Triangle based interpolation

Triangle based interpolation is introduced by an outline of two classical planar interpolation methods, viz. linear triangular facets and proximal polygons. These are shown to have opposite local bias. By applying cross products of triangles to obtain local gradients, a method designated “slant-top proximal polygon interpolation” is introduced that is intermediate between linear facets and polygonal interpolation in its local bias. This surface is not continuous, but, by extending and weighting the gradient planes, a C¹ surface can be obtained. The gradients also allow a roughness index to be calculated for each data point in the set. This index is used to control the shape of a blending function that provides a weighted combination of the gradient planes and linear interpolation. This results in a curvilinear, C¹,interpolation of the data set that is bounded by the linear interpolation and the weighted gradient planes and is tangent to the slant-top interpolation at the data points. These procedures may be applied to data with two, three, or four independent variables.     

Analysis of macroseismic and instrumental data for the study of the 19 November 1923 earthquake in the Aran Valley (Central Pyrenees)

The 19 November 1923 earthquake in the Aran Valley (Central Pyrenees), with observed maximum intensityI _max = VIII (MSK), has been studied through the compiling and reviewing of macroseismic information and collecting and processing early seismograms. Analysis of macroseismic data gives a focal depth ofh = 5 km and an anelastic attenuation coefficient 10^–3 km^–1. Analysis of early instrumental records allows the computation of estimate of magnitude (M _L = 5.6) and seismic moment (M _o = 1.1 × 10¹⁷ N × m) which are consistent with the values ofI _o andh obtained from macroseismic data.     

A geomathematic investigation of syntexis in a Gabbroic formation

Mathematical methods were employed in an investigation of syntexis in the Gabbro di Sondalo, a mafic igneous formation in the Central Alps (Italy). A set of 100 specimens was selected according to sampling criteria and analyzed by X-ray fluorescence for eight elements; Q-mode and R-mode factor analysis then was applied to these data. Results of the computations indicate that the general trend of differentiation is related to the syntexis of gneissic wall rocks (Gneiss del Monte Tonale) in the mafic melt. Ca and Mg are abundant in the gabbro, and Si and K in the gneiss; the independent variation trend of Fe and Ti may be related to a differentiation episode due to a magmatic phase. Spatial representation of multivariate data on differentiation also was performed on the main elements and a picture of the trend of syntexis was mapped.     

Rock media mathematical models processing

This paper reports on mathematical models of rock media processing and on their use in designing open pit coal mines. Spatial mathematical model of rock media was processed on a 25 km² model site, incorporating 918 borehole logs. The model is capable of providing information concerning the geological structure of every point of the investigated area by plotting geological cross-sections along given lines or by plotting contour lines of the surface or the base for thickness of chosen lithological strata. The computation of one point of a grid involves the following steps: Borehole logs are numericaly coded. The geological structure at an arbitrarily chosen pointP is computed as follows. All borehole logs inside the circle (P; R) are used to compute theZ-coordinate of the ground atP by some interpolation formula chosen from those contained in the program system. Next, we check what stratum occurs topmost at boreholes inside the circle and which is most probable as the top stratumC ₁ atP. TheZ-coordinate of theC ₁ stratum surface atP is computed. Then what strata occur underC ₁ stratum and which of them is the most probable stratumC ₂ is determined. The process of computation is repeated until a sequence of strataC _i atP andZ _i coordinates of their surfaces is ascertained. The interpolation formulas included in the system are proper linear combination of PAF (polynomial approximations formulas, linear or quadratic and weighted) and WAF (weighted average formulas). Among the various interpolation formulas, some proved more useful for tectonic fault lines, others for ordinary sedimentary surfaces.     

