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Published in Soil Sci. Soc. Am. J. 68:47-56 (2004).
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

DIVISION S-1—SOIL PHYSICS

The Effects of Salinity on the Accuracy and Uncertainty of Water Content Measurement

W. R. Hooka, T. P. A. Ferré*,b and N. J. Livingstona

a Dep. of Biology, Univ. of Victoria, Victoria, BC V8W 3N5, Canada
b Dep. of Hydrology and Water Resources, Univ. of Arizona, Tucson, AZ 85721-0011

* Corresponding author (ty{at}hwr.arizona.edu).


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 THEORY
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
We used an automatic network analyzer (ANA) operated in both time and frequency domain modes to investigate the measurement accuracy of metallic time domain reflectometry (TDR) probes operated in sands saturated with NaCl solutions of varying electrical conductivity (EC). We chose to use time domain transmission (TDT) measurements for this investigation to separate the effect of the bulk soil-probe interaction from the effect of the large reflection typically found at the air–soil boundary for a TDR configuration. Pulse travel times and their variability increase with increasing pore-water EC. The source of travel time variability arises from the extreme variability in pulse shape, thereby introducing a high degree of uncertainty to curve fitting routines used to determine travel times. Pulse shape distortion is due primarily to attenuation of high frequency components through conductive loss rather than by dispersion. There is generally good correspondence between pulse rise time and the average water content measurement error over a 0- to 40-dS m–1 EC range. For rise times <6 ns, measurement errors are <0.1 m3 m–3. Given that rise times can be determined easily, we recommend that they be reported routinely as indicators of data quality. Because the variability of travel time measurements at a given EC is on the same order as the average difference between the travel time measured at that EC and that measured at zero salinity, pulse rise times cannot be used to correct for individual travel time measurements at high salinity. Similarly, knowledge of the EC of the medium may allow for identification of erroneous water content measurements, but does not allow for error correction.

Abbreviations: ANA, automatic network analyzer • EC, electrical conductivity • FFT, fast fourier transform • GPR, ground penetrating radar • RF, radio frequency • TDR, time domain reflectometry • TDT, time domain transmission


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
THEORY
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
TIME DOMAIN ELECTROMAGNETIC METHODS, including TDR, TDT, and ground penetrating radar (GPR) provide effective rapid nondestructive means of determining volumetric water content in a wide variety of porous media. With its establishment as a widely used water content measurement technique, TDR is being extended to more challenging environments. One area of continuing limitation is the measurement of water content in media having significant attenuation of the desired signal (lossy media), such as in saline soils, certain clay soils, mine tailings, and compost. The ability to accurately measure water content in media that have salinity in the 10- to 40-dS m–1 range is important for salt-tolerant agricultural operations such as cotton farming, as well as for a variety of hydrogeologic and processing applications. Time domain reflectometry waveforms collected with standard uncoated continuous rod probes under such conditions show a decrease in the amplitude of the reflected pulse and corresponding degradation of the pulse shape. Water content measurement errors associated with these effects are inconsistent, with investigators showing results ranging from overestimates (Hook and Livingston, 1995; Wyseure et al., 1997; Sun et al., 2000) to no observed dependence (Topp et al., 1980; Dalton et al., 1984, Topp et al., 1988, Kelly et al., 1995). The introduction of probes employing shorting diodes and the use of waveform subtraction techniques (Hook et al., 1992) improve travel time determinations. However, water content measurement errors persist even when advanced diode shorting techniques are used (Sun et al., 2000). This suggests that these errors are due to fundamental changes in the TDR waveforms rather than incorrect waveform analysis.

The primary objective of this study is to determine the performance limits of uncoated TDR probes for measuring water content in high EC sands. Two considerations are made when addressing this objective. First, we establish whether pulse travel time increases systematically with increasing salinity and, if so, whether a particular property of TDR waveforms or independent knowledge of the pore-water EC could be used to identify and/or correct for these increases. Second, we establish whether a particular property of TDR waveforms could be used to quantify the uncertainty in travel time measurements, thereby allowing users to set a limit of allowable water content measurement error for specific applications.

