地表通量之監測與推估(1/2)

行政院國家科學委員會專題研究計畫 期中進度報告

地表通量之監測與推估(1/2)

期中進度報告(精簡版)

計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 96-2745-M-002-002- 執 行 期 間 ： 96 年 08 月 01 日至 97 年 07 月 31 日 執 行 單 位 ： 國立臺灣大學生物環境系統工程學系暨研究所 計 畫 主 持 人 ： 謝正義 處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 97 年 05 月 30 日

行政院國家科學委員會專題研究計畫成果報告

1

地表通量之監測與推估 (1/2)

2

Estimation and Monitoring of surface fluxes

3

計畫編號：NSC 96-2745-M-002-002

4

執行期限：96 年 8 月 1 日至 97 年 7 月 31 日

5

主持人：謝正義 國立台灣大學 生物環境系統工程系

6

hsieh@ntu.edu.tw

7

計畫參與人員：賴玫君 黃丞瑋 台灣大學 生物環境系統工程所

8 9 Abstract 10

Soil heat flux may play an important role in surface energy balance. In this study, 11

we examined the performances of two methods for predicting soil heat flux from 12

single layer time series data of soil temperature. The first one is the traditional 13

method, which an analytical solution of soil heat flux can be obtained by assuming the 14

surface soil temperature varies sinusoidally. The second one is the connection 15

between surface soil temperature and soil heat flux derived by half order 16

derivative/integral, and is based on a simple model of heat transfer described by a 17

one-dimensional diffusion equation with a constant heat diffusivity. Good 18

agreements between measured and predicted soil heat fluxes were found for both 19

methods. However, it was shown that the half order derivative method has a better 20

capability to capture flux accuracy and trend than the traditional method for long-term 21

soil heat flux estimation. 22

1. Introduction

1

In a well-watered and full vegetation covering surface, in fact, soil heat flux may 2

be the same order as sensible heat flux. When the vegetation is becoming senescence, 3

soil heat flux may be the same order as latent heat flux. Moreover, there is a 4

considerable portion of solar radiation which contributes to the soil surface heat flux 5

during the early growth period ( Kustas and Daughtry, 1990 ). The ratio of the soil 6

heat flux to the net radiation is found to be 0.5 and 0.3 for dry soils and wet soil 7

respectively ( Idso et al., 1975 ) . Based on many research about the Land-atmosphere 8

interaction, the surface soil heat flux, therefore, becomes a vital impact factor in the 9

energy balance system. 10

The influence of soil heat flux on chemical reactions and microclimate are 11

self-evident. Thus, both for the plant growth models and meteorological models, the 12

soil heat flux plays an important rule in the energy balance between air and soil ( van 13

Loon, Bastings and Moors, 1998 ). The determinations of the human-induced

14

greenhouse gas (radiative) forcing and energy budgets in climate model demonstrate 15

that the energy balance and heat flux are also significant factors (Delworth and 16

Knutson, 2000). And the acquirement of soil heat flux information should be more

17

applicable so that we could modify the climate model development with more detailed 18

soil heat flux data (Beltrami et al., 2006 ). 19

As the soil ground heat flux is one of the important parts of the energy balance at 20

the interface between land and atmosphere, the method possess the most appropriate 21

approximations with fewer limitations is the first thing to be figured out. Many 22

methods have been built to do the indirect measurements of the ground heat flux, and 23

most of the indirect measurements need that the soil temperature profile be measured 1

at the same time. It is not convenient and it may cost a lot while doing large area 2

measurement and conducting high frequency sampling. 3

As a matter of fact, solutions to the one-dimensional diffusion equation with a 4

constant diffusivity can exhibit the behavior of soil temperature in space and time. 5

The analytical solutions derived under the condition, however, which has many 6

restrictions, and do not represent real soil environment very well. There are still many 7

numerical methods for solving the equation and simulating the real condition, but 8

numerical methods are difficult to understand the behavior of the system ( Campbell 9

and Norman, 2000 ). This paper compares two different methods of forecasting soil

