Confidence Intervals of Quantiles in Hydrology Computed by an Analytical Method

Fitting probability distributions to hydrologic data samples is widely used for quantile estimation purposes. The estimated quantile (X^_T) is related to a return period (T). The confidence interval associated with each of the estimates has been calculated empirically, up until now, supposing that the quantile estimator is normally distributed. In this study, it is shown that the confidence interval follows a normal distribution only in the central part of the distribution. The real confidence limits are computed analytically, by defining and integrating the probability density function of the confidence interval. The results with an important number of hydrologic samples show that the upper confidence limits are significantly underestimated towards the tail of the distribution, when determined using the normality approximation for the quantile estimator.     

Morphometry of gibber gravel at Mt Sturt,New South Wales

Mt Sturt is a duricrusted residual, from which particles of silcrete are discharged on to flanking slopes and surrounding pediments. Dimensional analysis of the particles yields log‐normal curves, bimodal on the crest but unimodal elsewhere, with an orderly downslope decrease in median dimensions. The rate of decrease appears insufficient to reduce the particles to sand size within conceivable limits of pediment length. Analysis of shape indices gives normal distributions with no significant down‐slope change of index, except for the roundness index which tends to assume distributions of the Poisson type and to increase in value downslope. The general correspondence between observed distributions and theoretical curves is close, although it seems likely that gilgai can disturb to some extent the distribution of dimensions in particular samples.     

Grain sorting in gravel-bed streams and the choice of particle sizes for flow-competence evaluations

Flow-competence assessments of floods have been based on the largest particle sizes transported, and yield either the mean flow stress, mean velocity, or discharge per unit flow width. The use of extreme particle sizes has potential problems in that they may have been transported by debris flows rather than by the flood, it may be difficult to locate the largest particles within the flood deposits, and there are questions concerning how representative one or a few large particles might be of the transported sediments and therefore of the flood hydraulics. Such problems would be eliminated for the most part if competence evaluations are based on median grain sizes of transported sediments, or perhaps on some coarse percentile that is established by a reasonable number of grains. In order to examine such issues, the gravel-transport data of Milhous from Oak Creek, Oregon, and of Carling from Great Eggleshope Beck, England, have been analysed in terms of changing grain-size percentiles with varying flow stresses. A comparison between these two data sets is of added interest because the bed material in Oak Creek is segregated into well-developed pavement and subpavement layers, while such a layering of bed materials is largely absent in Great Eggleshope Beck. The analyses show that the trend of increasing sizes of the largest particles in the bedload samples (diameter D_m) with increasing flow stresses is consistent with similar dependencies based on sieve percentiles ranging from the medians (D₅₀) to the 95th percentiles (D₉₅). This indicates that the largest particles are an integral part of the overall distributions of bedload grain sizes, and respond to changing flow hydraulics along with the rest of the size distribution. In Oak Creek, the median grain size shows the largest change with increasing flow stresses, followed by D₆₀, and so on to D₉₅ which shows the smallest change. The variations in D_m continue this trend, and are similar to those for D₉₅. This systematic variation of grain-size percentiles in Oak Creek is consistent with changes in the overall distributions which tend to be symmetrical and Gaussian for low discharges, but become skewed Rosin distributions for high discharges. In contrast, in Great Eggleshope Beck the several percentiles and D_m show the same rate of shift to coarser sizes as flow stresses increase. This results in part from differences in sampling techniques wherein the bedload samples from Great Eggleshope Beck represent a complete flood event, while shorterterm samples at a specific flow stage were obtained in Oak Creek. As a result of the integrated sampling in Great Eggleshope Beck, the bedload grain-size distributions are more complex, commonly with a bimodal pattern. However, after accounting for differences in sampling schemes in the two streams, contrasting patterns in changing grain-size distributions remain, and these are concluded to reflect grain sorting differences as the bedload grain-size distributions approach the distributions of the bed materials. It is surprising that if criteria commonly employed to demonstrate the equal mobility of different grain sizes are used in the comparison, then Great Eggleshope Beck is far closer to this condition in spite of its minimal development of a pavement. It is concluded that the respective shapes of the bed-material grain-size distributions, in particular their degrees of skewness, are more important to the observed sorting patterns than are the effects of a pavement layer regulating grain entrapment to produce an equal mobility of different grain sizes. Therefore, the comparison has established that flow-competence relationships will differ from one stream to another, depending on the pattern of grain sorting which is a function of the bedmaterial grain-size distribution.     

