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a Inst. of Terrestrial Ecology, Swiss Federal Inst. of Technology Zurich, Grabenstrasse 11, CH-8952 Schlieren, Switzerland (currently, Inst. of Environmental Physics, Univ. of Heidelberg, D-69120 Heidelberg, Germany)
b Lab. for Electromagnetic Fields and Microwave Electronics, Swiss Federal Inst. of Technology Zurich, Gloriastrasse 35, CH-8092 Zurich, Switzerland
c Inst. of Terrestrial Ecology, Swiss Federal Inst. of Technology Zurich, Grabenstrasse 11, CH-8952 Schlieren, Switzerland
* Corresponding author (benedikt.oswald{at}iup.uni-heidelberg.de)
Received 1 December 2003.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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. As opposed to most or all previous studies, in this study we investigated a TDR probe that employs only one single metallic rod for measuring . The probe is based on the concept of a Sommerfeld wire. The electromagnetic behavior of the probe is analyzed and experimental results are presented that provide evidence of the probe's applicability and its working principles. Dielectric properties, and hence water contents, obtained with the single rod probe were compared with those of a standard two-wire TDR probe, and with gravimetrically determined water contents. Additionally, we calculated the probe's region of influence (RoI), which was considerably larger than that of standard two-wire probes. We also tested the RoI experimentally. The new probe is easier to install, robust, and has a larger sphere of influence over which the dielectric permittivity distribution is averaged.
Abbreviations: RoI, region of influence • SRP, single-rod probe • TDR, time domain reflectometry • TEM, transverse electric-magnetic • TM, transverse-magnetic • TRP, twin-rod probe
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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To sense the dielectric properties of the medium that surrounds the conductor, the SRP relies on a transverse-magnetic (TM) mode instead of a TEM mode. Due to the characteristics of electromagnetic mode propagation of the SRP, this probe type has a relatively large RoI. The RoI is defined as the radius r70 of the cylinder transporting 70% of the electromagnetic power of the dominant TM mode. In this study we will calculate r70 for different frequencies and dielectric permittivities, and show that the RoI is larger than that of a TRP, but with the electromagnetic power being contained within a cylinder of well-defined radius. We will also investigate the range of the SRP experimentally The RoI of the probe is an instrumental characteristic, different from the concept of sensitivity of a TDR probe, which has been investigated extensively (Knight, 1992; Ferré et al., 1996; Knight et al., 1997; Ferré et al., 1998, 2000).
We will use a semianalytical model for analyzing the TM mode. The model gives the theoretical background of the SRP, including methods for calculating the phase velocity, the attenuation of the propagating signal, and the TM mode fields. In particular, we will calculate the radius of fractional power rfrac, which is the radius within which a certain percentage of the TM mode's total power is contained for various frequencies and dielectric permittivities. The radius of fractional power is discussed with respect to the dimensions of the experimental setup. Experimental results are then presented that demonstrate the nature of the TDR signals measured with SRP in various sand–water mixtures. Measurements using TRP will be shown for comparison. Several unresolved questions with respect to the measurement function of the SRP are discussed at the end of this paper. A list of symbols is given in Appendix 1.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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, and magnetic, µ, properties of the medium. The dielectric permittivity can be converted into water content, , given an appropriate calibration function (e.g., Roth et al., 1990; Robinson et al., 2003). Therefore, for a given probe length 
and a measured two-way travel time t0 of the signal, one can calculate the signal speed based on the properties of the electromagnetic mode that is propagating on the probe. With a TRP (TEM mode) the velocity v of the signal equals the velocity c of an electromagnetic plane wave in a medium with dielectric permittivity and magnetic permeability µ given as (Ramo et al., 1984)
 | [1] |
The relative dielectric permittivity r is then obtained from
 | [2] |
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The SRP employs a TM mode without a magnetic field component in the longitudinal direction of the probe. Its characteristics are shown in Fig. 1 .
