Estimating Hydraulic Conductivity Using Pedotransfer Functions

Hydraulic conductivity is an important soil physical property, especially for modeling water flow and solute transport in soil, irrigation and drainage design, groundwater modeling and other agricultural and engineering, and environmental processes. Due to the importance of hydraulic conductivity, many direct methods have been developed for its measurement in the field and laboratory (Libardi et al., 1980; Klute and Dirksen, 1986). Interestingly, comparative studies of the different methods have shown that their relative accuracy varies amongst different soil types and field conditions (Gupta et al., 1993; Paige and Hillel, 1993; Mallants et al., 1997). No single method has been developed which performs very well in a wide range of circumstances and for all soil types (Zhang et al., 2007). Direct measurement techniques of the hydraulic conductivity are costly and time consuming, with large spatial variability (Jabro, 1992; Schaap and Leij, 1998; Christiaens and Feyen, 2002; Islam et al., 2006). Alternatively, indirect methods may be used to estimate hydraulic conductivity from easy-to-measure soil properties. Many indirect methods have been used including prediction of hydraulic conductivity from more easily measured soil properties, such as texture classes, the geometric mean particle size, organic carbon content, bulk density and effective porosity (Wösten and van Genuchten, 1988) and inverse modeling techniques (Rasoulzadeh, 2010; Rasoulzadeh and Yaghoubi, 2011). In recent years, pedotransfer functions (PTFs) were widely used to estimate the difficult-to-measure soil properties such as hydraulic conductivity from easy-to-measure soil properties. The term PTFs were coined by Bouma (1989) as translating data we have into what we need. PTFs were intended to translate easily measured soil properties, such as bulk density, particle size distribution, and organic matter content, into soil hydraulic properties which determined laboriously and costly. PTFs fill the gap between the available soil data and the properties which are more useful or required for a particular model or quality assessment (McBratney et al., 2002). In the other hand PTFs can be defined as predictive functions of certain soil properties from other easily, routinely, or cheaply measured properties. PTFs can be categorized into three main groups namely class PTFs, continuous PTFs and neural networks. Class PTFs calculate hydraulic properties for a textural class (e.g. sand) by assuming that similar soils have similar hydraulic properties; continuous PTFs on the other hand, use measured percentages of clay, silt, sand and organic matter content to provide continuously varying hydraulic properties across the textural triangle (Wösten et


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. In fact, continuous PTFs predict soil properties as a continuous function of one or more measured variables. Neural networks are an "attempt to build a mathematical model that supposedly works in an analogous way to the human brain" and were developed to improve the predictions of empirical PTFs. In brief, a neural network consists of an input, a hidden, and an output layer all containing "nodes". The number of nodes in input (soil bulk density, soil particle size data) and output (soil hydraulic properties) layers corresponds to the number of input and output variables of the model (Schaap and Bouten, 1996). PTFs must not be used to predict something that is easier to measure than the predictor. For example, If we measure the water retention curve only to predict saturated hydraulic conductivity (K s ), this is not an efficient PTF, as the cost of measuring a water retention curve is greater than measuring Ks itself (McBratney et al., 2002).

Pedotransfer functions for estimating saturated hydraulic conductivity
PTFs have become a 'white-hot' topic in the area of soil science and environmental research. PTFs which used to estimate saturated hydraulic conductivity (K s ) were developed using texture classes, the geometric mean particle size, organic carbon content, bulk density and effective porosity as predictor variables. Many PTFs were presented to predict K s . Here the following PTFs for K s are considered. All PTFs give K s in m.s -1 . Wösten et al. (1997) presented a function for determining K s as follows: where BD is bulk density in g.cm 3 , Clay is the percentage of clay, CS is the sum percentage of clay and silt, and Om is percent organic matter. Wösten et al. (1999) where BD is bulk density in g.cm 3 , Clay and Silt are the percentage of clay and silt, respectively, Topsoil is a parameter that is set to 1 for topsoils and to 0 for subsoils, and Om is percent organic matter.

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Estimating Hydraulic Conductivity Using Pedotransfer Functions 147 The Cosby's pedotransfer function (Cosby et al., 1984)  where b is an empirical parameter of Campbell's soil water retention function. The coefficient b is derived from the geometric mean particle diameter (mm), d g , and the standard deviation of mean particle diameter σ g :

Rosetta
Som e PTF s h ave b een in c o rpora te d int o stan da lon e c om puter pr ogram s l ike Ro sett a (Schaap et al., 2001). Rosetta uses a neural network and bootstrap approach for parameter prediction and uncertainty analysis respectively. Rosetta is able to estimate the van Genuchten water retention parameters (van Genuchten, 1980) and saturated hydraulic conductivity, as well as unsaturated hydraulic conductivity parameters, based on Mualem's (1976) pore-size model (Schaap et al., 2001). Here, Rosetta was used to estimate saturated hydraulic conductivity.