Our test media was sand saturated with distilled water or solutions of NaCl with electrical conductivities ranging from 0 to 40 dS m–1. Clean sand was selected to minimize the effects of surface conduction, bound water, and temperature dependence on the measurements (Wraith and Or, 1999). The volumetric water content of the sand was maintained at full saturation for two reasons. First, one of our objectives was to investigate the variability in travel time measurements under the most controlled conditions. Saturated conditions give rise to the most spatially uniform distribution of soil water for any homogeneous medium, minimizing the effects of variability in packing on the measured results. This provides the most conservative measure of the effects of small-scale heterogeneities in soil packing on trial-to-trial variability. Second, given that the EC is a function of the water content, spatial variability of the water content within the measurement volume of the probes will lead to a poorly defined bulk EC for a given pore-water salinity; saturated conditions minimize this potential error.

We used an ANA in the transmission mode as our primary measurement instrument. The ANA is equipped with a fast fourier transform (FFT) processor to simulate the response of a TDR-type step input. The combined time and frequency domain data provided by the ANA allow for greater insight into the effects of pore-water salinity on the response of uncoated rod TDR systems in saline soils. Operation in the transmission mode greatly reduces uncertainties associated with interference from reflections. For a reflection mode measurement, the reflection at the probe head interferes with the desired reflection from the end of the rods, which itself is dramatically reduced in amplitude with increasing salinity. Additionally, the superior signal/noise ratio capability of an ANA allows for measurements in more extreme saline environments than is possible with conventional analog time domain instruments. The ANA used in this study is an HP 8752C Network Analyzer (Hewlett Packard, Santa Rosa, CA), that makes measurements in the frequency range from 300kHz to 3 GHz.

We have included a derivation of the equations used for the calculation of water content from transmission measurements, as well as material on the theory of operation of the ANA, pulse distortion as related to dispersion, and pulse distortion as related to high frequency amplitude loss and the use of rise time to estimate such loss. In all equations we use the simple and intuitive variables of time interval, phase shift and amplitude loss, all of which are directly measured by the ANA.

The concept of an apparent dielectric constant (Ka) and the extensive analytic work that has flowed from this concept cannot be used for saline soils because this concept assumes there is no loss and thus the imaginary part of the dielectric constant is negligible. To apply our results to a theoretical or empirical model, it is necessary to employ the general dielectric constant equations, where both the real and the imaginary parts contain loss factors and time interval factors as well as frequency factors. The use of such complex models is beyond the scope of this paper.


   THEORY
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Relationship Between Water Content and Time Interval for the Transmission Mode
A linear equation for calculating volumetric water content, {theta} (m3 m–3) from a measured travel time in an ideal waveguide application, where it is assumed the waveguide is entirely surrounded by a porous medium such as soil, is (Hook and Livingston, 1996):

[1]
where tm is the time taken for an electromagnetic wave to pass through the soil, Tair is a normalizing factor, and ts is the time taken for a electromagnetic wave to pass through the same soil when it is oven dry ({theta} = 0). For a TDR application, Tair = 2L/c, and for a TDT application, Tair = L/c, where c is the speed of light in a vacuum and L is the length of the waveguide. In both cases, ts/Tair = 1.756 for the sand used in our experiments, and is about 1.5 to 1.6 for most agricultural soils (Hook and Livingston, 1996).

In standard TDR applications, the two-way travel time of an electromagnetic step pulse along a wave guide is determined from the time of arrival of characteristic reflections from the beginning (t1) and end of the probe (t2), and thus tm = t2 – t1, where t2 = tm + tcables and t1 = tcables, and where tcables is the two-way travel of the coaxial cables connecting the TDR instrument to the probe. For TDT the situation is more complex, since the effect of the cables cannot be directly subtracted. To isolate the travel time along the waveguide, two one-way travel time measurements are made: one with the waveguides embedded in the medium, tb, and a second with the waveguides in air, ta. These two travel times can be expressed as:

[2]

[3]
where tair is the time taken for the electromagnetic wave to pass through the air. For a waveguide closely approximating an ideal waveguide, such as the three-rod guide used in our experiments, tair may be closely approximated by the theoretical time Tair = L/c, and thus for a TDT application, the volumetric water content of the sand used in our experiments as a function of the measured travel time becomes:

[4]

For all water content calculations in this paper, Eq. [4] is used, and the definition of measured time interval is (tb – ta).

