A Fast and Accurate Two Orders of Scattering Model to Account for Polarization in Trace Gas Retrievals From Satellite Measurements

We calculate the reflection matrix for the first two orders of scattering in a vertically inhomogeneous, scattering-absorbing medium. We take full account of polarization, and perform a complete linearization (analytic differentiation) of the reflection matrix with respect to both the inherent optical properties of the medium and the surface reflection condition. Further, we compute a scalar-vector correction to the total intensity due to the effect of polarization; this correction is also fully linearized. The intensity correction is meant to be combined with a scalar intensity calculation (with all orders of scattering included) to approximate the intensity with polarization effects included. An approximate spherical treatment is given for the solar and viewing beam attenuation, enabling accurate computations for the range of viewing geometries encountered in practical radiative transfer applications. Sensitivity studies have been performed to evaluate the errors resulting from the use of our second-order scattering model to compute polarization in simulations of backscatter measurements of spectral bands by space-based instruments such as that on the Orbiting Carbon Observatory (OCO). It was found that the errors in the top of the atmosphere (TOA) radiance were less than 0.1% in most cases. The computation time was two orders of magnitude less than that for an exact vector computation. A linear error analysis study of simulated measurements from the OCO absorption bands shows that errors due to the two-order scattering approximation are much lower than the smoothing and measurement noise errors. This is in contrast to the observation that the retrieval error budget may be dominated by polarization if a scalar approximation to the total intensity is used. The second-order model is also an order of magnitude faster than a full multiple scattering scalar radiative transfer computation, making it feasible to be implemented in operational retrieval algorithms as an adjunct model to deal with polarization effects.