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Results for “Applied Mathematical Modelling”

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At least 19 recordsLinked to original sources
USGS1976

Numerical models of wind-driven circulation in lakes

The state-of-the-art of numerical modelling of large-scale wind-driven circulation in lakes is presented. The governing equations which describe this motion are discussed along with the appropriate numerical techniques necessary to solve them in lakes. The numerical models are categorized into three large primary groups: the layered models, the Ekman-type models, and the other three-dimensional models. Discussions and comparison of models are given and future research directions are suggested.

Applied Mathematical Modelling↗
USGS2018

Application of a luminescence‐based sediment transport model

Quantifying the transport history of sand is a challenging but important goal in geomorphology. In this paper, we take a simple idea that luminescence is bleached during transport and regenerates during storage, and use this as a basis to re‐envision luminescence as a sediment tracer. We apply a mathematical model describing luminescence through an idealized channel and reservoir system and then compare this idealized model to real rivers to see if luminescence can reproduce known sediment transport data. We provide results from application of this luminescence method in three rivers from the mid‐Atlantic region of the United States. This method appears promising. However, as a river system diverges from idealized conditions of the mathematical model, the luminescence data diverge from model predictions. We suggest that spatial variation in the delivery of sediment from hillslopes can be reflected in the channel sediment luminescence and that luminescence acts as a function of landscape dynamics.

Geophysical Research Letters↗
USGS2021

Multivariate classification of the crude oil petroleum systems in southeast Texas, USA, using conventional and compositional data analysis of biomarkers

Chemically, petroleum is an extraordinarily complex mixture of different types of hydrocarbons that are now possible to isolate and identify because of advances in geochemistry. Here, we use biomarkers and carbon isotopes to establish genetic differences and similarities among oil samples. Conventional approaches for evaluating biomarker and carbon isotope relative abundances include statistical techniques such as principal component and cluster analysis. Considering that proportions of the different hydrocarbon molecules are relative parts of a laboratory sample, the data are compositional in nature, thus requiring the use of log-ratio approaches for adequate mathematical modeling. We apply both traditional and compositional modeling approaches to crude oil samples from an onshore area of about 50,000 square miles in southeast Texas. The data comprise 177 crude oil samples from producing oil fields that include key biomarkers, elemental, and isotopic values commonly used in source rock correlation studies. Our results indicate that compositional modeling has higher discriminating power and lower uncertainty than the traditional approach, allowing the identification of up to 16 clusters. Each cluster represents one oil family from a source rock organofacies ranging from Carboniferous to Paleogene. The families provide new insights into important petroleum systems in the Texas onshore region of the Gulf of Mexico sedimentary basin.

Texas↗
USGS2006

Biodegradation of organic chemicals in soil/water microcosms system: Model development

The chemical interactions of hydrophobic organic contaminants with soils and sediments may result in strong binding and slow subsequent release rates that significantly affect remediation rates and endpoints. In order to illustrate the recalcitrance of chemical to degradation on sites, a sorption mechanism of intraparticle sequestration was postulated to operate on chemical remediation sites. Pseudo-first order sequestration kinetics is used in the study with the hypothesis that sequestration is an irreversibly surface-mediated process. A mathematical model based on mass balance equations was developed to describe the fate of chemical degradation in soil/water microcosm systems. In the model, diffusion was represented by Fick's second law, local sorption-desorption by a linear isotherm, irreversible sequestration by a pseudo-first order kinetics and biodegradation by Monod kinetics. Solutions were obtained to provide estimates of chemical concentrations. The mathematical model was applied to a benzene biodegradation batch test and simulated model responses correlated well compared to measurements of biodegradation of benzene in the batch soil/water microcosm system. A sensitivity analysis was performed to assess the effects of several parameters on model behavior. Overall chemical removal rate decreased and sequestration increased quickly with an increase in the sorption partition coefficient. When soil particle radius, a, was greater than 1 mm, an increase in radius produced a significant decrease in overall chemical removal rate as well as an increase in sequestration. However, when soil particle radius was less than 0.1 mm, an increase in radius resulted in small changes in the removal rate and sequestration. As pseudo-first order sequestration rate increased, both chemical removal rate and sequestration increased slightly. Model simulation results showed that desorption resistance played an important role in the bioavailability of organic chemicals in porous media. Complete biostabilization of chemicals on remediation sites can be achieved when the concentration of the reversibly sorbed chemical reduces to zero (i.e., undetectable), with a certain amount of irreversibly sequestrated chemical left inside the soil particle solid phase. ?? 2006 Springer Science + Business Media B.V.

