P -T paths from anatectic pelites

A relatively simple petrogenetic grid for partial melting of pelitic rocks in the NCKFMASH system is presented based on the assumption that the only H₂O available for melting is through dehydration reactions. The grid includes both discontinuous and continuous Fe-Mg reactions; contours of Fe/(Fe+Mg) for continuous reactions define P-T vectors along which continuous melting will occur. For biotite-bearing assemblages (garnet+biotite + sillimanite + K-feldspar + liquid and garnet + biotite + cordierite + K-feldspar + liquid), Fe/(Fe+Mg) contours have negative slopes and melting will occur with increasing temperature or pressure. For biotite-absent assemblages (garnet + cordierite + sillimanite + K-feldspar + liquid or garnet + cordierite + orthopyroxene + K-feldspar + liquid) Fe/(Fe + Mg) contours have flat slopes and melting will occur only with increasing pressure. The grid predicts that abundant matrix K-feldspar should only be observed if rocks are heated at P < 3.8 kbar, that abundant retrograde muscovite should only be observed if rocks are cooled at P > 3.8 kbar, and that generation of late biotite + sillimanite replacing garnet, cordierite, or as selvages around leu- cosomes should be common in rocks in which melt is not removed. There is also a predicted field for dehydration melting of staurolite between 5 and 12 kbar. Textures in migmatites from New Hampshire, USA, suggest that prograde dehydration melting reactions are very nearly completely reversible during cooling and crystallization in rocks in which melt is not removed. Therefore, many reaction textures in “low grade” migmatites may represent retrograde rather than prograde reactions. Received: 5 March 1998 / Accepted: 7 August 1998     

Eclogite-facies fluid infiltration: constraints from δ18O zoning in garnet

In situ analysis reveals that eclogite-facies garnets are zoned in δ¹⁸O with lower values in the core and rims that are ~1.5 to 2.5 ‰ higher. This pattern is present in 9 out of 12 garnets analyzed by SIMS from four orogenic eclogite terranes, and correlates with an increase in the mole fraction of pyrope and Mg/Fe ratio from core to rim, indicating prograde garnet growth. At the maximum temperatures and the time-scales experienced by these garnets, calculated intragranular diffusion distances for oxygen are small (<5 μm), indicating that δ¹⁸O records primary growth zoning and not diffusive exchange. The oxygen isotope gradients are larger than could form due to temperature changes during closed-system mineral growth. Thus, gradients reflect the compositions of fluids infiltrating during prograde metamorphism. Values of δ¹⁸O in garnet cores range from ?1 to 15 ‰, likely preserving the composition of the eclogite protoliths. Two garnet cores from the Almenningen eclogite in the Western Gneiss Region, Norway, have δ¹⁸O ~?1 ‰ and are the first negative δ¹⁸O eclogites identified in the region. In contrast with orogenic eclogites, seven high δ¹⁸O garnets (>5 ‰) from two kimberlites are homogeneous in δ¹⁸O, possibly due to diffusive exchange, which is possible for prolonged periods at higher mantle temperatures. Homogeneity of δ¹⁸O in garnets outside the normal mantle range (5–6 ‰) may be common in kimberlitic samples.     

The P–T–t paths of high-grade gneisses,Kaoko Belt,Namibia: Constraints from mineral data,U–Pb allanite and monazite and Sm–Nd/Lu–Hf garnet ages and garnet ion probe data

The Damara Orogeny is a late Neoproterozoic to Cambrian (ca. 570–480 Ma) intracratonic event that affected the Kaoko Belt, the inland branch of the Damara orogen and the Gariep Belt in Namibia and South Africa. This study focuses on the Pan-African evolution of part of the Kaoko Belt between the Puros shear zone and the Village mylonite zone which consists of Mesoproterozoic migmatitic para- and orthogneisses with minor granulite and amphibolite. Pseudosection modeling combined with thermobarometric calculations indicate that the para- and orthogneisses equilibrated at about 670–800 °C and ca. 0.6–0.8 GPa. Some garnets display a pronounced bell-shaped Ca, HREE, Y and Sr zoning, flat zoning profiles of Mn and Fe and concave upward concentration profiles of Sm and Nd. Pressure–temperature estimates obtained on these garnets reveal similar temperatures of 700–750 °C but slightly higher pressures of ca. 0.9 GPa. The preservation of distinct major and trace element zoning in garnet and the existence of broadly similar (near prograde) Sm–Nd and Lu–Hf garnet–whole rock ages of ca. 525 Ma obtained on the same sample indicate an extremely fast cooling path. Retrograde conditions persisted until ca. 490 Ma indicating a slow, late stage near isobaric cooling path. The resulting clockwise P–T–t path is consistent with crustal thickening through continent–continent collision followed by post-collisional extension and suggests that the upper amphibolite to granulite facies terrain of the central Kaoko Belt formed initially in a metamorphic field gradient of ca. 25–35 °C km^− 1 at moderately high pressures.     