Automatic Network Analyzer
Automatic network analyzers are used to determine the reflection and transmission characteristics of devices and networks. They can measure either the reflection or transmission coefficients of a test device, both in amplitude and phase, as a function of frequency. A simplified diagram of an ANA transmission measurement system is shown in Fig. 1a . Such a system consists of a signal source, a signal separation unit, a processor, and a display unit. The ANA utilizes a built-in synthesized source to generate an alternating electromagnetic wave of a specific frequency to excite the test sample. In the transmission mode a sinusoidal signal is separated and fed through both a transmission line loaded with a soil sample, and a bypass route. The signal from the transmission line is called the transmitted signal and the signal from the bypass route is called the incident signal. The split signals are processed through a detector and a mixer for amplitude and phase measurements and filtered to increase the signal/noise ratio. Then the amplitude (Atrans) of the transmitted signal relative to the amplitude of the incident signal (Ainc), and the phase difference ({phi}, radians) between the transmitted and incident signal are converted from analog to digital signals. Both amplitude and phase information are measured simultaneously as a function of frequency.



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  		Fig. 1. (a) Schematic diagram of the ANA in transmission mode. A(f) and {phi}(f) are the amplitude and phase measured at each frequency. The primed variables represent digital samples of these results. A'(t) is the simulated TDR response. (b) Transmission line model of a TDR probe of length L.

 
The attenuation, A (dB), is defined as the ratio of the wave amplitudes (Fig. 2) :

[5]



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  		Fig. 2. Incident and transmitted sine waves with period, T. The time difference t between the phase of the incident wave and the phase of the transmitted wave is shown, as well as the linear amplitudes, Ainc and Atrans.

 
The phase is defined as the ratio of the one-way travel time, t (s), along the transmission line to the period T (s cycle–1) as:

[6]
where f is the frequency (cycle s–1). The measurement is repeated over a specified set of frequencies.

A number of discussions of the relationship between frequency and time domain measurements have been presented (Brisco et al., 1992; Heimovaara, 1994; de Winter et al., 1996; and Heimovaara et al., 1996). The HP 8752C employs a real-time FFT to convolve a digitized description of the stimulating pulse with a digitized description of the circuit or medium, resulting in a digitized time domain description of the resultant pulse. We used a step pulse to simulate the pulse transmitted from a conventional TDR instrument. All the time domain results presented in this paper have been derived from artificial waveforms reconstructed using FFT methods applied to frequency specific measurements.

The method used here differs from the approach of previous investigators (e.g., Heimovaara, 1994) who have digitized the time domain response from a sampling oscilloscope and performed an inverse FFT to construct the frequency domain response.

This inverse transformation approach is far less accurate than the direct use of an ANA primarily because with an ANA the data is processed and digitized using low frequency highly filtered heterodyne techniques (intermediate frequency filter bandwidth of 3 kHz in our case) whereas the sampling oscilloscope must employ very wide-band sample-and-hold circuits (a typical bandwidth is 2 GHz) to digitize the pulse. These wide-band circuits introduce much more distortion and noise than those used in the frequency domain. Our approach also differs from Campbell (1990) and Heimovaara (1996) who used an ANA in the reflection mode, which is not suitable for measurements in high loss soils.

Dispersion (Phase Distortion)
Dispersion, also called phase distortion, describes the source of degradation of the shape of a pulse due to differences in the time delay experienced by the different frequency components of a pulse. In the radio frequency (RF) portion of the spectrum, pulse degradation is most often seen in frequency filters, which are typically composed of such energy storage components as capacitors and inductors. Degradation is also seen under certain atmospheric transmission conditions, and at the edge of RF absorption bands such as with water at about 20 GHz as shown in the Debye relaxation curves. In the optical region, dispersion is the cause of the rainbow pattern seen with glass prisms, and is the principal cause of pulse degradation in high-speed glass fiber optic communication networks.

For a step pulse of the sort used for TDR measurements, which can be most easily represented and analyzed as a square wave, the Fourier representation (Thomas and Rosa, 1998) of the pulse is composed of only the odd harmonics as:

[7]
where f1 is the repetition rate of the pulse train, typically 10 kHz, and the highest frequency of interest, nf1, is typically 1500 MHz. Thus there are typically 150000 spectral components. When such a pulse is passed through a transmission line having the same travel time for each frequency component, each component experiences a phase shift {phi} = 2{pi}f t (Eq. [6]), and a plot of phase versus frequency is linear. A phase variation from linear is defined as the dispersion or phase distortion (Pozar, 1993; Budak, 1974; Taylor and Huang, 1997). The dispersion must be significantly greater than ±0.167{pi} radians (±30°) to have an effect on the pulse shape.

