HOW DOES LIGHT MOVE THROUGH THE ATMOSPHERE?

We are interested in the question of how light moves through the atmosphere for two reasons:
  • We want to understand how changes in the atmosphere affect the amount of light of various wavelengths that reaches the Earth's surface. Changes in radiation received at or near the Earth's surface can have environmental or biological consequences.

    •  
  • We need to understand how light moves through the atmosphere in order to interpret radiation measurements to derive information about the state of the atmosphere.

Introduction
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, the 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.

Simple, flat-atmosphere model of the radiative transfer problem for the 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.
Two of the atmospheric layers from the above figure. Just for illustration, we put a little empty space between them.
This 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.

The 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.
 
 

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 the following figure, we illustrate all the possible trajectories for photons entering a 7-layer "atmosphere," in a layer-by-layer fashion.

Basic radiative transfer model of the atmosphere. 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.

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 the 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 the following diagram.

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.