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ORIGINAL RESEARCH

Laboratory Characterization of a Commercial Capacitance Sensor for Estimating Permittivity and Inferring Soil Water Content

^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.

	ABSTRACT

TOP EXECUTIVE SUMMARY ABSTRACT INTRODUCTION BACKGROUND AND THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 
Ring-capacitor sensors are used widely for real-time estimation of volumetric soil water content from measured resonant frequency f_r, which is directly affected by the bulk soil permittivity . However, the relationship f_r() 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 f_r 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 f_r 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

	INTRODUCTION

TOP EXECUTIVE SUMMARY ABSTRACT INTRODUCTION BACKGROUND AND THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 
CAPACITANCE SENSORS have been developed commercially and are being used globally for estimating soil water content. The relative permittivity , 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 .

	BACKGROUND AND THEORY

TOP EXECUTIVE SUMMARY ABSTRACT INTRODUCTION BACKGROUND AND THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 
Design and Deployment of the EnviroSMART Probe
The Sentek EnviroSMART probe and capacitance sensors have been designed to determine soil water content in the field with or without local surface access. Figure 1a shows field installation of the plastic access tube using a hand auger to ensure contact between the soil and outer surface of the tube. Sensors are attached to a probe (Fig. 1b "sensor stick") with integrated circuits for signal processing and analog to digital conversion. The probe is inserted in the access tube, and a five-wire cable connects each probe with multiple sensors to a datalogger using digital communication. Thus, there is no signal degradation between the probes and the datalogger. Probes have been operated successfully up to 450 m from a datalogger in Colorado, USA, and longer distances may be possible. In addition, probe installation is more efficient for soil profile installations using an access tube for the capacitance probe, instead of trenching or drilling multiple holes for TDR.

View larger version (116K): [in this window] [in a new window]   Fig. 1. (a) Installation of the access tube in the field; (b) EnviroSMART soil water content probe with capacitance sensors; (c) sensor with symbolized field lines; (d) sensor electronic board; and (e) equivalent circuit diagram. The dashed line in (e) represents the sensor electronics board, neglecting capacitors and resistors on the board.

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]

The inverse relationship is:

[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, f_r, of the sensor in a given environment, R (25000 < R < 37000) is converted to f_r using the linear relationship (Sentek technical support, personal communication, 2004):

[4]

where f_r has units of megahertz. Furthermore, f_r is understood to be the resonant frequency of the LC oscillator given by the inductance L of the coil mounted on the electronic board and the total capacitance C connected to the sensor electronics:

[5]

Electrical resistance is neglected here. The total capacitance C includes the series capacitance of the plastic tube C_acc plus any air gap between the rings and inner diameter of the tube, as well as an internal capacitance C_int 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]

where C = C() is the capacitance associated with the medium outside the access tube, which is the quantity of interest. Theoretically, C_int varies with C due to coupled effects on the inner and outer electrical fields of the ring capacitor. Also, C_int comprises both the capacitance of the physical inner space of the rings and the intrinsic capacitance on the sensor board. In the following derivation, C_int 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 f_r^–2 for known external capacitances C_SMD that this is a reasonable assumption.

Now it will be shown that a change C of the capacitance C is proportional to the difference (f_r^–2) = f_rA^–2 – f_rB^–2 of inverse squared values of two resonant frequencies f_rA and f_rB.

[7]

The proportionality factor dC/d(f_r^–2) can be expressed, using the chain rule, as:

[8]

The derivatives d(f_r^–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(f_r^–2) = (4²L)^–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 C_int 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 R^k 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 R^k are normalized to be:

[10]

R_a^k and R_w^k are the air and water readings recorded with sensor k installed in the access tube, surrounded by air and immersed in pure water at 25°C. The normalized sensor reading N^k is a dimensionless number having the values N^k = 0 for an air reading and N^k = 1 for a measurement in pure water of 25°C.

Using the relationship (Eq. [5]) between f_r and C, the normalized sensor reading N can be expressed by the sensor capacitances C_a, C_w, 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 d_w/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).

