A general approach to estimate soil water content from thermal inertia

Agricultural and Forest Meteorology 149 (2009) 1693–1698

Contents lists available at ScienceDirect

Agricultural and Forest Meteorology journal homepage: www.elsevier.com/locate/agrformet

A general approach to estimate soil water content from thermal inertia Sen Lu a, Zhaoqiang Ju b, Tusheng Ren a,*, Robert Horton c a

Department of Soil & Water Sciences, China Agricultural University, Beijing 100193, China Center for Agricultural Resources Research, Institute of Genetics and Development Biology, Chinese Academy of Sciences, Shijiazhuang 050021, China c Department of Agronomy, Iowa State University, Ames, IA 50011, USA b

A R T I C L E I N F O

A B S T R A C T

Article history: Received 17 December 2008 Received in revised form 15 May 2009 Accepted 18 May 2009

Remote sensing is a promising technique for obtaining information of the earth’s surface. Remotely sensed thermal inertia has been suggested for mapping soil water content. However, a general relationship between soil thermal inertia and water content is required to estimate soil water content from remotely sensed thermal inertia. In this study, we propose a new model that relates soil thermal inertia as a function of water content. The model requires readily available soil characteristics such as soil texture and bulk density. Heat pulse measurements of thermal inertia as a function of water content on nine soils of different textures were made to generate a universal Kerstan function. Model validation was performed independently in both laboratory and field, and the retrieved soil water contents from the new model were compared with previous models. Laboratory evaluation on an Iowa silt loam showed that the RMSE of the new model was 0.029 m3 m3, significantly less than [Murray, T., Verhoef, A., 2007. Moving towards a more mechanistic approach in the determination of soil heat flux from remote measurements. I. A universal approach to calculate thermal inertia. Agric. For. Meteorol. 147, 80– 87] model (0.109 m3 m3) and [Ma, A.N., Xue, Y., 1990. A study of remote sensing information model of soil moisture. In: Proceedings of the 11th Asian Conference on Remote Sensing. I. November 15-21. International Academic Publishers, Beijing, pp. P-11-1–P-11-5.] model (0.105 m3 m3). Similar results were obtained in a field test on a Chinese silt loam: the RMSE of the new model, [Murray, T., Verhoef, A., 2007. Moving towards a more mechanistic approach in the determination of soil heat flux from remote measurements. I. A universal approach to calculate thermal inertia. Agric. For. Meteorol. 147, 80–87] model, and [Ma, A.N., Xue, Y., 1990. A study of remote sensing information model of soil moisture. In: Proceedings of the 11th Asian Conference on Remote Sensing. I. November 15-21. International Academic Publishers, Beijing, pp. P-11-1–P-11-5.] model were 0.018, 0.071, and 0.159 m3 m3, respectively. Additionally the model was validated using literature data in which soil thermal properties were estimated from in situ temperature measurements. The mean errors of estimated water content were generally less than 0.02 m3 m3. We concluded that the new model was able to provide accurate water content predictions from soil thermal inertia. ß 2009 Elsevier B.V. All rights reserved.

Keywords: Soil water content Thermal inertia Remote sensing Model

1. Introduction Soil thermal inertia (P) is a property that characterizes soil resistance to surrounding temperature change (Verstraeten et al., 2006). The thermal inertia method has been used to estimate soil moisture from thermal infrared and visible bands for bare soil or sparsely vegetated regions (Price, 1977; Cai et al., 2007). It has the advantages of providing large spatial coverage and temporal continuity using air- or space-borne platforms, and therefore has practical applications in irrigation management of agricultural fields. The application of the thermal inertia method generally involves two

* Corresponding author. Tel.: +86 10 62733594; fax: +86 10 62733596. E-mail address: [email protected] (T. Ren). 0168-1923/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.agrformet.2009.05.011

steps: calculating or mapping thermal inertia from remote sensing data, and retrieving soil water content (u) from models that describe thermal inertia as a function of water content (the P(u) model). Many studies have been conducted to develop models that obtain thermal inertia from remote sensing data. Watson et al. (1971) first estimated thermal inertia from infrared data for geologic interpretation. Kahle et al. (1975) produced the first thermal inertia imagery. Later Price (1977) developed a general theory to calculate thermal inertia from remote sensing data, and applied the model to construct the thermal inertia map of southwestern USA. This model, however, required many parameters (e.g., solar constant, atmospheric transmittance, surface albedo, wind speed, day–night temperature difference) to estimate soil thermal inertia and was not convenient to use in practice (Cai et al., 2007). In 1978, NASA launched the Heat Capacity Mapping

