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Dielectric measurement techniques are used widely for estimation of water content in environmental media. However, factors such as temperature and salinity affecting the readings require further quantitative investigation and explanation. Theoretical sensitivities of capacitance sensors to liquid salinity and temperature of porous media were derived and computed using a revised electrical circuit analogue model in conjunction with a dielectric mixing model and a finite element model of Maxwell's equation to compute electrical field distributions. The mixing model estimates the bulk effective complex permittivities of solid-water-air media. The real part of the permittivity values were used in electric field simulations, from which different components of capacitance were calculated via numerical integration for input to the electrical circuit analogue. Circuit resistances representing the dielectric losses were calculated from the complex permittivity of the bulk soil and from the modeled fields. Resonant frequencies from the circuit analogue were used to update frequency-dependent variables in an iterative manner. Simulated resonant frequencies of the capacitance sensor display sensitivities to both temperature and salinity. The gradients in normalized frequency with temperature ranged from negative to positive values as salinity increased from 0 to 10 g L^{-1}. The model development and analyses improved our understanding of processes affecting the temperature and salinity sensitivities of capacitance sensors in general. This study provides a foundation for further work on inference of soil water content under field conditions.

Capacitance-based electronic instruments are gaining common application in the field for estimation of soil water content from bulk dielectric properties. Such instruments are generally sensitive to several factors, including soil temperature and electrical conductivity, affecting the effective complex permittivity of environmental media surrounding the probe. Temperature-dependence of the measured bulk permittivity and apparent water content has been observed in the laboratory [

Baumhardt et al. [^{-1}. They demonstrated that the capacitance probes were sensitive to both temperature and electrical conductivity, where a fluctuation of approximately 15°C caused a fluctuation in the apparent soil water content of 0.04 m^{3} m^{-3} based on the default calibration of the Sentek EnviroSCAN^{™}^{3} m^{-3} K^{-1} using factory and soil-specific calibrations.

Despite advances in the theory [

There remains a practical need for methods to compensate for temperature effects in soils of various bulk electrical conductivities, especially in shallow applications where temperature and water content fluctuations can be significant. Physics-based theory and methods as presented in this work are needed to predict a range of environmental conditions where temperature and electrical conductivity can affect instrumental measurements used to estimate soil water content.

The present theoretical developments are meant to be as generic as possible, while the capacitance sensor geometry is specific to a commercial device: Sentek capacitance sensors used on EnviroSMART^{™}^{™}

The normalized sensor reading _{r} is a dimensionless number (also known as “scaled frequency”):

where _{r,a} and _{r,w} are the resonant frequencies measured in air and pure water at the reference temperature _{ref} = 20°C with corresponding environmental permittivities _{a} = 1 and _{w}′ = 80.2 (real part valid for 100 - 250 MHz). According to the above definition, the normalized reading is _{ref}.

The relation

The experiments leading to this empirical relation were performed in almost lossless mixtures of water and dioxane (1,4-diethylene dioxide: C_{4}H_{8}O_{2}) with real permittivities in the range of 2.4 ≤ _{m}′ ≤ 78.4; and the fitting parameters were determined as _{2} = 1.12819 and _{m}′ < 40), such that _{m}′ ≈ _{r} measured with the capacitance sensor. However, in a conducting medium the measurement based _{m}′

The resonant frequency _{r}(_{m}) of a capacitance sensor is affected by the complex permittivity _{m} = _{m}′ + i _{m}″ of the medium (soil) surrounding the probe. The positive imaginary part used in this notation is the consequence of the negative time factor exp(-i _{m}, volumetric water content _{m}′_{m}″. Because the salinity _{m}, the resonant frequency _{r} response of the capacitance probe is sensitive to

The approach to predict _{r}(_{m}(_{m}(_{r}. If the bulk soil is considered as a dispersive medium with frequency-dependent _{m}(_{r} has to be determined iteratively as illustrated in

The frequency response of the capacitance sensor is modeled using a detailed circuit analogue with more components of capacitance than the one used by Kelleners et al. [

The electrical source and the inductance ^{2}_{tot} is composed of inner and outer capacitances with respect to the ring electrodes acting in parallel. These partial capacitances are associated with different volumes in the vicinity of the sensor electrodes.

_{in} is caused by the sensor volume with _{acc,in} = 1 inch (25.4 mm) inside the access tube. This volume includes the axisymmetric electrode holder and also the electronic board and wiring, which are not axisymmetric. The capacitance caused by the access tube volume _{acc,in} < _{acc,out} is separated into a parallel part _{acc,p} and a serial part _{acc,s} with respect to the electrical source. The volume outside the access tube (_{acc,out}) is represented by the capacitance _{m} with an electrical resistance _{m} in parallel for taking into account the conductivity losses of the medium (soil).

