powell et al-2008-journal of metamorphic geology

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On thermobarometry

R. P O WELL1 A N D T . J . B . H O L L A N D21School of Earth Sciences, The University of Melbourne, Victoria 3010, Australia ([email protected]) 2Department of Earth Sciences, University of Cambridge, Cambridge, UK

AB ST R ACT Thermobarometry, the estimation of the pressure–temperature (P – T ) conditions of metamorphism, canbe undertaken by using pseudosection calculations as well as by conventional methods. Conventionalthermobarometry uses only the equilibrium thermodynamics of balanced reactions between end-members of minerals, combined with the observed compositions of minerals. In contrast, pseudosections

involve a forward calculation of mineral equilibria for a given rock composition. When related toobserved rock data such as mineral assemblages, mineral proportions and mineral compositions,pseudosections have the power to provide valuable additional thermobarometric information. This isbecause the rock composition provides added constraints on P – T , unavailable in conventional

thermobarometry, such as when minerals in the mineral assemblage are no longer stable, or whenadditional minerals join the mineral assemblage. Considering both conventional and pseudosection

thermobarometry, a minimum requirement is that they use the same thermodynamic data and activity– composition models for the minerals involved. A new THERMOCALCTHERMOCALC facility is introduced that allowspseudosection datafile coding to be used for conventional thermobarometry. Guidelines are given andpitfalls discussed relating to pseudosection modelling and conventional thermobarometry. We arguethat, commonly, pseudosection modelling provides the most powerful thermobarometric tools.

Key words: ideal analysis; pseudosection; thermobarometry; THERMOCALCTHERMOCALC.

I NT R O DUCT I O N

Twenty years ago we wrote a paper (Powell & Holland,1988) that set out a new way of addressing thermo-barometry problems, and provided a software tool,

THERMOCALCTHERMOCALC, for performing them. That paper wasbased at least in part on Powell (1985), and the ap-proach was more fully explained in Powell & Holland(1994). Now, 20 years later, we wish to review thestatus of thermobarometry and to make additionalproposals regarding improving the quality andreporting of thermobarometric results.

Thermobarometry, the determination of the P – T conditions of formation of mineral assemblages, isunderpinned by a view of how metamorphism works.Accepting that the approach adopted is the applicationof equilibrium thermodynamics, a case must be madethat the mineral assemblage to be used in thermoba-rometry represents a preserved equilibrium from some(small) part of the P – T path that the rock followed(that part of the P – T path being the conditions of formation of the mineral assemblage). Preservationmust involve not only the mineral assemblage but alsothe mineral compositions. Much of the metamorphiccommunity has implicitly adopted an equilibriummodel of metamorphism, as summarized and discussedin Powell et al. (2005) (based on Guiraud et al., 2001and White & Powell, 2002) for specific aspects relatingto the preservation of metamorphic mineral assem-blages). Although critical to thermobarometry, dis-

cussion of the application of the equilibrium model of metamorphism to rocks, and thus the selection of whatmight be interpreted to be a preserved equilibriummineral assemblage and mineral compositions, is be-yond the scope of this paper.

Experimentally determined phase equilibria lie at theheart of nearly all thermobarometric methods. How-ever it is not uncommon for such methods to involvean extrapolation in P, T and/or mineral compositionfrom the experimental conditions to those that themineral assemblage experienced (e.g. see discussion of garnet-clinopyroxene Fe-Mg exchange below). Equi-librium thermodynamics then provide equations of anappropriate shape to fit the experimental data so thatthey can be extrapolated. This applies not only tomineral end-member data (e.g. Holland & Powell,1998), but also to activity–composition (a – x) data (e.g.Holland & Powell, 2003). Both are aided by calori-metric data, and also heuristics that guide the sign andmagnitudes of the parameters required. For examplethe value of the entropy of a mineral end-member thathas not been determined calorimetrically is likely to liewithin quite a small range, determined via volume-adjusted additivity constraints (as used in Holland &Powell, 1998).

Although the quality of the internally consistentthermodynamic data sets of mineral end-memberproperties has improved considerably in the last20 years (compare Holland & Powell, 1996, and Ber-man, 1988, with the currently available Holland &

J. metamorphic Geol., 2008, 26, 155–179 doi:10.1111/j.1525-1314.2007.00756.x

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Powell data set, upgrade 5.5, tcds55.txt, fromNovember 2003, based on Holland & Powell, 1998),knowledge of the a – x relationships for minerals, sili-cate liquid and fluid has been completely transformed,particularly in terms of our ability to handle multi-component phases that also involve order–disorder(Powell & Holland, 1993; Holland & Powell, 1996;

Powell & Holland, 1999; Holland & Powell, 2003). Asignificant part of this has been driven by improve-ments in our ability to calculate phase equilibria, notonly in terms of calculation methods but also in thesoftware available to do the calculations (e.g. THER-THER-MOCALCMOCALC: Powell & Holland, 1988; Powell et al., 1998;Perplex: Connolly, 1990). These a – x developmentshave been made largely to allow better and more reli-able petrological calculations.

In fact, conventional thermobarometry has notbeen the main focus of petrological calculation indriving recent thermodynamic development. Instead,it has been calculated P – T pseudosections (e.g.

Powell & Holland, 1990, fig. 5; Powell et al., 1998).Clearly, such thermobarometric calculations on min-eral assemblages and calculated pseudosections forrocks both require a – x relationships for minerals ingeologically realistic systems. On the one hand, thosewho perform thermobarometry with the averageP – T (or avPT) method of Powell & Holland (1994)use a – x relationships that are based on those used forpseudosection calculations, using the software AXAX(tjbh url: http://www.esc.cam.ac.uk/astaff/holland/index.html) to calculate activities from mineralchemical analyses. On the other hand, many ther-mobarometers use a – x relationships, as well asimplied end-member properties, that are less defen-

sible as they are inconsistent with establishedthermodynamic properties. It certainly makes senseto use precisely the same and best available a – xrelationships (and the same end-member properties)in thermobarometry as are used in pseudosectioncalculations. Such consistency would seem to be aminimum requirement for petrological calculations.In this contribution, various aspects of thermo-barometry are discussed, in the light of the progressmade in the last two decades.

T HER MO B AR O MET R I C MET HO DS

Equilibration volume

With the widespread adoption of an equilibrium modelof metamorphism, it has become commonplace to referto the equilibrium mineral assemblage, as though thisis something we can somehow prove. However this hasalways been a geological interpretation; there is no wayto prove it, although there may be an absence of fea-tures that would indicate that equilibrium was not at-tained. Therefore, it is more appropriate to use a formof words like the minerals that are interpreted to haveonce been in equilibrium with each other. It is in this

context that we refer to equilibration volume, thescale on which it is plausible to suggest that the min-erals in the mineral assemblage being considered forthermobarometry were in equilibrium with each other.In assessing an equilibration volume all the normalcaveats apply: different elements have differentmobilities; the cores of growth-zoned grains are ex-

cluded, as are replacive minerals that are intepreted tohave grown after the main mineral assemblage, andso on. Even for a correctly established equilibrationvolume, the observed mineral compositions maynot correspond to those originally present if diffusivere-equilibration has occurred during cooling.

Effectively, all thermobarometry starts with aninterpretation of the equilibration volume. For ther-mobarometric methods, like avPT, that use just themineral compositions, the equilibration volume is usedonly in the sense that minerals chosen to be in themineral assemblage that is used in the calculationsshould be in the same equilibration volume. For

compositionally homogeneous rocks, this need noteven be the case, as compositional homogeneity alsoimplies homogeneity in mineral assemblage and min-eral chemistry.

