Estimation of depth-resolved profiles of soil thermal diffusivity from temperature time series and uncertainty quantification

Improving the quantification of soil thermal and physical properties is key to achieving a better understanding and prediction of soil hydro-biogeochemical processes and their responses to changes in atmospheric forcing. Obtaining such information at numerous locations and/or over time with conventional soil sampling is challenging. The increasing availability of low-cost, vertically resolved temperature sensor arrays offers promise for improving the estimation of soil thermal properties from temperature time series, and the possible indirect estimation of physical properties. Still, the reliability and limitations of such an approach needs to be assessed. In the present study, we develop a parameter estimation approach based on a combination of thermal modeling, sliding time-windows, Bayesian inference, and Markov chain Monte Carlo simulation to estimate thermal diffusivity and its uncertainty over time, at numerous locations and at an unprecedented vertical spatial resolution (i.e., down to 5 to 10 cm vertical resolution) from soil temperature time series. We provide the necessary framework to assess under which environmental conditions (soil temperature gradient, fluctuations, and trend), temperature sensor characteristics (bias and level of noise) and deployment geometries (sensor number and position) soil thermal diffusivity can be reliably inferred. We validate the method with synthetic experiments and field studies. The synthetic experiments show that in the presence of median diurnal fluctuations ≥ 1.5 °C at 5 cm below the ground surface, temperature gradients > 2 °C m−1, and a sliding time-window of at least 4 days, the proposed method provides reliable depth-resolved thermal diffusivity estimates with percentage errors ≤ 10 % and posterior relative standard deviations ≤ 5 % up to 1 m depth. Reliable thermal diffusivity under such environmental conditions also requires temperature sensors spaced precisely (with few-millimeter accuracy), with a level of noise ≤ 0.02 °C, and with a bias defined by a standard deviation ≤ 0.01 °C. Finally, the application of the developed approach to field data indicates significant repeatability in results and similarity with independent measurements, as well as promise in using a sliding time-window to estimate temporal changes in soil thermal diffusivity, as needed to potentially capture changes in carbon or water content.

On a problems related to a concept of soil thermal diffusivity and estimation of its dependence on soil moisture

Two problems in the theory of soil thermal conductivity are considered. First, the concept of the thermal diffusivity coefficient is discussed. It was shown that this coefficient can be used for model predictions only in a certain special cases. In the general case (when the soil thermal capacity and thermal conductivity vary in space and/or in time), the thermal diffusivity does not naturally appear. It could be artificially introduced into the heat equation but, in any case, to solve this equation (i.e., to calculate the dynamics of the soil temperature), this one parameter is not sufficient. It is necessary to set both the heat capacity and thermal conductivity as a functions of spatial and temporal coordinates or as a functions of environmental factors (e.g. soil moisture) depending on these coordinates. In this regard, the widespread misconception of the supposed sufficiency of one parameter (soil thermal diffusivity as a ratio of soil thermal conductivity to thermal capacity) for solving the heat equation using numerical methods is discussed. The examples of the common difference schemes used in computational practice show that this is not the case. Secondly, the condition number for the problem of parameters identification for the dependence of the soil thermal diffusivity coefficient on humidity for one well-known equation is considered. It is shown on real examples, that this problem is often ill-conditioned when solved by the least-squares method. However, sometimes its stability can be significantly improved if simple constraints are set for certain parameters (least-squares method with constraints). В работе рассматриваются две проблемы, возникающие в теории теплопроводности почв. Во-первых, обсуждается понятие коэффициента температуропроводности в свете того, что оно появляется только в отдельных весьма частных случаях, а в общем случае (когда теплоемкость и теплопроводность изменяются по пространству и/или с течением времени) коэффициент температуропроводности естественным образом вообще не возникает. Для такой среды с переменными (по пространству и во времени) свойствами он может быть искусственно введен в уравнение динамики температурного поля, но, в любом случае, для решения этого уравнения (т.е. для расчета динамики температурного поля) недостаточно одного параметра необходимо задать и теплоемкость, и теплопроводность как функции пространственной и временной координат или как функции факторов среды (например, влажности), зависящих от этих координат. В связи с этим обсуждается и распространенное заблуждение о якобы достаточности одного параметра (коэффициента температуропроводности как отношения теплопроводности к теплоемкости) при решении вышеуказанного уравнения численными методами. На примерах основных разностных схем, применяемых в вычислительной практике, показано, что это не так. Во-вторых, рассматривается число обусловленности задачи идентификации параметров одного изветного уравнения зависимости коэффициента температуропроводности от влажности. На конкретных примерах показано, что данная задача при ее решении обычным методом наименьших квадратов часто является плохо обусловленной. Однако иногда ее обусловленность удается существенно улучшить при наложении простейших ограничений на искомые параметры (метод наименьших квадратов с ограничениями). Текст статьи на русском языке см. на вкладке Дополнительные файлы