1694

S. Lu et al. / Agricultural and Forest Meteorology 149 (2009) 1693–1698

Satellite that had the ability to derive soil thermal inertia maps. A limitation of the common thermal inertia model based on remote sensing data is the requirement of many field-measured parameters. Xue and Cracknell (1995) developed a model that required only one field parameter, the time that the surface reaches maximum daily temperature. Cai et al. (2007) further extended the Xue and Cracknell (1995) method for regions where satellite overpass time was not coincident with the time of maximum and minimum temperatures, and applied the model to generate thermal inertia image for the North China Plain. Studies on the relationship between soil thermal inertia and water content are scarce. Pratt and Ellyett (1979) calculated P, ffi pffiffiffiffiffiffi pffiffiffiffi P ¼ lC ¼ l= a, using the soil thermal conductivity (l) and volumetric heat capacity (C) models of de Vries (1963). The results showed that the magnitude of soil thermal inertia was affected by many factors, e.g., soil porosity, texture, and water content. They described the thermal inertia and soil water content relationship in several figures that were not practically convenient for retrieving water content. Price (1980) extended the work of Price (1977) and estimated u using an empirical linear equation. Due to its sitespecific characteristics, the equation parameters were valid only on the soil studied. Ma and Xue (1990) reported a relationship between soil thermal inertia and water content. This P(u) model (denoted MX model hereafter) had two variables, solid particle density and water content. Recently the MX model was applied to calculate u from P data in the North China Plain (Cai et al., 2007). To eliminate the MX model requirement of solving u iteratively, Cai et al. (2007) provided a lookup table where the u values could be read directly from known values of P. Murray and Verhoef (2007) adopted the concept of normalized thermal conductivity (Johansen, 1975; Coˆte´ and Konrad, 2005; Lu et al., 2007) and developed a more mechanistic model to estimate P (MV model hereafter). In this model, soil porosity and texture were included as driving variables, but the contribution of the soil solid phase composition to soil thermal physical properties was not considered. In addition, some researchers have used linear, logarithm, and exponential functions to describe the P(u) relationship (Cai et al., 2007). These functions, however, are empirical and have the drawback of being site- and time-specific. As a result, none of them are general enough to have been applied extensively. Recently, the heat pulse technique has been advanced for measuring soil l, C, and a simultaneously (Bristow et al., 1994; Kluitenberg et al., 1995; Ren et al., 1999). The method has the advantages of causing only a small disturbance to natural heat and water regimes, easy operation in both laboratory and field conditions, and providing in situ soil temperature and thermal properties. The heat pulse technique opens a new opportunity to investigate relationships between soil thermal parameters and other physical properties (e.g., bulk density, water content, temperature). The objective of this study is to develop a model that can describe the P(u) relationship from saturation to ovendryness based on soil thermal physical measurements from the heat pulse technique. A general approach for retrieving soil u from P is introduced, and the performance of the new model is evaluated by independent measurements in both laboratory and field conditions. 2. Modeling soil thermal inertia as a function of water content 2.1. The MX model Ma and Xue (1990) established the following P(u) relationship: n o 1:20:02ds u g 0:007ðds ug 20Þ2 ð0:8þ0:02ds ug Þ 1=2 P ¼ 2:1ds e þ ds   rffiffiffiffiffiffiffiffiffiffiffiffi 0:2d2s ug 1    41:8 1000 100

where P (J m2 K1 s1/2) was the soil thermal inertia as a function of gravimetric water content ug (%), and ds was soil particle density taken as 2.65 Mg m3. 2.2. The Murray and Verhoef (2007) model Murray and Verhoef (2007) proposed a method to calculate soil thermal inertia based upon the normalized theory of soil thermal conductivity (Johansen, 1975; Coˆte´ and Konrad, 2005; Lu et al., 2007): P ¼ Pdry þ ðP sat  Pdry ÞK p 2

(2) 1/2

where Pdry (kJ m K s ) was the thermal inertia of dry soil, Psat (kJ m2 K 1 s1/2) was the soil thermal inertia at saturation, and Kp was the Kersten function. Murray and Verhoef (2007) calculated Pdry and Psat from soil porosity (n) using the following empirical equations: P dry ¼ 1:0624n þ 1:0108

