reflected back into space.

We have so far ignored the angular distribution of the light entering and leaving the different layers. The direction the photon is going when it enters a layer very much affects whether anything happens to it. If a photon enters in a direction perpendicular to the layer's surface, it only has to traverse the thickness of the layer, while a photon that enters at a very steep angle must traverse a much greater distance before coming out the bottom, and so has a proportionately greater chance of having something happen to it first.

As seen in Figure 4.14, light incident on a layer in the normal direction (a=0, left) only has to traverse the thickness of the layer (d). Light that enters from an off-normal direction a>0, right) has a much greater distance to pass through the layer and so will be more likely to be either scattered or absorbed by the molecules in the layer.

The term zenith refers to the direction straight up from the earth at a particular point. In our model "flat" atmosphere, the zenith is always straight up from the horizontal surfaces of the layers. We describe the direction the light is travelling by its zenith angle, that is, the angle the direction of propagation makes with the zenith direction. In the figure above, this angle is labelled a. When the Sun is straight overhead the solar zenith angle is 0o, and at the horizon it is 90o. The photons entering the top of the atmosphere are coming directly from the sun. The angle at which these photons enter the top of the atmosphere largely determines the total probabilities that they will be absorbed, or hit the surface, or be reflected into space. Hence, the solar zenith angle is a very important parameter in radiative transfer studies of the Earth's atmosphere.

To completely specify the direction a photon is moving, one needs to specify a second angle. (This is analogous to the fact that, to specify a position on Earth's surface, you need to give both the longitude and the latitude.) We call the second angle the azimuth. It is measured from the direction to the Sun, as shown in Figure 4.15. The azimuth angle is the angle measured in the plane shown. This plane is perpendicular to the zenith direction.

To build our radiative transfer model of the atmosphere, we write down some equations for each layer, which basically say "If light of a certain intensity enters this layer (call it N, again) from a direction (szaN,in, azimuthN,in), then a certain fraction of that intensity will exit the layer in a direction specified by the angles (szaN,out, azimuthN,out).

The next step is to couple these layer equations to each other. To do this, you only need to write down the relationships between the outgoing sza and azimuth for layer N and the incoming szas and azimuths for layers N-1 and N+1.

What you are left with is a set of equations whose variables are the intensities that are passed from layer to layer, at different angles. What you do with these depends on what problem you are studying:

Using that information, you eliminate all the variables you are not explicitly interested in, and solve the remaining equations for the intensities you are interested in.

5.2 Radiation Scattering

The Sun emits electromagnetic radiation of all sorts; not only visible light, but also radio waves, infrared, ultraviolet light, and X-rays. The spectrum of the Sun's radiation (Figure 4.16) is complicated, but it is generally dominated by the characteristics of blackbody radiation: a broad distribution of intensity with respect to wavelength, whose maximum intensity is in the visible region. Superimposed on this broad distribution is a multitude of peaks and valleys that are due to the chemical constituents of the Sun's atmosphere. Figure 4.16 shows a very wide region (extreme ultraviolet to the left, far infrared to the right) of the spectrum of the solar light reaching the top of Earth's atmosphere.

Electromagnetic radiation interacts with matter (atoms and molecules) in a wide variety of ways. For example, some of the light passing through a medium can be absorbed by molecules, such as when white light passes through water with some food coloring; the eye perceives the colors that are not absorbed by the food coloring. Light can also be deflected, or scattered from one direction into another direction. For example, if you shine a beam of white light into a glass of milk, the light will exit the milk in all directions, and, viewed from any angle, the outgoing light will appear white. This is due to the scattering of the light by the tiny globules of fat that are suspended in the milk (they can be easily observed under a microscope). This is also similar to the effect of clouds (another suspension of very small particles) on light, which makes the clouds appear white. The scattering of light by particles that are much larger than molecules is called Mie (pronounced "mee") scattering, after one of the first people to study this phenomenon.

