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DIVISION S-1—SOIL PHYSICS

Predicting the Dielectric Constant–Water Content Relationship Using Artificial Neural Networks

^a Department of Water Resources Engineering, Lund University, Box 118, SE-221 00 Lund, Sweden
^b Department of Land, Air and Water Resources, University of California, Davis, CA 95616, USA
^c Department of Crop Physiology and Soil Science P.O. Box 50 DK-8830 Tjele, Denmark

* Corresponding author (magnus.persson{at}tvrl.lth.se)

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Accurate measurements of soil water content () are important in various applications in hydrology and soil science. The time domain reflectometry (TDR) technique has been widely used for measurements during the last two decades. The TDR utilizes the apparent dielectric constant (K_a) for estimations of . The K_a– relationship has been described using both empirical and physical models. Universal calibration equations that fit a wide range of different soil types have been developed. However, to achieve high accuracy, a soil-specific calibration needs to be conducted. In the present study, we use an artificial neural network (ANN) to predict the K_a– relationship using soil physical parameters for ten different soil types. The parameters that give the most significant reduction in the root mean square error (RMSE) are bulk density, clay content, and organic matter content. The K_a– relationship for each soil type is predicted using the other nine for calibration. It is shown that ANN predictions are as good as a soil specific calibration with comparable coefficient of determination and RMSE. Thus, by using ANN, highly accurate data can be obtained without need for elaborate soil specific calibration experiments.

Abbreviations: ANN, artificial neural network • K_a, dielectric constant • RMSE, root mean square error • TDR, time domain reflectometry • , water content

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
TIME DOMAIN REFLECTOMETRY has become an important and often used measuring technique for observations in soils. Topp et al. (1980) used a third-order polynomial equation to relate the TDR measured apparent K_a to . This equation, often called the Topp equation, gave good predictions of for several mineral soils and was not greatly affected by temperature, soil salinity, or bulk density. Subsequent studies, however, have found small but significant effects of soil type (e.g., Roth et al., 1992), percentage of organic matter and bulk density (e.g., Jacobsen and Schjønning, 1993; Malicki et al., 1994; Hook and Livingston, 1996), electrical conductivity (e.g., Dalton, 1992; Sun et al., 2000; Persson et al., 2000), and temperature (e.g., Pepin et al., 1995; Persson and Berndtsson, 1998; Or and Wraith, 1999; Wraith and Or, 1999). Other calibration models have also been developed, for example, empirical models based on an almost linear relationship between K^0.5_a and (Ledieu et al., 1986) and physically based dielectric mixing models (Birchak et al., 1974; de Loor, 1964).

The Topp equation, particularly in situations when no soil-specific calibration is available, is still widely used. It is important to note, however, that using the Topp equation in a mineral soil with moderate clay content may lead to errors of 0.02 m³ m^-3 or more. Thus, in studies where high accuracy is needed, a soil-specific calibration is normally required. This is, however, often an elaborate procedure, even if some recent studies have provided more efficient calibration methods (Young et al., 1997; Nissen et al., 2000). The data obtained from soil specific calibration experiments can be used to obtain best fit parameters of a third-order polynomial equation or a linear relationship between K^0.5_a and . To avoid such an elaborate, time-consuming procedure, the soil physical parameters can be used to obtain the K_a– relationship, without need for soil-specific calibration. However, no such study has been presented so far and, therefore, forms the basis for the present investigation, where an ANN is used.

The development of ANN began several decades ago. The ANN is conceived to mimic the functioning of the human brain by acquiring knowledge through a learning process and finding optimum weights for the different connections between the individual nerve cells. Mathematically, an ANN can be treated as a universal approximator. The ability to ‘train’ and ‘learn’ the output from a given input makes ANN capable of describing large scale complex problems such as forecasting stock markets, managing water resources, etc. During the last decade, ANN has been applied, with success, to various hydrological processes, such as rainfall-runoff modeling (e.g., Hsu et al., 1995), rainfall forecasting (e.g., French et al., 1992), water quality modeling (e.g., Maier and Dandy, 1996), reservoir operation, streamflow modeling, river basin classification, etc. (Govindaraju, 2000). Artificial neural networks have also been used to predict water retention characteristics from other, more easily measured soil variables like particle-size distribution and bulk density (Pachepsky et al., 1996; Schaap and Bouten, 1996; Koekkoek and Booltink, 1999). The ANN is, thus, a useful tool for achieving accurate data without cumbersome calibration. Recently, Persson et al. (2001) used ANN to calibrate TDR measurements. They showed that an ANN gave better prediction of the K_a– relationship than other commonly used models. However, they used only a single soil type (sand) and, thus, effects of different soil textures on the K_a– relationship were not investigated. With the encouraging results obtained using ANNs for water retention characteristics and estimations using soil physical parameters in mind, it seems possible that the same procedure could be used to predict the K_a– relationship.

