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Results for “Mathematics of Computation”

Search indexed USGS publications on groundwater, aquifers, geologic maps, mineral resources and earthquakes. Explore source records by subject and place.

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At least 73 records · Page 4Linked to original sources
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↗
USGS1989

Predicting earthquakes by analyzing accelerating precursory seismic activity

During 11 sequences of earthquakes that in retrospect can be classed as foreshocks, the accelerating rate at which seismic moment is released follows, at least in part, a simple equation. This equation (1) is {Mathematical expression},where {Mathematical expression} is the cumulative sum until time, t, of the square roots of seismic moments of individual foreshocks computed from reported magnitudes;C and n are constants; and tfis a limiting time at which the rate of seismic moment accumulation becomes infinite. The possible time of a major foreshock or main shock, tf,is found by the best fit of equation (1), or its integral, to step-like plots of {Mathematical expression} versus time using successive estimates of tfin linearized regressions until the maximum coefficient of determination, r2,is obtained. Analyzed examples include sequences preceding earthquakes at Cremasta, Greece, 2/5/66; Haicheng, China 2/4/75; Oaxaca, Mexico, 11/29/78; Petatlan, Mexico, 3/14/79; and Central Chile, 3/3/85. In 29 estimates of main-shock time, made as the sequences developed, the errors in 20 were less than one-half and in 9 less than one tenth the time remaining between the time of the last data used and the main shock. Some precursory sequences, or parts of them, yield no solution. Two sequences appear to include in their first parts the aftershocks of a previous event; plots using the integral of equation (1) show that the sequences are easily separable into aftershock and foreshock segments. Synthetic seismic sequences of shocks at equal time intervals were constructed to follow equation (1), using four values of n. In each series the resulting distributions of magnitudes closely follow the linear Gutenberg-Richter relation log N=a-bM, and the product n times b for each series is the same constant. In various forms and for decades, equation (1) has been used successfully to predict failure times of stressed metals and ceramics, landslides in soil and rock slopes, and volcanic eruptions. Results of more recent experiments and theoretical studies on crack propagation, fault mechanics, and acoustic emission can be closely reproduced by equation (1). Rate-process theory and continuum damage mechanics offer leads toward understanding the physical processes. ?? 1989 Birkha??user Verlag.

Pure and Applied Geophysics PAGEOPH↗
USGS1982

Two BASIC computer programs for the determination of in situ stresses using the CSIRO hollow inclusion stress cell and the USBM borehole deformation gage

The mathematical method of determining in-situ stresses by overcoring, using either the U.S. Bureau of Mines Borehole Deformation Gage or the Commonwealth Scientific and Industrial Research Organisation Hollow Inclusion Stress Cell, is summarized, and data reduction programs for each type of instrument, written in BASIC, are presented. The BASIC programs offer several advantages over previously available FORTRAN programs. They can be executed on a desk-top microcomputer at or near the field site, allowing the investigator to assess the quality of the data and make decisions on the need for additional testing while the crew is still in the field. Also, data input is much simpler than with currently available FORTRAN programs; either English or SI units can be used; and standard deviations of the principal stresses are computed as well as those of the geographic components.

Open-File Report↗
USGS1953

A modification in the technique of computing average lengths from the scales of fishes

In virtually all the studies that employ scales, otollths, or bony structures to obtain the growth history of fishes, it has been the custom to compute lengths for each individual fish and from these data obtain the average growth rates for any particular group. This method involves a considerable amount of mathematical manipulation, time, and effort. Theoretically it should be possible to obtain the same information simply by averaging the scale measurements for each year of life and the length of the fish employed and computing the average lengths from these data. This method would eliminate all calculations for individual fish. Although Van Oosten (1929: 338) pointed out many years ago the validity of this method of computation, his statements apparently have been overlooked by subsequent investigators.

