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a Institute of Terrestrial Ecosystems (ITES), Swiss Federal Institute of Technology (ETH), CHN E29, Universitätstr. 16, CH-8092 Zürich, Switzerland
b USDA-ARS, Great Plains Systems Research Unit, Fort Collins, CO, USA
c Institute of Applied Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
d Laboratory for Electromagnetic Fields and Microwave Electronics, ETHZ, ETZ K 88, Gloriastrasse 35, CH-8092 Zürich, Switzerland
e Institute of Terrestrial Ecosystems (ITES), Swiss Federal Institute of Technology (ETH), CHN F 28.1, Universitätstr. 16, CH-8092 Zürich, Switzerland
* Corresponding author (mike.schwank{at}env.ethz.ch)
1 The EnviroSMART capacitance sensors evaluated here were designed and manufactured by Sentek Pty. Ltd., Australia. Use of such commercial products does not constitute endorsement by the ETHZ, USDA-ARS, or University of Bern. Sentek did not provide any financial assistance (cash or in-kind) for the project. 
Received 13 January 2006.
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ABSTRACT |
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 TOPEXECUTIVE SUMMARY ABSTRACT INTRODUCTIONBACKGROUND AND THEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSREFERENCES |
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from measured resonant frequency fr, which is directly affected by the bulk soil permittivity 
. However, the relationship fr() requires improved quantification. We conducted laboratory experiments to characterize the response of the Sentek EnviroSMART sensor system for a full range of values from air to water and a range of temperatures. Water–dioxane mixtures were placed into a solvent-resistant container equipped with custom tools for heating and mixing the fluid, removing air bubbles from sensitive surfaces, measuring permittivity in situ, and creating an axisymmetric metal disturbance to the electric field. Total capacitance C was measured using a vector network analyzer (VNA) connected to one sensor, while four other sensors provided replicated fr readings. The measured temperature response of free water permittivity was linear with a negative slope, which is qualitatively consistent with theory. A precise nonlinear relationship between and normalized fr was derived. The instrumental error in was RMSE = 0.226 (for 3 < < 43), which corresponds to a measurement precision in () derived from Topp's equation of RMSE= 0.0034 m3m–3. Axisymmetric numerical simulations of the electric field supplemented the experimental results. The characteristic length scale for the distance measured radially from the access tube is 12.5 mm, meaning that 80 and 95% of the signal are sensed within approximately 20 and 37 mm of the access tube, respectively. The results are crucial for scientific applications of the investigated sensor type to environmental media.
Abbreviations: FEP, fluorinated ethylene-propylene • SMD, surface mounted device
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INTRODUCTION |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONBACKGROUND AND THEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSREFERENCES |
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, or "dielectric constant," acts as a proxy for the volumetric soil water content in units of cubic meters per cubic meter. Capacitance, or frequency domain, sensors are designed to measure resonant frequency rather than directly measuring capacitance, and the permittivity of a medium is most directly related to the effective capacitance. Such capacitance sensors have been evaluated previously on the basis of measured water contents (Baumhardt et al., 2000; Evett and Steiner, 1995; Paltineanu and Starr, 1997). With the exception of Kelleners et al. (2004a, 2004b), the relationship between the sensor reading (or resonant frequency) and added capacitance or permittivity of the measured medium has not been well characterized. Kelleners et al. (2004a, 2004b) focused on the effects of ionic conductivity and dielectric losses rather than on quantifying a relationship between soil permittivity and the sensor reading for a full range of soil permittivity values (i.e., no data for the approximate range of 3 < < 20). Available field data from near-surface sensors at 30- to 60-cm depths measured with the Sentek EnviroSMART (Sentek Sensor Technologies, Stepney, SA, Australia) sensor system display variations at diurnal and other time scales associated with measured temperature fluctuations (Green et al., 2004).1 The exact causes and quantification (i.e., correction) of these temperature effects on apparent water content measurements are currently unknown, although the phenomenon has been observed in the laboratory (Baumhardt et al., 2000) and theories have been postulated (Or and Wraith, 1999; Robinson et al., 2003).
