Overview

Now that you are familiar with the principles of atmospheric radiation, we can apply them to help us better understand weather and climate. Climate is related to weather, but the concepts used in predicting climate are very different from those used to predict weather.

For climate, we need to understand the global energy budget, which is comprised of solar radiation coming into the Earth’s atmosphere and infrared radiation leaving the atmosphere to go into space. We will see that, when averaged over the Earth and over sufficient time, the energy associated with infrared radiation emitted to space by the Earth’s surface and atmosphere essentially always balances the energy associated with solar radiation absorbed by the Earth’s surface and atmosphere. By increasing atmospheric concentrations of CO₂ and other greenhouse gases during the industrial era we have slightly perturbed this balance such that less infrared radiation is currently leaving the Earth system as compared to solar radiation being absorbed by it. This leads to additional energy being deposited into the Earth system that has been exhibited, in part, as a rise in surface air temperatures. At Earth’s surface the energy budgets of both downwelling solar and downwelling longwave radiation at short (second to minute to hour) timescales depends strongly on the presence of gases that absorb, emit, and scatter radiation in the atmosphere. Thus, Earth’s local surface temperature is exquisitely sensitive to the amounts and radiative properties of those gases and particles. We will do some simplified radiation calculations to show you how the Earth’s atmosphere affects the surface temperature.

For weather, we make predictions using models that consist of the equations of thermodynamics, motion, and microphysics. We initialize the models with observations and then let the model calculate the air motions going into the model future, thus giving weather forecasts. The models are good, but not so good that they can run for many days and continue to make accurate forecasts. So periodically, the models are adjusted by adding more observations, a process called data assimilation, in order to correct them and keep the forecasts accurate. Increasingly, satellite observations are being assimilated into the models to improve weather forecasts.

Satellite instruments observe atmospheric radiation: both visible sunlight scattered by Earth’s surface, clouds, and aerosols; and infrared radiation emitted by Earth’s surface and many of its atmospheric constituents. What the satellites measure depends on the wavelengths at which they collect radiation coming up to them. Typically, satellites observe in different wavelength bands, some of which cover wavelengths at which water vapor absorption is much stronger than for others. Taken together, the radiation in these different bands tells us much about the atmosphere’s temperature and moisture structure, which is just the kind of information that the models need to assimilate. You will learn how to interpret satellite observations of atmospheric radiation in support of applications such as vertically resolved temperature and moisture retrievals.

Learning Objectives

By the end of this lesson, you should be able to:

• demonstrate the effects of infrared absorbers on Earth’s temperature using a simple model
• explain the concept of radiative–convective equilibrium
• determine what a satellite is seeing by interpreting the observed spectrum of upwelling infrared radiation

Questions?

If you have any questions, please post them to the Course Questions discussion forum. I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.

7.1 Applications of Atmospheric Radiation

Let’s use what you learned in Lesson 6 to examine two applications of atmospheric radiation. The first application involves the role of atmospheric radiation and greenhouse gases in Earth’s climate. The second application is the interpretation of upwelling infrared radiation spectra measured by satellite instruments in space with an eye on improving weather forecasting. These two applications use the principles of atmospheric radiation in quite different ways, but understanding both is critical to you becoming a competent meteorologist or atmospheric scientist.

Earth’s atmosphere is essentially always in radiative energy balance, which is also called radiative equilibrium. By this, I mean that, when averaged over the whole Earth, the total amount of solar radiation energy per second that is absorbed by the Earth’s surface and atmosphere is about equal to the total amount of infrared radiation energy per second that leaves the Earth’s surface and atmosphere to go into space. There can be periods when this balance is not exact because changes in atmospheric or surface composition can alter the absorption or scattering of radiation in the Earth system. It can take a little while for all of the temperatures of all of the Earth system's parts to adjust, but if the changes stop, the Earth system will adjust its temperatures to come back into balance. Right now we are in a period where atmospheric CO₂ concentrations are increasing due to industrialization, the outgoing infrared radiation is slightly less than the incoming absorbed solar radiation, and the Earth system's temperatures are adjusting (by increasing) to try to bring the outgoing infrared radiation into balance with the incoming absorbed solar radiation. For most of the following discussion, we will use this concept of radiative equilibrium even though the current balance is not exact.

Always keep in mind that atmospheric radiation moves at the speed of light and that all objects are always radiating. Moreover, as soon as an object absorbs radiation and increases its temperature, its emitted radiation will increase. Thus energy is not “trapped” in the atmosphere and greenhouse gases do not “trap heat.” We will see instead that greenhouse gases act like another radiation energy source for Earth’s surface.

Before we do any calculations, let's summarize how different parts of the Earth system affect visible and infrared radiation (see table below). Earth's surface either absorbs or scatters both visible and infrared radiation, while the atmosphere mostly transmits the visible radiation, with a little scattering; and the atmosphere mostly absorbs infrared radiation, with a little transmission. Clouds, an important part of the Earth system, strongly absorb infrared radiation and both scatter and absorb visible radiation.