For thermobarometry involving pseudosections, theequilibration volume plays a more central role in that itis the chemical composition of the equilibration vol-ume that is used in the calculations. This compositionmay be obtained by XRF analysis, for rocks that arehomogeneous on the appropriate scale, possibly withcalculation to exclude cores of growth-zoned mineralsand adjustment if the pseudosection is in a modelsystem that excludes elements that are significant in theactual rock. Alternatively the composition of the

equilibration volume may be obtained from mineralcomposition maps, that also allow straightforwardexclusion of mineral cores (Clarke et al., 2001; Marmoet al., 2002). In addition, the equilibration volume maydepend on a geological inference, for example, to allowfor the presence of a fluid or melt and its volume andcomposition.

Pseudosection thermobarometry

Given the composition of an equilibration volume andthe appropriate thermodynamic data, a P – T pseudo-section can be calculated. For the mineral assemblageof the equilibration volume, thermobarometric infor-mation is contained in: (i) the position in P – T space of the field that corresponds to that mineral assemblage,and (ii) the proportions and compositions of the min-erals in that field. This can be seen in Fig. 1, anexample for the mineral assemblage garnet + cordie-rite + sillimanite + plagioclase + alkali feldspar +quartz + ilmenite, using an actual aluminous pelitecomposition in the system NCKFMASHTO (RH011from White et al., 2004, Table 1). The mineral pro-portion contours are shown in Fig. 1a and the mineralcomposition contours in Fig. 1b.

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What can be seen on such pseudosections is thatfields corresponding to particular mineral assemblagesmay be small or large. If the relevant field is small,thermobarometry is effectively complete, althoughestablishing that the calculated mineral compositionsmore or less match those in the rock is clearly worth-while. It remains to evaluate the dependence of the

position (and size) of the field on the assumptionsmade in drawing the pseudosection, primarily relatingto how well established the composition of the equili-bration volume is.

For large fields – and the field of the ascribedassemblage in Fig. 1 can be considered to be a mod-erately large one – it is appropriate to constrain theposition of the rock in the field via mineral proportionand mineral composition contours (e.g. Vance &Mahar, 1998). Not surprisingly some contours changerapidly with P – T whereas others change little. Thegarnet and cordierite proportions change rapidly withpressure, whereas the remainder changes slowly. Sim-

ilarly, and correlated with the corresponding propor-tions, the xcd and xg contours change rapidly withpressure, with

xk ¼ Fe

Fe þ Mg

k

:

In contrast, the remainder change slowly. Thus theusefulness of the thermobarometric information in thecontours varies considerably. Furthermore, the con-tour values of compositions that may not have closedat the conditions of formation are not useful forthermobarometry, for example feldspar composition(Fuhrman & Lindsley, 1988; Whitney, 1991).

What constitutes a small or a large field? The scalefor this depends in a sense on what is considered to beachievable with more conventional thermobarometry(including avPT). At least in avPT the answer comeswith the uncertainties on the calculated P – T , as well asrfit (i.e. the square root of MSWD as used in geo-chronology), used to assess whether the data should becombined. Although potentially realistic uncertaintiesare given as part of the thermobarometric calculation,this is commonly not true of other such methods. Inthe latter, P – T uncertainties are routinely reported thatare unrealistically small, especially in published garnet-biotite and garnet-pyroxene thermometry. This isunfortunate as it obscures the possibility of recognizingthe situation where there is, say, effectively no P – T information in a mineral assemblage (as discussedbelow).

The authors prejudice is that quoted uncertainties inconventional thermobarometry of less than ±50 Cand ±1 kbar should be treated with suspicion (± is2r, here and below). The difficulty is that the uncer-tainties are hard to establish, and that potentiallythe most damaging source of uncertainty is bias(see below). On this basis the greatest majority of fields on pseudosections have as much or more

Fig. 1. An NCKFMASHTO pseudosection for an aluminousmetapelite composition: H2O ¼ 5.65; SiO2 ¼ 65.47; Al2O3 ¼14.49; CaO ¼ 0.49; MgO ¼ 2.55; FeO ¼ 6.66; K2O ¼ 2.8;Na2O ¼ 1.17; TiO2 ¼ 0.65, and O ¼ 0.07 (molar). In (a), con-tours are for mineral proportions (e.g. g14 is the 0.14 garnetproportion contour), and, in (b), contours are compositionisopleths (e.g. xcd44 is the xcd ¼ 0.44 contour), for the targetmineral assemblage, garnet-cordierite-sillimanite-plagioclase-K-feldspar-ilmenite-quartz-melt. Mineral proportions are molar,on a one oxide basis, as normally output by THERMOCALCTHERMOCALC (to besimilar to volume proportions). Abbreviations: mu, muscovite;bi, biotite; sill, sillimanite; g, garnet; cd, cordierite; sp, hercyniticspinel; pl, plagioclase; ksp, K-feldspar; q, quartz; and liq, silicatemelt; xg ¼Fe

2+ ⁄ (Fe2+ + Mg)| g and xcd ¼ Fe

2+ ⁄ (Fe2+ +

Mg)|cd. The notation, for example -pl, means that pl (plagio-clase), although in the in excess list of phases at the top of thediagram, is not present in this part of the diagram.

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thermobarometric information in them (not even usingmineral proportions and compositions) than conven-tional thermobarometers.

Pseudosections do not merely give the conditions of formation of a mineral assemblage. They commonlyallow sections of P – T paths to be deduced from theway mineral proportions and mineral compositions

are interpreted to have evolved during the texturalevolution of a rock, when compared with phaserelationships in the corresponding pseudosection.Pseudosections allow the observed assemblage in aparticular rock to be placed in the context of alterna-tive mineral assemblages, in terms of other mineralsbecoming stable, as well as minerals being lost from theassemblage. Moreover the fact that pseudosections arenot dependent on establishing original mineral com-positions (as required for conventional thermobarom-eters) is an important added advantage, particularly intexturally complex rocks.

Having said this, pseudosections are not without

their limitations. In particular, it is not alwaysstraightforward to determine an appropriate equili-bration volume composition to use. In the presence of growth-zoned minerals, with some elements preferen-tially sequestered into mineral cores, the differencebetween the whole-rock composition and the equili-bration volume composition can be significant,although an attempt can be made to account for this(e.g. Marmo et al., 2002; Zuluaga et al., 2005). Assequestering takes place, the equilibration volumechanges, implying that no one pseudosection can beexpected to reflect all of a rock’s history. In fact it isimplicit in most uses to which pseudosections havebeen put that the diagrams are to be used only for the

prograde history where fluid (or melt) is present(Guiraud et al., 2001; White & Powell, 2002). Inhigher-grade rocks in which textural complexitydevelops via the production of domains duringdecompression/cooling, representing the equilibria inthe domains is difficult, not least because the domainsare commonly not closed systems (White et al., 2008).

In situations where the observed mineral assemblagein an equilibration volume is not matched by a field inthe pseudosection, this may indicate that the composi-tion of the equilibration volume has been incorrectlyassigned, but there are other possible causes. An obvi-ous explanation is that the thermodynamic models usedfor the minerals are not good enough. Appropriatemodels need to exist for all relevant phases in a systemof interest. But a silicate melt model that allows partialmelting of metabasic rocks to be considered has notbeen developed yet; there is no model for stilpnomelane;no clinopyroxene model has been developed that allowsconsideration of the whole range of mineral assem-blages from those formed at higher temperature, wherethe Ca-Tschermak end-member is important, to thoseformed at lower temperature where the jadeite substi-tution is important, and so on. Recently, a model foramphibole in NCFMASHO (Diener et al., 2007) has

been developed, but this needs to be extended to the K-and Ti-bearing systems so that granulite facies mineralassemblages can be considered properly. Mn end-members, apart from those for garnet and ilmenite,need to be readdressed, as the current models (based onMahar et al., 1997) involve ideal mixing for chlorite,biotite, etc. in the natural assemblage calibration, and

have been superceded. The quality of models also var-ies, with chlorite and sapphirine in FMAS due forreassessment in the light of Holland & Powell (2006).Regardless of the success of the pseudosection ap-proach so far, it is important to realize that the devel-opment of thermodynamic models is a work inprogress.