(3)

P sat ¼ 0:7882n1:29

(4)

For the Kersten function (Kp), Murray and Verhoef (2007) used the formula developed by Lu et al. (2007): h  i g d (5) K p ¼ exp g 1  Sr where g and d were soil texture dependent model parameters and Sr (=u/n) was the degree of saturation. The parameters given by Murray and Verhoef (2007) were g = 1.78 and d = 2.0 for coarse soils with sand content (fs) larger than 0.8; g = 0.93 and d = 1.5 for fine textured soils with fs less than 0.4, and g = 3.84 and d = 4.0 for soils with intermediate textures. Therefore, the Murray and Verhoef (2007) model estimates soil P from the information of soil texture, porosity, and water content using (Eqs. (2)–(5)). 2.3. The new P(u) model Soil thermal inertia is defined as the square root of the product of thermal conductivity and volumetric heat capacity. Therefore, any error in l and C measurements or estimation will produce uncertainties in the results of P. It has been demonstrated that the accuracy of l is largely influenced by the quartz content in the soil solid phase (Bristow, 1998), since quartz has thermal conductivity significantly larger than the other minerals (de Vries, 1963). Pratt and Ellyett (1979) showed that larger errors in P were produced if the solid composition was ignored in the P model. At a given soil porosity, the influence of soil solid composition became more significant with increasing water content (Pratt and Ellyett, 1979, Fig. 4). In other words, minimum errors on Pdry and maximum errors on Psat are expected when the information of solid composition is not included in soil thermal inertia modeling. To reduce the uncertainties in estimating Psat using Eq. (4), we apply the following procedure to calculate Psat by taking soil solid composition into account: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P sat ¼ lsat C sat (6) where lsat and Csat are the thermal conductivity and volumetric heat capacity, respectively, of saturated soil. The geometric mean equation is used to estimate lsat (Johansen, 1975; Coˆte´ and Konrad, 2005; Lu et al., 2007): 1n

lsat ¼ ðlqq l1q Þ o (1)

1

lnw

(7)

where lq and lw are the thermal conductivity of quartz (7.7 W m1 K 1) and water (0.594 W m1 K 1) respectively.

S. Lu et al. / Agricultural and Forest Meteorology 149 (2009) 1693–1698

continuously monitored using the thermo-time domain reflectometry (thermo-TDR) techniques (Liu et al., 2008). The sand fraction (fs) can be obtained from soil databases, site-specific texture analysis, or derived from soil maps (Murray and Verhoef, 2007). Alternatively, Peters-Lidard et al. (1998) estimated sand content by taking the median value of each texture category in the texture triangle.

The quartz content of total solids (q) is taken as the fraction of sand (fs) (Peters-Lidard et al., 1998; Murray and Verhoef, 2007). The thermal conductivity of other minerals (lo) was taken as 2.0 W m1 K1 for soils with q > 0.2, and 3.0 W m1 K 1 for soils with q  0.2 (Johansen, 1975). The volumetric heat capacity of saturated soils was calculated from a modified model of de Vries (1963) where heat capacity of the bulk soil was considered as the sum of heat capacities of different phases (Ren et al., 2003): C sat ¼ rb cs þ rw cw n

4. Soil thermal inertia measurements using the heat pulse technique

(8)

To establish a general Kersten function, we conducted heat pulse measurements on nine soils with a wide range of texture. Table 1 lists the sample location, land use, and basic soil characteristics of the soil samples. The samples were air-dried, ground and sieved through a 2 mm screen. Soil particle size distribution was determined by the pipette method (Gee and Or, 2002). Soil organic matter content was measured by the Walkely– Black titration method (Nelson and Sommers, 1982). Various amounts of water were added to the soil and mixed thoroughly. The moist soil was then packed into columns (50.2 mm inner diameter and 50.2 mm high) with desired bulk densities (Table 1). Theory for the heat pulse technique assumes a homogeneous isotropic soil medium. The heat conduction equation in radial coordinates is (Bristow, 2002):

where rb was the soil bulk density (Mg m3), cw was the specific heat of water (4.18 kJ kg1 K1), and rw was the density of water (1.00 Mg m3). A range of values of the specific heats of soil solids (cs) had been summarized by Ren et al. (2003). In this study, we used a value of 0.8 kJ kg1 K1. For the dry soil, Pdry was dominated by the porosity and the influence of mineralogical composition was small (Quiel, 1975; Pratt and Ellyett, 1979; Lu et al., 2007; Murray and Verhoef, 2007). Therefore Eq. (3) was applied to calculate Pdry in the new model. Following the model for soil shrinkage curve (Groenevelt and Grant, 2004), we developed the following Kp function to describe the normalized thermal inertia: m K p ¼ exp½eð1  S r Þ

