Soil Science Society of America Journal 66:753-759 (2002)
© 2002 Soil Science Society of America
DIVISION S-1—SOIL PHYSICS
Spatial and Statistical Similarities of Local Soil Water Fluxes
Bing Cheng Si*
Dep. of Soil Science, University of Saskatchewan, Saskatoon, SK. Canada S7N 5A8
* Corresponding author (bing.si{at}usask.ca)
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ABSTRACT
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Understanding the spatial and statistical distribution of soil water flux in a field is fundamental for stochastic modeling soil water flow and chemical transport in spatially variable soils. The objective of this study was to examine the persistence of the spatial pattern and statistical distribution of local soil water flux for different application rates during constant flux rainfall infiltrations. A series of constant flux-infiltration experiments were conducted in a spatially variable field. The local soil water fluxes for each of the 0- to 0.2-, 0- to 0.4-, 0- to 0.6-, and 0- to 0.8-m depths were determined from the change of water storage as a function of time before the wetting front passes the end of vertically installed time domain reflectometry (TDR) probes. The spatial similarity (persistent spatial pattern) of the measured soil water flux for different application rates at four depths was examined using Spearman rank correlation coefficient. Results showed that there was no persistent spatial similarity among measured soil water fluxes for the 0- to 0.2-, 0- to 0.4-, 0- to 0.6-, and 0- to 0.8-m depths. This indicates that transient infiltration experiments with different application rates have different flow pathways for each of the 0- to 0.2-, 0- to 0.4-, 0- to 0.6-, and 0- to 0.8-m depths. The statistical similarity (persistent statistical distribution) of soil water flux for different application rates was examined using histograms. Chi-square tests indicated that the histograms of soil water flux for different application rates were different for each of the 0- to 0.2-, 0- to 0.4-, 0- to 0.6-, and 0- to 0.8-m depths, suggesting the stochastic convective flow model may not be used to predict flow and transport in this field for different application rates.
Abbreviations: CV, coefficient of variance • TDR, time domain reflectometry
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INTRODUCTION
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EXPERIMENTAL STUDIES indicate water flow in most fields exhibits strong spatial variability (Nielsen et al., 1973). The knowledge of spatial variability of soil properties is important for predicting water and solute transport through spatially variable soils. Characterization of the spatial variability of soil water flux is of particular importance because variability in soil water flux increases travel time variation and enhances the possibility of nutrients leaching out of the root zone and contaminants reaching groundwater.
Many methods have been developed to determine soil water flux. A soil water-flux meter provides direct measurement of soil water flux (Dirksen, 1974). This method is subject to problems including the disruption of soil during installation and interruption of the normal soil water flow pattern (Wagenet, 1986). Soil water flux can be estimated through Darcy's law and the measurement of unsaturated hydraulic conductivity and hydraulic gradient. However, the unsaturated hydraulic conductivity is usually difficult and time-consuming to measure and the measurement of hydraulic gradient is error-prone (Nielsen et al., 1973; Simmons et al., 1979). Parkin et al. (1995) used TDR probes installed vertically at the soil surface beneath a constant-rate rainfall simulator to measure cumulative water storage with time. These authors estimated the local water flux from the slope of water storage versus time during early times before the wetting front reached the bottom of the TDR probes. The method was applied in field experiments and was used to determine the effective unsaturated hydraulic conductivity of the field soils (Parkin et al., 1995; Si et al., 1999). Strong spatial variations in field soil water flux were found even though the water application rate was uniform at the soil surface. As a result, local soil water flux is different from the application rate or the rainfall rate on a soil surface. Si and Kachanoski (unpublished data, 2001) utilized continuous application of a conservative tracer during steady-state solute transport experiments to determine soil water flux. These methods were applied to field experiments and compared well with the transient method of Parkin et al. (1995) and Si et al. (1999). All these authors assumed that flow is one-dimensional.
Generally speaking, one-dimensional flow may only occur in repacked soil columns. Even in relatively uniform soils, Poletika and Jury (1994) indicated that there was a redistribution of soil water below the surface, and that the water fluxes collected at different locations from the bottom of a soil column (0.3 m in height) were different. Butters et al. (1989) found that flow in the top 3 m of the soil can be described by one-dimensional flow models. Parkin et al. (1995) and Si et al. (1999) found that soil water storage increased linearly with time before the wetting front passed the end of the TDR rods, suggesting water flux was constant vertically at a location.