The use of linear programming in the analysis of petrological mixing problems

Petrological mixing problems such as modal analysis, magma mixing, and liquid line of descent calculations, can be solved using the methods of linear programming. If estimates of the standard error of the chemical data are introduced as weights into the set of equations, it is possible to assign confidence limits to the solutions which are obtained and to apply formal statistical tests to geological hypotheses based on the mixing model. This approach is applied to petrological data previously analysed by Wright and Doherty (1970) using a combination of linear programming and least squares methods. It is shown that some of the geological inferences which they drew were based on an overoptimistic assessment of the confidence limits on their solutions, and cannot be regarded as proven.     

Generalized Bootstrap Method for Assessment of?Uncertainty in Semivariogram Inference

The semivariogram and its related function, the covariance, play a central role in classical geostatistics for modeling the average continuity of spatially correlated attributes. Whereas all methods are formulated in terms of the true semivariogram, in practice what can be used are estimated semivariograms and models based on samples. A generalized form of the bootstrap method to properly model spatially correlated data is used to advance knowledge about the reliability of empirical semivariograms and semivariogram models based on a single sample. Among several methods available to generate spatially correlated resamples, we selected a method based on the LU decomposition and used several examples to illustrate the approach. The first one is a synthetic, isotropic, exhaustive sample following a normal distribution, the second example is also a synthetic but following a non-Gaussian random field, and a third empirical sample consists of actual raingauge measurements. Results show wider confidence intervals than those found previously by others with inadequate application of the bootstrap. Also, even for the Gaussian example, distributions for estimated semivariogram values and model parameters are positively skewed. In this sense, bootstrap percentile confidence intervals, which are not centered around the empirical semivariogram and do not require distributional assumptions for its construction, provide an achieved coverage similar to the nominal coverage. The latter cannot be achieved by symmetrical confidence intervals based on the standard error, regardless if the standard error is estimated from a parametric equation or from bootstrap.     

Statistics for Modeling Heavy Tailed Distributions in Geology: Part I. Methodology

Many natural phenomena exhibit size distributions that are power laws or power law type distributions. Power laws are specific in the sense that they can exhibit extremely long or heavy tails. The largest event in a sample from such distribution usually dominates the underlying physical or generating process (floods, earthquakes, diamond sizes and values, incomes, insurance). Often, the practitioner is faced with the difficult problem of predicting values far beyond the highest sample value and designing his system either to profit from them, or to protect against extreme quantiles. In this paper, we present a novel approach to estimating such heavy tails. The estimation of tail characteristics such as the extreme value index, extreme quantiles, and percentiles (rare events) is shown to depend primarily on the number of extreme data that are used to model the tail. Because only the most extreme data are useful for studying tails, thresholds must be selected above which the data are modeled as power laws. The mean square error (MSE) is used to select such thresholds. A semiparametric bootstrap method is developed to study estimation bias and variance and to derive confidence limits. A simulation study is performed to assess the accuracy of these confidence limits. The overall methodology is applied to the Harvard Central Moment Tensor catalog of global earthquakes.     

Error estimation in stratigraphic correlation

Correlation in coalfields depends on the recognition of suitable reference points in the succession. In the East Midlands coal seams and marker horizons provide the most dependable reference points recognizable in borehole records. Distinctive marker horizons provide especially useful reference points as they may be considered to mark points of absolute time equivalence, but their use is limited owing to their irregular distribution. A graphic technique is used to formulate ‘confidence limits of correlation’ between marker horizons where correlation is uncertain. Functions are derived which describe the confidence limits. Coal-seam correlation data from a set of boreholes are used to derive confidence curves in a number of cases. It is concluded that one set of curves may be used for all the Middle Coal Measures of the East Midlands Coalfield. Use of these curves as general tools in correlation is demonstrated.     