Light can also be deflected by individual molecules in a process called Rayleigh scattering, which is discussed more deeply in Section 5.2.1. This process is different from Mie scattering in a number of ways. First, if there are enough particles, Mie scattering results in exiting radiation whose intensity is pretty much the same, regardless of what angle you look at it from. However, Rayleigh scattering has a very strong dependence on the viewing angle, and the degree of this angular dependence depends upon the wavelength; the shorter the wavelength (in the visible region, this means the bluer the light), the stronger the scattering. This is why the sky appears blue. When the sun is low in the sky, light that passes over our heads would not be seen by us at all if it were not scattered by molecules in the atmosphere. The fact that we look up and see blue light is due to the fact that the blue-violet end of the visible part of the spectrum is Rayleigh scattered more strongly than light in the red-orange part of the spectrum. (If you look at a picture of astronauts on the moon, you see there is no visible "sky" beyond them, since there is no atmosphere to scatter the Sun's light.)

There are many kinds of processes in which matter absorbs or scatters light, and a few in which both of these occur. In some of these the scattering or absorbing particle (atom, molecule, or larger assembly of molecules) is essentially unchanged by the interaction with the light, but in others, the particle is very much changed. For example, radiation in the extreme ultraviolet and x-ray regions of the spectrum would be extremely harmful to life on this planet (and is of great concern for astronauts and spacecraft that travel outside the atmosphere). However, oxygen molecules (O2) high in the atmosphere, absorb this very energetic radiation and, in the process, split into individual oxygen atoms. Since there is a lot of oxygen in the atmosphere, a high altitude layer that is rich in oxygen atoms forms, and below this layer there is very little of this very harmful radiation.

Lower in the atmosphere, there is still a small amount of this oxygen dissociating radiation. However, because there is less of it, not as much of the oxygen is dissociated. In this part of the atmosphere, individual oxygen atoms can combine with oxygen molecules to form ozone, O3. Ozone absorbs radiation in the middle of the ultraviolet region of the spectrum, protecting us from that radiation which is not absorbed by the oxygen molecules higher in the atmosphere. When ozone does this, it dissociates back into an oxygen atom and an oxygen molecule, but the atom will quickly recombine with another oxygen molecule to make another molecule of ozone.

The top panel of Figure 4.17 shows the efficiencies of oxygen (O2) and ozone (O3) at absorbing photons, as a function of the photon wavelengths. Note that the O2 curve has been multiplied by 104 to put it on the scale of the graph. Molecule for molecule, ozone is much more efficient at absorbing light than oxygen. However, there is much more oxygen than ozone (by a factor of about 105).

The bottom panel shows the fluxes of radiation, as a function of wavelength, at the top of the atmosphere, at an altitude of 30 km, and at sea level. At 30 km, from 200 nm to 225, about as much radiation has been absorbed by oxygen as by ozone. However, at the surface, the radiation has been effectively completely absorbed, and mostly by ozone as the radiation passes through the ozone layer.

Because of the scattering of light by the atmosphere, some of the light that falls on the top of the atmosphere ends up being redirected upward, and goes into space. In addition to this backscattering from the atmosphere, some of the light that reaches Earth's surface is reflected back upward and leaves the atmosphere. When you look at Earth from space (from the Space Shuttle, for example), you see clouds and the surface. If you look at the edge (or limb) of Earth, you see what almost appears to be a sliver of bluish or reddish glow. This is the light backscattered from the atmosphere. By measuring this light from space, either from the Space Shuttle or from permanently orbiting satellites, we can measure the concentrations of many of the chemicals that are found in the atmosphere, which play an important role in controlling the radiation at the surface (including ozone and other species).

There are no atmospheric constituents that absorb significant amounts of light in the near ultraviolet and visible regions of the spectrum. Plants and animals on Earth have adapted themselves to be able to make use of this radiation; plants use the Sun's energy to turn carbon dioxide into organic molecules via photosynthesis, and animals use this radiation to see.

