Dielectric models for moisture determination of soils with variable organic matter content

An accurate calibration function relating soil dielectric permittivity to its volumetric water content is essential for reliable measurements of soil moisture with the use of dielectric sensors. In the operational frequency range of popular capacitance and impedance sensors, various soil properties may influence the relation between the soil moisture and dielectric permittivity, which may worsen the measurement accuracy of a sensor. The aim of this paper is to investigate the relations among the organic matter content, dry bulk density, volumetric water content and dielectric permittivity in the 10 – 500 MHz frequency range, which covers the frequency range of operation of common soil moisture dielectric sensors operating in the frequency domain. Samples of soil material of belonging to sand, sandy loam and silt loam texture classes with varied organic matter content were measured with the use of a coaxial-like open-ended seven-rod probe connected to a vector-network-analyzer. A new simplified dielectric model is presented in order to accurately describe the obtained spectra. Next, several models connecting the real part of dielectric permittivity and volumetric water content, taking account of the impact of bulk density, were examined. It was found that two linear segmented models fitted to the obtained data better in the case of soil materials of sandy loam and silt loam textures. The best segmented model exhibited RMSE of about 0.023 m 3 m (cid:0) 3 in comparison to about 0.04 m 3 m (cid:0) 3 in the case of a simple linear model without density correction.


