**3. Results**

### *3.1. The Variations of Soil Moisture*

Figure 2 shows the variation of soil moisture (θ) over 2014–2016 at the BJ, QOMS, and NADORS. At BJ, θ varied greatly for most months except autumn and winter (October– December) 2015 and winter and spring (January–May) 2016. During non-winter periods, θ varied with rainfall; while during other periods, θ fluctuations were mainly attributed to soil thawing and freezing processes with temperature (e.g., spring (March–April), 2014; autumn (November), 2016). Soil temperatures fluctuated around the freezing temperature (Figure S1), resulting in a water phase transition between liquid and ice. The minimum θ values were similar for 2014–2016, around 0.05 m<sup>3</sup> m<sup>−</sup><sup>3</sup> at BJ, while the maximum θ values differed for the three years and occurred in summer, with about 0.33 m<sup>3</sup> m<sup>−</sup><sup>3</sup> in July–August 2014, 0.28 m<sup>3</sup> m<sup>−</sup><sup>3</sup> in August 2015, and 0.25 m<sup>3</sup> m<sup>−</sup><sup>3</sup> in July 2016, respectively.

**Figure 2.** The variations of hourly soil moisture (θ, m<sup>3</sup> m<sup>−</sup>3) at the depth of 0.10 m at (**a**) BJ, (**b**) QOMS, and (**c**) NADORS for 2014–2016. Note that θ was measured at 0.04 m depth at BJ 2014, θ at NADORS were the arithmetic mean of soil moisture measured at the depths of 0.00 m and 0.20 m, and θ were directly measured at the 0.10 m depth for other cases.

Compared to BJ, the soil at QOMS and NADORS had distinct wet and dry cycles. The θ varied greatly in the summer and remained relatively constant during other periods. Besides, no large fluctuations in the θ were measured in winter at the two sites. The reason was that the θ decreased to a low value before winter, therefore, no large fluctuations in the

θ occurred as soil temperature varied around the freezing temperature. The θ ranged from 0.01 m<sup>3</sup> m<sup>−</sup><sup>3</sup> to 0.20 m<sup>3</sup> m<sup>−</sup><sup>3</sup> at QOMS, and 0.05 m<sup>3</sup> m<sup>−</sup><sup>3</sup> to 0.30 m<sup>3</sup> m<sup>−</sup><sup>3</sup> at NADORS.

Overall, the soil at QOMS was driest, followed by NADORS and BJ. Increases in the θ were sharp after rain, and decreases in the θ were relatively slow as soil water evaporated.

### *3.2. The Variations of Soil Apparent Thermal Diffusivity*

The hourly soil temperature amplitude (A) and phase (Φ) at two depths were extracted from soil temperature time series using DHR. For simplicity, only the 2014 results at the three sites are presented as illustrated in Figure 3, and the other results are shown in Figures S2–S4. The data were removed as A at 0.20 m depth was less than the corresponding threshold value at each site.

**Figure 3.** The variations of hourly soil temperature amplitude (A, ◦C) and phase (Φ, rad) at two depths of 0.04 m (or 0.0 m) and 0.2 m at (**<sup>a</sup>**,**b**) BJ, (**<sup>c</sup>**,**d**) QOMS, and (**<sup>e</sup>**,**f**) NADORS for 2014.

It is obvious that A varied with time, and its fluctuation at the shallow depth was much larger than that at the deeper depth (Figure 3a,c,e). As soil is heated by solar radiation at daytime and soil temperature generally decreases exponentially with depth [21], as does A. Compared to other periods, A was relatively small in winter and wet periods. Compared with A, the fluctuation of Φ was relatively small, and the deeper the depth, the larger the Φ. Since soil temperature phase shifts vary linearly with depth [21].

Among the sites, the differences in A and Φ at the two depths were larger at QOMS and NADORS than those at BJ. The soils at QOMS and NADORS were dry (corresponding to relatively small k, see the following results). Thus, soil temperature changes were transmitted slowly through the soil, resulting in large differences in A and Φ between the two depths at the two sites. Tong et al. [58] derived the relationship between conduction–convection k and ln(*A*1/*A*2) and (Φ2 − Φ1) by taking partial derivatives, finding that when ln(*A*1/*A*2) is constant, k increases as (Φ2 − Φ1) decreases; when (Φ2 − Φ1) is constant, k increases (decreases) with increasing ln(*A*1/*A*2) when (Φ2 − Φ1) > ln(*A*1/*A*2) [(Φ2−Φ1) < ln(*A*1/*A*2)].

