American Mineralogist, Volume 76, pages 128-137, 1991

Error propagation for barometers: 1. Accuracy and precision ofexperimentally located end-member reactions

Mlrrnnw J. KonN. Fru.Nr S. SpplnDepartment of Geology, Rensselaer Polytechnic Institute, Troy, New York 12180, U.S.A.

AssrRAcr

A new procedure for calculating the precision and accuracy of end-member reactions ispresented that accounts for the kinetics of reaction progress and possible displacementsbetween the experimental and true P-7 reference frames. Application to recently collectedexperimental data indicates that the estimated precision (lo) of the position of an end-member reaction at metamorphic temperatures (-900 K) using a single set of experimentaldata may be + 100 to + 310 bars and that a minimum estimate of accuracy (lo) may beon the order of +270 to +410 bars, depending on how precisely the reaction is bracketedand assuming lo potential systematic errors in pressure and temperature of +250 barsand +5 eC, respectively. The accuracy of the grossular-anorthite-kyanite-quartz (GASP)reaction position using five different studies is estimated to be between +350 and +380

bars (lo) at 900 K, depending on whether the individual studies were calibrated usingindependent standards.

INrnooucrrox

The accuracy and the precision of end-member reac-tions are of fundamental importance for understandingthe uncertainty of both barometer calibrations and pres-sure estimates made on natural rocks. Although severalmethods have been presented that purport to estimatethe precision of reaction positions (e.g., Demarest andHaselton, l98l; Hodges and McKenna, 1987; Powell andHolland, 1985; Berman et al., 1986), none of these meth-ods simultaneously accounts for experimental bracket size,the reproducibility of experimental conditions, and thekinetics of disequilibrium reaction progress. Similarly, al-though Hodges and McKenna (1987) and McKenna andHodges (1988) have described a method whereby the ac-curacy of an end-member reaction may be estimated, theirresults are inconsistent with the magnitudes of the poten-tial systematic errors that have been reported for exper-imental devices.

Generally there are three types of errors that must beconsidered: (l) the precision with which a reaction hasbeen bracketed, (2) systematic errors that result from un-certainty in the position of the standard reactions usedfor calibration, and (3) interlaboratory, procedural, andstarting material variability. Each one of these error prop-agates to substantial pressure uncertainties.

The purposes of this paper are (l) to present a newmethod for calculating the precision of an experimentallylocated reaction constrained by a single set of data thatis more in accord with bracket sizes and kinetic theory,(2) to describe the minimum uncertainty of an end-mem-ber reaction constrained by only a single set of data bypropagating systematic experimental uncertainties re-ported in the literature, and (3) to present a technique fordetermining a best-fit position of a reaction and assigning

0003-004x/9 I /0 I 02-0 I 28$02.00 128

realistic uncertainties to that position when there are sev-eral sets of experimental data representing different lab-oratories and starting materials.

In this paper, we have followed the general terminologyof Hodges and McKenna (1987), and the term "preci-sion" will refer to reproducibility or randomly distributederrors, as contrasted to "systematic errors." The term"accuracy" will be reserved to describe the combinationof precision and systematic uncertainties.

PnncrsroN oF REAcrroNS: srNGr,E DATA sETS

Kinetic theory

Two different approaches have been typically used toestimate the precision of reaction positions: linear (ormathematical) programming and linear regression. Lin-ear programming relies on the assumption that for anysingle experimental bracket, the position ofthe reactionis equally likely at any point within the bracket. That is,experiments only contain information about the directionof equilibrium, not its distance from the experimentalconditions. Use of linear regression implies that the mostlikely position for a reaction is in the middle of a bracket.

Transition state theory (e.g., Fisher and Lasaga, 198 l;Lasaga, l98lb) provides a useful theoretical frameworkwithin which to evaluate the assumptions inherent in thetwo techniques for estimating reaction precisions. An im-portant aspect ofthis theory revolves around the math-ematical description of the means by which reactant istransformed to product. For many reactions, this trans-formation occurs via an activated complex, which has ahigher Gibbs free energy than either the reactant or prod-uct. The difference between the energy of the activatedcomplex and that of the reactant is referred to as theactivation energy.

Because we know ofno studies that have investigatedquantitatively experimental reaction kinetics for pressuresensitive equilibria, we will assume that the activatedcomplex for the forward and reverse reactions is the same.If this assumption is made, then transition state theorypredicts that the reaction rate will have the form:

( la)

where R,., is the net reaction rate in units of concentrationper time, R* and R are the absolute reaction rates inthe forward and reverse directions, RZis the gas constanttimes temperature in kelvins,,4 is the reaction affinity, orthe absolute Gibbs free energy change between reactantand product that drives the reaction, and n is a constantof order one (Fisher and Lasaga, l98l; Lasaga, 1986). Wenote that R"., may have a different functional dependenceon I if defects dominate the kinetics (e.g., Lasaga, 1986),but that data collected by A. Koziol (personal commu-nication) for the reaction grossular-kyanite-quartz-pla-gioclase (GASP) favor Equation la. In the case of thebracketing ofpressure sensitive equilibria (i.e., at a spe-cific temperature), A equals the absolute value of ApAZ,where AZis the volume difference between reactants andproducts and AP is the difference between the experi-mental and equilibrium pressures:

r29

Although few brackets meet these qualifications, Equa-tion lb also predicts that any two experiments conductedat different pressures with reaction progress less than I 000/oshould uniquely define the position of the equilibrium.The true imprecision in the position of a reaction basedon such experiments then depends on how well reactionprogress can be measured, and as experimentalists be-come better at differentiating between small amounts ofreaction progress, reaction positions should become in-creasingly more precise. It is worth noting, however, that(l) if one of the reactants or products has been entirelyconsumed during the experiment, then no unique solu-tion for the reaction position is obtainable, and (2) be-cause R+ begins to deviate from constancy for large de-grees of reaction progress (>40o/o to 75o/o) at a givenpressure, temperature, and composition, estimating theinitial R,", in such a case may require extra experimentsfor different amounts of time.