Electromagnetic radiation carries energy, and the amount of energy is related to the wavelength of the radiation. For convenience of discussion, the spectrum (that is, the range of all possible wavelengths) is divided into a number of major regions: x-ray, ultraviolet, visible light, infrared, and radio. Radiation in the ultraviolet and x-ray regions is much more energetic than light in the visible region. As a result, radiation in those regions can have a greater impact on the atoms and molecules it encounters, causing such phenomena as the loss of electrons and dissociation into smaller molecules and atoms. And just as these phenomena can occur when such energetic light encounters molecules in the atmosphere, so too can they occur when that light encounters molecules that make up animal, vegetable, and mineral at Earth's surface. Thus, the shielding effect of Earth's atmosphere is important to protect us from the harmful effects of far-ultraviolet and x-rays. Electromagnetic radiation in the infrared and radio regions is much less energetic than in the x-ray and ultraviolet regions, and so generally it does not have as great an effect on the molecules it encounters. Although there are parts of the infrared region where the atmosphere is quite strongly absorbing (this is very important for understanding the Greenhouse Effect), the infrared radiation that does get through has a very small biological effect. The main biological effect of infrared radiation is indirect, via the heating of the biosphere.

The lower curve (marked "Surface" in Figure 4.17) for wavelengths longer than 290 nm, shows the spectrum of the light that penetrates through the atmosphere and falls on Earth's surface. Note that there is very little radiation that has wavelength shorter than about 315 nm in the uvb region, or the regions where the energy the light is carrying is greater than this (shorter wavelengths). This light has been filtered out by the atmosphere, the more energetic having been absorbed by oxygen molecules (which thereby dissociate to form oxygen atoms), and the somewhat less energetic (up to about 315 nm) having been absorbed by ozone. Though there are atmospheric constituents that absorb some of the light in the visible region of the spectrum (even ozone does this) they do not absorb very much light. In the infrared region, there is a great deal of absorption by atmospheric constituents, though there are some regions where the atmosphere is essentially transparent. The intensity of the Sun's light falls off rapidly into the far infrared. The Sun is also a powerful source of radio waves (wavelengths from millimeters to meters), but most of the radio emissions are associated with storms on the surface of the sun, and vary considerably with the number of sunspots and other indicators of solar activity. In contrast to this, the radiation output of the Sun in the uv region (from 290 nm to 360 nm) is relatively constant as the solar storms come and go.

Because the atmosphere scatters and absorbs light, by studying the light that reaches a place on Earth's surface, we can learn much about the chemical composition of the atmosphere above that place. But the radiation that is sent back toward space, from atmospheric scattering, by reflection from Earth's surface, and by thermal radiation from both the surface and the atmosphere, is also affected by the composition of the atmosphere. So we can also find out about the state of Earth's atmosphere by measuring the radiation from space using satellite borne instruments. This can be done over most any location on Earth.

5.2.1 Rayleigh scattering -- For simplicity, let's think of a single atom. As long as they stay within a small spherical region around the atom's nucleus, an atom's electrons are fairly free to move around (Figure 4.18). Along comes a photon, with its electric field oscillating, tracing out a sine wave in the "plane of polarization." Because the electrons have an electrical charge, they respond to the photon's electric field, feeling a force that, at any instant of time, is proportional to the magnitude of the electric field (Figure 4.19). So as the field oscillates, the electrons are forced up and down. Now, it is a law of electrodynamics that an electrically charged particle that accelerates produces an electromagnetic wave that propagates outward from the charged particle, like water waves radiate from where a pebble is thrown into a pond. At one instant the electron is going upward, and then it slows down, stops, and starts going downward. Thus, during this phase of its motion, it is accelerating downward. At some later time, it slows down, stops, and starts going upward, and so it must be accelerating upward. The frequency of this up and down oscillation is just the frequency of the photon, so the electromagnetic wave it produces has just the same frequency (and energy and wavelength) as the original photon. Also, the polarization of the outgoing wave is in the same direction as the original photon. Figure 4.20 depicts the outgoing scattered electromagnetic wave after the interaction of the electron with the incoming electromagnetic wave. The atom is at the center of the figure.

