Bayesian methods for fitting spectral models to noisy, sparse, proxy data

The background continuum of climate variability recorded in proxy records is often modelled using parametric spectral models, such as power-laws, auto-regressive processes, or stochastic differential equations.However, fitting such models to proxy data is usually done in an ad-hoc manner, such as by using least-squares fitting in log-log space.Here I will discuss two formal Bayesian methods for fitting parametric stochastic models to proxy data. One is a spectral-domain approach based the Whittle likelihood. The other is a time-domain approach based on Gaussian Processes.In both cases, I show how the standard approaches can be modified to account for some of the ways in which climate proxies alter spectral slopes: measurement error, time uncertainty, uneven sampling, and smoothing (e.g. from diffusion or bioturbation). Finally, I use synthetic data generated from power-law and Matern processes, and proxy-system models, to show expected skill of the two approaches for different proxies.I find that these formal approaches provide significant bias reduction relative to typical ad-hoc approaches, allowing for much more accurate calibration of stochastic models of climate variability across scales.