The extreme variability of cloud and rain fields poses serious problems in quantitative use of remotely sensed satellite and radar data. We show how to characterize this variability using scale invariant (sensor resolution independent) codimension functions which are exponents characterizing the probability distributions. These codimension functions in turn form a three parameter universality class. We review the properties of these multifractal measures and empirically evaluate the codimension functions as well as the universality classes for infrared and visible satellite cloud images using the new probability distribution/multiple scaling technique, refining previously published results and relating these to the established lognormal rain and cloud phenomenologies. We then show how to solve the radar observers' problem for multifractal radar reflectivity factors and to estimate the codimension function of rain from the radar. Finally, we reexamine some earlier (monofractal) analysis techniques in the light of our findings.