Reconstruction of soil thermal field from a single depth measurement Zhi-Hua Wang ⇑ School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, AZ 85287, USA article info Article history: Received 21 May 2012 Received in revised form 29 July 2012 Accepted 31 July 2012 Available online 8 August 2012 This manuscript was handled by Peter K. Kitanidis, Editor-in-Chief, with the assistance of J. Simunek, Associate Editor Keywords: Green’s function approach Soil thermal field Heat conduction Duhamel’s principle Surface energy balance Soil water content summary Soil field experiments usually consist of measurements of soil temperatures, heat fluxes and soil water contents. Accurate determination of the soil thermal field, in particular, prediction of the soil surface tem- perature and the ground heat, contains the signature to the surface energy partitioning, and is therefore critical to the surface energy balance closure problem. In this paper, we develop a numerical procedure to reconstruct the entire soil thermal field from a single depth measurement of either temperature or heat flux. The new algorithm is based on Green’s function approach by using the fundamental solution of heat conduction in semi-infinite soils and Duhamel’s integral for incorporation of general boundary condi- tions. It is highlighted that the new approach is capable of accurately reproducing results of some existing numerical approaches, with a more general setting and treatment of the heat diffusion problem, and hence provides a possible unified framework for the estimation of thermal field in soils. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction The transport of heat underneath the Earth’s surface and the subsequent determination of the soil thermal field (viz. soil tem- perature and soil heat flux) are critical in regulating the subsurface and surface physical processes. In particular, as all major surface energy budgets (net radiative, sensible, latent and ground heat fluxes) are strong functions of the surface temperature, the subsur- face heat transport largely dictates the partitioning of the available energy on the land surface (net radiation) into the dissipative heat budgets (sensible, latent and ground heat). A recent study by Bat- eni and Entekhabi (2012) showed that the land surface tempera- ture can implicitly contain the signature to the surface energy partitioning through linear stability analysis. The accurate deter- mination of the soil thermal field, therefore, is essential in estab- lishing and assessing the surface energy balance closure and has attracted extensive research effort concerned with climate, weath- er and atmospheric dynamics (de Silans et al., 1997; Foken, 2008; Heusinkveld et al., 2004; McCumber and Pielke, 1981). The subsurface thermal field can be constructed by solving the coupled heat, liquid water and vapor transport equations using ad- vanced numerical techniques, such as the finite element method (FEM) (Bittelli et al., 2008; Vogel et al., 2011). On the other hand, numerous analytically-based approaches have been developed in past decades, to predict soil temperature and/or soil heat flux based on the one-dimensional (1D) heat diffusion, with applica- tions to a wide range of areas including agronomy, meteorology, hydrology and ecology (Gao et al., 2003; Guaraglia et al., 2001; Holmes et al., 2008; Horton and Wierenga, 1983; Nunez et al., 2010). These models made use of analytical solutions of heat con- duction in semi-infinite soils, and are numerically more economic and provide deeper insight into the subsurface physics as com- pared to FEM. In these models, certain time series of measured soil thermal properties, temperatures and heat fluxes at various depths are required as auxiliary data to complete estimations of the soil thermal quantities. Alternatively, the force-restore method, origi- nally proposed for the derivation of prognostic surface tempera- ture equations (Bhumralkar, 1975), together with its improved forms (Arya, 2001; Deardorff, 1978; Gao et al., 2008), were proven a powerful tool for soil thermal predictions. Instead of solving the second order partial differential equation, the heat diffusion pro- cess is simplified and represented as first order ordinary differen- tial equations in the force-restore method, such that standard integration technique can be directly applied to obtain solutions of soil thermal field. However, some theoretical aspects regarding the heat conduc- tion process in soils remain obscure among researchers, leading to confusion in problem definition (Wang and Bou-Zeid, 2011), overdesign of auxiliary measurements (Wang and Bou-Zeid, 2012) and unnecessary restriction in model applicability (Wang and Bras, 1999). As a consequence, there are some major limita- tions inherent in most numerical models in the literature: 0022-1694/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jhydrol.2012.07.047 ⇑ Tel.: +1 480 727 2933; fax: +1 480 965 0557. E-mail address: zhwang@asu.edu Journal of Hydrology 464–465 (2012) 541–549 Contents lists available at SciVerse ScienceDirect Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol