An examination of the use of the logratio transformation for the testing of endmember hypotheses

Given a compositional dataset in the absence of any prior information on any mixing process which may have formed it, a complete analysis of mixtures determines three distinct types of estimates in order. These are: (i) the estimate of the number of endmembers or fixed source compositions, of which all the sample compositions of the dataset must be approximate mixtures; (ii) the estimated compositions for each of these chosen number of endmembers; and (iii) the estimated contributions of each of these endmember estimates to each sample. Traditionally, the estimate for the number of endmembers has been assessed either by mapping or by inspection of the coefficients of determination between the observed and estimated variables. Mapping entails the plotting on a map of the region from which the samples were taken, either the contours of the contributions of each endmember to each sample, or some other portrayal of the distribution of endmember abundances. Because it requires the complete analysis, assessment by this method is too elaborate except for final confirmation and display. Alternatively, choosing a number of endmembers, which result in suitability high coefficients of determination for all or most variables, may account for elements which are not part of the conjectured mixing process or, worse, may result in the identification of endmembers which may never in fact have existed. Such an error is similar to overspecifying a multiple regression model. So, the obvious starting point from which to assess the validity, or otherwise choice of endmember numbers, is to examine the matrix of residuals. The differences between the logratio-transformed observed and estimated data form an array of residual logratios. A linear combination of these may be formed for each sample, which, under a random perturbation assumption, should follow a univariate normal distribution. Whether or not this scalar is normal can be readily tested. It can also be examined graphically for such desirable qualities as symmetry when the test for normality may be too severe. This procedure is employed to assess the decompositions of the U.S.G.S. Mid-Pacific data and the Nazca Plate Surface sediments.This paper was presented at the 18th Geochautauqua, Newark, Delaware, 13–14 October 1989.     

Temperature dependence of the cation distribution in nickel aluminate (NiAl2O4) spinel: a powder XRD study

NiAl₂O₄ is a largely inverse spinel, which in detail shows increasing randomisation with temperature of Ni and Al between the octahedral and tetrahedral cation sites of the spinel structure. We have used powder XRD to determine this cation distribution in various samples of NiAl₂O₄ quenched after annealing between 700 and 1400° C. The inversion parameter (x) can be measured with a precision of ± 0.004 (one standard deviation), and a comparison of different methods of synthesis, X-ray diffraction and refinement techniques, suggests a probable accuracy of better than 0.01. The results are supported by some preliminary single crystal refinements on flux-grown samples.Below 800° C the rate of cation ordering becomes very slow, and, despite reaching an apparently steady state, it is doubtful if our samples attained complete internal equilibrium. Above 1250° C the cation redistribution becomes so fast that the quenching method becomes unreliable. Between 800 and 1250° C inclusive, the degree of inversion changes smoothly from 0.87 at 800° C to 0.79 at 1250° C, and is accompanied by linear changes in u, the oxygen parameter, from 0.2555 to 0.2563 (±0.0002), and a₀, the lattice parameter, from 8.0462 to 8.0522 Å (±0.0002 Å).     

Sediment budgeting: A case study in the Katiorin drainage basin,Kenya

The primary objective of this study was to compute a detailed budget for a small semiarid tropical drainage basin in Kenya. Results indicated that transfer of sediments (‘inputs’) from primary source areas was minor in comparison to changes in storage. The major sediment source area within the Katiorin drainage basin was the colluvial hillslope zone. The net change in storage within this zone was approximately 2100 Mg yr^?1. Surface wash and rilling were the dominant transport processes responsible for the remobilization of colluvial sediments. Sediment storage within the in-channel reservoir increased by 60 Mg yr^?1, which was minor when compared to the total store of sediment in this reservoir. During 1986, the channel network stored only a small fraction ( < 3 per cent) of the sediment delivered from the hillslope subsystem. Therefore, the in-channel reservoir had limited influence on sediment conveyance to the basin outlet. These data indicate that a static equilibrium condition cannot be assumed within the Katiorin drainage basin. Such an assumption would result in erosion estimates of approximately 5.5 mm yr^?1 for the entire basin (based on a sediment output of 7430 Mg km^?2 yr^?1 and a measured bulk density of 1.35 Mg m^?3). However, this masked the actual rates of 1.2 to 7.1 mm yr^?1 in subbasin primary source areas, and rates of 0.6 to 17 mm yr^?1 for colluvial material in the various subbasins. The extreme accelerated erosion rates resulted from minimal ground vegetation, steep slopes, soil crust formation, an erodible substrate, and a well-integrated drainage network for rapid conveyance of sediments from the hillslope subsystem to the basin outlet.     

