estimating the soil heat flux from observations of soil temperature near the surface1

Estimating the Soil Heat Flux from Observations of Soil Temperature Near the Surface1

R. HORTON AND P. J. WlERENGA2

ABSTRACTA method was described for estimating soil heat flux by measuring

soil moisture content and soil temperature at two or three depths nearthe soil surface. The method utilizes an analytical expression of heatflux for a semi-infinite homogeneous soil profile for which the surfacetemperature is described by a Fourier series. For relatively homogeneoussoil, the estimates of soil heat flux, based on observation of temperatureat two depths, compared well with values of heat flux determined withthe temperature integral method. Estimates were best when one of themeasurement depths was very near the surface (< 1 cm) and the otherbetween the 10- and 20-cm depths. When soil profiles were nonhomo-geneous it was found that three rather than two depths of temperatureobservation should be considered. Although the analytical solution forestimating the heat flux strictly does not apply to nonhomogeneous soils,heat flux values obtained with the proposed method compared favorablywith those computed with the temperature integral method.

Additional Index Words: soil heat transfer, soil heat flux density,conduction, analytical solution, smoothing temperature data.

Horton, R., and P. J. Wierenga. 1983. Estimating the soil heat fluxfrom observations of soil temperature near the surface. Soil Sci. Soc.Am. J. 47:14-20.

THE SURFACE soil heat flux is an important compo-nent of the energy balance at the earth's surface,

which may be writtenRn = LE + H + G, [1]

where Rn is the net radiation, LE the energy involvedwith evaporation of water, H the energy exchange withair, and G the energy transfer with the soil. The net daily

1 Journal Art. no. 926, Agricultural Experiment Station, New Mex-ico State University, Las Cruces, NM 88003. Received 16 Mar. 1982.Approved 20 Sept. 1982.

2 Former Graduate Student, now Assistant Professor, Dep. of Agron-omy, Iowa State University, and Professor, Dep. of Agronomy, NewMexico State University.

soil heat flux usually does not exceed 10 to 15% of Rn.However, soil heat flux can be a much larger part of thehourly heat balance, particularly for dry desert soils.Consequently, increased precision is required in estimat-ing the heat flux for hourly or shorter periods than for24-h or longer periods.

One method to measure soil heat flux is to use a heatflux meter, which is a thin flat plate placed in the soilnormal to the direction of heat flow. Only a single mea-surement of the temperature difference across the plateis required. However, the meter interferes with both liq-uid and vapor movement in soil, which limits its use todepths greater than about 10 cm. The meter requirescalibration in the medium in which measurements aredesired (Philip, 1961). Horton and Wierenga (1983)found that separate in-situ calibrations were required foreach placement of a meter.

Soil heat flux can also be measured with the temper-ature gradient method (Tanner, 1963). In this case G iscomputed from the apparent thermal conductivity, X, andthe gradient in temperature, T, with depth, z, as in Eq.[2]:

G = -\(BT/dz). [2]Both X and dT/dz must be known at the depths whereG is desired. The temperature gradient at a specific depthbelow the surface can be evaluated numerically usingvalues of temperature measured at closely spaced inter-vals at and/or near the specific depth. The value of X isdifficult to measure accurately, especially at or near thesoil surface, limiting application of the temperature gra-dient method.

With the temperature integral or calorimetric method,soil heat flux is computed from the change in heat storagein the soil profile over a given time interval (Lettau andDavidson, 1957). As well as the need to measure tem-perature at several depths, this method requires a deter-mination of the soil volumetric heat capacity at several

HORTON & WIERENGA : ESTIMATING THE SOIL HEAT FLUX FROM OBSERVATIONS OF SOIL TEMPERATURE 15

depths. The volumetric heat capacity is usually estimatedwith measurements of bulk density and water contentusing the following equation by de Vries (1963):

c = OA6Xm + 0.60X0 + Xw, [3]where Xm, X0, and Xw are the volume fractions of min-erals, organic matter, and water, respectively. Since therewill be diurnal changes in temperature only to about 50cm in moist mineral soils, the measurements may be con-fined to a relatively shallow zone. Even so, the numberof measurements required for accurate determination ofthe soil heat flux can be large.

