How to sum uncertainties, systematic and random I apologize for the simplistic questions. I have a retrieval process that has a set of random and systematic uncertainties associated with it. I'm assuming that these are all independent. The goal is to find the total error for this process. I have two questions related to this:
1) To find the total systematic uncertainty can I sum all the systematic uncertainty components in quadrature? Assuming that they are independent, it is unlikely that they will all contribute in the same direction and it seems to make sense to add them in quadrature. However, by definition they are not random. Does this mean they must be added linearly?
2) To find the total uncertainty (total_random + total_systematic), can the total_random and total_systematic be added in quadrature or must they be added linearly?
Thank you in advance for you help!
 A: 
1) To find the total systematic uncertainty can I sum all the systematic uncertainty components in quadrature? Assuming that they are independent, it is unlikely that they will all contribute in the same direction and it seems to make sense to add them in quadrature. However, by definition they are not random. Does this mean they must be added linearly?

I think what you mean here is that all the systematic uncertainty components are functions of the same underlying systematic uncertainty. Mathematically, there is some underlying systematic uncertainty random variable $S$, and each systematic component is some constant, or weight, $s_i$ times $S$. The $i$th system component can then be expressed as follows.
$$ S_i = s_iS$$
Without loss of generality, let the variance of $S$ be 1. If the variance of each systematic component is $v_i$, then $s_i=\sqrt{v_i}$, and the total systematic variance is given by
\begin{align}
\text{Var}\left(\sum_i S_i\right)
&= \text{Var}\left( \sum s_i S\right)\\
&= \left( \sum_i s_i \right)^2 \text{Var}(S)\\
&= \left( \sum_i s_i \right)^2\\
&= \left( \sum_i \sqrt{v_i} \right)^2
\end{align}
If your uncertainties had been specified in terms of standard deviations, then $s_i$ would be the standard deviation of each component and the answer would be simply be $\sum_i s_i$, i.e. they must be added linearly.

2) To find the total uncertainty (total_random + total_systematic), can the total_random and total_systematic be added in quadrature or must they be added linearly?

They are independent, and for any two random variables $X,Y$ that are independent, the variance of the sum is the sum of the variances, i.e. $\text{Var}(X+Y)=\text{Var}(X) + \text{Var}(Y)$. So if the total random and total system uncertainties are given as variances, you simply add them together. (Assuming that individual random uncertainties are independent of one another, the same applies to forming the total random uncertainty from individual random uncertainties.)
If your uncertainties are specified in terms of standard deviations, then you need to add in quadrature.
