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USGS1990

Stability analysis of Eulerian-Lagrangian methods for the one-dimensional shallow-water equations

In this paper stability and error analyses are discussed for some finite difference methods when applied to the one-dimensional shallow-water equations. Two finite difference formulations, which are based on a combined Eulerian-Lagrangian approach, are discussed. In the first part of this paper the results of numerical analyses for an explicit Eulerian-Lagrangian method (ELM) have shown that the method is unconditionally stable. This method, which is a generalized fixed grid method of characteristics, covers the Courant-Isaacson-Rees method as a special case. Some artificial viscosity is introduced by this scheme. However, because the method is unconditionally stable, the artificial viscosity can be brought under control either by reducing the spatial increment or by increasing the size of time step. The second part of the paper discusses a class of semi-implicit finite difference methods for the one-dimensional shallow-water equations. This method, when the Eulerian-Lagrangian approach is used for the convective terms, is also unconditionally stable and highly accurate for small space increments or large time steps. The semi-implicit methods seem to be more computationally efficient than the explicit ELM; at each time step a single tridiagonal system of linear equations is solved. The combined explicit and implicit ELM is best used in formulating a solution strategy for solving a network of interconnected channels. The explicit ELM is used at channel junctions for each time step. The semi-implicit method is then applied to the interior points in each channel segment. Following this solution strategy, the channel network problem can be reduced to a set of independent one-dimensional open-channel flow problems. Numerical results support properties given by the stability and error analyses.

Applied Mathematical Modelling↗

USGS2025

Are equilibrium shoreline models just convolutions?

Yes. Equilibrium shoreline models, which simulate wave-driven cross-shore erosion and accretion, are mathematically equivalent to a discrete convolution (i.e., a weighted, moving average) of a time series of wave-forcing conditions with a parameterized memory-decay kernel function. The direct equivalence between equilibrium shoreline models and convolutions reveals key theoretical aspects of equilibrium behavior. Convolutions (representing quasi-low-pass filter operations) provide an intuitive theoretical description of shoreline erosion and accretion behavior in response to waves: that is, shoreline position often mirrors the weighted moving average of wave time series. Model-convolution equivalence also provides a conceptual basis to interpret, evaluate, and construct data-driven Machine-Learning/Deep-Learning (ML/DL) models that use convolutions to extract features from data and then apply them for prediction (e.g., Convolutional Neural Networks (CNNs)). Finally, our findings provide a methodological pathway (based on Fourier transforms) for future understanding of wave-driven shoreline change, which can be used to interpret the coherence between the frequency spectrum of the processes of waves and shoreline change and construct more computationally efficient and effective shoreline-modeling approaches.

JGR Earth Surface↗

USGS2008

Dynamics of seismogenic volcanic extrusion resisted by a solid surface plug, Mount St. Helens, 2004-2005

The 2004-5 eruption of Mount St. Helens exhibited sustained, near-equilibrium behavior characterized by nearly steady extrusion of a solid dacite plug and nearly periodic occurrence of shallow earthquakes. Diverse data support the hypothesis that these earthquakes resulted from stick-slip motion along the margins of the plug as it was forced incrementally upward by ascending, solidifying, gas-poor magma. I formalize this hypothesis with a mathematical model derived by assuming that magma enters the base of the eruption conduit at a steady rate, invoking conservation of mass and momentum of the magma and plug, and postulating simple constitutive equations that describe magma and conduit compressibilities and friction along the plug margins. Reduction of the model equations reveals a strong mathematical analogy between the dynamics of the magma-plug system and those of a variably damped oscillator. Oscillations in extrusion velocity result from the interaction of plug inertia, a variable upward force due to magma pressure, and a downward force due to the plug weight. Damping of oscillations depends mostly on plug-boundary friction, and oscillations grow unstably if friction exhibits rate weakening similar to that observed in experiments. When growth of oscillations causes the extrusion rate to reach zero, however, gravity causes friction to reverse direction, and this reversal instigates a transition from unstable oscillations to self-regulating stick-slip cycles. The transition occurs irrespective of the details of rate-weakening behavior, and repetitive stick-slip cycles are, therefore, robust features of the system’s dynamics. The presence of a highly compressible elastic driving element (that is, magma containing bubbles) appears crucial for enabling seismogenic slip events to occur repeatedly at the shallow earthquake focal depths (<1 km) observed during the 2004-5 eruption. Computations show that fluctuations in magma pressure accompanying such slip events are <3 kPa, indicating that deviations from mechanical equilibrium are slight and that coseismic force drops are <10 8 N. These results imply that the system’s self-regulating behavior is not susceptible to dramatic change--provided that the rate of magma ascent remains similar to the rate of magma accretion at the base of the plug, that plug surface erosion more or less compensates for mass gain due to basal accretion, and that magma and rock properties do not change significantly. Even if disequilibrium initial conditions are imposed, the dynamics of the magma-plug system are strongly attracted to self-regulating stick-slip cycles, although this self-regulating behavior can be bypassed on the way to runaway behavior if the initial state is too far from equilibrium.

