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

All operators (e.g. analytical data) are well represented by normal distributions.
We are considering uncertainties of independent variables (no covariance).
The uncertainty of the result is small enough to be well represented by a normal distribution. This means that the result is roughly linear in the area of the uncertainty and therefore the resulting distribution is not asymmetrical.

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 of common operations

\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}

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 σ_t 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 σ_C and σ_P 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, R_i is the initial ratio, λ₁₀ and λ₂₆ are the radioactive decay constants, and σ_R and σ_R_i 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/R_i>0.8) or too old samples (R/R_i<0.2). Therefore, in this case, it is highly recommendable to calculate the extreme values of t (t+σ_t and t–σ_t) 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+σ_t and t–σ_t) 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 σ_RSampleand σ_N20 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 R_Sample>>R_Air. Also, for samples where R_Sample values are close to R_Air, 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.

Operation	Formula	Uncertainty
General	$f(a,b)$	$\sqrt{\left( \sigma_a \frac{\delta f(a,b)}{\delta a}\right) ^2+\left( \sigma_b \frac{\delta f(a,b)}{\delta b}\right)^2}$
Addition	$a+b$	$\sqrt{\sigma_a^2+\sigma_b^2}$
Subtraction	$a-b$	$\sqrt{\sigma_a^2+\sigma_b^2}$
Multiplication	$a \cdot b$	$\sqrt{\left( \sigma_a \cdot b\right) ^2+\left( \sigma_b \cdot a\right) ^2}$
Division	$a/b$	$\sqrt{\left( \frac{\sigma_a}{b}\right) ^2+\left( \sigma_b \cdot \frac{a}{b^2} \right) ^2}$
Power	$a^b$	$\sqrt{\left( \frac{\sigma_a \cdot a^{b} \cdot b}{a}\right) ^2+\left( \sigma_b \cdot a^b \cdot \ln{a} \right) ^2}$
Natural logarithm	$a \cdot \ln(b)$	$\sqrt{\left( \sigma_a \cdot \ln(b)\right) ^2+\left( \sigma_b \cdot \frac{a}{b} \right) ^2}$
Logarithm to base 10	$a \cdot \log_{10}(b)$	$\sqrt{\left( \sigma_a \cdot \log_{10}(b) \right) ^2+\left( \sigma_b \cdot \frac{a}{b \cdot \ln(10)} \right) ^2}$

Theory

Examples of common operations

Examples applied to cosmogenic isotope data

Calculating a 10Be age

26Al/10Be burial age

Calculating a 21Ne concentration

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Calculating a ¹⁰Be age

²⁶Al/¹⁰Be burial age

Calculating a ²¹Ne concentration