Soil Science Society of America Journal 67:730-736 (2003)
© 2003 Soil Science Society of America
DIVISION S-1—SOIL PHYSICS
Measurement of Local Soil Water Flux during Field Solute Transport Experiments
Bing Cheng Si*,a and
R. Gary Kachanoskib
a Dep. of Soil Science, University of Saskatchewan, Saskatoon, SK, Canada
b Dep. of Renewable Resources, University of Alberta, Edmonton, AB, Canada
* Corresponding author (bing.si{at}usask.ca)
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ABSTRACT
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Accurate measurement of local soil water flux, at the same location as solute concentration, remains a challenge, and the lack of these measurements limits our understanding of field solute transport. The objective of this paper is to develop and test methods for measuring local soil water flux under steady-state constant flux infiltration, typical of field solute transport experiments. The methods use vertically installed time domain reflectometry (TDR) probes to measure local solute mass flux from the change in TDR measured impedance after either a step increase or decrease (flushing) in electrolyte (tracer) concentration in the applied water. The local soil water flux was calculated directly from the local solute mass flux. The methods assume that water flow and solute transport are one-dimensional and that TDR estimates of bulk electrical conductivity are not sensitive to the vertical distribution of the applied tracer in the pore water. Field experiments indicate estimates of local soil water flux from step increase and step decrease of tracer concentration were very similar (R2 
0.63) and average water flux was similar to the application rate. There was moderate correlation between local water fluxes measured using these steady-state methods, and a previously reported method based on early time measurements of water storage during the initial (transient) states of water infiltration. The methods can be used to estimate field average hydraulic conductivity as a function of soil water content.
Abbreviations: ECa, electrical conductivity • TDR, time domain reflectometry
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INTRODUCTION
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NUMEROUS MODELS have been developed to describe, simulate, and predict chemical transport in homogeneous and heterogeneous soils (Coats and Smith, 1956; van Genuchten and Wierenga, 1976; Dagan and Bresler, 1979; Simmons, 1982; Jury and Roth, 1990). Three approaches that have been employed to describe field-scale solute transport include (Jury and Flühler, 1992): (i) the traditional convective-dispersive model; (ii) a stochastic-continuum model that uses covariance functions for random local scale transport parameters (Dagan, 1984); and (iii) a stochastic-convective stream tube model that views the field as a series of independent vertical soil columns (Dagan and Bresler, 1979; Jury and Roth, 1990). These models are distinguished by the degree of lateral solute mixing (Jury and Flühler, 1992; Toride and Leij, 1996). Applications of these models require the measurement or estimates of soil water flow and solute transport properties at different scales.
A standard method for estimating solute transport properties is to conduct a steady-state solute transport experiment in fields with prescribed flux boundary conditions and to measure solute flux concentrations in the soil or breakthrough curves at a certain depth. If soil water flux at a specific location is known, solute-transport properties of the soil can be inversely determined by fitting a model to the measured breakthrough curves. However, there is significant spatial variability of local soil water fluxes. Local soil water fluxes can be different from the application rate (Nielsen et al., 1973; Si et al., 1999). Accurate determination of local scale solute transport properties of soils requires knowledge of local soil water flux.
Solute concentration, depending on the problem and measurement methods, can be classified as flux concentration and resident concentration (Jury and Roth, 1990). Outflow concentration from a soil column is flux concentration and solute concentration measured by soil coring or horizontally installed TDR probes is resident concentration. The local flux concentration in the field can be measured by solution samplers (Butters et al., 1989) or vertically installed TDR probes (Kachanoski et al., 1992). In practices, our interests are in the field-scale solute concentration, which is required for validation of models and characterization of solute transport in spatially variable fields. However, only point-scale measurements can be obtained through TDR probes and solution samplers. Therefore, there is a need to scale up the point measurements. Accurate estimates of field scale flux concentration should be obtained by averaging across the local flux concentration weighted by the local water flux. Thus, knowledge of the latter is required at discrete sampling locations but is generally not available (Jury and Scotter, 1994).
