Calculating the Ångstrom turbidity coefficient and other quantities related to aerosol optical thickness

Optical thickness is a measure of the amount of direct sunlight reaching a detector that responds (theoretically) to a single wavelength of light. (In practice, all detectors respond to a range of wavelengths.) Optical depth is another commonly used name for the same measure -- these two terms are interchangeable.

Optical thickness (or optical depth) is affected by molecular (Rayleigh) scattering, gaseous absorption, and absorption or (mostly) scattering by aerosols. The portion of optical thickness due to aerosols is called aerosol optical thickness or aerosol optical depth, τ.

τ can easily be related to percent transmission of direct sunlight, T, which may be conceptually easier to understand than τ itself:

(1)             T = 100•exp(-τ)

Wavelength, optical thickness, and atmospheric turbidity (haziness) are related through Ångstrom's turbidity formula:

(2)             τ = β•λ^-α

where β is Ångstrom's turbidity coefficient, λ is wavelength in microns, and α is the Ångstrom exponent. α and β are independent of wavelength, and can be used to describe the size distribution of aerosol particles and the general haziness of the atmosphere. For two different wavelengths,

(3a)            τ₁ = β•λ₁^-α
                  τ₂ = β•λ₂^-α

from which

(3b)            τ₁/(λ₁^-α) = τ₂/(λ₂^-α)

Solving for α:

(3c)             α = ln(τ₁/τ₂)/ln(λ₂/λ₁)

A typical range for α is 0.5-2.5, with an average for natural atmospheres of around 1.3±0.5. Larger values of α, when the τ value for the larger wavelength is much smaller than the τ value for the smaller wavelength, imply a relatively high ratio of small particles to large (r > 0.5 µ) particles. As τ for the larger wavelength approaches the τ for the smaller wavelength, larger particles dominate the distribution and α gets smaller. It is not physically reasonable for the τ value of the larger wavelength to equal or exceed the τ value of the smaller wavelength.

Now calculate β from either wavelength:

(4)             β = τ₁•λ₁^α = τ₂•λ₂^α

where λ must be expressed in microns (500 nm = 0.500 μ). β values of less than 0.1 are associated with a relatively clear atmosphere, and values greater than 0.2 are associated with a relatively hazy atmosphere.

Given τ at two different wavelengths, the τ at a third wavelength can be inferred for the same atmospheric conditions. Rewrite (4) and solve for τ₃ using either the first or second wavelength:

(5)             ln(λ₃/λ₁)α = ln(τ₁/τ₃) = ln(τ₁) - ln(τ₃)
                  ln(τ₃) = ln(τ₁) - ln(λ₃/λ₁)α
                  τ₃ = exp[ln(τ₁) - ln(λ₃/λ₁)α]

This calculation is useful when τ values determined with one instrument must be compared to values from another instrument that uses different wavelengths.

Here's a worked-out example for wavelengths used in the GLOBE sun photometer:

λ₁ = 505 nm, λ₂ = 625 nm
τ₁ = 0.185, τ₂ = 0.155
α = ln(0.185/0.155)/ln(625/505) = 0.8299
Using the first wavelength, β = 0.185•0.505^0.8299 = 0.1049
Find τ for a wavelength of 550 nm:
τ₅₅₀ = exp[ln(0.185) - ln(550/505)•0.8299] = 0.1723

(author unknown) WMO Background Air Pollution MONitoring (BAPMON) Network Information Manual, TD-9789, September, 1990.

Iqbal, Muhammad. An Introduction to Solar Radiation. Academic Press, Toronto, 1983.