Thermal conductivity in a vadose zone: A novel spectral solution and application by the implication of significant coherence of thermal condition in the shallow soil water layer

In the context of the heterogeneity in the unsaturated or vadose zone, accurately representing the analytical mechanisms and in‐situ water content within the soil layer poses a significant challenge. Particularly in shallow layers, thermal conditions exhibit rapid changes in response to evolving surface temperatures. This study proposes a hypothesis suggesting that the in situ heat mechanism may notably impact the soil water layer. The research introduces an innovative approach to theoretically uncover thermal conditions, including soil temperature, soil temperature gradients, and heat flux, within the shallow Quaternary gravel layer at various depths through spectral analysis of temporal observations. The study presents a stochastic inverse solution to estimate thermal conductivity by leveraging spectral analysis of soil heat flux and temperature gradients. The findings reveal that thermal conditions exhibit the most prominent periodic fluctuations during the diurnal process over a 24‐hour cycle. The soil temperature gradients and heat flux measurements at depths of 0.1, 0.3, 0.6, and 1.2 m demonstrate their ability to capture changes in soil temperature and air temperature to a certain extent within the frequency domain. Furthermore, the analysis highlights the intrinsic uncertainty and sensitivity of estimating thermal conductivity in heterogeneous soil environments. The wide variability observed in thermal conductivity values, coupled with their dependence on soil type and environmental conditions, underscores the need for careful consideration of these factors in future studies and modeling efforts. Applying the derived inverse spectral solution allows for determining thermal conductivity throughout the soil‐water system across depths ranging from 0.1 to 1.2 m. As a result, this research demonstrates the feasibility and practicality of assessing the thermal conductivity of the soil layer in conjunction with heat flux and temperature gradients through spectral analysis.

energy partitioning between sensible heat (temperature change) and latent heat (evaporation or condensation of water).Understanding this balance is critical for modelling and predicting hydrological processes, including soil water content changes and groundwater recharge.Thermal conductivity affects the movement of water vapour and liquid water in the vadose zone.Differences in thermal conductivity between soil layers can create preferential flow paths for water movement, influencing infiltration rates, drainage, and the redistribution of water within the soil profile.
Thermal conductivity is also vital in vadose zone water management, influencing irrigation, evapotranspiration, soil water content monitoring, contaminant transport, groundwater recharge, soil structure, and adaptation to changing climate conditions.It is essential for making informed decisions promoting efficient water use, sustainable agricultural practices, and responsible environmental stewardship.
Thermal conductivity-based sensors can be used to estimate soil water content.These sensors provide indirect information about soil moisture levels by measuring the temperature changes over time.
Accurate monitoring of soil water content is essential for making informed decisions regarding water application.In vadose zone water management, understanding the movement of contaminants through soil is crucial for preventing groundwater pollution.Thermal conductivity affects the rate of contaminant movement and dispersion within the soil matrix.Knowledge of thermal conductivity helps predict the potential spread of contaminants and design remediation strategies.It also influences the downward movement of water from the vadose zone to the groundwater table.Proper groundwater recharge management requires accurately assessing water infiltration rates controlled by thermal conductivity.This knowledge helps optimize land use and water resource allocation.Thermal conductivity can be indicative of soil texture, structure, and porosity.These soil properties impact water movement and retention.Understanding the relationship between thermal conductivity and soil structure aids in managing soil compaction, aeration, and drainage, which are critical for healthy plant growth.Changes in temperature patterns due to climate change can affect soil moisture dynamics.Thermal conductivity influences soil response to temperature changes, impacting plant water availability.
Managing these changes requires thoroughly understanding how thermal conductivity influences soil-water-plant interactions.
