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Published online 8 March 2006
Published in Vadose Zone J 5:341-355 (2006)
DOI: 10.2136/vzj2005.0026
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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SPECIAL SECTION: FROM FIELD- TO LANDSCAPE-SCALE VADOSE ZONE PROCESSES

Spatial Association among Soil Hydraulic Properties, Soil Texture, and Geoelectrical Resistivity

Ole Wendrotha,*, Sylvia Koszinskib and Eugenia Pena-Yewtukhiva

a Dep. of Plant and Soil Sciences, Univ. of Kentucky, Lexington, KY 40546-0312
b Institut für Bodenlandschaftsforschung, ZALF, Eberswalder Str. 84, 15374 Müncheberg, Germany
* Corresponding author (owendroth{at}uky.edu)

COA-Agricultural Experiment Station Number: 05-06-066.


Received 15 February 2005.



   ABSTRACT
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Spatial variability of soil hydraulic properties causes considerable variations in water and solute flow and transport processes. It remains a difficult task to determine and describe the spatial pattern of soil physical properties for modeling landscape-scale vadose zone processes. Strategies that involve measurements of relevant variables and appropriate spatial modeling tools need to be identified for this purpose. The objective of this study was to investigate a nested sampling approach for soil hydraulic properties and associated variables, such as soil texture, in a highly variable soil landscape in northeastern Germany dominated by soils derived from glacial till. Additionally, spatially highly resolved and nondestructive measurements of geoelectrical resistivity were examined for their spatial relation to soil physical and transport properties and their applicability in spatial process models of soil hydraulic property coefficients. Two pedotransfer function (PTF) approaches were applied to gain insight into their capacity to estimate the mean and the spatial variation of soil hydraulic property coefficients, although pedotransfer functions are originally not developed for this purpose. State-space models estimated the spatial process of hydraulic conductivity at a soil water pressure head of –50 cm and the van Genuchten parameter n with relatively narrow confidence bands, whereas for {alpha} confidence was larger. Geoelectrical resistivity supported the estimation of the spatial distribution of soil hydraulic property coefficients. Despite the fact that without calibration, pedotransfer variable estimates did not always represent the mean of the respective variable, there was generally little spatial fluctuation reflected. However, considerable horizontal and vertical variability in soil textural properties existed within the investigated landscape. The findings suggest that state-space models incorporating intermittent observations and spatial autocovariance behavior provide more flexibility in the estimation of the spatial process of hydraulic functional coefficients than regression type functions only.

Abbreviations: ANN, Artificial neural network • DPTF, continuous pedotransfer function • PTF, pedotransfer function


   INTRODUCTION
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
ADEQUATE DESCRIPTION of water and solute flow and transport processes in field soils requires appropriate consideration and characterization of the spatial variability of soil hydraulic properties (Banton et al., 1997). Campbell (1978) observed the spatial behavior of sand content and soil pH value and suggested that the problem of estimating variability can be solved if it is possible to relate spatial point-to-point variation to other soil variables and landscape characteristics.

In this study, spatial variability of soil physical and hydraulic properties is considered as a type of spatial process. Their gradual changes in space should help identify their underlying driving forces and their interactions (DeAngelis and White, 1994; DeAngelis, 1994). Identifying not only random fluctuations but gradual changes requires the quantification of the spatial variability structure of observations taken in space. Depending on the variability of the data, this can require a high number of samples or individual observations of the variable of interest. Variograms can be employed to identify an appropriate sampling scheme (Nielsen and Alemi, 1989). To reduce the sampling effort, nested sampling strategies can be applied, with nonunique sampling distances (Warrick and Myers, 1987). Once the spatial variability structure is identified, measurements at the instrumental scale can be employed to estimate processes at the plot, field, or watershed scale (Ellsworth and Boast, 1996).

Ünlü et al. (1990) found spatial correlation lengths for lognormal transformed saturated hydraulic conductivity on the order of 15 m, and 40 m for pore size distribution. In a very intensive field investigation, Shouse et al. (1995) found spatial ranges of correlation between 10 and 20 m for shape parameters of the water retention curve. Transport coefficients are known to exhibit a short range of spatial structure, especially if they are dominated by soil structure. Saturated hydraulic conductivity Ks was shown to be correlated over 1 m only, and K at a soil water pressure head h = –15 cm over 7 to 17 m (Logsdon and Jaynes, 1996).

