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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↗

USGS1979

Mathematical model for simulating discharges on the Sabine River between Tatum and Ruliff, Texas

A mathematical model for simulating discharges on the Sabine River between Tatum and Ruliff, TX., was developed to evaluate the effects of release schedules on discharges from the Toledo Bend Reservoir compared to discharges under natural conditions. Using the discharge at Tatum, TX., the rainfall over the basin, and the discharge release schedule for the reservoir, discharge hydrographs for the natural and reservoir-controlled conditions can be computed.

Texas↗

USGS1994

Data requirements for simulation of hydrogeologic effects of liquid waste injection, Harrison and Jackson Counties, Mississippi

Available literature and data were reviewed to quantify data requirements for computer simulation of hydrogeologic effects of liquid waste injection in southeastern Mississippi. Emphasis of each review was placed on quantifying physical properties of current Class I injection zones in Harrison and Jackson Counties. Class I injection zones are zones that are used for injection of hazardous or non-hazardous liquid waste below a formation containing the lowermost underground source of drinking water located within one-quarter of a mile of the injection well. Several mathematical models have been developed to simulate injection effects. The Basic Plume Method was selected because it is commonly used in permit applications, and the Intercomp model was selected because it is generally accepted and used in injection-related research. The input data requirements of the two models were combined into a single data requirement list inclusive of physical properties of injection zones only; injected waste and well properties are not included because such information is site-specific by industry, which is beyond the scope of this report. Results of the reviews of available literature and data indicated that Class I permit applications and standard-reference chemistry and physics texts were the primary sources of information to quantify physical properties of injection zones in Harrison and Jackson Counties. With the exception of a few reports and supplementary data for one injection zone in Jackson County, very little additional information pertaining to physical properties of the injection zones was available in sources other than permit applications and standard-reference texts.

Mississippi↗

USGS2019

Analog experiments of lava flow emplacement

Laboratory experiments that simulate lava flows have been in use by volcanologists for many years. The behavior of flows in the lab, where “eruption” parameters, material properties, and environmental settings are tightly controlled, provides insight into the influence of various factors on flow evolution. A second benefit of laboratory lava flows is to provide a set of observations with which numerical models of flow emplacement can be tested. Models of lava flow emplacement vary in mathematical approach, physical assumptions, and computational cost. Nonetheless, all models require thorough testing and evaluation, and laboratory experiments produce an excellent test for models. This paper provides a primer on modern analog laboratory lava flow experiments. It reviews scaling con- siderations and provides quantitative information meant to guide future experimentalists in designing their experiments to be relevant to natural processes. Traditional and novel laboratory techniques are described, including a discussion of current limitations. New insights from recent experiments highlight the impact of topographic conditions and highlight the importance of considering bed roughness, major obstacles, and slope breaks. The influence of episodic or non-uniform effusion rate is demonstrated through recent experi- mental works. Lastly, the paper discusses several open questions about lava flow emplacement and the ways in which future improvements in experimental methods, such as the ability to utilize three-phase suspensions and materials with complex rheologies and to image the interior of flows could help answer these.

Annals of Geophysics↗

USGS2021

Computational methodology to analyze the effect of mass transfer rate on attenuation of leaked carbon dioxide in shallow aquifers

Exsolution and re-dissolution of CO 2 gas within heterogeneous porous media are investigated using experimental data and mathematical modeling. In a set of bench-scale experiments, water saturated with CO 2 under a given pressure is injected into a 2-D water-saturated porous media system, causing CO 2 gas to exsolve and migrate upwards. A layer of fine sand mimicking a heterogeneity within a shallow aquifer is present in the tank to study accumulation and trapping of exsolved CO 2 . Then, clean water is injected into the system and the accumulated CO 2 dissolves back into the flowing water. Simulated exsolution and dissolution mass transfer processes are studied using both nearequilibrium and kinetic approaches and compared to experimental data under conditions that do and do not include lateral background water flow. The mathematical model is based on the mixed hybrid finite element method that allows for accurate simulation of both advection- and diffusion- dominated processes.

