An analytical model for air pollutant transport and deposition from a point source

Ermak, D.L.

doi:10.1016/0004-6981(77)90140-8

Cited by 105 publications

(59 citation statements)

References 14 publications

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“…Generalize (3.23) to the case where the diffusivities K y (x) and K z (x) are not equal. This case is also considered by Ermak and an outline of his derivation can be found in the appendix of [10].…”

Section: Other Generalizationsmentioning

confidence: 99%

“…An equivalent formulation of this problem can be found by eliminating the source term from the PDE and instead introducing a delta function term into the boundary condition [10]:…”

Section: S( X) = Q δ(X) δ(Y) δ(Z −mentioning

confidence: 99%

“…The earliest analytical solution to the Gaussian plume equations with deposition was derived in [36]; however, Ermak [10] was the first to consider pollutant dispersion with both deposition and settling. Ermak applied Laplace transform methods to (3.21), (2.5d), and (3.22) and obtained the solution…”

Section: Other Generalizationsmentioning

confidence: 99%

“…Find Ermak's paper in your library, and show that his formula for the contaminant concentration (Eq. (5) from [10] c(r, y, z…”

Section: Other Generalizationsmentioning

confidence: 99%

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The Mathematics of Atmospheric Dispersion Modeling

Stockie¹

2011

SIAM Rev.

284

161

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Abstract. The Gaussian plume model is a standard approach for studying the transport of airborne contaminants due to turbulent diffusion and advection by the wind. This paper reviews the assumptions underlying the model, its derivation from the advection-diffusion equation, and the key properties of the plume solution. The results are then applied to solving an inverse problem in which emission source rates are determined from a given set of groundlevel contaminant measurements. This source identification problem can be formulated as an overdetermined linear system of equations that is most easily solved using the method of least squares. Various generalizations of this problem are discussed, and we illustrate our results with an application to the study of zinc emissions from a smelting operation.

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Section: Other Generalizationsmentioning

confidence: 99%

“…An equivalent formulation of this problem can be found by eliminating the source term from the PDE and instead introducing a delta function term into the boundary condition [10]:…”

Section: S( X) = Q δ(X) δ(Y) δ(Z −mentioning

confidence: 99%

Section: Other Generalizationsmentioning

confidence: 99%

“…Find Ermak's paper in your library, and show that his formula for the contaminant concentration (Eq. (5) from [10] c(r, y, z…”

Section: Other Generalizationsmentioning

confidence: 99%

See 2 more Smart Citations

The Mathematics of Atmospheric Dispersion Modeling

Stockie¹

2011

SIAM Rev.

284

161

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“…∂t where t is time, and the diffusion coefficients, K x,y,z , represent turbulent (eddy) diffusivity rather than molecular diffusion Assuming the source strength, Q, as a forcing term, a solution to this equation in a cartesian coordinate system was provided by Ermak et al [20] C(x, y, z)…”

Section: Gaussian Plume Modelmentioning

confidence: 99%

Greenhouse Gas Emissions and Dispersion 3. Reducing Uncertainty in Estimating Source Strength and Location through Plume Inversion Models

Prasad¹,

Pintar²,

Hu³

et al. 2015

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Recent development of accurate instruments for measuring greenhouse gas concentrations and the ability to mount them in ground-based vehicles has provided an opportunity to make temporally and spatially resolved measurements in the vicinity of suspected source locations, and for subsequently estimating the source location and strength. The basic approach of us ing downwind atmospheric measurements in an inversion methodology to predict the source strength and location is an ill-posed problem and results in large uncertainty. In this report, we present a new measurement methodology for reducing the uncertainty in predicting source strength from downwind measurements associated with inverse modeling. In order to demon strate the approach, an inversion methodology built around a plume dispersion model is de veloped. Synthetic data derived from an assumed source distribution is used to compare and contrast the predicted source strength and location. The effect of introducing various levels of noise in the synthetic data or uncertainty in meteorological variables on the inversion method ology is studied. Results indicate that the use of noisy measurement data had a small effect on the total predicted source strength, but gave rise to several spurious sources (in many cases 8-10 sources were detected, while the assumed source distribution only consisted of 2 sources). Use of noisy measurement data for inversion also introduced large uncertainty in the location of the predicted sources. A mathematical model for estimating an upper bound on the un certainty, and a bootstrap statistical approach for determining the variability in the predicted source distribution is demonstrated. The new measurement methodology, which involves using measurement data from two or more wind directions, combined together as part of a single inversion process is presented. Results of the bootstrap process indicated that the uncertainty in locating sources reduced significantly when measurements are made using the new proposed measurement approach. The proposed measurement system can be significant in determining emission inventories in urban domains at a high level of reliability, and for studying the role of remediation measures.

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‘Bayesian source detection and parameter estimation of a plume model based on sensor network measurements’ by C. Huang et al.: Discussion 2

Holan

Wikle

2010

Appl Stoch Models Bus & Ind

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Bayesian source detection and parameter estimation of a plume model based on sensor network measurements' by C. Huang et al.: Discussion 2The problem of source detection and parameter estimation for plume models based on sensor network measurements is timely and important. The authors are to be congratulated for going beyond the product-form model to a model motivated by an advection-diffusion PDE model. From a historical perspective, plumes have often been modeled using analytic solutions to various diffusion PDEs, leading to such formulations as the so-called Gaussian plume model (see Ermak [1], Lushi and Stokie [2], and the references therein). Relatively few efforts (by comparison) have been made to 'fit' these models from a rigorous statistical perspective and to perform statistical inference. Therefore, we believe that the authors' contribution is also rooted in raising the awareness of the statistics community to the problem of rigorously modeling plumes in the presence of uncertainty.In spite of the authors' careful and detailed coverage of the problem, there are a few places we believe that more extensive treatment/exposition might be illuminating. First, a more comprehensive simulation study would be informative. In this direction, there would be several avenues for extending the simulation (e.g. varying the spatial covariance, the design, and examining model mis-specification, among others). In the absence of a real data example, the simulations become critically important. In this case, it is unclear how effective the model will perform when applied to real sensor network data. In future research (by the authors and/or other researchers) we look forward to seeing the authors' model applied in practice.The authors suggest using an approximate (quasi-) likelihood, instead of the true likelihood, that may prove to be useful in extremely high dimensions. Although the simulation example presented does not necessitate such an approximation, real-world applications might require such approximations for real-time implementation. The approach, taken by the authors, assumes independent measurement errors and, as a result, produces an MCMC algorithm in which samples do not come from the 'target' distribution. Instead, samples are taken from a, potentially biased, approximation to the target distribution. Although this technique may be preferable from a purely computational efficiency perspective, it is philosophically appealing (and potentially more accurate) to use the exact likelihood. One possible alternative, that can be viewed as a compromise between the authors' approach and using the exact likelihood, would be to use a Whittle formulation [3]. In this context one would also avoid determinant calculations and matrix inversions.While the authors demonstrate the 'robustness' for their approximate approach, the simulation study is limited. The one MCMC simulation with 50 samples examines the independent measurement error case. Thus, in the future, it will be of interest to further investigate the accuracy of...

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An analytical model for air pollutant transport and deposition from a point source

Cited by 105 publications

References 14 publications

The Mathematics of Atmospheric Dispersion Modeling

The Mathematics of Atmospheric Dispersion Modeling

Greenhouse Gas Emissions and Dispersion 3. Reducing Uncertainty in Estimating Source Strength and Location through Plume Inversion Models

‘Bayesian source detection and parameter estimation of a plume model based on sensor network measurements’ by C. Huang et al.: Discussion 2

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