In 1957, Philip and D. A. (1957) developed a theory that accounted for soil water content under the influence of apparent vapour transfer, net moisture transfer, and latent heat.The temperature gradient can significantly impact soil water movement, particularly under nonisothermal conditions (Bach, 1992).Several studies have explored the coupled heat and water transfer in the soil by examining temperature and other soil physical properties, specifically thermal conductivity (Alrtimi et al., 2016;Evett et al., 2012;Hamdhan & Clarke, 2010;Hiraiwa & Kasubuchi, 2000;Lu et al., 2014;Miyazaki, 2005;Peng et al., 2017;Sato & Iwasa, 2000).From these studies, it has been observed that thermal conductivity resulting from latent heat transfer can be isolated.The volumetric water content of the soil surface changes due to sunlight and can be quantified by analysing temperature gradient and conduction by convection.Changes in heat storage are based on variations in soil temperature, and estimating soil heat capacity relies on soil water content.Empirical formulations for thermal conductivity can be expressed as functions of water content and porosity.The gradient method can also determine soil heat flux at a specific depth as the product of soil thermal conductivity and temperature gradient.Although in situ measurement of thermal conductivity is challenging, it has been suggested to support continuous, long-term monitoring of soil temperature and water content in the unsaturated zone.
The advanced utilization of thermal conductivity in heat transfer within soil water layers, including considerations of thermal diffusivity and other coupled effects, deserves attention (Brunetti et al., 2022;Nowamooz & Assadollahi, 2022;Stepanenko et al., 2021;Wu et al., 2021).The thermal diffusivity coefficient can be employed to analyse thermal conduction and other thermal transfers.Changes in soil temperature, water content distribution, and water flux variations yield valuable information for understanding soil behaviour during freezethaw cycles.Bayesian inference and Markov chain Monte Carlo simulation techniques can be applied to estimate thermal diffusivity and its associated uncertainty by incorporating soil temperature and gradient data.
Heat plays a pivotal role in influencing soil temperatures, which, in turn, impacts crop growth and yield.Understanding heat conduction in soil water layers allows farmers and agricultural scientists to decide when to plant, irrigate, or harvest crops.(Balota et al., 2008;Li et al., 2021;Rodríguez et al., 2015) Efficient water use is a global concern.Studying heat conduction in the soil allows for the determination of how various soil types retain or release heat, consequently impacting the moisture content of the soil; this knowledge can help develop precise irrigation strategies.The information is invaluable for conserving water while maintaining crop productivity in regions with limited water resources, such as arid climates.(Abu-Hamdeh & Reeder, 2000;Farouki & Farouki, 1981) Soil health is a fundamental aspect of sustainable agriculture.Soil microorganisms crucial for nutrient cycling and plant growth are sensitive to temperature.Excessive heat can harm these microorganisms.Studying heat conduction contributes to strategies for maintaining soil health, promoting sustainable farming practices, and reducing the need for chemical fertilizers.(Ayangbenro & Babalola, 2017;Wong, 2003).
Understanding heat conduction in soil water layers provides insights into how water interacts with the surrounding soil matrix.This knowledge is crucial for managing water resources effectively, especially in regions where water availability is a concern.It aids in estimating water movement through the vadose zone, a critical factor for groundwater recharge and the sustainability of water supply.(Wu et al., 2023;Xu & Singh, 1998) Heat conduction studies in soil water layers are relevant to agriculture.Farmers can make informed decisions about irrigation scheduling and water conservation based on soil temperature and heat flux data.This knowledge can lead to more efficient water usage in agriculture, reducing water wastage and improving crop yields.(Chen et al., 2020;Shang et al., 2004) Heat conduction studies can aid in assessing the environmental impact of various activities, such as land development, construction, or contamination incidents.Changes in soil temperature can affect the behaviour of contaminants and the biogeochemical processes that occur in the vadose zone.(Cardenas & Wilson, 2007;Gutiérrez & Jones, 2006) Monitoring heat conduction helps in the early detection and mitigation of potential environmental risks.
Radionuclides in shallow soil water layers can be highly mobile, especially in areas with varying temperatures.The vital mission aims to use controlled heat conduction to reduce contaminant mobility and prevent their migration into deeper soil layers or groundwater.By understanding and controlling heat conduction, one can manipulate the temperature distribution to inhibit the movement of radionuclides, thereby containing the contamination within the shallow layer (Burton et al., 2005;Mulligan et al., 2001).When the study delves into optimizing energy input for heat conduction-based decontamination, a practical approach reduces energy consumption and related expenses, rendering it a more sustainable and economically efficient solution.
