A framework based on 2-D Taylor expansion for quantifying the impacts of subpixel reflectance variance and covariance on cloud optical thickness and effective radius retrievals based on the bispectral method

The bispectral method retrieves cloud optical thickness (t) and cloud droplet effective radius (re) simultaneously from a pair of cloud reflectance observations, one in a visible or near-infrared (VIS/NIR) band and the other in a shortwave infrared (SWIR) band. A cloudy pixel is usually assumed to be horizontally homogeneous in the retrieval. Ignoring subpixel variations of cloud reflectances can lead to a significant bias in the retrieved t and r_e. In the literature, the retrievals of t and r_e are often assumed to be independent and considered separately when investigating the impact of subpixel cloud reflectance variations on the bispectral method. As a result, the impact on t is contributed only by the subpixel variation of VIS/NIR band reflectance and the impact on r_e only by the subpixel variation of SWIR band reflectance. In our new framework, we use the Taylor expansion of a two-variable function to understand and quantify the impacts of subpixel variances of VIS/NIR and SWIR cloud reflectances and their covariance on the t and r_e retrievals. This framework takes into account the fact that the retrievals are determined by both VIS/NIR and SWIR band observations in a mutually dependent way. In comparison with previous studies, it provides a more comprehensive understanding of how subpixel cloud reflectance variations impact the t and r_e retrievals based on the bispectral method. In particular, our framework provides a mathematical explanation of how the subpixel variation in VIS/NIR band influences the r_e retrieval and why it can sometimes outweigh the influence of variations in the SWIR band and dominate the error in r_e retrievals, leading to a potential contribution of positive bias to the r_e retrieval. We test our framework using synthetic cloud fields from a large-eddy simulation and real observations from Moderate Resolution Imaging Spectroradiometer. The predicted results based on our framework agree very well with the numerical simulations. Our framework can be used to estimate the retrieval uncertainty from subpixel reflectance variations in operational satellite cloud products and to help understand the differences in t and r_e retrievals between two instruments.