The transition state theory as applied to experimentalreaction progress has profound implications on whetherlinear programming or linear regression is better suitedfor estimating the precision of a reaction position fromexperimental brackets. If experimental brackets representl00o/o reaction progress, then no best estimate of the re-action position is obtainable based on kinetic theory, anda linear programming approach is probably better. Onthe other hand, if at least two experiments that representpartial reaction progress are performed at the same tem-perature but at different pressures, then an estimate ofthe most likely reaction position is possible from kinetictheory, and linear regression is probably better.

Because experimental data collected in the last few yearsfor pressure-sensitive equilibria increasingly representseveral experiments at a single temperature that exhibitonly partial reaction progress, it seems likely that linearregression analysis will become the method of choice forcalculating individual reaction positions and their preci-sions. However, to justify this approach it is importantfor experimentalists to report kinetic data, includingmasses and average grain sizes ofall phases in the startingmixture and products, along with the conditions and du-ration of the experiments. Furthermore, investigations ofreaction rate as a function of pressure overstepping arerequired either to verify that transition state theory is anadequate model or to develop a better kinetic model forpressure-sensitive equilibria. In the remainder of this pa-per, we will rely on linear regression for estimating re-action positions and their uncertainties. For simplicity werestrict discussion to experimental reversal data, in whicheach reversal is composed of two experiments conductedat the same temperature that show opposite reaction di-rections.

An example using hansition state theory toestimate reaction position

Suppose we have collected the hypothetical experimen-tal data for the GASP reaction as listed in Table I andwish to calculate the most likely reaction position and its

KOHN AND SPEAR: ACCURACY AND PRECISION OF BAROMETERS

R"o :R*+R:* . [ r - * r (# ) ]

(lb)

R* is generally a complicated function of concentrationand temperature, so that R,", depends on the duration ofthe experiment. This means that the magnitude of A andhence of the difference between experimental and equi-librium P-?"conditions is best judged from the initial R".,,which is usually estimated using many experiments con-ducted at the same P-Z conditions for different lengthsof time. Fortunately, several experimental studies (e.g.,Schramke et al., 1987; Heinrich et al., 1989) indicate that,even for relatively large changes in the number of molesof the phase used to monitor reaction progress, R* isnearly constan! that is, the extrapolated initial R"", isoften equal to the measured R".,. R* will be assumed tobe essentially constant for the small to moderate amountsof reaction considered below.

We know of no theoretical or empirical considerationsthat suggest that pressure has a significant effect on R*;therefore, for a given 7 and with the assumptions aboveregarding the degree ofreaction progress, the magnitudeof n, and the nature of the activated complex, the mag-nitude ofR"., is simply dependent on the amount of over-stepping (APAQ and will be symmetric about the truereaction position for brackets of pressure-sensitive equi-libria. This means that the most likely position for a re-action that is bracketed by two experiments that (l) showequal but opposite degrees of reaction progress and (2)were conducted for the same amount of time, will be atthe midpoint of the bracket.

n.",:n.fr -".r(Y;1

130 KOHN AND SPEAR: ACCURACY AND PRECISION OF BAROMETERS

Op

r(K) P(kbar) (kbar)

TABLE 1. Hypothetical experimental data for GASP reaction of starting material, then the terms ,flp,od.i,i,, l, and S arenot required, as they do not afect the magnitude of "4.

An example of the application of Equation 3b to a setof hypotheticaldata is given in Table l. We first assurnedthat there were stoichiometric proportions of grossularand anorthite in the starting material, the initial moles ofgrossular (mo,"..,^,,) was 2 x tg-c (90 mg), the initial peakheight intensity ratio (r,",.) was 1.0, the durations (r) were20 h, and the surface area of grossular (S) was 10 cm2(average grain radius of 7 5 pm). Using these values andthe hypothetical data in Table I for T, P, ro^u, and o,nnu,,the measured reaction rates and their uncertainties maybe calculated from Equation 3b, and are tabulated in Ta-ble l. From the measured reaction rates, Equation lb canbe solved using a Newton-Raphson procedure to obtainthe equilibrium pressure, which is 20527 bars. The best-fit reaction position is at a sligfutly higher pressure thanthe midpoint of the hypothetical bracket because the re-action rate was faster for the lower pressure reaction;therefore, this fictive experiment must have been fartherfrom the equilibrium position than the higher pressureexperiment.

Uncertainties in the reaction position may be estimat-ed using a Monte Carlo approach (Anderson, 1976) ornumerical error analysis (Roddick, 1987). For the data inTable l, either of these procedures results in an estimateduncertainty of +145 bars (la), ofwhich about +70 barsrepresents the contribution of o", while approximately+ 125 bars represents the effect of o,n"., (Table l).

The calculated uncertainty is much smaller than thatestimated using the method described by Demarest andHaselton (1981), which considers the reaction positionuncertainty to be a function of the full bracket width aswell as the experimental pressure precision. For the datain Table l, the approach of Demarest and Haselton re-sults in an estimated reaction position of 20500 bars withan uncertainty of +305 bars (lo). Thus, transition statetheory can allow a much more precise estimate of thereaction position than would be suggested by Demarestand Haselton (1981).

A physical depiction of the results above is shown inFigure I for the hypothetical data. This figure shows thedata and uncertainties in the the pressure and reactionrate plotted on a normalized reaction rate vs. pressurediagram. The three curves show the best-fit and boundingfunctions predicted for the data by transition state theory.The 2o confidence limits for the equilibrium pressure arecalculated from the intersection of the bounding curvesand the horizontal line defined by R,.,/R* : 0. The 2oconfidence limits calculated from the technique of Dema-rest and Haselton ( I 9 8 I ) are also presented and are clearlymuch larger than those estimated using transition statetheory.

Regression analysis: an example

Regression analysis ofthe precision ofbracketed equi-libria has been described by several different workers (e.9.,Demarest and Haselton, l98l; Powell and Holland, 1985;Hodges and McKenna, 1987). Many workers choose the

o.

R*t(mole/cm'.h)

(mole/cm, h)

1223 21.OO1223 20.00

1.3 0 .10.75 0.06

1 . 3 x 1 0 7 4 x 1 0 '- 1 . 4 x 1 0 7 4 x 1 O - 8

0.10.1

Note; The following assumptions were made: duration of each experi-ment was 20 h, initial moles of grossular was 2 t tg-r (90 mg), the initialpeak height intensity ratio was 1.0, and the surface area of grossular was10 cm, (average grain radius of 75 pm). The term lhd is the measuredpeak height intensity ratio (Grs/An) of the product, and d.* is the standarddeviation of the measurement of /ih"r. Bd is calculated from Equation 3b;o* calculated from propagation of o.,,, through Equation 3b.

uncertainty. For the initial molar quantities of reactantand product (ffi,.',n, and mo,-.,",,), the rate of change inthe number of moles of the phase 7 may be calculatedfrom the equation:

net rate of production or consumption of product

(rn"" - r,.,,)v.,.o/l(2)

(v,"rr rn r/ m,",^rr) * (v."6rs"J rfl o,"a,, r)

where 2.", and zo.oo are the absolute values of the stoichio-metric coefficients of the reactant and product in the re-action ofinterest (i.e., 3 and I for anorthite and grossularin the GASP reaction), I is the duration of the experi-ment, and r0.., and 4.i ?re the peak height intensity ratios(product/reactant) of the final product and the initialstarting material. We note here that although we haveassumed in our example that reaction progress is moni-tored using X-ray diffraction methods, the ratios in Equa-tion 2 are perfectly general, and other techniques ofmea-suring reaction progress may also be used (e.g., SEMimaging).

Equation 2 simplifies significantly if the starting ma-terials are mixed in stoichiometric proportions (i.e., ra*.,nn: v,.rm o,o6,1n1r/ u o,o6)i

net rate of production ot _(rr,", - rioi)mo,oo;^ir/t /,_\

consumption of product :

rr, + r-r,- (Ja''

Net reaction rates in terms of changes of concentrationper time may be calculated by normalizing Equation 3ato an extensive property of the system. For heterogeneousreactions (Lasaga, l98la) such as the GASP reaction, theimportant parameter is the surface area (,S) of one of thephases in the experiment, usually one with a low solubil-itv:

net rate ofproduction orconsumption of product (rr-" - r;,Jzro.o6,i"irlS

(/,,,, + rn"u)l(3b)

If (l) only the reaction position and its uncertainty aredesired, and (2) the composition of the starting mix andduration are the same for all the experiments, and notingthat surface areas are linearly proportional to the amount

,s

KOHN AND SPEAR: ACCURACY AND PRECISION OF BAROMETERS l 3 r

midpoints of brackets when regressing experimental dataand then weight each point according to the size of thebracket. The differences among various regression ap-proaches described in the literature often lies in how un-certainties are assigned to a single bracket (e.g., Demarestand Haselton, l98l; Powell and Holland, 1985) andwhether or not uncertainties in temperature are account-ed for (e.g., Hodges and McKenna, 1987). Other differ-ences include the choice of regression type, some workerspreferring a weighted least squares method (e.g., powelland Holland, 1985), while others have used a Monte Car-lo approach (e.g., Hodges and Crowley, 1985). As shownbelow, the variations in the assignment of uncertainty toeach datum have a nontrivial effect on the calculated re-action position.

We believe it is quite important that for many experi-mental data sets, the bracket midpoints are much morecolinear than is consistent with the error assigned to eachbracket according to the bracket width and experimentalreproducibility. In such cases, estimates of the reactionposition uncertainty based on the scatter ofthe data aboutthe best-fit line will be erroneously small, and should becalculated from the uncertainties assigned to the datathemselves (see Draper and Smith, 1981, p. 83). Thisapproach has been used for several ofthe data sets con-sidered below.

As an exercise in regression analysis, we considered theeffects ofvarious regression and weighting approaches onthe calculated slope, intercept, and their uncertainties forthe data of Koziol and Newton (1988). Specifically, wecompared the results of an ordinary weighted least squaresregression using the uncertainties calculated from tran-sition state theory (TST, see below) with those obtainedusing the weights suggested by Demarest and Haselton(1981: DH) and by Powell and Holland (1985: pH), aswell as with those of a York (1969: YK) regression rec-ommended by Hodges and McKenna (1987) (Table 2).