Generally, the one-dimensional flow assumption is justified for soils where the vertical variability is small compared with the horizontal one (Rubin and Or, 1993). Further, in many cases, although variability may exist in the vertical direction, the determination of soil hydraulic properties through field methods such as drainage experiments (Libardi et al., 1980) gives effective parameters, thus eliminating the variability in the vertical direction in the practical sense (Rubin and Or, 1993)
The one-dimensional flow model (convective flow) has been used to estimate the field average and variance of solute concentration during steady-state transport experiments (Dagan and Bresler, 1979) of soil water content during infiltration and drainage (Dagan and Bresler, 1983; Rubin and Or, 1993), solute travel time probability density functions (Jury, 1982; Jury and Roth, 1990), field average solute mass flux (Destouni, 1992; Jury and Scotter, 1994; Toride and Leij, 1996), and wetting front velocity (Young et al., 1999). Destouni (1992) related the probability density function of solute velocity to that of saturated hydraulic conductivity by adopting the theory of Dagan et al. (1992). She assumed that the variability of solute velocity is dominated by the spatial variability of soil saturated hydraulic conductivity. Therefore, for different flow rates, the region of fast flow is always fast and the region of slow flow is always slow. In other words, the spatial flow patterns are always the same for different application rates. Jury (1982) suggested the transfer function approach for modeling transport in heterogeneous soils. Once the breakthrough curve for a spike input of the solute on soil surface is measured at a certain depth, the transfer function of the system for the experimental condition is known and breakthrough curves can be predicted for all other solute boundary and initial conditions for the same water flux boundary condition (Jury and Scotter, 1994). For a different water application rate, however, a separate experiment has to be done to determine its transfer function. Jury (1982) utilized the fluid coordinate system and expressed the transfer function in terms of the cumulative infiltration rate. By doing so, Jury (1982) was able to use the transfer function derived for one set of boundary conditions and initial values to another. However, the success of the approaches of Destouni (1992) and Jury (1982) hinges on one of the following two assumptions: (i) fast flow locations are always fast and slow flow locations always slow for different application rates (Jury and Scotter, 1994). In another word, there is persistent flow pattern of soil water flux; (ii) for different application rates, the fast flow locations at one rate may not have fast flow at another rate; however, the proportion of relatively fast flow locations out of all measurement locations is always the same for different application rates. We refer the first one as spatial similarity and the second as statistical similarity. Few studies on similarity of soil water flow in heterogeneous soils have been conducted.
The objective of this study is to characterize the spatial and statistical similarity of measured local soil water flux for different application rates during constant flux rainfall infiltrations. The spatial similarity was examined using Spearman rank correlation and the statistical similarity was examined using histograms of measured local soil water fluxes in the field.
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MATERIALS AND METHODS
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Description of the Site and Experiments
Field infiltration measurements were conducted in 1995 and 1996 at the Canadian Forces Base Borden, Ontario, Canada. Extensive hydro-geological research, including a large scale, natural-gradient tracer test and forced gradient tests has been conducted by the University of Waterloo on this coarse-textured site (Sudicky, 1986). The spatial layout of the TDR probes and infiltration experiments have been described in detail by Si et al. (1999). Water was applied to a transect (2 m wide by 9 m long, covered by a portable greenhouse). Using a hanging track and nozzle system, the application rate was controlled through a pressure regulator. At the center of the transect, an area (0.3 m wide by 7.5 m long) was selected for TDR probe installation to avoid edge effects. Fifty TDR probes, each consisting of one solid steel rod and a hollow stainless steel rod, with a length of 0.2 m were installed vertically along the side edge of the 0.3 by 7.5 m transect at intervals of 0.15 m. These multipurpose TDR rods can be used to measure soil water content, soil matric potential, and solution concentration (Baumgartner et al, 1994). Similar TDR probes of 0.4-, 0.6-, and 0.8-m length were installed in parallel transects 0.1, 0.2, and 0.3 m away from the array of 0.2-m-long TDR probes, respectively. Thus, four arrays of a total of 200 TDR probes were installed along the 0.3 by 7.5 m2 instrumented area at the center of the 2-m-wide wetting area. Five different water application rates (0.9, 1.5, 2.6, 3.3, and 6.2 cm h-1) were used. The application rates were extremely uniform with coefficient of variance (CV) <3% (Si et al., 1999). After each infiltration experiment, the soil was allowed to drain sufficiently before another infiltration experiment. For all experiments, only the one with the application rate of 6.2 cm h-1 had observable ponding and water redistribution above the soil surface. Thus, this experiment was discarded for further analysis. Soil water content was measured using the TDR method of Topp et al. (1980). The readings were taken manually from the display screen of two precalibrated Tektronix (1502 B) metallic cable testers (Tektronix, Wilsonville, OR) by four operators. The readings were taken just prior to the start of water application. Four minutes were needed to scan 50 TDR probes using the two cable testers. When the wetting front was passing through a certain depth, the readings were taken every 5 to 10 min. Otherwise, the readings were taken every 20 to 40 min. depending on infiltration rate and rate of change of volumetric water content, for the 200 multipurpose TDR probes.