Geometry of a convergent zone, Central British Columbia, Canada

The Canadian Cordillera is separable into two major northerly trending tectonic units—the Pacific Orogen and the Columbian Orogen, with the latter further separated into the Omineca Crystalline Belt and the easterly Rocky Mountain Fold and Thrust Belt. Synkinematic metamorphism of Jurassic age within the Omineca Belt is thought to be associated with accretion of westerly terranes of the Pacific Orogen—more specifically the Quesnellia terrane—that was thrust easterly over the Omineca Belt towards the craton. Mylonitic rocks mark the margin between these two belts and this margin is well-exposed near Crooked Lake, central British Columbia.Structural analysis across the zone of convergence between these two terranes indicates that the cratonic basement and the accreted cover sequences have several phases of deformation and metamorphism in common. The initial common phase of deformation, wherein convergence is accomplished, is characterized by easterly verging folds that are superposed by a second common phase having westerly verging folds that deform the zone of convergence and control the present regional map pattern. A final common phase of deformation produced easterly verging folds.Change in vergence direction is interpreted as resulting from change in direction of transport related to subduction process: first obduction of Quesnellia onto the Omineca craton, followed later by easterly subduction of an oceanic Quesnellia below the craton.All evidence of transport direction(s) points to convergence occurring at very high angles to the zone of convergence. There appears to be no evidence of transport parallel with the strike of the zone. If transport has taken place parallel to the strike of the zone, then this transport occurred before convergence or evidence of this motion has been destroyed during the convergence.     

Al,Si ordering in cordierite using “magic angle spinning” NMR

A method of determining the number of Al-O-Al bonds per unit cell from ²⁹Si nuclear magnetic resonance (NMR) data of synthetic cordierites with increasing Si, Al order is described. The number of Al-O-Al bonds is found to vary linearly with the logarithm of the annealing time. This may be correlated with previously published heat of solution data on similar samples (Carpenter et al. 1983) to determine the enthalpy change Δh, associated with a single Al?Si interchange in cordierite. Δh is found to be 8.1 kcal/mole. The NMR data show that the short range Al, Si order cannot be described in terms of twin domains of ordered orthorhombic cordierite. An ordering model derived from group theoretical constraints on possible Al, Si distributions within the hexagonal symmetry of the cordierite is found to provide a better fit to the NMR data.     

Weighting of Observed Heads for the Inverse Problem

In using least-squares parameter estimation techniques to solve for hydrogeologic parameters, one may use a weighting function to reflect differing reliabilities of head measurements. In studies published to date, the weighting function has been used in an ad boc manner or not at all. The inverse square of the observed hydraulic gradient, adjusted to reflect the modeler's perception of geologic heterogeneity and data reliability, is typically an appropriate weighting function.     

Compositional and thermal convection in magma chambers

Magma chambers cool and crystallize at a rate determined by the heat flux from the chamber. The heat is lost predominantly through the roof, whereas crystallization takes place mainly at the floor. Both processes provide destabilizing buoyancy fluxes which drive highly unsteady, chaotic convection in the magma. Even at the lowest cooling rates the thermal Rayleigh number Ra is found to be extremely large for both mafic and granitic magmas. Since the compositional and thermal buoyancy fluxes are directly related it can be shown that the compositional Rayleigh number Rs (and therefore a total Rayleigh number) is very much greater than Ra. In the case of basaltic melt crystallizing olivine Rs is up to 10⁶ times greater than Ra. However compositional and thermal buoyancy fluxes are roughly equal. Therefore thermal and compositional density gradients contribute equally to convection velocities in the interior of the magma. Effects of thermal buoyancy generated by latent heat release at the floor are included.The latent heat boundary layer at the floor of a basaltic chamber is shown to be of the order of 1 m thick with very low thermal gradients whereas the compositional boundary layer is about 1 cm thick with large compositional gradients. As a consequence, the variation in the degree of supercooling in front of the crystal-liquid interface is dominated by compositional effects. The habit and composition of the growing crystals is also controlled by the nature of the compositional boundary layer. Elongate crystals are predicted to form when the thickness of the compositional boundary layer is small compared with the crystal size (as in laboratory experiments with aqueous solutions). In contrast, equant crystals form when the boundary layer is thicker than the crystals (as in most magma chambers). Instability of the boundary layer in the latter case gives rise to zoning within crystals. Diffusion of compatible trace elements through the boundary layer can also explain an inverse correlation, observed in layered intrusions, between Ni concentration in olivine and the proportion of Ni-bearing phases in the crystallizing assemblage.     

Patterns in seasonal abundance,growth and biomass of the atlantic silverside,Menidia menidia,in a New England estuary

Patterns in seasonal abundance (no. per m² surface area), growth and biomass (g per m² surface area) of an annual fish, the Atlantic silverside, Menidia menidia (L.) were investigated in a marsh and more seaward bay region of Essex Bay, Massachusetts from August 1976 to May 1978 using a quantitative beach seining technique. Silverside abundance varied greatly by season and year class during the study period. Abundance was high in 1976 but winter mortality (99%) left an adult density of only .01 per m² surface area in the marsh during spring 1977. Resultant 1977 year class density in the marsh was 1.88 per m² by late fall 1977 but winter mortality again produced an adult density of .01 per m² in spring 1978. Abundance was generally higher in the marsh than in the bay region especially during spring and late fall when catches in the bay were negligible. Based on catch rate comparisons, the summer and fall juvenile abundance of the 1976 year class was much higher than the juvenile abundance of the 1977 year class. Coincidentally, mean lengths and condition of the abundant 1976 year class in the late fall were significantly lower than those of the 1977 year class, suggesting density dependent population regulation. In both years, juveniles grew rapidly and reached full adult size by November when an offshore movement to deeper waters outside Essex Bay occurred. Biomass peaked in the marsh region in late fall 1977 at 7.8 g per m² wet weight. Winter mortality was size selective, favoring larger individuals. The annual life history design of M. menidia including an offshore winter movement and high winter mortality suggests that silversides represent an important pathway of energy flow from marsh to offshore trophic systems.