Efforts to partition thermal energy transfer between

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1 IN-SITU DETERMINATION OF SOIL THERMAL CHARACTERISTICS B. W. Avermann, M. J. McFarland, D. W. Hill MEMBER ASAE ABSTRACT For variations of the least-sqares regression procedre proposed by Letta (1971) were sed to define a soltion envelope containing the soil thermal diffsivity fnction, a(z), for a soil with a heterogeneos moistre profile. An explicit finite difference model of the one-dimensional heat eqation was sed to evalate the for variations of Letta's procedre based pon agreement between measred and simlated temperatres over a 24-h temperatre cycle. The application of Letta's regression procedre to a database containing only temperatre data from the cooling cycle portion of the dimal temperatre wave was shown to most faithflly reprodce field temperatre measrements. KEYWORDS. Soil, Temperatre. INTRODUCTION Efforts to partition thermal energy transfer between the soil srface and the air into its sensible and latent components depend pon independent determination of the soil heat flx, an important component of the srface energy balance. Soil heat flx may be measred in sit sing flat-plate heat flx transdcers (Philip, 1961), or it may be estimated by combining laboratory determinations of the soil thermal condctivity, X, with field measrements of the soil temperatre gradient, 5T/8z. Empirical methods of estimating the soil heat flx have been proposed by Oliver et al. (1987); Horton and Wierenga (1983); Novak and Black (1983); and Chodhry et al. (1987). The se of the heat flx transdcer can be impractical, especially in instances in which the soil thermal characteristics, soil moistre regimes, and incident solar radiation regimes vary spatially. Heat flx transdcers are difficlt to install and calibrate properly, and tend to interfere with the migration of soil moistre; when the condctivity of the flx plate material differs significantly from the condctivity of the soil in which it is installed, the transdcer also disrpts the soil heat flx. A method to determine soil thermal condctivity in sit sing simpler instrmentation wold facilitate the measrement of soil heat flx. The classical approach to the analysis of soil heat flx (Carslaw and Jaeger, 1959; Jackson and Kirkham, 1958; and Van Wijk, 1963) combines Forier's Law: q=-;isr 5z with the Principle of Conservation of Energy, (1) Qi A -q A =SS. (2) where q = heat flx in the z-direction (W/cm2) X, = soil thermal condctivity (W/cm C) 8T/8z = temperatre gradient in the z-direction (C/cm) qi = heat flx entering a control volme of soil (W/cm2) qo = heat flx leaving a control volme of soil (W/cm2) Ai = srface area throgh which incoming heat flows (cm2) AQ = srface area throgh which otgoing heat flows (cm2) 5S/5t = time rate of change of heat storage in a control volme of soil (W) Cp = soil volmetric heat capacity {Hew? C) If heat flx is assmed to occr in only one direction, z, then a combination of Forier's Law and the Principle of Conservation of Energy yields the one-dimensional heat eqation, a parabolic partial differential eqation of the form: 8t 8z V 8z / When the partial derivative on the right-hand side of eqation 3 is expanded, the reslting form of the heat eqation is: (3) Article has been reviewed and approved for pblication by the Soil and Water Div. of ASAE. Presented as ASAE Paper No The athors are Brent W. Avermann, Extension Associate/Water Qality, Marshall J. McFarland, Professor, and Dennis W. Hill, Gradate Research Assistant, Dept. of Agricltral Engineering, Texas A&M University, College Station. C 5T = x^+ ^^ ^ 8t 5z2 8z 5z Eqation 4 may then be rearranged to yield: (4) VOL. 35(3): MAY-JUNE American Society of Agricltral Engineers / 92 /

2 a:=a5>r^psr 8t 8z2 5z (5) in which a is defined as the soil thermal diffsivity in cm2/s: APPROXIMATION OF PARTIAL DERIVATIVES The first task in developing the finite difference model of eqation 5 was the algebraic approximation of the partial derivatives. The first partial of temperatre with respect to time was expressed as follows (sperscripts denote time step, sbscripts denote space indices): a _ X (6) 8T-Tf'-T 5t' At (10) and P is defined as: - 1 8X P = CpSz Eqation 5 is the form of the heat eqation sed as Letta's regression model. The two spatial derivatives of temperatre, 6T/8z and 52T/5z2, were treated as the independent variables, 8T/8t as the dependent variable, and a and p as the regression parameters. The objective of this stdy was to evalate the tility of for variations of the least-sqares regression procedre proposed by Letta (1971) for estimating the soil thermal condctivity as a fnction of depth, sing the following tools: A model of the