Prograde garnet growth along complex <Emphasis Type="Italic">P–T–t</Emphasis> paths: results from numerical experiments on polyphase garnet from the Wölz Complex (Austroalpine basement)

Garnet in metapelites from the Wölz Complex of the Austroalpine crystalline basement east of the Tauern Window characteristically consists of two growth phases, which preserve a comprehensive record of the geothermal history during polymetamorphism. From numerical modelling of garnet formation, detailed information on the pressure–temperature–time (P–T–t) evolution during prograde metamorphism is obtained. In that respect, the combined influences of chemical fractionation associated with garnet growth, modification of the original growth zoning through intragranular diffusion and the nucleation history on the chemical zoning of garnet as P and T change during growth are considered. The concentric chemical zoning observed in garnet and the homogenous rock matrix, which is devoid of chemical segregation, render the simulation of garnet growth through successive equilibrium states reliable. Whereas the first growth phase of garnet was formed at isobaric conditions of ～3.8 kbar at low heating/cooling rates, the second growth phase grew along a Barrovian P–T path marked with a thermal peak of ～625°C at ～10 kbar and a maximum in P of ～10.4 kbar at ～610°C. For the heating rate during the growth of the second phase of garnet, average rates faster than 50°C Ma^?1 are obtained. From geochronological investigations the first growth phase of garnet from the Wölz Complex pertains to the Permian metamorphic event. The second growth phase grew in the course of Eo-Alpine metamorphism during the Cretaceous.     

Trace element zoning in garnet from the Kwoiek Area,British Columbia: disequilibrium partitioning during garnet growth?

Trace element zoning in garnets from two contact-metamorphosed rocks from the Kwoiek area, British Columbia (Hollister 1969a), was measured with an ion microprobe. Zoning profiles have three distinct parts with chemical breaks defined by co-variation of major and trace elements. Important features of the trace element zoning profiles are: (1) roughly bell-shaped zoning profiles for Y and the HREEs, (2) an abrupt increase in Ti at a point midway through each garnet with inflections in the zoning profiles of other elements (Li, Na, Cr, V, Y, Zr, and the HREE), and (3) irregular Cr and V profiles. Unlike Mn zoning, the zoning profiles of most other trace elements cannot be easily modeled using simple Rayleigh fractionation models. Ti activity in the two samples is buffered by phase relations with ilmenite. Garnets from a continuously heated contact metamorphic environment should display continuous Ti zoning profiles if equilibrium was maintained and provided the Ti buffering assemblage did not change during garnet growth. The irregular Ti profiles suggest disequilibrium behavior. Several elements (Cr, V) may indicate breakdown of a phase enriched in trace elements during metamorphism. The source for the mass excess of these elements is probably the refractory cores of ilmenite grains. Either differing matrix transport rates of trace lements or interface kinetic controlled segregation could explain the unusual trace element behavior at the element inflection point. The preferred explanation involves segregation of elements at the interface of the garnet that were trapped during episodes of rapid garnet growth.     

Compositional zoning of garnet porphyroblasts from the polymetamorphic Wölz Complex, Eastern Alps

