where the first term contains scattered diffuse light; the second term contains directly scattered sunlight; and the third term contains thermally emitted radiation. Equation (6.73) in its entirety may not be solved analytically, but must instead be solved numerically. There are a number of ways of doing this and one of the most common is the matrix-operator (or doubling-adding) method. This technique basically applies a Gaussian quadrature technique to integration over zenith angle and the Fourier method to integration over azimuth angle and is well described by Goody and Yung (1989), Hansen and Travis (1974), and Plass et al. (1973). Alternative techniques include discrete ordinates (e.g., Hanel et al., 2003) and successive orders (Hansen and Travis, 1974). However, a particularly useful technique for cases where the scattering optical depth is small is the single-scattering approximation, which assumes that thermal emission is negligible and that the probability of a photon being scattered more than once is so small that only the second term on the right-hand side of Equation (6.73) need be considered. This leads to the directly integrable equation (where we have assumed that the optical depth of the atmosphere varies from zero at the top to infinity at the base)
A further particularly simple approximation to scattering, which is applicable for atmospheres with thin cloud layers, is the so-called reflecting layer approximation. Here, gas absorption spectra are used to calculate the transmission of a path from the Sun to a particular level in the atmosphere and back to the observer. If a thin cloud exists at that pressure level, then the observed spectrum may be approximated by multiplying this transmission by the effective reflectivity of the cloud layer. Several clouds may be approximated by summing the "reflections" from a number of levels and the technique is closely related to the single-scattering approximation.
6.6.2 Spherical atmospheres and limb viewing: Monte Carlo simulations
The scattering equations just derived relate to plane-parallel atmospheres and thus are only applicable when the zenith angles are not too near to 90°. Nearer the limb, and for limb-viewing geometries, the equations are unusable and thus more complicated approaches must be used, of which the most general is the Monte Carlo method.
As the name suggests, the Monte Carlo technique basically "fires" a large number of model photons into an atmosphere and tracks where they go using scattering probability functions and a random number generator. The technique is computationally expensive and slow to converge. However, if enough photon paths are simulated, accuracy is as good as the more conventional techniques, and the technique has the advantage of being able to model any geometry.
From Beer's law of absorption we know that the probability that a photon will pass through a slab of optical thickness r is given simply by exp(—r). Hence, by inversion we know that the optical thickness traveled by a random photon before
absorption or scattering is given simply by r = -log(R) (6.75)
where R is a random number between 0 and 1. From a given starting position and direction we may thus calculate the new position of the photon given this random optical thickness. At the new position, the probability of scattering is simply the single scattering albedo to and thus the photon is scattered if m > R, where R is a new random number between 0 and 1. If the photon is scattered, then the new photon direction may be calculated from the phase function, where the scattering angle 90 is given by
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