After obtaining hourly A and Φ values for soil temperature at the first depth and the 0.20 m depth, hourly k was determined with Equation (2). The daily k was also obtained by averaging the hourly values over a day. The variations of the hourly and daily k for 2014–2016 at the three sites are shown in Figure 4.

**Figure 4.** The variations of hourly (marked with dot) and daily (marked with square) soil apparent thermal diffusivity (k, m<sup>2</sup> s<sup>−</sup>1) of the 0.0 m to 0.20 m layer at (**a**) BJ, (**b**) QOMS, and (**c**) NADORS for 2014–2016.

Generally, k varied with time to varying degrees at different sites and years. At BJ and QOMS, k had an obvious seasonal variation (Figure 4a,b), and it roughly varied with θ during the wetting periods (compare Figure 2a,b to Figure 4a,b).

At BJ, k values in 2014 were larger than those in 2015 and 2016 except for some time in June. The 2014 k varied around 1.0 × 10−<sup>6</sup> m<sup>2</sup> s<sup>−</sup>1, and the minimum and maximum values were about 0.5 × 10−<sup>6</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> and 1.9 × 10−<sup>6</sup> m<sup>2</sup> s<sup>−</sup>1, respectively. For 2015–2016, the k ranged from 0.3 × 10−<sup>6</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> to 0.9 × 10−<sup>6</sup> m<sup>2</sup> s<sup>−</sup>1. For QOMS, k fluctuated between 1.0 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> to 4.0 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> over 2014–2016, and except for summer time, k varied slightly and was relatively small at most time. The k varied almost exclusively with θ, namely, large k corresponded to large θ (compare Figure 2b to Figure 4b).

Compared to BJ and QOMS, the differences in k at NADORS among the three years were relatively small, and k varied slightly during the non-winter periods, although the θ obviously varied. From January to mid-March, k tended to decrease from 3.3 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> to 1.2 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup>1; while it was relatively stable in spring–mid autumn (May-October), ranging from 2.5 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> to 3.2 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup>1.

Figure 4 shows that k is not always constant throughout a day, and it can change drastically when the soil is wetted (e.g., DOY 218 in 2015 at BJ; DOY 150 in 2014 at QOMS).

The monthly variations of k and θ for 2014–2016 at the three sites are further examined in Figure 5. The values of the monthly k (mean ± one standard deviation) are listed in Table 3.

At BJ, the monthly median k fluctuated around 1.0 × 10−<sup>6</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> in 2014. While the 2015 monthly k was around 7.0 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> in May–September and decreased to about 4.0 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> in October–December, the monthly k was around 4.0 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> in February–May and increased to 7.5 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> after May in 2016. For QOMS, the monthly k peaked in August 2014-2015, with a median value of 3.2 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup>1, and varied around 2.0 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> for most other months. In July and August 2016, the monthly k value was the largest, at 3.2 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup>1, after which it decreased almost linearly until December (2.3 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup>1). The monthly k was relatively stable at around 1.8 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> before July.

**Figure 5.** The monthly variations (25th, median, and 75th percentiles) of soil apparent thermal diffusivity (k, m<sup>2</sup> s<sup>−</sup>1) and moisture (θ, m<sup>3</sup> m<sup>−</sup>3) at (**a**) BJ, (**b**) QOMS, and (**c**) NADORS for 2014–2016. Data are deleted when there are less than 15 days available in a month.

**Table 3.** The monthly (mean ± one standard deviation) k × 10<sup>7</sup> (m<sup>2</sup> s<sup>−</sup>1) at each site.


1 indicates there is no data.

Differing from BJ and QOMS, the monthly k at NADORS varied little, about 3.0 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> for most of the non-winter period (April-October), although θ did vary during this period. It decreased from January to February over the 3-year period, ranging from 3.0 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> to 1.0 × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup>1.

Overall, the variation trends of monthly k were roughly similar to those of θ, except for BJ 2014 and NADORS 2016. The monthly k at BJ ranged from 0.4(±0.0) × 10−<sup>6</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> to 1.1(±0.2) × 10−<sup>6</sup> m<sup>2</sup> s<sup>−</sup>1, from 1.7(±0.0) × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> to 3.3(±0.2) × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> at QOMS, and from 2.1(±0.3) × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> to 3.1(±0.1) × 10−<sup>7</sup> m<sup>2</sup> s<sup>−</sup><sup>1</sup> at NADORS (Table 3).

### *3.3. The Relationship between Soil Apparent Thermal Diffusivity and Moisture*

Figure 6 shows how k varies with θ on an hourly timescale in 2014–2016 at the sites. Overall, the trends of unfrozen soil k versus θ at the three sites were roughly similar, i.e., k increased rapidly to a maximum value with increasing θ and then tended to be constant or decrease slightly as θ increased further. The values of θ corresponding to peak k values were different, and the θ values at QOMS were less than those at BJ.