Another cause of a mismatch between mineralassemblage and pseudosection is if the modelling isbeing performed in a smaller system than that con-trolling the equilibria. Note that there is no generalthermodynamically consistent way of extrapolating theequilibration volume composition to a smaller system,

although it may be possible in simple cases in practice.If a rock needs to be represented in NCKFMASHTO,say, it will generally not be good enough to use aKFMASH or a NCKFMASH pseudosection (asillustrated by White et al., 2007). What is close? Howmuch Zn needs to be in a rock before a pseudosectionthat we draw in a Zn-free system (in the absence of models for Zn-bearing minerals) can no longer beexpected to relate to the observed phase relationships?The question then is whether the presence of stauroliteor spinel in the rock depends on the Zn? This is diffi-cult to answer. In a situation where we can actually dothe calculations and compare results in full and smallersystems, the appearance of garnet with changing P – T

is certainly strongly dependent on the presence of evena small amount of Mn (Tinkham et al., 2001).

Recognizing that pseudosections need to account forall of the real system for thermobarometric use, con-ventional thermobarometry might then be consideredfor calculating P – T in rocks where this cannot be done.However, in practice, the activities of end-members forwhich thermodynamic data still need to be calculated,but in the presence of components of unknown influ-ence. Continuing the Zn example, and considering aspinel-bearing assemblage, to use the activities of MgAl2O4 and FeAl2O4 in thermobarometry requiresthat the effect of Zn on these activities be estimated.

Conventional thermobarometry

Here, conventional thermobarometry is considered toinclude all those thermobarometric methods that arebased on balanced reactions written between the end-members of phases in the equilibration volume. Foreach such reaction, an equilibrium relationship can bewritten, 0 ¼ DGo + RT ln K . Generally each equilib-rium relationship provides a relationship betweenpressure and temperature, the observed mineral com-positions having been substituted into the equilibrium




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constant. The equilibrium relationships may addi-tionally be a function of the composition of a phaseinterpreted to have once been present, for exampleaH2O. In the following, pressure and temperature aretreated as the only unknowns, with the knowledge thatthe methods can be generalized to solve for additionalunknowns.

The number of reactions that can be written betweenthe minerals in an assemblage depends on the end-members for which there are thermodynamic data (forcalculation of DGo), and the phases for which there areappropriate activity–composition (a – x) models. Werefer to this combination of end-member data and a – xmodels as thermodynamic descriptions of phases. Fora set of thermodynamic descriptions involving n end-members in a chemical system of m components (e.g.oxides), there are at most nCm+1 reactions, of whichn ) m are independent (in the sense that the remainingreactions can be written as linear combinations of those in an independent set). Note that nCm+1

n )

m.Given a set of thermodynamic descriptions and theaccompanying uncertainities, as well as the composi-tions of the phases and their uncertainties, the optimalthermobarometric result is given by the minimal (insome statistical sense) adjustment of the data, such thatthe P – T lines for the equilibria intersect at one P – T – the optimal P – T . As all the P – T information is held inan independent set of equilibria, only such an inde-pendent set needs to be considered (Powell & Holland,1988, Appendix B). So if the data are adjusted so thatthe P – T lines for the equilibria in an independent setintersect at a point, then by definition all the reactionsthat can be written also intersect at that point.

There is a set of possible optimal thermobarometricmethods, depending on the approach used to adjust thedata. Average P – T (Powell & Holland, 1994) is one of these: the data are adjusted via a weighted least squaresscheme in which the sum of the squares of the adjust-ments of the data, normalized to their uncertainties, isminimized. This is the form of optimal thermobarome-try that is implemented in THERMOCALCTHERMOCALC [Powell & Hol-land, 1988; now at version 3.3 (tc330); see also Gordon,1992]. Alternative schemes can be envisaged, involvingfor example minimizing the maximum adjustment of any of the data, weighted to their uncertainties. A morepromising development is one that involves a robuststatistical approach in which outlying data are recog-nized and down-weighted, allowing an optimal ther-mobarometric result to be calculated that is notdegraded by the presence of outliers (e.g. Powell et al.,2002, and the references to the statistics literaturetherein, particularly Hampel et al., 1986).

From a practical point of view, regardless of whichoptimal method is followed, assigning uncertainties canbe a problem. For mineral compositions, analyticaluncertainties should suffice, although compositionalvariability may make an additional contribution. Forend-member properties, if the Holland and Powell

internally consistent data set is adopted, then theuncertainties derived from the generation of the dataset are available for use (as in the THERMOCALCTHERMOCALC imple-mentation of avPT and other calculations). Howeverthe uncertainties associated with a – x relationships areless easy to assess (as discussed further below). Theextant avPT implementation uses a generalized way of

introducing an uncertainty on the activity of an end-member in a mineral, taking account of the number of sites across which mixing takes place and the size of theactivity (Powell & Holland, 1988, Appendix C). Theadditional facility in the THERMOCALCTHERMOCALC implementationof avPT for assigning uncertainties to activities intro-duced below gives more flexibility and is an improve-ment on the original approach.

Other conventional thermobarometric methods arenot optimal. TWEEQU-type methods, based onlooking at all the equilibria that can be written betweenthe end-members of the phases in the mineral assem-blage (Berman, 1991; Lieberman & Petrakakis, 1991),

do not use any uncertainties or correlations betweenthe equilibria. The P – T result is gained by eye from theplot of the equilibria, with no information on whetherequilibria are well constrained or quite uncertain. Yetit is obvious that there is a huge range in theseuncertainties. Furthermore, the non-statistical natureof the approach means that no uncertainty on the P – T result is accessible. The problems with such methodsare discussed in detail in Powell & Holland (1994).

An even less optimal method is the use of two, com-monly directly calibrated equilibria to estimate P – T .The data on which such direct experimental calibrationsare based have been used in the generation of internallyconsistent thermodynamic data sets, so their equilibria

are subsumed in methods like avPT. Nothing is gainedby the direct calibration approach, and informationfrom the other independent equilibria is lost. Moreover,using a direct approach means that the data are not putthrough a thermodynamic filter (in terms of functionalform, as well as in relation to other related experimentaldata), thus potentially further decreasing the reliabilityof the results produced. Combining relatively uncertainequilibria with well-constrained ones degrades results,but in a statistical approach (like avPT) this cannothappen. As the directly calibrated approach is notcommonly undertaken in a statistical context, it can bethat the equilibria chosen do not have as much P – T information in them as others in an independent set (e.g.if one is Fe-Mg exchange between garnet and biotite, orgarnet and clinopyroxene), or certainly in the indepen-dent set as a whole.

I NFO R MAT I O N AND T O O LS FO RT HER MO B AR O MET R Y

Thermodynamic and other data

The end-member properties of minerals that are usedin most mineral equilibria calculations are from inter-




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nally consistent thermodynamic data sets (e.g. Holland& Powell, 1998, and upgrades), as used by THERMOCALCTHERMOCALCand also commonly by other software. The Holland &Powell data set is derived from a combination of calorimetric data and experimental data, with the keystep (giving the internal consistency) being a weightedleast squares analysis of the data to determine the

enthalpies in the data set, given the entropies, volumes,heat capacities, etc. As new calorimetric data andexperimental data are published, they are incorporatedinto the data set, with upgrades released every fewyears (the current release is 5.5 of November 2003).New releases also often coincide with improvements inmethodology relating to the formulation of the thermo-dynamic properties as a function of P – T .

