1 Introduction

Known as the “Third Pole of the Earth,” the Qinghai-Tibet Plateau (QTP) plays an essential role in Asian monsoon system and climate change (Ma et al. 2011; Tang et al. 1979). It has the highest elevation, widest area, and thickest permafrost in the mid- and low-latitude regions, accounting for approximately 42.4% (1.06 × 106 km2) of the QTP (Zhao et al. 2017; Zou et al. 2017). Permafrost refers to soil, rocks, or sediments that have a temperature at or below 0 °C for at least two consecutive years and is a key component of the cryosphere (Riseborough et al. 2008). Compared with the high-latitude regions, the permafrost on the QTP is more sensitive to climate change due to their relative thin thickness, high ground temperature, uneven distribution, and poor stability (Qin et al. 2020; Xu and Wu 2019). These characteristics exacerbate the degradation of the permafrost, which is manifested as a rise in soil temperature, a thickening of the active layer, and the occurrence of freeze–thaw disasters such as thermal thaw subsidence and slump (Hjort et al. 2022; Ni et al. 2020; 2022). In recent years, considerable efforts have gone into investigating the changes of the permafrost and predicting its future. One of the important aspects is the study of soil thermal conductivity (STC) in permafrost. STC, which characterizes the heat transfer capacity of the soil, affects water and energy exchange between surface and deep soil (Cui et al. 2020; De Vries 1963). Under the global warming, STC can affect the response of permafrost to external thermal disturbance and plays an important role in the study of soil freeze–thaw depth and engineering stability, such as the embankments, the tunnels, and the pipelines (Chen et al. 2021; Mu et al. 2020; Pei et al. 2019; 2022). Moreover, as a key input parameter of land surface models, accurate STC value significantly affects the simulation results of soil temperature and surface energy balance (Dai et al. 2019; Li et al. 2019; Yang et al. 2021). However, the research in this area is still insufficient and needs to be further explored.

Direct observation and model calculation can both estimate the STC. At present, direct observations can obtain accurate values on the point scale by steady-state or transient methods (Tao and Zhang 1983; Yu et al. 2014). However, direct observations are generally time-consuming and labor-intensive and also can be easily affected by the observing environment. Especially in extreme-cold areas such as the QTP, the lack of oxygen combined with the harsh nature of the environment makes it difficult to achieve detailed field observations on large scales. So, it is difficult to systematically conduct a series of studies on hydrothermal dynamics because of the limited observational data (Li et al. 2019). Therefore, a simple and accurate calculation model of STC based on field observations is necessary to understand the changes in soil moisture (\({\theta }_{\mathrm{w}}\)) and heat transport in the permafrost regions of the QTP. Recently, considerable efforts have been devoted to characterizing the relationship between various factors and STC and then developing STC models under distinct conditions.

De Vries (1963) regarded soil as a mixture of ellipsoidal particles in a continuum and established a theoretical STC model under unsaturated and unfrozen conditions. This model has been widely used in STC modeling with superior accuracy. However, it is mostly suitable for unfrozen soil with a saturation level of 10–20% (Nikolaev et al. 2013) due to the selection of critical \({\theta }_{\mathrm{w}}\) and form factor in the model calculation (Ochsner et al. 2001). Tian et al. (2016) adjusted the parameters in the De Vries model and redefined the “continuum,” which improved the simulated accuracy (He et al. 2021a). But the Tian model would underestimate STC under conditions of low \({\theta }_{\mathrm{w}}\) (Yan et al. 2019). Furthermore, more empirical and semi-empirical models have also been developed. Based on the results of over 1000 tests on 19 different soil types, Kersten (1949) established an empirical formula for STC that was simple to calculate but generally not suitable for use at low \({\theta }_{\mathrm{w}}\) (Zhao et al. 2019a). Based on the Kersten model and a large number of experimental studies, Johansen (1975) proposed the concept of normalized STC (Ke) and established a STC model based on the relationship between Ke and saturation. In subsequent studies, the Johansen model has been widely applied and evaluated for different conditions and, in general, is superior to other schemes, especially in permafrost regions (Dai et al. 2019; Peters-Lidard et al. 1998). However, the Johansen model is sensitive to quartz content, soil porosity, and other soil parameters (Balland and Arp 2005; He et al. 2019). Therefore, improvements have been made to improve the accuracy of STC calculations under different ranges of soil parameters (Balland and Arp 2005; Côté and Konrad 2005; Farouki 1981; Lu et al. 2007; Zhao et al. 2019a; Zhao and Si 2019). In addition to the development of these STC models, machine learning methods have also been used to calculate STC, such as artificial neural network, K-nearest neighbors, extreme learning machine, multilayer perceptron, support vector machine, and decision tree, which provide new light on STC (Dong et al. 2022; Li et al. 2023; Sargam et al. 2021; Wang et al. 2023; Zhang et al. 2020a; 2020b).

Although the accuracy of the improved models has been enhanced to a certain extent compared with the original, most of them still cannot give accurate STC values for permafrost (He et al. 2021a). This is mainly because the dynamic mechanisms of change for STC are highly complicated under the influence of \({\theta }_{\mathrm{w}}\), solid ice content, soil temperature, mineral composition, particle size composition, porosity, and many other factors (He et al. 2017; 2021b; Malek et al. 2021; Nikoosokhan et al. 2016). Among them, \({\theta }_{\mathrm{w}}\) plays a crucial role (Farouki 1981; Johansen 1975; Yan et al. 2019). Moreover, the presence and content of solid ice in the soil can affect the mobility of soil liquid water and drastically change the thermal properties of the soil during the frozen period. This is because there is a large difference between the thermal conductivity of ice (2.29 Wm−1K−1) and that of water (0.57 Wm−1K−1). In the negative temperature range near 0 °C, liquid water has a drastic phase transition zone, which can significantly change the thermal properties of frozen soil (Bao et al. 2016; Hu et al. 2020a; Ochsner and Baker 2008; Tian et al. 2020; Zhao et al. 2019b). Consequently, it is necessary to study the STC at different negative temperatures.

