Understanding the Impact of Boundary and Initial Condition Errors on the Solution to a Thermal Diffusivity Inverse Problem

In this work, we consider simulation of heat flow in the shallow subsurface. As sunlight heats up the surface of soil, the thermal energy received dissipates downward into the ground. This process can be modeled using a partial differential equation known as the heat equation. The spatial distribution of soil thermal conductivities is a key factor in the modeling process. Prior to this study, temperature profiles were recorded at different depths at various times. This work is motivated by trying to match these temperature profiles using a simulation-based approach and analytic approaches in the context of an inverse problem. Specifically we determine soil thermal conductivities using derivative-free optimization to minimize the nonlinear-least square errors between simulation and data profile. Here, we conduct two sets of studies, assuming homogeneous and heterogeneous soil environments respectively. We also study how errors in the initial and boundary conditions propagate over time using both a numerical approach and an analytical method