Oxygen isotopes in mantle related and geothermally altered magmatites of the Transhimalayan (Gangdese) ranges

In closed magma systems SiO₂ approximately measures differentiation progress and oxygen isotopes can seem to obey Rayleigh fractionation only as a consequence of the behaviour of SiO₂. The main role of ¹⁸O is as a sensitive indicator of contamination, either at the start of differentiation ( ¹⁸O_init) or as a proportion of fractionation in AFC. Plots of ¹⁸O vs SiO₂-allow to determine initial ¹⁸O values for different sequences for source comparison. For NBS-28=9.60, the ¹⁸O at 48% SiO₂-varies between a high 6.4 for Kiglapait (Kalamarides 1984), 5.9 for Transhimalaya, 5.8 for Hachijo-Jima (Matsuhisa 1979), 5.6 for Koloula (Chivas et al. 1982) and a low 5.3 for the Darran Complex, New Zealand. The Transhimalayan batholiths (Gangdese belt) were emplaced in the Ladakh-Lhasa terrane, between the present-day Banggong-Nujiang, and Indus-Yarlung Tsangbo suture zones, after its accretion to Eurasia. The gradient of the least contaminated continuous ( ¹⁸O vs SiO₂-igneous trend line is similar to that of Koloula, and AFC calculations suggest a low secondary assimilation rate of less than 0.05 times the rate of crystallisation. Outliers enriched in ¹⁸O are frequent in the Lhasa, and apparently rare in the Ladakh transsect. Low- ¹⁸O (5.0–0) granitoids and andesites on the Lhasa-Yangbajain axis are the result of present day or recent near-surface geothermal activity; their quartzes still trace the granitoids to the Transhimalaya ¹⁸O trend line, but the distribution of low total rock or feldspar ¹⁸O values could be a guide to more recent heat flow and thermally marked tectonic lineaments. Two ignimbrites from Maqiang show hardly any ¹⁸O-contamination by crustal material.     

Possible non-equilibrium oxygen isotope effects in mantle nodules,an alternative to the Kyser-O'Neil-Carmichael18O/16O geothermometer

Kyser, O'Neil, and Carmichael (1981, 1982) measured the ¹⁸O values of coexisting minerals from peridotite nodules in alkali basalts and kimberlites, interpreting the nodules as equilibrium assemblages. Using Ca-Mg-Fe element-partition geothermometric data, they proposed an empirical¹⁸O/¹⁶O geothermometer: T(°C)=1,151–173–68 ², where is the per mil pyroxene-olivine fractionation. However, this geothermometer has an unusual crossover at 1,150 °C, and in contrast to what might be expected during closed-system equilibrium exchange, the most abundant mineral in the nodules (olivine) shows a much greater range in ¹⁸O (+4.4 to +7.5) than the much less abundant pyroxene (all 50 pyroxene analyses from spinel peridotites lie within the interval +5.3 to +6.5). On ¹⁸O-olivinevs. ¹⁸O-pyroxene diagrams, the mantle nodules exhibit data arrays that cut across the ¹⁸O=zero line. These arrays strongly resemble the non-equilibrium quartzfeldspar and feldspar-pyroxene ¹⁸O arrays that we now know are diagnostic of hydrothermally altered plutonic igneous rocks. Thus, we have re-interpreted the Kyser et al. data as non-equilibrium phenomena, casting doubt on their empirical geothermometer. The peridotite nodules appear to have been open systems that underwent metasomatic exchange with an external, oxygen-bearing fluid (CO₂, magma, H₂O, etc.); during this event, the relatively inert pyroxenes exchanged at a much slower rate than did the coexisting olivines and spinels, in agreement with available exchange-rate and diffusion measurements on these minerals. This accounts for the correlation between ¹⁸O pyroxene-olivine and the whole-rock ¹⁸O of the peridotites, which is a major difficulty with the equilibrium interpretation.Contribution No. 3978, Publications of the Division of Geological and Planetary Sciences, California Institute of Technology     

Fitting straight lines when both variables are subject to error.