By comparing the applicability of several calculation models in permafrost regions, Du et al. (2020) selected the STC models with the best performance in frozen and thawed periods, respectively. However, when we concatenated the results of Du et al. (2020) during the frozen and thawed periods, errors in the calculations near 0 °C still existed (Fig. 1), which is consistent with the results of previous studies as well (He et al. 2021a; Kojima et al. 2016). That is, due to the large difference between the thermal conductivity of ice and water, the STC can be significantly affected by the extent of the ice-water phase change with temperature at different stages of the freeze–thaw cycle. However, the current STC models were mainly calculated by 0 °C division. This division leads to the neglect of the drastic fluctuation of STC caused by the rapid ice-water phase transition (Li et al. 2019), which affects the accuracy of the STC simulation. Hence, it is necessary to propose a more reasonable model to evaluate the STC characteristics of the permafrost on the QTP.

Fig. 1
figure 1

Comparison between soil thermal conductivity values of observed and estimated by the best schemes based on Du et al. (2020) at WDL and FHS sites

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This research aimed to (a) divide a temperature range where liquid water changes from rapid phase transitions to stable changes, (b) develop a new model under different temperature ranges, (c) produce the dataset of STC in the permafrost regions of the entire QTP, and (d) analyze the temporal and spatial characteristics of STC for the permafrost regions. It is meant to provide basic input parameter for land surface model, thus, ultimately contributing to a better understanding of the degradation of frozen soils.

2 Data and methods

2.1 Data

2.1.1 Observational sites and data

In this study, six observational sites were chosen in the permafrost area of the central QTP: Xidatan (XDT), Kunlunshan (KLS), Tedaqiao (TDQ), Qingshuihe (QSH), Wudaoliang (WDL), and Fenghuoshan (FHS) (Fig. 2). The study area covered the main types of underlying surface, such as alpine meadow, alpine swamp meadow, alpine steppe, alpine desert, and alpine desert steppe in the permafrost region of the QTP. With an average elevation of more than 4500 m, the study area belongs to the sub-frigid and semi-arid climate zones (Lin and Wu 1981), which are generally representative of the QTP. The average annual near-surface (2 m) temperature is about − 5.5 to − 3.1 ℃. Besides, the annual average relative humidity ranges from 55.2 to 53.3%, and precipitation is mainly concentrated from May to September. The characteristics of the observational sites are listed in Table 1.

Fig. 2
figure 2

The spatial distribution of the observational sites used in the study. The distribution maps of the permafrost and non-permafrost zone were derived from Zou et al. (2017)

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Table 1 Information for the observational sites used in the study (Yue et al. 2017)
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At each observational site, five soil parameters were collected, including STC, soil temperature, \({\theta }_{\mathrm{w}}\), dry bulk density of the soil (ρd), and particle size.

  1. 1.

    The first three data parameters were collected by the Cryosphere Research Station on the QTP, Chinese Academy of Sciences, from October 1, 2016, to September 30, 2017. The STC measurements were determined by a TP01 instrument with an accuracy of ± 0.05 Wm−1K−1. Soil temperature was measured using 105T Thermocouple Probe with a precision of ± 0.1 °C. The \({\theta }_{\mathrm{w}}\) was measured using a Hydra soil moisture sensor with an accuracy of ± 3%. All data were recorded every 30 min using a CR1000 datalogger (Li et al. 2019; Zhao et al. 2021). In this study, owing to a large number of invalid \({\theta }_{\mathrm{w}}\) at deep layers, the research was carried out at depths of 10–50 cm.

  2. 2.

    The remaining two parameters were obtained through laboratory experiments. Soil samples were collected vertically along the test pit profiles by employing cutting rings at different depths. Then, the soil samples were weighed, oven-dried at 105 °C for 24 h in the laboratory, and weighed again to calculated the ρd. The dry soil samples were then ground and sieved (Duan et al. 2021, 2022; Yan et al. 2021). Soil particle sizes were determined using a Malvern Mastersizer 2000 particle size analyzer (http://www.malvern.com). Lastly, the soil textures at different depths at observational sites were classified based on the classification provided by the United States Department of Agriculture (USDA).

Before the development of the new model, the soil at different depths in a given observation site had been randomly divided into a training portion and a testing portion, ensuring that both portions contained soil samples within a depth of 10–50 cm. Thus, the 10 cm depth of QSH, 20 cm depth of KLS, and 50 cm depth of FHS were randomly selected and classified as the testing portion which was utilized to evaluate the accuracy of the new model. The remaining depths were used for training portion which was applied to develop the new model.

2.1.2 ERA5 reanalysis product

The European Centre for Medium-Range Weather Forecasts (ECMWF) Reanalysis v5 (ERA5) global reanalysis product, an alternative to the ERA-Interim, is the fifth generation of meteorological reanalysis dataset published by the ECMWF and covers historical data from 1981 to the present. Compared to the ERA-Interim product (Dee et al. 2011), ERA5 significantly improves data accuracy by using several integrated forecasting systems (Czernecki et al. 2019; Hersbach et al. 2019). Additionally, the ERA5 data has higher spatial–temporal resolution and more product parameters. It is found that the soil temperature and water data from ERA5 have better performance in the QTP (Cheng et al. 2019; Yang et al. 2020; Zhao et al. 2019c). To meet the requirements of the QTP simulations, we made the following adjustments to the data:

  1. 1.

    Since the data collection of ERA5 products adopts universal time, we adjusted the time to Beijing time, i.e., universal time + 8 h.

  2. 2.

    The data of the first two-layer depths of the dataset (i.e., 0–7 cm and 7–28 cm) were weighted and averaged to obtain the data of 0–28 cm depth;

  3. 3.

