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USGS1985

Subsurface storage of freshwater in South Florida; a digital model analysis of recoverability

As part of a study of the feasibility of recovering freshwater injected and stored underground in south Florida, a digital solute-transport model was used to investigate the relation of recovery efficiency to the variety of hydrogeologic conditions that could prevail in brackish artesian aquifers and to a variety of management alternatives. The analyses employed a modeling approach in which the control for sensitivity testing was a hypothetical aquifer considered representative of permeable zones in south Florida that might be used for storage of freshwater. Parameter variations in the tests represented possible variations in aquifer conditions in the area. The applicability of the analyses to south Florida limestone aquifers required the assumption that flow nonuniformities in those aquifers are small on the scale of volumes of water likely to be injected, and that their effect could be represented as hydrodynamic dispersion. Generally, it was shown that a loss of recovery efficiency is caused by (1) processes causing mixing of injected freshwater with native saline water (hydrodynamic dispersion), (2) processes causing the more or less irreversible displacement of the injected freshwater with respect to the well (buoyancy stratification, background hydraulic gradients, and interlayer dispersion), or (3) processes causing injection and withdrawal flow patterns to be dissimilar (directionally biased well-bore plugging, and dissimilar injection and withdrawal schedules in multiple-well systems). Other results indicated that recovery efficiency improves considerably with successive cycles, providing that each recovery phase ends when the chloride concentration of withdrawn water exceeds established criteria for potability (usually 250 milligrams per liter), and that freshwater injected into highly permeable or highly saline aquifers (such as the 'boulder zone') would buoy rapidly. Many hydrologic conditions were posed for model analysis. To have obtained comparable results with operational testing would have been more costly by orders of magnitude. The tradeoff is that the validity of results obtained from computer modeling is somewhat less certain. In particular, results must be qualified with observations that (1) the complex set of processes lumped as hydrodynamic dispersion is represented with a somewhat simplified mathematical approximation, and (2) other flow processes in limestone injection zones are as yet incompletely understood. Despite such reservations, the study is considered a practical example of the use of transport models in ground-water investigations.

Water Supply Paper↗

USGS1989

A direct method for calculating instrument noise levels in side-by-side seismometer evaluations

The subject of determining the inherent system noise levels present in modem broadband closed loop seismic sensors has been an evolving topic ever since closed loop systems became available. Closed loop systems are unique in that the system noise can not be determined via a blocked mass test as in older conventional open loop seismic sensors. Instead, most investigators have resorted to performing measurements on two or more systems operating in close proximity to one another and to analyzing the outputs of these systems with respect to one another to ascertain their relative noise levels. The analysis of side-by-side relative performance is inherently dependent on the accuracy of the mathematical modeling of the test configuration. This report presents a direct approach to extracting the system noise levels of two linear systems with a common coherent input signal. The mathematical solution to the problem is incredibly simple; however the practical application of the method encounters some difficulties. Examples of expected accuracies are presented as derived by simulating real systems performance using computer generated random noise. In addition, examples of the performance of the method when applied to real experimental test data are shown.

Open-File Report↗

USGS1986

Satellite orientation and position for geometric correction of scanner imagery.

The USGS Mini Image Processing System currently relies on a polynomial method for geometric correction of Landsat multispectral scanner (MSS) data. A large number of ground control points are required because polynomials do not model the sources of error. In order to reduce the number of necessary points, a set of mathematical equations modeling the Landsat satellite motions and MSS scanner has been derived and programmed. A best fit to the equations is obtained by using a least-squares technique that permits computation of the satellite orientation and position parameters based on only a few control points.-from Author

Photogrammetric Engineering and Remote Sensing↗

USGS1965

Mathematical models of catchment behavior

After an examination of trends in the modeling of hydrologic systems, a review of some recent studies is given. The authors' preliminary studies on the feasibility and efficiency of the automatic evaluation of catchment model parameters by use of a digital computer are described and some results presented.

