Published in Soil Sci. Soc. Am. J. 68:784-788 (2004).
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
DIVISION S-1—NOTES
ANALYTICAL SOIL–TEMPERATURE MODEL
CORRECTION FOR TEMPORAL VARIATION OF DAILY AMPLITUDE
Elimoel A. Elias*,a,
Rogerio Cichotac,
Hugo H. Torrianid and
Quirijn de Jong van Lierb
a Programa de Pós-graduação em Fisica do Ambiente Agricola, Departamento de Ciencias Exatas, Escola Superior de Agricultura "Luiz de Queiroz" (ESALQ), Universidade de São Paulo (USP), Caixa Postal 9, 13418-900, Piracicaba, SP, Brazil
b Departamento de Ciencias Exatas, Escola Superior de Agricultura "Luiz de Queiroz" (ESALQ), Universidade de São Paulo (USP), Caixa Postal 9, 13418-900, Piracicaba, SP, Brazil
c 394, Botanical Rd, Palmerston North-5301, New Zealand
d Departmento de Matématic, IMECC, UNICAMP, Caixa Postal 6065, Campinas, SP 13081-970, Brazil
* Corresponding author (eaelias{at}terra.com.br).
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ABSTRACT
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The exponential-sinusoidal one-dimensional analytical model reasonably describes soil temperature profile. Surface temperature may be represented by the sum of two sinusoids, one related to annual and the other to daily temperature variations, each one having constant amplitude. We introduce here a correction for the temporal variation of daily amplitude, and demonstrate that the heat equation can still be solved analytically. We compared temperature predictions obtained from the novel and from the usual analytical solution, using the root mean squared error (RMSE). For individual days, at a depth z = 0.10 m, the maximum value was RMSE = 0.30°C; for whole months, the maximum value was RMSE = 0.29°C, and the minimum was RMSE = 0.06°C. The only additional information required to apply our novel equation is information on temporal variation of daily amplitude, so we suggest our equation should be preferred when such data are readily available.
Abbreviations: DOY, day of year • RMSE, root mean squared error
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INTRODUCTION
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MANY PHYSICAL, CHEMICAL, AND BIOLOGICAL processes that occur in soil are influenced by temperature. Thus, the objective of several studies has been to find a function T = T(z,t) describing the soil temperature profile, where T (°C) is soil temperature at time t (s) and depth z (m). This function can be found by modeling soil energy movement, basing on physical laws and on some empirical parameters. Van Wijk and de Vries (1963) proposed a one-dimensional vertical analytical model. In this model, soil temperature is assumed to satisfy the heat equation:
 | [1] |
where K (m2 s–1) is the soil thermal diffusivity, assumed to be constant with depth and time. In addition, two boundary conditions must be satisfied. The first is:
 | [2] |
which indicates that the surface temperature varies sinusoidally with time, having a time-average value Ta (°C), amplitude A (°C), radial frequency 
(rad s–1), and phase constant 
(rad). The second boundary condition is:
 | [3] |
The solution satisfying Eq. [1], [2], and [3] is (van Wijk and de Vries, 1963; Kirkham and Powers, 1972, p. 468–473):
 | [4] |
where D (m) is the damping depth, defined as:
 | [5] |
Equation [4] has been used to describe either daily or annual temperature variations. For such purposes, the variables Ta, A, D, , and are replaced by Tad, Ad, Dd, d, and d for daily variations, or by Tay, Ay , Dy, y, and y for annual variations. However, to describe both variations, another equation, Eq. [10], has been used (Van Wijk and de Vries, 1963; Steenhuis and Walter, 1986), which will now be derived. Modification of Eq. [2] yields the corresponding boundary conditions:
 | [6] |
 | [7] |
In Eq. [6] the daily average surface temperature, Tad, is assumed to be constant throughout the period of interest, for example, days or weeks. In reality, Tad is time dependent. An expression for time dependence of Tad can be developed using Eq. [7]. In Eq. [7], T(0,t) is (approximately) the average surface temperature for the day corresponding to time t, so it can be written as:
 | [8] |
Upon substituting Eq. [8] into Eq. [6], we obtain:
 | [9] |
In Eq. [9], the term [Tay + Ay sin(yt + y)] is (approximately) the average temperature for the day corresponding to time t, and the daily temperature oscillates around it according to the term [Ad sin(dt + d)]. Thus, Eq. [9] can be regarded as a correction for Eq. [6], by taking into account the seasonal variation of Tad. A linear combination of independent solutions of Eq. [1] is also a solution of Eq. [1]. Thus the solution of the problem given by Eq. [1], [3], and [9] is:
 | [10] |
which was presented by Van Wijk and de Vries (1963).