Introduction
Accurate determination of soil volumetric water content θ is important in many fields such as agriculture, environmental monitoring, hydrological and climate studies (Robinson et al., 2008). Many commercial and experimental field soil water content sensors operate on the basis of dielectric measurements (Robinson et al., 2008;Vaz et al., 2013;Majcher et al., 2020;Hardie, 2020). Generally, these devices determine apparent dielectric permittivity ε a in the time domain (time-domain reflectometry, TDR or time-domain transmissometry, TDT) or relative permittivity in the frequency domain (frequency-domain reflectometry, FDR). Depending on the specific measurement method and the type of the sensor, their frequencies of operation lie in the range from several MHz to several GHz. Typically, FDR sensors operate at frequencies not higher than 300 MHz.
Complex relative dielectric permittivity ε * consists of the real ε ′ and imaginary ε ′′ parts: where j is the imaginary unit. In the case of soil, ε * values depend mainly on θ and also on the frequency of the electric field. The relation between soil dielectric permittivity and its water content is frequently modeled either by a polynomial function (like the Topp's equation (Topp et al., 1980), obtained for TDR-measured apparent permittivity ε a ) or by a linear equation dependent on the square root of ε a or ε ′ , in the general which can be applied in a wide frequency range of ε ′ measurement (Skierucha and Wilczek, 2010;Vaz et al., 2013;Szypłowska et al., 2019).
In the case of impedance/capacitance sensors providing a voltage output, a direct relation between the sensor's readout and θ also can be used. The relations between θ and ε ′ , ε a or the raw sensor output are also called calibration functions.
In the frequency range of operation of the soil moisture dielectric sensors, especially the FDR sensors, ε * may be affected not only by water content, but also by several other soil-specific properties, such as temperature, density, salinity or clay content, which could deteriorate the soil-moisture measurement accuracy of dielectric sensors if the calibration model does not account for their influence. Dielectric dispersion stemming from the bound water relaxation and other interphase phenomena such as polarization of the electrical double layer and the Maxwell-Wagner effect influence the soil dielectric-permittivity spectrum in the MHz range (Kelleners et al., 2005;Chen and Or, 2006;Wagner et al., 2011;Loewer et al., 2017). These phenomena specifically depend on the soil physical and physicochemical properties. Therefore, manufacturers of low-frequency devices often provide several calibrations valid for various soil types, usually differentiating among organic and mineral soils or by soil texture (Vaz et al., 2013;Delta-T Devices Ltd, 2017;Systems, 2018). Custom calibrations also can be performed by the user. It was observed that soil bulk density also affects the relation between volumetric water content and soil dielectric permittivity. The influence of density on TDR-measured apparent permittivity and its implication on θ(ε a ) relation was investigated in Malicki et al. (1996), where the following relation dependent on soil dry bulk density ρ was In terms of ̅̅̅̅ ̅ ε a √ (θ, ρ) this formula is written as follows: ̅̅̅̅ ̅ ε a √ (θ, ρ) = 0.819 + 0.168ρ + 0.159ρ 2 + (7.17 + 1.18ρ)θ.
