Haar wavelet in estimating depth profile of soil temperature
Introduction
The application of reaction–diffusion equations is day by day increasing. Its application in determining the depth profile of soil temperature was first discussed by Carslaw and Jaeger [1] and the analytical technique for obtaining the diurnal variation of soil temperature with depth by solving one-dimensional Fickian diffusion equation is well documented in literature [13], [14], [16], [17], [18], [19]. Since this method involved so many assumptions, which made the applications so restrictive, Dutta and Daoo [2] used an upwind finite difference technique, solved the same problem and some improvement in the results were reported to literature. Lepik [7], [20] used the Haar wavelet method for solving nonlinear integral and differential equations. The authors used the same boundary conditions, which were used in the analytical method, and good agreements in the results with the observed data were recorded. However the mean percentage of errors were observed to be 25.1 at 5 cm depth, 15.7 at 10 cm and 9.4 at 20 cm. Dutta in his paper used 5 cm spacing for z and 1 h for Δt. In the sensitivity analysis, it has been shown a small change in soil depth z < 5 cm (similarly a small change in Δt < 15 min) affect the soil temperature. In this paper we applied Haar wavelet technique to solve this problem in order to (i) minimize the error propagation as soil depth increases and (ii) have computational complexity under control. We were able to arrive better results mainly because, by nature, the surface temperature and the soil temperature at any depth follow the diurnal pattern of the ground heat flux, which is of sinusoidal waveform. This paper is organized as follows. In Section 2, we present the derivation of one-dimensional unsteady state partial differential equation for heat diffusion in soil related boundary conditions and its analytic solution. In Section 3, we explain the fundamentals of Haar wavelet and the construction of operational matrices. In Section 4, we explain the solution by wavelet transform.We discuss the results by comparing with observed data in Section 5. Sensitivity analysis is carried out in Section 6. Concluding remarks are given in Section 7.
Section snippets
Heat diffusion equation in soil
Hydraulic properties of unsaturated soils are adequately described by Van Geunuchten and Nielsen [3]. Kimball et al. [4] describes soil heat flux determination through temperature gradient method with computed thermal conductivities Messman [5] corrected errors associated with soil heat flux measurements. Dutta et al. [2] constructed a numerical model for determining depth profile of soil temperature. A brief description about the soil characteristics and methods determining soil heat flux are
Haar wavelet preliminaries
The orthogonal set of Haar wavelets hi(t) is a group of square waves [6] with magnitude +1 or −1 in some intervals and zeros elsewhere
Just these zeros make Haar wavelets be local, faster than other square functions such as Walsh and very useful in solving stiff systems.
Any function y(t) which is square integrable in the interval [0, 1) can be expanded in a Haar series with an infinite number of terms
Solution of diffusion equation by Haar wavelet
In the Haar domain, we assume that can be expanded as a Haar series asIntegrating (26) and applying the integration matrix P of Section 3, we haveUsing (26), (27) in (7), we haveThe differential equation (29) is solved using the conventional method. After dropping all the eigen values of (29) with positive real parts we haveLet us set
Results and discussion
Soil temperature measurements at five specific depths namely 0, 5, 10, 20 and 40 cm at Trombay were made during January 2001 by Daoo and subsequently used by Dutta [2]. Here the same set of data is taken into account for comparison with the Haar wavelet solution for time ranging from 1 to 24 h and z values ranging from 0 to 45 cm. The values of k, u0 and Δu0 calculated from the data set for one day are used for comparison of the predicted soil temperature using Haar wavelet procedure at the
Sensitivity analyses
Sensitivity analyses were used to determine the significance of: (1) the use of approximate thermal properties and (2) the use of fixed k values with respect to temperature. A sensitivity analysis of the Haar scheme was also performed with respect to the parameters k, u0 and Δu0 in order to substantiate their relative importance. We found that changes in thermal conductivity had little effect on u0 but caused some differences in u(z, t) in deep layers whereas changes in Δu0 (amplitude of soil
Conclusion
The use of Haar wavelets in the engineering produces exciting results. The characteristics of wavelet transforms make them particularly useful for the approximation of functions with steep gradients or sharp spikes. Certainly the orthogonal and orthonormal properties of wavelet basis allow us to simplify the calculation of integrals. The Haar wavelet model described in this study provides reasonable estimates of soil temperature at any depth and time. If few more parameters like specific heat
Acknowledgement
Our Thanks are due to Dr. T.R. Neelakandan, Professor of Civil Engineering, SASTRA University, Thanjavur 613 402, India for the useful discussions in the subject.
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