The null-alignment method to determine G is basedupon measurements of temperature and volumetric heatcapacity in the upper 20 or 30 cm of soils. This methodcombines the concepts of the temperature gradient andtemperature integral methods. The method includes aprocedure for determining X at a subsurface depth so thatthe inclusion of the temperature gradient method is notdetrimental. However, soil heat flux can be accuratelydetermined only when temperature and volumetric heatcapacity are determined at several depths. For example,Kimball and Jackson (1975) measured soil temperatureat 0.5-cm intervals above the 10-cm depth and at 2 cm-intervals between the 10- and 30-cm depths. Althoughthe methods which utilize measurement of temperatureto determine soil heat flux offer several advantages overthe use of a heat flux meter, the large number of mea-surements required for their accurate use is often pro-hibitive.

Ideally, methods of determining soil heat flux wouldnot require calibration, would not interfere with moisturemovement, and would require only a few convenient mea-surements at each time interval. Suomi (1957) and Hor-ton and Wierenga (1983) described such a method, whichmade use of resistance thermometers or thermocouplesconnected in parallel to determine changes in mean soiltemperature. However, the method was shown to be mostappropriate for soil with uniform volumetric heat capac-ity, and highly sensitive equipment was required to ac-curately determine the small changes in mean soil tem-perature at short time intervals (Horton and Wierenga,1983). Thus, use of the method is limited with regard tosoil conditions and instrumentation. A simple yet accu-rate method of estimating soil heat flux for a wide rangeof soil conditions is still desirable.

The purpose of this study is to discuss the estimationof soil heat flux with a method which requires observa-tions of temperature and moisture only near the soil sur-face. The estimated results will be compared with valuesof heat flux determined using the temperature integralmethod.

THEORETICALAn equation describing one-dimensional heat transfer in a

homogeneous medium isdT d*T

and

where T is the temperature, t the time, z the depth, and a theapparent thermal diffusivity. With boundary conditions

sin(na" [5]

lim T(z,t) = T,,Z—>OO

[6]

the solution of Eq. [4] is (Carslaw and Jaeger, 1959)

T(z,() =n-l

sin(mof [7]where T, is the temporal average soil temperature, assumed tobe the same at all depths, M the number of harmonics, Aon and$„„ the amplitude and phase angle, respectively, of the »'* har-monic for the upper boundary temperature, and o> the radialfrequency equal to 2ir/P with P being the period of the fun-damental cycle.

The soil heat flux can be obtained by differentiating Eq. [7]with respect to z and combining the result with the Fourier heatequation, Eq. [2], with A = etc (Horton, 1982). After differ-entiating Eq. [7] with respect to z, dT/dz is given by

Z'dz

[sin(mor ++ cos(no>r + $on — z\/n<o/2a)]} . [8]

Equation [8] can be written in a more compact form as

«X*,0 _dz

n-l

[9]When Eq. [9] is substituted into Eq. [2], an expression is ob-tained for the soil heat flux at all depths and times:

M

G(z,t) = exp(— z\/«o;/2a) sin [mo/ +

[10]+ (»/4) -Equation [10] represents the heat flux, positive downward, ina homogeneous soil profile, with the temperature at the surfacedescribed by a Fourier series and with a constant temperatureT, deep in the profile. To calculate the soil heat flux with Eq.[10] one has to know the values for Aon and 4>on for the tem-perature at one depth, as well as a and c for the soil.

EXPERIMENTALTwo separate analyses were made in this study. In the first

analysis, soil heat flux was considered for a relatively uniformprofile under grass. In the second analysis, soil heat flux wasconsidered for nonuniform profiles with bare surfaces. The firstanalysis considered heat flux with clear sky conditions, and thesecond analysis was made for partly cloudy conditions. In bothcases soil heat flux estimated with Eq. [10] was compared withheat flux determined by the temperature integral method.

The temperature integral method was used to determine thesoil heat flux, G, calorimetrically according to

where Q, the heat storage, is computed at a given time, t, as

[12]Q = c(z) T(z) dz,z\

where volumetric heat capacity, c, and temperature, T, are bothfunctions of depth, z. Q was determined at times ̂ and t2 using

16 SOIL SCI. SOC. AM. J., VOL. 47, 1983

VOLUMETRIC HEAT CAPACITY (co l /cm 3 C).30 .35 40 .45 .50 .55 .60

28

10

20

30

40Eo-50l-o.060

70

80

90

100

~^t i i *v 1 1\ X.

