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Results for “Mathematical and Computer Modelling”

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At least 127 records · Page 7Linked to original sources

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↗

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↗

USGS2020

Probabilistic application of an integrated catchment-estuary-coastal system model to assess the evolution of inlet-interrupted coasts over the 21st century

Inlet-interrupted sandy coasts are dynamic and complex coastal systems with continuously evolving geomorphological behaviors under the influences of both climate change and human activities. These coastal systems are of great importance to society (e.g., providing habitats, navigation, and recreational activities) and are affected by both oceanic and terrestrial processes. Therefore, the evolution of these inlet-interrupted coasts is better assessed by considering the entirety of the Catchment-Estuary-Coastal (CEC) systems, under plausible future scenarios for climate change and increasing pressures due to population growth and human activities. Such a holistic assessment of the long-term evolution of CEC systems can be achieved via reduced-complexity modeling techniques, which are also ably quantifying the uncertainties associated with the projections due to their lower simulation times. Here, we develop a novel probabilistic modeling framework to quantify the input-driven uncertainties associated with the evolution of CEC systems over the 21 st century. In this new approach, probabilistic assessment of the evolution of inlet-interrupted coasts is achieved by (1) probabilistically computing the exchange sediment volume between the inlet-estuary system and its adjacent coast, and (2) distributing the computed sediment volumes along the inlet-interrupted coast. The model is applied at three case study sites: Alsea estuary (United States), Dyfi estuary (United Kingdom), and Kalutara inlet (Sri Lanka). Model results indicate that there are significant uncertainties in projected volume exchange at all the CEC systems (min-max range of 2.0 million cubic meters in 2100 for RCP 8.5), and the uncertainties in these projected volumes illustrate the need for probabilistic modeling approaches to evaluate the long-term evolution of CEC systems. A comparison of 50 th percentile probabilistic projections with deterministic estimates shows that the deterministic approach overestimates the sediment volume exchange in 2100 by 15–30% at Alsea and Kalutara estuary systems. Projections of coastline change obtained for the case study sites show that accounting for all key processes governing coastline change along inlet-interrupted coasts in computing coastline change results in projections that are between 20 and 134% greater than the projections that would be obtained if only the Bruun effect were taken into account, underlining the inaccuracies associated with using the Bruun rule at inlet-interrupted coasts.

Frontiers in Applied Mathematics and Statistics↗

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↗

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↗

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↗

USGS2021

Periodic oscillation and tri-stability in mutualism systems with two consumers

This paper considers mutualistic interactions between two consumers, in which one consumer can consume a resource only by exchange of service for service with the other. By rigorous analysis on the one-resource and two-consumer model with Holling-type I response, we show periodic oscillations and tri-stability in the mutualism system: when their initial densities decrease, the consumers' interaction outcomes would change from coexistence in periodic oscillation, to persistence at a steady state, and to extinction. Under certain conditions, we also show two types of bi-stability in the system: the consumers would change from coexisting in periodic oscillation (resp. at a steady state) to going to extinction when their initial densities decrease. Then we analyze a modified system with Holling-type II response. Based on theoretical analysis and numerical computation, we show that there also exist tri-stability and two types of bi-stability in this system. Moreover, it is shown that varying the degree of obligation can lead to transition of interaction outcomes between coexistence in periodic oscillation (resp. at a steady state) and extinction of both consumers. These results are important in understanding complexity in mutualism.

Journal of Mathematical Analysis and Applications↗

USGS2022

Connecting regional-scale tree distribution models with seed dispersal kernels

Regional scale forest distribution models are important tools for biogeography and understanding the structure of forest communities in space. These models take climate and geographic variables as input and are therefore helpful for long-term decision support and climate adaptation planning. Generally, local processes of tree germination and seedling survival are resolved probabilistically with explanatory variables such as elevation, latitude, exposure, soil type, moisture availability, climate and weather inputs and `trained’ using landscape and regional presence-absence data and machine learning techniques. How seeds are distributed in these models, that is, determining the dispersal kernel, is far more problematic. The challenge is that variables conditioning vertebrate seed dispersal (motility and probability of utilization or caching in response to cover type) are not represented in large scale distribution models, and in fact vary on scales (10-100 meters) that are much smaller than the smallest pixel size for the distribution model (1-10 kilometers). We present a homogenized seed digestion kernel (HSDK) which incorporates this scale separation. Homogenization naturally links highly variable small-scale processes (like seed foraging and caching by birds and rodents) with large scale effects (like dispersal of seeds over tens of kilometers). We develop a homogenization strategy to predict seed dispersal on landscape scales, analytically linking small-scale variables (landscape fraction cover by tree type, gut residence times and cover type utilization by frugivorous birds) with large scale behaviors. Closed form approximations are developed in two dimensions for two limiting cases of seed handling behavior, and the approach is illustrated using landscape data and piñon-pine dispersal in a 630,000 square kilometer region in the southwestern US.

