A symmetrical exponential model of soil temperature in temperate steppe regions of China

Hui Zhang; Huishi Du; Shuangyuan Sun; Yitong Wang; Ting Wang; Linghui Li

doi:10.1515/geo-2022-0523

Article Open Access

A symmetrical exponential model of soil temperature in temperate steppe regions of China

Hui Zhang , Huishi Du , Shuangyuan Sun , Yitong Wang , Ting Wang and Linghui Li

Published/Copyright: August 11, 2023

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From the journal Open Geosciences Volume 15 Issue 1

Abstract

Global warming has caused changes in various ecological processes and has potential to change ecosystems’ stability. In spite of comprehensive studies to investigate air temperatures under global warming, much less is known about changes in soil temperatures, particularly in deep layers. Herein, we used 30 years of soil temperature data from a temperate steppe region to assess vertical characteristics and their changes in soil temperature from the surface to a depth of 3.20 m. We determined, apparently for the first time, that the soil temperature is the lowest at 0.2 cm layer at an annual level. Furthermore, the vertical variation of soil temperature (temperature variation with soil depth) strictly conformed to composite exponential function curves, and there were two composite exponential function curves that are symmetric to each other, to represent soil temperature in a pair of months with a difference of 6 months. Parameters in the functions changed as the soil warmed over 30 years. This finding explored the pattern of soil temperature in deep layers depending on the mathematics model. Model building and understanding is beneficial for predicting vertical and temporal extensions of soil temperature and their impact on below-ground processes in regional ecosystem.

Keywords: temperate steppe; soil temperature; vertical variation; exponential function; soil depth

1 Introduction

Global warming influences a series of ecological processes and ecosystems’ stability. Comprehensive studies have been performed to investigate the rising air temperature and its impact on various ecosystem processes at global and regional levels [1,2,3,4,5,6]. According to the Fifth IPCC Assessment Report [7], the global average temperature increased by about 0.85°C from 1880 to 2012. Since the beginning of the 21st century, the temperature has continued to rise. According to the Sixth IPCC Assessment Report, the global average annual air temperature increased by 0.99°C in the 21st century (2001–2020) compared with the pre-industrial era (1850–1900) [8]. All the above findings have proved that global warming has a certain impact on ecosystems changes.

In terrestrial ecosystems, soil plays an important role in both natural environment and human beings; hence, research of soil temperature is very valuable, and large-scale soil temperature has been studied since the beginning of this century. Global mean trends in ground temperature significantly increased for all datasets over multi-decadal periods [7,8]. On the national level, the analysis of the data from 292 stations in the United States showed that the soil temperature (depth of 10 cm) increased by 0.32°C over 10 years [9,10]. Qian et al. found that the soil temperature in Canada in the 20th century experienced significant warming, with the complex response process of soil temperature to air temperature and precipitation changes having an obvious impact on climate change [11]. The research results of Russia, the United Kingdom, the Netherlands, and other countries and regions in regional level showed that the soil temperature generally presents an increasing trend in recent decades [12,13,14,15,16,17]. In conclusion, soil temperatures in parts of the world are rising in various degrees.

In China, Zhang et al. determined that soil surface temperature increased 31% more than air temperature, and the soil temperature rise was predicted to have increased soil respiration by up to 28%, reinforcing climate warming and extending the potential growing season by up to 20 days across China, highlighting the importance of directly using soil temperature to assess and predict soil processes, rather than using air temperature instead of using soil temperature [18]. Further studies explained asymmetry between day and night in terms of increases in soil surface temperature in China. The soil diurnal temperature range (SDTR) decreased at most stations (average rate of −0.025°C/year), with the most profound decrease in winter (−0.08°C/year), from 1962 to 2011. The solar duration was positively related to SDTR and is regarded as the key underlying cause of the decreasing SDTR [19]. The latest finding indicated that the annual average 0 cm ground temperature increased at rates of 0.41 from 1961 to 2020 in China, with the greatest in winter and the northern China [20]. Although sufficient studies have been carried out to reveal the spatiotemporal soil temperature characteristics and changes, less is known about the deep layers soil, on account of missing data that have limited the quality of these values. Zhang et al. apparently first reported the soil temperature changes to a depth of 3.2 m analyzed across China; however, the aforementioned results are just numerical demonstrations, without corresponding function expression and curve fitting based on correlation at each layer level based on vertical pattern [19].