    The humidity data from ERA5 refers to the total soil water content, i.e., the \({\theta }_{\mathrm{w}}\) in the thawed period, and the sum of the \({\theta }_{\mathrm{w}}\) and the solid ice content in the frozen period. Additionally, this study required a value for the \({\theta }_{\mathrm{w}}\). Hu et al. (2020b) evaluated the applicability of various formulas for calculating the \({\theta }_{\mathrm{w}}\) under the negative temperature and found that the scheme of Zhang et al. (2017a) performed relatively well. Therefore, we used Zhang et al. (2017a) to calculate the \({\theta }_{\mathrm{w}}\) from the ERA5 data as follows:

    $${\theta }_{w}\mathrm{=}\overline{\theta }\mathrm{(1-}{\mathrm{(}\frac{{\mathrm{T}}_{\mathrm{f}}-{\mathrm{T}}}{\mathrm{273.15+}{\mathrm{T}}_{\mathrm{f}}}\mathrm{)}}^{\beta }\mathrm{)}, \mathrm{ T} < {\mathrm{T}}_{\mathrm{f}}$$
    (1)

where \(\overline{\theta }\) is the mean \({\theta }_{\mathrm{w}}\) when the temperature is greater than 0 ℃; \({\mathrm{T}}_{\mathrm{f}}\) is the freezing point, which is regarded as 0 ℃ in the study; \({\mathrm{T}}\) refer to the soil temperature; \(\beta\) is the fitting parameter; and the suggested value is 0.94 based on the fitting result of the hydrothermal observation data of the active layer (Hu et al. 2020b).

2.1.3 Soil dataset of QTP

The soil dataset used in this study to simulate the STC in the permafrost regions of the entire QTP was from the Genetic Soil Data Environment (GSDE; a Chinese dataset of soil properties for land surface modeling, http://westdc.westgis.ac.cn/data) developed by Shangguan et al. (2013). The GSDE offers basic soil property information such as particle-size distributions, organic carbon, and nutrients at a resolution of 0.00833° with a depth of 0–3 m using a linkage method based on the world soil map by Shi et al. (2004). Shangguan et al. (2013) used the Harmonized World Soil Database (HWSD) to evaluate the GSDE and found that the GSDE had higher spatial accuracy and more reasonable spatial magnitudes. For this study, we selected the soil texture and ρd from the GSDE. To meet the requirements of the simulation on the QTP, data from the first four depths of the GSDE, namely, 0–4.5 cm, 4.5–9.1 cm, 9.1–16.6 cm, and 16.6–28.9 cm, were weighted and averaged to obtain soil data at 0–28 cm depths. Moreover, to match the ERA5 reanalysis data, the resolution of GSDE was resampled to 0.1°.

2.2 Existing models for soil thermal conductivity

In this study, the development of the new model was based on the STC model by Johansen (1975). Using experimental data (Kersten 1949; Smith and Byers 1938; Smith 1942), Johansen (1975) first introduced the normalized thermal conductivity concept, Ke (i.e., the Kersten’s number) and proposed a model based on the relationship between the STC, Ke, and soil saturation level (Sr) for unsaturated soils:

$$\lambda \mathrm{=}\left\{\begin{array}{c}{\mathrm{(}\lambda }_{\mathrm{sat}}-{\lambda }_{\mathrm{dry}}\mathrm{)}{\mathrm{K}}_{\mathrm{e}}\mathrm{+}{\lambda }_{\mathrm{dry}} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\mathrm{S}}_{\mathrm{r}}>{1}\times {10}^{-{5}}\\ {\lambda }_{\mathrm{dry}} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\mathrm{S}}_{\mathrm{r}}\le {1}\times {10}^{-5}\end{array}\right.$$
(2)
$${\mathrm{K}}_{\mathrm{e}}\mathrm{=}\left\{\begin{array}{c}\mathrm{0.7}{\mathrm{log}}{\mathrm{S}}_{\mathrm{r}}\mathrm{+}\mathrm{1.0} \, \, \, \, \, \, \, \mathrm{0.05}\mathrm{<}{\mathrm{S}}_{\mathrm{r}}\mathrm{<}\mathrm{0.1}\mathrm{,} \, {\mathrm{c}}{\mathrm{o}}{\mathrm{a}}{\mathrm{r}}{\mathrm{s}}{\mathrm{e}} \, {\mathrm{s}}{\mathrm{o}}{\mathrm{i}}{\mathrm{l}}\mathrm{,} \, {\mathrm{u}}{\mathrm{n}}{\mathrm{f}}{\mathrm{r}}{\mathrm{o}}{\mathrm{z}}{\mathrm{e}}{\mathrm{n}}\\ {\mathrm{log}}{\mathrm{S}}_{\mathrm{r}}\mathrm{+}\mathrm{1.0}\mathrm{,} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \mathrm{0.1}\mathrm{<}{\mathrm{S}}_{\mathrm{r}}\mathrm{,} \, {\mathrm{f}}{\mathrm{i}}{\mathrm{n}}{\mathrm{e}} \, {\mathrm{s}}{\mathrm{o}}{\mathrm{i}}{\mathrm{l}}\mathrm{,} \, {\mathrm{u}}{\mathrm{n}}{\mathrm{f}}{\mathrm{r}}{\mathrm{o}}{\mathrm{z}}{\mathrm{e}}{\mathrm{n}}\\ {\mathrm{S}}_{\mathrm{r}}\mathrm{,} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\mathrm{f}}{\mathrm{r}}{\mathrm{o}}{\mathrm{z}}{\mathrm{e}}{\mathrm{n}} \, {\mathrm{s}}{\mathrm{o}}{\mathrm{i}}{\mathrm{l}}\end{array}\right.$$
(3)
$${\lambda }_{\mathrm{dry}}\mathrm{=}\frac{\mathrm{0.135*}{\rho }_{\mathrm{d}}\mathrm{+}\mathrm{0.0647}}{{\rho }_{\mathrm{s}}-\mathrm{0.947*}{\rho }_{\mathrm{d}}}$$
(4)
$${\lambda }_{\mathrm{sat}}\mathrm{=}\left\{\begin{array}{c}{\lambda }_{\mathrm{s}}^{{1}-{\theta }_{\mathrm{sat}}}{\lambda }_{\mathrm{w}}^{{\theta }_{\mathrm{sat}}}\mathrm{,} \, \, \, \, \, \, \, \, {\mathrm{T}}\ge {\mathrm{T}}_{\mathrm{f}}\\ {\lambda }_{\mathrm{s}}^{{1}-{\theta }_{\mathrm{sat}}}{\lambda }_{\mathrm{w}}^{{\theta }_{\mathrm{w}}}{\lambda }_{\mathrm{ice}}^{{\theta }_{\mathrm{sat}}-{\theta }_{\mathrm{w}}}\mathrm{,} \, \, \, {\mathrm{T}}\mathrm{<}{\mathrm{T}}_{\mathrm{f}}\end{array}\right.$$
(5)

