A First Principles Critique of the Back Calculation Method: Understanding and Assessing the Alteration of Atmospheric Gases Trapped in Ancient Fluid Inclusions

Justin G Park, Morgan F Schaller

The extraction of atmospheric gases from fluid inclusions has emerged as an extremely promising approach for directly constraining the composition of Earth's ancient atmospheres. However, the veracity of data obtained from these inclusions critically depends on how well one can account for the effects of physical chemistry and post-depositional alteration. The Back Calculation Method (BCM) attempts to discern unaltered atmospheric compositions from variably modified inclusion gases, primarily O2 and CO2, but fails to account for the effects of gas solubility. Moreover, the method lacks an explicit mathematical formulation, and limited articulation of its implicit assumptions and limitations has introduced ambiguity in the interpretation of reconstructed atmospheric compositions. From first principles, we derive the formulation underlying the BCM and use it to formally evaluate the limitations of the method. We expand this framework to include the variable effects of gas solubility and incorporate recent techniques to partition inclusion volatiles trapped in the gaseous and aqueous phases. In re-examining modern inclusion gases, we find evidence of heterogeneous entrapment of gas and brine, as well as variable alteration extents and pathways. These observations violate the core assumptions of the BCM, suggesting the technique has been applied inappropriately to ancient samples and that their interpreted atmospheric compositions require thorough reevaluation. We conclude by demonstrating carbon isotope measurements of inclusion CO2 can be incorporated to assess the extent of gas alteration and provide an outline for future work intending to reconstruct atmospheric compositions from these inclusions.