Although ideally the a – x relationships to be usedwould be independent of end-member properties, formany end-members this is not possible. For exampleexperimental data that allow properties of Tschermakend-members to be determined necessarily involve

solid solutions. In such cases, a – x relationships haveto be in place for the end-member properties to bedetermined, or the data analysis must generate both.Inevitably the interaction energies in the a – x relation-ships will be correlated with the enthalpies of theend-members involved.

What are being calibrated in such an approach arethe thermodynamic descriptions of the phases, notthermobarometers. This is appropriate because theform of the P – T dependence and to a lesser extent thecomposition dependence of the thermodynamicdescriptions are rather well established. Thus thermo-dynamics can be used to filter the data, and the processof generating the data set of end-member properties,

commonly with the implication of consistency with a – xrelationships, provides internal consistency. Amongother things it means that the results of experimentalstudies, that for whatever reason are not consistentwith the remainder of the calorimetric and experi-mental data, can be identified. On removal, they arenot then able to degrade the overall data set. Thusthermobarometers are best made via internally con-sistent data sets, and not calibrated directly.

This overall approach, as well as covering theinterpolation of the data involved, also should allowextrapolation from the P – T conditions and composi-tions of the experiments to those where the data will beapplied. Even if direct calibration of thermobarome-ters is attempted, it is unwise not to use internallyconsistent data set information. The fact that the g-cpxFe-Mg exchange thermometer of Krogh Ravna (2000)involves a DV and a DS (obtained by regression of theexperimental data) that are rather different from thedata set values suggests that the P – T dependence of theequilibria has not been properly distributed betweenDV , DS and the implied interaction energies in the a – xrelationships, and this makes extrapolation to condi-tions outside the P – T range of the experiments suspect.In this case another problem arises relating to the fact

that solid solutions in the experiments involve one wayof substituting Al in cpx (Ca-Tschermak molecule),whereas in many rocks it is substituted via the jadeiteend-member (and with the cpx having a differentstructure at lower temperature, in omphacite).

It is important to realize that the overall approach isa powerful one but that we are not yet able to do

everything we would like to do. As usual more exper-imental data are needed. Even at the methodologicallevel, problems are still being recognized. Recently, akey simplifying (and regularizing) assumption made byall authors has been shown not to behave as intended.This assumption, involving site Fe/Mg ratios taken tobe the same as the Fe/Mg ratio for the whole phase(the equipartition assumption) has bad and indefensi-ble energetic implications (Holland & Powell, 2006).We are in the process of revising all these models toremove the equipartition assumption (already, cpx:Green et al., 2007; amphibole: Diener et al., 2007).

In addition to thermodynamic data, for conven-

tional thermobarometry (and for pseudosection ther-mobarometry in comparing calculated mineralcompositions with observed values), a critical step isthe recalculation of mineral analyses. The conventionalway of considering mineral formula uncertainties thatstem from analytical uncertainties is to perform anerror propagation (e.g. Powell, 1978). Whereas this canbe helpful, particularly in showing how uncertaintiescan magnify in dealing with, for example, charge bal-ance calculated Fe3+ (Cawthorn & Collerson, 1974;Droop, 1987), or calculation of x

opxAl;M1 (Carswell, 1991),

an alternative approach can be useful in manipulatingmineral analyses for mineral equilibria applications.This has been termed the best analysis approach

(Carson & Powell, 1997; S ˘ tı ´pska ´ & Powell, 2005), butwill be referred to here as the ideal analysis approach.The idea is that close (in some sense) to the mineralanalysis by the electron microprobe is an analysis thathas some desired properties, for example exact stoi-chiometry and a specifed amount of ferric iron. Theeasiest way to perform such a calculation is for closeto be calculated in a least squares sense, with the idealanalysis being the one that minimizes the sum of thesquares of the displacements of the analysed wt%,weighted by their analytical uncertainties (see Appen-dix 1).

The ideal analysis approach is a tool with usesacross a range of calculations that start with mineralanalyses. For example, in the new form of data inputfor avPT with THERMOCALCTHERMOCALC using pseudosectiondatafile coding, the approach allows an analysis to bein a form that corresponds to the datafile coding, forexample it can be made to be stoichiometric. Theapproach is particularly useful if calculations are to beundertaken to investigate the consequence of varyingthe proportion of iron in an analysis that is consideredto be ferric (Carson & Powell, 1997; S ˘ tı ´pska ´ & Powell,2005). In the latter study, considering g-cpx Fe-Mgexchange thermometry for an eclogite, the cpx analyses




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were recalculated so that they were stoichiometric at arange of specified ferric values, to investigate thedependence of calculated temperature on this ferrous– ferric conversion. Because the calculation is a leastsquares one, the rfit of the analysis allows an appraisalof whether the ideal analysis with the specified ferriccontent is consistent statistically with the original

analysis given the analytical uncertainties (via a v2constraint, e.g. Mikhail, 1976). In both papers (Carson& Powell, 1997; S ˘ tı ´pska ´ & Powell, 2005), the approachallows an appreciation of the bias (see next section)introduced by assuming that no ferric iron is involvedin the mineral analysis (in opx and cpx, respectively).

Uncertainties

An elementary requirement of any calculation is theassessment of uncertainties on results that arise fromuncertainties on the input information, as has beenemphasized in a mineral equilibria context by Powell

(1978, 1985) and Powell & Holland (1988, 1994). It israrely possible to do a thorough job of this, but thiscannot be an excuse not to try. If the uncertainties arealready large, as commonly arises in the estimation of ferric iron in minerals in the context of Fe-Mg ex-change thermometry, then it is evident that final (butunknown) uncertainties can only be larger. Because of the incomplete nature of the propagation, it is impor-tant to realize that quoted uncertainties should bethought of as minima.

There are several aspects of such minimal uncer-tainties that need to be borne in mind. The first con-cerns whether P – T results on related rocks are likely tobe different or not (see also Worley & Powell, 2000).

The second concerns the fact that minor changes inassumptions or data will cause results to move aroundwithin such uncertainties (in a way that can be alarm-ing, but is understandable statistically). The third,which is a key issue in conventional thermobarometry,is whether a particular mineral assemblage has anytemperature and/or pressure information in it. Thisquestion should be asked in a context: in an ultrahighpressure (UHP) eclogite quite possibly 800 ± 100 C isa good result, whereas for an amphibolite or granulitefacies metapelite such a temperature range might bededuced directly from the hand-specimen. In apseudosection context, a scale is provided by the size of the field corresponding to the mineral assemblage.Commonly such a size is smaller than the estimateduncertainty. The possibility of an assemblage beingthermobarometrically non-informative (by conven-tional thermobarometry) can be thought of in apseudosection context: the field for an assemblage maybe well constrained in P – T by the presence/absence of minerals (i.e. the size of its field), yet within the field thecompositions of all the minerals change very little.

Why are uncertainties not all available to be prop-agated through calculations? Analytical uncertainties(related to how the electron microprobe is set up),

uncertainties that relate to the regression of experi-mental data in the generation of an internally consis-tent data set (e.g. Holland & Powell, 1998), or thefitting of experimental data to get interaction energiesfor a – x relationships are available for propagation andshould certainly be propagated. But uncertaintiesrelated to structural assumptions cannot be sensibly

propagated. Suppose an assumption in the recalcula-tion of a mineral is wrong, related for example tostoichiometry. Suppose the measured entropy of amineral end-member, assumed to be good within itsuncertainty, is wrong. Suppose the formulation of thea – x relationships is wrong. Such things give rise towhat was called systematic uncertainty in Powell(1985), but is better referred to as bias.

