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ORIGINAL RESEARCH PAPER

Laboratory Evaluation of the Dual-Probe Heat-Pulse Method for Measuring Soil Water Content

^a Plant and Soil Science Department, Texas Tech University, Lubbock, TX 79409
^b Department of Agronomy, Kansas State University, Manhattan, KS 66505
^c Rocky Mountain Research Station, U.S. Forest Service, 240 W. Prospect Rd., Fort Collins, CO 80526
^d Department of Biological and Agricultural Engineering, Kansas State University, Manhattan, KS 66505
^* Corresponding author (gjk{at}ksu.edu).

Contribution no. 03-204-J from the Kansas Agric. Exp. Stn., Manhattan, KS.

Received 7 December 2002.

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

 
The dual-probe heat-pulse (DPHP) method provides a means of estimating volumetric soil water content () and change in volumetric water content () from measurements of volumetric heat capacity. The purpose of this investigation was to characterize the accuracy and precision that can be achieved in measuring and with the DPHP method. Tempe pressure cells fitted with DPHP sensors were used to conduct desorption experiments in which DPHP-based estimates of and were compared with values estimated by the gravimetric method. For water contents corresponding to soil water pressure potentials below -100 kPa, comparisons were made by packing the pressure cells with soil wetted to known water contents. The investigation was conducted with seven soil materials representing a wide range of physical properties for mineral soils. The DPHP sensors slightly overestimated at low water contents, but it was shown that the bias could be removed by using an empirical calibration equation, = 1.09 _DPHP - 0.045. This relationship appears to be general inasmuch as it was shown to be applicable for all seven soil materials and for water contents ranging from 0.02 to 0.59 m³ m^-3. The general calibration equation was also shown to be effective in removing bias in estimates. Pooled regression analysis (all soil materials) showed that can be measured with a root mean square error (RMSE) of 0.022 m³ m^-3. Greater precision can be achieved with measurements (RMSE = 0.012 m³ m^-3); however, the results indicated a decrease in precision with increasing magnitude of .

Abbreviations: DACS, data acquisition and control system • DPHP, dual-probe heat-pulse • PVC, polyvinyl chloride • RMSE, root mean square error

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

 
THE DPHP METHOD provides an effective means of measuring soil water content and changes in soil water content (Campbell et al., 1991; Bristow et al., 1993; Noborio et al., 1996; Tarara and Ham, 1997; Song et al., 1998; Bristow, 1998; Song et al., 1999; Bristow et al., 2001; Campbell et al., 2002). The method, first proposed by Campbell et al. (1991), utilizes a sensor to obtain measurements of the soil volumetric heat capacity. Volumetric water content () and change in volumetric water content () are then estimated from the linear relationship between heat capacity and water content.

Dual-probe heat-pulse sensors have been widely used for routine and measurement in field experiments (Bremer et al., 1998; Ham and Knapp, 1998; Bremer and Ham, 1999; Bremer et al., 2001; Campbell et al., 2002); however, few investigations have been conducted to evaluate the level of accuracy and precision that can be achieved with the DPHP method. Tarara and Ham (1997) conducted a laboratory experiment in which DPHP-based estimates were compared with values obtained by the gravimetric method. They found that the two methods agreed to within 0.03 and 0.04 m³ m^-3 for two different soil materials over a water content range of 0.10 to 0.45 m³ m^-3. Estimates of for the two methods agreed to within 0.01 m³ m^-3 for both soils. Song et al. (1998)(1999) evaluated the DPHP method in greenhouse experiments involving containers that were fitted with multiple DPHP sensors. Their results showed that the DPHP method measured average to within 0.02 to 0.03 m³ m^-3 and average to within 0.01 m³ m^-3.

Campbell et al. (2002) and Noborio et al. (1996) also evaluated the DPHP method for measuring and . Campbell et al. (2002) inserted DPHP sensors in undisturbed peat cores obtained from two peat bogs in New Zealand. The DPHP-based estimates were compared with values obtained by the gravimetric method for a water content range of approximately 0.15 to 0.90 m³ m^-3. Regression analysis with estimates obtained by the two methods showed the DPHP method to be unbiased. The analysis also revealed that excellent precision was achieved with the DPHP method; however, precision was not quantified using a regression RMSE. The evaluation performed by Noborio et al. (1996) revealed poor precision in estimates obtained by the DPHP method, but it is likely that their estimates were adversely affected by excessive deflection of the probes of their DPHP sensor.

The objective of this experiment was to provide a thorough assessment of the accuracy and precision of the DPHP method for measuring and . To maximize the applicability of the results, experiments were conducted with soil materials having a range of physical properties. Specifically, soil materials were chosen to yield a range of textures, organic matter contents, bulk densities, and specific heats. The experiment was patterned after the laboratory experiment of Tarara and Ham (1997); however, improved techniques were used in the design and construction of both the DPHP sensors and the data acquisition and control system (DACS).