Nevertheless, it is not enough to derive univariate spatial correlation behavior, but it is essential to quantify the spatial covariance behavior of pairs of variables to derive efficient coregionalization schemes and a better understanding of the processes underlying soil hydraulic properties (Jury et al., 1987). Greminger et al. (1985) found sand content and soil water retention parameters to be cross correlated for distances <10 m. Interestingly, even small changes in sand content caused significant differences in water retention behavior in their study.

It remains a challenge in applied landscape research to develop sampling strategies and to identify simply observable variables in combination with spatial modeling approaches (Buchter et al., 1991). Nondestructive measurements at the land surface have shown considerable promise in assessing soil spatial variability at the landscape scale (Feddes et al., 1993; Rea and Knight, 1998; Chaplot et al., 2000). For example, geophysical measurements have been applied to determine the spatial variability of soil texture, K, porosity from the soil surface (Banton et al., 1997), soil hydraulic properties, and aquifer geometry (Hubbard et al., 1999). In this study, we examined the spatial cross correlation structure between geoelectrical resistivity ({rho}el) and soil hydraulic function parameters.

Within the last two decades, autoregressive approaches have been applied in an increasing number of fields of soil science and agronomy (Morkoc et al., 1985; Anderson and Cassel, 1986; Buchter et al., 1991; Wendroth et al., 1992, 2003), especially for state-space analysis. This technique identifies the spatial association of variables. Even if the spatial density of observations differs among the variables considered, state-space models provide adequate representation of the spatial process of a variable of interest, the confidence of estimation, and the magnitude of measurement and model errors (Nielsen and Wendroth, 2003). There are only a few applications of autoregressive state-space analysis of soil hydraulic properties (e.g., Buchter et al., 1991), and even those have not been applied to spatial modeling of soil physical properties in combination with geoelectrical resistivity. The idea of considering the relation between different variables and the state vector in the spatial neighborhood will be applied in this study to various soil physical variables that can be obtained with varying intensity of labor and effort.

Numerous PTFs have been derived to calculate soil hydraulic properties on the basis of soil textural data and other basic soil information (Vereecken et al., 1992). Pedotransfer functions have been successfully used for estimating general transport behavior in large regions with different soil types. The prediction quality depends on the underlying database including the range of soil textural classes, quality of the calibration data (Schaap and Leij, 1998) and the locations where data were obtained. Moreover, if there is a variety of measurement methods involved in obtaining the data, the quality of PTFs can be negatively affected by methodological differences (Minasny et al., 2004). From the variety of existing PTF approaches, no one is favored over others, although for many European soils, the PTFs derived by Wösten (1997) seem to work very well (Wagner et al., 2000). Artificial neural network analysis (ANN) is used frequently, and the ROSETTA algorithm developed by Schaap et al. (2001) is also helpful.

Schaap et al. (2001) emphasized that it is virtually impossible to perform enough measurements in many vadose zone studies, where a substantial spatial variability in soil hydraulic properties is present. Pedotransfer functions are developed for large areas with a considerable variability in soil properties to provide average estimates of transport behavior or maps of transport properties (Ferrer Julià et al., 2004). Nevertheless, under certain field conditions (Scheinost et al., 1997; Sinowski et al., 1997), a broad range in soil texture and soil hydraulic behavior within a 10-ha site is usually observed, which is typical of much larger regions. This requires alternative approaches for efficient characterization of spatial variability of soil hydraulic properties similar to those offered by PTFs at larger field scales.

A state variable is a variable that changes continuously through space or time due to an underlying process. In the context of this study, a hydraulic property state variable is a variable that reflects in any regard the continuous process of hydraulic properties through space. Hence, parameters for the empirical description of soil hydraulic properties, as well as hydraulic conductivity coefficients, can be considered hydraulic property state variables.

The overall objective of this study was to determine the spatial variability structure of soil hydraulic properties in a glacial till field soil in a moraine landscape of northeast Germany. The specific objectives were as follows:

  1. Based on the spatial covariance of soil hydraulic properties and other soil state variables, describe the spatial process of hydraulic property state variables with autoregressive state-space models.
  2. Examine the ability of geoelectrical resistivity to add to the spatial description of soil hydraulic properties.
  3. Compare measurements with state-space model estimations and with two different PTF approaches, the ANN approach (Schaap et al., 2001) and a continuous PTF (CPTF) (Wösten, 1997).


   MATERIALS AND METHODS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Site and Sampling Design
The study site is located in Northeast Brandenburg (Germany) approximately 80 km north of Berlin (53° 23' 25'' N, 13° 39' 23'' E). The site was under arable land use and planted to cereal crops. The region of this site is dominated by glacial deposits as the main soil forming substrates. The terrain is hummocky and located within a groundmoraine with underlying glacial till. Soils are sandy loamy in texture and classify as Alfisols and Inceptisols with udic and aquic moisture regimes (i.e., Aquepts located in depressions). The average annual rainfall at the site is approximately 520 mm. Average annual evapotranspiration is 616 mm. Average annual temperature is 8.2°C.

