The Tau Model for Data Redundancy and Information Combination in Earth Sciences: Theory and Application

Many decision-making processes in the Earth sciences require the combination of multiple data originating from diverse sources. These data are often indirect and uncertain, and their combination would call for a probabilistic approach. These data are also partially redundant with each other or with all others taken jointly. This overlap in information arises due to a variety of reasons—because the data arises from the same geology, because they originate from the same location or the same measurement device, etc. The proposed tau model combines partially redundant data, each taking the form of a prior probability for the event being assessed to occur given that single datum. The parameters of that tau model measure the additional contribution brought by any single datum over that of all previously considered data; they are data sequence-dependent and also data value-dependent. Data redundancy depends on the sequence in which the data is considered and also on the data values themselves. However, for a given sequence, averaging the tau model parameters over all possible data values leads to exact analytical expressions and corresponding approximations and inference avenues. Information on multiple-point connectivity of permeability arrives from core data, well-test data and seismic data which are defined over varying supports with complex redundancy between these information sources. In order to compute these tau weights for determining connectivity, one needs a model of data redundancy, here expressed as a vectorial training image (Ti) constructed using a prior conceptual knowledge of geology and the physics of data measurement. From such a vectorial Ti, the tau weights can be computed exactly. Neglecting data redundancy leads to an over-compounding of individual data information and the possible risk of making extreme decisions.     

Group Time Delay in the Reflection of P-Polarized Electromagnetic Plane Waves at a Stratigraphic Interface

The group time delay (GTD) formulae of quasi-total reflection of inhomogeneous P-polarized electromagnetic waves (P-PEW, whose electric field parallels to the plane of incidence), caused by the Goos–Hänchen effect, is derived by the phase shift of the wave. A numerical example where the frequency equals to 1 GHz is given. The curves of calculating results show that there are three discontinuous points at the critical angle of phase shift vector, the critical angle of attenuation vector and 90^○. When the angle of incidence is equal to one of these three angles, the GTD will become infinite, and the electromagnetic wave will propagate along the interface. The GTD will be very large, when the angle of incidence is close to one of these three angles. The results indicate that there is an important relationship between the two critical angles and the conductivity and permittivity of the two strata, and that the GTD has an important relationship with the angle of incidence. These results can be used to determine the lithology of the strata and to delimit the interfaces of strata more effectively. It is suggested that this research may prove useful in electromagnetic logging analysis and, perhaps, in the design of logging instruments.     

Stability Analysis and Numerical Simulation of Differential Frost Heave

Differential frost heave is often implicated in the formation of patterned ground in regions subject to recurrent freezing and thawing. A linear stability analysis (LSA) indicates that a continuum model of frost heave is linearly unstable under typical natural freezing conditions of silty-clay soils. A two-dimensional non-linear numerical analysis corroborates the frozen time LSA results, and also indicates the importance of non-linear and time-dependent terms that ultimately lead to a preferred mode, which the LSA fails to predict. Instability of the one-dimensional solution occurs at shallow freezing depths and near-zero surface loads when positive perturbations in the ice content at the freezing front lead to a concomitant increase in thermomolecular pressure and upward ice velocity. Differential frost heave can then occur because of the increased heat flux from the perturbed surfaces. A three-dimensional model using random initial surface perturbations indicates that regular surface patterns will evolve with a length scale in the order of 2–4 meters, which corresponds quite closely with naturally-occurring non-sorted patterned ground.     

Non-Linear Theory and Power-Law Models for Information Integration and Mineral Resources Quantitative Assessments

Singular physical or chemical processes may result in anomalous amounts of energy release or mass accumulation that, generally, are confined to narrow intervals in space or time. Singularity is a property of different types of non-linear natural processes including cloud formation, rainfall, hurricanes, flooding, landslides, earthquakes, wildfires, and mineralization. The end products of these non-linear processes can be modeled as fractals or multifractals. Hydrothermal processes in the Earth’s crust can result in ore deposits characterized by high concentrations of metals with fractal or multifractal properties. Here we show that the non-linear properties of the end products of singular mineralization processes can be applied for prediction of undiscovered mineral deposits and for quantitative mineral resource assessment, whether for mineral exploration or for regional, national and global planning for mineral resource utilization. In addition to the general theory and framework for the non-linear mineral resources assessment, this paper focuses on several power-law models proposed for characterizing non-linear properties of mineralization and for geoinformation extraction and integration. The theories, methods, and computer system discussed in this paper were validated using a case study dealing with hydrothermal Au mineral potential in southern Nova Scotia, Canada.     