To gain further insight into dielectric processes in soils from long-term time series of data collected with the capacitive EnviroSMART system, the relation between the sensor reading and the soil permittivity is necessary. For that purpose we performed the laboratory characterization using this specific sensor type. However, the procedures presented here are adaptable for characterizing similar new measuring systems before their field application.
Knowing the exact relation between the soil permittivity and the sensor reading is also required for validating and improving dielectric mixing models used for soil moisture estimation from proxy data . The scope of the present study is limited to inferring from based on previous work (e.g., Topp et al., 1980), noting that quantification of is not sufficient to determine the accuracy of in real soils due to complex losses associated with the imaginary part of .
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BACKGROUND AND THEORY |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONBACKGROUND AND THEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSREFERENCES |
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) is a function of the permittivity of the medium surrounding the access tube.
Basic Dielectric Theory
The relative permittivity = ' – i'' of a material is a dimensionless complex number (relative to a 
1 for air). The real part ' describes the ability of the material to interact with an external electric field in terms of energy storage (or wave propagation velocity). The imaginary part '' is the sum of a conductivity term describing the ohmic losses and a relaxation term representing the relaxation losses of the material. The formal expression for the imaginary part '' includes both electrical conductivity (in the numerator) and (effective) measurement frequency (in the denominator). Consequently, any effect of conductivity, including temperature effects, changes over the frequency range of the instrument as the soil water content changes.
The relative permittivity of air and the solid phase (matrix) of a soil are a = 1 and m 3–5, respectively. The permittivity of pure free water at 25°C and frequencies <1 GHz is w 78. Due to the large contrast between w, a, and m, the overall permittivity of a wet soil is a strong function of the volumetric water content . Thus, bulk permittivity is an appropriate proxy quantity for determining soil water content.
The bulk permittivity of a soil is a function of volumetric water content, salinity, temperature, electromagnetic frequency or wavelength, volume fraction of bound and free water related to the specific soil surface area, soil bulk material, and the shapes of the water inclusions (Dobson et al., 1985). Consequently, cannot be calculated as the linear weighting of the permittivities of the constituents according to their volume fraction. Therefore, more sophisticated dielectric mixing approaches considering the morphology of the dielectric constituents have to be used.
Empirical Dielectric Mixing Models
Various empirical and semiempirical models have been used to relate permittivity to water content of different soil types. The most commonly used empirical model (Topp et al., 1980) is:
 | [1] |
 | [2] |
The semiempirical mixing model of (Roth et al., 1990) is based on an empirical power-law dielectric mixing approach considering the aqueous, solid, and gaseous soil phases with corresponding permittivities w, s, and a and volumetric fractions , (1 – 
), and ( – ):
 | [3] |
The soil porosity has to be determined experimentally (here = 0.46), and the exponent 
= 0.46 ± 0.007 was determined from a nonlinear regression applied to measured data.
The semiempirical model of Wang and Schmugge (1980) considers textural effects (clay and sand content) in terms of an adjustable transition point t dividing the water content range into two domains. The transition point t is greater for soils with high clay content (high specific surface area) than for soils with high sand content. A linear three-phase mixture (paracrystalline water, solid, and gaseous phases) is applied for < t. Thereby, the permittivity of the paracrystalline water is reduced due to molecular interactions with the proximate solid soil phase. The fourth dielectric soil phase representing the water that is not affected by the solid phase (free water) is considered for > t.
