1. Introduction and Overview. 2. Detecting Influential Observations and Outliers. 3. Detecting and Assessing Collinearity. 4. Applications and Remedies. 5. Research Issues and Directions for Extensions. Bibliography. Author Index. Subject Index.

Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

A review of the geologic history of the Himalayan-Tibetan orogen suggests that at least 1400 km of north-south shortening has been absorbed by the orogen since the onset of the Indo-Asian collision at about 70 Ma. Significant crustal shortening, which leads to eventual construction of the Cenozoic Tibetan plateau, began more or less synchronously in the Eocene (50–40 Ma) in the Tethyan Himalaya in the south, and in the Kunlun Shan and the Qilian Shan some 1000–1400 km in the north. The Paleozoic and Mesozoic tectonic histories in the Himalayan-Tibetan orogen exerted a strong control over the Cenozoic strain history and strain distribution. The presence of widespread Triassic flysch complex in the Songpan-Ganzi-Hoh Xil and the Qiangtang terranes can be spatially correlated with Cenozoic volcanism and thrusting in central Tibet. The marked difference in seismic properties of the crust and the upper mantle between southern and central Tibet is a manifestation of both Mesozoic and Cenozoic tectonics. The form...

Geologic Evolution of the Himalayan-Tibetan Orogen

The thermodynamic properties of 154 mineral end-members, 13 silicate liquid end-members and 22 aqueous fluid species are presented in a revised and updated data set. The use of a temperature-dependent thermal expansion and bulk modulus, and the use of high-pressure equations of state for solids and fluids, allows calculation of mineral–fluid equilibria to 100 kbar pressure or higher. A pressure-dependent Landau model for order–disorder permits extension of disordering transitions to high pressures, and, in particular, allows the alpha–beta quartz transition to be handled more satisfactorily. Several melt end-members have been included to enable calculation of simple phase equilibria and as a first stage in developing melt mixing models in NCKFMASH. The simple aqueous species density model has been extended to enable speciation calculations and mineral solubility determination involving minerals and aqueous species at high temperatures and pressures. The data set has also been improved by incorporation of many new phase equilibrium constraints, calorimetric studies and new measurements of molar volume, thermal expansion and compressibility. This has led to a significant improvement in the level of agreement with the available experimental phase equilibria, and to greater flexibility in calculation of complex mineral equilibria. It is also shown that there is very good agreement between the data set and the most recent available calorimetric data.

An internally consistent thermodynamic data set for phases of petrological interest

Amphibole thermodynamics are approximated with the symmetric formalism (regular solution model for within-site non-ideality and a reciprocal solution model for cross-site-terms) in order to formulate improved thermometers for amphibole-plagioclase assemblages. This approximation provides a convenient framework with which to account for composition-dependence of the ideal (mixing-on-sites) equilibrium constants for the equilibria:
 For A and B all possible within-site and cross-site interactions among the species □−K−Na−Ca−Mg−Fe2+−Fe3+−Al−Si on the A, M4, M1, M3, M2 and T1 amphibole crystallographic sites were examined. Of the 36 possible interaction energy terms, application of the symmetric formalism results in a dramatic simplification to eight independent parameters. Plagioclase nonideality is modelled using Darken's quadratic formalism. We have supplemented an experimental data set of 92 amphibole-plagioclase pairs with 215 natural pairs from igneous and metamorphic rocks in which the pressure and temperature of equilibration are well constrained. Regression of the combined dataset yields values for the eight interaction parameters as well as for apparent enthalpy, entropy and volume changes for each reaction. These parameters are used to formulate two new thermometers, which perform well (±40°C) in the range 400–1000°C and 1–15 kbar over a broad range of bulk compositions, including tschermakitic amphiboles from garnet amphibolites which caused problems for the simple thermometer of Blundy and Holland (1990). For silica-saturated rocks both thermometers may be applied: in silica-undersaturated rocks or magmas thermometer B alone can be applied. An improved procedure for estimation of ferric iron in calcic amphiboles is presented in the appendix.

