Description: Understanding the uncertainties associated with proxy-based reconstructions of past climate is critical if they are to be used to validate climate models and contribute to a comprehensive understanding of the climate system. Here we present two related and complementary approaches to quantifying proxy uncertainty. The proxy forward model (PFM) “sedproxy” bitbucket.org/ecus/sedproxy numerically simulates the creation, archiving and observation of marine sediment archived proxies such as Mg/Ca in foraminiferal shells and the alkenone unsaturation index UK’37. It includes the effects of bioturbation, bias due to seasonality in the rate of proxy creation, aliasing of the seasonal temperature cycle into lower frequencies, and error due to cleaning, processing and measurement of samples. Numerical PFMs have the advantage of being very flexible, allowing many processes to be modelled and assessed for their importance. However, as more and more proxy-climate data become available, their use in advanced data products necessitates rapid estimates of uncertainties for both the raw reconstructions, and their smoothed/derived products, where individual measurements have been aggregated to coarser time scales or time-slices. To address this, we derive closed-form expressions for power spectral density of the various error sources. The power spectra describe both the magnitude and autocorrelation structure of the error, allowing timescale dependent proxy uncertainty to be estimated from a small number of parameters describing the nature of the proxy, and some simple assumptions about the variance of the true climate signal. We demonstrate and compare both approaches for time-series of the last millennia, Holocene, and the deglaciation. While the numerical forward model can create pseudoproxy records driven by climate model simulations, the analytical model of proxy error allows for a comprehensive exploration of parameter space and mapping of climate signal re-constructability, conditional on the climate and sampling conditions.

Description: Marine sedimentary records are a key archive when reconstructing past climate; however, mixing at the seabed (bioturbation) can strongly influence climate records, especially when sedimentation rates are low. By commingling the climate signal from different time periods, bioturbation both smooths climate records, by damping fast climate variations, and creates noise when measurements are made on samples containing small numbers of individual proxy carriers, such as foraminifera. Bioturbation also influences radiocarbon-based age-depth models, as sample ages may not represent the true ages of the sediment layers from which they were picked. While these effects were first described several decades ago, the advent of ultra-small-sample $^{14}$C dating now allows samples containing very small numbers of foraminifera to be measured, thus enabling us to directly measure the age-heterogeneity of sediment for the first time. Here, we use radiocarbon dates measured on replicated samples of 3-30 foraminifera to estimate age-heterogeneity for five marine sediment cores with sedimentation rates ranging from 2-30 cm kyr$^{-1}$. From their age-heterogeneities and sedimentation rates we infer mixing depths of 10-20 cm for our core sites. Our results show that when accounting for age-heterogeneity, the true error of radiocarbon dating can be several times larger than the reported measurement. We present estimates of this uncertainty as a function of sedimentation rate and the number of individuals per radiocarbon date. A better understanding of this uncertainty will help us to optimise radiocarbon measurements, construct age models with appropriate uncertainties and better interpret marine paleo records.