How to combine albedos I have estimations of the Earth's surface albedo for a region (0.13), as well as an estimation of the atmosphere albedo (0.3 at a solar zenith angle of 1 rad). My question is, how do I find the combined albedo of that region? Is it as simple as adding them?
 A: The Albedo describes the fraction of radiation reflected for a given area of the surface.


*

*If you are combining albedos for different parts of the surface of a body, you need to weight them by their areas, for example consider albedos $\alpha_1$ and $\alpha_2$ corresponding to areas $A_1$ and $A_2$, then the effective albedo for the total area is:


$$\alpha = \frac{\alpha_1 \cdot A_1 + \alpha_2 \cdot A_2}{A_1 + A_2}$$


*

*If you are trying to combine the effects of different components of the body's surface in the same area (for example including both the ground and clouds above it) then the problem becomes much more complex, but in general you can not simply add them (although in some situations that can be an okay approximation).

A: I've worked out a formula that assumes no absorption in the atmosphere, and that light is only backscattered. It takes into account the backscattering of the light reflected off the surface.
In the diagram below:


*

*TOA is Top Of Atmosphere

*$\tau_a$ and $A_a$ are the transmittance and albedo of the atmosphere such that $\tau_a+A_a=1$

*$A_s$ is the albedo of the surface

*$A_T$ is the combined albedo

*The $F$s represent fluxes, with the blue one being the only one we don't know.

*$F_1$ is the TOA solar radiation

*$F_2$ the solar radiation that gets transmitted through the atmosphere

*$F_3$ is the radiation reflected off the surface

*$F_4$ is the overall radiation reflected from the Earth

Firstly, we have
$$F_2 = \tau_a F_1$$
$$F_3 = A_s F_2$$
$$F_4 = A_T F_1 = A_a F_1 + \tau_a F_3$$
Rearranging the last one we get
$$A_T = A_a + \tau_a \frac{F_3}{F_1}$$
$$= A_a + \tau_a A_s \frac{F_2}{F_1}$$
$$= A_a + \tau_a^2 A_s$$
$$A_T = A_a + (1 - A_a)^2 A_s$$
Evidently, if $A_a$ is large, $A_T \approx A_a$, regardless of the value of $A_s$.
I've plotted this below

N.B. this ignores multiple reflections between the atmosphere and the surface.
