Uncertainties in differential thermodynamic (Gibbs' method) P-T paths

A combined analytical and numerical evaluation of the uncertainties in P-T paths is made for three assemblages that propagates errors in the parameters: initial pressure, initial temperature, initial composition, change of composition of monitor parameters, endmember entropy, and endmember volume. Propagated errors along an isobaric heating path (ΔT=77°C) for assemblage 1 (Grt-Bt-Pl-Qtz-Ms-Chl-H₂O), using as monitor parameters the mole fractions of almandine, spessartine, grossular, and anorthite, and ignoring uncertainties in thermodynamic properties, are approximately ±320 bars (1σ) and ±8.3°C (1σ) if rim compositional uncertainties of 5% in major cations are assumed, or ±2.5°C (1σ) and ±50 bars (1σ) if electron microprobe analytical uncertainties are assumed for compositions. The largest source of uncertainty is from the errors in the monitor parameters, and P-T path uncertainties can depend critically on which monitor parameters are used. If the mole fraction of annite is used as a monitor parameter in place of the anorthite content of plagioclase, then propagated uncertainties are worse than ±29°C (1σ) and ±5800 bars (1σ). P-T path uncertainties also depend on assemblage. 1σ precisions in assemblages 2 (Grt-Bt-Pl-Qtz-Ms-Sil-H₂O) and 3 (Grt-Bt-Qtz-Kfs-Sil-H₂O) using as monitor parameters the mole fractions of almandine, spessartine, grossular, and anorthite are calculated to be ±267 bars and ±43.4°C and ±372 bars and ±8.2°C respectively. Estimates of the accuracies in P-T paths that include potential errors in endmember entropy of ±1 J/mol·K and in endmember volume of ±1 cm³/mol are: ±324 bars and ±8.5°C (assemblage 1), ±341 bars and ±48.6°C (assemblage 2), and ±388 bars and ±9.7°C (assemblage 3). Use of different garnet, plagioclase, and muscovite activity models can change the length of a P-T path by as much as 15%, but does not typically change directions in P-T space significantly. Models that incorporate changes of fluid composition shorten P-T paths in assemblages 1 and 3 but do not change trajectories significantly. Assemblage 2 is virtually unaffected by fluid phase models. For the mineral assemblages considered here and using appropriate monitor parameters, propagated errors are small compared to the total path length, suggesting that the differential thermodynamic approach is a precise and accurate method for determining amounts of heating or thickening during metamorphism, and hence for interpreting orogenic processes.