Inductor-Capacitor (LC) Circuit Theory
Because a sensor reading R is proportional to the corresponding resonant frequency, fr, of the sensor in a given environment, R (25000 < R < 37000) is converted to fr using the linear relationship (Sentek technical support, personal communication, 2004):
 | [4] |
 | [5] |
Electrical resistance is neglected here. The total capacitance C includes the series capacitance of the plastic tube Cacc plus any air gap between the rings and inner diameter of the tube, as well as an internal capacitance Cint acting in parallel to C, comprised of the on-board sensor capacitance and the contribution caused by the materials inside the capacitor rings. As illustrated in Fig. 1e, these capacitances are in series and in parallel to each other (Kelleners et al., 2004b). Therefore, the total C connected to the sensor electronics is:
 | [6] |
= C() is the capacitance associated with the medium outside the access tube, which is the quantity of interest. Theoretically, Cint varies with C due to coupled effects on the inner and outer electrical fields of the ring capacitor. Also, Cint comprises both the capacitance of the physical inner space of the rings and the intrinsic capacitance on the sensor board. In the following derivation, Cint is treated as a constant. This approximation is invoked here only to correct for the effect of the fluorinated ethylene-propylene (FEP) coating around the plastic access tube on normalized sensor readings. It will be shown from the linear relation between measured fr–2 for known external capacitances CSMD that this is a reasonable assumption.
Now it will be shown that a change 
C of the capacitance C is proportional to the difference (fr–2) = frA–2 – frB–2 of inverse squared values of two resonant frequencies frA and frB.
 | [7] |
The proportionality factor dC/d(fr–2) can be expressed, using the chain rule, as:
 | [8] |
The derivatives d(fr–2)/dC and dC/dC are calculated from Eq. [5] and [6] leading to:
 | [9] |
This simple derivation shows that the proportionality factor dC/d(fr–2) = (4
2L)–1 in Eq. [7] does not depend on C if the equivalent circuit diagram sketched in Fig. 1e is used. The validity of this equivalent diagram with a constant value of Cint is confirmed experimentally using known external capacitors below.
Definition of Normalized Sensor Reading
The characteristics of individual EnviroSMART sensors are not perfectly uniform. As a consequence, the readings Rk of different sensors k are different even for identical materials with permittivity surrounding the sensors. To eliminate the effect of these sensor-specific differences, sensor readings Rk are normalized to be:
 | [10] |
Using the relationship (Eq. [5]) between fr and C, the normalized sensor reading N can be expressed by the sensor capacitances Ca, Cw, and C if it is embedded in air, water, and an arbitrary material with permittivity :
 | [11] |
Default Calibration for Water Content
A power-law calibration function was given by the vendor (Sentek, 2001) for relating N to . The empirical default calibration function N() with the three parameters a = 0.1957, b = 0.404, and c = 0.02852 is commonly used to estimate of sands, loams, and clay loams from N:
 | [12] |
The inverse relationship is thus
 | [13] |
Temperature Dependence of Environmental Permittivity
Below the relaxation frequency (<10 GHz), the permittivity of pure free water, w, decreases with increasing temperature T. Qualitatively, this is explained by increasing thermal distortion of the dipoles with increasing temperature, which hinders water molecules from aligning with the applied electric field. Meissner and Wentz (2004) provided a semiempirical model for measured values of w in terms of an approach based on two Debye relaxation frequencies. The estimated negative temperature gradient is approximately dw/dT –0.36 K–1 for frequencies smaller than 500 MHz.
However, positive temperature gradients have been observed under field conditions for high cation exchange capacity soils in which the relaxation frequency of colloid-bound water is often lower than the measurement frequency of the applied measurement technique (de Loor, 1983). In this regard, investigating the temperature dependency of the permittivity (T) of environmental material is of particular importance (Baumhardt et al., 2000; Evett et al., 2006; Logsdon and Laird, 2004; Wraith and Or, 1999).
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MATERIALS AND METHODS |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONBACKGROUND AND THEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSREFERENCES |
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Surface mounted devices (SMDs) of known capacitance values CSMD were soldered to one sensor electronics board, with and without the ring capacitor attached. Values of CSMD were selected such that the measured frequency range with the ring capacitor disconnected fell between sensor measurements in air and water (i.e., 1.03 pF 
CSMD 21.25 pF).
All measurements with SMD capacitors CSMD were taken in air without the plastic access tube. Thus, sensor readings with no added capacitance (CSMD = 0) exceeded the air count Ra when the probe was inserted into the access tube (i.e., standard procedure for sensor calibration (Sentek, 2001)). Subsequently, the ring capacitor was disconnected from the sensor electronics, such that CSMD was the only variable.