Accounting for bias is a major challenge. If we arelucky, such bias will affect the rocks that we are inter-ested in more or less equally, so that differences betweenthe calculated P – T values of rocks are maintained, butall rocks are displaced to higher or lower T , say.

Something like this might apply if the mineral equilibriaare consistent with experiment at high temperature(where equilibration occurs in the laboratory in finitetime), but the extrapolation to rock temperatures isinappropriate: all estimated temperatures are either toohigh or too low. But such systematics may well not be astransparent. More cogently, the problem with bias isthat generally it is unknown. Only when it becomesapparent is it relevant to consider systematics, in thecontext of evaluating previously produced results.

A situation where bias can be illustrated is where theimplementation of a thermobarometer involves regu-larization. Regularization arises when a simplificationis made that reduces the variability in results, appar-

ently reducing the uncertainty on a result, but in theprocess introduces bias. The first example involvesthe Al-in-orthopyroxene (opx) thermobarometer forthe assemblage opx + garnet. The calculated octahe-dral and tetrahedral Al site occupancies can be regu-larized by taking each of them to be the totalrecalculated Al divided by 2 (in the Na- and Cr-freecase). In the context of the actual charge balance,AlVI + Fe3+,VI ¼ AlIV, the regularization simplyignores the ferric iron. Although this might be a rea-sonable approximation in processing the originalexperimental data (if they were done at low aO2), it isunlikely to be the case when considering rocks. Theeffect of removing this regularization is considered byCarson & Powell (1997). The thermobarometricequation involves the product of the octahedral andtetrahedral occupancies. Calculating the charge bal-ance properly, the product decreases as ferric ironincreases, so the pressure increases (at constant tem-perature). The magnitude of the pressure bias of reg-ularization obviously depends on the composition of the opx, as discussed by Carson & Powell (1997).

The second example involves Fe-Mg exchangethermometry. As is obvious, and has been spelt outby various authors, e.g. Proyer et al. (2004), the




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means that THERMOCALCTHERMOCALC generates the equilibrium stateof order in minerals to be used in calculations, at theP – T corresponding to the estimated conditions of formation suggested initially by the user. Whenuncertainties on the mineral compositions are in-cluded, the minerals maintain stoichiometry. ForavPT, given datafile coding for minerals, and the

mineral analyses, it remains to:1 Recalculate the mineral analysis to put it into a

form that corresponds to the chemical system andcomposition variables used in the coding of the a – xrelationships. Arguably the best way of doing thisis via the ideal analysis approach introduced above.

2 Assign uncertainties to the composition variables,arising from analytical uncertainties, and depend-ing on how the composition variables have beencalculated. Uncertainties are available for thethermodynamic properties of the end-members thatarise from the generation of the data set used.

Propagation of analytical uncertainties through an

ideal analysis calculation (Appendix 1) is straightfor-ward. In the past, when the activities of the end-members were entered into THERMOCALCTHERMOCALC directly foravPT calculations, default uncertainties on theseactivities were provided by THERMOCALCTHERMOCALC. These wereformulated to cover not only composition uncertain-ties but also interaction energy uncertainties (Powell &Holland, 1988, Appendix D), with the latter involvingan uncertainty of about 2 kJ mol)1 (on a one-sitebasis). Now, with datafile coding of a – x relationships,the uncertainties on the compositional parameters areentered separately. The default uncertainty on inter-action energies used is 2 kJ mol)1 (on a one-site basis),a default being necessary, as generally they are not well

constrained. This default can be easily overridden, as itmust be when the uncertainties are actually known(e.g. for cpx in Green et al., 2007). The old defaultactivity uncertainties can still be specified, and are stillused if old style activity input (from AXAX) is used. Thenew uncertainty structure is available with THERMO-THERMO-CALCCALC 3.26 (tc326) and more advanced versions.

Of the statistical information and diagnostics given byTHERMOCALCTHERMOCALC (Powell & Holland, 1994), rfit is central inthat it allows an assessment of whether the data beingcombined should indeed be combined. For example, foravP involving a set of 10 independent reactions, rfitshouldbe less than1.35for the data to belongtogether atthe 95% significance level. [The value comes from

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2r1;0:05=ðr 1Þ

q ;

where r is the number of independent reactions.] Con-sidering the analogy of a set of 10 points defining alinear trend (no outliers), for rfit to be of the order of 1.35 implies that the average of the departure of thepoints from the line, each divided by its r, would beabout 1. From this it is clear that if the r on the data ischanged, then rfit is changed. In fact, if all the r on thedata is increased by the same proportion, the rfit is

decreased by the same proportion. In a situationwhere the data involve uncertainties that are not wellknown, in this case particularly the interaction energies,it is usual to multiply the uncertainty on calculated P – T by rfit if it fails the v

2 test, as done by THERMOCALCTHERMOCALC (andas done in geochronology in so-called York model 2:York, 1969). This results in a more realistic assignment

of uncertainty, but implicitly makes everything moreuncertain (by the factor rfit). This then hides the pos-sibility that just one or two data items are moreuncertain than originally specified, rather than all of them being more uncertain. Considering the diagnos-tics given by THERMOCALCTHERMOCALC that are used to indicate thepossibility of outlying data, data may be outlying onlybecause a particular input has been assigned anuncertainty that is unrealistically small.

As the uncertainties on the interaction energies areincreased, rfit decreases, until the usual v

2 test is pas-sed. But these large uncertainties are propagated touncertainties on P – T , so at least this is not a way to

make results look better than they are. If uncertaintieson interaction energies are decreased, rfit increases,with propagated uncertainties on P – T decreasing, atleast until the v2 test fails. Realistic assignment of uncertainties on interaction energies is therefore rele-vant. The tuning constants in the original defaultuncertainty on activities (Powell & Holland, 1988, p.198) were chosen based on our experience of analyticaland calorimetric uncertainties. With these, avPT forrocks that were considered to be good candidates forgiving good results (i.e. for which the v2 test would beexpected to pass) actually do pass the v2 test, with rfitaround 1. Therefore the default uncertainty on inter-action energies is maintained at 2 kJ mol)1 (on a one-

site basis).Regarding the P – T results and their uncertainties

derived by THERMOCALCTHERMOCALC, it is relevant to note that theresults may vary of the order of the rP and rT onvarying input assumptions, for example relating to thephases included or the input uncertainties, at the samerfit). If THERMOCALCTHERMOCALC gives 10 ± 2 kbar (± is 2rP), it isto be expected that a calculated pressure may be any-where within 9–11 kbar, say, with minor change of inputs.

An illustration: conventional thermobarometry in eclogitesusing THERMOCALCTHERMOCALC

Eclogites present particular challenges for thermo-barometry. When they contain hydrous minerals inaddition to phengite, for example amphibole(s), apseudosection approach has the capacity to be the mosteffective (e.g. Carson et al., 1999; Wei et al., 2003). Butwhen they are not involved, because temperatures arehigher, or because aH2O is lower, the garnet-omphacite-phengite ± kyanite ± quartz/coesite mineral assem-blage is stable over a huge range of temperature andpressure. Given that such rocks occur most commonlyin subduction settings, estimating the P – T of formation




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of such mineral assemblages has important geodynamicimplications (e.g. Hacker, 2006).

Using conventional thermobarometry to considerthe mineral assemblage, garnet-omphacite-phengite-kyanite-quartz/coesite, it is now widely recognized thatequilibria that involve Fe end-members of minerals, inparticular in garnet-omphacite Fe2+-Mg exchange

thermometry, will be unreliable unless Fe3+ can bemeasured in the minerals (Galazka-Friedman et al.,1998; Sobolev et al., 1999; Schmid et al., 2003; Liet al., 2005). Thus Krogh Ravna & Terry (2004)advocated using the intersection of reactions amongthe mineral end-members: grossular, pyrope, diopside,muscovite, celadonite, kyanite and quartz (or coesite).Here we look at these equilibria, and others involvingend-members of these phases, using THERMOCALCTHERMOCALC andthe a – x datafile codings in Appendix 2, with the aim of undertaking thermobarometry in this mineral assem-blage.