Heat conduction for decontaminating radionuclide releases in shallow soil water layers offers a promising solution to mitigate the environmental and public health risks of such contamination.This study contributes to safer nuclear facility operations and sustainable environmental practices by addressing these challenges through innovative research.
Thermal conductivity plays a crucial and significant role in evaluating water movement within a shallow vadose zone.For a hypothetical and hybrid study site (HHS) established for a long-term nuclear research mission, it is essential to implement procedures for assessing and controlling potential leakage incidents.Soil water content often plays a critical role in the release of radionuclides.However, in cases where the soil layer is directly influenced by external factors like sunlight and air temperature, these factors must be considered essential variables.Therefore, investigating the spectral coherence among thermal conditions, such as soil temperature, temperature gradient, and heat flux, and estimating thermal conductivity can greatly enhance future remediation control measures.
The primary objective of this study is to investigate the relationship between heat transfer conditions of soil water in both the time and frequency domains.In most cases, temporal changes exhibit nonstationary evolution due to the heterogeneity of layers and uncertain external influences.The heat flux within the soil-water coupling system encompasses thermal conductivity through conduction and latent heat transfer resulting from vapour heat transfer.Estimating latent heat in a heterogeneous material, such as a mixture of water and soil, is challenging.Assessing vapour's temporal and spatial variations considering heat transfer becomes a complex task that cannot be solely consolidated through temporal evolution.This study addresses the primary scientific issue of how thermal conditions, precisely soil temperature, soil temperature gradient, and heat flux, influence the thermal conductivity within the soil-water system.The research focuses on understanding the intricate relationship between these parameters and thermal conductivity in diurnal and semidiurnal temperature variations.
Sunlight is considered a significant and stable heat source in an open field under rainless conditions, typically exhibiting a periodic pattern (Shih, 2022).By analysing spectral coherence, significant associations between the studied parameters, including temperature, temperature gradient, and heat flux, can be detected in the frequency domain.It allows for identifying substantial components in the power spectrum, providing insights into the mechanisms within the studied system.Consequently, accurately identifying the variation of vapour in the soil layer becomes feasible by discovering the significant transmission patterns of thermal conditions.The improved twin heat probe method, which entails constructing a complex system, is used to measure the sample's thermal conductivity.However, verifying in-situ thermal conductivity is both costly and uncertain.Even using this equipment to obtain latent heat is challenging.By separating the latent heat transfer from the soil-water system, the thermal conductivity already represents a resultant (lumped) effect of other potential factors in the final estimation.Installing a single direct temperature or heat flux probe on the in-situ soil-water layer is much simpler than using twin heat probes.
The second objective of this study is to derive a pure spectral representation for analysing the thermal conductivity of the soil water system using Fourier's law.It requires a conduction mechanism system with net heat radiation represented as the heat flux at the ground surface as an inlet and temperature and heat flux at various soil depths as endpoints.Spectral coherence is collected for all parameters, including temperature, temperature gradient, and heat flux.Significant or relevant components in the frequency domain can represent the primary thermal transmission within the system.Furthermore, this automatically removes non-representative parts induced by non-stationary variations or external noise.

| Conceptual design and site description
The processes governing water flow in the subsurface layer include surface water infiltration into the soil, which results in the formation of water content, unsaturated flow through the soil, and saturated flow through soil or rock strata.The capillary fringe effect begins from the water table within the unsaturated layer and extends to the limit of capillary rise based on soil formation and hydraulic conductivity.As a result, the soil water properties at the shallow surface soil scale, within a few 10 cm, are more complex than those in the saturated zone.The Richards equation describes water movement in unsaturated soils, which accounts for nonlinear water flow (Richards, 1931).Hiraiwa and Kasubuchi (2000) have explored the phenomenon of water vapour movement under temperature gradients in the soil, emphasizing its significance in the context of heat and water interaction.They have also investigated the temperature dependence of soil thermal conductivity.
Soil heterogeneity in the subsurface layer typically governs water distribution in unsaturated aquifers.Dealing with decontaminating chemical pollutants entering the geological environment becomes complex, especially during accidental industrial facility operations.