Indeed, understanding the vertical variation of soil temperature can not only speculate the soil temperature below the standard detector of the weather station over a large area, potentially leading to more exploration of temperature-driven soil processes, such as respiration and microbial activity [21,22,23], but also help to realize the relationship between crop phenology and soil temperature at each layer, especially for harvest data of yearly plants and overwintering capacity of perennial plants [22,24]. It is significant to implore the vertical characteristics and functional distribution of soil temperature along the soil layer.

Hence, this study analyzed daily observation data from 1962 to 2011 in the temperate steppe region of China. The associated science questions are: (1) annual and seasonal soil temperature changes, (2) to simulate vertical characteristics of soil temperatures with a high fit model, and (3) how these functions show the changes in soil temperature characteristics during different periods and their potential implications. This is apparently the first report on the parametric quantification of vertical characteristics in soil temperature to a depth of 3.2 m using non-linear models.

2 Materials and methods

2.1 Study region

This study investigated China’s temperate steppe region (Figure 1), which is mainly distributed in northern China and dominated by an arid and semi-arid climate [21]. The elevation of this zone ranges from 60 to 4,441 m, which is generally lower from west to east. The annual average temperature in this region varies from −5 to 10°C, and the annual precipitation ranges from 350 to 530 mm [22,23].

Figure 1

Geographical distribution of the nine weather stations used in this study.

2.2 Data source and collation

Soil temperature data were obtained from the China Meteorological Administration (CMA), and all obtained data were measured (CMA standard protocols) in the national CMA weather station network. Soil temperature was measured at eight depths: 0, 0.05, 0.10, 0.20, 0.40, 0.80, 1.60, and 3.20 m.

The initial quality control of the soil temperature data was performed by the CMA, and the details are as follows:

First, a qualified dataset was established for available stations in China’s temperate steppe region. For all soil temperature variables (eight layers) and any given month to be included, two-thirds of daily records must have been available. This criterion reduced the number of stations (with a total of 12 months of valid data) to 360. Therefore, daily soil temperature records from nine stations were used to calculate monthly average soil temperatures. For any month and station, if the calculated average temperature differed from the 30-year average (1982–2011) for that month by at least three times the standard deviation from the mean, that value was treated as an outlier and was replaced by the average of the three nearest and evenly distributed stations for that month and year. After this quality check, the average monthly values of soil temperatures in each layer and each station were used for the following assessments.

Then, for convenience and consistency with other studies, the seasons here comprise whole months: winter is December, January, and February (DJF); spring is March, April, and May (MAM); summer is June, July, and August (JJA); and autumn is September, October, and November (SON). Hence, monthly soil temperature records from nine stations were used to calculate seasonal/annual soil temperatures for each station, and then, average seasonal/annual values in soil temperatures of the nine stations were calculated for the whole study region.

2.3 Soil temperature change

Trends of changes in soil temperature were analyzed using the monthly averages across 30 annum together with linear regression and the Mann–Kendall test. The rate of change per year was derived as the slope of the linear regression line (k), and the total change in 30 annum was calculated as k × 30. For each season, the seasonal average was calculated from those of 3 months first, and then, seasonal absolute changes were obtained using the same regression process for monthly and annual trends.

