Technical note: A Bayesian mixing model to unravel isotopic data and quantify trace gas production and consumption pathways for time series data – Time-resolved FRactionation And Mixing Evaluation (TimeFRAME)
Authors: Harris Eliza , Fischer Philipp , Lewicki Maciej P. , Lewicka-Szczebak Dominika , Harris Stephen J. , Perez-Cruz Fernando 
Published: August 2024

Abstract. Isotopic measurements of trace gases such as N2O, CO2, and CH4 contain valuable information about production and consumption pathways. Quantification of the underlying pathways contributing to variability in isotopic time series can provide answers to key scientific questions, such as the contribution of nitrification and denitrification to N2O emissions under different environmental conditions or the drivers of multiyear variability in atmospheric CH4 growth rate. However, there is currently no data analysis package available to solve isotopic production, mixing, and consumption problems for time series data in a unified manner while accounting for uncertainty in measurements and model parameters as well as temporal autocorrelation between data points and underlying mechanisms. Bayesian hierarchical models combine the use of expert information with measured data and a mathematical mixing model while considering and updating the uncertainties involved, and they are an ideal basis to approach this problem. Here we present the Time-resolved FRactionation And Mixing Evaluation (TimeFRAME) data analysis package. We use four different classes of Bayesian hierarchical models to solve production, mixing, and consumption contributions using multi-isotope time series measurements: (i) independent time step models, (ii) Gaussian process priors on measurements, (iii) Dirichlet–Gaussian process priors, and (iv) generalized linear models with spline bases. We show extensive testing of the four models for the case of N2O production and consumption in different variations. Incorporation of temporal information in approaches (i)–(iv) reduced uncertainty and noise compared to the independent model (i). Dirichlet–Gaussian process prior models have been found to be most reliable, allowing for simultaneous estimation of hyperparameters via Bayesian hierarchical modeling. Generalized linear models with spline bases seem promising as well, especially for fractionation estimation, although the robustness to real datasets is difficult to assess given their high flexibility. Experiments with simulated data for δ15Nbulk and δ15NSP of N2O showed that model performance across all classes could be greatly improved by reducing uncertainty in model input data – particularly isotopic end-members and fractionation factors. The addition of the δ18O additional isotopic dimension yielded a comparatively small benefit for N2O production pathways but improved quantification of the fraction of N2O consumed; however, the addition of isotopic dimensions orthogonal to existing information could strongly improve results, for example, clumped isotopes. The TimeFRAME package can be used to evaluate both static and time series datasets, with flexible choice of the number and type of isotopic end-members and the model setup allowing simple implementation for different trace gases. The package is available in R and is implemented using Stan for parameter estimation, in addition to supplementary functions re-implementing some of the surveyed isotope analysis techniques.