The model above described is a simplification of real field conditions, as K is generally not constant, and the boundary conditions can only be satisfied as a first approximation. In fact, the boundary condition given by Eq. [9] does not account for changes resulting for variable weather, although this approximation may be reasonable for averages calculated from long series of observations (Van Wijk and de Vries, 1963, p. 102, 113). Despite such simplifications, the model has been very useful because it "...has provided physical insight to soil heat transfer, a means for verifying numerical models and a theory allowing for the heat estimation of soil thermal properties ... and soil heat flux ..." (Kluitenberg and Horton, 1990, p. 1197). Thus, simplified models, however inexact, are very important. Also, they can be gradually corrected, at increasing levels of complexity.
Although the model of Van Wijk and de Vries (1963) accounts for time variation of the daily average temperature, it assumes that the daily amplitude is constant in time. In reality, the amplitude is not generally constant. So, the constant Ad in Eq. [9] and [10] is actually the annual average of the daily amplitude. The general objective of this study was to improve the analytical model described above, correcting still further the surface boundary condition, so that the model accounts for temporal variation of daily amplitude of soil surface temperature. This general objective has been divided in three specific objectives: (i) to demonstrate that the daily amplitude varies sinusoidally over the course of a year; (ii) to derive a new analytical solution for Eq. [1], when the boundary condition given by Eq. [9] has the constant value Ad replaced by the time-dependent daily amplitude; and (iii) to compare predictions from the novel solution with prediction from Eq. [10].
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Improved Sinusoidal Model for Soil Surface Temperature
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Temperature data were averaged over a series of 10 yr, and were obtained from the following sources:- IAG—Estação Meteorológica, Instituto de Astronomia, Geofísica e Ciências Atmosféricas, USP (Universidade de São Paulo), São Paulo, SP, Brazil. Latitude: 23°39'S, longitude 46°37'W, elevation: 800 m above mean sea level. Soil surface temperature was measured by mercury thermometers.
- AWS—Automatic Weather Station, Atmospheric Science, Division of Environmental and Life Sciences, Macquarie University, Sydney, Australia. Latitude: 33°46'S, longitude: 151°7'E, elevation: 55.0 m above mean sea level. Soil surface temperature was measured by a Campbell Scientific, Inc. (Logan, UT) TCAV Averaging Soil Thermocouple Probe. Data directly available from the internet home page: http://atmos.es.mq.edu.au/~aws2/variables.html (verified 20 Jan. 2004).
- IAC—Estação Meteorológica, Instituto Agronômico, Ribeirão Preto, SP, Brazil. Latitude: 21°11' S, longitude: 47°48' W, elevation: 621 m above mean sea level. Soil near-surface temperature was measured at a 2-cm depth by mercury thermometers.