Acta Polytechnica↗

USGS2015

PRMS-IV, the precipitation-runoff modeling system, version 4

Computer models that simulate the hydrologic cycle at a watershed scale facilitate assessment of variability in climate, biota, geology, and human activities on water availability and flow. This report describes an updated version of the Precipitation-Runoff Modeling System. The Precipitation-Runoff Modeling System is a deterministic, distributed-parameter, physical-process-based modeling system developed to evaluate the response of various combinations of climate and land use on streamflow and general watershed hydrology. Several new model components were developed, and all existing components were updated, to enhance performance and supportability. This report describes the history, application, concepts, organization, and mathematical formulation of the Precipitation-Runoff Modeling System and its model components. This updated version provides improvements in (1) system flexibility for integrated science, (2) verification of conservation of water during simulation, (3) methods for spatial distribution of climate boundary conditions, and (4) methods for simulation of soil-water flow and storage.

Techniques and Methods↗

USGS2017

Documentation for the “XT3D” option in the Node Property Flow (NPF) Package of MODFLOW 6

This report describes the “XT3D” option in the Node Property Flow (NPF) Package of MODFLOW 6. The XT3D option extends the capabilities of MODFLOW by enabling simulation of fully three-dimensional anisotropy on regular or irregular grids in a way that properly takes into account the full, three-dimensional conductivity tensor. It can also improve the accuracy of groundwater-flow simulations in cases in which the model grid violates certain geometric requirements. Three example problems demonstrate the use of the XT3D option to simulate groundwater flow on irregular grids and through three-dimensional porous media with anisotropic hydraulic conductivity. Conceptually, the XT3D method of estimating flow between two MODFLOW 6 model cells can be viewed in terms of three main mathematical steps: construction of head-gradient estimates by interpolation; construction of fluid-flux estimates by application of the full, three-dimensional form of Darcy’s Law, in which the conductivity tensor can be heterogeneous and anisotropic; and construction of the flow expression by enforcement of continuity of flow across the cell interface. The resulting XT3D flow expression, which relates the flow across the cell interface to the values of heads computed at neighboring nodes, is the sum of terms in which conductance-like coefficients multiply head differences, as in the conductance-based flow expression the NPF Package uses by default. However, the XT3D flow expression contains terms that involve “neighbors of neighbors” of the two cells for which the flow is being calculated. These additional terms have no analog in the conductance-based formulation. When assembled into matrix form, the XT3D formulation results in a larger stencil than the conductance-based formulation; that is, each row of the coefficient matrix generally contains more nonzero elements. The “RHS” suboption can be used to avoid expanding the stencil by placing the additional terms on the right-hand side of the matrix equation and evaluating them at the previous iteration or time step. The XT3D option can be an alternative to the Ghost-Node Correction (GNC) Package. However, the XT3D formulation is typically more computationally intensive than the conductance-based formulation the NPF Package uses by default, either with or without ghost nodes. Before deciding whether to use the GNC Package or XT3D option for production runs, the user should consider whether the conductance-based formulation alone can provide acceptable accuracy for the particular problem being solved.

Techniques and Methods↗

USGS2004

Use of RORA for Complex Ground-Water Flow Conditions

The RORA computer program for estimating recharge is based on a condition in which ground water flows perpendicular to the nearest stream that receives ground-water discharge. The method, therefore, does not explicitly account for the ground-water-flow component that is parallel to the stream. Hypothetical finite-difference simulations are used to demonstrate effects of complex flow conditions that consist of two components: one that is perpendicular to the stream and one that is parallel to the stream. Results of the simulations indicate that the RORA program can be used if certain constraints are applied in the estimation of the recession index, an input variable to the program. These constraints apply to a mathematical formulation based on aquifer properties, recession of ground-water levels, and recession of streamflow.