Equipment for Dielectric and Capacitance Measurements
Dielectric Measurements
Permittivities of solutions with different dioxane–water mixing ratios and temperatures, plus permittivities of plastic sensor components, were measured separately. The permittivity of a material was deduced from the measured reflection coefficient determined by the permittivity of the material in contact with the coaxial electrodes of a dielectric probe. A commercially available dielectric measurement system (Hewlett-Packard model HP 85070M, Hewlett-Packard Company, Palo Alto, CA) was used to measure the intrinsic electrical properties of materials in the radio- and microwave-frequency bands. The system included an HP 85070B high-temperature dielectric probe, VNA HP 8753E, software and all necessary accessories to measure the complex permittivity of liquids and semisolids. More details are available from the manufacturer (Agilent Technologies, Palo Alto, CA; http://www.home.agilent.com/USeng/nav/-536894858.536879746/pd.html [verified 12 July 2006]).
Before measuring the permittivity of a material, the system was calibrated by a three-step procedure: (i) probe in air (open ended), (ii) inner and outer electrode of the probe shorted with a metallic conductor, (iii) probe immersed in distilled water at 25°C. A Sucoflex 104 cable (HUBER+SUHNER, Essex Junction, VT) was used to connect the VNA with the dielectric probe representing an open-ended coaxial line. Due to the influence of mechanical tension on the phase-response of the cable, it was important to avoid moving this cable after calibrating the system.
We recorded 101 data in the frequency range 100 MHz f 2 GHz. However, the permittivities of the water–dioxane solutions were calculated as the average over the limited frequency range 300 MHz f 500 MHz. These frequencies were higher than the resonant frequency band of the EnviroSMART sensor (100 MHz < fr < 160 MHz). However, the measured real part of a given permittivity can be expected to remain essentially unchanged in free water at these lower frequencies (the permittivity of pure water at 25°C and 500 MHz is 78.38 and at 100 MHz it is 78.40; Meissner and Wentz, 2004).
The lower limit of 300 MHz was chosen because of increasing instrumental uncertainties below this frequency, and the upper limit of 500 MHz was chosen to avoid relaxation effects causing a reduction of the water permittivity. Increasing uncertainties at frequencies <300 MHz are caused primarily by the small measurement volume of the dielectric sensor leading to an electrode polarization (Schwan, 1992). Such sensors are generally more accurate at higher frequencies (Shang et al., 1999), whereas devices with a larger measurement volume are more accurate at lower frequencies.
Permittivity of Access Tube
The permittivity of the access tube material acc was measured with the reflection method described above, such that acc = 3.35 for frequencies not exceeding 1 GHz. Because the plastic material comprising the access tube is identical with the electrode holder material, the same permittivity hold = acc = 3.35 is used in the electromagnetic numerical simulations presented below.
Permittivities of Dioxane–Water Mixtures
Permittivities of samples with volumes of 100 mL and volumetric mixing ratios 
d ranging from pure water (d = 0) to pure dioxane (d = 1) were measured at 25°C. The measured values of the mixtures deviated significantly from the linear weighting of the permittivities w and d of the water and dioxane constituents. The main nonlinearity was quantified using a semiempirical power-law fitting approach (Sihvola, 1999):
 | [14] |
The optimized parameter ß = 0.813 describes the main deviation from the linear dielectric mixing. w = 78.38 and d = 2.2 are the literature values of the permittivities of pure water and dioxane, respectively, which are in agreement with our measurements within the given uncertainty. d is the volumetric dioxane mixing ratio:
 | [15] |
Capacitance Measurements
The capacitance between the two ring electrodes of one sensor was measured with the VNA HP 8751A. The connection from the VNA to the ring electrodes (load) was comprised of a Sucoflex 104 cable connected to a short (10 cm), flexible 50-
cable soldered directly to the inside of the ring electrodes. The reference plane (zero phase) was adjusted to the end of the 50- cable before measurement. The Sucoflex 104 cable was held in the same position for each measurement.