Progressive Fish-Culturist↗
USGS1994

An experimental study of zinc chloride speciation from 300 to 600 °C and 0.5 to 2.0 kbar in buffered hydrothermal solutions

The solubility of sphalerite (ZnS) was measured in KCl-HCl-H 2 O solutions at 300–600°C and 0.5–2.0 kbar. The silicate assemblage K-feldspar-muscovite (or andalusite)-quartz was used to buffer the solution to acid conditions, resulting in the total solubility reaction 2 K + + KAl 2 AlSi 3 O 10 ( OH ) 2 + 6 SiO 2 + ZnS + nCl − = ZnCl n (2− n ) + 3 KAlSi 3 O 8 + H 2 S . (muscovite) (quartz) (sphalerite) (K-feldspar) A computer retrieval technique was used to derive average chloride ligand numbers for chlorozinc species at 0.25–2.0 molal total chloride. This technique mathematically solves for the average ligand number using a series of pertinent chemical relations at P and T . Mono- and di-chlorozinc species were found to predominate throughout the pressure-temperature-composition range investigated. The logarithms of the first and second dissociation constants for ZnCl 2 0 were evaluated over the P - T range; for example, at 1 kbar, the values −0.41 and −1.42 were computed for the logarithm of the first dissociation constant, while −7.62 and −10.57 were computed for the logarithm of the second dissociation constant, for 400 and 500°C, respectively. Results are compared to past studies conducted at subcritical conditions and differ in that we find no evidence for more highly coordinated chloro-zinc species except possibly for ZnCl 3 − at 600°C, 1 and 2 kbar. Our results are consistent with electrostatic theory, which favors lower charged to neutral molecules in low dielectric-constant media.

Geochimica et Cosmochimica Acta↗
USGS1977

A computer program to calculate the resistivity and induced polarization response for a three-dimensional body in the presence of buried electrodes

Three-dimensional induced polarization and resistivity modeling for buried electrode configurations can be achieved by adapting surface integral techniques for surface electrode configurations to buried electrodes. Modification of. the surface technique is accomplished by considering the additional mathematical terms required to express-the changes in the electrical potential and geometry caused by placing the source and receiver electrodes below the surface. This report presents a listing of a computer program to calculate the resistivity and induced polarization response from a three-dimensional body for buried electrode configurations. The program is designed to calculate the response for the following electrode configurations: (1) hole-to-surface array with a buried bipole source and a surface bipole receiver, (2) hole-to-surface array with a buried pole source and a surface bipole receiver, (3) hole-to-hole array with a buried, fixed pole source and a moving bipole receiver, (4) surface-to-hole array with a fixed pole source on the surface and a moving bipole receiver in the borehole, (5) hole-to-hole array with a buried, fixed bipole source and a buried, moving bipole receiver, (6) hole-to-hole array with a buried, moving bipole source and a buried, moving bipole receiver, and (7) single-hole, buried bipole-bipole array. Input and output examples are given for each of the arrays.

Open-File Report↗
USGS2004

Extracting transient Rayleigh wave and its application in detecting quality of highway roadbed

This paper first explains the tau-p mapping method of extracting Rayleigh waves (LR waves) from field shot gathers. It also explains a mathematical model of physical character parameters of quality of high-grade roads. This paper then discusses an algorithm of computing dispersion curves using adjacent channels. Shear velocity and physical character parameters are obtained by inversion of dispersion curves. The algorithm using adjacent channels to calculating dispersion curves eliminates average effects that exist by using multi-channels to obtain dispersion curves so that it improves longitudinal and transverse resolution of LR waves and precision of non-invasive detection, and also broadens its application fields. By analysis of modeling results of detached computation of the ground roll and real examples of detecting density and pressure strength of a high-grade roadbed, and by comparison of shallow seismic image method with borehole cores, we concluded that: 1 the abnormal scale and configuration obtained by LR waves are mostly the same as the result of shallow seismic image method; 2 an average relative error of density obtained from LR waves inversion is 1.6% comparing with borehole coring; 3 transient LR waves in detecting density and pressure strength of a high-grade roadbed is feasible and effective.

Conference Paper↗
USGS1989

A general method for generating bathymetric data for hydrodynamic computer models

To generate water depth data from randomly distributed bathymetric data for numerical hydrodymamic models, raw input data from field surveys, water depth data digitized from nautical charts, or a combination of the two are sorted to given an ordered data set on which a search algorithm is used to isolate data for interpolation. Water depths at locations required by hydrodynamic models are interpolated from the bathymetric data base using linear or cubic shape functions used in the finite-element method. The bathymetric database organization and preprocessing, the search algorithm used in finding the bounding points for interpolation, the mathematics of the interpolation formulae, and the features of the automatic generation of water depths at hydrodynamic model grid points are included in the analysis. This report includes documentation of two computer programs which are used to: (1) organize the input bathymetric data; and (2) to interpolate depths for hydrodynamic models. An example of computer program operation is drawn from a realistic application to the San Francisco Bay estuarine system. (Author 's abstract)

Open-File Report↗
USGS1996

A three-dimensional method-of-characteristics solute-transport model (MOC3D)

