Question

It is known from observations that, on average, 30% of incoming solar radiation is reflected back to space, and only about 50% of it reaches the Earth

Answer 1

From law of conservation of energy:

$a_\lambda I_\lambda =I_\lambda -r_\lambda I_\lambda-\tau_\lambda I_\lambda$

$a_\lambda+r_\lambda+\tau_\lambda =1$

This is shows a process of absorption, reflection, and transmission account for all incident radiation.

When $a_\lambda = 0.5I$ and $r_\lambda = 0.3I$

Thus, $\tau_\lambda =1-a_\lambda-r_\lambda = 1-0.5-0.3 = 0.2$ .

The relation between transmittance is related to optical depth and to absorbance as,

$T=e^{-\tau}$

Therefore, the optical depth is

$T=e^{-\tau} =e^{-0.2}= 0.82$