Recognizing the vertical profile distribution of soil water content can be crucial in remediation efforts.Typically, weather factors such as sunlight and rainfall significantly influence soil water content.The thermal conditions also impact water and vapour within the soil layer.
In subtropical areas, surface temperature variations cause substantial changes in thermal conditions throughout the soil layers, whether or not there is sunlight.As a result, the soil water content experiences daily fluctuations.The temperature gradient in the soil acts as a driving force for the movement of both water vapour and liquid.The daily temperature variations may affect water movement in the soil.Therefore, it is crucial to estimate the temporal characteristics of thermal conditions accurately.During the period following rainfall, known as solarization, water may stagnate or move more slowly through the soil layer, making the timing estimation challenging.
Due to the challenges of measuring water vapour, this study neglects the coupling effect between soil vapour and thermal conditions.Instead, it focuses solely on pure thermal transfer within the soil depths.The thermal conditions observed include air temperature, temperature, temperature gradient, and heat flux at targeted depths of 0.1, 0.3, 0.6, and 1.2 m.The measured thermal conditions are assumed to exhibit uniformity across the soil material, water, and air within the soil texture.An observation station was responsible for data collection, and Figure 1 presents the collected data.Additionally, Table 1 showcases a selected set of parameters.
The studied site, designated as HHS, was specifically designed for researching and developing nuclear energy technologies, with a strong focus on reliable protection measures.It is located in the northern region of Taiwan.The surface of the site is covered with wild grass, which is regularly maintained at a fixed height of approximately 0.1 m.The geological composition of the site consists of recent alluvium with local thin sediments and a layer of Quaternary lateritic terrace gravel mixed with clay, extending from the surface down to the bedrock.The thickness of the gravel layer is approximately 24 m, and the water table is about 16 to 17 m below the surface.A sandstone bed is also found near the bottom of the gravel layer.Considering the proximity to the surface and the exposure to sunlight, it becomes crucial to analyse the in situ soil thermal conditions at the studied site.

| Thermal conductivity
The coupled heat and mass transfer in the soil can be complicated when examining the temperature dependence of thermal conductivity.Without considering latent heat transfer, the heat does not interact with vapor.For the scale of a few 10 cm of the soil layer, it may determine the thermal conductivity from temperature measurements using the one-dimensional linear equation in a vertical direction, which follows Fourier's law (Baron Fourier, 1878) where q is heat flux (W/m 2 ), T is the temperature ( C), κ is the thermal conductivity due to conductance (W/(m C)) and is the constant in the specific layer of soil, z is the vertical axis (m), d denote the derivative of variables.
It suggests specifying a domain for the soil layer as the controlled layer where the heating distributes uniformly in each layer.The heat flows upwards or downwards from the interior of the soil layer.The backward finite difference scheme for Equation (1) in each layer is where subscription index p and p À 1 indicate the controlled layer and the upper layer adjacent to the controlled layer, respectively.In a controlled layer, q p t ð Þ is uniform heat flux, T p t ð Þ is the uniform temperature, κ p is the constant thermal conductivity, z is the vertical axis (m), Δ denotes the increment, h p is the thickness of the soil layer, and contrast to the time-based approach, this study proposes a novel method that utilizes spectral analysis to infer thermal conductivity.
The key concept is to transform the time series data into the frequency domain and examine the power spectrum.By identifying significant components in the spectral density, it becomes possible to estimate the thermal conductivity associated with specific frequency bands.The heat flow patterns revealed in the power spectrum can reflect significant contributions from dominant frequencies while filtering out unknown sources of noisy fluctuations, which are challenging to eliminate through conventional methods.This approach offers a more reliable means of evaluating thermal conductivity than relying solely on regressive analysis of heat flow and temperature gradient observations.

| Spectral representation
and its inversed formula where Φ f ð Þ and Γ f ð Þ are Fourier transforms of ϕ t ð Þ and γ t ð Þ, f is the circular frequency (cycles/time), t is time, and i is ffiffiffiffiffiffi ffi À1 p .