The dispersion, d{phi}, as a function of travel time variation, dt, is:

[8]

Therefore, variations in the travel time at high frequencies have a much larger effect on the pulse shape than those at lower frequencies. Significant dispersion creates a distortion of the pulse shape even if the transmission medium does not affect the amplitudes of the higher frequency components. For the particular step pulse described above, the phase of all the components must be zero when the travel time is zero. An intuitive understanding of this process may be obtained by plotting the first few harmonics, and observing how the sharp leading edge is built up as more harmonics are added. If the proper phase relationship is not maintained, the shape of the leading edge is degraded. A network that has a linear phase is also described as a constant time delay network, emphasizing the fact that all frequency components undergo the same time delay, and thus the original phase relationship is maintained. If significant dispersion is observed in a soil mixture, it is evidence that the mixture contains RF energy storage mechanisms more complicated than those associated with free water. Equation [8] can be rearranged to show that the change in travel time due to a change in phase is an inverse function of frequency, and thus the higher frequency components are more important in determining the overall time interval accuracy.

Amplitude Loss and the Definition of the Rise Time of a Time Domain Pulse
The second major source of TDR or TDT pulse shape degradation is the loss of the high frequency components, Bn in Eq. [7]. This is due to absorption of the RF energy by the surrounding medium rather than dispersion. Given that the travel time can be determined more precisely using the higher frequency components, it is useful to determine the frequency content of a transmitted time domain pulse. A convenient measure of the degree of smoothing of a time domain pulse, and hence the amount of high frequency loss, is the rise time (tr, ns). In this paper, we define the rise time as the difference between the pulse arrival time (tn, ns) and the time at which the amplitude reaches one half of its eventual maximum (t0.5, ns), as shown on Fig. 3 . This definition was chosen because straight-line curve fitting procedures, such as that shown to determine tn on Fig. 3, are common in TDR applications and t0.5 can be easily determined from TDR waveforms. Therefore, using this definition of rise time, commonly applied TDR curve fitting procedures can be adapted easily for rise time determinations. Based on the Tektronix user's manual for TDR's for cable testing (Tektronix Inc. Metallic TDR's For Cable Testing. Application Note, Tektronix Inc, Beaverton, OR) a good approximation of the 3-dB upper frequency limit (fmax) of a step TDR pulse is:

[9]



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  		Fig. 3. A typical transmission mode waveform collected with 0.28-m long rods in sand saturated with 10-dS m–1 NaCl solution. The fitted tangent lines used to determine the one-way pulse arrival time through the medium and cables (tn = 20.4 ns) are shown. The rise time, tr = 7 ns, is the time delay between tn and the time at which the waveform amplitude is half of its eventual maximum, t0.5 = 27.4 ns. The one-way travel time through the cables and an air-filled container is tn-air.

 
This formula applies only in the absence of significant dispersion. The term, fmax, assumes the signal has been passed through a medium, which removes the higher frequency components in a smooth and monotonic fashion. fmax is the frequency at which the power of a component has fallen to one-half that of the low frequency components, and is often referred to as the 3-dB point. The action of the medium is often referred to as a low-pass filtering effect.

Probe Configurations
Zegelin et al. (1989) described a three-rod TDR probe designed to closely approximate a coaxial probe for soil water content measurement while retaining the ability to be inserted into a soil sample with minimal disturbance. The probes used in our study were constructed following this model for direct comparison with standard time domain measurements and were comprised of three stainless steel rods, 0.32 m long, 0.03 m in diameter, separated by 0.085 m. The rods were aligned in a horizontal plane. The rods extended through two opposite faces of a 0.28 by 0.17 by 0.19 m Plexiglas box, allowing for connection to the ANA in a transmission configuration through 50-ohm RG-58 coaxial cables. In the transmission mode used for this investigation, the traveling wave traverses a distance equal to the length of the transmission line, L. Therefore, a transmission line of length 2L was used in this study to model the response of a TDR transmission line of length L, as shown on Fig. 1b. This transmission mode configuration accurately models both the reflection losses in a TDR probe as well as losses due to the interaction of the surrounding media with the traveling RF wave.