Transition state theory may be applied to the data ofKoziol and Newton (1988) for the GASP reaction to pro-vide a set of nominal reaction positions and their uncer-tainties on which regressions can be performed. We firstassumed that (l) grossular and anorthite were initiallymixed in stoichiometric proportions (Koziol and New-ton, I 988), so that Equations 3a and 3b could be applied,and (2) any dissolution ofgrossular and anorthite into themelt phase present in the experiments did not affect theirrelative molar abundances. Based on grossular and an-orthite X-ray data for the experiments conducted at twotemperatures and general descriptions of all experiments(A. Koziol, personal communication), we concluded thatmost bracketing experiments represented between 100/oand 40o/o reaction progress. The assumption of stoichio-metric proportions of the starting material does notstrongly affect the reaction progress calculations, and theminimum acceptable change in X-ray peak intensity ra-tios (Koziol and Newton, 1988) implies reaction progressof at least 130/0. Finally, we assumed that the pressures ofthe experiments could be reproduced within + 100 bars;a poorer reproducibility is difficult to reconcile with the

P (kbar)

Fig. 1. Plot of R".,/R* vs. pressure for the hypothetical darain Table l. The curves show the functions that were fit to themidpoints of the data (middle curve) and the 2o bounds on thosedata using transition state theory (TST). Also shown are theuncertainties calculated using TST and the technique of Dema-rest and Haselton (1981) (DH).

internal consistency ofthe data. A nominal reaction po-sition and its uncertainty were then calculated for eachbracket as was done above for the hypothetical data.

Powell and Holland (1985) suggest assigning the brack-et midpoint as the nominal reaction position and thenassigning an uncertainty (lo) equal to the bracket quar-terwidth, although it should be understood that they wereonly attempting to obtain the reaction intercept position,not its slope; Hodges and McKenna (1987) suggest thattemperature imprecisions may be important when esti-mating slope and intercept uncertainties. Because Hodgesand McKenna (1987) did not state their preferred weight-ing scheme for pressures, we simply used the uncertain-ties calculated from transition state theory; this allows adirect evaluation of the importance of accounting for ex-perimental temperature imprecisions for this data set.Temperature imprecisions (lo) were assumed to be +5"C (A. Koziol, personal communication); this value istwice that suggested by Hodges and McKenna (1987). Anordinary weighted least squares method was used for thePH and DH regressions.

As can be seen in Table 2, the TST, PH, and YK stan-dard errors for slope and intercept are quite similar, andeach regression indicates similar uncertainties in the best-fit reaction position. Ignoring temperature irnprecisions(TST vs. YK) causes an underestimation of the reactionposition uncertainty of less than +30 bars over mid tolower crustal temperatures; if we had assumed the same2.5 'C uncertainty as suggested by Hodges and McKenna(1987), then the YK regression would actually have in-dicated smaller uncertainties than the TST. Thus, we seelittle practical advantage in accounting for temperatureimprecisions with data collected on modern experimentaldevices for pressure sensitive reactions (slopes less thanabout 25 bars/"C), although for later analysis, we accept

1.0

- F n t

S0)

20.720.319.9

.?,s. -----l' t ( D H ) l

F_ 20 __-{t (TST) |

r32 KOHN AND SPEAR: ACCURACY AND PRECISION OF BAROMETERS

TABLE 2. Selected regression results for GASP data sets a 2 i

Intercept + 1dCorrelation

Slope + 1o coefficient

Koziol and Newton (1980)-7253 + 817 22.70 + 0.59TST:

YK:DH:PH:

-7222 + 889-6128 t 1215

-5689 + 1535-5743 + 1710

-7373 + 2845-7438 + 2835

22.68 + 0.6521.94 + 0.88

-o.9977-0.9977-0.9973-0.9980-5873 + 826 21.76 + 0.60

Gasparik (1985)-3617 + 3038-3641 + 3256

19.97 + 1.98 -0.998719.98 + 2.11 -0.9987

DH:YK:

DH:YK:

DH:YK:

DH:YK:

Gasparik (1984)

Goldsmith (1980)

21.31 + 0.99 -0.997921.35 + 1.11 -0.9979

22.43 ! 1.80 -0.998022.48 + 1.80 -0.9980

-! ) 1 <JZ

Hay3 (1966)-7481 + 11496 23.18 + 7.28 -0.9996-7964 + 11664 23.50 !7.40 -0.9996

Notej TST, DH, and PH refer to regressions weighted according totransition state theory, Demarest and Haselton (1981), and Powell andHolland (1985) respectively. YK refers to regression using the t€chniqueof York (1969).

the YK results in Table 2 as the best estimate of theuncertainty in reaction position. The DH standard errorsare approximately 1.5 times larger than those calculatedusing the other approaches as a result of the larger un-certainties assigned to each datum. Because the PH andDH slopes and intercepts are somewhat different fromthe results of the other two regressions, predicted pres-sures differ by about 100-300 bars at the temperaturesof the experiments, and by about 350-650 bars at crustaltemperatures; the predicted pressures using the PH andTST regression are the most different ofthose considered.

Precision of reaction position

The precision of the reaction position may be calculat-ed by propagating the uncertainties in the regressed slopeand intercept and constructing corresponding 2o confi-dence limits. For a line defined by the equation:

Y : m X + b ( 4 )

where m is the slope and D is the intercept, the propagateduncertainty in Y may be calculated as:

o2v: oZ + (Xo-)'z 't 2p^6o6Xo- (5)

where o, and o^are the uncertainties in the intercept andslope respectively, and p-u is the correlation coemcientbetween m and b.

The 2o confidence limits for the reaction position re-sulting from the propagated YK regression uncertaintiesand correlation coefficient are shown in Figure 2 for thedata of Koziol and Newton (1988) (inner bounds) withthe experimental brackets for reference. As can be seen,the limits are well within seven of the brackets and areslightly inconsistent (although within assigned experi-mental reproducibility) with only one very narrow brack-et. The estimated precision of the reaction position usingthe YK regression is about +60 bars (1o) at 1350 K, nearthe midpoint of the data and +310 bars (lo) at 900 K.