Some limitations in the geostatistical evaluation of ore deposits

Conclusions The foregoing discussion indicates that geostatistical estimation of ore deposits is not local; it is not objective; it is not sensitive to local data trends; and it is not unrestrained by the range of data values.Kriging, as an interpolation method, is a variant of IDW least squares linear fit. As such, it suffers from the limitations of all IDW linear interpolation methods that employ only data values.The estimation variance, currently used to calculate the confidence limits of values for individual mining blocks, is hypothetical and globally derived. It is more closely related to sampling density than to local variation in the data set.Geostatistical methods, of course, have a real place in ore deposit assessment, e.g. global, comparative evaluation to assist decisions on development and investment. What is questioned here is the validity of employing a global method to assess detail (mining blocks) within an ore deposit.     

A comparison of procrystal and ab initio model representations of the electron-density distributions of minerals

 The procrystal calculation of the electron density is a very rapid procedure that offers a quick way to analyze various bonding properties of a crystal. This study explores the extent to which the positions, number, and properties of bond-critical points determined from the procrystal representations of the electron density for minerals are similar to those of first-principles ab initio model distributions. The purpose of the study is to determine the limits imposed upon interpretation of the procrystal electron density. Procrystal calculations of the electron density for more than 300 MO bonds in crystals were compared with those previously calculated using CRYSTAL98 and TOPOND software. For every bond-critical point found in the ab initio calculations, an equivalent one was also found in the procrystal model, with similar magnitudes of electron density, and at similar positions along the bonds. The curvatures of the electron densities obtained from the ab initio and the procrystal distributions are highly correlated. It is concluded that the procrystal distributions are capable of providing good estimates of the bonded radii of the atoms and the properties of the electron-density distributions at the bond-critical points. Because the procrystal model is so fast to compute, it is especially useful in addressing the question as to whether a pair of atoms is bonded or not. If the Bader criteria for bonding are accepted, then the successful generation of the bond-critical points by the procrystal model demonstrates that bonding is an atomic feature. The main difference between the critical-point properties of the procrystal and the ab initio model is that the curvature in the electron density perpendicular to the bond path of the ab initio model is sharper than for the procrystal model. This is interpreted as indicating that the electrons that migrate into a bond originate from its sides, and not from the regions closer to the nuclei. This observation also suggests that ab initio optimization routines could see an improvement in speed if the parameters relating to the angular components of atomic wave functions were to vary before the radial components. Received: 6 August 2001 / Accepted: 21 November 2001     

Galveston Bay: Temporal changes in the concentrations of trace organic contaminants in national status and trends oysters (1986–1994)

The temporal distributions for six classes of trace organic contaminants (chlordanes, DDTs, dieldrin, PAHs, PCBs, and butyltins) in oysters from six Galveston Bay sites from National Oceanic and Atmospheric Administration’s National Status and Trends (NS&T) Mussel Watch Program are compared with other NS&T sites from the Gulf of Mexico as well as all NS&T sites of the United States (East Coast, West Coast, and Gulf of Mexico). Decreases in the median for the Gulf-wide concentration of chlordanes, dieldrin, and butyltins occurred during 1986–1994. The Gulfwide median concentrations of DDTs, PAHs, and PCBs exhibited a strong cyclic distribution with time. For Galveston Bay oysters, “high” concentration is defined as the concentration greater than the median plus one standard deviation for all Gulf of Mexico sites. The percentage of sites having high concentrations during 1986–1994 for Galveston Bay oysters are 49% for dieldrin, 45% for butyltins, 40% for chlordanes, 38% for PCBs, 30% for PAHs, and 21% for DDTs. For PCBs, 43% of Galveston Bay oyster samples analyzed over the first 9 yr have concentrations high enough for potential biological effects to be observed in oysters. The percentages in other agents were chlordanes (22%), butyltins (22%), dieldrin (5%), and PAHs (4%). National Academy of Science-proposed regulatory limits for oysters were exceeded in only 2% of Galveston Bay samples for DDTs and 1% for PCBs. All other contaminants were below proposed NAS limits.     