Experimental Methods
Measurements were made in clean sand saturated with distilled water or NaCl solutions with electrical conductivities of 10, 25, and 40 dS m–1. Solution conductivities were measured with a conductivity meter (Model # 4020, Jenway Limited, Essex, UK) at an average temperature of 21°C. Each saturated sand mixture was packed into the testing box for measurement. Based on Archie's relationship (Archie, 1942) for the sand used in this experiment, the bulk electrical conductivities associated with these pore-water electrical conductivities are 0.34, 0.85, and 1.35 dS m–1, respectively. The sand was packed to a height of at least 0.05 m above the horizontal plane in which the three rods were located to ensure that the sample volume of the probes did not extend beyond the soil surface. Solution was ponded to a depth of at least 0.02 m on the soil surface to assure full saturation. Measurements were made using different box and cabling configurations; measurements made with 21 varying sand packing sequences are presented in this study. The water content of the saturated sand, determined gravimetrically, was 0.38 m3 m–3.

Measurement parameters were set and data was retrieved from the ANA using a desktop computer and HP VEE software (Hewlett Packard, Santa Rosa, CA). Frequency domain data and the results of the FFT were collected as data files for numerical analysis. Graphical output was also produced from the FFT analysis for manual waveform interpretation. Three independent researchers conducted the experiments and interpreted the results to reduce the potential for operator bias. In a further attempt to minimize these errors, analyses were performed both manually and using automated analysis software.

Probes were connected to the ANA by 50-ohm RG-58 coaxial cables. The ANA was configured for measurements in the transmission mode, measuring at 401 points in the frequency range of 3.74 to 1500 MHz. The phase and amplitude of the transmitted signal were measured first with the line empty and then with the line loaded with the test material. The FFT capability of the ANA was utilized to convert the frequency domain data to the time domain. Then the travel time and rise time of the signal were computed from the resulting FFT step-pulse using manual and automated straight-line curve-fitting techniques as shown in Fig. 3. The ANA factory calibration was found to be adequate, because all measurement sequences were calibrated by operating the probes in air.


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Typical output results from the ANA are shown in Fig. 4 . Figure 4a shows the relative amplitude of the transmitted wave as a function of its frequency for sand saturated with a 10-dS m–1 NaCl solution. For the very low frequencies, the input signal level is reduced by 15 dB at the output of the 0.28-m transmission line to an amplitude equal to 0.178 times that of the input. At 1000 MHz, the amplitude is reduced by 40 dB to an amplitude of 0.01 times that of the input. The –40-dB level shown on the figure represents the feed-through cut off for the ANA; data below this level are considered by Hewlett Packard to be unreliable because of various feed-through effects which bypass the probe/soil path. The frequency at which the amplitude falls below –40 dB is defined here as the –40-dB cutoff frequency. It is our experience that usable data can be obtained down to a level of –45 to –60 dB, depending on the circumstance, where we define usable data as that showing continuous phase changes with increasing frequency.



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  		Fig. 4. Automatic network analyzer (ANA) results collected for a sand saturated with a 10-dS m–1 NaCl solution. (a) Relative amplitude as a function of frequency. The dashed line represents the minimum amplitude for reliable measurements based on the manufacturer's reported feed-through cutoff. (b) Linearized phase as a function of frequency for the 10-dS m–1 and air cases and the difference between these two measurements.

 
Phase data obtained from the ANA have ambiguities at ±180°, because a phase measurement system cannot determine the difference between 0°, 360°, 720°, etc. To resolve this problem, the data may be converted to a linear format using a phase linearization program which takes advantage of the fact that the phase is known to increase continuously and monotonically. A typical representation of the linearized phase is shown in Fig. 4b. The phase measured with the probe in air is also shown in Fig. 4b. The difference between these two phases, also shown on the figure, is used to calculate the travel time at a particular frequency (Eq. [6]), thus eliminating the effect of the cables connected to the probe.

Time Domain Results
Figure 5 shows a series of typical time domain waveforms collected in air and in saturated sand using the ANA FFT function. Figure 5a shows the waveforms on a common scale. The results clearly demonstrate that the final waveform amplitude (Atrans/Ainc) decreases with increased salinity, potentially increasing measurement uncertainty due to the reduced signal/noise ratio. In Fig. 5b, the waveforms are normalized to the amplitude measured at 75 ns to emphasize differences in the rise times. The results show that the rise time increases with increasing salinity.



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  		Fig. 5. (a) Time domain reflectometry (TDR) traces collected from the air-filled container, and in sand saturated with distilled water (0 dS m–1) and three different NaCl solutions with electrical conductivities ranging from 10 to 40 dS m–1. (b) Amplitude normalized to the amplitude measured at 75 ns as a function of time for the solutions shown in Fig. 5a.

