I want to write a model for estimating incoming solar radiation for a specific latitude on earth but I am struggling to find an appropriate source which shows the required equations for doing so. Would anyone be able to provide me with a link to where I can find equations for estimating solar radiation (irradiance) given a specific cloud cover, latitude, time of day, and day of year?

• Are you only interested in the direct irradiance or do you also want model scattered light? Are you interested in a specific wavelength region (or spectral irradiance) or the total irradiance in the full spectrum? – jkej Mar 26 '13 at 0:05
• @Kate This is of interest to people studying and manufacturing photovoltaic solar cells. They talk about solar radiation at 1A (one atmosphere), 1.5A (one an a half atmospheres) etc. The following link can be a good starting point, but you might have to download the whole article in order to read it. mendeley.com/catalog/… – JKL Mar 26 '13 at 0:50
• Take a look at the SMARTS atmospheric model and associated documentation, it might have the clues you need. nrel.gov/rredc/smarts – boyfarrell Mar 26 '13 at 3:27
• @jkej I am interested in direct irradiance, where the mainaim of the model will be to estimate short wave radiation with wavelengths in the visible (VIS), near-ultraviolet (UV), and near-infrared (NIR) spectra. – KatyB Mar 26 '13 at 8:31

Ok, I'm still not sure on what level you want to do this, but I will start you off with some basics. The most important factor is probably the solar elevation angle, $\theta$. As described on the wiki-page it can be calculated using this formula:

$$\sin\theta=\cos h\cos\delta\cos\Phi+\sin\delta\sin\Phi$$

where $h$ is the hour angle, $\delta$ is the solar declination and $\Phi$ is the latitude. The trickiest to calculate of these is the solar declination. A few different formulas to calculate is can be found here. Which formula you use will depend on the accuarcy you need. I suggest starting with this formula:

$$\delta=-\arcsin(0.39789\cos(0.98565(N+10)+1.914\sin(0.98565(N-2))))$$

where $N$ is the day of year beginning with $N=0$ at 00:00:00 UTC on January 1 (prefereably calulate $N$ as a decimal number to increase accuracy). Note that this formula uses degree-based trigonometric functions.

Now, if we totally ignore atmosperic effects, total solar irradiance (of all wavelengths) incident on a horizontal surface will be:

$$E=A\sin\theta$$

where $A$ is the solar constant which approximatley has the value 1360 W/m$^2$ (on average, it varies by roughly 7% over the year due to the ellipticity of Earth's orbit).

Since this ignores atmospheric effects, the actual irradiance on the ground will be lower due to scattering and absorption. These effects will also depend on the solar elevation angle, since a lower angle gives a longer light path through the atmosphere.

Maybe, starting from this, you can explain what further aspects you need to model.

If you are interested just in the direct irradiance you can neglect the emmision and scattering terms in the Radiative Transfer Equation RTE wich can in this case be simplified to allow only for absorbtion and is known under the name Beer-Bougert-Lambert's law of absorption

$$\cos\theta_{0}\,\frac{\partial}{\partial p}S^{i} = -\frac{\kappa^i}{g}\,S^{i}$$

Here I assumed pressure coordinates to parameterize the height above ground, $i$ is the index of the band if you have more than one, and $S$ is the solar flux in Wm$^{-2}$. The zenith angle $\theta_0(\lambda,\phi,t)$ varies with longitude, latitude, and time of the year which can be taken into account, by

$$\cos\theta_0 = \cos\phi \cos\delta\cos h + \sin\phi \sin\delta$$

where $\delta$ is the declination angle and $h$ the hour angle.

The Absorption coefficient con be calculated for example as the mixing ratio $\rho_a/\rho$ of the absorber times the band strength $K^i$

$$\kappa^i = \frac{\rho_a}{\rho}K^i$$

The only thing left to do for a simple parameterization of the solar radiation is to specify the upper boundery condition for each band you want to consider, for example

$$\begin{eqnarray} S^1(p=0) & = & C_{sun}\,\:UV_{O3}\,\cos \theta_{0}\\ S^2(p=0) & = & C_{sun}\, \:VIS_{O3}\,\cos \theta_{0}\nonumber \\ S^3(p=0) & = & C_{sun}\,\:VIS_{H2O}\, \cos \theta_{0} \nonumber \\ S^4(p=0) & = & C_{sun}\, \:UV_{O2}\,\cos \theta_{0} \nonumber \\ S^5 & = & C_{sun}\,\cos \theta_{0} \, - \,\sum\limits_{i=1}^4 S^i (p=0) \nonumber \end{eqnarray}$$

Where $C_{sun}$ is the solar "constant" of about 1362 Wm$^{-2}$, $UV_{O3}$ for example is the relative part of the incoming solar energy that goes into the ozone UV absorber band, and the fifth band contains the radiation that is directly transmitted to the surface. With this simple parameterization you can calculate the solar radiation at the surface by integration the absorption law downward.

More comprehensive methods to solve the radiation problem in the shortwave regime include scattering on aerosols, clouds, and air molecules and allow for reflection at the surface too. Non-LTE (most important in for the longwave regime in the middle atmosphere for example) in the SW regime can be established by considering appropriate efficiencies for the conversion of absorbed solar radiation to kinetic energy of the atmospheric constituents. In the approach of discrete ordinates, the appropriate RTE is solved for different discrete zenith directions. The principle of invariance and the adding method apply some kind of ray tracing for the incoming solar beam inside an atmospheric layer in order to calculate the emerging radiation at the upper and lower boundary of the layer.

These more comprehensive methods are explained for example here where you can find a more detailled explanation and treatment of both, longwave and shortwave radiative transfer in the atmosphere.

• Good explanation of how absorption can be treated. However, can scattering terms really be neglegted? The fact that the sun is red at sunset is a clear indicator that scattering is important, at least for low sun angles. Also, your link to explain the calculation of zenith angle goes to the wiki-page on the solar azimuth angle. – jkej Mar 26 '13 at 12:55
• @jkej yes thanks, I updated my answer a little bit. – Dilaton Mar 26 '13 at 14:47

Heinrich Morf, A stochastic solar irradiance model adjusted on the Ångström–Prescott regression, Solar Energy, Volume 87, January 2013, Pages 1-21, ISSN 0038-092X, 10.1016/j.solener.2012.10.005. Keywords: Ångström–Prescott regression; Cloud enhancement; Photovoltaic fleet; Solar irradiance; Stochastic modeling

Heinrich Morf, The Stochastic Two-State Cloud Cover Model STSCCM, Solar Energy, Volume 85, Issue 5, May 2011, Pages 985-999, ISSN 0038-092X, 10.1016/j.solener.2011.02.015. Keywords: Climatology; Cloud cover; Meteorology; Solar energy; Stochastic modeling; Vertical visibility

Heinrich Morf, The Stochastic two-state solar irradiance model (STSIM), Solar Energy, Volume 62, Issue 2, February 1998, Pages 101-112, ISSN 0038-092X, 10.1016/S0038-092X(98)00004-8.

If you need to estimate solar radiation from the cloud cover in Europe you can look at the R-package 'sirad'. It contains a functon su() which implements the Supit-Van Kappel model. The model requires daily cloud cover and daily temperature range for input.

I studied physics in the old days but then had a career in engineering. I think it important to have experimental data to correlate with formulae and hypothesis. Perhaps astronomers studying solar radiation have data over several frequency ranges, times of year, latitudes etc. I think also interesting to know the amount of polarisation of the radiation according to the distance traversed, and its angle, through the atmosphere. IR reflective mirrors could redirect incident radiation vertically so that it would be lost to space with a minimum absorption and reflection by the atmosphere.