	MATERIALS AND METHODS

TOP EXECUTIVE SUMMARY ABSTRACT INTRODUCTION BACKGROUND AND THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 
Resonant Frequency Measurements using Capacitance Devices
To gain an improved understanding and characterization of the instrument, it is desirable to measure the resonant frequency response to known values of added capacitance. "Added" capacitance is emphasized, because the sensor electronic circuit has its own internal capacitance, which acts in parallel to the ring capacitance (Kelleners et al., 2004b).

Surface mounted devices (SMDs) of known capacitance values C_SMD were soldered to one sensor electronics board, with and without the ring capacitor attached. Values of C_SMD 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 C_SMD 21.25 pF).

All measurements with SMD capacitors C_SMD were taken in air without the plastic access tube. Thus, sensor readings with no added capacitance (C_SMD = 0) exceeded the air count R_a 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 C_SMD 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 < f_r < 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]

where V_d and V_w are dioxane and water volumes before mixing, respectively.

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 + i_i 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 Z_L 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 Z_L of the ring electrodes of the sensor. Because the load was assumed to be exclusively capacitive,

[17]

combining relation [16] with [17] and solving for the capacitance C allowed us to derive the frequency response of the sensor capacitance C.

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:

• solvent-resistant materials for containing the dioxane
  
• a volume large enough to avoid signal disturbance at the container boundaries, yet as small as possible to reduce the amount of dioxane used
  
• robust containment of toxic, volatile liquids
  
• ability to cool and heat the liquids for measurements at various temperatures up to 50°C;
  
• complete chemical and thermal mixing capability
  
• visibility for monitoring and removing any accumulation of air bubbles on sensor surfaces
  
• near-simultaneous measurement of permittivity and capacitance sensor readings
  

View larger version (54K): [in this window] [in a new window]   Fig. 2. Sketch of the solvent-resistant container with the tools used. The heater, dielectric probe, the thermometer, and the tools for removing air bubbles from the access tube and the dielectric probe are retracted during the capacitive measurements and the sensor readings. Dimensions are not to scale.

• Teflon valve and glass stem at the bottom to drain liquids
  
• sealable access port on top for adding fluids;
  
• submersible glass 300-W heater (commonly used in fish tanks) (The commercial thermostat was shorted to obtain a higher temperature range (up to 50°C for our experiments), and a variable load regulator (i.e., "dimmer switch") was installed to control the heating power. A hot air blower provided additional heating to the sidewalls as needed.)
  
• submersible, solvent-resistant housing for the permittivity sensor
  
• stainless-steel thermistor and digital thermometer
  
• magnetic mixer, including a machined cavity to hold the mixing stir bar in place
  
• solvent-resistant, submersed brush for cleaning the reflection probe
  
• collar suspended around the access tube for removing air bubbles and providing additional vertical mixing during heating
  

View larger version (112K): [in this window] [in a new window]   Fig. 3. Picture of the experimental setup used for characterizing the EnviroSMART soil water content sensor.

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 f_r as discussed by Kelleners et al. (2004b).

Sensor raw counts R_i^k 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 N_i^k of the sensors (Eq. [10]) were derived from four sensor raw counts R_i^k (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 D_min = 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 D_max = 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.

View larger version (75K): [in this window] [in a new window]   Fig. 4. Cylindrical metal sheet disturbance used for investigating the sampling volume of the sensor.

[18]

The mean distance D = (D₁ + D₂)/2, absolute difference D = D₁ – D₂ and relative difference D = D/D for each of the nine sizes (D_k > 0) of the cylinder are given in Table 1.

View this table: [in this window] [in a new window]   Table 1. Mean distance D between the access tube and the inner border of the coaxial metal disturbance. D and D are the absolute and the relative deviation resulting from the ellipsoidal shape.

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 .

View larger version (39K): [in this window] [in a new window]   Fig. 5. Rotation symmetrical model setup used for simulating the electric field E caused by a potential difference between the ring electrodes of the sensor. Capacitance C is calculated from the field distribution E.