Within the site, along a 500-m-long transect (Fig. 1 ), apparent geoelectrical resistivity was measured every 50 cm during three measurement periods (April 2000, October 2000, March 2001) using a device explained below. The three different measurement dates yielded similar and temporally stable patterns of geoelectrical resistivities. Therefore, only one date (March 2001) was chosen for the analysis presented here. At the same time, gravimetric water content was measured every 5 m at the five soil depths.


Figure 1
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  		Fig. 1. Aerial photograph of the transect and nests within the investigation site. The contour lines denote elevation isolines. The river Peege leads from southwest to northeast and is surrounded by a lowland with peaty soils. The dark area in the western part is a forest area.

 
To obtain small-scale variability structure of soil hydraulic properties, four nests within the 500-m-long transect were chosen, each being 21 m long. At alternating sampling distances of 4 and 1 m, 6-cm-long undisturbed soil cores (8-cm i.d.) were taken at five depths: 7 to 13, 27 to 33, 47 to 53, 67 to 73, and 87 to 93 cm. Within the nests, samples for soil texture, organic carbon, and gravimetric water content were taken every 1 m, and at the same five depth increments as in the transect. Hence, soil core and textural samples were assumed to represent the same depth increment.

Soil Properties
Soil texture was determined using the Gee and Bauder (1986) method: sand fractions were determined from wet sieving; clay, fine silt, and medium silt fractions were analyzed by the pipette method. The limits between individual textural analyses were chosen according to the German soil classification system (i.e., clay: < 2 µm, fine silt: 2–6.3 µm, medium silt: 6.3–20 µm, coarse silt: 20–63 µm, etc.) For applying these textural data to pedotransfer functions based on the international soil classification scheme (international classification system: specific class limits at 5, 50, and 500 µm; corresponding German classification system at 6.3, 63, and 630 µm, respectively) cumulative frequency distributions of soil texture were considered. To translate the fine soil particle fraction to the international system (<50 µm), the results from the German classification system (<63 µm) were adjusted by a log-linear interpolation between the cumulative textural fractions of <20 and <63 µm. This interpolation in most cases yielded a cumulative fraction of <50 µm approximately 3 g (100 g)–1 lower than the cumulative fraction of particles <63 µm. Soil organic carbon was determined for the upper two depth increments, 0 to 20 and 20 to 40 cm.

Soil water retention curve in the range –10 > h > –500 cm and unsaturated hydraulic conductivity K(h) in the range –20 > h > –500 cm were determined with the modified Wind (1968) method described in Wendroth et al. (1993). The evaporation method provides a set of average heads, h, average water content, {theta}, and average hydraulic conductivity values, K, for each time interval during the experiment. Since values of h cannot be predetermined, K(h) data pairs were interpolated at a double-log-linear basis to obtain a value for K at a specific h. From evaporation experiments, K50 was applied in the spatial analysis, representing K at h = –50 cm. In the procedure used by Wind (1968), the water retention curve is iteratively estimated. For this estimation, the water retention curve was parameterized according to van Genuchten (1980)

Formula 1[1]
where {theta} is the volumetric water content, and {theta}s and {theta}r denote the saturated and residual water content, respectively. The parameters {alpha}, and n determine the shape of the retention curve close to saturation and at the dry end, respectively, and m = 1 – 1/n. After the evaporation experiment, the dry bulk density {rho}b was measured gravimetrically.

Geoelectrical Resistivity
Soil electrical conductivity or resistivity depends on the soil textural and structural characteristics of the material and is particularly sensitive to the water content and fluid characteristics like concentration and mobility of electrolytes (Sheets and Hendrickx, 1995; Banton et al., 1997; Michot et al., 2003). In this study, geoelectrical measurements were taken at the end of the winter (March), when soil moisture variation along the horizontal plane in the nests was considered to be small (CV < 14.7%) compared with clay content variation (CV < 45.8%). Hence, the effect of soil water content on geoelectrical resistivity was considered small compared with the impact of soil textural variation.