 
Our experiments were performed as six test series performed by three investigators over a six-month period, and included 21 repacked samples of saturated sand. Each series was analyzed using either a manual or an automatic curve-fitting procedure, both of which were based on the curve-fit method shown in Fig. 3. The primary purpose of these measurements was to determine the effects of increased salinity on travel time measurements. The difference between the travel time measured at a given salinity and the travel time measured in the sand saturated with deionized water for each of the six test series was calculated and is defined as the travel time difference, {Delta}T. This difference is shown on Fig. 6 as a function of salinity. The corresponding water content data was determined from the travel time using Eq. [4]. The water content error is defined as the difference between the water content determined for a given salinity and the water content determined in the sand saturated with deionized water for each test series, and is also shown in Fig. 6. The mean and the standard deviations for the normalized data shown in Fig. 6, and for the absolute values of the water content data, are shown in Table 1.



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  		Fig. 6. Travel time difference ({Delta}t) as a function of the pore-water electrical conductivity (EC) in saturated sand. The travel time difference is converted to a water content measurement error on the second y axis.

 

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  		Table 1. Time interval and water content statistics for time domain data where SD is the standard deviation. Twenty-eight centimeter long rods in a sand saturated with distilled water (0 dS m–1) or NaCl solutions with electrical conductivities ranging from 10 to 40 dS m–1. Each of six individual test runs has been normalized to its value at zero pore-water EC.

 
Figure 6 shows a consistent increase in the average measured travel time as a function of the pore-water EC. This result is consistent with the findings of Sun et al. (2000) who showed that for a probe placed in a sand flushed with different saline solutions, there is a consistent increase in travel time as a function of the EC of the saturating solution and that this increase is well described by a polynomial equation. However, Fig. 6 and Table 1 also shows that there is a pronounced variation in the measured travel time among repackings of the same medium for the same probe and that this variation approaches the travel time measurement difference at all salinities. The same relationship is found for the mean water content error and it's variation. These data also suggests that at relatively low EC there is approximately equal contribution to the total variation due to operator and test condition differences, and due to increased EC. Under higher salinity conditions, EC effects dominate the variability of travel time measurements. The source of the large observed variation in measured travel time is attributed to the extreme variation in pulse shape for the different trials, as shown in Fig. 7 . This effect of increased EC on pulse shape is consistent with the finding of Weerts et al. (2001). The effect of this pulse distortion is to introduce a degree of uncertainty into the curve fitting results used to determine the travel times. We propose that local variations in soil packing along the probe give rise to the observed differences.



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  		Fig. 7. Time domain transmission (TDT) waveforms collected with 0.28-m rods in four repackings of sand saturated with 40-dS m–1 NaCl solution. Each waveform is normalized to the amplitude measured at 75 ns.

 
The increased variability of travel time measurements with increasing EC has significant implications for any attempts to devise a correction for water content measurements made in saline field soils. Specifically, these results suggest that the differences in travel time measurements collected at a series of different locations on a field site will vary in an unpredictable manner that cannot be described using a single polynomial correction, and will have a variation approximately as large as the average difference. Thus, a unique polynomial correction will be required for each probe location to determine the absolute volumetric water content from independent measurements of the travel time and EC. This problem will be confounded if the EC differs among measurement points, which may limit the ability to quantify changes in water content.

Error Detection
We found a strong correlation between the water content measurement error and the measured rise time of the transmitted pulse, as shown in Fig. 8 . In the region of 0- to 6-ns rise time, this is consistent with the findings of a previous paper in which Hook and Livingston (1995) showed the root mean square (RMS) trial-to-trial random error could be approximated as 0.1 of the rise time of the pulse. Our experimental results yield differences worse than those predicted by the simple Hook and Livingston (1995) formula at larger rise times, reaching a difference of 0.25 times the rise time at 15 ns. This is not unexpected considering the extreme degradation shown by the transmitted pulses (Fig. 7).



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  		Fig. 8. Water content measurement error as a function of rise time.

 
The data shown in Fig. 8 suggest a practical method to estimate the expected average increase in travel time based on a simple determination of the pulse rise time. As an example, for our experimental conditions rise times <6 ns gave rise to measured travel time increases <0.74 ns. This corresponds to a water content measurement error of 0.10 m3 m–3 or less, which might be set as an error limit.