relationship between the soil volmetric heat capacity, Cp, and soil moistre content, 0. An explicit finite difference model of the onedimensional expanded heat eqation. MODEL DEVELOPMENT ESTIMATING THE SOIL VOLUMETRIC HEAT CAPACITY, Cp The soil volmetric heat capacity was determined sing a techniqe proposed by devries (1963) and simplified by Campbell (1985): where Cm (7) c^e^of)=cii-(^f)+c,e (8) volmetric heat capacity of the mineral component of the soil (assmed to be 2.39J/cm3C) volmetric heat capacity of water (assmed tobe4.18j/cm3c) soil porosity ( )f = The volmetric heat capacities of the mineral and organic fractions are very similar, typically 2.39 and 2.50 J/cm3 C, respectively, so the two fractions have been combined in the term (1 - < )f). The soil porosity was calclated from soil blk density measrements sing the eqation: The first and second partials of temperatre with respect to space were defined by the explicit approximations: 5t ^ i = ^ 8z^ Azj 2AZj +(AZi^, + AZi_,) (Tl.-Ti) (TI-TU AZ;+ AZ; i+1 Azj_j + AZj (11) (12) The explicit finite difference eqation that reslts from the above approximations is: TJ+i = TJ+ 2aAt AZ; + 2pAt (Ti-Tj) _ (Tj-Tg AZ;+ AZ; i+1 AZi_i + Azj (Tli-TiJ 2AZi+AZi^i+AZi_i (13) ESTIMATING THE PARTIAL DERIVATIVES FROM FIELD TEMPERATURE MEASUREMENTS Using the expanded form of the heat eqation as the regression model reqired the estimation of 8T/8t, 8T/8z, and 82T/8z2 from field temperatre measrements. To determine the two spatial derivatives, the depthtemperatre profile at each time interval was fit with a cbic spline interpolating fnction composed of (n - 1) cbics of the form: T.(z) = ajz3+ biz^+ CjZ+ dj (14) The end conditions selected for the spline model specified 8T/8z = 0 and 82T/8z2 = 0 at 95 cm. The first and second derivatives in space, then, were calclated from the following expressions: (,^=LOO-(BD] VPD/ (9) 8T L8z = 3a;Z2+ 2b;Z+ C; (15) where BD is the soil blk density (g/cm^), and PD is the soil particle density, assmed to be 2.65 g/cm^, a typical vale for qartz and clay minerals (Campbell, 1985). 8^ 8z2 = 6ajZ+ 2bj (16) 834 TRANSACTIONS OF THE ASAE

3 The cbic spline model of T(z) preserved the continity of the first and second derivatives throghot the space domain. The time derivative of temperatre was expressed with a finite difference approximation, for which (i) is the depth index: THERMOCOUPLE OUTPUT VOLTAGES GND VI V2 V3 V4 V5 V6 V7 V8 V9 VIO Q Q Q Q Q Q Q Q Q Q ST_T,(t+At)-T,(t) 5t At (17) BOUNDARY AND INITIAL CONDITIONS The soltion of a one-dimensional parabolic differential eqation sch as eqation 4 on the domain 0 < z < 95 reqires the specification of the initial condition, T(z,t = 0), and two bondary conditions, at z = 0 and z = 95. The initial condition was specified with the measred soil temperatres at t = 0. The srface bondary condition was described as a Dirichlet bondary: T(z = 0,t) = T,(t) (18) for which To(t) was defined by measred soil temperatres at the soil srface, taken at 15-min intervals throghot the test. Intermediate vales of TQ(t) were represented by linear interpolation of adjacent 15-min measrements. The lower bondary condition was described as a Dirichlet bondary: T(z = 95,t) = T(z = 95,t = 0) (19) which is consistent with the docmented experimental observation that the temperatre at a sfficiently deep soil layer is time-invariant (Campbell, 1985). METHODS AND MATERIALS SITE DESCRIPTION The stdy was performed at the Texas A&M University Farm, 11 km west-sothwest of College Station, Texas, between 15 March and 16 April The Texas A&M University Farm is in the 100-yr flood plain of the Brazos River, a region consisting primarily of cattle rangeland and cotton farmland. Typical soils in the area are Norwood silt loams, Norwood silty clay loams, Weswood silt loams, and Miller clays, on 0 to 1% slopes. The data was collected in a lysimeter 133 cm in diameter and 133 cm in depth. The soil in the lysimeter was a Weswood silt loam (fine-silty, mixed, thermic family of flventic Ustochrepts). The soil srface was protected from precipitation throghot the stdy. EQUIPMENT Soil temperatres were measred sing the copperconstantan (type T) thermocople. Thermocoples were monted on a wooden rod at 0, 2, 4, 7,12,19, 28, 39, 55, 75, and 95 cm, sing the thermocople jnction at 95 cm as the reference jnction (JO), as shown in figre 1. Soil temperatres (relative to the soil temperatre at 95 cm) were measred at 900 s (15 min) intervals throghot the dration of the stdy. Soil moistre measrements were made from 5 cm to 95 cm (at 10-cm intervals) on days 1,4, 5, and 11 of the stdy Jl J2 J3 J4 J5 J6 ]7 J8 J9 JIO THERMOCOUPLE JUNCTIONS Constantan Wire Copper Wire Figre l-schematic diagram of the thermocople