10

heat flux. One is to evaluate the analytical solution of soil heat flux when assume that 11

the surface soil temperature varies sinusoidally ( Carslaw and Jaeger, 1986; Campbell 12

and Norman, 2000 ). Another is to simulate the diffusion equation with half order

13

derivative/ integral method (Wang and Bras,1998; Wang and Bras,1999; Wesson et al. 14

2001 ). Both methods get the simplicity of indirect measurements to estimate soil heat

15

flux with only single-level temperature. This study provides real environmental 16

scenario to quantitatively make careful assessment of the capability of these two 17

methods. It is the first to collect long-term, continuous, supporting observation data, 18

which are necessary for the validation and assessment. 19 20

2. Theory

21 2.1. Traditional method 22

According to the thought of heat conduction in solid ( Carslaw and Jaegr, 1986 ), 23

the soil properties are assumed to be uniform over entire soil profile, and the soil 1

temperature varies sinusoidally. Given these assumptions the temperature can be 2

formed as a function of time t (sec) and depth z (m): 3

 

, ave

 

0 exp /



sin

 

0 /

T z t T A z D _t t z D_ (1)

4

where A



0 (oC) is the amplitude of the temperature fluctuations; Tave(

o

C) is the 5

average temperature over a temperature circle(one day); t (sec) is the phase shift.₀

6

And  (1/sec) and D (m) represent for the angular frequency and damping depth, 7

respectively, which are calculated as: 8 2    (2) 9 0 2D D   (3) 10

where is the time over a temperature circle; ₀ s s k D c   (m2 s-1) is the thermal 11

diffusivity and k (W m-1K-1) is the thermal conductivity,  (Mg m_s -3) is the soil 12

density, c_s (J g-1 K-1) is the soil specific heat, and _{s s}c (MJ m-3 K-1) is the 13

volumetric heat capacity. In equation (1), damping depth shows two important 14

concepts about the hypothetical phenomenon of soil temperature. First, the amplitude 15

of temperature variation attenuating with depth depends on the damping depth. Also, 16

the damping depth affects the temperature circle shifting with depth. In a real soil 17

environment, the physical behaviors of soil temperature have similar characteristics 18

mentioned above. 19

In order to obtain the soil heat flux, Fourier’s law for heat transport is used to 20

computetheheatflow passing through a soillayerto the nextone.TheFourier’s law 21

for heat transport is described as: 22

dT

G k

dz

 (4)

1

where G is the soil heat flux (W m-2); T (oC)is the soil temperature and z (m) is 2

the depth below the ground surface. Apply equation (1) to the Fourier’s law and set 3

0

z , the surface soil heat flux can be computed from: 4

 

2



0 sin

 

0 / 4 0, A k t t G t D    _  _     (5) 5

Equation (5) indicates that soil heat flux also varies sinusoidally with time and its 6

mean equals to zero. When apply the traditional method, only single layer time series 7

data of soil temperature and soil properties are needed for estimating soil heat flux. 8

9

2.2. Half order derivative/integral method 10

In addition to the Fourier’s law, applying the continuity equation to construct the 11

diffusion equation is required, which can depict physical phenomenon of heat transfer 12

completely. The continuity equation is shown as: 13 s s T G c t z      (6) 14

The right hand side and left hand side present their physical meanings, the rate of heat 15

storage in a soil layer and rate of change of soil heat flux, respectively. 16

Combining the Fourier’s law and continuity equation and assuming the thermal 17

diffusivity D₀ is constant derive: 18 2 0 2 T T D t z  _    (7) 19

Practically, one-dimensional heat transfer over vertically homogeneous soil can be 20

described by Equation (7) with a constant diffusivity. For solving the differential 21

the variation of soil temperature far below the ground surface is smaller than the 1

variation of soil temperature near the ground surface and the initial soil temperature 2

profile is uniform are defined as two boundary conditions for the differential equation 3

(7). Therefore, the boundary condition for the soil temperature can be presented as: 4 0 T  ,T for t 0,z0 (8) 5 0 T  ,T for t 0,z  (9) 6