This equation takes the form of a monotonically increasing function of θ and ρ. Investigating the influence of ρ on soil dielectric permittivity is of practical importance for accurate θ measurement, since ρ for a given soil in a given field may change in time, e.g. under the influence of a tillage method (Jaskulska et al., 2020). Soil dielectric permittivity is also affected by the organic matter (OM) content (Jones et al., Jan 2002;Liu et al., 2013;Bircher et al., 2016;Park et al., 2019). Therefore, accurate determination of θ on the basis of soil dielectric properties, with the use of in situ sensors as well as microwave remote sensing, requires detailed knowledge on the influence of OM content on soil dielectric properties. In Liu et al. (2013) the impact of the soil OM on dielectric permittivity in 0.5 -40 GHz range was examined with the use of an open-ended probe. It was reported that due to the OM ability to bound water, the free water fraction in the soil is decreased, which in turn decreases ε ′ in the GHz region. Also, the decrease of bulk density with the increase of the OM content was linked to the decrease in soil permittivity. The influence of the OM content (OM, in %) on dry bulk density ρ (in g cm − 3 ) examined in Yang et al. (2014) resulted in the following relation, with reported R 2 = 0.5859: The change in soil aggregation caused by OM, influencing bulk density, porosity and the wilting point, in addition to the change in bound water/ free water fractions, is indicated in Park et al. (2019) as a source of the decrease in soil dielectric permittivity in the frequency range of operation of the TDR devices. The relation from Eq. (5) was applied in Park et al. (2019) in the dielectric mixing model accounting for the influence of soil organic matter. On the other hand, it was reported that OM tends to increase ε ′ in the MHz range (Bobrov et al., 2010).
Accurate retrieval of soil moisture from microwave remote sensing data in regions with organic soils also requires research on the impact of OM content on soil dielectric permittivity at GHz frequencies. In Mironov et al. (2010), a dielectric mixing model for an Arctic soil with very high organic matter content was developed in the frequency range 1 -16 GHz. In , a dielectric mixing model for an Arctic soil with about 50% of OM content was developed in the 0.05-15 GHz frequency range that includes low-frequency range of operation of many FDR soil moisture sensors in comparison to the previous model. The work of  presented a dielectric mixing model for tundra organic-rich soil (80-90% of OM content), at a single frequency of 1.4 GHz used by SMOS (Soil Moisture and Ocean Salinity) satellite. In Bircher et al. (2016), complex dielectric permittivity data measured in a resonance cavity at 1.26 GHz were used for evaluation of dielectric models for a wide range of organic soils. In Park et al. (2019), a dielectric mixing model taking account of the OM content was developed and validated with the use of dielectric permittivity measured by TDR meters.
The aim of the present paper was to analyze the relations between θ and ε ′ measured in the frequency range from 10 to 500 MHz of mineral soils with various OM content. This frequency range covers the operational frequencies of common FDR soil moisture measurement devices. The obtained spectra were modeled with the use of a new simplified dielectric model CC-ABC, developed in order to account for the out-ofband free-water dielectric relaxation pole with the use of the restricted frequency data, as well as the dielectric dispersion stemming from the low-frequency interfacial phenomena. The relations between θ and ε ′ for soil materials of various OM content in the stated frequency range were examined, compared to the existing relations from the literature and modeled, taking into account the impact of soil bulk density. Finally, a new segmented model was formulated, which exhibited the best fit to the experimental data.

