## Beerlambert Law And Optical Depth

Consider the propagation of radiation of wavelength X through a layer of thickness dx perpendicular to a beam of intensity F(X). The extinction of radiation on traversing an infinitesimal pathlength dx is linearly proportional to the amount of matter along the path

where b(x,X) (units of inverse length) is called the extinction coefficient and is proportional to the density of material in the medium. Extinction includes both absorption and scattering, each of which removes photons from the beam. In contrast to absorption, the radiant energy scattered remains in the form of radiation, but its direction is altered from that of the incident radiation. Dividing (4.17) by dx and letting dx —> 0 gives

which is known as the Beer-Lambert law.

Absorption and scattering occur simultaneously because all molecules (and particles) both absorb and scatter. The extinction coefficient b(x,X) is the sum of absorption and scattering:

BEER-LAMBERT LAW AND OPTICAL DEPTH 109 The optical depth (dimensionless) at wavelength X between points x\ and x2 is defined as

From (4.19), the total optical depth is a sum of the optical depth due to absorption and that due to scattering:

The most frequently used form of optical depth is that in which the two points are at different altitudes because one is often interested in how solar radiation is attenuated as it traverses the atmosphere. By definition, x = 0 at the top of the atmosphere (TOA), increasing, iike pressure, monotonically from zero at TOA to its value at any altitude z:

Let us integrate (4.18) from the TOA to an altitude z. Since z is decreasing in the direction of propagation of radiation, we need to add a minus sign on the left-hand side (LHS) of (4.18):

Integrating from ztoa to z and FTOa(X) to F(zX), we obtain

At the altitude at which t(z,x) = 1, the radiative flux is reduced to 1/e of its value at the top of the atmosphere. The transmittance of the atmosphere at wavelength X at height z is defined as the ratio F(z,/-)/^toaP-):

Although, as noted, molecules scatter as well as absorb radiation, our concern at the moment is with molecular absorption. The absorption coefficient for a gas A is proportional to the number of molecules of A per unit volume, nA (molecules cm-3). The absorption coefficient divided by the number density is called the molecule's absorption cross section:

<7a(^.) can be regarded as the effective cross-sectional area of a molecule for absorption of photons, and is a measure of the ability of a molecule to absorb photons. Absorption cross sections vary from zero to about 1 x 10-'6 cm2 molecule-1. A value of the absorption cross section of 10-18-10-17 cm2 molecule-1 is considered to be large.

Therefore, the optical depth of the atmosphere at the surface as a result of absorption by species A is rzroA

The integral in (4.27) presumes that the path of the solar beam is perpendicular to the Earth's surface. This is, of course, the case only when the sun is directly overhead.

The angle measured at the Earth's surface between the Sun and the zenith is called the solar zenith angle; it is denoted by the symbol 0o. When the sun is exactly overhead, 0o =0°; at the horizon 0o = 90°. As 0o increases, the pathlength of the solar beam through the atmosphere increases. If the relative pathlength of the solar beam from the TOA to the surface when the Sun is directly overhead (0O = 0°) is taken as 1.0, then the pathlength m at 0o, neglecting the sphericity of the Earth, is m = ——t— = sec 0O (4.28)

cos 00

The optical depth of the atmosphere increases as the pathlength increases since the solar beam transsects a longer section of the atmosphere, and there is proportionately more opportunity for extinction to occur. Therefore, the more general form of (4.27) is

Equation (4.28) holds for 0O less than about 75°. For larger values of 0o, m has to be computed, taking into account the path through the spherical atmospheric layers, the

TABLE 4.1 Relation between the Slant Path Optical Depth m and sec 80 for a Standard Rayleigh Atmosphere

0o sec 80 m

TABLE 4.1 Relation between the Slant Path Optical Depth m and sec 80 for a Standard Rayleigh Atmosphere

0o sec 80 m

 0 1 1 30 1.15 1.15 60 2 1.99 70 2.92 2.9 75 3.86 3.81 80 5.76 5.59 85 11.47 10.32 87 19.11 15.16 89 57.3 26.26 90 0 38.09

Source: Kasten and Young (1989).

Source: Kasten and Young (1989).

vertical profile of absorbing and scattering species, and the curvature of the optical rays as a result of refraction. Values of m are given in Table 4.1 for the molecular atmosphere, that is, the Rayleigh scattering optical depth.3