Usual methods for fitting a straight line, Y = + X,to data fail if the independent variable Xis subject to error. The problem is further complicated if there is no strong reason for selecting one of the two variables as independent; neither of the two lines may be correct. This review article discusses the maximum likelihood estimators of and under functional and structural models. These models involve differing assumptions about the statistical distributions of the dependent and independent variables. In addition, least-squares procedures are also considered. All these methods lead to the same result, a quadratic equation which can be solved to give an estimate of . This result requires knowledge of the ratio of the error variances, = ²/², where ² is the variance of the Yresiduals and ² is the variance of the X residuals. If ² and ² are unknown, estimates of can be difficult to obtain. If replicate sampling was employed, estimates of the variances can be made, and then of .Contribution Number 1 of the series of review articles by the Mathematical Geologists of the United States.     

Silicate melts at magmatic temperatures: in-situ structure determination to 1651°C and effect of temperature and bulk composition on the mixing behavior of structural units

The abundance of coexisting structural units in K-, Na-, and Li-silicate melts and glasses from 25° to 1654°C has been determined with in-situ micro-Raman spectroscopy. From these data an equilibrium constant, K_x, for the disproportionation reaction among the structural units coexisting in the melts, Si₂O₅(2Q³)SiO₃(Q²)+SiO₂(Q⁴), was calculated (K_x is the equilibrium constant derived by using mol fractions rather than activities of the structural units). From ln K_x vs l/T relationships the enthalpy (H_x) for the disproportionation reaction is in the range of-30 to 30 kJ/mol with systematic compositional dependence. In the potassium and sodium systems, where the disproportionation reaction shifts to the right with increasing temperature, the H_x increases with silica content (M/Si decreases, M=Na, K). For melts and supercooled liquids of composition Li₂O·2SiO₂ (Li/Si=1), the H_x is indistinguishable from 0. By decreasing the Li/Si to 0.667 (composition LS3) and beyond (e.g., LS4), the disproportionation reaction shifts to the left as the temperature is increased. For a given ratio of M/Si (M=K, Na, Li), there is a positive, near linear correlation between the H_x and the Z/r² of the metal cation. The slope of the H_x vs Z/r² regression lines increases as the system becomes more silica rich (i.e., M/Si is decreased). Activity coefficients for the individual structural units, _i, were calculated from the structural data combined with liquidus phase relations. These coefficients are linear functions of their mol fraction of the form _i=a lnX _i+b, where a is between 0.6 and 0.87, and X _i is the mol fraction of the unit. The value of the intercept, b, is near 0. The relationship between activity coefficients and abundance of individual structural units is not affected by temperature or the electronic properties of the alkali metal. The activity of the structural units, however, depend on their concentration, type of metal cation, and on temperature.     

The development of volcanic hosted massive sulfide and barite–gold orebodies on Wetar Island, Indonesia

Wetar Island is composed of Neogene volcanic rocks and minor oceanic sediments and forms part of the Inner Banda Arc. The island preserves precious metal-rich volcanogenic massive sulfide and barite deposits, which produced approximately 17 metric tonnes of gold. The polymetallic massive sulfides are dominantly pyrite (locally arsenian), with minor chalcopyrite which are cut by late fractures infilled with covellite, chalcocite, tennantite–tetrahedrite, enargite, bornite and Fe-poor sphalerite. Barite orebodies are developed on the flanks and locally overly the massive sulfides. These orebodies comprise friable barite and minor sulfides, cemented by a series of complex arsenates, oxides, hydroxides and sulfate, with gold present as <10 m free grains. Linear and pipe-like structures comprising barite and iron-oxides beneath the barite deposits are interpreted as feeder structures to the barite mineralization. Hydrothermal alteration around the orebodies is zoned and dominated by illite–kaolinite–smectite assemblages; however, local alunite and pyrophyllite are indicative of late acidic, oxidizing hydrothermal fluids proximal to mineralization. Altered footwall volcanic rocks give an illite K–Ar age of 4.7±0.16 Ma and a ⁴⁰Ar/³⁹Ar age of 4.93±0.21 Ma. Fluid inclusion data suggest that hydrothermal fluid temperatures were around 250–270°C, showed no evidence of boiling, with a mean salinity of 3.2 wt% equivalent NaCl. The ³⁴S composition of sulfides ranges between +3.3 and +11.7 and suggests a significant contribution of sulfur from the underlying volcanic edifice. The ³⁴S barite data vary between +22.4 and +31.0, close to Miocene seawater sulfate. Whole rock ⁸⁷Sr/⁸⁶Sr analyses of unaltered volcanic rocks (0.70748–0.71106) reflect contributions from subducted continental material in their source region. The ⁸⁷Sr/⁸⁶Sr barite data (0.7076–0.7088) indicate a dominant Miocene seawater component to the hydrothermal system. The mineral deposits formed on the flanks of a volcanic edifice at depths of ~2 km. Spectacular sulfide mounds showing talus textures are localized onto faults, which provided the main pathways for high-temperature hydrothermal fluids and the development of associated stockworks. The orebodies were covered and preserved by post-mineralization chert, gypsum, Globigerina-bearing limestone, lahars, subaqueous debris flows and pyroclastics rocks.     

