Published in Vadose Zone Journal 2:444-475 (2003)
© 2003 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SPECIAL SECTION - ADVANCES IN MEASUREMENT AND MONITORING METHODS
A Review of Advances in Dielectric and Electrical Conductivity Measurement in Soils Using Time Domain Reflectometry
D. A. Robinson*,a,
S. B. Jonesb,
J. M. Wraithc,
D. Ord and
S. P. Friedmane
a U.S. Salinity Laboratory, USDA-ARS, 450 W. Big Springs Road, Riverside, CA 92507
b Dep. Plants, Soils and Biometeorology, Utah State University, Logan, UT 84322-4820
c Land Resources & Environmental Sciences Dep., Montana State University, Bozeman, MT 59717-3120
d Dep. of Civil and Environmental Engineering, University of Connecticut, 261 Glenbrook Road, Unit 2037, Storrs, CT 06269
e Institute of Soil, Water and Environmental Sciences, The Volcani Center (ARO), Bet Dagan 50250, Israel
* Corresponding author (darobinson001{at}yahoo.co.uk)
Received 21 November 2002.
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ABSTRACT
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Substantial advances in the measurement of water content and bulk soil electrical conductivity (EC) using time domain reflectometry (TDR) have been made in the last two decades. The key to TDR's success is its ability to accurately measure the permittivity of a material and the fact that there is a good relationship between the permittivity of a material and its water content. A further advantage is the ability to estimate water content and measure bulk soil EC simultaneously using TDR. The aim of this review is to summarize and examine advances that have been made in terms of measuring permittivity and bulk EC. The review examines issues such as the effective frequency of the TDR measurement and waveform analysis in dispersive dielectrics. The growing importance of both waveform simulation and inverse analysis of waveforms is highlighted. Such methods hold great potential for obtaining far more information from TDR waveform analysis. Probe design is considered in some detail and practical guidance is given for probe construction. The importance of TDR measurement sampling volume is considered and the relative energy storage density is modeled for a range of probe designs. Tables are provided that compare some of the different aspects of commercial TDR equipment, and the units are discussed in terms of their performance and their advantages and disadvantages. It is hoped that the review will provide an informative guide to the more technical aspects of permittivity and EC measurement using TDR for the novice and expert alike.
Abbreviations: EC, electrical conductivity • TDR, time domain reflectometry
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INTRODUCTION
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WATER IS REQUIRED in some way by all living things; it is a fundamental constituent of life on our planet. Our survival as well as that of other organisms depends on a supply of water both to our own bodies and to the flora and fauna on which we live. One of the best ways to regulate water consumption is to know the quantity available and to manage the resource with prudence and stewardship (Hillel, 1991). To achieve this aim, techniques are preferred that can be used to measure a physical quantity closely related to the amount of water contained in a porous material, be it rock, soil, or an artificial medium.
The revolution in electronics in the latter half of the last century made the measurement of the electrical properties of materials more accessible than ever before. Measurement of the dielectric permittivity (dielectric constant) of a material emerged as an elegant method of estimating water content in porous materials. For the first time the same physical property (permittivity) could be measured for a range of scales and used to estimate water content. Electromagnetic methods, whether TDR (localized measurement), ground penetrating radar (two-dimensional profile), or active microwave remote sensing (land surface), all estimate water content based on the permittivity of the target medium. A further advance was the development of analysis methods using TDR. Time domain reflectometry was adapted to estimate both soil water content (Hoekstra and Delaney, 1974; Topp et al., 1980) and soil bulk EC simultaneously (Dalton et al., 1984). In spite of decades of research, we are only beginning to efficiently utilize electrical technology that ranges from satellite and airborne radar to ground penetrating radar and localized sensors such as TDR and impedance probes.
The underlying success of these techniques can be considered in two parts, the first of which is the equipment's ability to accurately measure the bulk dielectric permittivity and EC of a material. The second is the close relationship between the measured permittivity and the volumetric water content, or the ionic concentration and the bulk EC of the material. This review concentrates on the first stage, the accurate measurement of bulk permittivity and EC, and we confine ourselves to the use of TDR but acknowledge that other devices such as impedance probes (Dean et al., 1987; Hilhorst et al., 1993; Gaskin and Miller, 1996; Paltineanu and Starr, 1997) may also be used for this purpose. Time domain reflectometry has become a large topic in soil physics, primarily because of its adaptability and the continued development of novel applications. The focus of this review is on the measurement of bulk permittivity and EC, and thus some topics are dealt with only briefly or omitted. One of the strengths of the TDR measurement method is that many probes can be monitored almost simultaneously using a multiplexer (Baker and Allmaras, 1990; Heimovaara and Bouten, 1990; Herkelrath et al., 1991). This review discusses the measurement of bulk EC; however, we don't go any further to examine the interpretation of this in terms of soil solution conductivity. The literature on this aspect of TDR application is large, and the reader is referred to a recent publication for further reference (Dane and Topp, 2002). A further topic omitted from this review is the use of coated TDR probes. Coated probes have been proposed as a way to extend the working range of TDR in saline soils (Kelly et al., 1995; Nichol et al., 2002). However, the studies of Ferre et al. (1996) and Knight et al. (1997) indicate a strongly reduced sampling volume and questionable accuracy. Perhaps further investigation is needed to fully understand measurements made with such probes.
