Published online 27 October 2005
Published in Soil Sci Soc Am J 69:1922-1930 (2005)
DOI: 10.2136/sssaj2005.0022
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
Soil Physics
Determining Soil Hydraulic Properties from Tension Infiltrometer Measurements
Fuzzy Regression
Bing Cheng Si and
Waduwawatte Bodhinayake
Dep. of Soil Science, University of Saskatchewan, Saskatoon, SK, Canada
* Corresponding author (Bing.Si{at}usask.ca)
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ABSTRACT
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Tension infiltrometer measurements have been used to measure steady-state infiltration rates at applied tensions. The measurements can be used to determined soil hydraulic properties through linear or nonlinear regression. Such regression methods are often based on a few imprecise field measurements, and the traditional regression analysis may not yield valid estimates and reliable predictions. The objective of this study is to introduce fuzzy linear regression as an alternative to statistical regression analysis in determining hydraulic properties from tension infiltrometer measurements. Using a tension infiltrometer, in situ steady-state infiltration rates [q
(h)] were measured at six different tensions (h) between 3 and 22 cm of water on silty loam and clay loam soils. Hydraulic properties (i.e., field saturated hydraulic conductivity, Kfs, and inverse macroscopic capillary length scale, 
G) and their confidence intervals were estimated following the fuzzy least-square linear regression procedure with the minimum fuzziness criterion and linear least-squares method (traditional statistical method). Both calculation procedures yielded the same mean hydraulic properties. A comparison between fuzzy and statistical ln q(h) relationship indicated that the confidence bands resulting from both procedures enveloped all the measurement points, but fuzzy regression estimates offered a tighter fit around the midpoint values (least-square estimates). Fuzzy linear regression is more reliable and may be used as a complement or an alternative to statistical linear regression analysis for determining hydraulic properties from tension infiltrometer measurements.
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INTRODUCTION
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SOIL HYDRAULIC CONDUCTIVITY and inverse macroscopic capillary length scale are important hydraulic properties for the prediction of water flow and solute transport in soil. Tension infiltrometers have become valuable instruments that offer a simple, fast, and convenient means of determining these hydraulic properties in situ (Ankeny et al., 1991; Zhang, 1997), macroporosity (Wilson and Luxmoore, 1988; Bodhinayake et al., 2004a, 2004b) but methodology varies.
A number of calculation procedures exist for determining hydraulic properties from tension infiltrometer measurements, such as the sorptivity method (Smettem and Clothier, 1989), the two tension method (Ankeny et al., 1991), the multi-tension/disc size method (Logsdon and Jaynes, 1993), the inverse procedure (
imunek and van Genuchten, 1996, 1997; Schwartz and Evett, 2003), and early time analysis (Zhang, 1997). Inverse procedures allow estimation of parameters and their uncertainty in a statistical sense. However, inverse procedures require more measurements (usually one has to measure transient state infiltration rates) and non-uniqueness and numerical instability can be problems. Therefore, the nonlinear least-squares regression method introduced by Logsdon and Jaynes (1993) has been widely used, owing to its simplicity. They estimated saturated hydraulic conductivity (Kfs) and inverse macroscopic capillary length scale () from steady-state infiltration rate as a function of applied water tension through nonlinear least-squares regression. Classical linear or nonlinear regression assumes that the measurement errors are normally distributed and independent of each other. Since one needs a lot of samples to determine a probability distribution, linear or nonlinear regression requires at least 8 to 30 measurements or observations to obtain valid estimates of hydraulic parameters (i.e., Kfs and ) (Bardossy et al., 1990). However, as commonly found in most field experiments in hydrology, the number of steady-state infiltration measurements for a site is limited and varies from three to six due to the higher cost of longer field times. Moreover, infiltration measurements under tension are subject to different kinds of uncertainties. These arise from (a) measurement errors due to human and instrument imprecision, (b) an enormous spatial variability of soil hydraulic properties (Russo and Bouton, 1992), and (c) the presence of macropores that make the model invalid. Under these circumstances classical linear or nonlinear regression may not yield valid estimates for parameters. In particular, a confidence band estimated with a few data points is very wide and may not provide information that is useful for predictive purposes. Fuzzy regression technique may be a helpful tool to overcome these problems. It has been used in hydrology (Bardossy et al., 1990), paleoclimatic research (Boreux et al., 1997), radial tree-growth modeling (Boreux et al., 1998), solute transport (Uddameri, 2004), and prediction of the partition coefficient of persistent organic pollutants (Uddameri and Kuchanur 2004). Nevertheless, fuzzy linear regression has not been utilized to the best of our knowledge to estimate hydraulic properties from tension infiltrometer measurements. Furthermore, there has been no analysis of the estimated parameters' uncertainty obtained from regression analysis and its effect on prediction.
