() Laboratory Characterization of a Commercial Capacitance Sensor for Estimating Permittivity and Inferring Soil Water Content Mike Schwank,* Timothy R Green, Christian Mätzler, Hansruedi Benedickter[.]
Published online August 24, 2006 Laboratory Characterization of a Commercial Capacitance Sensor for Estimating Permittivity and Inferring Soil Water Content Mike Schwank,* Timothy R Green, Christian Maătzler, Hansruedi Benedickter, and Hannes Fluăhler 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 , e , 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 e 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 e and the sensor reading is also required for validating and improving dielectric mixing models used for soil moisture estimation from proxy data e The scope of the present study is limited to inferring u from e based on previous work (e.g., Topp et al., 1980), noting that quantification of e is not sufficient to determine the accuracy of u in real soils due to complex losses associated with the imaginary part of e Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved ABSTRACT Ring-capacitor sensors are used widely for real-time estimation of volumetric soil water content u from measured resonant frequency fr , which is directly affected by the bulk soil permittivity e However, the relationship fr(e) requires improved quantification We conducted laboratory experiments to characterize the response of the Sentek EnviroSMART sensor system for a full range of e 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 e and normalized fr was derived The instrumental error in e was RMSEe 0.226 (for , e , 43), which corresponds to a measurement precision in u(e) derived from Topp’s equation of RMSEu 0.0034 m3 m23 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 C have been developed commercially and are being used globally for estimating soil water content The relative permittivity e, or “dielectric constant,” acts as a proxy for the volumetric soil water content u 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) APACITANCE SENSORS BACKGROUND AND THEORY Design and Deployment of the EnviroSMART Probe M Schwank, Institute of Terrestrial Ecosystems (ITES), Swiss Federal Institute of Technology (ETH), CHN E29, Universitaătstr 16, CH-8092 Zuărich, Switzerland; T.R Green, USDA-ARS, Great Plains Systems Research Unit, Fort Collins, CO, USA; C Maătzler, Institute of Applied Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland; H Benedickter, Laboratory for Electromagnetic Fields and Microwave Electronics, ETHZ, ETZ K 88, Gloriastrasse 35, CH-8092 Zuărich, Switzerland; H Fluăhler, Institute of Terrestrial Ecosystems (ITES), Swiss Federal Institute of Technology (ETH), CHN F 28.1, Universitaătstr 16, CH-8092 Zuărich, Switzerland Received 13 Jan 2006 *Corresponding author (mike.schwank@env.ethz.ch) 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 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 Published in Vadose Zone Journal 5:1048–1064 (2006) Original Research doi:10.2136/vzj2006.0009 ª Soil Science Society of America 677 S Segoe Rd., Madison, WI 53711 USA Abbreviations: FEP, fluorinated ethylene-propylene; SMD, surface mounted device 1048 Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved www.vadosezonejournal.org 1049 Fig (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 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 Figures 1c through 1e provide a close-up view of the ring-capacitor sensor assembly with schematic electric field lines (torus-like, axisymmetric pattern), the electronic circuit board inside the rings, and an equivalent circuit diagram The external capacitance C(e) is a function of the permittivity e of the medium surrounding the access tube Basic Dielectric Theory The relative permittivity e e9 ie0 of a material is a dimensionless complex number (relative to ea » for air) The real part e9 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 e 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 e0 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 ea and em » 3–5, respectively The permittivity of pure free water at 258C and frequencies ,1 GHz is ew » 78 Due to the large contrast between ew, ea, and em, the overall permittivity e of a wet soil is a strong function of the volumetric water content u Thus, bulk permittivity e is an appropriate proxy quantity for determining soil water content The bulk permittivity e 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, e cannot be calculated as the linear weighting of the permittivities of the constituents 1050 VADOSE ZONE J., VOL 5, AUGUST 2006 according to their volume fraction Therefore, more sophisticated dielectric mixing approaches considering the morphology of the dielectric constituents have to be used Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved Empirical Dielectric Mixing Models Various empirical and semiempirical models have been used to relate permittivity e to water content u of different soil types The most commonly used empirical model (Topp et al., 1980) is: e 3:03 9:3u 146:0u 76:7u [1] The inverse relationship is: uTopp (e) 4:3 1026 e3 5:5 1024 e2 2:92 1022 e 5:3 1022 [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 ew, es , and ea and volumetric fractions u, (1 h), and (u h): e [ueaw (1 h)eas (h u)eaa ]1/a [3] The soil porosity h has to be determined experimentally (here h 0.46), and the exponent a 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 ut dividing the water content range into two domains The transition point ut 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 u , ut 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 u ut 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 (25 000 , R , 37000) is converted to fr using the linear relationship (Sentek technical support, personal communication, 2004): fr R 4:096 1023 MHz C5 1 Cint 21 Cacc Ce21 [6] where Ce C(e) is the capacitance associated with the medium outside the access tube, which is the quantity of interest Theoretically, Cint varies with Ce 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 fr22 for known external capacitances CSMD that this is a reasonable