[19]

defines the capacitance C between two conductors. The capacitance C between the electrodes of the EnviroSMART sensor was calculated from the computed electric field E and the corresponding dielectric displacement D = ·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 C_ij between electrodes i and j as described above. Thereby, the capacitances C_i,j are given in terms of a capacitance matrix [C_{i j}]. The solver calculates C_{i, 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 [C_i,j] is a 3 x 3 matrix. The element C_1,2 is associated with the contribution of the ring electrodes, and C_1,3 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]

View larger version (56K): [in this window] [in a new window]   Fig. 15. Electric field strength |E| resulting from the potential difference U = 2 V between the ring electrodes calculated using the model configuration with the metal foil disturbance at distance D = 27 mm from the access tube (acc = hold = 3.35 and = 20).

To estimate a characteristic distance of influence at the investigated permittivities, measured and modeled data [D, N] were fitted using the following approach:

[23]

where N₀ = N(D = 0) and N = 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]

where _i = _i – (N_i), is the slope of Topp's equation (Eq. [2]) at _i, _i values are measured using the reflection method, (N_i) values are estimated from normalized sensor readings N_i, 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.

	RESULTS AND DISCUSSION

TOP EXECUTIVE SUMMARY ABSTRACT INTRODUCTION BACKGROUND AND THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 
Surface Mounted Devices
One sensor was used for measurements with known capacitance C_SMD values as described above. The results are shown in Fig. 6 , where the almost perfectly linear responses in terms of f_r^–2 are expected for LC circuits (Eq. [5]), but the nonzero intercept even without the ring capacitor connected indicates additional capacitance in the electronics. The difference between the linear regression lines in terms of capacitance is approximately 6.3 pF. Because the SMD and ring capacitors were soldered in parallel, this difference is the total ring capacitance in air, including the effects of the sensor electronics and plastic sensor stick (Fig. 2) inside the ring, but without the access tube.

View larger version (20K): [in this window] [in a new window]   Fig. 6. Sensor responses fr–2 to surface mounted device (SMD) added capacitance CSMD with and without the ring capacitor attached to the instrument.

[25]

where C has units of pF and f_r 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: N_i = (N_i¹ + N_i ² + N_i ³ + N_i 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 N_i – N(_i) representing measurements under field conditions (where no FEP shrink fit is present) were calculated from the laboratory measurements N_i (i = 1–22) using the replacement:

[26]

where N() is the correction function to be derived below. Thus, the black circles in Fig. 7 are the readings N_i 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 N_i^k 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.

View larger version (22K): [in this window] [in a new window]   Fig. 7. Measured relation between permittivity of the dioxane–water mixture outside the access tube and normalized sensor reading N. The light gray data are the laboratory measurements; the data represented by the black circles are corrected for the calculated effect N() of the fluorinated ethylene-propylene (FEP) coating. The solid lines are the approximations using Eq. [29] and is the standard deviation between Nik measured with the sensors k = 1 to 4.

[27]

As will be shown, this approach provides a macroscopically accurate representation of the data within the entire range 0 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]

that eliminate two of the four fitting parameters a₀, a₁, a₂, and k in Eq. [27]:

[29]

The computed values of the remaining fitting parameters a₂ and k representing the approximation of the uncorrected data [N_i, _i] are a₂ = 1.15008 and k = 6.66056. For the data [N_i – 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 R_a and R_w 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 N_i – N(_i) and the approximation (N) evaluated at N= N_i – 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.

View larger version (13K): [in this window] [in a new window]   Fig. 8. Measured sensor capacitance C versus permittivity of the dioxane–water mixture outside the access tube.

View larger version (55K): [in this window] [in a new window]   Fig. 9. Electric field strength |E| resulting from the potential difference U = 2 V between the ring electrodes calculated using permittivities of access tube, electrode holder, and environmental material of acc = hold = 3.35 and = 20, respectively.

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 [C_{i j}]. Corresponding to the two conducting elements (two ring electrodes) comprised in the undisturbed model (Fig. 5), the computed [C_{i j}] is a 2 x 2 matrix. For the model parameters used here, the capacitance matrix [C_{i j}] is (in units of pF):

[30]

The matrix element C_1,2 = 18 pF is the capacitance between the two ring electrodes and thus interpreted as the sensor capacitance C. The asterisks in Fig. 10a show sensor capacitances C_model computed for a range of permittivities (1 78.38).