This report presents a model, MOC3D, that simulates three-dimensional solute transport in flowing ground water. The model computes changes in concentration of a single dissolved chemical constituent over time that are caused by advective transport, hydrodynamic dispersion (including both mechanical dispersion and diffusion), mixing (or dilution) from fluid sources, and mathematically simple chemical reactions (including linear sorption, which is represented by a retardation factor, and decay). The transport model is integrated with MODFLOW, a three-dimensional ground-water flow model that uses implicit finite-difference methods to solve the transient flow equation. MOC3D uses the method of characteristics to solve the transport equation on the basis of the hydraulic gradients computed with MODFLOW for a given time step. This implementation of the method of characteristics uses particle tracking to represent advective transport and explicit finite-difference methods to calculate the effects of other processes. However, the explicit procedure has several stability criteria that may limit the size of time increments for solving the transport equation; these are automatically determined by the program. For improved efficiency, the user can apply MOC3D to a subgrid of the primary MODFLOW grid that is used to solve the flow equation. However, the transport subgrid must have uniform grid spacing along rows and columns. The report includes a description of the theoretical basis of the model, a detailed description of input requirements and output options, and the results of model testing and evaluation. The model was evaluated for several problems for which exact analytical solutions are available and by benchmarking against other numerical codes for selected complex problems for which no exact solutions are available. These test results indicate that the model is very accurate for a wide range of conditions and yields minimal numerical dispersion for advection-dominated problems. Mass-balance errors are generally less than 10 percent, and tend to decrease and stabilize with time.

Water-Resources Investigations Report↗
USGS1988

Experiments with microcomputer-based artificial intelligence environments

The U.S. Geological Survey (USGS) has been experimenting with the use of relatively inexpensive microcomputers as artificial intelligence (AI) development environments. Several AI languages are available that perform fairly well on desk-top personal computers, as are low-to-medium cost expert system packages. Although performance of these systems is respectable, their speed and capacity limitations are questionable for serious earth science applications foreseen by the USGS. The most capable artificial intelligence applications currently are concentrated on what is known as the "artificial intelligence computer," and include Xerox D-series, Tektronix 4400 series, Symbolics 3600, VAX, LMI, and Texas Instruments Explorer. The artificial intelligence computer runs expert system shells and Lisp, Prolog, and Smalltalk programming languages. However, these AI environments are expensive. Recently, inexpensive 32-bit hardware has become available for the IBM/AT microcomputer. USGS has acquired and recently completed Beta-testing of the Gold Hill Systems 80386 Hummingboard, which runs Common Lisp on an IBM/AT microcomputer. Hummingboard appears to have the potential to overcome many of the speed/capacity limitations observed with AI-applications on standard personal computers. USGS is a Beta-test site for the Gold Hill Systems GoldWorks expert system. GoldWorks combines some high-end expert system shell capabilities in a medium-cost package. This shell is developed in Common Lisp, runs on the 80386 Hummingboard, and provides some expert system features formerly available only on AI-computers including frame and rule-based reasoning, on-line tutorial, multiple inheritance, and object-programming. ?? 1988 International Association for Mathematical Geology.

Mathematical Geology↗
USGS1987

On the functional optimization of a certain class of nonstationary spatial functions

Procedures are developed in order to obtain optimal estimates of linear functionals for a wide class of nonstationary spatial functions. These procedures rely on well-established constrained minimum-norm criteria, and are applicable to multidimensional phenomena which are characterized by the so-called hypothesis of inherentity. The latter requires elimination of the polynomial, trend-related components of the spatial function leading to stationary quantities, and also it generates some interesting mathematics within the context of modelling and optimization in several dimensions. The arguments are illustrated using various examples, and a case study computed in detail. ?? 1987 Plenum Publishing Corporation.