The example chosen to illustrate the ideas involved

is sample SM93, a Dabie Shan eclogite studied byProyer et al. (2004). Critically, they used Mo ¨ ssbauerto determine ferric iron in their omphacite (andgarnet). The mineral asemblage involves garnet-om-phacite-phengite-kyanite-coesite, and additionallytalc, and in their avPT calculations it was assumedthat the mineral assemblage was water-saturated.They showed that the calculations are much moreconsistent once the relatively large conversion of FeOto Fe2O3 is made (0.54 ± 0.1) (also see Appendix 1below). As an aside, a subtly different conclusion canbe drawn in favour of the avPT approach: the effectof using all Fe-as-FeO in the omphacite degrades thestatistics (compare rfit ¼ 1.38, Table 3, with 0.48,

Table 6) but in fact does not significantly change theresult (compare 589 C and 29.9 kbar with 585 Cand 30.8 kbar; P – T results are quoted here and be-low to 0.1 kbar and 1 C, for comparative purposes;for reporting thermobarometric results commonlyquoting to 0.5 kbar and 5 or 10 C is usuallyappropriate). This is because the default activityuncertainty used by THERMOCALCTHERMOCALC in their calculations(Proyer et al.) sufficiently downweighted the inher-ently more sensitive Fe2+ end-member-bearing equi-libria (particularly the garnet-omphacite Fe-Mgexchange reaction) such that their initial results wereeffectively unaffected by using the wrong clinopy-roxene composition.

In the following, the presence of talc in the assem-blage (and the inferred original presence of a H2Ofluid) will be ignored so that focus can be on themineral assemblage of interest: garnet-omphacite-phengite-kyanite-coesite. The calculations are madewith THERMOCALCTHERMOCALC 3.26 (Powell et al., 1998; most recentupgrade), using the thermodynamic data set 5.5(Powell et al., 1998; most recent upgrade), and the a – xcoding in the datafile in Appendix 2. The compositionvariables for the minerals are given in the datafile inthe Appendix. First, some equilibria for this assem-

blage are considered, including the Krogh Ravna &Terry (2004) ones, with particular reference to theuncertainties that are involved, in the light of the newfacilities available in THERMOCALCTHERMOCALC. The following arethe results for this rock using the Mo ¨ ssbauer-deter-mined ferrous–ferric compositions for omphacite andgarnet:

T (C)/P (kbar) 24 26 28 30 32 34 36

py + gr + 2coe ¼ 3di + 2ky 779 730 685 642 601 562 525

3cel + 4ky ¼ py + 3mu + 4coe 558 589 620 649 678 706 732

gr + 3cel + 2ky ¼ 3di + 3mu + 2coe 448 517 586 653 719 784 848

alm + gr + 2coe ¼ 3hed + 2ky 461 525 588 651 715 778 842

3fcel + 4ky ¼ alm + 3mu + 4coe ) 459 611 759 899 + +

gr + 3fcel + 2ky ¼ 3hed + 3mu + 2coe 404 501 596 688 778 864 948

3hed + py ¼ 3di + alm 617 627 636 646 656 666 676

py + 3fcel ¼ alm + 3cel 614 618 622 625 629 633 637

di +vfcel ¼ hed + cel 601 575 549 524 500 476 452

and, for the intersection involving the first three reac-tions (as used by Krogh Ravna & Terry, 2004),

29.8 kbar and 646

C. The second set of three reac-tions are for the equivalent Fe2+ equilibria, with anintersection at 27.5 kbar and 571 C. The third set isfor the corresponding Fe-Mg exchange reactions.

If the calculation is performed with all Fe-as-FeO inthe omphacite (and garnet), the first intersection moveslittle, to 31.2 kbar and 666 C, well within the uncer-tainties on the above result (see next). This supportsthe assumption made by Krogh Ravna & Terry (2004)and Hacker (2006) that just using the Fe-free equilibriaminimizes the problems stemming from ferric iron notbeing known for microprobe analyses of minerals.Looking at the Fe-Mg garnet-omphacite exchangereaction, the temperature at 30 kbar moves from 646

to 952 C, if it is assumed that the omphacite is ferric-free (as would be deduced from a simple charge-bal-ance calculation, Appendix 1). The second intersectionmoves to very high temperatures and pressures if noferric iron is used.

The uncertainties involved in the equilibria stemfrom the mineral analyses and from thermodynamics.However if several eclogites are considered, and theirrelative positions in P – T are the main issue, thenuncertainties on the mineral analyses (randomuncertainties) can be considered alone, giving rise torelative uncertainties on results (see also Worley &Powell, 2000). However if one eclogite needs to beplaced in P – T , then all uncertainties need to be com-bined, giving absolute uncertainties on P – T (that ingeneral are much larger).

To start to evaluate the random uncertainty contri-butions to the uncertainties on P – T , the likely maincontributor is omphacite. Using only the Monte Carlo-derived uncertainties on the datafile coding variablesx ¼ Fe2+/(Fe2+ + Mg), j ¼ Na/(Na + Ca) and f ¼Fe3+/(Fe3+ + Al) (Fig. 3 in Appendix 1) gives29.8 ± 3.2 kbar and 646 ± 52 C (with a strong po-sitive correlation between them), stemming just fromthe resulting uncertainty on the activity of the diopside




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end-member, the only omphacite end-member in-volved. Including additional uncertainties from thecompositions of garnet and muscovite increase theseuncertainties by


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the current state of knowledge of thermodynamicmodels of phases, there are certainly rocks that cannotbe modelled reliably.

Pseudosection modelling has to be undertaken inthe context of the processes envisaged to have beeninvolved in the formation of the rocks. The morecomplex the metamorphic history, the more the

pseudosections that are likely to be needed to under-stand each rock. Separate pseudosections might beneeded to understand different parts of the progradeand retrograde history as the equilibration volumecomposition changes; separate pseudosections to con-sider assumptions relating to loss/gain of fluid/melt,with P – x and T – x pseudosections being particularlyrelevant, and so on. Modelling is not restricted to P – T ,P – x and T – x pseudosections. Variables to use as axeson pseudosections come in conjugate pairs of intensiveand extensive variables (e.g. P or V , T or S , and lk orxk, for each component, k), Powell et al. (2005).Depending on the processes envisaged to be involved

in the formation of a mineral assemblage, it may wellbe that P, T and x are not the most informative choiceof axes for a pseudosection. Even when knowledge of P – T – x is sufficient to understand rocks, bearing inmind that rocks do not form at a point on a P – T path,but are a more or less encoded record of a larger orsmaller section of the path, pseudosections can be usedto try and understand the preservation of the observedmineral assemblage and the compositions and zoningof the minerals.

In terms of interpreting the results of pseudosectionmodelling, assessment of uncertainties is important,and insufficient attention has been paid to this up tonow. As noted earlier, pseudosection calculations can

be devised to help, for example T – x and P – x pseudo-sections spanning the estimated equilibration volumecomposition. Such diagrams are particularly importantif in fact the observed mineral assemblage is not foundon the pseudosection, allowing an assessment of whe-ther it is a misassignment of composition (but withinlikely uncertainty) that is responsible. A subtle conse-quence of slightly inappropriate equilibration volumecomposition (or thermodynamic models) can be that amineral that would be present only in a minor amountis either present when it should not be, or absent whenit should be. Such a problem is common with hydrouseclogites in which garnet, omphacite and quartz makeup a substantial proportion of the rock, the remainderof the composition varying widely for small changes inthe equilibration volume composition. As a conse-quence these small changes can result in markedchanges in the remaining phases that nevertheless arepresent in small proportions.