Journal of the Hydraulics Division↗

USGS2024

Preliminary implications of viscoelastic ray theory for anelastic seismic tomography models

The recent developments in general viscoelastic ray theory provide a rigorous mathematical framework for anelastic seismic tomography. They provide closed‐form solutions of forward ray‐tracing and simple inverse problems for anelastic horizontal and spherical layered media with material gradients. They provide ray‐tracing computation algorithms valid for all angles of incidence that account for changes in wave speed, attenuation, and trajectory of anelastic P and S body waves induced by anelastic boundaries. They account for theoretical predictions that seismic waves refract as inhomogeneous waves across anelastic boundaries for all angles of incidence, which in turn accounts for energy carried by plane waves along seismic boundaries at head wave critical angles and wide‐angle refracted (WAR) ray paths that are not predicted by elastic models. Exact viscoelastic ray‐tracing numerical results for various models provide examples that illustrate the effects of anelastic boundaries on the travel times and amplitudes of seismic waves. They show the effects are strongly dependent on angle of incidence. For near‐critical and wide angles of incidence the anelastic effects on travel times and amplitudes can be large and are not explained by elastic ray theory, but the effects on travel times can be relatively small and difficult to distinguish from those for elastic media for pre‐near‐critical angles of incidence. The results for some models indicate that reflected anelastic WAR waves may be observable at the surface and possibly account for some prominent seismic arrivals not explained by elasticity. These preliminary results suggest that the application of exact viscoelastic ray‐tracing computation algorithms to exploration and teleseismic data sets can reveal new insights regarding the properties and distribution of anelastic materials in the Earth.

Bulletin of the Seismological Society of America↗

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↗

USGS2011

Fitting a structured juvenile-adult model for green tree frogs to population estimates from capture-mark-recapture field data

We derive point and interval estimates for an urban population of green tree frogs ( Hyla cinerea ) from capture–mark–recapture field data obtained during the years 2006–2009. We present an infinite-dimensional least-squares approach which compares a mathematical population model to the statistical population estimates obtained from the field data. The model is composed of nonlinear first-order hyperbolic equations describing the dynamics of the amphibian population where individuals are divided into juveniles (tadpoles) and adults (frogs). To solve the least-squares problem, an explicit finite difference approximation is developed. Convergence results for the computed parameters are presented. Parameter estimates for the vital rates of juveniles and adults are obtained, and standard deviations for these estimates are computed. Numerical results for the model sensitivity with respect to these parameters are given. Finally, the above-mentioned parameter estimates are used to illustrate the long-time behavior of the population under investigation.

Bulletin of Mathematical Biology↗

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↗

USGS2000

Methods for estimating low-flow statistics for Massachusetts streams

Methods and computer software are described in this report for determining flow duration, low-flow frequency statistics, and August median flows. These low-flow statistics can be estimated for unregulated streams in Massachusetts using different methods depending on whether the location of interest is at a streamgaging station, a low-flow partial-record station, or an ungaged site where no data are available. Low-flow statistics for streamgaging stations can be estimated using standard U.S. Geological Survey methods described in the report. The MOVE.1 mathematical method and a graphical correlation method can be used to estimate low-flow statistics for low-flow partial-record stations. The MOVE.1 method is recommended when the relation between measured flows at a partial-record station and daily mean flows at a nearby, hydrologically similar streamgaging station is linear, and the graphical method is recommended when the relation is curved. Equations are presented for computing the variance and equivalent years of record for estimates of low-flow statistics for low-flow partial-record stations when either a single or multiple index stations are used to determine the estimates. The drainage-area ratio method or regression equations can be used to estimate low-flow statistics for ungaged sites where no data are available. The drainage-area ratio method is generally as accurate as or more accurate than regression estimates when the drainage-area ratio for an ungaged site is between 0.3 and 1.5 times the drainage area of the index data-collection site. Regression equations were developed to estimate the natural, long-term 99-, 98-, 95-, 90-, 85-, 80-, 75-, 70-, 60-, and 50-percent duration flows; the 7-day, 2-year and the 7-day, 10-year low flows; and the August median flow for ungaged sites in Massachusetts. Streamflow statistics and basin characteristics for 87 to 133 streamgaging stations and low-flow partial-record stations were used to develop the equations. The streamgaging stations had from 2 to 81 years of record, with a mean record length of 37 years. The low-flow partial-record stations had from 8 to 36 streamflow measurements, with a median of 14 measurements. All basin characteristics were determined from digital map data. The basin characteristics that were statistically significant in most of the final regression equations were drainage area, the area of stratified-drift deposits per unit of stream length plus 0.1, mean basin slope, and an indicator variable that was 0 in the eastern region and 1 in the western region of Massachusetts. The equations were developed by use of weighted-least-squares regression analyses, with weights assigned proportional to the years of record and inversely proportional to the variances of the streamflow statistics for the stations. Standard errors of prediction ranged from 70.7 to 17.5 percent for the equations to predict the 7-day, 10-year low flow and 50-percent duration flow, respectively. The equations are not applicable for use in the Southeast Coastal region of the State, or where basin characteristics for the selected ungaged site are outside the ranges of those for the stations used in the regression analyses. A World Wide Web application was developed that provides streamflow statistics for data collection stations from a data base and for ungaged sites by measuring the necessary basin characteristics for the site and solving the regression equations. Output provided by the Web application for ungaged sites includes a map of the drainage-basin boundary determined for the site, the measured basin characteristics, the estimated streamflow statistics, and 90-percent prediction intervals for the estimates. An equation is provided for combining regression and correlation estimates to obtain improved estimates of the streamflow statistics for low-flow partial-record stations. An equation is also provided for combining regression and drainage-area ratio estimates to obtain improved e