The objective of this study is to investigate whether fuzzy linear regression (Tanaka et al., 1982) would result in smaller parameter uncertainty than classical regression and to examine the effect of the uncertainty of the estimated parameters on predicting effective porosity from the estimated hydraulic parameters.
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THEORY
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Linear regression method (Statistical Method)
Gardner (1958) proposed the exponential dependence of hydraulic conductivity, K, on soil suction or tension, h:
 | [1a] |
where Kfs is the field saturated hydraulic conductivity (LT–1) and G is the inverse macroscopic capillary length scale (L–1). Wooding (1968) presented an approximate solution for unconfined steady infiltration rate at a given water tension, q(h)(LT–1), from a shallow circular water source and Gardner's (1958) exponential hydraulic conductivity function (Eq. [1a]) as given by,
 | [1b] |
where rd is the radius of the tension infiltrometer disc (L) and h is the applied water tension (L). Equation [1b] has two unknown parameters, Kfs and G. These parameters and their confidence intervals can be estimated through nonlinear least square regression between q(h) and h or linear regression between ln q(h) and h (Eq. [2])
 | [2] |
In the nonlinear least square regression, 
2 is minimized, while in linear regression, 
2 = 
2 is minimized, where q*(h) and q(h) are measured and predicted infiltration rates, respectively. Therefore, linear regression minimizes the log-transformed ratio of measured to predicted values. Generally, the nonlinear regression is more accurate than linear regression in term of sum of squared error (SSE), because former minimizes SSE and the latter does not. However, when the infiltration rates at different tensions vary considerably, the difference between the predicted and measured values tends to be smaller for smaller measurement values. Therefore, the nonlinear model will fit larger values better than smaller values, resulting in poor correspondence to infiltration rates at higher tensions. Linear regression minimizes the log-transformed ratio of measured to predicted values, which will not result in poor correspondence to small values. Infiltration rates are larger in magnitude near saturation and smaller at higher tensions, thus nonlinear regression will result in better correspondence to the measurements near saturation than linear regression, while the latter will accommodate the measurements at higher tension better than the nonlinear regression. In this case we choose the linear regression because (1) field soils are rarely at saturation in semiarid environments and it is more important to accommodate the measurements at higher tensions, and (2) there is higher spatial variability in measured infiltration rates at saturation than at higher tensions; therefore, measured infiltration rates near saturation should have less weight than that at higher tensions. In addition, linear regression results in exact parameter uncertainty, while nonlinear regression results in approximate uncertainty in the estimated parameters to the first order. However, both methods should produce similar results if proper weights are selected for each measured infiltration rate. For simplicity and because weights are generally assigned arbitrarily, we choose the linear regression with equal weights for all log-transformed measured infiltration rates.
Fuzzy Linear Regression Method
Fuzzy linear regression, introduced by Tanaka et al. (1982), provides an attractive approach for handling regression problems for practical situations where data are uncertain or exhibit large variability. Fuzzy regression operates with fuzzy subsets and fuzzy numbers and is based on fuzzy set theory. Basic definitions on fuzzy sets can be found in Dubois and Parde (1980). The definitions that are necessary to the understanding of the paper will be briefly reviewed.
Let Z be a set of elements (universe). S is called a fuzzy set or subset, denoted by 
, if S is a set of ordered pairs:
 | [3] |
where µ is the degree of membership of z in . The closer µ is to one, the more z belongs to ; the closer it is to zero, the less it belongs to . Fuzzy numbers are special fuzzy subsets. is a fuzzy number if and only if (i) the universe of Z is the set of real numbers, (ii) at least one element of z of the support (the open interval from its smallest to its largest value) has a membership grade equal to one (normal assumption), and (iii) the membership function has no local extrema (convex assumption).
In this study, the regression equation is developed using either fuzzy or non-fuzzy (crisp) inputs and outputs, but the regression coefficients are treated as triangular fuzzy numbers, that is, numbers that belong to a given set (interval) with a certain degree of membership only. Since the regression coefficients are fuzzy numbers, the estimated dependent variable 
is also a fuzzy number. For a bivariate regression analysis, the fuzzy regression model is represented as
 | [4] |
where Ã0 and Ã1 are the fuzzy intercept coefficient and the fuzzy slope coefficient, respectively. X is the independent variable and is a crisp number. If the intercept and the slope are assumed to have symmetrical triangular membership function, they can be represented using fuzzy center mj and fuzzy half-width cj [i.e., Ãj = (mj,cj) for j = 0, 1].
