Smoothing and interpolation by kriging and with splines

Let scalar measurements at distinct points x₁, , x_n be y₁, , y_n.We may look for a smooth function f(x)that goes through or near the points (x_i, y_i).Kriging assumes f(x)is a random function with known (possibly estimable) covariance function (in the simplest case). Splines assume a definition of the smoothness of a nonrandom function f(x).An elementary explanation is given of the fact that spline approximations are special cases of the solution of a kriging problem.     

An objective replacement method for censored geochemical data

Geochemical data are commonly censored, that is, concentrations for some samples are reported as less than or greater than some value. Censored data hampers statistical analysis because certain computational techniques used in statistical analysis require a complete set of uncensored data. We show that the simple substitution method for creating an uncensored dataset, e.g., replacement by3/4 times the detection limit, has serious flaws, and we present an objective method to determine the replacement value. Our basic premise is that the replacement value should equal the mean of the actual values represented by the qualified data. We adapt the maximum likelihood approach (Cohen, 1961) to estimate this mean. This method reproduces the mean and skewness as well or better than a simple substitution method using3/4 of the lower detection limit or3/4 of the upper detection limit. For a small proportion of less than substitutions, a simple-substitution replacement factor of 0.55 is preferable to3/4; for a small proportion of greater than substitutions, a simple-substitution replacement factor of 1.7 is preferable to4/3, provided the resulting replacement value does not exceed 100%. For more than 10% replacement, a mean empirical factor may be used. However, empirically determined simple-substitution replacement factors usually vary among different data sets and are less reliable with more replacements. Therefore, a maximum likelihood method is superior in general. Theoretical and empirical analyses show that true replacement factors for less thans decrease in magnitude with more replacements and larger standard deviation; those for greater thans increase in magnitude with more replacements and larger standard deviation. In contrast to any simple substitution method, the maximum likelihood method reproduces these variations. Using the maximum likelihood method for replacing less thans in our sample data set, correlation coefficients were reasonably accurately estimated in 90% of the cases for as much as 40% replacement and in 60% of the cases for 80% replacement. These results suggest that censored data can be utilized more than is commonly realized.     

High-pressure X-ray and Raman study of a ferrian magnesian spodumene

The high-pressure behaviour of a synthetic P2₁/c ferrian magnesian spodumene, ^M2 (Li_0.85Mg_0.09Fe²⁺ _0.06)^M1(Fe³⁺ _0.85Mg_0.15)Si₂O₆, has been investigated using in situ single-crystal X-ray diffraction and Raman spectroscopy. No phase transition has been observed within the pressure range investigated. The isothermal equation of state up to 7 GPa was determined. V₀, K_T0 and K, simultaneously refined with a Murnaghan equation of state, are: V₀= 415.66(7) ų, K_T0=83(1) GPa and K=9.6(6). The magnitudes of the principal unit-strain coefficients were calculated and their ratios 1:2:3=1.00:1.85:2.81 at P=6.83 GPa indicate a very strong anisotropy. Monitoring of the intensity of b-type reflections (h+k= 2n+1) confirms that from room conditions up to 7 GPa the primitive lattice is maintained. Raman spectra have been collected up to 7.4 GPa. No change in the number of observed vibrational modes occurs in the pressure range investigated. At high frequency, the Raman doublet relative to the Si–O–Si vibrations of the two distinct tetrahedral chains is a broad band at room pressure, however, the frequency difference between the two modes increases with increasing pressure.Operating system: Windows NT     

Über Protolithionit und Zinnwaldit

Zusammenfassung Aus der Literatur entnommene 142 chemische Analysen aller Arten von Lithiumglimmern werden nach dem Umrechnungsverfahren vonStevens (1938) in die vier Molekular-Komponenten Polylithionit—Paucilithionit—Muscovit—Siderophyllit zerlegt und in einem Konzentrationstetraeder dargestellt. Es ergeben sich vier geschlossene Felder: Lepidolithe (einschließlich Polylithionit und Kryophyllit), Zinnwaldit, Protolithionit (mit Lithium-Siderophyllit), lithiumhaltiger Muscovit. Die Punkte der chemischen Analysen werden als Projektion auf die Tetraederflächen reproduziert. Die Diskussion der Mischungslücken ergibt, daß meist wohl die Art der Strukturvariante maßgebend für die Unmischbarkeit sein dürfte.Für Lepidolithe, lithiumhaltige Muscovite und Protolithionite werden Diagramme der optischen Variabilität (n _z;) gegeben, für Zinnwaldit nur die arithmetischen Mittelwerte der optischen Daten.

