Corresponding author: J. K. Koestel, Department of Soil and Environment, Swedish University of Agricultural Sciences (SLU), PO Box 7014, 750 07 Uppsala, Sweden. (firstname.lastname@example.org)
 It is known that solute transport through soil is heterogeneous at all spatial scales. However, little data are available to allow quantification of these heterogeneities at the field scale or larger. In this study, we investigated the spatial patterns of soil properties, hydrologic state variables, and tracer breakthrough curves (BTCs) at the field scale for the inert solute transport under a steady-state irrigation rate which produced near-saturated conditions. Sixty-five undisturbed soil columns approximately 20 cm in height and diameter were sampled from the loamy topsoil of an agricultural field site in Silstrup (Denmark) at a sampling distance of approximately 15 m (with a few exceptions), covering an area of approximately 1 ha (60 m × 165 m). For 64 of the 65 investigated soil columns, we observed BTC shapes indicating a strong preferential transport. The strength of preferential transport was positively correlated with the bulk density and the degree of water saturation. The latter suggests that preferential macropore transport was the dominating transport process. Increased bulk densities were presumably related with a decrease in near-saturated hydraulic conductivities and as a consequence to larger water saturation and the activation of larger macropores. Our study provides further evidence that it should be possible to estimate solute transport properties from soil properties such as soil texture or bulk density. We also demonstrated that estimation approaches established for the column scale have to be upscaled when applied to the field scale or larger.
 Downward solute transport through soil is generally heterogeneous. It mostly takes place along preferential flow paths instead of homogeneously distributed through the whole soil pore space. Ever-increasing evidence has accumulated in recent decades that this is true at all spatial scales from the pore scale [e.g., Carminati et al., 2008], over the Darcy or column scale [e.g., Bloem et al., 2009; Kasteel et al., 2005; Schotanus et al., 2012], the plot scale [e.g., Bouma and Dekker, 1978; Flury et al., 1994], and the field scale [e.g., Bronswijk et al., 1995; Butters et al., 1989; Wild and Babiker, 1976] up to the regional scale [e.g., Kurunc et al., 2011; Shaffer et al., 1995; Iversen et al., 2011] and larger. It is crucial to take the spatial heterogeneity at all scales into account for obtaining quantitative predictions of water flow and solute transport through soils, which are predominantly needed at the field scale or larger, namely, for managing environmental and agricultural resources. However, the direct measurement of soil solute transport properties is time consuming and cumbersome. Due to time and budget limitations they are almost exclusively restricted to small scales (Darcy scale and smaller). It is not possible to map solute transport properties at larger scales (regional scale and larger) by direct measurements.
 The heterogeneities in solute transport fronts or plumes are subject to the soils' pore structure, namely, the abundance, distribution, and connectivity of pores of different sizes [e.g., Vogel, 2000]. Meanwhile, it is possible to create detailed 3-D images pore spaces of undisturbed soil by means of X-ray tomography, providing resolutions in the micrometer scale [e.g., Schlueter et al., 2011; Wang et al., 2012]. With the help of X-ray tomography and similar noninvasive methods as neutron beam scattering [e.g., Carminati et al., 2008] or positron emission tomography [e.g., Boutchko et al., 2012], it is to expect that relationships between the soil pore-network geometry and the water flow and solute transport will be quantifiable within the near future. However, high-resolution imaging methods like X-ray tomography are to our knowledge restricted to sample sizes of a few decimeters and are presently not applicable in the field. It is therefore neither possible to deduce solute transport properties from detailed pore-network images at large scales. Alternative methods are needed to infer to them.
 A way out may be offered by using proxy variables like soil texture, bulk density, or land use to estimate soil solute transport properties. The main assumption of this approach is that specific soil pore structures, and therefore also solute transport properties, develop not at random but are conditional to soil properties, land use, climatic conditions, etc., in other words the proxy variables stated above. Similar approaches have been referred to as pedotransfer functions since the late 1980s [Bouma, 1989; Wösten et al., 2001]. They have been widely applied with reasonable success for estimating bulk densities [e.g., Rawls, 1983; Tranter et al., 2007] or soil hydraulic properties [e.g., Schaap et al., 1998; Vereecken et al., 1989].
 The potential of pedotransfer functions for soil solute transport properties have been much less researched. This is not surprising since solute transport depends on the hydraulic conductivity which proved to be difficult to estimate from proxy variables [Vereecken et al., 2010]. The corresponding studies available in peer-reviewed literature therefore focus on relatively simple experimental conditions. These are predominantly experiments on small soil columns (Darcy scale) with an inert tracer and steady-state hydraulic initial and boundary conditions. Table 1 contains a nonexhaustive list of respective publications. It can be seen that foremost the clay content was found to be related to solute transport properties. In addition, soil aggregate or macrostructure [Vervoort et al., 1999; Shaw et al., 2000; Vanderborght et al., 2001], flow rate [Vanderborght et al., 2001; Koestel et al., 2012], and scale of experiment [Vanderborght and Vereecken, 2007; Koestel et al., 2012] were always found to be important when investigated. Besides soil bulk density, organic carbon content and soil texture in general were reported to be important (Table 1), whereas it is to note that the hydropedological proxies used by Jarvis et al. [2009, 2012] include all the three. For two of the latter three soil properties, contrasting results have been published stating that bulk density and organic carbon content appeared to rather have a subordinate influence on solute transport [Koestel et al., 2012].