assumption Now it will be shown that a change DC of the capacitance C is proportional to the difference D(fr22) frA22 frB22 of inverse squared values of two resonant frequencies frA and frB DC D (fr22 ) dC d( fr22 ) [7] The proportionality factor dC/d(fr22) can be expressed, using the chain rule, as: [ dC d(fr22 ) dCe d( fr22 ) dCe dC ] 21 [8] The derivatives d(fr22)/dCe and dCe /dC are calculated from Eq [5] and [6] leading to: 4p2 L dCe d(fr22 ) (1 Ce /Cacc )2 [9] and dCe dC (1 Ce /Cacc ) This simple derivation shows that the proportionality factor dC/d(fr22) (4p2L)21 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 [4] where fr has units of megahertz Furthermore, fr 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: pffiffiffiffiffiffiffi [5] 2p LC Electrical resistance is neglected here The total capacitance C includes the series capacitance of the plastic fr 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 Ce , 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: Definition of Normalized Sensor Reading The characteristics of individual EnviroSMART sensors are not perfectly uniform As a consequence, the readings Rek of different sensors k are different even for identical materials with permittivity e surrounding the sensors To eliminate the effect of these sensor-specific differences, sensor readings Rek are normalized to be: Nk [ Rka Rke Rka Rkw [10] 1051 Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved www.vadosezonejournal.org Rak and Rwk are the air and water readings recorded with sensor k installed in the access tube, surrounded by air and immersed in pure water at 258C The normalized sensor reading Nk is a dimensionless number having the values Nk for an air reading and Nk for a measurement in pure water of 258C Using the relationship (Eq [5]) between fr and C, the normalized sensor reading N can be expressed by the sensor capacitances Ca, Cw, and Ce if it is embedded in air, water, and an arbitrary material with permittivity e: 1 pffiffiffiffiffiffi pffiffiffiffiffiffi Ce Ca N5 1 pffiffiffiffiffiffi pffiffiffiffiffiffi Ca Cw [11] 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 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 Default Calibration for Water Content A power-law calibration function was given by the vendor (Sentek, 2001) for relating N to u The empirical default calibration function N(u) with the three parameters a 0.1957, b 0.404, and c 0.02852 is commonly used to estimate u of sands, loams, and clay loams from N: N a (100 u)b c The inverse relationship is thus N c 1=b uSentek (N) 0:01 a [12] [13] Temperature Dependence of Environmental Permittivity Below the relaxation frequency (,10 GHz), the permittivity of pure free water, ew, 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 ew in terms of an approach based on two Debye relaxation frequencies The estimated negative temperature gradient is approximately dew/dT » 20.36 K21 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 e(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 Resonant Frequency Measurements using Capacitance Devices To gain an improved understanding and characterization of the instrument, it is desirable to measure the resonant frequency 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 e 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 (HewlettPackard 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 258C A Sucoflex 104 cable (HUBER1SUHNER, 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 # 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 258C 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 sen- 1052 VADOSE ZONE J., VOL 5, AUGUST 2006 sors are generally more accurate at higher frequencies (Shang et al., 1999), whereas devices with a larger measurement volume are more accurate at lower frequencies tween Z0 and the impedance ZL of the ring electrodes of the sensor Because the load was assumed to be exclusively capacitive, ZL Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved Permittivity of Access Tube The permittivity of the access tube material eacc was measured with the reflection method described above, such that eacc 3.35 for frequencies not exceeding GHz Because the plastic material comprising the access tube is identical with the electrode holder material, the same permittivity ehold eacc 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 fd ranging from pure water (fd 0) to pure dioxane (fd 1) were measured at 258C The measured e values of the mixtures deviated significantly from the linear weighting of the permittivities ew and ed of the water and dioxane constituents The main nonlinearity was quantified using a semiempirical power-law fitting approach (Sihvola, 1999): e(fd ) [fd ebd (1 fd )ebw ]1=b [14] The optimized parameter b 0.813 describes the main deviation from the linear dielectric mixing ew 78.38 and ed 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 fd is the volumetric dioxane mixing ratio: 2i 2pfC [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 ea We selected dioxane (1,4-Diethylene dioxide: C4H8O2) with ed » 2.2 (Maurel and Price, 1973) Dioxane has a boiling point of 1018C, melting point of 128C, and flash point of 118C 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 Vd fd Vd Vw [15] where Vd and Vw 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-V cable soldered directly to the inside of the ring electrodes The reference plane (zero phase) was adjusted to the end of the 50-V 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 GL Gr iGi 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 lowfrequency sensor capacitance C The resulting low-frequency values of C were measured at the same environmental permittivities # e # 78.38 as the normalized sensor readings The frequency spectrum of the total capacitance C between the electrodes was calculated from GL, measured for a load with impedance ZL connected behind the reference plane (line impedance Z0 50 V): GL Gr iGi ZL Z0 ZL Z0 [16] The VNA setup allowed us to interpret the measurements GL as the result of the reflection caused by the mismatch be- The criteria for designing the measurement container depicted in