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.     

Lidar observations of turbulent vortex shedding by an isolated topographic feature

Scanning measurements by a single wavelength lidar (1.06 m) were made downwind of the Pt. Sur rock, an isolated hill (height 110 m) along the California coast. Turbulent eddies (approximate diameter 50 m) were observed detaching from a stationary aerosol feature above the rock and moving downwind. Under conditions with a high Froude number [1.8], the Strouhal number [0.22] of vortex shedding was close to that observed in tank experiments with a Reynolds number of 200.     

Paleohydrology and paleochemistry of Lake Manitoba,Canada: the isotope and ostracode records

Lake Manitoba, the largest lake in the Prairie region of North America, contains a fine-grained sequence of late Pleistocene and Holocene sediment that documents a complex postglacial history. This record indicates that differential isostatic rebound and changing climate have interacted with varying drainage basin size and hydrologic budget to create significant variations in lake level and limnological conditions. During the initial depositional period in the basin, the Lake Agassiz phase (12–9 ka), ¹⁸O of ostracodes ranged from –16 to –5 (PDB), implying the lake was variously dominated by cold, dilute glacial meltwater and warm to cold, slightly saline water.Candona subtriangulata, which prefers cold, dilute water, dominates the most negative ¹⁸O intervals, when the basin was part of proglacial Lake Agassiz. At times during this early phase, the ¹⁸O of the lake abruptly shifted to higher values; euryhaline taxa such asC. rawsoni orLimnocythere ceriotuberosa, and halobiont taxa such asL. staplini orL. sappaensis are dominant in these intervals. This positive covariance of isotope and ostracode records implies that the lake level episodically fell, isolating the Lake Manitoba basin from the main glacial lake.¹⁸O values from inorganic endogenic Mg-calcite in the post-Agassiz phase of Lake Manitoba trend from –4 at 8 ka to –11 at 4.5 ka. We interpret that this trend indicates a gradually increasing influence of isotopically low (–20 SMOW) Paleozoic groundwater inflow, although periods of increased evaporation during this time may account for zones of less negative isotopic values. The ¹⁸O of this inorganic calcite abruptly shifts to higher values (–6) after 4.5 ka due to the combined effects of increased evaporative enrichment in a closed basin lake and the increased contribution of isotopically high surface water inflow on the hydrologic budget. After 2 ka, the ¹⁸O of the Mg-calcite fluctuates between –13 and –7, implying short-term variability in the lake's hydrologic budget, with values indicating the lake varied from outflow-dominated to evaporation-dominated. The ¹³C values of Mg-calcite remain nearly constant from 8 to 4.5 ka and then trend to higher values upward in the section. This pattern suggests primary productivity in the lake was initially constant but gradually increased after 4.5 ka.This is the sixth in a series of papers published in this issue on the paleolimnology of arid regions. These papers were presented at the Sixth International Palaeolimnology Symposium held 19–21 April, 1993 at the Australian National University, Canberra, Australia. Dr A. R. Chivas served as guest editor for these papers.     

Fractional crystallization and zoning in igneous feldspars: ideal water-buffered liquid fractionation lines and feldspar zoning paths

Diagrams giving plagioclase and sanidine fractionation paths and liquid fractionation lines under conditions of ideal water-buffered fractional crystallization in the ternary feldspar system were constructed graphically using topological reasoning, and experimental data and calculated phase relationships from the literature. The liquidus lines and solidus or solvus paths are unique at constant P and a _{H 2}O. The composition of a liquid evolves with time and moves along a fractionation line by removal of successive crystal fractions, whereas the compositions of each of the crystal fractions lie on and define a solidus or solvus path. Most but not all such water-buffered lines and paths differ only slightly from those in which water is free to build up during crystallization and a _{H 2}O to increase, as in many rocks. Liquid compositions lying along liquidus fractionation lines are not normally preserved, unless erupted as aphyric lavas. The solidus or solvus paths may be preserved either as overgrowth zones in crystals (zoning paths) or as a series of crystal fractions in layered intrusions. The topologies of the lines and paths depend mainly on the nature of the two-feldspar boundary line separating the plagioclase and sanidine fields which is a function of P _{H 2}O or a _{H 2}O at constant P; increases in either progressively lower the liquidi and solidi and cause larger intersections of the solidi with the solvus. One-feldspar solidus paths at high P and a _{H 2}O are simple, whereas they are complex and may bend back on themselves at low P _{H 2}O or low a _{H 2}O at high P. Two-feldspar paths may be simultaneous (cotectic) or sequential (peritectic). The former are simple and do not meet at high P and a _{H 2}O, the critical solution line lying in the gap; they are complex and may bend back or overlap at low P _{H 2}O or low a _{H 2}O at high P, the position of the critical solution line being hard to determine. Liquids which have simultaneously fractionated two feldspars may fractionate only one towards the end, crystallization changing from subsolvus to hypersolvus. Sequential paths may involve overgrowth of an early feldspar by a later one, usually sanidine overgrowths on plagioclase, but plagioclase overgrowths on sanidine occur. These complexities explain in part the difficulties of unravelling the textural and compositional relationships of ternary feldspars in water-poor felsic igneous rocks (even in the absence of alteration or complex magma dynamics) and of trying to deduce phase relationships from natural occurrences of feldspars.C.R.P.G. contribution number 948     

Brønsted reactions on oxide mineral surfaces and the temperature-dependence of their dissolution rates

The pH-dependence of oxide dissolution rates is controlled by Brønsted acid-base reactions at the mineral surface. These reactions are rapid but depend explicitly on temperature, as do the subsequent slow rates of bond hydrolysis. The net result is that dissolution rates vary in a complicated fashion with temperature and solution pH. The enthalpy changes of acid-base reactions on oxide materials are sufficiently similar, however, that general statements can be made about their contribution. The enthalpy changes from proton adsorption to a hydroxyl functional group (SOH), or to a deprotonated functional group (SO ^–), are generally exothermic. The enthalpy changes become increasingly endothermic, however, as charge accumulates on the mineral surface and the charged species interact electrostatically. The result is that mineral dissolution rates are least sensitive to temperature, as measured with an Arrhenius-like rate law, at pH conditions near the Point of Zero Net Proton Charge.     

Monitoring changes of clouds

An analysis of the spatial and temporal scales of cloud variability and their coupling provided by the results from existing cloud observing systems allows us to reach the following conclusions about the necessary attributes of a cloud monitoring system. (1) Complete global coverage with uniform density is necessary to obtain an unbiased estimate of cloud change and an estimate of the reliability with which that change can be determined. (2) A spatial sampling interval of less than 50 km is required so that cloud cover distributions will generally be homogeneous, or statistically homogeneous, within a sample. (3) A sampling frequency of at least six times a day ensures not only that the diurnal and semi-diurnal cycles are not aliased into long term mean values, but also that changes in them can be monitored. (4) Since estimated climate changes are only evident on a decadal time-scale, unless cloud monitoring is continuous with a record length greater than 10 years and has very high precision ( 1%) instrument calibration with overlapping observations between each pair of instruments, it will not be possible either to detect or to diagnose the effects of cloud changes on the climate.