Watch this video (52 seconds) to learn more:

Image of Earth in the visible. Clouds are scattering radiation at all visible wavelengths back out to space, and thus appear to be white, while Earth's surface selectively scatters visible radiation at only some wavelengths and absorbs the rest.

Credit: NASA

Image showing outgoing longwave (infrared) radiation emitted by the Earth and atmosphere during the European heatwave of 2003. The blue and white colors are clouds, which are radiating at the lower temperatures of the upper troposphere, while the yellows are the Earth's surface and the lower troposphere, which are radiating at higher temperatures.

Credit: NASA [1]

7.2 Atmospheric Radiation and Earth’s Climate

Let’s first look at the general energy balance—the radiative equilibrium—of the Earth system (see figure below). The solar irradiance is essentially composed of parallel radiation beams (or radiances) that strike half the globe. At the same time, outgoing infrared radiation is emitted to space in all directions from both the sunlit and dark sides of the globe. At the top of the atmosphere, the difference of the incoming solar radiation energy minus the amount of solar radiation energy that is scattered back to space (this difference being the amount of solar radiation energy absorbed by the Earth system) must balance the emitted infrared radiation energy for radiative equilibrium to hold. The total amount of solar radiation energy striking Earth per second is equal to the solar irradiance, F (W m^–2), times the Earth’s cross sectional area, $\pi R_{Earth}{}^2 ({\text{m}}^2)$. Some of the solar radiation energy is reflected by clouds, aerosols, snow, ice, and the land surface back to space and is not absorbed, hence does not contribute energy to raise Earth’s temperature. The fraction that is reflected is called the albedo, and we can account for it by subtracting the albedo from 1 and multiplying $F\pi R_{Earth}{}^2$ times the difference: $F\pi R_{Earth}{}^2 \left(1-a\right)$. The albedo has been estimated to be 0.294 (Stephens et al., 2012, Nature Geoscience 5, p. 691). On the other hand, Earth and its atmosphere radiate in all directions and the radiation can be described by the Stefan–Boltzmann Law, which, recall, is the integral of the Planck function over all wavelengths. Thus the emitted infrared energy per unit area (or emitted infrared irradiance) out the top of the atmosphere is $\sigma T_{top}{}^4 (W{m}^{-2})$, where we have assumed an emissivity of 1 for the atmosphere at all emitted infrared radiation wavelengths. To get the total energy we must multiply this irradiance by the Earth’s total surface area, $4\pi R_{Earth}{}^2$. The top of the atmosphere is at an altitude of ~50–100 km above the surface, compared to Earth’s radius of 6400 km, so we will ignore this small difference.

Distribution of solar radiation into the Earth system and Earth infrared radiation out of the Earth system. The Sun’s rays are roughly parallel when they reach Earth and deposit more energy per unit area on Earth’s surface in the tropics than near the poles. Earth is a little warmer in the tropics than at the poles, so it radiates in all directions, though a little stronger in the tropics than at the poles.

Credit: Help Save the Climate [2]

See the video (1:37) below for a more detailed explanation:

Equating the solar radiation energy absorbed by the Earth system to the infrared radiation energy emitted by the Earth system to space gives the equation:

But what is the temperature at the top of the atmosphere, T_top? Put in the values F = 1361 W m^–2, a = 0.294, and σ = 5.67 x 10^–8 W m^–2 K^–4. Therefore,

The temperature at the top of the atmosphere is 255 K, which equals –18 ^oC or 0 ^oF. It is substantially less than Earth’s average surface temperature of 288 K, which equals 15 ^oC or 59 ^oF. This top-of-the-atmosphere temperature is the same as what the Earth’s surface temperature would be if Earth had no atmosphere but had the same albedo. It is clear from these calculations that the atmosphere, modeled with an emissivity (and hence absorptivity) of 1 over all emitted infrared radiation wavelengths, is creating a difference between the temperature at the top of the atmosphere and the temperature at Earth’s surface.

In particular, let’s look at only the vertical energy balance averaged over the entire globe. We will think of everything in terms of the SI units of irradiance (or energy per second per unit area), which is W m^–2. Consider two idealized cases first before examining the actual atmosphere: (1) the no-atmosphere model and (2) the infrared-opaque model.

Let’s build a simple, flat atmosphere with all solar and infrared radiation energy moving only vertically (see figure below). For both models we assume:

• solar irradiance is mostly in the visible (even though in reality half the solar irradiance is in the infrared with wavelengths greater than 0.7 µm);
• Earth's surface emits radiation in the infrared;
• radiation energy flowing up must equal radiation energy flowing down at the surface.

For the opaque-infrared model, we make the following additional assumptions:

• the atmosphere emits radiation in the infrared;
• the atmosphere is transparent to visible radiation;
• the atmosphere is opaque (that is, has an absorptivity of 1) to infrared radiation;
• radiation energy flowing up must equal radiation energy flowing down at every level in the atmosphere.