125 gN loo0 1100 12rm l3o0 14oo l50o 1600 lioo

T(K)

Fig.2. P-T diagram showing the experimental reversals re-ported by Koziol and Newton (1988) for the GASP reaction,with the best-fit reaction position and 2o precision (inner enve-lope) and accuracy (outer envelope) confidence limits. Inset showstwo brackets with reaction position and confidence limits forclarity. Precision of the reaction position for each bracket wasestimated using transition state theory.

As a final note on precision, we would like to point outthat the kinetics of some reactions are fast enough so thatvery precise reaction positions are available from thebracket positions themselves. For example, the equilibriainvestigated by Bohlen et al. (1980, 1983a, 1983b) andBohlen and Liotta (1986) are extremely tightly bracketedand have precisions of better than +100 to +200 barsover experimental and metamorphic temperature ranges.Thus, applying kinetic theory to some experimental datacan substantially improve the precision of the reactionlocation (e.g., the Koziol and Newton data set), whileother reactions are kinetically so rapid that transition statetheory provides little additional constraint.

Accuru,cv oF REAcrroNs: STNGLE DATA sET

Method

It is an unfortunate fact that experimental devices aresubject to systematic errors, and that these errors may belarge compared to the precision to which a reaction isbracketed. As was the case with experimental reactionkinetics, the magnitude of systematic errors over P-Ispacefor different experimental devices is not particularly well

KOHN AND SPEAR: ACCURACY AND PRECISION OF BAROMETERS r33

known, but some estimates have been summarized byHodges and McKenna (1987), and the interested readeris referred to that paper and to the references therein.Typical reported values for potential systematic pressureand temperature uncertainties (lo) for piston cylinder de-vices using salt cell assemblies are +250 bars and +2.5-5'C, respectively.

The effect of systematic elrors on experimental data isquite different from that of imprecisions, as depicted inFigure 3, which shows the brackets for a hypotheticalreaction as well asthe 2o confidence limits correspondingto the propagated precision for the reaction. The easiestway to deal with systematic errors in pressure is to assignthem as an uncertainty in the reference pressure. That is,the pressure axis in Figure 3,A, has a systematic uncer-tainty (o.r,.,). Similarly, systematic uncertainties in tem-peratuie contribute to the error in pressure as the productof the slope of the reaction (m) and the uncertainty intemperature (orrr",: Fig. 3B). As noted by Hodges andMcKenna (1987) the reaction slope is independent of theaccuracy of the experimental device, and so is availablefrom the regression of the experimental data for the pre-cision of the bracketed reaction.

Once systematic errors in pressure and temperaturehave been assigned and the precision of the slope andintercept have been estimated from the regression oftheexperimental data, the accuracy ofthe reaction positionmay be calculated from the equation:

a?: 4.r* * o?.s"* t (mor,"r*.)z (6)

where op,s"", and or.."", are the estimates of systematic pres-sure and temperature uncertainties and or** is the pre-cision ofthe reaction position as calculated using Equa-tion 5.

Applications

The accuracy of the data of Koziol and Newton (1988)using the YK regression (Table 2) after assigning esti-mated systematic experimental pressure and temperatureuncertainties of +250 bars and +5. C is

o,2 : (889)' + (0.657), - 2(0.9977)(889X0.652)

+ (250F + (5X22.68)l '

where the first three terms on the right side of the ex-pression represent the precision of the reaction positionas calculated using Equation 5. Because the slope ofthereaction is rather shallow in P-T space (22.68 bars/K),the systematic uncertainty in temperature does not sig-nificantly affect the accuracy ofthe reaction position, asindicated by the small magnitude of the last term.

The resulting total 2o confidence limits for the reactionare also plotted in Figure 2 (outer bounds). As can beseen, the relative contribution of the systematic errors tothe uncertainty in the reaction position is greatest nearthe midpoint of the data and has diminishing importanceas pressures are extrapolated outside the range ofthe data,and reaction precision becomes worse.

The accuracies represented by the bounds ofFigure 2

oP,syst

Temperature

TemperatureFig. 3. Schematic P-Idiagrarns showing effects of systematic

errors. (A) Systematic errors in pressure (o.""",) are equivalent toan uncertainty in the reference pressure. (B) Systematic errors intemperature (o.".,) introduce an uncertainty in pressure (dP) equalto the product of the slope (m) and o." ,.

are substantially smaller than those estimated by Mc-Kenna and Hodges (1988) for the data of Koziol andNewton (1988). For example, at 900 K, our propagateduncertainty (lo) is about a4l0 bars while McKenna andHodges (1988) estimate the uncertainty to be about +2400bars (lo) and virtually independent of temperature. Infact the discrepancy between our results and thein issomewhat larger than these numbers might at first indi-cate because we chose to omit a theoretical bracket at923 K when performing our regressions. If this bracketis included in the regression, then our method predictsan accuracy of about +375 bars (lo) at 900 K. We arenot able to explain how McKenna and Hodges derivedsuch large uncertainties, but we note: (l) when we applythe Hodges and McKenna technique (tlodges and Mc-Kenna, 1987; McKenna, personal communication) as-suming systematic uncertainties as described above, wemerely reproduce our estimates of the accuracy, and (2)the accuracies described by McKenna and Hodges (1988)

()Lr

cr)tt)()H

c.)L<

tt)(t)c)L<

dP

t34

for the GASP data are physically inconsistent with thediscussion in Hodges and McKenna (1987) regarding ex-perimental uncertainties. Possibly, McKenna and Hodges(1988) used different values or measures for systematicexperimental errors.