Determination of Soil Water Flux from Transient Infiltration Experiments
Local soil water flux during the transient phase of constant flux infiltration was determined using the method of Parkin et al. (1995) and Si et al. (1999). The cumulative storage of water (m3 m-2) to depth L, W(L,t), is measured by vertically installed TDR probes and is given by
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where is the average water content (m3 m-3) over the probe length, L (m). Before the wetting front reaches L, the derivative of cumulative storage of water measured by TDR with respect to time should approximately equal the local water flux at the soil surface, q (Parkin et al., 1995; Si et al., 1999). Assuming conservation of mass, one-dimensional flow, and that the applied water has not yet reached depth L, then
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Si et al. (1999) found that calculated soil water flux accounted for 90 to 110% of application rate (Si et al., 1999). Therefore, lateral water flow is not considered as significant for this study.
Spearman Correlation Coefficient
Vachaud et al. (1985) used the Spearman rank correlation coefficient to assess the temporal stability of soil water content in two fields. In this study, we use the Spearman rank correlation coefficient to identify the spatial similarity of the spatial pattern of soil water flux. Let Ni,j be the rank of the local soil water fluxes at two different rates at observation location i for i = 1,2,...n (n = 50, the number of the observations) for application rate j. The Spearman rank correlation coefficient is calculated by:
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The Spearman rank correlation coefficients were calculated using the SAS procedure CORR (SAS Institute, 1994). In this study, we assume that consistent spatial patterns exist only when the Spearman rank correlation coefficient between soil water fluxes at two different application rates is larger than 0.8.
Histograms of Local Soil Water Flux
The relationships between soil water flux and the saturated hydraulic conductivity can be described by the Brooks and Corey (1965), van Genuchten (1980), or Broadbridge and White (1988) equations. For all these equations, the relationship between soil water flux and the saturated hydraulic conductivity at unit gradient (or gravity dominant flow) is given as
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where Ks is the saturated hydraulic conductivity (cm h-1) and 
is a function of soil water content and other hydraulic parameters.
In the following, the frequency distribution of soil water flux will be derived from that of Ks. For simplicity, we assume that the variation in soil water content and other shape parameters is secondary to that of Ks. Many studies (Dagan and Bresler, 1983; Destourni, 1992) have showed that this assumption is valid. Therefore, can be treated as a constant for an application rate. For a known probability density functions (pdf) of Ks, fKs(Ks), the pdf of q, fq(q), is
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To facilitate comparison between histograms of the soil water flux at different application rates, the pdf of soil water flux needs to be standardized. Assume that the measured soil water-flux values are between maximum and minimum at 95% confidence level or the maximum and minimum are approximately two standard deviations away from the mean. Thus,
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where 
, Ksmaxand Ksmin are standard deviation, maximum, and minimum of the saturated hydraulic conductivity, respectively. Then, utilizing Eq. (5), we have
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where K s and q are the means of saturated hydraulic conductivity and local soil-water flux, and qmax and qmin are the maximum and minimum of local soil water flux. The variable, , is the standard deviation of Ks. Therefore, plots of frequencies vs.