Table 1. A Nonexhaustive Overview for Literature Reporting Correlations of Soil Properties, Site Factors, and Experimental Conditions With Solute Transport Properties
Type of Samples
Number of Samples
Transport Distance (cm)
Lateral Scale (cm)
Most Prominent Observed Relationship
World Reference Base for Soil Resources (WRB) classification.
Approximately 110 samples were also included in Koestel et al. .
Contains three samples from Shaw et al.  and six samples from Bedmar et al. .
Contains all samples from Jarvis et al. , approximately 110 samples from Vanderborght and Vereecken , and samples from Goncalves et al. , Poulsen et al. , and Vervoort et al. .
Relative 5%-arrival time with clay content and water flux
 It is reasonable for establishing relationships between soil properties and solute transport properties to focus on small scales since the small-scale relationships are very likely fundamental to corresponding relationships at larger scales [Vogel and Roth, 2003]. It must however not be forgotten that properties that are valid on small scales need to be scaled up if they are to be generalized for larger scales [e.g., Kolenbrander, 1970; Mallants et al., 1996]. Therefore, studies relating local solute transport properties to larger areas are likewise important. However, such studies are very scarce. We are only aware of two that investigate flux concentrations. One of them was undertaken by Lennartz et al. . They conducted bromide leaching experiments under steady-state irrigation rates of approximately 0.033 cm h−1 on the undisturbed soil columns collected from two fields near Kiel (Germany) from a 15 m × 15 m grid (N1=24 and N2=36) with the average clay contents of 10%. They found that 25% of all columns exhibited preferential flow taking the so-called “mobility index” with values smaller than 0.35 as an indicator. The mobility index MI was obtained by dividing the piston-flow velocity by the velocity obtained from fitting the convection-dispersion equation (CDE) [Glueckauf et al., 1949; Nielsen and Biggar, 1962] to breakthrough curves (BTCs). The spatial distribution of MI over each of the fields was not random, and regions with preferential flow were clustered. The spatial correlation length appeared to be less than the sampling distance of 15 m. The histogram of MI was for both investigated field sites' bimodal with one peak around 0.6 (denoted as nonpreferential flow) and another peak around 0.15 (denoted as preferential flow); the distribution of the dispersivity was highly right-skewed with the mean values of 2.67 and 1.05 cm for the two investigated fields, respectively, and coefficients of variation of approximately 100% for both fields.
 Another similar experiment was published by de Jonge et al.  and Poulsen et al. . They investigated 42 undisturbed soil columns taken from a field in Røgen (Denmark) with an average clay content of 13.4% with a sampling distance of 5 m. Bromide was applied under steady-state irrigation of 1 cm h−1. Poulsen et al.  did not calculate the mobility index MI. However, they fitted the mobile-immobile model [Coats and Smith, 1964; van Genuchten and Wierenga, 1976] which provides a similar parameter, namely, the ratio between the mobile and the total water content which is often denoted as β. By multiplying β with q q−1 where q (cm h−1) is the water flow rate, it can be seen that β bears formal similarity to MI. Poulsen et al.  found that the bromide transport velocity (derived from fitting the mobile-immobile transport model to the BTCs) as well as β were proportional to the clay content and the bulk density which were also mutually correlated. In addition, de Jonge et al.  inferred from colloid transport properties that preferential macropore transport was only dominant in soil samples with clay contents larger than 13%.
 In this paper, we present data on the spatial distribution of local soil solute transport properties over an approximately 1 ha large field site, which will contribute toward the long-term goal of establishing pedotransfer functions for the solute transport properties of soil. Nonparametric measures describing the shape of tracer BTCs under constant rate irrigation were contrasted and compared with soil texture, organic carbon content, and bulk density. Besides spatial correlations, we also investigated the upscaled solute transport properties of the entire field.