For the dara sers ofBohlen et al. (1980, 1983a, 1983b)and Bohlen and Liotta (1986), the accuracy of the end-member reactions at 900 K is estimated to be between+270 and +330 bars (lo) for the same assumed system-atic uncertainties in experimental pressures and temper-atures.

We emphasize Ihat any realistic error propagationanalysis for single data sets requires a priori knowledgeof potential systematic uncertainties in the pressure andtemperature of the experiments, because any indicationof their magnitudes is precluded using data collected onthe same experimental device with the same standardsand operating procedures. Only by comparing the resultsobtained on a single reaction using different types ofex-perimental devices (e.g., Hays and Bell, 1973; Bohlen,1984) or using different calibrations of the same type ofapparatus (e.9., Johannes et al., l97l) can any statisticalestimate of systematic experimental uncertainties andhence ofreaction position accuracy be made.

Mur,rrpr,n DATA sETS

While the description above is valid when propagatingerrors for single data sets, some reactions have been stud-ied by several experimentalists. Therefore, it is desirableto develop a method whereby multiple data sets can beused together to obtain a best-fit reaction position and toassign uncertainties to that position. Furthermore, it isby comparing different data sets that estimates of the ef-fects of interlaboratory and starting material variabilitycan be made.

If different data sets are collected with different system-atic errors, they will be displaced from each other whenplotted together. The effect of the displacement of theexperimental reference frames relative to each other leadsto very real difficulties in estimating both the line thatbest fits experimental data collected on more than oneapparatus and the errors corresponding to that best-fitline. For example, it is generally not justified to regressseveral different data sets together to estimate slope andintercept because of systematic pressure and temperatureerrors between different data sets and because the re-gressed best fit line will be most strongly controlled bythe tightest reversals, even ifthose reversals occur in dif-ferent data sets. Clearly, simultaneously regressing morethan one set of experimental data can result in an erro-neous slope and intercept.

Another important concern that must be addressed isthe standards used in each study. If the same referencestandards have been used, then discrepancies should re-flect variability in reactant and product materials, or someinterlaboratory variation not related to the uncertainty ofthe standard reactions; therefore, this variability consti-tutes precision, and additional terms, representing poten-tial systematic errors would be required to determine ac-

KOHN AND SPEAR: ACCURACY AND PRECISION OF BAROMETERS

curacy. If different standards have been used, then thevariability in a bracketed reaction position between stud-ies may be taken as a measure of interlaboratory varia-tion (including starting materials and standard reactions),and so the variability may be simply taken as the accu-facy.

Realistically, it may be difficult or impossible to dem-onstrate the independence of the standards used in dif-ferent experiments. Some standards, such as the albite-jadeite-quartz reaction and salt melting curves, are usedalmost globally for calibrating piston cylinder devices,and it is unlikely that many ofthe experiments conductedwithin the last few years are truly independent with re-spect to their standards. Consequently, it is probably moregenerally correct to calculate uncertainties in reaction po-sition assuming the same standards for different data sets;as is shown below, this approach tends to maximize cal-culated uncertainties for the data on GASP.

One approach to calculating slopes and intercepts formultiple data sets that preserves individual internal con-sistency is to regress each set ofdata separately for slopeand intercept pressure. All individual slopes and pres-sures may then be averaged, weighting each value in ac-cordance with the uncertainty of each parameter. Follow-ing Young (1962), a mean value and a minimum limiton its uncertainty when individual data are of varyingquality are given by the equations:

t / - \ l / l / r \ lu:12l+l l / l> l+l l e)

L \o ;,/ ) / L \o ;,,/l

and

where trr is the mean value and x, and o., are the individualvalues and their uncertainties. Another useful quantity isthe (weighted) standard deviation of the observationsabout the mean:

6,,: > {t(a - fi/o",1'1f )lr/oi,l. (8b)

Whereas Equation 8a represents a lower limit on the un-certainty in the mean value, Equation 8b is a direct mea-surement of the variability of the observations about themean. Note that the mean value (p) and its uncertainty(o,) must be explicitly evaluated at each value of x,. Forexperimental data such as for the GASP reaction, thisimplies that the best-fit reaction position may not be astraight line.

Applications: accuracy of the GASP reaction position

Experimental determinations of the end-member GASPreaction at different temperatures have been determinedin seven studies (Hays, 1966; Hariya and Kennedy, 1968;Schmid et al.,1978; Goldsmith, 1980; Gasparik, 1984,1985; and Koziol and Newton, 1988). We have chosento exclude the two studies of Hariya and Kennedy (1968)and Schmid et al. (1978) from consideration because ofthe difficulties of interpreting their data in terms of the

(8a).t:, /, ("r,9

KOHN AND SPEAR: ACCURACY AND PRECISION OF BAROMETERS 1 3 5

restrictive requirements for the technique developed abovefor single data sets. The remaining five studies may notbe completely independent ofeach other because all havebeen calibrated in part to the albite-jadeite-quartz equi-librium and because the corrections to the data of Gas-parik (1985) are partly based on the positions ofreactionsdetermined by his earlier study (Gasparik, 1984). How-ever, the accepted positions of standard reactions havechanged over the last 25 years, and not all studies usedthe same set of standards. We will first assume that thestandardizations were truly independent, and the possi-bility oflack ofstandard independence will be discussedlater.