Washington↗

USGS1988

Multi-model approach to petroleum resource appraisal using analytic methodologies for probabilistic systems

The geologic appraisal model that is selected for a petroleum resource assessment depends upon purpose of the assessment, basic geologic assumptions of the area, type of available data, time available before deadlines, available human and financial resources, available computer facilities, and, most importantly, the available quantitative methodology with corresponding computer software and any new quantitative methodology that would have to be developed. Therefore, different resource assessment projects usually require different geologic models. Also, more than one geologic model might be needed in a single project for assessing different regions of the study or for cross-checking resource estimates of the area. Some geologic analyses used in the past for petroleum resource appraisal involved play analysis. The corresponding quantitative methodologies of these analyses usually consisted of Monte Carlo simulation techniques. A probabilistic system of petroleum resource appraisal for play analysis has been designed to meet the following requirements: (1) includes a variety of geologic models, (2) uses an analytic methodology instead of Monte Carlo simulation, (3) possesses the capacity to aggregate estimates from many areas that have been assessed by different geologic models, and (4) runs quickly on a microcomputer. Geologic models consist of four basic types: reservoir engineering, volumetric yield, field size, and direct assessment. Several case histories and present studies by the U.S. Geological Survey are discussed. ?? 1988 International Association for Mathematical Geology.

Mathematical Geology↗

USGS2021

Implications of aggregating and smoothing daily production data on estimates of the transition time between flow regimes in horizontal hydraulically fractured Bakken oil wells

The level to which data are aggregated or smoothed can impact analytical and predictive modeling results. This paper discusses findings regarding such impacts on estimating change points in production flow regimes of horizontal hydraulically fractured shale oil wells producing from the middle member of the Bakken Formation. Change points that signal transitions in flow regimes are important because they subsequently affect estimates of ultimate recovery from wells producing from shale plays. Extending our earlier work, we employ two different statistical approaches, Bacon–Watts Bayesian regression and nonlinear constrained least squares regression, and a designed computational experiment to estimate the time of transition from the transient to the boundary-dominated flow regime for 14 different wells using daily production data rather than aggregated monthly data, as previously considered. The daily data were also smoothed to reduce noise. Computational experiments suggest that both statistical approaches can lead to plausible estimates of the transition point under different data aggregation or smoothing regimes, but that daily data are likely too granular to produce credible estimates. Although the expected value of transition points using smoothed daily data and monthly disaggregated data are generally comparable, the confidence intervals bounding the estimates based on smoothed daily data are generally wider. Our results not only inform the operational practices of oil producers engaged in economic evaluation of their shale resources and additional play development activities, but also the activities of petroleum research groups, government agencies, and financial organizations seeking to improve the trustworthiness of resource projections.