The objective of this paper is to develop and test methods for measuring the steady-state soil water flux at specific locations during steady-state solute transport experiments. Two new methods are presented. One method utilizes continuous application of a conservative tracer during steady-state solute transport experiments (step input). The second method uses a soil with an initially uniform distribution of solute concentration and then flushes the soil with tap water on a soil surface (flushing). These methods are applied to field experiments and compared with the transient method of measuring local water flux given by Parkin et al. (1995) and Si et al. (1999).
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THEORY
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Determination of Soil Water Flux from Transient Infiltration Experiments
During constant flux infiltration, soil water content in soil profile changes with time. However, as wetting front has completely passed a certain depth, soil water content, and soil water flux do not change with time anymore. We refer to the first stage when soil water content changes with time as transient phase of constant flux infiltration and second stage when soil water content and soil water flux are constant, as steady-state phase. Soil water flux during the transient phase of constant flux infiltration was determined using the method of Parkin at al. (1995) and Si et al. (1999). Briefly, the cumulative storage of water (m) to depth L, W(L,t), is measured by vertically installed TDR probes and is given by
 | [1] |
where 
(t) 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, qw|0, at the soil surface (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
 | [2] |
Equation [2] allows us to calculate the local water flux from soil water storage measurements before wetting front reaches the end of TDR rods.
Calibrations of Solute Concentration-Impedance Relationship
Kachanoski et al. (1992) showed that the average concentration of an added electrolyte tracer along a TDR probe could be measured from TDR-based estimates of bulk electrical conductivity, ECa. A linear relationship is generally observed between the resident solute tracer concentration, C and ECa for constant water contents ranging from relatively low to saturation, and for salinity levels ranging from 0 to approximately 50 dS m-1 (Ward et al., 1994; Mallants et al., 1996):
 | [3] |
where 
and ß are calibration constants at a given water content value.
Consider the case of steady-state constant flux infiltration with a step increase in electrolytic tracer with concentration C = C0 at t = t0. At large time, tF, after the tracer dispersion front has passed the end of the TDR probe, the solute will be distributed uniformly in the length of TDR rods, and the resident concentration along the length of vertically installed TDR probes will equal the step input concentration C0. Assuming relatively small solute concentration before the application of solute, the relationship between TDR measured ECa and the tracer concentration can be given by (Ward et al., 1994)
 | [4] |
where ECa(L,tF) is the TDR ECa measured at large time tF with the length of TDR probe, L, ECa(L,ti) is the TDR ECa measured before the application of solute on a soil surface, and is the calibration constant for a fixed probe length. Since C0 is known, the value of the calibration coefficient is easily calculated from the initial (before tracer application) and large time TDR measurements. With ßL known, the average concentration over the length of a TDR probe, 
, at any time, t, after the step application of solute can be estimated from
 | [5] |
where ECa(L,t) is the bulk electrical conductivity measured at any time t. Equations [3] through [5] hold for a number of methods of estimating ECa using TDR. For example, ECa (dS m-1) can be related to the TDR-measured long-time impedance, R, of an electromagnetic wave that travels through the soil using the following equation (Topp et al., 1988; Heimovaara et al., 1995).
 | [6] |
where Kc is the cell constant of the TDR probes, and Rcable(
) is the resistance associated with the cable, connectors, and cable tester. The cell constant, Kc, can be measured or treated as part of calibration constant ßL. For salinity levels <30 dS m-1, Rcable << R in Eq. [3] (Mallants et al., 1996). In this study, soil bulk electrical conductivity is much <30 dS m-1; thus, we assume Rcable = 0.
Solute and Water Flux during Solute Transport Experiments
The cumulative storage of solute (kg m-2) to depth L, S(L,t), can be measured by vertically installed TDR probes and is given by
 | [7] |
where 
, 
and L are the average solute concentration (kg m-3), average volumetric soil water content (m m-1), and the probe length, L (m) at time t.