2.4 Exponential/logarithmic model

We used scatter plots in the coordinate system to describe the relationship between soil depth and soil temperature, and we then found that composite exponential functions exactly fitted those plots. A set of strict symmetric composite exponential function curves were used to reflect the vertical variation of soil temperature in 2 months with an interval of half a year, e.g., January and July, April, and October. A set of composite exponential functions can be expressed as:

y = a + b ( 1 − e − c x ) ,

y = a + b ( 1 − e − c ( d − x ) ) ,

where a, b, c, and d are the parameters.

x = d was used to express the axis of symmetry of coupled functions.

In order to emphasize the temporal changes in vertical variation of soil temperature and check the feasibility of the models, we divided the study period into two stages: 1982–1995 and 1996–2011, using the same data processing, analysis, and simulation as mentioned earlier.

3 Results

3.1 Soil temperature change at each depth

The mean soil temperature at each depth fluctuated in a range of 10.32–10.62°C over the 30 annum interval of the study (Table 1 and Figure 2). The surface soil temperature was the highest (10.62°C) among the eight depths. The soil temperature showed a clear decrease as the soil deepened until reaching the 0.2 m layer, and then, the temperature increased further from 0.2 to 3.2 m (10.57°C). In short, the soil temperature was lowest at 0.2 m on the annual level. The range of soil temperature change between the eight layers had a clear seasonal pattern, higher temperature in two seasons, winter (15.72°C) and summer (14.24°C), and lower temperature in another seasons, spring (5.85°C) and autumn (4.17°C). The soil temperature increased at deeper layers in autumn and winter gradually; in winter, the soil temperature increased from −0.68 to 0.82°C, from the surface down to 0.8 m below the surface, and further to 9.64°C at 3.2 m, for a total change of nearly 16°C. Autumn soil temperature increased from 9.70 to 13.87°C from the surface down to 3.2 m below the surface, with a negligible decrement from 1.6 to 3.2 m. The temperature decreased at deeper soil layers in spring and summer gradually; spring soil temperature declined from 13.11 to 7.26°C, from the surface to 3.2 m below the surface, with a negligible increment from 1.6 to 3.2 m. Summer soil temperature declined from 25.74 to 11.50°C, from the surface to 3.2 m below the surface, for a decrement of 14.24°C.

Table 1

Average value of soil temperature in each layer on annual and seasonal scales in temperate steppe regions of China, 1982–2011 (°C)

Season	0 cm	0.05 m	0.1 m	0.15 m	0.2 m	0.4 m	0.8 m	1.6 m	3.2 m	Range
Annual	10.62	10.32	10.34	10.35	10.32	10.45	10.42	10.50	10.57	0.30
Winter (DJF)	–6.08	–4.66	–4.08	–3.61	–3.20	–1.79	0.82	4.99	9.64	15.72
Spring (MAM)	13.11	11.70	11.16	10.68	10.23	9.48	7.76	6.49	7.26	5.85
Summer (JJA)	25.74	24.13	23.67	23.30	22.89	21.74	19.47	15.81	11.50	14.24
Autumn (SON)	9.70	10.10	10.62	11.04	11.34	12.35	13.62	14.69	13.87	4.17

DJF, December, January, February; MAM, March, April, May; JJA, June, July, August; SON, September, October, November.

Figure 2

Thirty-year average soil temperature (°C) in each layer on annual and seasonal scales in temperate steppe regions of China, 1982–2011.

3.2 Annual trends

Averaged across all stations, the annual trend of soil temperature at eight different depths is similar, except for the significant increase in temperature at individual depths in the second half of the research period (Table 2 and Figure 3). Soil warming was greatest at the surface (0.62°C/10 annum), followed by shallower layers (0.38–0.41°C/10 annum), and was least in deeper layers (0.31–0.32°C/10 annum). There were decrements in soil temperature during 1995 and then a rapid increase after 1996 in all depths; over the next 4 years, not only did the soil surface temperature experience a 2.3°C increment, but the shallower and deeper layers experienced increments of 2.0 and 1.5°C, respectively. Table 2 shows that there were no obvious annual soil temperature trends in each layer during the first stage (1982–1996).