Mathematical Geosciences↗

USGS2007

Effective groundwater model calibration: With analysis of data, sensitivities, predictions, and uncertainty

Methods and guidelines for developing and using mathematical models Turn to Effective Groundwater Model Calibration for a set of methods and guidelines that can help produce more accurate and transparent mathematical models. The models can represent groundwater flow and transport and other natural and engineered systems. Use this book and its extensive exercises to learn methods to fully exploit the data on hand, maximize the model's potential, and troubleshoot any problems that arise. Use the methods to perform: Sensitivity analysis to evaluate the information content of data Data assessment to identify (a) existing measurements that dominate model development and predictions and (b) potential measurements likely to improve the reliability of predictions Calibration to develop models that are consistent with the data in an optimal manner Uncertainty evaluation to quantify and communicate errors in simulated results that are often used to make important societal decisions Most of the methods are based on linear and nonlinear regression theory. Fourteen guidelines show the reader how to use the methods advantageously in practical situations. Exercises focus on a groundwater flow system and management problem, enabling readers to apply all the methods presented in the text. The exercises can be completed using the material provided in the book, or as hands-on computer exercises using instructions and files available on the text's accompanying Web site. Throughout the book, the authors stress the need for valid statistical concepts and easily understood presentation methods required to achieve well-tested, transparent models. Most of the examples and all of the exercises focus on simulating groundwater systems; other examples come from surface-water hydrology and geophysics. The methods and guidelines in the text are broadly applicable and can be used by students, researchers, and engineers to simulate many kinds systems.

Book↗

USGS2014

A depth-averaged debris-flow model that includes the effects of evolving dilatancy: II. Numerical predictions and experimental tests.

We evaluate a new depth-averaged mathematical model that is designed to simulate all stages of debris-flow motion, from initiation to deposition. A companion paper shows how the model’s five governing equations describe simultaneous evolution of flow thickness, solid volume fraction, basal pore-fluid pressure, and two components of flow momentum. Each equation contains a source term that represents the influence of state-dependent granular dilatancy. Here we recapitulate the equations and analyze their eigenstructure to show that they form a hyperbolic system with desirable stability properties. To solve the equations we use a shock-capturing numerical scheme with adaptive mesh refinement, implemented in an open-source software package we call D-Claw. As tests of D-Claw, we compare model output with results from two sets of large-scale debris-flow experiments. One set focuses on flow initiation from landslides triggered by rising pore-water pressures, and the other focuses on downstream flow dynamics, runout, and deposition. D-Claw performs well in predicting evolution of flow speeds, thicknesses, and basal pore-fluid pressures measured in each type of experiment. Computational results illustrate the critical role of dilatancy in linking coevolution of the solid volume fraction and pore-fluid pressure, which mediates basal Coulomb friction and thereby regulates debris-flow dynamics.

Proceedings of the Royal Society A↗

USGS1985

Comparative study of aggregations under different dependency assumptions for assessment of undiscovered recoverable oil resources in the world

The U.S. Geological Survey assessed all significant sedimentary basins in the world for undiscovered conventionally recoverable crude-oil resources. Probabilistic methodology was applied to each basin assessment to produce estimates in the form of probability distributions. Basin probability distributions were computer aggregated to produce resource estimates for the entire world. Aggregation was approximated by a three-parameter lognormal distribution by combining the first three central moments of basin distributions. For purposes of experiment and study, world aggregation was conducted under four different sets of assumptions. The four cases are (1) dependent assessments of all basins, (2) dependent assessments within continental areas, but independent assessments among continental areas, (3) dependent assessments within countries, but independent assessments among countries, and (4) independent assessments of all basins. Mean estimate remained the same in all four cases, but the width of interval estimate formed using the 95th and 5th fractiles decreased with reduced dependency in going from first to fourth case. ?? 1985 Plenum Publishing Corporation.

Journal of the International Association for Mathe↗

USGS1988

Linear solvation energy relationships for toxicity of selected organic chemicals to Daphnia pulex and Daphnia magna