Changes in stable isotope ratios of metapelites and marbles during regional metamorphism,Mount Lofty Ranges,South Australia: implications for crustal scale fluid flow

The Mount Lofty Ranges comprises interlayered marbles, metapsammites, and metapelites that underwent regional metamorphism during the Delamarian Orogeny at 470–515 Ma. Peak metamorphic conditions increased from lowermost biotite grade (350–400°C) to migmatite grade (700°C) over 50–55 km parallel to the lithological strike of the rocks. With increasing metamorphic grade, ¹⁸O values of normal metapelites decrease from 14–16 to as low as 9.0, while ¹⁸O values of calcite in normal marbles decrease from 22–24 to as low as 13.2 These isotopic changes are far greater than can be accounted for by devolatilisation, implying widespread fluid-rock interaction. Contact metamorphism appears not to have affected the terrain, suggesting that fluid flow occurred during regional metamorphism. Down-temperature fluid flow from synmetamorphic granite plutons (¹⁸O=8.4–8.6) that occur at the highest metamorphic grades is unlikely to explain the resetting of oxygen isotopes because: (a) there is a paucity of skarns at granite-metasediment contacts; (b) the marbles generally do not contain low-XCO₂ mineral assemblages; (c) there is insufficient granite to provide the required volumes of water; (d) the marbles and metapelites retain a several permil difference in ¹⁸O values, even at high metamorphic grades. The oxygen isotope resetting may be accounted for by along-strike up-temperature fluid flow during regional metamorphism with time-integrated fluid fluxes of up to 5x10⁹ moles/m² (10⁵ m³/m²). If fluid flow occurred over 10⁵–10⁶ years, estimated intrinsic permeabilities are 10^-20 to 10^-16m². Variations in ¹⁸O at individual outcrops suggest that time-integrated fluid fluxes and intrinsic permeabilities may locally have varied by at least an order of magnitude. A general increase in XCO₂ values of marble assemblages with metamorphic grade is also consistent with the up-temperature fluid-flow model. Fluids in the metapelites may have been derived from these rocks by devolatilisation at low metamorphic grades; however, fluids in the marbles were probably derived in part from the surrounding siliceous rocks. The marble-metapelite boundaries preserve steep gradients in both ¹⁸O and XCO₂ values, suggesting that across-strike fluid fluxes were much lower than those parallel to strike. Up-temperature fluid flow may also have formed orthoamphibole rocks and caused melting of the metapelites at high grades.This paper is a contribution to IGCP Project 304 Lower Crustal Processes     

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.     