Massachusetts↗

USGS2012

Estimating landscape carrying capacity through maximum clique analysis

Habitat suitability (HS) maps are widely used tools in wildlife science and establish a link between wildlife populations and landscape pattern. Although HS maps spatially depict the distribution of optimal resources for a species, they do not reveal the population size a landscape is capable of supporting--information that is often crucial for decision makers and managers. We used a new approach, "maximum clique analysis," to demonstrate how HS maps for territorial species can be used to estimate the carrying capacity, N(k), of a given landscape. We estimated the N(k) of Ovenbirds (Seiurus aurocapillus) and bobcats (Lynx rufus) in an 1153-km2 study area in Vermont, USA. These two species were selected to highlight different approaches in building an HS map as well as computational challenges that can arise in a maximum clique analysis. We derived 30-m2 HS maps for each species via occupancy modeling (Ovenbird) and by resource utilization modeling (bobcats). For each species, we then identified all pixel locations on the map (points) that had sufficient resources in the surrounding area to maintain a home range (termed a "pseudo-home range"). These locations were converted to a mathematical graph, where any two points were linked if two pseudo-home ranges could exist on the landscape without violating territory boundaries. We used the program Cliquer to find the maximum clique of each graph. The resulting estimates of N(k) = 236 Ovenbirds and N(k) = 42 female bobcats were sensitive to different assumptions and model inputs. Estimates of N(k) via alternative, ad hoc methods were 1.4 to > 30 times greater than the maximum clique estimate, suggesting that the alternative results may be upwardly biased. The maximum clique analysis was computationally intensive but could handle problems with < 1500 total pseudo-home ranges (points). Given present computational constraints, it is best suited for species that occur in clustered distributions (where the problem can be broken into several, smaller problems), or for species with large home ranges relative to grid scale where resampling the points to a coarser resolution can reduce the problem to manageable proportions.

Ecological Applications↗

USGS1980

Study plan for the regional aquifer-system analysis of alluvial basins in south-central Arizona and adjacent states

The alluvial basins in the Southwestern United States constitute a major source of ground water and are relied upon extensively for agricultural, industrial, and public water supplies. Large-scale depletion of ground water is directly related to pumping that has occurred in the past few decades and is continuing today. The U.S. Geological Survey has started a 4-year study of the alluvial basins in south-central Arizona and parts of California, Nevada, and New Mexico. The study is designed to document and describe the hydrologic setting in the basins, the ground-water resources available, and the effects of historical development on the ground-water system. To aid in the study, mathematical models of selected basins will be developed for appraising the local and regional flow systems. Major tasks necessary to accomplish the study objectives include accumulating existing data on ground-water quantity and quality, entering the data into a computer file, identifying data deficiencies, and developing a program to remedy the deficiencies by collection of additional data. The approach to the study will be to develop and calibrate models of selected basins for which sufficient data exist and then to develop interpretation-transfer techniques whereby general predevelopment and postdevelopment conceptual models of the hydrologic system in other basins may be synthesized. The results will be applied to selected basins for testing, calibration, and modification. The end result of the project will be a better definition of the hydrologic parameters and a better understanding of the workings of the hydrologic system in the alluvial basins. The results will include models that can be used to study the effects of management alternatives and water-resources development on the hydrologic system.