circit. sing a netron probe (Troxler Laboratories, model 3221). Average soil blk density was determined in the laboratory sing the standard paraffin-coated aggregate method, sing seven soil samples taken at different depths within the soil profile. Soil blk density data are shown in the next section, "Reslts and Discssion." FOUR REGRESSION PROCEDURES The procedres sed to estimate the soil thermal characteristics are for variations of a regression approach proposed by Letta (1971). In Letta's method, eqation 5 is represented with a mltilinear regression model in which the two spatial derivatives of on the right-hand side of eqation 5 are the independent variables, the time derivative on the left-hand side of eqation 5 is the dependent variable, and the parameters a and p are the linear estimators. Table 1 gives a description of the for variations of Letta's regression approach evalated in this stdy. Variation 1 is identical to Letta's method, sing the entire dirnal temperatre cycle as the database for a twoparameter regression procedre. Variation 2 assmes that the soil thermal condctivity is independent of depth, reslting in a one-parameter linear regression. Variations 3 and 4 divide the dirnal temperatre cycle into its heating and cooling components and perform regression analyses on the two smaller datasets. RESULTS AND DISCUSSION SOIL VOLUMETRIC HEAT CAPACITY The soil moistre profiles measred on days 1, 4, 5, and 11 of the stdy are shown in Table 2. The migration of soil Variation TABLE 1. Description of for variations of Letta's regression approach Description 1 Two-parameter regression, fll dirnal cycles 2 One-parameter regression assming, p = 0, fll dirnal cycles 3 Two-parameter regression, heating cycle data only (5T/6t>0) 4 Two-parameter regression, cooling cycle data only (6T/6t<0) VOL. 35(3): MAY-JUNE

4 TABLE 2. Volmetric soil moistre profiles from netron probe measrements Depth (cm) Day 5 Day 11 Average t Day 7 moistre over time was assmed to be negligible throghot the stdy, and the average of the for soil moistre profiles was sed for the estimation of Cp. The average blk density of the soil in the lysimeter was determined from natral soil peds taken from the soil profile sing the standard paraffin-clod method. The reslting average blk density was 1.53 g/cm3, which is consistent with pblished vales for similar soils (Colwell, 1983). The sample standard deviation was 0.05 (n = 7) for the blk density measrements. Figre 2 shows the soil volmetric heat capacity as determined from blk density and soil moistre measrements. The field data were fit (r^ = 0.98) with a third-order polynomial for ease in interpolating at the thermocople depths. SOIL THERMAL DIFFUSIVITY Estimates of the soil thermal diffsivity were made sing the for variations of Letta's mltiple regression procedre. Figres 3 throgh 6 show the estimated vales of a at each depth between 2 and 75 cm. The increase in the variability of the estimates of a is primarily de to the decreased resoltion in the temperatre measrements at the lower depths. Below 39 cm, the Figre 3-Estiniates of a from variation 1 of Letta's regression dirnal temperatre variation was so slight that the analog/digital converter was often nable to resolve small temperatre changes over the 900 s time interval; conseqently, at the lower depths, many of the 5T/5t estimates took on a vale of zero, casing the matrix eqation from the regression analysis to approach singlarity. Other possible sorces of erratic behavior of the diffsivity estimates involve the violation of one or more of the fndamental assmptions made with the leastsqares regression procedre. The for assmptions, along with the terms commonly sed to describe them (where applicable), are listed below: The regression model is linear; that is, the independent variables are not correlated with one another. (Mlticollinearity describes the condition in which this assmption is violated.) All vales of the dependent variable are independent of one another. (Atocorrelation describes the condition in which this assmption is violated.) The vales of the dependent variable are normally distribted. > 1 Day? I Figre 2~Variation of soil volmetric heat capacity with depth. Figre 4-Estimates of a from variation 2 of Letta's regression 836 TRANSACTIONS OF THE AS AE