Marine hdrothermal alteration at a Kuroko ore deposit,Kosaka, Japan

Isotopic compositions were determined for quartz, sericite and bulk rock samples surrounding the Uwamuki no. 4 Kuroko ore body, Kosaka, Japan. ¹⁸O values of quartz from Siliceous Ore (S.O.), main body of Black Ore B.O.) and the upper layer of B.O. are fairly uniform, +8.7 to +10.5. Formation temperatures calculated from fractionation of ¹⁸O between sericite and quartz from B.O. and upper S.O. are 250° to 300° C. The ore-forming fluids had ¹⁸O values of +1 and D values of –10, from isotope compositions of quartz and sericite.Tertiary volcanic rocks surrounding the ore deposits at Kosaka have uniform ¹⁸O values, +8.1±1.0 (n=50), although their bulk chemical compositions are widely varied because of different degrees of alteration. White Rhyolite, which is an intensely altered rhyolite occurring in close association with the Kuroko ore bodies, has also uniform ¹⁸O values, +7.9±0.9 (n=19). Temperatures of alteration are estimated to be around 300° C from the oxygen isotope fractionation between quartz and sericite. Paleozoic basement rocks phyllite and chert, have high ¹⁸O values, +18 and +19. The Sasahata formation of unknown age, which lies between Tertiary and Paleozoic formations, has highly variable ¹⁸O, +8 to +16 (n=4). High ¹⁸O values of the basement rocks and the sharp difference in ¹⁸O at their boundary suggest that the hydrothermal system causing Kuroko mineralization was mainly confined within permeable Tertiary rocks. D values of altered Tertiary volcanic rocks are highly variable ranging from –34 to –64% (n=12). The variation of D does not correlate with change of chemical composition, ¹⁸O values, nor distance from the ore deposits. The relatively high D values of the altered rocks indicate that the major constituent of the hydrothermal fluid was sea water. However, another fluid having lower D must have also participated. The fluid could be evolved sea water modified by interaction with rocks and the admixture of magmatic fluid. The variation in D may suggest that sea water mixed dispersively with the fluid.     

Ultra-high temperatures from oxygen isotope thermometry of a coesite-sanidine grospydite

The oxygen isotope compositions of coesite, sanidine, kyanite, clinopyroxene and garnet were measured in an ultra-high pressure-temperature grospydite from the Roberts Victor kimberlite, South Africa. The ¹⁸O values (per mil v. SMOW) of each phase and (1 ) are as follows: coesite, 8.62 (0.31); sanidine, 8.31 (0.02); kyanite, 7.98 (0.08); pyroxene, 7.63 (0.11); garnet, 7.53 (0.03). In situ analyses of the coesite with the laser extraction system are ¹⁸O=9.35 (0.08), n=4, demonstrating that the coesite is homogeneous. The coesite has partially inverted to polycrystalline quartz and the pyroxene is extensively altered during uplift. The larger scatter for the mineral separate coesite and pyroxene data may be due to partial reequilibration between the decompression-related breakdown products of these two phases. The anomalously high ¹⁸O value of the grospydite (¹⁸O_wholerock=7.7) is consistent with altered oceanic crust as a source rock. Temperature estimates from a linear regression of all the data to three different published calibrations correspond to an equilibrium temperature of 1310±80°C. The calculated isotopic pressure effect is to lower these estimates by about 40°C at 40 kb. The estimated temperature based on Al–Si disorder in sanidine is 1200±100°C and that from Fe–Mg exchange thermometry between garnet and clinopyroxene is 1100±50°C. Given the large errors associated with thermometry at such high temperatures, it is concluded that the xenolith equilibrated that 1200±100°C. Pressure estimates are 45±5 kb, based on dilution of the univariant equilibria albite = jadeite + coesite and 2 kyanite + 3 diopside = grossular + pyrope + 2coesite. Zoning in the outer 20 m of the feldspar from Ab_0.8 to Ab₁₆ indicates rapid decompression to 25 kb or less. The isotopic temperature estimates are the highest ever obtained and combined with the high degree of Al–Si disorder in sanidine require rapid cooling from ultra-high temperatures. It is inferred that the xenolith was sampled at the time of equilibration, providing a point on the upper Cretaceous geotherm in the mantle below South Africa.     