\ NONIRRIGATED ^v IRRIGATED

^X >v

\ \\ \\\

\\\

U \\

\\

\

\\\

\\\

\

THERMAL CONDUCTIVITY (col/cm hr C)3 4 5 6 7 8 9 l (

10

20

30

40"io-50t-o.lil

060

70

80

90

100

NONIRRIGATED N^ IRRIGATED^~-^ A

^X \X \

X \

\ \\ \\\\

\\\\\\\\\\\\\\\\

i \Fig. 1—Depth distributions of volumetric heat capacity, c, and thermal

conductivity, A, used to predict variations in subsoil temperature ofirrigated and nonirrigated soil profiles.

the trapezoidal rule for integration, and G was computed withEq. [11] over the time interval from f, to t2. This value of Gwas compared to an estimate of G, Eq. [10], at the time midwaybetween /[ and /2.

Experiment 1Soil temperature was measured at several depths in a grass-

covered field at the Plant Science Research Center near LasCruces, N.M. The soil, classified as Glendale clay loam (mixed,calcareous, thermic family of Typic Torrifluvents) consisted ofa 60-cm layer of clay loam material overlying a deep sand.Copper-constantan thermocouples placed at depths of 1, 3, 5,10, 15, 25, 35, 50, and 60 cm were connected to datalogging

24IEUJO.

22

20

28

26UJ(E

24ccbjO.

22

20

18

16

Icm

9 12 15TIME (h rs )

18 21 24

Fig. 2—Observed values of temperature at 1, 5, 15, and 60 cm undergrass cover at Las Cruces, N.M., for day 1 (A) and day 2 (B).

equipment (Autodata nine datalogger, ACUREX Autodata, 485Clyde Ave., Mountain View, CA 94042) which continuallyscanned and numerically averaged the output of each ther-mocouple. Mean hourly values were recorded on tape with acassette recorder (Techtran Industries, 580 Jefferson Road,Rochester, NY 15623).

A Fourier series of M harmonics, Eq. [5], was fitted to ob-served values of temperature at the 1-cm depth using standardmultiple regression techniques (Draper and Smith, 1966) todetermine values of Aon and <bon. Volumetric heat capacity ofthe soil was determined from Eq. [3] using field measurementsof bulk density and water content. The daily mean apparentthermal diffusivity of the soil was determined from a harmonicanalysis of the soil temperature as described in Horton et al.(1983). With the values of Aon and $„„, c and a so determined,Eq. [10] was then used to compute the soil heat flux at the 1-cm depth. The estimated soil heat flux was compared to soilheat flux determinations based on the temperature integralmethod.

Experiment 2The objective of this analysis was to consider heat flux in

nonuniform soil profiles. Values of soil temperature were gen-erated using a numerical solution of the equation of heat transferin a nonhomogeneous medium (Wierenga and de Wit, 1970).In order to numerically generate the temperature values at var-ious depths in a soil profile, the temperature at two boundarydepths and the initial temperature distribution of the profilemust be known. Also a distribution of A and c with depth mustbe assumed.

For this analysis two cases were considered: one representativeof a recently irrigated soil profile, and the other of a nonirrigatedprofile. In the numerical solutions, soil temperatures at the upper


0.12 -

3 6 if 9 12 15

12 15 ^18 21 24

Fig. 3—Comparison between the soil heat flux computed with Eq. [10],using observed values of temperature at depths of 1 and 5 cm, andvalues of heat flux determined with the temperature integral methodfor day I (A) and day 2 (B).

boundary depth (surface) for both cases were taken to be thesoil temperatures measured at a depth of 1 cm in a recentlyirrigated soil and in a nonirrigated soil [Wierenga (1968); dataof 2 Sept. 1967]. The lower boundary temperatures (100 cm)were taken as constant at 25 °C. Initial temperature distributionswere generated by putting the numerical solution through threecycles of the boundary conditions. The assumed distributionsof c and X for both irrigated and nonirrigated profiles are pre-sented in Fig. 1. The distributions are similar to actual measuredvalues presented by Wierenga (1968).

