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Results for “Applied Mathematics and Computation”

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USGS2011

Modeling hazardous mass flows Geoflows09: Mathematical and computational aspects of modeling hazardous geophysical mass flows; Seattle, Washington, 9–11 March 2009

A recent workshop at the University of Washington focused on mathematical and computational aspects of modeling the dynamics of dense, gravity-driven mass movements such as rock avalanches and debris flows. About 30 participants came from seven countries and brought diverse backgrounds in geophysics; geology; physics; applied and computational mathematics; and civil, mechanical, and geotechnical engineering. The workshop was cosponsored by the U.S. Geological Survey Volcano Hazards Program, by the U.S. National Science Foundation through a Vertical Integration of Research and Education (VIGRE) in the Mathematical Sciences grant to the University of Washington, and by the Pacific Institute for the Mathematical Sciences. It began with a day of lectures open to the academic community at large and concluded with 2 days of focused discussions and collaborative work among the participants.

Eos, Earth and Space Science News↗

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↗

USGS1998

Lake bed classification using acoustic data

As part of our effort to identify the lake bed surficial substrates using remote sensing data, this work designs pattern classifiers by multivariate statistical methods. Probability distribution of the preprocessed acoustic signal is analyzed first. A confidence region approach is then adopted to improve the design of the existing classifier. A technique for further isolation is proposed which minimizes the expected loss from misclassification. The devices constructed are applicable for real-time lake bed categorization. A mimimax approach is suggested to treat more general cases where the a priori probability distribution of the substrate types is unknown. Comparison of the suggested methods with the traditional likelihood ratio tests is discussed.

Applied Mathematics and Computer Science↗

USGS2001

A simple calculator of ballistic trajectories for blocks ejected during volcanic eruptions

During the past century, numerous observers have described the violent ejection of large blocks and bombs from volcanoes during volcanic explosions. Minakami (1942) mapped the locations of blocks ejected from Asama Volcano during explosions in 1937. He developed a mathematical expression relating initial velocity and trajectory angle of ejected blocks to the ejection distance, taking into account air drag and assuming a constant drag coefficient. In the late 1950’s, Gorshkov (1959) estimated ejection velocities at Bezymianny volcano during its sector-collapse eruption. Wilson (1972) developed the first mathematical algorithm for ballistic trajectories in the volcanological literature (earlier ones had been available for military applications) that considered variations in drag coefficient with Reynolds number. Fagents and Wilson (1993) advanced the method of Wilson (1972) by considering the effect of reduced drag near the vent. From the 1970’s through the 1990’s other papers, too numerous to mention, have estimated volcanic ejection velocities from ballistic blocks. Since the early 1990’s there has been a decrease in the number of published papers that quantify ejection velocities from ballistic trajectories. This decrease has resulted in part from the appreciation that ejection velocities cannot be uniquely determined by ejection distance due to uncertainties in initial trajectory angle and drag force. On the other hand, the decrease in usage has coincided with an increase in the ease with which ballistic calculations can be made, due to the vast improvement in computer power and in the user-friendliness of computers. During the 1970’s, only volcanologists with mathematical acumen or those who could collaborate with applied mathematicians were able to make such estimates. With 21st century computer power, ballistic computation should be available to anyone as a back-of-the-envelope indicator of explosive power; the only factor preventing such usage is the lack of a user-friendly computer program. In this paper, I describe a program that can be used for quick ballistics calculations. The program, Eject!, was written in Microsoft Visual Basic® and operates on any personal computer running Microsoft® Windows 95 or later. Instructions on installing Eject! version 1.4 1. Copy the file "eject_1-4.zip" to the hard drive of your Windows-based computer 2. Unzip the files in "eject_1-4.zip" 3. Double-click on the file "setup.exe". This will install the software on your computer. 4. Once it is installed, go to the "start" button, the to "programs." You will see the program in your program list.

Open-File Report↗

USGS1989

Mass-conserving method of characteristics for streamflow modeling

A robust numerical model is presented for the computation of unsteady streamflow on steep river slopes. The one-dimensional model uses the method of characteristics on a specified space-time grid to solve the Saint-Venant equations. An additional continuity equation requirement on each space-time element provides greatly improved conservation of mass over traditional implementations of the method of characteristics on a fixed grid. The space-time geometry of the problem is described in a finite element setting. Hermite interpolation of channel parameters is used to avoid numerical difficulties that may occur with steep slopes due to discontinuities in the derivatives of data such as channel top width. Manning's equation for friction slope can be modified by a factor to make the slope more appropriate for steep rivers. The standard Manning's friction slope can also be used, if preferred. The computer model is not restricted to steep slopes, and applies as well to gently sloping streams. Two numerical examples support the mathematical approach and computational algorithm. (USGS)

