Soil Science Society of America Journal 65:1074-1080 (2001)
© 2001 Soil Science Society of America
DIVISION S-1—SOIL PHYSICS
Predicting Temperature and Heat Flow in a Sandy Soil by Electrical Modeling
Dardo O. Guaragliaa,
Jorge L. Pousa*,b and
Leonardo Pilanc
a CONICET, Dep. de Hidráulica, Facultad de Ingeniería, UNLP, La Plata, Argentina
b CONICET, Lab. de Oceanografía Costera, Facultad de Ciencias Naturales y Museo, UNLP, Casilla de Correo 45, (1900) La Plata, Argentina
c CNR, Istituto per lo Studio della Dinamica delle Grandi Masse, Venezia, Italia
* Corresponding author (dguaragl{at}volta.ing.unlp.edu.ar)
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ABSTRACT
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A model based on an electrical analogy between a soil column and an electrical transmission line was developed to predict temperature and heat flow as functions of depth and time in a sandy soil, taking into account changes in soil thermal conductivity and volumetric heat capacity due to variations in water content. The model was excited alternatively by both measured soil temperature at the 1-cm depth and solar radiation [Sr(t)], and solved with available electrical analysis software. The results were compared with field data collected during a 35-d field experiment carried out in the Lido beach, Venice, Italy. A very simple transfer function was identified for using measured Sr(t) as the input signal. This transfer function turned out to vary inversely with Sr(t). When the model is excited by temperature, and soil water content corrected every 5 d, the root mean square error (RMSE) for the calculated temperature at the 5-cm depth is less than 1°C. When it is excited by Sr(t), the RMSE at the 1-cm depth is less than 2°C. Hourly temperatures at different depths were found to depend strongly on surface phenomena, and to a lesser extent on other factors like soil water content below the top layers.
Abbreviations: c, specific heat (J kg-1 °C-1) • Ce, electrical capacitance (F) • Ct, heat capacity per unit area of a soil layer (J °C-1 m-2) • f, frequency (Hz) • G, electrical conductance (
-1) • GT(t), function equivalent to an electrical conductance (W m-2 °C-1) • I1(t), heat flow at the surface and within the top centimeter of soil (W m-2) • I2(t), heat flow transmitted below the 1-cm depth (W m-2) • L, electrical inductance (H) • R-C, ladder of resistances and capacitances for modeling soil below the 1-cm depth • r, correlation coefficient • Re, electrical resistance () • Rt, thermal resistance per unit area of a soil layer (°C m2 W-1) • RT(t), time dependent transfer function for the surface and top centimeter of soil (W-1 m2 °C) • RMSE, root mean square error • Sr(t), solar radiation (W m-2) • t, time (s) • T1(t), soil temperature at depth of 1 cm (°C) • Z, electrical impedance () • 
T, temperature difference across a soil layer (°C) • z, thickness of a surface-parallel soil layer (m) • 
, soil water content (m3 m-3) • 
, thermal diffusivity (m2 s-1) • 
, thermal conductivity (W m-1 °C-1) • 
, density (kg m-3) • c, volumetric heat capacity (J m-3 °C-1) • 
, heat flux density (W m-2)
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INTRODUCTION
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MANY PHYSICAL MODELS have been developed in connection with agricultural studies to predict soil temperature with depth and time. Some of them require measurements of soil temperature at or near the soil surface (Wierenga and de Wit, 1970; Hanks et al., 1971). Other models use soil and air temperatures normalized with respect to daily maximum and minimum, and averaged over a given period of time to obtain temperature curves for predicting purposes (Gupta et al., 1981, 1982, 1983, 1984), or are based on averaged or filtered air and soil temperature information (Persaud and Chang, 1983, 1984; Parton, 1984; Kemp et al., 1992). These models predict maximum, minimum, and daily mean soil temperatures rather well. However, calculation of mean values during normalizing or filtering processes results in loss of high frequency information. According to the Nyquist theorem (Schwartz and Shaw, 1975), at least two input samples per hour are needed to reproduce a bandlimited random hourly soil temperature. Thus, models identified from input/output filtered data are not able to recover high frequency information because this information was not taken into account in the system (model) identification process. It can be concluded that these models are valid for predicting low frequency soil temperature fluctuations, but they do not seem to be the most adequate ones for predicting hourly temperature variations. Pikul (1991) and Katul and Parlange (1993) use a surface energy balance approach for predicting hourly surface temperature. Both models require information on air temperature, relative humidity, wind speed, and net radiation, in addition to rainfall (Pikul, 1991) and soil heat flux (Katul and Parlange, 1993).