where λ is the estimated STC (Wm−1K−1), and λsat and λdry are the saturated and dry STC (Wm−1K−1), respectively. \({\rho }_{\mathrm{d}}\) and \({\rho }_{\mathrm{s}}\) are the soil dry density (gcm−3) and soil particle density, respectively. \({\theta }_{\mathrm{sat}}\) is the soil saturated water content (m3 m−3). λw and λice are the thermal conductivity of water (0.57 Wm−1K−1) and ice (2.29 Wm−1K−1), respectively. λs are the thermal conductivity of solid by \({\lambda }_{\mathrm{s}}\mathrm{=}{\lambda }_{\mathrm{q}}^{\mathrm{q}}{\lambda }_{\mathrm{o}}^{{1}-{\mathrm{q}}}\); λq is the thermal conductivity of quartz (7.7 Wm−1K−1); λo is the thermal conductivity of other minerals; it is suggested to be 2.0 Wm−1K−1 for the quartz content (q) > 0.2 and 3.0 Wm−1K−1 for q ≤ 0.2 (Johansen 1975); q is assumed to be 50% of the sand content (Du et al. 2020; He et al. 2021a).

$${\mathrm{S}}_{\mathrm{r}}={\theta }_{\mathrm{tot}}/{\mathrm{n}}$$
(6)

where n is porosity, and \({\theta }_{\mathrm{tot}}\) means the total water content, which implies the \({\theta }_{\mathrm{w}}\) at the positive temperature and the sum of the \({\theta }_{\mathrm{w}}\) and solid ice contents (θice) at negative temperatures, respectively. For frozen soil, assume that once frozen, the liquid water in the soil no longer migrated, and θice at a certain depth under frozen state (Tarnawski and Wagner 1993) was expressed as

$${\theta }_{\mathrm{ice}}\mathrm{=}\left({\theta }_{\mathrm{ini}}-{\theta }_{\mathrm{w}}\right)\frac{{\rho }_{\mathrm{w}}}{{\rho }_{\mathrm{ice}}}$$
(7)

where ρw and ρice are the density of water and ice, respectively. θini is the initial value during the frozen period (m3·m−3), which refers to \({\theta }_{\mathrm{w}}\) at the depth on the date of freezing (Zhao et al. 2004).

Validation of the new model was obtained by comparing with the Johansen model, Campbell model (Campbell, 1985), C&K model (Côté and Konrad 2005), He model (He et al. 2017), and Bao model (Bao et al. 2016). The first four models performed better in the evaluation study by Du et al. (2020). The Bao model (Bao et al. 2016) was derived by incorporating the influence of λice into the Johansen model and Yang model (Yang et al. 2005).

2.3 Evaluation metrics

Four evaluation indices were used to evaluate the performance of the model: (1) the root mean square error (RMSE), (2) the mean absolute error (MAE), (3) the mean bias error (MBE), and (4) the coefficient of determination (R2). They were calculated as follows:

$${\mathrm{RMSE}}\mathrm{=}\sqrt{\frac{{\sum }_{{\mathrm{i}}\mathrm{=}{1}}^{\mathrm{N}}{\left({\mathrm{E}}_{\mathrm{i}}-{\mathrm{O}}_{\mathrm{i}}\right)}^{2}}{\mathrm{N}}}$$
(8)
$${\mathrm{MAE}}\mathrm{=}\frac{{\sum }_{{\mathrm{i}}\mathrm{=}{1}}^{\mathrm{N}}\left|{\mathrm{E}}_{\mathrm{i}}-{\mathrm{O}}_{\mathrm{i}}\right|}{\mathrm{N}}$$
(9)
$${\mathrm{MBE}}\mathrm{=}\frac{{\sum }_{{\mathrm{i}}\mathrm{=}{1}}^{\mathrm{N}}\left({\mathrm{E}}_{\mathrm{i}}-{\mathrm{O}}_{\mathrm{i}}\right)}{\mathrm{N}}$$
(10)
$${\mathrm{R}}^{2}\mathrm{=}{(\frac{{\sum }_{\mathrm{i=1}}^{\mathrm{N}}\left({\mathrm{E}}_{\mathrm{i}}-\overline{E }\right)\left({\mathrm{O}}_{\mathrm{i}}-\overline{O }\right)}{\sqrt{{\sum }_{\mathrm{i} = {1}}^{\mathrm{N}}{\left({\mathrm{E}}_{\mathrm{i}}-\overline{E }\right)}^{2}}\sqrt{{\sum }_{\mathrm{i} = {1}}^{\mathrm{N}}{\left({\mathrm{O}}_{\mathrm{i}}-\overline{O }\right)}^{2}}})}^{2}$$
(11)

where Ei and Oi are the estimated and observed values, respectively; Oi is the mean value of the observed data; and N means the number of observations in the dataset.

3 Results and discussion

3.1 Development of the new STC model

It is well-known that \({\theta }_{\mathrm{w}}\) is an exponential function of soil temperature in the frozen state (Farouki 1981; Hu et al. 2023). In addition, soil temperature always affects the variability of \({\theta }_{\mathrm{w}}\) and STC during the freeze–thaw cycle. At a negative temperature near 0 °C, there is a phase where the \({\theta }_{\mathrm{w}}\) changes dramatically. As shown in Fig. 3, when the soil temperature begins to fall below 0 °C, a large amount of liquid water will rapidly condense into solid ice, and the \({\theta }_{\mathrm{w}}\) will decrease rapidly. Even a slight change in the soil temperature during this period could induce a noticeable change in \({\theta }_{\mathrm{w}}\). After reaching a certain temperature node, the \({\theta }_{\mathrm{w}}\) remains fundamentally constant, fluctuating only within a small range, even when the soil temperature changed dramatically. This variation of the \({\theta }_{\mathrm{w}}\) with the soil temperature was essentially the same at different soils, but these temperature nodes varied. In this study, “the capture percentage” of the temperature node was defined as the percentage of the rapid changing period to the actual rapid changing period of the moisture content. We selected – 1 °C, − 1.5 °C, − 2 °C, − 2.5 °C, and – 3 °C as temperature nodes and compared their capture percentage. As shown in Table 2, the capture percentage of different temperature nodes traces a parabolic curve reaching a peak at − 2.5 °C. Hence, − 2.5 °C was identified as the temperature node where \({\theta }_{\mathrm{w}}\) changed from rapid to stable states at negative temperatures.