Water, Air, & Soil Pollution↗
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↗
USGS2000

Estimating formation properties from early-time oscillatory water levels in a pumped well

Hydrologists often attempt to estimate formation properties from aquifer tests for which only the hydraulic responses in a pumped well are available. Borehole storage, turbulent head losses, and borehole skin, however, can mask the hydraulic behavior of the formation inferred from the water level in the pumped well. Also, in highly permeable formations or in formations at significant depth below land surface, where there is a long column of water in the well casing, oscillatory water levels may arise during the onset of pumping to further mask formation responses in the pumped well. Usually borehole phenomena are confined to the early stages of pumping or recovery, and late-time hydraulic data can be used to estimate formation properties. In many instances, however, early-time hydraulic data provide valuable information about the formation, especially if there are interferences in the late-time data. A mathematical model and its Laplace transform solution that account for inertial influences and turbulent head losses during pumping is developed for the coupled response between the pumped borehole and the formation. The formation is assumed to be homogeneous, isotropic, of infinite areal extent, and uniform thickness, with leakage from an overlying aquifer, and the screened or open interval of the pumped well is assumed to fully penetrate the pumped aquifer. Other mathematical models of aquifer flow can also be coupled with the equations describing turbulent head losses and the inertial effects on the water column in the pumped well. The mathematical model developed in this paper is sufficiently general to consider both underdamped conditions for which oscillations arise, and overdamped conditions for which there are no oscillations. Through numerical inversion of the Laplace transform solution, type curves from the mathematical model are developed to estimate formation properties through comparison with the measured hydraulic response in the pumped well. The mathematical model is applied to estimate formation properties from a singlewell test conducted near Waialua, Oahu, Hawaii. At this site, both the drawdown and recovery showed oscillatory water levels in the pumped well, and a step-drawdown test showed that approximately 86% of the drawdown is attributed to turbulent head losses. Analyses at this site using late-time drawdown data were confounded by the noise present in the measured water levels due primarily to nearby irrigation wells and ocean tides. By analyzing the early-time oscillatory recovery data at the Waialua site, upper and lower bounds were placed on the transmissivity, T, storage coefficient, S, and the leakance of the confining unit, K′/B′. The upper and lower bounds on T differ by a factor of 2. Upper and lower bounds on S and K′/B′ are much larger, because drawdown stabilized relatively quickly after the onset of pumping.

Journal of Hydrology↗
USGS2016

Simulating groundwater flow in karst aquifers with distributed parameter models—Comparison of porous-equivalent media and hybrid flow approaches