We employ garnet isopleth thermobarometry to derive the P–T conditions of Permian and Cretaceous metamorphism in the Wölz crystalline Complex of the Eastern Alps. The successive growth increments of two distinct growth zones of the garnet porphyroblasts from the Wölz Complex indicate garnet growth in the temperature interval of 540°C to 560°C at pressures of 400 to 500 MPa during the Permian and temperatures ranging from 550°C to 570°C at pressures in the range of 700 to 800 MPa during the Cretaceous Eo-Alpine event. Based on diffusion modelling of secondary compositional zoning within the outermost portion of the first garnet growth zone constraints on the timing of the Permian and the Eo-Alpine metamorphic events are derived. We infer that the rocks remained in a temperature interval between 570°C and 610°C over about 10 to 20 Ma during the Permian, whereas the high temperature stage of the Eo-Alpine event only lasted for about 0.2 Ma. Although peak metamorphic temperatures never exceeded 620°C, the prolonged thermal annealing during the Permian produced several 100 µm wide alteration halos in the garnet porphyroblasts and partially erased their thermobarometric memory. Short diffusion profiles which evolved around late stage cracks within the first garnet growth zone constrain the crack formation to have occurred during cooling below about 450°C after the Eo-Alpine event.     

Multiple P–T–d–t paths reveal the evolution of the final Nuna assembly in northeast Australia

The final assembly of the Mesoproterozoic supercontinent Nuna was marked by the collision of Laurentia and Australia at 1.60 Ga, which is recorded in the Georgetown Inlier of NE Australia. Here, we decipher the metamorphic evolution of this final Nuna collisional event using petrostructural analysis, major and trace element compositions of key minerals, thermodynamic modelling, and multi-method geochronology. The Georgetown Inlier is characterised by deformed and metamorphosed 1.70–1.62 Ga sedimentary and mafic rocks, which were intruded by c. 1.56 Ga old S-type granites. Garnet Lu–Hf and monazite U–Pb isotopic analyses distinguish two major metamorphic events (M1 at c. 1.60 Ga and M2 at c. 1.55 Ga), which allows at least two composite fabrics to be identified at the regional scale—c. 1.60 Ga S1 (consisting in fabrics S1a and S1b) and c. 1.55 Ga S2 (including fabrics S2a and S2b). Also, three tectono-metamorphic domains are distinguished: (a) the western domain, with S1 defined by low-P (LP) greenschist facies assemblages; (b) the central domain, where S1 fabric is preserved as medium-P (MP) amphibolite facies relicts, and locally as inclusion trails in garnet wrapped by the regionally dominant low-P amphibolite facies S2 fabric; and (c) the eastern domain dominated by upper amphibolite to granulite facies S2 foliation. In the central domain, 1.60 Ga MP–medium-T (MT) metamorphism (M1) developed within the staurolite–garnet stability field, with conditions ranging from 530–550°C at 6–7 kbar (garnet cores) to 620–650°C at 8–9 kbar (garnet rims), and it is associated with S1 fabric. The onset of 1.55 Ga LP–high-T (HT) metamorphism (M2) is marked by replacement of staurolite by andalusite (M2a/D2a), which was subsequently pseudomorphed by sillimanite (M2b/D2b) where granite and migmatite are abundant. P–T conditions ranged from 600 to 680°C and 4–6 kbar for the M2b sillimanite stage. 1.60 Ga garnet relicts within the S2 foliation highlight the progressive obliteration of the S1 fabric by regional S2 in the central zone during peak M2 metamorphism. In the eastern migmatitic complex, partial melting of paragneiss and amphibolite occurred syn- to post- S2, at 730–770°C and 6–8 kbar, and at 750–790°C and 6 kbar, respectively. The pressure–temperature–deformation–time paths reconstructed for the Georgetown Inlier suggest a c. 1.60 Ga M1/D1 event recorded under greenschist facies conditions in the western domain and under medium-P and medium-T conditions in the central domain. This event was followed by the regional 1.56–1.54 Ga low-P and high-T phase (M2/D2), extensively recorded in the central and eastern domains. Decompression between these two metamorphic events is ascribed to an episode of exhumation. The two-stage evolution supports the previous hypothesis that the Georgetown Inlier preserves continental collisional and subsequent thermal perturbation associated with granite emplacement.     