With the numerical procedure temperature values were gen-erated at one-half-h time intervals at uniform depth incrementsof 1 cm from the surface to a depth of 20 cm, and of 5-cmincrements from the 20-cm depth to a depth of 100 cm. Thesegenerated temperatures were used in considering the heat fluxfor both profile conditions as follows: A Fourier series of Mharmonics, Eq. [5], was fitted to the surface temperature valuesfor each profile [values measured by Wierenga (1968)] to de-termine Aon and *„„. The daily mean apparent thermal diffu-sivities of the soils were determined with the harmonic analysismethod (Horton et al., 1983). The values of Am, $on, c, and aso determined were inserted in Eq. [10], and the surface soilheat flux calculated for both profiles. The calculated values (Eq.[10]) were compared to soil heat flux determinations obtainedwith the temperature integral method using the same temper-ature data and the assumed variation of c with depth (Fig. 1).

RESULTSExperiment 1

The observed values of soil temperature at 1, 5, 15,and 60 cm for 2 d are presented in Fig. 2. The valueswere taken under clear sky conditions. For each day thevalues of temperature at the 1-cm depth were fitted to

0.12 -

g 0.08C\J

E

^ 0.04

5 0.00

<-0.04x

5 -0.08in

E 0.08

i 0.04

5 0.00

<-0.04I

O -0.08 -tn

o INTEGRAL METHOD—— EQUATION 10

9 12 15TIME (hrs)

Fig. 4 — Comparison between the soil heat flux computed with Eq. [10],using observed values of temperature at depths of 1 and 15 cm, andvalues of soil heat flux determined with the temperature integral methodfor day 1 (A) and day 2 (B).

Eq. [5]. Values of Aon and $on obtained by fitting thedata to Eq. [5] are presented in Table 1. Values of a forthe 1- to 5-, 1- to 15-, and 5- to 15-cm layers were de-termined as 3.50, 4.92, and 5.78 X 10~3 cm2^"1 for thefirst day, respectively, and of 3.17, 4.22, and 4.78 X 10~3

cm2-s~' for the second day. Inasmuch as the grass fieldwas irrigated prior to the temperature measurements, thevolumetric heat capacity was approximately uniform withdepth and found to be 0.5 cal/cm3 C. Equation [10] wasused to estimate the soil heat flux at the depth of 1 cmfor both days using the above values of a for the 1- to5- and for the 1- to 15-cm layers. Values of the soil heatflux are compared to the values determined with the tem-perature integral method in Fig. 3 and 4. In all cases theestimates are reasonable, but overall the agreement inFig. 4 was better.

The soil heat flux can also be estimated by consideringtwo soil layers — the 1- to 5- and the 5- to 15-cm layers.To do this the observed temperature fluctuation at 5 cmwas fitted to Eq. [5] and values of Aon and $on werecalculated (see Table 1). The heat flux for the finite layerof soil, 1 to 5 cm, was first calculated using the valuesof Aon and 4>on obtained from the soil temperature dis-tribution at 1 cm, and the c and a values for the 1- to5-cm layer. Using Eq. [10] the heat flux for the layerwas found by computing the difference in heat flux atthe top (e.g., at 1 cm) and bottom (e.g., at 5 cm) of thelayer. The heat flux at the 5-cm depth was obtained fromthe values of Aon and 4>on at 5 cm, and the values of cand a for the 5- to 15-cm layer with Eq. [10]. The sumof the heat flux for the 1- to 5-cm layer and for the soilat the 5-cm depth was taken to be the heat flux at the1-cm depth. This estimation of heat flux is compared with


Table 1—Values of amplitude, A, and phase angle, 0, fortemperature observed at 1- and 5-cm depths

under a grass cover.

Day 1

(B)

12345

6789

10

1 cm

A

ct4.191.080.250.110.11

0.060.050.060.050.06

*

rad3.710.583.043.312.06

2.302.192.461.591.68

5cm

A

C2.730.510.130.100.06

0.040.050.05

<t>

rad3.34

-0.092.602.812.04

2.552.322.07

Day 2

1 cm

A

C3.331.240.040.050.05

0.050.06

0

rad3.870.632.762.991.77

2.29-1.39

5cm

A

C2.080.680.030.020.03

<t>

rad3.470.091.082.380.60

3 6o

9 12 ISTIME (hrs)

o INTEGRAL METHOD——— EQUATION 10

21 24

t C = centigrade; rad = radian.

the measured values of heat flux in Fig. 5. For times >0300 h the agreement is excellent.