It adopts a finite record length T r for considered time series.The one-sided spectral density, for f > 0, is defined as where the expected-value operator E denotes an averaging value over a sampling period and Ã is the complex conjugate.S ϕϕ and S γγ are auto spectral density for processes ϕ t ð Þ f gand γ t ð Þ f g, respectively.It represents the cross-spectral density as where C ϕγ f ð Þ is the coincident spectral density and Q ϕγ f ð Þ is the quadrature spectral density (Bendat & Piersol, 2000).

| The spectral representation of thermal conductivity
Considering the thermal mechanism in Equation ( 1), let Z ψ and Z q are Fourier components of temperature gradient (ψ) and heat flux (q) for each controlled soil layer, respectively.It has By substituting Equation ( 7) into Equation (2), it is Moreover, it has Using Equation ( 5), it transforms Equation ( 9) to Therefore, thermal conductivity is Once S qq and S ψψ is observed, it can estimate the thermal conductivity, κ, using Equation (11).

| Spectral analysis
In the time-frequency domain, spectral analysis converts a time series with periodic fluctuations into the power spectrum in the frequency domain.Assuming a random variable in the time domain, u t ð Þ defines the complex Fourier components as where u is a time-dependent random variable, U is the complex Fourier components in the frequency domain, i ¼ ffiffiffiffiffiffi ffi À1 p , t is the elapsed time, f is the circular frequency.Let the finite time be in Equation ( 12), and it rewrites as where τ is finite time sequence.The component U, is the limited time length and also a function of frequency.
A stationary time series of a length of τ, it divides into n d contiguous segments, with each segment length of τ s , it estimates the twosided auto spectral density for each part by By averaging each resultant component, it obtains a final smooth twosided auto spectral density and should satisfy the statistical significance.
Given Δt as the sampling rate in time, the discrete frequency is where N denotes the length of segments.The smoothed, one-sided auto spectral density is expressed as It computes the raw estimate cross-spectral density for each subrecord for two different time series, for example, Auto spectral density of the temperature, gradient, and heat flux at various soil depths.
where the complex conjugate of The smooth computation of cross-spectral density and associated quantities are where Cuv It gives a 95% confidence interval (CI) of the auto spectra The coherence with a 95% confidence interval (CI) is Shih et al. (1999) derived the 95% non-zero coherence significance level.

| RESULTS
This study collected comprehensive data on air temperature, soil water temperature, as well as temperature gradients and heat flux specific to soil water to introduce a novel procedure.Figure 2 demonstrates the application of spectral analysis via the Fourier transform to identify periodic fluctuations over time and depict them as spectral density in the frequency domain.Notably, the data displayed notable deviations from the anticipated stationary pattern, prompting the need for pre-processing through time blocking to smooth the spectral density within the frequency band, ensuring a reliable estimation.The widely used least-square method eliminated linear trends within each time block, while the Hanning window addressed leakage issues during spectral density estimation (Bloomfield, 2004).1) and ( 2) and the subsequent derivation.
The spectral calculation results uncover diurnal and semidiurnal components within specific frequency bands (Figures 3 and 4A-E).This alignment with natural daily time cycles is evident in Figure 2, and the specific values are provided in Tables 2 and 3.