Applied Mathematics and Computation↗

USGS1988

The geometric signature: Quantifying landslide-terrain types from digital elevation models

Topography of various types and scales can be fingerprinted by computer analysis of altitude matrices (digital elevation models, or DEMs). The critical analytic tool is the geometric signature, a set of measures that describes topographic form well enough to distinguish among geomorphically disparate landscapes. Different surficial processes create topography with diagnostic forms that are recognizable in the field. The geometric signature abstracts those forms from contour maps or their DEMs and expresses them numerically. This multivariate characterization enables once-in-tractable problems to be addressed. The measures that constitute a geometric signature express different but complementary attributes of topographic form. Most parameters used here are statistical estimates of central tendency and dispersion for five major categories of terrain geometry; altitude, altitude variance spectrum, slope between slope reversals, and slope and its curvature at fixed slope lengths. As an experimental application of geometric signatures, two mapped terrain types associated with different processes of shallow landsliding in Marin County, California, were distinguished consistently by a 17-variable description of topography from 21??21 DEMs (30-m grid spacing). The small matrix is a statistical window that can be used to scan large DEMs by computer, thus potentially automating the mapping of contrasting terrain types. The two types in Marin County host either (1) slow slides: earth flows and slump-earth flows, or (2) rapid flows: debris avalanches and debris flows. The signature approach should adapt to terrain taxonomy and mapping in other areas, where conditions differ from those in Central California. ?? 1988 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↗

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↗

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

USGS Digital Spectral Library splib06a

Introduction We have assembled a digital reflectance spectral library that covers the wavelength range from the ultraviolet to far infrared along with sample documentation. The library includes samples of minerals, rocks, soils, physically constructed as well as mathematically computed mixtures, plants, vegetation communities, microorganisms, and man-made materials. The samples and spectra collected were assembled for the purpose of using spectral features for the remote detection of these and similar materials. Analysis of spectroscopic data from laboratory, aircraft, and spacecraft instrumentation requires a knowledge base. The spectral library discussed here forms a knowledge base for the spectroscopy of minerals and related materials of importance to a variety of research programs being conducted at the U.S. Geological Survey. Much of this library grew out of the need for spectra to support imaging spectroscopy studies of the Earth and planets. Imaging spectrometers, such as the National Aeronautics and Space Administration (NASA) Airborne Visible/Infra Red Imaging Spectrometer (AVIRIS) or the NASA Cassini Visual and Infrared Mapping Spectrometer (VIMS) which is currently orbiting Saturn, have narrow bandwidths in many contiguous spectral channels that permit accurate definition of absorption features in spectra from a variety of materials. Identification of materials from such data requires a comprehensive spectral library of minerals, vegetation, man-made materials, and other subjects in the scene. Our research involves the use of the spectral library to identify the components in a spectrum of an unknown. Therefore, the quality of the library must be very good. However, the quality required in a spectral library to successfully perform an investigation depends on the scientific questions to be answered and the type of algorithms to be used. For example, to map a mineral using imaging spectroscopy and the mapping algorithm of Clark and others (1990a, 2003b), one simply needs a diagnostic absorption band. The mapping system uses continuum-removed reference spectral features fitted to features in observed spectra. Spectral features for such algorithms can be obtained from a spectrum of a sample containing large amounts of contaminants, including those that add other spectral features, as long as the shape of the diagnostic feature of interest is not modified. If, however, the data are needed for radiative transfer models to derive mineral abundances from reflectance spectra, then completely uncontaminated spectra are required. This library contains spectra that span a range of quality, with purity indicators to flag spectra for (or against) particular uses. Acquiring spectral measurements and performing sample characterizations for this library has taken about 15 person-years of effort. Software to manage the library and provide scientific analysis capability is provided (Clark, 1980, 1993). A personal computer (PC) reader for the library is also available (Livo and others, 1993). The program reads specpr binary files (Clark, 1980, 1993) and plots spectra. Another program that reads the specpr format is written in IDL (Kokaly, 2005). In our view, an ideal spectral library consists of samples covering a very wide range of materials, has large wavelength range with very high precision, and has enough sample analyses and documentation to establish the quality of the spectra. Time and available resources limit what can be achieved. Ideally, for each mineral, the sample analysis would include X-ray diffraction (XRD), electron microprobe (EM) or X-ray fluorescence (XRF), and petrographic microscopic analyses. For some minerals, such as iron oxides, additional analyses such as Mossbauer would be helpful. We have found that to make the basic spectral measurements, provide XRD, EM or XRF analyses, and microscopic analyses, document the results, and complete an entry of one spectral library sample, all takes about

Data Series↗

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.