View larger version (23K): [in this window] [in a new window]   Fig. 10. (a) Modeled and measured capacitances Cmodel and Cexp between the ring electrodes as a function of the permittivity of the environmental material. The permittivity of the access tube and sensor holder are acc = hold = 3.35. (b) Modeled normalized sensor readings Nmodel and normalized readings Nexp measured in the laboratory (with the FEP shrink tube on the access tube) together with the interpolation function (Nexp).

The corresponding experimental data C_exp, (circles), N_exp (circles), and the data fit (N_exp) (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 value for = 1 relative to the total C at greater values. Further explanation and quantification of the discrepancy is left for future investigation. Here, it suffices to note the resulting difference in the shapes of the simulated and experimental N() curves in Fig. 10b.

Correction for Coating the Access Tube
Fluorinated ethylene-propylene heat-shrink tubing was placed around the plastic access tube to protect it from the dioxane solvent (Fig. 2). The influence of the FEP coating (permittivity _FEP 2, thickness 0.5 mm) on the measured permittivity was estimated using the electromagnetic field simulation software.

Similar effects of nonmetallic components covering the rods of TDR probes were investigated previously (Ferré et al., 1996). In their study, the sensitivity of the travel time with respect to the soil water content was enhanced by the presence of dielectric coatings.

Figure 11a shows capacitance values C_model() computed for 1 78.38 with the FEP coating (hollow circles)and without it (solid squares). As shown in Fig. 12a , the absolute value of the difference C() = C_{model with FEP}() – C_{model without FEP}() is small when the contrast between and _FEP is small but increases for larger . Furthermore, the difference C() is positive for _FEP = 2 and negative for _FEP. This is in accordance with expectations that capacitance is increased by the presence of the FEP coating for _FEP and vice versa.

View larger version (26K): [in this window] [in a new window]   Fig. 11. (a) Calculated sensor capacitances Cmodel with FEP() and Cmodel without FEP() for the case where the access tube is coated (hollow circles) and not coated (solid squares) with the 0.5 mm thick the FEP shrink fit with permittivity FEP = 2. (b) Normalized sensor readings Nmodel with FEP() and Nmodel without FEP() derived from Cmodel, with FEP() and Cmodel, without FEP() using Eq. [11] for the case with (hollow circles) and without (solid squares) the shrink fit on the access tube.

View larger version (21K): [in this window] [in a new window]   Fig. 12. (a) Difference C() between simulated capacitances Cmodel with FEP() with the FEP coating on the access tube and Cmodel without FEP() without the FEP coating taken from Fig. 11a. (b) Difference N() between corresponding sensor readings Nmodel with FEP() and Nmodel without FEP() taken from Fig. 11b.

The effect of the FEP coating is expressed as _RMS calculated from the difference (N) = (N_{model with FEP}) – (N_{model without FEP}) computed from the data shown in Fig. 11b:

[31]

The integration interval [N_min, N_max] is related to corresponding permittivity and soil moisture regimes [_min, _max] and [_min, _max]. The interval [_min, _max] is calculated from the boundaries _min and _max using the Topp model (Eq. [2]), and N_min, N_max are calculated from evaluating the interpolation N_{model without FEP}() from Fig. 11b at the permittivities _min and _max. For a realistic soil moisture regime [_min, _max] = [0.05, 0.5] these intervals are [_min, _max] = [3.85, 34.59] and [N_min, N_max] = [0.475, 0.943], respectively. The difference calculated for the above permittivity range using Eq. [31] is _RMS = 0.596.

The corresponding correction of the normalized readings N_i is represented by the modeled deviation N() = N_{model with FEP}() – N_{model without FEP}() plotted in Fig. 12b, which reaches a maximum of approximately 11.5 x 10^–3 at 6.2. This exceeds the maximum standard deviation between the normalized readings N_i^k measured with the four sensors k = 1–4 (right axes of Fig. 7) and is a systematic bias rather than a random error.

Sensor readings were taken with and without the FEP coating in water and air. Corresponding readings R averaged over the four sensors and the resonant frequencies f_r computed with Eq. [4] are listed in Table 2.