Schematics of two simple radiation energy balance models: the no-atmosphere model (left panel) and the opaque-infrared model (right panel). Some of the solar radiation energy is reflected back to space without affecting the Earth system (second yellow arrow in each panel). The net solar radiation energy that is absorbed by the surface (third yellow arrow in each panel) fuels the Earth system. Each of the red arrows, regardless of its length, is equal to an amount that represents the net solar radiation energy that is absorbed by the surface. When summing the arrows, use the net solar radiation energy that is absorbed by the surface (third yellow arrow in each panel), not the incoming solar and the reflected radiation (first and second yellow arrows, respectively).

Credit: W. Brune

Please watch the following video (2:18)

In the no-atmosphere model, the only radiating bodies are the Sun and the Earth. (By the way, if Earth had a pure nitrogen atmosphere, the results would be very similar to the no-atmosphere scenario.) The solar radiation passes through the altitude levels where a stratosphere and troposphere would be and the fraction 1 – a of the radiation is absorbed by the Earth’s surface. We assume that Earth’s albedo is still 0.294 so that 0.706, or 70.6%, of the solar radiation is absorbed at the surface with the rest reflected back to space. The Earth’s surface radiates infrared radiation energy back out to space with no absorption at the levels where the stratosphere and troposphere would be. The surface temperature in this model is such that the infrared radiation energy leaving the surface balances the incoming solar radiation energy absorbed by the surface. In terms of the arrows in the figure, there is one down arrow and one up arrow at every level.

So what would the temperature at Earth’s surface be if there was no atmosphere? Equation [7-2] applies to the no-atmosphere case and hence the Earth with no atmosphere has a surface temperature of 255 K. This temperature is the same as the radiating temperature at the top of our Earth with an atmosphere whose absorptivity (and hence emissivity) is 1 at all emitted infrared radiation wavelengths. The surface would be so cold that any water on it would freeze and stay frozen.

Now consider the opaque-infrared model. The atmosphere in this model is identical to the atmosphere modeled in the derivation of Equation [7-2]. Now, however, we will be paying attention to the temperature of Earth’s surface under such an atmosphere. As before, this atmosphere is transparent to all solar radiation energy coming down to Earth’s surface and is opaque to all infrared radiation. “Opaque” means that the infrared radiation is completely absorbed over very short distances (i.e., the absorptivity and emissivity are 1, and the absorption optical depth is great, so by Beer’s Law, very little infrared radiation is transmitted). The atmosphere itself is strongly emitting in all directions, both up and down, and the only infrared radiation that does not get absorbed is that emitted out the top of the stratosphere to space.

We know that the infrared radiation leaving the Earth system must come close to balancing the solar radiation absorbed by the Earth system. Otherwise, the temperatures of Earth’s surface and atmosphere would adjust until this condition was true. So, we will assume radiative equilibrium. Our model is a two-layer model—an upper layer and a lower layer—with a solid Earth beneath them. We are assuming that each layer is at a constant temperature and absorbs all infrared radiation energy impinging on it, and then emits infrared radiation out its top and its bottom in equal amounts (because the layer emits infrared radiation energy in both directions equally). The amount of infrared radiation energy emitted by the layer is determined by its temperature only because its emissivity is set to 1 at all infrared wavelengths. Thus between the upper layer and space, we have one arrow going down and one arrow going up: the outgoing emitted infrared radiation energy exactly balances the incoming solar radiation energy that is absorbed.

The upper layer thus also emits one arrow of infrared radiation down. So, at the interface between the upper and lower layer, the solar radiation and the upper layer's infrared radiation are going down (two arrows). To be in radiative equilibrium there must be enough upwelling infrared radiation from the lower layer to equal the incoming solar radiation energy that is absorbed and the downward infrared radiation emitted by the upper layer (two arrows). But that means that the lower layer must also be emitting the same amount of infrared radiation down to Earth’s surface (two arrows).

At Earth’s surface, there is the incoming solar radiation energy that is absorbed and the tropospheric downward emitted infrared radiation, equivalent to three times the incoming solar radiation energy that is absorbed. Thus Earth’s surface must be radiating upwelling infrared radiation energy equivalent to this incoming energy to maintain radiative equilibrium. So, in this simple model Earth’s surface is radiating three times the energy that the model without the atmosphere does. But to emit this larger amount of radiation the surface must be much warmer than the surface in the model without an atmosphere. We can calculate the surface temperature that would be required using equation [7-2], but adding the downward emitted infrared radiation energy from the troposphere to the solar radiation energy. One way to look at this situation is that the lower layer is providing a source of radiation energy at the Earth’s surface in addition to the solar radiation energy.