(9)

"

where Sr is the degree of saturation, e and m are parameters that control the shape of function. The determination of e and m from thermal inertia measurement will be illustrated later.

#

(11)

where T is temperature (8C), t is time (s), r is radial distance (m) and a is thermal diffusivity (m2 s1). When a heat pulse with duration to (s) is applied to a line heat source, the solution of Eq. (11) is (de Vries, 1952; Kluitenberg et al., 1993; Bristow et al., 1994):

For the MX model, an iteration method is required to retrieve ug from remotely sensed P. Alternatively, a lookup table of P versus ug can be used (Cai et al., 2007). Soil volumetric water content u (m3 m3) can then be calculated as the product of ug and soil bulk density (rb, Mg m3). In this study, we applied the following procedure to calculate u from remotely sensed P data. First the values of Pdry and Psat are calculated using Eqs. (3) and (6), respectively. Then Kp is determined by introducing the remotely sensed P into Eq (2). Finally Eq. (9) is rearranged to calculate u (m3 m3) from the remotely sensed thermal inertia:

DTðr; tÞ ¼

   2 

q0 r 2 r  Ei Ei 4pC a 4aðt  t 0 Þ 4at

t > t0

(12)

where DT (8C) is the temperature change at r, q’ is the energy input per unit length of heater per unit time (W m1), C is the volumetric heat capacity (J m3 K1), and Ei (x) is the exponential integral. Differentiating Eq. (12) with respect to t and setting the result to zero, the solution of thermal parameters is (Bristow et al., 1994):

Normally the particle density of a mineral soil is taken as 2.65 Mg m3 and soil porosity is simply estimated from soil bulk density rb (i.e., n = 1  rb/2.65). Therefore, as long as the Kersten function is developed, only two parameters, rb and fs, are required in the new P(u) model. As a basic physical parameter, field soil rb can be determined easily (Grossman and Reinsch, 2002) or



½1=ðt m  t 0 Þ  1=t m ln½t m =ðt m  t 0 Þ

a¼

r2 4

C¼

   

r 2 r 2  Ei Ei 4aðt m  t 0 Þ 4at m 4paDT m

(10)

e



@T @2 T 1 @T ¼a þ @t @r2 r @r

3. Retrieving soil water content from the new P(u) model

 1=m lnK p u ¼n 1

1695

(13)

q0

(14)

where tm (s) is the time at which the maximum temperature rise

DTm (8C) occurs. In practice, the values of tm and DTm can be obtained by analyzing the temperature change versus time data using a nonlinear regression technique (Welch et al., 1996).

Table 1 Sampling location, land use, texture (USDA classification), particle size distribution, organic matter (OM) content, and bulk density of the soils. Soil no.

1 2 3 4 5 6 7 8 9

Location

Inner Mongolia, China Hebei, China Ningxia, China Hebei, China Beijing,China Hebei, China Hebei, China Heilongjiang, China Beijing, China

Land use

Desert River bank Grassland Grassland Wheat field Corn field Corn field Soybean field Wheat field

Texture

Sand Sand Sandy loam Loam Silt loam Silt loam Silty clay loam Silty clay loam Clay loam

Particle size distribution (%) 2–0.05 mm

0.05–0.002 mm

<0.002 mm

94 93 67 50 27 11 19 8 32

1 1 21 41 51 70 54 60 38

5 6 12 9 22 19 27 32 30

OM (%)

rb (Mg m3)

0.09 0.07 0.86 0.25 1.19 0.84 0.39 3.02 0.27

1.60 1.60 1.39 1.38 1.33 1.31 1.30 1.30 1.29

S. Lu et al. / Agricultural and Forest Meteorology 149 (2009) 1693–1698

1696

Fig. 1. Soil thermal inertia as a function of water content as indicated by (a) P(u) relationship: soil thermal inertia (P) as a function of water content (u); (b) Kp(Sr) relationship: the normalized thermal inertia (Kp) as a function of degree of saturation (Sr) for selected soils.