The fitting of the fuzzy model is accomplished by minimizing one or more criteria that measure the fuzziness in the model. In this study, the fuzzy least-squares linear regression with minimum fuzziness criterion (Eq. [5]), introduced by Savic and Pedrycz (1991), is utilized. This approach consists of two steps. The first step uses ordinary least squares to find fuzzy center values of fuzzy regression coefficients (i.e., input and output data are considered as non-fuzzy data). Once the center value of the parameter is determined, the half-width value is sought in the second step through minimizing the fuzziness criterion. For the second step, the problem is to determine the fuzzy half-width c such that the membership value of Yi to its fuzzy estimate Y*i = Ã0 + Ã1Xi is at least H. The fuzzy coefficients are determined such that the estimated fuzzy output i has the minimum fuzzy width cj, while satisfying a target degree of belief H. The term H can be viewed as a measure of goodness of fit or a measure of compatibility between the regression model and data. Each of the observed data sets, which can be fuzzy 
or crisp datum Yi, must fall within the estimated i at H levels as shown in Fig. 1
. The value of H is between 0 and 1. An H of 0.0 indicates that the assumed model is extremely compatible with the data, while an H of 1.0 indicates the assumed model is extremely incompatible with the data. H is generally selected by a decision-maker. For the membership of Y*, the measured data value falls in the interval with smallest membership = H (Fig. 1). The maximum distance away from the center value equals ±
cj|Xij|, that is, it is equal to the distance from zero-cut of the fuzzy interval to the center value as shown in Fig. [1]. For the H cut of fuzzy membership of Y*, the maximum distance from the center value equals ±
cj|Xij| for a triangular membership function of Y*. We require all measurements Yi to fall in the interval of the H-cut of a fuzzy interval, which is mathematically equivalent to Eq. [7] through [8]. For the bivariate case, the half-widths of the fuzzy regression coefficients are determined by solving the following optimization problem (Tanaka et al., 1982; Chang and Ayyub, 2001):
 | [5] |
Subject to the following constraints:
 | [6] |
 | [7] |
 | [8] |
where T is the total fuzziness criterion of the regression model, n is the number of data points, Xij is the jth independent variable (in this study, X0 = 1 and X1 = h) for the ith sample, and Yi is the fuzzy center of the dependent variable (in this study, the log-transformed infiltration rate).
The approach proposed by Savic and Pedrycz, (1991), integrates minimum fuzziness criterion into the ordinary least-squares regression, This approach is appealing because it incorporates both least-squares and minimum fuzziness criterion in the development of the equation and has been shown to resolve some of the limitations associated with using minimum fuzziness criterion alone (Savic and Pedrycz, 1991). As stated earlier, this combined approach consisted of two steps. In the first step, the estimated output data were treated as crisp data (i.e., ci = 0) and the mid-points of regression coefficients were obtained by performing least-squares linear regression. Equation [2] can be expressed in the form of Eq. [4] with the centerline as:
 | [9] |
where = lnq(h), m0 = ln[Kfs(1+4/
rd)], and m1 = G.
In the second step, the results of the first step were used as center values of the fuzzy regression coefficients. Symmetric triangular membership functions are assumed for the regression coefficients. The half-widths of the fuzzy coefficients (c0 and c1) for degree of belief H = 0.75 were obtained by solving Eq. [5] to [8] (Savic and Pedrycz, 1991). The lower and upper confidence levels of fuzzy linear regression (FZ_L and FZ_U, respectively; Savic and Pedrycz, 1991) and 95% lower and upper confidence limits of the classical least-squares linear regression (LL and UL, respectively; Myers, 1989) were calculated as follows:
 | [10] |
 | [11] |
 | [12] |
 | [13] |
Uncertainty analysis
Effective porosity or water conducting porosity is important for evaluating the effect of soil management on soil pores and hydraulic properties. Here we will evaluate the effect of parameter uncertainty from fuzzy and statistical regression on soil effective porosity. Bodhinayake et al. (2004a) derived the following analytical expression for effective porosity based on infiltration rates from a tension infiltrometer:
 | [14] |
where 
is the surface tension of water, 
is the density of water, g is the gravity, e = 
and 
(a,b) is the macroporosity between pore radius a and b.
