Propagating data uncertainty through smoothing spline fits

Smoothing splines have been used extensively in the analysis of gas concentration data from bubbles in ice cores. Since the fit is a linear projection of the data, propagation of data uncertainty through the fitting process is formally straightforward and, as we demonstrate, readily achievable from pre-existing spline-fitting procedures. The uncertainty propagation can be extended to determining both uncertainties in derivatives and uncertainties in quantities that reflect rates of input to the atmosphere. As an example, we apply the technique to 1000 yr of methane data from a Law Dome ice core.