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

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

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)