Open-File Report↗

USGS1992

Mass-conserving method of characteristics for streamflow modeling

Water Supply Paper↗

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↗

USGS1980

Channel and dynamic flow characteristics of the Chattahoochee River, Buford Dam to Georgia Highway 141

Detailed flow and cross-section data for a 17-mile reach of the Chattahoochee River in northeast Georgia are described and summarized. Flow data include measurements of highly dynamic stage and discharge at five stations during the period March 21-23, 1976. Flow data were collected at 5-minute intervals and are listed accordingly. Coordinate data for 39 cross sections in the study reach are also listed. A mathematical model is developed and applied whereby stage data collected at a single station can be used to compute highly dynamic discharge at the station. The model is based on the continuity and momentum equations that describe unsteady, one-dimensional flow in open channels. Both equations are transformed to a single quadratic equation which describes mean flow velocity at a single station. Flow-geometry parameters used by the model are computed using cross-section coordinates and the equation which describes the area of an irregular polygon. Use of the model in conjunction with highly dynamic stage data collected on March 23, 1976, provided close agreement between measured and computed discharges. The model was also used to investigate the sensitivity of highly dynamic discharge to channel and flow parameters. Computed discharge was most sensitive to changes in channel roughness and slope.

Georgia↗

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↗

USGS1978

Temperature computation for temperate lakes

No abstract available.

Applied Mathematical Modelling↗

USGS2021

The application of ensemble wave forcing to quantify uncertainty of shoreline change predictions

Reliable predictions and accompanying uncertainty estimates of coastal evolution on decadal to centennial time scales are increasingly sought. So far, most coastal change projections rely on a single, deterministic realization of the unknown future wave climate, often derived from a global climate model. Yet, deterministic projections do not account for the stochastic nature of future wave conditions across a variety of temporal scales (e.g., daily, weekly, seasonally, and interannually). Here, we present an ensemble Kalman filter shoreline change model to predict coastal erosion and uncertainty due to waves at a variety of time scales. We compare shoreline change projections, simulated with and without ensemble wave forcing conditions by applying ensemble wave time series produced by a computationally efficient statistical downscaling method. We demonstrate a sizable (site-dependent) increase in model uncertainty compared with the unrealistic case of model projections based on a single, deterministic realization (e.g., a single time series) of the wave forcing. We support model-derived uncertainty estimates with a novel mathematical analysis of ensembles of idealized process models. Here, the developed ensemble modeling approach is applied to a well-monitored beach in Tairua, New Zealand. However, the model and uncertainty quantification techniques derived here are generally applicable to a variety of coastal settings around the world.

JGR Earth Surface↗

USGS2005

Parallel Fortran-MPI software for numerical inversion of the Laplace transform and its application to oscillatory water levels in groundwater environments

A parallel Fortran-MPI (Message Passing Interface) software for numerical inversion of the Laplace transform based on a Fourier series method is developed to meet the need of solving intensive computational problems involving oscillatory water level's response to hydraulic tests in a groundwater environment. The software is a parallel version of ACM (The Association for Computing Machinery) Transactions on Mathematical Software (TOMS) Algorithm 796. Running 38 test examples indicated that implementation of MPI techniques with distributed memory architecture speedups the processing and improves the efficiency. Applications to oscillatory water levels in a well during aquifer tests are presented to illustrate how this package can be applied to solve complicated environmental problems involved in differential and integral equations. The package is free and is easy to use for people with little or no previous experience in using MPI but who wish to get off to a quick start in parallel computing. ?? 2004 Elsevier Ltd. All rights reserved.

Environmental Modelling and Software↗

USGS1982

Hydrologic maps of the High Plains aquifer, January 1981, southwestern Kansas

The U.S. Geological Survey furnishes maps depicting saturated thickness of the High Plains aquifer to the Southwest Kansas Groundwater Management District No. 3 through a continuing cooperative agreement. The Groundwater Management District uses the maps to calculated a planned depletion rate of 40 percent of the saturated thickness of the aquifer in 25 years for a 13-county area. Because the information contained on the maps is used in management decisions, the Groundwater Management District is concerned with the accuracy, reliability, and consistency of the hydrologic information that the maps show. A mathematical technique, called kriging, has been used in the form of a computer program to interpolate hydrologic data based on a network of measured values (Karlinger and Skrivan, 1980). The technique has been applied to geohydrologic data in west-central Kansas with good results (Dunlap and Spinazola, 1981). The program calculates the values at the center of each 1-mile section in a predefined region and facilitates contouring of selected values. The kriging technique was used to produce the hydrologic maps in this report.