Arizona↗

USGS2007

OPR-PPR, a computer program for assessing data importance to model predictions using linear statistics

The OPR-PPR program calculates the Observation-Prediction (OPR) and Parameter-Prediction (PPR) statistics that can be used to evaluate the relative importance of various kinds of data to simulated predictions. The data considered fall into three categories: (1) existing observations, (2) potential observations, and (3) potential information about parameters. The first two are addressed by the OPR statistic; the third is addressed by the PPR statistic. The statistics are based on linear theory and measure the leverage of the data, which depends on the location, the type, and possibly the time of the data being considered. For example, in a ground-water system the type of data might be a head measurement at a particular location and time. As a measure of leverage, the statistics do not take into account the value of the measurement. As linear measures, the OPR and PPR statistics require minimal computational effort once sensitivities have been calculated. Sensitivities need to be calculated for only one set of parameter values; commonly these are the values estimated through model calibration. OPR-PPR can calculate the OPR and PPR statistics for any mathematical model that produces the necessary OPR-PPR input files. In this report, OPR-PPR capabilities are presented in the context of using the ground-water model MODFLOW-2000 and the universal inverse program UCODE_2005. The method used to calculate the OPR and PPR statistics is based on the linear equation for prediction standard deviation. Using sensitivities and other information, OPR-PPR calculates (a) the percent increase in the prediction standard deviation that results when one or more existing observations are omitted from the calibration data set; (b) the percent decrease in the prediction standard deviation that results when one or more potential observations are added to the calibration data set; or (c) the percent decrease in the prediction standard deviation that results when potential information on one or more parameters is added.

Techniques and Methods↗

USGS1984

U.S. Geological Survey probabilistic methodology for oil and gas resource appraisal of the United States

Probabilistic methodology used by the U.S. Geological Survey is described for estimating the quantity of undiscovered recoverable conventional resources of oil and gas in the United States. A judgmental probability distribution of the "quantity of resource" and its properties is determined for a geologic province or basin. From this distribution, point and interval estimates of the quantity of undiscovered resource are obtained. Distributions and their properties are established for each of the following resources: (1) oil and nonassociated gas from estimates of the probability of the resource being present and the conditional probability distribution of the quantity of resource given that the resource is present, (2) associated-dissolved gas from its corresponding oil distribution, (3) total gas, (4) oil and total gas in two or more provinces. Computer graphics routines are illustrated with examples from the U.S. Geological Survey Circular 860. ?? 1984 Plenum Publishing Corporation.

Journal of the International Association for Mathe↗

USGS1977

State of stress in the lithosphere: Inferences from the flow laws of olivine

The experimental flow data for rocks and minerals are reviewed and found to fit a law of the form {Mathematical expression} where {Mathematical expression} This law reduces to the familiar power-law stress dependency at low stress and to an exponential stress dependency at high stress. Using the material flow law parameters for olivine, stress profiles with depth and strain rate are computed for a representative range of temperature distributions in the lithosphere. The results show that the upper 15 to 25 km of the oceanic lithosphere must behave elastically or fail by fracture and that the remainder deforms by exponential law flow at intermediate depths and by power-law flow in the rest. A model computation of the gravitational sliding of a lithospheric plate using olivine rheology exhibits a very sharp decoupling zone which is a consequence of the combined effects of increasing stress and temperature on the flow law, which is a very sensitive function of both. ?? 1977 Birkha??user Verlag.

