# Calculate transparency from Beer-Lambert Law

I want to calculate a transparency parameter for a superposition of several volumes containing media using the Beer-Lambert Law, which states that

$$I/I_0=exp(-\tau)$$

where $I$ is the transmitted intensity, $I_0$ is the incident radiation intensity, and $\tau$ is the optical depth of the medium.

My question here is: Can I replace $\tau$ with $\tau_1 + \tau_2 + ... + \tau_n$ when dealing with n media of same/different optical depths? Does it matter whether the volumes overlap (assuming that they do not interact with each other)?

If the light passes first through medium $\tau_1$, then through medium $\tau_2$, the optical depths add up to $\tau = \tau_1 + \tau_2$ , as you can see from applying the Beer-Lambert-Formula twice.
The optical depth comprises the factors $\tau = \sigma N l$, the cross section $\sigma$ of the absorbers (molecules, atoms, ..) [unit m$^2$], the density of the absorbers $N$ [m$^{-3}$], and the optical path $l$ [m]. For example, if two media ($\sigma_1, N_1, \sigma_2, N_2$) overlap within the optical path, the effect of this absorbers just adds up, $\sigma N = \sigma_1 N_1+ \sigma_2 N_2$, where the product $\sigma N$ is the inverse absorption length.