# Basic error propagation and cosmogenic isotopes

This is a description of how we usually calculate the uncertainties of concentrations and ages and some practical examples on cosmonuclide analysis.

##### Theory

Considering that

the error propagation should be performed by considering the partial derivatives of the result respect the operators:

$\sigma_{f(a,b)} = \sqrt{\left( \sigma_a \frac{\delta f(a,b)}{\delta a}\right) ^2+\left( \sigma_b \frac{\delta f(a,b)}{\delta b}\right) ^2}$

where $a \pm \sigma_a$ and $b \pm \sigma_b$ are the operators within a standard deviation (one sigma) uncertainties.

##### Examples applied to cosmogenic isotope data
###### Calculating a 10Be age

The apparent 10Be age t of a surface sample with a 10Be concentration of C is

$t = \frac{-\ln\left(1-\frac{C \cdot \lambda}{P}\right)}{\lambda}$

where P is the production rate and λ is the radioactive decay constant. According to the general equation, the uncertainty of the apparent age σt should be calculated as

$\sigma_t = \sqrt{\left( \frac{\sigma_C}{P-C \cdot \lambda} \right) ^2 + \left( \frac{\sigma_{P} \cdot C}{P^2-P \cdot C \cdot \lambda} \right) ^2}$

where σC and σP are the uncertainties of the concentration and the production rate, respectively. However, as $t \simeq C/P$ for $C<, the following approximation is correct for most of the typically calculated exposure ages:

$\sigma_t = t \cdot \sqrt{\left( \frac{\sigma_C}{C} \right) ^2 + \left( \frac{\sigma_{P}}{P} \right) ^2}$

###### 26Al/10Be burial age

If we calculate a 26Al/10Be burial age t as

$t = \frac{\ln(\frac{R}{R_i})}{\lambda_{10}-\lambda_{26}}$

where R is the 26Al/10Be ratio, Ri is the initial ratio, λ10 and λ26 are the radioactive decay constants, and σR and σRi are the considered independent one-sigma uncertainties, the error of the burial age should be:

$\sigma_t = \sqrt{\left( \frac{\sigma_R}{R \cdot (\lambda_{10}-\lambda_{26})} \right) ^2 + \left( \frac{\sigma_{Ri}}{R_i \cdot (\lambda_{10}-\lambda_{26})} \right) ^2}$

In this case, this equation is often not realistic because the previous equation does not fit the third assumption (asymmetrical error) for typical R uncertainties from too young (R/Ri>0.8) or too old samples (R/Ri<0.2). Therefore, in this case, it is highly recommendable to calculate the extreme values of t (t+σt and tσt) by varying the burial equation as follows:

$\frac{\ln(\frac{R+\sigma_R}{R_i})}{\lambda_{10}-\lambda_{26}} \quad ; \quad \frac{\ln(\frac{R-\sigma_R}{R_i})}{\lambda_{10}-\lambda_{26}} \quad ; \quad \frac{\ln(\frac{R}{R_i+\sigma_{Ri}})}{\lambda_{10}-\lambda_{26}} \quad ; \quad \frac{\ln(\frac{R}{R_i-\sigma_{Ri}})}{\lambda_{10}-\lambda_{26}}$

to check the asymmetry and, if necessary, calculate the positive and negative errors of t (t+σt and tσt) from the maximum and minimum values obtained from the last equations.

###### Calculating a 21Ne concentration

If we consider that a cosmogenic 21Ne concentration is calculated as:

$[^{21}Ne] = \frac{\left( R_{Sample}-R_{Air} \right) \cdot N_{20}}{M}$

where R are the measured 21Ne/20Ne ratios, N20 is the total number of 20Ne atoms, M is the mass of the sample and we are considering σRSample and σN20 as independent errors, the uncertainty of the 21Ne concentration should be

$\sigma_{[^{21}Ne]} = \sqrt{\left( \frac{\sigma_{RSample} \cdot N_{20}}{M} \right) ^2 + \left( \frac{\sigma_{N20} \cdot (R_{Sample}-R_{Air})}{M} \right) ^2}$

that can be also expressed as

$\sigma_{[^{21}Ne]} = [^{21}Ne] \cdot \sqrt{\left( \frac{\sigma_{RSample}}{R_{Sample}-R_{Air}} \right) ^2 + \left( \frac{\sigma_{N20}}{N_{20}} \right) ^2 }$

and it is always larger than

${[^{21}Ne] \cdot \sqrt{\left( \frac{\sigma_{RSample}}{R_{Sample}} \right) ^2 + \left( \frac{\sigma_{N20}}{N_{20}} \right) ^2 }} \quad \neq \quad \sigma_{[^{21}Ne]}$

which is not a realistic way of calculating the uncertainty of [21Ne] unless RSample>>RAir. Also, for samples where RSample values are close to RAir, the uncertainty of [21Ne] might be asymmetrical, and it is highly recommendable to check the shape of the [21Ne] uncertainty as explained in the burial section.