Evaluation of graphical and multivariate statistical methods for classification of water chemistry data

A robust classification scheme for partitioning water chemistry samples into homogeneous groups is an important tool for the characterization of hydrologic systems. In this paper we test the performance of the many available graphical and statistical methodologies used to classify water samples including: Collins bar diagram, pie diagram, Stiff pattern diagram, Schoeller plot, Piper diagram, Q-mode hierarchical cluster analysis, K-means clustering, principal components analysis, and fuzzy k-means clustering. All the methods are discussed and compared as to their ability to cluster, ease of use, and ease of interpretation. In addition, several issues related to data preparation, database editing, data-gap filling, data screening, and data quality assurance are discussed and a database construction methodology is presented. The use of graphical techniques proved to have limitations compared with the multivariate methods for large data sets. Principal components analysis is useful for data reduction and to assess the continuity/overlap of clusters or clustering/similarities in the data. The most efficient grouping was achieved by statistical clustering techniques. However, these techniques do not provide information on the chemistry of the statistical groups. The combination of graphical and statistical techniques provides a consistent and objective means to classify large numbers of samples while retaining the ease of classic graphical presentations. Electronic Publication     

Diffusive dissolution in ternary systems: analysis with applications to quartz and quartzite dissolution in molten silicates

Exact solutions to equations governing isothermal diffusive dissolution of a crystalline slab in a ternary liquid were obtained to include the effect of coupled chemical diffusion in the liquid. These analytical results, supplemented by approximate solutions valid for slow dissolving, provide new insights into the characteristics of diffusive dissolution in ternary systems. Dissolution rate is proportional to square root of time in diffusive dissolution. The coefficient of proportionality is a function of diffusion coefficients, liquidus relation, melt composition at the crystal–melt interface, and compositions of the dissolving crystal and starting melt. In the limit of slow dissolving, the dissolution rate can be written in terms of three dimensionless parameters that are functions of the aforementioned parameters. Dissolution rate is proportional to the diffusion rate of the slow eigen component in the melt when the diffusion rate of the minor eigen component is much slower than the diffusion rate of the major eigen component.Laboratory experiments of diffusive dissolution of single crystals and polycrystalline aggregates of quartz in a haplodacitic melt (25 wt.% CaO, 15 wt.% Al₂O₃, and 60 wt.% SiO₂) were conducted at 1500°C and 1 GPa. Measured dissolution distances (X_b, in microns) are proportional to the square root of experimental run time (t, in seconds), X_b = −0.620 (±0.019) √t. Chemical concentration profiles measured from quenched melts are invariant with time when displayed against the distance (measured from the crystal–melt interface) normalized by the square root of time. The melt compositions at the crystal–melt interface, extrapolated from the measured diffusion profiles in the quenched melts, are within 0.2 wt.% of the independently measured quartz liquidus in the ternary CaO–Al₂O₃–SiO₂ at 1500°C and 1 GPa. These results suggest that crystal and melt are in chemical equilibrium at their interface shortly after the onset of dissolution. Diffusive dissolution of quartz and quartzite is characterized by slow dissolving. Using quartz liquidus as one of the boundary conditions, it has been shown that the calculated dissolution distances and concentration profiles are in good agreement with the experimentally measured ones. Coupled diffusion played an essential role in quartz and quartzite dissolution in haplodacitic to haplobasaltic melts, and is likely to play an important role in diffusion-limited kinetic processes such as crystal growth and dissolution in natural melts of basaltic–rhyolitic compositions.     

Diffusion control of quartz and forsterite dissolution rates

An established engineering model is used to identify conditions where diffusion controls the dissolution of quartz and forsterite in packed beds. The model shows that diffusion control is favored at low advection flux, large grain size, high temperature, and high pH (if the reaction consumes H⁺). Quartz dissolution is chemical reaction controlled for most geochemically reasonable combinations of temperature, grain size, and flow rate. On the other hand, forsterite dissolution rates can be diffusion controlled for typical advection fluxes, grain sizes, temperatures, and pH’s. The apparent activation energy for diffusion-controlled reactions in a packed bed is much higher than the <∼20 kJ/mol value that is often used to identify diffusion controlled reactions. The models are quite general and can be adapted to deal with other mineral dissolution and precipitation reactions.     