The history of using relative permittivity to estimate water content is confined to the last century, principally due to the measurement constraints imposed by the availability of instrumentation. It was recognized early on that the use of radio frequencies might be utilized to estimate water content. For example, Smith-Rose (1933)( 1935) and Thomas (1966) gave accounts of early attempts to estimate moisture. However, not until the aftermath of the Second World War did the use of high-frequency electrical measurements in basic research really begin to expand. Dielectric theory had been well established, with Debye (1929) winning the Nobel Prize for his work on polar molecules. However, measurements had not kept pace with the advancing theory. The work of Hasted among others (Hasted et al., 1948; Ritson and Hasted, 1948; Hasted, 1973) stands out for pioneering work on the high-frequency measurement of permittivity, mostly in liquids. The pioneering work of Nelson et al. (1953) initiated a 50-yr contribution of research relating dielectric measurements to water content in vegetables, grains, and other composite porous media. Two distinct paths for permittivity measurement were seen to emerge in the 1960s. First, capacitance probes could be constructed following the development of small, high quality electrical components, (Thomas, 1966; Wobschall, 1978; Dean et al., 1987) and could be used for routine measurement in soils. Second, Fellner-Feldegg (1969) suggested the use of TDR for measuring permittivity, which was taken up in soil science by Hoekstra and Delaney (1974) and in the seminal work by Topp et al. (1980). Compared with capacitance probes, TDR's reduced susceptibility to signal interference, due to probe geometry and bulk EC as well as the minimal soil disturbance involved using multiple rods has led to its acceptance as a practical technique for measuring the permittivity of porous media (Cassel et al., 1994; Topp and Reynolds, 1998; Noborio, 2001; Dane and Topp, 2002). In the last 20 yr the earth sciences have successfully developed, applied, and expanded the use of TDR as a method for measuring permittivity and estimating water content, and it is now also being applied to measurements such as slope stability in geotechnical engineering (Dowding and O'Connor, 1999).
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The Permittivity of Porous Media
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Porous materials of interest in the earth sciences are usually composed of three components: the solid matrix, a gaseous phase, and the liquid water phase. The liquid water phase is sometimes subdivided further into free water and bound water, which is restricted in its mobility by adsorption on surfaces. The relative permittivity of air is 1, while those for common minerals in soils and rocks lie in the range 4.5 to 10 (Keller, 1989; Robinson and Friedman 2003), while water has a permittivity of 78.5 at 25°C. Thus the permittivity of any water bearing porous material is strongly influenced by its water content. The origin of the permittivity is the asymmetry of charge in the water molecule (Fig. 1), which leads to a small displacement of the positive and negative charge centers creating a permanent dipole of 6.216 x 10-30 C m. When placed in an alternating electric field the molecules overcome their random thermal motion and align with the field (Fig. 1). The process of alignment stores electrical energy, which is released once the application of the field is stopped. This alignment of the molecules manifests itself as the real part of the relative permittivity (
'r). However, materials are rarely pure and usually contain some actual charge carriers such as ions. The loss of energy due to ionic conductivity is described by the imaginary part (''r), termed dielectric loss. Another source of loss occurs when the molecules being aligned by the alternating field can no longer keep up with the speed of field alternation. The molecules are said to relax and energy is dissipated as heat. These properties are conveniently written as
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Fig. 1. (a) The dipole moment of a water molecule. (b) Water molecules randomly aligned (left) and being aligned by an external field (right). This alignment causes the storage of energy described as the real part of the permittivity.
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Here the relative permittivity, r is the ratio of the permittivity of the material, (F m-1) to that of free space, o (8.854 x 10-12 F m-1). For a list of variables, see the Appendix.
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The complex relative permittivity *r describes energy storage and energy loss. The real part is designated 'r and associated with energy storage, and the imaginary components are associated with energy dissipation. Losses are associated with two main processes, molecular relaxation (''relax) and electrical conductivity (
dc), where f is frequency and j is the imaginary number 
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MEASUREMENT PRINCIPLES
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Permittivity Measurement Using Travel Time Analysis along Transmission Lines
A transmission line forms the sensor for the TDR measuring system. Some classic designs are presented in Fig. 2. Their design and construction will determine the quality of the measurements made using the TDR technique; hence, we devote a major part of this review to designing and constructing probes to achieve optimal measurements. Transmission line theory is covered in detail in the literature (Kraus, 1984; Lorrain et al., 1988; Ibbotson, 1999), so in this section only an overview of the principles is given.
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Fig. 2. Three TDR probe designs. Left to right, three-rod probe, two-rod probe, and parallel plate probe.
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Time domain reflectometry measures the propagation velocity of a step voltage pulse with a bandwidth of around 20kHz to 1.5 GHz (Heimovaara, 1994). The velocity of this signal is primarily a function of the permittivity of the material through which it travels with potential modification by conductive losses. It is often convenient to consider the analogy of the propagation velocity of an electromagnetic plane wave that depends on the materials electromagnetic properties through which it travels:
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where c is the velocity of light (3 x 108 m s-1), r is the relative permittivity, µo is the magnetic permeability of vacuum (1.257 x 10-6 H m-1), and µr is the relative magnetic permeability. The relative magnetic permeability is unity in most earth materials, with the exception of some iron oxides (Robinson et al., 1994; Sharma, 1997).
