Modeling zenith luminance on Madrid partly cloudy skies from diffuse illuminance on a horizontal surface and solar altitude

Abstract

The present work deals with zenith luminance estimation for Madrid partly cloudy skies (cloud cover from 1 to 7 oktas) from solar altitude and diffuse illuminance on a horizontal surface. Two different approaches are followed, as in our recent work for cloudless skies. The first approach has been to compare the experimental zenith luminance values with the corresponding values estimated using the equations for the five standards for partly cloudy skies recently proposed by Kittler et al. The dependence of zenith luminance on solar altitude is similar for both the best fit of the experimental data with a 5th degree polynomial and the best fit of the estimated values obtained with the mentioned sky standards. The zenith luminance vs. solar altitude curve for sky standard IV.3 lies closest to the best fit of the experimental data. Next, a multiple linear correlation based on the equation for sky standard IV.3 is developed, and excellent agreement is found with the best fit of experimental data. Three empirical coefficients and the two constants given in the equation for sky standard IV.3 are needed in this approach, and the equation of the model is not a simple one. The second approach has been to try different fits of the experimental data, looking for a simple equation to be used as a model. As in our recent investigation for the cloudless skies case, it is observed that a simple inverse equation relating the ratio between zenith luminance and diffuse illuminance to the cosine of the solar altitude, containing only two empirically determined coefficients, gives an adequate fit to the experimental data. From this equation, Lz can be estimated if diffuse illuminance on a horizontal surface and solar altitude are known with approximately the same values for the statistical estimators as if the multiple linear correlation obtained with the first approach is used. As a consequence, the proposed model is given by the simple inverse equation. © 2003 Published by Elsevier Ltd.