The series capacitors _{acc,s} and _{m} are interpreted as distinct capacitances by virtue of a double layer of electrical charges accumulated at the dielectric discontinuity at the outer access tube boundary _{acc,out} associated with the unequal polarization within and beyond the access tube. The polarization _{acc,out} is continuous if the permittivities _{acc} and _{m} of the access tube and the medium are equal, so by definition _{acc,s} and _{m} do not exist in this case (_{m} → _{acc}: _{acc,s} = _{m} = 0 pF). However, the capacitance between the ring-electrodes caused by the volume _{acc,in} comprising the access tube and the medium, is certainly not zero for _{acc} = _{m}. For this purpose, the capacitor _{acc,m} is introduced to account for the parallel capacitance at _{acc}=_{m} caused by the volume confined by _{acc,in}.

A symbolic justification of the necessity for introducing _{acc,m} is sketched in _{1}, _{2} within a composite plate capacitor is illustrated. The large circles with the plus and minus signs symbolize electrical charges on the conducting electrodes of the capacitor associated with _{1} = _{2}, whereas the corresponding small circles are the charges of the double-layer accumulated at the non-conducting dielectric discontinuity. As it is depicted, the capacitance of the composite capacitor is represented by a capacitor with uniform dielectric _{1} = _{2} (corresponding to _{acc,m}) in parallel with the series of two capacitors with permittivities _{1} and _{2} (corresponding to _{acc,s} and _{m}).

The resonant frequency _{r} of the circuit analogue is derived by applying Complex AC-Circuit Theory [_{R}, _{L} and _{C} are used:

Consequently, the impedance _{Rm}, _{L}, _{Cacc,s}, _{Cm}, and _{Cp} where _{p} = _{in} + _{acc,p} + _{acc,m} is in parallel to the electrical source:

The circuit is operated at its resonant frequency _{r} if the energy uptake is maximal, _{r}) takes a real value. The physically meaningful solution of the corresponding equation _{r}) = 0 is:

The above expression _{r}(_{p}, _{acc,s}, _{m}, _{m}) is used to calculate the resonance of the circuit analogue (_{m} → ∞) and a highly conducting (_{m} → 0) medium the expression

Thereby, the operative capacitance for the lossless case (_{m} → ∞) is _{p} + (_{acc,s} · _{acc,s} + _{m}) representing the total input-capacity of the circuit diagram without the presence of the resistor _{m}. In the limit _{m} → 0 one finds the larger value _{p} + _{acc,s} representing the corresponding operative capacitance for hot-wired _{m}. These considerations already show that _{r} is expected to increase with _{m} between the two limits for _{m} → 0 and _{m} → ∞.

The effective permittivity _{m}_{m}′ + i _{m}″ and the electrical DC-conductivity _{m} of the media surrounding the sensor are described on the basis of a physical dielectric mixing approach. Thereby, the effective permittivity _{m} is defined using the relation <_{m} <_{m} of a multiphase mixture comprising

Thereby, only the volume fraction _{j}_{j}_{e} of the host appear in the mixing formula. In fact, the assumption that the spheres in the mixture be of the same size can be relaxed as long as all spheres are small compared with the wavelength of the operating electrical field. Because this effective medium approach is derived directly from basic electromagnetism, it can be applied for complex permittivity values _{j}_{e} without any restrictions.

We applied the mixing rule _{m} = _{m}′ + i _{m}″ of the porous medium representing the bulk soil with porosity _{g} and _{a} = 1 embedded in the water phase with permittivity _{w}. Accordingly,

The above mixing approach is best for sandy soils at relatively high water contents, where the water phase is the host of nearly spherical air bubbles. Complex air-water geometries at low water contents might be not described properly by ^{2}g^{-1}) the above mixing model is improper due to relaxation phenomena like Maxwell-Wagner or due to the increasing fractional amount of bound water becoming a further dielectric phase to be considered [

The constant value _{g} = 5.5 + i 0.2 is reasonable for the permittivity of silicon dioxide (SiO_{2}, quartz) [_{w} = _{w}′ + i _{w}″ is computed from a fit of measurements performed at frequency _{w}(_{w}. The _{w}(

The validity of the semi-empirical model for _{w}(_{w}(

The DC-conductivity _{m} of a medium can be deduced from _{m} by expressing the spectral properties of _{m} as the sum of a conductivity term comprising _{m}, the constant _{m ∞} representing the high-frequency limit, and a Debye relaxation term. The Debye term considers the rotational relaxation frequency _{rel} ≫ 1 GHz (_{m DC} and _{m ∞}, respectively. The conductivity term describes the steadily increasing (hyperbolic) losses for frequencies _{m}″. The frequency spectrum of _{m}(_{0} = 8.85 10^{-12} F m^{-1}):

