Nicholas Dudley Ward

Robust identification in the disc algebra using rational wavelets and orthonormal basis functions

Worst-case identification using corrupted frequency response measurements is considered in a framework which allows one to consider a variety of rational basis functions; these include Laguerre models, Kautz models and rational wavelets. Conditions are given for model sets to be complete in the disc algebra and examples are given. Explicit error bounds are provided for approximate modelling using specific basis functions.

A Construction of Rational Wavelets and Frames in Hardy--Sobolev Spaces With Applications to System Modeling

Using the Daubechies wavelet theory we establish rational wavelet decompositions of the Hardy--Sobolev classes on the half-plane. The decay of wavelet coefficients is analyzed and error bounds for approximation are given. We give applications to the modeling of linear systems and to the model reduction of infinite-dimensional systems.

Setting Risk‐Informed Environmental Standards for Bacillus Anthracis Spores

In many cases, human health risk from biological agents is associated with aerosol exposures. Because air concentrations decline rapidly after a release, it may be necessary to use concentrations found in other environmental media to infer future or past aerosol exposures. This article presents an approach for linking environmental concentrations of Bacillus. anthracis (B. anthracis) spores on walls, floors, ventilation system filters, and in human nasal passages with human health risk from exposure to B. anthracis spores. This approach is then used to calculate example values of risk-informed concentration standards for both retrospective risk mitigation (e.g., prophylactic antibiotics) and prospective risk mitigation (e.g., environmental clean up and reoccupancy). A large number of assumptions are required to calculate these values, and the resulting values have large uncertainties associated with them. The values calculated here suggest that documenting compliance with risks in the range of 10(-4) to 10(-6) would be challenging for small diameter (respirable) spore particles. For less stringent risk targets and for releases of larger diameter particles (which are less respirable and hence less hazardous), environmental sampling would be more promising.

Rational wavelet decompositions of transfer functions in hardy-sobolev classes

The model approximation of transfer functions using rational wavelets (or molecular decompositions) is considered. By using techniques from Hardy-Sobolev spaces it is shown that Hilbert space methods such as a modified matching-pursuit algorithm and least-squares technique can be employed to obtain good approximations in bothH2 andH∞ norms. Several theoretical results are given on rates of convergence when the methods are applied to delay systems and fractional filters.

Stream Depletion from Pumping a Semiconfined Aquifer in a Two-Layer Leaky Aquifer System

We consider the case of pumping a semiconfined aquifer overlain by a phreatic aquifer hydraulically connected to an infinitely long and straight stream. The hydraulics of the system are described by a linearized model in which it is assumed that flow in the aquifers is essentially horizontal, and that drawdown in the phreatic aquifer is small enough to allow the water table to be sufficiently well described by the Theis equation. Fourier-Laplace transforms of drawdown are obtained and numerically inverted. Comparisons are made with another well-known analytic model of Hunt, who considered a single aquifer/aquitard system. It is observed that while both models may have excellent agreement during the initial phase of pumping, the longer-term consequences of sustained pumping are quite distinct.

Identification of Aquifer Parameters from Pumping Test Data with Regard for Uncertainty

AbstractWhen fitting hydraulic models of groundwater flow to pumping test data, Bayesian inference provides a framework for quantifying the posterior uncertainty of aquifer parameters estimated from data and the most likely range of parameters that are consistent with the data. In this study, noise-perturbed drawdown data is measured. For clarity, groundwater models with few parameters are considered and Markov chain Monte Carlo is used to quantify uncertainty of transmissivity, storativity, and leakage parameters. These models exhibit many of the features typically encountered in much higher dimensional computational groundwater models like multimodality, failure of least squares algorithms, and poorly determined parameters. For comparison, Bayesian inference is contrasted with least squares model fitting.

Simulation of a Multilayer Leaky Aquifer with Stream Depletion

AbstractSemianalytic formulas are obtained for flow to a well screened in a leaky aquifer that was overlain by an aquitard and phreatic aquifer and underlain by a second aquitard and leaky aquifer. Formulas are also obtained for the scenario in which the phreatic aquifer is hydraulically connected to a rectilinear stream. Adaptive mesh refinement is used to obtain highly resolved simulations for a meandering stream depletion problem that is both numerically challenging and has physically interesting features. The associated inverse problem is also considered, and a simple example is presented of synthetic pumping test data, which demonstrates that it can be very difficult to quantify the actual effects of pumping. The conclusion is that a framework for uncertainty quantification of aquifer parameters is necessary for objective determination of the parameters.

Estimation of aquifer dimensions from passive seismic signals with approximate wave propagation models

Recently, it has been proposed that spontaneous seismic activity could be used in the estimation of hydrological parameters of aquifers such as permeability and storage. Approximate wave propagation models such as ray tracing, which are commonly used in hydrological parameter estimation with active sources and backscattering geometry, are not feasible with passive seismological imaging. With respect to full wave propagation models, the most accurate known model for aquifers is the poroelastic model while bedrock is usually modelled as an elastic medium. Using a poroelastic model in the forward model can be a computationally impractical choice. In this paper, we carry out a feasibility study in which we attempt to estimate the aquifer depth and water table using a highly approximate elastic model also for the aquifer. We adopt the Bayesian approximation error approach in which a statistical model is constructed for the errors that are induced by using model approximations such as sparse meshing and simplified physical models. We consider the problem in a simple two-dimensional geometry and show that straightforward adoption of approximate models leads to inconsistent parameter estimates, that is, the true parameters have essentially vanishing posterior density. On the other hand, using the Bayesian approximation error approach, the parameter estimates are consistent.

Distribution of stream depletion.

In this paper, we consider Hunts stream depletion model, which models the effect of pumping adjacent to an infinitely long stream. In this model it is assumed that drawdown is described by an extension of the Theis equation. We derive a formula for the distribution of stream depletion and comment on the use of this formula in pump test design.

Uncertainty Quantification for Stream Depletion Tests

AbstractThis study considers the problem of quantifying stream depletion from pumping test data. Bayesian inference is used to quantify the posterior uncertainty of parameters for a simple vertically heterogeneous aquifer model, in which the pumped semiconfined aquifer is separated by an aquiclude from a phreatic aquifer hydraulically connected to a stream. This study investigates the effects of using different data sets and shows that a single pumping test is generally not sufficient to determine stream depletion within reasonable limits. However, uncertainty quantification conducted within a Bayesian context reveals that by judicious design of aquifer tests, stream depletion can be accurately determined from data.