Fig 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 508C; 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 Components and Tools The functionality required above was achieved using the following special tools and components (Fig and 3): 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 508C 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 Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved www.vadosezonejournal.org 1053 Fig 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 solvent-resistant, submersed brush for cleaning the re- flection probe collar suspended around the access tube for removing air bubbles and providing additional vertical mixing during heating The heater, permittivity sensor, thermistor–thermometer, cleaning brush, and collar were retracted during frequency/ capacitance measurements to avoid signal disturbance The container (Fig 2) was mounted on three legs, one of which is the mixer, on a table with rollers (Fig 3) for transportation to and from a cold room Simultaneous Measurements of Permittivity and Frequency/Capacitance The functional relation e(N) between the permittivity e of a material around the soil moisture probe and the normalized reading N was deduced by measuring the permittivity of the en- compassing 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 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 e For quality control, samples were taken after completing the measurements at each dioxane–water mixing ratio fd,i to measure the corresponding ei 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 e between capacitance measurements without recalibrating the system Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved 1054 VADOSE ZONE J., VOL 5, AUGUST 2006 Fig Cylindrical metal sheet disturbance used for investigating the sampling volume of the sensor Fig Picture of the experimental setup used for characterizing the EnviroSMART soil water content sensor 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-V cable was soldered to the inside of the ring electrodes The VNA depicted in the left side of Fig 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 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 e 1, 16.4, 20.3, and 78.38, each with 10 distances # 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 mm was realized by wrapping the brass foil tightly around the access tube The resulting shapes of the brass roll were not perfectly cylindrical, as can be seen in Fig 4d From the main axes d1 and d2 the corresponding distances D1 and D2 from the access tube with diameter 57.4 mm were calculated from: Dk 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 u and soil permittivity Here, we focus on the relation between the real part (hereafter denoted simply as e) 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 ei (i 1–22) of the liquid encompassing was varied Sensor readings for ei between the dioxane permittivity ed » 2.2 and the water permittivity ew » 78.38 at 258C were investigated with volumetric dioxane–water mixing ratios fd from (pure dioxane) to (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 ei 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 dk 57:4 mm k 1,2 [18] The mean distance D (D1 D2)/2, absolute difference DD D1 D2 and relative difference dD DD/D for each of the nine sizes (Dk 0) of the cylinder are given in Table Axisymmetric Electromagnetic Simulations The electric field distributions resulting from the two electrodes of the sensor were simulated using the commercially available finite element software Maxwell 2D, (Ansoft, Pittsburgh, PA; http://www.ansoft.com/maxwellsv/ [verified 12 July 2006]) The software computes a two-dimensional field solution, and a full three-dimensional solution can be calculated for an axially symmetric problem 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, Table Mean distance D between the access tube and the inner border of the coaxial metal disturbance DD and dD are the absolute and the relative deviation resulting from the ellipsoidal shape Position i Di DDi dDi 2.5 2.5 3.5 4.5 9.5 % 55 22 22 12 10 13 12 mm 4.55 9.3 13.8 20.55 35.05 47.3 66.55 81.05 96.3 1055 Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved www.vadosezonejournal.org 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 (ea 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 e 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 Calculation of Sensor Capacitances The surfaces of two electrodes represented by ideal conductors are equipotential with potential difference U, which varies linearly with the total charge 6Q accumulated onto the electrode surfaces, such that [19] Q [ CU 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 eE, which allows for calculating the storage of the electrical field energy f within a volume (vol): f5 # E Ddv vol [20] This field energy f 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 df Udq qdq/C Consequently, the energy f used to transfer the total charge Q from one conductor to the other is Q Q Q2 CU f # df # qdq 2C C0 [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 Ci j 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 V applied to the electrode i and grounding all other electrodes i p j Such matrices have been calculated for the model setup depicted in Fig (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 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 C C1,2 C1,3 [22] This value of C is more than the value calculated for the model without the metal disturbance Twelve finite distances # D # 162 mm were simulated for various permittivities (e 1, 5, 10, 20, 30, 40, 78.38) To estimate a characteristic distance of influence L at the investigated permittivities, measured and modeled data [D, N] were fitted using the following approach: Fig 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 D N(D) N¥ (N0 N¥ ) exp L [23] 1056 VADOSE ZONE J., VOL 5, AUGUST 2006 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 L for different e values were derived from measurements and simulations and then compared with each other Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved Analysis of Errors