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Introduction

There are several methods for determining soil moisture content. Conventional Time Domain Reflectometry (Topp et al., 1980) and drying methods are commonly used to estimate water content in homogeneous soils at shallow depths (Yoshikawa et al., 2004; Boike and Roth, 1997). Moreover, accurate measurement of the total water content can be accomplished by thermalization of neutrons and by gamma ray attenuation, but transportation of radioactive equipment into Arctic regions is impracticable (Boike and Roth, 1997). A review of state-of-the-art methods for measuring soil water content can be found in (Topp and Ferre, 2002; Gardner, 1986). However, the above-mentioned well-established methods rely on installation of special equipment in the field, or on laboratory experiments, and hence are not applicable for recovering soil water content from various soil temperature records.

Thermal properties can be also measured in the laboratory or in-situ experiments including the Needle Probe (Herzen and Maxwell, 1959), Divided Bar (Birch, 1950), borehole relaxation (Wilhelm, 1990), non-linear fitting (Da-Xin, 1986), thermal pulse (Silliman and Neuzil, 1990), and estimation from thermal gradients (Somerton, 1992) methods. Reviews of some of these methods can be found in (Beck, 1988). Similar to methods measuring water content, the methods for determining thermal properties are not applicable for recovering thermal properties from typical temperature measurements, i.e. temperature records at different depths. Methods that estimate thermal properties from temperature records include the Simple Fourier Methods (Carson, 1963), Perturbed Fourier Method (Hurley and Wiltshire, 1993), and Graphical Finite Difference Method (Zhang and Osterkamp, 1995; McGaw et al., 1978; Hinkel, 1997). They estimate coefficients in the heat equation and yield accurate results for the thermal diffusivity (ratio of the thermal conductivity to the heat capacity) only when the phase change of water does not occur.

One alternative capable of estimating both thermal properties and water content of soil is variational assimilation of temperature observations into a model of soil freezing and thawing. A goal of the variational assimilation is to adjust/optimize a set $\mathscr{C}$ of model parameters in order to minimize a difference

\begin{displaymath}
J(\mathscr{C})\approx\Vert T_o-T\Vert^2
\end{displaymath}

between the observed $T_o$ and modeled $T$ temperatures. The set $\mathscr{C}$ includes parameters related to thermal conductivity, soil porosity and coefficients determining unfrozen water content for partially frozen soil. Beck (1964) and Nagler (1965) have applied the least square variational approach to estimate thermal properties in a heat conduction problem without phase change. In this article, we compute soil temperature, $T$, by employing the 1-D heat equation with phase change of water (Carslaw and Jaeger, 1959) and minimize the discrepancy

$J(\mathscr{C})$ by optimizing $\mathscr{C}$.

Results from (Alifanov, 1994; Alifanov et al., 1996; Beck et al., 1985) include a detailed mathematical and theoretical analysis of variational temperature assimilation for the heat equation without explicit phase change terms. Some analysis of parameter estimation in heat conduction problems with phase change can be found in (Ouyang, 1992; Permyakov, 2004; Pavlov et al., 1980). However, it is hard to find discussion of variational assimilation being used to recover soil properties from in-situ temperature measurements in the active layer and permafrost.

We apply a variational technique to estimate thermal conductivity, porosity, and coefficients describing unfrozen water content at four locations along the Dalton Highway in Alaska. To evaluate the thermal properties, we use daily temperature measurements and a once-a-year temperature profile in 60 meter boreholes. Also, we conduct several numerical experiments and explore robustness of recovering the thermal properties. The recovered properties that are associated with the minimum of $J(\mathscr{C})$ are sought by an iterative method. Since there could be several local minima, a thoughtful selection of an initial approximation of $\mathscr{C}$ as well as a certain regularization is necessary (Nicolsky et al., 2007). We add to $J(\mathscr{C})$ a regularization that incorporates a priori estimate of values in the optimal control vector. Besides the regularization, we assume that the soil properties are constants within each soil horizon. This assumption decreases a hundred fold the number of variables in the control vector on which the cost function depends, and hence simplifies the problem.




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