Alliance for Lighting Information

Rayleigh Scatter Index (RSI)

by Jefferey F. Knox and David M. Keith

Rayleigh Scatter Index is a simple metric that incorporates the aspects of radiant energy that are fundamental to Rayleigh scatter - how much radiant energy at each wavelength and the value for that wavelength. These two aspects are combined acccording to Lord Rayleigh's equation, to produce a metric for comparative evaluation of the radiation in uplight. With "all other things being equal", the amount of radiant energy S(w) and its wavelength w are what concern us. Including a factor to scale the number to a useful value, we end up with a metric called Rayleigh Scatter Index (RSI), defined as:

Eq: RSI = F * Summation [ S(w) * (1/w)4 ]

This equation shows how to calculate RSI: for each wavelength the amount of radiant energy at that wavelength is multiplied by the value of the wavelength raised to the negative fourth power, and then a summation is made over the range of 360 to 770 nanometers, which corresponds to the visible portion of the electro-magnetic spectrum. When the factor F is set to 5.0E11, the S(w) is scaled to 1 Watt total radiant power over the spectrum and the wavelengths are in nanometers, the resulting RSI values typically are between zero and ten.

The RSI evaluation at each wavelength RSI(w) can be summed over the entire spectrum or used as SPDs of the radiation "reflected" by the sky. In all cases the resulting values are only suitable for comparisons and the absolute values should not be considered as having merit in themselves, because the constant used in these RSI calculations is chosen strictly for convenience and has no other numerical basis. However, using the analysis basis of "all other things being equal", by starting with "equal amounts" and treating them all the same, the values do have merit for comparisons.

As verification of the suitability of RSI for estimating atmospheric scattering, the SPD for a blackbody radiator at 6500K - which closely represents sunlight reaching the atmosphere - was evaluated for its own chromaticity coordinates (x = 31, y=32) and corresponding color. For color comparison purposes, the corresponding color is shown at luminance L = 75 out of 100, which makes that sample look gray - typically the luminance would be so high that any source with that SPD would appear white.

The blackbody SPD was also evaluated using RSI to develop an SPD for the "scattered" radiation, which "should" closely represent daylight. In fact the resuting SPD has significantly more very short (near UV) radiation which is typically absorbed by atmospheric elements. The "scattered" radiation was also evaluated for chromaticity coordinates (x = 23, y=22) and corresponding color. The sample is also shown at luminance L = 75.

The following CIE chromaticity diagram shows the locations of the blackbody and "scattered" radiation.

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