The purpose of the present study is to investigate the use of ANN to understand (or derive) the K_a– relationship from physical parameters of the soil. The performance of ANNs is compared with the ones achieved using common calibration techniques. An attempt is also made to study the influence of different soil physical parameters on the behavior of ANN.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Water Content Measurements Using Time Domain Reflectometry
The TDR is an electromagnetic technique in which the TDR instrument sends a signal through a cable to a probe buried in the soil. From the reflected signal, the propagation velocity can be calculated. Topp et al. (1980) were among the first to use TDR for moisture content measurements in soil. They introduced the apparent dielectric constant K_a which is related to the propagation velocity. The K_a of air, soil particles, and water (at 20°C) is 1, 2 to 5, and 80 respectively, which makes the measured bulk K_a highly dependent on . Topp et al. (1980) used a third-order polynomial equation to describe the K_a– relationship:

[1]

Ledieu et al. (1986) used a linear relationship between K^0.5_a and . Some dielectric mixing models have also been used for determination of from K_a. In these models, the bulk soil K_a is calculated from K_a and volumetric content of each phase in the soil, normally soil particles, water, and air. Birchak et al. (1974) presented a semi-empirical dielectric mixing model.

[2]

where _air and _s are the volumetric content of air and solid particles, and K_air, K_s, and K_w are the dielectric constant of air, solids, and water, respectively. The -value can be optimized for the actual data set and is normally found to be in the range of 0.4 to 0.8 (see e.g., Jacobsen and Schjønning, 1995). An -value of 0.5 corresponds to refractive mixing and it also yields a linear relationship between K^0.5_a and . The validity of this has been proven by several researchers (e.g., Ledieu et al., 1986; Hook and Livingston, 1996).

The de Loor model (de Loor, 1964), is a physical model based on the concept that water and air represent disc shaped inclusion in a host medium (solid soil). The de Loor model can be written as

[3]

The de Loor model has been proven to make reliable predictions of . Even though the model does not contain any fitting parameters, several studies have shown that it gives more accurate predictions than the Birchack model (Dirksen and Dasberg, 1993; Bohl and Roth, 1994; Jacobsen and Schjønning, 1995; Persson et al., 2001). To use Eq. [2] and [3], the porosity (1 - _s) and K_s have to be known. The K_s is difficult to measure directly, normally it is estimated from the K_a reading in oven dry soil. Another option is to use table values for K_s (see e.g., Hook and Livingston, 1996). The porosity can be estimated from the bulk density by measuring (or assuming) the particle density. Both dielectric mixing models can be modified to include bound water, however, this is not considered in the present study since it has been shown that this does not significantly improve the calibration for this particular data set (Jacobsen and Schjønning, 1995).

It should be noted that Eq. [1] through [3] are valid only under certain circumstances. Especially, the dielectric constants are in reality not constant but depending on the frequency. For example, the K_w decreases dramatically at high frequencies due to relaxation. The TDR signal covers a rather broad spectrum of frequencies. It has been shown that within the TDRs frequency range (50 MHz–1 GHz), the K_a of all major soil components is fairly constant (Weast, 1986; Campbell, 1990). The actual frequency range is dependant on the TDR instrument, cable length, experimental setup, etc. Therefore, care should be taken when results from different experiments are compared. Another concern is the dielectric and ohmic losses. These are, however, small for most soils (e.g., Roth et al., 1992).

Neural Networks
The development of ANNs began several decades ago (McCulloch and Pitts, 1943), inspired by the biological neural network structure observed in human brain. The ANN is a computational approach inherently suited to problems that are mathematically difficult to describe (French et al., 1992), that is, when the equations involved in the physical processes are not well known. Consequently, ANN is a mathematical structure that is capable of representing arbitrarily complex nonlinear processes that relate the inputs and outputs of any system (Hsu et al., 1995). However, it should be kept in mind that using ANNs do not always guarantee good results.

In Fig. 1 , an example of a back propagation three layer feed forward ANN is shown. This is only one type of neural network, there are several other types. The three layers are called input, hidden, and output layer, respectively. The bias nodes will add a random input noise to each weight. This noise makes the network weights constantly ‘shake’ as the training progresses. The purpose of this noise is to help the network jump out from a gradient direction that leads to local minima on the weight surface.

View larger version (20K): [in this window] [in a new window]   Fig. 1. Schematic description of a 2-4-2 ANN.