A statistical analysis of exploration geochemical data for uranium

This paper describes a statistical analysis of reconnaissance exploration geochemical data for uranium. Three sets of data were analyzed, as they related to: (a) Charlotte-Winston-Salem and (b) Charlotte two-degree NTMS quadrangles of the south-eastern U.S.A. The coefficient of variation for uranium in each of the three sets of data was less than unity and hence no transformation of the original variable was needed. These data were subjected to correlogram analysis. A first-order Markovian model of the type: Y₀ exp (-a |p|) was fit by the least-squares method to serial correlation coefficients of these data using the method proposed by Deming (1948). The model was tested by computing the variance-volume relationship for assumed individual blocks of unit length. The noise in the input (record) was eliminated by the application of an optimum bilateral exponential smoothing technique developed by Agterberg. A comparison of spectral density estimates obtained by a maximum entropy method employing Yule-Walker equations and the Burg algorithm was also made. The prediction error coefficients needed to determine the order of the autoregressive process and hence the spectral densities were determined in both cases for the three sets of data.On leave from National Geophysical Research Institute, Hyderabad-7, India.     

A scale of dissolution for quartz and its implications for diagenetic processes in sandstones

During diagenesis quartz grains undergo selective dissolution, controlled in location by the surface energy characteristics of the individual grains. Experimental etching in HF of isolated quartz grains reproduces comparable textures to those of natural occurrences. Some experimental results illustrate the specific effects of surface textures on their initial dissolution rates, so demonstrating the control surface energy variation has over dissolution. A hierarchy of grain surface characteristics, according to surface energies, provides a useful guide to the relative rates of dissolution during decomposition.     

Modern statistical analysis and optimal estimation of geotechnical data

A common problem in geotechnical engineering is to estimate the parameters of a soil deposit from scattered measurements for use in deterministic models, finite elements for example. In this article two major sources of soil uncertainty are considered: measurement errors and spatial variability. A stochastic framework is then developed, in terms of the theory of random functions, to clarify dependencies and interactions between these uncertainties and the expected soil behaviour. This improves the engineering judgement leading to optimum design decisions. Spatial multivariable models describing geological processes in terms of covariances, parameter increments, recursive equations, empirical relationships or combinations thereof, improve the statistical inferences of the underlying random functions, and are of use both qualitatively and quantitatively. “Best” estimators in a well-established stochastic sense are derived from easily programmed procedures, and are tested in a number of common geotechnical applications, in an attempt to understand their properties and investigate their potential uses.     

A numerical model for porosity modification at a sandstone-mudstone boundary by quartz pressure dissolution and diffusive mass transfer

In many sandstones quartz cements significantly reduce porosity. The origin of these cements is often unclear. This paper investigates a possible mechanism for the generation of silica for quartz cement by pressure dissolution in interbedded mudstones. Theoretical models of quartz pressure dissolution, including the effects of silica precipitation kinetics, show that the concentration of dissolved silica in the pore fluids of a compacting sediment increases with decreasing grain size and silica precipitation constant. Quartz precipitation is strongly inhibited by the presence of small amounts of clay within a sediment, suggesting that siltstones and quartz-rich mudstones which are undergoing pressure dissolution may act as a source of dissolved silica for export to nearby, coarser sediments. A computational model for the diagenetic modification of a sandstone-mudstone interface due to pressure dissolution is described. Both sandstone and mudstone layers are assumed to be actively compacting by pressure dissolution, and mass transport by molecular diffusion is considered. As quartz precipitation in the mudstone layer is relatively slow compared to that in the sandstone, significant amounts of dissolved silica become available in the mudstone, and may be exported into the adjacent sandstone. In the absence of pore-fluid advection, this may result in the formation of extensive secondary quartz within the sandstone, close to the interface. The volume of silica exported from the mudstone is limited by the length scale over which diffusion through the mudstone is effective. This is typically 3–5 m. The volume of silica available therefore suggests that extensive porosity modification within the adjacent sandstone can only occur close to the mudstone. Thus it is possible that thin sandstones could become cemented by slow diffusive transfer of silica, but that in thicker sandstones the silica may become dispersed by pore-fluid advection.     