The spectral density of the temperature, temperature gradient, and heat flux analysis reveals that the diurnal component with a 24-h period exhibits higher significance than the semidiurnal component with a 12-h period (Figure 3).However, it is observed that the coherence between soil depths of 0.1 versus 0. Figure 5A shows the time series of air temperature, soil water temperature, temperature gradient, and heat flux specific to the soil water layer in a diurnal pattern, filtered using a bandpass filter at the frequency range of 0.04067 to 0.04267 cph and the corresponding period of 24.590 to 23.437 h.The distinct periodic variation is quite apparent, with the diurnal component exhibiting higher significance than the semidiurnal component (Figure 5B).In Table 2, the time lag between air and soil temperature at a depth of 0.1 m is 5.6 h.At soil depths ranging from 0.1 to 1.2 m, the time lag for temperature varies between 0.71 and 5.6 h, for temperature gradient between 1.13 and 6.01 h, and for heat flux between 7.43 and 9.15 h.Similarly, in Table 3, for the semidiurnal pattern filtered using a bandpass filter with a frequency range of 0.08233 to 0.08433 cph and the corresponding period of 12.15 to 11.86 h (Figure 5B), the time lag between air and soil temperature at a  2 and Table 3).These findings raise questions about the underlying factors contributing to such variations in thermal conductivity.Soil depth plays a crucial role, with thermal conductivity decreasing as depth increases.However, these results also underscore the inherent uncertainty in estimating thermal conductivity in heterogeneous soil environments.T A B L E 2 Spectral analysis and thermal conductivity of the diurnal component.T A B L E 3 Spectral analysis and thermal conductivity of the semidiurnal component.agriculture, and environmental conservation.The related fields can be soil moisture dynamics, evapotranspiration, groundwater recharge, soilwater interaction, climate modeling, irrigation management, ecological monitoring, and water quality (Anderson et al., 2012;Bittelli et al., 2008;Dong et al., 2015;Glynn & Plummer, 2005;Kumar & Dey, 2011;Kurylyk et al., 2014;Rossman et al., 2014;Shrestha et al., 2015).This research offers significant findings for comprehending heat conduction at the HHS site.The estimated heat conductivity, obtained through the timefrequency spectral approach, is reliable in eliminating unknown noises The results reveal that the spectral density and coherence around the 0.3 m depth are less significant, hinting at the possible presence of heterogeneous depositions or artificial materials in that particular location.For future investigations, the methodology employed in this study can be applied to examine vertical variations of soil thermal conditions in both the time and frequency domains.This approach can be precious when focusing on aspects such as soil vapor or soil water thermal diffusivity and convection effects.Furthermore, additional research is needed to understand better the influence of various factors on soil thermal conditions within the soil layer.Quantifying the associated uncertainties resulting from the omission of these factors will also be a critical area of investigation.
AbbreviationParameter UnitAIRTMPAir temperature C SOT010 Temperature at a soil depth between 0 and 0.1 m is the temperature gradient.Equation (2) presents a temporal process in the controlled soil layer.However, various unknown factors often influence temperature and heat flux observations, including natural and artificial effects.In F I G U R E 2 Time series of the air temperature, soil temperature, temperature gradient, and heat flux.
Cross-spectral density of the temperature at various soil depths.(B) Cross-spectral density of the temperature gradient at various soil depths.(C).Cross-spectral density of the heat flux at various soil depths.(D) Cross-spectral density of the temperature and temperature gradient at various soil depths.(E) Cross-spectral density of the temperature and heat flux at various soil depths.where χ 2 n;α is the Chi-square distribution for a percentage α so that the probability χ 2 n > χ 2 n; α h i ¼ α with degrees of freedom n, and n ¼ 2n d .
The frequency resolution was determined as 3.205E-03 cycles per hour (cph) for a subrecord, with a length of 312 hour out of the total 624 hour (h).A 95% confidence interval (CI) revealed lower and upper levels of 0.359 and 8.257 for auto spectral density (Figure3).Notably, assessing the significance of the auto spectral density in a logarithmic scale allowed for identifying spectral peaks surpassing the upper bound.The non-zero coherence (NZC) significance level for crossspectral density stands at 0.781.Instead of relying on the confidence interval, using NZC to assess the significance of cross-spectral density F I G U R E 4 (Continued) is reasonable.Figures 4A-C depict the cross-spectral density of temperature, temperature gradient, and heat flux at various depths, showing distinct significance in specific frequency bands.Figures 4D,E also demonstrate the cross-spectral density of temperature versus temperature gradient and temperature versus heat flux, respectively, at various depths.These findings support the requirements outlined in Equations (

Figures
Figures 5A,B thus show the generation of bandpass time series corresponding to these components.Figure 6 illustrates the cross-spectral density and coherence of temperature gradient and heat flux at the target depths, addressing their relationship for estimating soil water conductivity using Equation (11).
Figure 4E, and Plot b of Figure 4E.These findings suggest that the thermal processes occurring in the vicinity of the soil depth of approximately 0.3 m are complex or hindered.