Here, relaxation losses can be neglected, because the frequencies of the electric field (100 – 160 MHz) excited by the considered capacitance sensor are much smaller than 1 GHz (_{rel}). Thus, the DC-conductivity σ_{m}(_{m}″, _{m}″(

As a matter of clarification, the fact that the above expression involves _{m} is frequency dependent, because _{m}″ is inversely proportional to

Schwank et al. [

The model geometry implemented in the numerical simulation for computing _{r}, _{z}) in radial and tangential components. The simulations were performed within the volume confined by 0 ≤ _{max} and -_{max} ≤ _{max} with _{max} = 8 inches (203 mm) and _{max} = 4 inches (102 mm) for the potential difference _{±}_{acc} = _{h} = 3.35 in accordance with measured values for _{m}′ in the range 3.35 = _{acc} ≤ _{m}′ ≤ 80. It has been proven numerically that an imaginary part ε_{m}″ ≠ 0 of _{m} does not alter the values of

The recursion proceeded until reaching the targeted change of 0.01 % of the total field energies of subsequent iterations. Meeting this stopping criterion typically required 10 grid points in the adaptive finite-element mesh. The modeled fields _{in}(_{m}′), _{acc,p}(_{m}′), _{acc,m}, _{acc,s}(_{m}′), _{m}(_{m}) and _{m}(_{m}′, _{m}″) for the permittivity range 3.35 = _{acc} ≤ _{m}′ ≤ 80.

The total capacitance _{tot} was first decomposed into two basic parallel capacitance components associated with the volumes _{in} and _{out} confined by _{acc,in} and _{acc,in} (compare

Based on the circuit analogue, the capacitance _{out} caused by the axisymmetric volume _{out} is:

As will be explained below the capacitances _{acc,m}, _{acc,s}, _{m}, _{acc,p}, and _{out} were computed from the field _{m}′. These capacitance values ultimately were used for computing _{r}(_{p}, _{acc,s}, _{m}, _{m}) from _{p} used in

Because the field caused by the asymmetric wiring and the electronic board cannot be simulated with the two-dimensional finite element software, _{in} was derived from the difference between the measured total capacitance _{tot} and the simulated capacitance _{out}. Simulated and measured values of _{tot}(_{m}′) deviate significantly from each other as shown in Schwank et al. [_{tot}(_{m}′) used in _{m}′ ≤ 78.4) and corrected for the effect of the Fluorinated Ethylene-Propylene (FEP) coating (

The capacitance _{out}(_{m}′) due to the volume outside the access tube is calculated from the field energy _{out} within the volume _{out} given by _{acc,in}. Thereby, _{out} equals the volume integral of the scalar product of the

Furthermore, _{out} is related to the electrode potential difference _{±} and the capacitance _{out}:

Combining _{out}(_{m}′) from the field _{r}, _{z}) simulated for _{m}′. The special case when _{m}′ = _{acc} = 3.35 yields the value _{acc,m} = _{out}(_{m}′ = _{acc}) = 3.63 pF.

_{m}(_{m}′) and _{acc,s}(_{m}′) can not be calculated using _{m} and _{acc,s} can still be computed from _{r}, _{z}) if understood to be the parallel connection of infinitesimal capacitances _{m}(_{ac}_{c,s}(_{acc,out}. Thus, for ring surfaces with radius _{acc,out} located at ±

The infinitesimal capacitances _{m}(_{acc,s}(_{m}(_{acc,s}(

The factor 1/2 is due to the fact that _{acc,s} is the series of two identical capacitances joined at the symmetry plane.

According to Kirchhoff s voltage law, _{m}(_{acc,s}(_{±} and the voltage _{+}_{P}_{2}(_{2}(_{acc},_{out},

_{+P}_{2}(_{r}, _{z}) from a point _{1}_{acc,in}, _{+}) at _{+} within the surface of the positive electrode to the point _{2}(_{acc,out}, _{1} with _{2}(

For calculating the surface charge density at a dielectric discontinuity, it is appropriate to separate the total volume charge density _{u} and _{p} produced by unpaired (u) and paired (p) charges accumulated via charges that can move far away from their partners of opposite sign and by charges accumulated via displacements of paired charges [_{p}_{u} of the free charges is given by the flux density _{0}_{u}

Integrating _{u} and _{p} expressed by the normal components of the fields _{1} and _{2} at both sides of the interface (

Because _{u} = 0, whereas the polarization charge density _{p} occurs at a dielectric discontinuity. Therefore, _{p} at the interface separating the access tube from the environmental medium with permittivities _{m} = _{acc} can be calculated from _{r}, _{z}) with respect to the _{p} was determined by the difference between _{r}(_{acc,out} − _{acc,out} +

The resulting infinitesimal charge _{p}·_{acc,out}_{2}(_{acc,out},

_{m}(_{m}′) and _{acc,s}(_{m}′) for _{m}′ ≠ _{acc} (see methods above for _{m}′ = _{acc}).