Non-ideal interactions in calcic amphiboles and their bearing on amphibole-plagioclase thermometry

https://www.perplex.ethz.ch/papers/connolly_epsl_05.pdf

Computation of phase equilibria by linear programming: A tool for geodynamic modeling and its application to subduction zone decarbonation

The thermodynamic properties of 254 end-members, including 210 mineral end-members, 18 silicate liquid end-members and 26 aqueous fluid species are presented in a revised and updated internally consistent thermodynamic data set. The PVT properties of the data set phases are now based on a modified Tait equation of state (EOS) for the solids and the Pitzer & Sterner (1995) equation for gaseous components. Thermal expansion and compressibility are linked within the modified Tait EOS (TEOS) by a thermal pressure formulation using an Einstein temperature to model the temperature dependence of both the thermal expansion and bulk modulus in a consistent way. The new EOS has led to improved fitting of the phase equilibrium experiments. Many new end-members have been added, including several deep mantle phases and, for the first time, sulphur-bearing minerals. Silicate liquid end-members are in good agreement with both phase equilibrium experiments and measured heat of melting. The new dataset considerably enhances the capabilities for thermodynamic calculation on rocks, melts and aqueous fluids under crustal to deep mantle conditions. Implementations are already available in thermocalc to take advantage of the new data set and its methodologies, as illustrated by example calculations on sapphirine-bearing equilibria, sulphur-bearing equilibria and calculations to 300 kbar and 2000 °C to extend to lower mantle conditions.

An improved and extended internally consistent thermodynamic dataset for phases of petrological interest, involving a new equation of state for solids

We discuss the use of regression diagnostics combined with nonlinear least-squares to refine cell parameters from powder diffraction data, presenting a method which minimizes residuals in the experimentallydetermined quantity (usually 20hkt or energy, Ehkt). Regression diagnostics, particularly deletion diagnostics, are invaluable in detection of outliers and influential data which could be deleterious to the regressed results. The usual practice of simple inspection of calculated residuals alone often fails to detect the seriously deleterious outliers in a dataset, because bare residuals provide no information on the leverage (sensitivity) of the datum concerned. The regression diagnostics which predict the change expected in each cell constant upon deletion of each observation (hkl reflection) are particularly valuable in assessing the sensitivity of the calculated results to individual reflections. A new computer program, implementing nonlinear regression methods and providing the diagnostic output, is described. I~YWORDS: powder diffraction, regression diagnostics, lattice parameters, computer program.

Unit Cell Refinement from Powder Diffraction Data: The Use of Regression Diagnostics

For petrological calculations, including geothermobarometry and the calculation of phase diagrams (for example, P–T petrogenetic grids and pseudosections), it is necessary to be able to express the activity–composition (a–x) relations of minerals, melt and fluid in multicomponent systems Although the symmetric formalism—a macroscopic regular model approach to a–x relations—is an easy-to-formulate, general way of doing this, the energetic relationships are a symmetric function of composition We allow asymmetric energetics to be accommodated via a simple extension to the symmetric formalism which turns it into a macroscopic van Laar formulation We term this the asymmetric formalism (ASF) In the symmetric formalism, the a–x relations are specified by an interaction energy for each of the constituent binaries amongst the independent set of end members used to represent the phase In the asymmetric formalism, there is additionally a "size parameter" for each of the end members in the independent set, with size parameter differences between end members accounting for asymmetry In the case of fluid mixtures, for example, H2O–CO2, the volumes of the end members as a function of pressure and temperature serve as the size parameters, providing an excellent fit to the a–x relations In the case of minerals and silicate liquid, the size parameters are empirical parameters to be determined along with the interaction energies as part of the calibration of the a–x relations In this way, we determine the a–x relations for feldspars in the systems KAlSi3O8–NaAlSi3O8 and KAlSi3O8–NaAlSi3O8–CaAl2Si2O8, for carbonates in the system CaCO3–MgCO3, for melt in the melting relationships involving forsterite, protoenstatite and cristobalite in the system Mg2SiO4–SiO2, as well as for fluids in the system H2O–CO2 In each case the a–x relations allow the corresponding, experimentally determined phase diagrams to be reproduced faithfully The asymmetric formalism provides a powerful and flexible way of handling a–x relations of complex phases in multicomponent systems for petrological calculations

Activity–composition relations for phases in petrological calculations: an asymmetric multicomponent formulation