Statistical uncertainty analysis. In the following, we utilize first-order perturbation analysis (Dagan, 1984; Gelhar, 1993) to obtain the mean and variance of water-conducting porosity. Since Ks is lognormally distributed, we write y = ln(Ks). We assume that y and G are composed of a constant deterministic term and a random term that has an expectation of zero:
 | [15] |
where E[ ] represents expectation and bar represents the mean. Similarly. For a function of y and G, f(y, G), the first-order perturbation can be written as:
 | [16] |
Taking the expectation of Eq. [16] to the first order yields,
 | [17] |
where f1
= 
and f2 = 
+ 
+ 
. The perturbation of (a,b), '(a,b) can be obtained through '
= – 
. Therefore,
where
and
The variance of ', 
2 = E
, which can be obtained analytically to the first order as:
 | [18] |
where COV
= E
'G,y'
is the covariance between 'G and y'. Since statistical linear regression gives the mean, standard deviation of, and the covariance between, y and G, we can calculate the mean and variance of (a,b).
Fuzzy uncertainty analysis: In the following, we assume that Kfs and G are imprecise and are represented by fuzzy numbers. As a result, the effective or water conducting porosity is also a fuzzy number. There are no direct mathematical operations on fuzzy sets, when the relationship between two fuzzy numbers is nonlinear. However, the application of the extension principle can be performed at different -level cuts. The -level cut (set) of a fuzzy subset is the set of those elements that have at least membership:
 | [19] |
For every chosen level, each fuzzy parameter is represented by a closed interval. For the effective porosity, interval boundaries for the given level could mathematically be formulated as the following optimization problem (Dubois and Parde, 1980; Schulz and Huwe, 1997):
 | [20] |
subject to the constraints:
 | [21] |
where 
and 
are the left and 
and 
the right interval boundaries of the -level cuts on and . By minimizing and maximizing the dependent variable subject to constraints given by the fuzzy input variable, the minimum and maximum values of the unknown soil effective porosity for a given -level cut are provided. To generate the complete membership function of fuzzy soil effective porosity, the procedure has to be repeated for several different cuts.
Generally, nonlinear optimizing routines are necessary to solve minimizing/maximizing problems under constraints. However, fuzzy soil effective porosity can be solved analytically.
Due to the continuity of dependent variables and its strict monotonicity relative to their input parameters Kfs and G within the domain considered, the solution of Eq. [20] could be simply calculated by:
 | [22] |
Therefore, the membership function can be constructed for different -cut levels.
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MATERIALS AND METHODS
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To compare the classical least-square log-linear regression and fuzzy linear regression for determining Kfs and G, two field experiments were performed using a tension infiltrometer on a farm field in Laura, Saskatchewan, Canada (51°52' N lat., 107° 18' W long) and on a second farm at the St. Denis National Wildlife Area in Central Saskatchewan, Canada (106°06' W, 52°02' N; 545–560 m above sea level). The soil at the Laura site is described as an Elstow associaton: Dark Brown Chernozems (Typic Haplustolls) developed on a glacio-lacustrine parent material with a silt loam texture in Ap (0–14 cm) soil horizon. This site has been under a crop-fallow rotation dominated by wheat (Triticum aestivum L.) with some barley (Hordeum vulgare L.) since 1966. The soil at the St. Denis site is dominantly Orthic Dark Brown Chernozems (fine-loamy, mixed, frigid, Typic Haplustolls; Soil Survey Staff, 1999) developed from moderately to fine textured unsorted glacial till (Bodhinayake and Si, 2004).
To prepare for the infiltration measurements, the straw residue was removed from the surface. Any vegetation present was carefully trimmed to the soil surface using a pair of scissors and then removed. Infiltration measurements were performed using a tension infiltrometer with a 20-cm diam. disk (Soil Measurement Systems, Tuscon, AZ). A thin layer (
5 mm) of fine testing sand was applied over the prepared surface in a circular area with a diameter equal to the infiltration disk. This smoothed out any irregularities of the soil surface and ensured good hydraulic contact between the soil surface and the infiltrometer disk. The infiltration rates were measured at 3, 6, 10, 13, 17, and 22 cm water tensions. The tension infiltrometer, preset at 22 cm tension, was gently placed on the sand and the amount of water infiltrating into the soil, measured by the water level drop in the graduated reservoir tower, was recorded as a function of time. When the amount of water entered into the soil did not change with time (generally take about 30 min) for three consecutive measurements taken at 5-min intervals, steady-state flow was assumed and the steady-state infiltration rate was calculated based on the last three measurements. The water tensions were then set sequentially to 17, 13, 10, 6, and 3 cm and the corresponding steady-state infiltration rates were obtained.