Water-Resources Investigations Report↗

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↗

USGS2008

SEAWAT Version 4: A Computer Program for Simulation of Multi-Species Solute and Heat Transport

The SEAWAT program is a coupled version of MODFLOW and MT3DMS designed to simulate three-dimensional, variable-density, saturated ground-water flow. Flexible equations were added to the program to allow fluid density to be calculated as a function of one or more MT3DMS species. Fluid density may also be calculated as a function of fluid pressure. The effect of fluid viscosity variations on ground-water flow was included as an option. Fluid viscosity can be calculated as a function of one or more MT3DMS species, and the program includes additional functions for representing the dependence on temperature. Although MT3DMS and SEAWAT are not explicitly designed to simulate heat transport, temperature can be simulated as one of the species by entering appropriate transport coefficients. For example, the process of heat conduction is mathematically analogous to Fickian diffusion. Heat conduction can be represented in SEAWAT by assigning a thermal diffusivity for the temperature species (instead of a molecular diffusion coefficient for a solute species). Heat exchange with the solid matrix can be treated in a similar manner by using the mathematically equivalent process of solute sorption. By combining flexible equations for fluid density and viscosity with multi-species transport, SEAWAT Version 4 represents variable-density ground-water flow coupled with multi-species solute and heat transport. SEAWAT Version 4 is based on MODFLOW-2000 and MT3DMS and retains all of the functionality of SEAWAT-2000. SEAWAT Version 4 also supports new simulation options for coupling flow and transport, and for representing constant-head boundaries. In previous versions of SEAWAT, the flow equation was solved for every transport timestep, regardless of whether or not there was a large change in fluid density. A new option was implemented in SEAWAT Version 4 that allows users to control how often the flow field is updated. New options were also implemented for representing constant-head boundaries with the Time-Variant Constant-Head (CHD) Package. These options allow for increased flexibility when using CHD flow boundaries with the zero-dispersive flux solute boundaries implemented by MT3DMS at constant-head cells. This report contains revised input instructions for the MT3DMS Dispersion (DSP) Package, Variable-Density Flow (VDF) Package, Viscosity (VSC) Package, and CHD Package. The report concludes with seven cases of an example problem designed to highlight many of the new features.

Techniques and Methods↗

USGS1983

Ground-water models for water resource planning

In the past decade hydrogeologists have emphasized the development of computer-based mathematical models to aid in the understanding of flow, the transport of solutes, transport of heat, and deformation in the ground-water system. These models have been used to provide information and predictions for water managers. Too frequently, ground-water was neglected in water resource planning because managers believed that it could not be adequately evaluated in terms of availability, quality, and effect of development on surface-water supplies. Now, however, with newly developed digital ground-water models, effects of development can be predicted. Such models have been used to predict hydrologic and quality changes under different stresses. These models have grown in complexity over the last ten years from simple one-layer models to three-dimensional simulations of ground-water flow, which may include solute transport, heat transport, effects of land subsidence, and encroachment of saltwater. Case histories illustrate how predictive ground-water models have provided the information needed for the sound planning and management of water resources in the USA. ?? 1983 D. Reidel Publishing Company.

GeoJournal↗

USGS1972

Hydrograph simulation models of the Hillsborough and Alafia Rivers, Florida: a preliminary report

