A Fast-Converging Kernel Density Estimator for Dispersion in Horizontally Homogeneous Meteorological Conditions

Gunther Bijloos,Johan Meyers

doi:10.3390/atmos12101343

Abstract

Kernel smoothers are often used in Lagrangian particle dispersion simulations to estimate the concentration distribution of tracer gasses, pollutants etc. Their main disadvantage is that they suffer from the curse of dimensionality, i.e., they converge at a rate of 4/(d+4) with d the number of dimensions. Under the assumption of horizontally homogeneous meteorological conditions, we present a kernel density estimator that estimates a 3D concentration field with the faster convergence rate of a 1D kernel smoother, i.e., 4/5. This density estimator has been derived from the Langevin equation using path integral theory and simply consists of the product between a Gaussian kernel and a 1D kernel smoother. Its numerical convergence rate and efficiency are compared with that of a 3D kernel smoother. The convergence study shows that the path integral-based estimator has a superior convergence rate with efficiency, in mean integrated squared error sense, comparable with the one of the optimal 3D Epanechnikov kernel. Horizontally homogeneous meteorological conditions are often assumed in near-field range dispersion studies. Therefore, we illustrate the performance of our method by simulating experiments from the Project Prairie Grass data set.

Highlights

In atmospheric dispersion modeling, one describes the transport of tracer gasses or other scalar quantities under given meteorological conditions and release characteristics.A Lagrangian approach is often used to model atmospheric dispersion
Small deviations from the theoretical convergence rate that are present as well in the PI–finite-difference estimate (FD) estimator are most likely due to numerical errors in the estimation of the second derivative and the integration used for the mean integrated squared error (MISE)
A new kernel density estimator derived from the Langevin equation has been presented for dispersion assuming local Gaussian turbulence and horizontally homogeneous meteorological conditions

Summary

Introduction

One describes the transport of tracer gasses or other scalar quantities under given meteorological conditions and release characteristics. A Lagrangian approach is often used to model atmospheric dispersion. In this approach, a stochastic differential equation (SDE) describes the trajectories of the pollutant particles. The objective is to obtain the distribution of the particle positions from this SDE. Nonparametric density estimation has the advantage that no pre-specified functional form for the distribution is assumed. Examples of such methods are histograms, orthogonal series estimators, restricted maximum likelihood estimators, etc. We will focus on the kernel smoothing approach [2,3]

Results

Discussion

Conclusion