The three-dimensional steady-state infiltration rates obtained at different water tensions were used to estimate hydraulic properties (Kfs and G) following linear regression method (statistical method) as well as fuzzy linear regression method (Tanaka et al., 1982; Savic and Pedrycz, 1991). For the classical statistical regression method, a linear least-squares regression of log-transformed steady-state infiltration rate, q(h), as a function of water tension (Eq. [1]) was performed to estimate ln(Kfs) and . In fuzzy linear regression, the combined approach proposed by Savic and Pedrycz, (1991), which integrates minimum fuzziness criterion into the ordinary least-squares regression, was used to estimate ln (Kfs) and G. The minimization of the fuzziness criterion Eq. [5] subject to Eq. [6] through [8] was performed using MathCad 2000 (MathSoft, Cambridge). The lower and upper confidence levels of fuzzy linear regression (FZ_L and FZ_U) and 95% lower and upper confidence limits of classical least-squares linear regression (LL and UL) were calculated using Eq. [12] and [13]. Confidence intervals obtained from the classical least-squares linear regression and fuzzy least-squares linear regression with the minimum fuzziness criterion (Savic and Pedrycz, 1991) were then compared. Fuzzy membership for estimated Kfs and (a,b) are calculated using Eq. [19] through [22].
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RESULTS AND DISCUSSION
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Figures 2 and 3
show measured infiltration rates at different tensions (filled circles) for the grassland and cultivated soils. As expected, the infiltration rate decreases with increase in tension. However, the six measured points did not fall on the straight line of the logarithm of q(h) as a function of h (Eq. [2]). The deviation from the linear relationship may be a result of measurement errors, imprecision in Gardner's model Eq. [1a] to describe this soil, and imprecision in Eq. [1b] for describing steady-state infiltration.
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Fig. 2. Measured and predicted steady-state infiltration rate q as a function of tension (h) and their 95% confidence interval, and fuzzy upper and lower limits with a degree of belief H = 0.75 for the grassland soil.
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Fig. 3. Measured and predicted steady-state infiltration rate q as a function of tension (h) and their 95% confidence interval, and fuzzy upper and lower limits with the degree of belief H = 0.75 for the cultivated soil.
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The steady-state infiltration rates obtained from the tension infiltrometer fitted closely to Eq. [2] with a correlation coefficient >0.98 for both soils. The estimated ln(Kfs) value (standard deviation) was –8.75 (0.064) [ln(cm s–1)] for the grassland and –9.75 (0.078) [ln(cm s–1)] for the cultivated land. The estimated G values and standard deviations were 0.054 (0.0097) cm–1 for the grasslands and 0.08 (0.01) cm–1 for the cultivated land (Table 1). However, the confidence intervals are large for both values. The large confidence interval can be attributed to: (1) limitation of the tension infiltrometer in covering a wider range of tensions; tension infiltrometers can only measure infiltration rate at tensions from 0 to 20 cm because at high tensions, the infiltration rate will be too small to be accurate and take an unrealistically long time to make a measurement; (2) traditional linear regression generally requires more than 8 to 30 measurements and a limited number of steady-state infiltration rates at different tensions can be measured in the range of tension (Bardossy et al., 1990); and (3) there may be autocorrelation between steady-state infiltration measurement errors at different tensions. This autocorrelation means that infiltration rates measured at different tensions carry common information on the linear relationship, further reducing the effective number of measurements. Therefore, more data points do not mean better estimation. However, increasing data points that are dissimilar or adding different kinds of data (such as water content, instead of infiltration rate) would reduce the autocorrelation between data points, thus resulting in narrow confidence intervals. All these violate the basic assumptions of the statistical regression, thus leading to more uncertainty. Therefore, a different paradigm is needed to deal with the tension infiltrometer measurements.
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Table 1. Estimated parameters and their covariances using statistical linear regression and first-order perturbation.
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As expected, the estimated ln(Kfs) and G value obtained using the fuzzy linear regression method were same to the ln(Kfs) and G values obtained from the statistical linear regression procedure. For a degree of belief (H) set equal to 0.75, the fuzzy linear relationship between q(h) and h along with the 95% confidence intervals obtained for both fuzzy linear regression and the linear version of the statistical linear regression method (Eq. [2])) is presented in Fig. 2. The centerline of the fuzzy regression model for the grassland land soil is given by ln q(h) = ln[Kfs(1 + 4/rd)] – h = (–7.538, 0.776) – 0.054 h and the centerline of the fuzzy regression model for the cultivated land is given by lnq = ln[Kfs(1 + 4/rd)] – h = (–8.797, 0.847) – 0.080 h.