Persaud and Chang (1983)(1984) state that temperature changes in the soil profile are mainly due to the transference of heat energy produced at the soil surface from incident Sr(t). Gupta et al. (1981) report that soil surface temperature has a rapid response to changes in weather conditions (e.g., cloud cover), but that air temperature responds slowly. Therefore, to reproduce the rapid variations of the soil temperature, it would be reasonable to think of measured Sr(t) as one of the most important inputs to a model. Solar radiation data is available at many weather stations.
The objective of this work is to present a model for reproducing the rapid variations of soil temperature. The model is based on an electrical analogy between a soil column and an electrical transmission line, and was applied to a bare sandy soil to identify the factors that influence most the conversion of Sr(t) into temperature. Calculations required 5 min on a PC with an AMD586 microprocessor of 133 MHz, with 32 Mb RAM, and running under Windows ‘95 OSR2. The manuscript has been organized as follows: (i) A soil thermal model analog to an electrical transmission line is proposed as a generalization of a ladder of resistive-capacitive cells. (ii) The model is validated with experimental data by exciting it with soil temperature measured at the 1-cm depth. (iii) A simple transfer function between Sr(t) and soil temperature at depth of 1 cm is identified and used to calculate hourly soil temperature profiles.
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MODEL DESCRIPTION
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For a soil column divided into a series of horizontal, surface-parallel layers of thickness z (m) and constant heat flux density and thermal properties, the thermal resistance Rt (°C m2 W-1) and heat capacity Ct (J °C-1 m-2) (both per unit area) of a single layer are given by,
 | (1) |
where T (°C) is the temperature difference across the layer, (W m-2) the heat flux density, (W m-1 °C-1) the thermal conductivity, (kg m-3) the density, and c (J kg-1 °C-1) the specific heat. c (J m-3 °C-1) and = /c (m2 s-1) are the volumetric heat capacity and thermal diffusivity of the layer, respectively.
From the analogy between heat flow in soil and current flow in an electrical R-C transmission line of distributed parameters, Eq. [1] allows constructing an electrical model for this multi-layer soil column geometry by replacing the set of Rt and Ct with the set of their electrical analogue resistance, Re () and capacitance, Ce (F), connected in the form of an R-C ladder (Guaraglia and Pousa, 1999). In this model, heat flow is replaced by current (A) and temperature by voltage (V). Thus, an infinitely deep soil column must be represented by an infinite R-C line. In practice, however, any electrical model has a finite number of R-C cells, but this number should be large enough to prevent undesirable finite-length effects. A useful concept to test for these effects is the impedance of a transmission line (see Appendix). A detailed derivation of the electrical analogy between linear heat flow and the transmission-line equations can be found in Pipes (1958)(p. 446–454 and 496–498).
When soil temperature measured at a given depth is used to calculate temperature and heat flow at other depths, the electrical analogy requires that the model be excited with an ideal voltage source (no internal serial resistance) proportional to the measured temperature. If soil heat flow or Sr(t) are used to excite the model, an ideal current source (no internal parallel resistance) should be used. Figure 1 shows the complete model from the soil surface and with measured Sr(t) as the input signal, where t is the time. To convert electric current (radiation) into voltage (temperature), it is necessary to add a resistance in parallel with the current source. This resistance should be time dependent (see below), and is denoted by RT (t), where the subscript T stands for temperature and t is the time. The meaning of the remaining elements in Fig. 1 is explained below.