2. Materials and Methods
2.1. Site and Soil Sampling
 Our experiments were carried out on soil sampled from a field site in Silstrup, northern Denmark (56°55′56.16″N, 8°38′43.91″E). Two soil profiles have been described at this site, one of which was classified as an Alfic Argiudoll and the other as a Typic Hapludoll, according to the U.S. Department of Agriculture (USDA) soil classification system [Lindhardt et al., 2001; Kjær et al., 2007]. The soil texture varies between the silty loam and the loam (USDA texture classification). Below the plowed topsoil, the upper meter of the subsoil is heavily fractured and bioturbated and contains 100–1000 biopores m−2. The field had been plowed in November 2008 to a depth of 23 cm and harrowed twice to a depth of 5 cm in March 2009. On 4 April 2009, slurry had been injected to a depth of 10 cm and to a depth of 4–5 cm in September 2009. In October 2010, 65 undisturbed columns (height 17.2–20 cm, diameter 20 cm) were collected from the topsoil. At the time of sampling the field was cultivated with red fescue (Festuca rubra L.). The majority of the sampling locations were arranged in an equidistant grid with a spacing of 15 m covering an area of approximately 1 ha. The exact sampling locations are shown in Figure 1. The sampling points in the grid were placed either at the grid intersects or 1 m away from the grid intersects in random directions, avoiding wheel tracks. This was also done because the sampling grid was aligned with the plow direction and plowing is expected to cause periodical bias [Koestel et al., 2009; Petersen et al., 1997] in soil properties that we sought to avoid. The columns were pushed into the topsoil with a hydraulic press from a tractor until the edge of the column was level with the soil surface. The columns were carefully excavated by hand, trimmed and sealed with plastic caps, and brought to the laboratory. The texture of bulk soil samples was determined according to Gee and Or  by a combined sieve and hydrometer method. Eight texture fractions were determined (Table 2). In addition, the silt and sand fractions were lumped into superclasses (Utot and Stot) whose grain-size boundaries correspond to the ones defined by the USDA. The organic carbon content OC was determined on a LECO analyzer coupled with an infrared CO2 detector (Thermo Fisher Scientific Inc., USA). It was recently hypothesized that the ratio between the clay content Ctot and organic carbon content OC is a fundamental control of the structure of the agricultural topsoil and therefore also for the soil solute transport properties [Dexter et al., 2008; de Jonge et al., 2009]. In the following, we refer to it as the n-index nD. We also calculated the organic matter content OM from the organic carbon content OC, assuming a conversion factor of 1.7. The bulk density ρ (g cm−3) was determined from the weights of the columns before and after drying at 105°C. The soil porosity φ (cm3 cm−3) was estimated from the bulk density ρ assuming the average densities of 2.7 and 1.47 g cm−3 for the solid soil mineral and organic phases, respectively. Supplementary information about the Silstrup field site, soil and column handling, and additional analysis carried out during the column leaching experiments can be found in Norgaard et al. .
Table 2. Texture Fractions Investigated in Our Study and the Corresponding Abbreviations and Definitions
Range of Effective Grain Sizes
Very fine sand
Very coarse sand
2.2. Breakthrough Experiments
 Prior to the BTC experiments, the 65 columns were saturated from the bottom using artificial soil water (see Table 3) for approximately 3 days and then drained through a ceramic plate to a matrix potential of −10 cm at the base of the column (approximately 3 days). The columns were then placed on a steel grid with a mesh size of 1 mm. No suction was applied to the lower boundary. The columns were irrigated with artificial rain water (see Table 3) with an intensity of 1 cm h−1 from a rotating irrigation head equipped with 44 needles placed randomly to ensure a homogenous application. No ponding was observed during the onset of the experiments except for column 60, where the ponding height remained minimal and no overflow occurred. The effluent was collected through a funnel leading down to 24 plastic bottles rotating automatically. After steady-state flow had been established, a 10 min pulse of tritium-spiked (on average 957.58 kBeq/L) artificial rainwater was applied. The effluent was collected in the plastic bottles on the turntable at sampling intervals of 10–30 min. The effluent samples were analyzed for their tritium concentration using liquid scintillation (TRI-CARB 2250 CA). After the breakthrough experiments, the volumetric water contents θ (cm3 cm−3) attained during steady-state flow were determined by weighing the wet and air-dried soil columns. The water saturation S (cm cm−3) was defined as the ratio between the volumetric water content θ and the soil porosity φ. Note that the here presented water saturation S is only a proxy for the true water saturation since the water content θ but not the soil porosity φ was explicitly measured.
Table 3. Specifications of the Water Used for Saturating and Irrigating the Soil Columns
Electrical Conductivity (mS cm−1)
Artificial soil solution
Artificial rain water
2.3. Deconvolution of the BTCs
 We first assigned the average time of two consecutive measurements tm−1 and tm to the radioactivity C (kBeq/L) detected at tm:
 This corresponds to approximating the change in tritium concentration between the two measurements as being linear. Next, we inserted virtual measurements with a temporal discretization of dt=0.033 h (2 min). The respective tritium concentrations C were obtained by linear interpolation between each two consecutive measurements. The insertion of the virtual measurements was included to constrain the deconvolution of the BTC which is described in the following.
 We numerically fitted the convolution integral
where C0 (kBeq/L) is the radioactivity of the tracer pulse in the irrigation water, Cin (kBeq/L) is the time series of the radioactivity in the irrigation water which is C0 during the first 10 min of each BTC experiment and zero for the remaining time. Furthermore, t (h) and τ (h) are time variables, and f (h−1) is the transfer function [Jury and Roth, 1990], also denoted as the transport-time probability density function (PDF) in the following. We applied a mixture of two log-normal distribution functions [see Koestel et al., 2011] to estimate f according to
where w1 and w2 are the weighting factors that sum up to one, and g1 (h−1) and g2 (h−1) are the log-normal distribution functions of the form
where μ is the log-normalized mean, σ is the log-normalized standard deviation of g, and k is a correction factor for the tracer mass, which depends not only on the mass balance error but also on the temporal discretization dt.