Schematic diagrams of the TDR unit and a section of transmission line are presented in Fig. 3. A step voltage is applied between the conductors at the pulse generator. The signal propagates down the line and is reflected from the end of the probe; the returning signal is sampled in the TDR device. The velocity of the signal in a perfect dielectric is therefore
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where l is the length (m) and t is the time (s) for a round trip (back and forth). Equating Eq. [4] and rearranging gives the round trip propagation time (t) of the wave as a function of both the length of transmission line (l) and the permittivity of the material:
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Fig. 3. A schematic diagram of the TDR main components. The window on the right illustrates two waveforms, one in air and one in water. The dip is caused by an electrical marker in the head of the TDR probe so that the software can locate the start point for travel time analysis.
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Hence it follows that the permittivity can be determined by measuring the time it takes the wave to traverse the probe. Waveforms for air and water are presented in Fig. 3 to demonstrate that the one-way travel time (e.g., the Tektronix [Beaverton, OR] cable tester divides the two-way travel time in half) is measured from the place marked "start" to the points marked "end reflection," and show that the travel time increases as the permittivity of the material increases.
Having determined how the overall system works it is helpful to examine how a plane wave propagates along a transmission line. A single-frequency, sinusoidal wave of angular frequency, 
(i.e., 2
f), can be considered. A useful approach is to use the circuit diagram presented in Fig. 4. The transmission line can be analyzed as a circuit with series impedance (Zs):
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and a shunt admittance (Y), which is the inverse of the parallel impedances (Zp):
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where L is the line inductance in series with a resistance R, which stems from the skin effect along the rod; C is the capacitance of the transmission line per unit length, dependent on material between and geometry of the TDR probe; and G is the transmission line conductance (Fig. 3). The line is then said to have a propagation constant, 
, which in general is a complex number, which is = 
= 
+ jß. The real () and imaginary parts (ß) are named the attenuation and phase constants respectively. From electromagnetic theory (Kraus, 1984) the phase velocity vp is determined from the phase constant ß, the imaginary part of the propagation constant , according to
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Fig. 4. Schematic diagram representing the electrical circuit analogy of a transmission line. L, R, C, and G are the line inductance, conductor skin resistance, medium capacitance, and medium conductance per unit length  z.
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In the case of a transmission line without losses Eq. [8] is often abbreviated to
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It can be seen that if the wave is propagating along a real transmission line with a geometric factor g (m), a capacitance (C = gor), and inductance [L = (1/g)µoµr], then the phase velocity in vacuum is the speed of light (c = 3 x 108 m s-1), and in any other material the equation for the velocity relative to that of light is according to Eq. [3]. In the case of a transmission line with some losses (where G is not <<C; Kraus, 1984) across the dielectric, the conductance term in Eq. [8] cannot be neglected and thus the velocity of the wave is modified. This reduces the velocity of the wave through the medium relative to that of light according to (Von Hippel, 1954)
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This means that the permittivity determined from the travel time analysis is no longer equivalent to the real part but to an apparent permittivity Ka (Topp et al., 1980; White et al., 1994). It is also a function of the dissipation across the rods caused either by relaxation losses (''relax) or by electrical conductivity [dc/(o)]. The effect of losses can be included into the equation for the measured apparent permittivity by combining Eq. [4] and [10] such that Ka is
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As the losses become more significant the propagation time increases, and thus higher apparent permittivity values are measured (Fig. 5). The difficulty in applying an equation such as Eq. [11] is that it is formulated for a plane wave at a single frequency; TDR is a broadband technique and a waveform is composed of many frequencies. However, the data presented in Fig. 5 tend to confirm that the general impact of low ionic conductivity (
10 dS m-1) values can be described by Eq. [11]. In the case of the two independent data sets presented, one for KCl solution and the other for KCl saturated glass beads, 400 MHz was found to be an appropriate effective frequency. This effective frequency is unlikely to be universal. It will depend on the TDR device used, the construction of the probe, and the dispersive nature of the dielectric.
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Fig. 5. Left, The apparent permittivity as a function of conductivity for two frequencies, 250 and 750 MHz. Permittivities of 80, 40, and 25 represent water, saturated clay, and saturated sand, respectively. Right, TDR measurements in KCl solutions and KCl saturated glass beads, Eq. [11] fitted using a frequency of 400 MHz as the effective frequency.
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The loss tangent (tan2
) refers to the ratio of the imaginary to real permittivity.
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One aim of designing probes used solely for measuring water content is to minimize electrical conductance across the probe. This is achieved by having rods with a low geometric factor, g, (reciprocal of the commonly used cell constant, m-1), which is discussed below in the section on probe design. Assuming that the imaginary permittivity and conductivity terms are small means Ka and 'r can be considered equivalent. An important aspect of the measurement that impacts both the real and imaginary permittivity is what might be termed an effective frequency (f in Eq. [12]). This is important because according to Eq. [12] the magnitude of the loss tangent due to EC will depend on this effective frequency. Figure 6 illustrates loss tangents for several electrical conductivities and clearly demonstrates how they increase in the lower frequencies, resulting in increasing imaginary permittivity.