Soil thermal conductivity and thermal inertia are then calculated from the following equations (Pratt and Ellyett, 1979; Verhoef, 2004; Verstraeten et al., 2006):

l ¼ aC P¼

pffiffiffiffiffiffiffi lC

(15) (16)

We used the thermo-TDR probe to conduct the heat pulse measurements. The thermo-TDR sensor has three parallel needles in a plane with a spacing of 6 mm between adjacent needles. Each needle is 40 mm in length and 1.3 mm in diameter. A thermocouple is placed in the middle of the outer needles for measuring temperature. The middle needle hosts a resistance wire to generate heat pulse. Details of the thermo-TDR probe are presented in Ren et al. (1999), Ochsner et al. (2001), and Heitman et al. (2007). The thermo-TDR probe was inserted into the surface of each soil column. During each measurement, a DC current was applied to the middle heater for 15 s to generate a heat pulse. The temperature change in the outer needles was recorded by a data logger (CR23X, Campbell Scientific, Logan, UT, USA) with 1 s interval. The measurement was repeated three times for each sample. Soil thermal parameters (a, l, C) were calculated by using a non-linear regression technique (Welch et al., 1996) from the temperature increase vs. time in the outer needles. Soil thermal inertia was then calculated (Eq. (16)). Finally, the soil samples were oven-dried at 105 8C, and the bulk density and water content were determined.

5. Determination of parameters e and m of the Kersten function The heat pulse measurements of P and u were used to derive parameters in the Kersten function (Eq. (9)). Measured P data were normalized using Eq. (2) to obtain Kp, and the degree of saturation Sr was calculated from u and soil porosity. Fig. 1a shows the measured P as a function of u for four soils of different textures, and Fig. 1b is the Kp and Sr relationship for the selected soils. After the normalization process, the Kp(Sr) relationships fell into two distinct texture groups: the coarse soils with fs larger than 0.4 and fine soils with fs less than 0.4, similar to the normalized soil thermal conductivity–water content relationships reported by Lu et al. (2007). By fitting Eq. (9) to the heat pulse data of the two groups independently, we determined the parameters of e and m for the two groups of soils: 2.95 and 0.16 for the coarse soils (soils 1–4), and 0.60 and 0.71 for the fine soils (soils 5–9). Fig. 2 shows the model results from Eq. (9) along with the measured data for the two groups of soils. The newly developed Kerstan functions are able to represent the Kp(Sr) relationship accurately. 6. Model validation The performance of the new model, along with the previous models, was first evaluated using independent data from both laboratory and field measurements. The soil used in the laboratory study was a silt loam (2% sand and 25% clay) collected from

Fig. 2. The normalized soil thermal inertia (Kp) as a function of degree of saturation (Sr) for (a) coarse soils with sand content larger than 0.40, and (b) fine soils with sand content less than 0.40.

S. Lu et al. / Agricultural and Forest Meteorology 149 (2009) 1693–1698

1697

Fig. 3. Laboratory and field evaluation of the performance of the Ma and Xue (1990) model (MX model), Murray and Verhoef (2007) model (MV model), and the new model on (a) a USA silt loam soil from Treynor, Iowa; and (b) a Chinese silt loam from Hebei.