Mathematical (digital) models that simulate flood hydrographs from rainfall records have been developed for the following gaging stations in the Hillsborough and Alafia River basins of west-central Florida: Hillsborough River near Tampa, Alafia River at Lithia, and north Prong Alafia River near Keysville. These models, which were developed from historical streamflow and and rainfall records, are based on rainfall-runoff and unit-hydrograph procedures involving an arbitrary separation of the flood hydrograph. These models assume the flood hydrograph to be composed of only two flow components, direct (storm) runoff, and base flow. Expressions describing these two flow components are derived from streamflow and rainfall records and are combined analytically to form algorithms (models), which are programmed for processing on a digital computing system. Most Hillsborough and Alafia River flood discharges can be simulated with expected relative errors less than or equal to 30 percent and flood peaks can be simulated with average relative errors less than 15 percent. Because of the inadequate rainfall network that is used in obtaining input data for the North Prong Alafia River model, simulated peaks are frequently in error by more than 40 percent, particularly for storms having highly variable areal rainfall distribution. Simulation errors are the result of rainfall sample errors and, to a lesser extent, model inadequacy. Data errors associated with the determination of mean basin precipitation are the result of the small number and poor areal distribution of rainfall stations available for use in the study. Model inadequacy, however, is attributed to the basic underlying theory, particularly the rainfall-runoff relation. These models broaden and enhance existing water-management capabilities within these basins by allowing the establishment and implementation of programs providing for continued development in these areas. Specifically, the models serve not only as a basis for forecasting floods, but also for simulating hydrologic information needed in flood-plain mapping and delineating and evaluating alternative flood control and abatement plans.

Florida↗

USGS1987

Documentation of a deep percolation model for estimating ground-water recharge

A deep percolation model, which operates on a daily basis, was developed to estimate long-term average groundwater recharge from precipitation. It has been designed primarily to simulate recharge in large areas with variable weather, soils, and land uses, but it can also be used at any scale. The physical and mathematical concepts of the deep percolation model, its subroutines and data requirements, and input data sequence and formats are documented. The physical processes simulated are soil moisture accumulation, evaporation from bare soil, plant transpiration, surface water runoff, snow accumulation and melt, and accumulation and evaporation of intercepted precipitation. The minimum data sets for the operation of the model are daily values of precipitation and maximum and minimum air temperature, soil thickness and available water capacity, soil texture, and land use. Long-term average annual precipitation, actual daily stream discharge, monthly estimates of base flow, Soil Conservation Service surface runoff curve numbers, land surface altitude-slope-aspect, and temperature lapse rates are optional. The program is written in the FORTRAN 77 language with no enhancements and should run on most computer systems without modifications. Documentation has been prepared so that program modifications may be made for inclusions of additional physical processes or deletion of ones not considered important. (Author 's abstract)

Open-File Report↗

USGS1989

Conversion and comparison of the mathematical, three-dimensional, finite-difference, ground-water flow model to the modular, three-dimensional, finite-difference, ground-water flow model for the Tesuque aquifer system in northern New Mexico

The three-dimensional finite-difference groundwater model (using a mathematical groundwater flow code) of the Tesuque aquifer system in northern New Mexico was converted to run using the U.S. Geological Survey 's modular groundwater flow code. Results from the final versions of the predevelopment and 1947 to 2080 transient simulations of the two models are compared. A correlation coefficient of 0.9905 was obtained for the match in block-by-block head-dependent fluxes for predevelopment conditions. There are, however, significant differences in at least two specific cases. In the first case, a difference is associated with the net loss from the Pojoaque River and its tributaries to the aquifer. The net loss by the river is given as 1.134 cu ft/sec using the original groundwater model, which is 38.1% less than the net loss by the river of 1.8319 cu ft/sec computed in this study. In the second case, the large difference is computed for the transient decline in the hydraulic head of a model block near Tesuque Pueblo. The hydraulic-head decline by 2080 is, using the original model, 249 ft, which is 14.7% less than the hydraulic head of 292 ft computed by this study. In general, the differences between the two sets of results are not large enough to lead to different conclusions regarding the behavior of the system at steady state or when pumped. (USGS)

Open-File Report↗

USGS2023

Calibration of imperfect geophysical models by multiple satellite interferograms with measurement bias