2.4. Evaluation of the Solute BTCs
 First, we forward modeled the parameter sets obtained from fitting equation (2) to the data with a temporal discretization of 10(−15, −15.05, −15.1,…, 14.95, 15) h. Next, we set all the values of the PDFs f(t) corresponding to an effluent concentration smaller than the detection limit, namely, , to zero. We then numerically derived five nonparametric BTC shape measures from the reconstructed transfer functions [Koestel et al., 2011]. In the following we give a short overview of the investigated shape measures. A more detailed description of how the five shape measures are calculated is given by Koestel et al. . First, we calculated the (nonparametric) average transport velocity v (cm h−1) and the apparent dispersivity λapp (cm). The former is defined as the ratio of the column length L (cm) to , the normalized first moment of f, i.e., the arithmetic mean of the transport time. The latter is calculated as
where μ2 is the second central moment of f, i.e., the variance of the transport time; and the ratio of and is the coefficient of variation of f [Jury and Roth, 1990]. The temporal moments of the PDFs f(t) were obtained by using the trapezoidal rule.
 We also calculated the ratio of the piston-flow velocity to the average transport velocity which is here denoted as η:
where q is the water flow rate, and θ is the volumetric water content, and v (cm h−1) is the average tracer velocity, which is defined by . Note that the definition of the mobility index MI [Lennartz et al., 1997] and piston-flow-to-transport-velocity ratio η is formally identical. What distinguishes η as defined here from MI is the fact that different types of transfer functions were used to fit the data: Lennartz et al.  used the transfer function corresponding to the CDE, whereas we used a double log-normal transfer function.
 The fourth and fifth shape measures investigated in this study are the relative arrival time of the first 5% of the tracer mass p0.05 and the holdback H. As calculating these two shape measures requires a good temporal resolution between we forward-modeled f(t) a second time, this time with a resolution of 0.033 h (2 min).
 The relative 5%-arrival time is derived from the normalized arrival times T,
and the normalized transport-time PDF, denoted as fn,
 The relative arrival time of the first 5% of the tracer p0.05 is calculated from the cumulative distribution function (CDF) Fn which is obtained by integrating fn,
as such that
 The 5% arrival time p0.05 is negatively correlated with the strength of preferential transport [Knudby and Carrera, 2005].
 Another shape measure for the degree of preferential transport is the holdback factor H defined as the volume of tracer-free water which is still in the soil column after the passage of one effective water-filled pore volume. It was introduced by Danckwerts  but has only been infrequently applied in the soil science community by Rose . In contrast to p0.05, the holdback H is positively correlated with the strength of preferential transport. Koestel et al.  found that H resolves strong to very strong preferential transport well but offers less resolution than p0.05 when preferential solute transport is weaker. In this sense, p0.05 and H complement each other, whereas p0.05 has the advantage that it appears to be more robust to the choice of transfer function type used in equation (2) [Koestel et al., 2011].
2.5. Spatial Interpolation
 We interpolated the soil properties, the hydrologic state variables, and the BTC shape measures to facilitate visual recognition of spatial patterns. The sampling distance was approximately 15 m. It follows that the interpolated images only show heterogeneities larger than this distance, i.e., the sampling distance acts as a low-pass filter. We advise the reader against interpreting the data between the sampling points as known. As the sampling grid used in this study is slightly distorted in the X direction, linear or nearest neighbor interpolations lead to interpolations that falsely suggest undulating patterns aligned to the sampling grid distortions. We therefore used natural neighbor interpolation [Sibson, 1981] which resulted in a more isotropic representation of the sampled data. Finally, we calculated semivariograms [Christakos, 2000] for all predictors, hydrologic state variables, and BTC shape measures for estimating the respective correlation lengths.
2.6. Regression Analyses
 We built simple predictive relationships for the five shape measures using linear ridge regressions with the available texture data, i.e., Ctot, Utot, fU, cU, Stot, vfS, fS, mS, cS, and vcS (Table 2), the organic matter content OM, the n-index nD, and the bulk density ρ as predictors. Prior to evaluating the regression functions, the predictors were normalized to their respective means and standard deviation. If the corresponding shape measure was log-normal distributed, the logarithmized values were estimated rather than the original value.
 We first applied a bootstrap approach to select predictor subsets as promising candidates for the ridge regressions. This step was included to identify weak predictors with little contribution to the regression. Excluding the weak predictors greatly shortened and simplified the final regression functions without significantly reducing their performance. For each bootstrap sample nBS we carried out a forward-stepwise predictor selection. In each step we added or dropped the predictor which minimized the Akaike information criterion (AIC) until a local minimum was reached [Hastie et al., 2009]. In this fashion a best-subset regression function was obtained for each bootstrap sample. The performance of each regression function was appraised by the leave-one-out (LOO) error. The 25 regression functions with the smallest LOO errors were selected for ridge regression, thereby implementing a nongreedy subset selection. The bootstrap approach was also used for appraising the importance of each of the 13 predictors. This was quantified as the percentage of all (bootstrap) regression functions for which the predictor was retained.