In Equation (8), the initial soil temperature profile uniformly distribute through every 7

layers, the assumption implies that the iterative computation of soil heat flux is started 8

when the soil heat flux is zero. Equation (9) is the other boundary condition for 9

solving the differential equation (7) and it means that the variation of soil temperature 10

is nearly to zero when the depth exceeds the damping depth. 11

Moreover, the process for acquiring analytical solution of equation (7) is tedious 12

and has many limitations. Use of the half order time derivative method (Wang and 13

Bras, 1998) facilitates the derivation. Main advantage of the method is to establish a

14

relationship linking the soil heat flux to the soil temperature with minimum 15

limitations. When the heat transfer mechanism is delineated by equation (7)-(9), the 16

half order time derivative method derives that the vertical gradient of temperature can 17

be a function of weighted average of soil temperature time series according to the 18

concept of fractional calculus. Then, the approximation model used to estimate the 19

soil heat flux G by the soil temperature T at any depth can be built (Appendix A). 20

The result is as follows: 21

 

, 0

 

, G z t D T z t z    22

 

0 tdT z s, D 

_

(10) 23

where s is the integration variable. As the equation (10) represents, only measuring

1

single layer time series data of soil temperature at certain depth and the soil property 2

information are required to estimate soil heat flux at the same depth. 3

4

2.3. Application 5

Assessing the applicability of these two methods with continuous, long-term 6

measured information, therefore, are trustworthy. In the process of verifying the 7

methods, no matter which method has been chosen to use, the soil temperature T 8

always is the dominant required measured variable. And both of these two methods 9

also need the soil properties to be the input parameters, such as the thermal 10

conductivity k, the soil density  and soil specific heat_s c ._s

11

According to the basic assumption of the traditional methods, the soil 12

temperature varies sinusoildally, which can be approximately determined by the 13

amplitude of the temperature fluctuations A(0) when daily maximum and minimum 14

soil temperature are picked from real soil temperature time series. Moreover, in order 15

to get the more appropriate sinusoidal fitting for the temperature time series, the phase 16

shift must be taken into account. 17

The completeness of the soil temperature time series information has 18

considerable influence on the modeling results, especially when the half order 19

derivative/intergral method is applied. The shortage of temperature time series 20

information leads to disrupt integral computation. In other words, the more complete 21

temperature time series information can be collected, the more accurate estimated soil 22

heat flux can be evaluated. Furthermore, when the half order derivative/integral 23

method is operating, the starting time point of the integration should coincide with the 1

time when soil heat flux equals to zero. In fact, it is difficult to determine the initial 2

time point of integration, because the exact time point when the soil heat flux equals 3

zero remains unknown without measured soil heat flux data. A proper way to deal 4

with this problem is to set the starting time of integration as the beginning of night 5

when the soil heat flux approximately approach to zero (not exact equal to zero). 6

Therefore, the approximation still has its own satisfactory accuracy, which can be 7

elucidated in this study. 8

9

3. Experiments

10

The measurements of the soil heat flux and the soil temperature were used to 11

assess the performance of these two methods —Traditional method and Half order 12

derivative/integral method. Also, the measurements were conducted for this study 13

through whole year 2002 and the experiment site was located in fertilized grassland in 14

Co. Cork in southern Ireland (Latitude: 52.14 ° N, Longitude: 8.66 ° W). In the 15

experiment site, the dominant plant species was perennial ryegrass, and the grassland 16

type was moderately high quality pasture and meadow. Besides, the grass height in the 17

experiment site varied from 0.1 to 0.2 meter. Additionally, the soil properties are 18

essential parameters, and some soil property information is summarized in Table 1 19

and Table 2. 20

Different from other complicated experiments for determining soil heat flux, this 21

study only set up single level equipment to do the indirect measurement. The soil heat 22

flux plate (HUKSELFFLUX) was buried parallel to the top of the soil surface at the 23

depth of 5 centimeters and the temperature probe was buried at the same depth near 1

the soil heat flux plat. Also, the net radiometer was erected to collect the net radiation 2

data for confirming the processes of energy transport. The soil heat flux, soil 3

temperature and net radiation data were collected every 30 minutes by a datalogger 4

during whole study year 2002. 5

6

4. Results and discussion

7

When estimate the soil heat flux by the traditional method and half order 8

derivative/intergral method, the two parameters — volumetric heat capacity _{s s}c