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142 chemical analyses of the different species of lithium micas that are given in the literature are transformed into the four molecular components polylithionite — paucilithionite —muscovite — siderophyllite according to the conversion method ofStevens (1938). They are presented in a concentration tetrahedron. The density of their representing points is reproduced by projecting them on the tetrahedron faces; they show four closed compartments: lepidolites (including polylithionite and kryophyllite), zinnwaldite, protolithionite (with lithium siderophyllite), lithian muscovite. A discussion on the gaps of immiscibility indicates that the kind of polymorphic modification must probably be responsible for the proved immiscibilities.The optical variability (n ;n -n for the lepidolites, lithian muscovites and protolithionites is given on diagrams, whereas for zinnwaldites only the arithmetical mean values of the optical data are given.

     

Toward an expert system for identification of minerals in thin section

Minerals may be identified by optical inspection and x-ray diffraction analysis. Full and correct identification, however, requires experience and extensive knowledge of mineral characteristics and association. Computer systems designed to approach levels of human expertise in similarly complex identification tasks have become increasingly effective with the application and refinement of various Artificial Intelligence (AI) techniques. These knowledge-based systemsuse the skills, knowledge, and rules of thumb that distinguish the expert from the knowledgeable layman to emulate human expertise. They also may be modified to serve a tutorial role whereby a nonexpert's approach to the task may be compared with that of the expert (system), and criticized accordingly. Such a knowledge-based system capable of identifying minerals from their optical characteristics is being developed at the University of Reading.     

Sapphirine/kornenipine-bearing rocks and crustal uplift history of the Limpopo belt,Southern Africa

Sapphirine/kornerupine-bearing rocks occur within the anorthosites of the Messina layered intrusion in the Limpopo mobile belt of Zimbabwe. The X_Mg range of the major minerals is as follows: cordierite (0.98-0.93); enstatite (0.97-0.86); chlorite (0.98-0.92); phlogopite (0.98-0.90); sapphirine (0.98-0.86); kornerupine (0.94-0.88); gedrite (0.96-0.85); spinel (0.92-0.78). There are four rock types, the constituent minerals of which have different values, which decrease in the above mineral order; other minerals are corundum, sillimanite and relict kyanite. We recognise twenty reactions without phlogopite and nine reactions involving phlogopite. The textural relations and the plots of the microprobe data of coexisting minerals in the MgO-Al₂O₃-SiO₂-(H₂O) system are consistent with the following sequence of main reactions: (1) enstatite+corundum cordierite+sapphirine; (4) sapphirine+sillimanite cordierite+corundum; (8) kornerupine+corundum cordierite+sapphirine; (13) kornerupine cordierite+sapphirine+enstatite; (15) enstatite+spinel chlorite+sapphirine; (18) cordierite+sapphirine chlorite+corundum; (20) sapphirine chlorite+corundum+spinel. The early reactions are shown by coarse-grained reaction intergrowths, kornerupine and gedrite breakdown is shown by finer-grained symplectites, and the latest reactions by very fine-grained products in micro-fractures. These selected reactions illustrate a remarkably steep trajectory from thePT peak close to 10 kbar and 800° C to the minimum observable at 3.5–4.5 kbar and 700° C as indicated by the pure MASH system. Very rapid uplift took place under nearly isothermal conditions. The protolith of this material was possibly sedimentary, derived from altered volcanic rocks. The bulk composition is close to the composition of kornerupine or to a mixture of alunite, chlorite and pyrophyllite. These texturally and mineralogically complex rocks contain a wealth of relevant data for documenting crustal uplift history.