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The propagation of guided waves is described with Maxwell's equations. For a TM mode on a Sommerfeld wire, these equations can be considerably simplified (Sommerfeld, 1899). The resulting set of three equations is:
 | [3a] |
 | [3b] |
 | [3c] |
is the tangential component of the magnetic field. Sommerfeld (1899) solved these equations using the assumption that the functions Er, Ez, and H are the product of a common factor exp(–j
t + jhx) and functions depending on radius r only; that is,
 | [4a] |
 | [4b] |
 | [4c] |
is the angular frequency, and h the complex propagation constant. When inserted into Eq. [3a] through [3c] and accounting for the boundary conditions at the interface between the metal conductor and the dielectric at radius r = a, the following transcendental equation results:
 | [5] |
is its derivative with respect to 
, J0(L) is the Bessel function of the first kind of order 0, and J'0
is its derivative with respect to L. Equation [5] must be solved for the propagation constant h to obtain the speed of propagation. While and L are functions of radius r in a cylindrical coordinate system, and thus depend on the material properties of the regions, the propagation constant h by definition must be equal in both regions. For r 
a inside the wire, the dimensionless radius L is given by
 | [6] |
r 
exterior to the wire
 | [7] |
Hence, at r = a we have = a
and L = a
. The general expression for the wave number k in Eq. [5] is
 | [8] |
In the exterior region the wave number is k, and the conductivity 
= 0. In the region inside the wire of the SRP, the wave number is kL while corresponds to the ohmic conductivity of the metallic conductor of the SRP. The dielectric permittivity ' inside the wire is assumed to be the value of vacuum 0. Once the propagation constant h is computed, we can express the wavelength 
TM of the TM mode through
 | [9] |
denoting the real part of a complex number, and hence
 | [10] |
and µ must be used instead of 0 and µ0.
Transverse-Magnetic Mode Propagation Constant h
We developed a Mathematica 4 program (Wolfram, 1991) for solving the eigenvalue Eq. [5] to better understand the dependence of h on permittivity, SRP radius, and frequency. In particular, we computed the propagation constant for the configuration analyzed by Sommerfeld (1899) to compare his manual solution with our results. Sommerfeld (1899) calculated the propagation constant h for a copper wire with a radius r = 0.001 m, based on a value of 
= µ0/2 = 4.7 x 105 m–2, and used this value in his approximate solution of the nonlinear eigenvalue equation. His value of corresponds to the conductivity of Cu ( = 5.5955 x 107 S m–1), as listed in Table 1 (Case 1). We used a value of = 5.882 x 107 S m–1 for the conductivity of Cu and obtained h and Le, the distance after which the amplitude was reduced by a factor of e = exp(1), as given by Case 2 of Table 1. The distance Le = 770 m obtained by Sommerfeld (1899) agrees well with the value from our calculations, given his approximate calculation. In general, attenuation decreases with increasing probe diameter, and weakly depends on the dielectric permittivity around the SRP's metallic core. Furthermore, attenuation need not be considered when the dielectric has no losses (i.e., tan
diel = 0). This approximation is acceptable for sand–water mixtures but not when the SRP is inserted into tap water, as will be shown below. Additionally, once h is available, we can also evaluate the field components (see Table 2 and the next section).
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, normalized with respect to the value at the boundary of the SRP at r = rSRP = 0.005 m, at a frequency of 1 GHz, a relative dielectric permittivity r = 1 corresponding to air, and with Cu as the conducting material. The absolute value of Ez is about three orders of magnitude smaller than the other components, Er and H. The plots of the TM mode field components in Fig. 2 are instructive in that they explain the nature of the mode. However, one must also consider the power transported by the mode to assess the RoI of the SRP.
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 | [11] |
Given the expressions for the electric and magnetic field components of the TM mode (Table 2) we can calculate the Poynting vector S. Integrating S over the cross-sectional area of a cylinder with radius rc, coaxial to the SRP configuration, leads to the power Prc transported by the TM mode within this cylinder, excluding the portion of the cylinder occupied by the SRP metallic core
 | [12] |
 | [13] |
The ratio Pfrac between Prc and Ptot is defined as the fractional power Pfrac transported by the TM mode within a cylinder of radius rc:
 | [14] |
Figure 3
shows plots of the fractional power for r = 1, 2.8, and 20, and the frequencies f = 0.3, 1, 2, and 3 GHz. These values were selected to find the upper and lower limits of the RoI corresponding to the expected situation in the experimental setup. The lowest frequency of 0.3 GHz was selected since this is the lower limit of the passband of the widening structure that is used to couple the TEM mode from the coaxial cable into the TM mode of the SRP. Lower frequencies are mostly blocked by this structure. Values of the radius r70 are tabulated in Table 3. Notice from Fig. 3 and Table 3 that r70 generally decreases with increasing frequency f while r70 generally decreases with increasing dielectric permittivity r. Also, the SRP has a cylindrically shaped measurement volume, given by the radius of the RoI and the length of the SRP; this measurement volume is generally much larger than that of a TRP with comparable lateral dimensions. The data in Table 3 further imply that the measurement volume depends on the dielectric permittivity, and hence on the water content itself.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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vs. (), derived by Roth et al. (1990), and data obtained from the travel time analysis. We note that the travel time of the signal in tap water was difficult to extract due to the considerable attenuation.