9

and thermal conductivity k, are required which depend on the thermal properties. 10

The volumetric heat capacity is therefore computed as the sum of the heat capacities 11

of the three parts of soil components — minerals, water and organic matter (Zhang et 12 al., 2007): 13 s sc m m mc w wc o o oc     (11) 14

where  (m_m 3/m3) is the volume fraction of minerals, (m_o 3/m3) is the volume 15

fraction of organic matter; and (m3/m3) is the volumetric soil moisture. During the 16

year 2002, the measured volumetric soil moisture  ranged from 0.22 to 0.61, the 17

volume fraction of organic matters  was small to be negligible and the volume_o 18

fraction of minerals was 0.4. When the heat capacities of minerals and water were 19

2.31 (MJ m-3K-1) and 4.81 (MJ m-3K-1) respectively, the volumetric heat capacity_{s s}c

20

can be derived to be 1.84 to 3.47 (MJ m-3K-1). 21

The other required input parameter — thermal conductivity k, is reasonably 22

estimated to be 1.1 (W m-1K-1) by using the first 50％ soil temperature and soil heat 23

flux data and applying the half order derivative/integral method to minimize the root 1

mean squire error (RMSE) between observed soil heat flux Goand estimated soil flux

2

Ge. The mathematical presentation can be written as:

3





12 2 1 : _{e o} Minimize G G N  _    



 (12) 4

where N is the quantities of data. Remember that Ge contains a variable k and the

5

goal is to search for the optimum k that can minimize the RMSE. 6

7

4.1. traditional method 8

There was no appropriate equipment for measuring surface temperature precisely 9

and successively until now. Therefore, only the soil temperature at the depth 5 10

centimeters can be measured in this study. When complete soil properties information 11

was collected, the traditional method and half order derivative/intergral method were 12

pushed to estimate the soil heat flux just at a certain depth due to the lack of integral 13

soil surface temperature. But heat flux at the energy exchange interface — soil 14

surface, is more important in the energy balance system. Although the surface 15

temperature may be estimated by the linear extrapolation of the soil temperature at 16

deeper layers and can also be used to estimate the surface soil heat flux, the estimated 17

surface temperature still has a bias against the real surface soil temperature. For the 18

soil temperature is not a linear function of depth and has a phase shift with depth, 19

solving this problem through the relationship between the surface soil heat flux Gs 20

and the soil heat flux at deeper layer Gz is a better alternative. The difference between 21

the surface soil heat flux Gs and the soil heat flux Gz (estimated heat flux) at a certain 22

depth z is caused by the heat storage term S within the layer. It can be derived from 1

equation (6) and expressed as 2 s z G  G S (13) 3 Where 4 0 z s s T S c dz t    



(14) 5

Fig.1 shows the observed soil temperature time series at depth z equals to 5 6

centimeters through all the study year 2002. Follow the equation (5), (13) and (14), 7

Fig.2 shows estimation versus observation of soil surface heat flux by the traditional 8

method and the solid line is 1:1 and R2=0.762 in Fig.2. The result is a good illustration 9

to explain that the traditional method has acceptable ability to catch the flux accuracy 10

and trend, although the soil temperature is fitted by a sinusoidal function instead of 11

real soil temperature. For clarity, choose successive thirty days (day 70 to day 100) to 12

plot the estimated and observed time series of soil surface heat flux in Fig.3. It reveals 13

that the magnitude of soil heat flux transport upward is overestimated during 14

nighttime (The value of soil heat flux is negative). This kind of error was caused, 15

because the variation of soil temperature cannot be fitted well due to the assumption 16

of the traditional method—soil temperature varies sinusoidally with time and depth. 17

But the use of the traditional method has an advantage, it does not need successive 18

and integral soil temperature data. Contrarily, only daily maximum and minimum soil 19

temperature data are required. It is to mean that the lost of temperature data influences 20

the results little by the traditional method and the results of estimated surface soil heat 21

flux are good enough. 22

1

4.2. half order derivative/integral method 2

Apply the observed soil temperature data showed in Fig.1 to the equation (10) and 3

follow the equation (13) and (14), Fig.4 shows the relationship between observed and 4

estimated surface soil heat flux by the half order derivative/integral method. In Fig.4, 5