The parallel part _{acc,p}(_{m}′) of the access tube capacitance is calculated from the charge _{acc,z=0} that would occur on a virtual conductor at the symmetry plane _{acc,z=0} to the positive ring electrode:

Because this virtual conductor coincides with the equipotential at _{acc,z=0} = _{±} / 2, it does not affect the field _{acc,p} is the series of two identical capacitors comprised of the positive ring electrode and the virtual electrode at _{r} = 0 at _{acc,z=0} can be deduced from _{z}(_{0} · _{z}(_{acc,in} ≤ _{acc,out} of the access tube:

Values of the resistance _{m}(_{m}′, _{m}″) of a porous medium outside the access tube (soil) with conductivities _{m}(_{m}″) were derived from _{r}, _{z}) computed for _{m}′ in the corresponding volume _{out} (_{acc,out}). For this purpose a relationship between _{m} and _{m} is deduced below.

In accordance with _{R} and _{C} can be expressed by corresponding local infinitesimal resistor

The corresponding absolute values of the ohmic and the capacitive impedances |_{Rm}| and |_{Cm}| of the medium outside the sensor are:

In general, the ohmic and the capacitive impedances associated with an arbitrary arrangement of dielectric and conductive components are determined by the arising configuration of the field _{R}| / |_{C}| between the local impedances is the same as the ratio |_{Rm}| / |_{Cm}| between the ohmic and capacitive impedances caused by the volume _{out}. Thus _{m} can be expressed as a function of _{m} and _{m} using the relations

Furthermore, the geometric factor _{m}(_{m}′) of the volume filled with the medium outside the access tube can be calculated from _{m}(_{m}′):

Thus, _{m} is inversely related to both _{m}(_{m}″) and _{m}(_{m}′).

Because the water permittivity _{w}(_{m}(_{r}(_{p}, _{acc,s}, _{m}, _{m}) are not known _{r} for a dispersive medium.

The capacitances _{acc,s}(_{m}′(^{j}_{m}(_{m}′(^{j}_{p}(_{m}′(^{j}_{in}(_{m}′(^{j}_{acc,p}(_{m}′(^{j}_{acc,m} used in _{m}(_{m}′(^{j}_{m}″(^{j}^{1} is calculated with the circuit diagram elements associated with _{m}(^{0}) corresponding to ^{0} = <_{r}> = 130 MHz. Higher-order approximations ^{j}_{m}(^{j}^{-1}) at frequency ^{j}^{-1}. For a broad range of starting frequencies ^{0}, this procedure converged within three iterations to an accuracy of |^{j}^{j}^{-1}| ≤ 1 kHz yielding the final resonant frequency _{r} of the sensor encompassed in a dispersive dielectric medium.

_{m}(_{m}′(_{m}″(_{g} = 5.5 + i 0.2 and _{a} = 1 of the spherical grains and air-bubbles are assumed to be non-dispersive, so the frequency dependence of _{m}(_{w}(_{w}′(_{w}″(_{m}′(_{m}″(^{-3}. The porosity was _{ref}=20°C was chosen.

The real part _{m}′(_{m}′ / ^{-5} MHz^{-1} within the frequency range 100 MHz ≤ _{m}″(_{m}″(_{rel} ≈ 16 GHz corresponding to the relaxation frequency of non-saline pure water at _{ref} = 20°C. The imaginary part _{m}″(_{rel} the gradient _{m}′(_{m}′(_{m}″(

The resonance frequency band _{r,min} ≤ _{r,max} of the sensor with the boundaries _{r,min} = 100 MHz and _{r,max} = 160 MHz corresponds with the excited frequencies of the _{m}(_{m}″ is determined solely by _{m}.

Furthermore, the necessity of using an iterative approach to derive _{r} becomes obvious by inspecting the variability of θ _{m}″(_{r,min} ≤ _{r,max}, the maximum variation is rather pronounced: _{m}″(_{r,min}) - _{m}″(_{r,max}) ≈ 23 for _{m} calculated from _{m}′(_{r,min}) - _{m}′(_{r,max}) < 10^{-3} for _{m}′ does not require _{r} to be determined iteratively.

