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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

Projected effects of proposed chloride-control projects on shallow ground water– Preliminary results for the Wichita River basin, Texas

The U.S. Army Corps of Engineers' plan to control the natural chloride pollution in the Wichita River basin includes the construction of Truscott Brine Lake on a tributary of the North Wichita River. In connection with the proposed brine lake, the U.S. Geological Survey was requested to: (1) Define the existing ground-water conditions in the shallow fresh-water system of the project area; and (2) project the post-construction effects of the proposed lake on the fresh-water aquifer, especially in relation to hydraulic-head changes but also with respect to possible changes in the chemical quality of the ground water. The fresh-water aquifer in the project area is a shallow water-table system with relatively fresh water (generally a calcium sulfate type) that contains approximately 500 to 5,000 milligrams per liter of dissolved solids. The aquifer consists of Permian rocks with very small values of hydraulic conductivity and overlies a brine system (sodium chloride type) that is even less permeable. A thin transition zone separates the fresh-water aquifer and the brine system. Small quantities of infiltration from precipitation throughout the area's watershed constitute the recharge to the aquifer. Discharge from the aquifer consists of the small base flow along creeks; well discharge is negligible. Two-dimensional mathematical computer models were developed for aquifer simulation of: (1) Steady-state conditions in a fresh-water system and (2) transient conditions in a brine- fresh-water system where the density effects of the brine are considered. The main results 'of projecting the effects of the proposed Truscott Brine Lake on the fresh-water aquifer are: (1) Hydraulic head rises of 5 to 40 feet would be confined to areas near the proposed dam and along the lake shoreline, and (2) migration of salt water downstream from the dam generally would be limited to less than 1 mile and apparently would not reach equilibrium during the 100-year duration of the project. The modeling efforts did not include possible effects related to hydrodynamic dispersion in the brine- fresh-water system. Possible changes in the hydraulic conductivity of the aquifer, due to physical and chemical interactions in the brine and fresh-water environments, also were not considered.

Texas

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

On regional geomagnetic charts

When regional geomagnetic charts for areas roughly the size of the United States were compiled by hand, some large local anomalies were displayed in the isomagnetic lines. Since the late nineteen sixties, when the compilation of charts using computers and mathematical models was started, most of the details available in the hand drawn regional charts have been lost. One exception to this is the Canadian magnetic declination chart for 1980. This chart was constructed using a 180 degree spherical harmonic model. It managed to show considerable detail, but even more detail might be useful. Suggestions are made about how more detail might be displayed in regional charts when adequate data are available.

Journal of Geomagnetism & Geoelectricity

Water-discharge determinations for the tidal reach of the Willamette River from Ross Island Bridge to Mile 10.3, Portland, Oregon

Water-discharge, velocity, and slope variations for a 3.7-mile-Iong tidal reach of the Willamette River at Portland, Oreg., were defined from discharge measurements and river stage data collected between July 1962 and January 1965. Observed water discharge during tide-affected flows, during floods, and during backwater from the Columbia River and recorded stages at each end of the river reach were used to determine water discharge from two mathematical models. These models use a finite-difference method to solve the equations of moderately unsteady open-channel streamflow, and discharges are computed by an electronic digital computer. Discharges computed by using the mathematical models compare satisfactorily with observed discharges, except during the period of backwater from the annual flood of the Columbia River. The flow resistance coefficients used in the models vary with discharge; for one model, the coefficients for discharges above 30,000 cfs (cubic feet per second) are 12 and 24 percent less than the coefficient used for discharges below 30,000 cfs. Daily mean discharges were determined by use of one mathematical model for approximately two-thirds of the water year, October 1963 through September 1964. Agreement of computed with routed daily mean discharges is fair; above 30,000 cfs, average differences between the two discharges are about 10 percent, and below 30,000 cfs, computed daily discharges are consistently greater (by as much as 25 percent) than routed discharges. The other model was used to compute discharges for the unusually high flood flows of December 1964.

Water Supply Paper

Can machine learning accelerate process understanding and decision-relevant predictions of river water quality?

The global decline of water quality in rivers and streams has resulted in a pressing need to design new watershed management strategies. Water quality can be affected by multiple stressors including population growth, land use change, global warming, and extreme events, with repercussions on human and ecosystem health. A scientific understanding of factors affecting riverine water quality and predictions at local to regional scales, and at sub-daily to decadal timescales are needed for optimal management of watersheds and river basins. Here, we discuss how machine learning (ML) can enable development of more accurate, computationally tractable, and scalable models for analysis and predictions of river water quality. We review relevant state-of-the art applications of ML for water quality models and discuss opportunities to improve the use of ML for emerging computational and mathematical methods for model selection, hyperparameter optimization, incorporating process knowledge into ML models, improving explainablity, uncertainty quantification, and model-data integration. We then present considerations for using ML to address water quality problems given their scale and complexity, available data and computational resources, and stakeholder needs. When combined with decades of process understanding, interdisciplinary advances in knowledge-guided ML, information theory, data integration, and analytics can help address fundamental science questions and enable decision-relevant predictions of riverine water quality.