The solid line is 1:1 and R2=0.984. The result shows that use of the half order 6

derivative/integral method has a better capability to predict soil surface heat flux than 7

the traditional method. In fact, the iterative computation of soil heat flux cannot be 8

exactly started at the time point that the real soil heat flux equals to zero. It means that 9

the undetermined starting iterative time point may leads to error. In Fig.4, the 10

integration is started only once according to the completeness of the soil temperature 11

time series information. The soil heat flux nearly approach to zero at the beginning of 12

night, because the absence of solar radiation in the nighttime and the intensive solar 13

radiation in the daytime make the quantities of soil heat flux passes upward is much 14

smaller than the quantities of soil heat flux passes downward. This is the reason why 15

the starting iterative time has been chosen at the beginning of night. Furthermore, the 16

lost of temperature data will increase the number of restarting iterative time point and 17

reduce the accuracy by the half order derivative/integral method. It is to mean that 18

accumulated errors depend on the number of restarting iterative time point. For clarity, 19

the estimated and observed heat flux data within the same period (day 70 to day 100) 20

as Fig.2 are plotted in Fig.5. It reveals that even though the half order 21

derivative/integral method has uncertainties for predicting soil heat flux, it still keeps 22

good precision. 23

There are some more information that cannot be detected in Fig.4 and Fig.5. 1

Thus, the observed and estimated cumulative soil heat fluxes through the study year 2

were used to examine the performance of half order derivative/integral method in 3

detail and plotted in Fig.6. It shows that the cumulative values of estimated soil heat 4

flux increased slowly during the first 75 days of the year, but the values in the real 5

state of affairs decreased rapidly. And the estimated cumulative soil heat flux has quite 6

good accuracy after the first 75 days. At the same time, the net radiation data also 7

shows the same phenomenon as the observed cumulative soil heat flux. It is because 8

the near ground air temperature is lower than soil temperature and it usually happens 9

in colder weather. The results and discussion above show that the values of heat flux 10

transport upward were underestimated by the half order derivative/integral method 11

during the first 75 days of the year. After the first 75 days of the year, the estimation 12

error of soil heat flux may be neutralized with increasing computational time because 13

daily variation of soil heat flux alternately goes up to positive values (heat transports 14

downward) and goes down to negative values (heat transports upward). Furthermore, 15

we can conclude that the undetermined starting iterative time point indeed produces 16

this error and the results are not so good as the heat transports upward more, but the 17

error attenuates with computational time. Also, the daytime ratio of the soil heat flux 18

(measured or estimated) to net radiation ranging from 0.2 to 0.5 agrees with the 19

results proposed by Clothier et al. (1986) and Ogee et al. (2000). It indicates that 20

measured data are reliable and estimated results are satisfactory. 21

Not only the soil heat flux but also the soil temperature can be estimated through 22

the half order derivative/integral method. The soil temperature can be expressed as the 23

integration of soil heat flux using the half order derivative/integral method adversely 1

(Wang and Bras, 1999). With complete soil heat flux information, soil temperature 2

can be computed by the equation below: 3



0 ₀



1 t g s s G s ds T t T k C t s    



(15)

Follow the equation (15), Fig.7 shows the estimation versus observation of soil 4

temperature through the study year. In Fig.7, The solid line is 1:1 and R2=0.94. And 5

Fig.8 shows the observed and estimated soil temperature time series. It is clear that 6

although the trend of the estimated soil temperature is fairly good as the observed soil 7

temperature, the soil temperature is considerably underestimated. It indicates that the 8

estimation results of soil temperature are not as well as soil heat flux. When the first 9

75 days of the study year are skipped, Fig.9 presents the estimation versus observation 10

of soil temperature. Also, Fig.10 shows the observed and estimated soil temperature 11

time series. In Fig.9, The solid line is 1:1 and R2=0.9384. Therefore, the results are 12

better than the estimation including the first 75 days data. There are many reason can 13

be observed to explain why the errors and bias exist. Theoretically, the undetermined 14

starting time point for soil heat flux estimation and undetermined initial temperature 15