Mathematically, we can account for this extra energy near Earth's surface by simply multiplying the solar radiant energy by an IR multiplier, multiplier_IR = 3, in equation [7-2]:

This temperature (336 K = 63 ^oC = 145 ^oF) is deadly and much higher than Earth’s actual surface temperature, 288 K. So this model also fails to simulate the real Earth. The no-atmosphere model is too cold while the model with a two-layer, infrared-opaque atmosphere is too hot. So we can guess that something in between might be just right.

Indeed this is the case! If you look at the infrared absorption spectrum in Lesson 6, you will recall that there are some wavelengths at which all the infrared is absorbed and others, called windows, at which only a small fraction of the infrared radiation is absorbed. So, we find that a mix of total absorption, partial absorption, and no absorption at various wavelengths gives an atmosphere that allows Earth’s surface to radiate much radiation directly to space at some wavelengths but not at other wavelengths, where troposphere absorption is strong. But a large absorptivity implies a large emissivity so that at those wavelengths for which there is strong absorption there is also emission; however, given that the troposphere is cooler than the surface, the troposphere emits less upwelling infrared radiation energy than it absorbs from the warmer surface underneath. But irrespective of wavelength, emission by the troposphere is downwards as well as upwards, and provides another radiation energy source to heat Earth’s surface. This is called the greenhouse effect, which is poorly named because a greenhouse warms the Earth by suppressing heat loss by convection whereas the troposphere warms the Earth by emitting infrared radiation.

A study by Kiehl and Trenberth (1997, Bulletin of the American Meteorological Society 78, p. 197) determined the contributions to the greenhouse effect. It was shown that 81% of the greenhouse effect is due to greenhouse gases and 19% is due to clouds. Of the greenhouse effect resulting from gases, 60% is contributed by water vapor, 26% by carbon dioxide, and 14% by ozone, nitrous oxide, and methane. Though clouds are an important contributor to the greenhouse effect, they are actually more effective at reflecting solar radiation back to space: clouds cause a greenhouse warming of 30 W m^–2 but a reflective cooling of 50 W m^–2 for a net radiative cooling of 20 W m^–2 (National Research Council, 2008 [3]).

In parts of the spectrum where water vapor, carbon dioxide and other gases absorb more weakly, the atmosphere is less opaque. However, if the amounts of these gases are increased, then they will absorb more strongly and thus start emitting more strongly, thus increasing the radiation emitted by the atmosphere to the surface and thus increasing the surface temperature in order for the surface to come into radiative equilibrium. Remember that the energy going out of the top of the atmosphere is still essentially the same as the solar radiation energy coming into the atmosphere that is absorbed. In a sense, by adding carbon dioxide and other greenhouse gases to the atmosphere, we are moving Earth’s surface temperature from being closer to the lower value of the no-atmosphere model to being closer to the higher value of the infrared-opaque model.

In summary, the greenhouse effect has a very dramatic impact on Earth's climate by warming the surface by 33 ^oC. The greenhouse effect is mostly natural and keeps the Earth habitable. Recall from Lesson 6 that the greenhouse effect probably kept the Earth at a habitable temperature early in Earth's history, when the energy output of the Sun was much lower. Despite water vapor being the most important greenhouse gas, it is condensable and thus has an extremely short lifetime in the atmosphere (about 8 days, easily calculated as the global total precipitable water divided by the global precipitation rate). It is the longer-lifetime greenhouse gases, particularly carbon dioxide, that ultimately have the bigger impact on climate. In fact, climate model simulations show that when carbon dioxide is completely removed from the atmosphere, the Earth's surface temperature drops by more than 30 ^oC within 30 years, consistent with our simple estimations of the greenhouse effect (Lacis et al., 2010, Science 330, pg. 356). The large temperature drop is possible because the initial cooling due to carbon dioxide reduces the amount of water vapor in the atmosphere, as expected from the Clausius–Clapeyron equation. A positive feedback, called water-vapor feedback, then ensues, leading to further cooling and even lower water vapor, until after 30 years, the water vapor content of the atmosphere is only 10% of its initial value. Water vapor thus follows carbon dioxide, which is why Penn State glaciologist and climate scientist, Richard Alley, refers to carbon dioxide as the "control knob [4]" of the Earth's climate.

7.3 What does the energy balance of the real atmosphere look like?

The real atmosphere's energy balance includes not only radiation energy but also energy associated with evaporation and convection (see figure below). However, the atmosphere is still very close to total energy balance at each level.

The average vertical energy balance of the actual atmosphere. All energies are represented as a percentage of the incoming solar irradiance at the top of the atmosphere (340.2 W m^–2 = 100 units). Solar irradiance is on the left (yellow arrows), infrared radiation is in the middle (red arrows), and convection (5 units) and evaporation (24 units) are on the right (blue arrows).

Credit: W. Brune (after D. Hartmann)

First, let’s go through each set of arrows to see what is happening. The average solar irradiance at the top of the atmosphere is 340.2 W m^–2, which we will represent as being 100 units and then compare all other energy amounts to it.