_{m}′ and _{m}″ computed from _{r}> = 130 MHz corresponding to an intermediate sensor resonance. The four different line types are results for liquid salinities _{m}′(^{3}m^{-3}. Bound water was ignored here, which may have affected the results at low values of

The real part _{m}′(_{m}″(_{m}′ and _{m}″, and increasing _{m}′ but increases _{m}″ (compare lines for equal _{m}′/_{m}″/^{-1} for ^{3}m^{-3} to _{m}′/^{-1} for ^{3}m^{-3}. The temperature gradients _{m}″/_{m}″ are close to zero and _{m}″/

The same set of _{m}(_{m}(_{m} increases with _{m} ≈ 400 mS m^{-1} were modeled for ^{3}m^{-3}, and _{m} evaluated here is reasonable for the electrical conductivity of field soils. However, bulk electrical conductivity values of field soils are not necessarily the exclusive result of soil-water salinity. The implied relationship between _{m} is limited to “ideal” granular media, but many soils with clay minerals exhibit enhanced electrical conductivity even for non-saline soil water. Nevertheless, the relationship in _{m}(_{m}″) remains valid.

_{acc} = 3.35 ≤ _{m}′ ≤ 80. _{tot}(_{m}′) (solid squares) are the measured values of total capacitance; and _{in}(_{m}′) values (stars) were derived from _{out}(_{m}′) values (diamonds) computed from the field energy _{out} within the volume _{out} using the relations _{m}(_{m}′) (open circles) and the serial part of the access tube capacitance _{acc,s}(_{m}′) (up-triangles) caused by the double-layer of charges at the outer access tube boundary were calculated using the relations _{acc,m} (dashed line) representing the capacitance due to the volume _{out} for _{m}′ = _{acc} = 3.35 was computed as _{acc,m} = _{out}(_{m}′ = _{acc}) = 3.63 pF. The parallel part _{acc,p}(_{m}′) (down-triangles) was computed using

The values of _{tot}, _{in}, and C_{out} increase nonlinearly with _{m}′ taking values of 12.5 pF ≤ _{tot} ≤ 26.9 pF and 8.8 pF ≤ _{in} ≤ 16.5 pF, and 4.1 pF ≤ _{out} ≤ 10.5 pF for 3.35 = _{acc} ≤ _{m}′ ≤ 80. Both _{in}(_{m}′) and C_{out}(_{m}′) increase at similar rates, even though there is no change in permittivity of the inner volume (_{acc,in}), which indicates that the electrical field inside the capacitor rings is affected by _{m}′ outside the rings and access tube. The values of _{acc,p}(_{m}′) < 0.22 pF are clearly small compared with the other parallel capacitances of the circuit diagram (_{m}′ approaching _{acc} = 3.35, the values of the two polarization-induced capacitances _{acc,s}(_{m}′) and _{m}(_{m}′) approach zero, because no double-layer of charges exists at the outer access tube boundary _{acc,out}.

Values of the geometric factor of the medium _{m}(_{m}′) (open circles) were calculated from _{m}(_{m}′) as shown in _{m}(_{m}′) varies substantially and nonlinearly over the permittivity range corresponding to soils from dry to wet conditions. Another geometric factor _{out}(_{m}′) (diamonds) associated with _{out}(_{m}′) is plotted together with _{m}(_{m}′), because _{out}(_{m}′) includes the parallel influence of _{acc,m}, which together with _{acc,s}(_{m}′) and _{m}(_{m}′) causes _{out}(_{m}′) to decrease with _{m}′. Evett et al. [^{3}m^{-3} and 0.179 m^{3}m^{-3}), which concurs with measured and simulated results in dielectric liquids [_{m}′. These studies offer experimental evidence for the fact that the relative distribution of the field energies within the sensor and within the volume proximate to the sensor changes with water content. The dependency of the computed values of _{m} and _{out} on _{m}′ corroborates these experimental findings qualitatively, but the quantitative relationship between geometric factors and the measurement volume is unknown.

_{m}(_{m}′, _{m}) evaluated for _{m} = 5, 20, 100, 500 mS m^{-1} and 3.35 ≤ _{m}′ ≤ 80. There is a dramatic decrease of _{m} with increasing _{m}′ for _{m}′ < 10 (noting the log scale for _{m}) corresponding to low water contents _{m}(_{m}′) in this range. However, as already mentioned and discussed in the next section, for _{m} < 10 Ω and for _{m} > 1000 Ω, the effects of the resistor on _{r} are asymptotic to no resistance and perfect insulation, respectively.