Scientific Investigations Report↗

USGS1985

Three-dimensional trend mapping from wire-line logs

Mapping of lithofacies and porosities of stratigraphic units is complicated because these properties vary in three dimensions. The method of moments was proposed by Krumbein and Libby (1957) as a technique to aid in resolving this problem. Moments are easily computed from wireline logs and are simple statistics which summarize vertical variation in a log trace. Combinations of moment maps have proved useful in understanding vertical and lateral changes in lithology of sedimentary rock units. Although moments have meaning both as statistical descriptors and as mechanical properties, they also define polynomial curves which approximate lithologic changes as a function of depth. These polynomials can be fitted by least-squares methods, partitioning major trends in rock properties from finescale fluctuations. Analysis of variance yields the degree of fit of any polynomial and measures the proportion of vertical variability expressed by any moment or combination of moments. In addition, polynomial curves can be differentiated to determine depths at which pronounced expressions of facies occur and to determine the locations of boundaries between major lithologic subdivisions. Moments can be estimated at any location in an area by interpolating from log moments at control wells. A matrix algebra operation then converts moment estimates to coefficients of a polynomial function which describes a continuous curve of lithologic variation with depth. If this procedure is applied to a grid of geographic locations, the result is a model of variability in three dimensions. Resolution of the model is determined largely by number of moments used in its generation. The method is illustrated with an analysis of lithofacies in the Simpson Group of south-central Kansas; the three-dimensional model is shown as cross sections and slice maps. In this study, the gamma-ray log is used as a measure of shaliness of the unit. However, the method is general and can be applied, for example, to suites of neutron, density, or sonic logs to produce three-dimensional models of porosity in reservoir rocks. ?? 1985 Plenum Publishing Corporation.

Journal of the International Association for Mathe↗

USGS1993

Relationship of geological and geothermal field properties: Midcontinent area, USA, an example

Quantitative approaches to data analysis in the last decade have become important in basin modeling and mineral-resource estimation. The interrelation of geological, geophysical, geochemical, and geohydrological variables is important in adjusting a model to a real-world situation. Revealing the interdependences of variables can contribute in understanding the processes interacting in sedimentary basins. It is reasonably simple to compare spatial data of the same type but more difficult if different properties are involved. Statistical techniques, such as cluster analysis or principal components analysis, or some algebraic approaches can be used to ascertain the relations of standardized spatial data. In this example, structural configuration on five different stratigraphic horizons, one total sediment thickness map, and four maps of geothermal data were copared. As expected, the structural maps are highly related because all had undergone about the same deformation with differing degrees of intensity. The temperature gradients derived (1) from shallow borehole logging measurements under equilibrium conditions with the surrounding rock, and (2) from non-equilibrium bottom-hole temperatures (BHT) from deeper depths are mainly independent of each other. This was expected and confirmed also for the two temperature maps at 1000 ft which were constructed using both types of gradient values. Thus, it is evident that the use of a 2-point (BHT and surface temperature) straightline calculation of a mean temperature gradient gives different information about the geothermal regime than using gradients from temperatures logged under equilibrium conditions. Nevertheless, it is useful to determine to what a degree the larger dataset of nonequilibrium temperatures could reflect quantitative relationships to geologic conditions. Comparing all maps of geothermal information vs. the structural and the sediment thickness maps, it was determined that all correlations are moderately negative or slightly positive. These results are clearly shown by the cluster analysis and the principal components. Considering a close relationship between temperature and thermal conductivity of the sediments as observed for most of the Midcontinent area and relatively homogeneous heat-flow density conditions for the study area these results support the following assumptions: (1) undifferentiated geothermal gradients, computed from temperatures of different depth intervals and differing sediment properties, cannot contribute to an improved understanding of the temperature structure and its controls within the sedimentary cover, and (2) the quantitative approach of revealing such relations needs refined datasets of temperature information valid for the different depth levels or stratigraphic units. ?? 1993 International Association for Mathematical Geology.

Mathematical Geology↗

USGS2018

Real-time decision-making during emergency disease outbreaks

In the event of a new infectious disease outbreak, mathematical and simulation models are commonly used to inform policy by evaluating which control strategies will minimize the impact of the epidemic. In the early stages of such outbreaks, substantial parameter uncertainty may limit the ability of models to provide accurate predictions, and policymakers do not have the luxury of waiting for data to alleviate this state of uncertainty. For policymakers, however, it is the selection of the optimal control intervention in the face of uncertainty, rather than accuracy of model predictions, that is the measure of success that counts. We simulate the process of real-time decision-making by fitting an epidemic model to observed, spatially-explicit, infection data at weekly intervals throughout two historical outbreaks of foot-and-mouth disease, UK in 2001 and Miyazaki, Japan in 2010, and compare forward simulations of the impact of switching to an alternative control intervention at the time point in question. These are compared to policy recommendations generated in hindsight using data from the entire outbreak, thereby comparing the best we could have done at the time with the best we could have done in retrospect. Our results show that the control policy that would have been chosen using all the data is also identified from an early stage in an outbreak using only the available data, despite high variability in projections of epidemic size. Critically, we find that it is an improved understanding of the locations of infected farms, rather than improved estimates of transmission parameters, that drives improved prediction of the relative performance of control interventions. However, the ability to estimate undetected infectious premises is a function of uncertainty in the transmission parameters. Here, we demonstrate the need for both real-time model fitting and generating projections to evaluate alternative control interventions throughout an outbreak. Our results highlight the use of using models at outbreak onset to inform policy and the importance of state-dependent interventions that adapt in response to additional information throughout an outbreak.

PLOS Computational Biology↗