In the Laurentian Great Lakes, more than 300 contaminants have been identified in fish, other biota, water, and sediment. Current hazard assessment of these chemicals by the National Fisheries Research Center-Great Lakes is based on their toxicity, occurrence in the environment, and source. Although scientists at the Center have tested over 70 chemicals with the crustacean Daphnia pulex , the number of experimental data needed to screen the huge array of chemicals in the Great Lakes exceeds the practical capabilities of conducting bioassays. This limitation can be partly circumvented, however, by using mathematical models based on quantitative structure-activity relationships (QSAR) to provide rapid, inexpensive estimates of toxicity. Many properties of chemicals, including toxicity, bioaccumulation and water solubility are well correlated and can be predicted by equations of the generalized linear solvation energy relationships (LSER). The equation we used to model solute toxicity is Toxicity = constant + m VI/100 + s (π* + dδ) + b βm + a αm where VI = intrinsic (Van der Waals) molar volume; π* = molecular dipolarity/polarizability; δ = polarizability 'correction term'; βm = solute hydrogen bond acceptor basicity; and αm = solute hydrogen bond donor acidity. The subscript m designates solute monomer values for α and β. We applied the LSER model to 48-h acute toxicity data (measured as immobilization) for six classes of chemicals detected in Great Lakes fish. The following regression was obtained for Daphnia pulex (concentration = μM): log EC50 = 4.86 - 4.35 VI/100; N = 38, r2 = 0.867, sd = 0.403 We also used the LSER modeling approach to analyze to a large published data set of 24-h acute toxicity for Daphnia magna ; the following regression resulted, for eight classes of compounds (concentration = mM): log EC50 = 3.88 - 4.52 VI/100 - 1.62 π* + 1.66 βm - 0.916 αm; N = 62, r2 = 0.859, sd = 0.375 In addition we developed computer software that identifies chemical structures, estimates the LSER parameters, and predicts toxicity. The LSER models promise to be effective in differentiating between reactive and nonreactive toxicity behavior where other models have failed. Contaminants with reactive behavior are generally the most toxic and rank highest in hazard assessment of environmental chemicals.

Proceedings of the QSAR↗

USGS2013

The age of scarplike landforms from diffusion-equation analysis

The purpose of this paper is to review developments in the quantitative modeling of fault-scarp geomorphology, principally those since 1980. These developments utilize diffusionequation mathematics, in several different forms, as the basic model of fault-scarp evolution. Because solutions to the general diffusion equation evolve with time, as we expect faultscarp morphology to evolve with time, the model solutions carry information about the age of the structure and thus its time of formation; hence the inclusion of this paper in this volume. The evolution of fault-scarp morphology holds a small but special place in the much larger class of problems in landform evolution. In general, landform evolution means the evolution of topography as a function of both space and time. It is the outcome of the competition among those tectonic processes that make topography, erosive processes that destroy topography, and depositional processes that redistribute topography. Deposition and erosion can always be coupled through conservation-of-mass relations, but in general deposition occurs at great distance from the source region of detritus. Moreover, erosion is an inherently rough process whereas deposition is inherently smooth, as is evident from even casual inspection of shaded-relief, digital-elevation maps (e.g., Thelin and Pike, 1990; Simpson and Anders, 1992) and the current fascination with fractal representations oferoding terrains (e.g., Huang and Turcotte, 1989; Newman and Turcotte, 1990). Nevertheless, large-scale landform-evolution modeling, now a computationally intensive, advanced numerical exercise, is generating ever more realistic landforms (e.g., Willgoose and others, 1991a,b; Kooi and Beaumont, 1994; Tucker and Slingerland, 1994), although many of the rate coefficients remain poorly prescribed

Book chapter↗

USGS1983

Characteristic analysis-1981: Final program and a possible discovery

The latest ornewest version of thecharacteristicanalysis (NCHARAN)computer program offers the exploration geologist a wide variety of options for integrating regionalized multivariate data. The options include the selection of regional cells for characterizing deposit models, the selection of variables that constitute the models, and the choice of logical combinations of variables that best represent these models. Moreover, the program provides for the display of results which, in turn, makes possible review, reselection, and refinement of a model. Most important, the performance of the above-mentioned steps in an interactive computing mode can result in a timely and meaningful interpretation of the data available to the exploration geologist. The most recent application of characteristic analysis has resulted in the possible discovery of economic sulfide mineralization in the Grong area in central Norway. Exploration data for 27 geophysical, geological, and geochemical variables were used to construct a mineralized and a lithogeochemical model for an area that contained a known massive sulfide deposit. The models were applied to exploration data collected from the Gjersvik area in the Grong mining district and resulted in the identification of two localities of possible mineralization. Detailed field examination revealed the presence of a sulfide vein system and a partially inverted stratigraphic sequence indicating the possible presence of a massive sulfide deposit at depth. ?? 1983 Plenum Publishing Corporation.

