Methods

Introduction

Cosmogenic nuclides are typically used to either constrain an exposure age, a burial age, or an erosion rate. Constraining the landscape history and past erosion rates in previously glaciated terrains is, however, notoriously difficult because it involves a large number of unknowns. This tool uses an approach based on the Markov Chain Monte Carlo (MCMC) technique. The model framework currently incorporates any combination of the following terrestrial cosmogenic nuclides (TCNs) 10Be, 26Al, 14C, and 21Ne in order to constrain a two-stage glacial/interglacial history at the site of sampling.

The MCMC technique is used to simulate TCN concentrations associated with a large number of different glacial-interglacial histories, including highly varying glacial and interglacial erosion rates. Based on comparisons to measured concentrations, it is possible to determine the most likely landscape history and associated uncertainties.

The model approach approximates the values and uncertainties of four output parameters; interglacial erosion rate (εint), glacial erosion rate (εgla), time of last deglaciation (tdegla), and the climate threshold value in the marine oxygen-isotope record (δ18Othreshold) at the site of sampling.

In the following we give a basic overview of the applied methods and their application. For a full description see the open-access publication by Knudsen et al. (2015).

Markov-Chain Monte Carlo (MCMC) basics

The inversion problem of turning observed TCN concentrations into erosion histories is handled using a conventional Metropolis-Hastings MCMC approach. The model parameters are constrained between fixed model parameter bounds specified by the user. Erosion rates (εint, εgla), which may vary over several orders of magnitude, are tested with uniform probability across the logarithmic parameter interval. The temporal parameter (tdegla) and climate record threshold value (δ18Othreshold) are tested with uniform probability across the linear parameter interval.

When model parameters (εint, εgla, tdegla, δ18Othreshold) are varied within specified limits, they can be thought of as being orthogonal axes spanning a coordinate system in four-dimensional space. Each position in this model space is associated with a unique set of model parameter values.

Given a single value of model parameters (εint, εgla, tdegla, δ18Othreshold) within the specified limits, the TCN concentration after the duration of e.g. the entire Quaternary period in a sample can be computed. This forward model describes a possible history of exhumation and TCN production in a sample volume as it experiences the variable physical environment of the Quaternary.

Two-stage glacial-interglacial forward model

The forward model builds on the assumption of "two-stage uniformitarianism", meaning that the processes that operated during the Holocene also operated during earlier interglacials with comparable intensity. Likewise, the erosion rate during the past glacial periods is assumed to be comparable.

The model approach assumes that glacial periods were characterized by 100% shielding and no exposure, which would require more than 10 m of ice thickness for production due to spallation (>50 m for muons). Interglacial periods are assumed to have been characterized by 100% exposure and zero shielding. The production of TCNs takes place during the interglacials, while erosion removes the land surface at different rates during the glacials and interglacials.

The forward model switches between glacial and interglacial state when the selected climate record crosses a threshold value. The provided climate records are based on a benthic δ18O record, smoothed by various degrees, implying that climate at the site of sample is correlated to the global state.

What is a MCMC walker?

A MCMC walker is in this context a numerical entity which sequentially explores the model parameter space in order to obtain the closest match between the forward model and the observational dataset of TCNs. During each iteration the walker takes its current position in model space, plugs the parameter value into the forward model, and evaluates if the output result matches the observational record better or worse than the output at its previous position in model space. If the new results better matches the observed dataset, it continues walking in the same direction in model space.

Starting at a random place inside the model space, a burn-in phase of 1000 iterations is first used to make a crude search of the entire model space. The burn-in phase is followed by a similar but more detailed and local search of the model space, based on the best-fit model parameters from the burn-in phase. The weighted least-squared misfit to observed TCN concentrations is used to evaluate the likelyhood for the combinations of model parameter values. The MCMC walker continues exploring the model space until it is sufficiently satisfied with the best model parameter estimate it has found.

For a given observational data set more than one set of model parameters may produce forward models which sufficiently satisfy the MCMC walker. In this case the solution is non-unique. Even worse, a single MCMC walker may find an area in model space which seemingly is in good correspondence with the observational data set, but the walker is missing a much better set of model parameters since they are located somewhere entirely different in the model space. In order to mitigate these issues, MCMC inversions are often performed using several MCMC walkers. The starting point of each MCMC walker is chosen at random, resulting in unique walks through the model space. If a single walker is caught in an area of non-ideal solutions, chances are that the other walkers will find the area of better model parameters.

The computational time depends on the number of MCMC walkers. When casually trying out the calculator we recommend using low numbers of MCMC walkers (1 to 2) in order to obtain fast results and reduce load on the server. When attempting to produce high-quality and reliable results, the number of walkers should be increased (3 to 4).

Citing the MCMC cosmo calculator

If you use the results generated by this tool in a scientific publication, please acknowledge this fact by citing:

Knudsen, M.F., Egholm, D.L., Jacobsen, B.H., Larsen, N.K., Jansen, J.D., Andersen, J.L., Linge, H.C., 2015.
A multi-nuclide approach to constrain landscape evolution and past erosion rates in previously glaciated terrains.
Quaternary Geochronology 30, 100-113, doi:10.1016/j.quageo.2015.08.004.

You may use the following BibTeX entry:

    @article{Knudsen2015,
        author = "Knudsen, M. F. and Egholm, D. L. and Jacobsen, B. H.
            and Larsen, N. K. and Jansen, J. D. and Andersen, J. L.
            and Linge, H. C.",
        title = "A multi-nuclide approach to constrain landscape
            evolution and past erosion rates in previously glaciated
            terrains",
        journal = "Quaternary Geochronology",
        volume = "30, Part A",
        number = "",
        pages = "100--113",
        year = "2015",
        issn = "1871-1014",
        doi = "http://dx.doi.org/10.1016/j.quageo.2015.08.004",
    }