 Ridge regression is a shrinkage method which trades bias for variance, i.e., the predictive relationships obtained are slightly biased but more robust against outliers, or in other words, overfitting is prevented by imposing a smoothness constraint on the regression [Hastie et al., 2009]. It has the form
where y = (y1, y2, y3,…, ym)T is the vector of the target variable, i.e., either one of v, λapp, p0.05, H, or η, for the m data sets; X is an m × n matrix of the n normalized predictor vectors xn = (x1,n, x2,n, x3,n,…, xm,n)T which contain m data sets corresponding to the size of one bootstrap sample; b = (b1, b2, b3,…, bn)T is the vector of the n regression coefficients; I is the n × n identity matrix; and b0 is the intercept which is set to the mean value of y. For ridge regression the regression coefficients b are calculated as
where α is the penalty term which determines the degree of shrinking applied to the regression. The penalty term α was determined by fivefold cross validation. The final regression functions were found by applying equation (12) to the complete data set with the size m=65, using the 25 best predictor subsets that had been identified during bootstrapping. Out of these 25 regression functions (25 for each investigated shape measure), the one with the smallest AIC was selected. Figure 2 illustrates the above described approach.
2.7. Upscaling of the Column-Scale BTCs to the Field Scale
 We numerically upscaled the column-scale normalized PDFs to the field-scale PDF un(t) by taking the arithmetic mean of all m (=65) data sets:
 As the irrigation rate was constant for all columns and ponding was negligible, weighting of the normalized PDFs by the water flux can be dispensed with. The upscaling approach assumes that each column-scale BTC was representative for the area in the field where the column had been collected. Each area would correspond to approximately 0.015 ha (65−1 ha). It should also be noted that the upscaling approach only approximates the field-scale transport characteristics with respect to the lateral scale. The upscaled PDF is therefore only an estimate for the field-scale solute transport through the topsoil.
 The field-scale shape measures were derived from un(t) as described in section 'Materials and Methods'.'Materials and Methods'. We applied the regression relationships obtained for the local-scale BTC shape measures on the arithmetic mean soil properties to obtain estimates of the field-scale shape measures. The deviations between the shape measures from the upscaled PDF un(t) and the estimated shape measures were discussed.
3. Results and Discussion
3.1. Soil Properties and Hydraulic State Variables
 Figure 3a shows that the Silstrup field site exhibits a slight gradient in the clay content Ctot with a larger clay content (approximately 0.18 g g−1) in the northern half of the investigated area than in the southern half (approximately 0.15 g g−1). The total sand fraction Stot and the total silt fraction Utot are relatively constant between 0.5 and 0.53 g g−1 and 0.29 and 0.33 g g−1, respectively (Figures 3b and 3c). The only major variation in the total sand content Stot is observed in the northwestern area of the field where it varies between 0.45 and 0.55 g g−1 and is negatively correlated with the clay content. For the total silt content Utot there were only two sampling locations in the north of the investigated area where it was slightly smaller with the values of 0.24 g g−1. The bulk density varies between 1.4 and 1.6 g cm−3 (Figure 4a). The regions with lower bulk density are located in the center of the investigated field. The organic matter is distributed relatively homogeneously and varies between 0.029 and 0.037 g g−1 with its minima and maxima neighboring in the northwestern quadrant of the field (Figure 4b). The pattern of the n-index nD follows the distribution of the clay content with larger values in the northern than in the southern half of the field (Figure 4c).
 Figure 5 depicts the distribution of the volumetric water content θ and the degree of saturation S under the experimental conditions during the tracer experiments, i.e., steady-state irrigation of 1 cm h−1. The water content θ varied between 0.38 and 0.43 cm3 cm−3, and its pattern exhibited little resemblance to the soil properties discussed above. The water saturation S, however, closely followed the distribution of the bulk density ρ. All soil columns were close to the water saturation (S>0.84). In approximately half of them the water saturation exceeded 0.93. The water content θ and the water saturation S are shown in Table 4.
Table 4. Volumetric Water Contents θ, Water Saturation S, Transport Velocities v, Apparent Dispersivity λapp, Relative 5%-Arrival Time p0.05, Holdback H, Ratio Between Piston-Flow and Transport Velocity η, Coefficient of Determinations From Fitting the Transfer Functions to the Data, and Mass Balance (MB).
θ (cm3 cm−3)
v (cm h−1)
The predicted field-scale shape measures using the arithmetic mean values of the local-scale soil properties and the regression relationships are listed in Table 5.
 Figure 6 shows the four examples of the BTCs with the respective fits of the transfer function (see equations (1), (2), (4)) as well as the corresponding transport-time PDFs and CDFs. Bimodal BTCs like the one sampled from column 29 were only observed in three cases. The column-scale BTC shape measures are listed in Table 4. The histograms of three of the considered BTC shape measures, namely, the column-scale velocity v, and apparent dispersivity λapp, and the 5% arrival time p0.05, resembled log-normal distributions, whereas the histogram of the piston-flow-to-transport-velocity ratio η and the holdback H were quasi-normally distributed (not shown).