Now we have to think about energy conservation. If a photon, having some definite energy, encounters an electron bound in a molecule, the response of the electron produces an outgoing electromagnetic wave of the same energy. The electron, and the molecule or atom it is in, are not sources of energy. In fact, once this photon-electron encounter is over, the molecule is in no physical way different from what it was before the encounter. That means that this process must make the initial photon disappear! It also means that, from a single photon-electron encounter, only a single photon may leave. But we have the notion that a photon is a particle that travels in some direction, and this would seem to contradict the notion that the oscillation of the electron produces circular outgoing electromagnetic waves. Here we run headlong into one of the deep mysteries of the interaction of light with matter: Some phenomena of light must be explained in terms of the photon concept, the photon traveling in a definite direction. Other phenomena must be explained in terms of the wave concept, where a wave may be curved, just like water waves.

Here's how we resolve this apparent contradiction. In any single photon-electron encounter, the original photon disappears, and is replaced by a single outgoing photon, which proceeds away from the location of the encounter in a straight line path whose direction lies more or less in the plane perpendicular to the direction of the polarization of the original photon. The next photon that encounters the same electron will elicit the same response, producing yet another single outgoing photon, which heads off in a straight line path, in the same plane, but in some other direction. Another photon comes by, and another, and another. Each is scattered in some essentially random direction in this plane. Actually, choose any two directions in this plane, and an outgoing photon is equally likely to scatter into one as into the other (even right back in the opposite direction of the original photon). So if you look at a single photon-electron encounter, you will see single photons leaving the encounter in definite directions, but if you look at the total of all the photons that leave, they form the circularly symmetric distribution of outgoing electromagnetic radiation suggested by the analogy with the pond water waves.

Let's do a thought experiment. If you sit in the plane of the outgoing photons, and look at the electron as it is responding to an incoming photon, you "see" the electron moving up and down. And when an outgoing photon comes past you, its plane of polarization is in that same direction. If you now move upward, out of the plane, just a bit, and look again, you can still see the electron moving up and down. But now, because of the fact you are looking at the electron from an angle, it does not seem to move as much as it did when seen from in the plane. The higher up you go, the less it appears to move. If you look down at the electron from the top, you can not tell it's moving at all. What does this signify?

If, when looking at the moving electron from a point at some distance away from it, you see it moving up and down, then some of the electromagnetic radiation its motion is producing is coming toward you. Because the amplitude of the motion is less the further out of the scattering plane you sit, the intensity of that radiation is less. In fact, it is less by a factor that's the cosine of the angle the line from you to the electron makes with the plane. When you are 90° out of the plane (i.e. looking down on the electron), you see no motion. The cosine of 90° is 0, so this says that no radiation is scattered in that direction.

Now, we take the same step we did before to get from the electromagnetic wave concept to the photon concept. We observe that the electromagnetic wave is propagating away from the moving electron not only in the plane, but in almost all other directions as well. As for the photons, an individual photon can only be going in one direction away from the moving electron. If we watch lots and lots of photons undergoing Rayleigh scattering from the electron, we will see some leave pretty much in the scattering plane, we will see others (somewhat fewer) that are leaving along lines that make a 20° angle with the scattering plane. We will see somewhat fewer still leaving along lines that make 80° angles with the scattering plane. The relative numbers that leave at some angle are actually proportional to the square of the cosine of that angle. Figure 4.21 shows the amplitude of Rayleigh scattering by an atom or molecule (at the center of the "doughnut"). The electromagnetic waves are coming in from the left, and they are polarized with their electric fields in the vertical direction. There is no scattering into the vertical direction. Choose a point on this surface, and draw a line from that point to the center. The distance between the points is proportional to the intensity of the radiation scattered in the direction of the chosen point. The intensity is largest in any direction in the plane perpendicular to the plane of polarization.

