Basic error propagation and cosmogenic isotopes

This is a description on 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 on common operations
OperationFormulaUncertainty
Generalf(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}
Additiona+b \sqrt{\sigma_a^2+\sigma_b^2}
Subtractiona-b \sqrt{\sigma_a^2+\sigma_b^2}
Multiplicationa \cdot b \sqrt{\left( \sigma_a \cdot b\right) ^2+\left( \sigma_b \cdot a\right) ^2}
Divisiona/b \sqrt{\left( \frac{\sigma_a}{b}\right) ^2+\left( \sigma_b \cdot \frac{a}{b^2} \right) ^2}
Powera^b \sqrt{\left( \frac{\sigma_a \cdot a^{b} \cdot b}{a}\right) ^2+\left( \sigma_b \cdot a^b \cdot \ln{a} \right) ^2}
Natural logarithma \cdot \ln(b) \sqrt{\left( \sigma_a \cdot \ln(b)\right) ^2+\left( \sigma_b \cdot \frac{a}{b} \right) ^2}
Logarithm to base 10a \cdot \log_{10}(b) \sqrt{\left( \sigma_a \cdot \log_{10}(b) \right) ^2+\left( \sigma_b \cdot \frac{a}{b \cdot \ln(10)} \right) ^2}
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 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<<P/\lambda, 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 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 were 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.

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