Journal of the International Association for Mathe↗

USGS2010

Approaches to highly parameterized inversion: A guide to using PEST for model-parameter and predictive-uncertainty analysis

Analysis of the uncertainty associated with parameters used by a numerical model, and with predictions that depend on those parameters, is fundamental to the use of modeling in support of decisionmaking. Unfortunately, predictive uncertainty analysis with regard to models can be very computationally demanding, due in part to complex constraints on parameters that arise from expert knowledge of system properties on the one hand (knowledge constraints) and from the necessity for the model parameters to assume values that allow the model to reproduce historical system behavior on the other hand (calibration constraints). Enforcement of knowledge and calibration constraints on parameters used by a model does not eliminate the uncertainty in those parameters. In fact, in many cases, enforcement of calibration constraints simply reduces the uncertainties associated with a number of broad-scale combinations of model parameters that collectively describe spatially averaged system properties. The uncertainties associated with other combinations of parameters, especially those that pertain to small-scale parameter heterogeneity, may not be reduced through the calibration process. To the extent that a prediction depends on system-property detail, its postcalibration variability may be reduced very little, if at all, by applying calibration constraints; knowledge constraints remain the only limits on the variability of predictions that depend on such detail. Regrettably, in many common modeling applications, these constraints are weak. Though the PEST software suite was initially developed as a tool for model calibration, recent developments have focused on the evaluation of model-parameter and predictive uncertainty. As a complement to functionality that it provides for highly parameterized inversion (calibration) by means of formal mathematical regularization techniques, the PEST suite provides utilities for linear and nonlinear error-variance and uncertainty analysis in these highly parameterized modeling contexts. Availability of these utilities is particularly important because, in many cases, a significant proportion of the uncertainty associated with model parameters-and the predictions that depend on them-arises from differences between the complex properties of the real world and the simplified representation of those properties that is expressed by the calibrated model. This report is intended to guide intermediate to advanced modelers in the use of capabilities available with the PEST suite of programs for evaluating model predictive error and uncertainty. A brief theoretical background is presented on sources of parameter and predictive uncertainty and on the means for evaluating this uncertainty. Applications of PEST tools are then discussed for overdetermined and underdetermined problems, both linear and nonlinear. PEST tools for calculating contributions to model predictive uncertainty, as well as optimization of data acquisition for reducing parameter and predictive uncertainty, are presented. The appendixes list the relevant PEST variables, files, and utilities required for the analyses described in the document.

Michigan↗

USGS1998

Singularity and Nonnormality in the Classification of Compositional Data

Geologists may want to classify compositional data and express the classification as a map. Regionalized classification is a tool that can be used for this purpose, but it incorporates discriminant analysis, which requires the computation and inversion of a covariance matrix. Covariance matrices of compositional data always will be singular (noninvertible) because of the unit-sum constraint. Fortunately, discriminant analyses can be calculated using a pseudo-inverse of the singular covariance matrix; this is done automatically by some statistical packages such as SAS. Granulometric data from the Darss Sill region of the Baltic Sea is used to explore how the pseudo-inversion procedure influences discriminant analysis results, comparing the algorithm used by SAS to the more conventional Moore-Penrose algorithm. Logratio transforms have been recommended to overcome problems associated with analysis of compositional data, including singularity. A regionalized classification of the Darss Sill data after logratio transformation is different only slightly from one based on raw granulometric data, suggesting that closure problems do not influence severely regionalized classification of compositional data.

Mathematical Geology↗

USGS2015

Water Quality, Cyanobacteria, and Environmental Factors and Their Relations to Microcystin Concentrations for Use in Predictive Models at Ohio Lake Erie and Inland Lake Recreational Sites, 2013-14