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

 
Campbell et al. (1991) proposed a sensor with two parallel, cylindrical probes. One probe contained a thermocouple and the other contained enamel-coated resistance wire that was used to introduce a heat impulse. By assuming that the sensor approximates instantaneous heating of an infinite line source in isothermal, homogeneous soil, they developed an inverse relationship between the maximum temperature rise at the temperature probe and the volumetric heat capacity of the medium. The relationship is

[1]

where C is the volumetric heat capacity (J m^-3 °C^-1), q is the heat input per unit length of heater (J m^-1), and T_m is the maximum temperature rise (°C) observed at a radial distance r from the heat source (m). The constant r is treated as an apparent rather than actual probe spacing. Apparent probe spacing for each sensor is obtained by making measurements in a medium of known heat capacity and calculating r. Only measurements of q and T_m are then needed to calculate C from Eq. [1].

Campbell et al. (1991) also suggested that measurements of C might be useful for measuring soil water content. Upon assuming that the heat capacity of the soil gas phase is negligible, C becomes a weighted sum of the heat capacities of soil water and soil solid constituents (Kluitenberg, 2002)

[2]

where C_w is the volumetric heat capacity of water (J m^-3 °C^-1), is the volumetric water content (m³ m^-3), _b is the bulk density (kg m^-3), and c_s is the specific heat of the soil solid (mineral and organic) constituents (J kg^-1 °C^-1). Substituting Eq. [2] into Eq. [1] and rearranging gives

[3]

which indicates that can be estimated from measurements of q and T_m, provided that _b and c_s are known. Bristow et al. (1993) showed that change in water content () can be obtained by using

[4]

where the subscript 0 refers to the initial reading and the subscript i refers to the ith reading taken at some later time. This relationship has the obvious advantage that measurements of _b and c_s are not required; however, Eq. [4] is valid only if _b and c_s remain constant in time.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

 
Description of Soils
Samples were obtained from the A horizon of six soils. These soils were mapped as Wymore silty clay loam (fine, smectitic, mesic Aquertic Argiudolls), Chase silty clay loam (fine, smectitic, mesic Aquertic Argiudolls), Pawnee clay loam (fine, smectitic, mesic Oxyaquic Vertic Argiudolls), Olmitz loam (fine-loamy, mixed, superactive, mesic Cumulic Hapludolls), Haynie sandy loam (coarse-silty, mixed, superactive, calcareous, mesic Mollic Udifluvents), and Kahola silt loam (fine-silty, mixed, mesic Cumulic Hapludolls). A United States Golf Association root zone greens mix was also sampled. The greens mix was being used in the construction of the Colbert Hills golf course (Manhattan, KS) and is identified herein as Colbert Hills sand. It consisted of approximately 90% sand and 10% sphagnum peat on a volume basis.

All soil materials were air-dried, ground, and passed through a 2-mm sieve. Particle-size distribution was obtained by the hydrometer method (Gee and Bauder, 1986). Organic matter content was determined by the Walkley–Black method (Combs and Nathan, 1997). Measurements of c_s were performed by the Thermophysical Properties Research Laboratory, Inc., West Lafayette, IN, using a standard Perkin-Elmer (Wellesley, MA) Model DSC-2 Differential Scanning Calorimeter with sapphire as the reference material (ASTM, 1994).

Sensor Construction
Sixteen DPHP sensors were constructed for this experiment. See Kluitenberg (2002) for a detailed schematic diagram of a sensor. Housings for the heater and temperature probes were 35.6-mm-long sections of 18-gauge, stainless-steel tubing (1.27-mm o.d., 0.84-mm i.d.) with a single flared end (fabrication by Small Parts, Inc., Miami Lakes, FL). Heater probes were constructed by threading enamel-coated resistance wire (79-µm-diam., 205 m^-1, Nichrome 80 Alloy, Pelican Wire Co., Naples, FL) through a housing four times, resulting in two loops within the housing and a resistance of 820 m^-1 of heater probe. The total resistance of completed heater probes was approximately 33 . Temperature probes were constructed by placing a thermistor (Model 10K3MCD1, 0.46-mm-diam., 10 k at 25°C, Betatherm Corp., Shrewsbury, MA) inside a housing such that the thermistor was positioned 13.5 mm from the unflared end. The heater and temperature probes were filled with an epoxy (Omegabond 101, Omega Engineering, Stamford, CT) that has a high thermal conductivity and provides excellent electrical insulation. The lead wires extending from the heater and thermistor probes were soldered to a 6-conductor, 24-American wire gauge, ribbon cable that was used to wire the completed sensor to the DACS. The heater and temperature probes were held parallel with a spacing of approximately 6 mm by inserting them into polyvinyl chloride (PVC) blocks. The unflared ends of the heater and temperature probes protruded 27 mm from the PVC block. The lead wires of the heater and temperature probes were trimmed so that the connection to the ribbon cable was contained within the cavity of the PVC block. Upon filling the cavity with epoxy (Omegabond 101), the finished sensors were waterproof and electrically insulated.