Understanding karst aquifers, for purposes of their management and protection, poses unique challenges. Karst aquifers are characterized by groundwater flow through conduits (tertiary porosity), and (or) layers with interconnected pores (secondary porosity) and through intergranular porosity (primary or matrix porosity). Since the late 1960s, advances have been made in the development of numerical computer codes and the use of mathematical model applications towards the understanding of dual (primary [matrix] and secondary [fractures and conduits]) porosity groundwater flow processes, as well as characterization and management of karst aquifers. The Floridan aquifer system (FAS) in Florida and parts of Alabama, Georgia, and South Carolina is composed of a thick sequence of predominantly carbonate rocks. Karst features are present over much of its area, especially in Florida where more than 30 first-magnitude springs occur, numerous sinkholes and submerged conduits have been mapped, and numerous circular lakes within sinkhole depressions are present. Different types of mathematical models have been applied for simulation of the FAS. Most of these models are distributed parameter models based on the assumption that, like a sponge, water flows through connected pores within the aquifer system and can be simulated with the same mathematical methods applied to flow through sand and gravel aquifers; these models are usually referred to as porous-equivalent media models. The partial differential equation solved for groundwater flow is the potential flow equation of fluid mechanics, which is used when flow is dominated by potential energy and has been applied for many fluid problems in which kinetic energy terms are dropped from the differential equation solved. In many groundwater model codes (basic MODFLOW), it is assumed that the water has a constant temperature and density and that flow is laminar, such that kinetic energy has minimal impact on flow. Some models have been developed that incorporate the submerged conduits as a one-dimensional pipe network within the aquifer rather than as discrete, extremely transmissive features in a porous-equivalent medium; these submerged conduit models are usually referred to as hybrid models and may include the capability to simulate both laminar and turbulent flow in the one-dimensional pipe network. Comparisons of the application of a porous-equivalent media model with and without turbulence (MODFLOW-Conduit Flow Process mode 2 and basic MODFLOW, respectively) and a hybrid (MODFLOW-Conduit Flow Process mode 1) model to the Woodville Karst Plain near Tallahassee, Florida, indicated that for annual, monthly, or seasonal average hydrologic conditions, all methods met calibration criteria (matched observed groundwater levels and average flows). Thus, the increased effort required, such as the collection of data on conduit location, to develop a hybrid model and its increased computational burden, is not necessary for simulation of average hydrologic conditions (non-laminar flow effects on simulated head and spring discharge were minimal). However, simulation of a large storm event in the Woodville Karst Plain with daily stress periods indicated that turbulence is important for matching daily springflow hydrographs. Thus, if matching streamflow hydrographs over a storm event is required, the simulation of non-laminar flow and the location of conduits are required. The main challenge in application of the methods and approaches for developing hybrid models relates to the difficulty of mapping conduit networks or having high-quality datasets to calibrate these models. Additionally, hybrid models have long simulation times, which can preclude the use of parameter estimation for calibration. Simulation of contaminant transport that does not account for preferential flow through conduits or extremely permeable zones in any approach is ill-advised. Simulation results in other karst aquifers or other parts of the FAS may differ from the comparison demonstrated herein.

Florida↗
USGS2014

Taking the mystery out of mathematical model applications to karst aquifers—A primer

Advances in mathematical model applications toward the understanding of the complex flow, characterization, and water-supply management issues for karst aquifers have occurred in recent years. Different types of mathematical models can be applied successfully if appropriate information is available and the problems are adequately identified. The mathematical approaches discussed in this paper are divided into three major categories: 1) distributed parameter models, 2) lumped parameter models, and 3) fitting models. The modeling approaches are described conceptually with examples (but without equations) to help non-mathematicians understand the applications.

Conference Paper↗
USGS2023

Ecotoxicological studies indicate that sublethal and lethal processes limit insect-mediated contaminant flux

Merolimnic insects can accumulate and transport considerable amounts of aquatic contaminants to terrestrial systems. The rate of contaminant biotransport, termed insect-mediated contaminant flux (IMCF), depends on emergent insect biomass and contaminant accumulation, both functions of environmental concentration. Here we develop a mathematical model of IMCF and apply it to three ecotoxicological studies obtained through the U.S. Environmental Protection Agency's ECOTOX database to determine at which concentration maximum IMCF occurs. Model results demonstrate that the maximum IMCF depends on competing rates of biomass loss and contaminant accumulation and does not necessarily occur at the highest insect or environmental contaminant concentration. Additionally, modeling results suggest that sublethal contaminant effects (e.g., decreased growth) on insect biomass can be an important and potentially underappreciated control on IMCF.

Environmental Toxicology and Chemistry↗
USGS2015

Fecal indicator organism modeling and microbial source tracking in environmental waters: Chapter 3.4.6

Mathematical models have been widely applied to surface waters to estimate rates of settling, resuspension, flow, dispersion, and advection in order to calculate movement of particles that influence water quality. Of particular interest are the movement, survival, and persistence of microbial pathogens or their surrogates, which may contaminate recreational water, drinking water, or shellfish. Most models devoted to microbial water quality have been focused on fecal indicator organisms (FIO), which act as a surrogate for pathogens and viruses. Process-based modeling and statistical modeling have been used to track contamination events to source and to predict future events. The use of these two types of models require different levels of expertise and input; process-based models rely on theoretical physical constructs to explain present conditions and biological distribution while data-based, statistical models use extant paired data to do the same. The selection of the appropriate model and interpretation of results is critical to proper use of these tools in microbial source tracking. Integration of the modeling approaches could provide insight for tracking and predicting contamination events in real time. A review of modeling efforts reveals that process-based modeling has great promise for microbial source tracking efforts; further, combining the understanding of physical processes influencing FIO contamination developed with process-based models and molecular characterization of the population by gene-based (i.e., biological) or chemical markers may be an effective approach for locating sources and remediating contamination in order to protect human health better.