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Fig. 1. Model for measured solar radiation Sr(t) as the input signal. RT(t) = time dependent transfer function for the surface and the top centimeter of soil; I1(t) = heat flow at the surface and within the top centimeter of soil; I2(t) = heat flow transmitted below the 1-cm depth; T1(t) = soil temperature at depth of 1 cm; R-C = ladder of resistances and capacitances for modeling soil below the 1-cm depth.
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Following Wu et al. (1997), it was considered a transfer function representing both the surface and the top centimeter of soil. These authors describe a transfer function model in the time domain as an input-output filter-type model that relates two time series in such a way that observations of the input time series may be used to estimate the output time series. The transfer function used in this work can be conveniently identified on its physical grounds by considering Sr(t) as the cause of (i) the physical processes at the soil surface plus the heat flow in the top 1 cm layer [both represented by I1(t) in Fig. 1, where the subscript 1 stands for the top 1 cm layer and t is the time], and (ii) the heat flow transmitted beyond the 1-cm depth [represented by I2(t) in Fig. 1, where 2 denotes the soil column below the 1-cm depth]. This somewhat arbitrary way of splitting up the processes produced by Sr(t) was adopted because the uppermost measured temperature available is at depth of 1 cm. A function GT(t) (W m-2 °C-1) (equivalent to a conductance in the electrical analogy) can be introduced as the ratio of I1(t) (W m-2) to temperature at the 1-cm depth, T1 (t) (°C) (t being the time),
 | (2) |
where the subscript T in GT(t) stands for temperature (voltage). The transfer function for the top 1 cm of soil is now defined as RT(t) = 1/GT(t) (W-1 m2 °C). The complete model is thus made up of an ideal current source Sr(t) in parallel with RT(t) (representing the top centimeter of soil and all surface phenomena), and an R-C ladder representing the soil below the 1 cm-depth (Fig. 1).
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MATERIALS AND METHODS
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Field Experiment
Soil water content (m3 m-3), soil temperature T (°C), solar radiation Sr (W m-2) at 1.5 m above the ground, air temperature Ta (°C) at 2.2 m above the ground, and rainfall (mm) were recorded as functions of time in a natural environment at Lido Beach, Venice, Italy, from 24 June to 28 July 1996. The experiment was carried out within a plastic circular container 1 m in radius and 0.9 m deep, full of initially saturated sand (porosity 0.4) and almost totally buried in the beach. The average mineralogical composition of the sand is 70% carbonates (calcite and dolomite) and 30% silicates (quartz and feldspar) (Bonardi and Tosi, 1995, 1997). The analysis of the grain size showed that 98.6% have a diameter between 0.2 and 1 mm (Soika, 1961). Soil water content and temperature were measured at depths of 5, 20, 35, and 50 cm; temperature was also measured at the 1-cm depth. Water content was determined with Time Domain Reflectometry (TDR) (Noborio et al., 1996), and temperature measurements with PT100 platinum resistances. Temperature and Sr(t) data were collected every 20 minutes, and water content every 2 hours. Although during the period of field experiment several mild days with remarkable Sr(t) favored evaporation, there were also 6 rainfall events totaling 40 mm which stopped the drying process.
Electrical Model
Linear interpolation between measured values was used for calculating the temperature and water content profiles to estimate every R-C cell of the model. The c and of each layer were calculated from through the following relations (Campbell, 1985, p. 31–32):
 | (3) |
The initial temperature profile was introduced into the model as an electric charge stored in the capacitor of each R-C cell. To test the influence of the soil layer thickness on the results, the sandy soil was modeled with layers 0.5, 1, 2.5, 3, 5, 10, 20, and 40 mm thick. In every case, the model was excited with the temperature measured at the 1-cm depth, and the temperature at the 5-cm depth was calculated for the first 10 d, keeping a constant water content. It was found that 3-mm layers allow a good compromise between acceptable RMSE and reasonable computing time. Root mean square error of 0.74°C was determined using
 | (4) |
where the subscripts c and m refer to calculated and measured temperatures, respectively, and N is the number of observations.
