## 1. Introduction

Soil apparent thermal diffusivity

## 2. Materials and methods

### a. Site

Data were collected at the Taklimakan Desert Atmosphere and Environment Observation Experiment Station located at Tazhong, China (hereinafter referred to as the Tazhong station), from day of year (DOY) 98 to 104 in 2011. This site was located at 38.98°N, 83.64°E, with an altitude of 1103 m. The ground surface was bare and relatively flat. The soil at the site was predominantly fine sand. The site had an arid climate zone with maximum and minimum air temperatures of 319 and 240 K, respectively. The mean annual air temperature and precipitation were 285 K and 24 mm, respectively. The site had 2690 h of sunshine and 263 frost-free days per year (averaged from 1996 to 2010).

Soil temperatures were measured at the surface and at 0.10-, 0.20-, and 0.40-m depths by temperature probes (model 109L, Campbell Scientific Inc.), with an accuracy of ±0.2 K. Soil volumetric water contents (VWCs) were measured at 0.025-, 0.10-, 0.20-, and 0.40-m depths by soil moisture sensors (model CS616, Campbell Scientific Inc.), with an accuracy of ±2.5%. The soil temperature and moisture sensors were sampled each second, and all of the sensor outputs were averaged over 30-min time periods and recorded. Soil sensor information is provided in Table 1. Standard micrometeorological measurements were also made at the site, including four radiation components, wind speed, wind direction, air temperature, relative humidity, air pressure, and precipitation.

Soil temperature and VWC sensors at the Tazhong station.

### b. Theory

#### 1) The mathematical interrelationships among the three algorithms for calculating apparent thermal diffusivity

Relationships among the three thermal diffusivity algorithms were derived mathematically. Based on the derived relationships, it can be concluded that when

#### 2) The mathematical relationships of and to and

Partial derivatives were calculated to obtain the variations of

is closer to than to , except for the case of = 0; - if
> 0, ; - if
< 0, ; - when the phase shift
is constant, increases (decreases) with increasing if and reaches a maximum value at and increases with increasing ; - when
is constant, decreases with increasing , and increases (decreases) with increasing when .

The details of the derivation process for sections 2b(1) and 2b(2) are presented in the appendix.

#### 3) Data processing

Sine functions, *i* = 1, 2, and 3.

## 3. Results

The interrelationships among

The max, min, and avg values of air temperature (TA), air relative humidity (RH), vapor pressure *e*, wind speed (WS), and air pressure (Pa) at the Tazhong station from DOY 98 to 104 in 2011.

Figure 1 shows temporal variations in the soil temperature values measured at the surface and at 0.10-, 0.20-, and 0.40-m depths and volumetric water contents measured at 0.025-, 0.10-, 0.20-, and 0.40-m depths. The soil temperature values fluctuated diurnally. Soil temperature amplitudes decreased with depth, and soil temperatures shifted phase as soil depth increased. The maximum (minimum) surface soil temperature during this 7-day period was 330.8 K (268.5 K), recorded on DOY 104. It can be seen in Fig. 1a that the vertical temperature difference between the surface and the 0.10-m depth was 35.4 K at 1300 local time on DOY 102.

Figure 1b shows the temporal variation in the volumetric soil water content. The volumetric water content also changed diurnally, and the diurnal variations at shallow depths were larger than those at deeper depths. The volumetric water content was greatest at the 0.025-m depth, and the largest diurnal variation occurred at the surface. The magnitudes of the volumetric water content were similar at the 0.10- and 0.20-m depths.

To verify the mathematical conclusions for the relative magnitudes of the

The diurnal and average values of ^{−7} (1.96 × 10^{−7}) m^{2} s^{−1}, 1.79 × 10^{−7} (2.35 × 10^{−7}) m^{2} s^{−1}, and 2.22 × 10^{−7} (1.96 × 10^{−7}) m^{2} s^{−1} at the shallower (deeper) layer, respectively, showing that ^{−7} m s^{−1}), and it was negative (−4.82 × 10^{−7} m s^{−1}) for the deeper layer (0.10–0.20 m).

The diurnal and avg values of soil apparent thermal diffusivity ^{2} s^{−1}) determined by Eqs. (6), (7), and (3), and ^{−1}) determined by Eq. (4) for the 0.00–0.10 and 0.10–0.20 m soil layers at the Tazhong station from DOY 98 to 104 in 2011.

To illustrate the mathematical conclusions of Eqs. (3) and (4) given in section 2b(2), Figs. 3 and 4 present the variations of

Both

## 4. Discussion

Gao et al. (2008) presented two-dimensional figures of

To further understand the variations of

Equations (3) and (4) demonstrated that

Variable

The relationships of

The

This study did not account for water phase changes in soil, although such phase changes (i.e., water vaporization, vapor condensation, ice freezing, and ice thawing) could at times play an important role in the surface energy budget. Future studies should further investigate the role of soil water phase changes on estimation of soil thermal properties.

## 5. Conclusions

This study extends earlier findings on soil apparent thermal diffusivity

This study was supported by the Ministry of Science and Technology of China (JFYS2016ZY01002213), National Natural Science Foundation of China (Grant 41275022), the State Scholarship Fund of China Scholarship Council (201608320197), Jiangsu Province graduate education innovation project (KYLX16_0943), and International S&T Cooperation Program of China (ISTCP; Grant 2011DFG23210). We are very grateful to Dr. Aili Maimaitiming (Institute of Desert Meteorology, China Meteorological Administration) for providing data from the Tazhong station. We appreciate the valuable comments of Profs. Zhihua Wang (Arizona State University), Yubin Li (Nanjing University of Information Science and Technology), and Beyong Lee (Nanjing University of Information Science and Technology). We are very grateful to two anonymous reviewers for their careful review and valuable comments, which led to substantial improvement of this manuscript. The code (in MATLAB format) used in this paper can be obtained from the corresponding author and first author.

# APPENDIX

## Details of the Derivation Process

### a. The mathematical interrelationships among the three algorithms for calculating apparent thermal diffusivity

We assume that the logarithmic amplitude attenuation

Now we consider in turn Eqs. (A8) and (A9) for the conditions

- When
, . From Eqs. (A8) and (A9), we get,Thus, is smaller than and , , and are all positive, so that is closer to than to . - When
< 0, , thenAgain, is smaller than , and is closer to than to because , , and are all positive.

Therefore, combining the results in (i) and (ii), we conclude that, except for the case of

### b. The mathematical relationships of and to and

#### 1) Variation of with and

- When the phase shift
is constant,Because and are both positive real numbers, , and the sign of depends on the relative magnitudes of and , thus, - when
, that is, , increases with increasing ; - when
, that is, , decreases with increasing ; and - when
, that is, , is maximized.

- when
- When the logarithmic amplitude attenuation
is constant, the following can be shown from Eq. (A12):

#### 2) Variation of with and

- When the phase shift
is constant,Note that and , so , implying that increases with increasing . - When the logarithmic amplitude attenuation
is constant, the following can be obtained from Eq. (A15):Note that , , and the sign of depends on the relative magnitudes of and , thus, - when
, that is, , increases with increasing , and - when
, that is, , decreases with increasing .

- when

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