• Leftmost two columns of yellow arrows: Of the solar irradiance coming into the atmosphere, most of the solar ultraviolet irradiance, about 3 units, is absorbed in the stratosphere and warms it, leaving 97 units to make it to the troposphere. 17 units, mostly at wavelengths just longer than solar visible wavelengths, are absorbed in the troposphere and another 30 units are scattered back out to space by bright objects, such as clouds, non-absorbing aerosols, snow, ice, and the land surface, leaving 50 units to be absorbed at Earth’s surface.
• First column of red upward arrows: The Earth’s surface emits upwelling infrared irradiance of 110 units, only 12 units of which are transmitted through the troposphere into the stratosphere, and 10 of these 12 units are subsequently transmitted through the stratosphere to space.
• Second column of red upward arrows: The troposphere radiates 89 units down and 60 units up; 54 of these 60 units escape to space. Unlike our simple two-layer model in which we assumed that the troposphere emitted equally up and down, the real troposphere is more complex and the downward radiation exceeds the upward radiation because of the vertical distribution of temperature (with temperature decreasing with height through the troposphere), water vapor, and carbon dioxide.
• Third column of red upward arrows: The stratosphere radiates 5 units downward and 6 units upward.
• Rightmost blue columns: There is significant non-radiation vertical energy transport at the surface. Of the net 29 units of irradiance absorbed at the Earth’s surface, 24 units go into latent heat. Latent heat quantifies the amount of irradiance necessary to evaporate liquid water (mostly seawater) at Earth’s surface to water vapor. This water vapor is transported upward by convection to form clouds, which releases this energy into the troposphere, warming it. The remaining 5 units of net irradiance absorbed by the surface goes into sensible heat. Sensible heat is the conduction of energy between the warmer Earth’s surface and the cooler tropospheric air, thus warming the air and causing it to become less dense (higher virtual temperature) than its surrounding air, followed by convection, which moves warmer air upward.

At each level, the amount of energy going down must equal the amount of energy going up. Thus, at the top of the stratosphere, 100 units cross into the stratosphere from space, and to balance this downward energy are 30 units of reflected solar irradiance upward to space and 70 units upward emitted infrared radiation that makes it to space. At the top of the troposphere, the downwelling of 97 units of solar irradiance and 5 units of infrared irradiance is balanced by the upwelling of 30 units of reflected solar irradiance and 72 units of infrared irradiance. At Earth’s surface, the downward fluxes of solar irradiance (50 units) and infrared irradiance (89 units) balance the upward fluxes of 110 units infrared irradiance, the 24 units of latent heat, and the 5 units of sensible heat.

In reality, the Earth’s surface and atmosphere are not in simple radiative equilibrium, but are instead in radiative–convective equilibrium. Furthermore, the atmosphere is in radiative–convective equilibrium globally, but not locally (see figure below). The absorbed solar irradiance is much greater near the equator than the poles because that is where the surface is most perpendicular to the incoming solar irradiance. The radiative and convective net upward energy transport is greatest at the equator as well (because Earth’s surface is warmer there than at the poles). Overall, there is significant net incoming radiation energy between 30^oS and 30^oN latitude and a net outgoing radiation energy poleward of 30^o in both hemispheres.

This uneven distribution of incoming and outgoing radiation results in a flow of energy from the tropics to the poles (see figure below). It unleashes forces that cause warm air to move poleward and cold air to move equatorward. The poleward motion of warmer air, coupled with the Coriolis force that curves moving air to the right in the Northern Hemisphere and to the left in the Southern Hemisphere, causes the atmosphere’s basic wind structure, and thus its weather. We'll talk more about these forces and the resulting motion in the next few lessons when we discuss atmospheric motion (kinematics) and the forces (dynamics) that cause the motion that results in weather.

The uneven distribution of incoming solar radiant energy and outgoing radiant energy and the resulting net incoming energy near the equator and net outgoing radiant energy toward the poles.

Credit: NOAA [5]

7.4 Applications to Remote Sensing

A second application of the principles of atmospheric radiation is satellite remote sensing (see figure below).

NOAA composite images of the Northern Hemisphere on 25 January 2015. Left: visible (0.55–0.75 μm) radiances, where white indicates bright (large) values and black indicates dark (low) values; center: infrared window (10.2–11.2 μm) radiances, where white indicates cold (low) values and black indicates warm (high) values; right: water vapor (6.5–7.0 μm) radiances, where white indicates cold (low) values and black indicates warm (high) values. See the WeatherTAP website [6] for satellite tutorials.

Credit: NOAA

Description of the Major NOAA Geostationary Satellite (GOES) Data Products

The visible channel (0.55–0.75 μm) records reflected sunlight radiances, where whiter shades are more reflected light and darker shades are less, just like in a black-and-white photograph. Land reflects more light than oceans and lakes; clouds and snow cover reflect more light than land. The visible channel goes dark at night.

