Quantification of proxy records obtained from geological archives is key for extending the observational record to estimate the rate, strength, and impact of past climate changes but also for validating climate model simulations, improving future climate predictions. SCUBIDO (Simulating Climate Using Bayesian Inference with proxy Data Observations) is a new statistical model for reconstructing palaeoclimate variability and its uncertainty using Bayesian inference on multivariate non-biological proxy data. We have developed the model for annually laminated (varved) lake sediments, as they provide a high temporal resolution to reconstructions with precise chronologies. This model uses non-destructive X-ray fluorescence core scanning (XRF-CS) data (chemical elemental composition of the sediments) because it can provide multivariate proxy information at a near-continuous, sub-millimetre resolution, and, when applied to annually laminated (varved) lake sediments or sediments with high accumulation rates, the reconstructions can be of an annual resolution. Whilst this model has been built for this proxy type, its flexibility means that the model could be applied to other multivariate proxy datasets.

SCUBIDO uses a calibration period of instrumental climate data and overlapping µXRF-CS data to learn about the direct relationship between each geochemical element (reflecting different depositional processes) and climate but also the covariant response between the elements and climate. The understanding of these relationships is then applied to the rest of the record to transform the proxy values into a posterior distribution of palaeoclimate with quantified uncertainties. In this paper, we describe the mathematical details of this Bayesian approach and show detailed walk-through examples that reconstruct Holocene annual mean temperature from two varved lake records from central England and southern Finland. We choose to use varved sediments to demonstrate this approach, as SCUBIDO does not include a chronological module; thus the tight chronology associated with varved sediments is important. The out-of-sample validation for both sites shows a good agreement between the reconstructed and instrumental temperatures, emphasising the validity of this approach. The mathematical details and code have been synthesised into the R package, SCUBIDO, for simplification and to encourage others to use this modelling approach and produce their own reconstructions. Whilst the model has been designed and tested on varved sediments, µXRF-CS data from other types of sediment records that record a climate signal could also benefit from this approach.