Vol. 8, No. 3, 2015

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Parameter identification and sensitivity analysis to a thermal diffusivity inverse problem

Brian Leventhal, Xiaojing Fu, Kathleen Fowler and Owen Eslinger

Vol. 8 (2015), No. 3, 385–400

The solution to inverse problems is an application shared by mathematicians, scientists, and engineers. For this work, a set of shallow soil temperatures measured at eight depths between 0 and 30 cm and sampled every five minutes over 24 hours is used to determine the diffusivity of the soil. Thermal diffusivity is a modeling parameter that impacts how heat flows through soil. In particular, it is not known in advance if the subsurface region is homogeneous or heterogeneous, which means the thermal diffusivity may or may not depend on depth. To this end, it is not clear which assumptions may apply to represent the physical system embedded within the parameter estimation problem. Analytic methods and a simulation based least-squares approach to approximate the diffusivity are compared to fit the temperature profiles to different heat flow models. The simulation is based on a spatially dependent, nonsteady-state discretization to a partial differential equation. To complete the work, a statistical sensitivity study using analysis of variance is used to understand how errors that arise in the modeling phase impact the final model. We show that for the analytic methods, errors in the initial fitting of the temperature data to sinusoidal boundary conditions can have a strong impact on the thermal diffusivity values. Our proposed framework shows that this soil sample is heterogeneous and that modeling depends significantly on data used as top and bottom boundary conditions. This work offers a protocol to determine the soil type and model sensitivities using analytic, numerical, and statistical approaches and is applicable to modifications of the classic heat equation found across disciplines.

inverse problems, subsurface flow, sensitivity analysis
Mathematical Subject Classification 2010
Primary: 35K05, 49N45, 62J10
Secondary: 35Q93
Received: 28 April 2011
Revised: 11 November 2013
Accepted: 20 December 2013
Published: 5 June 2015

Communicated by Suzanne Lenhart
Brian Leventhal
Department of Psychology in Education
University of Pittsburgh
5930 Wesley W. Posvar Hall
Pittsburgh, PA 15260
United States
Xiaojing Fu
Department of Civil and Environmental Engineering
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
United States
Kathleen Fowler
Department of Mathematics
Clarkson University
8 Clarkson Avenue
Potsdam, NY 13699
United States
Owen Eslinger
3909 Halls Ferry Road
Vicksburg, MS 39180
United States