Harmful cyanobacterial “algal” blooms (cyanoHABs) and associated toxins, such as microcystin, are a major water-quality issue for Lake Erie and inland lakes in Ohio. Predicting when and where a bloom may occur is important to protect the public that uses and consumes a water resource; however, predictions are complicated and likely site specific because of the many factors affecting toxin production. Monitoring for a variety of environmental and water-quality factors, for concentrations of cyanobacteria by molecular methods, and for algal pigments such as chlorophyll and phycocyanin by using optical sensors may provide data that can be used to predict the occurrence of cyanoHABs. To test these monitoring approaches, water-quality samples were collected at Ohio recreational sites during May–November in 2013 and 2014. In 2013, samples were collected monthly at eight sites at eight lakes to facilitate an initial assessment and select sites for more intensive sampling during 2014. In 2014, samples were collected approximately weekly at five sites at three lakes. Physical water-quality parameters were measured at the time of sampling. Composite samples were preserved and analyzed for dissolved and total nutrients, toxins, phytoplankton abundance and biovolume, and cyanobacterial genes by molecular methods. Molecular assays were done to enumerate (1) general cyanobacteria, (2) general Microcystis and Dolichospermum ( Anabaena ), (3) mcyE genes for Microcystis , Dolichospermum ( Anabaena ), and Planktothrix targeting deoxyribonucleic acid (DNA), and (4) mcyE transcripts for Microcystis , Dolichospermum ( Anabaena ), and Planktothrix targeting ribonucleic acid (RNA).The DNA assays for the mcyE gene provide data on cyanobacteria that have the potential to produce microcystin, whereas the RNA assays provide data on cyanobacteria that are actively transcribing the toxin gene. Environmental data were obtained from available online sources. Quality-control (QC) samples were collected and analyzed for all constituents to characterize bias and variability; however, QC data for molecular assays were examined in more detail than for the other constituents. The QC data for molecular assays suggested that sampling variability and qPCR variability were small in comparison with the combined variability associated with sample filtering, extraction and purification, and the matrix itself. A total of 46 water-quality samples were collected during 2013 at 8 beach sites—Buck Creek, Buckeye Crystal, Deer Creek, Harsha Main, Maumee Bay State Park (MBSP) Inland (negative control site), MBSP Lake Erie, Port Clinton, and Sandusky Bay. Microcystin was detected in 67–100 percent of samples at all sites except for MBSP Inland, where microcystin was detected in only 20 percent of samples. Microcystin concentrations ranged from <0.10 to 48 micrograms per liter (µ/L), with the widest range found at MBSP Lake Erie and the highest concentrations found at Buckeye Crystal. Saxitoxin was detected in five samples, and cylindrospermopsin was not detected in any samples. A total of 65 water-quality samples were collected during 2014 at 5 sites on 3 lakes—Buckeye Fairfield and Onion Island, Harsha Main and Campers, and MBSP Lake Erie beach. Four of the sites were bathing beaches and one site, Onion Island, was an offshore boater swim area. Concentrations of microcystin ranged from <0.10 to 240 µ/L and, as in 2013, the widest range was found at MBSP Lake Erie. At Buckeye Lake, microcystin concentrations were consistently high (greater than 20 µ/L), ranging from 23 to 81 µ/L. At Harsha Main and Campers, microcystin concentrations ranged from <0.10 to 15 µ/L. Saxitoxin was detected in four samples collected at MBSP Lake Erie. Throughout the 2014 season, the cyanobacterial community, as determined by molecular and microscopy methods, and the dominance associated with the highest microcystin concentrations were unique to individual lakes. At Buckeye Lake, Planktothrix dominated the cyanobacterial community throughout the season and Planktothrix DNA and RNA were found in 100 percent of samples; Microcystis mcyE DNA was found in low concentrations. At Harsha Lake, Dolichospermum and Microcystis were a substantial percentage of the community from late May through August, and the highest microcystin concentrations occurred in June and July. At MBSP Lake Erie, Microcystis generally dominated from mid-July through early November, and the highest microcystin concentrations occurred in August. Spearman’s correlation coefficient (rho) was computed to determine the relations between environmental and water-quality factors and microcystin concentrations at four sites—Buckeye Fairfield, Buckeye Onion Island, Harsha Main, and MBSP Lake Erie. Factors were evaluated for use as potential independent variables in two types of predictive models—daily and long-term models. Easily or continuously measured water-quality factors and available environmental data are used for daily predictions that do not require a site visit. Data from factors used in daily predictions and results from samples collected and analyzed in a laboratory are used for long-term predictions (a few days to several weeks). A few statistically significant correlations (p ≤ 0.05) between microcystin concentrations and factors for both daily and long-term predictions were found at Buckeye Onion Island, and many were found at Harsha Main and MBSP Lake Erie. There were only a few statistically significant factors for daily predictions at Buckeye Fairfield, likely because of the lack of variability in microcystin concentrations. Among factors for daily predictions, phycocyanin had the highest Spearman’s correlation to microcystin concentrations (rho = 0.79 to 0.93) at all sites except for Buckeye Fairfield. Turbidity, pH, algae category, and Secchi depth were significantly correlated to microcystin concentrations at Harsha Main and MBSP Lake Erie. Algae categories were observational categories from 0 (none) to 4 (extreme). Several discharge variables (Maumee River at Waterville, river mouth is approximately 3.5 miles from the beach) at MBSP Lake Erie were promising environmental factors for daily predictions. In addition to discrete water-quality measurements recorded at Harsha Main at the time of sampling, many manipulated measurements (factors derived from mathematical manipulation of time-series data) available from a nearby continuous monitor were strongly correlated to microcystin concentrations; the highest correlation was found for the relation between microcystin concentrations and the antecedent 7-day average phycocyanin (rho = 0.98). For long-term predictions, the most highly correlated molecular assays were Planktothrix mcyE DNA at Buckeye Onion Island and Microcystis mcyE DNA at Harsha Main and MBSP Lake Erie. Concentrations of several nutrient constituents were significantly correlated to microcystin concentrations including total nitrogen at Buckeye Onion Island, ammonia and nitrate plus nitrite (both negatively correlated) at Harsha Main and MBSP Lake Erie, and total phosphorus at MBSP Lake Erie. The results of this study showed that water-quality and environmental variables are promising for use in site-specific daily or long-term predictive models. In order to develop more accurate models to predict toxin concentrations at freshwater lake sites, data need to be collected more frequently and for consecutive days in future studies.