Journal of Optimization Theory and Applications↗
USGS2005

MODFLOW-2005 : the U.S. Geological Survey modular ground-water model--the ground-water flow process

This report presents MODFLOW-2005, which is a new version of the finite-difference ground-water model commonly called MODFLOW. Ground-water flow is simulated using a block-centered finite-difference approach. Layers can be simulated as confined or unconfined. Flow associated with external stresses, such as wells, areal recharge, evapotranspiration, drains, and rivers, also can be simulated. The report includes detailed explanations of physical and mathematical concepts on which the model is based, an explanation of how those concepts are incorporated in the modular structure of the computer program, instructions for using the model, and details of the computer code. The modular structure consists of a MAIN Program and a series of highly independent subroutines. The subroutines are grouped into 'packages.' Each package deals with a specific feature of the hydrologic system that is to be simulated, such as flow from rivers or flow into drains, or with a specific method of solving the set of simultaneous equations resulting from the finite-difference method. Several solution methods are incorporated, including the Preconditioned Conjugate-Gradient method. The division of the program into packages permits the user to examine specific hydrologic features of the model independently. This also facilitates development of additional capabilities because new packages can be added to the program without modifying the existing packages. The input and output systems of the computer program also are designed to permit maximum flexibility. The program is designed to allow other capabilities, such as transport and optimization, to be incorporated, but this report is limited to describing the ground-water flow capability. The program is written in Fortran 90 and will run without modification on most computers that have a Fortran 90 compiler.

Techniques and Methods↗
USGS1989

Stream temperature investigations: field and analytic methods

This document provides guidance to the user of the U.S. Fish and Wildlife Service’s Stream Network Temperature Model (SNTEMP). Planning a temperature study is discussed in terms of understanding the management objectives and ensuring that the questions will be accurately answered with the modeling approach being used. A sensitivity analysis of SNTEMP is presented to illustrate which input variables are most important in predicting stream temperatures. This information helps prioritize data collection activities, highlights the need for quality control, focuses on which parameters can be estimated rather than measured, and offers a broader perspective on management options in terms of knowing where the biggest temperature response will be felt. All of the major input variables for stream geometry, meteorology, and hydrology are discussed in detail. Each variable is defined, with guidance given on how to measure it, what kind of equipment to use, where to obtain it from another agency, and how to calculate it if the data are in a form other than that required by SNTEMP. Examples are presented for the various forms in which water temperature, discharge, and meteorological data are commonly found. Ranges of values for certain input variables that are difficult to measure of estimate are given. Particular attention is given to those variables not commonly understood by field biologists likely to be involved in a stream temperature study. Pertinent literature is cited for each variable, with emphasis on how other people have treated particular problems and on results they have found. Model calibration, verification, and validation steps are defined and outlines, with measures of "goodness-of-fit" given for comparing simulated stream temperatures with observed values. The question of how good is good enough is explored, and attention is given to the kinds of simulation and data reduction errors that one should be alert for. Some special cases dealing with ice and reservoir temperature are mentioned. Special attention is given to understanding micro-thermal habitats that act as important thermal refugia under low flow conditions; their causes, extent, and management implications are discussed. Alternative public domain stream and reservoir temperature models are contrasted with SNTEMP. A distinction is made between steady-flow and dynamic-flow models and their respective capabilities. Regression models are offered as an alternative approach for some situations, with appropriate mathematical formulas suggested. Appendices provide information on State and Federal agencies that are good data sources, vendors for field instrumentation, and small computer programs useful in data reduction.

Report↗
USGS2004

Acidity and alkalinity in mine drainage: Theoretical considerations

Acidity, net acidity, and net alkalinity are widely used parameters for the characterization of mine drainage, but these terms are not well defined and are often misunderstood. Incorrect interpretation of acidity, alkalinity, and derivative terms can lead to inadequate treatment design or poor regulatory decisions. We briefly explain derivations of theoretical expressions of three types of alkalinities (caustic, phenolphthalein, and total) and acidities (mineral, CO 2 , and total). Theoretically defined total alkalinity is closely analogous to measured alkalinity and presents few practical interpretation problems. Theoretically defined “CO 2- acidity” is closely related to most standard titration methods used for mine drainage with an endpoint pH of 8.3, but it presents numerous interpretation problems, and it is unfortunately named because CO 2 is intentionally driven off during titration of mine-drainage samples. Using the proton condition/massaction approach and employing graphs for visualization, we explore the concept of principal components and how to assign acidity contributions to solution species, including aqueous complexes, commonly found in mine drainage. We define a comprehensive theoretical definition of acidity in mine drainage on the basis of aqueous speciation at the sample pH and the capacity of these species to undergo hydrolysis to pH 8.3. This definition indicates the computed acidity in milligrams per liter (mg L -1 ) as CaCO 3 (based on pH and analytical concentrations of dissolved Fe III , Fe II , Mn, and Al in mg L -1 ): Acidity computed = 50. (10 (3-pH) + 3.C FeIII /55.8 + 2.C FeII /55.8 + 2.C Mn /54.9 + 3.C Al /27.0) underestimates contributions from HSO 4 - and H + , but overestimates the acidity due to Fe 3+ . These errors tend to approximately cancel each other. We demonstrate that “net alkalinity” is a valid mathematical construction based on theoretical definitions of alkalinity and acidity. We demonstrate that, for most mine-drainage solutions, a useful net alkalinity value can be derived from: 1) alkalinity and acidity values based on aqueous speciation, 2) measured alkalinity - computed acidity, or 3) taking the negative of the value obtained in a standard method “hot peroxide” acidity titration, provided that labs report negative values . We recommend the third approach; i.e., Net alkalinity = - Hot Acidity.