 The spatial patterns of all the five investigated shape measures (Figure 7) followed the spatial distributions of the bulk density ρ (Figure 4) and the water saturation S (Figure 5b). The piston-flow-to-transport-velocity ratio η varied between 0.19 and 1.41. The median of 0.92 indicates that the tracer was accelerated relative to the water flow in the majority of the soil columns. This may be explained by preferential flow which caused bypassing of the tracer of a considerable fraction of the pore space [e.g., Vanderborght and Vereecken, 2007]. The values of η were clearly larger than the values of the mobility index MI reported by Lennartz et al. . However, just the fact that Lennartz et al.  used a different transfer-function type than in our study makes it difficult to relate η and MI. Furthermore, it has to be taken into account that Lennartz et al.  used an anionic tracer which favors bypassing of pore space due to anion exclusion [Thomas and Swoboda, 1970; Rose et al., 2009].
 The occurrence of preferential flow is supported by the relatively large apparent dispersivities λapp (median of 19.28 cm) and also by the 5% arrival times p0.05 and the holdback factors H. Both p0.05 and H indicated a strong preferential transport (Figures 7d and 7e). The 5% arrival time p0.05 in our study had a median value of 0.08, which was clearly below the median p0.05 found for undisturbed soil in the meta-analysis of Koestel et al. , which was approximately 0.25. If a p0.05 of less than 0.25 is chosen as a threshold for preferential transport, then only one column did not exhibit preferential transport (column 73 which yielded a p0.05 of 0.35).
3.3. Correlations Among Soil Properties, Hydraulic State Variables, and BTC Shape Measures
 The observed similarity between the patterns of water saturation S and all investigated BTC shape measures was consistent with large mutual correlation coefficients (Figure 8). The correlations were such that with increasing S, the tracer arrived earlier, the effective water content decreased, and the spread of the BTCs became larger. Figure 8 shows that mainly five soil properties were significantly correlated with the investigated BTC shape measures. These were the bulk density ρ, the fractions of very fine sand vfS and very coarse sand vcS, the organic matter OM, and the n-index nD. The spatial distributions of the two sand fractions are depicted in Figure 9. The bulk density ρ exhibited the strongest correlations with the BTC shape measures with absolute values between 0.44 and 0.67 and was positively correlated with the strength of preferential transport. Notably, ρ was also very strongly correlated with the water saturation S (r=0.82). The very fine sand fraction vfS was only significantly correlated with λapp, p0.05, and H and in contrast negatively correlated with the strength of preferential transport with absolute Spearman rank correlation coefficients between 0.36 and 0.42. Its correlation coefficient with the water saturation was −0.44. The very coarse sand fraction vcS was positively correlated with the transport velocity v (r=0.47) and negatively with the piston-flow-to-transport-velocity ratio η (r=−0.47), i.e., the transport was by trend accelerated in the columns containing larger amounts of very coarse sand. The opposite was observed for the column with larger organic matter contents OM (Figure 8), with Spearman correlation coefficients of −0.41 and 0.4, respectively. Notably, OM was negatively correlated with the very coarse sand fraction vcS with a Spearman correlation coefficient of −0.52. The correlations of the n-index nD with the transport velocity v and the piston-flow-to-transport-velocity ratio η were opposite to the ones found for OM (Figure 8). This is consistent with the fact that no correlation between the clay content Ctot and any shape measure was observed as nD is defined as the ratio between clay content and organic carbon content.
 In conclusion, the water saturation S was under the experimental conditions of steady-state irrigation of 1 cm h−1 strongly correlated with the bulk density ρ. The water saturation S was, in turn, significantly positively correlated with the strength of preferential transport. The relationship among the bulk density ρ, the water saturation S, and the strength of preferential transport portrayed by the holdback H is also illustrated in Figure 10. It can be seen that the spatial distributions of ρ, S, and H roughly matched. An intuitive explanation for the observed correlations is the following: the bulk density was related to reductions of the near-saturated hydraulic conductivity, which lead to larger water saturations under the constant water flow rate of 1 cm h−1 and to the activation of larger macropores and therefore stronger macropore flow and stronger preferential transport characteristics. A definitive proof of this hypothesis would however require 3-D images of the soil pore space or measurements of the near-saturated hydraulic conductivity which both were not at our disposal.
 It is to point out that the meta-analysis of 733 BTC experiments of Koestel et al.  yielded a negative correlation between bulk density ρ and the strength of preferential transport. This is in contrast to our results from the Silstrup field site, which strongly suggests that the relationships at the Silstrup site cannot be generalized in a simple fashion for other soils.
