In a non-homogeneous medium, these parameters can vary with altitude and location along the path, formally making these terms n(s), σλ(s), T(s), and Iλ(s).
Since emission can occur in all directions, atmospheric radiative transfer (like Planck's Law) requires units involving a solid angle, such as W/sr/m2.
The same phenomena makes the absorptivity of incoming radiation less than 1 and equal to emissivity (Kirchhoff's law).
The vibrational and rotational excited states of greenhouse gases that emit thermal infrared radiation are in LTE up to about 60 km.
[7] Radiative transfer calculations show negligible change (0.2%) due to absorption and emission above about 50 km.
When radiation is scattered (the phenomena that makes the sky appear blue) or when the fraction of molecules in an excited state is not determined by the Boltzmann distribution (and LTE doesn't exist), more complicated equations are required.
When these parameters are first measured with a radiosonde, the observed spectrum of the downward flux of thermal infrared (DLR) agrees closely with calculations and varies dramatically with location.
[9][10] Where dI is negative, absorption is greater than emission, and net effect is to locally warm the atmosphere.
By repeated approximation, Schwarzschild's equation can be used to calculate the equilibrium temperature change caused by an increase in GHGs, but only in the upper atmosphere where heat transport by convection is unimportant.
Schwarzschild's equation can be derived from Kirchhoff's law of thermal radiation, which states that absorptivity must equal emissivity at a given wavelength.
Setting absorptivity equal to emissivity affords: The total change in radiation, dI, passing through the slab is given by: Schwarzschild's equation has also been derived from Einstein coefficients by assuming a Maxwell–Boltzmann distribution of energy between a ground and excited state (LTE).
The absorption cross-section (σλ) is empirically determined from this oscillator strength and the broadening of the absorption/emission line by collisions, the Doppler effect and the uncertainty principle.
This is usually the case when working with a laboratory spectrophotometer, where the sample is near 300 K and the light source is a filament at several thousand K. If the medium is homogeneous, nσλ doesn't vary with location.
Integration over a path of length s affords the form of Beer's Law used most often in the laboratory experiments: 'If no other fluxes change, the law of conservation of energy demands that the Earth warm (from one steady state to another) until balance is restored between inward and outward fluxes.
Schwarzschild's equation is used to calculate the outward radiative flux from the Earth (measured in W/m2 perpendicular to the surface) at any altitude, especially the "top of the atmosphere" or TOA.
These increments are numerically integrated from the surface to the TOA to give the flux of thermal infrared to space, commonly referred to as outgoing long-wavelength radiation (OLR).
The net downward radiative flux of thermal IR (DLR) produced by emission from GHGs in the atmosphere is obtained by integrating dI from the TOA (where I0 is zero) to the surface.
In "line-by-line" methods, the change in spectral intensity (dIλ, W/sr/m2/μm) is numerically integrated using a wavelength increment small enough (less than 1 nm) to accurately describe the shape of each absorption line.
The HITRAN database contains the parameters needed to describe 7.4 million absorption lines for 47 GHGs and 120 isotopologues.
A variety of programs or radiative transfer codes can be used to process this data, including an online facility, SpectralCalc.
[15] To reduce the computational demand, weather forecast and climate models use broad-band methods that handle many lines as a single "band".
Then the upward and downward intensities are integrated over a forward hemisphere, a process that can be simplified by using a "diffusivity factor" or "average effective zenith angle" of 53°.
Schwarzschild's equation provides a simple explanation for the existence of the greenhouse effect and demonstrates that it requires a non-zero lapse rate.
[19] Rising air in the atmosphere expands and cools as the pressure on it falls, producing a negative temperature gradient in the Earth's troposphere.
According to Schwarzschild's equation, the rate of fall in outward intensity is proportional to the density of GHGs (n) in the atmosphere and their absorption cross-sections (σλ).
Both observations and calculations show a slight "negative greenhouse effect" – more radiation emitted from the TOA than the surface.
[22] In the absence of thermal emission, wavelengths that are strongly absorbed by GHGs can be significantly attenuated within 10 m in the lower atmosphere.
This has led some to falsely believe that Schwarzschild's equation predicts no radiative forcing at wavelengths where absorption is "saturated".
This fallacy involves reaching erroneous conclusions by focusing on energy exchange near the planetary surface rather than at the top of the atmosphere (TOA).
[24]: 413 The radiative forcing from doubling carbon dioxide occurs mostly on the flanks of the strongest absorption band.