Geoelectrical resistivity was measured with a Half-Wenner array method (Peschel, 1967), implemented in a multi-electrode device IMPETUS 12Fs. In our case, 11 electrodes were lined up at a horizontal spacing of 0.5 m, and the second current electrode was offline (Fig. 2 ). The off-set electrode B has to be positioned at a distance from the array 100 times the spacing of the array electrodes (Big-M, 2000); that is, electrode B had to be at a 50-m distance from the array. According to Telford et al. (1996), the location of an electrode at infinity requires that it has very little influence on the rest of the array. With this special case of a pole-dipole array, it is only necessary to move one potential electrode (Telford et al., 1996). Electrical current is first injected to Electrode 1 (A5) and B (Fig. 2). The difference potential is measured between Electrode 6 (M) and 11 (N5) realizing the Half-Wenner AMN configuration for the greatest soil depth with a spatial distance between electrodes being 2.5 m. The electrodes are then commutated to measure Half-Wenner NMA by injecting the current to Electrode N5. Hence, for the NMA array, the functional roles of Electrodes A5 and N5 are substituted by the commutation. Half-Wenner AMN and NMA are averaged for focused imaging (Kampke, 1999) and result in the apparent resistivity at depth 5 (2.5 m) for the first profile. The next step is to inject the current between Electrode 2 (A4) and B to measure the difference potential between M and N4 (Fig. 2) and to commutate the electrodes again for averaging the apparent resistivity for Soil Depth 4 (2 m). This procedure is repeated for the remaining three soil depths. While measuring the resistivities for the upper soil depths, Electrode 1 is switched to the end of the cable and fixed to the next electrode contact. The resistivity measurements for the next profile are started immediately after finishing the measurements for Soil Depth 1 (0.5 m) of Profile 1 (P1). The configuration is "rolling" along the transect. A comparison of the Half-Wenner array with other possible electrode configurations is discussed in Kampke (1999). One of the conclusions in that article is that, "the half-Wenner arrays are a good choice. They achieved the best results among the examined methods of focused imaging in the quality of reconstruction the anomaly in shape and resistivity contrast." Other studies have shown that the pole-dipole array helped overcome difficulties associated with the use of other arrays (Barker, 1981).


Figure 2
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  		Fig. 2. Schematic of the Half-Wenner Array for the measurement of geoelectrical resistivity.

 
The resistivity is measured for a volume roughly the dimensions of the electrode separation. If the soil were homogeneous, the measured apparent resistivity would be equal to the true resistivity and would be constant for any current and electrode arrangement. However, since the soil is layered and inhomogeneous, the apparent and the true resistivity differ and vary. For the measured apparent resistivity, data processing is required to reconstruct the distribution of true resistivities for different layers and positions. This procedure is known as inversion; different techniques for solving the inverse problem are known. The true resistivity can be used for comparison with other soil properties, determined for a distinct volume of soil. In this study, an inversion procedure was performed based on a Simultaneous Iterative Reconstruction Technique (Weller et al., 1996; Olayinka and Weller, 1997; Kampke, 1999).

Statistical Analysis
Variogram and Cross-Variogram Analysis
To quantify the range of spatial representativity of measurements, the spatial variability structure of the different variables was quantified with variogram analysis for each of the five layers individually. The result should indicate whether a variable was determined with sufficient accuracy (i.e., representing a local domain of influence), or if the analytical error exceeded the signal that would result in a pure nugget effect. The experimental variogram {gamma}(l) is calculated according to Journel and Huijbregts (1978):

Formula 2[2]
where the difference between variable Ai at location xi and the same variable being separated by lag distance l from that location (xi + l) is calculated. In this study, experimental semivariograms were calculated for properties measured at individual depths but not across depths.

As for univariate data sets, the variability structure between two regionalized variables Ai(xi) and Bi(xi) is quantified similarly using covariogram or cross-variogram analysis with Yates and Warrick (2002) and Nielsen and Wendroth (2003):

Formula 3[3]
where the cross-variance {Gamma}(l) changes with l until a plateau is reached. Depending on the sign of the correlation between two variables, the cross-variogram can be positive or negative. The shape of the cross-variogram indicates up to what spatial distance the two variables are spatially related. Together with the semivariogram, the cross-variogram is a key function for spatial coregionalization using co-kriging (Yates and Warrick, 2002).