 Some previous studies have suggested that organic matter may reduce preferential transport in a cultivated topsoil [Roulier and Jarvis, 2003; Jarvis, 2007]. By trend, the soil organic matter content was negatively correlated with the transport velocity v and positively correlated with the piston-flow-to-transport-velocity ratio η. Hence, larger organic matter contents supported retardation of tritium. This may be interpreted as a reduction in preferential transport. The same holds for the ratio between clay and organic carbon, nD, which is also thought to exert an important control on the aggregate structure developed under cultivation [de Jonge et al., 2009]. However, our study provides no evidence that nD may be a superior indicator for soil susceptibility to retarded transport than OM. It is to note that neither OM nor nD where significantly correlated with any shape measure related to the spread and earliness of the tracer arrival. This may be due to the fact that the variability of the organic matter content at the Silstrup field site (2.9%–3.7%) was probably too small to lead to significant differences in the respective features of the investigated BTCs. It is moreover a further indication that the piston-flow-to-transport-velocity ratio η carries different information on the solute transport process than λapp, p0.05, and H, as already observed by Koestel et al. .
 The clay content had no directly observable effect on the BTC shapes at Silstrup. However, this is not a contradiction to the empirical notion that the clay content is an important driver of preferential flow and transport. The results of the meta-analysis of Koestel et al.  suggest that the occurrence of preferential transport depends in a threshold-like manner on the clay content. They found a threshold of approximately 8% clay above which preferential transport was possible. Above this threshold, the strength of preferential transport did not depend linearly on the clay content. Similar thresholds were also reported by Quisenberry et al.  and de Jonge et al.  for much smaller data sets, at clay contents of 8% and 13%, respectively. These clay contents roughly coincide with that necessary to form stable aggregates [Horn et al., 1994]. As the clay content at Silstrup (14%–19%) was well above these thresholds, we suggest that the strong preferential transport characteristics which can be attributed to almost all investigated BTCs were in part caused by soil aggregation and in part also by the presence of biopores in combination with the irrigation rate that lead to water contents close to saturation. Notably, column 73, which was the only column for which preferential transport was not observed, still had a clay content of 15.6% but otherwise possessed soil properties that according to the observed correlation acted against the promotion of both near-saturated conditions and preferential flow, namely, a small bulk density ρ of 1.42 g cm−3, a large very fine sand fraction vfS (0.074 g g−1), and a small very coarse sand fraction vcS (0.065 g g−1). It should be noted that the bottom boundary condition which was a seepage face leads to saturated conditions near the outflow. Saturated conditions promote preferential transport in general, and the strong preferential characteristics that were found in our study are to an uncertain extent affected by the choice of the bottom boundary condition.
 We also investigated the spatial correlation of the predictors and BTC shape measures. Only 4 of the 13 investigated predictor variables had a correlation length larger than the sampling distance (Figure 11a). These were the total clay content Ctot, the total sand content Stot, the organic matter content OM, and the very coarse sand content vcS. We found that only the sand content Stot exhibited a correlation length within 15 m (sampling distance) and 100 m (average scale the field). It was approximately 50 m (Figure 11a). The nearly linear variograms of the other three soil properties suggest strong trends of the respective soil properties over the Silstrup field site. Of these, only the vcS was significantly correlated with the BTC shape measures, albeit only moderately. It is therefore not surprising that the majority of the BTC shape measures also show correlation lengths of less than 15 m (Figure 11b). Only the transport velocity v and the piston-flow-to-transport-velocity ratio η had correlation lengths larger than the sampling distance, but they also had a large nugget with a semivariance of >60% (Figure 11b). We conclude that a better spatial resolution of sampling sites (<15 m) would be beneficial to fully characterize the spatial dependence of solute transport properties at the Silstrup field site under near-saturated hydraulic conditions.
3.4. Regression Analysis
 The results of the bootstrap and regression analyses support the conclusions drawn from the correlation between soil properties and BTC shape measures. All regression equations contained the bulk density (Table 5) which was to be expected given its large correlations to all investigated shape measures. Notably, all regression relationships except the one for the piston-flow-to-transport-velocity ratio η include the very fine or the very coarse sand fraction as a predictor (Table 5). This suggests that more detailed information on the soil texture than the three USDA texture fractions clay, silt, and sand may be useful to predict solute transport properties from soil texture. Table 5 illustrates that the ridge-regression approach yielded the coefficients of determination R2 between 0.41 and 0.44 for the apparent dispersivity λapp and the holdback H. Taking into account the relatively small variability of soil texture and organic matter content as well as BTC shape measures in this study, a coefficient of determination of larger than 0.4 is still a fairly good result. The transport velocity v, the relative 5% arrival time p0.05, and the piston-flow-to-transport-velocity ratio η were less well predicted from the investigated soil properties with R2 of 0.31, 0.31, and 0.28, respectively.
Table 5. Regression Relationships Between Soil Properties and the Five Investigated BTC Shape Measures Using Ridge Regressiona
In addition, the respective coefficients of determinations are shown.
 In the following, we used H in our examples which illustrate typical details for the predictor importance and the regression performance found for the investigated BTC shape measures. Figure 12a depicts the frequency with which each of the 13 predictors were selected as a best subset in the 5000 bootstrap samples used for the nongreedy best-subset evaluation. The more frequent a predictor was selected, the greater was its importance. Figure 12a illustrates that the bulk density ρ was by far the most important predictor, followed by the very fine sand fraction vfS. Consequently, regression functions containing only these two variables also proved to perform best in the cross validation. Figure 12b shows how the function in Table 5 predicts the holdback factor, while Figures 12c and 12d illustrate how it reproduced the spatial pattern of H as compared to the measured values. Finally, Figure 12e shows the respective residuals as a percentage of the measured holdback. Due to the smoothness constraint used in the ridge regression, the maximum values were generally underestimated, whereas the minimum values were overestimated.