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Fig. 6. The loss tangent (tan2) as a function of frequency for three differing bulk electrical conductivities.
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INTERPRETING AND MODELING WAVEFORMS
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Dispersive Media and Effective TDR Frequency
Waveform interpretation can be best understood when an understanding of what is being measured is clear. The TDR waveform reflects the electrical properties of the material through which it travels and is determined by the amount of power at each frequency measured that combine to make the waveform. Friel and Or (1999) took a direct series of measurements of TDR power output with a sample connected from the TDR to a spectrum analyzer, showing an inverse square decrease in power as a function of frequency (
1/f2) in air. Results for air, water, and an electrolyte solution are presented in Fig. 7 and show that most of the power is below 500 MHz. This graph is useful in demonstrating how a dielectric with a high permittivity like water (80) reduces the power at all frequencies but especially the higher ones. The results are presented up to 2GHz; above 1.5 GHz the signal is mostly noise (i.e., lower than -120 dBm). The introduction of a salt into the water further reduces the power at each frequency. This graph provides useful insight when we discuss the issue of effective frequency raised at the end of this section. An effective frequency can be defined in several ways, and we shall examine each of these and its implications.
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Fig. 7. Spectrum analyzer measurements for air, water, and a saline solution (data from Friel and Or, 1999). The TDR was connected through a coaxial cell to a spectrum analyzer. The results give the TDR power response as a function of frequency.
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Or and Rasmussen (1999) defined the effective frequency as the highest frequency component of the signal passing through the dielectric without being filtered. Using tangent lines fitted to the TDR waveform, they measured the permittivity of ethanol–water mixtures and then compared them with the real permittivity measured as a function of frequency using a network analyzer. They fitted a curve to give the effective frequency (f*) as a function of the TDR measured permittivity:
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where f
= 2.34 GHz and = 0.0216. They suggested frequencies ranging between about 550 MHz at low permittivity values and high dispersion and 1.8 GHz at the permittivity of water (80) with low dispersion. To understand this a little better we examine each of these steps in the time and frequency domains for a dispersive dielectric.
Careful TDR calibration can result in accurate (±0.1) measurement of permittivity. However, many materials, especially soils, can be dispersive, which makes waveform interpretation and permittivity measurement more difficult (Heimovaara, 2001). A dielectric material is dispersive if it suffers from relaxation in the measurement bandwidth (0.001–1.75 GHz). For example, air is nondispersive and water is effectively nondispersive, as its relaxation frequency (17 GHz) is well outside the TDR's frequency bandwidth. However, most alcohols and many soils exhibit relaxations under 1 GHz. The cause of this relaxation in soils can be due to the reduced mobility of water near surfaces Maxwell Wagner relaxation, and it is especially pronounced in the high surface area clay soils. When such a material demonstrates relaxation within the TDR measurement bandwidth, it causes dispersion. The input signal of the TDR is composed of many frequencies, and at the start these are all in phase. In a dispersive medium, the different frequency components of the input signal travel at different speeds. This is caused by the real permittivity changing as a function of frequency and is presented diagrammatically in Fig. 8 using a measurement made in propanol as an example. The arrows indicate that a frequency of 200 MHz will effectively "see" a permittivity of 19.0, while a frequency of 1 GHz will "see" a permittivity of 8.4. Since waves travel faster in materials with low permittivity, the signal no longer travels in phase but spreads out. This can be observed by examining the TDR waveforms in Fig. 9. The waveforms are for mixtures of propanol and water, Waveform 1 is propanol with a little absorbed water, and the proportion of water increases from waveform 2 to 9. The slope of the second reflection increases as more water is added to the propanol; the waveform becomes sharper and more distinct. Figure 10 shows the corresponding real part of the permittivity in the frequency domain for the 9 propanol–water mixtures corresponding to the waveforms in Fig. 9. The arrows indicate the method of Or and Rasmussen (1999) using the measured TDR permittivity by fitting tangent lines (A in Fig. 9) and using this to obtain a frequency f* corresponding to the highest frequency component in the measurement. From inverse analysis of waveforms (Heimovaara, 2001; Weerts et al., 2001) we know that the point at which the tangent lines intersect for a waveform represents the fastest moving part of the signal. We can therefore see that this frequency, f*, changes a lot as dispersion increases.
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Fig. 8. The real permittivity of propanol with some absorbed water. The arrows demonstrate that a 200-MHz signal will "see" a permittivity of 19.0, whereas a signal with a frequency of 1000 MHz will "see" a permittivity of 8.4. The high-frequency signal will therefore travel faster than the low-frequency signal which sees a higher permittivity.
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Fig. 9. Waveforms collected with a 0.18-m coaxial cell for air, water, and propanol–water mixtures. Waveforms 1 through 9 have increasing amounts of water. Location A shows how tangent lines were fitted to the waveforms.
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Fig. 10. The real permittivity of the propanol–water mixtures corresponding to Fig. 9. The arrows indicate the permittivity values measured from the waveforms in Fig. 9 used to determine the frequency to which they correspond.