Fig. 3
figure 3

Variation in the soil liquid moisture under the negative soil temperature

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Table 2 The capture percentage of the temperature node for rapid changing period
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In the STC calculation models, Ke was mostly calculated as a function of Sr under positive temperature which was an improvement on the Johansen model and was assumed to be equal to Sr under negative temperature. In other words, the relationship between Ke and Sr was explored independently at positive and negative temperatures. In the study, we assumed 0 °C and − 2.5 °C as the temperature segments to analyze the relationships between Ke and Sr under different temperature ranges (Fig. 4). It should be noted that, as Ke is not an inherent property of soil but a concept of normalized STC, the value of Ke cannot be obtained through actual measurement. In this study, the Ke value of soil samples was inversely derived under Eq. (2), Eq. (5), and Du-λdry scheme (Du et al. 2022). It can be seen that when soil temperature > 0 °C, Sr increased with the occurrence of soil thawing and rainfall, and the soils consistently displayed a logarithmic increasing trend in Ke. When the soil temperature was less than 0 °C, the Sr was fundamentally unchanged, but the Ke value fluctuated within a certain range, especially in the − 2.5 ~ 0 °C temperature range, where the fluctuation reached up to 0.6. Hence, setting Ke equal to Sr under negative soil temperatures would lead to large calculation errors and eventually to an inaccurate STC during the frozen period, especially in the temperature range of − 2.5 ~ 0 °C. We then performed a stepwise regression of the effect of soil properties on Ke in different temperature ranges (Table 3). When soil temperature ≥ 0 °C, the change in Ke is deeply affected by the \({\theta }_{\mathrm{w}}\) and ρd, while, under negative temperatures, \({\theta }_{\mathrm{tot}}\) and soil clay content determine Ke, but the influence degree of the above two factors is different in 0 ~  − 2.5 °C range and less than − 2.5 °C, respectively. Then, the formula for Ke in different temperature ranges was established as follows:

$${\mathrm{K}}_{\mathrm{e}}\mathrm{=}\left\{\begin{array}{c}\mathrm{0.173}\mathrm{+}\mathrm{0.544}{\mathrm{e}}{\mathrm{x}}{\mathrm{p}}\mathrm{(}{\theta }_{\mathrm{w}}\mathrm{)}-\mathrm{0.064}{\mathrm{e}}{\mathrm{x}}{\mathrm{p}}\mathrm{(}{\rho }_{\mathrm{d}}\mathrm{)}\mathrm{,} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, T\ge 0^\circ{\rm C} \\ \mathrm{5.507}-\mathrm{6.777}{\mathrm{e}}{\mathrm{x}}{\mathrm{p}}\mathrm{(}{\mathrm{\%}}_{\mathrm{clay}}\mathrm{)}\mathrm{+}\mathrm{1.444}{\mathrm{e}}{\mathrm{x}}{\mathrm{p}}\mathrm{(}{\theta }_{\mathrm{tot}}\mathrm{)}\mathrm{,} \, \, \, \, \, \, \, \, \, -2.5^\circ{\rm C} \le T<0^\circ{\rm C} \\ \mathrm{0.005}-\mathrm{14.103}{\mathrm{(\%}}_{\mathrm{clay}}\mathrm{)}\mathrm{+}\mathrm{2.594}{\theta }_{\mathrm{tot}}\mathrm{,} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, T<-2.5^\circ{\rm C} \end{array}\right.$$
(12)

where \({\theta }_{\mathrm{tot}}\) is the total water content under the negative temperature, which means the sum content of moisture content (\({\theta }_{\mathrm{w}}\)) and solid ice content, and \({\mathrm{\%}}_{\mathrm{clay}}\) is the soil clay content.

Fig. 4
figure 4

The relationship between Ke and degree of saturation under different ranges of soil temperature

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Table 3 Errors of the Ke parameter of the fitting group under different influencing factors
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Hence, based on the studies of Johansen (1975) and Du et al. (2022) and the new Ke model, a new model for STC has been proposed. The λsat scheme was proposed by Johansen (1975). Additionally, Du et al. (2022) developed a new model for λdry with different calculation methods under a demarcation point of \({\rho }_{\mathrm{d}}\) = 1.4 gcm−3. The new STC model was established as follows:

