Least-squares regression methods are mathematically powerful, conceptually and computationally simple, and widely used in many fields. However, none of the commonly-used flavors of least-squares regression, such as York regression or Generalized Least Squares (GLS), take into account the full set of covariances between all observed (x,y) values. Here we describe the Omnivariant Generalized Least Squares (OGLS) method to fit a model of the form y=f(x), accounting for the full error correlation structure of the (x,y) data, based on a first-order linear propagation of the uncertainties in all variables into errors in y residuals, followed by minimizing the vector of y residuals with respect to the Mahalanobis norm defined by its covariance matrix. This approach may be described as a generalization of both York regression and GLS. It is mathematically exact for straight-line fits, and is also suitable for many non-linear models. Here we describe the principles of OGLS regression and discuss its properties, caveats, and practical use, and provide two consistent open-source implementations in Python and R. To illustrate how various fields of geochronology and stable-isotope geochemistry may benefit from this new method, we discuss how OGLS may specifically apply to 40Ar/39Ar dating and how it provides robust mathematical evidence that Δ47 carbonate calibrations in the recently defined I-CDES metrological scale are statistically indistinguishable, effectively solving long-standing methodological discrepancies.

Clumped-isotope measurements in CO2 and carbonates (Δ47) present a number of technical challenges and require correcting for various sources of analytical non-linearity. For now we lack a formal description of the analytical errors associated with these correction steps, which are not accounted for in most data processing methods currently in use. Here we formulate a quantitative description of Δ47 error propagation, fully taking into account standardization errors and their properties. We find that standardization errors are highly sensitive to the isotopic compositions (δ47, Δ47) of unknown samples relative to the standards used for analytical corrections, and in many cases constitute a non-negligible source of uncertainty, causing true measurements errors to exceed traditionally reported error estimates by a factor of 1.5 (typically) to 3.5 (in extreme cases). Using Monte Carlo simulations based on the full InterCarb data set, we find that this model yields accurate error estimates in spite of small non-Gaussian effects which remain entirely negligible in practice. We also describe various standardization strategies, along with the assumptions they rely on, in the context of this model, and propose a new, “pooled” standardization approach designed to yield more robust/accurate corrections. Among other uses, the mathematical framework described here may be helpful to improve standardization protocols (e.g., anchor/unknown ratios) and inform future efforts to define community reference materials. What’s more, these models imply that the inter-laboratory scatter (N = 5329) observed in the InterCarb exercise [Bernasconi et al., 2021] can be entirely explained as the effects of current standardization procedures. Based on these findings, we recommend that future studies systematically report full analytical uncertainties taking standardization errors into account. In line with this recommendation, we provide user-friendly online resources and an open-source Python library designed to facilitate the use of these error models.

Increased adoption and improved methodology in carbonate clumped isotope thermometry has greatly enhanced our ability to interrogate a suite of Earth-system processes. However, interlaboratory discrepancies in quantifying Increased use and improved methodology of carbonate clumped isotope thermometry has greatly enhanced our ability to interrogate a suite of Earth-system processes. However, inter-laboratory discrepancies in quantifying carbonate clumped isotope (Δ47) measurements persist, and their specific sources remain unclear. To address inter-laboratory differences, we first provide consensus values from the clumped isotope community for four carbonate standards relative to heated and equilibrated gases with 1,819 individual analyses from 10 laboratories. Then we analyzed the four carbonate standards along with three additional standards, spanning a broad range of δ47 and Δ47 compositions, for a total of 5,329 analyses on 25 individual mass spectrometers from 22 different laboratories. Treating three of the materials as known standards and the other four as unknowns, we find that the use of carbonate reference materials is a robust method for standardization that yields inter-laboratory discrepancies entirely consistent with in-laboratory analytical uncertainties. Carbonate reference materials, along with measurement and data processing practices described herein, provide the carbonate clumped isotope community with a robust approach to achieve inter-laboratory agreement as we continue to use and improve this powerful geochemical tool. We propose that carbonate clumped isotope data normalized to the carbonate reference materials described in this publication should be reported as Δ47 (I-CDES) for Intercarb-Carbon Dioxide Equilibrium Scale.

The potential for carbonate clumped isotope thermometry to independently constrain both the formation temperature (T∆47 ) of carbonate minerals and fluid oxygen isotope composition allows insight into long-standing questions in the Earth sciences, but remaining discrepancies between calibration schemes hamper interpretation of T∆47 measurements. To address discrepancies between calibrations, we designed and analyzed a sample suite (41 total samples) with broad applicability across the geosciences, with an exceptionally wide range of formation temperatures, precipitation methods, and mineralogies. We see no statistically significant offset between sample types, although comparison of calcite and dolomite remains inconclusive. When data are reduced identically, the regression defined by this study is nearly identical to that defined by four previous calibration studies that used carbonate-based standardization; we combine these data to present a composite carbonate-standardized regression equation. Agreement across a wide range of temperature and sample types demonstrates a unified, broadly applicable clumped isotope thermometer calibration.