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The issue of the importance of an effective frequency was raised by Topp et al. (2000) as it is required if Eq. [11] is to be used to try and obtain a real permittivity from the apparent permittivity (Ka) measured by the TDR. In the case of this effective frequency it is likely that it should reflect the frequency where the majority of the energy of the signal is contained. Thus, an alternative determination of an effective frequency was proposed by Topp et al. (2000). They suggested using what they termed the maximum passable frequency for the sensor, which they denoted fmax. They suggested that an estimate of this value can be obtained from the rise time of the reflection from the end of the TDR probe after it has traveled through a dielectric material. This method has been utilized by previous workers (Hilhorst, 1998; Sun et al., 2000), and an outline of the method is given in Appendix C of the Tektronix application note entitled, "TDR's for Cable Testing." The rise time is measured as the time between 10 and 90% of the signal magnitude illustrated in Fig. 11. An example of the analysis required to determine the rise time value (tr) is presented in this figure. The value of fmax is then calculated according to (Strickland, 1970)
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this simplifies to fmax (Hz) = 0.35/tr, where tr is measured in seconds. The above method was used by Topp et al. (2000) to estimate the effective frequency for TDR measurements made in soils with clays; they suggested frequency bandwidths between 100 and 400 MHz.
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Fig. 11. Demonstration of the method used to estimate the highest passable frequency for a given material. tr is the time measured between the signal rising 10 to 90% of the signal magnitude.
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In Fig. 12 we compare results using the two methods for the propanol–water mixtures presented in Fig. 9. The value obtained for the frequency using Eq. [14] is consistently about 0.45 of the value obtained using the Or and Rasmussen (1999) method. This frequency value probably reflects a location where most of the energy of the signal is located. It would be interesting to see in future work if this determined value corresponds with the location of the group velocity of the signal, which is the speed of the wave packet. The terminology used by Topp et al. (2000) of the maximum passable frequency is perhaps misleading and should be dispensed with in light of the comparison with the method of Or and Rasmussen (1999). With regard to terminology, the frequency obtained by Or and Rasmussen (1999) should be considered the maximum frequency, f*, and that obtained from Eq. [14] perhaps an effective frequency, feff. Interestingly, in nondispersive media both methods give similar values for the highest frequency. The line at the top of the graph indicates the frequency (1.8 GHz) obtained in water using the Or and Rasmussen (1999) method. This is in close agreement with the value of 1.75 GHz given by the Tektronix application note entitled, "TDR's for Cable Testing." The diagram also indicates that the frequency determined by either method will reduce if the material is dispersive in the TDR bandwidth (arrows and dashed line). The curve presented by Or and Rasmussen (1999) indicates values of f* to be expected with water–alcohol mixtures. However, this is likely to vary depending on the permittivity of the material and its relaxation time.
This has interesting implications for measurements made in heavy clay and mineral soils. Figure 13 presents dielectric spectra of water-saturated quartz sand and bentonite clay at two water contents, 0.05 and 0.30 m3 m-3. These spectra indicate that for a granular material like saturated quartz grains the real permittivity doesn't change much in the TDR frequency bandwidth. However, the real part of the permittivity for the bentonite changes dramatically below 500 MHz. Behavior like this will undoubtedly affect TDR waveforms and permittivity measurements. In the TDR soil literature, measurements made on bentonite (Dirksen and Dasberg, 1993) showed a sharp rise in the apparent permittivity at water contents above 0.25 m3 m-3. The permittivity values measured rose above the calibration of Topp et al. (1980). This behavior is in agreement with the description of measurement in dispersive media. For the case of dispersive clayey soils we can expect the reverse behavior of the propanol–water mixtures presented in Fig. 12. The increasing real permittivity observed for bentonite in Fig. 13 was also observed to increase as the water content increased (data not shown). Thus, at higher water contents more dispersion occurs, with which we would expect lower frequencies (<500 MHz). With lower frequencies at higher water contents, we'd expect to see an increase in permittivity, which is what has been observed in the data of Dirksen and Dasberg (1993) for bentonite. An important point to make is that not all clay soils will be dispersive. It is likely that the soils used in Topp's original work (Topp et al., 1980) were nondispersive, hence their optimistic suggestion of a universal calibration for soils.
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Fig. 13. The real permittivity measured for saturated quartz sand and bentonite clay at two water contents. The figure demonstrates the sharply changing real permittivity of moist bentonite below 500 MHz.
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These findings raise the issue of what is an appropriate permittivity measurement in a dispersive soil. Measurement frequency in dispersive soils will change as a function of water content, as will the real part of the permittivity. The use of permittivity determined by travel time analysis may not be the most appropriate value. It is preferable to have a value for a fixed frequency, preferably above the frequencies where relaxation strongly affects the real part of the permittivity. Some workers have suggested the use of alternative methods of interpreting waveforms for dispersive media. Huisman et al. (2002) suggested fitting a simulated waveform with frequency-dependent parameters to the captured waveform. By doing this, a frequency could be chosen at which to determine the permittivity from the best-fitted relaxation equation, *(f). They tested this method in a sandy soil, which unfortunately had insufficient relaxation to test this idea thoroughly. Lin (2003b) used both Debye and volumetric mixing models to describe the frequency-dependent characteristics of the soil so that simulated waveforms could be fitted to the measured waveforms. He concluded that both methods appeared to give a reasonable description of the frequency response of the soils. This approach opens a new way to interpreting TDR information and along with inverse analysis of waveforms could lead to more comprehensive understanding of the soil permittivity–water content relationship.