F
I G U R E 4 (Continued)depth of 0.1 m is 4.51 h.At soil depths ranging from 0.1 to 1.2 m, the time lag varies from 0.05 to 2.68, 1.43 to 5.9, and 3.91 to 5.78 h for the temperature, temperature gradient, and heat flux, respectively.It is important to highlight that Figures5A,Bunderscore the greater significance of the diurnal component than the semidiurnal one.The coherence between soil temperature gradient and soil heat flux is high for the diurnal component at 24 h for the target depths, except at a soil depth of 0.3 m (Plots a, c, and d of Figure 6, and Plot b of Figure 6).However, the coherence does not reach the NZC level for the semidiurnal component, except at a depth of 0.1 m (Plots b, c, and d of Figure 6, and Plot a of Figure 6).It determines the thermal conductivity in the soil layer using a stochastic inverse solution of Equation (11), selecting the most significant components at the diurnal and semidiurnal bands.The results reveal that the thermal conductivity at soil depths of 0.1, 0.3, 0.6, and 1.2 m is 3.61, 0.44, 0.03, and 0.07 W/(m C), respectively, for the diurnal component, while it is 7.53, 0.72, 0.03, and 1.88 W/(m C) for the semidiurnal component (see Table

F
I G U R E 4 (Continued) F I G U R E 5 (A) Time series of the soil temperature, temperature gradient, and heat flux by a diurnal bandpass filter.(B).Time series of the soil temperature, temperature gradient, and heat flux by a semidiurnal bandpass filter.To further contextualize these findings, we compare them with previous studies.Ghuman and Lal (1985) suggested a thermal conductivity range of 0.15 to 0.17 W/(m C) for sandy clay loam, while Nikolaev et al. (2013) reported a wide thermal conductivity range of approximately 0.2 to 4.5 W/(m C) for various types of sandy soil.Zhang and Wang (2017) provided thermal conductivity values for different materials, indicating a value of 1.28 W/(m C) for clay.The estimation of soil thermal conductivity using the inverse spectral solution reveals values ranging from approximately 7.53 to 0.03 W/(m C) across different soil depths, from 0.1 to 1.2 m.It demonstrates the feasibility and effectiveness of employing spectral analysis to assess soil thermal conductivity concerning heat flux and temperature gradient.The substantial variability in thermal conductivity values observed in our study, in conjunction with the wide range reported in previous research, highlights the sensitivity of this parameter to soil type and environmental conditions.This sensitivity introduces a level of uncertainty in thermal conductivity estimations that must be carefully considered in future studies and modeling efforts.However, due to foreseeable uncertainties, there is a need for further exploration to enhance our understanding of how other factors, such as external water input or other chemical and physical mechanisms, influence soil thermal conductivity within the soil water layer.It is possible to have uncertainty of estimated heat conductivity due to ignoring factors, such as soil water thermal diffusivity or convection.Studying heat conduction in soil water layers is essential for gaining insights into various hydrological processes.It helps researchers and hydrologists model and predict water movement, assess water availability, and make informed decisions about water resource management, F I G U R E 6 Cross-spectral density of the temperature gradient and heat flux at various soil depths.
and isolating apparent and periodic variations of interest.The observed variations in thermal conductivity across different soil depths emphasize the need for a comprehensive understanding of the factors influencing this parameter.Acknowledging the uncertainty and sensitivity associated with thermal conductivity estimates is essential for accurate modeling and predictions in soil science and related fields.5 | CONCLUSIONThis study explored the auto and cross spectra of temperature, temperature gradient, and heat flux within the shallow Quaternary gravel formation.By analyzing thermal conditions, it was observed that temperature gradients and heat flux measurements at depths of 0.1, 0.3, 0.6, and 1.2 m effectively capture the frequency patterns of both soil and air temperature variations.Notably, these thermal conditions exhibit dominant periodic fluctuations on both a diurnal scale and a semidiurnal scale.The diurnal component's thermal conductivity values are approximately 3.61, 0.44, 0.03, and 0.07 W/(m C) at depths of 0.1, 0.3, 0.6, and 1.2 m, respectively.Meanwhile, the semidiurnal component yields thermal conductivity values of approximately 7.53, 0.72, 0.03, and 1.88 W/(m C) at the same depths.