A new formulation of garnet–clinopyroxene geothermometer based on accumulation and statistical analysis of a large experimental data set

Published experimental data including garnet and clinopyroxene as run products were used to develop a new formulation of the garnet–clinopyroxene geothermometer based on 333 garnet–clinopyroxene pairs. Only experiments with graphite capsules were selected because of difficulty in estimating the Fe³⁺ content of clinopyroxene. For the calibration, a published subregular‐solution model was adopted to express the non‐ideality of garnet. The magnitude of the Fe–Mg excess interaction parameter for clinopyroxene (W_FeMg^Cpx), and differences in enthalpy and entropy of the Fe–Mg exchange reaction were regressed from the accumulated experimental data set. As a result, a markedly negative value was obtained for the Fe–Mg excess interaction parameter of clinopyroxene (W_FeMg^Cpx = ? 3843 J mol^?1). The pressure correction is simply treated as linear, and the difference in volume of the Fe–Mg exchange reaction was calculated from a published thermodynamic data set and fixed to be ?120.72 (J kbar^?1 mol^?1). The regressed and obtained thermometer formulation is as follows: where T = temperature, P = pressure (kbar), A = 0.5 X_grs (X_prp ? X_alm ? X_sps), B = 0.5 X_grs (X_prp ? X_alm + X_sps), C = 0.5 (X_grs + X_sps) (X_prp ? X_alm), X_prp = Mg/(Fe²⁺ + Mn + Mg + Ca)^Grt, X_alm = Fe/(Fe²⁺ + Mn + Mg + Ca)^Grt, X_sps = Mn/(Fe²⁺ + Mn + Mg + Ca)^Grt, X_grs = Ca/(Fe²⁺ + Mn + Mg + Ca)^Grt, X_Mg^Cpx = Mg/(Al + Fe^total + Mg)^Cpx, X_Fe^Cpx = Fe²⁺/(Al + Fe^total + Mg)^Cpx, K_D = (Fe²⁺/Mg)^Grt/(Fe²⁺/Mg)^Cpx, Grt = garnet, Cpx = clinopyroxene. A test of this new formulation to the accumulated data gave results that are concordant with the experimental temperatures over the whole range of the experimental temperatures (800–1820 °C), with a standard deviation (1 sigma) of 74 °C. Previous formulations of the thermometer are inconsistent with the accumulated data set; they underestimate temperatures by about 100 °C at >1300 °C and overestimate by 100–200 °C at <1300 °C. In addition, they tend to overestimate temperatures for high‐Ca garnet (X_grs ≈ 0.30–0.50). This new formulation has been tested against previous formulations of the thermometer by application to natural eclogites. This gave temperatures some 20–100 °C lower than previous formulations.     

Some methods for statistical analysis of multimodal distributions and their application to grain-size data

Some of the methods used in the resolution of mixed normal distributions are discussed under three headings: analytical, graphical, and numerical methods. Attention is given to their applicability in the analysis of grain-size data as derived from sieving. Comparisons are made by applying several methods to published data. It is concluded that the numerical methods offer most scope, especially the method of nonlinear least squares. Some analyses of beach sediments, using this method, are presented. The adoption of a convention for the number of individuals in the sample increases ease of interpretation.     