Interpretation of Waveforms in Layered Media
An important phenomenon encountered in many natural porous media such as soils and sediments is the presence of sharp wetting and drying fronts of contrasting zones of wet and dry layers, which have corresponding contrasting permittivity values. Because TDR rods are often placed vertically downward, these layers are likely to be bisected. The impact this has on the waveform is presented in Fig. 14. Both waveforms were obtained with 50% water and 50% air; however, one has water over air and the other air over water. What is apparent from these waveforms is that they have different shapes depending on whether air (low permittivity) or water (high permittivity) comes first, but the travel time is still the same.
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Fig. 14. Waveforms from layered water and air with the respective order changed. The shape of the waveforms changes but the travel time remains the same.
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The issue of TDR measurements across a wetting or drying front was examined in a number of articles (Topp et al., 1982a; Nadler et al., 1991; Dasberg and Hopmans, 1992; Feng et al., 1999; Timlin and Pachepsky, 2002). Topp et al. (1982a) produced a two-layer model based on summing the travel times of the TDR signal in the layers.
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A similar experiment considering the two soil layers as two separate sections of a transmission line allowed Feng et al. (1999) to model the TDR waveforms of layered systems with realistic results.
Chan and Knight (1999)(2001) demonstrated that averaging of the propagation velocity through a layered media changes depending on the ratio wavelength/layer thickness (Fig. 15). In terms of permittivity, the averaging changes from refractive index averaging, which they term ray theory, to arithmetic averaging, termed effective medium theory, with the transition zone occurring at a wavelength to layer thickness ratio of about 4. This value corresponds to the layers being one-quarter wavelength thick. When the layers are becoming thin, and effectively invisible to the traveling wave, the situation is similar to that of making the Ka the arithmetic mean of the fractions of solid and water. This is analogous to the case when the electric field is parallel to the layering of an anisotropic medium made of infinitely thin disk-shaped particles (Jones and Friedman, 2000)
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Fig. 15. Measurements from Chan and Knight (2001) for the change in velocity averaging. As the wavelength/layer thickness ratio increases above 4, the permittivity averaging changes from refractive index to arithmetic.
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Chan and Knight (2001) demonstrated this with TDR measurements in layers of dry coarse sand and fine wet sand, presented in terms of normalized velocity (vnormalized) in Fig. 15:
 | [18] |
In the figure the wavelength divided by layer thickness is presented. However, in light of the discussion about effective frequency and the difficulty of assigning an effective frequency or wavelength to TDR measurements, Schaap et al. (2003) proposed that it was more appropriate to plot the normalized velocity as a function of layer thickness. Measurements made with alternating disks of water (78.5) and Plexiglas (3.5) are presented in Fig. 16. They demonstrate the two averaging regimes for TDR measurements in layered media, refractive index (Eq. [19]) and arithmetic (Eq. [20]). The arrow points in the direction of increasing wavelength/thickness ratio.
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Fig. 16. Refractive index and arithmetic averaging of permittivity for layers of Plexiglas and water. As the wavelength/layer thickness ratio increases above 4, the averaging moves from refractive index to arithmetic, as indicated by the arrow (Schaap et al., 2003).
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 | [19] |
 | [20] |
In these equations, 
is the fractional water length (volumetric water content) and Kwater and KPlexiglas denote the permittivity of the water and Plexiglas, respectively.
The potential of waveform simulation was suggested in the above discussion concerning dispersive media. Again it offers great potential for understanding the response in layered materials (Feng et al., 1999). Timlin and Pachepsky (2002) showed how fitting simulated waveforms to the measured waveforms with vertical probes could be used to obtain infiltration rates for the soil. Simulated and measured waveforms from Schaap et al. (2003) for the data shown in Fig. 16 are presented in Fig. 17. In the simulated waveforms the probe head was omitted from the modeling, as it was unnecessary for the qualitative comparison. Information obtained from this approach allowed Schaap et al. (2003) to demonstrate the frequency dependence of the permittivity averaging of the layering in their paper. They observed that at the lower frequencies (<100 MHz) all layers studied needed to be averaged using the arithmetic averaging. At higher frequencies thin layers remained in the arithmetic-averaging regime, while thicker layers were averaged according to the refractive index averaging regime. The refractive index averaging regime will be appropriate for most soils and TDR probe lengths. An experiment using coarse sand was used to test this. Seven hundred–micrometer quartz sand was wetted from the base of a 0.18-m-tall cell with a 0.15-m-long parallel plate probe in it. The data presented in Fig. 18, measured with the TDR, confirm that refractive index averaging is appropriate.
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Fig. 17. Modeled and measured waveforms for layers of Plexiglas and water. (A) Modeled waveforms for homogeneous dielectric. (B) Modeled waveforms for layered dielectrics. (C) Measured waveforms for layered dielectrics corresponding to those modeled in Fig. 17B. (From Schaap et al., 2003).
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Fig. 18. Measurements of the permittivity of coarse grained (700 µm) monosized quartz sand. The data and modeling suggest that for these measurements the averaging remains in the refractive index regime.