Further investigation is required to determine if this correlation could be extended to conditions of partial saturation, other soils, and other pore-water chemistries with the possibility of forming an empirical method for predicting the accuracy of water content measurements using time domain methods. However, based on these preliminary results, we strongly recommend that TDR systems and analysis software include the simple modifications necessary to determine the rise time to potentially identify waveforms that will give rise to unacceptable water content measurement uncertainties.

In regards the application of this approach to a variety of commercially available TDR instruments and systems, we took data over the frequency range of 0.3 to 1500 MHz, and thus the effective rise time for the ANA (Eq. [9]) was 0.23 ns. We propose a practical limit of 6 ns, and thus these results can be applied to any instrument having a pulse rise time of 2 ns or less. This would include almost all commercial TDR instruments.

Frequency Domain Results
Figure 9 shows the frequency domain results for saturated sand. Examination of the magnitude as a function of frequency (Fig. 9a) shows two clear results. First, the zero salinity (0 dS m–1) case shows ripples from multiple reflections that are damped out at 10 dS m–1 and higher salinities. Second, for solutions with electrical conductivities >10 dS m–1, magnitudes drop sharply to very low values, leaving only low frequency components above the –40-dB level. These latter two magnitude plots show discontinuities above about 100 Mhz. We believe these are due to breakdown of the ANA signal processing in this region, rather than to physical RF absorption bands. Figure 9b shows that the slope of the phase difference curve at a given low frequency increases with the salinity of the pore water. From Eq. [6] this indicates that there is an increase in the travel time for low frequency components with increasing salinity. Note that although all phase data above the frequency at which the amplitude falls below the –40-dB level are considered to be unreliable, the phase remains relatively linear for frequencies as high as 100 MHz.



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  		Fig. 9. (a) Magnitude as a function of frequency for sand saturated with distilled water (0 dS m–1) and NaCl solutions of three electrical conductivities (EC) ranging from 10 to 40 dS m–1. The dashed line represents the manufacturer reported feed-through cutoff. The 25- and 40-dS m–1 cases have –40-dB cutoff frequencies of 22.8 and 64 MHz, respectively. (b) Linearized phase difference as a function of frequency for sand saturated with distilled water and the three NaCl solutions shown in (a). For clarity, the results are only shown for the lowest 200 MHz. The cutoff frequencies for the higher EC cases are shown as dashed lines.

 
Comparison of Frequency Domain and Time Domain Travel Time Determinations
We determined the slope of the phase curves of Fig. 9 using linear regression over the frequency interval from 0 to the –40-dB cutoff frequency. The –40-dB cutoff frequencies for the four salinity levels are tabulated in the third row of Table 2, and the corresponding slopes translated into travel times using Eq. [6] are tabulated in the fifth row. These travel times may be compared with those determined from the time domain waveforms as tabulated in row two of Table 2. The time domain measurements are consistently lower than those determined from individual frequency components with magnitudes higher than –40 dB. This implies that the travel times determined in the time domain are dominated by high frequency components of the time domain pulse that experienced attenuation too great to be measured reliably in the frequency domain. We verified this by measuring the travel time in the time domain for the 0 and 25-dS m–1 cases using only the FFT frequency components within the –40-dB cutoff frequency range shown in Row 3 of Table 2. These results are shown in the sixth row of Table 2, and match very closely the travel times determined from the phase data. These results illustrate the power of making measurements simultaneously at many frequencies over a wide frequency spectrum, as is characteristic of time domain techniques using a step-pulse stimulation. Note also that these results were obtained by an FFT technique, and thus further implies that the amplitudes of individual spectral components may be far below that specified for the ANA, and yet the transformed result of thousands of these weak components may be strong enough to give reliable measurements. Apparently the effect of internal ANA instrument and external interfering reflections on the measured amplitude and phase of individual components are averaged out in the FFT process.


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  		Table 2. Comparison of time domain and frequency domain results and measured and calculated parameters for 0.28-m long rods in a sand saturated with distilled water (0 dS m–1) or NaCl solutions with electrical conductivities ranging from 10 to 40 dS m–1.

 
We also calculated the 3-dB upper frequency limit, fmax, based on the rise time using Eq. [9] and have tabulated that quantity in the fourth row of Table 2. These frequencies show the same trend as the –40-dB cutoff frequencies for the two higher salinity cases. Finally we calculated the deviation from linear for the four curves of Fig. 9b, and found a maximum deviation of ±34° (Table 2, Row 7). This demonstrates that dispersion is minimal even for highly saline sands, and thus the main cause of wave shape distortion is the attenuation of the high frequency components.