Hydrological Processes

One-dimensional steady-state stream water-quality model

A computer program, based on a one-dimensional mathematical model which predicts the stream water-quality response characteristics from waste source inputs, is described and documented. Variables predicted include dissolved oxygen, biochemical oxygen demand, nitrogen forms, total and fecal-coliform bacteria, orthophosphate-phosphorus, and various conservative substances. The model is based primarily on the well known Streeter-Phelps oxygen-sag equation. Special options of the program include the capability of handling nonpoint source waste inputs and anoxic conditions. The model formulation is based on a steady-state assumption which requires constant flow rate of waste and stream discharges and associated parameters. To achieve a problem solution, each reach of a stream system is broken into a given number of subreaches, generally defined by locations of waste or tributary inflow points. All waste constituents are assumed to be completely mixed within any cross section.

Water-Resources Investigations Report

Computing maximum-likelihood estimates for parameters of the National Descriptive Model of Mercury in Fish

This report presents the mathematical expressions and the computational techniques required to compute maximum-likelihood estimates for the parameters of the National Descriptive Model of Mercury in Fish (NDMMF), a statistical model used to predict the concentration of methylmercury in fish tissue. The expressions and techniques reported here were prepared to support the development of custom software capable of computing NDMMF parameter estimates more quickly and using less computer memory than is currently possible with available general-purpose statistical software. Computation of maximum-likelihood estimates for the NDMMF by numerical solution of a system of simultaneous equations through repeated Newton-Raphson iterations is described. This report explains the derivation of the mathematical expressions required for computational parameter estimation in sufficient detail to facilitate future derivations for any revised versions of the NDMMF that may be developed.

Open-File Report

A multiple-objective optimal exploration strategy

Exploration for natural resources is accomplished through partial sampling of extensive domains. Such imperfect knowledge is subject to sampling error. Complex systems of equations resulting from modelling based on the theory of correlated random fields are reduced to simple analytical expressions providing global indices of estimation variance. The indices are utilized by multiple objective decision criteria to find the best sampling strategies. The approach is not limited by geometric nature of the sampling, covers a wide range in spatial continuity and leads to a step-by-step procedure.

Mathematical and Computer Modelling

A simple approach to nonlinear estimation of physical systems

Recursive algorithms for estimating the states of nonlinear physical systems are developed. This requires some key hypotheses regarding the structure of the underlying processes. Members of this class of random processes have several desirable properties for the nonlinear estimation of random signals. An assumption is made about the form of the estimator, which may then take account of a wide range of applications. Under the above assumption, the estimation algorithm is mathematically suboptimal but effective and computationally attractive. It may be compared favorably to Taylor series-type filters, nonlinear filters which approximate the probability density by Edgeworth or Gram-Charlier series, as well as to conventional statistical linearization-type estimators. To link theory with practice, some numerical results for a simulated system are presented, in which the responses from the proposed and the extended Kalman algorithms are compared.

Mathematical and Computer Modelling

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

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

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

Simulating groundwater flow in karst aquifers with distributed parameter models—Comparison of porous-equivalent media and hybrid flow approaches