Conference Paper↗
USGS1976

Flood-frequency analysis for small natural streams in Georgia

Flood information from 104 project basins (0.1 to 20 square miles) and data available from 170 larger basins (20 to 1,500 square miles) are analyzed to provide planners and designers with relations for estimating the magnitude and frequency of flood-peak discharges on small natural streams in Georgia. More than 90 percent of the project activity was directed toward the acquisition and processing of data for more than two thousand rainfall-runoff periods occurring during 1964-74 at the 104 project stations. Storm rainfall data were computed at 5-minute intervals from records of six long-term (about 75 years) National Weather Service Stations. The U.S. Geological Survey mathematical rainfall-runoff model is used to synthesize long-term flood records for about 80 percent of the project stations. Station flood-frequency data are developed, using the log-Pearson Type III distribution with regionalized skew coefficients. Multiple regression analysis is used to define relations between flood-frequency station data for small and large streams and 10 physical and climatological basin characteristics. The analysis indicates the drainage-basin size is the most significant variable. Five regions having distinct flood-peak characteristics are delineated. The developed relations, expressed as equations and nomographs, are considered usable for virtually any site in Georgia where the drainage area is between 0.1 and 20 square miles, and the flow is natural. (Woodard-USGS)

Open-File Report↗
USGS2014

A computer program for uncertainty analysis integrating regression and Bayesian methods

This work develops a new functionality in UCODE_2014 to evaluate Bayesian credible intervals using the Markov Chain Monte Carlo (MCMC) method. The MCMC capability in UCODE_2014 is based on the FORTRAN version of the differential evolution adaptive Metropolis (DREAM) algorithm of Vrugt et al. (2009), which estimates the posterior probability density function of model parameters in high-dimensional and multimodal sampling problems. The UCODE MCMC capability provides eleven prior probability distributions and three ways to initialize the sampling process. It evaluates parametric and predictive uncertainties and it has parallel computing capability based on multiple chains to accelerate the sampling process. This paper tests and demonstrates the MCMC capability using a 10-dimensional multimodal mathematical function, a 100-dimensional Gaussian function, and a groundwater reactive transport model. The use of the MCMC capability is made straightforward and flexible by adopting the JUPITER API protocol. With the new MCMC capability, UCODE_2014 can be used to calculate three types of uncertainty intervals, which all can account for prior information: (1) linear confidence intervals which require linearity and Gaussian error assumptions and typically 10s–100s of highly parallelizable model runs after optimization, (2) nonlinear confidence intervals which require a smooth objective function surface and Gaussian observation error assumptions and typically 100s–1,000s of partially parallelizable model runs after optimization, and (3) MCMC Bayesian credible intervals which require few assumptions and commonly 10,000s–100,000s or more partially parallelizable model runs. Ready access allows users to select methods best suited to their work, and to compare methods in many circumstances.

Environmental Modelling and Software↗
USGS2018

A snapshot of women of the U.S. Geological Survey in STEM and related careers

Introduction The term “STEM” has been used to group together the fields of science, technology, engineering, and mathematics and to describe education and professions related to these fields. The professional fields connected to STEM education are thought of as engineering, medicine, and computer technology. Yet these professional fields are merely the tip of the iceberg. Numerous opportunities in these fields encompass environmental research. The possibilities range from predicting the next earthquake to saving polar bears from extinction to developing a vaccine for salmon measles. The science of natural systems is complex and often requires people from a variety of fields of expertise to make headway with a solution. To that end, the U.S. Geological Survey (USGS) has long recognized the need for a diversity of STEM expertise to address the Nation’s environmental research needs and the vision to facilitate integration of these fields. We are team builders! In this book, we point out the many facets of research carried out by USGS STEM scientists in an effort to show career options and pathways not typically pursued. The women portrayed were selected by USGS associate and regional directors as representative of particular fields and to inspire future generations.

Circular↗