Inversely-Mapped Analytical Solutions for Flow Patterns around and within Inclined Elliptic Inclusions in Fluid-Saturated Rocks

In this paper, an inverse mapping is used to transform the previously-derived analytical solutions from a local elliptical coordinate system into a conventional Cartesian coordinate system. This enables a complete set of exact analytical solutions to be derived rigorously for the pore-fluid velocity, stream function, and excess pore-fluid pressure around and within buried inclined elliptic inclusions in pore-fluid-saturated porous rocks. To maximize the application range of the derived analytical solutions, the focal distance of an ellipse is used to represent the size of the ellipse, while the length ratio of the long axis to the short one is used to represent the geometrical shape of the ellipse. Since the present analytical solutions are expressed in a conventional Cartesian coordinate system, it is convenient to investigate, both qualitatively and quantitatively, the distribution patterns of the pore-fluid flow and excess pressure around and within many different families of buried inclined elliptic inclusions. The major advantage in using the present analytical solution is that they can be conveniently computed in a global Cartesian coordinate system, which is widely used in many scientific and engineering computations. As an application example, the present analytical solutions have been used to investigate how the dip angle of an inclined elliptic inclusion affects the distribution patterns of the pore-fluid flow and excess pore-fluid pressure when the permeability ratio of the elliptic inclusion is of finite but nonzero values.     

A geophysical study of the granites and the sedimentary basins of the Xanthi area (N. Greece)

Intrusive features of varying size can be interpreted from the aeromagnetic map of the Xanthi area in N. Greece.The Xanthi pluton, which outcrops north of the city of Xanthi, seems to have the shape of a truncated pyramid. This feature has relatively large areal extent and reaches an approximate depth of 7 km. Another, relatively large magnetic body is buried under the sediments at the estuary of the Nestos River.3-D models of several smaller intrusions were constructed and the produced effect was compared to the observed. Some of these intrusions seem to be detached branches of the large Xanthi pluton.The basement in the outer part of the basin of the Nestos River seems to be buried at about 4 km depth. This figure is obtained by the Multiple Source Werner Deconvolution estimates and it is in agreement with the results of former geophysical studies and deep industrial boreholes.A 3-D model of the Xanthi-Komotini basin suggests that this basin is about 0.4 km deep at its southern part. The depth at its northern boundary is about 1.8 km while the boundary itself is formed by the large Kavala-Xanthi-Komotini fault.The Tertiary basin of the Nestos River and the observed magmatism are consistent with the idea of an older extensional tectonic regime in the area.     

Transition from eclogite to amphibolite-facies metamorphism in the Austroalpine Ulten Zone

Summary The Ulten Zone of the Austroalpine crystalline basement south-west of Meran (Italy) contains metapelitic schists and granoblastic paragneisses, leucocratic orthogneisses, migmatites (in both gneiss-lithologies), metabasites and ultramafic lenses. Metamorphic textures of the metapelitic schists and granoblastic paragneisses indicate two different metamorphic events, characterized by two mineral assemblages, which differ in mineral chemistry: (1) an eclogite facies mineral assemblage (M1) comprising Grt-Ky I-Bt. Ms-Kfs-PI-Qtz-Rt, and (2) an amphibolite facies mineral assemblage (M2) comprising Grt-KyII-Bt-Ms-PI-Qtz-Ilm±St. For the M1 event, pressures of at least 15kbar and temperatures of about 700°±50°C can be estimated. The later amphibolite facies overprint occurred at pressures of 6 to 8kbar and about 600°±50°C. The M1 and M2 assemblages belong to a continuous clockwise metamorphic evolution during the Variscan orogeny. Evidence for Alpine metamorphism can only be detected by sericite rims around kyanite and reset biotite ages. The migmatites, which contribute about 15–30vol.% of all rocks in the investigated area, were formed on the prograde path during the M1 event. Dissolution of H₂O in the melted part of the migmatites resulted in a CO₂dominated fluid, which was trapped in primary kyanite (M1) fluid inclusions. Secondary H₂O-rich fluid inclusions are found in quartz grains and may represent the fluid which enabled a pervasive equilibration during M2.