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
The application of an ANA with a built-in FFT processor, operated in the transmission mode, provides a unique opportunity to study the performance of time domain electromagnetic methods for measuring volumetric water content under highly saline conditions. Our time domain measurements show that there is an increase in the travel time of a pulse with increasing pore-water salinity and that the variability of travel time measurements also increases with the pore-water EC. Over the range of EC studied, this variability leads to uncertainties in the measured water content that approach the average water content measurement error due to increased salinity. We propose that this variability is the result of small-scale differences in the soil packing among replicate installations of the probe in the same saturated sand. This suggests that, under spatially variable field conditions, while analytic or empirical corrections may be possible for a single probe installation, it would be impossible to form a general correction for the effects of salinity on the measured volumetric water content even if the soil EC is known.

The rise time of a transmitted step pulse increases with pore-water EC. For our experimental conditions, the rise time measured directly from the time domain waveform is shown to be an effective measure of the uncertainty in travel time measurements. Therefore, we recommend the standard use of curve-fit techniques to calculate the rise-time so that this information can be used to identify potentially large water content measurement errors. We conclude that for most scientific and monitoring applications, rise-times >6 ns will produce water content data with questionable accuracy. Future work will address the applicability of these results to differing water contents, soils, probe configurations and soil water chemistries.

Using frequency-domain measurements, we observed greatly increased attenuation of the higher frequency components of a step pulse with increased salinity. We observed a corresponding increase in the travel time for the lower frequency components as salinity increased. The travel time measured in the time domain from the aggregate pulse is far lower than the travel times of these low frequency components. Both observations suggest that high frequency components, which are too highly attenuated to be measured individually even with a sophisticated ANA in transmission, dominate the travel time of the time domain pulse. We also have determined that dispersion is minimal even for highly saline sands, and thus does not appear to be a significant contributor to reflected (TDR) or transmitted pulse shape degradation.

In our experiments, the effects of salinity on water content error were not apparent until the salinity of the sands under test reached 25 dS m–1. The only means by which we were able to obtain reliable data at this salinity was through the use of an ANA in transmission mode. We suggest that previous investigators using conventional analog instruments in reflection could not obtain reliable data at such a high level of salinity due to the much poorer signal/noise ratio of such analog instruments, and the multiple reflection interference characteristic of TDR techniques. This may have contributed to the inconsistent results reported in the literature. We believe that our results and the general use of the techniques described in this paper will aid in attempts to develop a theoretical explanation of the mechanisms underlying the interaction between RF waves and saline soils.


APPENDIX

List of Symbols

Symbol
 Units
 Description
A dB Attenuation
Atrans — Amplitude of the transmitted signal
Ainc — Amplitude of the incident signal
Bn — Amplitude of the nth harmonic in a Fourier series
c m ns–1 Speed of light in a vacuum
f cycle s–1 Frequency
fmax cycle s–1 3 dB upper frequency limit
Ka — Apparent dielectric constant
L m Length of waveguide
t ns Time
tm ns Travel time in medium alone
t0.5 ns Time at which the amplitude of the transmitted
  signal is one half of its eventual maximum
t1 ns Round trip travel time to beginning of probe
  (TDR)
t2 ns Round trip travel time to end of probe (TDR)
ta ns Total travel time in cables and probe with probe
  in air (TDT)
tair ns Travel time through probe with probe in air alone
  (TDT)
tb ns Total travel time in cables and probe with probe
  in medium (TDT)
tcables ns Travel time in cables alone (two way for TDR,
  one way for TDT)
tn ns Total pulse arrival time, Fig. 3
tn-air ns Total pulse arrival time in air, Fig. 3
tr ns Rise time
ts ns Travel time for soil alone with {theta} = 0
T ns Period of a sine wave
Tair ns Normalizing factor equal to the theoretical travel
  time in air for a waveguide of length L
{Delta}t ns Travel time measurement difference from that
  measured at zero salinity
{phi} radians Phase difference between the transmitted and
  incident signals
{theta} m3 m–3 Volumetric water content


   ACKNOWLEDGMENTS
 
We thank Marija Stojkovic for her careful laboratory measurements and Kris Caputa for his technical support related to ANA measurements. This research was supported by grants from the Natural Science and Engineering Research Council of Canada. We also thank the five anonymous reviewers who provided numerous constructive comments on our original manuscript.

Received for publication October 10, 2002.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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