Models that solve the equation of heat conduction in soil require knowledge of some boundary conditions that many times have to be estimated (Persaud and Chang, 1984; Campbell, 1985). Although the analog electrical model of this work does not need specification of any boundary condition at the end of the R-C line, it is necessary to know its minimum length if finite-length effects are to be avoided (see Appendix). The model allows using a depth and/or time dependent thermal diffusivity. Software for electric circuit analysis was used to calculate soil temperature and heat flow as functions of depth and time (Guaraglia and Pousa, 1999).
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RESULTS
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Temperature Input
The sandy soil of Lido beach was modeled from 1 to 103 cm with 342 R-C cells, each representing a 3-mm layer. In the first test, the model consisted only of the R-C ladder (Fig. 1), and was excited with an ideal voltage source proportional to the measured temperature at the 1-cm depth. Soil water content was corrected every 5 d. Figure 2 illustrates model results and measured values at depths of 5 cm (Fig. 2a) and 50 cm (Fig. 2b) for 15 d. Small jumps in calculated temperature are apparent (e.g., from the fifth to the sixth day) (Fig. 2b). They occur when is changed in steps, while the upper layers, initially saturated, are undergoing a rapid and continuous drying process. If is kept constant, model results begin to deviate from field data from the fifth day on, approximately. Root mean square errors for calculated temperatures at depths of 5, 20, 35, and 50 cm are 0.90, 0.25, 0.28, and 0.18°C, respectively. This suggests that the R-C ladder models heat diffusion adequately. Although heat flux was also calculated by the model, the RMSE could not be estimated because field data was not available.
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Fig. 2. Measured (light line) and calculated (dark line) temperatures at depths of 5 cm (a) and 50 cm (b) with the model excited by measured temperature at the 1-cm depth. Soil water content variable with depth and corrected every 5 d.
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Solar Radiation Input
Heat flow transmitted below the 1-cm depth was calculated with the model excited by temperature at the 1-cm depth as described above. Figure 3 shows that the resulting GT(t) of Eq. [2] has a shape similar to that of measured Sr(t), though it never becomes zero. It was found that GT(t) could be fitted by a linear function of Sr(t),
 | (5) |
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Fig. 3. Solar radiation, Sr(t), and the function equivalent to an electrical conductance, GT(t), calculated with field data from 25 June to 2 July.
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During daytime, the quantity a accounts for the transformation of part of the incoming Sr(t) into heat in the top 1 cm of soil. At night, it accounts for heat transfer from the soil to the environment. The quantity b represents the rate of change of GT(t) with Sr(t). For the period 25 June to 2 July, a = 2.64 (W m-2 °C-1) and b = 0.0298 (°C-1), with a correlation coefficient r = 0.982 and a RMSE of 1.657 W m-2 °C-1.
With Sr(t) as the input, the complete model of Fig. 1 was run for the same 8 d period used to determine a and b. Root mean square errors between measured and calculated temperatures at depths of 1, 5, 20, 35, and 50 cm were 1.06, 1.01, 0.68, 0.17, and 0.14°C, respectively. Figure 4 shows model results and field measurements at the 1-cm depth for the first 5 d. This very simple transfer function attached to the R-C ladder allows calculation of soil temperature and heat flow as functions of depth and time using measured Sr(t) as the input signal.
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Fig. 4. Measured (light line) and calculated (dark line) temperatures at the 1-cm depth for the first 5 d. Model excited by measured solar radiation Sr(t). The R-C ladder of Fig. 1 remained constant with time.