Table 2

Trends of soil temperature in each layer during different periods in temperate steppe regions of China (°C/10 annum)

Stage	0 cm	0.05 m	0.1 m	0.15 m	0.2 m	0.4 m	0.8 m	1.6 m	3.2 m
1982–2011	0.62*	0.41*	0.38*	0.38*	0.34*	0.32*	0.32*	0.31*	0.33*
1982–1996	0.08*	0.05	0.05	0.06*	0.05	–0.01	0.03	0.04*	0.11*
1997–2011	0.69*	0.38*	0.26*	0.32*	0.3*	0.14	0.24*	0.22*	0.23*

* p < 0.05.

Figure 3

Time series of soil temperature in each layer in temperate steppe regions of China, 1982–2011.

As for the second stage (1997–2011), there was a significant increase in soil temperature in each layer; soil warming was greatest at the surface (0.69°C/10 annum), followed by shallower layers (0.26–0.38°C/10 annum) and least in deeper layers (0.14–0.24°C/10 annum).

3.3 Exponential model

Take the annual average soil temperature values of nine stations as a whole for a study period of 30 years. Figure 4 depicts the vertical structure of soil temperature in the temperate steppe region. It is observed that there were considerable differences in surface soil temperature over 12 months; the order of magnitude was January > December > February > November > March > October > April > September > May > June > August > July (range, –9 to 28°C). The temperature range diminished with soil deepened; at the deepest soil layer, 3.2 m, the annual temperature variation range narrowed to 10°C.

Figure 4

Vertical structure of soil temperature of each month in temperate steppe regions of China, 1982–2011.

Furthermore, in a new discovery, the variation of soil temperature with the depth of soil in each month can be fitted by composite exponential functions.

For a composite logarithmic function curve used to reflect the vertical variation of soil temperatures in January (Figure 4), the functional equation is y = –13.51 + 14.28(1 − e^{−0.11(21−x)}); for a composite exponential function curve used to reflect the vertical variation of soil temperatures in July (Figure 4), the functional equation is y = –13.51 + 14.28(1 − e^−0.11x). In the two equations, the values of intercept and linear coefficient terms depend on surface temperature; they are –13.51 and 14.28, respectively. The logarithmic coefficient term depending on variation between soil layers is –0.11. The two functions are symmetric with respect to x = 10.5, and going down the two curves, they intersect at about 4 m. This means that at a depth of 4 m, the soil temperature in January and July was the same (10.5°C).

For a composite logarithmic function curve used to reflect the vertical variation of soil temperature in February (Figure 4), the functional equation is y = –11.93 + 14.96(1 − e^{−0.07(21−x)}); for a composite exponential function curve used to reflect the vertical variation of soil temperature in August (Figure 4), the functional equation is y = –11.93 + 14.96(1 − e^−0.07x). In the two equations, the values of the intercept and linear coefficient terms depend on surface temperature; they are –11.93 and 14.96, respectively. The logarithmic coefficient term, depending on variation between soil layers, is –0.07. The two functions are symmetric with respect to x = 10.5, and going down the two curves, they intersect at about 5 m. This means that at a depth of 5 m, the soil temperature in February and August was the same (10.5°C).

For a composite logarithmic function curve used to reflect the vertical variation of soil temperature in December (Figure 4), the functional equation is y = –15.2 + 15.43(1 − e^{−0.15(21−x)}); for a composite exponential function curve to reflect the vertical variation of soil temperature in June (Figure 4), the functional equation is y = –15.2 + 15.43(1 − e^−0.15x). In the two equations, the values of intercept and linear coefficient terms depend on surface temperature; they are –15.2 and 15.43, respectively. The logarithmic coefficient term, depending on variation between soil layers, is –0.15. The two functions are symmetric with respect to x = 10.5; they intersect at about 2.8 m. This means that at a depth of 2.8 m, the soil temperature in December and June was the same (10.5°C).