Soilpar 2
Soilpar 2 provides 15 PTF procedures to estimate soil parameters. The PTFs procedures are classified as point pedotransfer and function pedotransfer. Point PTFs estimate some specific points of interest of the water retention characteristic and/or saturated hydraulic conductivity. Two of these methods also estimate bulk density. Soilpar 2 uses PTFs of Jabro (1992), Jaynes and Tyler (1984), Puckett et al. (1985), and Campbell (1985) to estimate saturated hydraulic conductivity. Function PTFs, which estimate the parameters of retention functions are implemented: Rawls and Brakensiek (1989), to estimate the Brooks and Corey (1964) function parameters; Vereecken et al. (1989), to estimate the van Genuchten (1980) function parameters; Campbell (1985) to estimate the Campbell function parameters (Campbell, 1974)); Mayr and Jarvis (1999), to estimate the parameters of the Hutson and Cass (1987) modification of the Campbell function. All these methods require as input soil particle size distribution and bulk density. The Mayr and Jarvis, and Vereecken et al. methods also require organic carbon content (Acutis and Donatelli, 2003).

Pedotransfer functions for estimating unsaturated hydraulic conductivity
One of the most popular analytical functions for predicting unsaturated hydraulic conductivity (K(θ)) is the van Genuchten-Mualem model which is Combination of soil water retention function of the van Genuchten (1980) and Mualem's (1976) pore-size model as follows: and S e , is where θ(ψ) is the measured volumetric water content (cm 3 .cm −3 ) at suction ψ (cm-water); θ r and θ s are residual and saturation water content (cm 3 .cm −3 ) respectively, the dimensionless n is the shape factor, and K s is the saturated hydraulic conductivity. Other popular function for predicting K(θ) is the Brooks and Corey (1964) model as follows: where λ is the pore size index. Campbell (1985) proposed a function for determining K(θ) as: where K s is saturated hydraulic conductivities, θ is measured volumetric water content(cm 3 .cm −3 ), θ s is the saturation water content (cm 3 .cm −3 ), and b is the slope of ln ѱ vs ln θ in the soil water retention curve. The unsaturated hydraulic conductivity function is particularly difficult and timeconsuming to measure directly. So, in many model applications, reliance is often placed on predictions of unsaturated conductivity based on measurements of soil water retention and K s . Direct measurements of soil water retention and K s are time-consuming and costly, too. So here, PTFs are used to estimate unsaturated hydraulic conductivity. Rawls and Brakensiek (1985) provided equations for the estimation of van Genuchten, Brooks -Corey and Campbell parameters as follows: where LAM is pore size index, pc is percent clay, ps is percent sand, por is the porosity. The unsaturated hydraulic conductivity of van Genuchten parameter (n) is then calculated from the above relations as follow: Campbell's parameter (b) is estimated as follows: In the Brooks and Corey function λ is equal to LAM.

Rosetta
The van Genuchten parameters, θ r , θ s , and n were estimated from measured particle size and bulk density using Rosetta software (Schaap et al., 2001).

Soilpar 2
Using measured particle size and bulk density data, Campbell model parameter value (b) was estimated using the Soilpar 2 (Acutis and Donatelli, 2003). Note that, for estimating unsaturated hydraulic conductivity, measured value of θ s and K s in the lab were used.

PTFs of saturated hydraulic conductivity
Two following statistical criteria were used for the evaluation of PTFs to estimate saturated hydraulic conductivity based on the approach presented by Tietje and Hennings (1996). Geometric mean error ratio (GMER) and geometric standard deviation of the error ratio (GSDER) were calculated from the error ratio ε of measured saturated hydraulic conductivity (K s ) m vs. predicted saturated hydraulic conductivity (K s ) p values: The GMER equal to 1 corresponds to an exact matching between measured and predictive saturated hydraulic conductivity; the GMER<1 indicates that predicted values of saturated hydraulic conductivity are generally underestimated; GMER>1 points to a general overprediction. The GSDER equal to 1 corresponds to a perfect matching and it grows with deviation from measured data. The best PTF will, therefore, give a GMER close to 1 and a small GSDER. Also, other statistical criterion named deviation time (DT) was used to evaluate PTF S as follows: www.intechopen.com The DT equal to 1 shows an exact matching between measured and predictive saturated hydraulic conductivity.

PTFs of unsaturated hydraulic conductivity
Estimated unsaturated hydraulic conductivity using PTFs were compared by calculating modified index of agreement d' (Legates and McCabe, 1999): where O i is the individual observed unsaturated hydraulic conductivity (K(θ)) value at θ i , S i is the individual simulated value at θ i , O' is the mean observed value and n is the number of paired observed-simulated values. The value of d' varies from 0.0 to 1.0, with higher values indicating better agreement with the observations. The interpretation of d' closely follows the interpretation of R 2 for the range of most values encountered (Legates and McCabe, 1999).