5 Day! 4.00e-4 - % J5 Day? 2.00e^ - O.OOe e-4 - ^ ^f^ o D 1^ GEi D El % 1 Q >«" QB ^ % Figre 5~Estimates of a from variation 3 of Letta's regression The variance of the dependent variable is the same for all vales of the independent variables. (When this condition is satisfied, the data set is said to be homoscedastic) The independent variables in eqation 5, the first and second spatial derivatives of temperatre, are closely related to one another, making the regression model inherently mlticollinear. In simple terms, that means that the regression model is "nable to separate ot the effect of each individal variable" (Hoshmand, 1988) on the dependent variable, 5T/5t. Conseqently, each regression coefficient will reflect not only the inflence of its corresponding independent variable, bt also the inflence of the other (presmably nrelated) independent variables. Hoshmand (1988) also states that atocorrelation is often observed when time series data are sed in regression analysis. Sch is the case with the regression procedre sed in this stdy. Figre 7, a plot of residals as a fnction of time from a two-parameter regression performed on data at 7 cm on day 7, shows an obvios time-dependence of the residal, and is an example of a graph which sggests the presence of atocorrelation. In this case, the timeinterdependence of one or more independent variables is contribting to the model error. ^.OOe Figre 7-Tinie dependence of the residal from the regression procedre at 7 cm, day 7. In order to simplify the evalation of the for options, the daily estimates for a from each regression approach were averaged, yielding the smmary in figre 8. It is evident from figres 5 and 8 that variation 3 gives more erratic reslts than any of the other three variations at all depths. The extreme and npredictable behavior of variation 3 at the pper depths is de to the relatively small size of the database generated by selecting only those ordered triples for which 6T/5t > 0; while a fll dirnal regression with At = 900 s generated 96 data points, a heating-cycle regression at the pper depths with the same time interval generated between 30 and 40 data points, depending pon the shape of the srface temperatre wave. (Heating-cycle fractions of sch dimal waves as might be encontered on hot, snny days with low hmidity may contain 30% or fewer of the dimal data points.) Sch a limitation wold be remedied, of corse, by decreasing the time interval between temperatre measrements. For the prposes of this stdy, however, the predictions of the heating-cycle regression were deemed nreliable. Day 7 a> CA g a* V) «j ^ CIL4 < Figre 6-Estimates of a from variation 4 of Letta's regression Figre 8-A smmary of the estimates of a from the for regression procedres. VOL. 35(3): MAY-JUNE

6 Field Data Modeled Data IH B Figre lo-measred and modeled temperatres at 12 cm, day Figre 9-SoiI temperatre at 12 cm, as predicted by the finite difference model on day 7. Evalation of the remaining three regression options was based pon the agreement between field temperatres and temperatres predicted by the finite-difference model. Figre 9 shows that the modeled temperatres sing the diffsivity and condctivity fnctions estimated by the cooling-cycle regression (variation 4) most accrately reprodce field measrements, which is to be expected since the latent heat flxes, for which the model does not accont, are small dring the cooling phase. The oneparameter regression (variation 2) reprodced field conditions least accrately. Average vales for the soil thermal condctivity, calclated as a prodct of the soil thermal diffsivity (a, estimated from the cooling-cycle regression) and the volmetric heat capacity (Cp, interpolated from field measrements shown in fig. 2), are shown in Table 3. Figres 10 and 11 show the measred and modeled temperatres at 12 cm on days 1 and 3 sing the average diffsivity fnction from the cooling-cycle regressions in the nmerical model. The agreement in both cases was good; however, the amplitdes of the modeled temperatre waves were consistently smaller than those of the field temperatres, indicating that the nmerical model tended to nderestimate the transfer of thermal energy between soil layers. This indicated that the actal field diffsivities were probably higher than those sed in the nmerical model. The nmerical model was then rn with the diffsivity fnction estimated on day 1 by the cooling-cycle regression. Figres 12 and 13 show the measred and modeled temperatres at 19 cm for days 7 and 8. This diffsivity fnction, which consistently overestimated the thermal transfer between soil layers, was determined to be a sitable pper bondary for a "soltion envelope" in which the tre diffsivity fnction wold be expected. SUMMARY AND CONCLUSIONS A method for in sit determination of the soil thermal characteristics for nonhomogeneos moistre conditions was presented. Soil volmetric heat capacity was determined from the average soil blk density and soil moistre profiles sing a weighted-average techniqe proposed by devries (1963) and simplified by Campbell (1985). For variations of a mltiple regression techniqe proposed by Letta (1971) for estimating the soil thermal diffsivity were evalated. Field temperatre data were compared with temperatres predicted by a finite difference model of the expanded heat eqation to determine which variation prodced the best estimates of a. The pper and lower bondaries were described by Dirichlet bondary conditions (time-variant at the srface, constant-temperatre at the lower bondary). In all cases, variability in estimates of the regression coefficients increased with depth de to the decreased resoltion of the temperatre measrements; 39 cm appeared to be the practical limit for all of the for variations of Letta's regression Conseqently, comparisons among the for based pon estimates of thermal diffsivity below 39 cm were deemed inconclsive. Above 39 cm, however, the application of Letta's techniqe sing only temperatre data from the cooling-cycle (5T/5t < 0) portion of the dirnal temperatre TABLE 3. Smmary of estimated soil thermal properties sing the cooling-cycle regression Depth (cm) Cp (J/cm^Q a (cm /sec) X (W/cmC) ^ e s H o on Field Data Modeled Data Figre 11-Measred and modeled temperatres at 12 cm, day TRANSACTIONS OF THE ASAE

7 Field Data Modeled Data a, S Figre 12-Measred and modeled temperatres at 19 cm, day 7. wave was shown to generate the best reslts. There are at least two apparent reasons for that reslt: The physical model sed to describe the process of thermal transfer within a soil-air system did not accont for the transfer of latent heat throgh moistre flx, especially vapor flx; in essence, the model assmes zero vapor flx, which wold be most nearly tre dring the cooling cycle at the srface. Therefore, the se of only cooling-cycle data removes the data which vapor flxes wold affect the most. The presence of higher-order (than dirnal) harmonics directly affected the accracy of a nmerical procedre which discretizes the domain. Finite-difference discretization in time obscred important short-term temperatre variations which tended to arise most prominently dring the heating cycle. In addition, the crvatre of the timetemperatre relationship tended to be less dring the cooling cycle than dring the heating cycle; conseqently, the cooling cycle wold be most sited to the finite-difference method of estimating 6T/5t. While the primary predictive vale of the regression approach seems to have been affirmed, the statistical shortcomings of the approach mst not be ignored, especially the violation of fndamental assmptions for a regression model. Since the terms 5T/6z and 52T/5z2 are correlated by the derivative operation, the treatment of 8T/8Z and 82T/8z2 as independent variables in a regression procedre is statistically invalid. Also, the raw data from which the predictors arise are tainted by serial correlation; that condition, however, might be remedied throgh the se of atoregressive techniqes. The tre vale of the regression model discssed in this article is limited to the prediction of the thermal characteristics, and no conclsions with respect to casality, confidence intervals, or other statistical inferences shold be made Figre 13-Measred and modeled temperatres at 19 cm, day 8. REFERENCES Campbell, G. S Soil Physics With Basic: Transport Models for Soil-Plant Systems. New York: Elsevier Science Pblishers. Carslaw, H. S. and J. C. Jaeger Condction of Heat in Solids, London: Oxford University Press. Chodhry, B. J., S. B. Idso and R. J. Reginato Analysis of an empirical model for soil heatflxnder a growing wheat crop for estimating evaporation by an infrared-temperatre based energy balance eqation. Agri. and For. Met. 39(4): Colwell, R. N Manal of Remote Sensing. Falls Chrch, VA: American Society of Photogrammetry. devries, D. A Thermal properties of soils, In Physics of Plant Environment, ed. W. R. Van Wijk. Amsterdam: North-Holland Pbl. Co. Horton, R. and P. J. Wierenga Estimating the soil heat flx from observations of soil temperatre near the srface. Soil Sci.Soc. Am. J. 47(m4'20. Hoshmand, A. R Statistical Methods for Agricltral Sciences. Portland, OR: Timber Press. Jackson, R. D. and D. Kirkham Method of measrement of the real thermal diffsivity of moist soil. Soil Sci. Soc. Am. Proc. 22(6): Letta, B Determination of the thermal diffsivity in the pper layers of a natral grond cover. Soil Science 112(3): Novak, M. D. and T. A. Black The srface heat flx density of a bare soil. Atmosphere-Ocean 21(4): Oliver, S. A., H. R. Oliver, J. S. Wallace and A. M. Roberts Soil heatflxand temperatre variation with vegetation, soil type, and climate. Agri. and For. Met. 39(3): Philip, J. R The theory of heatflxmeters. /. Geophys. Res. 66(2): Van Wijk, W. R., ed Physics of Plant Environment. Amsterdam: North-Holland Pbl. Co. VOL. 35(3): MAY-JUNE