Estimation of Thermal Diffusivity for Greenhouse Soil Temperature Simulation

In greenhouse energy balance models, the soil thermal parameters are important for evaluating the heat transfer between the greenhouse air and the soil. In this study, the soil thermal diffusivity was estimated from greenhouse soil temperature data using the amplitude, phase-shift, arctangent, logarithmic, and min-max methods. The results showed that the amplitude method and the min-max method performed well in estimating the soil thermal diffusivity. The obtained soil thermal diffusivity was input into a sinusoidal model to determine the greenhouse soil temperature at different soil depths. For greenhouse applications, the daily average soil temperature at different depths was predicted according to the temperature at the surface and the annual mean soil temperature. The model was validated using soil temperature data from summer and winter, when the greenhouse was cooled and heated, respectively.

Estimating Soil Thermal Diffusivity Using Pedotransfer Functions with Nonlinear Regression

Background and Objective:Pedotransfer Functions (PTFs) are widely used for estimating soil thermal diffusivity. Some attempts have been made to indirectly predict soil thermal diffusivity from the easy available fundamental soil physics properties. The aim of the work was to validate usage PTFs with Nonlinear Regression (NLR) for estimating soil thermal diffusivity (KD), moreover was to select the best predictor variables used for determination of PTFs.Materials and Methods:Soil thermal diffusivity was measured at different values of water content using Kondratieff method. The parameters of the quadratic equation, which described the relation between thermal diffusivity and water content, were determined by the fitting curve and using PTFs (exponential equations) based on soil physical properties. The Combination of different soil physical properties used as PTF model’s independent variables was tested. Three classes of PTFs were proposed using NLR to estimate KDwere: KDPTF-1 (Sand+ Silt+ Clay), KDPTF-2 (Sand+ Silt+ Clay + Bulk density), and KDPTF-3 (Sand+ Silt+ Clay+ Bulk density + Organic matter).Results:The best class of PTF could be used for calculating the parameters of the quadratic equation and soil thermal diffusivity, was KDPTF-1 which taking into account the percentage of sand, silt and clay, RMSE=2.94×10-8m2/s, and GMER =1.05.Conclusion:The quadratic and exponential equations were representing the nonlinear regression equations, which could be used for estimating soil thermal diffusivity at different values of water content from easily available data on soil texture, bulk density, and organic matter content.

An Empirical Model for Estimating Soil Thermal Diffusivity from Texture, Bulk Density, and Degree of Saturation

Soil thermal diffusivity κ is an essential parameter for studying surface and subsurface heat transfer and temperature changes. It is well understood that κ mainly varies with soil texture, water content θ, and bulk density ρb, but few models are available to accurately quantify the relationship. In this study, an empirical model is developed for estimating κ from soil particle size distribution, ρb, and degree of water saturation Sr. The model parameters are determined by fitting the proposed equations to heat-pulse κ data for eight soils covering wide ranges of texture, ρb, and Sr. Independent evaluations with published κ data show that the new model describes the κ(Sr) relationship accurately, with root-mean-square errors less than 0.75 × 10−7 m2 s−1. The proposed κ(Sr) model also describes the responses of κ to ρb changes accurately in both laboratory and field conditions. The new model is also used successfully for predicting near-surface soil temperature dynamics using the harmonic method. The results suggest that this model provides useful estimates of κ from Sr, ρb, and soil texture.