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common derivations
==================
#. time dependent from time independent variable:
.. math::
\forall t: x(t,:) = x(:)
#. total uncertainty from random and systematic uncertainty:
=================== ================================== ========================= =======================================
symbol description unit variable name
=================== ================================== ========================= =======================================
:math:`\mu_{x}` total uncertainty for a variable x same as that of :math:`x` `<variable>_uncertainty {:}`
:math:`\mu^{r}_{x}` uncertainty due to random effects same as that of :math:`x` `<variable>_uncertainty_random {:}`
for a variable x
:math:`\mu^{s}_{x}` uncertainty due to systematic same as that of :math:`x` `<variable>_uncertainty_systematic {:}`
effects for a variable x
=================== ================================== ========================= =======================================
The pattern `:` for the first dimensions can represent any combination of dimensions for which `x {:}` exists.
.. math::
\mu_{x} = \sqrt{{\mu^{r}_{x}}^2 + {\mu^{s}_{x}}^2}
#. total uncertainty from vertical covariance:
================== ================================== =========================== =============================================
symbol description unit variable name
================== ================================== =========================== =============================================
:math:`S_{x}(i,j)` covariance for a variable x square of that of :math:`x` `<variable>_covariance {:,vertical,vertical}`
:math:`\mu_{x}(i)` total uncertainty for a variable x same as that of :math:`x` `<variable>_uncertainty {:,vertical}`
================== ================================== =========================== =============================================
The pattern `:` for the dimensions can represent any combination of dimensions for which `x {:,vertical}` exists.
.. math::
\forall i: \mu_{x}(i) = \sqrt{S(i,i)}
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