$$\uplambda \mathrm{=}\left\{\begin{array}{c}{\mathrm{(}\lambda }_{\mathrm{sat}}-{\lambda }_{\mathrm{dry}}\mathrm{)}{\mathrm{K}}_{\mathrm{e}}\mathrm{+}{\lambda }_{\mathrm{dry}} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\mathrm{S}}_{\mathrm{r}}>{1}\times {10}^{-{5}}\\ {\lambda }_{\mathrm{dry}} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\mathrm{S}}_{\mathrm{r}}\le {1}\times {10}^{-{5}}\end{array}\right.$$
(13)
$${\lambda }_{\mathrm{sat}}\mathrm{=}\left\{\begin{array}{c}{\lambda }_{\mathrm{s}}^{{1}-{\uptheta }_{\mathrm{sat}}}{\lambda }_{\mathrm{w}}^{{\theta }_{\mathrm{sat}}}\mathrm{,} \, \, \, \, \, {\mathrm{T}}\ge {0}^\circ{\rm C} \\ {\lambda }_{\mathrm{s}}^{{1}-{\theta }_{\mathrm{sat}}}{\lambda }_{\mathrm{w}}^{{\theta }_{\mathrm{w}}}{\lambda }_{\mathrm{ice}}^{{\theta }_{\mathrm{sat}}-{\theta }_{\mathrm{w}}}\mathrm{,} \, {\mathrm{T}}<{0}^\circ{\rm C} \end{array}\right.({\mathrm{Johansen}}, 1975)$$
(14)
$${\lambda }_{\mathrm{dry}}\mathrm{=}\left\{\begin{array}{c}{\mathrm{e}}{\mathrm{x}}{\mathrm{p}}\mathrm{(}-\mathrm{1.432}/{\rho }_{\mathrm{d}}-\mathrm{0.494}\mathrm{)}\mathrm{,} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\rho }_{d}<1.4 g {{\mathrm{cm}}}^{-3}\\ \frac{\left({\kappa }_{s}{\lambda }_{\mathrm{s}}-{\lambda }_{\mathrm{a}}\right){\rho }_{\mathrm{d}}\mathrm{+}{\lambda }_{\mathrm{a}}{\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{s}}-\mathrm{(1}-{\kappa }_{s}\mathrm{)}{\rho }_{\mathrm{d}}}\mathrm{,} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\rho }_{d}\ge 1.4 g {{\mathrm{cm}}}^{-3}\end{array}\right.(\mathrm{Du et al}., 2022)$$
(15)
$${\mathrm{K}}_{\mathrm{e}}\mathrm{=}\left\{\begin{array}{c}\mathrm{0.173}\mathrm{+}\mathrm{0.544}{\mathrm{e}}{\mathrm{x}}{\mathrm{p}}\mathrm{(}{\theta }_{\mathrm{w}}\mathrm{)}-\mathrm{0.064}{\mathrm{e}}{\mathrm{x}}{\mathrm{p}}\mathrm{(}{\rho }_{\mathrm{d}}\mathrm{)}\mathrm{,} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, T\ge 0^\circ{\rm C} \\ \mathrm{5.507}-\mathrm{6.777}{\mathrm{e}}{\mathrm{x}}{\mathrm{p}}\mathrm{(}{\mathrm{\%}}_{\mathrm{clay}}\mathrm{)}\mathrm{+}\mathrm{1.444}{\mathrm{e}}{\mathrm{x}}{\mathrm{p}}\mathrm{(}{\theta }_{\mathrm{tot}}\mathrm{)}\mathrm{,} \, \, \, \, \, \, \, \, \, -2.5^\circ{\rm C} \le T<0^\circ{\rm C} \\ \mathrm{0.005}-\mathrm{14.103}{\mathrm{(\%}}_{\mathrm{clay}}\mathrm{)}\mathrm{+}\mathrm{2.594}{\theta }_{\mathrm{tot}}\mathrm{,} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, T<-2.5^\circ{\rm C} \end{array}\right.$$
(16)

3.2 Validation of the new model for STC

Figure 5 compares the measurements and estimates by the previous models and the new STC model in the testing portion, and Table 4 shows the performance metrics for each model. The results showed that the new model produced the most accurate predictions, with only a slight underestimation. At 10 cm depth, the C&K model resulted in best predictions, followed by Campbell model and the new model. At 20 cm depth, the new model had the smallest RMSE, MAE, and MBE. And at the depth of 50 cm, the errors of the new model were significantly reduced than those in other models.

Fig. 5
figure 5

Comparison of the estimated STC with the measured values at depth of 10–50 cm in the testing portion

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Table 4 The error values of the estimated STC by different models of the testing portion
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To further ascertain the relative performance of the above models over different temperature ranges, the mean values of the models were calculated at depths of 10–50 cm, and then, the values of RMSE and R2 were given (Fig. 6). The results showed that the RMSE value of the new model was the smallest over any temperature range, which means a significant improvement compared with other models. Furthermore, the R2 of the new model was best in the range of − 2.5 ~ 0 °C and also ranked well in other temperature ranges. In other words, the new model with demarcation points of 0 °C and − 2.5 °C achieved the best fit within a wide temperature range. Thus, the new model could reflect changes in the STC at depths of 10–50 cm in permafrost regions with favorable simulation performance.

Fig. 6
figure 6

RMSE (a) and R2 (b) of different STC models under different ranges of soil temperature

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3.3 Temporal and spatial characteristics of STC in the permafrost region of the QTP

Based on the new model, the daily STC product in the permafrost region of the QTP at the depths of 0–28 cm was estimated by combining the ERA5 product and the GSDE dataset. Then, the temporal and spatial characteristics of STC in the permafrost region from 1982 to 2020 were analyzed. Note that the ERA5 product used ended on December 31, 2020. Thus, the research stage of the frozen period runs until 2019, which was the period from October 1, 2019, to April 30, 2020.

As illustrated in Fig. 7, the average STC value of the permafrost was about 0.495 Wm−1K−1, and those in the frozen period (from October of that year to April of the following year) and thawed period (from May to September) were 0.262 Wm−1K−1 and 0.823 Wm−1K−1, respectively. In recent 39 years, the interannual variation of STC in the permafrost region of QTP showed a tiny increasing trend, which was the same in both the thawed and frozen periods. Overall, the STC in the permafrost region of the QTP increased slowly from 1982 to 2020.