P–V–T equation of state of Mn3Al2Si3O12 spessartine garnet

The thermoelastic parameters of synthetic Mn₃Al₂Si₃O₁₂ spessartine garnet were examined in situ at high pressure up to 13 GPa and high temperature up to 1,100 K, by synchrotron radiation energy dispersive X-ray diffraction within a DIA-type multi-anvil press apparatus. The analysis of room temperature data yielded K ₀ = 172 ± 4 GPa and K ₀ ^′  = 5.0 ± 0.9 when V _0,300 is fixed to 1,564.96 Å³. Fitting of P–V–T data by means of the high-temperature third-order Birch–Murnaghan EoS gives the thermoelastic parameters: K ₀ = 171 ± 4 GPa, K ₀ ^′  = 5.3 ± 0.8, (?K _0,T /?T) _P  = ?0.049 ± 0.007 GPa K^?1, a ₀ = 1.59 ± 0.33 × 10^?5 K^?1 and b ₀ = 2.91 ± 0.69 × 10^?8 K^?2 (e.g., α _0,300 = 2.46 ± 0.54 × 10^?5 K^?1). Comparison with thermoelastic properties of other garnet end-members indicated that the compression mechanism of spessartine might be the same as almandine and pyrope but differs from that of grossular. On the other hand, at high temperature, spessartine softens substantially faster than pyrope and grossular. Such softening, which is also reported for almandine, emphasize the importance of the cation in the dodecahedral site on the thermoelastic properties of aluminosilicate garnet.     

Constraints on the early cooling rates of Mt. Dabie from diffusion modeling of chemical zoning of garnet

Diffusion modeling of zoning profiles in eclogite garnets from three different tectonic units of Mt. Dabie, UHPM unit, HPM unit and northern Dabie, was used to estimate the relative time span and cooling rates of these rocks. Modeling result for the Huangzhen eclogite garnet shows that the maximal time span for the diffusion-adjustment process is about 22 Ma since the peak-temperature metamorphism, which is the maximum time span from amphibolite facies metamorphism to greenschist facies metamorphism. The Bixiling eclogites had subjected to a cooling process at a rate of - 10℃/Ma from 750℃ to 560℃ during 20 Ma. The second cooling stage of the Raobazhai eclogite following granulite-facies metamorphism is an initial fast cooling process at a rate of about 25℃/Ma and then slowed down gradually. All these belong to a coherent Dabie collision orogen with differences in subduction depth and exhumation/uplifting path.     

Reconstruction of the metamorphic <Emphasis Type="Italic">P–T</Emphasis> path from the garnet zoning in aluminous schists from the Tsogt Block,Mongolian Altai

The paper presents original authors’ data on aluminous schists in the Tsogt tectonic plate in the Southern Altai Metamorphic Belt. The nappe includes a medium-temperature/medium-pressure zonal metamorphic complex, whose metamorphic grade varies from the greenschist to epidote-amphibolite facies. The garnet and garnet–staurolite schists contain three garnet generations of different composition and morphology. The P–T metamorphic parameters estimated by mineralogical geothermometers and geobarometers and by numerical modeling with the PERPLEX 668 software provide evidence of two successive metamorphic episodes: high-gradient (of the andalusite–sillimanite type, geothermal gradient approximately 40–50°/km) and low-gradient (kyanite–sillimanite type, geothermal gradient approximately 27°/km). The P-T parameters of the older episode are T = 545–575°C and P = 3.1–3.7 kbar. Metamorphism during the younger episode was zonal, and its peak parameters were T = 560–565°C, P = 6.4–7.2 kbar for the garnet zone and T = 585–615°C, P = 7.1–7.8 kbar for the staurolite zone. The metamorphism evolved according to a clockwise P–T path: the pressure increased during the first episode at a practically constant temperature, and then during the second episode, the temperature increased at a nearly constant pressure. Such trends are typical of metamorphism related to collisional tectonic settings and may be explained by crustal thickening due to overthrusting. The regional crustal thickening reached at least 15–18 km.     

Variable P–T paths and HP-UHP metamorphism in a Precambrian terrane,Gridino, Russia: Petrological evidence and geodynamic implications

The Gridino Complex represents one of the oldest eclogite-facies terranes on Earth. It consists of blocks, boudins and lenses of eclogites, pyroxenites, and epidosites as well as deformed eclogitized dikes within biotite-amphibole gneisses. Detailed petrological studies of the pyroxenites and different types of eclogites reveal considerable diversity in metamorphic pressure (P) – temperature (T) conditions (from 1.3 GPa at 660 °C to 3.0 GPa at 660 °C) and fluid regimes (wet vs. dry) experienced by these rocks. Dike-related rocks escaped prograde metamorphism and reached higher pressures than the lenses and blocks that experienced considerable prograde metamorphic reworking. The variability in P–T conditions and the shapes of P–T paths are in agreement with the results of thermomechanical modeling and data from (U)HP metamorphic rocks exhumed during continent–continent collision in the Phanerozoic. The T/P ratio estimated for an eclogitized dike from Eclogitovii Island of the Gridino Complex corresponds to the gradients of < 350 °C/GPa attributed to high-pressure (HP)–ultrahigh-pressure (UHP) metamorphic belts, which are often considered as representative of modern style plate tectonics operating in the Phanerozoic Eon. The data presented in this paper suggest that occurrences of HP-UHP metamorphic terrains might be extended back towards the time of either the assembly of Columbia in the Paleoproterozoic or Kenorland in the Neoarchean.     