All connections between nodes represent weights. At each node in the hidden layer, all input data, multiplied with its respective weight, are summarized and then used as input in a transformation. Different transfer functions can be used, both linear and nonlinear, but the most common is the sigmoid transfer function

[4]

where O_K is the output for the hidden layer node K and S_K is the sum of inputs multiplied with respective weight. The same procedure is repeated for the next layer. That is, the output from each hidden node is multiplied with a weight and summarized and used as input in Eq. [4].

Initially, the weights are chosen randomly, for subsequent iterations the values of weights are optimized in an iterative calibration procedure, called training, using a back propagation algorithm (Rumelhart et al., 1986). The performance of the ANN is determined using the RMSE, and the coefficient of determination (r²) values. Using these two parameters, both the ability of the ANN to reproduce the variance of the measured data as well as actual errors for the predictions are considered. During the training procedure, the ANN model should be used to produce output for an independent data set, this is called validation. If the RMSE of the validation data gets significantly higher than for the calibration data, the ANN is ‘overtrained’. An overtrained ANN ‘learns’ the noise as well as the signal.

	MATERIALS AND METHODS

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Time Domain Reflectometry Measurements
A detailed calibration of the K_a– relationship was carried out in soils collected at five different locations in Denmark. From each location, soil was sampled from both the plough layer and the subsoil, resulting in ten different soil textures (see Table 1). The soil was air dried, sieved and then divided into subsamples that were mixed with water. The soil was then packed into polyvinyl Cl (PVC) cylinders (0.18 m long and 0.19 m in diameter) and TDR measurements were taken using two rod probes connected to a Trace system I, model 6050X1 (Soilmoisture Equipment Corp., Santa Barbara, CA). In each subsample, ten TDR measurements were taken and averaged. In total, 189 subsamples were analyzed. The number of subsamples for each soil texture (n) is given in Table 1. The bulk density varied between 1.24 to 1.80 Mg m^-3 and was measured for each subsample. Gravimetrical determination of water content was carried out by drying the soil at 110°C during 48 h. The range in and K_a was 0.01 to 0.36 m³ m^-3 and 3.00 to 26.5, respectively. All TDR measurements were taken at a constant temperature of 15°C. More information concerning the measurements can be found in Jacobsen and Schjønning (1993).

The dependency of each of the six inputs (K_a, bulk density, organic matter, clay, silt, and sand content) was investigated using the ANN. This was previously done for the same data set using traditional regression analysis (Jacobsen and Schjønning, 1993). First, a 1-9-1 ANN using only K_a as input was constructed and the RMSE was determined. Then, each of the other five inputs was added and, thus, five 2-9-1 ANN were run and the RMSEs can be compared. The simulation was halted after 100000 iterations. Three simulations using different initial weights were conducted for each ANN. Using this approach, we can estimate the improvement of the ANN performance for each parameter. It is important, however, to also consider the correlation between the parameters when interpreting the result of this analysis.

Finally, the ANN model was used to predict the K_a– relationship for each of the ten soil types using the other nine for calibration. As input, K_a and the three soil physical parameters that gave the most significant improvement were chosen. Thus, ten 4-9-1 ANN were run.

Comparison of ANN with Other Models
In the present analysis, we use ANN to predict the K_a– relationship for each of the ten soil types using the other nine soils for calibration. For example, for the Jyndevad topsoil, the validation data set consists of 25 measurements for this soil texture and the other 164 data points are used for calibration. This approach is similar to the case where an ANN has been calibrated against a number of measurements of K_a and from several different soil types with different soil physical parameters. Then, a new soil type, independent from the others, is to be examined. Instead of conducting an elaborate soil specific calibration, the ANN is used to predict the K_a– relationship. In other words, we are extrapolating the calibrated ANN model to independent data.

The results are compared with traditional calibration models to investigate the performance of the K_a– relationship predicted by ANN. First, the ANN output is compared with the output obtained from the common Topp equation. Jacobsen and Schjønning (1993) showed that the Topp equation did not give accurate predictions when was larger than 0.20 m³ m^-3. This deviation might be the result of differences in some soil properties between the soil used and the soils for which the Topp equation was optimized, or from systematic deviations depending on the TDR system, experimental setup, calibration procedure, etc. Jacobsen and Schjønning (1993) also optimized the parameters of Eq. [1] using all ten soil types, the relationship obtained can be regarded as the optimum calibration equation for the actual experimental setup and soils studied. In the following we will use the term ‘system specific’ calibration for this relationship.

The de Loor model is also used to predict from TDR measured K_a. The bulk density of each sample is used to calculate _s, the K_s is set to 3.5. The porosity was calculated using the particle density, both values measured for each soil type and an assumed value (2.6 Mg m^-3) were used.