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In a second stage, an attempt was made to reproduce temperature measurements for the whole 35-d experiment, which involves different soil water content conditions. This time GT(t) was fitted by a polynomial of the third degree whose coefficients, in increasing powers of Sr(t), were 2.1659, 4.1939 x10-2, -2.8786 x 10-5 and 1.2833 x10-8. The correlation coefficient and RMSE associated with this polynomial fit were 0.978 and 1.781 W m-2 °C-1, respectively. Soil thermal diffusivity for this test was allowed to vary with depth but not with time, and was assumed to have the profile corresponding to that of the second day of the experiment. The R-C ladder of Fig. 1 remained thus constant, RT(t) being the only variable of the model. Root mean square errors for the whole experiment were 1.92, 2.00, 1.02, 0.69, and 0.52°C for the depths of 1, 5, 20, 35, and 50 cm, respectively.
The above results can be compared with those obtained with other models. Parton (1984) shows that for a shortgrass prairie, during the growing season, the RMSEs of his model for the hourly temperature were 4.42, 3.14, 2.87, and 1.64°C, respectively, for the soil surface, 4, 10, and 20 cm depths. Kemp et al. (1992) calculate hourly temperatures for five summer days with RMSEs of 3.99, 2.39, 1.59, and 0.81°C for 1, 10, 20, and 50 cm soil depths, respectively. Katul and Parlange (1993) determine soil surface temperature with an RMSE of 2.94°C. In the model of Pikul (1991), the standard deviation of the differences between predicted surface temperature and 2.5-mm soil temperature was 1.98°C for a period of soil drying. The electrical model shows similar results to those obtained with the last two models, but requires less input data.
To study the influence of soil water content below the 1-cm depth the model was run several times, changing Re and Ce according to measured values of , but keeping them constant for the whole experiment. The polynomial coefficients of GT(t) were the same as above. All of the runs gave similar results. In particular, for the R-C ladder corresponding to 23 July—a day of relatively low (0.09 at depth of 5 cm) when compared with the high value of the first day (0.38 at depth of 5 cm)—the RMSEs were 1.86, 1.90, 1.05, 0.80, and 0.60°C for depths of 1, 5, 20, 35, and 50 cm, respectively. This suggests that for a sandy soil such as that of the Lido beach, the hourly soil temperature does not depend on the soil water content below the first centimeters so much as it does on Sr(t).
The model in Fig. 1 reproduces fairly well the high and low frequency soil temperature variations, but only during the first 13 d (Fig. 5). Between Days 13 and 14, a 2-mm rainfall occurred and a drop in minimum daily temperature of about 4°C was verified at the 1-cm depth. In fact, the mean soil temperature decreases for some days after a rainfall or a persistent high speed wind. When the top layers begin to dry after a rainfall, the maximum soil temperature at depth of 1 cm increases with a slope of about 1.25°C/d during sunny days. This increment was modeled with a capacitor of 22 J m-2 °C-1 in series with RT(t) (not shown in Fig. 1). This would indicate that the slow increase in soil temperature depends on the accumulated Sr(t). If this capacitor were discharged with rainfall or high speed winds, the aforementioned temperature drops might be reproduced. However, with the available data it was not possible to find a function that linked the capacitor discharge together with rainfall and wind for the whole 35-d period.
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Fig. 5. Measured (light line) and calculated (dark line) temperatures at the 1-cm depth for the first 16 d with the model excited by measured solar radiation Sr(t). The model failed to reproduce a sudden drop in measured temperature on Day 13.