For a composite logarithmic function curve to reflect the vertical variation of soil temperature in November (Figure 4), the functional equation is y = –31 + 31.1(1 − e^{−0.26(21−x)}); for a composite exponential function curve to reflect the vertical variation of soil temperature in May (Figure 4), the functional equation is y = –31 + 31.1(1 − e^−0.26x). In the two equations, the values of intercept and linear coefficient terms are –31 and 31.1, respectively, and the logarithmic coefficient term is –0.26. The two functions are symmetric with respect to x = 10.5; they intersect at about 1.6 m. This means that at a depth of 1.6 m, the soil temperature was the same in November and May (10.5°C).

For a composite power function curve to reflect the vertical variation of soil temperature in October (Figure 4), the functional equation is y = –8.5 + 9(1 − 0.79^(21−x)); for the composite power function curve to reflect the vertical variation of soil temperature in April (Figure 4), the functional equation is y = –8.5 + 9(1 − 0.79^(21−x)). At a depth of 0.2 m, the soil temperature was the same in October and April (10.5°C). Soil temperature was different from the expected values in the 1.6–3.2 m soil layer, resulting in a small fitting degree, and the two curves are not completely symmetrical. However, the general trend is still a composite power curve.

In September, the soil temperature gradually increased at soil depth greater than 0.8 m and gradually decreased at soil depth less than 0.8 m. In April, the soil temperature also showed different trends above and below 0.8 m depth (Figure 4). This results in soil temperature failing to match the exponential or logarithmic function in September as well as April. However, ignoring the surface temperature anomalies in September, these two groups of scattered points are still considered as a set of symmetric functions about x = 10.5.

With the exception of March and September, the curves of soil temperature change with depth in most months can be fitted by composite exponential functions (Table 3). y = a + b(1 − e^−cx) and y = a + b (1 − e^−c(d−x)) are used to describe the vertical characteristics of soil temperature.

Table 3

Exponential models and their parameters used to simulate vertical characteristic of soil temperature for each month in temperate steppe regions of China, 1982–2011

Month	Function	Axis	R
January	y = –13.51 + 14.28(1 − e^{−0.11(21−x)})	y = 10.5	0.9603
July	y = –13.51 + 14.28(1 − e^−0.11x)	y = 10.5	0.9994
February	y = –11.93 + 14.96(1 − e^{−0.07(21−x)})	y = 10.5	0.9756
August	y = –11.93 + 14.96(1 − e^−0.07x)	y = 10.5	0.9906
March	NA	NA	NA
September	NA	NA
April	y = –8.5 + 9(1 − 0.79^x)	y = 10.5	0.9012
October	y = –8.5 + 9(1 − 0.79^(21−x))	y = 10.5	0.8856
May	y = –31 + 31.1(1 − e^−0.26x)	y = 10.5	0.9876
November	y = –31 + 31.1(1 − e^{−0.26(21−x)})	y = 10.5	0.9871
June	y = –15.2 + 15.43(1 − e^−0.15x)	y = 10.5	0.9983
December	y = –15.2 + 15.43(1 − e^{−0.15(21− x )})	y = 10.5	0.9902

As mentioned earlier, we found that there were obvious soil temperature vertical characteristics in winter and summer, with a high degree of simulation (Table 4, Figure 5). In winter, soil temperature at each depth gradually increased at deeper layers, presenting a curve of composite exponential function y = a + b (1 − e^−cx). In summer, soil temperature at each depth gradually decreased at deeper layers, presenting a curve of composite exponential function y = a + b (1 − e^−c(d−x)). In addition, the two functions depicting 2 months (defined in the Methods section) are symmetric, and the axis of symmetry is x = 10.5.