Consistent estimates of the slope and its possible un-certainties for the combination of all five data sets maybe calculated from the individual slopes and their uncer-tainties listed in Table 2 (YK), using Equations 7, 8a, and8b; the results are independent of both Z and assump-tions regarding systematic errors, and are listed at thebottom of Table 3. Use of Equations 8b indicates thatthe scatter of the individual slopes about the weightedmean is approximately 1.5 times the statistical minimumcalculated using Equation 8a.

Assuming standards are independent. The pressures ofthe position of an individual reaction at a given temper-ature may be calculated from individual regressed slopesand intercepts (Table 2 and Eq. 4). However, the propa-gation of uncertainties in pressure depends on whetherthe standards used by the different studies are indepen-dent. If they are independent, then the precisions andaccuracies of pressure for each set of experimental dataat the temperature of interest may be calculated by usingEquation 5 and then Equation 6, and a consistent meanand its possible uncertainties may be estimated by usingEquations 7, 8a, and 8b; that is, the calculations followthe sequence: Equations 5, 6,7, and then 8. This proce-dure accounts for systematic uncertainties prior to thecalculation ofp and o,.

Assuming that the standardizations of the five GASPdata sets are independent, the weighted mean pressureand its uncertainties for a variety of temperatures are thevalues listed in columns 2. 3. and 4 of Table 3. Thesewere calculated by using the YK regression results of Ta-ble 2 and by assuming systematic pressure and temper-ature uncertainties of +250 bars and t5 'C for each ofthe four most recent data sets, and +500 bars and + lOeCfor the data ofHays (1966). The uncertainty in pressurecalculated using Equation 8b (column 4) is larger thanthat calculated using Equation 8a (column 3), except inthe range 800- I 300 K. This is because the two most pre-cise data sets (Gasparik, 1984; Koziol and Newton, 1988)converge in this temperature rang€. The larger of the twouncertainties tabulated in the third and fourth columnsof Table 3 ranges from about +200 bars at the experi-mental temperatures to about +350 bars at crustal tem-peratures.

Assuming standards are the same. Although it mightbe argued that if the standards are the same, then thedata from the diferent studies can be regressed together,

Tlele 3, Average pressures, slope, and unc€rtainties for GASPreaction

Average pressure and its uncertainty at different temperatures

Assuming difierenlstandards: Assuming same standards:

Uncertainties UncertaintiesP(bars) (bars) P(bars) (bars)

r ( $ Eq .7 Eq .8a Eq .8b Eq .7 Eq .8a Eq .8b Eq .6(1) (2) (3) (4) (s) (6) (7) (8)

0 -6756 764 957750 9941 402 408800 11055 380 374850 12169 359 342900 13284 3d!8 310950 14398 317 280

1000 15512 298 2521100 17738 261 2051200 19954 229 1781300 22152 201 1791400 24320 176 2091500 26469 155 2701600 28646 157 3672000 37554 288 646

Average slope andits uncertainty

(independent ot f)

-6780 7399910 348

11025 32212140 29613256 27014373 24415491 21717731 16s19980 11422240 7024473 5926515 6928635 8037505 327

937 976365 456328 427293 403258 384224 367192 349135 31997 29697 290

140 307262 379353 447658 713

Slope Uncertainties(bars/K) (bargK)

Eq .7 Eq .8a Eq .8b

22.22 o.52 0.78

/Vofe: Average slope and its uncertainties are independent of interpre-tation of experimental calibration standards and temperature. Accuracies(column 8) were calculated using Equation 6 from the maximum of theEquation 8a and 8b pressure uncertainties (columns 6 and 7) and fromassumed J250 bars and +5'C systematic pressure and temperatureuncertainties.

we believe that this approach is only justifiable in theextremely unlikely event that exactly the same startingmaterials and laboratory procedures were used. Becauseof the variability of fluxes used in the GASP studies, thefact that most of the studies were conducted in differentlaboratories, and the long period of time over which thedifferent studies were conducted, we do not believe thata simultaneous regression of all GASP data is warranted,even if the standard reactions were identical. Conse-quently, we favor application ofEquations 7, 8a, and 8b.

Assuming the same reference reactions for the differentstudies, uncertainties in the position of an individual re-action may be calculated from Equation 5. With these o*,new values of pr and its uncertainties may be calculatedby using Equations 7,8a, and 8b. These calculations arereported in Table 3 for the GASP data in the fifth, sixth,and seventh columns. The accuracy of the reaction po-sition may then be estimated from Equation 6 by assum-ing values for the systematic uncertainties in pressure andtemperature and by noting thal or.r*" is the uncertainty inp at the temperature of interest; that is, the equations areapplied in the sequence 5,7,8, and then 6, and so p andits precision are calculated before accounting for system-atic uncertainties in pressure and temperature. For theGASP data, we assumed systematic pressure and tem-

o

a

I

tr

Hays (1966)

Goldsmith (1980)

Gaspuik (1984)

Gaspaik (1985)

Koziol md Newton (198E)

136 KOHN AND SPEAR: ACCURACY AND PRECISION OF BAROMETERS

tsg 22.s

t25-- - 900 1000 ll0o 12c0 1300 1400 1500 1600 1700

T(K)

Fig. 4. P-T diagram showing the experimental reversals re-ported for the GASP reaction in five studies (Hays, 1966; Gold-smith, 1980; Gasparik, 1984, 1985; and Koziol and Newton,I 988), with the best-fit reaction positions and 2o confidence lim-its. Inset shows brackets, reaction positions (central two curves;lower assumes that different experimental calibration standardswere used for each experimental data set, upper assumes samestandards), confidence limits assuming different calibration stan-dards for each experimental data set (intermediate two curves),and confidence limits assuming the same standards for each dataset (outer two curves).

perature uncertainties of +250 bars and +5 oC, respec-tively, and applied Equation 6 using the maximum ofprecisions for multiple data sets reported in Table 3 (col-umns 6 and 7); the results are tabulated in column 8 ofTable 3. For this case, the accuracy of the GASP reactionposition at 900 K is estimated to be approximately + 380bars (lo).

