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a George E. Brown Jr., Salinity Lab., 450 W. Big Springs Rd., Riverside, CA 92507
b The Institute of Soil, Water and Environmental Sciences, (ARO) The Volcani Center, Bet Dagan, Israel
* Corresponding author (mschaap{at}ussl.ars.usda.gov)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Abbreviations: EC, electrical conductivity • EMT, effective medium theory • GPR, ground-penetrating radar • TDR, time domain reflectometry
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Early studies that applied TDR to determine the effective permittivity of layered materials showed encouraging results. For example, Birchack et al. (1974) presented a model based on layers, as did Topp et al. (1982), and the method was later named the refractive index mixing method (Whalley, 1993). Topp's derivation is a simple model for two layers based on the summation of propagation times, describing the apparent, measured permittivity (Ka) as:
 | [1] |
Where, L is the total length of the measurement, L1 is the length of the first layer with permittivity K1 and L2 is the length of the second layer with permittivity of K2. This has been further investigated by Nadler et al. (1991) who found reasonable correspondence with Topps model (Topp et al., 1980). However, both Nadler et al. (1991) and Dasberg and Hopmans (1992) pointed out the difficulty that arose with interpreting their waveforms from TDR data in layered media.
Recently Chan and Knight (1999)(2001) demonstrated that the concept of simply summing propagation times in layers is not always correct and that the averaging of propagation velocity changes with the ratio of effective wavelength (
) to layer thickness (t) (Fig. 1). They suggested (Chan and Knight, 2001) that a transition zone exists where the ratio is about 4; this corresponds to a layer thickness of a quarter wavelength. Chan and Knight (2001) used an equation for which they adopted the term ray theory for the situation where the ratio is smaller than 4:
 | [2] |
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= 
f1 + 
f2 (Eq. [1]) where K1 is the permittivity of that section.
Above values of 4 what they termed effective medium theory (EMT) is valid:
 | [3] |
Which is equivalent to the arithmetic averaging of permittivity Ka = K1f1 + K2f2.
The choice between refractive and arithmetic averaging has important consequences for the interpretation of water content from TDR or related measurements of apparent permittivity in layered media. In this study we investigate which averaging regime should be used for layered materials, and under what circumstances. To this end we performed measurements on two-, three-, and multi-layered systems. We will also present computed results based on electromagnetic wave propagation theory for the multi-layer system and compare these with the measurements and discuss the causes of a transition between refractive and arithmetic averaging.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Characterization of Averaging in Air and Water
Experiments were conducted with three-rod TDR probes with a water–air interface forming two layers. Both the permittivity and the electrical conductivity (EC) were measured with the cable tester. In the first experiment the probe was initially held parallel to the water–air interface and was then rotated about the interface until it was vertically immersed in the water (Fig. 2). In a second experiment, the 20-cm probe was inserted perpendicular to the air–water interface 1 cm at a time until the probe was fully immersed in the water (Fig. 2).
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Multi-Layer Experiment
This experiment was conducted using a two-rod probe with 2-mm thick electrodes, 30.24 cm long at 1.2 cm center spacing. Holes were drilled in 5-cm diameter acrylic disks corresponding with the probe electrodes; the disks were 0.378 cm thick and 80 disks filled the length of the probe. Figure 3 shows a photograph of the probe with 40 disks placed on it, separated by an equal distance in which de-aired water was placed and held using an outer sheath. The disks could be manipulated to give layers of Acrylic and water of differing thickness. Layers corresponded to thicknesses of 0.38, 0.76, 1.52, 1.90, 3.04, 3.80, 7.60, and 15.20 cm. The waveforms retrieved from a multiple-layer medium are noisy and difficult to interpret. To ensure correct interpretation the waveguide was shorted across the end after the waveforms were collected so the end point could be determined with some certainty. The waveforms were printed off and the waveform analysis was done by visually fitting the tangents.
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 | [4] |
R(f), Vo(f), and S11(f) are all complex quantities that can either be given as real and imaginary parts, or as magnitude and phase.
Composite Scatter Function for a Segmented Probe
Heimovaara (1994) used a special probe design that required only the scatter function of the sensor to be known. In this study we consider a probe that consists of multiple segments, each with a separate scatter function. Feng et al. (1999) showed how to compute effective scatter functions for such segmented systems. In essence their theory involves determining the impedances, propagation constants, and reflection coefficients for individual segments, after which the effective scatter function can be computed with
 | [5] |
for k = N, N -1, ...3, 2, and finally 1. The scatter function for k = N is computed by setting SN+111 equal to the end reflection, which is 1 and -1 for open-ended and shorted probes, respectively (see also Feng et al., 1999). The reflection coefficient 
ks is computed according to
 | [6] |
 | [7] |
is the propagation constant, which is computed as
 | [8] |
The quantities Z, L, and C are dependent on the magnetic (L) and dielectric properties (C) of the medium and also depend on the geometry of the probe. For two-rod probes the following expressions hold (Davidson, 1978):
 | [9] |
 | [10] |
 | [11] |
*, is frequency-dependent and was computed with the Debeye equation
 | [12] |
s and 
are the static permittivity and permittivity at infinite frequency, respectively, frel is the relaxation frequency of water. For the Acrylic * was assumed constant for all frequencies.