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r values calculated for the two different probe types differ considerably. The measurement volume of the SRP is a cylinder with a diameter of approximately 0.2 to 0.3 m, while the TRP has a RoI diameter of about 0.03 m and a distance between the center of the rods of 0.028 m. |
DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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A few experimental difficulties were encountered during our tests of the SRP concept. To exclude the influence of the container walls on the SRP traces, the container with the sand–water mixture had to be relatively large, both in diameter and height. Having a large container inevitably leads to rapid redistribution of water in the vertical direction. This problem is less critical at the lower water contents. Also, for the same reason, samples for measuring gravimetric water content cannot have relatively high degrees of saturation. The calibration relationship () (Roth et al., 1990), shown in Fig. 7, is reliable up to a volumetric water content of about 0.2. The determined () may still be applicable at higher volumetric water contents, but with more uncertainty. To determine a reliable calibration relationship for the wet range, a different experimental configuration may be required. We emphasize that while the SRP principle works up to complete saturation, the experimental setup to derive the calibration function in the wet range will need to be improved.
In this study we performed measurements using metallic rods, inserted parallel to the SRP, to investigate the range of the SRP. These measurements cannot be compared directly with the RoI calculations since the metallic rod produces a point-like perturbation. Nevertheless, the RoI gives evidence on the far-reaching nature of the TM mode. Therefore, the sand container with a diameter of 0.4 m appeared to be large enough to not disturb the measured TDR traces of the SRP. This is further supported by the fact that the results obtained with the TRP and the SRP were similar. We also conclude from the theoretical calculations that the RoI of an SRP is considerably larger than that of a TRP with comparable rod diameter.
The present theoretical analysis of the SRP neglects ohmic and dielectric losses ( = tandiel = 0). While the involved model works for such media as sand–water mixtures, further investigations will need to consider the conductivity of the dielectric surrounding the metallic core of the SRP (i.e., a more dissipative dielectric material, such as a fine-textured soil. The effect of ohmic losses on the SRP trace is shown in Fig. 6 for the tap water trace.
While the current experimental setup uses a TDR instrument (Tektronix 11801) with a very short rise time (28 ps), there is no fundamental obstacle to using a more widespread, standard Tektronix 1501b/c cable tester as long as only travel-time measurements are required. Due to its longer rise times, longer SRPs will yield better results. With respect to practical applications the same types of connectors as used for standard TDR probes can be employed. We also note that the coupling structure used in our study may benefit from improvements with respect to both size, weight, and lower cutoff frequency.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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. This type of probe uses a TM mode for signal propagation. The SRP was first analyzed theoretically in terms of the dominating TM mode of propagation. The propagation constant h was obtained from the solution of a nonlinear eigenvalue equation, and expressions for the electric and magnetic field components Er, Ez, and H were evaluated. We also calculated the RoI of the SRP as a function of the frequency and dielectric permittivity of the material surrounding the SRP. Results show that the measurement volume of the SRP depends on the dielectric permittivity (i.e., the water content). Future investigations will need to consider the conductivity of the dielectric material surrounding the metallic core of the SRP (i.e., a dissipative dielectric material).
We also experimentally tested the SRP for different probe lengths (150, 200, and 300 mm). Travel times were extracted from the measured TDR traces, converted to dielectric permittivities r, and compared with travel times obtained with a standard TRP. The relationship between and r was interpreted using the mixture model of Roth et al. (1990). We used a relatively large container for the SRP measurements to reduce potential interference by the walls of the container. Outliers in the () relationships may have been caused by the fact that the volumetric water content was not entirely uniform in the vertical direction of the container, despite our rapid gravimetric sampling immediately after packing the sand–water mixture. We conclude that it is best to use the longer probes, such as the SRP300 and SRP200, and to avoid the shorter SRP150. In particular, from an electromagnetic point of view, it is advisable to use longer probes so that the TM mode has a longer section where it can fully develop after leaving the coupling structure.