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Übergang von eklogit-zu amphibolitfazieller Matamorphose in der austroalpinen Ultenzone

Zusammenfassung Die Ulten Zone, ein Teil des ostalpinen kristallinen basements, südwestlich von Meran, wird aus Metapeliten and granoblastischen Paragneisen, leukokraten Orthogneisen, Migmatiten (in beiden Lithologien), Metabasiten and ultramafischen Linsen aufgebaut. Metamorphe Texturen der Metapelite und granoblastischen Paragneise lassen auf zwei verschiedene metamorphe Ereignisse schließen, die durch unterschiedliche Mineral-chemismen und Paragenesen charakterisiert sind: (1) eine eklogitfazielle Paragenese (M1), bestehend aus Grt-KyI-Bt-Ms-Kfs-P1-Qtz-Rt und (2) eine amphibolitfazielle Paragenese (M2), bestehend aus Grt-KyII-Bt-Ms-P1-Qtz-Ilm±St. Für M1 konnten Minimaldrucke von 15kbar und Temperaturen von 700°±50°C abgeleitet werden. Die spätere amphibolitfazielle Überprägung fand bei 6 bis 8kbar und 600°±50°C statt. M1 und M2 gehören einer kontinuierlichen Metamorphoseentwicklung während der variszischen Orogenese an.Die Migmatite, ungefähr 15–30vol.% der Gesteine im untersuchten Gebiet, wurden am prograden Pfad während des M1 Ereignisses gebildet. Aufgrund der höheren Löslichkeit von H₂0 in der Schmelze, blieb ein CO₂, reiches Fluid zurück, das im primären Kyanit (M1) eingeschlossen wurde. Wässrige Flüssigkeitseinschlüsse können in Quarzkörnern gefunden werden. Dieses Fluid ist wahrscheinlich für die Reequilibrierung zu amphibolitfaziellen Bedingungen verantwortlich.

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Calorimetric study of the coesite-stishovite transformation and calculation of the phase boundary

High temperature drop-solution calorimetry in molten 2 PbO · B₂O₃ at 1044 K for coesite and stishovite polymorphs of silica was carried out to determine the enthalpy of the coesite-stishovite transition. These experiments were performed on high-purity, single-phase samples of coesite and stishovite. Our new value for the enthalpy of the coesitestishovite transition (ΔH ₂₉₈ ⁰ ) is 29.85 ± 0.78 kJ/mol, which is about 35% lower than previously reported by Akaogi and Navrotsky (1984) and Holm et al. (1967), but which compares well with new measurements by Akaogi et al. (1994b). Using these new data, we have calculated the equilibrium phase boundary between coesite and stishovite and obtained a slope, dP/dT=0.0031 (2) GPa/K. This calculated slope is in good agreement with that determined [0.0026 (2) GPa/K] from the in-situ X-ray diffraction study of Zhang et al. (1996).     

Trace Element and Isotope Geochemistry of the Volcanic Rocks of Bequia, Grenadine Islands, Lesser Antilles Arc: a Study of Subduction Enrichment and Intra-crustal Contamination