Setting seagrass depth, coverage, and light targets for the Indian River Lagoon system, Florida

Seagrass protection and restoration in Florida’s Indian River Lagoon system (IRLS) is a mutual goal of state and federal programs. These programs require, the establishment of management targets indicative of seagrass recovery and health. We used three metrics related to seagrass distribution: areal coverage, depth limit, and light requirement. In order to account for the IRLS’s spatial heterogeneity and temporal variability, we developed coverage and depth limit targets for each of its 19 segments. Our method consisted of two steps: mapping the union of seagrass coverages from all availabe mapping years (1943, 1986, 1989, 1992, 1994, 1996, and 1999) to delineate wherever seagrass had been mapped and determining the distribution of depth limits based on 5,615 depth measurements collected on or very near the deep-edge boundary of the union coverage. The frequency distribution of depth limits derived from the union coverage, along with the median (50th percentile) and maximum (95th percentile) depth limits, serve as the seagrass depth targets for each segment. The median and maximum depth targets for the IRLS vary among segments from 0.8 to 1.8 and 1.2 to 2.8 m, respectively.Halodule wrightii is typically the dominant seagrass species at the deep-edge of IRLS grass beds. We set light requirement targets by using a 10-yr record of light data (1990–1999) and the union coverage depth limit distributions from the most temporally stable seagrass segments. The average annual light requirement, based on the medians of the depth limit distributions, is 33 ± 17% of the subsurface light. The minimum annual light requirement, based on of the 95th percentile of the depth distributions, is 20 ± 14%; the minimum growing season light requirement (March to mid September) is essentially the same (20 ± 13%). Variation in depth limits and light requirements, is probably due to factors other than light that influence the depth limit of seagrasses (e.g., competition, physical disturbance). The methods used in this study are robust when applied to large or long-term data sets and can be applied to other estuaries where grass beds are routinely monitored and mapped.     

The use of slump structures as palaeoslope orientation estimators

The two main methods of estimating palaeoslope direction from slump structure orientation data are reviewed. The mean axis method takes the downslope direction as the perpendicular to the mean slump fold axis. The separation are method estimates the downslope direction by the bisector of a planar separation angle between groups of slump folds with opposite downplunge asymmetry. The advantages and disadvantages of each method are considered in detail, by reference first to idealized slump fold distributions and then to a compilation of sixty natural slump fold axis distributions, mostly from published data. Among the advantages of the mean axis method are the ease of deriving confidence limits for estimates and its applicability where fold asymmetry data are unavailable. The main advantage of the separation are method is its validity for strongly skewed fold axis samples. The statistical robustness of the mean axis method, however, due to its reliance on average properties of data, makes it more generally reliable than the separation arc method which depends on extreme properties of data. A procedure for determining palaeoslope directions is suggested which reflects the respective strengths of the two methods.     

Application of uncertainty analysis to groundwater pollution modeling

Prediction and evaluation of pollution of the subsurface environment and planning remedial actions at existing sites may be useful for siting and designing new land-based waste treatment or disposal facilities. Most models used to make such predictions assume that the system behaves deterministically. A variety of factors, however, introduce uncertainty into the model predictions. The factors include model and pollution transport parameters and geometric uncertainty. The Monte Carlo technique is applied to evaluate the uncertainty, as illustrated by applying three analytical groundwater pollution transport models. The uncertainty analysis provides estimates of statistical reliability in model outputs of pollution concentration and arrival time. Examples are provided that demonstrate: (a) confidence limits around predicted values of concentration and arrival time can be obtained, (b) the selection of probability distributions for input parameters affects the output variables, and (c) the probability distribution of the output variables can be different from that of the input variables, even when all input parameters have the same probability distribution     