in Estimated Water Content If Topp’s equation (Eq [2]) is assumed to characterize the relationship u(e) for an ideal soil, the instrumental root mean squared error in u (RMSEu) can be computed directly from the errors in e determined from the present experiments: O[ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n du(ei ) Dei RMSEu [ n i51 de ] [24] duðei Þ where Dei ei e(Ni), is the slope of Topp’s equation de (Eq [2]) at ei, ei values are measured using the reflection method, e(Ni) values are estimated from normalized sensor readings Ni, and n is the number of measurements Topp’s equation is widely used with e values estimated from TDR data, and estimates of e tend to decrease with increasing measurement frequency Thus, u(e) and its first derivative used in Eq [24] can vary with the instrument used, making the RMSEu 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 e values RESULTS AND DISCUSSION Surface Mounted Devices One sensor was used for measurements with known capacitance CSMD values as described above The results are shown in Fig 6, where the almost perfectly linear responses in terms of fr22 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 Fig Sensor responses fr22 to surface mounted device (SMD) added capacitance CSMD with and without the ring capacitor attached to the instrument and plastic sensor stick (Fig 2) inside the ring, but without the access tube Inverting the linear regression equation with the ring capacitor attached leads to a useful expression for the change in capacitance DC with change in measured resonant frequency D( fr22) fr,A22 fr,B22 under two different conditions A and B: DC 2:982 105 D( fr22 ) [25] where DC 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 e are presented below Measured C and N values are then compared with simulated values Measured Normalized Sensor Readings Figure shows the measured relation between the permittivity ei of the dioxane-water mixture encompassing the access tube and the normalized reading averaged over the four sensors: Ni (N1i Ni2 Ni3 Ni4)/4 The light gray circles in Fig are the data measured with the equipment shown in Fig Consequently, these data were not corrected for the calculated effect of the FEP coating Corrected normalized readings Ni DN(ei) 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: Ni i Ni DN(ei ) [26] where DN(e) is the correction function to be derived below Thus, the black circles in Fig are the readings Ni corrected for the calculated effect DN(e) of the FEP shrink tube, which is not applied in the field On the right axes of Fig 7, the standard deviations s between Nik measured with the four sensors (k 1–4) are plotted As Fig 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 DN(e) of the fluorinated ethylene-propylene (FEP) coating The solid lines are the approximations using Eq [29] and s is the standard deviation between Nik measured with the sensors k to Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved www.vadosezonejournal.org 1057 can be seen, the maximum s is ,4 1023 for # N # 1, allowing for sensor-independent examinations if normalized sensor readings are used The solid lines in Fig represent the least mean square approximations to the data [Ni, ei] (light gray circles) and the corrected data [Ni DN(ei), ei] (black circles) (i 1–22) The fitting approach considers a quadratic and an exponential factor: e(N) (a0 a1 N a2 N )exp ðkNÞ [27] As will be shown, this approach provides a macroscopically accurate representation of the data within the entire range # N # As indicated by the dashed portion of the fitted lines and as discussed below, the fit near N may be less accurate than at steeper portions of the curve In accordance with the definition of N, Eq [27] must fulfill two constraints: e(N 0) ea and e(N 1) ew 78:38 [28] that eliminate two of the four fitting parameters a0, a1, a2, and k in Eq [27]: e(N) [ea (ew e2k 2 a2 )N a2 N ] exp ðkNÞ [29] The computed values of the remaining fitting parameters a2 and k representing the approximation of the uncorrected data [Ni, ei] are a2 1.15008 and k 6.66056 For the data [Ni DN(ei), ei] 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 e(N) from field-measured data N From the sensor reading (raw count) Re of a sensor placed in an environmental material with unknown e, 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, e of the environmental material can be estimated using relation [29] For the greatest accuracy, relation [29] should be applied only to permittivities e , 40 because of decreasing sensitivity of N with respect to e 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 DN(ei) and the approximation e(N ) evaluated at N Ni DN(ei) and the measured permittivities ei is RMSEe 0.859 for , e , 80 and only RMSEe 0.226 for the measured range of , e , 43 pertaining to soil water The overall RMSEe of 0.859 is affected primarily by one large deviation at e(N 0.9925) 77.12 If this is excluded the RMSEe is 0.314 for , e , 64, which is more indicative of the expected deviations Fig Measured sensor capacitance C versus permittivity e of the dioxane–water mixture outside the access tube measured in soils is instrumental (affected by the sensor electronics, inner ring capacitance, and the access tube) Second, C(e) is very nonlinear in this range, so the geometric factor relating C to e is not a constant, which indicates large changes in the electromagnetic field pattern with changes in the environmental e values These factors are related to the instrumental design and determine the measurement sensitivity and accuracy Calculated Capacitances and Normalized Sensor Readings The experimental results presented in the previous sections were modeled using the procedure described above Figure shows a cross section of the field strength |E| in the sensor region The permittivity of the environmental material is e 20, and the potential difference between the two ring electrodes is U V The field is concentrated within a narrow range around the sensor electrodes The highest field strength |E| actually occurs within the sensor structure (electrode holder and access tube) Immediately outside the access tube, the field |E| is