The half-width of the intercept for the grassland is 0.776, meaning that the y values vary between –8.314 and –6.762, and all the values of 
must fall in the interval and the values outside of the interval have zero membership. Similarly, for the cultivated land, the half-width of the intercept is 0.847, meaning that the y values are between –9.644 and –7.950. The half-width for the slope or G was zero for both soils, indicating that the coefficient was essentially crisp and did not contribute to the overall fuzziness in the predicted values. The reason is that we seek fuzzy coefficients that possess minimum fuzziness through Eq. [6] to [8]. Therefore, the slope of the relationship is crisp and the fuzziness in the relationship comes from the intercept that is only a function of Kfs. Based on operation on intervals, ln
= m – ln
. Substitution of G and rd leads to ln(Kfs) = m – 1.21 for the grassland land and ln(Kfs) = m – 0.95 for the cultivated land. Substitution of the center and half-width for m, resulting in the center and half-width of ln(Kfs), are –8.793 and 0.776 for the grassland and –9.750 and 0.847 for the cultivated land. The membership function for Kfs can be obtained from the center and half-width of ln(Kfs) through the cuts and is shown in Fig. 4
. Even though the membership function for ln(Kfs) is triangular, that of Kfs is not triangular because of the nonlinear relationship between ln(Kfs) and Kfs. The grassland soil had higher Kfs because the grassland had smaller bulk density, high organic matter content, and more root channels than the cultivated lands, even though the grassland had higher clay content (Table 2). In addition, the grassland soil had a higher spread in Kfs. This is because cultivated soil is coarser in texture and may have a uniform pore-size distribution, thus this soil is best described by Gardner's equation (Eq. [1a]); the grassland soil had more organic matter and higher clay content, which will not best represented by Eq. [1a]. Furthermore, it can be seen from Fig. 2 that both fuzzy and statistical regression confidence intervals envelop all the data that are used to develop the equation. However, the fuzzy interval provides tighter binding of the data, and thus fuzzy linear regression is more reliable and useful for estimating parameters and their uncertainty.
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Fig. 4. Estimated fuzzy membership function for the saturated hydraulic conductivity of the grassland and cultivated soils.
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The uncertainty associated with the estimated hydraulic parameters has significant effects on the uncertainty of effective porosity. With the hydraulic parameters estimated from the traditional statistical regression, the 95% confidence interval for (a,b) (Eq. [18]) is between-3.13 x 10–3 to 3.13 x 10–3 for the grassland and –4.18 x 10–4 to 4.19 x 10–4 for the cultivated soil. The membership function for (a,b) calculated from the fuzzy regression, is shown in Fig. 5
. Clearly all values for (a,b) fall between 3.5 x 10–7 and 1.7 x 10–6 for the grassland soil and 3.4 x 10–7 and 1.8 x 10–6 for the cultivated soil. The 95% confidence interval given by the statistical regression is much wider and encompasses the interval derived from fuzzy regression. In addition, the possible values given by the confidence interval can be negative, which are unrealistic. For the fuzzy regression, the possible values given by the membership function of (a,b) fall in a narrow interval and are all positive. Although it is difficult to prove that fuzzy regression is more accurate, fuzzy regression gives more realistic values or uncertainty than the traditional statistical regression.
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Fig. 5. Estimated fuzzy membership function for the effective porosity of the grassland and cultivated soils.
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There are many varieties of fuzzy linear regression methods (Chang and Ayyub, 2001). In this paper, we adopted the two-step fuzzy regression, for the following reasons: (1) the estimated hydraulic parameter values from fuzzy regression are the same to those of traditional statistical regression; (2) the two-step fuzzy regression is more intuitive and has the same minimization criterion; and (3) research showed that the one step fuzzy regression can be less accurate than the traditional regression (Kim et al., 1996). The method presented in this study for fuzzy linear regression can be easily extended to nonsymmetrical membership function and to fuzzy nonlinear regression (Bardossy et al., 1993).
We used a degree of belief of 0.75, which is arbitrary, just like the selection of 95% confidence level for statistical regression. Experience indicated that a degree of belief of 0.50 or 0.75 is a reasonable level (Tanaka et al., 1982). The upper and lower bounds do not change much from H = 0.75 to H = 0.5. The fuzzy interval of fuzzy regression is quite different from the confidence interval in statistical regression. For example, a 95% confidence interval for the mean regression estimates implies that if many dependent samples are taken at the same levels of an independent variable, then 95% of the interval that contains all samples, will encompass the true value of the mean of the dependent variable. Therefore, the confidence interval should be interpreted in relation to repeated sampling or future prediction. However, a fuzzy interval corresponding to H = 0.75 means that the H-cut of the fuzzy interval will include all observations. Thus, the focus of the fuzzy interval is entirely on the given observations, not on future sampling or predictions. Therefore, contrary to statistical regression, fuzzy regression is not suitable for making predictions (Kim et al., 1996). However, fuzzy regression is well-suited for parameter estimation from given observations. There are established theory to obtain fuzzy set based uncertainty and statistics based uncertainty (as we did for effective macroporosity in this study) and to allow decision-making based on fuzzy uncertainty or statistical uncertainty. Therefore, for these purposes, we can use either fuzzy regression or statistical regression and compare the two techniques.
