Application of small angle approximation in circumsolar irradiance modelling

Abstract

Circumsolar irradiance is the part of diffuse or scattered irradiance within a small angle (2.5° for typical pyrheliometers) around the sun which is measured, together with direct normal irradiance, by pyrheliometers and reported as part of the direct normal irradiance. This could create an inconsistency as much as 300 W/m2 between radiative transfer modelling and pyrheliometer measurements, having further ramifications for the validation of surface solar irradiance data derived from various sources such as sky imaging camera, satellite and numerical weather prediction models. In addition, accurate modelling of the angular pattern of solar irradiance within a very small angle from the centre of the sun is important for solar energy generation using concentrating (thermal or photovoltaic) technologies. This article presents an analytical model for the calculation of circumsolar radiation, based on the ‘small angle approximation’. Validation against full radiative transfer calculation shows that the approximate method is accurate for common pyrheliometers used operationally. The error is less than 0.5 W/m2 for opening half angle of 2.5°, and less than 2.0 W/m2 for opening half angle of 5°, for typical cloud and aerosol types and the full range of optical depths. This small angle approximation method is relatively quick to calculate (when compared to full radiative transfer modelling calculations times) and can underpin enhanced modelling of solar radiation for satellite validation, solar energy generation and numerous other applications across the globe.