**Figure 6.** The variation of soil apparent thermal diffusivity (k, m<sup>2</sup> s<sup>−</sup>1) with soil moisture (θ, m<sup>3</sup> m<sup>−</sup>3) at 0.10 m depth on an hourly timescale for 2014 (in the 1st column), 2015 (in the 2nd column) and 2016 (in the 3rd column) at (**<sup>a</sup>**–**<sup>c</sup>**) BJ, (**d**–**f**) QOMS, and (**g**–**i**) NADORS, respectively. The color bar indicates the value of soil temperature at the depth of 0.20 m (Ts0.2m), and the larger the temperature, the redder the color. The marker size suggests the amplitude of Ts0.2m (A0.2m), and the larger the value of A0.2m, the larger the marker. The correlation coefficients (r) between k and θ are provided for all of the data and for the data when Ts0.2m > 0 ◦C, respectively. "\*\*" indicates *p*-value < 0.01. The probability distributions of k and θ are shown on the y-axis and x-axis sides, respectively.

Interestingly, the relationship between k and θ did not appear to be significant when Ts0.2m < 0 ◦C and θ < 0.1 m<sup>3</sup> m<sup>−</sup>3, i.e., k fluctuated greatly within a narrow range of θ (e.g., see the blue points in Figure 6a,c,g–i). The correlation coefficients (r) between k and θ for all of the data ranged from 0.66 to 0.80 except for BJ 2014 (r = 0.37) and NADORS 2016 (r = 0.46). Under the condition of Ts0.2m > 0 ◦C, the r coefficients changed to different degrees at each site and year, and the changes depended on the location of the data of Ts0.2m ≤ 0 ◦C on the curve. Overall, without including data for Ts0.2m ≤ 0 ◦C, the r coefficients decreased at NADORS, increased at BJ, and changed slightly at QOMS, respectively.

Note that the probability distributions of k were different, especially at different sites. The same is true for θ. Compared to k, the probability distribution of θ for a given site was more consistent during 2014–2016. There was an obvious gap between 0.05 m<sup>3</sup> m<sup>−</sup><sup>3</sup> to 0.10 m<sup>3</sup> m<sup>−</sup><sup>3</sup> at NADORS (Figure 6g–i), since the data in the θ range were deleted according to the standard of A0.2m < 0.1 ◦C, as mentioned in Section 2.4.

The relationship between k and θ on a daily timescale (Figure S5) was similar to that on an hourly timescale. Few studies have investigated the relationship between k and θ for soil below and above the freezing temperature simultaneously.

We furthermore investigated the relationship between median k and θ on a monthly timescale, as shown in Figure 7. Similar to the hourly results, the monthly k tended to increase with θ when θ was relatively small, reached a peak value and then became relatively stable as θ increased further.

**Figure 7.** Same as Figure 6, but for the relationship between median k and θ on a monthly timescale. The number of months per year (N) and corresponding correlation coefficients (r) are given. "\*\*" indicates *p*-value < 0.01, and there is no label after r if r is not significant (*p*-value > 0.05). The data for a month are deleted when the number of days is less than 15.

Overall, monthly k was significantly correlated to θ regardless of site and year, except for 2014 BJ and 2016 NADORS. The significant (*p*-value < 0.01) r ranged from 0.80 (2016 BJ) to 0.92 (2015 QOMS) on a monthly timescale, and from 0.64 (NADORS) to 0.88 (QOMS) on an annual timescale. The r coefficients of k vs. θ on a monthly timescale were larger than those on an hourly timescale. This could be explained because the effect of frozen soil on the relationship between k and θ was greatly reduced on a monthly timescale.

In order to indirectly investigate the effect of soil moisture on k from the vegetation aspect, the relationship between monthly k and NDVI is also examined as shown in Figure 8. The ranges of NDVI were different at the three sites, and the maximum NDVI at BJ (0.51) was approximately 2.5 times that of QOMS (0.22) and NADORS (0.18).

**Figure 8.** Same as Figure 7, but for the relationship between k and NDVI. "\*" indicates *p*-value < 0.05, "\*\*" indicates *p*-value < 0.01, and there is no label after r if r is not significant (*p*-value > 0.05).

Interestingly, the monthly k had a similar correlation with NDVI as it did with θ. At QOMS and NADORS, the r coefficients of k vs. NDVI were close to those of k vs. θ (Δr < 0.08), while the r coefficients of k vs. NDVI were smaller than those of k vs. θ at BJ.