Pliocene volcanics on the island of Bequia comprise two interbeddedsuites of basalts and andesites. The isotopically homogeneoussuite (IHS) has a limited range of Sr—Nd—Pb isotopes(⁸⁷Sr/⁸⁶Sr 0.7040–0.7046, ¹⁴³ Nd/¹⁴⁴ Nd 0.5130 and ²⁰⁶Pb/²⁰⁴Pb 19.36–19.51), and mantle-like ¹⁸O values (5.5in clinopyroxene). The isotopically diverse suite (IDS) is characterizedby much wider ranges of radiogenic isotopes (⁸⁷ Sr/⁸⁶Sr 0.7048–0.7077,¹⁴³ Nd/¹⁴⁴ Nd 0.5128–0.5123 and ²⁰⁶ Pb/²⁰⁴ Pb 19.7–20.2),in which all of the Sr and Pb ratios are higher and Nd ratiosare lower than those of the IHS. The IDS is also characterizedby high ¹⁸ O values, up to 7.6 in clinopyroxene. The Sr andPb isotope ratios are too high, and the Nd isotope ratios aretoo low in the IDS for any of these lavas to be derived fromunmodified depleted mantle. Both suites are petrologically very similar and their majorelement compositions and phenocryst contents suggest that theywere formed largely by fractional crystallization of a hydroustholeiitic melt at pressures <3 kbar. The isotopic ratiosand enrichments in large ion lithophile elements (LILE), andto some extent light rare earth elements (LREE), as comparedwith mid-ocean ridge basalts (MORB), of the IHS lavas suggestthat they were derived from a depleted mantle source which hadbeen re-enriched by the addition of 1–4% of a subductioncomponent. This component probably comprised a mixture of dehydrationfluids, and perhaps minor siliceous melts, released from subductingsediments and mafic crust. The extreme isotopic ranges, largeenrichments in incompatible elements, more fractionated LREEpatterns and higher ¹⁸ O values of the IDS lavas are interpretedas resulting from 10–55% assimilation—fractionalcrystallization of sediments, derived from the Guyana Shield,which are present in the arc crust, by IHS type melts. KEY WORDS: trace elements; radiogenic isotopes; arc lavas; Lesser Antilles ^*Corresponding author.     

Distribution of Ferric Iron in some Upper-Mantle Assemblages

The distribution of ferric iron among the phases of upper-mantlerocks, as a function of pressure (P), temperature (T) and bulkcomposition, has been studied using ⁵⁷Fe Mssbauer spectroscopyto determine the Fe³⁺/Fe ratios of mineral separates from 35peridotite and pyroxenite samples. The whole-rock Fe³⁺ complementof a peridotite is typically shared approximately evenly amongthe major anhydrous phases (spinel and/or garnet, orthopyroxeneand clinopyroxene), with the important exception of olivine,which contains negligible Fe³⁺. Whole-rock Fe³⁺ contents areindependent of the T and P of equilibration of the rock, butshow a well-defined simple inverse correlation with the degreeof depletion in a basaltic component. Fe³⁺ in spinel and inboth pyroxenes from the spinel Iherzolite facies shows a positivecorrelation with temperature, presumably owing to the decreasein the modal abundance of spinel. In garnet peridotites, theFe³⁺ in garnet increases markedly with increasing T and P, whereasthat in clinopyroxene remains approximately constant. The complexnature of the partitioning of Fe³⁺ between mantle phases resultsin complicated patterns of the activities of the Fe³⁺ -bearingcomponents, and thus in calculated equilibrium f_O2, which showlittle correlation with whole-rock Fe³⁺ or degree of depletion.Whether Fe³⁺ is taken into account or ignored in calculatingmineral formulae for geothermobarometry can have major effectson the resulting calculated T and P. For Fe-Mg exchange geothermometers,large errors must occur when applied to samples more oxidizedor reduced than the experimental calibrations, whose f_O2 conditionsare largely unknown. Two-pyroxene thermometry is more immuneto this problem, and probably provides the most reliable P—Testimates. Accordingly, the convergence of P—T valuesderived for a given garnet peridotite assemblage may not necessarilybe indicative of mineral equilibrium. The prospects for thecalculation of accurate Fe³⁺ contents from electron microprobeanalyses by assuming stoichiometry are good for spinel, uncertainfor garnet, and distinctly poor for pyroxenes. KEY WORDS: mantle; oxidation; partitioning; peridotite; thermobarometry ^*Corresponding author. Present address: School of Earth and Ocean Sciences, University of Victoria, P.O. Box 1700, Victoria, B.C., V8W 2Y2, Canada