already reduced by a factor of 10 compared with |E| in the electrode holder between the positive and the negative ring electrode Measured Sensor Capacitances The capacitance C between the electrodes of one sensor is computed from VNA measurements of the complex reflection coefficient GL The relative values of C and the shape of the C(e) curve in Fig give the following insights First, C(e 1) is approximately one-half of the value of C(e 25), which may represent a relatively wet soil Thus, approximately one-half of the total capacitance Fig Electric field strength |E| resulting from the potential difference U V between the ring electrodes calculated using permittivities of access tube, electrode holder, and environmental material of eacc ehold 3.35 and e 20, respectively Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved 1058 VADOSE ZONE J., VOL 5, AUGUST 2006 The field energy f 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 p 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 matrix For the model parameters used here, the capacitance matrix [Ci j] is (in units of pF): 18 [Ci;j ] [30] 18 The matrix element C1,2 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 Cmodel computed for a range of permittivities (1 # e # 78.38) The asterisks in Fig 10b show normalized sensor readings Nmodel calculated from the modeled capacitance for # e # 78.38 using Eq [11], with Ca 4.62 pF for e ea and Cw 24.65 pF for e ew 78.38 (at 258C) Fig 10 (a) Modeled and measured capacitances Cmodel and Cexp between the ring electrodes as a function of the permittivity e of the environmental material The permittivity of the access tube and sensor holder are eacc ehold 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 e(Nexp) The corresponding experimental data Cexp, (circles), Nexp (circles), and the data fit e(Nexp) (solid line) already presented in Fig and Fig 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 e relative to the total C at greater e 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(e) 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 eFEP » 2, thickness » 0.5 mm) on the measured permittivity e was estimated using the electromagnetic field simulation software Similar effects of nonmetallic components covering the rods of TDR probes were investigated previously (Ferre´ 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 Cmodel(e) computed for # e # 78.38 with the FEP coating (hollow circles) and without it (solid squares) As shown in Fig 12a, the absolute value of the difference DC(e) Cmodel with FEP(e) Cmodel without FEP(e) is small when the contrast between e and eFEP is small but increases for larger e Furthermore, the difference DC(e) is positive for e # eFEP and negative for e $ eFEP This is Fig 11 (a) Calculated sensor capacitances Cmodel with FEP(e) and Cmodel without FEP(e) 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 eFEP (b) Normalized sensor readings Nmodel with FEP(e) and Nmodel without FEP(e) derived from Cmodel, with FEP(e) and Cmodel, without FEP(e) using Eq [11] for the case with (hollow circles) and without (solid squares) the shrink fit on the access tube 1059 Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved www.vadosezonejournal.org Fig 12 (a) Difference DC(e) between simulated capacitances Cmodel with FEP(e) with the FEP coating on the access tube and Cmodel without FEP(e) without the FEP coating taken from Fig 11a (b) Difference DN(e) between corresponding sensor readings Nmodel with FEP(e) and Nmodel without FEP(e) taken from Fig 11b in accordance with expectations that capacitance is increased by the presence of the FEP coating for e # eFEP and vice versa Figure 11b shows normalized sensor readings Nmodel, with FEP(e) and Nmodel without FEP(e) derived from Cmodel with FEP(e) and Cmodel without FEP(e) for the case with (hollow circles) and without (solid squares) the shrink fit on the access tube These Nmodel values were computed using Eq [11] with the calculated air and water capacitances Cmodel with FEP, a 4.6909 pF, Cmodel with FEP, w 20.758 pF and Cmodel without FEP, a 4.6207 pF, Cmodel without, w 24.649 pF as labeled in Fig 11a The effect of the FEP coating is expressed as DeRMS calculated from the difference De(N) e(Nmodel with FEP) e(Nmodel without FEP) computed from the data shown in Fig 11b: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uNmax u u # {De(N)}2 dN u t Nmin DeRMS [31] Nmax Nmin The integration interval [Nmin, Nmax] is related to corresponding permittivity and soil moisture regimes [emin, emax] and [umin, umax] The interval [emin, emax] is calculated from the boundaries umin and umax using the Topp model (Eq [2]), and Nmin, Nmax are calculated from evaluating the interpolation Nmodel without FEP(e) from Fig 11b at the permittivities emin and emax For a realistic soil moisture regime [umin, umax] [0.05, 0.5] these intervals are [emin, emax] [3.85, 34.59] and [Nmin, Nmax] [0.475, 0.943], respectively The difference calculated for the above permittivity range using Eq [31] is DeRMS 0.596 The corresponding correction of the normalized readings Ni is represented by the modeled deviation DN(e) Nmodel with FEP(e) Nmodel without FEP(e) plotted in Fig 12b, which reaches a maximum of approximately 11.5 1023 at e » 6.2 This exceeds the maximum standard deviation s between the normalized readings Nik 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 fr computed with Eq [4] are listed in Table Modeled capacitance Cmodel values in Table are the marked data (arrows) from Fig 11a for ea and ew These values were used to determine the changes in modeled capacitance with and without the FEP coating (DC under “Model” in Table 2) Because the experimental DC determines DR, and consequently DN, only the relative changes must be simulated properly Here, we assume that DC(e) values fall between the end members for air and water Thus, if the model simulates DC accurately for air and water, it can be assumed to represent the change correctly for all permittivity values The experimental change in capacitance can be calculated directly from Eq [25] using the change D(fr22) fr, with FEP22 fr, without FEP22 in resonant frequency