During the period before the solute dispersion front first reaches depth L, for a step increase of solute at the surface under constant water flux and steady-state conditions, the derivative of the cumulative storage of solute measured by TDR with respect to time should approximately equal the local solute flux at the soil surface, Js|0. Assuming conservation of mass, one-dimensional water flow and solute transport, and the applied solute has not yet reached depth L, then
 | [8] |
Substitution of Eq. [5] into Eq. [8] leads to
 | [9] |
Equation [9] allows calculation of the local solute flux during the early stage of application of a step increase of tracer solution. By measuring the change in TDR ECa with time, the local water flux during a step input experiment for a known input solution concentration C0 and solute flux Js|0, is
 | [10] |
Substitution of Eq. [9] into Eq. [10] leads to
 | [11] |
Once ßL is known, Eq. [11] can be used to calculate local soil water flux for a step input solute transport experiment from early time measurement of bulk soil electrical conductivity. There are many methods for obtaining ßL. For a coarse-textured soil, one of the methods is the step input experiment as used by Ward et al. (1994) and Mallants et al. (1996). Substitution of Eq. [4] into Eq. [11] leads to
 | [12] |
Thus, at early time the local soil water flux is directly proportional to the slope of the TDR measured ECa versus time. The slope can be determined through simple linear regression of ECa with time using early TDR ECa measurements. This gives local soil water flux at the soil surface. The advantage of Eq. [12] is that local soil water flux can be calculated from a single step input solute transport experiment, without additional need for calibration of relationships between ECa and solute concentration C.
After the entire tracer dispersion front from the step increase has passed the end of the TDR probe, the tracer solution concentration along the length of the probe will be equal to C0 (Note: this may need a long time for a fine-textured soil). At this time, a step decrease in tracer concentration (to background levels in the water source) can be imposed and a second measurement of qW|0 obtained from the flushing of the tracer along the TDR probe. For this flushing experiment, solute flux at depth L, Js|L is
 | [13] |
and soil water flux at the soil surface is
 | [14] |
In addition, similar to Eq. [4], C0 can be expressed as
 | [15] |
Where ECa(L,tFi) is the ECa just before flushing (i.e., when the soil solution concentration is C0). ECa(L,tFF) is the ECa at the end of the flushing experiment, when all the tracer is flushed out of the full length of the TDR rods (Note: for a conservative tracer, ECa(L,tF) and ECa(L,ti) should equal ECa(L,tFi) and ECa(L,tFF), respectively). Substitution of Eq. [13] and [15] into Eq. [14] leads to
 | [16] |
Equations [12] and [16] provide direct methods for measurement of local soil water flux at the scale of a single TDR probe during step input and flushing solute transport experiments. Interestingly, calculating the water flux does not require the input tracer concentration to be known. Only a measurable change in ECa is required. Furthermore, Eq. [12] and [16] need the average soil water content, not the transport volume. Therefore, the knowledge of mobile soil water content does not affect the estimation of local soil water flux, because only the total water content is involved in the calculation of local soil water flux. However, an implicit assumption in Eq. [12] and [16] is that the distribution of tracer in the soil water pore space does not greatly influence the TDR measured ECa. For a step increase or flushing solute transport experiment, this assumption likely holds, particularly when the input solution concentration is low and drastic change of solute concentration within the length of TDR probes does not exist, which is the case for this study.
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MATERIALS AND METHODS
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Field infiltration measurements were conducted in 1995 and 1996 at the Canadian Forces Base Borden, ON, Canada. Extensive hydrogeological research, including a large-scale natural-gradient tracer test and forced gradient tests have 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 vertically 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 vertically 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 for infiltration experiments. The application rates were uniform with coefficient of variation (CV) <3% (Si et al., 1999).