Is there a role for conventional thermobarometry?Yes, in certain circumstances, if carefully applied. It isinteresting to look at the literature for the last decadeand see that in the majority of studies that haveadopted pseudosection modelling, conventional ther-mobarometry is not used (but see S ˘ tı ´pska ´ & Powell,

2005, for an exception). It is tempting to suggest thatonce metamorphic geologists adopt the pseudosectionapproach, conventional thermobarometry is seen asless useful. In fact we are not aware of any study wherethe two approaches have been used in competition witheach other (but see Coggon & Holland, 2002, for cal-culations on the Parigi garnet quartzite). It would be

interesting to know if there were examples where thepseudosection approach produced less useful resultsthan conventional thermobarometry. Such a casewould need to be made using a realistic assignment of uncertainties (as above, in this paper), as well as byusing the same thermodynamic models for the miner-als. It does not seem appropriate – now, in 2008 – touse just conventional thermobarometry in the absenceof reasons not to use the pseudosection approach.

A potentially important use for conventional ther-mobarometry is when the pseudosection approach isineffective. It might be useful for rocks in which thereare significant additional phases and/or elements such

that pseudosections cannot be calculated, but suchrocks are commonly accompanied by rocks that areamenable to pseudosection modelling. A situationwhere avPT might be valuable is where the rocks in-volve sufficient unconstrained factors, for examplerelating to presence/absence of fluid and to unknownactivities of H2O and CO2, in which a pseudosectionapproach would be difficult to apply usefully. In factPowell & Holland (1994) have an example where avPThas been used, omitting the end-members H2O andCO2 in the generation of the independent set of reac-tions for a carbonate-bearing rock (RP13), so that theresults are independent of fluid and its end-members.But corresponding pseudosections have not been

calculated for this example.Conventional thermobarometry may also have a

role to play where the field of interest on a pseudo-section is large and the mineral proportions andcompositions are slowly varying in P – T , for example inessentially anhydrous eclogites (e.g. Krogh Ravna &Terry, 2004; Hacker, 2006). However this is preciselythe situation where conventional thermobarometryfails, given that the spacing of isopleth (and mineralproportion) contours is intimately related to the P – T dependence in thermobarometers. Faced with such asituation, S ˘ tı ´pska ´ & Powell (2005) used g-cpx Fe-Mgexchange thermometry to get a temperature of 760 Cat 18 kbar with an uncertainty of the order of 100C,for a quarter-space pseudosection field that opens tohigher pressure and temperature from 17 kbar and700 C. In this circumstance, against suggestions of much higher temperature, this is a useful result for thisBohemian Massif eclogite. With a better thermody-namic model for clinopyroxene (see above), P – T couldmost likely be better constrained by contours of cal-cium Tschermak molecule, calcium Eskola molecule,etc., rather than us having to resort to exchangethermometry. In the context of such conventionalthermobarometry complementing pseudosection




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thermobarometry it is appropriate to consider the ap-proaches being integrated, and the possibility of doingthat is discussed below (as hybrid methods).

Another circumstance in which conventional ther-mobarometry has the capacity to complementpseudosection modelling is via chemical systems whichare effectively orthogonal to the systems that have

been used for pseudosection modelling. The recentexperimental calibration of various trace elementthermometers provides methods that, for example,involve the solubility of Ti in zircon (in equilibriumwith rutile and quartz), Zr in rutile (in equilibrium withzircon and quartz), and Ti in quartz (in equilibriumwith rutile) (Wark & Watson, 2006; Watson et al.,2006; Tomkins et al., 2007). These methods need notbe standalone: there is no reason why the equilibriainvolved cannot be incorporated into pseudosectionmodelling.

Considering standalone conventional thermoba-rometry, it is crucial that uncertainties are handled

realistically. This is because they constrain the esti-mated P – T , whereas commonly in pseudosectionthermobarometry it is the existence of the fields of alternative mineral assemblages that constrains P – T .As shown in the illustration above the situation isparticularly serious with Fe-Mg exchange thermome-try. In our experience, in the absence of constraintsfrom H2O-bearing reactions, temperature is not wellconstrained by thermobarometry, such that avP usinggeological prejudice for temperature is more useful (assuggested in Powell & Holland, 1994). With theexception of uncertainties arising from estimatingferric iron, uncertainties on a – x relationships arecommonly the most important uncertainties in avP

(and avPT).Conventional thermobarometry tends to have large

or relatively large uncertainties on individual P – T estimates (absolute uncertainties). However if relativethermobarometry is the purpose of studying a se-quence of rocks, and if they have the same mineralassemblage, the uncertainties on the P – T differencestend to be much smaller (i.e. the delPT approach of Worley & Powell, 2000). This is because the uncer-tainties deriving from the internally consistent data setand the a – x relationships tend to cancel, leaving justthe uncertainties stemming from the analysis of theminerals.

Thermobarometry of granulite facies and UHT rocks

As this is a Special Issue on Granulites and Granulites,this section considers some specific aspects of thermo-barometry on such rocks. Thermobarometry of granu-lite facies rocks and, even more so, UHT rocks, presentsparticular difficulties. They stem partly from the rolethat partial melting commonly plays, but mainly fromthe fact that diffusion, being strongly temperature-dependent, becomes notably faster into the granulitefacies and, even more so, into UHT conditions.

Partial melting plays a critical role in granulite faciesmetamorphism – in creating structural complexity vialeucosome development, and in controlling the pres-ervation of mineral assemblages via melt loss (e.g.White & Powell, 2002). Both are relevant in the contextof thermobarometry, the former in making it moredifficult to determine equilibration volumes, the latter

in relation to the P – T conditions that are representedin the preserved mineral assemblage.

Partial melting and melt loss also play a more subtlerole in changing rock composition and in loweringaH2O. If the prograde history of a rock that has in-volved melt loss is to be modelled using pseudosec-tions, then the addition of melt back into the observedrock composition needs to be attempted (e.g. Whiteet al., 2004). In thermobarometry of mineral assem-blages formed at subsolidus conditions, assumingfluid-present conditions, i.e aH2O ¼ 1, is commonly justifiable (at least when CO2 or CH4 is not impli-cated). In thermobarometry of mineral assemblages

formed above the solidus, aH2O is treated as an addi-tional unknown, along with P – T , in conventionalthermobarometry. In pseudosection modelling it isreasonable to assume that the rocks were just H2O-saturated at the solidus, for first-cycle metasedimentsthat were H2O-saturated throughout their progradehistory. However this is unlikely to be a goodassumption for subsequent cycle metasediments (i.e.ones being reworked), if they were substantially driedout in a previous metamorphism, and not then rehy-drated. Such concerns may apply to any orthogneissthat may never have been H2O-saturated.

Faster diffusion at higher temperatures introduces adifferent set of problems. At lower temperature, the

preservation of a mineral assemblage when a rock usesup its fluid starting to retrogress (e.g. Guiraud et al.,2001) also appears to coincide with the preservation of mineral compositions. Particularly in the absence of ametamorphic fluid, temperature is effectively below theclosure temperature of the various exchange reactionsthat might continue to change the mineral composi-tions during cooling. Into the granulite facies this is nolonger true. Feldspar and Fe-Ti oxides do not recordtheir peak metamorphic compositions. Continued Fe-Mg exchange between minerals occurs during cooling.In such circumstances, in which the mineral composi-tions cannot be used reliably, conventional ther-mobarometry is of little use, unless original mineralcompositions can be reconstructed (e.g. Pattison &Be ´ gin, 1994; Pattison et al., 2003).