for different application rates should have the same shape and peak location. Note that is cancelled out in Eq. [7]. Even though values of are different for different application rates, the relationships between the standardized frequency distribution of Ks and q do not depend on . Fifteen equal intervals were obtained by dividing qmax - qmin by 15 and intervals were ranked in ascending order and numbered consecutively from 1 to 15 as bin number. The frequency was counted as the number of soil water flux measurements that fall in each interval. The similarity of binned distributions of local soil water fluxes measured at different application rates was tested using the chi-square test. Suppose that Ri is the number of events observed in the ith bin for the local soil water fluxes at one rate and Si the number of events in the same bin i for the local soil water fluxes at another rate. Then the chi-square statistics, with the degree of freedom of NB - 1, is (Press et al., 1992)
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where NB is the number of bins.
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RESULTS AND DISCUSSION
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Figure 1
shows the measured soil water fluxes along the transect during constant flux infiltration experiments for four application rates for the 0- to 0.2-m depth. Considerable spatial variability existed in measured soil water flux for all four application rates, even though the application rate on the soil surface was extremely uniform (CV < 3%), suggesting a three-dimensional water flow in this field (Table 1). However, the increase of soil water storage with time at a certain location was constant vertically (linear relationships between soil water storage and time), much like flow in a stream tube. This formation of vertical stream-tube-like path in a spatially variable field was believed to result from water redistribution in the first few centimeters below the soil surface and subsequent establishment of a vertical flow pathway (Si et al., 1999). The existence of horizontal spatial variation in measured soil water flux was also supported in Fig. 2 through 4
for 0- to 0.4-, 0- to 0.6-, and 0- to 0.8-m depths, respectively. The standard measurement error of TDR for water content is 0.013 for soils with different textures (Topp et al., 1980). Relative measurement error using TDR, as would be relevant here, would be significantly lower. Accordingly, the maximum standard measurement error for water storage for depth L is w = 0.013 x L. Water flux being the slope of water storage as a function of time, has a variance (Myers, 1989),
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where ti is the time when water-storage measurements were taken before the wetting front passed the end of TDR rods. For the 0.2-m depth, the variance of measurement error for soil water storage, and soil water-storage measurements were taken between 0 to 1 h from the start of transient infiltration experiments. The summation of squared measurement time is much larger than 1. Consequently, 2q < 0.07 cm2 h-2 (Eq. [9]) for each location. For the transect with 50 measurement locations, the standard deviation of measured local soil water flux should be less than [0.07/(50-1)]1/2 = 0.04 cm h-1, which is much less than the standard deviation of measured water flux (Table 1). Therefore, measurement error from TDR is not the dominant term for the spatial variability of measured soil water flux for the 0 to 0.2 m. For greater depths (0 to 0.4 and 0 to 0.6 m), measurement error because of TDR measurement error is larger than that of the 0- to 0.2-m depth. However, the time for the wetting front to pass the end of TDR rods and number of measurements of soil water storage per location also increase. As a result, the measurement error because of TDR was smaller than that of the 0- to 0.2-m depth. Therefore, for all depths, the variability in measured soil water flux (Table 1) is not mainly caused by TDR-measurement error for all depths. This is consistent with the past studies showing spatial variability of soil hydraulic properties dominates measurement error (Nielsen et al., 1973; Dagan and Bresler, 1983).
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Fig. 1. Measured local soil water flux (solid line) and application rate (dashed line) during constant flux infiltration experiments for application rates of (a) 0.9, (b) 1.5, (c) 2.6, and (d) 3.3 cm h-1 for 0.2-m depth.
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Fig. 2. Measured local soil water flux (solid line) and application rate (dashed line) during constant flux infiltration experiments for application rates of (a) 0.9, (b) 1.5, (c) 2.6, and (d) 3.3 cm h-1 for 0.4-m depth.
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Fig. 4. Measured local soil water flux (solid line) and application rate (dashed line) during constant flux infiltration experiments for application rates of (a) 0.9, (b) 1.5, (c) 2.6, and (d) 3.3 cm h-1 for 0.8-m depth.
