Given aerosol optical depth (AOD), Angstrom alpha and precipitable water (AKA total column water vapor) calculate Linke turbidity factor using Molineaux (1998), Bird-Hulstrom (1980), Berk (1996) and Kasten (1980).

1. calculate Angstrom turbidity alpha exponent if not known, from AOD at two wavelengths, lambda1 and lambda2:

    alpha0 = -log(aod1 / aod2) / log(lambda1 / lambda2)
   

   Example with lambda1 = 1240nm and lambda2 = 550nm

    alpha0 = -log(aod1240nm / aod550nm) / log(1240 / 550)
   

2. get aod at 700[nm] from alpha

    aod700 = aod550* ((700 / 550) ^ (-alpha0))
   

3. From numerically integrated spectral simulations done with Modtran (Berk, 1996), Molineaux (1998) obtained for the broadband optical depth of a clean and dry atmopshere (fictious atmosphere that comprises only the effects of Rayleigh scattering and absorption by the atmosphere gases other than the water vapor) the following expression where M is airmass.

    delta_cda = - 0.101 + 0.235 * M ^ (-0.16)
   

4. The broadband water vapor optical depth where pwat is the precipitable water vapor content of the atmosphere in [cm]. The precision of these fits is better than 1% when compared with Modtran simulations in the range 1 < M < 6 and 0 < w < 5 cm.

    delta_w = 0.112 * M ^ (-0.55) * pwat ^ (0.34);
   

5. Aerosol

   either using (Molineaux 1998)

    delta_a = aod700
   

   or using (Bird-Hulstrom 1980)

    delta_a = 0.27583*aod380 + 0.35*aod500
   

6. Using the Kasten pyrheliometric formula (1980), the Linke turbidity at M

        TL = -(9.4 + 0.9*M) * log(exp(-M * (delta_cda + delta_w + delta_a))) / M
   

This derivation was developed in collaboration with Armel Oumbe @aoumbe

references:

1. A new airmass independent formulation for the Linke turbidity coefficient, Ineichen, Perez
2. Equivalence of pyrheliometric and monochromatic aerosol optical depth at a single key wavelength, Mlineaux, Ineichen and O'Neill