with and without FEP The resulting values in Table are very close to the simulated changes for air and water, where the errors (DCmodel DCexp) 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 ew(T), pure dioxane ed(T), and a mixture of 98% dioxane and 2% water edw98(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 [Ni, ei] Figure 13a shows ew(T) measured in situ with the reflection method (solid dots) and ew(T) deduced from the EnviroSMART sensor readings (hollow circles) for # T # 508C Both water permittivities ew(T) show a negative gradient dew/dT The linear regression of ew(T) measured with the VNA electromagnetic reflection method yields a gradient of de/dT » 20.36 K21, which quantitatively agrees with the Debye model, but the capacitance sensor data yield de/dT » 20.59 K21 One could correct the slope dew/dT based on the fit of N(e) near ew 80 by adjusting the parameter values in Table Averaged experimental sensor readings ^R& and resonant frequencies fr with and without the fluorinated ethylenepropylene (FEP) coating on the access tube showing corresponding capacitance changes DC compared with modelderived DC quantities for air and water environments Exp Condition Air Air Water Water without FEP with FEP with FEP without FEP ^R& fr 36473.5 36436.3 25919.5 24172.5 MHz 149.4 149.2 106.2 99.0 Model DC 0.0273 23.962 C pF 4.6207 4.6909 20.758 24.649 DC 0.0702 23.891 1060 VADOSE ZONE J., VOL 5, AUGUST 2006 20.012 K21, which is close to dedw98/dT 20.009 K21 estimated from the measurements Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved Characteristic Distance of Influence The electric field caused by the two ring electrodes at the potential U 61 V is concentrated in a narrow region around the electrodes (Fig 9) Consequently, the capacitance is influenced predominantly by the permittivity e 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 L 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 L does not imply that the sensor sampling volume is cylindrical Ferre´ 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 Fig 13 (a) Temperature dependency ew(T) of the permittivity of pure water (fd 0) measured with the VNA dielectric sensor (solid dots) and derived from the capacitance sensor readings (hollow circles) (b) Temperature dependencies ed(T) and edw98(T) of pure dioxane (fd 1, hollow squares) and a solution with fd 0.98 (hollow diamonds) deduced from the capacitance sensor readings Eq [29] Another option is to find a different mathematical form for the calibration equation, which is beyond the present scope However, dew/dT is negative and constant even without additional fitting The temperature dependencies of pure dioxane ed(T) and of the 98% dioxane–water mixture edw98(T) for 12 # T # 258C are depicted in Fig 13b Dioxane permittivities ed(T) display no temperature dependence within the accuracy of our measurements The literature value for the permittivity of dioxane is 2.20 0.11 (Maurel and Price, 1973) This is consistent with the average ed 2.089 0.008 deduced from the nine sensor readings at the temperature between 12.5 and 25.48C The average permittivity edw98 and the temperature gradient dedw98/dT between 13.7 and 22.48C are edw98 2.679 0.026 and dedw98/dT 20.009 K21, respectively The permittivity of the solution estimated from the volumetric mixing ratio fd 0.98 using the dielectric power law (Eq [14]) is edw98 3.07, which is within the uncertainty De » 0.15 at fd 0.98 of the model (Maurel and Price, 1973) Assuming a linear mixing effect on the temperature response, the estimated slope would be A concentric metal sheet with changeable diameter was installed around the access tube as shown in Fig 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 Nexp(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 Nexp(D 0) N0 » 1.02 were the same for all e Furthermore, N0 is due to the fact that the capacitance for D exceeds the capacitance measured in water at 258C The measurements Nexp(D 96.3 mm) were taken at the largest possible distance limited by the diameter of the container, and Nexp(D ¥) N¥ are the undisturbed normalized readings The diameter of the container is large enough to ensure that the outer container wall does not affect the sensor readings To estimate L representing the characteristic distance of influence at the investigated permittivities, the experimental data [D, Nexp] were fitted using Eq [23] with N0 1.02 and N¥ 0, 0.752, 0.804, shown as solid lines in Fig 14a The calculated least square fit parameter values for L and the corresponding RMSE values between the approximations and the measured values are listed in Table for the data measured at e 1, 16.4, 20.3, 78.38 1061 Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved www.vadosezonejournal.org Fig 15 Electric field strength |E| resulting from the potential difference U V between the ring electrodes calculated using the model configuration with the metal foil disturbance at distance D 27 mm from the access tube (eacc ehold 3.35 and e 20) 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 e (ea 1, 16.4, 20.3, ew 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 and 162 mm, and the permittivity e was between the air and the water value The solid lines are the best fits with the exponential model (Eq [23]) The data measured at e 16.4 and 20.3 are approximated well by the simple exponential approach (Eq [23]), whereas the exponential approximations of the data measured at e and 78.38 are not adequate However, realistic permittivities e of natural soils are typically between and 40 for very dry and watersaturated soils, respectively The characteristic distance of influence L within this range (e 16.4 and 20.3, bold solid lines) is approximately 12.5 mm Calculated Distance of Influence Figure 15 shows the field strength |E| calculated for potentials U 61 V applied to the ring electrodes The coaxial metal disturbance increased |E| between the Table Fitting parameters and standard deviations s between the approximation and the measured data plotted in Fig 14a according to approximation [23] e ea (air) 16.4 20.3 78.38 (water @ 25°C) [mm] 4.321 12.259 13.176 55.957 23 s [10 