T0 may cause the errors. In other aspect, it makes the cumulated error increase

16

continuously and let the bias between observed and estimated soil temperature 17

become larger because the behavior of soil temperature is different from soil heat flux. 18

Long-term estimations of soil heat flux and soil temperature by the half order 19

derivative/integral method shows that both of these two estimations reveal different 20

orders in errors, when the air temperature is higher than the soil temperature and the 21

soil heat flux transport upward. 1

2

5. Conclusion

3

We examined the performances of the traditional method and the half order 4

derivative/integral method for predicting soil heat flux. Also, we estimated the diurnal 5

average and annual average soil thermal conductivity by the half order 6

derivative/integral method. Based on our measurements and predictions, we have 7

demonstrated the following: 8

（1） Different from other indirect measurements, both of these two methods only 9

need single layer time series data of soil temperature. And it can reduce the cost 10

of experiment and it is more convenient than other indirect measurements. 11

（2） Because the variation of soil temperature cannot be fitted well by the traditional 12

method, the magnitude heat flux transport upward is overestimated and the 13

magnitude heat flux transport downward is underestimated. 14

（3） In order to obtain a good estimation through the half order derivative/integral 15

method, the successive and complete data information is required. In other 16

words, the broken data information is the vital fact to reduce the accuracy. And 17

the estimation error of soil heat flux can be eliminated with computational time. 18

（4） The half order derivative/integral method can be contrarily used to estimate soil 19

temperature from the soil heat flux. Different from the soil heat flux, the 20

estimation error of soil temperature cumulatively increases with time. 21

1

Acknowledgement

2

We thank the National Science Council, Taiwan for their support of this study. 3

Appendix A：The process for evaluating the soil heat flux by half order

1

derivative/intergral method

2 3

The algorithm for the half order derivative/intergral method has been proposed by 4

( Wang and Bras, 1998 ). First of all, define new variables z%z/ D0 and

5

0

T T

  , and make equation(7)(9)(10) transfer to: 6 2 2 t z  _  % (A1) 0  , for t0, z%0 (A2) 0  , for t0,z%  (A3)

We begin by taking the Laplace transform of both sides of the differential equation 7

(A1) according to the definition of Laplace transformL f t

  

s sF s f 0 t   _{  } _    , 8

and the left hand side becomes s%

     

z s%, %z, 0  s% %z s, because of the initial 9

condition (A2). Also, Laplace transform of the right hand side is still not changed 10

comparing to the original form because the Laplace transform is based on the variable 11 t . 12

 

, 22 s z s z     % % % % (A4)

where % is the Laplace transform of and its definition is: 13

 

   

0 , exp , z s st z t dt  % % 

_

 % _(A5)

Now, setting % ez% and applying it to (A4) turn out . It means that thes

14

general solution of equation(A4) is a linear combination: 15

  

z s, A s exp

 

z s B s



exp

 

z s

where A s



and B s



are arbitrary functions of s and can be determined by the

1

boundary conditions. B s



equals zero because of the boundary condition (A3) and 2

then equation (A6) becomes: 3

  

z s, A s exp

 

z s

% %  % _(A7)

Differentiating both sides with respect to zgives: 4

 

z s, s A s



exp

 

z s z

_{ }

%% % % (A8)

and substituting (A7) into (A8) yields 5

 

z s, s

 

z s, z     %% % % % (A9)

Depending on the equation derived by the fractional calculus ( Miller and Ross, 6 1993): 7









1 1



0 0 m v v m k k m v k L D f t s F s s D f        



(A10)

,the right hand side of (A9) becomes 8

 

 

1 2 , ₁ , 2 s z s L z t t                % % % _(A11)

The last term of (A10) is eliminated by the boundary condition f



0  . Replace0 9

the right hand side of (A9) by (A11). Thus, 10

 

 

1 2 1 2 , , z s L z t z t   _{ }  _     _{ }   % % % % (A12)

Inverting the Laplace transform of (A12) leads to 11

 

 

1 2 1 2 , , z t z t z t      % % _ % (A13)