All of the analyses above lead to the ability to predict the joint effects of salinity _{r} as functions of the electrical resistance _{m} and conductivity _{m}, then show the predicted temperature sensitivities for a range of water contents _{m}′ and

The sensor resonant frequencies _{r} presented below were calculated from _{m} is considered as an independent variable in the relationship _{r}(_{m}). The inductance _{acc,s}(_{m}′), _{m}(_{m}′), and _{p}(_{m}_{m}′ are listed in

For a realistic range of soil permittivities _{m}′ < 30, the resonant frequencies shown in _{m}′. This behavior is in accordance with the increasing resonant frequency of a simple LC-oscillator with decreasing capacitance (compare _{m}′ the computed _{r}(_{m}) shown in _{m} and reaches asymptotically higher values for large _{m}. This emanates from the circuit diagram (_{m} → ∞ when _{m}(_{m}′) is in series with _{acc,s}(_{m}′) and a maximum value for _{m} → 0 when _{m}(_{m}′) is bypassed. Thus, the resonances of basic LC-oscillators with different operative capacitances (see _{r}(_{m}) for low and high values of _{m}.

Likewise the dependence _{r}(_{m}) shown in _{m}(_{m}′, _{m}″) was calculated from _{m}(_{m}′) given by _{m} is considered the free parameter. A corresponding asymptotic behavior is observed for _{r}(_{m}). For each _{m}′ the maximum resonance _{r}(_{m} = 0) is equal to _{r}(_{m} → ∞) representing the lossless cases. Likewise, the values of _{r}(_{m} = 500 mS m^{-1}) are almost the same as the corresponding lower-limits _{r}(_{m} → 0) obtained for a highly lossy medium. The fact that the difference _{r}(_{m} = 0) - _{r}(_{m} = 500 mS m^{-1}) is very similar to the difference _{r}(_{m} → ∞) - _{r}(_{m} → 0) for any _{m}′ shows that the range 0 ≤ _{m} ≤ 500 mS m^{-1} yields values of _{m} (see also _{m} < 1000 Ω. This theoretical sensitivity of _{r} to _{m} coincides with observations of other investigators [_{r} may be sensitive to bulk electrical conductivity in the range 0 ≤ _{m} ≤ 500 mS m^{-1}.

Normalized readings _{m}) and _{m}) calculated from _{r}(_{m}) and _{r}(_{m}) using _{r,}_{k}_{m} = ∞, _{m} = 0) and thus simply calculated using _{tot,a} = 9.93 pF and _{tot,w} = 26.92 pF for air and pure water at the reference temperature _{ref} = 20°C. These values are in agreement with the measured total capacitances presented in Schwank et al. [_{r,a} = 166.6 MHz and _{r,w} = 101.2 MHz, respectively. By virtue of definition _{m} and _{m} vary in the opposite directions of those for the corresponding _{r}/_{m} and _{r}/_{m}.

Based on the simulated sensor responses to changes in electrical conductivity shown in _{w}′ = 80) to be very large. A simple experimental check of the model response to salinity is given in _{ref} = 20°C measured in the laboratory (symbols) and simulated (line). The simulated magnitude of response underestimated the measured response, but the shape of the curve, particularly the salinity at which

Permittivities _{m}′ as the result of electrical conductance _{m} occurring in natural moist soils. The sensitivity _{m} of the measured proxy _{m} is explored below. Thereby, _{m} is expressed as:

The partial derivative ∂_{m}′, _{m}) and the corresponding derivative ∂_{m} evaluated at _{m}′, _{m}) are computed from the modeled data shown in

The values of _{m} plotted in _{m}′, which would be measured for _{m} = 0. The distortion of a measurement within the realistic range _{m}′ < 30 (bold lines in _{m}′. For 30 < _{m}′ < 80 (thin lines in _{m} decrease with increasing _{m}′ because _{m} becomes very small for high _{m}′ (see _{m} do not occur at _{m}′ = 6.54 and _{m}^{-1} where the derivative _{m} = 0.81·10^{-3} mS m^{-1} is greatest (bold dot on the dashed line in

Simulated relative changes _{m}′ of permittivities _{m} = 0) and corresponding simulations for _{m} > 0 at the same water content _{m}′, _{m}) can be expressed using

The calculations reveal rather large positive values of _{m} > 50 mS m. This clearly shows that permittivities of electrically conducting media (soils) are expected to be overestimated if measured with a capacitance sensor. Other investigators [_{m} > 500 mS m^{-1}), which may not contradict the present model at relatively high soil water contents and associated permittivities (e.g., _{m}′ > 30). However, we do not expect _{m} to affect the sensor readings much at low _{m} > 500 mS m^{-1}.