Improvements in Grade Tonnage Curve Prediction via Sequential Gaussian Fractal Simulation

An application of sequential Gaussian fractal conditional simulation is presented, using actual sparse, irregularly spaced open cut gold data, to predict the grade tonnage curve associated with a single mining bench. It is shown that this constitutes an improvement over the prediction of the grade tonnage curves obtained via standard sequential Gaussian simulation and median indicator simulation. The fractal conditional simulation method uses the model of the covariance of increments of fractional Brownian motion together with the fractal co-dimension derived from a truncated power semivariogram model to create conditional simulations from irregularly spaced data. The grades and tonnages above several discrete cut offs are calculated for each of one hundred simulations, and the mean and variance of the grade and tonnage values for each cut off are computed to give an average grade tonnage curve with associated confidence limits.     

Some guidelines for the practical application of Fry's method of strain analysis

A systematic analysis of patterns in artificial and natural point distributions is presented in order to demonstrate how the nature and degree of anticlustering of point distributions affect the character of Fry diagrams derived from them. Ideally anticlustered artificial point distributions illustrate the nature of the dependence between the degree of anticlustering and the minimum number of points required in such a distribution if it is to reveal strain accurately by Fry's method. In addition, these distributions provide straightforward results which aid in the interpretation of less ideally anticlustered natural point distributions.The results from the artificial distributions may be used to estimate the minimum number of points required for strain determination in natural distributions of varying degrees of anticlustering, as long as the anticlustering is sufficiently strong. However, application of the results from the artificial distributions to less strongly anticlustered natural point distributions shows not only that the existence of a vacancy field is an insufficient criterion for determining whether or not Fry's method is appropriate for strain determination but also that the nature of the anticlustering in these natural distributions limits the effectiveness of enhancing strain definition by increasing the number of points composing the distribution.     

The chemistry of bottled mineral and spring waters from Norway,Sweden, Finland and Iceland

Twenty-two bottled mineral and spring waters from Norway, Sweden, Finland and Iceland have been analysed for 71 inorganic chemical parameters with low detection limits as a subset of a large European survey of bottled groundwater chemistry (N = 884). The Nordic bottled groundwaters comprise mainly Ca–Na–HCO₃–Cl water types, but more distinct Ca–HCO₃, Na HCO₃ and Na–Cl water types are also offered. The distributions for most elements fall between groundwater from Fennoscandian Quaternary unconsolidated aquifers and groundwater from Norwegian crystalline bedrock boreholes. Treated tap waters have slightly lower median values for many parameters, but elements associated with plumbing have significantly higher concentrations in tap waters than in bottled waters. The small dataset is able to show that excessive fluoride and uranium contents are potential drinking water problems in Fennoscandia. Nitrate and arsenic displayed low to moderate concentrations, but the number of samples from Finland and Northern Sweden was too low to detect that elevated concentrations of arsenic occur in bedrock boreholes in some regions. The data shows clearly that water sold in plastic bottles is contaminated with antimony. Antimony is toxic and suspected to be carcinogenic, but the levels are well below the EU drinking water limit. The study does not provide any health-based arguments for buying bottled mineral and spring waters for those who are served with drinking water from public waterworks. Drinking water from crystalline bedrock aquifers should be analysed. In case of elevated concentrations of fluoride, uranium or arsenic, most bottled waters, but not all, will be better alternatives when treatment of the well water is not practicable.     

Confidence limits for fission-track dating

The effect of random error on age reliability is investigated with respect to the externaldetector method of fission-track dating. Obtaining confidence limits to age is shown to be equivalent to obtaining confidence limits for products and ratios of Poisson expectations. Attention is drawn to the relevant methodology for calculating exact confidence limits, and some simple approximations to these limits are suggested for use when the number of track counts is large.