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TDR Waveform Theory and Modeling
The above discussion indicates the potential of both simulating waveforms and conducting inverse analysis on waveforms to obtain greater information content. Transitioning between time and frequency domains is of interest for extending the information content extracted from TDR measurements. Waveforms obtained for travel time analyses rely on proper identification of markers for accurate permittivity determination and subsequent water content estimation. Early efforts to streamline this process were aimed at automating waveform analysis techniques (Baker and Allmaras, 1990; Heimovaara and Bouten, 1990). In recent years, advanced means of analyzing waveforms have been developed that extend the information content beyond conventional travel-time analysis illustrated in Fig. 30. Information on the complex dielectric permittivity is embedded within the waveform captured in the time domain and may be extracted in the frequency domain using Fourier analysis. Inverse Fourier analysis may be used to fit frequency-dependent model parameters to waveforms in the time domain. In the time domain, the input signal, vo(t), describes the cable tester, cable, and probe head, and the response function, r(t), describes the entire system, including the sample being measured. These are obtained from TDR waveforms measured in air and in the sample material. The measurement in air may have the central conductor pin removed from the coaxial sample holder, or the input function may be modeled. The system response, s(t), contains information on the sample's dielectric permittivity, where r(t) is described by the following convolution integral (van Gemert, 1973):
 | [21] |
where 
is the variable of integration and s(t) describes how an input signal will be modified by the sample. The discrete fast Fourier transform of both vo(t) and r(t) reduce the unsolvable convolution theorem integral to a simple algebraic expression describing the scatter or system function, S11(f), in terms of the frequency-dependent response, R(f), and input, V0(f), functions, given by Lathi (1992) as
 | [22] |
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Fig. 30. An example of a waveform in water showing tangent lines fitted at the end of the waveform and the times used to calibrate a TDR probe (Eq. [45]).
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Alternative transmission line modeling approaches have been demonstrated which describe the waveform using a wave propagation model, accounting for multiple reflections of the segmented system (e.g., cable, head, and probe), and described in terms of input impedance (Lin, 2003a, 2003b; Feng et al., 1999). The determination of sample properties is (i.e., , dc, frel) generally performed by fitting model parameters to the discrete measured waveform. This fitting may take place in the time domain, or after Fourier transform, the fitting may take place in the frequency domain. The theoretical basis of many of these approaches relies on a coaxial probe geometry, which is only approximated by multirod probe designs. Data presented in both domains are illustrated in Fig. 19, where the discrete data from the waveform, Vi, measured in the time domain are presented in Fig. 19a. The reflection coefficient is determined from the measured waveform voltages, described as
 | [23] |
where Vo is the voltage in the cable before entering the probe and Vref is the voltage in the cable tester (i.e., at -0.6 m in the Tektronix 1502), sometimes assumed to be zero. These were transformed to frequency-dependent discrete data points making up the scatter function shown in Fig. 19b. Waveform transformation procedures and results are described by a number of authors (Heimovaara, 1994; Friel and Or, 1999; Weerts et al., 2001).
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Fig. 19. (a) TDR waveform in water. (b) The corresponding scatter function for water measured using TDR and modeled using the Cole–Cole (1941) relation.
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Time Domain Analysis
A multisection approach may be used to reconstruct waveforms in the time domain (Yanuka et al., 1988; Feng et al., 1999; Heimovaara, 2001; Robinson et al., 2003; Schaap et al., 2003). Variations of this approach attempt to describe the cable–head–sample system using a multisection scatter function, which may again be linked to the complex dielectric permittivity of the soil. Feng et al. (1999) demonstrated the usefulness of this multisection approach for extracting information from multilayered soils oriented perpendicular to the propagating electrical signal. Heimovaara (2001) demonstrated the potential for extracting information on signal dispersion and subsequently on the soil properties from inverse analysis. Greater information content is possible using optimization techniques coupled with dielectric mixing model parameters describing the frequency-dependent character of soil constituents (Heimovaara et al., 1996; Friel and Or, 1999; Lin, 2003a). Using inverse analysis of 2000 simulated TDR waveforms, Weerts et al. (2001) examined Debye model parameter sensitivity and determined correlations between model parameters and waveform characteristics. They found bulk EC to have the strongest influence on the waveform character. Both high-frequency and static permittivity parameters had a significant influence in waveform modeling, but each were capable of fitting waveforms under three possible scenarios dependent on the relaxation frequency selection. The three scenarios show the dispersion curve within the TDR frequency bandwidth being (i) cut off by the low frequency limit, (ii) cut off by the high frequency limit, or (iii) completely contained within the frequency bandwidth of the waveform data. Further refinement of these techniques should improve the information content and reliability of analysis for studying relaxation phenomena and complex permittivity determination.
