An improved pseudo spherical shell algorithm for vector radiative transfer

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The radiative transfer solution in a plane-parallel geometry is a good approximation for many applications in the Earth or other planetary systems as the Earth’s radius is quite large (∼6371 km). The plane-parallel geometry is however problematic in polar regions where the solar zenith angle is usually large (>60∘) and the spherical shell effect is significant. One simple solution is the so-called pseudo-spherical shell (PSS) approximation, which treats the solar beam attenuation exactly along the nadir in the spherical shell atmosphere while keeping the plane parallel geometry for multiple scattering calculation. The PSS approximation improves the solution for intermediately large solar zenith angles, though the error is still large for large viewing zenith angles. In order to further improve the treatment of a spherical shell geometry, we have developed an improved pseudo spherical shell (IPSS) approximation. In the method, we used the following techniques: I.) The single scattering solution is solved exactly for the spherical shell atmosphere; II.) The multiple to single scattering solution ratio is solved using the plane-parallel geometry with our radiative transfer code based on successive order of scattering method; III.) The ratio of the multiple to single scattering solution is assumed to be the same for both the plane parallel and spherical shell geometry. We tested the performance of IPSS with two benchmark cases involving the Rayleigh scattering matrix. If the Rayleigh optical thickness is 0.25, the error is smaller than 1% for most of the viewing directions (<70∘). If the Rayleigh optical thickness is 1.0, the error is bounded within ±2%. The error does not show obvious dependence on the viewing zenith angle. Our newly developed IPSS scheme is highly accurate and can be used in the remote sensing applications of the polar regions.