Soil material and preparation of samples
Materials used in the experiment consisted of river sand, black soil, garden soil and an artificial soil mixture of river sand with black soil, 50% by mass. For each soil material, the particle-size distribution was determined with the use of the laser diffraction method by a diffractometer Malvern (UK) Mastersizer 2000 with a Hydro G dispersion unit with soil aggregates dispersed by an ultrasound probe (Bieganowski et al., 2018;Bieganowski et al., 2013). Total organic carbon (TOC) was measured by a TOC V-CPH Analyzer (Shimadzu Corporation, Japan) with a solid sample combustion unit SSM-5000A. Texture and total organic carbon content of the soil materials are presented in Table 1.

Table 1
Texture (fractions of sand, silt and clay particles (in %) and the texture class according to USDA) and total organic carbon content (TOC, in %) of the measured soil materials.

Material
Sand ( Before the measurement, soil materials were air-dried and sieved by a 2 mm sieve. After air-drying, the soil material was stored in an airconditioned room at 20 ± 1 • C. Samples for 13 measurement series were preparedfour series from the river sand, black soil, garden soil and soil mixture without any additions, and nine series from the river sand, black soil and soil mixture with additions of OM of 2%, 5% and 10% of the total air-dry sample mass. The OM was added in the form of plant leaves dried and grounded in a mortar. The total OM content (OM t ) of the samples depended also on the initial amount of OM (OM i ), calculated by multiplying the TOC value by 1.72 (Stenberg et al., 2010). The values of OM t as a percent of dry sample mass are presented in Table 2.
Each measurement series consisted of five samples of various water content from air-dry to near saturation, with the exception of garden soil, for which there were seven water content levels, from air-dry to near saturation. Samples were prepared by mixing the prepared soil material with pure deionized water to the desired water content. All samples were then packed into a glass cylindrical container with sample volume of 30.8 cm 3 . Care was taken to ensure that there would be no visible air spaces nor significant density variations in the material under test, which was achieved by loading each sample layer by layer and pressing gently with a small flat rubber press. The density of the samples ranged between 0.86 to 1.89 g cm − 3 , with a mean value of 1.32 g cm − 3 . After the dielectric measurement, gravimetric water content was determined with the use of oven-drying at 105 • C for 24 h, and the volumetric water content was finally calculated. This standard temperature of oven drying is acceptable also for soils containing significant amounts of organic matter, according to O' Kelly and Sivakumar (2014). Each sample of a given soil material and a given water content was prepared and loaded independently, therefore the samples in a measurement series differed in dry bulk density, which ranged from 0.54 to 1.73 g cm − 3 , with a mean value of 1.04 g cm − 3 .

Dielectric spectra measurement
All complex dielectric permittivity spectra were measured with the use of a seven-rod open-ended-type probe presented in Szerement et al. (2020), which was based on a previous construction of Szerement et al. (2019). The probe was equipped with a central rod and six outer rods protruding from a polyacetal positioning element by 15 and 20 mm, respectively. The capacitance in air of the sensing element of the probe was about 0.4 pF. The dimensions of the probe are presented in Fig. 1. The probe was connected to a vector-network-analyzer PNA-L N5230C (Agilent Technologies). In Szerement et al. (2020), the performance of the probe was verified in the frequency range up to 500 MHz. In the present study, the measurements of the complex reflection coefficient were conducted in the frequency range from 10 to 500 MHz with a step of 10 MHz. Calibration of the probe and extraction of the complex dielectric permittivity spectra of the measured samples were performed with the use of the Open-Water-Liquid calibration procedure (Bao et al., 1994;Wagner et al., Feb. 2014) implemented as in Szerement et al. (2020), with methanol as the calibration liquid. The probe calibration was performed on each measurement day and verified on isopropanol. All measurements were taken in an air-conditioned laboratory at 20 ± 1 • C.