Fig. 7
figure 7

Interannual variation features of STC over the permafrost region on QTP during 1982–2020

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Figure 8 shows the spatial characteristics of the STC in the permafrost region of the QTP. Geographically, the STC was high in the southeast and low in the northwest (Fig. 8a). In general, the lower values occurred in high-altitude mountains, which have lower mean annual ground temperatures (MAGTs) (Ni et al. 2020). As illustrated in Fig. 8b and c, on the whole, in both frozen or thawed periods, STC values were high in the southeast and low in the northwest. Specifically, the STC values were concentrated in the 0.2 ~ 0.3 Wm−1K−1 range during the frozen period, with little spatial difference. High values of STC were found in the eastern and southeastern regions where with relatively high \({\theta }_{\mathrm{w}}\). In contrast, during the thawed period, the STC values were concentrated in the 0.6 to 1.0 Wm−1K−1range, with a large spatial spread and scatter, which was related to the spatial spread of precipitation. The differences in spatial distributions of STC in the permafrost regions of the QTP are the result of climate, soil temperature, \({\theta }_{\mathrm{w}}\), vegetation type, soil texture, and other factors. First, from east to west, the climate of the QTP gradually changes from sub-humid to extremely arid, with large variations. According to Yu et al. (2015), the high summer temperature values of the QTP usually occur in the southwest, southeast, and northeast regions of the QTP. Additionally, the \({\theta }_{\mathrm{w}}\) in the permafrost region of the QTP is higher in the southeast, but lower in the southwest and hinterland due to the spatial distribution of precipitation (Zhao et al. 2019c). This results in a better ecological environment and higher vegetation coverage in the eastern part of the QTP (Yang et al. 2009). On the contrary, the western part’s ecological environment is poor and its vegetation coverage is low (Wang et al. 2016). Generally speaking, well-developed vegetation can effectively increase soil moisture and organic matter content (Hu et al. 2022), further improving STC. In addition, a previous study found that the degree of soil development and the number of soil types were higher in the eastern part of the QTP than in the western part (Li et al. 2015). All these factors may lead to significant spatial differences in the STC. Our evaluation results in the permafrost region in the QTP differ slightly from previous results (Liu et al. 2023) during the frozen period. There are two reasons for these differences: (1) Our results are for the top 28 cm of soil, while Liu’s results are for the top 5 cm. (2) Our results are mainly calculated from parameterization model developed by the measured data, while those of Liu et al. (2023) are from XGBoost (a machine learning model). Moreover, there are some errors and differences in the results caused by different input data including some known errors in our data results, which are analyzed in Section 3.4.

Fig. 8
figure 8

Spatial distribution of STC across the permafrost regions on the QTP from 1982 to 2020. a Mean annual. b Frozen period. c Thawed period

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Figure 9 shows the spatial distribution of the mean STC in the permafrost region of the QTP from 1982 to 2020. From December to the following March, the soil was in the fully frozen stage, and the STC at this stage was essentially the same. After April, the STC in the boundary region of the permafrost began to rise, as the frozen soil began to thaw due to rising air temperatures (Fig. 13). In May, further increases in air temperature, with associated increases in precipitation from the onset of the summer monsoon (Zhang et al. 2017b), led to an increase in the \({\theta }_{\mathrm{w}}\), followed by an increase in STC, especially in the transition zone of the permafrost region and the seasonal permafrost region. It was found that the average annual ground temperature of the permafrost in this area was often close to 0 °C. This type of permafrost was extremely sensitive to climate change and was frequently the main constituent of the permafrost degradation areas (Ni et al. 2020). From June to August, the plateau experiences its maximum air temperature and precipitation (Fig. 13). The soil in the permafrost region thawed completely, and in addition, the surface solar radiation, \({\theta }_{\mathrm{w}}\), and soil temperature were large. In August, the STC in the permafrost region reached its yearly maximum. Notably, large values of STC appeared in the southeastern part of the QTP, mainly due to higher precipitation in this region under the influence of abundant water vapor from the Indian Ocean and the Bay of Bengal (Feng and Zhou 2012). In September, the summer monsoon gradually slowed and precipitation gradually decreased from the northwest to the southeast; the STC also decreased following the same pattern. Beginning in October, the air temperature began to drop, and the soil temperature fell accordingly. As liquid water turns into solid ice, the soil begins to freeze. The STC started to decrease first in the hinterland of the QTP. By November, the soil was almost fully frozen in most areas, except for the high \({\theta }_{\mathrm{w}}\) areas in the southern area, and the STC values mostly began to stabilize. In general, as the temperature increased, the STC in the permafrost region of the QTP increased from the permafrost boundary region to the central region and from the southeast to the northwest and then decreased in the opposite direction as the temperature decreased.

Fig. 9
figure 9

Monthly spatial distribution of STC over the permafrost on the QTP from 1982 to 2020

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Previous studies have pointed out that the variation of soil thermal properties (STC, soil heat capacity, etc.) and the heat conversion during the ice-water phase transition may cause abnormal soil temperature fluctuations (Yang and Wang 2019) and further affect the stability of the permafrost (Ni et al. 2022). Figure 10 shows the spatial change rate of the STC in the permafrost region from 1982 to 2020. We can see that the annual change rates were positive, except for in some sporadic areas. In other words, the STC in the permafrost region of the QTP mainly showed a consistent increasing trend, but the overall change was slight (0.008 Wm−1K−1/10a). It is noteworthy that STC increased rapidly in the southwest of the QTP, where annual average values increased in the range of 0.01 ~ 0.03 Wm−1K−1/10a, which might be related to its higher rate of temperature rise (Ran et al. 2018) and greater increasing trend in \({\theta }_{\mathrm{w}}\) (Zhao et al. 2019c).

Fig. 10
figure 10

Change rate of the STC across the permafrost regions on the QTP from 1982 to 2020

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Based on the mean annual ground temperature (MAGT) and GIPL2 model, permafrost areas on the QTP were generally divided into six types: extremely unstable permafrost, unstable permafrost, transition permafrost, sub-stable permafrost, stable permafrost, and extremely stable permafrost (Cheng and Wang 1982; Qin et al. 2017) (Fig. 13). Combined with the permafrost type, we can see that the higher STC values are typically distributed in the transition region between permafrost and seasonally frozen ground (Fig. 8), which belongs to the unstable permafrost type in danger of thaw settlement (Ni et al. 2022). Areas showing decreasing trend in STC were roughly distributed in sporadic regions along the north-western and southeastern margins (Fig. 10), most of which were considered extremely stable permafrost, and the rate of change in these areas was between − 0.033 and 0 Wm−1K−1/10a. Permafrost with STC increasing rates between 0 and 0.01 Wm−1K−1/10a was mostly identified as stable permafrost and sub-stable permafrost. Regions with relatively rapid STC growth rates were concentrated in the eastern, southeastern, and south-western regions and were mostly considered unstable and extremely unstable permafrost. As we all know, under the continued warming of the QTP, the increase of soil temperature and the drastic phase transition of the ice-water causing increase in STC. Higher STC would entail more heat entering the ground and hasten permafrost degradation (Mekonnen et al. 2021). This confirms the reliability of our simulation results to some extent. However, the factors causing permafrost degradation are very complex (Chang et al. 2022; Ding et al. 2019; Guo and Wang 2017; Li et al. 2021), and the specific degradation mechanism needs to be further studied. Overall, the spatial features of the rates of STC change in the permafrost regions of the QTP over the past 40 years showed significant increases in the southeastern, eastern, and south-western parts; a slight increase in the hinterland; and a significant decrease along the north-western margin.

