A Bayesian stormwater quality model and its application to water quality monitoring

Stormwater runoff is a topic of research that over the years has increasingly grown due to its impact on our water resources. Treatment systems have been developed to mitigate this impact by preserving the pre-development hydrologic and water quality characteristics of the drainage areas. Understanding of the systems\u27 treatment capabilities is required for stormwater management. The goal of this research was to study the application of a decay treatment model as a conceptual tool for understanding the pollutant removal characteristics of stormwater systems. Three systems were studied in this research: a sand filter, a gravel wetland, and a retention pond. The contaminants under consideration include: total suspended solids (TSS), total petroleum hydrocarbons - diesel range hydrocarbons (TPH-D), dissolved inorganic nitrogen (DIN, comprised of nitrate, nitrite, and ammonia), and zinc (Zn). The mathematical model was based on the mass balance principle and the assumption that an n-order decay model describes the complex processes of pollutant removal (for example sedimentation, biodegradation, filtration, plant uptake, and chemical precipitation). The model was defined by the parameters of removal rate (k) and the decay order (n). For each treatment system, a collection of storm events was monitored between 2004 and 2006. Monitoring of the treatment systems was performed in a side by side fashion so that each system received the same stormwater quantity and quality. This configuration made possible a comparison of the calibrated parameters obtained for each system. The best set of parameters of the decay model was determined by using a simulated annealing technique as part of the optimization process. Monte Carlo simulations were performed to describe the uncertainty of the estimated effluent concentrations. The gravel wetland achieved the highest median DIN and TSS removal rates. For TPH-D, the highest median removal rates were achieved by the retention pond and gravel wetland. The sand filter and the gravel wetland achieved the highest median Zn removal rates. First and second order decay models were more likely to describe the observed effluent concentrations. A Bayesian statistical approach for determining parameter uncertainty of the stormwater treatment model is presented. For this model, it was found that a second order decay model was more likely to reproduce estimated effluent concentrations. Mean removal rate values were computed from the posterior distributions. Specifically, for the gravel wetland: kTSS = 59, kZn = 2115, kTPH-D = 88, kDIN = 7; for the sand filter: kTss = 1.7, kZn = 1568, kTPH-D = 57, kDIN = 2; and for the retention pond: kTss = 0.8, kZn = 4645, kTPH-D = 68, kDIN = 8 (k in units of (mg/l)-1/day)

A perturbative approach to the spectral zeta functions of strings, drums and quantum billiards

We have obtained an explicit expression for the spectral zeta functions and for the heat kernel of strings, drums and quantum billiards working to third order in perturbation theory, using a generalization of the binomial theorem to operators. The perturbative parameter used in the expansion is either the small deformation of a reference domain (for instance a square), or a small variation of the density around a constant value (in two dimensions both cases can apply). This expansion is well defined even in presence of degenerations of the unperturbed spectrum. We have discussed several examples in one, two and three dimensions, obtaining in some cases the analytic continuation of the series, which we have then used to evaluate the corresponding Casimir energy. For the case of a string with piecewise constant density, subject to different boundary conditions, and of two concentric cylinders of very close radii, we have reproduced results previously published, thus obtaining a useful check of our method.Comment: 23 pages, 5 figures, 2 tables; version accepted on Journal of Mathematical Physic

Scalar transport in compressible flow

Transport of scalar fields in compressible flow is investigated. The effective equations governing the transport at scales large compared to those of the advecting flow are derived by using multi-scale techniques. Ballistic transport generally takes place when both the solenoidal and the potential components of the velocity do not vanish, despite of the fact that it has zero average value. The calculation of the effective ballistic velocity $V_b$ is reduced to the solution of one auxiliary equation. An analytic expression for $V_b$ is derived in some special instances, i.e. flows depending on a single coordinate, random with short correlation times and slightly compressible cellular flow. The effective mean velocity $V_b$ vanishes for velocity fields which are either incompressible or potential and time-independent. For generic compressible flow, the most general conditions ensuring the absence of ballistic transport are isotropy and/or parity invariance. When $V_b$ vanishes (or in the frame of reference moving with velocity $V_b$), standard diffusive transport takes place. It is known that diffusion is always enhanced by incompressible flow. On the contrary, we show that diffusion is depleted in the presence of time-independent potential flow. Trapping effects due to potential wells are responsible for this depletion. For time-dependent potential flow or generic compressible flow, transport rates are enhanced or depleted depending on the detailed structure of the velocity field.Comment: 27 pages, submitted to Physica

Optimal hedging of Derivatives with transaction costs

We investigate the optimal strategy over a finite time horizon for a portfolio of stock and bond and a derivative in an multiplicative Markovian market model with transaction costs (friction). The optimization problem is solved by a Hamilton-Bellman-Jacobi equation, which by the verification theorem has well-behaved solutions if certain conditions on a potential are satisfied. In the case at hand, these conditions simply imply arbitrage-free ("Black-Scholes") pricing of the derivative. While pricing is hence not changed by friction allow a portfolio to fluctuate around a delta hedge. In the limit of weak friction, we determine the optimal control to essentially be of two parts: a strong control, which tries to bring the stock-and-derivative portfolio towards a Black-Scholes delta hedge; and a weak control, which moves the portfolio by adding or subtracting a Black-Scholes hedge. For simplicity we assume growth-optimal investment criteria and quadratic friction.Comment: Revised version, expanded introduction and references 17 pages, submitted to International Journal of Theoretical and Applied Finance (IJTAF

Convergence Rates in L^2 for Elliptic Homogenization Problems

We study rates of convergence of solutions in L^2 and H^{1/2} for a family of elliptic systems {L_\epsilon} with rapidly oscillating oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L_\epsilon}. Most of our results, which rely on the recently established uniform estimates for the L^2 Dirichlet and Neumann problems in \cite{12,13}, are new even for smooth domains.Comment: 25 page

Reactive Turbulent Flow in Low-Dimensional, Disordered Media

We analyze the reactions $A+A \to \emptyset$ and $A + B \to \emptyset$ occurring in a model of turbulent flow in two dimensions. We find the reactant concentrations at long times, using a field-theoretic renormalization group analysis. We find a variety of interesting behavior, including, in the presence of potential disorder, decay rates faster than that for well-mixed reactions.Comment: 6 pages, 4 figures. To appear in Phys. Rev.

Exact solution of the one-dimensional ballistic aggregation

An exact expression for the mass distribution $\rho(M,t)$ of the ballistic aggregation model in one dimension is derived in the long time regime. It is shown that it obeys scaling $\rho(M,t)=t^{-4/3}F(M/t^{2/3})$ with a scaling function $F(z)\sim z^{-1/2}$ for $z\ll 1$ and $F(z)\sim \exp(-z^3/12)$ for $z\gg 1$. Relevance of these results to Burgers turbulence is discussed.Comment: 11 pages, 2 Postscript figure