The frequency response of the complex load reflection coefficient 
L = r + ii was measured for 201 frequencies f in the range 0.1 f 500 MHz. Due to resonance phenomena in the ring electrode circuit occurring at various frequencies, we assumed the average of the measurements in the frequency range 80 f 100 MHz to be representative for the low-frequency sensor capacitance C. The resulting low-frequency values of C were measured at the same environmental permittivities 1 78.38 as the normalized sensor readings.
The frequency spectrum of the total capacitance C between the electrodes was calculated from L, measured for a load with impedance ZL connected behind the reference plane (line impedance Z0 = 50 ):
 | [16] |
The VNA setup allowed us to interpret the measurements L as the result of the reflection caused by the mismatch between Z0 and the impedance ZL of the ring electrodes of the sensor. Because the load was assumed to be exclusively capacitive,
 | [17] |
Materials Selection, Container Design, and Fabrication
The experiments were designed to measure dielectric properties of different well-mixed liquids with permittivity values representative of a range of environmental soil–air–water permittivities. The ideal liquids are water and another liquid fully miscible in water with a permittivity near a = 1. We selected dioxane (1,4-Diethylene dioxide: C4H8O2) with d 2.2 (Maurel and Price, 1973). Dioxane has a boiling point of 101°C,melting point of 12°C, and flash point of 11°C. The low flash point and high volatility require special handling and ventilation, as well as the need to avoid direct localized heating. Dioxane is also a strong solvent, used as a cosolvent in the pharmaceutical industry, requiring solvent-resistant materials in contact with the liquid. Furthermore, the melting point and flash point limit the experimental temperature range. The full range of possible permittivities using various dioxane–water mixtures outweighed the difficulties of working with such a chemical.
Container Design Criteria
The criteria for designing the measurement container depicted in Fig. 2
were:
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Simultaneous Measurements of Permittivity and Frequency/Capacitance
The functional relation (N) between the permittivity of a material around the soil moisture probe and the normalized reading N was deduced by measuring the permittivity of the encompassing dioxane–water mixture simultaneously with taking EnviroSMART sensor readings. Furthermore, the capacitance C between the two sensor ring electrodes disconnected from the sensor electronics was measured under the same conditions.
Figure 3 shows the experimental setup used for these measurements. The VNA to the right, connected to the dielectric probe, was used for in situ measurements of the dioxane–water permittivity . For quality control, samples were taken after completing the measurements at each dioxane–water mixing ratio d,i to measure the corresponding i with the reflection method using a static probe setup. The dielectric sensor was taken out of the sensitive region of the EnviroSMART sensor while frequency readings were taken by placing the entire VNA on a lifting table mechanically connected with the holder of the dielectric probe. Alternating mechanical tension acting on the Sucoflex 104 cable after repositioning the probe was minimized with this setup, allowing for in situ measurements of between capacitance measurements without recalibrating the system.
The uppermost of the five capacitance sensors mounted on the sensor stick was used for direct capacitance measurements. The electronics of this sensor were disconnected from the electrodes and a short 50- cable was soldered to the inside of the ring electrodes. The VNA depicted in the left side of Fig. 3 was thus connected for measuring the capacitance C independently of the EnviroSMART proprietary electronics and signal processing. Altering mechanical tension acting on the Sucoflex 104 cable was minimized by disconnecting the short coaxial cable to the capacitor from the VNA, such that the Sucoflex cable was always in the same position during measurements.
Normalized Sensor Readings
Quantifying the relation between permittivity and the normalized sensor reading N is of high practical importance for quantitative soil moisture estimation using adequate models describing the relation between soil water content and soil permittivity. Here, we focus on the relation between the real part (hereafter denoted simply as ) of the environmental permittivity and N because the effect of the imaginary part of the permittivity on a reading N is expected to be minor under laboratory conditions. This assumption does not consider effects of electrical conductivity in the media on measured resonant frequency fr as discussed by Kelleners et al. (2004b).