Book chapter↗
USGS1977

Fluorite solubility equilibria in selected geothermal waters

Calculation of chemical equilibria in 351 hot springs and surface waters from selected geothermal areas in the western United States indicate that the solubility of the mineral fluorite, CaF 2 , provides an equilibrium control on dissolved fluoride activity. Waters that are undersaturated have undergone dilution by non-thermal waters as shown by decreased conductivity and temperature values, and only 2% of the samples are supersaturated by more than the expected error. Calculations also demonstrate that simultaneous chemical equilibria between the thermal waters and calcite as well as fluorite minerals exist under a variety of conditions. Testing for fluorite solubility required a critical review of the thermodynamic data for fluorite. By applying multiple regression of a mathematical model to selected published data we have obtained revised estimates of the p K (10,96), ΔG o f (−280.08 kcal/mole), ΔH o f (−292.59 kcal/mole), S° (16.39 cal/deg/mole) and C o P (16.16 cal/deg/mole) for CaF 2 at 25°C and 1 atm. Association constants and reaction enthalpies for fluoride complexes with boron, calcium and iron are included in this review. The excellent agreement between the computer-based activity products and the revised p K suggests that the chemistry of geothermal waters may also be a guide to evaluating mineral solubility data where major discrepancies are evident.

Geochimica et Cosmochimica Acta↗
USGS2015

Bayesian models: A statistical primer for ecologists

Bayesian modeling has become an indispensable tool for ecological research because it is uniquely suited to deal with complexity in a statistically coherent way. This textbook provides a comprehensive and accessible introduction to the latest Bayesian methods—in language ecologists can understand. Unlike other books on the subject, this one emphasizes the principles behind the computations, giving ecologists a big-picture understanding of how to implement this powerful statistical approach. Bayesian Models is an essential primer for non-statisticians. It begins with a definition of probability and develops a step-by-step sequence of connected ideas, including basic distribution theory, network diagrams, hierarchical models, Markov chain Monte Carlo, and inference from single and multiple models. This unique book places less emphasis on computer coding, favoring instead a concise presentation of the mathematical statistics needed to understand how and why Bayesian analysis works. It also explains how to write out properly formulated hierarchical Bayesian models and use them in computing, research papers, and proposals. This primer enables ecologists to understand the statistical principles behind Bayesian modeling and apply them to research, teaching, policy, and management. Presents the mathematical and statistical foundations of Bayesian modeling in language accessible to non-statisticians Covers basic distribution theory, network diagrams, hierarchical models, Markov chain Monte Carlo, and more Deemphasizes computer coding in favor of basic principles Explains how to write out properly factored statistical expressions representing Bayesian models

Book↗
USGS2013

Surveillance theory applied to virus detection: a case for targeted discovery

Virus detection and mathematical modeling have gone through rapid developments in the past decade. Both offer new insights into the epidemiology of infectious disease and characterization of future risk; however, modeling has not yet been applied to designing the best surveillance strategies for viral and pathogen discovery. We review recent developments and propose methods to integrate viral and pathogen discovery and mathematical modeling through optimal surveillance theory, arguing for a more targeted approach to novel virus detection guided by the principles of adaptive management and structured decision-making.

