Calculate reflectance from material properties and thickness

Question

I read about climate change and the Idealized greenhouse model. Albedo is crucial concept in this model. The albedo varies widely for different materials [2]. I got interested in cloud albedo -- especially, how the thickness of the cloud changes the albedo. Unfortunately, the linked Wikipedia article does not contain any quantitative formulas.

From a laboratory class I remembered the Beer-Lambert law and how it relates the attenuation of light to the material itself, the length the light travels through the material and the concentration. I thought of the existence of a similar law that relates material properties and thickness to reflectance -- which could be used to find the albedo. I could not find such a law.

Clouds are highly irregular, so I thought about a different setup. Imagine a rectangular cuboid filled with fog -- which I hope is a good approximation to a cloud -- and shine a light beam on one side. Experience with fog tells me

• the reflectance increases with thickness for a "given type of fog".

I tried to come up with a quantitative formula, and thougt: Slice the cuboid perpendicular to the beam of light to get many slices of thickness $dx$. Each slice reflects an amount of light $\alpha dx$, absorbs light $\beta dx$ and transmits light $\gamma dx$. ($\alpha + \beta + \gamma = 1$). The first slice is kinda easy. It reflects, absorbs and transmits. The second slice does the same. But it reflects to the back to the first slice, which in turn reflects back some amount. I would be drawing a lot of possible paths the fractions of the original beam might take. I knew that there are eventually only 3 possible options for the fractions of the light beam: reflection, absorption, transmission. Nevertheless, I was not able to write down a definitive formula. Is my approach flawed?

I think that this problem is very similar the following, which I suppose is probably esaier since one does not have to worry about droplet size: Imagine a cuboid of glass (assumed to be completely transparent) in which there is a certain concentration of titanium dioxide (used as white paint). How to calculate the diffuse reflectance, diffuse transmissivity and absorptance of this setup depending on the thickness of the cuboid and the concentration of titanium dioxide? I do not care about the angle at which the light is coming back. I'd be glad at any help.

Answer 1

You have jumped into a very complicated and difficult branch of research. There's a reason that people to this day use the all-Fortran progam MODTRAN . (there are some tools such as PCModWin which simplifies the user interface compared with MODTRAN's requirement to read text files written in exact Hollerith form). MODTRAN includes a few shittons of reference data tables to help the user set up specific atmospheres, specific materials, etc.

Answer 2

On a fundamental level, clouds are white because of Mie scattering. If you do programming in Python, there is a module called miepython, which might be interesting for you to play around a bit:

https://miepython.readthedocs.io/en/latest/

There is a small tutorial, the whole thing is easy to use. You can calculate extinction, scattering and backscattering. Just do it for different wavelengths as shown in the tutorial. Inputs are droplet size (just try different ones) and complex refractive index of water.

Might be an interesting task, but as @Carl Witthoft pointed out correctly, the whole thing is very complex. This suggestion here is just for the curious mind wanting to know what happens on the microphysical scale for single scattering events... if you are interested in climate change, go for sophisticated models, as Carl Witthoft suggested.