Calculation of TDR Waveforms
To compute modeled waveforms an input function Vo (Eq. [4]) is necessary. Heimovaara (1994) determined Vo by measuring the open-ended reflection of the probe head. However, determining an input signal in this way is somewhat problematic because even at the output port of a Tektronix 1502B cable tester the signal already has undergone some internal reflections that contaminate the signal. In addition cable connectors may also cause some small reflections. A relatively clean input signal can be determined by using a procedure outlined by Huisman et al. (2002). In this approach, open and short-circuited time-domain waveforms [Wo(t) and Ws(t)] were measured by connecting standard open and short calibration loads (Model 8550Q, Maury Microwave Corp., Ontario, CA) to the port of the cable tester. A waveform with minimal internal reflections is then computed by
 | [13] |
The Wo(t) and Ws(t) signals were acquired with a program developed by Heimovaara (1994) that was modified to measure 16 384 points from -0.5 m to approximately 65 m. The input signal Vo(f) was computed by taking the back difference of W(t), followed by a Fast Fourier Transform (Heimovaara, 1994). The input signal frequencies range from 0 to 37.1150 GHz, in steps of 4.5312 MHz. Inspection of the spectrum, however, indicates that the signal does not contain relevant information beyond 6 GHz (results not shown). Compared with Heimovaara et al. (1996) who presented an input signal with a bandwidth of 3.5 GHz our signal has a much wider frequency range. One reason for this is that it appears that the TEKTRONIX 1502C emits a signal with a broader bandwidth at higher resolutions (our input waveform was determined for a cable-tester resolution of 0.1 m div-1 vs. 0.5 m div-1 in Heimovaara, 1994 and Heimovaara et al., 1996). Further, we determined our input function directly at the port of the cable tester. We therefore had no cable losses, which are proportional to 
f and are especially strong at higher frequencies (Davidson, 1978).
Scatter functions for the multi-layer systems were computed with Eq. [5] through [12] using the relevant segmentation information listed above. We set the zero-frequency permittivity (s) of water at 25°C to 78.5 o (where o is the permittivity of vacuum); the permittivity of water at infinite frequency () was set at 4.22o, and the relaxation frequency frel was set at 17 GHz. For the permittivity of the Acrylic we assumed a value of 2.76o (Lide, 1993) for all frequencies.
Computed waveforms were calculated by Inverse Fourier Transforms of R(f) (Eq. [4]), followed by a cumulative sum to counteract the back-difference operation used for the computation of V0. Reflection times were calculated by computing waveforms with open-ended and shorted-end reflections. The time where the open and shorted waveforms diverged by more than 0.01 was taken as the end reflection. The beginning of the sensor was defined at the time where the waveforms diverged by 0.01 from the input waveform, W(t) (Eq. [13]). Reflection times of single segment waveforms computed in this way diverged <0.01 ns from reflection times computed directly from sensor length and permittivity (results not shown).
We would like to stress that, contrary to the measurements, we did not "attach" a cable to our modeled probe, nor did we include a probe head (cable to probe transition). We also did not include effects of a cut-off frequency for transversal wave propagation through a probe (Ramo et al., 1984). The modeled system is therefore a highly idealized version of the probe used for the multi-layer probe and we did not try to fit computed and measured waveforms beyond using known probe length and geometry.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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A follow up experiment was conducted with a probe inserted, perpendicular to a water bath. Figure 4 presents the results for a 20-cm probe inserted 1 cm at a time into the water. Both arithmetic average permittivity and refractive index average permittivity are plotted and the data clearly corresponds with the refractive index averaging regime.
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= 78.3) and 0.5 Acrylic ( = 2.76). Waveforms were calculated assuming a homogeneous material with the respective permittivity values; relaxation was assumed to occur at the same frequency as that of water (17 GHz). In Fig. 6B waveforms are presented for the same probe but now with thin multiple layers. Two layers represents one layer of Acrylic and one layer of water, 80 layers, 40 layers of each. A short was placed at the end of each probe in the computation to determine the exact end point for the reflection as the waveforms become more complex when more layers are present. Arrows are placed on the graph from the homogeneous media representing the refractive and arithmetic bounds. They indicate that for two layers the travel time is that for refractive index averaging where as for multiple layers the travel time agrees with arithmetic averaging of the permittivity.