Treynor, Iowa. The sample was air-dried, sieved, mixed with various amounts of water, and packed into soil columns with a bulk density of 1.20 Mg m3. Soil thermal inertia was then measured with the heat pulse technique as discussed previously, and soil water content was then determined gravimetrically. The field test was conducted in a fallow field located in the experimental farm of the Hebei Academy of Agricultural and Forestry Science, Shijiazhuang, China. The soil was a silt loam having 17% sand and 24% clay, with a bulk density of 1.33  0.05 Mg m3. Heat pulse measurements were initiated after a rainfall event on June 29, 2002, and were continued for 2–3-day intervals until July 23, 2002 when no apparent change in soil water content was observed. The measurement protocols were the same as the laboratory study with the thermo-TDR probe inserted into field surfaces. Immediately after heat pulse measurements, undisturbed soil cores (0–5 cm depth) were collected with a ring sampler (5 cm height and 5 cm in diameter) from the field surface at the measured spot, to determine soil bulk density and water content. For a given u, three repeated measurements were conducted. The new model was further tested using three data sets from Wierenga et al. (1969) and Kaune et al. (1993). Wierenga et al. (1969) measured the thermal conductivity of a Yolo silt loam at different water contents using the line heat source technique, and by applying the phase and amplitude equations, they estimated the apparent thermal diffusivity from soil temperature measurements. In this study, we used their 5 cm deep measurements to test the new model. Soil thermal inertia as a function of water content was calculated from the thermal property data (as presented in their Figs. 1 and 2). The values of Pdry and Psat were obtained using Eq. (3) and Eq. (7), respectively. Kaune et al. (1993) compared the thermal properties of disturbed and structured samples of a loess soil. They applied the harmonic method to estimate the daily mean apparent thermal diffusivity from soil temperature measurements at 0.2, 5, 10, 20, 35 and 50 cm. Soil volumetric heat capacity was obtained according to de Vries (1963) using bulk density and water content data, and thermal conductivity was then calculated as the product of thermal diffusivity and volumetric heat capacity. In this work, we calculated soil thermal inertia at different water contents for the disturbed and undisturbed soil samples. Soil thermal property data were taken from Fig. 3 of Kaune et al. (1993). Fig. 3 presents the measured P(u) data and the model results of the MX model, MV model, and the new model. The MX model showed large discrepancies for both laboratory and field soils. In general, it underestimated thermal inertia in the lower water content region, and over-predicted thermal inertia in the higher water content region. A P value of 0 was observed with the MX model at u = 0 m3 m3, which was pointless (Pratt and Ellyett,

1979). The MV model underestimated thermal inertia over the entire water content range for both soils. Our examination indicated that the errors of the MV model were caused mainly by the errors in Psat estimation, where soil mineral information was not considered (data not shown). In contrast to these models, the predictions of the new model closely matched the measurements over the entire water content range of both soils (Fig. 3). The RMSE of modeled P from the MX model, the MV model, and the new model were 0.633, 0.230, and 0.068 kJ m2 K1 s1/2 on the Iowa soil, and 0.764, 0.219 and 0.091 kJ m2 K1 s1/2 on the Chinese soil, indicating that the new model could describe the P–u relationship accurately. Fig. 4 compares the estimated u values from the new thermal inertia model versus actual u measurements. The mean error (ME) and root mean square error (RMSE) of u predictions from the Ma and Xue (1990) model, Murray and Verhoef (2007) model, and the new model are presented in Table 2. In general, the estimated u agreed well with the measurements, as indicated by the random distribution of the data points along the 1:1 line and the relatively large value of R2 (0.889) of the fitted line (Fig. 4). The RMSE of u estimates from the new model was in the range of 0.018– 0.051 m3 m3 which was less than that of the MV model (0.034-

Fig. 4. Comparison of estimated soil water contents from the new thermal inertia model and measured water contents using laboratory measurements on a silt loam soil from Treynor, Iowa USA, field measurements on a silt loam soil from Hebei, China, and literature data taken from Wierenga et al. (1969) and Kaune et al. (1993).

S. Lu et al. / Agricultural and Forest Meteorology 149 (2009) 1693–1698

1698

Table 2 Mean error (ME) and root mean square error (RMSE) of the Ma and Xue (1990) model (MX model), Murray and Verhoef (2007) model (MV model), and the new model. Laboratory measurements were on samples of the Iowa soil, and field measurements were conducted in Hebei, China. Soil

Iowa soil Chinese soil Wierenga et al. (1969) Kaune et al. (1993): disturbed soil Kaune et al. (1993): structured soil