Sensor raw counts Rik were recorded with the four sensors k = (1, 2, 3, 4) inside the access tube, while permittivity i (i = 1–22) of the liquid encompassing was varied. Sensor readings for i between the dioxane permittivity d 2.2 and the water permittivity w 78.38 at 25°C were investigated with volumetric dioxane–water mixing ratios d from 0 (pure dioxane) to 1 (pure water). The normalized readings Nik of the sensors (Eq. [10]) were derived from four sensor raw counts Rik (i = 1–22, k = 1–4) recorded for the permittivities i.
Cylindrical Metal Interference
To estimate the sampling volume of the sensor, we introduced a cylindrical disturbance at distance D from the access tube. The coaxial metallic disturbance surrounding the access tube was comprised of a brass foil of thickness 0.3 mm. Variable diameters were achieved by rolling up the foil and holding it in place with four removable rods per position.
The sensor mounted in the experimental container filled with water and supplemented with metal disturbance is shown in Fig. 4a
through 4c, where the coaxial metallic disturbance is shown in the photographs at mean distances D = 13.8, 35.5, and 96.3 mm, respectively. Measurements were taken at = 1, 16.4, 20.3, and 78.38, each with 10 distances 0 D 96 mm, and without any metal disturbance (i.e., D = 
). The experiment was started with the metal disturbance installed in the tightest position Dmin = 4.55 mm around the access tube. The larger distances were realized by subsequently removing four holding rods. All of the rods were removed at the largest distance Dmax = 96.3 mm at which point the foil was in contact with the container wall. The distance D0 = 0 mm was realized by wrapping the brass foil tightly around the access tube.
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 | [18] |
D = D1 – D2 and relative difference 
D = D/D for each of the nine sizes (Dk > 0) of the cylinder are given in Table 1.
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Numerical simulation of an electric field model for the spatial sensitivity of TDR probes was demonstrated by Knight et al. (1997), following previous analytical derivations (Knight, 1992). They analyzed the effects of fluid-filled gaps or dielectric coatings around the TDR rods on the ability to measure the water content of the surrounding porous media.
The present axisymmetric numerical model makes it possible to calculate the total capacitance C between two ring electrodes. At the margin of the simulation area, the boundary condition was set to "balloon" for the case in which the structure was infinitely faraway from all other electromagnetic sources. Furthermore, all of the dielectric components were assumed to be lossless, and the metallic components (ring electrodes and brass sheet) were represented by ideal conductors.
The model implementation did not consider the possibility of air gaps between the electrodes and the access tube nor between the access tube and the environmental material. Furthermore, the actual sensor includes an electronic board inside the electrode holder that was not considered due to its highly asymmetric geometry and unknown dielectric properties. The medium inside the electrode holder was assumed to be the same as air (a = 1). These two model restrictions may have led to errors in the computed C values.
Two versions of the electromagnetic model are presented for (i) computing the capacitance C between the two ring electrodes of the sensor embedded in homogeneous environmental media with permittivity and (ii) computing C as affected by an additional coaxial disturbance at distance D from the access tube of the sensor. The corresponding axisymmetrical model setup used for calculating the electric field E is depicted in Fig. 5
.
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 | [19] |
·E, which allows for calculating the storage of the electrical field energy within a volume (vol):
 | [20] |
This field energy has to be consistent with the energy spent for transferring the charge Q from one electrode to the other. This gives an alternative definition of capacitance C, which was used for computing C from the field E. The energy needed for transferring the infinitesimal charge dq over the potential difference U is d = Udq = qdq/C. Consequently, the energy used to transfer the total charge Q from one conductor to the other is
 | [21] |
Combining Eq. [20] with the last expression of Eq. [21] allows for calculating C from the electric field E.