The infrared window channel (10.2–11.2 μm) is over a wavelength band where the cloud-free atmosphere is transparent. As a result, it primarily records infrared radiation emitted from Earth’s surface and clouds, with emission and absorption by the gases in the atmosphere playing a secondary role. In the figure above, the greater the surface temperature (and hence the greater the radiance or radiation energy according to Equation [6-5]), the darker the shading. Thus cloud tops, which are at higher altitudes and thus colder, appear brighter.

NOAA thermal infrared (IR) channel temperature scale as a gray scale. Less radiant objects are colder and are given lighter shading.

Credit: NOAA

The water vapor channel (6.5–7.0 μm) covers a strong water vapor absorption band. Thus, radiation energy at this wavelength is strongly absorbed and the radiation energy recorded by the satellite for this channel must originate from the top of the highest moist layer. Within the moist layer, the absorptivity at this wavelength is effectively 1 and it is only near the top of the moist layer that the absorption optical thickness becomes small enough that the radiation energy can escape to space and be recorded by the satellite. Note that the higher the top of the moist layer, the lower the temperature and the less radiance recorded by the satellite. Lower radiances (and hence higher, colder moist layers) are given whiter shading; darker shading is given to higher radiances (and hence lower, warmer moist layers).

7.5 What is the math behind these physical descriptions of the GOES data products?

In Lesson 6, we derived an equation (Schwarzschild’s equation) for the change in radiance as a function of path between an infrared source and an observer:

where I is the directed beam of radiation (or radiance) along the path from the object to the observer, s is the distance along that path, P_e is the Planck function radiance at the temperature of the air (really the greenhouse gases in the air) along the path, and κ_a is the absorption coefficient of the air along the path.

Let’s apply this equation to the point-of-view of an Earth-observing satellite. Define $\tau$ (tau) as the optical path between the satellite $(\tau =0)$ and some arbitrary point along the optical path given by $\tau$ . We are not using Earth’s surface as the zero point as we often do, but instead, we are using the satellite as the zero point and letting the distance, s, and thus the optical path, change from there. The change in the optical path equals:

where the negative sign indicates that τ increases as s decreases.

Integrating both sides from the satellite to some distance s from the satellite:

To make it easier to understand what is going on, we will switch variables in [6.16] from actual distance s to optical path τ because it is the optical path, not the actual distance, that determines what the satellite detects.

This equation can be integrated to give the radiance observed by the satellite at an optical depth ${\tau }_i$ looking down at Earth:

So, what does this mean?

• The left-hand side is the radiance that the satellite observes.
• The first term on the right-hand side is a source’s radiance that is absorbed along the path according to Beer’s Law. $I(\tau )$ could be the radiance emitted by Earth’s surface and $exp(-{\tau }_i)$ the transmittance from Earth’s surface to the satellite.
• The second term on the right hand side is the emitted radiance of the atmosphere integrated over all points along the path, with transmission between each point of emission and the satellite accounted for by the exponential factor $exp(-\tau )$. For example, for the water vapor channel, P_e is the emission of radiance at the water vapor channel wavelength from some point along the path and $exp(-\tau )$ represents the transmission of that radiance through all the other water vapor along the path between the emission point and the satellite.
• The satellite simply will not detect much radiance from an object, solid or gas, if the optical path, $\tau$ , between it and the satellite is 3 or more because exp(–3) = 0.05.
• Remember that P_e depends on temperature (equation 6.4), so that P_e will be smaller at higher altitudes where the temperature is lower.

We have neglected scattering in these equations. Molecular scattering is insignificant at infrared and longer (for example, microwave) wavelengths. Cloud particle and aerosol scattering is important at visible and near-infrared (1–4 μm) wavelengths, but less so at thermal infrared (4–50 μm) wavelengths, where absorption dominates. In the thermal infrared, water clouds have an absorptivity, hence emissivity, close to 1 and emit according to the Planck function (equation 6.4).

Let's look back at a figure from Lesson 6 (reproduced below) and focus on terrestrial radiation in the atmosphere (the right side of the figure). We can see at which wavelengths the greenhouse gases in the atmosphere—mostly water vapor and carbon dioxide—absorb and at which wavelengths radiation mostly passes through the atmosphere. Note that much of Earth's infrared irradiance is absorbed by the atmosphere. However, there is a "window," extending from about 8 to 13 μm, in which most of the radiation from the Earth's surface passes through the atmosphere into space. One exception in this window is a fairly narrow band of absorption around 9.6 μm that is due to ozone.

Calculated solar and terrestrial irradiances in the atmosphere. Gases considered for absorption are water vapor, carbon dioxide, oxygen, ozone, and nitrous oxide. Scattering is due to all gases (but mainly nitrogen and oxygen). The top panel shows the downgoing solar radiation at the Earth's surface (black curve with red filling) and the upgoing terrestrial radiation at the top of the atmosphere (black curve with blue filling). Also shown in the top panel are the Planck spectral irradiances for representative solar and terrestrial temperatures. Note that all curves in the top panel have been scaled such that the peak values are roughly the same. The middle panel shows the total extinction (due to absorption and scattering) by gases in the atmosphere. The bottom panel show the absorptivity of the individual gases and the scattering.