State-Space Analysis
In state-space analysis, a variable vector Z at location i including p variables is described as a function of the same vector at the previous location i – 1. The relation between the two states is manifested in a p x p transition matrix {Phi}. State-space analysis is applied here as a first-order autoregressive model with the state equation (Morkoc et al., 1985):

Formula 3[4]
with {omega}i being zero-mean uncorrelated model error with covariance matrix Q. Since the true state of the system cannot be observed and the vector of observations {Upsilon}i always includes some error, the state equation is embedded in an observation equation (Shumway, 1988):

Formula 5[5]
through a measurement matrix Mi. The observation equation includes the zero-mean uncorrelated measurement error {upsilon}i with covariance matrix R. Both the state and observation equations are incorporated in the Kalman filter (Kalman, 1960), in which predictions are updated whenever an observation is available. In this analysis, a matrix of autoregression coefficients ({Phi}) that reflects the spatial coincidence of different variables observed along the transect is estimated with the EM algorithm (Shumway and Stoffer, 1982). The EM algorithm is a maximum likelihood–based optimization procedure for the estimation of transition coefficients, initial mean, spectral density, and measurement variance. The estimation is used in conjunction with the Kalman filter (i.e., updating and smoothing are incorporated in the parameter estimation). For further details, see Nielsen and Wendroth (2003).

In this study, the observation vector in the four nests consisted of observations of soil texture and geoelectrical resistivity taken every meter. Because of the alternating sampling distance of 1 and 4 m for soil cores that were taken for hydraulic property determination, two locations with known physical properties were followed by three locations without an observation of physical properties. Hence, updating the prediction in the stochastic filter occurred according to the sampling of hydraulic properties in an alternating pattern of 1 and 4 m, respectively. Depending on the impact of underlying processes, the confidence limit of estimation is expected to increase when no observations exist until the next available observation, where the prediction and its variance is updated (Katul et al., 1993; Nielsen and Wendroth, 2003).

Before the estimation of transfer coefficient matrices, the observed variables xi were normalized (xt) with respect to their mean x and variance {sigma}x according to Nielsen and Wendroth (2003):

Formula 6[6]
yielding normalized series with x = 0.5 and {sigma}x = 0.25. For the presentation of estimated results in comparison to results of pedotransfer functions, the estimations were back-transformed based on the original mean and standard deviation.

Pedotransfer Functions
Soil hydraulic property parameters {alpha}, n, (Eq. [1]), and K50 were estimated with two pedotransfer function approaches. One was a hierarchical neural network PTF implemented in the ROSETTA computer program (Schaap et al., 2001). The program is available through public domain. Soil hydraulic property parameters were calculated from sand, silt, and clay contents, and dry bulk density. Estimates from the neural network approach are denoted {alpha}ANN and nANN, and were obtained directly. The value for K50ANN was calculated from the resulting parameters and the following equation (Wösten and van Genuchten, 1988):

Formula 7[7]
for h = –50 cm. In this equation, K0 is not the saturated hydraulic conductivity, but the conductivity at a matching point, and {ell} is a fitted exponent.

The second approach applied in this study was a set of continuous pedotransfer functions (CPTF) derived by Wösten (1997). The equations for {alpha}, n, l, and Ks are:

Formula 8[8]

Formula 8

Formula 8

Formula 9[9]

Formula 9

Formula 9

Formula 10[10]
and

Formula 11[11]
For the CPTF, Ks is considered the hydraulic conductivity at saturation equal to the one at the matching point. The transforms denoted by "*" of the coefficients are {alpha}* = ln({alpha}), n* = ln(n – 1), {ell}* = ln[({ell} + 2)/(2 – {ell})], and Ks* = ln(Ks) (Wösten, 1997). The median of the sand particle fractions is denoted by M50. In Eq. [9], [10], and [11], the units are as follows (Wösten, 1997): SOM, Clay, and Silt: percent; {rho}b: grams per cubic centimeter; M50: micrometers. Estimates obtained from the continuous pedotransfer function (Wösten, 1997) are denoted {alpha}CPTF, nCPTF, and K50CPTF, respectively.


   RESULTS AND DISCUSSION
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Spatial Processes
Some examples of variable combinations and their spatial processes within the four nests are displayed in Fig. 3 through 9. For the 0- to 20-cm soil layer, the process of the {alpha} coefficient is plotted versus that of sand content in Fig. 3. The results obtained for {alpha} were noisier in Nest 2 than in the other nests. Except for Nest 3, there tended to be an inverse relation between {alpha} and sand content, which corresponds to results of Minasny et al. (2004) for a limited data range. There was also a general tendency for an inverse relation between {rho}el and clay at the same soil depth (Fig. 4 ). There was a tendency for an inverse relation between both variables in Nests 1 and 2. The smooth behavior of the {rho}el data is probably due to the influence of the numerical inversion of apparent geoelectrical resistivity data.


Figure 3
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  		Fig. 3. Spatial process of {alpha} (Eq. [1]) and sand content at the 0- to 20-cm depth increment within the four nests of the experimental site. Soil physical measurements yielding values for {alpha} were taken at the 7- to 13-cm depth.