View this table: [in this window] [in a new window]   Table 2. Averaged experimental sensor readings R and resonant frequencies fr with and without the fluorinated ethylene-propylene (FEP) coating on the access tube showing corresponding capacitance changes C compared with model-derived C quantities for air and water environments.

The experimental change in capacitance can be calculated directly from Eq. [25] using the change (f_r^–2) = f_{r, with FEP}^–2 – f_{r, without FEP}^–2 in resonant frequency with and without FEP. The resulting values in Table 2 are very close to the simulated changes for air and water, where the errors (C_model – C_exp) are approximately 0.04 pF for air and 0.07 pF for water. These are negligible relative to experimental uncertainties, and this result provides confidence in the correction given above.

Temperature Dependence of Environmental Permittivity
The temperature dependencies of pure water _w(T), pure dioxane _d(T), and a mixture of 98% dioxane and 2% water _dw98(T) were estimated from measured sensor readings, which were converted to permittivities using the new empirical function (Eq. [29]) with the fitting parameters a₂ = 1.15008 and k = 6.66056 derived for uncorrected data [N_i, _i].

Figure 13a shows _w(T) measured in situ with the reflection method (solid dots) and _w(T) deduced from the EnviroSMART sensor readings (hollow circles) for 5 T 50°C. Both water permittivities _w(T) show a negative gradient d_w/dT. The linear regression of _w(T) measured with the VNA electromagnetic reflection method yields a gradient of d/dT –0.36 K^–1, which quantitatively agrees with the Debye model, but the capacitance sensor data yield d/dT –0.59 K^–1. One could correct the slope d_w/dT based on the fit of N() near _w = 80 by adjusting the parameter values in Eq. [29]. Another option is to find a different mathematical form for the calibration equation, which is beyond the present scope. However, d_w/dT is negative and constant even without additional fitting.

View larger version (22K): [in this window] [in a new window]   Fig. 13. (a) Temperature dependency w(T) of the permittivity of pure water (d = 0) measured with the VNA dielectric sensor (solid dots) and derived from the capacitance sensor readings (hollow circles). (b) Temperature dependencies d(T) and dw98(T) of pure dioxane (d = 1, hollow squares) and a solution with d = 0.98 (hollow diamonds) deduced from the capacitance sensor readings.

Characteristic Distance of Influence
The electric field caused by the two ring electrodes at the potential U = ±1 V is concentrated in a narrow region around the electrodes (Fig. 9). Consequently, the capacitance is influenced predominantly by the permittivity of the environmental material closest to the access tube. Measuring the effect of a concentric dielectric disturbance located at distance D provides a means of estimating a characteristic distance of influence on the sensor.

The concentric arrangement allows for realizing a relatively simple experimental setup, which can be represented by the electromagnetic simulations. However, indicating a distance of influences does not imply that the sensor sampling volume is cylindrical. Ferré et al. (1998) noted that the shape and the size of a sampling volume of several different dielectric probes closely followed the distribution of the electric field, which was determined by the specific arrangement of the measuring electrodes and the permittivity of the media of investigation. From the calculated external field distribution plotted in Fig. 9, one can infer that the equipotential areas within the environmental media are reasonably represented by annular cylinders with heights corresponding approximately to the maximum electrode separation.

Measured Distance of Influence
A concentric metal sheet with changeable diameter was installed around the access tube as shown in Fig. 4. This setup enabled measurement of the change in sensor readings resulting from of an extreme coaxial disturbance in the environmental material.

Figure 14a shows the experimental values of N = N_exp(D) versus mean distance D between the access tube and the coaxial metal sheet. When the metal sheet was in contact with the access tube, the normalized sensor readings N_exp(D = 0) = N₀ 1.02 were the same for all . Furthermore, N₀ > 1 is due to the fact that the capacitance for D = 0 exceeds the capacitance measured in water at 25°C.

View larger version (31K): [in this window] [in a new window]   Fig. 14. Normalized sensor reading N versus distance D between the access tube and the metal sheet: (a) measured at 10 distances up to D = 96.3 mm for = (a = 1, 16.4, 20.3, w = 78.38); and (b) N(D) calculated from the electromagnetic model. The distance D between the access tube and the metallic cylinder is varied between 0 and 162 mm, and the permittivity was between the air and the water value. The solid lines are the best fits with the exponential model (Eq. [23]).

