Assume you are given a matrix U with $u_{i,j}$ being the value of U at location $i,j$ and $\delta u_{i,j}$ being the uncertainty of U at $i,j$ (uncertainty implying noise in the estimate). If I perform the following simple operation (for example) at each $i,j$ location, what would be the output uncertainty.
$$ o_{i,j} = \frac{(u_{i,j} - u_{i+1, j} + u_{i-1,j} - u_{i,j-1} + u_{i, j+1})}{k} $$
where $o_{i,j}$ is the output O at $i,j$ with $k$ being some constant.
I have the following the questions.
If I wish to perform uncertainty estimation on the output $\delta o_{i,j}$, how to perform for such an operation.
Also, if I wish to do proper uncertainties where I assume I know the input variance-covariance matrix which implies that I have input covariances between $u_{i,j}$ and $u_{i+1,j}$, $u_{i,j}$ and $u_{i,j+1}$ and so on.., then how does the output value vary at $\delta o_{i,j}$
How does the error propagate in both cases.
If by uncertainty you mean a variance then you can use the Sum of correlated variables rule (plus the scaling rule): $${\rm VAR}(o_{i,j}) = \frac{1}{k^2} \left({\rm VAR}(u_{i,j}) +{\rm VAR}(u_{i+1,j})+{\rm VAR}(u_{i-1,j})+{\rm VAR}(u_{i,j+1})+{\rm VAR}(u_{i,j}-1) - {\rm COV}(u_{i,j},u_{i+1,j}) + {\rm COV}(u_{i,j},u_{i-1,j}))- {\rm COV}(u_{i,j},u_{i,j-1}) + {\rm COV}(u_{i,j},u_{i,j+1})) \pm ...\right),$$
where the covariance terms go between all pairs and they carry the sign of the product terms (while all variance terms are positive).