  Motivated by a recent observation on the self-organized criticality of
cumulus clouds (Phys. Rev E 103, 052106, 2021) we study their connection to
self-similar rough surfaces, in which $f\equiv \log I$ plays the role of the
main field, where $I$ is the intensity of the received visible light. By
simulating the light scattering based on a coarse-grained phenomenological
model in a two-dimensional cloud, we argue the possible connection of $I$ to
the actual cloud thickness. Although in the vertical incident light $f$ is
proportional to the cloud thickness, in the general case it is complected. We
study the statistical properties of observational data for $f$ with a focus on
the conventional exponents of this scale-invariant rough surface. By
calculating the roughness exponents, and comparing them with other exponents
like the fractal dimension of loops, the distribution function of the radius of
gyration and loop lengths, and the exponent of the green function, we prove
that this surface is unconventional in the sense that it is the non-Gaussian
self-affine random surface which violates the Kondev hyper-scaling relations.