Credit: Robert A Rohde, Global Warming Art, via Wikimedia Commons [7]

Satellites observe radiance from both Earth's surface and from the atmosphere at different pressure levels. The example below shows the infrared spectrum observed by a special satellite, which captures the details of the full infrared spectrum. In contrast, weather satellites, such as GOES, look in only selected wavelength bands. The special satellite, in this case, observed the Earth under clear skies over the Western Pacific Ocean. Hence, the radiance observed between about 8 and 13 μm came from Earth's surface (the ocean, in this case) and had a temperature of about 295 K, or 22 ^oC. Hence, a GOES weather satellite in the IR band (10.2–11.2 μm, indicated in the figure below), would see the ocean surface. At wavelengths lower than 8 μm, note that the radiance is coming from sources that are colder. Specifically, the radiance is coming from water vapor with a temperature of about 260 K between 7 and 8 μm and 240 K between 6 and 7 μm. The temperature difference is due to the increase in water vapor absorptivity as wavelength decreases from 8 to 6 μm (see lower panel in figure above). Because lower temperatures are related to higher altitudes, the special satellite observed water vapor at lower altitudes near 8 μm and higher altitudes near 6 μm. Thus, satellites can observe radiance from different altitudes in the atmosphere by using different wavelengths. In this case, a GOES weather satellite in the water vapor band (6.7–7.0 μm, indicated in the figure below), would be seeing the atmosphere at an altitude where the temperature is about 250 K.

Another example from the figure below is the strong carbon dioxide and water vapor absorption near 15 μm. At wavelengths near 13 μm, the satellite is observing radiance mostly from CO₂ and H₂O from lower in the atmosphere because the emissivity of CO₂ is less at those wavelengths. At wavelengths nearer 15 μm, the CO₂ emissivity is much greater and the satellite is observing CO₂ and H₂O radiance from temperatures below 220 K and therefore much higher in the atmosphere, actually at the tropopause. Note the very narrow spike right in the middle of this strongly absorbing (and thus emitting) CO₂ absorption band. Why does the temperature go up? Answer: In this most-strongly absorbing part of the band the satellite is seeing the CO₂ radiance coming from the stratosphere, which is warmer than the tropopause. Note that the CO₂ and H₂O at lower altitudes are emitting in the 15 μm band, but all of that radiance is being absorbed; only the layer that has no significant absorption above it can be observed by the satellite.

Infrared spectrum of Earth observed by a satellite under clear conditions over the tropical Western Pacific Ocean. The spectrum extends from 6.0 to 25 μm. Dashed lines are the Planck functions for objects at different temperatures from 300 K to 200 K. Where the measured radiance matches the dashed line the radiance came from water vapor and carbon dioxide at that temperature. Thus, if you know the atmospheric temperature profile, then you can estimate the altitude from which the radiance is coming (on average). Also indicated on the figure are the wavelength bands of the GOES water vapor (WV) and infrared (IR) bands.

Credit: W. Brune (data from NOAA Star Center for Satellite Applications and Research)

Watch the following video (2:46) on infrared spectrum analysis:

Look at another scene, which is the top of a thunderstorm in the tropical western Pacific. Remember that reasonably thick clouds are opaque in the infrared and therefore act as infrared irradiance sources that radiate at the temperature of their altitude. The cloud's radiance was equivalent to Planck distribution function irradiance with a temperature of 210 to 220 K. These temperatures occur at an altitude just below the tropical tropopause, which means that this storm cloud reached altitudes of 14–16 km. Note that in the middle of the 15 μm CO₂ absorption band the satellite observed only the CO₂ in the stratosphere (there is essentially no water vapor in the stratosphere). We know this because the radiance temperature is higher and the absorption is so strong that the radiance must be coming from higher altitudes closer to the satellite.

Infrared spectrum of Earth observed by a satellite under thunderstorm conditions over the tropical Western Pacific Ocean. The spectrum extends from 6.0 to 25 μm. Dashed lines are the Planck functions for objects at different temperatures from 300 K to 200 K. Remember that clouds are opaque in the infrared and therefore radiate with the Planck distribution function spectral irradiance.

Credit: W. Brune (data from NOAA Star Center for Satellite Applications and Research)

Let’s put all of this together.