From observing soil surface and near-surface temperature data, we have realized that the daily amplitude, C(t) (°C), can be described reasonably well by a sinusoidal function of time:
 | [11] |
where B (°C) is a constant, related to the temporal variation of daily amplitude around its averaged value Ad, b (rad s–1) is the radial frequency, and ß (rad) is a phase constant. Eq. [11] was adjusted using the least-square method (Sen and Srivastava, 1990). For this adjustment, monthly arithmetic means of daily amplitudes were used, instead of daily values. Resulting parameters are shown in Table 1. A reasonable sinusoidal fit is observable in Fig. 1
; the straight line, representing constant daily amplitude, does not appear to be such a good fit. To quantify these graphic observations, predicted results, Pi, were compared with observed data, Oi, by the root mean squared error, or RMSE (Willmott, 1981):
 | [12] |
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Fig. 1. Temporal variation of daily amplitude of soil surface (a, b) and near-surface (c) temperature in three locations: (a) São Paulo, Brazil; (b) Sydney, Australia; and (c) Ribeirão Preto, Brazil. Dots refer to observed values; solid lines represent the adjusted sinusoidal equation, Eq. [11], fitted to observations, and dashed lines represent the mean value (Ad).
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In this case, Oi are the monthly means, so that n = 12. We made Pi = C(t), where C(t) is given by the adjusted Eq. [11]; and we also made Pi = Ad = constant. Numerical values are shown in Table 2; they are consistent with our observations about Fig. 1. We thus conclude that Eq. [11] is a better description of daily amplitude of soil surface temperature than a constant value, Ad.
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Table 2. Root mean squared error (RMSE) values from comparing daily amplitude of soil temperature data with the sinusoidal model, C(t) from Eq. [11], and with the constant, average value, Ad, in three locations.
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Improved Analytical Model for Soil Profile Temperature
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From results of the previous section, we inferred that the analytical model for soil profile temperature would become more realistic if another correction were made to the surface boundary condition, replacing Ad in Eq. [9] by C(t), so that Eq. [9] becomes:
 | [13] |
or
 | [14] |
To obtain the solution corresponding to Eq. [14], we note, once more, that a linear combination of independent solutions to Eq. [1] is also a solution to Eq. [1]. Unfortunately, the additional term in Eq. [14] involves a product, not a sum, of sinusoids. But it can be readily changed into a sum by using the trigonometric identities:
 | [15] |
 | [16] |
Combining Eq. [15] and [16] gives:
 | [17] |
From Eq. [17], it follows that Eq. [14] can be written as:
 | [18] |
The solution of Eq. [1], [3], and [18] now becomes:
 | [19] |
where D' and D'' are the damping depths, given by Eq. [5], for the two additional temperature waves, with ' = (d – b) and '' = (d + b).
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Comparison of the Analytical Models for Soil Profile Temperature
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A preliminary comparison of predictions from Eq. [10] and [19] can be readily made considering the damping depths involved. As d 
10–4 rad s–1 and b 10–7 rad s–1, we recognize that d >> b. Thus, the variables d, ', and '' have approximately the same numerical values. Due to Eq. [5], numerical values of Dd, D', and D'' are approximately the same, too. According to van Wijk and de Vries (1963)(p. 111): "In the common types of soil the diurnal variation does not penetrate below 50 cm and the annual variation not below 10 m." This is because Dd << Dy. As D' D'' Dd << Dy, only the first two terms on the right-hand side of Eq. [10] and [19] are important for z > 0.50 m, so that these equations become practically the same for such depths.
A comprehensive comparison of predictions from Eq. [10] and [19] was performed using Eq. [12]. In this case, Pi were temperature values, T(z,t), predicted from Eq. [10], and Oi were values from Eq. [19]. Hourly predicted temperature values were used to feed Eq. [12]. Thus, to calculate RMSE over a whole year, n = 8760; over whole months, n = 720; and over individual days, n = 24. Values of T(z,t) were calculated at z = 0.00, 0.10, 0.20, 0.40, and 1.00 m.
