Functional forms for approximating the relative optical air mass

Abstract

[1] This article constitutes a review and systematic comparison of functional forms for approximating the air mass from the zenith to the horizon. Among them, we find the most meaningful forms in atmospheric optics, geophysics, meteorology, and solar energy science, as well as several forms arising from the study of the atmospheric delay of electromagnetic signals, whose relationship with the air mass was recently proved by the authors. In total, we have compared 26 functional forms, and the fits have been done for three atmospheric profiles, an observer at sea level, and the median wavelength of the Sun's spectral irradiance (0.7274 mm). As a result, the best of the uniparametric forms has more than three centuries of history; the best of the biparametric forms was recently introduced by one of the authors; the best of the tri-and tetraparametric forms were originally proposed for modeling the atmospheric delay of radio signals; and the best of the forms with more than four parameters is used here for the first time. On the basis of these, for the 1976 U.S. Standard Atmosphere (USSA-76), we provide one-, two-, three-, four-, and five-parameter formulas whose maximum deviations are 1.70, 2.91 Â 10 À1 , 3.28 Â 10 À2 , 2.49 Â 10 À3 , and 3.24 Â 10 À4 , respectively.Citation: Rapp-Arrarás, Í., and J. M. Domingo-Santos (2011), Functional forms for approximating the relative optical air mass,

Global horizontal irradiance clear sky models : implementation and analysis.

“…Many formulas have been developed by various authors to fit measured data as a function of zenith angle. Rapp-Arraras and Domingo-Santos present a good overview of 26 different functional forms of published air mass models in [74]. Some of the most commonly used air mass models for solar radiation modeling are highlighted below.…”

Accurate analytic approximation for the Chapman grazing incidence function

“…For numerical calculations used in this article we adopted and extended the original Fortran code by Huestis ( 2001). The calculation of special functions was performed using the algorithms according to Press et al (1992), which were adapted for double precision calculations. The Fortran code (Fortran 90) for calculation of Chapman function and its analytic approximations in different forms is available along its Python wrapper (Python 3.8) at web address https:// gitlab.…”

Halo ratio from ground-based all-sky imaging

“…The fall-off of brightness for increasing z, associated with vignetting, takes place in nearly every digital photograph, in all optical lens systems, in particular wide-angle and ultrawide-angle lenses like fisheye lenses (Jacobs and Wilson, 2007). Optical and natural vignetting are associated with a smaller lens opening for obliquely incident light and the cos 4 law of illumination falloff, respectively, both inherent to any lens design (Ray, 2002). In general, vignetting increases with the aperture and decreases with the focal length.…”

“…Sky imaging finds application in determining fractional cloud cover (Johnson and Hering, 1987;Long and DeLuisi, 1998;Slater et al, 2001;Long et al, 2001;Berger et al, 2005;Kassianov et al, 2005;Cazorla et al, 2008b) and macrophysical cloud properties such as cloud brokenness, distribution, number and uniformity (Shields et al, 1997;Kegelmeyer, 1994;Long et al, 2006), in assessing the impact of cloud cover on surface solar irradiance (Pfister et al, 2003), in estimating cloud base height, either from lowcost digital consumer cameras (Seiz et al, 2002;Janeiro et al, 2010) or by means of paired whole sky cameras (Lyons, 1971;Rocks, 1987;Allmen and Kegelmeyer Jr., 1996), in cloud detection and classification (Calbó and Sabburg, 2008;Heinle et al, 2010;Ghonima et al, 2012), in short-term weather forecasting (Chow et al, 2011), in characterizing aerosol (Cazorla et al, 2008a) and in determining cloud-free lines of sight Lund, 1972, 1973;Lund, 1973;Lund et al, 1980). Sky imaging is not new in the field of cirrus investigation as testified by previous works on measurements of the 22 • halo intensity from photographic photometry (Lynch et al, 1985), on the effects of ice crystal structure on halo formation (Sassen et al, 1994) and on the characterization of cirrus through a combination of polarization lidar and photographic observations of cirrus optical displays (Sassen et al, 2003).…”