• Water vapor, carbon dioxide, and ozone have banded absorption in the thermal infrared due to vibrational–rotational transitions governed by quantum mechanical rules.
• As the absorption decreases, the emissivity decreases. Thus, weakly absorbing gases are also weakly emitting at the same wavelength.
• By looking at different wavelengths either inside, outside, or near absorption bands, a satellite can detect radiation emitted at different heights within the atmosphere.
• In the middle of an absorption band, where the absorption is greatest, the optical path is also the greatest; at these wavelengths a satellite detects only the emissions from the nearest (and highest) layers because the lower ones produce radiation that is absorbed before reaching the satellite.
• In wavelength “windows” between absorption bands, the absorption is small so that the satellite can detect radiation emitted all the way down to the Earth’s surface.
• On the edges of absorption bands, for which the absorption is weak but still significant, satellites detect radiation emitted from the middle troposphere but not from the surface .
• The total radiance is strongly dependent on temperature:
   
  $I_s=\sigma T^4$
  
   
• Thus, if a satellite detects only radiation emitted from the upper troposphere as a result of strong absorption below, its recorded radiance will correspond to temperatures of the upper troposphere (~200–220 K).
• If the satellite detects radiation emitted all the way down to Earth’s surface, then it will record a radiance with temperatures approaching that of the Earth’s surface.
• The water vapor (6.7 μm) channel contains wavelengths at which water vapor absorption is fairly strong, and so it records radiation emitted from the middle troposphere but not below because there is always enough water vapor to absorb irradiance emitted by Earth's surface or the water vapor near Earth's surface. Because the distribution of water vapor is highly variable in time, horizontal position, and vertical position, satellites detect radiance originating from different depths in the atmosphere at different times and places. Basically, with a knowledge of temperature profiles and recorded radiances across the water vapor channel, the optical depths that result from water vapor can be retrieved and the relative humidities determined.

As I said earlier, by observing the CO₂ radiance at different wavelengths, the satellite can be sampling CO₂ radiance from different altitudes (see figure below). The top panel is the radiance from 12 to 18 μm centered on the strong 15 μm CO₂ absorption band. Look at the wavelengths marked 1 through 4. The bottom left panel in the figure shows the absorptivity from the top of the atmosphere to a given pressure level as a function of pressure level at these four wavelengths. Note that for the most strongly absorbed wavelength, 1, the radiance of all the CO₂ and H₂O below a pressure level of about 150 hPa is completely absorbed. Thus, very little of the radiation received by the satellite comes from below this pressure level. On the other hand, very little of the radiance received from the satellite comes from above the 0.1 hPa pressure level because the absorptivity (and hence emissivity) there is zero. Thus, the radiance reaching space must primarily come from between the 150 and 0.1 hPa pressure levels. The panel on the lower right shows the relative contribution of each pressure level to the radiance that reaches space. For wavelength 1, we see that almost all radiance comes from the stratosphere (between about 100 and 1 hPa).

Look at equation 7.6 to see that the absorption of lower layers is exponential so that there are no sharp layers that emit radiance at each wavelength, but instead, the radiance the satellite observes at any wavelength comes from a band that has soft edges. If we look at the wavelength at 2, 3, and 4, we see that the CO₂ and H₂O radiance comes from further down in the atmosphere. For wavelength 4, the satellite is observing radiance from Earth's surface as well as from the CO₂ and H₂O below about 500 hPa, whereas for the wavelength marked 3, the radiance is only slightly from Earth's surface and mostly from CO₂ and H₂O in the middle troposphere.

Top panel: Infrared spectrum of Earth observed by a satellite between about 12 and 18 μm. Numbers 1 through 4 indicate wavelengths of decreasing absorption (i.e., 1 indicates a strongly absorbing wavelength and 4 indicates a weakly absorbing wavelength). Bottom left panel: Absorptivity from the top of the atmosphere to a given pressure level as a function of pressure level (hPa) for the four wavelengths indicated in the top panel. Bottom right panel: The weighting function for each of the four wavelengths, which gives the relative contribution of each pressure level to the radiance that makes it to space.

Credit: W. Brune (data from NOAA Star Center for Satellite Applications and Research)

Summary and Final Tasks

Two applications of the theory of atmospheric radiation have been presented. The most important concepts used are:

• everything radiates
• solar visible irradiance strikes Earth on one side, but Earth radiates in the infrared in all directions
• the total energy for solar visible radiation absorbed in the Earth system closely balances the total energy for the infrared radiation going out to space
• the atmosphere is highly transparent in the visible and weakly transparent in the infrared.

For climate, these principles mean that water vapor, carbon dioxide, and other gases radiate energy to Earth’s surface, keeping it warmer than it would be if the atmosphere did not have these gases. This downward infrared radiation is the greenhouse effect, a natural phenomenon that has been enhanced by human activity, mainly fossil fuel burning. Clouds contribute to the greenhouse effect, but they actually do more cooling by reflecting visible radiation from the sun back to space.

For satellite infrared observations, some wavelength bands are in windows, so that the satellites see radiation from Earth’s surface. Other bands are completely absorbed by water vapor or carbon dioxide, so that the infrared getting to the satellite comes from the top of the water vapor column. Clouds are opaque in the infrared, so the satellite sees their tops, which are radiating at the temperature of that altitude.