Kansas↗

USGS2004

Granular avalanches across irregular three-dimensional terrain: 2. Experimental tests

Scaling considerations indicate that miniature experiments can be used to test models of granular avalanches in which the effects of intergranular fluid and cohesion are negligible. To test predictions of a granular avalanche model described in a companion paper, we performed bench top experiments involving avalanches of dry sand across irregular basal topography that mimicked the complexity of natural terrain. The experiments employed a novel method of laser-assisted cartography to map the three-dimensional morphology of rapidly moving avalanches, thereby providing high-resolution data for comparison with model output. Model input consisted of two material properties (angles of internal and basal Coulomb friction of the sand), which were measured in independent tests, and of initial and boundary conditions that characterized the geometry of the experimental apparatus. Experimental results demonstrate that the model accurately predicts not only the gross behavior but also many details of avalanche motion, from initiation to deposition. We attribute this accuracy to a mathematical and computational formulation that conserves mass and momentum in three-dimensional physical space and satisfies the Coulomb equation in three-dimensional stress space. Our results support the hypothesis that a Coulomb proportionality between shear and normal stresses applies in moderately rapid granular flows and that complicated constitutive postulates are unnecessary if momentum conservation is strictly enforced in continuum avalanche models. Furthermore, predictions of our Coulomb continuum model contrast with those of a Coulomb point mass model, illustrating the importance of multidimensional modeling and model testing.

Journal of Geophysical Research Earth Surface↗

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↗

USGS2017

A general modeling framework for describing spatially structured population dynamics

Variation in movement across time and space fundamentally shapes the abundance and distribution of populations. Although a variety of approaches model structured population dynamics, they are limited to specific types of spatially structured populations and lack a unifying framework. Here, we propose a unified network-based framework sufficiently novel in its flexibility to capture a wide variety of spatiotemporal processes including metapopulations and a range of migratory patterns. It can accommodate different kinds of age structures, forms of population growth, dispersal, nomadism and migration, and alternative life-history strategies. Our objective was to link three general elements common to all spatially structured populations (space, time and movement) under a single mathematical framework. To do this, we adopt a network modeling approach. The spatial structure of a population is represented by a weighted and directed network. Each node and each edge has a set of attributes which vary through time. The dynamics of our network-based population is modeled with discrete time steps. Using both theoretical and real-world examples, we show how common elements recur across species with disparate movement strategies and how they can be combined under a unified mathematical framework. We illustrate how metapopulations, various migratory patterns, and nomadism can be represented with this modeling approach. We also apply our network-based framework to four organisms spanning a wide range of life histories, movement patterns, and carrying capacities. General computer code to implement our framework is provided, which can be applied to almost any spatially structured population. This framework contributes to our theoretical understanding of population dynamics and has practical management applications, including understanding the impact of perturbations on population size, distribution, and movement patterns. By working within a common framework, there is less chance that comparative analyses are colored by model details rather than general principles

Ecology and Evolution↗

USGS1977

Chemical variability in the Lakeview Mountains pluton, southern California batholith: A comparison of the methods of correspondence analysis and extended Q-mode factor analysis

An extended method of Q -mode factor analysis that has been described previously offers a number of important advantages over conventional Q -mode factor analysis and correspondence analysis when applied to compositional data. Among these are the ability to compute the compositions, in the original units of the data, represented by the reference axes. The only special requirement of the data is that the values for each observation sum to a constant. Mathematically derived reference axes, such as the principal-components and varimax axes, commonly represent compositions that are partly negative. Even where no negative values are present, these compositions are comparatively difficult to interpret in petrologic terms and impossible or inappropriate to include in petrogenetic models. A model has been developed for the Lakeview Mountains pluton which employs vectors representing the compositions of melanocratic schlieren, leucocratic schlieren, and inclusions as the reference axes. When the effect of the inclusions is removed, the compositional zoning in the pluton is like that found in otherwise similar plutons in the Sierra Nevada batholith. The genetic implication of the model is that the inclusions were separated from the magma at least in its marginal parts. In the central part of the pluton, the inclusions were either separated to a lesser extent than in the marginal parts, or were reincorporated into the magma.

California↗

USGS2000

Uncertainty estimation for resource assessment-an application to coal

The U.S. Geological Survey is conducting a national assessment of coal resources. As part of that assessment, a geostatistical procedure has been developed to estimate the uncertainty of coal resources for the historical categories of geological assurance: measured, indicated, inferred, and hypothetical coal. Data consist of spatially clustered coal thickness measurements from coal beds and/or zones that cover, in some cases, several thousand square kilometers. Our procedure involved trend removal, an examination of spatial correlation, computation of a sample semivariogram, and fitting a semivariogram model. This model provided standard deviations for the uncertainty estimates. The number of sample points (drill holes) in each historical category also was estimated. Measurement error in the thickness of the coal bed/zone was obtained from the fitted model or supplied exogenously. From this information approximate estimates of uncertainty on the historical categories were computed. We illustrate the methodology using drill hole data from the Harmon coal bed located in southwestern North Dakota. The methodology will be applied to approximately 50 coal data sets.

Mathematical Geology↗