Dielectric spectra modeling
In order to describe the dielectric relaxation in the samples under test, the obtained spectra were modeled at first with the use of a twopole Debye and a Cole-Cole-Debye model, which can be written in general form as where ω = 2πf and f is the frequency of the electric field, ε ∞ is the high frequency permittivity limit, σ b is the bulk electrical conductivity, ε 0 is vacuum permittivity, Δε n and τ n are relaxation amplitude and relaxation time of an n-th pole, respectively, β n is the stretching exponent of an n-th pole (β n = 1 for Debye-type poles). The relaxation time of an n-th pole is related to the relaxation frequency f rn of an n-th pole according to relation: The high frequency free-water dipole relaxation was described by a Debye pole, while for the low-frequency relaxation phenomena two variants were tested: (i) with one low-frequency Debye pole, the model will be designated D-D, and (ii) with one low-frequency Cole-Cole pole, the model will be designated CC-D. The poles in D-D and CC-D models are numbered from high to low frequencies.
The upper frequency limit of the measured spectra is 500 MHz. The free-water-dipole relaxation frequency of about 17 GHz lies far beyond the available band, which gives concern over the accuracy of the fitted parameters. Therefore, in order to accurately describe the obtained spectra, a simplified model for band-limited data could be useful, since we do not have sufficient information about the out-of-band pole. The developed model, which includes a Debye or Cole-Cole in-band poles and an out-of-band pole with simplified description, can be written in the form: where A, B, and C are parameters describing the high-frequency out-ofband free-water-dipole relaxation pole. Again, two versions of the model were tested: (i) with one low-frequency Debye pole, the model will be designated D-ABC, and (ii) with one low-frequency Cole-Cole pole, the model will be designated CC-ABC. The derivation of the general multipole simplified model and the method of the estimation of the model parameters was presented in the Appendix.

Overview of the data analysis
The first step of the data analysis was to examine the relations among the organic matter content, dry bulk density, real part of soil dielectric permittivity and volumetric water content. Next, at each frequency Table 2 Total organic matter content OM t in the respective measurement series (in %).  A. Szypłowska et al. separately, eight models connecting soil moisture and the real part of dielectric permittivity were examined, based on the form of Eq.
(2) and the Malicki-Plagge-Roth Eq. (4). This equation, which models the influence of bulk density on soil dielectric permittivity, can be generalized in the following polynomial form: where ε may stand as ε a or ε ′ for data obtained in the time-or the frequency domain, respectively. An analogous relation θ( ̅̅ ̅ ε √ , ρ) can also be considered: These equations were then fitted to the data with the use of linear regression. Next, new segmented models were proposed and fitted to the data using nonlinear-least-squares Levenberg-Marquardt algorithm.
The calculations were performed in the Matlab 2019a environment (MathWorks Inc., 2019).

Dielectric spectra measurement and modeling
Dielectric spectra of measurement series of river sand and river sand with 5% OM are presented in Fig. 2. As expected, ε ′ rose with the increase in θ. Adding extra OM modified the available range of water content. OM also increased the low-frequency dispersion, especially for river sand samples, indicated by a visible rise in ε ′ at low frequencies.
This effect was especially evident at frequencies below about 50 MHz. Above 100 MHz, the real part of soil dielectric permittivity remained virtually constant for a given sample of all tested soil materials. Also, the values of ε ′′ at low frequencies increased with the increase of OM content. For some samples, especially those of high water content, a slight rise in ε ′ at frequencies above 400 MHz is noticed, which is also visible in Fig. 2. Although the probe performance has been previously verified for frequencies up to 500 MHz, this behavior might be caused by a lowfrequency tail of an artifact occurring above the operational frequency range of the probe . Therefore, the further analysis of the relations between ε ′ and θ at various frequencies will be conducted with the use of ε ′ obtained from the fitted model, which mitigated the influence of non-physical artifacts. The model also effectively smoothed the spectra at all frequencies, lessening the impact of random measurement errors. The four dielectric models were fitted to the spectra. The maximum and minimum values of standard errors of the determined parameters, as well as minimum, maximum and mean values of root-mean-squared error (RMSE) for each model are presented in Table 3. All of the models under investigation fitted quite well to the data. However, models with the low-frequency Cole-Cole pole fitted better, with lower mean RMSE and maximum RMSE than the models with the lowfrequency Debye pole. The D-D model performed the worst in the terms of RMSE. On the other hand, the models with the Cole-Cole pole exhibited generally higher standard errors of estimated parameters than the models with a Debye low-frequency pole, which is explained by the addition of one more parameter (β) to the already quite well fitted models. In the terms of RMSE, CC-ABC and CC-D models were comparable. However, as expected, the high frequency pole parameters in the CC-D model were estimated with high standard errors. In the case of the CC-ABC model, parameters A and B were estimated with reasonable standard errors, while standard errors for parameter C were very high. Especially, the A parameter, whose physical meaning corresponds to the sum of ε ∞ and Δε 1 of the high-frequency pole of the D-D and CC-D models, was estimated with very low standard errors (and relative standard errors). Fig. 2 depicts sample dielectric spectra with fitted CC-ABC model, which was regarded as optimal. The measured raw ε ′ at selected frequencies, the ε ′ obtained by fitting the CC-ABC model and basic properties of all measured samples were presented in the Supplementary Table. 3.2. Influence of organic matter content and density on the real part of dielectric permittivity  at 20 and 100 MHz were found. Also, no direct significant correlation between the OM content and dielectric permittivity, independent of θ, was observed. The relation between organic matter content and dry bulk density was also examined and presented in Fig. 6. For river sand and soil mixture, ρ decreased with the organic matter content. In case of black soil, the average value of ρ sligthly increased after the first addition of extra OM and remained virtually constant with further additions. A linear function was fitted to the data for all soils combined: with the coefficient of determination R 2 = 0.60 and the root-meansquared error RMSE = 0.20. As in Eq. (5) from Yang et al. (2014), soil bulk density decreases with the OM content, but the obtained parameters slightly differ from those in Eq. (5).