Comparison of elemental composition of macerals determined by electron microprobe to whole-coal ultimate analysis data

The elemental composition of the individual macerals in a suite of Australian coals has been determined in polished sections using light-element electron microprobe techniques. The analyses of the individual macerals in each coal were combined with data on maceral abundance to produce an inferred chemical composition for the organic matter of the respective whole-coal samples, and this was compared, for each sample, to the respective whole-coal ultimate analysis data, corrected to a dry, ash-free (daf) basis. Except for slightly lower values in some lower-rank coals, the inferred percentages of whole-coal C estimated from the microprobe data were found to be very close to the respective whole-coal C percentages as determined by conventional ultimate analysis. The proportion of O in the coals indicated by the microprobe study, however, appears to be as much as 2% higher than that derived from the ultimate analysis data, especially in the lower-rank coal samples. The difference it may represent errors in calculating the O percentages in ultimate analysis, errors in the microprobe analysis due to difficulties in calibration or measurement, or increased proportions of O in the coals due to factors such as take-up with storage of the polished sections. The percentages of whole-coal N calculated from the microprobe data are up to 0.5% (absolute) below the proportion of N determined directly by whole-coal ultimate analysis. This may reflect the inherent difficulty of dealing with a light element at low concentrations by the microprobe technique, or it may indicate that some of the N occurs in the coals in mineral form. The percentages of whole-coal (organic) S calculated from the microprobe study are close to the percentages of organic S determined for each sample by more conventional techniques. With the exception of (organic) O, which may be affected by other factors, and also possibly of N, the electron microprobe technique appears from the study to provide results that are consistent with ultimate analysis over a wide rank range.     

Sedimentary facies of barataria bay,Louisiana determined by multivariate statistical techniques

A multivariate statistical strategy employing cluster analysis, discriminant analysis, and ordination was used to classify and interpret depositional environments of Barataria Bay, Louisiana, from grain-size data tabulated by Krumbein and Aberdeen (1937). Weight-percent whole-phi variables for 69 samples were tested for redundancy using R-mode cluster analysis. These samples were partitioned into five environmentally significant facies using Q-mode cluster analysis: (1) beach-ridge sand; (2) shallow wave-zone sand; (3) channel sand and silty sand; (4) channel-margin silty sand; and (5) low-energy sandy silt and clayey silt. Then the classification was extended with discriminant analysis to the remaining 29 samples which exhibited incipient flocculation and whose constituents did not total unity. Environments of deposition were associated with each facies following the method of Visher (1969) and using gradient analysis on a two-dimensional Q-mode ordination. Wilk's lambda and discriminant analysis tested the statistical significance of these facies; they were further tested by comparing their distribution with field relations and known environmental processes.     

Comparison and combination of a gravimetric quasigeoid with a levelled GPS data set by statistical analysis

For GPS levelling applications, it is convenient to express the height reference surface in a suitable geodetic reference system. This can be obtained through a set of levelled GPS points. Unfortunately, available data are sparse. A gravimetric geoid is often used to interpolate the height reference surface issued from GPS and levelling. Both surfaces do not coincide exactly with each other. At this point, one must compare two realisations of the geoid, detect outliers, retrieve (if possible) the causes of the discrepancies and finally combine the two kinds of data. The paper presents some practical solutions to these problems.     

Factor analysis of sphericity and roundness data of clastic quartz grains: Environmental significance

The first four sample statistics of two-dimensional projection sphericity and those of two-dimensional projection roundness of clastic quartz grains belonging to river, beach and eolian environments and to the three size grades 0.350 mm-0.297 mm, 0.297 mm-0.250 mm and 0.250 mm-0.210 mm were subjected to R-mode factor analysis. The results show that the particles of these size grades are carried largely as a traction carpet in the river and beach environments and are carried largely by saltation in the eolian environment. Sorting and abrasion are the causes of areal variation in the mean sphericity and mean roundness of a population, respectively. Selective sorting causes nonnormality in the sphericity distribution, whereas roundness distribution is originally nonnormal. Selective sorting is most effective in the eolian, less so in the beach and least so in the river environment, and consequently sphericity-nonnormality constitutes a valuable parameter for environmental discrimination. Intensity of rounding is highest in the beach, less in the eolian and least in the river environments. Due to its wide variation from environment to environment the mean roundness constitutes a valuable parameter for differentiating different environments. Sphericity variables contribute more to the discrimination of environments than roundness variables.     