To estimate representing the characteristic distance of influence at the investigated permittivities, the experimental data [D, N_exp] were fitted using Eq. [23] with N₀ = 1.02 and N = 0, 0.752, 0.804, 1 shown as solid lines in Fig. 14a. The calculated least square fit parameter values for and the corresponding RMSE values between the approximations and the measured values are listed in Table 3 for the data measured at = 1, 16.4, 20.3, 78.38.

View this table: [in this window] [in a new window]   Table 3. Fitting parameters and standard deviations between the approximation and the measured data plotted in Fig. 14a according to approximation [23].

Calculated Distance of Influence
Figure 15 shows the field strength |E| calculated for potentials U = ±1 V applied to the ring electrodes. The coaxial metal disturbance increased |E| between the access tube and the metal sheet compared with |E| at the corresponding location of the undisturbed situation shown, inFig. 9. The field outside the metal sheet was shielded (|E| = 0) as the result of the constant potential on the conducting disturbance.

The undisturbed values of C with D = were calculated without the metal sheet present. Modeled N values are plotted in Fig. 14b and interpolated using the fitting approach (Eq. [23]) along with the measured N(D). For the calculations with > 5, the exponential fit is better adapted than for 5. This is consistent with the analysis performed for the measured N. The values of the fitting parameter representing the characteristic distances of influence and the corresponding standard deviations between the exponential fit and the model results N are listed in Table 4. The range of characteristic distances (10 < < 13 mm) deduced from the electromagnetic calculations for 5 40 overlaps the range of 12 < < 13 mm deduced from the experiment for = 16.4 and 20.3.

View this table: [in this window] [in a new window]   Table 4. Fitting parameters and standard deviations between the exponential fit (Eq. [23]) and the calculated data plotted in Fig. 14b.

If there are heterogeneities in a soil system, the soil nearest the access tube has the greatest influence. Likewise, any soil disturbances (e.g., air gaps, stones) immediately outside the access tube will be weighted heavily, and the exponential model could be used to predict such effects, particularly when combined with Eq. [29] to predict effects on the apparent permittivity.

Analysis of Errors in Estimated Water Content
Instrumental Errors for Ideal Soils
If Topp's equation (Eq. [2]) is assumed to characterize the relationship () for an ideal soil, the RMSE can be computed directly from RMSE determined from the present experiments. Such errors should be viewed as instrumental precision, rather than actual measurement errors in real soil–water systems, where Topp's model may not represent () accurately for a given soil. The estimated errors in correspond to an instrumental precision of RMSE = 0.0034 m³ m^–3 for 3 < < 43. This indicates that small changes in can be detected, which agrees with our field experience.

Comparison with the Sentek Default Calibration for Water Content
As stated by the vendor, the default calibration (Eq. [13]) may not be adequate for all soil types. Separate calibration has to be performed for soils with very high specific surface area or different textural layers within the sensitivity volume of the sensor. Figure 16a shows versus N, where the default calibration is represented by the black solid line. For comparison, in Fig. 16a we calculated [(N)] using three different empirical dielectric mixing models for relating with from Eq. [29], which gives (N).

View larger version (25K): [in this window] [in a new window]   Fig. 16. Comparison between the default calibration proposed by the vendor (Sentek) and three different dielectric mixing models (Topp et al., 1980; Roth et al., 1990; Wang and Schmugge, 1980) with respect to: (a) the water content [(N)] calculated using relation [29] for (N) and (b) water content–permittivity relationships ().

The deviation between the water content derived from the Sentek default calibration and the water content calculated using the Topp model is defined as:

[32]

where (N) is given by Eq. [29].

The interval boundaries _min and _max are the minimum and maximum water contents for which the deviation _RMS was calculated. The N values N_min and N_max corresponding to _min and _max were computed using the Sentek default calibration function (Eq. [12]).