Meteoric hydrothermal origin of calcites in “Green Tuff” formations,Miocene age,Japan

Oxygen and carbon isotope compositions were determined for calcites from the Green Tuff formations of Miocene age in Japan. Values of ¹⁸O from 24 calcites in altered rocks from 5 districts range from –2 to +16_SMOW, in most cases from 0 to +8_SMOW. The low ¹⁸O values rule out the possibility of their low-temperature origin or any significant contribution of magmatic fluid in the calcite precipitation. These values, coupled with their mineral assemblages, suggest that the calcites formed from meteoric hydrothermal solutions which caused propylitic alteration after the submarine strata became emergent.Values of ¹³C from the calcites show a wide variation from –17 to 0_PDB. Calcites from different districts have different ranges of ¹³C values, indicating that there was no homogeneous reservoir of carbon at the time the calcite formed, and that the carbon had local sources. Carbon isotopic compositions of calcite within ore deposits in the Green Tuff formations range from –19 to 0_PDB, similar to those of calcite in the altered rocks in the same district, suggesting that the carbon in ore calcites was likely supplied from the surrounding rocks through activity of meteoric hydrothermal solutions.     

Calibration of Al/Si order variations in anorthite

New single crystal diffraction data for natural and heat-treated anorthite crystals (Angel et al. 1990) allow the determination of their states of Al/Si order in terms of a macroscopic order parameter,Q _OD , for the transition. Numerical values ofQ _OD obtained from estimates of site occupancies are shown to vary with the scalar spontaneous strain, _s , as _s Q _OD ² , and with the ratio of the sums of typeb (superlattice) reflections and typea (sublattice) reflections asI _b/I _a Q _OD ² . An empirical calibration for pure anorthite is obtained giving varies between 0.92 and 0.87 in samples equilibrated at T1300° C, but then falls off relatively rapidly with increasing temperature, reaching 0.7 near the melting point ( 1557° C). The observed temperature dependence does not conform to the predictions of the simplest single order parameter models; coupling ofQ _OD withQ of the transition is suspeeted.     

Oxygen and carbon isotopic study on the genesis of the Steirischer Erzberg siderite deposit (Austria)

The carbon and oxygen isotopic composition of Fe-carbonate ore and its calcitic to dolomitic Devonian host rocks at the Steirischer Erzberg siderite deposit (Greywacke zone, Upper Austroalpine Unit) were determined in order to constrain the source and nature of the Fe-rich mineralizing fluid. The ¹⁸O-values obtained for various Fe-carbonate generations and the carbonate host lie within a similar range between + 14.6 and + 21.6 (V-SMOW). No good correlation exists between the relative ages of the carbonate phases and their O isotopic composition. The variation in ¹⁸O-values is due to metamorphic recrystallization with locally variable fluid/rock ratios. The average ¹³C-value of the carbonate host is +0.5 ± 1.2 (PDB) which corresponds well to worldwide Phanerozoic marine carbonate values. The first Fecarbonate generation has slightly lower ¹³C-values, on average -1.4 ± 0.8 (PDB). Recrystallization of both the carbonate host minerals and the ankerite/siderite led to significantly lower ¹³C-values of -4.2 ± 0.6 and-4.7 ± 0.7, respectively. Within the basal breccia of the post-Hercynian transgression series matrix calcite/ dolomite shows an average ¹³C-value of -2.9 ± 0.7, and matrix siderite/ankerite an average value of-4.1 ± 0.4. These data, together with Sr isotope data published previously, strongly support a late-diagenetic or epigenetic first Fe-mineralization from convecting formation waters. They ascended along extension faults and were driven by an increased heat flow caused by crustal thinning during a Devonian rifting phase that initiated the separation of the Noric terrane from Africa. A potential source of the Fe could have been the underlying Ordovician acid volcanics. Regional metamorphism related to collision tectonics in the Late Carboniferous (Hercynian) and later during the Alpine orogeny, caused intensive recrystallization and partial mobilization of the various carbonate phases.     

