Beer-Lambert 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 

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  

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, like pressure, monotonically from zero at TOA to its value at any altitude z. 

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