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Fig. 3. Measured local soil water flux (solid line) and application rate (dashed line) during constant flux infiltration experiments for application rates of (a) 0.9, (b) 1.5, (c) 2.6, and (d) 3.3 cm h-1 for 0.6-m depth.
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There were no strong associations between fast flow (high water flux) or slow flow locations in measured soil water flux along the transect for different application rates for the 0- to 0.2-m depth (Fig. 1). This is confirmed by the Spearman rank correlation coefficients (Table 2). Of all the pairs, only two pairs have a significant correlation at a confidence level of 0.95. Furthermore, the correlation coefficients are weak, further indicating the weak associations in soil water flux among different application rates.
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Table 2. Spearman Rank correlation coefficients among soil water flux during constant flux infiltration experiments with four application rates for 0.2-, 0.4-, 0.6-, and 0.8-m depth.
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For the 0- to 0.4-m depth, a local pattern existed in soil water fluxes measured at application rates of 0.9 and 1.5 cm h-1 from locations 0 to 3.0 m. However, the same is not true from locations 3.0 to 7.5 m (Fig. 2). No consistent patterns existed across the whole transect in measured soil water fluxes for different application rates. Furthermore, the Spearman rank correlation coefficients between soil water fluxes measured at different application rates were weak (<0.6), even though four out of six correlation coefficients between soil water fluxes measured at different application rates are significant at a confidence level of 0.95 (Table 2). Therefore, <36% of variation in local soil water flux measured at one application rate can be explained by that at another water application rate. This weak correlation further indicated the weak spatial patterns in measured soil water flux at different application rates.
For the 0- to 0.6-m depth, some observable local patterns existed in local soil water fluxes along the transect measured during transient infiltrations at different application rates (Fig. 3). Again, no clear patterns existed across the whole transect for all application rates. Although all the correlations between local water fluxes measured during transient infiltrations under different application rates are significant at a confidence level of 0.95, the correlation coefficients were generally from weak to mild, explaining <50% of the spatial variation, except a strong correlation (rs = 0.85) existed between local soil water fluxes measured at application rates of 1.5 and 2.6 cm h-1.
For the 0- to 0.8-m depth, there were local patterns in soil water fluxes measured at different application rates (Fig. 4). At locations 0.75 to 1.5 m, the measured soil water flux increases from locations 0.75 to 1.05 m, and then decreases from locations 1.05 to 1.5 m for an application rate of 0.9 cm h-1. Soil water flux increases again from 1.5 to 1.8 m and then decreases from 1.8 to 1.95. Similar patterns also existed for application rates of 1.5 and 3.3 cm h-1. Note that a slow flow (small water flux) region existed at locations from 5.4 to 5.85 m for an application rate of 2.6 cm h-1, as well as for an application rate of 3.3 cm h-1. In addition, another slow flow region developed at locations from 3.6 to 4.05 at application rates of 2.6 and 3.3 cm h-1, which did not exist at application rates of 0.9 and 1.5 cm h-1. Therefore, there were no consistent patterns in local soil water fluxes across the transect for all application rates. This is supported by the weak, though significant at the 95% confidence level, correlation coefficients of local soil water fluxes for different application rates (Table 2).
The weak to mild correlations among measured local soil water fluxes at different application rates for the four depths suggested only weak spatial similarity in measured local soil fluxes. Therefore, accurate prediction of soil water flux from one rate to another is not possible. This is not unexpected because total porosity and pore-size distribution may be different for different locations. At low application rates, only these small pores are activated and locations with many small pores will have faster flow than locations with a small proportion of small pores. However, at higher application rates, large pores dominate flow and locations having a large proportion of large pores will have fast water flow. Therefore, to have consistent flow pattern at higher and low application rates requires that locations having a large volume of small pores, must also have a large volume of large pores. This is unlikely, and it is anticipated that different flow patterns exist at different water application rates for the same soil.
Arrays of TDR probes were installed in parallel transects 0.1 m apart for each of the four depths (0.2, 0.4, 0.6, and 0.8 m). Therefore, TDR probes for these four depths were at different spatial locations. Because of the spatial variability in soil water fluxes, no attempt is made to compare the spatial patterns in soil water fluxes along the transect between different depths at an application rate.