Data Acquisition and Control System
The DACS, designed to accommodate 16 DPHP sensors, consisted of a datalogger (Model CR10X, Campbell Scientific, Logan, UT), two multiplexers (Model AM416, Campbell Scientific, Logan, UT), a 1- current-sensing resistor (Model VPR5, 0.1% tolerance, Vishay Resistors, Malvern, PA), a 5-k bridge resistor (Model S102K, 0.1% tolerance, Vishay Resistors, Malvern, PA), and a 12-V battery. All heater probes were wired to one multiplexer, and all temperature probes were wired to the other. The datalogger was used to control the heat pulse and measure temperature, and the multiplexers were used to switch between probes. The switched 12-V terminal of the datalogger was used to apply power to each heater probe for 8 s. This yielded a heat input per unit length of q 0.9 kJ m^-1. Heat input was measured by placing the 1- resistor in series with the heater probe. The current-sensing resistor had two leads for carrying current and two leads for measuring voltage drop. Ohm's law was used to determine current from measured voltage drop. Current was measured at 0.5-s intervals during heating and integrated over time to yield q. The thermistors were measured using a four-wire half-bridge circuit that utilized the 5-k resistor as a reference. Resistance measurements were converted to temperatures using the Steinhart–Hart equation and factory coefficients (BetaTHERM Corp., 1994). Ham and Tarara (1998) showed that sufficient accuracy in temperature measurement can be achieved without using individual calibration curves for each sensor. Initial temperature was measured immediately before applying the heat input. Temperature was then monitored at 0.5-s intervals for 70 s after heating was terminated. T_m was calculated from the maximum temperature observed after heating.

Tempe Pressure-Cell Setup
Eight Tempe pressure cells (Model 1405, Soilmoisture Equipment Corp., Santa Barbara, CA) were used. Each was fitted with a 0.1-MPa high-flow ceramic plate (Model 1435B1M3, Soilmoisture Equipment Corp.), a 8.55 cm i.d. by 6 cm long brass cylinder (Model 1426L6, Soilmoisture Equipment Corp.), and two DPHP sensors. The ribbon cable for each sensor was passed through a horizontal slit, sealed with epoxy, 1.5 cm from the top of the brass cylinder. Sufficient lengths of ribbon cable were left inside the cylinder to allow horizontal placement of the sensors at 2 and 4 cm below the top of the ring. A connector was attached to each ribbon cable immediately outside of the cylinder so that the pressure cell could be disconnected from the DACS for weighing. The pressure cells and DACS were placed in a constant-temperature room maintained at 25°C.

Dual-Probe Heat-Pulse Sensor Calibration
Each sensor was calibrated in water-saturated glass beads. Dry beads were added to each cell in 1-cm increments. The cell was shaken from side to side following the addition of each increment to maximize density. The beads were saturated from below during a 12-h period to minimize air entrapment. The specific heat of the dry glass beads (0.794 kJ kg^-1 °C^-1) was determined by the Thermophysical Properties Research Laboratory, Inc., West Lafayette, IN. Measurements of q and T_m were collected at hourly intervals for 24 h, but the measurements were staggered so that there was a 30-min delay between measurements taken with sensors in the same pressure cell. This allowed the water-saturated beads to come to thermal equilibrium between measurements. Apparent probe spacing was calculated by solving Eq. [1] for r, with C = 2.82 MJ m^-3 °C^-1, the volumetric heat capacity of the water-saturated glass beads. Average apparent probe spacing for the 24 sensors was 5.88 mm, with a standard deviation of 0.11 mm.

Pressure-Cell Experiment
Each soil was mixed to a water content suitable for packing using a 5 mM CaSO₄ solution. Moist soil then was added to the cells in 1-cm increments to achieve the target bulk density for each soil (Table 1). After packing each increment, the soil surface was roughened to minimize layering effects. The sensors were placed in the soil horizontally at depths of 2 and 4 cm from the top of the cylinder. The pressure cells were wetted from below with 5 mM CaSO₄ solution, allowed to equilibrate for 24 h, and then attached to a manifold that was connected to a regulated air pressure source. Pressure was measured using water and Hg manometers. Each cell was desorbed for 24 h, weighed, and then desorbed at another pressure increment in the following increments: 2.5, 5.0, 7.5, 10.0, 12.5, 15, 25, 35, 50, 75, and 100 kPa. Measurements were obtained with the DPHP sensors in each cell at 2-h intervals. Each pressure cell was dismantled after it was desorbed to 100 kPa, and gravimetric soil water content (kg kg^-1), _g, was determined. The procedures of Reginato and van Bavel (1962) and Klute (1986) were used to convert the value of _g at 100 kPa to estimates for each pressure increment.