22.9 5.51 5.65 3.53 ] access tube and the metal sheet compared with |E| at the corresponding location of the undisturbed situation shown, in Fig 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 e 5, the exponential fit is better adapted than for e # This is consistent with the analysis performed for the measured N The values of the fitting parameter L representing the characteristic distances of influence and the corresponding standard deviations s between the exponential fit and the model results N are listed in Table The range of characteristic distances (10 , L , 13 mm) deduced from the electromagnetic calculations for # e # 40 overlaps the range of 12 , L , 13 mm deduced from the experiment for e 16.4 and 20.3 It is helpful to consider the percent difference induced by the metal border at different distances The relative errors in N(D) at D L, 2L, and 3L are 37, 14, and 5%, respectively For the fitted exponential (L » 12.5 mm), we expect 33, 67, 80, 95, and 99% of the signal to be based on material within approximately 5, 14, 20, 37, and 58 mm of the access tube, respectively Paltineanu and Starr (1997) reported that 99% of the signal corresponded to a distance of approximately 100 mm for soil Table Fitting parameters and standard deviations s between the exponential fit (Eq [23]) and the calculated data plotted in Fig 14b e (air) 10 20 30 40 78.38 (water @ 25°C) [mm] 6.678 10.09 11.27 12.07 12.44 12.61 13.12 23 s [10 33.68 11.05 6.007 2.914 1.992 1.455 1.009 ] Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved 1062 VADOSE ZONE J., VOL 5, AUGUST 2006 systems surrounded by air, but their Fig showed a similar value of 95% at approximately 40 mm In our case, use of a metallic cylinder affected the electrical field (Fig 15) differently than air, which may account for some of the differences 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 u(e) for an ideal soil, the RMSEu can be computed directly from RMSEe 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 u(e) accurately for a given soil The estimated errors in e correspond to an instrumental precision of RMSEu 0.0034 m3 m23 for , e , 43 This indicates that small changes in u 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 u versus N, where the default calibration is represented by the black solid line For comparison, in Fig 16a we calculated u[e(N)] using three different empirical dielectric mixing models for relating e with u from Eq [29], which gives e(N) The u values derived from the Sentek default calibration were biased relative to u values calculated from the three dielectric mixing models in combination with relation [29] Where Topp’s equation can be applied, the Sentek default calibration equation would underestimate u for the full range of typical field water contents (0.05 , u , 0.50 m3 m23) The deviation between the water content derived from the Sentek default calibration and the water content calculated using the Topp model is defined as: DuTopp (N) [ uSentek (N) uTopp [e(N)] [32] where e(N) is given by Eq [29] The u interval boundaries umin and umax are the minimum and maximum water contents for which the deviation DuRMS was calculated The N values Nmin and Nmax corresponding to umin and umax were computed using the Sentek default calibration function (Eq [12]) 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 u[e(N)] calculated using relation [29] for e(N) and (b) water content–permittivity relationships e(u) The deviation DuRMS for the interval [umin, umax] was solved numerically using: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uNmax u u # {Du u Topp (N)} dN t Nmin DuRMS [33] Nmax Nmin For 0.05 # u # 0.5, DuRMS equals 0.066 m3 m23, 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 u 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 e(u) The bold solid line was computed using the Sentek default calibration to calculate N from u first, and then calculate e from Eq [29] For u 0.30 m3 m23 the discrepancy between the dielectric mixing modelbased permittivities and e deduced from the Sentek de- 1063 Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved www.vadosezonejournal.org fault calibration increases rapidly The latter reaches unrealistically high values of e 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 u even near saturation (»0.45 m3 m23) The problem illustrated here involves the indirect use of Eq [12] to obtain e via Topp’s equation (Eq [2]) Given our present experimental results, one can now compute e directly from N using Eq [29] For e 25, the default calibration yields values more than 0.05 m3 m23 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 is estimated to be approximately 12.5 mm, and 95% of the signal is sensed within approximately 37 mm of the access tube Different models for u(e) were compared with the manufacturer’s default calibration, showing that the default calibration tends to underestimate u by 0.066 m3 m23 The mixing models may not represent u(e) 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 SUMMARY AND CONCLUSIONS The experiments described above led to the following main results: An ideal relationship between sensor resonant frequency fr and added SMD capacitance CSMD was established It confirmed a linear relationship (R2 » 1) between fr22 and CSMD, but with a nonzero intercept, implying a parallel capacitance in the circuit, even without having the ring capacitor connected Dioxane–water mixtures provided a range of e values representative of soils from dry to fully saturated (»4 , e , 40), which has not been demonstrated with other dielectric surrogates Capacitance sensors displayed appropriate qualitative temperature T dependence of the permittivity of free water ew, where ew(T) was very linear with a negative slope There was no apparent temperature sensitivity in pure dioxane A new empirical relationship has been derived between environmental permittivity e and sensor readings normalized using air and water readings (see Eq [29] with a2 1.1282 and k 6.6485) The RMSE of e between fitted and measured permittivities was 0.859 for the full range of measurements (1 , e , 80) and only 0.226 for the range expected for soil