The deviation _RMS for the interval [_min, _max] was solved numerically using:

[33]

For 0.05 0.5, _RMS equals 0.066 m³ m^–3, which reflects the negative bias in Fig. 16a. This result concurs with measurements in soils (Baumhardt et al., 2000, Fig. 2) where the default curve underestimated for N > 0.8. However, Fig. 16a also shows a bias (underestimation relative to the three mixing models) for N < 0.8, which was not observed by (Baumhardt et al., 2000, Fig. 2).

Another comparison between the default calibration and soil dielectric mixing models is presented in Fig. 16b in terms of (). The bold solid line was computed using the Sentek default calibration to calculate N from first, and then calculate from Eq. [29]. For > 0.30 m³ m^–3 the discrepancy between the dielectric mixing model-based permittivities and deduced from the Sentek default calibration increases rapidly. The latter reaches unrealistically high values of at very moist soil conditions.

This confirms the need for custom calibration of the power-law equation [12] as performed by Baumhardt et al. (2000), particularly for soils at very high water contents. In our experience with silty loams, however, the default calibration yields feasible values of even near saturation (0.45 m³ m^–3). The problem illustrated here involves the indirect use of Eq. [12] to obtain via Topp's equation (Eq. [2]). Given our present experimental results, one can now compute directly from N using Eq. [29]. For > 25, the default calibration yields values more than 0.05 m³ m^–3 below those expected from Topp's model. Users should be aware of these potentially substantial errors and avoid using Eq. [13] with the default parameters under such conditions.

	SUMMARY AND CONCLUSIONS

TOP EXECUTIVE SUMMARY ABSTRACT INTRODUCTION BACKGROUND AND THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 
The experiments described above led to the following main results:

1. An ideal relationship between sensor resonant frequency f_r and added SMD capacitance C_SMD was established. It confirmed a linear relationship (R² 1) between f_r^–2 and C_SMD, but with a nonzero intercept, implying a parallel capacitance in the circuit, even without having the ring capacitor connected.
   
2. Dioxane–water mixtures provided a range of values representative of soils from dry to fully saturated (4 < < 40), which has not been demonstrated with other dielectric surrogates.
   
3. Capacitance sensors displayed appropriate qualitative temperature T dependence of the permittivity of free water _w, where _w(T) was very linear with a negative slope. There was no apparent temperature sensitivity in pure dioxane.
   
4. A new empirical relationship has been derived between environmental permittivity and sensor readings normalized using air and water readings (see Eq. [29] with a₂ = 1.1282 and k = 6.6485). The RMSE of between fitted and measured permittivities was 0.859 for the full range of measurements (1< < 80) and only 0.226 for the range expected for soil water measurements (3 < < 43).
   
5. For the soil relevant range (3 < < 43), this RMSE for results in an instrumental error of estimated soil water content of RMSE = 0.0034 m³ m^–3 using Topp's equation (Eq. [2]) to represent ().
   
6. Approximately one-half of the total capacitance measured is instrumental (affected by the sensor electronics, inner ring capacitance, and the access tube), and C() is very nonlinear, 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.
   
7. Metal border experiments were performed and simulated using an axisymmetric numerical model. Using an exponential equation (Eq. [29]) to represent the signal disturbance, the characteristic annular distance is estimated to be approximately 12.5 mm, and 95% of the signal is sensed within approximately 37 mm of the access tube.
   
8. Different models for () were compared with the manufacturer's default calibration, showing that the default calibration tends to underestimate by 0.066 m³ m^–3. The mixing models may not represent () well for some soils, which may explain differences between the present theoretical results and previous measurements in real soils (e.g., Baumhardt et al., 2000).
   

The quantitative results from these laboratory experiments provide the information needed to analyze field measurements using EnviroSMART probes in terms of soil permittivity and its temperature dependence.

	ACKNOWLEDGMENTS

 
The following ETHZ staff members provided important technical assistance: Hanspeter Läser (container design and construction); Hans Wunderli (expansive metal design, construction, and digital photography); Hannes Wydler (frabrication of custom tools); Kurt Barmettler (chemical advice); René Saladin (Dioxane handling and safety); Jörg Leuenberger (infrastructure); and Peter Brühwiler (specialty machining). Comments from Steven Evett, Sally Logsdon, Ty Ferré, and an anonymous reviewer helped us improve the manuscript.

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