Ohio↗

USGS2013

Time-lapse analysis of methane quantity in Mary Lee group of coal seams using filter-based multiple-point geostatistical simulation

Coal seam degasification and its success are important for controlling methane, and thus for the health and safety of coal miners. During the course of degasification, properties of coal seams change. Thus, the changes in coal reservoir conditions and in-place gas content as well as methane emission potential into mines should be evaluated by examining time-dependent changes and the presence of major heterogeneities and geological discontinuities in the field. In this work, time-lapsed reservoir and fluid storage properties of the New Castle coal seam, Mary Lee/Blue Creek seam, and Jagger seam of Black Warrior Basin, Alabama, were determined from gas and water production history matching and production forecasting of vertical degasification wellbores. These properties were combined with isotherm and other important data to compute gas-in-place (GIP) and its change with time at borehole locations. Time-lapsed training images (TIs) of GIP and GIP difference corresponding to each coal and date were generated by using these point-wise data and Voronoi decomposition on the TI grid, which included faults as discontinuities for expansion of Voronoi regions. Filter-based multiple-point geostatistical simulations, which were preferred in this study due to anisotropies and discontinuities in the area, were used to predict time-lapsed GIP distributions within the study area. Performed simulations were used for mapping spatial time-lapsed methane quantities as well as their uncertainties within the study area. The systematic approach presented in this paper is the first time in literature that history matching, TIs of GIPs and filter simulations are used for degasification performance evaluation and for assessing GIP for mining safety. Results from this study showed that using production history matching of coalbed methane wells to determine time-lapsed reservoir data could be used to compute spatial GIP and representative GIP TIs generated through Voronoi decomposition. Furthermore, performing filter simulations using point-wise data and TIs could be used to predict methane quantity in coal seams subjected to degasification. During the course of the study, it was shown that the material balance of gas produced by wellbores and the GIP reductions in coal seams predicted using filter simulations compared very well, showing the success of filter simulations for continuous variables in this case study. Quantitative results from filter simulations of GIP within the studied area briefly showed that GIP was reduced from an initial ∼73 Bcf (median) to ∼46 Bcf (2011), representing a 37 % decrease and varying spatially through degasification. It is forecasted that there will be an additional ∼2 Bcf reduction in methane quantity between 2011 and 2015. This study and presented results showed that the applied methodology and utilized techniques can be used to map GIP and its change within coal seams after degasification, which can further be used for ventilation design for methane control in coal mines.