Table 4

Exponential models and their parameters used to simulate vertical characteristics of soil temperature for each season in temperate steppe regions of China, 1982–2011

Season	Function	Axis	R
Winter	y = −14.6 + 15.45(1 − e^{−0.12(21−x)})	y = 10.5	0.9997
Summer	y = −14.6 + 15.45(1 − e^–0.12x)	y = 10.5	0.9961
Spring	y = −16.5 + 17(1 − e^−0.3x)	y = 10.5	0.8977
Autumn	y = −16.5 + 17(1 − e^{−0.3(21−x)})	y = 10.5	0.8215

Figure 5

Seasonal vertical structure of soil temperature in temperate steppe regions of China, 1982–2011.

In spring and autumn, vertical variation of soil temperature was broadly in line with the composite function curve (Table 4, Figure 5), y = a + b (1 − e^−cx) (y = a + b (1 − c ^x)) and y = a + b (1 − e^−c(d−x)) (y = a + b (1 − c ^21−x)). The degree of fitting was highest in May and November and lowest in March and September, leading to relatively poorer regularity in vertical variation of soil temperature than in winter and summer. Table 4 shows the vertical characteristics of soil temperature at the seasonal scale.

During the entire study period, soil temperatures in the first half of the period (1982–1996) were higher than after 1996, and the increments of soil temperature from 1982–1996 to 1997–2011 were larger on the surface than in deeper layers (Table 2, Figure 3). For winter and summer, during 1982–1996, the soil temperature vertical variation was broadly in line with the composite exponential function curve, y = −14.7 + 15.54(1−e^−0.12x) and y = −14.7 + 15.54(1−e^{−0.12(20−x)}) (Table 5, Figure 6). These two functions are symmetric, and the axis of symmetry is x = 10, which means that the annual mean soil temperature was 10°C. After 1997, the soil temperature vertical variation was broadly in line with the composite exponential function curve (Table 5, Figure 7), y = –14.5 + 15.36(1 − e^−0.11x) in winter and y = –14.5 + 15.36(1 − e^{−0.11(22−x)}) in summer. These two functions also are symmetric, and the axis of symmetry is x = 11, which means the annual mean soil temperature was 11°C between 1997 and 2011.

Table 5

Exponential models and their parameters used to simulate vertical characteristics of soil temperature for each season in different periods in temperate steppe regions of China

Stage	Season	Function	Axis	R
1982–1996	Winter	y = −14.7 + 15.54(1 − e^{−0.12(20−x)})	y = 10	0.9987
	Summer	y = −14.7 + 15.54(1 − e^−0.12x)	y = 10	0.9995
	Spring	y = −16.8 + 17(1 − e^−0.4x)	y = 10	0.7709
	Autumn	y = −16.8 + 17(1 − e^{−0.4(20−x)})	y = 10	0.6512
1997–2011	Winter	y = −14.5 + 15.36(1 − e^{−0.11(22−x)})	y = 11	0.9956
	Summer	y = −14.5 + 15.36(1 − e^−0.11x)	y = 11	0.9992
	Spring	y = −16.2 + 17(1 − e^−0.35x)	y = 11	0.8202
	Autumn	y = −16.2 + 17(1 − e^{–0.35(22–x)})	y = 11	0.5924

Figure 6

Seasonal vertical structure of soil temperature in temperate steppe regions of China, 1982–1996.

Figure 7

Seasonal vertical structure of soil temperature in temperate steppe regions of China, 1997–2011.