MX model

MV model

New model

ME

RMSE

ME

RMSE

ME

RMSE

0.073 0.156 0.082 0.135

0.105 0.159 0.094 0.136

0.087 0.067 0.079 0.019

0.109 0.071 0.084 0.034

0.009 0.009 0.006 0.019

0.029 0.018 0.019 0.023

0.200

0.201

0.068

0.077

0.045

0.051

0.109 m3 m3) and the MX model (0.094–0.201 m3 m3) (Table 2). The new model performed better than the MV model and MX model, and it was capable of providing accurate estimates of u from soil thermal inertia data. Except for the structured soil from Kaune et al. (1993), the mean errors of estimated water content were generally smaller than 0.02 m3 m3. The relative larger error of the new model on the structured soil may arise from the uncertainty of the soil bulk density, as both l and C are sensitive to changes in soil bulk density. In Table 1 of Kaune et al. (1993), the bulk density of the structured soil had a standard error of 0.06 g cm3. A simple calculation indicated that the accuracy of estimation was increased significantly with increasing soil bulk density (data not shown). 7. Conclusion In this study, a general model for calculating soil water content from thermal inertia was developed. We adopted the normalized theory of soil thermal conductivity to estimate soil thermal inertia as a function of degree of saturation across from the information of soil texture and bulk density. The model parameters were generated from thermal inertia measurements on nine soils using the heat pulse technique. Additionally the new model was improved by including soil solid phase composition (i.e., sand content) to estimate thermal inertia at saturation. Sand content was neglected in the previous models. Independent evaluation of the new thermal inertia model was performed in laboratory and field, and using three data sets from the literature in which soil thermal inertia was estimated from in situ soil temperature measurements. The results showed that estimated water content from the new soil thermal inertia model agreed well with measured water content values. The new thermal inertia model is capable of providing accurate u from remotely sensed thermal inertia values. Acknowledgements The authors gratefully thank Dr. Guoyin Cai for his technical assistance. We also thank Dr. Moran (Associate Editor), and two anonymous reviewers for helpful comments to improve the manuscript. This research was supported by the Basic Research Development Program of China (973 Program) (No. 2009CB118607) and the China Postdoctoral Science Foundation Project (No. 20080440448). References Bristow, K.L., 1998. Measurement of thermal properties and water content of unsaturated sandy soil using dual-probe heat-pulse probes. Agric. For. Meteorol. 89, 75–84. Bristow, K.L., 2002. Thermal conductivity. In: Dane, J.H., Topp, G.C. (Eds.), Methods of Soil Analysis. Part 4. Physical Methods. SSSA, Madison, pp. 1209–1226.