Kinetics of dissolution of mechanically comminuted rock-forming oxides and silicates—I. Deformation and dissolution of quartz under laboratory conditions

The extensive evidence on the properties of ground quartz such as grain morphology and adherence and subsurface structural damage shows a consistent pattern, interpretable in terms of the competing processes of fracture and of local plastic deformation.The well-documented pattern of dissolution of ground quartz in aqueous solutions is also consistent. At large undersaturations and constant other environmental variables, the apparent dissolution rate of ground quartz decreases exponentially with the increasing thickness of the equivalent dissolved disturbed layer, and the rate of release of adherent tiny fragments shows roughly the same dependence on the latter. At small undersaturations distribution of solubilities also needs to be considered.     

Geobarothermometric properties of growth twins and mathematical analysis of quartz data for a broad range of temperatures and pressures

The thermoelastic analysis of the possible stable stress-free equilibrium states of growth twins is shown to be the basis for a new technique of geobarothermometry. The detailed study of a cyclic cross-shaped twin in low quartz exemplifies the procedure. To explicitly determine the barothermometric function, the simultaneous dependence upon the external pressure and temperature of the lattice-parameter ratios c/c ₀ and a/a ₀ of quartz has been calculated; we present a five-parameter, temperature-corrected version of the equations proposed by Thurston (1967) for extrapolating to high pressures the behavior of lattice parameters of crystalline solids in isothermal compression.     

The statistical study of hydraulic geometry: A new direction for compositional data analysis

Hydraulic exponents and unit hydraulic exponents are unit-sum constrained, which requires that they be analyzed by statistical methods designed for compositional data. Though uncertainties remain regarding selection of the best constraining operation and method of handling departures from the unit-sum constraint, neither category of uncertainty should be an impediment to the selection of the appropriate statistical methodology. In a small sample study, the hydraulic geometry of different types of streams were compared: (1) semi-arid: perennial vs. ephemeral; (2) tropical: Puerto Rico vs. West Malaysia; and (3) semi-arid vs. tropical (by pooling the previous data sets). All three comparisons revealed statistically significant differences in either logratio mean vectorsor logratio covariance matrices but not both. All six categories of data had logistic normal distributions. Because the derivatives at a given discharge of curvilinear hydraulic geometry relationships and hydraulic exponents on either side of the breakpoints of piecewise linear relationships are unit-sum constrained, they also can be studied by compositional methods. However, the compositional approach is limited in cases where distributions have large departures from logistic normality and for streams that have negative hydraulic exponents.     

Geochemical assessment of fluoride enrichment and nitrate contamination in groundwater in hard-rock aquifer by using graphical and statistical methods