Understanding karst aquifers, for purposes of their management and protection, poses unique challenges. Karst aquifers are characterized by groundwater flow through conduits (tertiary porosity), and (or) layers with interconnected pores (secondary porosity) and through intergranular porosity (primary or matrix porosity). Since the late 1960s, advances have been made in the development of numerical computer codes and the use of mathematical model applications towards the understanding of dual (primary [matrix] and secondary [fractures and conduits]) porosity groundwater flow processes, as well as characterization and management of karst aquifers. The Floridan aquifer system (FAS) in Florida and parts of Alabama, Georgia, and South Carolina is composed of a thick sequence of predominantly carbonate rocks. Karst features are present over much of its area, especially in Florida where more than 30 first-magnitude springs occur, numerous sinkholes and submerged conduits have been mapped, and numerous circular lakes within sinkhole depressions are present. Different types of mathematical models have been applied for simulation of the FAS. Most of these models are distributed parameter models based on the assumption that, like a sponge, water flows through connected pores within the aquifer system and can be simulated with the same mathematical methods applied to flow through sand and gravel aquifers; these models are usually referred to as porous-equivalent media models. The partial differential equation solved for groundwater flow is the potential flow equation of fluid mechanics, which is used when flow is dominated by potential energy and has been applied for many fluid problems in which kinetic energy terms are dropped from the differential equation solved. In many groundwater model codes (basic MODFLOW), it is assumed that the water has a constant temperature and density and that flow is laminar, such that kinetic energy has minimal impact on flow. Some models have been developed that incorporate the submerged conduits as a one-dimensional pipe network within the aquifer rather than as discrete, extremely transmissive features in a porous-equivalent medium; these submerged conduit models are usually referred to as hybrid models and may include the capability to simulate both laminar and turbulent flow in the one-dimensional pipe network. Comparisons of the application of a porous-equivalent media model with and without turbulence (MODFLOW-Conduit Flow Process mode 2 and basic MODFLOW, respectively) and a hybrid (MODFLOW-Conduit Flow Process mode 1) model to the Woodville Karst Plain near Tallahassee, Florida, indicated that for annual, monthly, or seasonal average hydrologic conditions, all methods met calibration criteria (matched observed groundwater levels and average flows). Thus, the increased effort required, such as the collection of data on conduit location, to develop a hybrid model and its increased computational burden, is not necessary for simulation of average hydrologic conditions (non-laminar flow effects on simulated head and spring discharge were minimal). However, simulation of a large storm event in the Woodville Karst Plain with daily stress periods indicated that turbulence is important for matching daily springflow hydrographs. Thus, if matching streamflow hydrographs over a storm event is required, the simulation of non-laminar flow and the location of conduits are required. The main challenge in application of the methods and approaches for developing hybrid models relates to the difficulty of mapping conduit networks or having high-quality datasets to calibrate these models. Additionally, hybrid models have long simulation times, which can preclude the use of parameter estimation for calibration. Simulation of contaminant transport that does not account for preferential flow through conduits or extremely permeable zones in any approach is ill-advised. Simulation results in other karst aquifers or other parts of the FAS may differ from the comparison demonstrated herein.

Florida

BatTool: Projecting bat populations facing multiple stressors using a demographic model

Bats provide ecologically and agriculturally important ecosystem services but are currently experiencing population declines caused by multiple environmental stressors, including mortality from white-nose syndrome and wind energy development. Analyses of the current and future health and viability of these species may support conservation management decision making. Demographic modeling provides a quantitative tool for decision makers and conservation managers to make more informed decisions, but widespread adoption of these tools can be limited because of the complexity of the mathematical, statistical, and computational components involved in implementing these models. In this work, we provide an exposition of the BatTool R package, detailing the primary components of the matrix projection model, a publicly accessible graphical user interface ( https://rconnect.usgs.gov/battool ) facilitating user-defined scenario analyses, and its intended uses and limitations (Wiens et al., US Geol Surv Data Release 2022; Wiens et al., US Geol Surv Softw Release 2022). We present a case study involving wind energy permitting, weighing the effects of potential mortality caused by a hypothetical wind energy facility on the projected abundance of four imperiled bat species in the Midwestern United States.

Methods in Ecology and Evolution

A final report on computed magneto-telluric curves for hypothetical models of crustal structure

Several mathematical models were investigated to determine the capa-bilities of the magneto-telluric method for determining the resistivity structure of the earth's crust. The model parameters were based on the crust model proposed by Keller (1963). The mathematical technique used was developed by Cagniard (1953). The investigations indicate that a three-layer model approximation of the crust and mantle is the most detailed model warranted in inter-preting the information provided by the magneto-telluric method about the lower crust. Only the thickness of the lower crust can be determined, and not the resistivity.

Crustal Studies Technical Letters

A frictional population model of seismicity rate change

We study models of seismicity rate changes caused by the application of a static stress perturbation to a population of faults and discuss our results with respect to the model proposed by Dieterich (1994). These models assume distribution of nucleation sites (e.g., faults) obeying rate-state frictional relations that fail at constant rate under tectonic loading alone, and predicts a positive static stress step at time to will cause an immediate increased seismicity rate that decays according to Omori's law. We show one way in which the Dieterich model may be constructed from simple general idead, illustratted using numerically computed synthetic seismicity and mathematical formulation. We show that seismicity rate change predicted by these models (1) depend on the particular relationship between the clock-advanced failure and fault maturity, (2) are largest for the faults closest to failure at to, (3) depend strongly on which state evolution law faults obey, and (4) are insensitive to some types of population hetrogeneity. We also find that if individual faults fail repeatedly and populations are finite, at timescales much longer than typical aftershock durations, quiescence follows at seismicity rate increase regardless of the specific frictional relations. For the examined models the quiescence duration is comparable to the ratio of stress change to stressing rate ????/??,which occurs after a time comparable to the average recurrence interval of the individual faults in the population and repeats in the absence of any new load may pertubations; this simple model may partly explain observations of repeated clustering of earthquakes. Copyright 2005 by the American Geophysical Union.

Journal of Geophysical Research B: Solid Earth

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