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CONCLUSIONS
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The model presented in this work proved to be a tool to simulate heat flow in a bare sandy soil as a purely diffusive phenomenon, provided that soil thermal characteristics and water content are known. The model was excited by temperature or Sr(t), alternatively. When measured Sr(t) was used as the input signal, a transfer function, RT(t), for the surface and the top 1 cm layer, was introduced. It linked Sr(t) with the soil column below the depth of 1 cm. Results from the model reproduced fairly well the soil temperature recorded during a 35-d field experiment, even though different water contents were used below the 1-cm depth. This would indicate that in this case, the only variable of the circuit, RT(t), played a more important role in determining soil temperature than the rest of the soil column did. However, the model did not simulate sudden temperature drops due to rain and wind. Although raining days during the experiment slightly increased soil water content at the 5-cm depth, the influence of rain on soil temperature was very important, suggesting that the water content of the top layer strongly influenced heat transfer from this layer to the environment.
The concept of electrical impedance is used to avoid specifying boundary conditions at the bottom of the soil column represented by the electrical analogy. The electrical impedance Z () looking into two nodes of a circuit is defined as the complex ratio between electromotive force (V) and current (A) (von Kármán and Biot, 1940; International Electrotechnical Comission, 1985). This definition is valid for circuits in sinusoidal steady-state condition for which the ratio of voltage to current phasors is constant for a given frequency. If R() is the resistance per unit length of the wires in an electric line, G(-1) the conductance per unit length between the two wires of the line, C(F) the capacitance per unit length of the two wires, and L (H) the inductance per unit length of the line, the characteristic impedance Zk () of the line is given by (Pipes, 1958)
 | (A1) |
where 
= 2
f, f (Hz) being the frequency, and j is the imaginary unit. For a perfectly insulated line (G = 0) with no distributed inductance (L = 0), Zk becomes
 | (A2) |
If the load impedance (Zo) at the output of a real, length-limited line is equal to Zk, all the energy delivered to the line at the input is transmitted to Zo and no energy is reflected. If Zo is not equal to Zk, a fraction of the energy arriving at the output will be reflected, and the current at any point in the line will depend on the boundary condition specified by Zo. Another point to take into account is the relationship between Zk and the impedance looking into the input of the line (Zi) called the input impedance. For an ideal infinite line, it holds that Zi = Zk, no matter where the line input is considered to be (Johnson, 1950).
Application to the Thermal Case
The simplified R-C line of Eq. [A2] is adequate to model heat diffusion phenomena in soil. The product RC is the time constant of the line, and in the electrical analogy of heat flow 1/RC is the soil thermal diffusivity (Pipes, 1958). The length of an R-C line intended for modeling a soil column of given depth within an unlimited soil should be carefully chosen, as otherwise undesirable finite-length effects will arise. It is found that the R-C line should be extended well beyond the deepest point of the soil column to be modeled, for truncation of the line at that point is tantamount to having a situation in which Zo 
Zk. If the thermal diffusivity of an unlimited soil is constant with depth (1/RC = constant) the input impedance Zi anywhere within the modeling line should also remain constant and equal to Zk. Therefore, when using a finite R-C line for modeling heat flow within an infinite soil of constant , any variation of Zi at any section of the line would indicate that it is not long enough to avoid finite-length effects. If the line is shortened, but Zi 
Zk, its length would be still good enough. This condition allows obtaining a compromise between minimum modeling errors and a manageable number of R-C cells.
Numerical Tests for the Line Length
Because data were recorded up to a depth of 50 cm, the model was first run with a test soil to find the minimum length to prevent finite-length effects for this depth. Thus a series of and values were taken from Jury et al. (1991) for a sand with a porosity of 0.60 and different values of . The electrical model was excited with a sinusoidal temperature of amplitude 1°C and a period of 1 d applied at a depth of 1 cm. It was found that the length of the R-C line had to be extended up to 96 cm to prevent the above mentioned finite-length effects. The same analysis was then performed with the sandy soil of Lido beach for 0 
0.38. As in the case of the test soil, the R-C line had to be approximately twice as long as the depth of interest.
Received for publication July 31, 2000.
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TOP
ABSTRACT
INTRODUCTION