 

Figure 4
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  		Fig. 4. Spatial process of geoelectrical resistivity {rho}el and clay content at the 0- to 20-cm depth increment within the four nests of the experimental site.

 
The process of K50 versus sand content in the four nests at the 20- to 40-cm depth is displayed in Fig. 5 , with both variables showing a positive relation. At the same depth, {rho}el and n are also closely related, as shown in Fig. 6 . It should be noted that from the same curve parameter estimation procedure, relatively large spatial fluctuations are obtained in Nest 2 for {alpha} (Fig. 3) but not for n.


Figure 5
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  		Fig. 5. Spatial process of hydraulic conductivity at h = –50 cm (K50) and sand content at the 20- to 40-cm depth increment within the four nests of the experimental site. Soil physical measurements yielding values for K50 were taken at the 27- to 33-cm depth.

 

Figure 6
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  		Fig. 6. Spatial process of geoelectrical resistivity {rho}el and n (Eq. [1]) at the 20- to 40-cm depth increment within the four nests of the experimental site. Soil physical measurements yielding values for n were taken at the 27- to 33-cm depth.

 
The relation between K50 and spring water content GWCf in the 40- to 60-cm layer is illustrated in Fig. 7 . In general, the average level of K50 is low compared with other studies (Wösten et al., 1987; Michiels et al., 1989). This low level of hydraulic conductivity is probably a consequence of the strong soil compaction present in this soil layer and throughout the soil profile, manifested by high values of {rho}b. Even soil temperatures below the freezing point during winter cannot cause considerable loosening of this soil compaction. Blake et al. (1976) observed similar phenomena, although for a Mollisol. The low hydraulic conductivity values are also in agreement with the results of Nielsen et al. (1960) reported for cemented sands, Schindler et al. (1985) for sandy soils in a geographical region close to the one in this study, and Nimmo and Akstin (1988) for compacted sandy soils. There is a drastic impact of even small increases in {rho}b causing considerable decreases in hydraulic conductivity of sandy soils (Miles et al., 1988).


Figure 7
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  		Fig. 7. Spatial process of hydraulic conductivity at h = –50 cm (K50) and spring field water content GWCf at the 40- to 60-cm depth increment within the four nests of the experimental site. Soil physical measurements yielding values for K50 were taken at the 47- to 53-cm depth.

 
The inverse relation between K50 and GWCf is probably physically caused by the impact of textural composition on water retention. With increasing clay content, more water is retained; however, it does not contribute to water transport at h = –50 cm. On the other hand, the relatively sandier spots exhibit increased K50.

For the 60- to 80-cm soil layer, {rho}el and {alpha} are very closely related (Fig. 8 ). Only in Nest 2 does this relation not show up as clearly as in the other three nests. Even fluctuations observed over short distances are reflected in both variables.


Figure 8
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  		Fig. 8. Spatial process of geoelectrical resistivity {rho}el and {alpha} (Eq. [1]) at the 60- to 80-cm depth increment within the four nests of the experimental site. Soil physical measurements yielding values for {alpha} were taken at the 77- to 83-cm depth.

 
Spatial series of {rho}el and clay content in the 80- to 100-cm soil depth are related inversely (Fig. 9 ). Some strong fluctuations in clay content in Nests 1 and 2 are not manifested in the variation of {rho}el.


Figure 9
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  		Fig. 9. Spatial process of geoelectrical resistivity {rho}el and clay content at the 80- to 100-cm depth increment within the four nests of the experimental site.

 
Spatial Variability Structure
In general, the semivariance increases from the shortest sampling distance up to a distance of approximately 20 m (Fig. 10 ). Hence, the latter manifests the range of spatially structured variance. This range corresponds to results of the autocorrelation length of Ks found by Russo and Bresler (1981), to the spatial range of soil hydraulic conductivity function parameters of Jury et al. (1987), to the spatial range of scaling factors for soil hydraulic properties determined by Lascano and Stroosnijder (1993), to the range for {alpha} and n identified by Shouse et al. (1995), and to results on the spatial ranges of {rho}b and soil water content at field capacity (Burden and Selim, 1989). However, ranges determined here were shorter than those obtained for pore size distribution parameters by Ünlü et al. (1990), but longer than those obtained for {alpha} and Ks by Mohanty et al. (1994) and Mallants et al. (1996), and for K close to soil water saturation by Logsdon and Jaynes (1996).