We present, as a progress report, a revised and much enlarged version of the thermodynamic dataset given earlier (Holland & Powell, 1985). This new set includes data for 123 mineral and fluid end-members made consistent with over 200 P–T–XCO2–fO2 phase equilibrium experiments. Several improvements and advances have been made, in addition to the increased coverage of mineral phases: the data are now presented in three groups ranked according to reliability; a large number of iron-bearing phases has been included through experimental and, in some cases, natural Fe:Mg partitioning data; H2O and CO2 contents of cordierites are accounted for with the solution model of Kurepin (1985); simple Landau theory is used to model lambda anomalies in heat capacity and the Al/Si order–disorder behaviour in some silicates, and Tschermak-substituted end-members have been derived for iron and magnesium end-members of chlorite, talc, muscovite, biotite, pyroxene and amphibole.



For the subset of data which overlap those of Berman (1988), it is encouraging to find both (1) very substantial agreement between the two sets of thermodynamic data and (2) that the two sets reproduce the phase equilibrium experimental brackets to a very similar degree of accuracy. The main differences in the two datasets involve size (123 as compared to 67 end-members), the methods used in data reduction (least squares as compared to linear programming), and the provision for estimation of uncertainties with this dataset. For calculations on mineral assemblages in rocks, we aim to maximize the information available from the dataset, by combining the equilibria from all the reactions which can be written between the end-members in the minerals. For phase diagram calculations, we calculate the compositions of complex solid solutions (together with P and T) involved in invariant, univariant and divariant assemblages. Moreover we strongly believe in attempting to assess the probable uncertainties in calculated equilibria and hence provide a framework for performing simple error propagation in all calculations in thermocalc, the computer program we offer for an effective use of the dataset and the calculation methods we advocate.

An enlarged and updated internally consistent thermodynamic dataset with uncertainties and correlations: the system K2O–Na2O–CaO–MgO–MnO–FeO–Fe2O3–Al2O3–TiO2–SiO2–C–H2–O2

mproved activity–composition relationships for biotite, garnet and silicate liquid are used to construct updated P–T grids and pseudosections for high-grade metapelites. The biotite model involves Ti charge-balanced by hydrogen deprotonation on the hydroxyl site, following the substitution , where HD represents the hydroxyl site. Relative to equivalent biotite-breakdown melting reactions in P–T grids in K2O–FeO–MgO–Al2O3–SiO2–H2O (KFMASH), those in K2O–FeO–MgO–Al2O3–SiO2–H2O–TiO2–O2 (KFMASHTO) occur at temperatures close to 50 °C higher. A further consequence of the updated activity models is that spinel-bearing equilibria occur to higher temperature and higher pressure. In contrast, the addition of Na2O and CaO to KFMASH to make the Na2O–CaO–K2O–FeO–MgO–Al2O3–SiO2–H2O (NCKFMASH) system lowers key biotite-breakdown melting reactions in P–T space relative to KFMASH. Combination of the KFMASHTO and NCKFMASH systems to make Na2O–CaO–K2O–FeO–MgO–Al2O3–SiO2–H2O–TiO2–O2 (NCKFMASHTO) results in key biotite-breakdown melting reactions occurring at temperatures intermediate between those in KFMASHTO and those in NCKFMASH. Given such differences, the choice of model system will be critical to inferred P–T conditions in the application of mineral equilibria modelling to rocks. Further, pseudosections constructed in KFMASH, NCKFMASH and NCKFMASHTO for several representative rock compositions show substantial differences not only in the P–T conditions of key metamorphic assemblages but also overall topology, with the calculations in NCKFMASHTO more reliably reflecting equilibria in rocks. Application of mineral equilibria modelling to rocks should be undertaken in the most comprehensive system possible, if reliable quantitative P–T information is to be derived.

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1525-1314.2007.00711.x

Tim Holland

Papers

An improved and extended internally consistent thermodynamic dataset for phases of petrological interest, involving a new equation of state for solids

Unit Cell Refinement from Powder Diffraction Data: The Use of Regression Diagnostics

Activity–composition relations for phases in petrological calculations: an asymmetric multicomponent formulation

An enlarged and updated internally consistent thermodynamic dataset with uncertainties and correlations: the system K2O–Na2O–CaO–MgO–MnO–FeO–Fe2O3–Al2O3–TiO2–SiO2–C–H2–O2

Progress relating to calculation of partial melting equilibria for metapelites