In practice the simulated _{m} on _{m}′ of the soil permittivity, which would be measured for _{m} = 0, is the proxy-quantity used to infer the soil water content _{m}′ corrected for electrical conductivity using such a correction function:

In this equation, temperature affects permittivity via its effect on electrical conductivity, but the direct effect of temperature on water permittivity is not included as it is not a sensor effect.

The temperature dependence of _{r}(_{m}″(_{m}(_{m}″(_{r} must be calculated iteratively following the procedure illustrated in

_{r}(_{g}^{3} m^{-3}. The corresponding predicted resonant frequencies are in the range of 109 MHz ≤ _{r} ≤ 136 MHz. This is approximately the same as the range from _{r}(_{m} → ∞) to _{r}(_{m} = 0) calculated for 6 ≤ _{m}′ ≤ 30 (

Depending on _{r}(_{m}″ are low, leading to values of _{m} which exceed the upper limit of the sensitivity range (_{r}. Under these circumstances, _{r} values are determined predominantly by the total capacitance _{tot}(_{m}′) which increases with _{m}′ (_{w}′/^{-1} [_{r}(_{r}/_{m} > 0 associated with charged particles. This difference points out a limitation of the simplified model, such that one needs to consider _{m} rather than merely

As _{r}(^{3} m^{-3} ≤ ^{3} m^{-3}. This behavior is in accordance with increasing losses leading to _{m} within the sensitive range (10 Ω < _{m} < 1000 Ω) affecting _{r}. Consequently, the gradients _{r}(_{m}/_{m}/_{m}/_{r}(_{r} decreases with decreasing _{m} (_{r}(_{r,a} = 166.6 MHz and _{r,w} = 101.2 MHz respectively. As a result, _{r}/

Finally, we estimated _{m}′ of the real part of the medium permittivity computed directly from the mixing model (_{m}′ simulated for ^{3} m^{-3} , _{g}_{ref} = 20°C and the corresponding simulation for _{ref}. As an extension to

Thereby, the temperature sensitivity of the proxy value

Again,

The real part of the medium permittivity _{m}′ used to estimate the relative deviation _{m}′ is computed from the dielectric mixing model _{ref} = 20°C, S = 0 ppt and _{r}> = 130 MHz. The frequency-dependence of these _{m}′ for 0 ppt ≤ _{r,min} ≤ _{r},_{max} is minor (see

The resulting values computed for the four water contents _{ref} and for _{m} is almost zero in this case for all _{ref} and for _{ref} is not caused by the effect of the medium conductivity on the sensor output

Consequently, the relative deviations _{m}′ with increasing _{w}′/^{-1} [_{ref} because the sensor output _{m} as a function of _{m}′/_{m} with _{ref} meaning that the negative gradients _{m}′^{3} m^{-3} the correspondingly high values _{m} exceed the sensitive range of _{m}′/^{3} m^{-3}), the positive effect of

The capacitance sensor investigated here is used to infer soil water content _{r} (after normalization by air and water counts) or via estimation of the real part of permittivity _{m}′ as a proxy for _{m}′). Here, we refined the theoretical understanding and quantification of the sensor responses to variable _{m}. This was achieved by combining a dielectric mixing model, electrical circuit analogue, and finite-element numerical model of the electrical field in and around the sensor for a range of environmental conditions. Within the mixing model, the complex permittivity of saline water was based on previous empirical studies [

The frequency dependence of _{m} is confirmed and quantified by the mixing model, showing that _{m}′ is relatively insensitive to _{m}″ can be highly sensitive to both _{r} < 160 MHz), this sensitivity is due to conductivity losses rather than high-frequency relaxation losses. The computed values of _{m}′(_{m}″(_{m}″/_{m}, given knowledge of the multivariate response of _{m}(_{m} with _{m} in the electrical circuit analogue. _{m}(_{m}′) was shown to be highly sensitive to _{m}′ only for _{m}′ < 10 (

In addition to _{m}, various components of the total sensor capacitance as functions of _{m}′ were computed from numerical simulations of the electrical field, where _{m}(_{m}′) was found to be nearly linear (_{m}(_{m}′) of the medium of interest was shown to be highly variable (0 < _{m} < 0.75) and nonlinear for _{m}′ < 25 (

Given all circuit component values as functions of the environmental variables (_{m} or _{m}. Values of _{m} for all values of _{m}′(_{m}′ = 6.54 (_{m}. Estimation errors for _{m}′ were computed as functions of _{m} and _{m}′ within a range of almost ± 10 % (

The physics-based theory presented here is of general validity for understanding the sensitivity of capacitance sensors to the electrical conductivity of the surrounding media. Because the soil electrical conductivity also depends on temperature, the resulting modeled instrumental sensitivity has to be taken into account before interpreting thermal effects in measured data. Future studies may use the present approach and results to identify functions to correct for temperature and salinity effects on the estimated soil water content. Such correction functions can then be compared with laboratory and field data. Based on the potential errors in frequency-based estimates of permittivity computed here, field data measured with such capacitance sensors will not provide reliable absolute estimates of water content without such correction functions.