4 Discussion

The findings show large-scale soil temperature variation in nonlinear regression, expressed with sets of exponential (power) function models, apparently for the first time. Consequently, after scientific consideration, the soil temperature data we use are as deep as 3.2 m, the soil temperature extending vertical variation to a depth of 3.2 m (monitoring limit in this study), as previous studies usually focused on shallow layers [10,11,12,20]. The simulation of the exponential model is more specific than that of the linear function [13,14], and the fitting degree is very satisfactory, which can estimate the soil temperature without probing any soil layer, even below 3.2 m. It can be used as a method and guidance for the study of soil ecological processes, regional ecometeorological simulations, and agricultural production practices, while mitigating the increase in air and soil temperatures due to global warming, etc. A series of studies will be beneficial to estimate extended soil temperature in vertical and time scales and to investigate temperature-dependent soil processes, e.g., decomposition of soil organic carbon and soil microbial activity, using detailed soil temperature data instead of estimated values from air temperature [25,26]. Seasonally, the fitting degree was great in winter and summer and poor in spring and autumn. Due to the particularity of heat transfer in spring and autumn, there is a transitional distribution of vertical changes in soil temperature, resulting in more complex vertical structural characteristics.

However, due to the monitoring soil layer limitations, we just analyzed the characteristics of the soil temperature from the surface to 3.2 m depth. Ultimately only nine effective stations were selected according to the strict data quality testing standards. These effective stations are evenly distributed throughout the study area; hence, we consider that they are geographically representative of the temperate steppe regions of China in this study. It is expected to expand the quantification method to national and even global scales, revealing the vertical distribution characteristics of soil temperature. We will divide China into many regions based on different patterns of soil temperature vertical variation and then analyze the relationships and differences between them, to explore the spatial patterns. In order to more accurately verify the fitting degree of composite exponential function to the vertical characteristics of soil temperature, we will design an on-site experiment, selecting two stations to explore deeper soil temperature as to 5 m.

5 Conclusions

The mean soil temperature at each depth fluctuated in a range of 10.3–10.7°C over the 30 annum interval of the study (1962–2011). The surface soil temperature is the highest, and the lowest is 0.2 m underground. The range of soil temperature change over eight layers was greater in winter and summer. Over the past three decades, the soil has warmed significantly, with a clear vertical pattern running from the surface to deeper layers, showing the increasing change trend from high to low. There were decrements in soil temperature during 1995 and then rapid increases after 1996 in all soil depths. It is suggested that the variation of soil temperature with soil depth can be fitted by the pair of symmetric composite exponential functions y = a + b(1 − e^−cx) and y = a + b(1 − e^−c(d−x)). Seasonally, soil temperature gradually increased at deeper soil layers in winter, presenting a curve of composite exponential function y = –14.6 + 15.45(1 − e^{−0.12(21−x)}); soil temperature gradually decreased at deeper soil layers in summer, presenting a curve of composite exponential function y = –14.6 + 15.45(1 − e^−0.12x); these two functions are symmetric, with an axis of symmetry of x = 10.5.

Based on a mathematical model, we explored the deep soil temperature change pattern in this work. The model’s development and comprehension are beneficial in forecasting vertical variations in soil temperature, which is crucial in predicting plant growth activities and microbiological changes.

Funding information: This research was funded by the National Natural Science Foundation of China (No. 41871022), Jilin Provincial Natural Science Foundation in China (No.YDZJ202301ZYTS217), and College Students’ Innovative Entrepreneurial Training Plan Program in China (No. S202210203047).
Author contributions: Conceptualization – H.Z. and H.D.; methodology and software – H.Z. and T.W.; validation – S.S. and Y.W.; data analysis – Y.W., S.S., and H.Z.; writing – original draft preparation – H.Z. and S.S.; writing – review and editing – H.Z., T.W., and L.L.; English writing revision – L.L.; supervision – H.D.; funding acquisition – H.D. All authors have read and agreed to the published version of this manuscript.
Conflict of interest: The authors declare no conflict of interest.
Institutional review board statement: Not applicable.
Data availability statement: Soil temperature data were obtained from the China Meteorological Administration (CMA), and all obtained data were measured (CMA standard protocols) in the national CMA Weather Station Network. At http://data.cma.cn/.

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Received: 2023-03-05

Revised: 2023-06-15

Accepted: 2023-07-17

Published Online: 2023-08-11

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/geo-2022-0523

Keywords for this article

temperate steppe; soil temperature; vertical variation; exponential function; soil depth

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