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 | [14] |
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Although the measured and computed waveforms in Fig. 6 show good qualitative agreement they do not provide information as to why a transition from refractive to arithmetic averaging occurs for multi-layered systems. Chan and Knight (2001) suggested that a characteristic wavelength/layer thickness ratio affected the averaging transition, with a transition value of about 4. However, because TDR is a broadband method that includes frequencies up to a few GHz (Heimovaara, 1994, and this study), it is difficult to explain the averaging transition in terms of one wavelength. We believe that the explanation for the transition should be sought in the fact that the signal does not solely propagate straight to the end of the probe and back again to the beginning. Instead, many (partial) reflections occur at the impedance mismatches at segment boundaries. The multiple reflections cause the average signal path to lengthen causing an increase in the travel time. Subsequently, a longer travel time is interpreted as a higher apparent permittivity—even though the water content does not change. The question now becomes whether this delay occurs for all frequencies or whether certain frequencies are favored because of constructive or destructive interference. To this end we further analyzed the modeled results.
The scatter function for a transmission line contains all relevant information for signal losses and signal delay. The scatter function computed in Eq. [5] presents this information in terms of real and imaginary numbers that have a computational interpretation but not directly a meaning in a physical sense. However, by converting the scatter function to a polar form (i.e., by computing a magnitude and a phase) more useful quantities can be derived. The magnitude (i.e., |S11|) of the scatter function provides information about the relative dielectric losses, but is not useful for the following discussion. Phase information contains information on how much a transmission line delays a signal and can be computed by taking the complex argument of the S11(f) scatter function {i.e., arctan[im(S11)/re(S11)]}. The phase difference of a sinusoidal signal entering and leaving a transmission line of length L is (2
fLr)/c radians, where one complete cycle of the signal is 2 radians (or 360 degrees), r is the relative permittivity and c the speed of light. For a perfectly matched cable-transmission line system with an open-ended reflection the phase difference is (4fLr)/c as the distance L is traveled twice. The delay of the signal (in seconds) is simply calculated by dividing the phase difference by 2f.
In Fig. 9 we plotted the delay (left axis) and square root of the apparent permittivity (right axis) versus frequency for a 30.24-cm probe with 2, 4, 16, 40, and 80 layers. Also shown are two horizontal lines that represent the calculated delay for refractive and arithmetic averaging (r equals 27.62 and 40.53, respectively). To produce the data for this figure we used the same probe and segmentation properties as used for Fig. 6B. However, to substantially simplify an otherwise very complicated graph we assumed that relaxation of water did not occur (i.e., the modeled system has no dielectric losses). Omission of free water relaxation does not substantially affect the following discussion.
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When broadband methods are used it can be expected that refractive averaging will be found for two or four layer systems because the average delay over a frequency range is similar to that of the refractive regime. However, narrow-band or single frequency dielectric methods for these systems are unlikely to exhibit refractive averaging. Regimes below and above refractive averaging are then possible. Subrefractive averaging can be explained by reflection of most of the signal out of the probe before it reaches the end of the sensor (permittivities would appear to be too low). Super-refractive averaging indicates that the multiple reflections tend to "lock-up" the signal at certain resonance frequencies (permittivities appear to be too high). We note that for low frequencies, the two and four layer systems exhibit more arithmetic-like averaging regimes, qualitatively confirming Chan and Knight (2001). Depending on the bandwidth of the method used it seems that multi-layer systems are more likely to exhibit arithmetic averaging, only at very high frequencies these systems will show refractive averaging. The arithmetic regime can be explained by the increased likelihood that multi-layer systems will exhibit signal resonance at the many impedance mismatches that exist within the probe, which tends to increase the delay and hence to increase the apparent permittivity.
We would like to note that the scenario presented in Fig. 6 through 9 represents a rather extreme case of permittivity contrasts (r equals 2.76 for Acrylic and around 80 for water). Real world applications in soils or sediments will most likely have much smaller contrasts such as a permittivity of 3 to 5 for completely dry soil to 25 or 35 for completely saturated media, but be smaller still in most cases. Nevertheless, the complex transition from refractive to arithmetic will most likely occur for such systems as well and will roughly scale with the root of the permittivities involved. Finally, we would like to note that the transition from refractive to arithmetic averaging is affected by many factors, such as sensor length and geometry (including impedance), permittivity of the porous material, thickness and periodicity in layering (or lack thereof), and the bandwidth of the dielectric method used. The results presented in Fig. 9 are but for one probe design and results are likely to be qualitatively and quantitatively different in other circumstances.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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ACKNOWLEDGMENTS |
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Received for publication July 26, 2002.
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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