For solute transport experiments, only two rates were selected: 3.3 and 0.9 cm h-1. We selected 3.3 cm h-1, because it was the highest rate with no ponding on soil surface during the infiltration experiment. Another infiltration rate 0.9 cm h-1 was selected because it is a lower rate but still have detectable change of ECa in a small time interval. For an application rate of 3.3 cm h-1, the water application system was shut off after steady-state infiltration was reached (soil water storage did not change with time for the 0.4-m probes). This allowed water to drain freely from the soil profile until soil water storage decreased slowly with time. The drainage data were used by Si and Kachanoski (2000) to validate theory to predict soil water storage during drainage from hydraulic parameters estimated from infiltration. Then the water application system was turned on with approximately the same rate (3.3 cm h-1). After steady state was again reached, the source of the water application system was switched to a KCl tracer solution (1 g L-1 KCl). The impedance load (R) was recorded before the application of KCl and every 5 to 10 min during the experiment until the KCl distributed uniformly within the length of TDR rods (L = 0.8 m) and the impedance load reading did not change with time. This step input experiment took about 36 h. Subsequently, the water source was switched back to tracer-free water again, initiating the flushing experiment. The increases of the impedance load were recorded every 5 to 10 min during the flushing experiment until the impedance load was restored to the same level as before the application of KCL. This flushing experiment took about 62 h. Soil water content during the step and flushing solute experiments was measured regularly using the TDR method of Topp et al. (1980). The arithmetic average value of these soil water content measurements during each experiment was used as the steady-state soil water content for the experiment. All the measurements (R and dielectric constant) were measured using two precalibrated Tektronix (1502B) metallic cable testers (Tektronix, Wilsonville, OR).
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RESULTS AND DISCUSSION
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An example of measured solute storage as a function of time for a step increase of solute on the soil surface for an individual TDR probe is shown in Fig. 1a
. Generally, solute storage to depth L, S(L,t), increased linearly with time t, for early time measurements (i.e., before the solute dispersion front reached the ends of the probe). This suggested that solute was added at a constant rate or the solute flux is constant at early time at this location. An example of measured solute storage as a function of time for a flushing tracer-free water application for a soil with uniform initial concentration in the soil profile for the same TDR probe is shown in Fig. 1b. As expected, S(L,t) decreased with time linearly (R2 = 0.999) for early time measurements. It is suggested that solute was removed at a constant rate, or the solute flux is constant at early time at a specific location. Therefore, solute transport was approximately one-dimensional for both the step and flushing experiments. Note that solute flux for the step experiment is smaller than the flushing experiment. This may be caused by measurement errors or the water application rate was higher during flushing experiments than that during step solute transport experiments.
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Fig. 1. An example of the linear relationship of soil solute storage verus time, at early time, for an application rate of 3.3 cm h-1 and TDR probe (0.2 m) at position 1.5 m during: (a) continuous solute application on soil surface (step input) and (b) flushing experiment. The first 14 points (t = 0 to 1 h) were used in both regressions.
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Local soil water flux was calculated for each of the probe locations across the transect using Eq. [12] and [16]. Figure 2
shows the measured soil water flux for each TDR probe along the transect from the three methods (i.e., transient infiltration, step experiment, flushing experiment) for the 0.2-m long TDR probes. There were significant spatial variations in solute and water flux along the transect despite very uniform water application. This finding is consistent with those of Parkin et al. (1995) and Si et al.(1999), who found that local water infiltration during transient infiltration approximates one-dimensional flow; further, local soil water flux is different from location to location, exhibiting horizontal spatial variability. This combination of locally vertical flow with horizontal variability in local water flux approximate the stream-tube assumption about solute and water flux in heterogeneous soils (Dagan and Bresler, 1979).
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Fig. 2. Measured soil water flux along the transect using transient infiltration, step and flushing experiments for an application rate of approximately 3.3 cm h-1 and a probe length of 0.2 m.
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For the application rate equal to 3.3 cm h-1 and a probe length of 0.2 m, the means (and coefficients of variation) of the measured water flux along the transect were 3.18 (0.15), 3.53(0.17), and 3.33(0.17) cm h-1 for the transient infiltration, step and flushing experiments, respectively. For probes at a length of 0.4 m and an application rate equal to 3.3 cm h-1, the averages (and coefficients of variation) of the measured water flux along the transect were 3.29(0.11), 3.23(0.14), and 3.41 (0.13) cm h-1 for transient infiltration, step solute transport experiment, and flushing solute transport experiment, respectively. For the application rate of 0.9 cm h-1 and a probe length of 0.2 m, the averages (and coefficients of variation) of the measured water flux along the transect were 0.84(0.08), 0.84(0.10), and 0.96 (0.09) cm h-1 for transient infiltration, step solute transport experiment, and flushing solute transport experiment, respectively. These mean values are close to the application rates with a mass recovery (average measured water flux across the transect vs. water application rate) within 90 to 110%. This good mass recovery suggests that the three methods were consistent with each other and provided similar average soil water flux along the transect with the application rate.