View this table: [in this window] [in a new window]   Table 1. Particle-size distribution, organic matter content (OM), and specific heat (cs) of soil materials used in this study. Also shown is the bulk density (b) to which soil materials were packed. Values of cs in parentheses were estimated using Eq. [7].

Two desorption sequences were required to complete all measurements. The first sequence included two cells each of the Olmitz, Pawnee, Wymore, and Chase soils. The second sequence included two cells each of Kahola, Colbert Hills sand, Haynie at _b = 1.25 Mg m^-3, and Haynie at _b = 1.35 Mg m^-3. Results for the Haynie at _b = 1.25 Mg m^-3 are not reported here due to excessive settling.

To evaluate the DPHP sensors at water contents lower than those achieved in the desorption sequences, soil was wetted to known water contents and packed into each pressure cell. The cells were then attached to the DACS, and automated water content readings were taken for 24 h. Soil samples for _g determination were collected as described above. New soil wetted to a different water content was then packed into each pressure cell and the process was repeated. Measurements of were not calculated because the DPHP sensors were not operated continuously during this time.

Error Analysis
It is evident from Eq. [3] that errors in measuring or estimating _b, c_s, q, T_m, and r will be manifested in computed values of . First-order error analysis (Kempthorne and Allmaras, 1986, p. 20) of Eq. [3] yields

[5]

The partial derivatives (sensitivity coefficients) in Eq. [5] can be obtained by differentiation of Eq. [3]. Thus, Eq. [5] becomes

[6]

This relationship can be used to approximate absolute errors in that result from absolute errors in measuring or estimating the input parameters.

	RESULTS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

 
Dual-Probe Heat-Pulse vs. Pressure-Cell Estimates
Water content measurements are shown for desorption sequences performed with the Chase and Wymore soils (Fig. 1). Similar results were obtained for the other soil materials. The lines represent measurements from individual DPHP sensors, and the points represent values calculated from weight changes of the pressure cells. Although there was a 2-cm difference in soil water potential between the two DPHP sensors in each pressure cell, a t-test (P = 0.05) showed that there was no statistically significant depth effect in the DPHP measurements. Water contents obtained by the two methods closely followed the same trends during all the desorption sequences. In general, the DPHP method overestimated throughout the desorption process for all soils, with the exception of the measurements taken at 2.5 kPa. There also appeared to be a clear trend for increased overestimation at the dry end of the desorption sequence for many of the soils.

View larger version (17K): [in this window] [in a new window]   Fig. 1. Volumetric water content () as a function of time from pressure-cell desorption measurements with (a) Chase and (b) Wymore soils. Lines represent measurements obtained with dual-probe heat-pulse sensors. Points represent measurements obtained using the pressure-cell method. Pressure steps ranged from 2 to 100 kPa at 24-h intervals. Results for other soils were similar.

View larger version (57K): [in this window] [in a new window]   Fig. 2. Comparison of volumetric water content () as determined by dual-probe heat-pulse (DPHP) and pressure-cell methods for the (a) Olmitz, (b) Pawnee, (c) Wymore, and (d) Chase soils. Additional results for regression analysis (lines not shown) are given in Table 2. Comparison of volumetric water content () as determined by dual-probe heat-pulse (DPHP) and pressure-cell methods for the (e) Kahola, (f) Colbert Hills, and (g) Haynie soils. Additional results for regression analyses (lines not shown) are given in Table 2.

View this table: [in this window] [in a new window]   Table 2. Results from statistical analysis of relationship between dual-probe heat-pulse and pressure-cell methods for soils used in the experiment. SEslope is the standard error of the slope parameter and SEintercept is the standard error of the intercept parameter. RMSE is the root mean square error for each model. SSerror is the sum of squares error for each model with n - 2 degrees of freedom.

Data for the seven soils were pooled to determine the overall agreement between the two methods of measuring (Fig. 3). There was good agreement between the two methods for a wide range of water contents (0.02–0.59 m³ m^-3). As in the analysis with the individual soils, the pooled data exhibit bias toward overestimation of with the DPHP method at lower water contents. Linear regression analysis with the pooled water content data (Table 2) yielded an RMSE of 0.023 m³ m^-3. A t-test (P = 0.05) indicated that the slope of the pooled model was not equal to one and the intercept was not equal to zero.