Fuzzy regression is not affected by the autocorrelation between measurements, because in the derivation of fuzzy regression, only membership of individual measurement is considered; in statistical regression, to derive the statistical uncertainty of the estimates, independent error distribution is not assumed. These characteristics of fuzzy regression and statistical regression determines that autocorrelation between measurements is important for statistical regression, but not for fuzzy regression.
In this study, we used the Wooding's solution for the whole range of h. The solution fit reasonably the measured q(h), even though there are some deviation from straight linear relationships between ln(q) and h. Logsdon's (1999) piecemeal fit with 3 to 6 points per section depending on shape, may result in better fit to the measurements, however, at expense of more uncertainty with the estimated parameters given a limited number of measurements of q(h).
For the statistical uncertainty analysis, we used the well-accepted first-order perturbation approaches (Dagan, 1984; Gelhar, 1993). The first-order perturbation is approximate and accurate when the standard deviations of log(Kfs) and G are less than one (Gelhar, 1993). In this study, the standard deviations of both variables are less than one and thus the uncertainty analysis is reliable.
The fuzzy or statistical regression model is conditioned on the tension range and extrapolation beyond this range is not suitable. Tension infiltrometers measure infiltration rates near saturation and will provide accurate estimates for Kfs. However, G is a parameter that highly depends on the hydraulic conductivity at high tensions. Estimation of G based on measurements near saturation may not apply to high-tension ranges, regardless of fuzzy or statistical regression techniques.
In this study, we considered the parameter uncertainty induced by estimation procedures. We assumed that the flux measurement of q(h) is accurate and has negligible uncertainty. Future studies may examine the combined uncertainties from the measurement of q(h), spatial variability, and estimation procedures.
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CONCLUSION
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A fuzzy least-squares linear regression with minimum fuzziness criterion and classical log-linear least squares regression procedures were used to estimate hydraulic properties (i.e., Kfs and G) from in situ tension infiltrometer measurements. The tension infiltrometer measurements used were limited to six imprecise steady-state infiltration measurements at six different water tensions. Both procedures resulted in the same mean hydraulic properties. Nevertheless, comparison between fuzzy and classical linear lnq(h) – h relationships indicated that the fuzzy regression model enveloped all the data points and offered a tighter fit around the mid-point values (least-square estimates). Uncertainty analysis indicated that the parameters obtained from fuzzy regression resulted in less uncertainty in effective porosity than those obtained from classical regression. Therefore, fuzzy linear regression seems more reliable and useful for estimating hydraulic properties from tension infiltrometer measurements for the two soils tested. More extensive tests are needed for a full evaluation of the two methods.
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ACKNOWLEDGMENTS
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Funding for this project was partially provided by the National Science and Engineering Research Council of Canada (NSERC).
Received for publication January 18, 2005.
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REFERENCES
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- Ankeny, M.D., M. Ahmed, T.C. Kaspar, and R. Horton. 1991. Simple field method for determining unsaturated hydraulic conductivity. Soil Sci. Soc. Am. J. 55:467–470.[Abstract/Free Full Text]
- Bardossy, A., I. Bogardi, and L. Duckstein. 1990. Fuzzy regression in hydrology. Water Resour. Res. 26:1497–1508.[CrossRef]
- Bardossy, A., I. Bogardi, and L. Duckstein. 1993. Fuzzy nonlinear regression analysis of dose–response relationships. Eur. J. Oper. Res. 66:36–51.[CrossRef]
- Bodhinayake, W., and B.C. Si. 2004. Near-saturated surface soil hydraulic properties under different land used in the St. Denis National Wildlife Area, Saskatchewan, Canada. Hydrol. Process. 18:2835–2850.[CrossRef]
- Bodhinayake, W., B.C. Si, and C. Xiao. 2004a. New methods for determining water-conducting Macro- and Mesoporosity from tension infiltrometer. Soil Sci. Soc. Am. J. 68:760–769.[Abstract/Free Full Text]
- Bodhinayake, W., B.C. Si, and K. Naborio. 2004b. Determination of hydraulic properties in sloping landscapes from tension and double-Ring infiltrometers. Vadose Zone J. 3:967–970.
- Boreux, J.-J., G. Pesti, L. Duckstein, and J. Nicolas. 1997. Age model estimation in paleoclimatic research: Fuzzy regression and radiocarbon uncertainties. Palaeogeograph, Palaeoclimatol., Palaeoecol. 128:29–37.
