# Plane-parallel versus pseudo-spherical versus spherical

In texts about radiative transfer one often encounters expressions stating that “plane-parallel geometry is assumed”, or sometimes, even “pseudo-spherical” geometry is assumed. The statement “plane-parallel geometry” is just another way of saying that the Earth is assumed to be flat. For most modern observers that is not correct. Still, flat Earth, or plane-parallel, geometry is often used when solving radiative transfer problems. The main reason is computational simplicity. And, the approximation works for rather large solar zenith angles. Before continuing with some examples, please note that what follows below may not be directly applicable to your problem. Hence, make your own tests and choose your approach thereafter.

The radiation reaching the Earth's surface may be decomposed into a direct part and a diffuse part. The figure below shows the ratio of the plane-parellel direct irradiance versus the spherical direct irradiance (dotted lines) for two wavelengths in the UV. The solid curves are pseudo-spherical/spherical direct irradiance ratios. For the shortest wavelength the plane-parallel approximation underestimates the direct irradiance by 2% at a solar zenith angle of 60°. Depending on wavelength and application the pseudo-spherical approximation may be acceptable for the direct irradiance up to about 80°.

For the total (=direct+diffuse) irradiance the similar ratios are shown in the figure below. When the diffuse irradiation is included both the plane-parallel approximation and especially the pseudo-spherical approximation, works well for larger solar zenith angles than the direct irradiance alone. At large solar zenith angles most of the radiation is diffuse. The diffuse radiation is driven by the direct radiation (remember the direct beam source term in the transfer equation for the diffuse radiation field). Hence, it is a little surprising that the total irradiance works for larger solar zenith angles than the direct irradiance. However, as the sun sets, most of the scattering of the direct to diffuse radiation takes place at higher altitudes where the effect of sphericity is smaller. Thus, a reasonable approach for large solar zenith angles is to treat the direct beam with the pseudo-spherical approximation and solve the diffuse radiative transfer equation as for the plane-parallel situation, but with the direct beam source replaced with its pseudo-spherical equivalent. This is the so-called pseudo-spherical approximation.

user_area/plane-parellel_versus_pseudo-spherical_versus_spherical.txt · Last modified: 2008/10/27 13:36 by arve 