Estimation of thermal properties and boundary heat transfer coefficient of the ground with a Bayesian technique

TL;DR

This work tackles the inverse problem of estimating the ground's thermal conductivity $\kappa$, volumetric (or preferably, appropriate) heat capacity $c_v$, and a time-varying surface heat transfer coefficient $h(t)$ from in-situ ground temperature measurements using a Bayesian framework. The forward model solves 1D transient heat conduction with a DuFort--Frankel scheme, and the inverse uses Markov chain Monte Carlo (Metropolis--Hastings) with Gaussian priors and, for Case C, a Gaussian Markov random-field prior on $h(t)$, including a Rayleigh hyperprior for the smoothness parameter $\gamma$. Case studies with real data show that the estimated parameters are close to literature values and yield smaller temperature residuals than using literature parameters; Case C, with a finer time discretization and GP prior on $h(t)$, offers the best fit and credible estimates. An urban-scale simulation demonstrates that incorporating a time-varying $h(t)$ substantially affects energy balance and long-wave radiation exchanges, emphasizing the practical value of this calibration for UHI modeling. Overall, the Bayesian approach provides robust parameter estimates and improved predictive capability for ground heat transfer in urban environments.

Abstract

Urbanization is the key contributor for climate change. Increasing urbanization rate causes an urban heat island (UHI) effect, which strongly depends on the short- and long-wave radiation balance heat flux between the surfaces. In order to calculate accurately this heat flux, it is required to assess the surface temperature which depends on the knowledge of the thermal properties and the surface heat transfer coefficients in the heat transfer problem. The aim of this paper is to estimate the thermal properties of the ground and the time varying surface heat transfer coefficient by solving an inverse problem. The Dufort--Frankel scheme is applied for solving the unsteady heat transfer problem. For the inverse problem, a Markov chain Monte Carlo method is used to estimate the posterior probability density function of unknown parameters within the Bayesian framework of statistics, by applying the Metropolis-Hastings algorithm for random sample generation. Actual temperature measurements available at different ground depths were used for the solution of the inverse problem. Different time discretizations were examined for the transient heat transfer coefficient at the ground surface, which then involved different prior distributions. Results of different case studies show that the estimated values of the unknown parameters were in accordance with literature values. Moreover, with the present solution of the inverse problem the temperature residuals were smaller than those obtained by using literature values for the unknowns.

Figures (17)

• Figure 1: Illustration of the physical model.
• Figure 2: Time variation of the air and ground temperatures (a), radiation flux (b), wind velocity and direction (c) and initial temperature (d).
• Figure 3: Sensitivity coefficients for each parameter
• Figure 4: Markov chains for: $c$(a) and $\kappa$(c), Histograms of the samples after the burn-in for: $c$(b) and $\kappa$(d) for both cases A and B
• Figure 5: Markov chains for: $\left\{h_i\right\}_{i=1}^{i=3}$(a). Histograms of the samples after the burn-in for: $h_1$(b), $h_2$(c), and $h_3$(d) for both cases A and B.