 

 

1 2 0 1 0 2 1 , , T z t T z t T z D t  _  _{  }     (A14)

by returning the new variables to the originals. Again, substituting a equation 1

introduced by the fractional calculus ( Miller and Ross, 1993) 2



 

0

_{ }



1 1 t d f t d f s ds dt dt _{t s}    _{ } 



(A15)

for the right hand side of (A14) derives that 3

 

 

0 0 , 1 , t T z s ds T z t z D s t s   _ 



 

 

0 0 , 1 t dT z s D t s   



(A16)

Finally, the prognostic result can be evaluated by applying the equation(A16) to the 4 Fourier’s law 5

 

, 0

 

, G z t D T z t z   

 

0 0 , t dT z s D t s   



(A17)

In practice, transforming equation(A17) into discrete form is easier to start the 6

integration. Equation (A17) also can be written down as 7

 

0

 

0 , , t T z s D ds G z t s t s     



(A18)

And its discrete form is 8 0 1 1 0 1 2 N i i N i N i i i i D T T G t t t t t t          _    _ 



(A19)

where N is the number of intervals . 9

References

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6

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10

Balland, V., P.A. Arp, 2005, Modeling soil thermal conductivities over a wide range of 11

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13

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17

Clothier, B.E., K.L. Clawson, P.J. Pinter,JR., M.S. Moran, R.J. Reginato and R.D. 18

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21

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3

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6

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9

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12

Kustas, W. P., and C. S. T. Daughtry, 1990, Estimation of the soil heat flux / net 13

radiation ratio from spectral data, Agric. Forest Meteo., 49, 205-223. 14

15

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22 23

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soil or of other bodies, Nature, 214. 2

3

van Loon, W. K. P., H. M. H. Bastings, E.J. Moors, 1998 Calibration of soil heat flux 4

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6

Wang, J., R. L. Bras, 1998, A new method for estimation of sensible heat flux from air 7

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9

Wang, J., R. L. Bras, 1999, Ground heat flux estimated from surface soil temperature, 10

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11 12

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17

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List of Figures

1 2

Figure 1. The observed soil temperature time series at depth z equals to 5 centimeters

3

through all the study year 2002 (17520 points). 4

5

Figure 2. Estimation versus observation of soil surface heat flux by the traditional

6

method (The solid line is 1:1 and R2=0.743) (17520 points). 7

8

Figure 3. Estimated and observed soil heat flux time series by the traditional method.

9

(Dot line is estimation and solid line is observation)(1440 points). 10

11

Figure 4. Estimation versus observation of soil surface heat flux by the half order

12

derivative/intergral method (The solid line is 1:1 and R2=0.984)(17288 points). 13

14

Figure 5. Estimated and observed soil heat flux time series by the half order

15

derivative/intergral method. (Dot line is estimation and solid line is observation)(1440 16

points). 17

18

Figure 6. Estimated and observed cumulative soil heat fluxes by the half order

19

derivative/intergral method through the study year(17288 points). 20

21

Figure 7. Estimation versus observation of soil temperature by the half order

22

derivative/intergral method through the study year (The solid line is 1:1 and 23

series by the half order derivative/intergral method through the study year. (17520 1

points) 2

3

Figure 9. Estimation versus observation of soil temperature by the half order

4

derivative/intergral method from day 75 to day 365 (The solid line is 1:1 and 5

R2=0.9384)(13923 points). 6

7

Figure 10. Estimated and observed soil temperature time series by the half order

8

derivative/intergral method from day 75 to day 365. (13923 points) 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Table1. Experiment site information in Cork through the study year 2002.

1

Soil properties (0 –10 centimeters)

Vegetation Humid grassland

Bulk density (g/cm3) 1.3

Clay fraction 0.06

Organic content at soil surface (kg C/kg) 0.05 Soil moisture (water filled porosity) 0.5

Porosity (％ volume) 70

2

Table2. Annual average soil properties in the experiment site.

3

Soil thermal conductivity (W m-1K-1) 1.07

Soil thermal diffusivity (m2s-1) 2.18×10-7

Damping depth (m) 1.48