We believe that the concept of the SRP is useful and offers several advantages over standard TDR probes. For example, because of its simple construction, a SRP can be inserted more easily into different types of media. It also has drawbacks, such as a relatively large RoI, which may compromise the accuracy of more localized measurements, and a more complicated analysis of travel times. Nevertheless, we have shown that a SRP can be successfully used for measurements of the water content if reliable calibration functions (Roth et al., 1990; Robinson et al., 2003) are available.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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c, speed of light (m s–1)
c0, speed of light in vacuum (m s–1)
cTM, speed of the TM mode (m s–1)
Er, radial component of the electric field vector E (V m–1)
E0r
, radial component function of the electric field vector E, depends only on radius r (V m–1)
Ez, axial component of the electric field vector E (V m–1)
E0z, axial component function of the electric field vector E, depends only on radius r (V m–1)
f, frequency (Hz)
h, propagation constant (m–1)
H, magnetic field vector (A m–1)
H, tangential component of the magnetic field vector H (A m–1)
H0, tangential component function of the magnetic field vector H, depends only on radius r (A m–1)
H0(r), Hankel function of the first kind of order 0
H'0, derivative of H0 with respect to r
J0(r), Bessel function of first kind of order 0
J'0, derivative of J0 with respect to r
k, wave number outside the metallic core (m–1)
kL, wave number inside the metallic core (m–1)
j, imaginary number 
Le, distance after which the amplitude has been reduced by a factor of e = exp(1) (m)
, length of a TDR probe (m)
Pfrac, fraction of total power Ptot contained within a cylinder of radius rc
Ptot, total power transported by TM mode (W)
Prc, power contained within cylinder with radius rc, coaxial to the single rod probe (W)
r, radius variable (m)
r70, radius of cylinder coaxial to the single rod probe, within which 70% of the electromagnetic power of the dominant TM mode is transported (m)
rc, cylinder radius (m)
rfrac, radius of cylinder coaxial to the single rod probe, within which a specific fraction of the electromagnetic power of the dominant TM mode is transported (m)
rSRP, radius of single rod probe (m)
S, the (electromagnetic) Poynting vector: average power density per unit area (vA m–2)
t, time (s)
trise, tr, rise time of an electric signal, defined as the time required for the signal to rise from 10 to 90% of its final value (s)
t0, two-way travel time of an electric signal on a transmission line (s)
tandiel, dielectric loss tangent of a material
v, velocity of an electric signal (m s–1)
x, spatial variable (m)
Greek Symbols
= r0, absolute complex dielectric permittivity (As V–1 m–1)
0, absolute dielectric permittivity of vacuum [(µ0c2)–1]
r, complex relative dielectric permittivity
a, relative permittivity of the air phase
c, effective relative permittivity of a composite medium
s, relative permittivity of the solid phase
w, relative permittivity of the aqueous phase
, porosity
µ0, magnetic permeability of vacuum 4
10–7 (Vs A–1 m–1)
µr = 
, complex relative magnetic permeability, equals 1 for considered soil materials
µ = µrµ0, magnetic permeability of a material (Vs A–1 m–1)
µ'r, real part of the complex relative magnetic permeability
µ''r, imaginary part of the complex relative magnetic permeability
, wavelength of an electromagnetic signal (m)
0, wavelength of an electromagnetic signal in vacuum or air (m)
TEM, wavelength of the TEM mode (m)
TM, wavelength of the TM mode, dominant on the single rod probe (m)
, angular frequency = 2f (rad s–1)
, dimensionless radius in the region outside the metallic core
L, dimensionless radius in the region inside the metallic core
, ohmic conductivity (S m–1)
, volumetric water content (m3 m–3)
, variable used in eigenvalue calculation (m–2)
Fracture Symbols
, imaginary part of a complex number
, real part of a complex number
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ACKNOWLEDGMENTS |
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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H. Vereecken, S. Hubbard, A. Binley, and T. Ferre Hydrogeophysics: An Introduction from the Guest Editors Vadose Zone J., November 1, 2004; 3(4): 1060 - 1062. [Full Text] [PDF]
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= {10 + j0.00001,300 + j0.02} for all cases.
= 0.52), and from traces measured with twin-rod probes (TRP) and single-rod probes (SRP) of different lengths. Top: SRP300 (300 mm); middle, SRP200 (200 mm); bottom, SRP150 (150 mm). Traces were recorded and analyzed when metallic rods with the same lengths were inserted parallel to the SRPs. The distance between the SRP and the metallic rod were 50 mm (circles), 100 mm (diamonds), and 200 mm (squares); the plus (+) symbols identify the TRP traces.