A more rigorous comparison of the three methods requires, in addition to comparison of the means of measured soil water flux, examination of one to one correspondence or the spatial patterns of measured local soil water fluxes by the three methods. The spatial variation of the measured soil water flux by the three methods had very similar spatial patterns for the application rate equal to 3.3 cm h-1 and a probe length equal to 0.2 m (Fig. 2). Higher values of local soil water flux measured during transient infiltration experiments were generally associated with higher values of local soil water flux measured by the two solute transport experiments. The same is true for other application rate and lengths of TDR probes (not shown). There was a strong linear correlation (R2 = 0.83) between measured soil water flux from the step and flushing experiments for the 3.3 cm h-1 application rate and the 0.2 m long TDR probes (Fig. 3a)
. The slope of the regression line is not statistically different from one (p < 0.01). The same is true for soil water flux measured using a probe length of 0.4 m (R2 = 0.84) (Fig. 3b). For an infiltration rate of 0.9 cm h-1 at a probe length of 0.2 m, there was also a significant correlation (R2 = 0.63) (Fig. 4)
. However, there was a significant deviation from the 1:1 correspondence between local soil water fluxes measured using the two steady state methods (Fig. 4). The local soil water fluxes during the step experiment were significantly smaller than that of the flushing experiments. This might be a reflection of a smaller application rate during the step experiment from reduced pressure from the pressure generator used to pump solute solution from a tank to the moving nozzle system. Interestingly, the slope is not statistically different from 1 (p < 0.01), which suggests that the spatial patterns of water flux measured during the two solute transport experiments for each of the two application rates was similar.
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Fig. 3. Relationship between soil water fluxes measured during step increase and flushing solute transport experiments for an application rate of 3.3 cm h-1 for a probe length of (a) 0.2 m and (b) 0.4 m.
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Fig. 4. Relationship between fluxes measured during step and flushing solute transport experiments for an application rate of 0.9 cm h-1 and a probe length of 0.2 m.
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The local soil water flux measured during the transient stage of constant flux infiltration, as shown in Si et al. (1999), was compared with the soil water flux measured using the step and flushing methods. There were significant positive correlations between local soil water fluxes measured from transient infiltration and step experiments (R2 = 0.54) and between local soil water fluxes measured from infiltration and flushing experiments (R2 = 0.56) for the application rate of 3.3 cm h-1 and 0.2-m length probes (Fig. 5a)
. Similar results were obtained for a probe length of 0.4 m with R2 = 0.44 and 0.47 for step and flushing respectively (Fig. 5b). The correlations between transient and steady-state measurement of local water flux, though significant (p < 0.001), were not as strong as the correlation between water fluxes measured from flushing and step experiments. The moderate correlation may reflect a subtle change of surface boundary conditions between flow experiments especially the surface soil microstructure, which results in different patterns of water redistribution near the soil surface. Alternatively, the steady-state transport experiments may be closer to true one-dimensional flow than the transient stage of constant flux infiltration. Therefore, flushing and step solute transport experiments had approximately the same water flow conditions, but are somewhat different from transient infiltration in water flow. For the same length of TDR probe (0.2 m), the correlation between the step measurements versus transient water flux measurements was poorer (R2 = 0.22) for the lower water application rate (0.9 cm h-1), than that (R2 = 0.54) of high water application rate (3.3 cm h-1). Subsequent work will explore the differences in the spatial pattern of water flux during transient versus steady-state conditions.
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Fig. 5. Relationship between soil water flux measured during step, flush, and transient infiltration experiments with an application rate of 3.3 cm h-1 and a probe length of (a) 0.2 m and (b) 0.4 m.