Model calibration consists of using experimental or field data to estimate the unknown parameters of a mathematical model. The presence of model discrepancy and measurement bias in the data complicates this task. Satellite interferograms, for instance, are widely used for calibrating geophysical models in geological hazard quantification. In this work, we used satellite interferograms to relate ground deformation observations to the properties of the magma chamber at K i¯lauea Volcano in Hawai‘i. We derived closed-form marginal likelihoods and implemented posterior sampling procedures that simultaneously estimate the model discrepancy of physical models, and the measurement bias from the atmospheric error in satellite interferograms. We found that model calibration by aggregating multiple interferograms and downsampling the pixels in the interferograms can reduce the computation complexity compared to calibration approaches based on multiple data sets. The conditions that lead to no loss of information from data aggregation and downsampling are studied. Simulation illustrates that both discrepancy and measurement bias can be estimated, and real applications demonstrate that modeling both effects helps obtain a reliable estimation of a physical model’s unobserved parameters and enhance its predictive accuracy. We implement the computational tools in the RobustCalibration package available on CRAN.

Technometrics↗

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↗

USGS2004

Section 1. Simulation of surface-water integrated flow and transport in two-dimensions: SWIFT2D user's manual

A numerical model for simulation of surface-water integrated flow and transport in two (horizontal-space) dimensions is documented. The model solves vertically integrated forms of the equations of mass and momentum conservation and solute transport equations for heat, salt, and constituent fluxes. An equation of state for salt balance directly couples solution of the hydrodynamic and transport equations to account for the horizontal density gradient effects of salt concentrations on flow. The model can be used to simulate the hydrodynamics, transport, and water quality of well-mixed bodies of water, such as estuaries, coastal seas, harbors, lakes, rivers, and inland waterways. The finite-difference model can be applied to geographical areas bounded by any combination of closed land or open water boundaries. The simulation program accounts for sources of internal discharges (such as tributary rivers or hydraulic outfalls), tidal flats, islands, dams, and movable flow barriers or sluices. Water-quality computations can treat reactive and (or) conservative constituents simultaneously. Input requirements include bathymetric and topographic data defining land-surface elevations, time-varying water level or flow conditions at open boundaries, and hydraulic coefficients. Optional input includes the geometry of hydraulic barriers and constituent concentrations at open boundaries. Time-dependent water level, flow, and constituent-concentration data are required for model calibration and verification. Model output consists of printed reports and digital files of numerical results in forms suitable for postprocessing by graphical software programs and (or) scientific visualization packages. The model is compatible with most mainframe, workstation, mini- and micro-computer operating systems and FORTRAN compilers. This report defines the mathematical formulation and computational features of the model, explains the solution technique and related model constraints, describes the model framework, documents the type and format of inputs required, and identifies the type and format of output available.

Techniques and Methods↗

USGS1989

Response of selected binomial coefficients to varying degrees of matrix sparseness and to matrices with known data interrelationships

Numerous departures from ideal relationships are revealed by Monte Carlo simulations of widely accepted binomial coefficients. For example, simulations incorporating varying levels of matrix sparseness (presence of zeros indicating lack of data) and computation of expected values reveal that not only are all common coefficients influenced by zero data, but also that some coefficients do not discriminate between sparse or dense matrices (few zero data). Such coefficients computationally merge mutually shared and mutually absent information and do not exploit all the information incorporated within the standard 2 ?? 2 contingency table; therefore, the commonly used formulae for such coefficients are more complicated than the actual range of values produced. Other coefficients do differentiate between mutual presences and absences; however, a number of these coefficients do not demonstrate a linear relationship to matrix sparseness. Finally, simulations using nonrandom matrices with known degrees of row-by-row similarities signify that several coefficients either do not display a reasonable range of values or are nonlinear with respect to known relationships within the data. Analyses with nonrandom matrices yield clues as to the utility of certain coefficients for specific applications. For example, coefficients such as Jaccard, Dice, and Baroni-Urbani and Buser are useful if correction of sparseness is desired, whereas the Russell-Rao coefficient is useful when sparseness correction is not desired. ?? 1989 International Association for Mathematical Geology.

Mathematical Geology↗