The finite element software calculates capacitances Cij between electrodes i and j as described above. Thereby, the capacitances Ci,j are given in terms of a capacitance matrix [Ci j]. The solver calculates Ci, j from the fields E resulting from U = 1 V applied to the electrode i and grounding all other electrodes i 
j. Such matrices have been calculated for the model setup depicted in Fig. 5 (with and without considering a coaxial metal disturbance) for calculating sensor capacitances C.
Calculated Distance of Influence
We used the Maxwell 2D finite element software to calculate the effect of the metallic disturbance at distance D from the access tube. Thereby, the electrical potential of the coaxial metal disturbance is defined to be floating (electrically not connected with a fixed potential, e.g., ground).
According to the three conducting components (two ring electrodes and the metal sheet) in the model configuration sketched in Fig. 5, the capacitance matrix [Ci,j] is a 3 x 3 matrix. The element C1,2 is associated with the contribution of the ring electrodes, and C1,3 is the contribution of the capacitance caused by the coaxial metal foil (looking ahead to Fig. 15). For the model parameters used in the field calculation, the sensor capacitance C was calculated according to
 | [22] |
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D 162 mm were simulated for various permittivities ( = 1, 5, 10, 20, 30, 40, 78.38).
To estimate a characteristic distance of influence 
at the investigated permittivities, measured and modeled data [D, N] were fitted using the following approach:
 | [23] |
= N(D = ) are measured and modeled data with the metal in contact with the access tube and with no metal disturbance, respectively. Characteristic distances for different values were derived from measurements and simulations and then compared with each other.
Analysis of Errors in Estimated Water Content
If Topp's equation (Eq. [2]) is assumed to characterize the relationship () for an ideal soil, the instrumental root mean squared error in (RMSE) can be computed directly from the errors in determined from the present experiments:
 | [24] |
i = i – (Ni), 
is the slope of Topp's equation (Eq. [2]) at i, i values are measured using the reflection method, (Ni) values are estimated from normalized sensor readings Ni, and n is the number of measurements.
Topp's equation is widely used with values estimated from TDR data, and estimates of tend to decrease with increasing measurement frequency. Thus, () and its first derivative used in Eq. [24] can vary with the instrument used, making the RMSE computed with Eq. [24] a relative indicator of the present instrumental errors. With this caveat, Eq. [24] is expected to provide a reasonable estimate of the instrumental error, not including effects of bulk electrical conductivity or other soil factors influencing the apparent values.
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RESULTS AND DISCUSSION |
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TOPEXECUTIVE SUMMARYABSTRACTINTRODUCTIONBACKGROUND AND THEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSREFERENCES |
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C with change in measured resonant frequency (fr–2) = fr,A–2 – fr,B–2 under two different conditions A and B:
 | [25] |
C has units of pF and fr has units of MHz. This empirical result is consistent with the theory above.
Normalized Sensor Readings and Capacitances
Normalized sensor readings N measured in environmental material with well-controlled permittivity are presented below. Measured C and N values are then compared with simulated values.
Measured Normalized Sensor Readings
Figure 7
shows the measured relation between the permittivity i of the dioxane-water mixture encompassing the access tube and the normalized reading averaged over the four sensors: Ni = (Ni1 + Ni 2 + Ni 3 + Ni 4)/4. The light gray circles in Fig. 7 are the data measured with the equipment shown in Fig. 2. Consequently, these data were not corrected for the calculated effect of the FEP coating. Corrected normalized readings Ni – N(i) representing measurements under field conditions (where no FEP shrink fit is present) were calculated from the laboratory measurements Ni (i = 1–22) using the replacement:
 | [26] |
N() is the correction function to be derived below. Thus, the black circles in Fig. 7 are the readings Ni corrected for the calculated effect N() of the FEP shrink tube, which is not applied in the field. On the right axes of Fig. 7, the standard deviations 
between Nik measured with the four sensors (k = 1–4) are plotted. As can be seen, the maximum is <4 x 10–3 for 0 N 1, allowing for sensor-independent examinations if normalized sensor readings are used.