Figure 4 shows the best-fit reaction position (fourth linefrom top), the individual brackets, and the 2o confidencelimits (second and fifth lines from top) for the assumptionthat the standards were different, and the best-fit reactionposition (third line from top) and 2o confidence limits(outer envelope) corresponding to the assumption that thestandards were the same for all data sets. For the latterassumption the data of Koziol and Newton (1988) aremuch more heavily weighted than the other sets of data,and so the best-fit reaction position and confidence limitsare bowed to higher pressure near the weighted meantemperature ofthese data. Because very similar standards

were used to calibrate the precise data of both Gasparik(1984) and Koziol and Newton (1988) (e.g., the albite-jadeite-quartz reaction location of Holland, 1980), we fa-vor the assumption that the standards were the same foreach data set and believe that the outer bounds ofFigure4 best represent the accuracy of the GASP reaction po-sition based on the five data sets. Furthermore, we be-lieve it is likely that the disparity ofpressures at the ex-perimental conditions (about +200 bars) representsdifferences in starting materials or unspecified interlabor-atory variability. That is, a measure of interlaboratory,procedural, and starting material variability using the dif-ferent GASP determinations may be approximately +200bars (lo).

As with the results for the single data set of Koziol andNewton (1988), our estimate of the accuracy is muchsmaller than that of McKenna and Hodges (1988). Usinga data set similar to that shown in Figure 4, McKennaand Hodges (1988) calculated a Ir accuracy at 900 K ofabout +1150 bars. As demonstrated in our study, theGASP data sets are simply much more consistent witheach other than the uncertainties calculated by McKennaand Hodges (1988) would imply.

DrscussroN

The small errors indicated for the precision and accu-racy of the position of the GASP reaction should not beassumed to apply to all experimentally determined equi-libria. The small propagated errors are a result of theconsistency of several high-quality experimental deter-minations of the reaction. Equilibria that have only beenstudied in a single set of experiments usually cannot fulfillthese conditions and may be correspondingly less accu-rate.

On the other hand, it should be recognized that a singleset of extremely precise data (e.g., GRAIL) may be justas accurate as several sets ofonly moderately precise data(e.g., the collective GASP data), especially if the moder-ately precise data sets are calibrated to the same stan-dards. For example, our analysis of the extremely tightlyreversed single data sets presented by Bohlen et al. (1980,1983a, 1983b) and Bohlen and Liotta (1986) all indicateprecisions of better than a 100 to +200 bars (lo) at meta-morphic temperatures. If potential systematic experi-mental pressure and temperature errors are assumed tobe +250 bars and +5 "C, and if interlaboratory and start-ing material variabilities are assumed to contribute +200bars of uncertainty (from the calculations above for theGASP reaction), then the accuracy of each reaction po-sition is about +340 to +380 bars (lo) at metamorphictemperatures. That is, the reactions studied by Bohlen etal. (1980, 1983a, 1983b) and Bohlen and Liotta (1986)may be just as accurately located as the GASP reaction,even though the GASP reaction has been more intenselystudied.

Although it may be argued that the systematic experi-mental uncertainties we have assumed are not correct, itis relatively straightforward to adjust our estimates for

KOHN AND SPEAR: ACCURACY AND PRECISION OF BAROMETERS r37

different values of systematic errors. We leave it to thosewho are skeptical (or at least uncertain) to perform anyrecalculations according to his or her belief as to the mag-nitude of such errors. Furthermore, we note that our anal-ysis only accounts for those errors about which we know(e.g., bracket widths) or can make assumptions (e.g., sys-tematic errors). There may, of course, be other unknownsources oferror that bias reaction positions, and we cau-tion that our estimates of uncertainty must be minima.

Finally, we emphasize that the preceding analysis doesnot represent the total uncertainty in the pressure ofequilibration of a natural sample. Additional sources oferror such as precision and accuracy of electron micro-probe analyses, uncertainty in activity coefficients, andinterpretation of compositional heterogeneity in naturalsamples must be considered, as is discussed in part 2(Kohn and Spear, l99l).

AcxNowr,rncMENTs

This study is part of the M.S. work of M.J.IC We thank S. Bohlen. T.Gasparik, K. Hodges, A. Koziol, L. McKenna, E.B. Watson, and E. youngfor reviewing early version of this paper, S. Roecker and A. Koziol fordiscussing error propagation and experimental techniques, and T. Hoisch,R. Powell, and J.C. Roddick for formal reviews; we panicularly appre-ciated R. Powell's correcting some "wrong-headed reasoning" on our part.We thank L. McKenna for supplying us with a printout of his york re-gression program and for spurring our interest in the York-type regressronapproach. We are very grateful to A. Koziol for supplying us with unpub-lished X-ray data of the GASP experiments; however, any misinterpre-tation or misuse of the data is entirely ow responsibility. This work wasfunded by NSF grants EAR-8708609 and EAR-8803785 to F.S.S. and aNational Science Foundation graduato fellowship to M.J.K.

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Me.r*uscmrr RECETVED J,Nr;,c.Rv 29, 1990Me.Nuscnrrr AccEprED Noveunen 19, 1990