View larger version (25K): [in this window] [in a new window]   Fig. 3. Comparison of volumetric water content () as determined by dual-probe heat-pulse (DPHP) and pressure-cell methods for all seven soils combined. Additional results for regression analysis (line not shown) are given in Table 2.

Dual-Probe Heat-Pulse vs. Pressure-Cell Estimates
A comparison of values obtained by the two methods is shown in Fig. 4. As noted above, water contents collected at 100 kPa were taken to be the initial water contents in computing values of . Thus, in general, smaller values correspond to water contents late in the desorption sequences; larger values of correspond to water contents early in the desorption sequences. The range of the measurements (Fig. 4) is considerably smaller than the range of the measurements (Fig. 3). This is because could be calculated only for measurements obtained during the desorption experiments.

View larger version (26K): [in this window] [in a new window]   Fig. 4. Comparison of change in volumetric water content () as determined by dual-probe heat-pulse (DPHP) and pressure-cell methods for all seven soils combined. Additional results for regression analysis (line not shown) are given in Table 3.

View this table: [in this window] [in a new window]   Table 3. Results from statistical analysis of relationship between dual-probe heat-pulse and pressure-cell methods for soils used in the experiment. SEslope is the standard error of the slope parameter and SEintercept is the standard error of the intercept parameter. RMSE is the root mean square error for each model. SSerror is the sum of squares error for each model with n - 2 degrees of freedom.

Linear regression analysis was performed with the data for each soil (Table 3). The RMSEs ranged from 0.006 to 0.014 m³ m^-3, values noticeably smaller than those for the water content regression analyses. Inferences regarding slopes and intercepts of the regression models were made with separate t-tests (P = 0.05). For all seven soils, it was concluded that slopes were not equal to one, and intercepts were not equal to zero.

Linear regression analysis with the pooled data (Fig. 4, Table 3) showed good agreement (RMSE = 0.012 m³ m^-3) between the two methods for the range in from 0.00 to 0.33 m³ m^-3. Inferences regarding slope and intercept of the regression model were made with a t-test (P = 0.05). It was concluded that slope was not equal to one, and intercept was not equal to zero. A model comparison test was also performed for the data, which also showed that the pooling of the data was not statistically justified.

First-Order Error Analysis
First-order error analysis can be used to determine how errors in measuring or estimating _b, c_s, q, T_m, and r are manifested in estimates of . Table 4 shows the sensitivity coefficients from Eq. [6] for typical values of input parameters: _b = 1.3 Mg m^-3, c_s = 0.8 kJ kg^-1 °C^-1, q = 0.9 kJ m^-1, and r = 6 mm. Because some of the sensitivity coefficients depend on , coefficients were computed for = 0.0 m³ m^-3, = 0.3 m³ m^-3, and = 0.6 m³ m^-3. Values of T_m corresponding to this set of constants were determined to be 2.81°C ( = 0.0 m³ m^-3), 1.28°C ( = 0.3 m³ m^-3), and 0.83°C ( = 0.6 m³ m^-3) by using Eq. [3] with C_w = 4.18 MJ m^-3 °C^-1.

View this table: [in this window] [in a new window]   Table 4. Sensitivity coefficients from Eq. [6] for typical values of input parameters: b = 1.3 Mg m-3, cs = 0.8 kJ kg-1 °C-1, q = 0.9 kJ m-1, r = 6 mm, Tm = 2.81°C ( = 0.0 m3 m-3), Tm = 1.28°C ( = 0.3 m3 m-3), and Tm = 0.83°C ( = 0.6 m3 m-3). Sensitivity coefficients are given as a function of because some of the coefficients are -dependent. Sensitivity coefficients were used to compute absolute errors in () for 5% positive biases in b, cs, q, and r, and a positive bias of 0.05°C in Tm.

A slightly different approach is needed to illustrate how is affected by bias in T_m. Because T_m takes on different values for different values of , bias in T_m was achieved by using T_m = 0.05°C for all values of T_m. This indicates a positive bias of 0.05°C in T_m, which corresponds to overestimates of approximately 2, 4, and 6% for T_m = 2.81, 1.28, and 0.83°C, respectively. For example, multiplying T_m = 0.05°C by the sensitivity coefficient -0.089°C^-1 yields -0.004 m³ m^-3 at = 0.0 m³ m^-3 (Table 4). Thus, a positive bias of 0.05°C in T_m results in a negative bias of 0.004 m³ m^-3 in water content at = 0.0 m³ m^-3. Like the sensitivity coefficients for q and r, the sensitivity coefficients for T_m increase in magnitude with increasing .

	DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

 
Dual-Probe Heat-Pulse Estimates of
The results (Fig. 2 and 3, Table 2) indicated good agreement between the DPHP method and the pressure-cell method for measuring , but there was consistent bias toward overestimation of at lower water contents. It is unlikely that bias in any one of the parameters _b, c_s, q, T_m, or r caused the consistent bias observed in Fig. 2 and 3. This can be concluded by examining the sensitivity coefficients obtained from the first-order error analysis (Table 4). The sensitivity coefficients for _b and c_s are constants (not dependent on ), so bias in the values of _b or c_s would result in offset error rather than the bias evident in Fig. 2 and 3. The sensitivity coefficients for q, T_m, and r, on the other hand, increase in magnitude with increasing . Thus, bias in the values of q, T_m, or r would cause bias in the data of Fig. 2 and 3. Regression slopes <1.0 would be obtained if values of q were negatively biased, or if values of T_m or r were positively biased. However, these biases in q, T_m, or r would also result in underestimates of at low water contents. That is, they would result in negative intercepts in the regression relationships. We conclude, therefore, that the bias observed in Fig. 2 and 3 is not the result of bias in any single input parameter. Although some combination of biased input parameters could cause the consistent bias observed in Fig. 2 and 3, this explanation is unlikely.

A more likely explanation for the bias observed in Fig. 2 and 3 is that errors in T_m increased proportionally with increasing T_m (decreasing ). Calculations with the same parameter values used to compute the sensitivity coefficients (Table 4) show that the bias in Fig. 2 and 3 can be reproduced if error in T_m increases nonlinearly from <1% at T_m = 0.83°C ( = 0.6 m³ m^-3) to approximately 14% at T_m = 2.81°C ( = 0.0 m³ m^-3). This error is distinctly different from the T_m error assumed in illustrating the utility of the first-order error analysis. Contact resistance offers one possible explanation for nonlinearly increasing error in T_m as water content decreases and T_m increases. It is reasonable to expect that contact resistance between the soil and temperature probe increases as water content decreases. Another possible explanation is nonideality of the temperature measurement because of differences between the thermal properties of the soil and the temperature probe. The mismatch in thermal properties will change with water content owing to the dependence of soil thermal properties on water content. Vapor distillation effects also may play a role. Noborio et al. (1996) used numerical simulations to investigate the contribution of thermally induced vapor movement for water contents ranging from 0.1 m³ m^-3 to saturation. Vapor distillation had a minimal influence on T_m for much of the water content range, but the authors reported that simulated values of T_m were appreciably greater as water contents approached 0.1 m³ m^-3. Clearly, additional work is needed to determine if any of these mechanisms cause nonlinear increases in T_m error with decreasing water content.

The bias observed in Fig. 2 and 3 is consistent with that reported by Tarara and Ham (1997) and Song et al. (1998). Bristow et al. (2001) also observed bias toward overestimation of at lower water contents, but the bias in their results was not as pronounced. These results and ours, all obtained with mineral soil materials, contrast with the results of Campbell et al. (2002), who evaluated the DPHP method in two peat soils. Although their evaluation did not include water contents lower than approximately 0.15 m³ m^-3, their results showed no evidence of bias in water content estimates. These contrasting observations regarding bias may be related to the vastly different physical properties of peat and mineral soils. All of the mechanisms that might cause nonlinear increases in T_m error with decreasing water content could be manifested differently in peat soils than in mineral soils. This issue is clearly deserving of attention in future experimental work with DPHP sensors.

It would be desirable for the DPHP method to be completely unbiased, but the presence of bias does not limit the utility of the method, if the bias is consistent. The pooled regression model (Fig. 3, Table 2) suggests that a single calibration relationship may hold for soil materials with a broad range of physical properties. Although it was concluded that pooling of the seven regression models is not justified statistically, the pooled model still may have practical utility. Rearranging the pooled regression model yields the general calibration relationship = 1.09 _DPHP - 0.045. It appears that this general relationship can be used to eliminate bias in DPHP measurements. It also eliminates the need for soil specific calibration.

An evaluation of the DPHP method also must take into account the measurement precision that can be achieved with the method. This requires an examination of the degree of scatter about the regression models shown in Fig. 2 and 3, which can be characterized by the RMSEs for the regression models (Table 2). The RMSEs ranged from 0.01 to 0.029 m³ m^-3 in the regression relationships for individual soils, and the pooled model had an RSME of 0.023 m³ m^-3. These RMSEs indicate that excellent precision can be achieved with the DPHP method.