Table 3
Minimum and maximum values of standard errors and median relative standard error (in %) of estimated parameters, and maximum, minimum and mean values of RMSE obtained for the four models fitted to all measured dielectric spectra. Values in square brackets are the estimated parameter values for which the respective standard errors were obtained. √ at 20 MHz vs. volumetric water content θ of all tested samples. The shape of the markers of the data points refer to the tested soils and the color of the markers correspond to the total OM content of the samples..

Relations between volumetric water content and the real part of dielectric permittivity
The relations between θ and ̅̅̅ ̅ ε ′ √ at four frequencies of 20, 50, 100 and 450 MHz for all measured samples at all OM content levels were presented in Fig. 7. At each examined frequency, a linear model in the form of Eq. (2), which will be referred to as the T1 model, was fitted to the data. Also, the reference calibrations from the literature were compared to the data measured by the seven-rod probe in the present study. These reference calibrations included relations obtained with the use of a coaxial transmission-line cell at all four frequencies , calibration of an experimental FDR sensor (Skierucha and Wilczek, 2010) presented at 450 MHz graph and the calibrations of three commercial soil moisture sensors: WET2 at 20 MHz (Delta-T Devices Ltd, 2019), HydraProbe at 50 MHz (Systems, 2018) and ML3 ThetaProbe at 100 MHz (Delta-T Devices Ltd, 2017). The calibrations for WET2 sensor for organic and sandy soils and for ML3 ThetaProbe for organic soils agree well with the obtained data. The calibrations for WET2 and ML3 ThetaProbe sensors for mineral soils, as well as the general calibration for the HydraProbe and for the experimental FDR sensor would underestimate the water content, to a varying degree. The highest discrepancy between the measured and modeled θ at the given In order to model the relation between dielectric permittivity and volumetric water content of the measured samples, and to examine the effect of density, eight models were fitted to the experimental datatwo models being linear functions ̅̅̅ ̅ ε ′ √ (θ) and θ( ̅̅̅ ̅ ε ′ √ ) (model labels: E1 and T1, respectively), three models being of the general form of Eq. (9): with non-zero B 1 parameter (E2 model), non-zero B 1 and C parameters (E3 model) and non-zero all parameters of Eq. (9) (E4 model), and three models being of the general form of Eq. (10): with non-zero b 1 parameter (T2 model), non-zero b 1 and c parameters (T3 model) and non-zero all parameters of Eq. (10) (T4 model). The frequency dependence of the root-mean-squared error of fitting RMSE and the coefficient of determination R 2 for the investigated models were presented in Figs. 8 and 9, respectively. It occurred that the best fits of the tested models were obtained at frequencies between 49 and 65 MHz. Description of the models, fitted parameters and statistics for ̅̅̅ ̅ ε ′ √ obtained at an arbitrary frequency of 60 MHz, were presented in Table 4. The frequency of 60 MHz was chosen because it was close to the best-fit frequency for all tested models.
Based on the obtained R 2 values, models in the form θ( T2, T3 and T4) performed better than respective models ̅̅̅ ̅ ε ′ √ (θ, ρ) (E2, E3 and E4), while models T1 and E1 were equivalent. The inclusion of ρ terms significantly improved the R 2 and RMSE of the fit. However, the inclusion of the quadratic term ρ 2 improved the fit only slightly, while the   respective parameters B 2 and b 2 exhibited high p-values (greater than 0.05). Therefore, model T3 was regarded as optimal for θ modeling, having satisfactory performance and with all parameters statistically significant. Fig. 10 presents the comparison of the volumetric water content estimation by T1 and T3 models at 60 MHz. The T3 model has significantly lower RMSE of 0.027 m 3 m − 3 than the T1 model with RMSE of 0.048 m 3 m − 3 , which corresponds with the positioning of the points related to the T3 model visibly closer to the 1:1 line on the graph than in the case of the T1 model. As a result of the dependence of ε ′ on frequency, the parameters of the fitted models also depended on frequency. The frequency dependence of the parameters of E2 and E3 models was depicted in Fig. 11. For both models, parameters A 0 and B 1 remain virtually constant with the exception of slight change at low frequencies.
A 1 parameter of both models and C parameter of the E3 model decreased with the increase in frequency, especially at frequencies below 100 MHz. For the E2 model, parameters A 1 and B 1 were positive in the whole frequency range, which confirmed that Eq. (9) fitted to the data was a monotonically increasing function of both θ and ρ as in (Malicki et al., 1996). In the case of the E3 model, the C parameter was negative, however, the value of the B 1 parameter for the E3 model enabled retaining the increasing character of Eq. (9) with respect to ρ in the whole examined range of volumetric water content. The frequency dependence of the parameters of the T3 model, which was the most promising model regarding θ estimation, was presented in Fig. 12. Additionally, parameters of the fitted models T1, T2, and T3 at selected frequencies, including the operational frequencies of dielectric sensors, the calibrations of which were used for reference, were presented in Table 5. The greatest changes with frequency were observed below 50 MHz. At frequencies higher than 100 MHz, the parameters' values changed very little with frequency.