water measurements (3 , e , 43) For the soil relevant range (3 , e , 43), this RMSE for e results in an instrumental error of estimated soil water content u of RMSEu 0.0034 m3 m23 using Topp’s equation (Eq [2]) to represent e(u) Approximately one-half of the total capacitance measured is instrumental (affected by the sensor electronics, inner ring capacitance, and the access tube), and C(e) is very nonlinear, so the geometric factor relating C to e is not a constant, which indicates large changes in the electromagnetic field pattern with changes in the environmental e values 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 The following ETHZ staff members provided important technical assistance: Hanspeter Laăser (container design and construction); Hans Wunderli (expansive metal design, construction, and digital photography); Hannes Wydler (frabrication of custom tools); Kurt Barmettler (chemical advice); Rene´ Saladin (Dioxane handling and safety); Joărg Leuenberger (infrastructure); and Peter Bruăhwiler (specialty machining) Comments from Steven Evett, Sally Logsdon, Ty Ferre´, and an anonymous reviewer helped us improve the manuscript REFERENCES Baumhardt, R.L., R.J Lascano, and S.R Evett 2000 Soil material, temperature, and salinity effects on calibration of multisensor capacitance probes Soil Sci Soc Am J 64:1940–1946 de Loor, G.P 1983 The dielectric properties of wet materials IEEE Trans Geosci Remote Sens GE-21:364–369 Dobson, M.C., F.T Ulaby, M.T Hallikainen, and M.A El-Rayes 1985 Microwave dielectric behavior of wet soil Part II: Dielectric mixing models IEEE Trans Geosci Remote Sens GE-23:35–46 Evett, S.R., and J.L Steiner 1995 Precision of neutron scattering and capacitance type soil water content gauges from field calibration Soil Sci Soc Am J 59:961–968 Evett, S.R., J.A Tolk, and T.A Howell 2006 Sensors for soil profile moisture measurement: Accuracy, axial response, calibration, precision and temperature dependence Vadose Zone J 5: (in press) Ferre´, P.A., J.H Knight, D.L Rudolph, and R.G Kachanoski 1998 The sample areas of conventional and alternative time domain reflectometry probes Water Resour Res 34:2971–2980 Ferre´, P.A., D.L Rudolph, and R.G Kachanoski 1996 Spatial averaging of water content by time domain reflectometry: Implications for twin rod probes with and without dielectric coatings Water Resour Res 32:271–279 Green, T.R., M Schwank, and H Fluăhler 2004 Temperature effects on capacitance measurements of soil water: Theory and field application In Annual Meetings Abstracts [CD-ROM] a04-3974 ASA CSSA SSSA, Madison, WI Kelleners, T.J., R.W.O Soppe, J.E Ayars, and T.H Skaggs 2004a Calibration of capacitance probe sensors in a saline silty clay soil Soil Sci Soc Am J 68:770–778 Kelleners, T.J., R.W.O Soppe, D.A Robinson, M.G Schaap, J.E Ayars, and T.H Skaggs 2004b Calibration of capacitance probe sensors using electric circuit theory Soil Sci Soc Am J 68:430–439 Knight, J.H 1992 Sensitivity of time domain reflectometry measurements to lateral variations in soil water content Water Resour Res 28:2345–2352 Knight, J.H., P.A Ferre´, D.L Rudolph, and R.G Kachanoski 1997 A numerical analysis of the effects of coatings and gaps upon relative dielectric permittivity measurement with time domain reflectometry Water Resour Res 33:1455–1460 Reproduced from Vadose Zone Journal Published by Soil Science Society of America All copyrights reserved 1064 VADOSE ZONE J., VOL 5, AUGUST 2006 Logsdon, S., and D Laird 2004 Cation and water content effects on dipole rotation activation energy of smectites Soil Sci Soc Am J 68:1586–1591 Maurel, P., and A.H Price 1973 A flow method for the measurement of the permittivity and dielectric loss factor of liquids J Phys [E] 6:1095–1097 Meissner, T., and F.J Wentz 2004 The complex dielectric constant of pure and sea water from microwave satellite observations IEEE Trans Geosci Remote Sens 42:1836–1849 Or, D., and J.M Wraith 1999 Temperature effects on soil bulk dielectric permittivity measured by time domain reflectometry: A physical model Water Resour Res 35:371–383 Paltineanu, I.C., and J.L Starr 1997 Real-time soil water dynamics using multisensor capacitance probes: Laboratory calibration Soil Sci Soc Am J 61:1576–1585 Robinson, D.A., S.B Jones, J.M Wraith, D Or, and S.P Friedman 2003 A review of advances in dielectric and electrical conductivity measurement in soils using time domain reflectometry Available at www.vadosezonejournal.org Vadose Zone J 2:444–475 Roth, K., R Schulin, H Fluăhler, and W Attinger 1990 Calibration of time domain reflectometry for water content measurement using a composite dielectric approach Water Resour Res 26: 2267–2273 Schwan, H.P 1992 Linear and nonlinear electrode polarization and biological materials J Biomed Eng 20:269–288 Sentek 2001 Calibration of Sentek Pty Ltd Soil Moisture Sensors Sentek, Stepney, SA, Australia Shang, J.Q., R.K Rowe, J.A Umana, and J.W Scholte 1999 A complex permittivity measurement system for undisturbed/compacted soils Geotech Test J 22:159–168 Sihvola, A 1999 Electromagnetic mixing formulas and applications The Institution of Electrical Engineers, London Topp, G.C., J.L Davis, and A.P Annan 1980 Electromagnetic determination of soil water content: Measurements in coaxial transmission lines Water Resour Res 16:574–582 Wang, J.R., and T Schmugge 1980 An empirical model for the complex dielectric permittivity of soils as a function of water content IEEE Trans Geosci Remote Sens GE-18:288–295 Wraith, J.M., and D Or 1999 Temperature effects on soil bulk dielectric permittivity measured by time domain reflectometry: Experimental evidence and hypothesis development Water Resour Res 35:361–369 ... data measured at e 16.4 and 20.3 are approximated well by the simple exponential approach (Eq [23]), whereas the exponential approximations of the data measured at e and 78.38 are not adequate... element software calculates capacitances Ci j 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... well as an internal capacitance Cint acting in parallel to Ce , comprised of the on-board sensor capacitance and the contribution caused by the materials inside the capacitor rings As illustrated