common derivations

  1. time dependent from time independent variable

    \[\forall t: x(t,:) = x(:)\]
  2. total uncertainty from random and systematic uncertainty

    symbol

    description

    unit

    variable name

    \(\mu_{x}\)

    total uncertainty for a variable x

    same as that of \(x\)

    <variable>_uncertainty {:}

    \(\mu^{r}_{x}\)

    uncertainty due to random effects for a variable x

    same as that of \(x\)

    <variable>_uncertainty_random {:}

    \(\mu^{s}_{x}\)

    uncertainty due to systematic effects for a variable x

    same as that of \(x\)

    <variable>_uncertainty_systematic {:}

    The pattern : for the first dimensions can represent any combination of dimensions for which x {:} exists.

    \[\mu_{x} = \sqrt{{\mu^{r}_{x}}^2 + {\mu^{s}_{x}}^2}\]
  3. total uncertainty from vertical covariance

    symbol

    description

    unit

    variable name

    \(S_{x}(i,j)\)

    covariance for a variable x

    square of that of \(x\)

    <variable>_covariance {:,vertical,vertical}

    \(\mu_{x}(i)\)

    total uncertainty for a variable x

    same as that of \(x\)

    <variable>_uncertainty {:,vertical}

    The pattern : for the dimensions can represent any combination of dimensions for which x {:,vertical} exists.

    \[\forall i: \mu_{x}(i) = \sqrt{S(i,i)}\]