The power incident on a lidar receiver as the result of scattering from an atmosphere containing particulate matter is given by:
where the optical depth, (r), is defined as:
Laser pulse energy, .
Speed of light, .
Collecting area of the receiver, .
Range to the scattering volume, .
Aerosol scattering cross section per unit volume, .
Scattering cross section per unit volume for air molecules, .
Extinction cross section per unit volume, .
Backscatter phase function,
This equation illustrates the difficulty of measuring optical properties with a standard lidar; the instrument provides a single measurement of power at each range which is dependent on both the scattering cross section at that range and the two-way attenuation on the path to the scattering volume. As a result the signal contains insufficient information to separately determine the scattering cross section and the optical depth. Much effort has been applied to solving for the extinction profile by making assumptions regarding relationships between the backscatter cross section and the extinction cross section(for example see: Klett, Applied Optics, Vol 20, pp 211). Unfortunately, both direct measurements of the extinction to backscatter ratio and calculations from Mie theory show that these relationships are highly variable. The relationship between extinction and backscatter is dependent on the size distribution of the scattering particles. Since size distributions are highly variable, the extinction cross sections derived using these techniques are unreliable.