Spearman rank correlation coefficients of local soil water fluxes between different application rates are stronger for 0- to 0.6- and 0- to 0.8-m depths than that of 0- to 0.2- and 0- to 0.4-m depths. This is an indication of increasing similarity with increases in sampling depths or sampling volume.
In the following, we examine if the pdf of measured soil water flux are the same for different rates at the same depth. This information is the basis of stochastic approaches for predicting solute travel time pdf with the stochastic approach (Destouni, 1992; Jury and Scotter, 1994; Toride and Leij, 1996). Chi-square tests were conducted to examine statistically if the histograms are drawn from the same distribution functions for each rate of the same depth. All tests rejected the null hypothesis that histograms of soil water fluxes for different application rates are similar at the 95% confidence level. Therefore, statistical similarity did not exist. This is consistent with the weak spatial similarity of local soil water fluxes for different application rates for the same depth, as shown earlier in this paper.
This conclusion has a significant implication for the application of stochastic approaches in modeling water flow and chemical transport in the field. The Lagrangian approach of Destouni (1992) assumed that pdf of soil water flux is determined by the pdf of saturated hydraulic conductivity through a linear transformation (Eq. [5]). Accordingly, pdfs of local soil water fluxes in a field should resemble the pdf of saturated hydraulic conductivity. This is in contrary to what was found for each of the 0- to 0.2-, 0- to 0.4-, 0- to 0.6-, and 0- to 0.8-m depths. For the stochastic convective flow, Jury (1982) presented the stochastic-convective lognormal transfer function and used the cumulative infiltration rate as the independent variable instead of time. In this way, Jury and Roth (1990) were able to predict solute breakthrough curves for other flow conditions in addition to the experimental conditions the transfer function was derived from. Since cumulative infiltration rate is the integration of water flux with respective to time, which is just a linear transformation of soil water flux requirement of same pdf of cumulative infiltration rate is equivalent to the requirement of the same soil water flux distribution for different application rates. This will require the pdf of local soil water flux to be similar for different application rates. Again, this is not satisfied for each of the 0- to 0.2-, 0- to 0.4-, 0- to 0.6-, and 0- to 0.8-m depths.
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SUMMARY AND CONCLUSIONS
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Examination of the spatial and statistical distribution of soil water flux in a field is fundamental for modeling soil water flow and chemical transport in spatially variable soils. A series of constant flux infiltration experiments were conducted and the soil water fluxes at depths of 0- to 0.2-, 0- to 0.4-, 0- to 0.6-, and 0- to 0.8-m were determined from the change of water storage as a function of time. The spatial similarity (persistent spatial pattern) of measured soil water flux for different application rates at four depths was examined using Spearman rank correlation coefficient. Results showed that there was no persistent spatial similarity among measured soil water fluxes for different application rates for each of the 0- to 0.2-, 0- to 0.4-, 0- to 0.6-, and 0- to 0.8-m depths. The statistical similarity (persistent statistical distribution) of soil water flux for different application rates was also examined using histograms. Chi-square statistics did not accept the null hypothesis that histograms for local soil water flux are similar for different application rates, suggesting the stochastic convective flow model may not be used to describe flow and transport for the 0- to 0.2-, 0- to 0.4-, 0- to 0.6-, and 0- to 0.8-m depths in this field.
Horizontal spatial variations of soil water flux measured at depths of 0.2, 0.4, 0.6, and 0.8 m have important implications. Stochastic analyses such as those of Dagan and Bresler (1983) and Rubin and Or (1993), generally assumed uniform soil water flux throughout the field. This assumption is in contrary with our experimental evidence, thus, interpretation of their analysis in fields should be cautious. To make stochastic analysis more pertinent to the field, spatial variation in soil water flux should be taken into account and the influence of spatial variation of local soil water flux examined in the stochastic analysis.
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ACKNOWLEDGMENTS
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Support for the experiments from Dr. R.G. Kachanoski is gratefully acknowledged. I thank Dr. E. de Jong for his helpful comments on the manuscript. The manuscript also benefited from the critiques of anonymous reviewers and the associate editor Dr. A. Ward.
Received for publication April 12, 2001.
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