We gratefully acknowledge Christian Mätzler (University of Bern) for his scientific advice, Hannes Flühler and Rainer Schulin (ETH Zürich) for facilitating and encouraging the transatlantic cooperation between authors, and Sander Huisman (Forschungszentrum Jülich) and Robert Schwartz and Steven Evett (USDA-ARS, Bushland, Texas) and an anonymous reviewer for helpful comments on the manuscript.

Circuit diagram used for modeling the frequency response of the capacitance sensor (a and b).

Total capacitance is composed of _{in} and _{out} due to volume inside and outside the inner access tube boundary. _{acc,p} and _{acc,s} are the parallel and the serial parts of the access tube capacitance, respectively. _{acc,m} accounts for the capacitance of the volume outside the inner access tube boundary if the dielectric contrast between the medium and the access tube disappears. The parallel connection of _{m} and _{m} is the electrical analogue of the conductive medium (soil) outside the access tube. The symbolic justification of _{acc,m} is sketched in panel c.

Flow-chart of the iterative approach for modeling the sensor resonance _{r} determined by the sensor properties (_{m} of the porous solid-water-air medium (soil) surrounding the probe. Relevant equation numbers are indicated (in parentheses).

a) Sensor with symbolized electric field

Schematic diagram of the structure of the porous medium used for modeling the effective medium permittivity _{m} representing the moist soil surrounding the sensor access tube. The grains with permittivity _{g} and the air-bubbles with permittivity _{a} are approximated by spherical inclusions in water with permittivity _{w}.

Frequency spectrum of _{m}(_{m}′(_{m}″(_{g}_{ref} = 20°C, ^{3}m^{-3}. The frequencies _{r,min} and _{r,max} are the boundaries of the resonance frequency band of the sensor.

a) Computed real parts _{m}′(_{m}″(_{g} = 5.5 + i 0.2) and air-bubbles (_{a} = 1) embedded in water, computed for ^{3}m^{-3}.

Calculated electrical conductivities _{m}(

Capacitances used in the circuit analogue for computing sensor resonance _{r}. _{tot} was based on measurements, _{in}, _{out}, _{acc,p}, _{acc,s}, and _{m} were calculated from the _{acc} ≤ _{m}′ ≤ 80, and _{acc,m} = 3.63 pF represents the capacitance for _{m}′ = _{acc} of the volume _{out}. b) Geometric factors _{m} and _{out} derived from _{m} and _{out}. c) Resistance _{m} of the medium derived from _{m} using (27) for four different conductivities _{m} = 5, 20, 100, 500 mS m^{-1}.

a) Resonance frequencies _{r} versus resistance _{m} of the medium and b) _{r} versus medium conductivity _{m}. c) Normalized sensor reading _{m} and d) _{m}.

Measured (symbols) and simulated (line) values of normalized sensor reading _{w} of water with salinity _{ref}

a) Gradients _{m} representing the sensitivities of measured permittivities _{m}. b) Simulated relative overestimations _{m}′ of measurements taken in conductive media.

a) Computed resonant frequencies _{r}(_{m}(_{m}(_{r}.

Relative deviations _{m}′ between simulated _{ref} = 20°C and _{ref}. The value _{m}′ is computed from (8) with _{ref} = 20°C, _{g}

Values of capacitances _{acc,s}(_{m}′), _{m}(_{m}′), and _{p}(_{m}_{in}(_{m}_{acc,p}(_{m}_{acc,m} + _{el} used in (5) for computing the resonant frequency _{r} of the circuit diagram assumed to represent the sensor resonance.

_{m}′ [-] |
_{acc,s} [pF] |
_{m} [pF] |
_{p} [pF] |
---|---|---|---|

| |||

4 | 1.22 | 0.52 | 12.83 |

6 | 3.38 | 2.06 | 13.39 |

6.54 | 3.75 | 2.47 | 13.51 |

8 | 4.50 | 3.46 | 13.97 |

10 | 5.19 | 4.93 | 14.62 |

12 | 5.66 | 6.33 | 15.18 |

15 | 6.13 | 8.39 | 15.91 |

20 | 6.61 | 11.84 | 16.77 |

30 | 7.09 | 18.65 | 18.03 |

40 | 7.34 | 25.43 | 18.75 |

60 | 7.59 | 38.98 | 19.74 |

80 | 7.71 | 52.53 | 20.19 |