Saturated hydraulic conductivity
Study area is located in northwest of Iran in Naghadeh county, Azarbaijanegharbi province, Iran (Fig. 1). The total area of the Naghadeh county is 52100 ha and is located at coordinates 36° 57′ N and 45° 22′ E. Ten locations in the Naghadeh county was considered and undisturbed soil samples were taken by using a steel cylinder of 100 cm 3 volume (5 cm in diameter, and 5.1 cm in height) from 0-15 cm depth to measure saturated hydraulic conductivity and bulk density. The samples were transported carefully to avoid disturbance. Also, disturbed soil samples were taken using plastic bags to measure particle density, soil texture and organic matter. Saturated hydraulic conductivity was measured by the constant-head method (Israelsen and Hansen, 1962). Samples (steel cylinders) of soil were oven dried at 105ºC and bulk density was calculated from cylinder volume and oven dry soil mass. Particle size distribution (sand, silt and clay percentages) was measured by the hydrometer method. Soil particle density was measured using a glass pycnometer; 10 g air-dried (<2 mm) soil sample was placed into the pycnometer and the displaced volume of distilled water was determined (Jacob and Clarke, 2002). Total porosity was calculated using bulk density (ρ b ) and particle density (ρ p ) according to the following equation: The organic matter was determined by Walkley and Black rapid titration method (Nelson and Sommers, 1996). The measured soil properties are shown in Table 1. Saturated hydraulic conductivities were estimated according to the above mentioned PTFs (Eqs. 1 to 16) as well as Rosetta and Soilpar 2 software and compared to measured K s of the 10 soils. Note that PTFs of Jabro, Jaynes and Tyler, Puckett et al. which are used in Soilpar 2 2, hereafter named Soilpar 2-Jabro, Soilpar 2-Jaynes -Tyler, and Soilpar 2-Puckett et al., respectively. Figures 2 and 3 show measured vs. estimated values for all models tested. With regard to Figures 2 and 3, it is clear that Soilpar 2-Jabro for estimating K s was in excellent agreement with the measured value. After Soilpar 2-Jabro, Rosetta could estimate K s with reasonable accuracy. Three statistical criteria (Eqs. 25 to 28) were used for the evaluation of PTFs which estimate saturated hydraulic conductivity. Calculated values of DT, GMER and GSDER were shown in Table 2. Soilpar 2-Jabro resulted in lower DT (2.91), GMER and GMER equal to 1.13 and 3.06, respectively, performed better than the others PTFs (  Table 2) these PTFs could not be able to estimate K s with reasonable accuracy. The organic matter content is an important variable when infiltration rates are estimated in non-saturated soils, but it has less influence in saturated soils. The main explanation is that organic matter mainly affects retention forces (matric potential), the type of forces that almost do not work in saturated soils where forces are basically affected by gravity. For this reason when estimating water retention parameters in soils, organic matter is a valuable variable to use in PTF (Wösten et al., 1999), but the contribution of organic matter content in estimating K s was very low and it was mainly limited to explain the relationship between soil structure and K s .

Unsaturated hydraulic conductivity
For comparison of the different models in predicting unsaturated hydraulic conductivity of soils, data sets, including data of K s , measured data of unsaturated hydraulic conductivity, fractions of sand, silt and clay, bulk density of 27 soils were selected from the UNSODA hydraulic property database (Names et al., 1999), and used in the study. The measured soil properties from the UNSODA which used in this study, summarized in Table 3.  Table 3. Mean, standard deviation (SD), Max and Min of soil samples parameters By using fractions of sand, silt and clay and bulk density, unsaturated hydraulic conductivities (K(θ)) were estimated according to the PTFs of Rawls and Brakensiek (1985) (Eqs. 21 to 24) as well as Rosetta and Soilpar 2 software and compared to measured K(θ) of the 27 soils. It is noted that PTFs of Rawls and Brakensiek (1985) which were used to estimate parameters of van Genuchten, Brooks -Corey and Campbell functions (Eqs. 17 to 20), hereafter named van Gen-R, B&C-R, and Cam-R, respectively. Also just Campbell model parameter value was estimated using the Soilpar 2, hereafter named Soilpar-Cam. Figure 4 shows measured vs. estimated K(θ) by mentioned PTFs. To facilitate comparison of the PTFs, mean value of modified index of agreement (d') for the same soil texture classes was calculated (Table 4). With regard to Figure 4 and Table 4, one could conclude that for sand, loamy sand, sandy clay loam, and clay textures, the van Gen-R had the bigger d', indicating its higher accuracy in predicting K(θ) as compared to the other PTFs. The best PTF for loam, sandy loam, and silty loam textures is the Soilpar-Cam. Wagner et al. (2001) found that the performance of the Campbell model could be improved when the particle size distribution data used in the determining the Campbell parameters are as detailed as possible, while knowledge of only three fractions (clay, silt, and sand) may reduce the function performance considerably.

Conclusions
Based on the results some of conclusions can be summarized as follows:  PTFs are a powerful tool to estimate saturated and unsaturated hydraulic conductivity. Because PTFs estimate hydraulic conductivity from easy-to-measure soil properties so they have the clear advantage that they are relatively inexpensive and easy to use.  The mean of error parameters DT, GMER and GSDER (  (Table 2).  The mean value of modified index of agreement (d') showed that for sand, loamy sand, sandy clay loam, and clay textures, the van Gen-R had the bigger d', indicating its higher accuracy in predicting K(θ) as compared to the other PTFs. One can be concluded that Gen-R was approximately good in describing the functional relationship between the soil moisture and unsaturated hydraulic conductivity for mentioned soils. The best