Rayleigh scattering and polarization -- In the discussion above, we have described the relationship of the polarization of the outgoing photon to that of the original photon. The sunlight falling on the top of Earth's atmosphere is almost completely unpolarized. That is, if you choose any two lines perpendicular to the light's direction of travel, there will be just as many photons aligned with one of the lines as with the other. However, due to the Rayleigh scattering in the atmosphere, the light that reaches Earth's surface, or that are scattered back into space, is partially polarized. That is, if we look at a large number of photons that are scattered into some direction, not all the photons will have exactly the same polarization. To quantify this, we discuss the degree of polarization in the Rayleigh scattered light.

Suppose we have a sample of a Rayleigh scattering material, and we introduce an incoming beam of light. If the scattering angle (i.e. the angle between the incoming and outgoing photon directions) is , then the degree of polarization is proportional to sin 2theta. This means that the degree of polarization is greatest when the angle between the incoming and outgoing photon directions is 90°.

As mentioned elsewhere, the short answer to the everlasting question, Why is the sky blue? is that there is Rayleigh scattering of the Sun's light by the molecules in the air. You can see the polarization effect for yourself. Find a pair of polaroid sunglasses or a polarizing filter such as is used in photography. On a clear day, when the Sun is low in the sky, and when the sky does not look very hazy (whitish glow all around the horizon), look at the sky through the polarizing filter or one of the lenses of the sunglasses. First, look straight up. Most of the light you will see has been scattered through an angle of roughly 90°. Now rotate the polarizer (the lens or filter) slowly, and notice what happens to the light. Note the orientation of the polarizer when the greatest amount of light is seen. Now look at different regions of the horizon, relatively near the Sun (don't look right at the Sun!), at right angles to the Sun, and in the opposite direction. Every time, choose a direction, look in that direction through the polarizer, and rotate the polarizer slowly, noting any brightening or darkening effect.

It is important to realize that the light that arrives at the top of the atmosphere from the Sun is unpolarized: Just as many photons have their planes of polarization in any direction as in any other direction. The degree of polarization of the light reaching the surface of Earth, or reaching some satellite instrument in space, is due to the Rayleigh scattering process in the atmosphere.

Some creatures, notably the hymenoptera family of insects (bees, wasps, and hornets), have eyes with built-in polarizers, and they use the polarization of the sky light to navigate. Because these creatures' eyes are responsive further into the ultraviolet than our own, and because the ultraviolet light is even more strongly Rayleigh scattered than the blue-violet light we can see, they actually see a greater contrast in sky brightness as they rotate than we see when we rotate the polarizer. (Their ultraviolet vision also helps them see flower pigments.)

Rayleigh scattering cross sections -- Not every photon that passes by a molecule, with its semimobile electrons, gets Rayleigh scattered. The mathematical theory of Rayleigh scattering must take into account the wavelength of the light, the size and shape of the region the semimobile electrons can move around in, how tightly bound they are to the atom's nucleus, and how many of them there are in a molecule. The development of this theory is beyond the level of this lecture. However, the result is essentially this: The cross section for scattering is most sensitive to the wavelength, and goes as 1/ lambda4.

Visible light has wavelengths in the range of about 400 nm to 700 nm. So, according to this law, blue light (at 400 nm) is roughly 9 times as strongly scattered as red light. Of course, infrared light is very weakly Rayleigh scattered, while ultraviolet is even more strongly scattered than any visible light.

Figure 4.22 shows a Rayleigh scattering cross section as a function of wavelength through the UVA, UVB, and visible regions of the spectrum. The greater the cross section, the more efficient the molecules are at scattering the light. The scattering cross section is about 9 times as great at the blue-violet end of the visible spectrum than at the red end. Over the UV region, the scattering cross section increases by a futher factor of about 3.