I have two rough surfaces A and B, each with standard height deviation $\sigma _a$ and $\sigma _b$ and mean height $z_a$ and $z_b$.
Now my question is:
How can I find a distance distribution between the two surfaces ?
I know that both surfaces can be modeled as Gaussian:
$$h(z) = \frac{1}{{\sigma \sqrt {2\pi } }}e^{{{ - \left( {z- \mu } \right)^2 } \mathord{\left/ {\vphantom {{ - \left( {z - \mu } \right)^2 } {2\sigma ^2 }}} \right.-} {2\sigma ^2 }}}$$
So to find the distance between the surfaces, can I just sum them, like so ?
$$h_a(z) + h_b(z)$$
Let the two profiles be denoted by $y_a(i)$ and $y_b(i)$. As they both come from a Gaussian distribution, the difference is also Gaussian (e.g., see wikipedia). Thus, if $y_a(i) \sim N(z_a, \sigma_a^2)$ and $y_b(i) \sim N(z_b, \sigma_b^2)$ with $$ N(z,\sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(y-z)^2}{2\sigma^2}}, $$ then the difference is $(y_a(i)-y_b(i)) \sim N(z_a-z_b, \sigma_a^2+\sigma_b^2)$.