A Fractal Toy Model of Opaque Cumulus Clouds for Information Content Analysis in Optical Tomography, and Two Complementary Random Walk Processes Therein

The cloudy pixel acquisition rate by NASA's MODIS sensors mandates operational cloud property retrievals one pixel at a time, using just two wavelengths to infer two quantities: optical thickness & effective particle radius. In spite of clear evidence in the images, the assumed cloud geometry is a uniform plane-parallel slab, hence 1D radiative transfer (RT) modeling. Some stratus look somewhat like that model, so operational retrievals can work for them, maybe, at least far enough from their edges. But, although key to weather and climate, convectively-driven 3D cumulus are all but forsaken by the mono-pixel/mono-view bi-spectral scheme. In contrast, computed optical cloud tomography (COCT) uses multi-angle images with multi-spectral or multi-polarimetric content, and is adapted specifically to vertically-developed 3D clouds. COCT is still in its infancy. After a successful demo with synthetic (LES + 3D RT) and real (AirMSPI) airborne data at high spatial resolution (~20 m pixels and 3D gridcells), we are exploring new ways of implementing COCT for space-based sensors with ~300 m pixels, where there is sub-pixel variability and potentially high opacity across the pixel. Both of these real-world facts of life violate core assumptions in the numerical 3D RT model at the heart of the current COCT algorithm.

To properly revamp COCT for space-based multi-angle sensor systems, both present (MISR+MODIS/Terra) and future (MAIA, PACE, AOS), we designed a 2D toy model to easily see the RT effects of cloud boundaries and internal variability. Looking down, clouds are fractals with a dimension of ≈4/3, consistent with scalars in turbulence (Lovejoy, 1982). Looking horizontally, clouds are shaped by multi-scale convective updrafts, so we adopt a Koch curve (Df = log4/log3 ≈ 1.26) for the outer shape. Internal turbulence is simulated with lognormal 2D fractional Brownian motion (5/3 spectral exponent). Finally, to mimic microphysics "101" in a cloud-scale convective updraft, we overlay a gradient adjusted to parcel-theory's prediction for extinction. With this model in hand, we uncovered two random-walk processes that are key to advancing COCT: a directional diffusion in the "outer shell," and a classic spatial diffusion inside the cloud's "veiled core" (where there is very little structural information to recover with COCT).