# 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.

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