Dating magmatic and hydrothermal processes using andradite-rich garnet U–Pb geochronometry

Andradite-rich garnet is a common U-bearing mineral in a variety of alkalic igneous rocks and skarn deposits, but has been largely neglected as a U–Pb chronometer. In situ laser ablation-inductively coupled plasma mass spectrometry U–Pb dates of andradite-rich garnet from a syenite pluton and two iron skarn deposits in the North China craton demonstrate the suitability and reliability of the mineral in accurately dating magmatic and hydrothermal processes. Two hydrothermal garnets from the iron skarn deposits have homogenous cores and zoned rims (Ad₈₆Gr₁₁ to Ad₉₈Gr₁) with 22–118 ppm U, whereas one magmatic garnet from the syenite is texturally and compositionally homogenous (Ad₇₀Gr₂₂ to Ad₇₇Gr₁₄) and has 0.1–20 ppm U. All three garnets have flat time-resolved signals obtained from depth profile analyses for U, indicating structurally bound U. Uranium is correlated with REE in both magmatic and hydrothermal garnets, indicating that the incorporation of U into the garnet is largely controlled by substitution mechanisms. Two hydrothermal garnets yielded U–Pb dates of 129 ± 2 (2σ; MSWD = 0.7) and 130 ± 1 Ma (2σ; MSWD = 0.5), indistinguishable from zircon U–Pb dates of 131 ± 1 and 129 ± 1 Ma for their respective ore-related intrusions. The magmatic garnet has a U–Pb age of 389 ± 3 Ma (2σ; MSWD = 0.6), consistent with a U–Pb zircon date of 388 ± 2 Ma for the syenite. The consistency between the garnet and zircon U–Pb dates confirms the reliability and accuracy of garnet U–Pb dating. Given the occurrence of andradite-rich garnet in alkaline and ultramafic magmatic rocks and hydrothermal ore deposits, our results highlight the potential utilization of garnet as a powerful U–Pb geochronometer for dating magmatism and skarn-related mineralization.     

Assessment of advective–dispersive contaminant transport in heterogeneous aquifers using a meshless method

Groundwater solute transport phenomena typically occur in water-bearing zones with heterogeneous solute dispersive characteristics and/or media hydraulic properties. A radial basis function collocation method (RBFCM)-based numerical method was developed in order to investigate the ability of RBFCM to accurately portray solute transport phenomena under heterogeneous conditions. Simulations were performed for 1-D and 2-D transport scenarios in which scale-dependent dispersivity fields were taken into consideration and compared with available analytical solutions. Different radial basis functions (RBFs) were employed for assessing the sensitivity of the present method on the selected RBFs. The simulation results were also compared with the results of MT3DMS which is a modular three-dimensional transport model with alternative solution schemes including the method of characteristics, the implicit central finite difference and the third order total variation diminishing finite volume. The proposed model was also used to simulate a real case condition where solute transport through a two-layer soil medium had been investigated experimentally. The results showed that RBFCM represented a powerful tool for predicting the solute transport occurrence under heterogeneous conditions with high accuracy.     