Experiment 2The temperature variation at the soil surface and the

numerically computed values of temperature at depthsof 5 and 20 cm for both irrigated and nonirrigated pro-files are presented in Fig. 6. For each profile, the tem-peratures at both 0 and 5 cm were fitted to Eq. [5] toobtain values of Aon and $on at each depth (Table 2).

0.12

g 0.08(SJ

E

i 0.04

0.00

< -0.04UJ

-0.08

-0.12

g 0.08C\JE

i 0.04

0.00

2 -0.04T

_J

5 -0.08

Fig. S—Comparison between the soil heat flux computed with Eq. [10]using observed values of temperature at depths of 1, 5, and 15 cm,and values of heat flux determined with the temperature integral methodfor day 1 (A) and day 2 (B).

B

9 12 15TIME (hrs)

Table 2—Values of amplitudes, A, and phase angles, <t>, fortemperature at the 0- and 5-cm depths for recently

irrigated and nonirrigated profiles.

Irrigated

In)

12345

6789

10

Ocm

A

ct11.14

2.750.541.440.52

0.770.550.510.200.05

4>rad3.890.99

-0.422.84

-1.162.014.591.803.532.12

5cm

A

C7.081.430.230.580.18

0.270.170.140.06

<t>

rad3.450.40

-1.322.004.11

0.953.360.642.41

NonirrigatedOcm

A

C16.644.550.771.980.81

1.200.890.750.38

<t>rad4.001.06

-0.132.92

-1.092.014.661.723.44

5cm

A

C8.401.750.240.560.20

0.250.190.120.06

<t>

rad3.410.26

-1.221.813.80

0.683.110.171.82

t C = centigrade; rad = radian.

The values of c considered for the three layers, 0 to 5, 0to 20, and 5 to 20 cm obtained from Fig. 1, were foundto be 0.465, 0.515, and 0.530 cal/cm3 ° C for the irrigatedprofile and 0.295, 0.342, and 0.358 cal/cm3 °C for thenonirrigated profile, respectively. Values of a for the threelayers, 0 to 5, 0 to 20, and 5 to 20 cm, were determinedto be 4.56, 4.67, and 4.72 X 10"3 cm2^-' for the irri-gated profile and 2.28, 3.39, and 3.94 X 10~3 cm2-s-'for the nonirrigated profile, respectively. Figure 7 pre-sents a comparison between surface heat flux estimatedwith Eq. [10], based upon temperatures at the surface

48

' 40

ui(T

0-Z 24UJ

26

0 cm

20 cm

IRRIGATED

9 12 15TIME ( h r s )

18 24

Fig. 6—Soil temperature variation at 0, 5, and 20 cm in irrigated andnonirrigated soil profiles.


and at the 20-cm depth, and heat flux computed with thetemperature integral method using all numerically gen-erated temperatures. For both profiles the heat flux com-puted with Eq. [10] closely follows the heat flux calcu-lated with the integral method. There is better agreementfor the irrigated profile. The irrigated soil varied less withdepth in both c and X than the nonirrigated soil. Figure8 shows a comparison between the heat flux computedwith the integral method and the heat flux estimated withEq. [10], based upon soil temperature data for the 0-,5-, and 20-cm depths. There is a slight improvement inagreement with results over those shown in Fig. 7 for therecently irrigated profile, but a large improvement forthe results from the nonirrigated profile.

DISCUSSIONThe results from Experiments 1 and 2 indicate that

for relatively uniform soil the heat flux could be estimatedfrom soil temperatures measured at only two depths.However, heat fluxes so computed agreed more closelywith the actual heat fluxes (based on the temperatureintegral method) when the 1- and 15- rather than the 1-and 5-cm depths were considered. If the lower depth is

deep in the profile (between 10 and 20 cm), estimates ofboth c and a are more representative of the soil profile,which apparently resulted in improved estimates of thesoil heat flux.

In cases where the soil profiles were nonhomogeneous,the estimations of soil heat flux improved when temper-ature observations at three rather than two depths wereused. In our examples the positions of temperature ob-servation were near the surface (< 1 cm), 5 cm, andsomewhere between 10 and 20 cm. When temperatureobservations at three depths are considered, the soil isactually segmented into two layers. The top layer is finitewhile the lower layer is semi-infinite. Although Eq. [10]does not strictly apply to such a soil, the surface heatflux estimations made with Eq. [10] in this manner werefound to closely agree with determinations using the tem-perature integral method.