Pseudosection thermobarometry, which does notrely on observed mineral compositions, can be used if the composition of an equilibration volume can beestablished. Even in texturally complex rocks, forexample when high-grade mineral assemblages havebeen affected by retrogression, such pseudosectionmodelling can be successful. However the retrogradepartitioning of coarse-grained rocks into domains bydiffusion in the earlier stages of decompression/cooling




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can make it difficult to establish an appropriate com-position for pseudosection modelling to consider peakmetamorphic conditions. Modelling the textural com-plexity itself is also possible but is likely to require theuse of chemical potential-chemical potential diagrams(e.g. White et al., 2008). This is because the domainscommonly cannot be considered as closed systems as

some elements are mobile on a scale larger than thedomains, being available as needed in various domains(complex textures of White et al., 2008).

Extreme melting and melt loss mean that consider-ation of a rock’s prograde history may be effectivelyimpossible. Even if an equilibration volume composi-tion can be established, recognition of the peak meta-morphic mineral assemblage itself can be difficult. Forexample, a pseudosection may have fields with thediagnostic sapphirine + quartz mineral assemblage,but it may not be clear whether the two minerals evercoexisted in the rock, with for example the sapphirineoccurring in domains that have become SiO2-under-

saturated early in the cooling history. Particularproblems may occur in orthogneisses in which igneousprecursor minerals are preserved, presenting the dan-ger of combining minerals with a quite different historyfor thermobarometry, generating misleading results(e.g. Racek et al., 2008).

In summary, pseudosections are likely to be the mosteffective form of thermobarometry for granulite faciesand UHT rocks, on taking into account the provisosand difficulties discussed above, and with the adoptionof the strategies for pseudosection modelling outlinedin the previous section.

Towards the future: hybrid methodsThermobarometry using pseudosections has beenessentially qualitative, once the composition of theequilibration volume is chosen, and the diagram drawn(the forward part of the modelling). The inverse partof the modelling involves a qualitative comparison of the observed mineral compositions and proportionswith the calculated equilibria. On the other hand,conventional thermobarometry is pure inverse model-ling, but only using the observed mineral composi-tions, not the composition of the equilibration volumenor the mineral proportions.

Hybrid methods can be envisaged that use more (orall) of the observations. The prerequisite for this is thatthe modelling can be done with a chemical system thatis sufficiently close to that in which the equilibrationvolume occurs (e.g. NCKFMASHTO). Then it isstraightforward to set up a weighted least squaresproblem in which the mineral compositions are to bematched with the calculated values, the mineral pro-portions with the calculated ones, and the equilibrationvolume composition with the calculated one (via thecalculated mineral proportions and compositions).Similar to avPT, this could involve minimizing in aleast squares sense the adjustments of the mineral

proportions, mineral compositions and the equilibra-tion volume composition to give the (optimal) P – T result.

Such a complete hybrid method has not yet beenimplemented. Compared with the qualitative use of pseudosections, it would have the merit of bringing intoplay the uncertainty in the equilibration volume com-

position, the effect of which can be difficult to assesscurrently (see below). It is worth noting that the vari-ance of the mineral assemblage in the model system isrelevant in considering the control of the equilibrationvolume composition on phase relationships, as well asthe dependence of mineral compositions on this com-position. For a divariant mineral assemblage, all themineral compositions are fixed by P – T . The equili-bration volume composition controls only the positionof the P – T field where the divariant assemblage occurs,not the mineral compositions. For progressively highervariance fields, the mineral compositions tend tobecome progressively more dependent on the equili-

bration volume composition, as well as on P – T . In suchhigher variance fields, thermobarometry, via calculatedmineral composition or mineral proportion isopleths,actually requires an input rock composition for deter-mining P – T . In a hybrid method, the imposed rockcomposition becomes a critical constraint, allowingP – T estimation in rocks where conventional thermo-barometry cannot be applied at all.

Regardless of the dependence of calculated mineralcompositions on the equilibration volume composition(as well as on P – T ), the advantage of pseudosections isthat calculated mineral compositions can be looked atdirectly. In contrast, in conventional thermobarometrythe mineral compositions are present only in the

encoded form of the equilibrium constant for theequilibria being considered. It is the nature of thisencoding that gives rise to the more or less strong cor-relations between the reaction lines in P – T . Because of this it is usually difficult to see what the consequence isof varying mineral compositions, or conversely what theimplied compositions are, having performed an avPTcalculation. It is possible to calculate the adjustedactivities implied by the avPT result, but generally notwhat are the implied mineral compositions.

An added strength of hybrid methods relates tohandling unknowns in thermobarometry in addition toP – T , like aH2O or aO2. The dependence of results on asuperimposed value of say a

O2can be established in

conventional thermobarometry, but only in the contextof observed (or estimated) ferric iron in the minerals.In hybrid methods that involve a model chemical sys-tem that approaches the rock one, there are moreinteresting possibilities that involve superimposing sayaO2 and then calculating the ferric iron contents of theminerals using it. These contents can then be comparedwith the observed or estimated values, or can be used

per se if these values are not known (as is commonlythe case). This then amounts to a generalization of theabove complete hybrid approach in which, addition-




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ally, some compositional variables of the minerals arecalculated as part of the solution, at specified values of intensive variables.

ACK NO WLEDGEMENT S

We thank the convenors of the Granulites and

Granulites meeting in Brasilia in 2006 for theopportunity to talk about thermobarometry there,and to contribute here. We would like to acknowl-edge discussions concerning most aspects of mineralequilibria over the last several years with R. White.J. Diener, R. White and E. Green are thanked forreading earlier versions of the manuscript. D. Tink-ham, C. Warren and Z. Page are thanked for helpfulreviews, and M. Brown for valuable editorial guid-ance in how to improve the manuscript.

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Received 06 May 2007; revision accepted 06 November 2007.

AP P ENDI X 1 : I DEAL ANALY SI S AP P R O ACH

The ideal analysis approach is a tool for handling mineralchemical analyses, first suggested by Carson & Powell (1997) andalso used by S ˘ tı ´pska ´ & Powell (2005). It involves calculating ananalysis which is closest (in some sense) to the real analysed onebut which has specified stoichiometric or other constraints. Thiscalculated analysis can be thought of as the ideal analysis thatmatches these constraints. The simplest way of doing this calcu-lation is to use weighted least squares: the sum of the squares of

the variations from the original analysis, weighted by the corre-sponding uncertainties, is minimized subject to the specified con-straints.

The ideal analysis approach complements a consideration of thepropagation of analytical uncertainties (stemming primarily fromthe Poisson counting statistics in electron probe microanalysis) (e.g.Hodges & McKenna, 1987; Kohn & Spear, 1991). For visual powerin considering the propagation of uncertainties, the best approach isa Monte Carlo one in which many virtual analyses (produced fromthe real one by varying the oxide wt% within their uncertainties)




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are recalculated and presented for example in histogram form. It iseasy in such an approach to exclude virtual analyses that conflictwith given stoichiometric constraints, particularly inequalities, forexample, Si > 2 in omphacite (recalculated on 6 oxygen). In fact,in combination with the ideal analysis approach, Monte Carlo canbe used to propagate analytical uncertainties with stoichiometricequalities obeyed identically.

The power of the ideal analysis approach comes when there are

unknowns in the mineral analysis, for example the proportion of all-Fe-as-FeO to be converted to Fe2O3, and the standard approach is touse a stoichiometric constraint to estimate it. Such so-called ferricfiddles are notoriously senstive to the quality of the mineral analysis.Sometimes th

powell et al-2008-journal of metamorphic geology

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