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i] (light gray circles) and the corrected data [Ni – N(i), i] (black circles) (i = 1–22). The fitting approach considers a quadratic and an exponential factor:
 | [27] |
N 1. As indicated by the dashed portion of the fitted lines and as discussed below, the fit near N = 1 may be less accurate than at steeper portions of the curve. In accordance with the definition of N, Eq. [27] must fulfill two constraints:
 | [28] |
 | [29] |
The computed values of the remaining fitting parameters a2 and k representing the approximation of the uncorrected data [Ni, i] are a2 = 1.15008 and k = 6.66056. For the data [Ni – N(i), i] corrected for the effect of the FEP coating one finds: a2 = 1.12819 and k = 6.64846. The latter values should be used for calculating permittivities (N) from field-measured data N.
From the sensor reading (raw count) R of a sensor placed in an environmental material with unknown , one can calculate the normalized reading N using the definition [10] with the air and water counts Ra and Rw of the sensor. From this, of the environmental material can be estimated using relation [29]. For the greatest accuracy, relation [29] should be applied only to permittivities < 40 because of decreasing sensitivity of N with respect to at higher permittivities (dashed line in Fig. 7). Permittivities of soils with realistic water contents are typically smaller than 40.
The resulting RMSE between the corrected data Ni – N(i) and the approximation (N) evaluated at N= Ni – N(i) and the measured permittivities i is RMSE = 0.859 for 1 < < 80 and only RMSE = 0.226 for the measured range of 3 < < 43 pertaining to soil water. The overall RMSE of 0.859 is affected primarily by one large deviation at (N = 0.9925) = 77.12. If this is excluded the RMSE is 0.314 for 1 < < 64, which is more indicative of the expected deviations.
Measured Sensor Capacitances
The capacitance C between the electrodes of one sensor is computed from VNA measurements of the complex reflection coefficient L. The relative values of C and the shape of the C() curve in Fig. 8
give the following insights. First, C( = 1) is approximately one-half of the value of C( = 25), which may represent a relatively wet soil. Thus, approximately one-half of the total capacitance measured in soils is instrumental (affected by the sensor electronics, inner ring capacitance, and the access tube). Second, C() is very nonlinear in this range, so the geometric factor relating C to is not a constant, which indicates large changes in the electromagnetic field pattern with changes in the environmental values. These factors are related to the instrumental design and determine the measurement sensitivity and accuracy.
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= 20, and the potential difference between the two ring electrodes is U = 2 V.
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The field energy in Eq. [20] needed for calculating a capacitance in Eq. [21] was computed for the total simulation volume. Consequently, a capacitance value comprises contributions of partial capacitances originating from different regions. The simulated contribution of the region inside the electrode holder is the most uncertain due to the electronic board, which is not considered in the simulation. An adequate representation of the electronic board would have required full three-dimensional capability of the finite element software. Neglecting the asymmetrical internal material might be the main reason for deviations (shown later) between measured and simulated sensor capacitances. Furthermore, internal and external fields are interdependent, such that the inner capacitance cannot be treated as a fixed, parallel capacitor here. A more rigorous investigation of these problems is beyond the scope of this work.
Capacitances between all the conducting elements i and j (i j) of the model were calculated and displayed as a capacitance matrix [Ci j]. Corresponding to the two conducting elements (two ring electrodes) comprised in the undisturbed model (Fig. 5), the computed [Ci j] is a 2 x 2 matrix. For the model parameters used here, the capacitance matrix [Ci j] is (in units of pF):
 | [30] |
78.38).
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78.38 using Eq. [11], with Ca = 4.62 pF for = a = 1 and Cw = 24.65 pF for = w = 78.38 (at 25°C).
The corresponding experimental data Cexp, (circles), Nexp (circles), and the data fit (Nexp) (solid line) already presented in Fig. 7 and Fig. 8 are also shown in Fig. 10 for comparison. The disparity between measured and simulated results is likely due to the two factors mentioned above, air gaps between the sensor (capacitor rings) and the access tube and neglecting the unknown internal capacitance due to the electronic board. Indeed, Fig. 10a shows a large C