3.5. Field-Scale BTC
 Figure 13 shows the normalized field-scale PDF un(t) in comparison to the column-scale PDFs . The field-scale BTC shape measures are listed in Table 4. The field-scale transport velocity v is upscaled by the harmonic mean of the local-scale velocities (Table 4). The differences between the harmonic mean of the local-scale velocities and the field-scale velocity was probably caused by numerical errors. The piston-flow-to-transport-velocity ratio η was similar on both scales. The three related shape measures apparent dispersivity λapp, relative 5%-arrival time p0.05, and holdback H indicate that the strength of preferential transport was stronger at the field than at the column scale. Table 4 also shows that using the column-scale-derived regression functions (Table 5) for estimating the field-scale BTC shape measures from the mean soil properties leads to values that are comparable to the median of the column-scale shape measures, an exception being the transport velocities. The field-scale transport velocity would thus be overestimated, whereas the strength of preferential characteristics would be underestimated.
 The field-scale transport-time PDF derived here applies only to the topmost 20 cm of the soil at the Silstrup field site. It may be speculated how the field-scale PDF would appear at a greater soil depth, e.g., 100 cm. We consider two scenarios in the following. First, if the transport velocities in transport direction are uncorrelated, the lateral solute mixing would approximate the convection-dispersion regime [e.g., Roth and Hammel, 1996]. Then, the apparent dispersivity would remain constant, whereas the relative 5%-arrival time would increase and the holdback decrease with transport distance. If however the transport velocities in the transport direction were correlated, the apparent dispersivity would increase [e.g., Flühler et al., 1996]. Examples would be the triggering of additional preferential transport paths at soil horizon boundaries or vertical earthworm channels over the whole soil horizon. The 5% arrival time (and holdback) would then either increase (decrease) to a lesser extent than in the above discussed scenario or remain constant. A mix of both the above described scenarios appears to be likely in the majority of cases [Flühler et al., 1996]. The field-scale solute BTC at 100 cm depth will then exhibit a similar velocity and piston-flow-to-transport-velocity ratio but a larger dispersivity and relative 5%-arrival time and a smaller holdback than at a depth of 20 cm.
 We investigated the tracer BTCs on the soil columns collected from the cultivated topsoil at the Silstrup field site under the near-saturated hydrologic conditions. Of the 65 investigated BTC experiments, 64 showed strong preferential transport features. The correlation between the strength of preferential transport and water saturation suggests that macropore transport was the dominant process. The observed variations in the preferential transport strength were also positively correlated with the bulk density. This may be explained as follows: larger bulk densities presumably led to a decrease in near-saturated hydraulic conductivities and as a consequence the activation of larger macropores. Correlations between the BTC shapes and the organic carbon content or between the ratio between clay and organic carbon content were only observed for the transport velocity and the piston-flow-to-transport-velocity ratio.
 Our study provides further evidence that it is possible to estimate soil solute transport properties from soil properties such as soil texture or bulk density. As a corollary, our results suggest that more detailed information on the soil texture than the three USDA texture fractions clay, silt, and sand may be useful in this context. We demonstrated that various BTC shape measures were relatively well predicted using linear regression approaches involving the bulk density and the very fine sand fraction, even though all BTCs exhibited strong preferential transport and the respective shape measures only showed relatively small variability.
 It must be noted however that the regression functions presented here for the BTC shape measures cannot be simply generalized to other field sites. This is strongly suggested by the metastudy of 733 BTC experiments of Koestel et al.  who found an opposite correlation between bulk density and strength of preferential transport to the one found in this study. An upscaling approach for estimating the field-scale BTC for the topsoil from the column-scale BTCs suggests that the regression relationships between soil properties and column-scale BTCs yield wrong field-scale shape measures if applied to average field-scale soil properties. The transport velocity would be overestimated, and the strength of preferential transport underestimated.
 It is notable that we also found that the spatial correlation length of the investigated BTC shape measures was smaller than the sampling distance of 15 m. In the future, studies aiming to establish predictive relationships for the soil solute transport properties should include data from noninvasive geophysical soil-mapping techniques like EM38 [Saey et al., 2009] or γ-ray spectroscopy [van Egmond et al., 2010] as a complement to traditionally measured soil properties. This would allow a better spatial resolution of the predictor variables and therefore also a better characterization of the variability of the soil solute transport properties. It would furthermore improve conditions for estimating solute transport properties at the field scale.
 This study was funded by the Danish Pesticide Leaching Assessment Program (www.pesticidvarsling.dk) and by the international project “Soil Infrastructure, Interfaces, and Translocation Processes in Inner Space (Soil-it-is)” which is funded by the Danish Research Council for Technology and Production Sciences (http://www.agrsci.dk/soil-it-is/).