It is important to note that the calculated RMSE values may underestimate the precision that can be achieved with the DPHP method. Standard linear regression theory is based on the assumption that the independent variable is known without error. The independent variable in this case is as determined by the pressure-cell method, but this method is subject to at least two possible sources of error. Weighing error is one possible source of uncertainty. Another possible source of uncertainty results from the fact that the soil volume characterized with the pressure-cell method (334 cm³) was more than two orders of magnitude greater than the soil volume characterized by the DPHP sensors (<1 cm³) installed in each pressure cell. Thus, the calculated RMSE values may be inflated due to the presence of error in the independent variable.

Dual-Probe Heat-Pulse Estimates of
The DPHP and pressure-cell methods for measuring agreed well (Fig. 4, Table 3), but there was bias toward underestimation of at higher values. As noted in the analysis of the results, this most likely resulted from errors in T_m. The pooled regression analysis (Fig. 4, Table 3) yielded a slope of 0.93 and an intercept of -0.005. These values are reasonably close to expected values. Because values are obtained by differencing values, the regression slope for the data in Fig. 4 is expected to be the same as the regression slope for the data in Fig. 3 (0.92). Likewise, a zero intercept is expected.

If the DPHP estimates had been "corrected" with the general calibration relationship, = 1.09 _DPHP - 0.045, before differencing, the expected calibration relationship would have a slope of 1.09 and a zero intercept. These values are also close to expected values obtained by rearranging the pooled regression model of Fig. 4 ( = 1.08 _DPHP + 0.005), thus supporting the validity of the general calibration relationship, = 1.09 _DPHP - 0.045, derived from Fig. 3.

Root mean square errors ranged from 0.006 to 0.014 m³ m^-3 in the regression relationships for individual soils, and the pooled model had an RMSE of 0.012 m³ m^-3 (Fig. 4, Table 3). Comparison of these values with the RMSEs for the regression analyses (Table 2) reveals that greater precision was achieved in measuring than in measuring . However, examination of Fig. 4 shows that the analysis does not extend over the entire range of possible values, and there appears to be a marked increase in scatter with increasing .

The fact that can be measured with greater precision than is noteworthy because there are many situations in which change in water content is the desired quantity of interest. An important example is water uptake by plant roots (Song et al., 1998, 1999). The excellent precision of the method, coupled with its relatively fine spatial resolution, suggests that it may be useful for measuring spatial patterns of soil water uptake by plant roots.

Implementation of the Dual-Probe Heat-Pulse Method
In this investigation, measured values of c_s were used in Eq. [3] to estimate with the DPHP method. For routine implementation of the DPHP method, it may not be practical to measure c_s. However, good estimates of c_s can be obtained for mineral soils if the organic matter content is known. This is because there is generally minimal variation in the specific heat of the mineral and organic constituents of soils. The specific heat of the soil solid constituents, c_s, can be estimated from (Kluitenberg, 2002):

[7]

where c_m and c_o are the specific heats (J kg^-1 °C^-1) of the mineral and organic soil constituents, respectively. The mass fraction (dry-weight basis) of the soil mineral constituents, _m, can be calculated from _m = 1 - _o, where _o is the mass fraction of the soil organic constituents, or the organic matter content.

Table 1 shows estimated values of c_s in parentheses, following the measured values. These values were obtained using Eq. [7] with c_o = 1.9 kJ kg^-1 °C^-1 (Kluitenberg, 2002) and c_m = 0.74 kJ kg^-1 °C^-1. The value for c_m was obtained by assuming a temperature of 25°C and employing the fact that specific heat varies linearly from 0.67 kJ kg^-1 °C^-1 at -18°C to 0.79 kJ kg^-1 °C^-1 at 60°C (Kluitenberg, 2002). Estimated specific heats were lower than measured specific heats for all soil materials except the Colbert Hills sand, but the differences were relatively small. Differences ranged from 0.3% for the Olmitz soil to 5.2% for the Wymore soil.

The expected error in estimates caused by error in c_s can be estimated by using the sensitivity coefficient of -0.31 kg °C kJ^-1 for c_s (Table 4) that was obtained from the first-order error analysis. The deviation between measured and estimated specific heats for the Olmitz soil was -0.002 kJ kg^-1 °C^-1. Multiplying this by the sensitivity coefficient yields an expected error of 0.0006 m³ m^-3. The deviation between measured and estimated specific heats for the Wymore soil was -0.044 kJ kg^-1 °C^-1, which yields an expected error of 0.014 m³ m^-3. These results suggest a likely range of water content error (0.0006–0.014 m³ m^-3) that can be expected when implementing the DPHP method with estimated, rather than measured, specific heats. Unfortunately, this approach has limited practical value for predicting error due to the use of estimated specific heat for a specific soil because it requires knowledge of the true (measured) specific heat.