Segmented linear models
Examination of the relations between soil dielectric permittivity and volumetric water content presented in Fig. 7 indicated that especially at low water content levels, there is a deviation from the linear relation of Eq.
(2) for all soil materials except the river sand. A hypothesis was formulated that a relation in the form of a segmented line with a breakpoint ( ̅̅̅̅ ε ′ c √ ,θ c ), presented in Eq. (12), might better model the θ(ε ′ ) relation: Because both lines have the common breakpoint, the equation above may also be reformulated with explicit use of the breakpoint permittivity in the second relation as follows: with four independent parameters: a s1 , a s2 , b s1 and the breakpoint value ̅̅̅̅ ε ′ c √ . We refer to this model as the segmented model. The presence of a characteristic water content point, at which a change in the slope of the relation between the TDR-measured dielectric permittivity and volumetric water content, was observed for degraded peatland muck soil in (Gnatowski et al., Dec 2018).
The segmented model was fit to all data series obtained for soil mixture, black soil and garden soil at all examined frequencies, with the use of a custom Matlab procedure, which estimated simultaneously the breakpoint and the rest of the model parameters. This procedure used Table 4 Coefficient of determination R 2 , root-mean-squared error of fitting RMSE, values of the fitted parameters with their standard errors and p-values for the tested models at 60 MHz (all soils).  Fig. 10. Volumetric water content estimated by T1 and T3 models (θ model ) at 60 MHz vs. volumetric water content obtained from oven-drying (θ grav ). Dashed line represents 1:1 relation.
nonlinear least squares fitting with constraints. The constraints were used to enforce the horizontal location of the breakpoint between the maximum and minimum value on the horizontal axis. The graph obtained for data at 20 MHz was presented in Fig. 13. The linear T1 model fitted quite well to the measurement series of the river sand only with RMSE = 0.020 m 3 m − 3 and R 2 = 0.979. On the other hand, the linear T1 model fitted to the measurement series of soil mixture, black soil and garden soil with RMSE = 0.036 m 3 m − 3 and R 2 = 0.957. At the frequency in question, the T1 model fitted to all data with RMSE = 0.049 m 3 m − 3 and R 2 = 0.917. The segmented model performed better than the linear T1 model fitted to all data and the linear T1 model fitted to the data without the river sand measurement series as well, with RMSE of the fit of the segmented model of 0.027 m 3 m − 3 and R 2 = 0.978. An inclusion of the linear term dependent on soil dry bulk density has also been considered. The segmented model with the density term was formulated as follows: which can be rewritten in the form analogous to Eq. (13): with five independent parameters: c s1 , c s2 , d s1 and the breakpoint value ̅̅̅̅ ε ′ c √ . This model was also fitted to all data with the exception of measurement series pertaining to the river sand. The coefficients of determination of the fitted segmented model and segmented model with density, as well as models E1-E4 and T1-T4 fitted to data without river sand measurement series were presented in Fig. 14(a). For all models, the highest R 2 values were obtained in the low-frequency range, below 50 MHz. At higher frequencies, coefficient of determination decreased with the increase in frequency. Models T1 and E1 were equivalent.    Models from the T group exhibited slightly better fit than the respective models from the E group. Models T3 and T4, had virtually identical R 2 , as well as models E3 and E4, which confirms that adding the quadratic density term was not necessary. The segmented model, without the inclusion of density, exhibited slightly better fit to the data than the best T3 and T4 models accounting for the influence of the density. However, inclusion of the linear density term in the segmented model with density provided the highest R 2 > 0.98 in the whole frequency range. The segmented model with the density term also had the smallest RMSE of the fit (below 0.025 m 3 m − 3 ) in comparison to the segmented model and T1-T4 models, as depicted in Fig. 14(b). The segmented model without the density term performed slightly better also in terms of RMSE than the best models from the T group. This might indicate that the segmented model could be preferred when the data on dry bulk density for a given soil are not available. The performance of the segmented model and segmented model with the density term at an arbitrary frequency of 20 MHz was further depicted in Fig. 15. As expected, points representing the segmented model with density were positioned closer to the 1:1 line than the points pertaining to the segmented model. Parameters of the both examined segmented models depended on the frequency, which was depicted in Fig. 16 and in Table 6. The highest changes in frequency of the parameters' values were observed at low frequencies below 50 MHz, while above 100 MHz the parameters were virtually frequencyindependent. The breakpoint value of ̅̅̅̅ ε ′ c √ corresponded to the soil dielectric permittivity value in the range of about 4.6-5.2 in the case of the segmented model and 4.8-5.5 in the case of the segmented model with density in the frequency range in question, which corresponded to the volumetric water content of about 0.2 m 3 m − 3 . In Gnatowski et al. (Dec 2018) the breakpoint was obtained at about 0.6 m 3 m − 3 . However, the peatland muck soil examined in that paper significantly differed from the soil material examined in the present study, e.g. in terms of bulk density. Also, the applied model differed in the choice of an independent variable (apparent dielectric permittivity vs. the square root of permittivity in the present study). In our previous paper utilizing soil complex dielectric permittivity spectra obtained with the use of a coaxial transmission-line cell, no breakpoint was observed . However, the samples examined in the previous study were more dense and there was only a small number of samples of dielectric permittivity lower than 5, which possibly prevented the observation of a potential breakpoint. The lack of observation of the breakpoint in studies in soil moisture sensor calibrations, e.g. in Vaz et al. (2013), might indicate that this is a density-dependent phenomena, highly sensitive to a sample loading procedure and/or soil disturbance during the probe insertion, which needs further research.