Pure and Applied Geophysics PAGEOPH↗

USGS2012

Slope-Area Computation Program Graphical User Interface 1.0—A Preprocessing and Postprocessing Tool for Estimating Peak Flood Discharge Using the Slope-Area Method

The slope-area method is a technique for estimating the peak discharge of a flood after the water has receded (Dalrymple and Benson, 1967). This type of discharge estimate is called an “indirect measurement” because it relies on evidence left behind by the flood, such as high-water marks (HWMs) on trees or buildings. These indicators of flood stage are combined with measurements of the cross-sectional geometry of the stream, estimates of channel roughness, and a mathematical model that balances the total energy of the flow between cross sections. This is in contrast to a “direct” measurement of discharge during the flood where cross-sectional area is measured and a current meter or acoustic equipment is used to measure the water velocity. When a direct discharge measurement cannot be made at a gage during high flows because of logistics or safety reasons, an indirect measurement of a peak discharge is useful for defining the high-flow section of the stage-discharge relation (rating curve) at the stream gage, resulting in more accurate computation of high flows. The Slope-Area Computation program (SAC; Fulford, 1994) is an implementation of the slope-area method that computes a peak-discharge estimate from inputs of water-surface slope (from surveyed HWMs), channel geometry, and estimated channel roughness. SAC is a command line program written in Fortran that reads input data from a formatted text file and prints results to another formatted text file. Preparing the input file can be time-consuming and prone to errors. This document describes the SAC graphical user interface (GUI), a crossplatform “wrapper” application that prepares the SAC input file, executes the program, and helps the user interpret the output. The SAC GUI is an update and enhancement of the slope-area method (SAM; Hortness, 2004; Berenbrock, 1996), an earlier spreadsheet tool used to aid field personnel in the completion of a slope-area measurement. The SAC GUI reads survey data, develops a plan-view plot, water-surface profile, cross-section plots, and develops the SAC input file. The SAC GUI also develops HEC-2 files that can be imported into HEC–RAS.

Fact Sheet↗

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↗

USGS1980

Temperature and solute-transport simulation in streamflow using a Lagrangian reference frame

A computer program for simulating one-dimensional, unsteady temperature and solute transport in a river has been developed and documented for general use. The solution approach to the convective-diffusion equation uses a moving reference frame (Lagrangian) which greatly simplifies the mathematics of the solution procedure and dramatically reduces errors caused by numerical dispersion. The solution procedure has the further advantages, relative to conventional Eulerian solution schemes, of being easy to understand in the physical sense, of being extremely stable numerically, and of providing an accounting system which is very useful for model calibration. The model documentation is presented as a series of four programs of increasing complexity. The conservative transport model can be used to route a single conservative substance, such as dye, through a reach of a river. The simplified temperature model is used to predict water temperature in rivers, either with or without thermal loading, when few meteorological data are available. Only equilibrium temperature and windspeed are required. It is suggested that air temperature can be used to approximate equilibrium temperature. The complete temperature model is highly accurate but requires rather complete meteorological data. Finally, the 10-parameter model can be used to route as many as 10 interacting constituents through a river reach. The mathematical description of the interaction between the constituents, which does not need to be linear, is generally up to the user to supply. An example problem is solved for a three-parameter system involving temperature, dissolved oxygen, and biochemical oxygen demand. For simplicity, all models are developed and presented assuming steady non-uniform flow. Generalization of the models to allow unsteady flow is extremely simple, involving the addition of no more than 18 cards to the program deck. The report is concluded by describing this generalization for any of the models. Before using the models with unsteady flow, a flow model must be used to calculate and store the necessary flow data at each cross section and time step. Such a flow model is available and documented.

Water-Resources Investigations Report↗

USGS1989

A vertically averaged spectral model for tidal circulation in estuaries

A frequency dependent computer model based on the two-dimensional vertically averaged shallow-water equations is described for general purpose application in tidally dominated embayments. This model simulates the response of both tides and tidal currents to user-specified geometries and boundary conditions. The mathematical formulation and practical application of the model are discussed in detail. Salient features of the model include the ability to specify: (1) stage at the open boundaries as well as within the model grid, (2) velocities on open boundaries (river inflows and so forth), (3) spatially variable wind stress, and (4) spatially variable bottom friction. Using harmonically analyzed field data as boundary conditions, this model can be used to make real time predictions of tides and tidal currents. (USGS)

Water-Resources Investigations Report↗