I have observed a vector of quantities $\vec y$. I wish to use these to constrain a vector of initial conditions $\vec x$ that are related to $\vec y$ through a non-linear (numerically evaluated) function $f$, i.e. I want to find $$\hat x = \underset{\vec x}{\arg\min} \; \left[ f(\vec x) - \vec y \right]^2.$$ However, $\vec y$ has been measured with uncertainties $\vec \sigma$. Therefore, I should minimize something like a reduced $\chi^2$: $$\hat x = \underset{\vec x}{\arg\min} \; \sum {\frac{\left[ f(\vec x) - \vec y \right]^2}{\sigma^2}}.$$ However, I fear the following two situations:
$f$ is nonlinear, so two elements of $f(\vec x)$ could provide identical information, and then my minimization would be biased towards that redundant information. The values $\vec y$ have been measured with covariance, but this covariance only accounts for the measurement error and does not account for the covariance in the model $f$.
If one of the variables is measured much more precisely than the other variables, then it will effectively be the only thing being fit by the minimization procedure.
Are there any ways around 1 and 2?