Conclusions
In the present study, the relations between the real part of soil dielectric permittivity and volumetric water content were examined in the 10-500 MHz frequency range for samples of four soil materials with organic matter content from 0.1 to 21.1%. Before the analysis, the dielectric spectra were modeled with four dielectric models, including a new model with low-frequency Cole-Cole pole and a simplified highfrequency polethe CC-ABC model, which was found to be optimal. It was confirmed that the organic matter content influenced soil dry bulk density. The examined θ( ) and ̅̅̅ ̅ ε ′ √ (θ) models indicated that taking into account the influence of soil dry bulk density on the relation between volumetric water content and the real part of dielectric permittivity significantly improved the fit of the calibration functions. However, the usage of these models in the field may not be possible, if dry bulk density of a tested soil is unknown.
It was observed that the river sand samples exhibited linear θ( ̅̅̅ ̅ ε ′ √ ) relation. However, in the case of other soil materials of sandy loam and silt loam textures, models consisting of two linear segments with a common breakpoint exhibited a better fit, especially after including the density-dependent term. Therefore, the segmented model with density is recommended for silt loam and sandy loam textures in the case when soil dry bulk density is known. In the case when soil dry bulk density is unknown, the segmented model without density is recommended, since it surpassed not only the classical linear θ( ) model dependent only on the square root of soil dielectric permittivity, but also performed as well as the classical θ( ̅̅̅ ̅ ε ′ √ , ρ) models with the density terms. Thus, the segmented model without density might be the most practical.
The samples analysed in the present study consisted of non-saline soil materials only. Variations of soil bulk electrical conductivity may significantly impact the performance of the models in question in the examined frequency range. Further research on a greater number of saline and non-saline soils, with careful consideration of sample loading and density influence, is needed in order to establish the soils for which the segmented model is the most beneficial. Then, a new group of practical models dedicated to the field use could be developed. Frequency dependence of (a) R 2 of the segmented model and segmented model with density (dash-dotted lines), T1-T4 models fitted without the river sand measurement series (solid lines) and E1-E4 models fitted without the river sand measurement series (dashed lines), and (b) RMSE of the segmented model and segmented model with density (dash-dotted lines), and T1-T4 models fitted without the river sand measurement series (solid lines).

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments
Funding: The measurement results were obtained at Tokai University during A. Szypłowska's JSPS Invitational Fellowship for Research in Japan (Long-term). This article has been supported by the Polish National Agency for Academic Exchange under Grant No. PPI/APM/2018/ 1/00048/U/001. Declarations of interest: none.

A.1. Formulation of the general simplified model
Complete N-pole Debye model for a lossy dielectric is given by: We assume that the measurement data is taken up to ω max . Let us assume that poles are sorted and the first N 1 poles lay below ω max while the rest N 2 = N − N 1 poles are beyond ω max . For those out-of-band poles we have ω max τ n ≪1, thus we can write the following second-order approximation: After grouping all of the terms, we obtain ε * (ω) ≈ ε ∞ + This can be further written in a simplified form as: Δε n 1 + jωτ n + σ b jωε 0 ,  Δε n τ n , Δε n τ 2 n . (20)

A.2. Estimation of the model parameters
The value of σ b is determined from low-frequency estimation. We then assume that for all poles ωτ n ≪1, which yields the following second-order approximation: Δε n − jω Multiplying by jωε 0 we obtain: Δε n τ 2 n ) .
This equation can be fitted in a least-squares sense to obtain an estimate of σ b . We estimate A, B, and C from high-frequency approximation. At high-frequencies we can assume that for in-band poles we have ωτ n ≫1. Thus, from the original model we obtain which can be rewritten as: This again can be fitted in the least squares to obtain estimates of A, B, and C. In this fitting we use the estimate of σ b from the previous step.