Alabama↗

USGS1993

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 by 3/4 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 using 3/4 of the lower detection limit or 3/4 of the upper detection limit. For a small proportion of "less than" substitutions, a simple-substitution replacement factor of 0.55 is preferable to 3/4; for a small proportion of "greater than" substitutions, a simple-substitution replacement factor of 1.7 is preferable to 4/3, 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. ?? 1993 International Association for Mathematical Geology.

Mathematical Geology↗

USGS1980

Scaling variables and interpretation of eigenvalues in principal component analysis of geologic data

The dominant feature distinguishing one method of principal components analysis from another is the manner in which the original data are transformed prior to the other computations. The only other distinguishing feature of any importance is whether the eigenvectors of the inner product-moment of the transformed data matrix are taken directly as the Q-mode scores or scaled by the square roots of their associated eigenvalues and called the R-mode loadings. If the eigenvectors are extracted from the product-moment correlation matrix, the variables, in effect, were transformed by column standardization (zero means and unit variances), and the sum of the p-largest eigenvalues divided by the sum of all the eigenvalues indicates the degree to which a model containing p components will account for the total variance in the original data. However, if the data were transformed in any manner other than column standardization, the eigenvalues cannot be used in this manner, but can only be used to determine the degree to which the model will account for the transformed data. Regardless of the type of principal components analysis that is performed-even whether it is R or Q-mode-the goodness-of-fit of the model to the original data is given better by the eigenvalues of the correlation matrix than by those of the matrix that was actually factored. ?? 1980 Plenum Publishing Corporation.

Journal of the International Association for Mathe↗

USGS2005

Estimating the magnitude and frequency of floods for streams in west-central Florida, 2001

Flood discharges were estimated for recurrence intervals of 2, 5, 10, 25, 50, 100, 200, and 500 years for 94 streamflow stations in west-central Florida. Most of the stations are located within the 10,000 square-mile, 16-county area that forms the Southwest Florida Water Management District. All stations had at least 10 years of homogeneous record, and none have flood discharges that are significantly affected by regulation or urbanization. Guidelines established by the U.S. Water Resources Council in Bulletin 17B were used to estimate flood discharges from gaging station records. Multiple linear regression analysis was then used to mathematically relate estimates of flood discharge for selected recurrence intervals to explanatory basin characteristics. Contributing drainage area, channel slope, and the percent of total drainage area covered by lakes (percent lake area) were the basin characteristics that provided the best regression estimates. The study area was subdivided into four geographic regions to further refine the regression equations. Region 1 at the northern end of the study area includes large rivers that are characteristic of the rolling karst terrain of northern Florida. Only a small part of Region 1 lies within the boundaries of the Southwest Florida Water Management District. Contributing drainage area and percent lake area were the most statistically significant basin characteristics in Region 1; the prediction error of the regression equations varied with the recurrence interval and ranged from 57 to 69 percent. In the three other regions of the study area, contributing drainage area, channel slope, and percent lake area were the most statistically significant basin characteristics, and are the three characteristics that can be used to best estimate the magnitude and frequency of floods on most streams within the Southwest Florida Water Management District. The Withlacoochee River Basin dominates Region 2; the prediction error of the regression models in the region ranged from 65 to 68 percent. The basins that drain into the northern part of Tampa Bay and the upper reaches of the Peace River Basin are in Region 3, which had prediction errors ranging from 54 to 74 percent. Region 4, at the southern end of the study area, had prediction errors that ranged from 40 to 56 percent. Estimates of flood discharge become more accurate as longer periods of record are used for analyses; results of this study should be used in lieu of results from earlier U.S. Geological Survey studies of flood magnitude and frequency in west-central Florida. A comparison of current results with earlier studies indicates that use of a longer period of record with additional high-water events produces substantially higher flood-discharge estimates for many gaging stations. Another comparison indicates that the use of a computed, generalized skew in a previous study in 1979 tended to overestimate flood discharges.

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