Fluid-assisted retrogression of garnet and P–T history of metapelites from HP/UHP metamorphic terrane (Pohorje Mountains, Eastern Alps)

Fluid inclusions in garnet combined with element X-ray mapping, phase equilibrium modelling and conventional thermobarometry have been used to constrain the metamorphic evolution of metapelitic gneiss from the HP/UHP metamorphic terrane of Pohorje Mountains in the Eastern Alps, Slovenia. Retrograde P–T trajectory from ~2.75 GPa and 780°C is constrained by the composition of matrix phengite (6.66 apfu Si) coexisting with garnet cores, kyanite and quartz. The intersection of the X _Prp = 0.25 isopleth for the garnet with the upper stability boundary for K-feldspar in the matrix indicates near-isothermal decompression to ~0.9 GPa at 720°C. Temperatures over 650°C during this stage are corroborated by the high degree of ordering of graphite inclusions associated with Zn, Mg-rich staurolite and phlogopite in the Mg-rich (X _Prp = 0.22–0.25) garnet cores. Majority of garnet porphyroblasts are depleted in Mg (down to X _Prp = 0.09) and enriched in Mn (up to X _Sps = 0.12) along cracks and at their margins. The associated retrograde mineral assemblage comprises Zn, Mg-poor staurolite, muscovite, biotite–siderophyllite, sillimanite and quartz. The onset of the retrogression and the compositional modification of the garnet porphyroblasts were accompanied by the addition of fluid-deposited graphite around older graphite inclusions, probably due to removal of water from a graphite-buffered COH fluid by dissolution in partial silicic melt. Instantaneous expulsion of water near the melt solidus (640°C, max. 0.45 GPa) caused dissolution of the graphite at redox conditions corresponding to 0.25–1.25 logfO₂ units below the QFM buffer, giving rise to a H₂O–CO₂–CH₄ fluid trapped in primary inclusions in Mn-rich, Mg-poor, almandine garnet that reprecipitated within the retrogressed domains. The absence of re-equilibration textures and consistent densities of the fluid inclusions reflect a near-isochoric cooling postdating the near-isothermal decompression. Bulk water content in the metapelite attained 2 wt% during this stage. The low-degree partial melting and extensive hydration due to the release of the internally derived, low-pressure aqueous fluids led to the reset of peak-pressure mineral assemblage.     

A new method for characterizing heterogeneities from a core image using local Hölder exponents

The two-dimensional multifractional Brownian motion (2D-mBm) provides a convenient model for image processing. However, the accurate estimation of its associated local regularity function presents a serious difficulty. Here, we suggest a new estimator of the local Hölder exponent (H) of 2D-mBm paths. The idea consists of extending the algorithm, as suggested by Peltier and Lévy-Véhél (1995) for quantifying regularity of mono-dimensional signals, to the 2D case. The resulting algorithm is first tested on artificial 2D-mBm paths shaped by the kriging method. The good accuracy of the obtained H estimations motivates us to implement it on digitalized image data of a core extracted from an Algerian borehole. The derived regularity map shows a significant correlation with the geological features observed on the investigated core. These lithological discontinuities are marked by local variations of the Hölder exponent value. Besides, it is shown that no clear relationship can be drawn between regularity and digitalized data. To conclude, the proposed method may be of high importance in characterizing heterogeneities from core images using the regularity exponents.     

Precise estimation of pressure–temperature paths from zoned minerals using Markov random field modeling: theory and synthetic inversion

The chemical zoning profile in metamorphic minerals is often used to deduce the pressure–temperature (P–T) history of rock. However, it remains difficult to restore detailed paths from zoned minerals because thermobarometric evaluation of metamorphic conditions involves several uncertainties, including measurement errors and geological noise. We propose a new stochastic framework for estimating precise P–T paths from a chemical zoning structure using the Markov random field (MRF) model, which is a type of Bayesian stochastic method that is often applied to image analysis. The continuity of pressure and temperature during mineral growth is incorporated by Gaussian Markov chains as prior probabilities in order to apply the MRF model to the P–T path inversion. The most probable P–T path can be obtained by maximizing the posterior probability of the sequential set of P and T given the observed compositions of zoned minerals. Synthetic P–T inversion tests were conducted in order to investigate the effectiveness and validity of the proposed model from zoned Mg–Fe–Ca garnet in the divariant KNCFMASH system. In the present study, the steepest descent method was implemented in order to maximize the posterior probability using the Markov chain Monte Carlo algorithm. The proposed method successfully reproduced the detailed shape of the synthetic P–T path by eliminating appropriately the statistical compositional noises without operator’s subjectivity and prior knowledge. It was also used to simultaneously evaluate the uncertainty of pressure, temperature, and mineral compositions for all measurement points. The MRF method may have potential to deal with several geological uncertainties, which cause cumbersome systematic errors, by its Bayesian approach and flexible formalism, so that it comprises potentially powerful tools for various inverse problems in petrology.