This systematic study was carried out with objective to delineate the various sources responsible for \(\hbox {NO}_{3}^{-}\) contamination and \(\hbox {F}^{-}\) enrichment by utilizing statistical and graphical methods. Since Central Ground Water Board, India, indicated susceptibility of \(\hbox {NO}_{3}^{-}\) contamination and \(\hbox {F}^{-}\) enrichment, in most of the groundwater, \(\hbox {NO}_{3}^{-}\) and \(\hbox {F}^{-}\) concentration primarily observed \({>}45\) and \({>}1.5~\hbox {mg/L}\), respectively, i.e., higher than the permissible limit for drinking water. Water Quality Index (WQI) indicates \({\sim }22.81\%\) groundwater are good-water, \({\sim }71.14\%\) groundwater poor-water, \({\sim }5.37\%\) very poor-water and 0.67% unsuitable for drinking purpose. Piper diagram indicates \({\sim }59.73\%\) groundwater hydrogeochemical facies are Ca–Mg–\(\hbox {HCO}_{3 }\) water-types, \({\sim }28.19\%\) Ca–Mg–\(\hbox {SO}_{4}\)–Cl water-types, \({\sim }8.72\%\) Na–K–\(\hbox {SO}_{4}\)–Cl water-types and 3.36% Na–K–\(\hbox {HCO}_{3 }\) water-types. This classification indicates dissolution and mixing are mainly controlling groundwater chemistry. Salinity diagram indicate \({\sim }44.30\%\) groundwater under in low sodium and medium salinity hazard, \({\sim }49.66\%\) groundwater fall under low sodium and high salinity hazard, \({\sim }3.36\%\) groundwater fall under very-high salinity hazard. Sodium adsorption ratio indicates \({\sim }97\%\) groundwater are in excellent condition for irrigation. The spatial distribution of \(\hbox {NO}_{3}^{-}\) indicates significant contribution of fertilizer from agriculture lands. Fluoride enrichment occurs in groundwater through the dissolution of fluoride-rich minerals. By reducing the consumption of fertilizer and stress over groundwater, the water quality can be improved.     

Multivariate statistical analysis of soil laboratory data from Montgomeryshire,Wales

This paper describes the application of multivariate statistical techniques in the study of the soils of Montgomeryshire, Wales. Principal components analysis is used to analyze data on 39 soil horizon properties for 150 soil profiles throughout the county. Six principal components which account for 62% of the total variance in the data are selected and used in the cluster analysis of the soil profiles using a non-hierarchical grouping technique. The analysis revealed certain patterns in the data and the grouping of the soil profiles which have important implications for describing and characterizing the soils in the area and for understanding their distribution and the relationship between them.     

The stability of annite+quartz: reversed experimental data for the reaction 2 annite+3 quartz=2 sanidine+3 fayalite +2 H2O

Reversals for the reaction 2 annite+3 quartz=2 sanidine+3 fayalite+2 H₂O have been experimentally determined in cold-seal pressure vessels at pressures of 2, 3, 4 and 5?kbar, limiting annite +quartz stability towards higher temperatures. The equilibrium passes through the temperature intervals 500–540°?C (2?kbar), 550–570°?C (3?kbar), 570–590°?C (4?kbar) and 590–610°?C (5?kbar). Starting materials for most experiments were mixtures of synthetic annite +fayalite+sanidine+quartz and in some runs annite+quartz alone. Microprobe analyses of the reacted mixtures showed that the annites deviate slightly from their ideal Si/Al ratio (Si per formula unit ranges between 2.85 and 2.92, Al^VI between 0.06 and 0.15). As determined by Mössbauer spectroscopy, the Fe³⁺ content of annite in the assemblage annite+fayalite +sanidine+quartz is around 5–7%. The experimental data were used to extract the thermodynamic standard state enthalpy and entropy of annite as follows: H ⁰ _f,?Ann =?5125.896±8.319 [kJ/mol] and S ⁰ _Ann=432.62±8.89 [J/mol/K] (consistent with the Holland and Powell 1990 data set), and H ⁰ _f,Ann =?5130.971±7.939 [kJ/mol] and S ⁰ _Ann=424.02±8.39 [J/mol/K] (consistent with the TWEEQ data base, Berman 1991). The preceeding values are close to the standard state properties derived from hydrogen sensor data of the redox reaction annite=sanidine+magnetite+H ₂ (Dachs 1994). The experimental half-reversal of Eugster and Wones (1962) on the annite +quartz breakdown reaction could not be reproduced experimentally (formation of annite from sanidine+fayalite+quartz at 540°?C/1.035?kbar/magnetite-iron buffer) and probable reasons for this discrepancy remain unclear. The extracted thermodynamic standard state properties of annite were used to calculate annite and annite+quartz stabilities for pressures between 2 and 5?kbar.