Figure 10
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  		Fig. 10. Semivariograms and cross variograms for selected variables at various soil depths within the four nests of the experimental site. Respective semivariance units are original units squared, and cross-variance units are the product of original units of both variables involved.

 
At lag distances l > 20 m, the variance is not structured anymore. For spatial sampling domains exhibiting a similar variance behavior, mapping of physical properties is prohibitive if based on a measurement scheme that is coarser than 20 m. For most variables shown in Fig. 10, the semivariance even reaches a peak around 20 m that is larger than the variance plateau in the range of random variability (i.e., at l > 20 m). There is not sufficient information to interpret this shape of the experimental variogram as a hole effect (Webster and Oliver, 2001). Especially the semivariograms of {rho}el at the different soil depths exhibit low variances at the shortest sampling distance (Fig. 10). This was expected as the original {rho}el data series fluctuate very smoothly (Fig. 4, 6, 8, and 9). In general, the spatial structure of variables presented here can be considered strongly to moderately structured if the criteria of Cambardella et al. (1994) for the magnitude of the nugget/sill ratio are applied (results not shown).

The cross variograms displayed in the right column of plots in Fig. 10 manifest the spatial covariance structure between the variable pairs. As for the individual variables, the experimental cross variograms {Gamma}(l) exhibit a range of spatial association of approximately 20 m. The resulting variograms and cross variograms in Fig. 10 would strongly support a spatial coregionalization such as co-kriging. For that interpolation technique, variogram and cross variogram models with identical structural components would need to be fitted to the experimental variograms and cross variograms, respectively (Deutsch and Journel, 1992; Yates and Warrick, 2002).

Spatial Description
In this section on spatial modeling analysis, first-order autoregressive state-space models are evaluated with respect to their representation of the spatial processes of soil hydraulic property-related state variables. Moreover, comparing measured and state-space modeled results with the results of the hierarchical (ANN, Schaap et al., 2001) and the continuous pedotransfer function (CPTF, Wösten, 1997) should reveal whether the average level and the spatial process of physical coefficients and parameters were conserved in results obtained from those PTFs. The latter are based on regression with soil textural, organic matter (CPTF), and dry bulk density information (ANN), but do not account for sampling coordinates and spatial covariance structure between observations.

With the exception of extreme fluctuations, the spatial process of {alpha} and its confidence interval is estimated with sufficient accuracy in the upper soil layer (0–20 cm) based on {rho}el and sand content (Fig. 11 ). However, at this depth, the autoregression coefficients for {rho}el and for Sand remain fairly small and indicate the lower contribution of {rho}el to the estimation of {alpha} compared with {alpha} at the previous location (Fig. 11). The overall average of {alpha} is reflected better by the ANN than the CPTF. The ANN appears to slightly overpredict {alpha}, whereas CPTF underpredicts it. Both pedotransfer approaches do not reflect the spatial variation within the four nests but rather remain on an average level, with ANN being higher than that of CPTF (Fig. 11).


Figure 11
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  		Fig. 11. Autoregressive state-space (95% confidence intervals), neural network (ANN), and pedotransfer (CPTF) estimation of {alpha} at the 0- to 20-cm depth increment within the four nests of the experimental site. Soil physical measurements yielding values for {alpha} were taken at the 7- to 13-cm depth.

 
The same variables are employed to estimate the spatial process of n in the 20- to 40-cm layer (Fig. 12 ). Considering spatial relations between n, Sand, and {rho}el and their spatial covariance structure yields a promising representation of the spatial fluctuations (Fig. 12). From the state-space model equation displayed in Fig. 12, the large impact of {rho}el compared with sand content and n at the previous location becomes obvious. The estimates of n obtained by ANN and CPTF are very close to each other (Fig. 11). Spatial variation in the data is just intimated by the PTFs. Nevertheless, the variation is not very pronounced. Again, the average level of n is estimated with reasonable accuracy.


Figure 12
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  		Fig. 12. Autoregressive state-space (95% confidence intervals), neural network (ANN) and pedotransfer (CPTF) estimation of n (Eq. [1]) at the 20- to 40-cm depth increment within the four nests of the experimental site. Soil physical measurements yielding values for n were taken at the 27- to 33-cm depth.

