Estimation of Spectral Power Laws in Time-Uncertain Series of Data with Application to the GISP2 \(δ^{18}\)O Record

Errors in the timing assigned to observations degrade estimates of the power spectrum in a complicated and nonlocal fashion. It is clear that timing errors will smear concentrations of spectral energy across a wide band of frequencies, leading to uncertainties in the analysis of spectral peaks. Less understood is the influence of timing errors upon the background continuum. We find that power law distributions of spectral energy are largely insensitive to errors in timing at frequencies much smaller than the Nyquist frequency, though timing errors do increase the uncertainty associated with estimates of power law scaling exponents. These results are illustrated analytically and through Monte Carlo simulation and are applied in the context of evaluating the power law behavior of oxygen isotopes obtained from Greenland ice cores. Age errors in layer counted ice cores are modeled as a discrete and monotonic random walk that includes the possibility of biases toward under- or overcounting. The δ\(^{18}O_{ice}\) record from the Greenland Ice Sheet Project 2 is found to follow a power law of \(1.40 \pm 0.19\) for periods between 0.7 and 50 ky, and equivalent results are also obtained for other Greenland ice cores.