Several studies have shown that obtaining best fit parameters of Eq. [1] would give a better estimate than the linear K^0.5_a– model, the Birchak, and the de Loor models (Jacobsen and Schjønning, 1995; Young et al., 1997; Persson et al., 2001). This is not surprising since Eq. [1] contains four empirical parameters while the other models have zero to two parameters. Thus, the most accurate measurements will be obtained by conducting a soil-specific calibration and optimizing the parameters of Eq. [1]. The best fit parameters of Eq. [1] are obtained for each of the ten soil types and the r² and RMSE are calculated.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
The RMSE values for the 6-k-1 ANNs, with different k values, for the standardized data set are presented in Fig. 2 . For k values between 2 and 6, the RMSE decreases and thereafter is more or less constant. This result is in agreement with previous studies (e.g., Schaap and Bouten, 1996; Persson et al., 2001). A k value of 9 was chosen. The RMSEs using the nonstandardized data sets were 0.007 and 0.008 m³ m^-3 for the calibration and validation data sets, respectively, when k = 9, indicating that the model was not overtrained. In all subsequent ANN models, k = 9 was used. Since these models contained 1 to 4 input nodes, this k value should be sufficient.

View larger version (14K): [in this window] [in a new window]   Fig. 2. Root mean square error using standardized data for different number of hidden nodes, k.

The analysis above shows that using ANN, an elaborate soil specific calibration can be avoided without loss in accuracy. Once the ANN is calibrated using different soil types it can be used to predict relationships for new soil types. Only the soil physical parameters (in our case clay content, organic matter content, and bulk density) have to be known beforehand. The best approach using conventional K_a– relationships seems to be Eq. [3]. In this case, bulk and particle densities need to be known. The RMSE values using Eq. [3] are, however, twice as high compared with the ANN. Even if a soil specific calibration is made, the accuracy is not improved compared with using ANN. Since the ANN is a purely data-learning method, good results are, however, not always guaranteed. When a new soil type is examined, it might be best to check the ANN performance against at least one known K_a– combination. If the ANN gives a bad estimate of for this combination, the ANN should not be used in this soil type.

We believe that the ANN prediction can be improved if a data set with a wider range of soil textures is used for the calibration set. Other parameters, e.g., electrical conductivity, specific surface, and temperature can also be incorporated in the model. It is important, however, that the soil for which the prediction is made is not too different compared with those used in the calibration data set. The ANN method presented in this study should, therefore, be regarded as a system-specific model since errors associated with different TDR systems, TDR frequency ranges, measurement temperature, soil salinity, and calibration procedures might be larger than the ones obtained using the ANN.

	CONCLUSIONS

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The TDR technique has been widely used during the nineties. The universal calibration equation presented by Topp et al. (1980) has been successfully applied to many different mineral soil types with relatively small errors (normally around 0.02 m³ m^-3). Few attempts, however, have been made to improve the K_a– calibration by including soil physical parameters. Thus, when accurate results are needed for a specific soil type, a specific calibration experiment has to be carried out.

The use of ANNs in hydrology and soil science has increased during the last decade and is now more and more used as an alternative to traditional regression analysis. The ANNs are capable of finding the best solution to complex problems. In the present study, ANNs were used to predict the K_a– relationship using soil physical parameters. The data set analyzed has previously been presented by Jacobsen and Schjønning (1993), and contains a detailed calibration of the K_a– relationship for 10 different soil textures including sand, loamy sand, sandy loam, sandy clay loam, and loam. Besides K_a, five different soil physical parameters were used, bulk density, clay, silt, sand, and organic matter content. By inclusion of one parameter at the time in a 2-9-1 ANN, it was found that the parameters that improved the calibration most were bulk density, followed by clay and organic matter content. These results are in agreement with the analysis originally made by Jacobsen and Schjønning (1993) for the same data set.

To show the applicability of using ANNs to predict the K_a– relationship for an independent data set, the measurements from one soil type were used for validation and the other nine soil types as calibration. In this analysis, K_a was used together with bulk density, clay, and organic matter content as input data in a 4-9-1 ANN. It was shown that the ANN provided significantly better prediction of than the Topp equation and Eq. [1] with optimized parameters for all ten soils (system specific calibration). Furthermore, the ANN prediction gave r² and RMSE in the same range, and in about half of the soil types better results, compared with a soil-specific calibration. Thus, by using ANN it is possible to obtain accurate results without the need for soil specific calibration experiments.

	ACKNOWLEDGMENTS

 
This study was funded by the Swedish Research Council for Engineering Sciences and the Swedish Natural Science Research Council. The second author wishes to thank the Nils Hörjel foundation for granting a scholarship for his stay at the Department of Water Resources Engineering, Lund University.

Received for publication July 17, 2000.

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TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

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