Langley Extrapolation

Langley extrapolation is a method for measuring the Sun's radiance with ground-based instrumentation, thereby removing the effect of the atmosphere. It is based on repeated measurements with a sun photometer operated at a given location for a cloudless morning or afternoon, as the Sun moves across the sky. It is named for American astronomer and physicist Samuel Pierpont Langley.

It is known from Beer's law that, for every instantaneous measurement, the direct-Sun radiance I is linked to the solar extraterrestrial radiance I₀ and the atmospheric optical depth τ by the following equation:

where m is a geometrical factor accounting for the slant path through the atmosphere, known as the airmass factor. For a plane-parallel atmosphere, the airmass factor is simple to determine if one knows the solar zenith angle θ: m = 1/cos(θ). As time passes, the Sun moves across the sky, and therefore and vary according to known astronomical laws.

By taking the logarithm of the above equation, one obtains

and if one assumes that the atmospheric disturbance τ does not change during the observations (which last for a morning or an afternoon), the plot of ln I versus m is a straight line. Then, by linear extrapolation to m = 0, one obtains I₀, i.e. the Sun's radiance that would be observed by an instrument placed above the atmosphere.

The requirement for good Langley plots is a constant atmosphere (constant τ). This requirement can be fulfilled only under particular conditions, since the atmosphere is continuously changing. Needed conditions are in particular: the absence of clouds along the optical path, and the absence of variations in the atmospheric aerosol layer. Since aerosols tend to be more concentrated at low altitude, Langley extrapolation is often performed at high mountain sites. Data from NASA Glenn Research Center indicates that the Langley plot accuracy is improved if the data is taken above the tropopause.