In many cases the estimations of heat flux with Eq.[10] were least accurate for the beginning and endingtime periods. This is a common phenomenon that occurswhen smoothing data or when differentiating smootheddata (Kimball, 1974, 1976). In order to improve the es-timations of heat flux at the beginning of, and at the endof the time period for which soil heat flux is to be esti-

0.36 -

0.24

x 0.12

0.00

-0.12

-0.24 L

0.36 -

IRRIGATED

9 12 15TIME ( h r s )

INTEGRAL METHODEQUATION 10

NONIRRGATED

-0.24 L

Fig. 7—Comparison between the soil heat flux computed with Eq. [10],using values of temperature at depths of 0 and 20 cm, and values ofheat flux determined with the temperature integral method.

° INTEGRAL METHOD——— EQUATION 10

-0.24 L

Fig. 8—Comparison between soil heat flux computed with Eq. [10], usingtemperatures at the surface, and at the 5- and 20-cm depths, andvalues of heat flux determined with the temperature integral method.


0.121-

| 0.08CJ

ui 0.04

x0

u.

2 -°-04I

O -0.08</>

o o 6 ° o

9 12 15TIME (hrs)

o o <TO-o INTEGRAL METHOD

—— EQUATION 10

21 24

fig. 9—Comparison between the soil heat flux computed with Eq. [10],using observed values of temperature at depths of 1 and 15 cm, andvalues of soil heat flux determined with the temperature integral method.Data from 3 h before and 3 h after the 24-h time period were consideredwhen fitting the Fourier series.

mated, several observations of soil temperature should beincluded before and after this time period when fittingdata to Eq. [5]. If the heat flux is to be estimated for aperiod of 24 h then temperature data occurring before,during, and after the 24-h period should be used whenfitting for Aon and $on. As an example, the soil heat fluxwas calculated for the first day of Experiment 1, usingtemperature observations at the 1- and 15-cm depths.The results presented in Fig. 9 show that the estimatedheat flux, for times < 0300 h, is closer to the heat fluxcalculated with the integral method (see also Fig. 4a).

The analytical method, Eq. [10], for estimating soilheat flux offers several advantages over previous methods.First, the mean apparent thermal diffusivity for a layerof soil is determined rather than the apparent thermalconductivity at a specific depth as required by the tem-perature gradient method. Also, the analytical method isnot as sensitive to inaccuracies in X as is the temperaturegradient method. The soil surface heat flux as determinedby the temperature gradient method is directly propor-tional to the value of X, while the soil surface heat fluxas determined by the analytical method is directly pro-portional to the square root of a. Second, the methodproposed here is based on temperature observations, whichcan be made with small sensors which can be installedwith minimal disturbance to the soil and which do notinterfere with the movement of water and heat. This lackof soil disturbance and interference is advantageous overheat flux meters used either alone or in the combinationmethod. Additionally, if thermocouples are used as thetemperature sensors, no separate individual calibrationwith additional apparatus is required. Third, with theanalytical method, temperature and water content mea-surements are required at only 2 or 3 depths, which ismuch fewer than that required by the temperature in-tegral and null-alignment methods. Finally, the analyticalmethod is advantageous over the mean temperature

methods because it requires measurements only in theupper 20 cm of soil, while the latter require measurementsto 50 cm or more. In fact, it is better for the presentmethod to use the temperature data from the upper soilprofile only, because changes in temperature are morepronounced, and the apparent thermal diffusivity can bedetermined more accurately. The result is that soil tem-perature measuring equipment can be less sensitive forthe analytic method than for the mean temperaturemethods, because the mean soil temperature for a thicklayer of soil changes much more slowly than temperatureat a single depth in the upper 20 cm.

A further advantage of using the analytical methodfor estimating soil heat flux is that estimates of the ap-parent thermal diffusivity are obtained. Values of a arenot only important for estimates of soil heat flux, but canalso be used with Eq. [7] to estimate the soil temperature(Horton et al., 1983). If a is known for a particular soilprofile, observations of soil temperature are required atonly one depth in order to estimate soil heat flux and/ortemperature.

A disadvantage of the analytical method is that morecomputations are required than with the mean temper-ature methods. However, with the availability of multipleregression computer packages, the computations shouldnot be difficult.

estimating the soil heat flux from observations of soil temperature near the surface1

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