3.4 Possible causes for simulation STC in the permafrost region of the QTP

To access the accuracy of the estimated STC product, we compared them with the observed data along the Qinghai-Tibet Highway (Fig. 11). As shown in Fig. 11, when the soil temperature was above 0 °C, the calculated STC corresponded nicely with the observed value, with a 13.8% error. However, at negative soil temperatures, the new model gave significantly lower STC values compared to those of the observations, resulting in 366.9% error. Farouki (1981) estimated that reasonable predictions were those that do not deviate more than about 25% from measured values. In other words, the calculated STC data based on the new model and the ERA5 data in the permafrost region of the QTP were feasible during the thawed period. Possible causes of the simulation errors are discussed below.

Fig. 11
figure 11

Comparison between STC values of observed and estimated along the Qinghai-Tibet Highway from October 1, 2016, to September 30, 2017

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3.4.1 The influence of the input data

The establishment of the new model was based on observations from field stations, ensuring both the reliability and continuity of the data. However, when developing the STC product in the permafrost regions of QTP, soil temperature and water data from ERA5 data and the soil data of GSDE were used as substitutes for input data due to the lack of field observation data.

There have been numerous studies on the applicability of ERA5 data to the QTP. Yang et al. (2020) pointed out that the soil temperature and water data from ERA5 performed well on the QTP during the thawed period. Figure 12 shows the variations in the observed \({\theta }_{\mathrm{w}}\) and those calculated from ERA5 data at temperatures below 0 °C at the XDT site. The difference between the calculated values and the observed values was large, with a clear overestimation. This inaccuracy also occurred at several other sites in this study and inevitably weakens the contribution of the ice content and the phase transition to the STC. This may be one of the main reasons for the underestimated values of the STC in the permafrost regions of the QTP during the frozen period.

Fig. 12
figure 12

Variations in the soil moisture content of observed and estimated by ERA5 below 0 °C at XDT site from October 1, 2016, to September 30, 2017

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In addition, there are some errors between the soil texture derived from the GSDE dataset by Shangguan et al. (2013) and the observed reality. Yang et al. (2020) compared this dataset and the laboratory measurements along the Qinghai-Tibet Highway and found that the GSDE overestimated the clay content and underestimated the sand content, which enhanced the water retention capacity of the soil and thus increased the \({\theta }_{\mathrm{w}}\). This would also increase the calculation error of the new model.

In summary, the quality of the STC product performed better during the thawed period than that during the frozen period. In the future, with the improvement of the accuracy of the input dataset, the accuracy of the STC product will be also improved.

3.4.2 The shortcomings of the new model

There are many factors affecting STC, such as \({\theta }_{\mathrm{w}}\), ice content, soil temperature, soil texture, organic matter content, salinity, and vegetation type (He et al. 2019; 2021b; Malek et al. 2021; Nikoosokhan et al. 2016). The improved model in this study considered primarily mineral soil, but delegated the effects of soil organic matter, gravel fraction, soil salinity, or other factors. For example, the salinity affects the freezing temperature of liquid water and then affects the variation of STC through soil temperature and \({\theta }_{\mathrm{w}}\). Moreover, soil organic matter affects the bulk density of the soil and the thermal resistance between particles, resulting in changes in soil porosity, \({\theta }_{\mathrm{w}}\) and other factors that have a great impact on STC. Hence, the inclusion of more influential factors in the calculation model may increase its reliability.

The choice of − 2.5 °C as a universal threshold may not be appropriate for all soil types, particularly given substantial variations in soil freezing characteristic curves between fine and coarse soils (Hu et al. 2020b; Teng et al. 2020). The soil types and other soil properties in the permafrost region of the QTP are not entirely consistent. Liu et al. (2022) indicated that the soil particles of the QTP are dominated by sand, especially in the northwest portion; the southeastern QTP has a high content of silt and clay. In this study, the soil particles of the soil samples from the Qinghai-Tibet Highway are mainly sand content, with an average content of 93%. That is, our new model is suitable for predicting STC for coarse soils. This might bring some calculation errors to the simulation of STC in the southeastern QTP. Hence, establishing different temperature nodes for different soil types should be further studied in future works.

4 Conclusions

This study divided three temperature range to develop a new STC model in the permafrost region of the QTP. Based on the new model, the daily STC product over the permafrost region of the QTP from 1982 to 2020 was estimated. The main conclusions were as follows:

  1. 1.

    In the temperature range of − 2.5 to 0 ℃, the moisture undergoes a rapid phase change with slight changes in soil temperature. When the temperature was below − 2.5 ℃, the moisture content remains basically unchanged; even if the temperature changes greatly, it changes smoothly.

  2. 2.

    The simulation accuracy of the new model had been improved in permafrost regions, and it could better reflect the changing properties of the STC, especially for soils of the 50 cm depth.

  3. 3.

    The average value of the STC in the permafrost regions on the QTP was about 0.495 Wm−1K−1 during 1982 to 2020, showing a spatial pattern of low STCs in the northwest and high STCs in the southeast.

  4. 4.

    From 1982 to 2020, the STC showed a relatively slow increasing trend with a change rate of 0.008 Wm−1K−1/10a. Spatially, the regions with the highest rates of increase were concentrated in the east, southeast, and south-west and included mostly unstable or extremely unstable permafrost.

  5. 5.

    The quality of the STC product performed better during the thawed period than that during the frozen period. In the future, with the improvement of the accuracy of the input dataset (e.g., soil moisture content, soil texture), the accuracy of the STC product will be also improved.