A new method for obtaining velocity and diffusivity from timedependent distributions of a tracer via the maximum likelihood estimator for the advectiondiffusion equation
Abstract
An inverse problem for the advectiondiffusion equation is considered, and a method of maximum likelihood (ML) estimation is developed to derive velocity and diffusivity from timedependent distributions of a tracer. Piterbarg and Rozovskii showed theoretically that the ML estimator for diffusivity is consistent ever in an asymptotic case of infinite number of observational spatial modes. In the present work, the ML estimator is studied based on numerical experiments with a tracer in a twodimensional flow under the condition of a limited number of observations in space. The numerical experiments involve the direct and the inverse problems. For the former, the time evolution of a tracer is simulated using the Galerkintype methodas a response of the conservation equation to stochastic forcing. In the inverse problem, the advectiondiffusion equation is fitted to the simulated data employing the ML estimator. It is shown that the ML method allows us a method to estimate diffusion coefficient components D{sub x} and D{sub y} based on a short time series of tracer observations. The estimate of the diffusion anistropy, D{sub x}/D{sub y}, is shown to be even more robust than the estimate of the diffusivity itself. A comparison with an estimation technique based on the finitedifferencemore »
 Authors:
 Kyushu Univ., Kasuga (Japan)
 Univ. of Southern California, Los Angeles, CA (United States)
 Publication Date:
 OSTI Identifier:
 535390
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 133; Journal Issue: 2; Other Information: PBD: 15 May 1997
 Country of Publication:
 United States
 Language:
 English
 Subject:
 66 PHYSICS; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; DIFFUSION; PARTIAL DIFFERENTIAL EQUATIONS; ADVECTION; NUMERICAL SOLUTION; TIME DEPENDENCE; MAXIMUMLIKELIHOOD FIT
Citation Formats
Ostrovskii, A.G., and Piterbarg, L.I. A new method for obtaining velocity and diffusivity from timedependent distributions of a tracer via the maximum likelihood estimator for the advectiondiffusion equation. United States: N. p., 1997.
Web. doi:10.1006/jcph.1997.5674.
Ostrovskii, A.G., & Piterbarg, L.I. A new method for obtaining velocity and diffusivity from timedependent distributions of a tracer via the maximum likelihood estimator for the advectiondiffusion equation. United States. doi:10.1006/jcph.1997.5674.
Ostrovskii, A.G., and Piterbarg, L.I. Thu .
"A new method for obtaining velocity and diffusivity from timedependent distributions of a tracer via the maximum likelihood estimator for the advectiondiffusion equation". United States.
doi:10.1006/jcph.1997.5674.
@article{osti_535390,
title = {A new method for obtaining velocity and diffusivity from timedependent distributions of a tracer via the maximum likelihood estimator for the advectiondiffusion equation},
author = {Ostrovskii, A.G. and Piterbarg, L.I.},
abstractNote = {An inverse problem for the advectiondiffusion equation is considered, and a method of maximum likelihood (ML) estimation is developed to derive velocity and diffusivity from timedependent distributions of a tracer. Piterbarg and Rozovskii showed theoretically that the ML estimator for diffusivity is consistent ever in an asymptotic case of infinite number of observational spatial modes. In the present work, the ML estimator is studied based on numerical experiments with a tracer in a twodimensional flow under the condition of a limited number of observations in space. The numerical experiments involve the direct and the inverse problems. For the former, the time evolution of a tracer is simulated using the Galerkintype methodas a response of the conservation equation to stochastic forcing. In the inverse problem, the advectiondiffusion equation is fitted to the simulated data employing the ML estimator. It is shown that the ML method allows us a method to estimate diffusion coefficient components D{sub x} and D{sub y} based on a short time series of tracer observations. The estimate of the diffusion anistropy, D{sub x}/D{sub y}, is shown to be even more robust than the estimate of the diffusivity itself. A comparison with an estimation technique based on the finitedifference approximation demonstrates advantages of the ML estimator. Finally, the ML method is employed for analysis of heat balance in the upper layer of the North Pacific in the winter. This application focuses on the heat diffusion anisotropy at the ocean mesoscale. 29 refs., 14 figs.},
doi = {10.1006/jcph.1997.5674},
journal = {Journal of Computational Physics},
number = 2,
volume = 133,
place = {United States},
year = {Thu May 15 00:00:00 EDT 1997},
month = {Thu May 15 00:00:00 EDT 1997}
}

The advectiondiffusion equation with time dependent velocity and anisotropic time dependent diffusion tensor is examined in regard to its nonclassical transport features and to the use of a nonorthogonal coordinate system. Although this equation appears in diverse physical problems, particularly in particle transport in stochastic velocity fields and in underground porous media, a detailed analysis of its solutions is lacking. In order to study the effects of the timedependent coefficients and the anisotropic diffusion on transport, we solve analytically the equation for an initial Dirac delta pulse. Here, we discuss the solutions to three cases: one based on powerlaw correlationmore »

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