Future Virology↗
USGS2003

Diagnostic tools for mixing models of stream water chemistry

Mixing models provide a useful null hypothesis against which to evaluate processes controlling stream water chemical data. Because conservative mixing of end‐members with constant concentration is a linear process, a number of simple mathematical and multivariate statistical methods can be applied to this problem. Although mixing models have been most typically used in the context of mixing soil and groundwater end‐members, an extension of the mathematics of mixing models is presented that assesses the “fit” of a multivariate data set to a lower dimensional mixing subspace without the need for explicitly identified end‐members. Diagnostic tools are developed to determine the approximate rank of the data set and to assess lack of fit of the data. This permits identification of processes that violate the assumptions of the mixing model and can suggest the dominant processes controlling stream water chemical variation. These same diagnostic tools can be used to assess the fit of the chemistry of one site into the mixing subspace of a different site, thereby permitting an assessment of the consistency of controlling end‐members across sites. This technique is applied to a number of sites at the Panola Mountain Research Watershed located near Atlanta, Georgia.

Water Resources Research↗
USGS2015

Assessing local population vulnerability to wind energy development with branching process models: an application to wind energy development

Quantifying the impact of anthropogenic development on local populations is important for conservation biology and wildlife management. However, these local populations are often subject to demographic stochasticity because of their small population size. Traditional modeling efforts such as population projection matrices do not consider this source of variation whereas individual-based models, which include demographic stochasticity, are computationally intense and lack analytical tractability. One compromise between approaches is branching process models because they accommodate demographic stochasticity and are easily calculated. These models are known within some sub-fields of probability and mathematical ecology but are not often applied in conservation biology and applied ecology. We applied branching process models to quantitatively compare and prioritize species locally vulnerable to the development of wind energy facilities. Specifically, we examined species vulnerability using branching process models for four representative species: A cave bat (a long-lived, low fecundity species), a tree bat (short-lived, moderate fecundity species), a grassland songbird (a short-lived, high fecundity species), and an eagle (a long-lived, slow maturation species). Wind turbine-induced mortality has been observed for all of these species types, raising conservation concerns. We simulated different mortality rates from wind farms while calculating local extinction probabilities. The longer-lived species types (e.g., cave bats and eagles) had much more pronounced transitions from low extinction risk to high extinction risk than short-lived species types (e.g., tree bats and grassland songbirds). High-offspring-producing species types had a much greater variability in baseline risk of extinction than the lower-offspring-producing species types. Long-lived species types may appear stable until a critical level of incidental mortality occurs. After this threshold, the risk of extirpation for a local population may rapidly increase with only minimal increases in wind mortality. Conservation biologists and wildlife managers may need to consider this mortality pattern when issuing take permits and developing monitoring protocols for wind facilities. We also describe how our branching process models may be generalized across a wider range of species for a larger assessment project and then describe how our methods may be applied to other stressors in addition to wind.

Ecosphere↗
USGS2007

Calculation of nonlinear confidence and prediction intervals for ground-water flow models

A method is derived to efficiently compute nonlinear confidence and prediction intervals on any function of parameters derived as output from a mathematical model of a physical system. The method is applied to the problem of obtaining confidence and prediction intervals for manually-calibrated ground-water flow models. To obtain confidence and prediction intervals resulting from uncertainties in parameters, the calibrated model and information on extreme ranges and ordering of the model parameters within one or more independent groups are required. If random errors in the dependent variable are present in addition to uncertainties in parameters, then calculation of prediction intervals also requires information on the extreme range of error expected. A simple Monte Carlo method is used to compute the quantiles necessary to establish probability levels for the confidence and prediction intervals. Application of the method to a hypothetical example showed that inclusion of random errors in the dependent variable in addition to uncertainties in parameters can considerably widen the prediction intervals.

Journal of the American Water Resources Associatio↗
USGS1973

Mathematical simulation of temperatures in deep impoundments: verification tests of the Water Resources Engineers, Inc. model - Horsetooth and Flaming Gorge Reservoirs

Successful use of predictive mathematical models requires verification of the accuracy of the models by applying them to existing situations where the prediction can be compared with reality. A Corps of Engineers' modification of a deep reservoir thermal stratification model developed by Water Resources Engineers, Inc., was applied to two existing Bureau of Reclamation reservoirs for verification. Diffusion coefficients used for the Corps' Detroit Reservoir were found to apply to Horsetooth Reservoir in Colorado, for which very food computer input data were available. The Detroit diffusion coefficients gave a reasonable simulation of Flaming Gorge Reservoir in Wyoming and Utah, which has very complex and variable physical characteristics and for which only average-quality computer input data were available.

Colorado↗