Introduction

Thermobarometry based upon relict phase equilibria preserved in igneous and metamorphic rocks is essential to petrology and geodynamics. In its earliest incarnation, observed mineral assemblages were compared directly to experimentally calibrated phase equilibria ([Goldschmidt1954], [Bowen1915]). Subsequently, with the advance of micro-analytic techniques, thermodynamic methods were used to account for the influence of chemistry on chemically simplified models of an observed mineral equilibrium ([Schmid1978], [Ghent1976]). The essence of these methods is that they satisfy a necessary condition for the equilibrium of the observed mineralogy. With the advent of extensive thermodynamic databases ([Helgeson1978], [Holland1985], [Berman1988]), it became practical, at least in principle, to solve all the necessary conditions for the equilibrium of an observed assemblage ([Powell1988], [Berman1991]). The advantage of the foregoing strategies, referred to here as classical thermobarometry (Slide 2), is that they rely only on the chemistry of those minerals that petrological argument suggests were in equilibrium at a point in time, i.e., the methods require only the existence of local equilibrium ([Korzhinskii1959], [Thompson1969]) among the minerals of interest.

The disadvantages of classical thermobarometry are: that it includes no test for the stability of the putative equilibrium assemblage; and that the probability of finding a single physicochemical condition that satisfies all the necessary conditions for equilibrium is vanishingly small. These disadvantages have the consequence that there is no guarantee that the inferred physicochemical conditions are consistent with the thermodynamic model from which they were derived. Recognition of this limitation gave rise to the use of forward thermodynamic models for thermobarometric purposes ([Powell1988]), an approach designated modern thermobarometry here (Slide 3). In these forward models a bulk, or effective bulk, composition is assumed, and the mineralogy is predicted by free energy minimization as a function of the physicochemical variables, usually pressure and temperature, of interest. The physicochemical conditions of equilibration for an observed mineral assemblage are taken to be those of the forward model when it most closely matches the observed mineralogy.

The process of finding the best match between observed and predicted mineralogy given a known, or partially known, bulk composition is an inverse problem that was formalized with the Bingo-Antidote algorithm ([Duesterhoeft2020]). Bingo-Antidote minimizes a quantitative measure of the misfit between observed and predicted mineralogy as a function of pressure, temperature, and any unmeasured volatile components. More recent contributions (LinaForma [Mackay2024], IntersecT [Nerone2025]) have adopted this approach to the simplified problem of minimizing the misfit between observed and predicted mineralogy in a calculated phase diagram section. An innovation of these latter contributions is that they account for the analytical uncertainty of the observed mineral chemistry.

Aside from various technical limitations, the fundamental limitation of modern thermobarometry is the assumption that equilibrium occurs on a finite spatial scale. This assumption conflicts with the notion of local equilibrium as equilibrium at a point ([Korzhinskii1959], [Thompson1969]).

It is widely appreciated that a polymetamorphic rock and/or a rock with chemically zoned minerals cannot represent a relict bulk equilibrium, which makes the application of modern thermobarometry to such rocks problematic. This problem is either ignored or circumvented by deducing an effective bulk composition for the minerals that are presumed to represent a relict equilibrium. Although this latter method is sometimes well-justified, it is costly and it cannot be applied in situations where compositional zoning of minerals is continuous. To avoid such difficulties, MC_fit employs an inversion strategy that relies, as in classical thermobarometry, only on the compositions of those minerals which are presumed to represent a relict equilibrium. The AvPT+ algorithm ([Green2025]) is also designed to invert for pressure and temperature solely from observed mineral compositions. Unlike MC_fit, AvPT+ relies on a local free energy minimizer (THERMOCALC, [Powell1988]), which has the consequence that it does not directly test for the stability of the observed mineralogy. AvPT+ makes no provision for unmeasured compositional information, and observed mineral compositions must be processed prior to inversion so that they can be perfectly reproduced by the thermodynamic model. Whether such pre-processing is useful is a matter of philosophy. Optionally, MC_fit can incorporate information on the bulk composition of a system, in which case it is functionally similar to Bingo-Antidote.

Bingo-Antidote, AvPT+, and MC_fit are fundamentally non-probabilistic; they identify best-fit solutions, but not necessarily the most probable solution. Free energy minimization has been employed in a probabilistic framework for geophysical problems ([Khan2006], [Afonso2013], [Dorn2015]). Forthcoming work ([Mackay2025]) aims to apply such a framework to thermobarometric problems. As in the case of the pre-processing of mineral compositions by AvPT+, whether a probabilistic approach is desirable in the context of thermobarometry is a matter for debate.

Key attributes of the four programs mentioned above are summarized in Table 1.