deletions | additions       diff --git a/section_Water_Vapor_Calibration_The__.tex b/section_Water_Vapor_Calibration_The__.tex  index 84a931a..017b115 100644  --- a/section_Water_Vapor_Calibration_The__.tex  +++ b/section_Water_Vapor_Calibration_The__.tex 
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The direct isotopologue calibration determines the instrument response function for each isotopologue used to calculate isotope ratios. This technique has the advantage that there is no need to charcterise both the VSMOW calibration and humidity-isotope response functions, as these are explicitly accounted for. Breifly, the instrument response function is characterised by supplying the instrument with known mixing ratios of each isotopologue (\textit{X} and \textit{X_i}) that covers the range expected to be observed. By then applying the determined calibration functions the corrected isotope ratios are calculated:  \[\delta=\bigg[\frac{\chi_i}{\chi}-1\bigg]\permil \]   $\permil$  \[\delta - \delta_0\ = \bigg[\frac{\alpha\epsilon}{1-\alpha(1-\epsilon)}-1\bigg] \times ln\bigg[\frac{\alpha l_0 + q - \alpha(1-\epsilon)(q - q_0)}{\alpha l_0 + q_0}\bigg]\]