Sulfur isotope ratios of Icelandic rocks

Sulfur isotope ratios have been determined in 27 selected volcanic rocks from Iceland together with their whole rock chemistry. The ³⁴S of analyzed basalts ranges from –2.0 to +0.4 with an average value of –0.8 Tholeiitic and alkaline rocks exhibit little difference in ³⁴S values but the intermediate and acid rocks analyzed have higher ³⁴S values up to +4.2 It is suggested that the overall variation in sulfur isotope composition of the basalts is caused by degassing. The small range of the ³⁴S values and its similarity to other oceanic and continental basalts, suggest that the depleted mantle is homogeneous in its sulfur isotope composition. The ³⁴S of the depleted mantle is estimated to be within the range for undegassed oceanic basalts, –0.5 to +1.0     

Interpolation by regularized spline with tension: I. Theory and implementation

Bivariate and trivariate functions for interpolation from scattered data are derived. They are constructed by explicit minimization of a general smoothness functional, and they include a tension parameter that controls the character of the interpolation function (e.g., for bivariate case the surface can be tuned from a membrane to a thin steel plate), Tension can be applied also in a chosen direction, for modeling of phenomena with a simple type of anisotropy. The functions have regular derivatives of all orders everywhere. This makes them suitable for analysis of surface geometry and for direct application in models where derivatives are necessary. For processing of large datasets (thousands of data points), which are now common in geosciences, a segmentation algorithm with a flexible size of overlapping neighborhood is presented. Simple examples demonstrating flexibility and accuracy of the functions are presented.On leave from the Department of Physical Geography and Cartography, Comenius University, Mlynská dolina, Bratislava, Czechoslovakia.On leave from the Institute of Physics, Dúbravská cesta 9, Bratislava, Czechoslovakia.     

A stable isotope study of retrograde alteration in SW Connemara,Ireland

Dalradian metamorphic rocks, Lower Ordovician meta-igneous rocks (MGS) and Caledonian granites of the Connemara complex in SW Connemara all show intense retrograde alteration. Alteration primarily involves sericitization and saussuritization of plagioclase, the alteration of biotite and hornblende to chlorite and the formation of secondary epidote. The alteration is associated with sealed microcracks in all rocks and planes of secondary fluid inclusions in quartz where it occurs, and was the result of a phase of fluid influx into these rocks. In hand specimen K-feldspar becomes progressively reddened with increasing alteration. Mineralogical alteration in the MGS and Caledonian granites took place at temperatures 275±15°C and in the MGS P_fluid is estimated to be 1.5 kbar during alteration. The °D values of alteration phases are:-18 to-29 (fluid inclusions),-47 to-61 (chlorites) and-11 to-31 (epidotes). Chlorite ¹⁸O values are +0.2 to +4.3, while ¹⁸O values for quartz-K-feldspar pairs show both positively sloped (MGS) and highly unusual negatively sloped (Caledonian granites) arrays, diverging from the normal magmatic field on a - plot. The stable isotope data show that the fluid that caused retrogression continued to be present in most rocks until temperatures fell to 200–140°C. The retrograde fluid had D -20 to-30 in all lithologies, but the fluid ¹⁸O varied both spatially and temporally within the range-4 to +7. The fO₂ of the fluid that deposited the epidotes in the MGS varied with its ¹⁸O value, with the most ¹⁸O-depleted fluid being the most oxidizing. The D values, together with low (<0) ¹⁸O values for the retrograde fluid in some lithologies indicate that this fluid was of meteoric origin. This meteoric fluid was probably responsible for the alteration in all lithologies during a single phase of fluid infiltration. The variation in retrograde fluid ¹⁸O values is attributed to the effects of variable oxygen isotope shifting of this meteoric fluid by fluid-rock interaction. Infiltration of meteoric fluid into this area was most likely accomplished by convection of pore fluids around the heat anomaly of the Galway granite soon after intrusion at 400 Ma. However convective circulation of meteoric water and mineralogical alteration could possible have occurred considerably later.