 
For the same depth, K50 is estimated based on Sand and {rho}el (Fig. 13 ). State-space estimates reflect the measurements except for some observed large fluctuations. As in the previous example, {rho}el plays a very important role for the estimation of K50, similar to that of Sand content as can be seen from the transition equation (Fig. 13). Both PTFs overestimate the measured K50, with CPTF being stronger than ANN (Fig. 13). This result may be due to the fact that the measured K50 is extremely small compared with results from sandy soils that were included in the data bases for the derivation of ANN and CPTF. Information from sandy soils underlying CPTF was preferentially obtained from northwest European soils with different formation factors and apparently larger hydraulic conductivity than those at the experimental site of the present study in northeast Germany. However, the variation in soil properties used in the PTFs did not yield substantial spatial variations of the result. Moreover, the level of K50 estimated with both PTFs depends on the matching of the relative hydraulic conductivity curve at saturation (Alexander and Skaggs, 1987).


Figure 13
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  		Fig. 13. Autoregressive state-space (95% confidence intervals), neural network (ANN), and pedotransfer (CPTF) estimation of hydraulic conductivity at h = –50 cm (K50) at the 20- to 40-cm depth increment within the four nests of the experimental site. Soil physical measurements yielding values for K50 were taken at the 27- to 33-cm depth.

 
For the 40- to 60-cm soil depth increment, K50 has a similar magnitude as in the overlying layer (Fig. 13 and 14 ), but still lower than results reported from other studies. Neither the average level nor the spatial variation of the process is reflected in the estimate based on ANN (Fig. 13). In another study, Gradwell (1979) showed a lack of reflection of measured hydraulic conductivity in calculated K values, although there was considerable soil variation. Since Jarvis and Messing (1995) observed overestimates of K, if the relative K function was matched at saturation, the average level of K estimation could perhaps be improved by matching under unsaturated conditions (Carvallo et al., 1976; Ehlers, 1976).


Figure 14
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  		Fig. 14. Autoregressive state-space (95% confidence intervals), neural network (ANN), and pedotransfer (CPTF) estimation of hydraulic conductivity at h = –50 cm (K50) at the 40- to 60-cm depth increment within the four nests of the experimental site. Soil physical measurements yielding values for K50 were taken at the 47- to 53-cm depth.

 
At the 60- to 80-cm soil depth (Fig. 15 ) the impact of {rho}el on {alpha} is clearly larger compared to the 0–20 cm-depth (Fig. 11). In this layer, although generally at a higher level than measured values, the estimation based on ANN yields slight spatial fluctuations, but not as pronounced as measurements and state-space results (Fig. 15). The estimation based on CPTF reflects well the average level of {alpha}, but does not exhibit a strong spatial fluctuation.


Figure 15
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  		Fig. 15. Autoregressive state-space (95% confidence intervals), neural network (ANN), and pedotransfer (CPTF) estimation of {alpha} (Eq. [1]) at the 60- to 80-cm depth increment within the four nests of the experimental site. Soil physical measurements yielding values for {alpha} were taken at the 67- to 73-cm depth.

 
Depending on the spatial correlation between various variables, the confidence of the state-space estimation is either narrow or wide. Estimations for K50 and n (Fig. 12, 13, and 14) yield relatively narrower 95% confidence bands than those of {alpha} (Fig. 11 and 15). The close spatial relation between the different variables becomes obvious from the fact that confidence bands do not increase substantially in those spatial zones, where no observations of the variable of interest were available except for the underlying variables (Morkoc et al., 1985). An exception to this stable confidence band may exist in the case of {alpha} estimated at the 60- to 80-cm soil depth.


   CONCLUSIONS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
In this study, spatial statistical analysis of soil hydraulic and physical properties and their description in state-space models were compared with PTF approaches that ignore spatial correlations between variables. The PTFs provided reasonable average estimates that can certainly be improved by further site-specific calibration. However, the PTFs did not provide sufficient flexibility to describe the spatial fluctuations observed in the field. On the other hand, in this study we presented an alternative, combining spatially nested measurements of different state variables, and how they can be incorporated in a simple state-space approach to estimate the spatial distribution of soil physical properties or parameters of soil hydraulic functions. Considering spatial locations and gradual changes of soil physical properties measured in a nested design proved to better support spatial estimations of ecologically relevant variables than PTFs because the PTF concept does not include spatial covariance structure. Therefore, PTFs are insufficient to describe the spatial pattern of relevant soil hydraulic functions at the field scale of heterogeneous landscapes like the one investigated in this study.


   ACKNOWLEDGMENTS
 
The authors acknowledge financial support of this research project by the German Research Foundation (DFG). The excellent technical support by Norbert Wypler, Evelyn Strehmann, Margret Scholz, Renata Hypscher, Anita Griegoleit, and Dagmar Schulz is greatly appreciated. The authors thank Dr. A.D. Karathanasis for his helpful comments on this manuscript.


   REFERENCES
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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