For the sake of comparing Eq. [10] and [19], it was sufficient to use a typical value of soil thermal diffusivity, K = 5.56 x 10–7 m2 s–1. Other parameter values used in the comparison were obtained from IAG data: Tay = 21.18°C, Ay = 3.51°C, Ad = 7.49°C, B = 0.95°C, Dy = 2.36 m, Dd = 0.12 m, D' = 0.12 m, D'' = 0.12 m, y = 1.99 x 10–7 rad s–1, d = 7.27 x 10–5 rad s–1, ' = 7.25 x 10–5 rad s–1, '' = 7.29 x 10–5 rad s–1, y = 5.02 rad, d = 1.85 rad, ß = 1.73 rad.
We found RMSE = 0.21°C for the whole year, at z = 0.10, so the overall correction introduced by our novel equation is small. However, RMSE values for individual days may be higher. For z = 0.10 m, the maximum value is RMSE = 0.30°C, and is achieved twice a year, at day of year (DOY) 145 and DOY 327. For both days, we found a maximum temperature prediction disagreement of 0.42°C, at Hours 2 and 14. Figure 2
shows hourly temperature for Day 145; disagreement between the two models is clearly observable. Root mean squared error values for whole months are shown in Table 3. At z = 0.10 m, values are considerable for May and November, although very small for February and August. We also observed that RMSE values are very small for z = 0.40 m and negligible for z = 1.00 m, for all months, as expected from our considerations above on damping depths.
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Fig. 2. Soil temperature at day of year (DOY) 145 at four selected depths, for the usual model, Eq. [10], and for the model corrected for temporal variation of daily amplitude, Eq. [19], in São Paulo, Brazil. The daily average temperature, calculated with Eq. [8], is also shown.
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Table 3. Root mean squared error (RMSE) values averaged over months, at four soil depths, in São Paulo, Brazil. RMSE values were obtained from predicted results from Eq. [10] and [19]. Greatest disagreements are indicated by underlining, and least disagreements in italic. Root mean squared error values averaged over the whole year are also shown.
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Conclusions
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For the studied data, daily amplitude can be better described by a sinusoidal function of time presented here, Eq. [11], rather than by a constant value.
From Eq. [11], a corrected equation, Eq. [13], was presented for describing the boundary condition for soil surface temperature, which is a more realistic option than Eq. [9], assumed in the cited literature. The heat equation, Eq. [1], can be easily solved for the new surface boundary condition. The novel equation obtained for soil profile temperature, Eq. [19], does not increase the complexity level of the problem studied, when compared with the old equation, Eq. [10]. From an educational point of view, anyone who can understand and apply Eq. [10] could also understand and apply Eq. [19]. The number of exponential-sinusoidal terms doubles, but this should not present any considerable difficulty in calculations.
According to the data plots (Fig. 2) and RMSE values (Table 3), Eq. [19] yields results different from Eq. [10]. The only additional information required to apply Eq. [19] is information on temporal variation of daily amplitude; so we suggest that Eq. [19] should be preferred over Eq. [10] when such data are readily available.
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ACKNOWLEDGMENTS
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Authors are grateful to Dr. Sergio Oliveira Moraes for comments related to this study, and to CAPES for financial support.
Received for publication May 7, 2003.
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REFERENCES
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- Kirkham, D., and W.L. Powers. 1972. Advanced soil physics. Wiley Interscience, Inc., New York, N.Y.
- Kluitenberg, G.J., and R. Horton. 1990. Analytical solution for two-dimensional heat conduction beneath a partial surface mulch. Soil Sci. Soc. Am. J. 54:1197–1206.[Abstract/Free Full Text]
- Sen, A., and M. Srivastava. 1990. Regression analysis. Spring-Verlag, New York.
- Steenhuis, T.S., and M.F. Walter. 1986. Will drainage increase spring soil temperatures in cool and humid climates? Trans. ASAE 29:1641–1645.
- Van Wijk, W.R., and D.A. De Vries. 1963. Periodic temperature variations in a homogenous soil. In Van Wijk, W.R. (ed.) Physics of plant environment. North Holland Publ. Co., Amsterdam.
- Willmott, C.J. 1981. On the validations of models. Physical Geography 2:184–194.
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ABSTRACT
INTRODUCTION