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Parkin et al. (1995) and Si et al. (1999) used TDR probes installed vertically at the soil surface beneath a constant-rate rainfall simulator to measure cumulative water storage with time. They estimated the local infiltration rate from the slope of water storage vs. time during early time, before the wetting front reaches the bottom of the TDR probe. Assuming a unit gradient and utilizing the spatial variability in local infiltration rate, they directly estimated the unsaturated hydraulic conductivity for a wide range of water contents, using two and five water application rates, by Parkin et al. (1995) and Si et al. (1999), respectively. Similarly, the measured local soil water flux from solute transport experiments was used to estimate field average hydraulic conductivity, K, as a function of water content, 
. Figure 6
depicts the calculated hydraulic conductivity, from measured local soil water fluxes during flushing solute transport experiments, as a function of local soil water content. As we mentioned earlier, the unit gradient and one-dimensional flow assumptions may be more appropriate in steady-state infiltration than in transient infiltration. Therefore, we expected that the steady state methods should be more accurate. However, the measured field effective soil hydraulic conductivity as a function of soil water content using the proposed methods has a good agreement with the K() estimated from the multi-infiltration experiments (Si et al., 1999) (Fig. 6), suggesting the three methods are equivalent.
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Fig. 6. Comparison of measured hydraulic conductivity as a function of soil water content from flushing experiments for the two application rates with K() reported in Si et al. (1999).
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The proposed methods for measurement of local soil water fluxes are fast and can be automated. Based on a commonly used solute transport experiment to calibrate TDR or to estimate solute transport parameters, these methods can be used to measure local solute flux and local water flux during steady state solute transport experiments in heterogeneous soils. In addition, the proposed methods combined with the method of Kachanoski et al. (1992) using vertically installed TDR probes, provide measurement of local soil water flux and relative local solute mass flux (and thus flux concentration) at the same spatial location. This allows field average flux concentration to be obtained from spatially averaging local flux concentration, which was not previously possible.
A limitation of the methods is that the stream-tube model is implied. This seems justified, given that lateral spatial variability in soil hydraulic properties is much larger than the vertical spatial variability for this study. Further, for solute transport, the stream-tube model is more likely to be accurate near the surface than the convective-dispersive models, provided that lateral redistribution is not occurring. This assumption is most likely to be valid at early times after solute addition and in applications where solute is applied over a wide area (e.g., agricultural fields), because the time required for mixing exceeds the time to reach depths near the surface (Jury and Scotter, 1994). As a result, solute transport in these experiments is governed primarily by advection, unless lateral mixing is enhanced by some mechanism such as a soil layer of low hydraulic conductivity.
As indicated by Mallants et al. (1996), calibration of TDR ECa and soil resident concentration from a step input solute transport experiment is very time-consuming to ensure that soil solution concentration equals input solution concentration, particularly for a fine-textured soil. In this study, we used a coarse-textured soil (Borden sand) and extra care was taken to ensure the equilibration of the soil solution with the input solution. However, for a fine-textured soil, a separate experiment such as a short-pulse experiment is needed for the calibration of ECa vs. soil resident concentration relationships (Mallants et al., 1996). In that case, Eq. [11] can be used to estimate local soil water flux. However, further research is needed for field verification of the methods in fine-textured soils.
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SUMMARY
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New methods for measuring local soil water flux during steady-state solute transport experiments were presented. The methods utilized vertically installed TDR probes to measure the change of solute storage as a function of time during step input or subsequent flushing experiments. The rate of change of solute storage at early time before the solute dispersion front reaches the end of the TDR probes was used to calculate the solute mass flux. Since the input concentration and steady-state water content are constant, the solute mass flux can be used to estimate local water flux. The methods were applied to field solute transport experiments. Calculated soil water flux based on these two methods are in good agreement and had moderate correlation with the water flux calculated from the transient phase of constant flux infiltration.
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ACKNOWLEDGMENTS
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The senior author thanks Dr. E. de Jong, Shelley Woods, and Miles Dyck for their helpful comments on the manuscript. The manuscript also benefitted from the critiques of anonymous reviewers. This study was partially supported by grants awarded to both authors by the Natural Sciences and Engineering Research Council of Canada (NSERC).
Received for publication August 27, 2001.
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