Bristow, K.L., Kluitenberg, G.J., Horton, R., 1994. Measurement of soil thermal properties with a dual-probe heat-pulse technique. Soil Sci. Soc. Am. J. 58, 1288–1294. Cai, G., Xue, Y., Hu, Y., Wang, Y., Guo, J., Luo, Y., Wu, C., Zhong, S., Qi, S., 2007. Soil moisture retrieval from MODIS data in Northern China Plain using thermal inertia model. Int. J. Remote Sens. 28 (16), 3567–3581. Coˆte´, J., Konrad, J.-M., 2005. A generalized thermal conductivity model for soils and construction materials. Can. Geotech. J. 42, 443–458. de Vries, D.A., 1952. A nonstationary method for determining thermal conductivity of soil in situ. Soil Sci. 73, 83–89. de Vries, D.A., 1963. Thermal properties of soils. In: Van Wijk, W.R. (Ed.), Physics of Plant Environment. North-Holland Publishing Company, Amsterdam, pp. 210– 235. Gee, G.W., Or, D., 2002. Particle-size analysis. In: Dane, J.H., Topp, G.C. (Eds.), Methods of Soil Analysis. Part 4. Physical Methods. SSSA, Madison, pp. 255–293. Groenevelt, P.H., Grant, C.D., 2004. Analysis of soil shrinkage data. Soil Till. Res. 79, 71–77. Grossman, R.B., Reinsch, T.G., 2002. Bulk density and linear extensibility. In: Dane, J.H., Topp, G.C. (Eds.), Methods of Soil Analysis. Part. 4. Physical Methods. SSSA, Madison, pp. 201–228. Heitman, J.L., Horton, R., Ren, T., Ochsner, T.E., 2007. An improved approach for measurement of coupled heat and water transfer in soil cells. Soil Sci. Soc. Am. J. 71, 872–880. Johansen, O., 1975. Thermal conductivity of soils. Ph.D. Thesis. Trondheim, Norway (CRREL draft translation 637, 1977). Kahle, A.B., Gillespie, A.R., Goetz, A.F.H., Addington, J.D., 1975. Thermal inertia mapping. In: Proceedings of the Tenth International Symposium on Remote Sensing of Environment, vol. 2. pp. 985–994. Kluitenberg, G.J., Bristow, K.L., Das, B.S., 1995. Error analysis of heat pulse method for measuring soil heat capacity, diffusivity, and conductivity. Soil Sci. Soc. Am. J. 59, 719–726. Kluitenberg, G.J., Ham, J.M., Bristow, K.L., 1993. Error analysis of the heat-pulse method for measuring the volumetric heat capacity of soil. Soil Sci. Soc. Am. J. 57, 1444–1451. Kaune, A., Turk, T., Horn, R., 1993. Alteration of soil thermal properties by structure formation. J. Soil Sci. 44, 231–248. Liu, X., Ren, T., Horton, R., 2008. Determination of soil bulk density with thermotime domain reflectometry sensors. Soil Sci. Soc. Am. J. 72, 1000–1005. Lu, S., Ren, T., Gong, Y., Horton, R., 2007. An improved model for predicting soil thermal conductivity from water content at room temperature. Soil Sci. Soc. Am. J. 71, 8–14. Ma, A.N., Xue, Y., 1990. A study of remote sensing information model of soil moisture. In: Proceedings of the 11th Asian Conference on Remote Sensing. I. November 15-21. International Academic Publishers, Beijing, P-11-1–P-11-5. Murray, T., Verhoef, A., 2007. Moving towards a more mechanistic approach in the determination of soil heat flux from remote measurements. I. A universal approach to calculate thermal inertia. Agric. For. Meteorol 147, 80–87. Nelson, D.W., Sommers, L.E., 1982. Total carbon, organic carbon, and organic matter. In: Page, A.L., et al. (Eds.), Methods of Soil Analysis. Part. 2. Agronomy Monograph 9. 2nd ed. ASA and SSSA, Madison, pp. 539–579. Ochsner, T.E., Horton, R., Ren, T., 2001. A new perspective on soil thermal properties. Soil Sci. Soc. Am. J. 65, 1641–1647. Peters-Lidard, C.D., Blackburn, E., Liang, X., Wood, E.F., 1998. The effect of soil thermal conductivity parameterization on surface energy fluxes and temperature. J. Atmos. Sci. 55, 1209–1224. Pratt, D.A., Ellyett, C.D., 1979. The thermal inertia approach to mapping of soil moisture and geology. Remote Sens. Environ. 8, 151–168. Price, J.C., 1977. Thermal inertia mapping: a new view of the earth. J. Geophys. Res. 82, 2582–2590. Price, J.C., 1980. The potential of remote sensed thermal infrared data to infer surface soil moisture and evaporation. Water Resour. Res. 16 (4), 787–795. Quiel, F., 1975. Thermal/IR in geology, some limitations in the interpretation of imagery. Photogramm. Eng. Remote Sens. 41 (3), 341–346. Ren, T., Noborio, K., Horton, R., 1999. Measuring soil water content, electrical conductivity and thermal properties with a thermo-time domain reflectometry probe. Soil Sci. Soc. Am. J. 63, 450–457. Ren, T., Ochsner, T.E., Horton, R., Ju, Z., 2003. Heat-pulse method for soil water content measurement: influence of the specific heat of the soil solids. Soil Sci. Soc. Am. J. 67, 1631–1634. Verhoef, A., 2004. Remote estimation of thermal inertia and soil heat flux for bare soil. Agric. For. Meteorol. 123, 221–236. Verstraeten, W.W., Veroustraete, F., van der Sande, C.J., Grootaers, I., Feyen, J., 2006. Soil moisture retrieval using thermal inertia, determined with visible and thermal spaceborne data, validation for European forests. Remote Sens. Environ. 101, 299–314. Watson, K., Rowen, L.C., Offield, T.W., 1971. Application of thermal modeling in the geologic interpretation of IR images. Remote Sens. Environ. 3, 2017–2041. Welch, S.M., Kluitenberg, G.J., Bristow, K.L., 1996. Rapid numerical estimation of soil thermal properties for a broad class of heat pulse emitter geometries. Meas. Sci. Technol. 7, 932–938. Wierenga, P.J., Nielsen, D.R., Hagan, R.M., 1969. Thermal properties of a soil based upon field and laboratory measurements. Soil Sci. Soc. Am. Proc. 33, 354–360. Xue, Y., Cracknell, A.P., 1995. Advanced thermal inertia modeling. Int. J. Remote Sens. 16, 431–446.z