An alternative approach for evaluating the effect of error in specific heat is to examine error in an average sense, for all soil materials pooled together. This was explored by using estimated instead of measured values of c_s in the DPHP method. That is, estimated values of c_s were used in Eq. [3] instead of measured values when calculating water contents. Comparison of the recalculated and original values (Fig. 5) shows that slight bias and offset errors were introduced by using estimated specific heats. The offset error is due to the fact that estimated specific heats were generally lower than measured specific heats. The bias results from the fact that the deviations between estimated and measured specific heats were generally greater for soils with greater specific heat (Table 1). The regression results (Fig. 5) show that the combination of bias and offset errors was relatively small (0.001 m³ m^-3) at = 0.0 m³ m^-3, but increased to 0.007 m³ m^-3 at = 0.6 m³ m^-3. For an arbitrary water content, the combination of bias and offset errors is 0.01 + 0.001 m³ m^-3. Using estimated specific heats also caused a loss of precision, characterized by the regression RMSE of 0.006 m³ m^-3. Thus, it appears that using estimated instead of measured specific heats in the DPHP method will degrade accuracy by 0.01 + 0.001 m³ m^-3 and will degrade precision by 0.006 m³ m^-3.

View larger version (20K): [in this window] [in a new window]   Fig. 5. Comparison of volumetric water content () as determined by the dual-probe heat-pulse (DPHP) method using estimated specific heats and the DPHP method using measured specific heats. See Table 1 for a comparison of measured and estimated specific heats.

The intent of the present investigation was to establish the accuracy and precision of the DPHP method under optimal conditions. To that end, DPHP sensors were placed in the soil during packing to minimize deflection of the heater and temperature probes, and measurements were conducted under isothermal conditions. Probe deflection is an issue of potential concern in a field setting if the sensors are pushed into undisturbed soil. It is well known that measurements of C, , and are sensitive to errors in probe spacing caused by probe deflection (Campbell et al., 1991; Bristow et al., 1993; Kluitenberg et al., 1993). However, the extent of probe deflection that can be expected in a field setting has not been established. Significant probe deflection will undoubtedly result in lower precision than that reported herein, and accuracy may be compromised if deflection introduces bias toward either larger of smaller probe spacings. The characteristically nonisothermal field environment is also an issue of potential concern because the theory underlying the DPHP method assumes isothermal conditions. To date, no attempt has been made to evaluate how the DPHP method is influenced by nonisothermal conditions. Clearly, additional work is needed on both of these fronts to establish the accuracy and precision that can be achieve with the method in field settings.

Other potential sources of concern in field implementation are loss of probe–soil contact and convective heat transfer caused by soil water movement. To date, our experience with field measurements has revealed no evidence of poor probe–soil contact. Soil water movement also appears to be an issue of minor importance, except in the presence of extremely high soil water flux densities (Kluitenberg and Heitman, 2002). Thus, we speculate that these issues will not significantly degrade the accuracy and precision that can be achieved with the DPHP method in the field.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

 
Estimates of obtained with DPHP sensors were slightly biased toward overestimation at low water contents, but it was shown that this bias can be removed by using an empirical calibration equation, = 1.09 _DPHP - 0.045. This relationship appears to be general inasmuch as it was derived from measurements for a wide range of water contents in mineral soil materials with a wide range of textures, organic matter contents, bulk densities, and specific heats. Thus, the empirical calibration equation allows for unbiased estimates of with the DPHP method and eliminates the need for soil specific calibration. The general calibration equation also was shown to be effective in removing bias in estimates.

The results also indicate that excellent precision can be achieved in measuring of and with the DPHP method. The pooled regression results suggest that can be measured with an RMSE of 0.023 m³ m^-3. Greater precision can be achieved with measurements (RMSE = 0.012 m³ m^-3); however, decreases in precision can be expected as the magnitude of increases.

For routine implementation of the DPHP method, it may not be practical to obtain measurements of soil specific heat. This is not an issue when using the method to measure because knowledge of the specific heat is not required. However, small losses in both accuracy and precision can be expected when estimated values of specific heat are used to measure . Our analysis shows that accuracy will degrade by 0.01 + 0.001 m³ m^-3, and precision will degrade by 0.006 m³ m^-3.

The intent of this investigation was to establish the accuracy and precision of the DPHP method under optimal conditions. Additional work is needed to evaluate the accuracy and precision that can be achieved with this method in a field setting. Issues of particular concern are the potential for probe deflection upon insertion of DPHP sensors and nonisothermal soil conditions.

	ACKNOWLEDGMENTS

 
Financial support for this research was provided by the Kansas Center for Agricultural Resources and the Environment (KCARE). The authors gratefully acknowledge Joshua Heitman for assisting with the laboratory experiment. We also thank Dr. Jan Hopmans for valuable comments on an earlier version of this manuscript.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS REFERENCES

 

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