Frequency Domain Analysis
The frequency content of the measured TDR waveform extends from about 20 kHz to roughly 1.5 GHz (Heimovaara, 1994; Fig. 7). The quality of the measurement equipment, the permittivity of the dielectric and its lossy nature, as well as probe geometry, may influence this range. Frequency-dependent information may be determined by fitting an appropriate model to the transformed scatter function in the frequency domain. One such model describes the multiple reflections of an open-ended coaxial transmission line, which can be modeled according to the scatter function given by Clarkson et al. (1977) as
 | [24] |
in which L (m) is the probe length and where 
is the reflection coefficient described as
 | [25] |
in which z = zc/zp is the impedance ratio of the cable, zc, and probe, zp, and where is the transverse electromagnetic mode propagation constant written as
 | [26] |
Heimovaara (1994) and Friel and Or (1999) adopted a modified form of the Debye (1929) model by Cole and Cole (1941) to fit *(f) to scatter function data, using parameters describing the static and high-frequency dielectric permittivity, relaxation frequency, and EC of different liquids:
 | [27] |
where, is the high-frequency limit of the real permittivity, s,cc is the static value of the real permittivity, f is the measurement frequency (Hz), frel is the mean relaxation frequency, ßcc is a parameter accounting for a spread in relaxation frequency, dc is the solution electrical conductivity (S m-1), and j is the imaginary number . Lin (2003b) built on this approach and modeled the frequency-dependent dielectric permittivity of soil constituents (i.e., solid, air, water, and bound water) using the Debye (1929) model. These dielectric components were incorporated into the dielectric mixing model of Dobson et al. (1985) and fitted in the frequency domain. Information lost due to attenuation under lossy conditions is less detrimental for a frequency domain analysis than the second reflection determination in a time domain analysis. Jones and Or (2001) coupled frequency domain analysis with the use of short (0.02–0.06 m) TDR probes to extend the TDR measurement range in saline soils. Permittivity determination was increased to bulk EC levels eight times greater than realized using conventional time domain analysis with a 0.15-m probe.
Electrical Conductivity Measurement Using TDR
One of the great strengths of TDR is that it can be used to measure bulk EC in addition to permittivity (Dalton et al., 1984; Topp et al., 1988; Dalton, 1992; Nadler et al., 1991; Heimovaara and de Water, 1993; Mojid et al., 1997). This section examines the principal way of measuring the EC from TDR waveforms and an alternative broadband conductivity method.
The Method of Giese and Tiemann
Giese and Tiemann (1975) are credited with the first determination of sample resistance using TDR waveforms. This is equivalent to measuring the low-frequency resistance across the sample between the probe rods and has been used by Nadler et al. (1991). Heimovaara (1993) also used the reflection coefficient at infinite time () as a method of calculating the sample resistance, incorporating it into software for TDR waveform analysis (Heimovaara and de Water, 1993) where Vo and Vmax (i.e., Vi in Eq. [23]) are illustrated in Fig. 20. The resistance across the rods is then calculated according to
 | [28] |
where Rtot is the total resistance (
) of the transmission line, Zc is the cable tester impedance (50 ), and is the reflection coefficient at infinite time on the waveform, or at a point where the reflection level has stabilized to a maximum value. The Tektronix TDR has a useful feature in which the resistance () value at any given apparent distance along the transmission line can be read directly at the location of the cursor. Friedman and Jones (2001) used this feature to characterize qualitatively the dependence of the electrical conductivity's anisotropic factor on distance in the time domain, which relates to frequency.
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Fig. 20. TDR waveforms from deionized water and from a KCl solution. The various voltages used to obtain the reflection coefficient are illustrated.
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Heimovaara and deWater (1993) further proposed that the total resistance was made up of two components, that of the cable (Rc) and that of the sample (Rs):
 | [29] |
However, more recently Castiglione and Shouse (2003) demonstrated that this intuitive relationship is inexact. They presented a new procedure to calculate the sample resistance (Rs), independent of contributions from cable and fittings, that requires the waveform to be scaled according to the reflection coefficient for an open circuit (open) in air and for a shorted circuit (short):
 | [30] |
The value of scaled is now used in Eq. [28] in place of to determine the sample resistance. Making good repeatable measurements for the open and short are difficult using conventional TDR probes, and specialized calibration kits have been used for precision work in this regard (Feldman et al., 1996; Huisman et al., 2002). The bulk EC for a given temperature depends on the cell constant, or geometric factor (g) of the probe conductors (discussed in more detail in the following section), and is
 | [31] |
In most circumstances, when Rc << Rs, Eq. [29] is adequate for measurements of EC in soils.
The Broad-Band Conductivity Method of Topp, Yanuka, Zebchuk, and Zegelin
The method described above allows one to obtain a measurement of the low-frequency EC. However, conductive losses occur at a range of frequencies. Thus, Topp et al. (1988) proposed an alternative method of estimating what is best described as a broad-band conductivity term. According to Topp et al. (1988) and Topp et al. (2000) the conductivity term (Topp) provides an estimate of the combined effects of both the dc conductivity (dc) and the imaginary permittivity (''r) due to relaxation losses.
 | [32] |
Topp et al.'s (1988) expression was given in terms of voltages in Nadler et al. (1991) as
 | [33] |
with l as the physical length of the probe (m). The voltages with reference to the waveform are illustrated in Fig. 20. One can see that for deionized water the signal is horizontal; however, the equivalent signal for KCl slopes down to V1. This divergence of the signal from the horizontal is considered to be caused by attenuation of the signal as it propagates along the length of the probe. As the attenuation depends on both the EC of the material and its dielectric relaxation properties, this broadband conductivity term may capture effects of both dielectric losses and conductivity for the TDR frequency bandwidth. Interpretation of the waveform in this way is interesting but should not be considered equivalent to measurements with low-frequency techniques such as a bridge or four-probe instruments.
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PROBE DESIGN, CONSTRUCTION, AND CALIBRATION
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TOP
ABSTRACT
INTRODUCTION
