Nonlocal independent pixel approximation: direct and inverse problems

The independent pixel approximation (IPA), which treats radiative properties of each pixel independently by using standard plane-parallel calculations preserves scale-invariance found in the analyses of the horizontal variability of liquid water in marine stratocumulus clouds. Several studies, however, report a violation of scale-invariance in LANDSAT cloud radiance fields that are much smoother than cloud structure on small scales. This shows a limitation of IPA on small scales: it is unable to simulate the smooth small-scale behavior that is due to the horizontal photon transport. This paper introduces a "nonlocal" independent pixel approximation (NIPA) that extends the IPA by incorporating empirically the smoothing effects of horizontal interpixel fluxes through a convolution product of the IPA and an approximate Green function for radiative transfer. The authors also address the inverse problem of cloud optical depth retrieval from satellite data, showing how NIPA can be used to overcome the limitations of current IPA-based methods at small scales.