- Boreux, J.-J., C. Gadbin-Henry, J. Guiot, and L. Tessier. 1998. Radial tree-growth modeling with fuzzy regression. Can. J. For. Res. 28:1249–1260.[CrossRef]
- Chang, Y.-H.O., and B.M. Ayyub. 2001. Fuzzy regression methods—A comparative assessment. Fuzzy Sets Syst. 119:187–203.[CrossRef]
- Dagan, G. 1984. Solute transport in heterogeneous porous formations. J. Fluid Mechanics. 145:151–177.[CrossRef]
- Dubois, D., and H. Parde. 1980. Fuzzy sets and systems: Theory and applications. Academic Press, New York.
- Gardner, W.R. 1958. Some steady-state solutions of unsaturated moisture flow equations with application to evaporation from a water table. Soil Sci. 85:228–232.
- Gelhar, L.W. 1993. Stochastic subsurface hydrology. Prentice Hall. Englewood Cliffs, N.J.
- Kim, K.J., H. Moskowitz, and M. Koksalan. 1996. Fuzzy versus statistical regression. Eur. J. Oper. Res. 92:417–434.[CrossRef]
- Logsdon, S.D., and D.B. Jaynes. 1993. Methodology for determining hydraulic conductivity with tension infiltrometers. Soil Sci. Soc. Am. J. 57:1426–1431.[Abstract/Free Full Text]
- Logsdon, S.D. 1999. Unsaturated soil hydraulic properties in the region near saturation. p. 579–586. In M.Th. van Genuchten et al. (ed.) Characterization and measurement of the hydraulic properties of unsaturated porous media. Proc. of the Int. Workshop, Riverside, CA, 22–24 Oct. 1997. Univ. of California, Riverside.
- Myers, R.H. 1989. Classical modern regression with applications. 2nd ed. Duxbury Press. Belmont, CA.
- Russo, D., and M. Bouton. 1992. Statistical-analysis of spatial variability in unsaturated flow parameters. Water Resour. Res. 28:1911–1925.[CrossRef]
- Savic, D.A., and W. Pedrycz. 1991. Evaluation of fuzzy linear regression models. Fuzzy Sets Syst. 39:51–63.
- Schulz, K., and B. Huwe. 1997. Water flow modeling in the unsaturated zone with imprecise parameters using a fuzzy approach. J. Hydrol. (Amsterdam) 201:211–229.
- Schwartz, R.C., and S.R. Evett. 2003. Conjunctive use of tension infiltrometers and time-domain reflectometry for inverse estimation of soil hydraulic properties. Vadose Zone J. 2:530–538.[Abstract/Free Full Text]
- imunek, J., and M.Th. van Genuchten. 1996. Estimating unsaturated soil hydraulic properties from tension disc infiltrometer data by numerical inversion. Water Resour. Res. 32:2683–2696.[CrossRef]
- imunek, J., and M.Th. van Genuchten. 1997. Estimating unsaturated soil hydraulic properties from multiple tension disc infiltrometer data. Soil Sci. 162:383–398.[CrossRef]
- Smettem, K.R., and B.E. Clothier. 1989. Measuring unsaturated sorptivity and hydraulic conductivity using multiple disc permeameters. J. Soil Sci. 40:563–568.[CrossRef]
- Soil Survey Staff. 1999. Soil taxonomy. A basic system of soil classification for making and interpreting soil surveys. Agric. Handb. No. 436. 2nd ed. NRCS, USDA, U.S. Gov. Print. Office, Washington, DC.
- Tanaka, H., S. Uejima, and K. Asai. 1982. Linear regression analysis with fuzzy model. IEEE Trans. Syst. Man Cybern. SMC-12:903–907.
- Uddameri, V. 2004. Relationships of longitudinal dispersivity and scale developed from fuzzy least-square regressions. Environ. Geology. 45:1172–1178.[CrossRef]
- Uddameri, V., and M. Kuchanur. 2004. Fuzzy QSAR for predicting log Koc of persistent organic pollutants. Chemosphere 54:771–776.[Medline]
- Wilson, G.V., and R.J. Luxmoore. 1988. Infiltration, macroporosity and mesoporosity distributions on two forested watersheds. Soil Sci. Soc. Am. J. 52:329–335.[Abstract/Free Full Text]
- Wooding, R.A. 1968. Steady infiltration from a shallow circular pond. Water Resour. Res. 4:1259–1273.
- Zhang, R. 1997. Determination of soil sorptivity and hydraulic conductivity from the disk infiltrometer. Soil Sci. Soc. Am. J. 61:1024–1030.[Abstract/Free Full Text]
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ABSTRACT
INTRODUCTION
